BOSTON UNIVERSITY COLLEGE OF ENGINEERING
Dissertation
RESONANT OPERATION OF NANOELECTROMECHANICAL SYSTEMS IN FLUIDIC ENVIRONMENTS
by
DEVREZ MEHMET KARABACAK B.S., Middle East Technical University, 2002 M.S., Boston University, 2005
Submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy 2008
c Copyright by ° DEVREZ MEHMET KARABACAK 2007
Acknowledgments First and foremost, I would like to thank my advisor Prof. Kamil L. Ekinci, whose unwavering support has enabled my research. He has tirelessly provided me with valuable guidance and advice throughout my graduate studies. I am forever grateful for his academic mentorship and relentless effort to raising the standards of my work. I am also thankful to Prof. Victor Yakhot for his endless enthusiasm to interpret the fluidic phenomena that emerged from this study and his belief in the significance of these results, which were sources of inspiration in the final phases of my research. Without his theoretical contributions, this thesis would have been incomplete. I want to thank my committee members Prof. Todd W. Murray and Prof. Paul E. Barbone for their valuable guidance and for providing numerous suggestions over the years. I am grateful to Prof. Taejoon Kouh, from whom I inherited a vast amount of practical knowledge regarding many aspects of NEMS. I would like to acknowl¨ u¨ who has contributed with both academic and logistic edge Prof. M. Selim Unl support in the optical sections of this work. I am thankful to Prof. P. Scott Carney and Dr. Stephen B. Ippolito, with whom I had the privilege of working briefly. They provided not only optical expertise and academic advice but also an enjoyable research experience during our collaboration. I want to thank the fellow graduate students at the Laboratory for Nanometer Scale Engineering; Utku Kemiktarak, Onur Bas¸arır, Chien-Chih Huang, Michael Y. Shagam, N. Onur Azak and Carl Hart, for their support of my work at various stages. I would like to particularly acknowledge Utku and Onur B., who were always available with their unique ideas, advice and humor throughout the years. Furthermore, their efforts in providing critical review of this thesis and my previiv
ous work stimulated the development of many ideas. I am especially grateful to ¨ ˙ Ozkumur ¨ u¨ for being at my side at times Ayc¸a Yalc¸ın, Emre I. and F. Hakan Kokl of success and difficulty, providing not only motivation but also making my graduate school life most enjoyable. I would also like to thank A. Nickolas Vamivakas and Dr. Oluwaseyi O. Balogun for setting aside the time to provide practical solutions to my problems in optics. ˙ Finally, I wish to acknowledge my mother and uncle, Inci and Enver Karabacak, to whom I dedicate this thesis. Without their never-ending support and devotion throughout my life, this accomplishment would not have been possible.
v
RESONANT OPERATION OF NANOELECTROMECHANICAL SYSTEMS IN FLUIDIC ENVIRONMENTS (Order No.
)
DEVREZ MEHMET KARABACAK Boston University, College of Engineering, 2008 Major Professor: Kamil L. Ekinci, Ph.D., Assistant Professor of Aerospace and Mechanical Engineering
ABSTRACT Nanoelectromechanical Systems (NEMS) are electromechanical systems with critical dimensions in the sub-micron range. NEMS are often operated in their fundamental resonant modes at high frequencies. As such, their active masses and intrinsic dissipation levels are usually very small. Resultantly, NEMS devices are rapidly being developed for a variety of sensing applications, high frequency electromechanical signal processing and time-keeping tasks. Most high-impact sensing applications require NEMS operation in fluids, where undesirable fluidic effects degrade device performance. Overcoming the challenges in operating NEMS devices in fluids requires a new understanding of fluid dynamics at high frequencies. Newtonian fluid dynamics, which has been used to describe similar problems in larger scale devices such as MEMS, requires characteristic length and time scales of the flow to be significantly larger than the vi
mean free path and relaxation time of the fluid, respectively. Both assumptions fail for NEMS devices in gaseous environments. In fact, NEMS resonators provide a unique opportunity to study this previously inaccessible flow regime. In an effort to understand this novel flow regime, doubly-clamped nanomechanical beam resonators spanning a wide range of size and resonance frequencies were fabricated. The resonant behavior of these devices was studied as a function of surrounding gas pressure in an optical interferometer, which was developed for characterizing sub-wavelength devices. Through analysis of resonance parameters, fluidic dissipation and inertial loading effects were extracted. The combination of varying gas pressure and resonator dimensions allowed for the exploration of a large parameter space. The experimental data were compared to various formulations of fluid dynamics developed for resonant structures. The observed transitions in fluidic dissipation as a function of frequency and gas pressure agreed closely with a recently developed non-Newtonian theory of fluid dynamics at high frequencies. The possibility of increasingly elastic response, and thus reduced fluidic dissipation, at high resonance frequencies may result in exciting opportunities for next-generation NEMS operating in fluids.
vii
Contents 1
Introduction to Nanoelectromechanical Systems (NEMS)
1
1.1
Overview of NEMS . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1
1.2
Mathematical Description . . . . . . . . . . . . . . . . . . . . . . . .
7
1.2.1
Beam Theory . . . . . . . . . . . . . . . . . . . . . . . . . . .
7
1.2.2
Frequency Domain Description of NEMS Response . . . . .
17
Fabrication of Nanomechanical Structures . . . . . . . . . . . . . . .
19
1.3.1
Overview of Semiconductor Fabrication Techniques . . . . .
19
1.3.2
Fabrication of Doubly-clamped Resonators . . . . . . . . . .
23
1.3
2
3
Optical Displacement Detection in Nanoelectromechanical Systems
29
2.1
Interferometric Methods . . . . . . . . . . . . . . . . . . . . . . . . .
31
2.1.1
Michelson Interferometry . . . . . . . . . . . . . . . . . . . .
31
2.1.2
Fabry-Perot Interferometry . . . . . . . . . . . . . . . . . . .
41
2.1.3
Modeling of Optical Interferometry in Nanoelectromechanical Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . .
44
2.2
Knife-edge Technique . . . . . . . . . . . . . . . . . . . . . . . . . . .
66
2.3
Darkfield Detection by Diffraction of Evanescent Waves . . . . . . .
76
2.4
Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
90
Characterization of High-Frequency Nanoflows using Nanomechanical Resonators 94 3.1
Theory of Oscillating Flow . . . . . . . . . . . . . . . . . . . . . . . . 101
viii
3.2
3.3
3.4 4
3.1.1
Newtonian Fluid Dynamics . . . . . . . . . . . . . . . . . . . 104
3.1.2
Non-Newtonian Fluid Dynamics . . . . . . . . . . . . . . . . 109
Measurement of Nanomechanical Resonator Characteristics . . . . 116 3.2.1
Experimental Setup . . . . . . . . . . . . . . . . . . . . . . . . 116
3.2.2
Device Actuation . . . . . . . . . . . . . . . . . . . . . . . . . 119
3.2.3
Analysis of Resonance Data in the Frequency Domain . . . 122
3.2.4
Thermal Effects due to Laser Absorption . . . . . . . . . . . 126
3.2.5
Pressure Effects on Resonator Performance . . . . . . . . . . 129
Analysis of Fluidic Interaction . . . . . . . . . . . . . . . . . . . . . . 130 3.3.1
Fluidic Dissipation Measurements and Theoretical Fits . . . 134
3.3.2
Inertial Load Measurements . . . . . . . . . . . . . . . . . . . 142
Implications on Device Design . . . . . . . . . . . . . . . . . . . . . 144
Conclusions and Future Directions
147
4.1
Summary of Achievements . . . . . . . . . . . . . . . . . . . . . . . 147
4.2
Suggestions on Future Directions . . . . . . . . . . . . . . . . . . . . 150
References
155
Curriculum Vitae
167
ix
List of Tables 1.1
Modal parameters for the first four modes of a cantilever beam. . .
13
1.2
Modal parameters for the first four modes of a doubly-clamped beam. 14
3.1
Device parameters, transition pressure P (Wi = 1) and the approximate lower pressure limit Pmin for accurate measurements for the devices used in the study. 1st harmonic mode was also employed for some AFM cantilevers. . . . . . . . . . . . . . . . . . . . . . . . . 132
x
List of Figures 1·1
Schematic of a typical electromechanical sensing element, where the structure is stimulated by an electrical control signal and the output is closely monitored for sensing of environmental changes (Ekinci and Roukes, 2005). . . . . . . . . . . . . . . . . . . . . . . . .
1·2
2
Fundamental mode shape of the doubly-clamped beam resonator. A beam of length l, width w and thickness h is suspended at a distance U0 above an infinite substrate and is displaced by U (y, t) during the harmonic motion of the resonator. . . . . . . . . . . . . . . .
1·3
7
Schematic of standard NEMS fabrication steps. (a) The process starts on a commercial silicon on silicon oxide wafer. (b) The structure pattern is created on the wafer using photolithography and electron beam lithography, each step followed by a thin film deposition of chromium. (c) The pattern is transferred to the underlying silicon layer through reactive ion etching. (d) The defined beams are released through wet chemical etch of the sacrificial oxide layer.
xi
24
1·4
Images from the fabrication of a NEMS sample with a set of 16 nanomechanical beams of identical length l = 5.6 µm, thickness h = 230 nm and varying width 200 < w < 1000 nm. (a) A post-liftoff optical microscope image of the contact pad structure defined through photolithography. (b) Optical microscope image of the reactive ion etch mask (chromium) after lift-off. The nanomechanical beams are defined within the center area of the contact pads by EBL. (c) The SEM image of the finished sample. The scale bars in each figure represents 50 µm. . . . . . . . . . . . . . . . . . . . . . . . . .
1·5
25
Oblique scanning electron microscope image of (a) a suspended and metal coated family of beams of length 3.6 ≤ l ≤ 9.6 µm and width 200 ≤ w ≤ 400 nm, with a device thickness of h = 200 nm, and (b) the suspended structure, showing the extent of the undercut in the silicon dioxide layer. . . . . . . . . . . . . . . . . . . . . .
2·1
27
(a) Schematic of a typical path-stabilized Michelson interferometer. (b) The intensity output as a function of static optical path length difference zs − zr . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2·2
32
(a) The photodetector signal Vpd during a typical sweep of the reference mirror, as function of the optical path length difference zs − zr , measured using a laser of wavelength λ = 633 nm at a power level of P0 ∼ 800 µW incident on a micromechanical cantilever sample of width w = 50 µm. The photodetector responsivity is RP D = 0.4 A/W. The point of highest sensitivity is determined to be at Vopt = 3.95 mV. (b) The low frequency components of Vpd stabilized at Vopt = 3.95 mV using PID feedback. . . . . . . . . . . . xii
38
2·3
The spectral density of thermal motion of a commercial AFM cantilever with dimensions (w × h × l) 50 µm ×2 µm ×460 µm, in its fundamental mode at atmospheric pressure. The solid line is the estimation from Eq. 2.8 using the experimentally determined quality factor Q = 47. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2·4
39
(a) Sketch of the typical Fabry-Perot configuration where multiple reflections from two partially reflective surfaces can be tuned to obtain very high transmissivity TF P or enhanced reflectivity RF P by varying the ratio of the cavity length U0 to wavelength λ. (b) Setup schematic of NEMS displacement detection using Fabry-Perot interferometry, where the reflection from the nanomechanical beamsubstrate cavity is collected by a high-NA objective lens (OL) and directed onto a photodetector (PD). . . . . . . . . . . . . . . . . . . .
2·5
(a) Laser spot focussed on beam (b) cross-sectional view of the multilayer structure within the optical spot. . . . . . . . . . . . . . . . . .
2·6
41
43
(0)
(a) Electric field amplitude Ey (x, z) of the incoming Gaussian beam at wavelength λ=633 nm, for a focusing objective lens of NA=0.5, (r)
and f =4 mm. (b) The electric field amplitude Ey (x, z) of the reflected-scattered EM field from a NEMS. The scattering NEMS beam has width w=170 nm, cavity length U0 =300 nm and silicon layer thickness h=200 nm (device not shown). This solution is obtained by finite element analysis. The substrate is positioned at the bottom of plot (b). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
xiii
49
2·7
The photodetector circuit model. The noise in the circuit arises from uncorrelated current and voltage sources shown in gray; the signal current source is in black. The RF amplifier has a 50 Ω input impedance and two uncorrelated noise sources. . . . . . . . . . . .
2·8
54
The photodetector current (upper plot) calculated as a function of the beam position (lower plot) using the presented model. The following parameters are used: w = 170 nm, h = 200 nm, U0 = 350 nm with a vibrational amplitude of u0 = 5 nm. The current is calculated at the photodiode located at Lp ≈ 15 cm from the sample. The detector responsivity is Rpd = 0.4 A/W and the probing laser beam power is P0 = 100 µW at λ = 633 nm. . . . . . . . . . . . . . .
2·9
56
Optical characteristics of a set of NEMS beams with w = 170, 300 and 400 nm and h = 200 nm as a function of U0 . The plots are for P0 = 100 µW and RP D = 0.4 A/W. (a) Device reflectivity R oscillates with a period of ∝ λ/2, as U0 is varied. Note that oscillation amplitude of R is highest for the smallest structure. (b) Displacement responsivity RFu P is the relevant quantity for displacement detection. It is calculated by numerically differentiating the data in part (a) with respect to U0 . . . . . . . . . . . . . . . . . . . . . . . . .
58
2·10 Displacement sensitivity results of Fabry-Perot interferometry, based (AT )
upon SI
≈ 100 pA2 /Hz, for a set of NEMS beams with w = 170,
300 and 400 nm and h = 200 nm as a function of U0 . The plots are for P0 = 100 µW and RP D = 0.4 A/W. . . . . . . . . . . . . . . . . .
xiv
59
2·11 (a) Reflectivity as a function of the beam width for metallic and metallized silicon NEMS beams. Here, silicon layer thickness is set to h = 200 nm, and the gap thickness is set to U0 = 400nm. A cavity resonance appears to become dominant at w ≈ 160nm in the metallized silicon beams. (b) Maximum displacement responsivity RFu P −max of the Fabry-Perot technique as a function of w. . . . . . .
60
2·12 Device reflectivity R as a function of λ, for three devices with width w = 170, 300 and 400 nm. Here, h =200 nm, U0 = 350 nm and parameters of the free space optics are kept constant for all the simulations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . √ 2·13 (a) The displacement responsivity RM Su of u and (b) noise floor
62
Michelson interferometry in NEMS. The back substrate is removed to eliminate any cavity effects. Here, P0 = 100 µW at λ = 633 nm, (AT )
SI
≈ 100 pA2 /Hz and RP D = 0.4 A/W. . . . . . . . . . . . . . . .
64
2·14 Schematic of doubly-clamped beam fabrication from nitride membranes. (a) The process starts by etching of an access window at the backside of a double-side coated silicon nitride-silicon structure. The window is defined using photolithography and the nitride layer is etched using RIE. (b) The exposed silicon layer is wet etched using KOH solution to create a thin nitride membrane. (c) The nanomechanical beam-gate structure is patterned by EBL and thermal deposition of a thin metal film. (d) The pattern is etched into the nitride layer using RIE to release the structure. . . . . . . .
xv
67
2·15 (a) Scanning electron micrograph of a doubly clamped silicon nitride beam with a side gate. Here, w × h × l = 200 nm×125 nm×14 µm; g = 130 nm. (b) An illustration of the experiment setup. The optical spot is focused at an offset of xs from the beam center of the NEMS resonator. The NEMS center displacement from equilibrium is u. The center of the beam is aligned to the origin. . . . . . . . . .
69
2·16 In-plane fundamental flexural resonance of the beam shown in Fig. 2·15(a), under varying drive amplitude VDC . The inset displays the change in the resonance frequency ω0 with VDC . . . . . . . . . . . .
70
2·17 Normalized optical responsivity of the NEMS devices, with dimensions of (a) w = 500 nm and g = 500 nm and (b) w = 200 nm and g = 100 nm. The inset shows the reflected power PR as a function of the spot position xs . Analytical calculation results for |∂PR /∂xs | are plotted as dashed lines. In (b), the solid line shows |∂PR /∂xs | without the contribution of the gate. . . . . . . . . . . . . . . . . . .
73
2·18 Peak responsivity RKE0 contour plot as a function of both beam width w and spot diameter d. . . . . . . . . . . . . . . . . . . . . . .
xvi
75
2·19 Schematic of the proposed evanescent detection approach. (a) In this method a typical doubly-clamped nanomechanical beam resonator is illuminated by evanescent waves formed by focusing a near-IR laser through an index matched numerical aperture increasing lens (NAIL) attached to the backside of the sample. (b) Crosssectional (x − z plane) view of the proposed detection. Here, an infinitely long strip of width w is suspended by U0 , and oscillates with amplitude u0 above a substrate of refractive index n. The evanescent illumination field Ei in the direction of ki is formed by the total internal reflection of the incoming wave in the direction of ks . . . .
78
2·20 The normalized distribution of the intensity (the squared magnitude of the electric field) as a function of angle in the plane normal to the longitudinal axis of the strip of width w = 50 nm. The plots are for different strip height U0 values are as indicated in the legend. The incident field in vacuum is taken to be of unit amplitude at the Si interface with wavelength λ = 1.3 µm and a wave vector of (2.2k0 , 0, 2.0ik0 ). The dashed lines indicate the results of numerical simulation and the solid lines the outcome of Eq. (2.48). The incident field is (a) TE-polarized (b) TM-polarized. The curves were normalized to the peak height of the analytic result at 100 nm and then scaled by a factor C
indicated by the curve for
display purposes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
84
2·21 As in Fig. 2·20, except that the strip width is taken to be 500 nm and the height is fixed at U0 = 200 nm. . . . . . . . . . . . . . . . . . . .
xvii
85
2·22 A proposed setup layout for displacement detection of nanomechanical beam motion through diffraction of evanescent waves. A collimated laser beam in the near-IR wavelength is passed through a low-NA (∼ 0.25) focussing lens and a NAIL attached to the back of the sample substrate to form an evanescent field incident on the NEMS structure. The scattered light is collected through a high-NA (∼ 0.7) objective lens from the front and directed onto a photodetector. The aperture ring ensures that the field is fully evanescent by blocking the central portion of the incoming laser beam. The structure is illuminated and imaged by a CCD from the front side for alignment purposes. . . . . . . . . . . . . . . . . . . . . . . . . .
87
2·23 The dynamic signal in the scattered field due to a focus superposition of evanescent waves with a FWHM spot size of d ∼ 250 nm centered on the strip, at height U0 = 100 nm, at an oscillation of amplitude u0 = 0.5 nm, for both TE and TM polarization of the incident field for (a) beams of varying width w at b = 0 nm air gap, and (b) varying NAIL-substrate air gap b for a beam of width w = 200 nm. The results were normalized with the optical input power P0 , determined at the entrance of the aperture ring shown in the setup layout in Fig. 2·22. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3·1
89
Knudsen number Kn = lmf p /L as a function of gas pressure P , for devices of varying length scales L indicated in the legend for nitrogen gas, dm = 0.38 nm. . . . . . . . . . . . . . . . . . . . . . . . . . .
xviii
98
3·2
Schematic of the Stokes’ second flow problem; an infinite plate oscillating with angular frequency ω sets up waves in the fluid in the normal (y) direction with amplitude v(y, t). . . . . . . . . . . . . . . 114
3·3
Optical measurement setup for characterization of micromechanical and nanomechanical structures under controlled pressure environments. In the inset, a typical CCD image of a family of doublyclamped nanomechanical beams with the focused laser spot is displayed. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
3·4
Resonance of a silicon doubly-clamped beam of width w = 500 nm, thickness h = 280 nm and length l = 11.2 µm at vacuum pressure level of P = 0.049 Torr. (a) Displacement |u|2 and phase θ in the frequency domain, along with the Lorentzian fit to |u|2 (solid line). (b) Quadrature plot of the real κ and imaginary χ components of the response, as defined by Eq. 3.52. The device quality factor was determined to be Qs ≈ 1530. . . . . . . . . . . . . . . . . . . . . . . . 123
3·5
Thermally-induced frequency shift due to laser absorption, as the incident laser power is increased, measured using a doubly-clamped resonator of dimensions (w × h × l) 230 nm ×200 nm ×9.6 µm with an undamped resonance frequency of 24.2 MHz. The measured resonance frequency ωd at each pressure level was normalized by the damped resonance frequency ωd,0 at the lowest power level, to remove inertial loading effect of the pressure increase from the comparison. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
xix
3·6
(a) Resonance of a silicon doubly-clamped beam of width w = 500 nm, thickness h = 280 nm and length l = 11.2 µm at various N2 pressures in the chamber: P = 0.049, 5.4, 32, 100, 302 and 942 Torr. (b) The extracted quality factor Q and normalized resonance frequency ω/ω0 of the same device as a function of pressure. Here, Qs ≈ 1530 and ω0 /2π = 18.1 MHz. . . . . . . . . . . . . . . . . . . . 130
3·7
(a) NEMS displacement urms at drive voltages Vdc = 4, 8, 16, 23 V, measured with Michelson interferometry at P = 500 Torr. The device dimensions are (w × h × l) 0.73 × 0.28 × 5.6 µm and the undamped resonance frequency is ω0 /2π = 58.6 MHz. (b) The effect of drive voltage Vdc on displacement urms and quality factor Q for the same nanomechanical beam. . . . . . . . . . . . . . . . . . . . . 131
3·8
Normalized fluidic dissipation γn as a function of pressure obtained from beams at indicated frequencies, as listed in Table 1.2. The lines are from models of molecular collision (Eq. 3.29), Sader (Eq. 3.24) and Yakhot (Eq. 3.43) using τ ≈ 1850/P . For results obtained from nanomechanical structures, squeeze film damping (Eq. 3.26) was also included. The turning point of Wi = 1 is marked with an arrow in each plot. Yakhot-Colosqui formulation of Eq. 3.43 was multiplied with a fitting constant of (a) 2.8 (b) 2.8 (c) 3.5 (d) 5.0. . . 135
3·9
As in Fig. 3·8, for the beams at frequencies as indicated on the plots. Yakhot-Colosqui formulation of Eq. 3.43 was multiplied with a fitting constant of (a) 3.8 (b) 3.0 (c) 3.8 (d) 2.5 (e) 3.6 (f) 2.9. . . . . . . . 136
xx
3·10 As in Fig. 3·8, for the beams at frequencies as indicated on the plots. The turning point of Wi = 1 is marked with an arrow in each plot. Yakhot-Colosqui formulation of Eq. 3.43 was multiplied with a fitting constant of (a) 3.2 (b) 2.0 (c) 3.7 (d) 3.2. . . . . . . . . . . . . . . 137 3·11 Relaxation time τ as a function of pressure. The points were extracted from fluidic dissipation measurements of 13 resonators, displayed in Figs. 3·8-3·10. The solid line is a least-mean-squares fit and indicates that τ ≈ 1850/P . The dashed line represents the theoretical calculation of bulk relaxation time from τ ∼ lmf p /cs . . . . . 138 3·12 Normalized fluidic dissipation γn as a function of the resonator frequency ω0 for several resonators at four different pressures. From top to bottom, τ ≈ 1850/P =2.3, 4.6, 9.2, 18.5 ns. The lines are fits calculated using Eq. 3.43. Wi ≈ 1 points are marked with an arrow for each pressure. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140 √ 3·13 The scaling of γn / ωµρf for all resonators with ωτ . Each symbol corresponds to an individual resonator and the solid line is Eq. 3.56. All the predictions were multiplied by the same fitting factor of 2.8. 141 3·14 Normalized frequency
∆ω ω0
for beams of identical widths w = 500 nm
and thicknesses h = 280 nm, but varying lengths and resonance frequencies. The pressure for Wi ≈ 1 is marked with an arrow. Dashed lines are least-mean-squares fits to ∆ω ∝ P 1/3 . . . . . . . . . 143
xxi
3·15 Normalized quality factor in fluid Qf n = ω0 /γn as a function of the resonator frequency ω0 for several resonators at four different pressures. From top to bottom, τ ≈ 1850/P = 18.5, 9.2, 4.6, 2.3 ns. The lines are fits calculated using Eq. 3.43. Wi ≈ 1 points are marked with an arrow for each pressure. . . . . . . . . . . . . . . . 145 4·1
(a) Displacement response of a 100 µm thick single crystal piezoelectric ceramic2 with 15 dBm actuation actuation power applied from the network analyzer. The displacement signal of the ceramic, measured using Michelson interferometry, was buried below the background signal beyond 35 MHz. (b) The resonance signal of a doubly-clamped
nanomechanical
resonator
of
dimensions (w × h × l) 590 nm ×220 nm ×4.6 µm, actuated with the same piezoelectric ceramic in P = 100 mTorr pressure, measured with
Fabry-Perot
interferometry
with
an
incident
optical
power level P0 ∼ 1 mW. The quality factor was determined to be Q ∼ 500. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152 4·2
An optical microscope image, obtained with the setup described in Fig. 3·3, of a family of suspended nanomechanical beams of identical length l = 5.6 µm, and thickness h = 230 nm and varying width 200 < w < 730 nm, obtained while being immersed in a glycerol:water (1:1) solution. The scanning electron microscope image of the same sample is presented in Fig. 1·4. The focused laser spot is visible in the center. . . . . . . . . . . . . . . . . . . . . . . . . . . 153
xxii
List of Abbreviations
AFM
.............
Atomic Force Microscope
BOE
.............
Buffered Oxide Etch
BS
.............
Beam Splitter
CCD
.............
Charge Coupled Device
CVD
.............
Chemical Vapor Deposition
DC
.............
Direct Current
EBL
.............
Electron Beam Lithography
FFT
.............
Fast Fourier Transform
FWHM
.............
Full-Width at Half-Maximum
HMDS
.............
Hexamethyldisilazane
MEMS
.............
Microelectromechanical Systems
MIBK
.............
Methyl isobutyl ketone
NA
.............
Numerical Aperture
NAIL
.............
Numerical Aperture Increasing Lens
xxiii
NEMS
.............
Nanoelectromechanical Systems
OL
.............
Objective Lens
PID
.............
Proportional Integral Differential
PMMA
.............
Polymethyl Methacrylate
PD
.............
Photodetector
RF
.............
Radio Frequency
RIE
.............
Reactive Ion Etching
SEM
.............
Scanning Electron Microscope
SIL
.............
Solid Immersion Lens
SIMOX
.............
Separation by Implantation of Oxygen
SOI
.............
Silicon on Insulator
TE
.............
Transverse Electric
TM
.............
Transverse Magnetic
UHV
.............
Ultra High Vacuum
UV
.............
Ultraviolet
xxiv
1
Chapter 1
Introduction to Nanoelectromechanical Systems (NEMS) 1.1
Overview of NEMS
For the past few decades, miniturization has been at the forefront of technological efforts in various engineering fields, providing mankind with previously unimaginable devices ranging from integrated circuits, minuscule flow systems to intelligent sensors that are now part of our everyday tasks. We have long surpassed the vision laid out by Richard Feynman in his now famous lecture (Feynman, 1960); yet the drive to create increasingly smaller devices with enhanced functionalities is greater than ever (Roukes, 2001b). With microelectromechanical systems (MEMS) well established in its commercialization path, the research community has recently turned its attention to next frontier of miniaturization: nanoelectromechanical systems (NEMS) (Ekinci and Roukes, 2005; Blencowe, 2005; Craighead, 2000). Electromechanical devices, in general, have been in development for a long time, dating back to the charge sensor built by Coulomb (Roukes, 2001a). The mechanical structure in electromechanical devices can be used as an actuator of a task, or more often, it is employed as a sensor of its surroundings. A typical electromechanical sensor element can be described by the schematic in Fig. 1·1 (Ek-
2 Electrical Input Signal
Input Mechanical Transducer (Actuator) Stimulus
Environmental Perturbation
Mechanical Element
Electrical Output Signal
Mechanical Response
Output Transducer (Detector)
Figure 1·1: Schematic of a typical electromechanical sensing element, where the structure is stimulated by an electrical control signal and the output is closely monitored for sensing of environmental changes (Ekinci and Roukes, 2005). inci and Roukes, 2005). The electromechanical system, regardless of scale, usually consists of multiple transducers and a mechanical element whose response is well characterized. The transducers enable the actuation and detection of mechanical motion, through optical or electrical methods. Here, for acurate measurement purposes, it is important to keep the actuation and sensing transducers in minimal interaction with each other. Furthermore, the structure can be employed either in its static or dynamic mode. In static systems, the device experiences only slight changes in deflection. In contrast, in dynamic mode the mechanical structure is under constant oscillation, often in its resonant mode where the response is significantly amplified. In this study the emphasis will be placed on resonant NEMS devices. When the electromechanical system is utilized as a sensor, the mechanical response (whether static or dynamic) is constantly monitored. Changes in the transducer signal are used as indication of variations in the environment. These changes can be used in probing many types of mechanical interactions; mass, pressure, acceleration, among others. Clearly, the sensitivity of the electromechanical system to environmental changes relies heavily on the size of the device at hand. As device dimensions
3 are reduced, previously unattainable characteristic properties are reached. While intrinsic mass and heat capacity of the device are reduced to extremely low values, the surface-to-volume ratio and fundamental resonance frequency increases significantly for miniscule devices. In fact, previously inaccessible mechanical vibration frequencies exceeding 1 GHz have recently been demonstrated as fundamental resonance frequencies for devices only a few microns in length (Huang et al., 2003; Peng et al., 2006). While RF-MEMS technology can achieve relatively high frequencies through construction of very stiff structures (k ∼ 106 N/m) (Clark et al., 2005), NEMS technology reaches similar levels with structures that are significantly more responsive, with stiffness values k ∼ 10 N/m. Additionally, NEMS structures typically demonstrate low intrinsic dissipation, where quality factors of 103 ≤ Q ≤ 104 (Mohanty et al., 2002) at room temperature conditions are easily attainable. More recently, high-Q nanowires with Q ∼ 105 were achieved through the use of high tensile stress nitride wafers (Verbridge et al., 2006b). Nanoelectromechanical devices are being rapidly developed for a wide range of applications, even at these early stages of the technology. Mass sensing is promising to be one of the early adaptations of NEMS technology. Single molecule sensitivity has been demonstrated to be within reach (Ekinci et al., 2004b), and quite recently, mass resolution in the zeptograms (10−21 g) was experimentally achieved (Yang et al., 2006). The prospect of such high sensitivity biomolecule detection is already generating much excitement and various device configurations are being analyzed (Ilic et al., 2004; Gupta et al., 2004; Burg et al., 2002; Lavrik and Datskos, 2003; Gupta et al., 2006). Additionally, force microscopy (Rugar et al., 2004) using ultra-thin soft cantilevers with attonewton force sensitivities (Stowe et al., 1997), high sensitivity electrometry using torsional nanomechanical
4 resonators (Cleland and Roukes, 1998), ultra-fast optomechanical (Sekaric et al., 2002b) and electromechanical (Erbe et al., 2000a) signal modulators are but a few of the early applications of NEMS technology. Beyond the exciting engineering applications that are rapidly emerging, nanomechanical structures are being considered as the ultimate tool in studying quantum mechanics (Schwab and Roukes, 2005). Already, some very interesting, novel physical phenomena are being investigated through measurements of quantized thermal conductance (Schwab et al., 2000) and motion near the quantum limit (LaHaye et al., 2004; Knobel and Cleland, 2003). Nanoelectromechanical systems clearly present many opportunities for exciting new applications and discoveries. However, significant challenges remain in truly exploiting the full potential behind NEMS. One of the major hurdles is the development of efficient, sensitive and broadband transducers for detection at the nanoscale (Ekinci, 2005). Many of the techniques commonly used in the MEMS domain are not truly scalable to nanoscale devices. Among the earlier demonstrations, magnetomotive displacement detection was prominent in NEMS applications (Cleland and Roukes, 1996; Yang et al., 2001). This technique relies on the detection of the electromotive force induced current generated during the cyclic motion of the nanomechanical beam. However, undesired coupling between the detection and actuation lines makes this method difficult to implement. Furthermore, the need for high magnetic field strengths (up to 10 Tesla) and cryogenic setups make magnetomotive detection scheme highly impractical for wide scale application at ambient conditions. Capacitive detection, while being extremely successful and widely used in MEMS applications, is another method that has met significant difficulties in scal-
5 ing. The device capacitances and motion amplitudes diminish rapidly as the structures are being scaled down in size. Hence, the capacitance variation during device oscillation rapidly falls below detectable limits, imposed by parasitic effects at high frequencies. However, the prospect of forming a fully integrated, all-electronic read-out scheme which can be multiplexed into simultaneous detection of many beams has motivated significant research in this area. Single electron transistors, with sensitivities near the quantum limit have been demonstrated in NEMS (Knobel and Cleland, 2003; LaHaye et al., 2004). More recently, impedance matching has been proposed as a more practical way of overcoming parasitic effects in the detection circuit, and its application to monitoring arrays of nanomechanical beams was successfully demonstrated (Truitt et al., 2007). Piezoresistive detection relies on the measurement of stress-induced resistivity change in the device layer during deformation of the device. This integrated detection scheme was initially demonstrated more than a decade ago in highlydoped silicon micro-cantilevers employed in atomic force microscopy (Tortonese et al., 1993). However, as the devices are reduced in size the characteristic resistance of the silicon bridge increases significantly, obscuring the detection signal. Recently, piezoresistive detection has been developed by using metal pads as piezoresistive lines, with detection enhancements through heterodyne downmixing of the detection signal (Bargatin et al., 2005; Li et al., 2007; Bargatin et al., 2007). Optical displacement detection is a widely applied transduction scheme for nanoelectromechanical systems (Ekinci et al., 2006). Many configurations have been demonstrated in optical measurements of nanomechanical motion; Michelson and Fabry-Perot Interferometry (Kouh et al., 2005b; Carr et al., 1998), confo-
6 cal microscopy (Meyer et al., 2003), knife-edge detection (Karabacak et al., 2006), gratings transducers (Keeler et al., 2004), and fiber interferometry (Shagam, 2006). Optical techniques provide the advantage of remote, broadband and nondestructive testing of devices. However, implementation of optical methods becomes challenging at the nanometer length scale, where strong diffraction effects dominate reflections from devices of sub-wavelength dimensions (Karabacak et al., 2005). Nonetheless, optics provides a flexible testing environment for characterizing the response of nanomechanical systems. Fabry-Perot interferometry has been utilized as the method of choice for this study of fluidic dissipation in nanomechanical devices. Chapter 2 presents some of the optical methods that have been developed for NEMS and discusses the issues involved in their application, with detailed analysis. Until recently, most nanoelectromechanical systems research was conducted within ultra-high vacuum (UHV) environments, to minimize fluidic dissipation effects and to obtain high quality factors (Ekinci et al., 2004a; Kouh et al., 2005b). However, this bulky and costly approach is very impracticle for applications outside the laboratory environment. Furthermore, most of the promising applications of NEMS, like biomolecule detection, require operation in fluidic environments. Recently, resonant operation of nanomechanical beams has been demonstrated to be feasible in both gases (Karabacak et al., 2007b; Li et al., 2007; Sekaric et al., 2002a) and liquids (Verbridge et al., 2006a). In Chapter 3, a thorough experimental study of the high frequency nanofluidic effects on resonant nanomechanical beams is presented. Evidently, oscillatory flow at such high-frequencies as those generated by NEMS involve interesting new physics. A theoretical understanding of these observations is laid out in detail in Chapter 3. Finally, discussions
7 of this enhanced understanding of nanofluidic effects is utilized in developing suggestions for the next generation of resonant NEMS for operation in fluidic environments.
1.2 1.2.1
Mathematical Description Beam Theory
The modal analysis of the nanomechanical resonator can be derived from the Euler-Bernoulli beam theory (Cleland, 2003). In this study, the structure of interest is the doubly-clamped beam. The geometry and variable definitions are presented in Figure 1·2, where the doubly-clamped beam structure is depicted in its fundamental mode shape.
z
x
l U(y,t)
y
U0
h
w
Figure 1·2: Fundamental mode shape of the doubly-clamped beam resonator. A beam of length l, width w and thickness h is suspended at a distance U0 above an infinite substrate and is displaced by U (y, t) during the harmonic motion of the resonator. Defining the position dependent displacement of the beam in the z-direction to be U (y, t), the force balance on the infinitesimal section of an undamped beam can be expressed as (Cleland, 2003) ∂ 2 U (y, t) ∂ 2 U (y, t) ∂2 EI + ρs Ac = f (y, t) , ∂y 2 ∂y 2 ∂t2
(1.1)
8 where E represents the modulus of elasticity, I is the moment of inertia, ρs is the density and Ac is the cross-sectional area of the beam. For a beam of rectangular cross-section with thickness h and width w, the moment of inertia becomes I = 1 3 hw 12
and its cross-sectional area is Ac = wh. It is assumed that the beam is
under a time-varying external load in the z-direction expressed by the function f (y, t). In Eq. 1.1, the first term represents the stress formation in the beam due to the deformation, whereas the second term comes from the inertial effect of the motion. From separation of variables, the displacement U (y, t) of the beam can be split into the time and position dependent components, and the result can be expressed as a summation of the motion in each mode n, U (y, t) =
X
qn (t) ϕn (y).
(1.2)
n
Here, ϕn (y) is the mode shape function based on the coordinate along the beam and qn (t) is the time-dependent amplitude of motion, for each mode n. Using Eq. 1.2, Equation 1.1 can be restated as EI
X n
qn (t)
X ∂ 2 ∂ 2 ϕn (y) ∂ 2 qn (t) + ρ A = f (y, t) . ϕ (y) s c n 2 ∂y 2 ∂y 2 ∂t n
(1.3)
In the above equation, it is assumed that the beam is isotropic and has a uniform cross-section. Examining the the homogeneous case of f (y, t) = 0 only, Eq. 1.3 can be written as 1 ∂ 2 qn (t) EI ∂ 2 ∂ 2 ϕn (y) = − = ωn2 , ρs Ac ϕn (y) ∂y 2 ∂y 2 qn (t) ∂t2
(1.4)
9 for each mode n. The first part of Equation 1.4 can be formulated as ∂ 4 ϕn (y) − βn4 ϕn (y) = 0, ∂y 4 µ ¶1/4 ρs Ac βn = ωn1/2 , EI
(1.5)
where βn is a constant, for each mode n, based on the geometry and material. The solution to Eq. 1.5 provides the beam deformation shape ϕn (y) of each mode. The above homogeneous partial differential equation has a solution of the form (Cleland, 2003) ϕn (y) = an cos (βn y) + bn sin (βn y) + cn cosh (βn y) + dn sinh (βn y) .
(1.6)
The constants in Equation 1.6 can be extracted by imposing the boundary conditions on the solution, based on the specific geometry. For a doubly-clamped beam, where the displacement and its derivative are zero on both ends, the boundary conditions of the problem can be expressed as ¯ ¯ dϕn ¯¯ dϕn ¯¯ = = 0. ϕn (0) = ϕn (l) = dy ¯y=0 dy ¯y=l
(1.7)
Applying these conditions on Eq. 1.6 yields ϕn (y) = an [cosh (βn y) − cos (βn y)] + bn [sin (βn y) − sinh (βn y)] , an (cosh (βn l) − cos (βn l)) = , bn (sin (βn l) + sinh (βn l)) cos (βn l) cosh (βn l) − 1 = 0.
(1.8)
From the above set of equations, non-trivial values for βn and the ratio an /bn can
10 be extracted as βn l = 4.73004, 7.8532, 10.9956, 14.1372... an = 1.01781, 0.99923, 1.0000, 1.0000... bn
(1.9)
The values in Eq. 1.9 can then be utilized in calculating the corresponding resonance frequency of an individual mode as sµ ωn =
βn2
¶ EI . ρs Ac
(1.10)
A similar solution can be employed for the cantilever structure using the appropriate clamped and free boundary conditions, ¯ ¯ ¯ dϕn ¯¯ d2 ϕn ¯¯ d3 ϕn ¯¯ ϕn (0) = = = = 0. dy ¯y=0 dy 2 ¯y=l dy 3 ¯y=l
(1.11)
Equation 1.3 can be multiplied by ϕm (y) and integrated along the beam length from y = 0 to y = l, such that
EI
Zl X 0
n
∂ 4 ϕn (y) qn ϕm (y) dy + ρs Ac ∂y 4
Zl X 0
n
Zl q¨n ϕn (y)ϕm (y)dy =
f (y, t)ϕm (y) dy. 0
(1.12)
Here, the q¨n notation represents the second time derivative of qn . By applying integration by parts twice, it is possible to expand the first term of
11 Eq. 1.12 as Zl 0
¯l Z l ∂ 4 ϕn (y) ∂ 3 ϕn (y) ¯¯ ∂ϕm (y) ∂ 3 ϕn (y) dy = ϕ (y) − dy, ϕm (y) m ∂y 4 ∂y 3 ¯0 ∂y ∂y 3 0 · ¸¯l 3 ∂ ϕn (y) ∂ϕm (y) ∂ 2 ϕn (y) ¯¯ = ϕm (y) − ¯ ∂y 3 ∂y ∂y 2 0 Zl 2 ∂ ϕm (y) ∂ 2 ϕn (y) + dy. (1.13) ∂y 2 ∂y 2 0
For typical beam boundary conditions of clamped, free, or pinned, the first two terms in the above equation can be demonstrated to be zero for each individual mode of the beam. Hence, the above integration by parts is reduced to Zl
∂ 4 ϕn (y) ϕm (y) dy = ∂y 4
0
Zl
∂ 2 ϕn (y) ∂ 2 ϕm (y) dy. ∂y 2 ∂y 2
(1.14)
0
From Equation 1.5, two separate modes n and m can be expressed as ∂ 4 ϕn (y) = βn4 ϕn (y) , ∂y 4 ∂ 4 ϕm (y) 4 = βm ϕm (y) . ∂y 4
(1.15)
Multiplying the upper equation with ϕm and the lower equation with ϕn , integrating along y and subtracting from each other, one obtains ¢ ¡ 4 4 βn − βm
Zl
Zl ϕn (y) ϕm (y)dy =
0
0
∂ 4 ϕn (y) ϕm (y) dy − ∂y 4
Zl ϕn (y)
∂ 4 ϕm (y) dy. (1.16) ∂y 4
0
The right hand side of the above equation can be reduced zero using Eq. 1.14.
12 Therefore, it is clear that ¡
¢ 4
ZL
βn4 − βm
ϕn (y) ϕm (y)dy = 0.
(1.17)
0 4 unless n = m, Eq. 1.17 essentially states that Since βn4 6= βm
Zl ϕn (y)ϕm (y) dy = 0, if n 6= m.
(1.18)
0
Eq. 1.18 is often referred to as the orthogonality condition of vibrational modes (de Silva, 2006). Using the orthogonality property expressed by Eq. 1.18, the summation in Eq. 1.12 can now be dropped and the equation can be formulated as Zl qn EI
∂ 4 ϕn (y) ϕn (y) dy + ρs Ac q¨n ∂y 4
0
Zl
Zl 2
[ϕn (y)] dy = 0
f (y, t)ϕn (y) dy.
(1.19)
0
The result of Eq. 1.14 can be applied here to simplify the first term in the above expression, resulting in the equation of motion of an undamped beam as Zl ·
Zl 2
ρs Ac q¨n
[ϕn (y)] dy + qn EI 0
∂ 2 ϕn (y) ∂y 2
¸2
Zl dy =
f (y, t)ϕn (y) dy.
(1.20)
0
0
The above equation of motion can be expressed in the more compact form of Zl m0n q¨n + kn0 qn =
f (y, t) ϕn (y)dy, 0
(1.21)
13 where the parameters m0n and kn0 are the modal mass and stiffness coefficients; Z1 m0n
[ϕn (ψ)]2 dψ,
= ρs Ac l 0
kn0
EI = l3
Z1 ·
∂ 2 ϕn (ψ) ∂ψ 2
¸2 dψ.
(1.22)
0
Here, ψ is the normalized coordinate along the beam. The complete set of modal device parameters of βn , ωn , m0n and kn0 for the fundamental and the first three harmonic modes for cantilevers and doubly-clamped structures are listed in Tables 1.1 and 1.2, respectively. Table 1.1: Modal parameters for the first four modes of a cantilever beam. n
0
1
2
3
βn l
1.875
4.694
7.855
10.996
l kn0 EI
3
3.088
121.3
952.2
3653
1 m0n ρs A cl
0.25
0.25
0.25
0.25
3.52
22.0
61.7
120.9
q ωn
ρs Ac l4 EI
Higher harmonics of micromechanical and nanomechanical structures can be studied through base excitation by piezoelectric drive (Ozdoganlar et al., 2005) or electrothermal actuation (Bargatin et al., 2007). However, in this study, the nanomechanical resonators are operated only at their fundamental modes for several practical reasons. Primarily, the devices already demonstrate very high frequencies and small displacement amplitudes at their fundamental modes. Actuation of the nanobeams at their higher modes would further increase the obser-
14
Table 1.2: Modal parameters for the first four modes of a doubly-clamped beam. n
0
1
2
3
βn l
4.730
7.853
10.996
14.137
l kn0 EI
198.56
1670
6327
17470
1 m0n ρs A cl
0.397
0.439
0.433
0.437
22.3
61.7
120.9
199.9
3
q ωn
ρs Ac l4 EI
vation frequency and reduce the vibrational amplitudes, making the already difficult displacement detection even more challenging, especially under non-vacuum conditions. Additionally, the electrostatic actuation mechanism used in this study creates a uniform load on the structure, making excitation possible only at the fundamental mode of the structure. Most often, in optical detection, the displacement at only a specific coordinate of the beam is measured, reducing the need for the complete solution. Logically, the observation point y0 is often selected to coincide with the position of maximum deformation. At the fundamental mode, y0 = l/2 for a doubly-clamped beam, and y0 = l for a cantilever. If the analysis is limited to only the first mode of the system (n = 0), then the displacement of the device at position y0 can simply be expressed from Eq. 1.2 as u(t) = U (y0 , t) = q0 (t) ϕ0 (y0 ).
(1.23)
Since q0 is known to satisfy Eq. 1.21, the equation of motion can be obtained by
15 multiplying Eq. 1.21 with ϕ0 (y0 ), Zl m00 u¨
+
k00 u
= ϕ0 (y0 )
f (y, t) ϕ0 (y)dy.
(1.24)
0
The continuation of the analysis now requires the definition of the forcing function. Most of the mechanical interactions on the beams can be modeled through either point force or distributed load approximations, or a combination of the two. The assumption of a point load can be mathematically expressed as f (y, t) = F (t) δd (y − y0 ), which translates into a time-varying force of magnitude F (t) being applied on the beam at position y0 . Here, δd (y) is the Dirac delta function. If point load approximation is used, then the effective structural mass ms and stiffness k can be extracted from Eq. 1.24 as m00 , [ϕ0 (y0 )]2 k00 k = . [ϕ0 (y0 )]2
ms =
(1.25)
However, the electrostatic actuation method used in this study is better modeled through the distributed uniform load formulation. Defining a spatially uniform force as F (t) = f¯ (t) l, effective constants become ms =
m00 , R1 ϕ0 (y0 ) ϕ0 (ψ)dψ 0
k =
k00 . R1 ϕ0 (y0 ) ϕ0 (ψ)dψ
(1.26)
0
Using the effective constants defined in Eq. 1.25 or 1.26, the equation of motion now takes the form of the widely analyzed simple harmonic oscillator model
16 (Cleland, 2003) ms u¨(t) + ku(t) = F.
(1.27)
For the doubly-clamped beams of rectangular cross-section, actuated in their fundamental mode by distributed loads, the effective parameters can be calculated from Eq. 1.26 using the values listed in Table 1.2 for n = 0, Ac = wh and I = h3 w/12, as s ω0 h = 1.03 2 2π l
E , ρs
ms = 0.76ρs hwl, k = 32
Eh3 w . l3
(1.28)
These results are in agreement with the published values in literature for the same problem description (Ekinci and Roukes, 2005; Weaver et al., 2001; Gere and Timoshenko, 1997). Intrinsic dissipation of the structure can also be integrated into the above derivation by modifying Equation 1.20 to include the structural damping. Assuming that the damping is of uniform amplitude, proportional to the device velocity, p = ξ u˙ n , the damped equation of motion can be expressed as (Salapaka et al., 1997) Zl ρs Ac u¨n
2
[ϕn (y)] dy + ξ u˙ n 0
Zl ·
Zl 2
[ϕn (y)] dy + un EI 0
∂ 2 ϕn (y) ∂y 2
¸2 dy
0 l Z
=
f (y, t)ϕn (y) dy.
(1.29)
0
An analysis similar to the one leading to Eq. 1.27 can be developed to include the damping term γsn u˙ n , where γsn expresses the modal structural damping. Since in this study only fundamental modes of the doubly clamped structure is of concern,
17 the modal subscript n is going to be dropped henceforth for simplicity. Therefore, the simple harmonic oscillator equation for a damped system becomes ms u¨ + γs u˙ + ku = F,
(1.30)
with effective structural mass ms , damping γs and stiffness k. 1.2.2 Frequency Domain Description of NEMS Response It was shown in the preceding section that the equation of motion of the nanoelectromechanical system can be well approximated through the one-dimensional damped harmonic oscillator description in Eq. 1.30, using the appropriate lumped constants for mass, stiffness and damping. The solution to this equation can be shown to have the complex functional form (Cleland, 2003) u (t) = u(ω)ei(ωt+φ) ,
(1.31)
where φ is determined through initial conditions. In the absence of external forcing on the structure, combining Eqs. 1.30 and 1.31 leads to ω 2 + iγs ω − ω02 = 0,
(1.32)
where ω0 is defined by Eq. 1.10. The above equation has a complex solution, which is equal to ω = ωd − iγs /2. Here, ωd is the damped resonance frequency, which can be expressed as
r ωd =
ω02 −
γs2 . 4
(1.33)
Rewriting Eq. 1.31 with these new definitions, one obtains u (t) = u(ωd )e−γt/2 eiωd t .
(1.34)
18 Equation 1.34 represents harmonic motion of frequency ωd , with a decay of the amplitude of motion to its 1/e value within a time of ∼ 2/γ. Often, the damping in harmonic oscillators is described by the quality factor Q, which is defined as the ratio of energy stored in the structure to energy dissipated per cycle of oscillation. Quality factor can be mathematically expressed as (Cleland, 2003) Q
−1
¯ ¯ ¯ γ/2 ¯ ¯ ¯= p γ ≡ 2¯ ωd ¯ ω02 − γ 2 /4
(1.35)
In NEMS, the structural dissipation γs is often very small, whereas typical resonance frequencies ω0 are high, such that γs ¿ ω0 . Therefore, Eq. 1.35 can be simplified to Qs ≈
ω0 , γs
(1.36)
where Qs is the structural quality factor. It is often more convenient, however, to detect the response of NEMS in the frequency domain. The equation of motion can be expressed in the frequency domain by substituting Eq. 1.31 into Eq. 1.30, ms
£¡
¢ ¤ ω02 − ω 2 + iωγs u (ω) = F (ω) .
(1.37)
Hence, the displacement response of the system can be described as u (ω) =
F (ω) 1 h i. ms (ω 2 − ω 2 ) + i ωω0 0
Qs
(1.38)
19
1.3
Fabrication of Nanomechanical Structures
1.3.1 Overview of Semiconductor Fabrication Techniques In fabricating nano-scale structures, two fundamentally different approaches are available; top-down and bottom-up. In the top-down approach, an increasingly small structure is fabricated, starting with macroscopic interconnects down to the nanoscale structure. In this approach, surface micromachining is employed where a multi-layer wafer is incrementally etched down through multiple mask definitions. The top-down fabrication steps are often well defined and the processes are highly controllable. However, the top-down methodology can be restrictive in device dimensions due to the resolution limitations of the lithography techniques, deposition methods or surface roughness in the layers. As the top-down techniques are starting to reach their fundamental limits, bottom-up approaches have been proposed as possible replacements in developing the next generation of nanomechanical structures. In bottom-up approach, in contrast to the topdown approach, incrementally larger structures emerge. In these techniques the construction of the devices often commences with synthesis processes, essentially building upwards from the molecular level. In most cases, the high-purity synthesized material is dispersed on a substrate, fixed down and accessed electrically or optically for macroscopic manipulation and observation purposes. In recent years, successful demonstrations of nanomechanical resonator fabrication has been demonstrated by electrochemical synthesis of platinum nanowires (Husain et al., 2003), growth of boron nanowires (Otten et al., 2002) and carbon nanotubes (Sazonova et al., 2004) by chemical vapor deposition. Despite the advantage of these techniques in providing high surface quality resonators with crosssectional dimensions below 100 nm, the complicated processes are not yet fully
20 controllable or scalable enough for successful parallel fabrication of a range of devices. Due its well established steps and repeatability, only top-down fabrication methods are utilized within this study. In the top-down methodology, the most critical step in creating devices with dimensions in the sub-micron scale is the high-resolution lithography process. The lithography techniques, mainly adapted from the semiconductor industry, are often the limiting factor in reduction of device dimensions. Photolithography is undoubtedly the most established pattern generation tool available. In photolithography, light is passed through a pre-defined mask pattern to expose a photo-sensitive polymer coated surface. The polymer on the exposed surface can then be selectively removed, essentially transferring the mask pattern onto the polymer layer. Photolithography is a highly parallel process where many devices can be defined simultaneously over a large area in an extremely short time period; making this process highly attractive for device fabrication. Unfortunately the optical diffraction limit of ∼ λ/2 makes photolithography impractical for fabricating devices in the nano-domain. Nonetheless, recent efforts of reducing the exposure wavelength λ has extended photolithography use into the sub-micron. Deep-UV photolithography (λ ∼ 240 nm) has been successfully adapted to fabrication of suspended nanoelectromechanical systems, with device resolution down to 500 ¨ nm (Forster et al., 2006). More commonly, however, electron beam lithography (EBL) has been selected as the method of choice in fabricating the new generation of mechanical structures with sub-micron dimensions. Instead of light, electrons are used as the exposure source. The process is often applied inside a modified scanning electron microscope, where the exposure position of the electron gun can be computer controlled
21 to a high precision. Electron beam lithography allows for the creation of high resolution features on a surface coated with a charge sensitive polymer. This maskless process has been widely applied in NEMS research due to its flexibility and high resolution (Craighead, 2000; Carr and Craighead, 1997; Huang et al., 2003). The EBL technique, however, is a series fabrication technique where each point in the pattern needs to exposed separately, leading to very long process times of hours even for the most simple geometries. Recently, a fabrication method combining the parallel aspect of photolithography with resolution of EBL was developed. Nanoimprint lithography relies on the creation of a solid imprint template, often defined through EBL, and its stamping on the polymer coating on a wafer, essentially transferring the mold pattern to the polymer (Chou et al., 1996). This technique, which is more suitable for larger scale parallel fabrication of a repetitive pattern, has been applied successfully to fabrication of suspended nanoelectromechanical structures (Huang and Ekinci, 2006). Material deposition is another widely used technique in microfabrication. There is a very wide range of commercially available deposition techniques developed for the semiconductor industry, ranging from epitaxial growth to chemical vapor deposition (CVD) methods. A majority of these are beyond the scope of this study where a simple bilayer structure was constructed on top of a commercially available silicon-on-insulator (SOI) wafer. Only thin-film metal deposition was performed in defining etch-resistant masking layers. Thin-film metal deposition can be performed either through thermal or electron beam evaporation. In thermal evaporation, the target material is resistively heated into vapor form and condensed upon the substrate. While this is simple in practice, it is highly limited to materials of low evaporation temperature. The material temperature must be
22 kept below the evaporation temperature of the heating element (often tungsten) to prevent contamination. In electron beam deposition, the target is evaporated by highly accelerated electrons emitted from a hot filament. Although electron beam evaporation provides higher quality layers from many materials, the process is more complicated and requires highly specialized equipment. Both evaporation processes are performed under high vacuum conditions to prevent contamination during layer deposition. In transferring the defined pattern to the underlying layer, etch processes of both dry and wet nature are used in surface micromachining. In both types of etchants, reactions can be selected, based on the desired geometry, to obtain isotropic or anisotropic removal of the material. In all etching processes, the critical parameter is the selectivity of the etchant between the protective masking layer and the material to be removed. Dry etching processes involve the exposure of the masked sample to reactive gas mixtures, often in low pressure plasma form. The reactive plasma is obtained by a strong RF field, which ionizes the gas mixture. The reactive ions are accelerated towards the target wafer, etching the exposed surface during collisions. If the reactions are purely chemical, the process is referred to as “plasma etching” (Maluf, 1999). On the other hand, in reactive ion etching (RIE), the kinetic energy of the ion bombardment is also exploited for material removal. The process is often highly anisotropic, although the pressure level can be adjusted to determine the undercut of the material to some extent. Nonetheless, in the RIE process, one cannot prevent the plasma from attacking the sidewalls of the trench being etched. For fabrication of high-aspect ratio trenches, deep-RIE has recently been developed (Maluf, 1999).
23 Wet etching relies on the surface chemistry of the liquid etchants to selectively remove material. Majority of wet etching processes are of isotropic nature. Nevertheless, significant anisotropy can be observed in silicon layers due to etch rate differences among crystal planes of the material. Wet etching processes require minimal equipment and are highly parallel, making them very cost efficient. The main drawback in wet etch is due to the highly corrosive nature of the etchants, which tend to erode the metal interconnects on the device surfaces. Hence, the etching chemistry tends to become a significant limiting factor in material selection. 1.3.2 Fabrication of Doubly-clamped Resonators The doubly-clamped nanomechanical beam resonators in this study were fabricated using the top-down approach outlined in the previous section. The critical steps of the process are displayed in Figure 1·3. The process starts on a commercial silicon-on-insulator (SOI) wafer. These heterostructure substrates, shown in Fig. 1·3(a), contain a buried silicon oxide (SiO2 ) layer acting as a sacrificial insulating layer between the bulk silicon handle and the surface device layer. The SOI wafers used in this study were created using the SIMOX1 process,(Anc, 2004) with typical sacrificial oxide layer of thickness U0 = 400 nm. The thickness of the top silicon layer determines the thickness of the resonator, which ranged from 200 nm≤ h ≤ 280 nm in this study. The top-down fabrication process of the resonators starts with the preparation of metal contact pads with photolithography. These pads provide a device area of approximately 0.15 mm ×0.15 mm at its center. These pads are enlarged on the 1
Separation by Implantation of Oxygen
24
Silicon Silicon
Silicon Oxide Oxide Silicon
(a)
(b)
(c)
(d)
Figure 1·3: Schematic of standard NEMS fabrication steps. (a) The process starts on a commercial silicon on silicon oxide wafer. (b) The structure pattern is created on the wafer using photolithography and electron beam lithography, each step followed by a thin film deposition of chromium. (c) The pattern is transferred to the underlying silicon layer through reactive ion etching. (d) The defined beams are released through wet chemical etch of the sacrificial oxide layer. outer edges to allow for electrical access to the structure by wire bonding. Furthermore, these pads act as markers for future optical alignment tasks. Initially, the cleaned wafer piece was spin coated at 2500 rpm for 15 seconds with the photoresist primer2 to assist in adhesion of the resist layer to the silicon. A positive photoresist3 layer of approximately 15 µm was then spun at the same rotation speed for 30 seconds. The coated wafer was baked on a hot-plate at 100 o C for 4 minutes. Photolithography4 was performed at a wavelength of λ = 405 nm, with an intensity level of 10 mW/cm2 for an exposure time of ∼ 10 seconds. The sample was then developed for ∼ 1 minute to remove the resist layer in the exposed areas. The pad fabrication was then completed with a thermal evaporation5 of a thin film of chromium and the lift-off of the metal coating from photoresist covered areas by acetone-methanol washing. At this stage, the remaining metal pads are of the same design as the transparent areas of the initial mask layer. The center device area of the pads is displayed in Figure 1·4(a). 2
Hexamethyldisilazane (HMDS), Polysciences, Inc. Microposit s1813 Photoresist, Shipley Company. 4 MJB-3 Mask Aligner, Karl Suss MicroTech AG. 5 Auto 306, BOC Edwards, Inc. 3
25
(a)
(b)
(c)
Figure 1·4: Images from the fabrication of a NEMS sample with a set of 16 nanomechanical beams of identical length l = 5.6 µm, thickness h = 230 nm and varying width 200 < w < 1000 nm. (a) A post-lift-off optical microscope image of the contact pad structure defined through photolithography. (b) Optical microscope image of the reactive ion etch mask (chromium) after lift-off. The nanomechanical beams are defined within the center area of the contact pads by EBL. (c) The SEM image of the finished sample. The scale bars in each figure represents 50 µm. The sample is then coated with charge sensitive polymer PMMA6 for electron beam lithography (EBL) process. The developer solution dissolves the low molecular weight PMMA layer at higher reaction rates. This property is often exploited by the preparation of a bi-layer structure of PMMA (with the low density PMMA at the bottom) to form an undercut, easing the lift-off of the second metal layer to be deposited. In the fabrication procedure used here, an initial layer of low molecular weight PMMA (495k) was spin coated on the wafer (at 1800 rpm for 45 seconds) followed by a oven baking at 185 o C for 15 minutes. A PMMA secondary layer of higher molecular weight (950k) was then applied (at 3000 rpm for 45 seconds) and the baking was repeated. The desired device pattern of a set of doubly-clamped beams, sketched in simple form in Fig. 1·3(b), was then transferred to the charge sensitive positive resist 6
Polymethyl Methacrylate, MicroChem Corp.
26 inside a scanning electron microscope (SEM),7 with direct patterning capabilities.8 The PMMA was exposed to a charge of ∼ 280 µC/cm2 , and the sub-micron features were patterned at a magnification of 850x at a current of 15 pA under an acceleration voltage of 40 kV. The exposed PMMA regions were selectively removed by a development solution,9 and the sample was metal coated through thermal evaporation of a thin layer of chromium. The remaining PMMA, along with the metal layer on top of it, was removed by acetone-methanol lift-off process. The remaining metal layer of the beam pattern serves as the mask in the reactive ion etching (RIE) process that follows. This layer, aligned to the center area of the contact pads, is shown in an optical microscope image in Fig. 1·4(b) and in more detail in an SEM image in Fig. 1·4(c). As shown in Figure 1·3(c), the sample pattern is transferred to underlying silicon device layer by reactive ion etching of uncoated areas.10 Tetrafluoromethane (CF4 ) and oxygen (O2 ) mixture11 was used as the etchant, and the timing was adjusted depending on the silicon layer thickness, with an approximate etch rate of 100 nm/min. Finally, the nanomechanical beams were released by wet etching of the sacrificial layer of the now exposed silicon oxide layer, as shown in Figure 1·3(d). A buffered oxide etchant (BOE)12 was used in removing the oxide layer, at an approximate rate of 80 nm/minute. The timing of the oxide etch process is determined by the thickness of the sacrificial layer or the width of the widest beam, whichever is higher. After the sample is taken out of the etchant solution, it is 7
JSM 6400, JEOL Ltd. Nano Pattern Generation System (NPGS), JC Nabity Lithography Systems. 9 Methyl isobutyl ketone (MIBK):Isopropyl Alcohol (1:3). 10 Plasma Therm 790, Unaxis, Inc. 11 CF4 :O2 (50:5 sccm) at 100 mTorr. 12 6:1 Buffered Oxide Etch, Mallinckrodt Baker Inc. 8
27
(a)
(b)
Figure 1·5: Oblique scanning electron microscope image of (a) a suspended and metal coated family of beams of length 3.6 ≤ l ≤ 9.6 µm and width 200 ≤ w ≤ 400 nm, with a device thickness of h = 200 nm, and (b) the suspended structure, showing the extent of the undercut in the silicon dioxide layer. dipped into deionized water bath and dried with caution. The softer structures (long and thin beams) may collapse due to the strong surface tension effects experienced at this length scale. This surface tension effect is often the limiting factor hindering the fabrication of high-aspect ratio devices. Special drying techniques, like supercritical drying with carbon dioxide may need to be pursued if high-aspect ratio structures are to be fabricated (Kim et al., 1998). While not significantly reacting with chromium, BOE was observed to corrode metals like aluminum at a significant speed, impeding their use in nanomechanical beams. Specialty etchants13 have been recently developed to provide higher etch selectivity between silicon oxide and aluminum. However, attempts to use the pad etch were unsuccessful as the corrosion rate of aluminum was still too high for this 13
Pad Etch 777, Fujifilm Electronic Materials, Inc.
28 application. As a concluding step, sequential layers of chromium and gold were evaporated to create a conductive film on the now exposed substrate, needed for electrostatic actuation purposes. This metal layer is also beneficial to the optical displacement methods outlined in Chapter 2. A SEM image of the completed structure is displayed in Figure 1·5(a) in an oblique view. In Fig. 1·5(b), a magnified image of the suspended nanomechanical beam of h = 200 nm thickness is shown. Also in view is the extent of the undesired undercut below the clamping pad. Prolonged etch times should be avoided to minimize this undercut which leads to increased clamping losses and lower device quality factor Q. Furthermore, over-etching can damage the suspended structure as buffered oxide etch starts to attack the silicon layer, albeit at a much slower rate.
29
Chapter 2
Optical Displacement Detection in Nanoelectromechanical Systems The non-contact, highly-localized and ultra-sensitive nature of optical probes provide significant motivation to pursue them in a variety of applications. Optical detection has been increasingly useful in a variety of fields ranging from particle image velocimetry (Keane et al., 1995), fluorescence microscopy (Rost, 1995), to surface acoustic wave detection (Balogun and Murray, 2006). The ability to perform remote measurements on an area limited only by the optical spot size make the optical techniques extremely attractive as motion transducers in microand nanomechanics. Furthermore, electrical and mechanical motion detection schemes with bandwidth limitations are difficult to implement for displacement detection in these novel devices with high resonance frequencies (Ekinci, 2005). Optical methods, on the other hand, are extremely broadband with a flat responsivity well into the GHz regime. Optical displacement detection techniques have been widely used in microelectromechanical systems (MEMS) (Yaralioglu et al., 1998), atomic force microscopy (Meyer and Amer, 1988; T. R. Albrecht and Smith, 1992), and detection of ultrasonic surface waves (Wagner, 1990). Several optical methods for the above mentioned applications have been developed such as Michelson and Fabry-Perot Interferometry, and optical beam deflection method. Despite the significant dif-
30 ferences in the configuration of all these methods, the displacement of the surface is detected by measurement of a variation in either phase or intensity. Phasemodulation (interferometric) measurements rely on the change of optical path length due to the out-of-plane motion of the surface. Since the phase information is lost in direct power or intensity measurements, phase modulation methods require a reference reflection path of constant phase to create pronounced interference effects from the variations of phase in the detection path. Intensity modulation methods, however, depend on the direct variation of reflectance or scatter due to the motion of the surface with respect to the stationary optical spot. Hence, they can be implemented to displacement detection of in-plane motion. There are significant challenges in scaling these techniques into the nano-scale devices due to diffraction (Kouh et al., 2005b; Karabacak et al., 2005). Sensitive optical displacement detection requires the motion of the structure under study to significantly alter the scattered optical field in either phase or intensity. Innovations in fabrication techniques have enabled mechanical structures of sub-micron dimensions, significantly below the diffraction limited spot size of d ≈ λ/2NA. Here, NA= nsin(θ) is the numerical aperture of the optical components and λ is the free-space wavelength of light. For optical detection on sub-micron devices, the optical field needs to be highly localized. There have recently been significant advances in forming or detecting optical signals in the sub-wavelength dimensions using oil and solid immersion lens (SIL) (Mansfield and Kino, 1990), nearfield microscopy (Hecht et al., 2000; Hartschuh et al., 2003) and numerical aperture increasing lens (NAIL) (Ippolito et al., 2001; Liu et al., 2005). However, most of these techniques require the approach of a relatively large probe to the nanomechanical device to a very small distance, determined by the evanescent wave de-
31 cay length. This technical challenge, requiring precise and complex equipment, hinders the utilization of the near-field techniques outside the laboratory environment. In this chapter, the application of two phase modulation measurement techniques will be introduced, namely Michelson and Fabry-Perot Interferometry, for measurement of the out-of-plane displacement of nanomechanical beams. Additionally, knife-edge displacement detection method, suitable for transduction of in-plane motion of resonators will be described in Section 2.2. As a novel alternative to the more traditional approaches, the diffraction of evanescent waves as a detection mechanism, will also be presented at its conceptual stage with a full analytical analysis, in Sec. 2.3.
2.1
Interferometric Methods
In optical detection of displacement in nanomechanical structures, the devices are often probed by a tightly focused laser beam, formed by a high-NA objective lens (Kouh et al., 2005b; Carr et al., 1998). The reflection of this probe beam returning from the moving NEMS is often collected by the same lens, and is directed onto a photodetector by beamsplitters. In Fabry-Perot interferometry (Fig. 2·4), the power fluctuations in the probe beam are monitored as the NEMS device moves. In path-stabilized Michelson interferometry (Fig. 2·1), the probe beam interferes with a reference beam upon the photodetector. 2.1.1 Michelson Interferometry Michelson interferometry, named after the inventor Albert Abraham Michelson, is a widely used detection method for surface displacements. In a typical setup,
32 shown in Fig. 2·1(a), a coherent laser beam is split into two arms (detection and reference), using a beamsplitter (BS). The detection arm is directed on to the sample, often being focussed into a small spot using a high-NA objective lens (OL). The beam reflecting from the sample surface interferes with the beam reflecting from the reference mirror (RM) on the photodetector (PD). The interference will be constructive or destructive, depending on the phase difference between the arriving waves. λ
PD
RM
I
A BS LPF OL NEMS
PI D
max
Relative Intensity
PZT
NA
I
Points of highest displacement detection sensitivity
min
(z -z ) s
(a)
r
(b)
Figure 2·1: (a) Schematic of a typical path-stabilized Michelson interferometer. (b) The intensity output as a function of static optical path length difference zs − zr . The electric fields reflecting from reference mirror Er and the sample surface Es can be expressed respectively as: Es = Er =
p
Is ei(ωt−k[zs −2u(t)]) ,
p
Ir ei(ωt−kzr )
(2.1)
where u(t) is the time dependent displacement of the sample surface from its equi-
33 librium point (as depicted in Fig. 1·2), k = 2π/λ is the wavenumber, zs and zr are the total static path length traveled by the signal and reference arms, respectively. Is and Ir are the intensities of the sample and reference reflections, respectively, such that Is = Es∗ Es , Ir = Er∗ Er ,
(2.2)
where “*” indicates complex conjugation. Upon interference, the intensity on the photodetector surface can be expressed as Ipd = |Es + Er |2 = (Es + Er ) (Es∗ + Er∗ ) p = Is + Ir + 2 Is Ir cos [k (zs − zr ) − 2ku (t)] √ · ¸ Is Ir = (Is + Ir ) 1 + 2 cos [k (zs − zr ) − 2ku (t)] Is + Ir
(2.3)
For nanomechanical structures, the deflection u(t) is significantly smaller than the optical detection wavelength, u(t) ¿ λ. In such a case, Equation 2.3 can be further expanded as ·
√
Is Ir Ipd ≈ (Is + Ir ) 1 + 2 cos [k (zs − zr )] Is + Ir √ ¸ Is Ir ku (t) sin [k (zs − zr )] . + 4 Is + Ir
(2.4)
This final form of Equation 2.4 allows for the separation of the static background (time-independent) intensity Ibg from the dynamic signal Is that is proportional to
34 the displacement of the sample, Ibg Is
√ · ¸ Is Ir = (Is + Ir ) 1 + 2 cos [k (zs − zr )] , Is + Ir p = 4 Is Ir ku (t) sin [k (zs − zr )] .
(2.5)
A plot of the photodetector intensity Ipd from Equation 2.3 is provided in Figure 2·1(b) as a function of the path difference zs −zr . It is apparent from Fig. 2·1(b), that at odd multiples of λ/2, destructive interference will lead to a minimum intensity as measured by the photodetector, while constructive interference will increase the measured power to its maximum. From Eq. 2.3, the maximum and minimum intensities in the plot can be calculated to be ½
Imin Imax
√
¾ Is Ir = (Is + Ir ) 1 − 2 , Is + Ir √ ¾ ½ Is Ir . = (Is + Ir ) 1 + 2 Is + Ir
(2.6)
The difference between the maximum and minimum intensity values incident on √ the photodetector is often referred to as modulation depth, M = Imax −Imin = 4 Is Ir . The responsivity of the Michelson setup configuration to small changes in optical path length is proportional to the change in optical intensity on the photodetector due to small changes in sample surface, such that R = ∂I/∂z. The point of highest sensitivity for a Michelson setup configuration is obtained when the slope of the relative intensity curve (Fig. 2·1(b)) attains its maximum value, at odd multiples of λ/4. The maximum responsivity of the Michelson configuration can be expressed as RM u
¯ p 2π ∂I ¯¯ = 4k M. Is Ir = = ¯ ∂z zs −zr =λ/4 λ
(2.7)
Equation 2.7 presents two significant implications leading to the optimization of
35 the dynamic signal. The first one is that increase in the total intensity of the laser will improve the measurement sensitivity of the setup linearly. Secondly, given a fixed amount of optical power that is going to be split into the measurement and reference arms of the interferometer, one can optimize the sensitivity by selecting the 50/50 beam splitter for the task, instead of other available ratios. For the setup of this study, a cube beam splitter was selected accordingly.1 In Michelson interferometry, if the reference and signal reflections are of identical wavefronts, the interference pattern will be of uniform intensity. However, achieving identical wavefronts is impossible due to several factors: wavefront abberations from the different optical components in the paths, imperfect collimation of the incoming laser beam, misalignment of the sample with respect to the focal plane of the high-NA objective lens and angular misalignments between the two reflecting surfaces in the setup. Therefore, more realistically, the interfering waves will possess curved wavefronts of unequal phase difference and the interference pattern will have circular rings of bright and dark intensity. The size of the central fringe will be directly proportional to the achievement of good alignment, and therefore should be maximized for improving the performance of the interferometer. The alignment procedure utilized in the construction of this setup starts with collimation of the output of the fiber. The collimation diameter was selected to slightly overfill the back aperture diameter of the objective lens.2 Once the desired collimation diameter is obtained, the beam splitter needs to be inserted to create the detection and reference arms of the interferometer, as demonstrated in Fig. 2·1(a). The angular alignment of the beam splitter, the sample position and reference mirror is then performed to maximize fringe size. 1 2
BS010 400-700 nm Broadband Non-polarizing Beam Splitter Cube, Thorlabs, Inc. M Plan APO SL 80x, Mitutoyo Corporation.
36 The objective lens is then inserted and the sample is brought into its focal plane to reobtain the fringes of similar size. Finally, a spatial filter of circular aperture was inserted in the recombination path leading to the photodetector, to selectively block all light outside the central fringe. The spatial filtering is necessary to prevent the averaging (“wash-out”) of the destructive interference effect in the outer ring and the constructive interference effect in the central ring or vice versa. The nature of the Michelson setup clearly dictates that optimal and reliable performance of the interferometer depends on the stabilization of the static path length difference between the reference and signal reflections. Path-stabilized Michelson Interferometry was proposed (Wagner, 1987) as a solution to suppressing the random fluctuations in the optical path and optimizing the static path length difference. Despite the use of air-damped optical tables, one cannot totally damp out low-frequency room vibrations or prevent air turbulence surrounding the setup. Hence, active feedback during sensitive displacement measurements is essential in the Michelson setup. For the measurements of nanomechanical devices in this study, active path-stabilization was performed by filtering and amplifying the low-frequency (< 5 kHz) components3 of the photodetector signal. The low frequency components were then passed through a commercial PID feedback circuit.4 The reference mirror was placed on top of a piezoelectric actuator stack and the output of the feedback circuit, amplified by piezoelectric actuator driver5 with a gain of 15 V/V, was supplied to actuator. At this stage, an initial sweep of the reference mirror position zr needs to be performed (by sweeping voltage applied across the piezoelectric stack) to obtain 3
SR560 Low Noise Preamplifier, Stanford Research Systems, Inc. SIM960 Analog PID Controller, Stanford Research Systems, Inc. 5 MDT694 Single Channel Piezo Driver, Thorlabs, Inc. 4
37 an intensity curve, similar to that displayed in Fig. 2·1. This intensity sweep allows for the extraction of two critical pieces of information needed for operation of Michelson interferometer. Keeping in mind that the intensity modulation curve will be identical when moving the reference mirror or the sample surface and using Equation 2.7, the modulation depth of the sweep can be used in obtaining the measurement responsivity of the setup to sample vibrations. This relatively easy calibration method is essential in determining the exact displacement value of the sample surface, and provides a significant motivation to choose Michelson interferometry over competing methods. Additionally, it is clear from Equation 2.4 that when the static path length difference zs − zr is adjusted to its optimal value, the voltage readout at the photodetector is going to be proportional to the mean of the sinusoidal intensity curve, such that Vopt ∝
Imin +Imax . 2
Once the Vopt
value is obtained from the reference mirror sweep, it can be used as the reference value in the feedback circuit. A typical interferometric sweep using the configuration described in Fig. 2·1 is presented in Fig. 2·2(a). For this measurement, point of maximum response is determined to be Vopt = 3.95 mV for an incident laser power of P0 ≈ 800 µW on an AFM cantilever of width w = 50 µm, when using a photodetector with a responsivity of RP D = 0.4 A/W.6 The corresponding peak displacement responsivity of the Michelson interferometer is determined from Eq. 2.7 as RM u = 29.8 µV/nm. Once optimum responsivity position of the reference mirror and the corresponding photodetector voltage Vopt is determined, its stability at this position needs to be ensured for the duration of the measurement of high-frequency displacements in the sample. The stability of the mirror can be monitored using the 6
DET210, High-Speed Silicon Detector, Thorlabs, Inc.
38
5.5
5.5
5.0
4.0
(m V)
3.5
V
4.5
4.5
4.0
pd
V
pd
(m V)
5.0
3.0
3.5
3.0
2.5
2.5
0
200
400
600
z -z (nm) s
(a)
r
800
0
200
400
600
800
t (s)
(b)
Figure 2·2: (a) The photodetector signal Vpd during a typical sweep of the reference mirror, as function of the optical path length difference zs − zr , measured using a laser of wavelength λ = 633 nm at a power level of P0 ∼ 800 µW incident on a micromechanical cantilever sample of width w = 50 µm. The photodetector responsivity is RP D = 0.4 A/W. The point of highest sensitivity is determined to be at Vopt = 3.95 mV. (b) The low frequency components of Vpd stabilized at Vopt = 3.95 mV using PID feedback. low-frequency components of the photodetector signal. A typical recording of Vpd during active PID feedback is presented in Fig. 2·2(b), stabilized at the highest responsivity point obtained from the sweep in Fig. 2·2(a). Such stability can be achieved by adjustments to the active feedback parameters Proportional, Integral and Differential gain. In the path-stabilization system, the proportional gain will create responses to the mirror fluctuations at their respective frequencies, whereas integral gain will compensate long term drifts in the mirror. Differential gain is beneficial in compensating for high-frequency, rapid motion. The adjustment of the feedback parameters can be performed by monitoring of the low-frequency components of the photodetector signal while the each gain parameter is incrementally increased. Too much gain will eventually lead the system into instability and the system will begin to oscillate. Once this point is reached, the gain should
39
6
4 3
u
2
S
th,1/2
(pm/Hz
1 /2
)
5
1 0 9.5
10.0
10.5 / 2
11.0
11.5
(kHz)
Figure 2·3: The spectral density of thermal motion of a commercial AFM cantilever with dimensions (w × h × l) 50 µm ×2 µm ×460 µm, in its fundamental mode at atmospheric pressure. The solid line is the estimation from Eq. 2.8 using the experimentally determined quality factor Q = 47. be lowered until system is once again stable. After a stable proportional feedback response is reached, the procedure can be applied to integral and differential gain, in that order. The measurement of the sample displacement can commence once a stable mirror position is obtained. To demonstrate the functioning of our setup, the thermomechanical noise of a commercial AFM cantilever with dimensions (w × h × l) 50 µm ×2 µm ×460 µm, was measured at its fundamental mode. The spectral p density of the thermal motion Su,th of the cantilever was extracted from the spectrum analyzer data using the Michelson interferometer responsivity obtained from Fig. 2·2 RM u = 29.8 µV/nm and the result is presented in Fig. 2·3. The quality factor of the cantilever was determined to be Q = 47 in atmospheric pressure. The spectral noise density of a resonator can be estimated using the response function
40 from Eq. 1.38 as (Ekinci and Roukes, 2005) p SF,th (ω)
q Su,th (ω) =
¯. ¯ ¯ d¯ ms ¯(ωd2 − ω 2 ) − i ωω Q ¯
(2.8)
The spectral density of thermal excitation SF,th (ω) at temperature T can be expressed as (Albrecht et al., 1991) SF,th =
4ms ωd kb T , Q
(2.9)
where kb is the Boltzmann constant. The thermal motion estimation from Eq. 2.8 is plotted as a solid line in Fig. 2·3. Here, the experimentally determined quality factor Q = 47 is used in Eq. 2.9 and the mass of the resonator was estimated from ms = k/ω02 , using stiffness k ≈ 0.3 N/m based on product specifications. Using the responsivity value RM u = 29.8 µV/nm, the detection sensitivity of the Michelson setup can also be estimated. In our setup, the measurements were (AT )
limited by the amplifier noise with spectral density SV ≈ 0.25 nV2 /Hz.7 The √ detection sensitivity Su can be determined based on this limitation, p
S u = RM u
p
pm SV = 0.017 √ , Hz
(2.10)
for an optical power level P0 ∼ 800 µW incident on the sample. As demonstrated here, Michelson interferometry is a very powerful and useful tool in absolute measurements of extremely small displacements. However, the stabilization and alignment issues described above make its application difficult in many environments. The readily available Fabry-Perot cavity configuration in NEMS devices makes Michelson a secondary choice, as discussed at the end of 7
Femto Amplifier HSA-Y-1-60 1 GHz High Speed Amplifier, FEMTO Messtechnik GmbH
41
λ
P0
PD
RFP
A BS
OL
U0
NA
NEMS
TFP
(a)
(b)
Figure 2·4: (a) Sketch of the typical Fabry-Perot configuration where multiple reflections from two partially reflective surfaces can be tuned to obtain very high transmissivity TF P or enhanced reflectivity RF P by varying the ratio of the cavity length U0 to wavelength λ. (b) Setup schematic of NEMS displacement detection using Fabry-Perot interferometry, where the reflection from the nanomechanical beam-substrate cavity is collected by a high-NA objective lens (OL) and directed onto a photodetector (PD). this chapter. 2.1.2
Fabry-Perot Interferometry
Fabry-Perot interferometry, named after its inventors Charles Fabry and Alfred Perot, depends on the interference between reflections from two integrated, parallel and highly-reflective surfaces. In such a geometry, an optical cavity is formed between these surfaces and is filled with standing waves due to multiple reflections from the two surfaces, as demonstrated in Fig. 2·4(a). The reflectivity RF P or transmissivity TF P of these cavities are highly dependent on the ratio of the cavity length to the effective wavelength λ = λ0 /|nc |. Here, nc is the complex refractive
42 index of the medium and λ0 is the free-space wavelength of the incident coherent light. For a cavity with large lateral dimensions, the transmission intensity of the cavity can be calculated by performing a series summation of an infinite number of reflections. For a standard Fabry-Perot interferometer where two mirrors of reflectivity r form a cavity with lateral dimensions larger than the optical spot size, the transmissivity of the cavity becomes (Wagner, 1990) TF P =
(1 − r)2 . (1 − r)2 + 4r sin2 (4πU0 cos (θ) /λ)
(2.11)
Here, θ is the angle of incidence of the incoming light. The transmission of the cavity can be significantly amplified by tuning the effective cavity length U0 cos (θ) to an odd multiple of the effective wavelength λ/2, such that all interferences are constructive. The cavity reflection can be obtained from RF P = 1 − TF P . This Fabry-Perot interference can be employed in obtaining wavelength specific transmission or reflection. Such applications of optical cavities are widely encountered in lasers, telecommunication and spectroscopy technologies. A second possibility for benefiting from this cumulative interference as a sensing mechanism is through changes in reflectance or transmissions due to variations in the cavity length. Therefore, Fabry-Perot interferometry can also be used as a displacement detection method by exploiting the sensitivity of the cavity response to the effective cavity length U0 . A significant advantage of using Fabry-Perot interferometry lies in its integrated nature. Often, the optical cavity is formed from integrated surfaces during fabrication such that the surfaces are guaranteed to be parallel and at specific separations. Additionally, at least one of the surfaces is kept stationary at all times providing stable cavity length, isolated from environmental vibration effects. This
43 configuration greatly reduces the complexity of the setup by removing the reference reflection arm needed in the Michelson setup. Hence, in the Fabry-Perot configuration displayed in Fig. 2·4(b), alignment and stabilization tasks are significantly simplified.
E
H
S pot si ze W0 ≈ λ / 2 N A
k x
depth of f ocus
y z
z0 =
M etal
π W0 2 λ
S i l i con
S acri f i ci al G ap
O pti cal C avi ty
S i l i con S ubstrate x z
(a)
(b)
Figure 2·5: (a) Laser spot focussed on beam (b) cross-sectional view of the multi-layer structure within the optical spot. Recently, Fabry-Perot interferometry has been exploited for displacement detection in NEMS (Kouh et al., 2005b; Carr et al., 1998). Integrated optical cavities are formed in NEMS devices when mechanical structures with high aspect ratios are suspended by chemical etching of the buried sacrificial layer (Carr and Craighead, 1997; Erbe et al., 2000b; Meyer et al., 2003). In these devices, the suspended mechanical structure is of sub-wavelength dimension in one direction, and direct optical access to the underlying substrate surface through a single spot is possible, as shown in Figure 2·5(a). In general, both surfaces are metal-coated and are within a close proximity of each other for electrostatic actuation. Hence, an optical cavity is formed from two highly reflective, parallel surfaces with a separation
44 distance smaller than the depth of focus of the detection optics. A cross-sectional view of this multi-layered structure is displayed in Figure 2·5(b). Unlike a traditional Fabry-Perot interferometer setup, one of the cavity mirrors is laterally limited in dimension. Nonetheless, as will be demonstrated in the following sections, this readily available, integrated optical cavity, where reflections from the substrate and the top surface of the beam interfere, can provide a strong displacement signal after simple alignment. 2.1.3 Modeling of Optical Interferometry in Nanoelectromechanical Systems As beam dimensions are reduced beyond the optical detection wavelength, diffraction effects become increasingly dominant. Efforts to perform Michelson interferometry are hindered as Fabry-Perot cavity effects dominate the signal (Kouh et al., 2005b). On the other hand, this strong cavity effect can be exploited to perform sensitive displacement detection with reduced optical setup complexity. Therefore, an understanding of cavity and diffraction effects become critical to understanding displacement detection of nanomechanical motion. Unlike standard Fabry-Perot interferometry, where lateral dimensions are irrelevant to the problem, an analysis of cavity effects in NEMS is complicated by diffraction effects from sub-wavelength structures. An accurate analytical interpretation of FabryPerot interferometry becomes challenging at this length scale. Furthermore, due to the layered nature of even the most simple nanomechanical devices, the electromagnetic (EM) field travels in various media, creating the need for a careful consideration of material properties. In fact, an accurate understanding of the interaction of light with structures of sub-wavelength dimensions requires a complete solution of the EM field equations in the vicinity of the device. However, the
45 numerical analysis of electromagnetic effects is computationally demanding due to the spatial resolution needed for accurate solutions. As a result, the numerical solution domain is often restricted in size to the near-field and correlation of the complete near-field solution to the far-field observations is only possible through certain approximations. To overcome the above mentioned challenges in analysis of optical displacement detection in NEMS, the problem was split into three sections. The foundation of the analysis is the electromagnetic modeling of interaction between the laser beam and the nanomechanical resonator. Here, EM solution in the cross sectional plane of a nanomechanical beam of doubly-clamped geometry is obtained through finite element analysis. Secondly, the near-field reflection was propagated to the photodetector, based on the parameters of the free-space optics. Finally, the noise analysis of the detection circuit, based on the incident photodetector intensity and corresponding detection sensitivity calculations are presented. The setup of the analysis allows for the combination of multiple sources of detection noise that allowed for the investigation of both fundamental and experimental limitations in the detection schemes of interest. Additionally, the model allows for analysis of both the Michelson and Fabry-Perot interferometry. Near-field Modeling of Electromagnetic Interaction with NEMS Many nanoelectromechanical devices, due to the high aspect-ratio structure at hand, present two significantly different length scales (Karabacak et al., 2005; Kouh et al., 2005b; Carr et al., 1998). The cross-section, shown in Fig. 2·5(b), is of dimensions in the sub-micron range, while the length of the structure is of order l ∼ 10 µm. The motion of the device is often in the responsive axis, in the
46 x − z plane as shown in Fig. 2·5(b). This geometry can be exploited to reduce the problem dimensions. In a typical optical setup for nanomechanical displacement detection, a tightly focussed laser spot is aligned onto a long beam at its point of highest deflection. For doubly-clamped beams, the structures of interest in this study, the point of largest deflection is the center y = l/2. Furthermore, due to the symmetry about the center in the fundamental out-of-plane mode (see Sec. 1.2.1), the deflection of the doubly-clamped structure U (y, t) is uniform across its central portion, i.e. derivative of the deflection at y = l/2 is zero. In our experiments, the focused laser spot is diffraction limited with spot diameter, 2W0 ∼ 1 µm, where W0 is the beam waist radius. Therefore, W0 ¿ l and U (y, t) ≈ u(t) for all practical purposes, in many of the first generation nanoelectromechanical devices. This observations allows for the approximation of reducing the problem to the cross-sectional (x − z) plane. While these conditions hold for all devices within this study, divergence from the results should be expected for smaller devices or larger spot sizes. Furthermore, it should be noted here that, in NEMS, the deformation in the system is significantly small, such that, the static cavity length U0 À u. Lastly, the substrate below the device was approximated to be a silicon handle layer of infinite dimensions. This is an accurate approximation as the silicon layer represents the wafer onto which the nanomechanical structures are fabricated, as described in Section 1.3, which is of thickness ≥ 500 µm. Having established the calculation domain of the optical interaction, the solution method can be selected with better judgement. A variety of numerical tools are becoming available for optical modeling, ranging from finite difference time domain (FDTD) to finite difference frequency domain (FDFD) (Liu and Kowarz, 1999) and integral formulation (Marx and Psaltis, 1997). However, due to the ge-
47 ometric complexity of the multilayered structure in this study and its ease of use, finite element method (FEM) was selected as the numerical approach of this simulation and a commercially available FEM solver was used for this application.8 In finite element tools the optical interactions are obtained by solution of the time harmonic vector field equation (Jin, 2002; Comsol, 2004) ∇ × (∇ × E)/µp − ωl2 εc E = 0
(2.12)
In the above equation, µp is the permeability and εc is the complex permittivity of the medium, such that εc = ε−iσc /ωl for a material of conductivity σc at an optical frequency of ωl . To solve Equation 2.12, the solution domain was meshed using a triangular grid, with all edge length values smaller than λef f /10, where λef f is the effective wavelength in each domain area demonstrated in Fig. 2·5. The solution of the discretized problem is obtained using a stationary, direct solver (Comsol, 2004). The incoming laser beam is going to be approximated using Gaussian optics. This approximation is valid in the paraxial limit of optics and cannot be applied in analyzing optical systems with a very high numerical aperture (Saleh and Teich, 1991). All the relevant parameters necessary for analyzing the beam profile can be uniquely determined by the focal length f and the numerical aperture NA of the objective lens, for a fixed probe wavelength λ. The electric field magnitude Ey of the normalized, focused Gaussian beam with TE-polarization, propagating along z-axis can be expressed as (Saleh and Teich, 1991) Ey0 (x, z) = A0 8
¡ ¢ W0 exp −x2 /W 2 (z) exp (−iφ(x, z)) . W (z)
Comsol Femlab Electromagnetics Module (3.0), Comsol, Inc.
(2.13)
48 Here, the depth of focus of the objective lens is defined as z0 = πW02 /λ and the phase φ(x, z) of the wavefront is φ (x, z) = kz + k
x2 −1 z 2 ¤ − tan z0 2z 1 + (z0 /z) £
The
converging Gaussian beam will £ ¤ R(z) = z 1 + (z0 /z)2 and beam width
have
a
radius
£ ¤1/2 W (z) = W0 1 + (z/z0 )2
(2.14) of
curvature
(2.15)
Given the above parameters, one can calculate the electric field of the Gaussian beam as demonstrated in Figure 2·6(a) for the beam waist of a laser spot formed by an objective lens of NA=0.5 with f =4 mm and λ=633 nm. Fig. 2·6(a) is calculated by providing an incoming laser beam at the upper boundary of the computation domain, with electric field magnitude defined by Eq. 2.13 and all the relevant parameters. All other boundaries are defined to sustain continuity across the boundary. The domain is defined to be a vacuum, such that ε = ε0 = 8.8542 × 10−12 F/m, µp = µp,0 = 1.2566 × 10−6 H/m and n = 1. The obtained solution in Fig. 2·6(a) demonstrates the converging Gaussian profile as expected. To simulate focusing the laser spot upon the suspended nanomechanical structure as shown in Fig. 2·5, the cross-sectional model of the beam was placed in the focal point of the converging Gaussian beam described by Eq. 2.13. Figure 2·6(b) (r)
displays the electric field amplitude Ey (x, z) of the reflected wave for the incoming electric field in part (a). In part (b), lower boundary of the domain was modeled as the silicon substrate while the side boundaries remained in the continuity boundary condition. The electromagnetic properties, such as permittivity εc and conductivity σc , of the materials need to be supplied to the solver for each
49 |Ey|
Max
x
W0 ≈ λ / 2NA
1 μm
z
Min
(a)
(b) (0)
Figure 2·6: (a) Electric field amplitude Ey (x, z) of the incoming Gaussian beam at wavelength λ=633 nm, for a focusing objective lens of NA=0.5, and (r) f =4 mm. (b) The electric field amplitude Ey (x, z) of the reflected-scattered EM field from a NEMS. The scattering NEMS beam has width w=170 nm, cavity length U0 =300 nm and silicon layer thickness h=200 nm (device not shown). This solution is obtained by finite element analysis. The substrate is positioned at the bottom of plot (b). simulation domain (Palik, 1985). The finite element solver is designed to compute the entire electromagnetic field based on a defined boundary condition according to the provided domain properties. However, electromagnetic field amplitude of the reflected wave is experimentally more relevant. Therefore, the incoming field calculation (displayed in Fig. 2·6(a)) was subtracted from the finite element com(r)
putation results in order to obtain Ey (x, z) displayed in Fig. 2·6(b). The finite element solution of the reflected wave in Fig. 2·6(b) is computed for a beam width of w = 170 nm, silicon layer thickness of h = 200 nm and vacuum gap of U0 = 300 nm. Certain important observations can be extracted from the simulation result (r)
of Ey (x, z). It is clear from Fig. 2·6(b) that the reflected beam is strongly affected by reference substrate below the suspended structure, as would be expected for a beam of w = 170 nm. Perhaps less obvious, yet equally important is the fact that the silicon layer of suspended mechanical structure acts as an effective Fabry-Perot cavity formation. Additionally, a clear first order diffraction of the incident laser
50 beam is observed in a wide reflection angle. This scatter is beyond the collection capabilities of the experimental setup, limited by a numerical aperture of NA=0.5 corresponding to a collection half angle of θ = 30. The dynamic solution, where the optical signal change due to the NEMS motion in the out-of-plane direction as a function of time t, can be formulated in the same manner. It is important to keep in mind that the NEMS displacement at its fundamental resonance frequency ω can essentially be treated in a static manner since ω ¿ ωl . As with any interferometer, the electromagnetic field reflecting from the suspended nanomechanical structure is a function of its position u(t)+U0 above the substrate. As established in Sec. 1.2.2, the motion of the suspended nanomechanical structure is of harmonic nature where u(t) = u0 eiωt . Therefore (r)
the position dependent electromagnetic field Ey (x, z) reflecting from the NEMS in motion will possess the harmonic time dependence of its motion, such that (r)
(r)
Ey (x, z, t) ≈ Ey (x, z) eiωt . Wave Propagation to the Far-field Once the reflected wave intensity is extracted from the finite element model of the near-field optics, it needs to be collimated and propagated to the photodetector. For this task, Fourier optics was used. Certain approximations were needed in going from the diffracted-reflected EM field solution to Fourier analysis. Foremost, spatial frequencies in the electromagnetic field subject to Fourier analysis should not exceed the inverse wavelength 1/λ (Saleh and Teich, 1991). However, the EM (r)
field solutions at the focal plane of the lens Ey (x, z ≈ 0) contains high spatial frequency fluctuations - due mostly to the strong diffraction of the sub-wavelength NEMS. Experimentally, these high-spatial frequency evanescent components de-
51 cay within the near-field. Hence, these components are not collected, and can be neglected in our far-field calculations. Secondly, only the diffracted-reflected EM field with wave vector lying within the angle of convergence, θ0 ≈ W0 /z0 , of the lens can be collected (see Fig. 2·6). This collection limit can be expressed ˆ ≤ θ0 , between the unit vector zˆ of optical as the cone angle, −θ0 ≤ cos−1 (ˆ z · k) axis and the unit wavevector kˆ = k/|k| of the reflected wave. By extracting the reflected wave at z ∼ 5λ from the finite element solution, these problems were circumvented. At this distance, beyond the near-field, the high spatial frequency evanescent components have strongly decayed, yet the crucial features in the EM field due to the NEMS are present. Furthermore, the effective collection angle of the optical setup was considered by adjustment of the spatial cut-off so that the solution possesses only the desired wave vectors. Additionally, this distance can still be considered close enough to the focal plane, since the depth of focus of the objective lens is z0 ≈ πλ. Therefore the simplification of assuming the extracted reflection intensity to be approximately at z ≈ 0 does not introduce significant error. The effect of the collection-collimation lens can be expressed through an inverse Fourier transform operation (Goodman, 1996). The collimated electric field at the observation plane z = f , can be obtained as
Ey(c)
(x) ≈
i h k 2 Z∞ exp i 4f x (iλf )1/2
dx
0
Ey(r)
· ¸ 2π 0 (xx ) (x ) exp i λf 0
(2.16)
−∞
In this equation, the integration is performed over the focal plane z = 0 of the lens with focal length f . (c)
The obtained collimated field Ey (x) is then propagated in free-space, along the optical (z) axis. The electric field at the photodetector, located at a distance
52 z = Lp from the collimation lens, can be represented as an integral of harmonic functions (Saleh and Teich, 1991), Z∞ Ey(p) (x, z = Lp ) =
Fcol (νx ) exp (−i2πνx x) exp (−ikz Lp ) dνx .
(2.17)
−∞ (c)
In Equation 2.17, Fcol (νx ) is the Fourier Transform of Ey (x) with spatial frequency of νx = kx /2π and wavenumber kx . The wavenumber along the optip cal axis of propagation is defined as kz = k 2 − kx2 . The propagation length Lp is much greater than any of the length scales in the nanomechanical structure and the near field calculation. The free space propagation calculation defined by Eq. 2.17 was implemented using Fast Fourier Transform (FFT) algorithm.9 To minimize aliasing effects from the numerical FFT, beam propagation calculation can only be performed in incremental propagation steps, in an iterative manner (Travis et al., 2000). In this calculation of free-space propagation, it was observed that incremental propagation steps of ∆L ∼ λ were sufficiently small to prevent aliasing. Spatial window of the discrete Fourier Transform is picked large enough to avoid the edge effects associated with numerical FFT (Davis, 2003). Detection Circuit As the final step of modeling, the optical power in the electromagnetic field returning from the nanomechanical structure is converted into a photodetector current I. For this purpose, the introduction of optical reflectivity RN EM S for the NEMS device is convenient. RN EM S is defined as the ratio of the reflected power to the incident power P0 on the device. The total reflected power is the integrated in(pd)
tensity profile of the electric field Ey 9
Matlab 6.5, MathWorks, Inc.
incident on the photodetector. Hence, the
53 reflectivity can be expressed as an integral over the photodetector surface A,
RN EM S =
1 P0
Z r A
¯ ¯ ¯ (pd) ¯2 E ¯ ¯ y ε0 dA. µp,0 2
(2.18)
Using Eq. 2.18, the photodetector current can be written as I = Rpd P0 RN EM S , where Rpd = ηe/~ωl is the photodiode responsivity (Wagner, 1990). Here, η is the quantum efficiency of the photodetector, e is the electronic charge and ~ is Planck’s constant. The constants ε0 and µp,0 are permittivity and permeability of free-space, respectively. As demonstrated in Fig. 2·4, in Fabry-Perot interferometry the electromagnetic field incident on the photodiode is simply the propagated wave. Therefore, the (pd)
EM incident on the photodetector Ey (pd)
from Eq. 2.17, such that Ey
is equal to the reflected wave, as obtained
(p)
= Ey . In Michelson interferometry, schematized in
Fig. 2·1, the intensity incident on the photodetector can be simulated by interfering (ref )
the propagated wave with a reference beam, Ey
(pd)
. Hence, Ey
(p)
(ref )
= Ey + Ey
needs to be inserted into Equation 2.18. Noise Analysis The noise model for the photodetector-amplifier circuit is presented in Figure 2·7. The noise sources, indicated in gray in Fig. 2·7, originating in the detection circuit are the shot noise and the dark current noise of the photodetector, and the electronic noise of the amplifier. These various sources may be expressed in terms of their equivalent current noise with power spectral density SI , with units of
54
Photodetector
vA
Amplifier 50 Ω AV
iSignal
iShot
iA
iDark
Figure 2·7: The photodetector circuit model. The noise in the circuit arises from uncorrelated current and voltage sources shown in gray; the signal current source is in black. The RF amplifier has a 50 Ω input impedance and two uncorrelated noise sources. (S)
A2 /Hz. For instance, the shot noise power spectral density SI Z r (S) SI
= Rpd A
ε0 µp,0
is formulated by
¿¯ ¯À ¯ (pd) ¯2 ¯Ey ¯ 2
dA.
(2.19)
(D)
The hi brackets denote the time averaging operation. The dark current noise SI
depends upon the reverse bias and the size of the active photodetector area (Saleh and Teich, 1991). The noise generated within the amplifier can be described by two uncorrelated noise sources: a voltage noise and a current noise source with (A)
power spectral densities SV
(A)
and SI
respectively, as displayed in Figure 2·7. In
radio-frequency (RF) amplifiers, the input impedance Rin = 50 Ω and therefore the total noise can be converted into a noise current where (AT )
SI
(A)
≤ SI
(A)
2 + SV /Rin .
(2.20)
In a typical photodetector-amplifier circuit, at the low optical power levels P0 ∼ (S)
100 µW used for nanomechanical motion detection (Kouh et al., 2005b), SI (D)
1pA2 /Hz, SI
(AT )
≈ 10−3 pA2 /Hz, and SI
≈
≈ 100pA2 /Hz. Therefore, optical dis-
55 placement detection in NEMS is limited by the amplifier noise at power levels at P0 ≤ 1 mW. Efforts to increase optical power have been hindered by strong absorption of silicon in the visible wavelengths which create significant thermal effects in nanomechanical beams (Sampathkumar et al., 2006; Ilic et al., 2005). These thermal effects lead to significant frequency shifts, which will be further discussed, and can create unstable response or even permanent damage to the devices. With the above definitions complete, the transfer function of the nanoelectromechanical system, that relates the nanomechanical displacement u to the resulting photodetector current I can be defined. The above described finite element analysis provides the tools to characterize dependence of the reflected EM field on the device displacement u. Complementing this, the intensity field profile incident upon the photodetector was determined from the collimation and propagation calculations. Concluding the analysis is the photodetector responsivity, which relates the electromagnetic field intensity on the photosensitive surface to the obtained current I. In Figure 2·8, a sample calculation of the photodetector current as a function of the beam center position upon the substrate using the complete model is presented. In this computation, a typical nanomechanical beam of width w = 170 nm and silicon layer thickness h = 200 nm with a chromium metal coverage of thickness 15 nm was used as the geometric factors. The incident field was taken to be of wavelength λ = 633 nm with the incident laser power P0 = 100 µW. The result of Fig. 2·8 clearly indicates a direct correlation between the position of the beam center and current modulation ∆i(t) obtained at the photodetector. The combination of all the components of the model allows us to numerically
56
∆i(t) (µA)
0.6 0.3 0.0 -0.3
U0+u(t) (nm)
-0.6
352
348
344
Time (a.u.) Figure 2·8: The photodetector current (upper plot) calculated as a function of the beam position (lower plot) using the presented model. The following parameters are used: w = 170 nm, h = 200 nm, U0 = 350 nm with a vibrational amplitude of u0 = 5 nm. The current is calculated at the photodiode located at Lp ≈ 15 cm from the sample. The detector responsivity is Rpd = 0.4 A/W and the probing laser beam power is P0 = 100 µW at λ = 633 nm. determine the system responsivity to NEMS displacements as ¯ ¯ ¯ ∂I ¯ 1 Ru (ω) = ¯¯ ¯¯ = Rpd ∂u P0
¯ ¯ ¯ ∂P ¯ ¯ ¯ ¯ ∂u ¯ .
(2.21)
The system responsivity can then be used in establishing the minimum detectable displacement signal. Minimum detectable displacement is commonly extracted from a signal-to-noise ratio of unity (SNR=1). In the optical detection scheme, the main sources of noise originate from the photodetector circuit, as described through Eq. 2.20, the laser source and the mechanical vibrations of the overall setup. Mechanical vibrations of the entire setup are at lower frequencies than the operation frequencies of NEMS. Nonetheless, for signal stabilization purposes, the mechanical fluctuations of the detection system was minimized by increasing
57 the setup compactness and rigidity. Furthermore, the sample was placed on a mechanical alignment stage with piezoelectric feedback capabilities and its position was fixed at the focal plane. The laser source exhibits intensity and phase fluctuations, and 1/f noise. Although the laser source fluctuations are not in anyway the limiting noise floor, they can be limited through temperature stabilization and back-reflection minimization. Hence, at the experimental power levels of this study, P0 < 1 mW, the remaining significant noise contribution is that of the amplifier circuit, with an experi(AT )
mentally determined noise floor of spectral density SI
≈ 100 pA2 /Hz.10 The
spectral density of the current noise can now be converted into minimum de√ √ tectable vibration amplitude, i.e. displacement noise, Su in units of m/ Hz, q p Su = (AT )
based on the dominant noise source SI
(AT )
SI
Ru
,
(2.22)
and system responsivity Ru , assuming
SNR=1. Fabry-Perot Interferometry Results The above described model can now be employed in studying the sensitivity limits of NEMS displacement detection based upon optical cavities (Kouh et al., 2005b; Carr et al., 1998). Although a very wide parameter space can be explored using the developed model, the emphasis will be placed on device geometry, which will both allow for understanding optical phenomena and assist in the development of future devices of enhanced sensitivity. The effect of varying width w, thickness h and sacrificial gap U0 will be studied. Additionally, the wavelength 10
Femto Amplifier HSA-Y-1-60 1 GHz High Speed Amplifier, FEMTO Messtechnik GmbH
58 λ will be scanned in an effort to obtain an improved understanding of the cavity effects. To allow for a reliable comparison, the reflectivity RN EM S will be plotted in each analysis. The displacement responsivity RFu P of the Fabry-Perot Interfer√ ometry and the displacement noise floor Su will also be obtained. In all of the following calculations RP D = 0.4 A/m and P0 = 100 µW will be used, motivated by the experimental conditions. 170 nm
0.8
0.7
400 nm
0.10
FP u
0.6
300 nm
0.15
( A/nm)
400 nm
R
170 nm
300 nm
0.05
0.5
0.00
0.4 0
100
200
300
400
500
100
200
300
(a)
400
500
(nm)
(nm)
(b)
Figure 2·9: Optical characteristics of a set of NEMS beams with w = 170, 300 and 400 nm and h = 200 nm as a function of U0 . The plots are for P0 = 100 µW and RP D = 0.4 A/W. (a) Device reflectivity R oscillates with a period of ∝ λ/2, as U0 is varied. Note that oscillation amplitude of R is highest for the smallest structure. (b) Displacement responsivity RFu P is the relevant quantity for displacement detection. It is calculated by numerically differentiating the data in part (a) with respect to U0 . Figure 2·9 demonstrates the effects of the sacrificial (vacuum) gap thickness U0 upon the optical characteristics of the system. In Fig. 2·9(a), the change in reflectivity RN EM S of power from the cavity is plotted as U0 is varied for three different beams with w = 170, 300 and 400 nm for a fixed beam thickness h = 200 nm and λ = 633 nm. A clear oscillation in power is observed from the simulation results with an oscillation period of ∼ λ/2, independent of the critical beam dimension
59
170 nm
1 /2
)
300 nm 400 nm
1
S
u
1/2
(pm/Hz
10
0.1 0
100
200
300
400
500
(nm)
Figure 2·10: Displacement sensitivity results of Fabry-Perot interferometry, (AT ) based upon SI ≈ 100 pA2 /Hz, for a set of NEMS beams with w = 170, 300 and 400 nm and h = 200 nm as a function of U0 . The plots are for P0 = 100 µW and RP D = 0.4 A/W. w. This observation is consistent with an interferometric effect, in this case forming between the light reflected from the top of the nanomechanical beam and that returning from the underlying substrate. The displacement responsivity RFu P , described in Eq. 2.21 can be extracted from Fig. 2·9(a) for Fabry-Perot interferometry by determining the rate of change of the photodetector signal with respect to the cavity length U0 . By taking a numerical derivative of Fig. 2·9(a), RFu P as a function of U0 can be obtained as illustrated in Figure 2·9(b). By utilizing the dominant (AT )
noise value SI
≈ 100 pA2 /Hz, the displacement sensitivity (noise floor) can also
be determined - as shown in Figure 2·10. In sub-wavelength devices, the width w of the NEMS beam essentially determines the reflectivity of the device. The effects of the beam width are already apparent in the plots of Fig. 2·9 and 2·10. In Fig. 2·11(a), the vacuum gap U0 and the beam thickness are kept constant at U0 = 400 nm and h = 200 nm, respectively while the beam width w is varied to determine the reflectivity of the device. The
60
1.0
0.6
Metal Metallized i
0.4
0.10
0.05 u
R
0.8
FP-max
( A/nm)
0.15
S
0.00 0
400
800
1200
0
400
800
w (nm)
w (nm)
(a)
(b)
1200
Figure 2·11: (a) Reflectivity as a function of the beam width for metallic and metallized silicon NEMS beams. Here, silicon layer thickness is set to h = 200 nm, and the gap thickness is set to U0 = 400nm. A cavity resonance appears to become dominant at w ≈ 160nm in the metallized silicon beams. (b) Maximum displacement responsivity RFu P −max of the Fabry-Perot technique as a function of w. calculation was repeated for metallized silicon NEMS as well as for metallic devices for comparison. A metallic device shows a monotonically decreasing RFu P as the optical spot size becomes larger than the device - until all the light starts to reflect from the substrate. The metallized silicon, in contrast, exhibits a peak in R around w ≈ 160 nm. The slight increase in the device reflectivity below w = 160 nm is possibly the result of an increase in the fraction of the power reflecting from the substrate, as well as a possible cavity effect due to the presence of silicon. In Fig. 2·11(b), maximum displacement responsivity RuF P −max as a function of w is plotted. For obtaining RFu P −max , plots similar to Fig. 2·9(b) were generated for each w and peak amplitudes were extracted. Finally, the incoming laser wavelength λ is another parameter that was studied. In this part of the analysis, the nanomechanical beam dimensions and the optical parameters such as the NA and f were fixed while the optical response as
61 a function of λ. Cavity resonances of light have been studied and exploited for device characterization in MEMS (Stievater et al., 2003) and in micron-scale optomechanical filters.(Tucker et al., 2002) In this study, the analysis was extended to sub-wavelength NEMS structures by employing a range of wavelengths from λ ≈ 400 nm up to λ ≈ 1.3 µm. Optical properties of silicon vary significantly within this spectrum and needed to be adjusted in the simulations with the correct permittivity values (Palik, 1985). When analyzing the optical cavity formed by the geometry displayed in Fig. 2·5, it is necessary to include the silicon layer as well as the sacrificial gap below the (ef f )
metal coated surface. Hence, the effective cavity length U0 (ef f )
U0
= U0 + nSi h,
can be defined as (2.23)
where nS i is the silicon refractive index at the appropriate wavelength. The reflectivity values of the cavities in suspended silicon beams with w =170, 300 and 400 nm are displayed in Figure 2·12, for constant beam height U0 = 400 nm and thickness h = 200 nm. Several resonances are apparent in each NEMS device analyzed here. In Fabry-Perot interferometry, enhanced reflections are obtained when the multiple reflections that coincide are in-phase to create constructive interference. Hence, based on the simple optical path length arguments, optical resonances are expected at (Vaughan, 1989) (ef f )
λq =
2U0 q
, for q = 1, 2, 3...
(2.24)
Here, q is the axial mode number and it is assumed that the wavefront that enters the cavity is perpendicular to the reflecting surface. The results in Fig. 2·12 for the w = 170 nm beam clearly show the expected resonances at their respective wave-
62
170 nm 300 nm
0.8
450 nm
R
0.6
0.4
0.2 400
600
800
1000 1200 1400
(nm)
Figure 2·12: Device reflectivity R as a function of λ, for three devices with width w = 170, 300 and 400 nm. Here, h =200 nm, U0 = 350 nm and parameters of the free space optics are kept constant for all the simulations. lengths, for mode number values of q = 2, 3 and 4 in the spectrum analyzed. In larger beam widths, such pronounced peaks are harder to locate. This weakness could be attributed to the reduced power flow into the cavity as w is enlarged, or due to increased absorptive losses in the material. In any case, the calculated optical quality factor values are highest in the w = 170 nm beam approaching Qcavity ≈ 10 at λ ≈ 640 nm; as the beam width is increased at the same wavelength, Qcavity decreases to Qcavity ≈ 8 for the w = 450 nm beam. Apparently, the quality factor values reported here are much lower than those measured in MEMS devices (Tucker et al., 2002). This is well expected as the sub-wavelength cavities demonstrate significant diffraction effects and hence loss of quality. Michelson Interferometry Results The above discussion can be extended to Michelson interferometry with minor adjustments in the model. Primarily, to assess the effectiveness of Michelson interferometry in NEMS, cavity effects ought to be removed from the analysis.
63 Physically, this corresponds to removing the substrate below the nanomechanical structure, sketched in Fig. 2·5 (Kouh et al., 2005b). The remainder of the analysis of Michelson interferometry is essentially the same as the analysis of Fabry-Perot (p)
interferometry. The only difference is in the final step, where the EM field Ey (ref )
returning from the NEMS device is interfered with a reference beam Ey . The ¯ ¯ ¯ (ref ) ¯ (ref ) reference beam can be defined as Ey = ¯Ey ¯ e−iϕref with an arbitrary phase ϕref , which can be adjusted by shifting the location of the reference mirror in a typical setup, as discussed in Sec. 2.1.1. The intensity of the interference profile ¯ ¯2 ¯ (p) (ref ) ¯ formed on the photodiode is proportional to ¯Ey + Ey ¯ , which is expanded as ¯ (p) ¯ ¯ ¯ ¯ ¯ ¯ ¯¯ ¯ ¯Ey + Ey(ref ) ¯2 ≈ ¯Ey(p) ¯2 + ¯Ey(ref ) ¯2 + 2 ¯Ey(ref ) ¯ ¯Ey(p) ¯ cos (∆ϕ) .
(2.25)
Here, ∆ϕ is roughly the phase difference between the reference and probe beams, and depends upon the NEMS position as well as ϕref . The photodetector current in Michelson interferometry is, therefore, Z r I (M ) = RP D A
¯2 ¯ ¯ (p) (ref ) ¯ + E E ¯ ¯ y y ε0 dA µp,0 2
(2.26)
In a similar fashion to Eq. 2.21, the displacement responsivity for small NEMS displacements can be extracted as RM u
¯ (M ) ¯ ¯ ∂I ¯ ¯. (ω) ≈ ¯¯ ∂u ¯
(2.27)
In experiments, one usually adjusts ϕref until a maximum responsivity is obtained, as described in Sec. 2.1.1. Mathematically, this can be compensated in the calculations through normalization of the obtained Michelson responsivity values to remove the effect of ϕref . RM u and the displacement detection noise floor
64
1.6
Metal 1/2
)
Metallized Si
(pm/Hz
Metal Metallized i
0.4
U
1/2
M( u
0.8
0.1
S
A/nm)
1.2
0.01
S
0.0 0
400
w
(a)
800
(nm)
1200
0
400
800
w
1200
(nm)
(b)
√ Figure 2·13: (a) The displacement responsivity RM Su of u and (b) noise floor Michelson interferometry in NEMS. The back substrate is removed to eliminate (AT ) ≈ 100 pA2 /Hz and any cavity effects. Here, P0 = 100 µW at λ = 633 nm, SI RP D = 0.4 A/W. √
Su for Michelson interferometry is shown in Figure 2·13(a) and (b), respectively.
This calculation is performed for the typical experimental values of U0 = 400 nm, h = 200 nm at λ = 633 nm and P0 = 100 µW and RP D = 0.4 A/W for metal and metallized silicon beams. As before, the dominant noise source value (AT )
of SI
≈ 100 pA2 /Hz was used. In the results, it can be clearly observed that
the responsivity decreases and the noise floor increases as the reflecting surface of the NEMS beam is reduced below the diffraction limited spot size. Also apparent is an interesting resonance in the silicon beams. Given that the substrate is removed in these calculations, we speculate that this resonance arises inside the silicon layer. Discussion In this section, a detailed numerical model was developed to study optical displacement detection in NEMS. This model was employed in obtaining the dis-
65 placement sensitivity limits as a function of various device parameters, optical parameters and interferometry type. The numerical results clearly indicate that devices with certain dimensions enable enhanced displacement detection. These results can be exploited to implement ultra-sensitive displacement detection in NEMS, through various optimization steps. At the initial stages of device design, one can select a SOI wafer with the appropriate layer thicknesses to enhance the response. At the measurement level, one can tune the vacuum gap U0 by flexing the beam towards the substrate using static electrostatic forces to obtain responsivity levels that approach Rmax . u Finally, the wavelength λ can be adjusted in the measurement stage to improve the optical response of the device. The developed analysis was used in comparison with previously obtained data (Kouh et al., 2005b). The finite element solution clearly predicts the observed trends in both Fabry-Perot and Michelson interferometry measurements. In Michelson interferometry, the increasing loss of sensitivity in sub-micron devices is both numerically and experimentally observed. Additionally, the increase in Fabry-Perot interferometry responsivity is also well-described the model (Kouh et al., 2005b). The observed discrepancy in the responsivity values extracted from the numerical and experimental observations can be attributed to significant signal losses in free-space optical setups. It is unfortunately quite challenging to incorporate these into calculations. These differences in amplitudes which are essentially scaling factors to the data, do not affect the overall conclusions that can be extracted from this theoretical study. The study of Michelson interferometry indicates that the displacement sensitivity of path-stabilized Michelson optical interferometry quickly deteriorates in
66 the NEMS domain, RM u decreases monotonically with decreasing beam width w without exception. This signal loss can be attributed to the strong diffractive effects that scatter the incident laser beam. On the other hand, Fabry-Perot interferometry response increases continually as device dimensions are reduced, within the current fabrication limitations. The signal enhancement for sub-wavelength devices are believed to be a result of increased reflection from the substrate as the device width w is reduced. Hence, Fabry-Perot interferometry becomes highly appealing for the NEMS domain.
2.2
Knife-edge Technique
Optical interferometry described in Section 2.1 is very useful and sensitive in detecting out-of-plane displacements at the nanometer scale. Interferometric measurement scheme relies on changes in the optical path length to create phase difference between the probe beam and reference beam. However, recently various nanomechanical structures have been developed as in-plane resonators, whose motion is perpendicular to the optical axis (Kouh et al., 2005a; Truitt et al., 2007; Meyer et al., 2003; Almog et al., 2007). To complement the interferometric detection schemes developed in our lab, and to couple optically to the motion of these novel structures, an optical knife-edge displacement detection method was developed (Karabacak et al., 2006). The technique relies on recording the variation in reflected optical power from the active mechanical structure, as the doubly-clamped beam moves inside a Gaussian optical spot. The mechanical structure essentially acts as a knife-edge by partially reflecting the incident optical beam. Knifeedges are widely used for characterizing optical spots (Khosrofian and Garetz, 1983; Firester et al., 1977) and have been previously implemented in optical beam-
67 deflection based displacement detection (Wagner, 1990). However, its adaptation to displacement detection in this configuration and its application to nanomechanical beams is novel. Although the underlying principles are significantly different, the component layout required by this methodology is identical to that of the Fabry-Perot interferometer, presented in Section 2.1.2. Hence, the optical knifeedge technique was easily implemented with the existing setup.
Silicon Nitride Silicon
(a)
(b)
(c)
(d)
Figure 2·14: Schematic of doubly-clamped beam fabrication from nitride membranes. (a) The process starts by etching of an access window at the backside of a double-side coated silicon nitride-silicon structure. The window is defined using photolithography and the nitride layer is etched using RIE. (b) The exposed silicon layer is wet etched using KOH solution to create a thin nitride membrane. (c) The nanomechanical beam-gate structure is patterned by EBL and thermal deposition of a thin metal film. (d) The pattern is etched into the nitride layer using RIE to release the structure. To demonstrate the concept, doubly clamped nanomechanical resonators fabricated from silicon nitride were employed (Kouh et al., 2005a; Truitt et al., 2007). The standard fabrication process of in-plane resonators, outlined in Fig. 2·14, starts with the etching of a window on one side of a double sided silicon nitride wafer. This window, with dimensions in ∼ 100 µm range, is defined optically and etched into the silicon nitride layer using reactive ion etching (RIE). The exposed silicon layer is then wet-etched in potassium hydroxide (KOH) solution until the opposite nitride layer is reached, as demonstrated in Fig. 2·14(b). This process forms a thin nitride membrane exposed on both sides, and its thickness is defined by the nitride thickness of the commercially obtained wafer. The lateral dimensions of
68 the membrane are determined by the size of the etch window and the etch time. The nanomechanical structure is then defined on top of this membrane through electron beam lithography (EBL) and thin-film deposition. The pattern is transferred to the underlying nitride layer by a secondary RIE process. As a result of the etching of the uncoated areas, the nanomechanical resonator is released. As shown in the artificially colored scanning electron microscope (SEM) image in Figure 2·15(a), an electrically isolated side gate is often fabricated next to the doubly-clamped beam for in-plane electrostatic actuation. The optical reflectivity was further enhanced with the thermally deposited thin layer of chromium and gold. The device displayed in Fig. 2·15(a) is of beam dimension (w × h × l) 200 nm×125 nm×14 µm and has a beam-gate separation g = 130 nm. The quality factor of the resonator was determined to be Q ∼ 800 in ultra-high vacuum (UHV) conditions. The knife-edge measurement configuration is illustrated in Fig. 2·15(b), showing the nanomechanical beam and the optical spot. The experimental measurements were performed with a He-Ne laser (λ=633 nm) which was tightly focused onto the nanomechanical beam by an objective lens with a numerical aperture of 0.5.11 This configuration resulted in a diffraction limited optical spot of diameter d = 1.2 µm (FWHM), determined by scanning the laser spot across a pad edge and fitting a Gaussian beam profile to the derivative of the measured reflected optical power (Khosrofian and Garetz, 1983). The center of the optical spot is offset by a distance xs from the equilibrium beam center position at x = 0 and the displacement of the beam center during actuation is denoted u. In the knife-edge technique, the detection responsivity RKE can be defined as 11
G Plan Apo 50x, Mitutoyo Corporation
69
u
(a)
(b)
Figure 2·15: (a) Scanning electron micrograph of a doubly clamped silicon nitride beam with a side gate. Here, w × h × l = 200 nm×125 nm×14 µm; g = 130 nm. (b) An illustration of the experiment setup. The optical spot is focused at an offset of xs from the beam center of the NEMS resonator. The NEMS center displacement from equilibrium is u. The center of the beam is aligned to the origin. the change in the reflected optical power Pr due to a change in nanomechanical beam deflection u, as denoted in Fig. 2·15(b), RKE =
1 ∂Pr . P0 ∂u
(2.28)
In Eq. 2.28, RKE is normalized with respect to the incident power P0 . To demonstrate the functionality of the optical knife-edge technique, the fundamental resonance mode of the beam displayed in Fig. 2·15(a) was measured with this method (Karabacak et al., 2006). The obtained response signal is displayed in Figure 2·16 for varying static actuation voltage VDC . The resonance measurements such as those shown in Fig. 2·16 were performed at the point of maximum available responsivity RKE0 . Such an optimal operation point xs0 can be found by shifting the optical spot position xs with respect to the nanomechanical beam while moni-
70
Figure 2·16: In-plane fundamental flexural resonance of the beam shown in Fig. 2·15(a), under varying drive amplitude VDC . The inset displays the change in the resonance frequency ω0 with VDC . toring the high frequency displacement measurement, and finally locking the relative position of the optical spot at the position of maximum signal. The obtained optical power response was calibrated to actual displacement values using the detection system responsivity RKE0 , using the procedure outlined below. This optical spot sweep approach has proven to be effective in obtaining the highest signal-to-noise ratio. However, for absolute measurements of the nanomechanical beam displacements, the responsivity of the optical setup RKE needs to be determined in absolute terms, in units of µm−1 . In Michelson interferometry, the same problem was resolved by determining the optical response to a controlled change in reference path length, as described in Section 2.1.1. Here, a similar method was devised. In the knife-edge technique, the variation in the optical reflection occurs due to a change in the position of the nanomechanical beam with respect to the center of the optical spot with a nonuniform intensity distribution. Exploiting this fact, it can be approximated that the same change in reflected optical power will be obtained if the optical spot
71 is shifted with respect to the beam. Based on this first-order approximation, a displacement u of the beam center relative to the optical spot ought to give rise to the same measured power change as a displacement xs of the spot relative to the stationary beam, if u = −xs . Using this formalism, the responsivity RKE expressed in Equation 2.28 can now be reformulated as, RKE ≈ −
1 ∂Pr . P0 ∂xs
(2.29)
Equation 2.29 provides an approximate RKE value can be determined by measuring Pr as a function of xs and extracting ∂Pr /∂xs . Therefore, peak responsivity RKE0 and most sensitive measurement position can then be obtained for a given spot diameter d and beam width w. The above approximation is only valid when the following conditions are satisfied: (i) the power reflected from the side gate is negligible, and (ii) the beam is long so that the curvature around the beam center during motion is small. The second condition is easily satisfied for in-plane nanomechanical structures currently being tested, whose length is l À u, as explained in Sec. 2.1.3 (Karabacak et al., 2006; Kouh et al., 2005a; Truitt et al., 2007). On the other hand, in some cases condition (i) is not fulfilled and its effects will be further discussed below. If the incident optical beam intensity profile is taken to be a Gaussian function, Pr (xs ) and knife-edge technique responsivity RKE can be analytically extracted. The optical intensity incident on the device, in the x − y plane, can be expressed as 2 /d2
Iinc (x, y) = I0 e−2(x−xs )
e−2y
2 /d2
,
(2.30)
where I0 is the peak intensity at center of the optical spot at (x = xs , y = 0). If diffraction effects are neglected, the reflection from the device can be calculated
72 by integrating the incident intensity in Eq. 2.30 over the reflecting portions of the device surface shown in Fig. 2·17(b) (Firester et al., 1977; Khosrofian and Garetz, 1983). Therefore, the reflected optical power becomes, Z+∞ Z+∞ Pr (xs ) ≈ Iinc (x, y)R(x, y)dxdy.
(2.31)
−∞ −∞
The function R(x, y) represents the reflectivity of the surface. For the simplest form, it can be taken as unity if the point is on the device or the gate, where the gold layer provides a high reflection, and zero everywhere else. The extraction of absolute responsivity can be easily implemented experimentally. Precision stages that allow for measurements of translation were utilized in scanning the optical spot over the devices, while recording Pr (xs ) in real-time. To explore the limitations of the above described method, such spot-scan measurements were performed on two representative devices with different dimensions. The devices, henceforth D1 and D2 , had identical thickness (h = 125 nm) and length (l = 14 µm), while their lateral geometry was varied: D1 had w = 500 nm and g = 500 nm; D2 had w = 200 nm and g = 100 nm. The important difference between the two devices comes from the fact that g ∼ d/2 for D1 , but g < d/2 for D2 . The inset of Fig. 2·17(a) displays Pr (xs ) for D1 . At large negative values of xs , no light is reflected since the optical spot is too far away for any laser portion of the laser beam to be incident on the device. As the spot is scanned over the beam, the reflected power increases due until the spot is completely over the gate (large positive xs ) and all of the incident light essentially reflects back. The main plot in Fig. 2·17(a) is the normalized numerical derivative of Pr (xs ) with respect to xs for D1 . Similar plots for D2 are presented in the inset and the main body
73 of Fig. 2·17(b). The results obtained from Eq. 2.29, using the reflected power estimated from Eq. 2.31, are presented as the dashed lines in both Figs. 2·17(a) and (b). The spot size of d = 1.2 µm used in the calculations was determined from separate knife-edge measurements. The data and the calculation are in good qualitative agreement. The discrepancy in the magnitude of the peaks can be attributed to diffraction effects at this scale. 1.0
0.5
0.5
0.4
s
-4
0.2
0
xs (
4
8
m)
r
r
0.0 -8
0.6
PR (norm.)
P / x (1/ m)
PR (norm.)
0.4
s
P / x (1/ m)
1.0
0.0 -8
-4
0.2
0
xs (
4
8
m)
0.0
0.0 -12
-8
-4
xs ( (a)
0
m)
4
-0.2 -12
-8
-4
x s(
0
4
m)
(b)
Figure 2·17: Normalized optical responsivity of the NEMS devices, with dimensions of (a) w = 500 nm and g = 500 nm and (b) w = 200 nm and g = 100 nm. The inset shows the reflected power PR as a function of the spot position xs . Analytical calculation results for |∂PR /∂xs | are plotted as dashed lines. In (b), the solid line shows |∂PR /∂xs | without the contribution of the gate. One significant difference between the results obtained from the two different geometries is immediately apparent in Figure 2·17. In part (a), with wide gatebeam gap g, the responsivity displays double peaks whereas the sweep across device D2 , displayed in Fig. 2·17(b), produced a single peak in |∂Pr /∂xs |. The same trend is exactly predicted by the analytical solution provided by Equation 2.29, shown as dashed lines. The physical explanation of this response can be provided as follows: as the optical spot is being scanned from large negative values
74 of xs to positive xs , the optical spot first crosses over the nanomechanical beam. This creates an increase in the reflected power. Once over the beam, the reflection decays as the spot crosses over the gate-beam gap. The second peak in |∂Pr /∂xs | arises as the peak of the Gaussian intensity profile crosses onto the gate area. The two separate peaks indicate that the gate contribution to the reflected power is negligible around xs0 . Thus, Eq. 2.29 is accurate in predicting RKE0 in devices such as D1 where g ∼ d/2 or greater. The extraction of RKE0 from devices with small gate-beam separation distance (g ¿ d/2) requires further analysis. In such geometries, the reflection contribution from the gate becomes significant enough to compensate for the decay in the reflection due to the gap, hence a single slope of reflection increase is observed during a scan. In structures such as D2 , the measured ∂Pr /∂xs is not only due to the increased illumination of the beam surface but due to a growth in the optical spot coverage of the gate area. This gate contribution, however, is irrelevant in estimating the sensitivity of the setup to displacements in the nanomechanical beam. Therefore, the contribution from the gate cannot be neglected when the spot is positioned at xs0 . In fact, this gate effect will lead to an overestimation of the measurement responsivity RKE0 , leading to an underestimation of the displacement value. In Fig. 2·17(b), the solid theoretical line for D2 is the result of the calculation of ∂Pr /∂xs from Eq. 2.31 with the near-by gate removed. This result indicates that the inaccuracy in RKE0 due to the gate can be as much as 30%. In short, the knife-edge technique is more accurate in absolute displacement detection in devices where the gate is far-away in comparison to the spot size. Otherwise a combination of theoretical and experimental analysis, as presented herein, can be used to obtain an estimate for the absolute displacement.
75
Figure 2·18: Peak responsivity RKE0 contour plot as a function of both beam width w and spot diameter d. The experimentally determined responsivity values were in the range RKE0 ∼ 0.2 µm−1 in sub-wavelength devices. This corresponded to a displacement senp √ sitivity of SuKE ∼ 1 pm/ Hz at a power P ≈ 300 µW. In our experiments, the (AT )
sensitivity was again limited by the amplifier noise of SI
≈ 100 pA2 /Hz as de-
scribed in Sec. 2.1.3. These values are comparable or better than those obtained in the results of reported optical interferometry measurements for out-of-plane motion, in Section 2.1.3. For a device that had w = 200 nm, under identical incident power and spot size values, Michelson interferometry and Fabry-Perot interfer√ ometry resulted in R0 ∼ 0.01 µm−1 corresponding to a noise floor of Su ∼ 20 √ pm/ Hz (Kouh et al., 2005b; Karabacak et al., 2005). To assist in designing the next generation of beams to be used with optical knife-edge displacement detection technique, Equation 2.29 was used in assessing the limits of the knife-edge detection technique. Two parameters ultimately determine the responsivity RKE0 of this technique, effective device size w and optical spot size d. Assuming the gate was significantly far away, the change in knife-edge responsivity within the w-d parameter space was calculated to assist in future de-
76 velopment of in-plane nanomechanical resonators. The results of this calculation are presented in Fig. 2·18, where equal responsivity contours as a function of w and d are plotted. The results clearly demonstrate that detection sensitivity will deteriorate as the device size is reduced or as the optical spot is enlarged. There are many advantages of developing NEMS resonators on membranes with in-plane motion. The device geometry provides the obvious advantage of providing an access window to the device from both the front and the back side of the wafer. This secondary access is missing in the devices constructed from SOI wafers displayed in Figs. 1·3 and 1·4, where the substrate remains after fabrication. This geometry can be exploited for optical access in real-time with material deposition from the opposite direction (Kouh et al., 2005a). The lack of back substrate can be of further value at large optical detection power. By removing the substrate, one can remove the background in the optical signal (Carr et al., 1998; Kouh et al., 2005b; Karabacak et al., 2005). In shot noise limited detection, this may result in an improvement in the signal-to-noise ratio. Furthermore, the removal of the resonant cavity formation, observed in NEMS with silicon substrates and described in Section 2.1.3, can reduce the thermal absorption, a significant issue in NEMS detection that is discussed at length in Sec. 3.2.4.
2.3
Darkfield Detection by Diffraction of Evanescent Waves
As discussed throughout this chapter, optical displacement detection NEMS requires highly localized fields, so that scattered signal from the moving nanomechanical beam is maximized while undesired reflections from the substrate are suppressed. In the interferometric techniques described in Sec. 2.1, this observed change was in the phase of the reflected field. On the other hand, the intensity
77 change was critical in the knife-edge detection method, described in Sec. 2.2. In all these cases, however, it was observed that the dynamic scattering from a moving sub-wavelength structure, such as a nanomechanical resonator, is usually weak as compared to the static background signal. Furthermore, in all of the above analyses, it was clear that the sensitivity of the setup was proportional to the incident power and was limited by this value. However, the total incident optical power must be reduced to ensure that the system remains minimally perturbed by the detection method and that the technique is, indeed, nondestructive. In the above described “traditional” methods, the device was illuminated with a highNA (∼ 0.5) objective lens, and the reflection was collected through the same path. The size of the probing optical field was therefore restricted by the diffraction limited spot size, d ∼ 1 µm. To overcome some of these issues, a dark-field microscopy approach is proposed in this section, where the device is illuminated by evanescent waves. Such a field with high localization can be produced using immersion lens techniques. In this proposed approach, the evanescent waves diffracted and scattered by the device are collected by the far-field optics. To analyze the feasibility of such a technique, diffraction theory was applied to evanescent waves in the experimental configuration (Karabacak et al., 2007a). This analytic diffraction theory solution is compared to numerical simulation at the end of the section, with good agreement. The geometry of the proposed detection scheme is illustrated in Fig. 2·19. The suspended doubly-clamped nanomechanical resonator is modeled as a thin strip of infinite length with width w. Consistent with earlier discussion, the resonator (strip) is positioned at a height U0 above the interface of the dielectric substrate half-space. As discussed in detail in Section 2.1.3, the infinite length assumption
78
zz w w
z
x
k
y
Ei
φi k i
uδΔ 0
Δ0 U xx
k ss φs TE TE
k0 (a)
TM TM
(b)
Figure 2·19: Schematic of the proposed evanescent detection approach. (a) In this method a typical doubly-clamped nanomechanical beam resonator is illuminated by evanescent waves formed by focusing a near-IR laser through an index matched numerical aperture increasing lens (NAIL) attached to the backside of the sample. (b) Cross-sectional (x − z plane) view of the proposed detection. Here, an infinitely long strip of width w is suspended by U0 , and oscillates with amplitude u0 above a substrate of refractive index n. The evanescent illumination field Ei in the direction of ki is formed by the total internal reflection of the incoming wave in the direction of ks . is valid for the nanomechanical devices of this study as they usually have a high aspect ratio with length much larger than the diameter of a tightly-focused optical spot. The planar dielectric boundary between the substrate and vacuum within the arrangement allows for the generation in a highly localized evanescent field around the structure. A tightly focused evanescent field can be generated with a numerical aperture increasing lens (NAIL) (Ippolito et al., 2001), in a configuration where the NEMS sample is attached to an index matched hemisphere, as illustrated in Fig. 2·19(a). Such configurations have been successfully demonstrated as tools of subsurface imaging of thermal emissions from integrated circuits (Ip-
79 polito et al., 2004) and quantum dots (Liu et al., 2005), but their application to darkfield microscopy is novel. In the proposed technique, the detection signal will form from the scatter of the evanescent field Ei by the nanomechanical device (Karabacak et al., 2007a). The analysis of this configuration starts with the formulation of a scattered field due to the diffraction of an incident evanescent wave from an infinite slit. The incident field, denoted by Ei and propagating in the ki direction, can be described as Ei (r) = E0 eiki ·r .
(2.32)
Here, the location of the slit is described by r0 , such that it extends infinitely in y 0 with a width of w in the x0 direction. The scattered field E must obey the field equations (Born and Wolf, 1999) ∇ × ∇ × E − k02 E = 4πS,
(2.33)
where S is the source in the domain. A Green function G(r, r0 ) can be constructed to obey the same boundary conditions and the differential equation as the field ∇ × ∇ × G(r, r0 ) − k02 G(r, r0 ) = 4πδ(r − r0 )I,
(2.34)
where I is the identity dyad. Taking the dot product of Eq. 2.33 by G(r, r0 ) and Eq. 2.34 by E(r0 ), subtracting from each other and integrating over r0 , one obtains the electric field description of Z E(r) =
1 d r G(r, r ) · S(r ) + 4π 3 0
0
0
Z d3 r0 [E(r0 ) · ∇ × ∇ × G(r, r0 ) −G(r, r0 ) · ∇ × ∇ × E(r0 )] .
(2.35)
80 Here, the source S can be set to zero for the problem at hand, since it lays beyond the domain of the integration. Using the second Green’s vector identity, the electromagnetic wave diffracted by an open slit B on a black surface A can be expressed from Eq. 2.35 as 1 E(r) = 4π
Z d2 r0 [Ei (r0 ) × ∇ × G(r, r0 ) − G(r, r0 ) × ∇ × Ei (r0 )] · n.
(2.36)
B
From Babinet’s principle, the diffraction from a strip is essentially the same as the diffracted wave from a slit of the same dimensions, subtracted from the incident field (Born and Wolf, 1999). Therefore, the propagated electric field becomes 1 E(r) = Ei (r) − 4π
Z d2 r0 [Ei (r0 ) × ∇ × G(r, r0 ) − G(r, r0 ) × ∇ × Ei (r0 )] · n. (2.37) A
Furthermore, since the incident field Ei (r) is taken to be evanescent in this analysis, it does not propagate beyond a very short decay length and can be dropped out of the far-field expressions. Hence, the resultant integral for the field E due to the diffraction from a strip area A is given by the expression (Karabacak et al., 2007a) 1 E(r) = − 4π
Z d2 r0 [Ei (r0 ) × ∇ × G(r, r0 ) − G(r, r0 ) × ∇ × Ei (r0 )] · n.
(2.38)
A
Here, G is the half-space dyadic Green function that satisfies the wave equation in the upper and lower half-spaces separately (except at r = r0 ) and obeys the boundary conditions that n×G and n×∇×G are continuous, commensurate with the condition on the tangential components of the electric and magnetic fields (Carney and Schotland, 2001). G can be expressed as an angular spectrum of plane waves. Since only the forward scattered field is relevant for this problem,
81 the Green dyad needs to be expressed for z > z 0 only, i G(r, r ) = 2π 0
Z 0
d2 kk eik·(r−r )
i 1 h 0 D(k) + e2ikz z R(k) . kz
(2.39)
q In Eq. 2.39, k = (kk , kz ), kz = k02 − kk2 , and D and R are the dyadic quantities which ensure the transversality of, respectively, (i) the plane-wave propagated directly from r0 to r, and (ii) the plane-wave propagated via reflection from the interface. k0 is the wavevector in vacuum, such that k0 =
2π λ0
and kk is the wavevector in
the x − y plane, as shown in Fig. 2·19(b). The dyads in Eq. 2.39, may be expressed as D(k) = uˆte (k)ˆ ute (k) + uˆtm (k)ˆ utm (k), ˜ + uˆtm (k)rtm uˆtm (k), ˜ R(k) = uˆte (k)rte uˆte (k)
(2.40)
where uˆte/tm are the unit vectors of the TE/TM basis relative to k, as demonstrated in Fig. 2·19(b). These unit vectors can be constructed as uˆte (k) = k × n/|k × n|, uˆtm (k) = k × uˆte /|k × uˆte |.
(2.41)
ˆ uˆte , uˆtm form an ordered orthonormal triple Here, kˆ = uˆte × uˆtm = k/|k| so that k, ˜ is defined to be the reflection with uˆte always parallel to the interface. The vector k ˜ = (kk , −kz ). The reflected waves for uˆte/tm (k) ˜ of k through the z = 0 plane, i.e., k can be defined in a similar fashion to Eq. 2.41. The quantities rte and rtm are the respective Fresnel coefficients for TE and TM waves, given by rtm = (n2 kz − kz0 )/(n2 kz + kz0 ), rte = (kz − kz0 )/(kz + kz0 ),
(2.42)
82 and kz0 =
q
n2 k02 − kk2 . Furthermore, from Equation 2.32, it can be shown that ∇ × Ei = iki × Ei .
(2.43)
Similarly from Eq. 2.39, 1 ∇ × G(r, r ) = 2π
Z
0
2
d kk e
ik·(r−r0 )
i 1 h 2ikz z 0 ˜ D(k) × k + e R(k) × k . kz
(2.44)
˜ = kk in conjunction with Equation 2.38 can be rearranged using the property k+ k Eqs. 2.43 and 2.44, 1 E(r) = 2 8π
Z
Z 0
2 0
d2 kk eik·(r−r )
dr A
1 iki ·r0 e E0 × [D(k) × (k + ki ) kz i 0 ˜ + ki ) · n. +e2ikz z R(k) × (k
(2.45)
Since the interest of the analysis lies in the far-field scattering, such that r À r0 , Eq. 2.45 can be integrated over kk asymptotically using the two dimensional stationary phase method (Borovikov, 1994). The result can be expressed as −ieik0 r E(r) ∼ 4πr
Z 0
d2 r0 e−i(k−ki )·r E0 × [D(k) × (k + ki ) A
i 0 ˜ + ki ) · n, +e2ikz z R(k) × (k
(2.46)
where k = k0 rˆ, that is k is parallel to r. The scattered waves now need to be integrated over the strip area A. For a strip of infinite length in the y 0 direction and width w in the x0 direction, the integration can be expressed as Z 0
d2 r0 e−ik·r = 2πδ(ky ) A
sin(kx w/2) −ikz U0 e . kx /2
(2.47)
83 It is important to note here that the solution can be adjusted to accommodate a finite strip length by replacing the δ(ky ) term with a sharp sinc function, similar to that defined for the x-direction in Eq. 2.47. Finally, inserting the strip geometry of Eq. 2.47 into Eq. 2.46 and using the dyadic functions defined through Eqs. 2.40 and 2.42, the diffracted field amplitude can be obtained as −ieik0 r sin[(kx − kix )w/2] δ(ky − kiy ) 2r (kx − kix )/2 £ e−i(kz −ki z)U0 E0 × uˆte/tm (k) × (k + ki ) i 2ikz U0 ˜ ˜ +e rte/tm uˆte/tm (k) × (k + ki ) · n.
Ete/tm (r) =
(2.48)
Here k k r and the te/tm subscript indicates either transverse electric or transverse magnetic fields relative to the direction of observation. Figure 2·20 displays the scattering intensity, calculated from Equation 2.48, for a nanomechanical beam of width w = 50 nm when illuminated by a single wave of wavevector ki = (2.2k0 , 0, 2.0ik0 ), for two height values of U0 = 100 and 200 nm. The result is presented as a function of observation angle in the x − z plane. The calculation was performed at the experimentally realistic value of λ = 1.3 µm, where silicon is non-absorptive (Palik, 1985) and NAIL operation through the wafer is possible (Ippolito et al., 2005). In Fig. 2·20, the TE-scatter demonstrates a single peak intensity aligned with the beam where as the TM-scatter is observed to be strong in the wider angles. This strong scatter in wide angles is believed to be due to the edge effects of the beam, and will be discussed further. To understand the limitations of the diffraction theory, the analytical results of Eq. (2.48) are compared with numerical solutions of Maxwell’s equations (Karabacak et al., 2007a). This is accomplished by solution of a domain integral equation for the electric field (Visser et al., 1999) which is constructed with the use of the
84
Intensity (arb. units)
100 nm
-90
200 nm
-60
-30
C
100 nm
TM
TE
=1
200 nm
C=4
C
=20
C=800
C
=20
C=200
0
30
Angle (degrees)
(a)
60
90
-90
-60
-30
0
30
60
90
Angle (degrees)
(b)
Figure 2·20: The normalized distribution of the intensity (the squared magnitude of the electric field) as a function of angle in the plane normal to the longitudinal axis of the strip of width w = 50 nm. The plots are for different strip height U0 values are as indicated in the legend. The incident field in vacuum is taken to be of unit amplitude at the Si interface with wavelength λ = 1.3 µm and a wave vector of (2.2k0 , 0, 2.0ik0 ). The dashed lines indicate the results of numerical simulation and the solid lines the outcome of Eq. (2.48). The incident field is (a) TE-polarized (b) TM-polarized. The curves were normalized to the peak height of the analytic result at 100 nm and then scaled by a factor C indicated by the curve for display purposes. electromagnetic Green tensor for the vacuum/substrate system. The equations are solved numerically by matrix inversion operations. As discussed earlier in Figure 2·5, the bi-layer structure of the NEMS device consists of a silicon layer with a thin metal coating. Since at λ = 1.3 µm, the silicon is totally transparent, the scattering structure is the metal layer. Hence, in this comparison the strip was modeled as a silver structure of thickness of 50 nm with complex index of refraction 0.385+i8.95 at λ = 1.3 µm. The numerical results are shown as dashed lines in Fig. 2·20. As with the analytical result, the peak of the pattern for TE incident field is centered for all results. For the TM case, the left peak in the analytic and numeric results is observed to differ by 2.7◦ for both heights. The right peaks differ by 8.6◦ for the
85
TE
Intensity (arb. units)
TM
-90
-60
-30
0
30
60
90
Angle (degrees)
Figure 2·21: As in Fig. 2·20, except that the strip width is taken to be 500 nm and the height is fixed at U0 = 200 nm. U0 = 100 nm case and 12◦ for the U0 = 200 nm situation. In simulation of scattering from wider strips (w = 500 nm) the numerical results and diffraction theory differ considerably for TM-polarized incident fields, as seen in Fig. 2·21.
In these calculations, the peak scatter angle of the
TM-polarized field calculated from Eq. 2.48 is observed to disagree with numerical results. This discrepancy is believed to occur for the wider strip because the strip-substrate system forms a resonant cavity for TM-polarized case but not the TE-polarized case. Multiple scattering in this cavity tends to randomize the transverse momentum of the field, moving the peak of the intensity to the middle of the angular range. For such wide structures, the use of Equation 2.48 should be avoided. The results in Figs. 2·20, 2·21 and Eq. 2.48 are for individual incident waves. Despite the theoretical value of the scattering problem of individual waves, from the experimentalist’s point of view, total scattered field and its dynamic oscillations due to device motion is much more important. The total field calculation requires the integration of the scattering from individual plane waves across all col-
86 lection angles based on the numerical aperture. Furthermore, the total scattered field power collected at the far-field, namely Ps , needs to include the scattering from the total illumination field, which can be formed by coherent superposition of plane waves at all angles of incidence. Defining the angles of incidence as Φ and the angular cone defined by the numerical aperture of collection lens as Θ, this collected power can be expressed through a coherent summation of Eq. 2.48 as
¯2 Z Z ¯¯ Ete/tm (r)¯ Ps = dΦdΘ. 2Z0
(2.49)
Θ Φ
Here, Z0 = 377 Ω is the characteristic optical impedance of free space (Saleh and Teich, 1991). Even more importantly, dynamic signal magnitude δPs can be obtained through multiple computations of this collected field for different beam height values. In the simplest case, this dynamic signal for a beam oscillation of U0 ± u can be approximated as δPs = |Ps |U0 +u0 − |Ps |U0 −u0 .
(2.50)
Experimentally, the above described detection method can be implemented through dark-field microscopy. In dark-field microscopy, the illumination field is constructed so as to prevent the transmission of any light to the collection plane in the absence of the sample. In the setup proposed here, the fully evanescent illumination field will be formed by total internal reflection of high angle waves at the silicon-vacuum boundary surface, as displayed in Fig. 2·19. In order to generate a tightly focussed evanescent field, numerical aperture increasing lens (NAIL) can be attached to the back of the sample substrate. Such setups have been demonstrated to achieve spot sizes with diameters ∼ λ/n2 , where n is the refractive index of the substrate and NAIL (Ippolito et al., 2001). Figure 2·22 details
87 Optical Fiber, Single Mode λ =1.3 µm
Ap e r tu r e Ring Glass Covers Collection Lens (NA~0.7)
F o c us i ng L e n s (NA~0.25) Sample in Vacuum Beam Splitter ( 5 0/ 5 0 ) CCD
Removable M ir ro r
High Speed Photodiode
Illumination
Figure 2·22: A proposed setup layout for displacement detection of nanomechanical beam motion through diffraction of evanescent waves. A collimated laser beam in the near-IR wavelength is passed through a low-NA (∼ 0.25) focussing lens and a NAIL attached to the back of the sample substrate to form an evanescent field incident on the NEMS structure. The scattered light is collected through a high-NA (∼ 0.7) objective lens from the front and directed onto a photodetector. The aperture ring ensures that the field is fully evanescent by blocking the central portion of the incoming laser beam. The structure is illuminated and imaged by a CCD from the front side for alignment purposes. a possible setup configuration for performing dark-field microscopy on NEMS samples. The index-matched hemisphere acting as the NAIL will be attached to the sample substrate and placed in pressure controlled environment. The near-IR waves at λ = 1.3 µm wavelength will focused on the structure through a combination of low-NA (∼ 0.25) lens and the NAIL. This configuration leads to a computed spot size (FWHM) of d ∼ 250 nm. To obtain a truly dark-field microscope, the incident field needs to consist of only evanescent components. For silicon (n = 3.45) and vacuum interface, this requirement correlates to incidence
88 angles of φi > sin( n1 ) ≈ 17◦ . Such an incident field can be achieved through aperture tailoring at the back focal plane of the focusing objective lens, by blocking of the central portion of the optical beam, as shown in Fig. 2·22. The scattered waves from the structure can be collected from the front through a high-NA (∼ 0.7) objective lens and detected. The sample can be illuminated from the front and imaged through a CCD camera for alignment purposes using a removable mirror mechanism, which is taken out during data acquisition to minimize signal losses. To explore the experimental feasibility of the setup depicted in Fig. 2·22, an analytical model was constructed using the evanescent wave diffraction formulations and NAIL transmission solutions (Ippolito, 2004). The analytical model allowed for the normalization of the scattered power Ps to the optical input power P0 , taken at the exit of the fiber. Losses and wave aberrations introduced by the glass covers were accounted for in the solution. Initially, the fluctuation in the scattered power is calculated as the strip vibrates in the z−direction with a conservative amplitude of u0 = 0.5 nm, using the numerical aperture values indicated in Fig. 2·22. Fig. 2·23(a) was obtained from Eq. 2.50 for beams of varying lateral dimension w but constant nominal height U0 = 100 nm. The results indicate an optimum strip width w for a given spot diameter, or conversely, an optimum spot diameter for a given w. This result may be counterintuitive since most optical methods used at this length scale display a monotonic improvement in the detected signal as the optical spot diameter is reduced. Two factors are contributing to this behavior: (i) wide strips push the scattered field into higher angles beyond the collection angle, and (ii) diffraction is dominated by the edges of the strip and a tightly focused field concentrates energy away from the edges. This result, however, is promising for NEMS devices
89
10
TE
x106
6
TM
0
4
P/P
P/P
0
x106
8
TE
6
TM
4
2
2 0
0
200
400
600
w (nm)
(a)
800
0
0
20
40
60
80
100
b (nm)
(b)
Figure 2·23: The dynamic signal in the scattered field due to a focus superposition of evanescent waves with a FWHM spot size of d ∼ 250 nm centered on the strip, at height U0 = 100 nm, at an oscillation of amplitude u0 = 0.5 nm, for both TE and TM polarization of the incident field for (a) beams of varying width w at b = 0 nm air gap, and (b) varying NAIL-substrate air gap b for a beam of width w = 200 nm. The results were normalized with the optical input power P0 , determined at the entrance of the aperture ring shown in the setup layout in Fig. 2·22. which typically have characteristic width around w ∼ 100 nm. Furthermore, the spot size can be adjusted to some extend by defocusing of the spot and optimizing the signal level. The results of Fig. 2·23(a) were obtained with the assumption of perfect transmission between the NAIL and the sample substrate. However, air gaps are unavoidable in configurations where the hemispherical lens is later attached to the sample, due to non-uniformities in both surfaces (Ippolito et al., 2005). In fact, air gap heights of b ∼ 50 nm are considered typical in such setups, with current fabrication methods. Therefore, the effect of the air gap on the detection signal was also analyzed in this study. The effect of the air gap can be formulated using stratified medium analysis, and a complete derivation can be found in Chapter
90 2.2 of (Ippolito, 2004), which will not be repeated here. The specific results of this gap on the scattering of evanescent waves, are presented in Fig. 2·23(b), demonstrating the significance of this effect. An expected air gap of b = 50 nm attenuates the obtainable signal by approximately an order of magnitude. Finally, for a crude comparison, assuming lossless transmission between the NAIL and device substrate, at an optical input power of P0 = 1 mW and U0 = 100 nm, the obtained dynamic signal amplitude from diffraction of evanescent waves is observed to be comparable to or higher than commonly used displacement detection schemes in NEMS described in this chapter. The promise of the proposed dark-field technique, however, lies in the fact that the background signal is largely eliminated and this can be a significant advantage, especially if the shot-noise level of the photodetector is the limiting factor in the measurement. However, diffraction of evanescent waves setup requires significant effort in constructing, where delicate attachment of the NAIL and precise alignment is essential. Furthermore, this detection scheme requires nanomechanical beam to be very close to the silicon substrate to benefit from the high intensity of rapidly decaying evanescent field. Fig. 2·23 was computed for beams of height U0 = 100 nm and the dynamic signal amplitudes will decrease significantly if the separation was increased.
2.4
Conclusions
In this chapter various optical methods with significantly different approaches to displacement detection in NEMS were outlined. Fabry-Perot interferometry was selected as the method of choice for the purpose of studying fluidic interactions in nanomechanical structures, due to experimental concerns outlined here. Initially, out-of-plane resonators as described in Sec. 1.3 were determined to be
91 more suitable for study of structures of varying geometry and hence, resonance frequency. The flexibility in fabricating a family of beams of varying dimensions, as shown in Fig. 1·5 on a single sample chip simplifies comparison of the beams. Constructing of such structures with varying length can be troublesome in the in-plane resonator geometry, shown in Fig. 2·15, where the beam length is determined by the membrane dimensions and the presence of a near-by gate is required for actuation purposes. Therefore, the use of knife-edge displacement detection was impractical for this study, despite its simplicity, stability and strong signal amplitudes. The use of diffraction of evanescent waves as motion detection method was also not implemented for various reasons. While the dark-field illumination approach can provide a significant advantage in shot-noise limited setups, it was determined that weak optical power levels used for sub-wavelength structures did not present such a need. In fact, for shot-noise limited operation in the NEMS domain, accounting for losses in the optical setups, one would require significantly high power levels of order P0 ∼ 10 mW to overcome amplifier noise levels discussed in Sec. 2.1.3. However, at such high power levels, laser absorption in the bilayer structures leads to significant thermal effects, even permanent structural damage. Furthermore, the alignment of hemispherical index matched spheres to substrates can be technically challenging (Ippolito, 2004) and fabrication of nanomechanical beams within the evanescent decay length requires SOI wafers with ultra-thin sacrificial oxide layers; commercial availability is often limited to U0 = 100 nm. Even then, wet etching of high aspect ratio beams with thin oxide layers leads to collapse of the beams due to surface tension of the etching liquid. Hence, fabrication steps would need to be adjusted for such tasks, perhaps
92 through implementation of supercritical drying (Kim et al., 1998). Fabry-Perot interferometry, as described in Sec. 2.1.2, has proven to be very useful in NEMS displacement detection through various studies (Kouh et al., 2005b; Carr et al., 1998). The use of the substrate as an inherent reference surface greatly simplifies the construction of the setup. Furthermore, external vibrations and possible misalignment issues are overcome by the integrated approach. The method, however, is highly sensitive to the positioning of the sample. Long term stability and repeatability can be achieved, provided that the sample is kept stationary with respect to the optical beam both in lateral directions and along the optical axis. The main handicap of Fabry-Perot interferometer is the lack of calibration capability for absolute displacement detection in NEMS. However, as will be discussed in Chapter 3, amplitude of motion is irrelevant to the discussions of fluidic damping and Fabry-Perot interferometer can be implemented for this study. An additional concern is that Fabry-Perot can only function if the reference reflection surface is within the depth of focus of the objective lens z0 ∼ 1 µm as well as the coherence length of the laser. This issue is a significant limitation to the geometry of devices, preventing the use of this technique for displacement detection in devices such as AFM cantilevers and nanomechanical membrane resonators. Michelson interferometry, through the use of an external reference mirror, overcomes the geometric limitations of Fabry-Perot method and provides a measure of absolute displacement. However, the stability and alignment of the reference surface can be challenging, requiring precise alignment of all components. This task is further complicated for sub-wavelength structures where the reflection pattern can be non-uniform. Additionally, active feedback is required to overcome
93 setup vibrations and long-term stability. Such complications can significantly degrade and distort the detection signal. Hence the use of Michelson interferometry method was limited in this study to obtaining reference displacement values and characterization of micron-sized AFM cantilevers, where reflected laser beams are highly uniform and intensity levels are strong.
94
Chapter 3
Characterization of High-Frequency Nanoflows using Nanomechanical Resonators As fabrication, actuation and detection mechanisms in NEMS are becoming established, the time is ripe for the exploration of the interaction between nanomechanical resonators and the surrounding fluidic environments. The study of flow at the nanometer scale has recently emerged as a vibrant research field, referred to as nanofluidics (Karniadakis et al., 2005). Until now, most nanofluidics work was concerned with liquid flow in nanoscale channels and remains strictly in the Newtonian regime. In contrast, the emerging field of resonant NEMS, provides access to a truly novel nanofluidic regime. The high-frequency nanomechanical resonators with frequencies reaching ω/2π ∼ 1 GHz (Huang et al., 2003; Peng et al., 2006), allow for probing oscillatory flow that were inaccessible by past experiments which employed vibrating mechanical structures (Tough et al., 1963; Rodahl et al., 1995). In fact, in this chapter, an analysis of dissipation in nanomechanical resonators that leads to direct observation of viscoelasticity in gas phase near atmospheric pressure levels will be presented. Beyond the fundamental value of researching this interesting flow regime, the development of the full potential of resonant NEMS depends on understanding the fluidic interactions under these new conditions. Already, measurements of
95 nanomechanical resonance in both air (Sekaric et al., 2002a; Li et al., 2007) and liquid (Verbridge et al., 2006a) have been demonstrated, albeit with significant damping effects. The importance of fluidic damping on increasingly miniaturized mechanical structures was underlined more than four decades ago, during the initial development of the tuning fork (Newell, 1968). Since then, various models for describing oscillating flows have been adapted for increasingly smaller length and time scales. In this study, the trend is extended to the recently accessed nanometer and nanosecond regime. Here, a systematic analysis of some of the fluidic issues in operating NEMS devices in fluidic environments will be provided. The emphasis of the chapter will be on analyzing high frequency oscillatory gas flows that are created by nanomechanical resonators in surrounding gaseous environments. To fully understand the flow regimes that are encountered by NEMS devices as well as the approximations that can be made in the analysis, an overview of the fundamentals of fluid dynamics is essential. Due to the complicated nature of most fluid dynamics problems, exact analytical solutions are seldom obtained. Hence, any flow analysis needs to start with certain approximations. The separation of dominant forces and negligible effects can be achieved using characteristic length L and time T scales. For oscillatory flows, the time scale can be taken as T ∼ 1/ω, where ω is the angular frequency and characteristic flow speed can be approximated as V ∼ ωL. Often, non-dimensional numbers are employed in comparing physical quantities. One of the most prominent dimensionless parameters is the Reynolds number, employed in determining the magnitude of the viscous effects with respect to the inertial forces in the flow. The inertial and viscous forces in oscillatory flows can be roughly estimated from the rate of change in kinetic energy K˙ and the rate of
96 energy dissipation D˙ due to viscous shear rate (Peters and Sumner, 2003). Using characteristic variables, K˙ and D˙ can be expressed as ¶ ρf L3 V 2 = ρf L3 V 2 ω, 2 ∂V D˙ ∼ V L2 µ = µV 2 L. ∂s
E˙ k ∼
∂ ∂t
µ
(3.1)
Here, ρf and µ are the density and dynamic viscosity of the fluid, respectively, and s is the distance along the streamline. Using Eq. 3.1, Reynolds number for oscillatory flows can be estimated by Re =
Inertial Forces E˙ k /V ρf L3 V 2 ω ρf ωL2 = = = . ˙ Viscous Forces µV 2 L µ D/V
(3.2)
At high Reynolds numbers of Re À 1, the flow is considered to be dominated by inertia and can be modeled as inviscid flow. Otherwise, the viscous effects need to be accounted for in the calculations. The flow generated by a typical nanomechanical structure (L ∼ w = 300) oscillating at ω/2π = 50 MHz, at atmospheric pressure, creates a flow in the low-Reynolds number region of Re < 1. Hence, the fluidic analysis in NEMS will need viscous flow analysis, and inviscid flow will not be discussed within the scope of this thesis. Another dimensionless number of importance is the Mach number. Ma represents the relative velocity of the flow, and is used in determining the compressibility of the flow. For oscillatory structures, a realistic value for the flow velocity v can be obtained from the amplitude of oscillation A and frequency ω, and Mach number is often expressed as Ma =
Aω v ∼ . cs cs
(3.3)
97 Here, cs is the speed of sound. For air at ambient temperature, cs ≈ 340 m/s. At Ma & 1, compressibility effects become non-negligible and appropriate considerations need to be included into the analysis. However, for nanomechanical structures, Mach number remains very low (Ma ¿ 1), hence the analysis can be safely limited to incompressible flow. In discussing nanofluidics, Knudsen number is a widely mentioned dimensionless measure of flow length scale (Karniadakis et al., 2005). Knudsen number compares the mean free path of the gas molecules lmf p to the characteristic length scale of the flow L, Kn =
lmf p . L
(3.4)
If the gas molecules are modeled as hard spheres with scattering cross-sections of πd2m , the mean free path of the gas molecules is derived to be (Bhiladvala and Wang, 2004) lmf p = 0.23
kB T . d2m P
(3.5)
For nitrogen gas molecules of diameter dm = 0.38 nm (Hirschfelder et al., 1964), the mean free path becomes lmf p = 65 nm at atmospheric pressure. Knudsen number is often used in determining the importance of considering the particulate nature of the flow. For Kn < 0.01, the fluid can be considered to be continuum and the well established Navier-Stokes formulations, with the no-slip boundary condition can be applied to describe the flow. For devices with dimensions well below the mean free path, Kn > 10, the interaction is in the so-called molecular flow regime, and momentum exchange during collisions can be used to obtain the fluidic force on the structure. The intermediate flow regime of 0.01 < Kn < 10, often labeled as transition flow regime, is not well understood and an accurate analytical description of the flow that is in agreement with the well-established
98
2 nm
Molecular
10
6
200 nm 2
Kn
10
10
10
10
m
4
2
0
Transition
-2
Continuum
10
-3
10
-2
10
-1
10
0
P (Torr)
10
1
10
2
10
3
Figure 3·1: Knudsen number Kn = lmf p /L as a function of gas pressure P , for devices of varying length scales L indicated in the legend for nitrogen gas, dm = 0.38 nm. Newtonian limit is lacking in literature (Bhiladvala and Wang, 2004; Blom et al., 1992). In Fig. 3·1, the change of Knudsen number as a function of gas pressure is plotted for flow length scales L = 2 nm, 200 nm and 20 µm, with each flow regime marked. Here, the critical parameter that determines the flow regime is the length scale of the flow L. In mechanical oscillators, this is often taken to be equal to the width of the beam, such that L ∼ w (Bhiladvala and Wang, 2004; Paul et al., 2006). This is based on the assumption that the dominant flow direction is around the width of the structure, which in fact is a disputable approximation due to the possibility of flow along the beam. In either case, it is clear that for nanomechanical beams the relevant flow length scale is 100 nm < L < 5 µm. From Fig. 3·1, it is seen that at this length scale and near atmospheric pressure levels, divergence from the classical continuum flow descriptions should be expected.
99 Fluids can be further classified as Newtonian and non-Newtonian based on their viscous response to shear stress. Newtonian fluids are those that demonstrate a linear relationship between the shear stress σ and velocity gradient perpendicular to the direction of shear. For an incompressible fluid with isotropic properties, µ σij = µ
∂vj ∂vi + ∂xj ∂xi
¶ .
(3.6)
The Navier-Stokes equations based upon the Newtonian approximation have been remarkably successful over the centuries in formulating solutions for relevant flow problems both in bulk and near solid walls (Landau and Lifshitz, 1987). This will be further discussed in Sec. 3.1. In the non-Newtonian fluids, Eq. 3.6 with constant viscosity µ becomes invalid. Such non-linear behavior is not uncommon in fluids; blood, honey, pudding are some of the more famous non-Newtonian fluids. Many sources of non-linearities in viscosity can create divergence from Newtonian behavior. More commonly, power-law liquids like blood demonstrate an effective viscosity that is dependent on the rate of shear (Faber, 1995). Or in Bingham plastics like mud, the linear relation between shear stress and shear rate appears only beyond a certain threshold of shear (Tanner, 2000). Another source of non-linearity in viscosity can arise due to the finite time necessary for the fluid to respond to shear. In general, this relaxation time τ , is on the order of nanoseconds for gases at atmospheric conditions, and τ ∼ 10−12 s for water. Therefore, the effect of the response time is often neglected when compared to the time scale of the flow. In other words, for excitations much slower than the relaxation time T À τ , the fluid reaches equilibrium rapidly and the “history” of the fluid is irrelevant to current state. If, however, the rate of shear becomes
100 rapid such that T . τ , non-equilibrium effects fluid need to be considered and all interactions that occur within the relaxation time become relevant to current response. In a very simple model, if the fluid is assumed to relax to its equilibrium with the response function e−T /τ , then the viscosity of the fluid can be extracted as (Faber, 1995) µ=
1 µ0 , 1 − iωτ
(3.7)
where ω ≈ 1/T corresponds to the angular frequency of the flow and µ0 is the viscosity of the fluid under quasistatic shear. For oscillatory flows, the non-dimensional Weissenberg number Wi = ωτ determines the nature of the response. From Eq. 3.7, two general observations can be drawn; as ωτ → 0, the Newtonian viscous fluid behavior dominates the flow whereas at the ωτ → ∞ limit the fluid begins to behave increasingly like an elastic solid. This combined behavior is referred to as viscoelasticity. The relaxation model leading to Eq. 3.7 is essentially equivalent to the response of a mechanical spring and dashpot connected in series and is often named after Maxwell, the first to describe such viscoelastic behavior. Typically, viscoelasticity is easily observed in substances with long relaxation times like polymer solutions or honey. In gaseous environments, the relaxation time scale is often too short for observation. The relaxation time of gases are often empirically defined as τ ∝ 1/P (Rodahl et al., 1995), and can be approximated from the mean free path as τ ≈ lmf p /cs (Yakhot et al., 2006). Based on this approximation, the relaxation time of gases is of order τ ∼ 10−9 s under atmospheric pressure. NEMS devices (ω ∼ 109 1/s) offer a unique opportunity to observe the same viscoelastic effects in gas state, at reasonable pressure levels. In fact, a direct observation of the divergence from the classical Newtonian solutions of Navier-
101 Stokes, which is derived based on the assumption of Wi ¿ 1, is possible by tuning the relaxation time through pressure sweeps. Based on the descriptions above, the conditions encountered by NEMS resonators near atmospheric pressure levels can be summarized as incompressible, non-Newtonian (viscoelastic) flow. In the following sections, analysis of some of the Newtonian and non-Newtonian flow models that are widely employed in analyzing fluidic effects in micro/nanomechanical resonators will be presented. Furthermore, a novel analysis incorporating viscoelastic effects is presented based on the high-frequency solution to the Stokes’ second flow problem (Yakhot and Colosqui, 2007). The parameter space of this study actually demonstrates the transition from Newtonian to non-Newtonian flow, which is only explained through this recently developed model.
3.1
Theory of Oscillating Flow
The fluidic interactions can be experimentally probed through the changes in the resonance characteristics of a simple beam as the flow conditions are varied. Returning to Sec. 1.2.2, the equation of motion of the beam was derived in the form of simple harmonic oscillator using effective mass ms , stiffness k and γs . From Eq. 1.37, the equation of motion can be recalled as ms
£¡
¤ ¢ ω02 − ω 2 + iωγs u (ω) = F (ω) .
(3.8)
The harmonic solution to Eq. 3.8 can be expressed as © ª u (t) = Re u0 eiωt = u0 cos(ωt),
(3.9)
102 where u0 is the amplitude of oscillation at frequency ω/2π. Until now, the external forcing term F (ω) was left undefined. In this section, the emphasis will be placed on developing a theoretical understanding of the fluidic drag force Ff on the surface of the oscillating mechanical structure. The surrounding fluid creates two significant effects on an oscillating structure; inertial and dissipative. While the dissipative force is proportional to the velocity, the inertial loading is created as a result of the acceleration. Hence, the fluidic drag force can be expressed as Ff = fd v + fi
∂v , ∂t
(3.10)
where v(t) is the velocity of the structure; v = u. ˙ Using Eq. 3.9, the relative velocity of the flow with respect to the surface becomes v = −u0 ωsin(ωt).
(3.11)
The fluidic dissipation in a single period of oscillation is equal to the work done by the drag force during the cycle, 2π/ω Z
W =
F · vdt.
(3.12)
0
Analyzing the individual components separately, the energy losses can be expressed as 2π/ω Z
2π/ω Z
fi u¨ (t)u˙ (t) dt = −fi u20 ω 3 0
0
2π/ω Z
2π/ω Z
fd u˙ (t)u˙ (t) dt = fd u20 ω 0
sin (ωt) cos (ωt) dt = 0,
ω sin2 (ωt) dt = πfd u20 ω. 0
(3.13)
103 As expected, the inertial interaction fi is demonstrated to be a conservative force and energy dissipation occurs due to fd only. Hereafter, the analysis of fluidic interaction will be split along these lines of dissipative and inertial effects. The total external force on the structure can be defined as a combination of the drive force F0 and fluidic drag Ff , such that F = F0 −Ff . Along with the definition in Eq. 3.10, the equation of motion (Eq. 3.8) can be reorganized as ¤ £ ms ω02 − (1 + βf ) ω 2 + iω (γs + γf ) u (ω) = F0 .
(3.14)
Here, βf = fi /ms and γf = fd /ms are the effective fluidic inertia and dissipation, respectively, normalized by the resonator mass ms . In fact, Eq. 3.14 can still be examined through the simple harmonic oscillator formulation of Eq. 3.8, with the slight change in variables, £ ¤ ms ωd2 − ω 2 + iωωd /Q u (ω) = F (ω) .
(3.15)
Here, the new effective device characteristics are the damped resonance frequency ωd and total quality factor Q, q ωd =
ω02 (1 − βf ),
(3.16)
and 1 γs + γf 1 1 = = + . Q ωd Qs Qf
(3.17)
In the above equation Qf is the quality factor of the resonator in fluidic environment. It is important to note here that this simplified variable switch was performed with the assumption of small inertial loading on the structure, βf ¿ 1. This assumption is demonstrated to be well-justified for nanomechanical res-
104 onators in Sec. 3.3. The definition of the drag force Ff now depends on the selected flow model. In the following sections, the effective interaction parameters βf and γf are extracted from the more commonly employed analytical fluid models. The analysis is split into Newtonian and non-Newtonian flow, based on the dimensionless characteristics described above. These formulations are further categorized in terms of geometry. Despite the rather simplistic rectangular cross-section of most mechanical resonators, the discontinous edges in this geometry presents significant challenges in obtaining analytical solutions. Hence, the flow analysis usually commences with certain approximations in geometry. Models based on oscillating sphere (Blom et al., 1992), cylinder (Sader, 1998) and plate geometries (Yakhot and Colosqui, 2007) have been developed to describe rectangular resonators, albeit through the use of geometric factors. 3.1.1 Newtonian Fluid Dynamics As previously mentioned, for nanoflows Re < 1 and Ma ¿ 1, and therefore viscous, incompressible flow analysis needs to be applied. In the Newtonian model, the linear relationship between shear rate and shear stress, described by Eq. 3.6 is taken to be valid. The Navier-Stokes equations, named after Claude-Louis Navier and George Gabriel Stokes, are well-established in describing the Newtonian flow conditions (Landau and Lifshitz, 1987). Their derivation, not presented here, can be obtained from the conservation of mass, momentum and energy for a given arbitrary control unit. In its most relevant form, the Navier-Stokes formulation
105 for incompressible flow and the continuity condition can be expressed as 1 ∂v µ + (v · ∇) v = − ∇P + ∇2 v, ∂t ρf ρf ∇ · v = 0.
(3.18)
However, there are inherent analytical difficulties in solving Eq. 3.18 for flow around a rectangular cross-section, requiring simplifications in geometry. Sphere Model One of the earlier propositions was based on the approximation of the beam as a sphere with an appropriate radius r (Blom et al., 1992). The viscous drag force on a sphere due to an oscillating flow has been previously formulated (Landau and Lifshitz, 1987). The fluidic drag force can be obtained from the integration of the shear stress across the spherical surface, ³ r´ , fd = 6πµr 1 + ¶ µ δ 2 3 9δ fi = πr ρf 1 + . 3 2r
(3.19)
Here, the frequency and pressure dependent penetration depth of the lateral waves is given by
s δ=
2µ . ρf ω
(3.20)
Here, the sphere radius was used as a fitting factor when describing the fluidic dissipation observed in cantilevers. It was initially determined that for cantilevers of width w the approximation r ∼ w would be suitable (Blom et al., 1992). Later, the approximation 2r ∼ w was proposed as being more accurate (Hosaka et al., 1995). Nonetheless, even in the relatively larger cantilever dimensions of w ∼ 0.2
106 mm, Eq. 3.19 was observed to fail at pressure levels of P < 1 Torr, when the Knudsen number reaches a relatively high value (Vignola et al., 2006). For lower pressure levels, molecular collisions theory (see discussions below) was proposed as suitable (Blom et al., 1992). While analytically very straightforward, the sphere model is considered a rough approximation of a rectangular beam. In high-frequency NEMS devices, dissipation predictions from Eq. 3.19 have demonstrated significant disparity to the experimental data. Even though it performed better in describing the dissipation effects upon larger scale AFM cantilevers, the predictions of the more refined cylinder model, discussed below, were significantly more accurate. Cylinder Model The crude approximation of a sphere in explaining the flow around a rectangular beam often fails in its description. Hence, a more realistic but analytically more challenging cross-flow around a cylinder model was developed for describing dissipation in micro-mechanical resonators (Sader, 1998; Paul and Cross, 2004). Similar to the sphere model, the oscillating cross-flow solution around a cylinder is obtained from Eq. 3.18 using the no-slip boundary condition and relative boundary velocity of sinusoidal form, as in Eq. 3.11. The solution of the surface force on the cylinder, originally derived by Stokes in 1851, can be expressed as (Paul et al., 2006) Ff (ω) =
ρf ms ω 2 Γ (ω, Re) u (ω) . ρs
(3.21)
In applying the cylinder approximation, it was assumed that the beam is essentially a cylinder with a diameter equal to the beam width w. The dimensionless hydrodynamic function Γ (ω, Re) can be obtained from the rigorous analytical so-
107 lution of Eq. 3.18 in the cylindrical coordinates for oscillating flow (Rosenhead, 1963),
³
´ 4iK1 −i iRe ³ √ ´. Γcirc (ω, Re) = 1 + √ iReK0 −i iRe √
(3.22)
Here, K0 and K1 are modified Bessel functions of the third kind. Reynolds number Re is as defined by Eq. 3.2, using the characteristic length of the flow as L = w/2. More recently, a numerically obtained correction function Ω(ω, Re) was proposed to improve the accuracy of Eq. 3.22 in its adaptation to rectangular geometry (Sader, 1998), Γ (ω, Re) = Ω(ω, Re)Γcirc (ω, Re) .
(3.23)
The Equations 3.10, 3.14 and 3.21 can be combined to determine the inertial and dissipative loads that form on the mechanical resonator, ρf Γr (ω, Re), ρs ρf = ωΓi (ω, Re). ρs
βf = γf
(3.24)
Here, Γr (ω, Re) and Γi (ω, Re) are the real and imaginary components, respectively, of the hydrodynamic function as described by Eq. 3.23. Eq. 3.24 has been demonstrated to be very successful in describing the resonant response in micron-scale AFM cantilevers in gaseous and liquid environments (Chon et al., 2000). However, as it will be demonstrated in Sec. 3.3, this Newtonian description diverges significantly from the dissipation data for high frequency nanomechanical beams, as the approximations underlying the NavierStokes equations fail.
108 Squeeze-Film Damping In geometries where resonator devices are in close proximity of a large, stationary surface squeeze-film damping has been demonstrated (T. Veijola and Ryhanen, 1995; Hutcherson, 2004). This phenomenon develops due to the air film between the moving device and the stationary substrate getting squeezed during the cyclic motion of the beam. As the beam flexes towards the substrate, the squeezed gas creates a positive pressure difference inside the gap. Inversely, a negative differential pressure forms on the film as the device moves away from the substrate. This results in a flow that is paralel to the substrate and beam surface that form the cavity, as the fluid is pushed away out of the gap. In the NEMS resonators tested in this study, presence of the substrate below the suspended structure, creates the possibility of squeeze-film damping effects to appear. In the continuum regime, the damping effect due to the squeeze film can be obtained from the solutions of Navier-Stokes equations. Often, the linearized isothermal Reynolds equation for compressible films is used in describe the pressure formation in the gap (Veijola et al., 1998; Hutcherson, 2004), P0 U02 2 ∇ 12µ
µ
P (x, y, t) P0
¶
∂ − ∂t
µ
P (x, y, t) P0
¶
∂ = ∂t
µ
u P0
¶ .
(3.25)
Here, P0 is the ambient pressure, U0 is the mean beam-substrate separation and u is the motion of the device from its equilibrium, as defined in Fig. 1·2. The solution of Eq. 3.25 has been developed for an oscillating rectangular beam (Blech, 1983). The non-dimensionalized damping force due to squeeze film damping can
109 be expressed through the series expansion (Hutcherson, 2004) γf =
m2 + (n/b)2 64σsf P0 wl X o n π 6 U0 ms ω m,n odd (mn)2 £m2 + (n/b)2 ¤2 + σ 2 ±π 4 sf
(3.26)
In this formulation, the beam aspect ratio is expressed as b = l/w and squeeze film number σsf is formulated as σsf =
12µl2 ω . U02 P0
(3.27)
The squeeze-film damping has been widely used in modeling fluidic dissipation in MEMS structures with air gap distances much smaller than lateral device dimensions, w À U0 (Gallis and Torczynski, 2004). However, this is not expected for the nanomechanical structures under study here, where the air gap height values are of the same order of magnitude with the air gap w ∼ U0 . In fact, it is demonstrated in Sec. 3.3, that the squeeze film damping significantly underestimates the experimentally observed fluidic losses in nanomechanical resonators. 3.1.2 Non-Newtonian Fluid Dynamics Molecular Flow At large Knudsen numbers, the flow becomes highly rarefied and the mean free path between the molecules becomes comparable to and even larger than the relevant device dimensions. In analyzing the fluidic interaction surrounding macroscopic structures, such rarefied flow conditions are only reached in the extremely low pressure levels, encountered in vacuum technology, high-altitude aeronautics and space exploration (Kogan, 1969). However, as demonstrated in Fig. 3·1,this limit is easily reached at intermediate vacuum levels for nanomechanical structures. In molecular flow regime Kn > 10, the descriptions of continuum can no longer be considered valid. At this low pressure limit, individual collisions of non-
110 interacting gas molecules with the structure need to be considered as the main interaction between the device and its environment. Hence, momentum exchange during random collisions between gas molecules and the structure becomes the dominant fluidic energy dissipation cause (Newell, 1968). Molecular collisions are categorized as specular and diffuse (Karniadakis et al., 2005). In specular reflections, the underlying assumption is that only the molecule velocity component normal to the surface is changed and that no momentum transfer in the tangential axis occurs during the collision of the molecule with the surface. Under the diffuse collision assumption, the incidence and reflectance angle of the colliding molecule are taken to be different and hence tangential force interaction needs to be accounted for. However, an analysis of diffuse collisions is analytically challenging and no such analysis has been adapted to mechanical resonators in rarefied environments. Furthermore, the simple specular collision model has been demonstrated to be sufficiently successful in high Knudsen number flows (Blom et al., 1992; Bhiladvala and Wang, 2004). In specular collisions, it is assumed that ideal momentum exchange occurs between two elastic structures.
The average velocity distribution of the gas
molecules at temperature T can be approximated from Maxwell-Boltzmann distribution as (Bhiladvala and Wang, 2004) r Vm =
3kB N T , mm n
(3.28)
where N is the Avogadro’s number, kB is the Boltzmann constant and mm is the mass of the gas molecule. Through conservation of momentum and energy, the damping force on the structure can be expressed as Ff = 2ρf Vm vlw. In this equation, due to the low density of molecular flow regime, inertial loading on the
111 structure is taken as negligible, i.e., fi ≈ 0. Therefore, the damping coefficient in molecular flow regime can be expressed as γf =
2ρf Vm lw . ms
(3.29)
As the density of the fluid ρf is proportional to the pressure P , Eq. 3.29 is predicting a dissipation increase as γf ∝ P . The validity of Eq. 3.29 is strictly restricted to high Knudsen number flow regimes, where molecular collisions are the dominant interaction. The experimental results obtained within the scope of this work confirm this result. The damping predictions of Eq. 3.29 are often relatively accurate at low pressure levels and fail at pressure levels approaching atmospheric levels, as demonstrated in Sec. 3.3. Stokes’ Second Flow Problem in the High Frequency Limit One of the fundamental approximations underlying the Navier-Stokes equation (Eq. 3.18) is the assumption that the fluid is in thermodynamic equilibrium. This assumption essentially requires that the flow perturbation time scale T is significantly larger than the molecular relaxation time τ . This condition of Wi = τ /T ¿ 1, has been successful in analyzing flow under most common conditions. In gaseous environments, with relaxation time scales in the nanoseconds, the assumption of Wi ¿ 1 is often taken for granted for all relevant pressure levels. However, nanomechanical resonators of ω ∼ 109 1/s can create oscillating flows at Wi ≥ 1 at reasonable pressure levels. As a result, the assumptions underlying the classical Navier-Stokes equation are no-longer valid and a re-evaluation is necessary for describing the viscous interactions surrounding nanomechanical structures.
112 Recently, in an effort to model NEMS in fluids, Stokes’ Second Flow Problem was extended to the high-frequency regime (Yakhot and Colosqui, 2007). The analysis was based on the fundamental Boltzmann kinetic equation (Chen et al., 2003), ∂f + vm · ∇f = C. ∂t
(3.30)
Here, f (x, vm , t) is the number density of particles at position x with molecular velocity vm , at a given time t. For small perturbations on the system, the collision integral C can be expressed as (Yakhot et al., 2006) C=
f − f eq , τ
(3.31)
where f eq represents the particle distribution at thermodynamic equilibrium. In Eq. 3.31, which is often referred to as the Boltzmann-BGK model, it is assumed that the system monotonically decays to equilibrium within a relaxation time τ after the perturbation (Bhatnagar et al., 1954). The relaxation time in bulk flow can be estimated from the mean free path lmf p as τ ≈ lmf p /cs where cs is the speed of sound (Yakhot et al., 2006). To describe the fluid interaction with the oscillating surface, the important parameter of analysis is the shear stress on the structure. The stress formation due to random velocity fluctuations in the flow can be formulated through the Reynold’s stress tensor, often encountered in analysis of turbulence (Mathieu and Scott, 2000). The stress tensor σij can be physically described as the net flux of i component of momentum being transferred in the j direction (Huang, 1987). Mathematically, the quantity being transferred is mm (vim − vi ) and the effective flux is n(vjm − vj ), where mm is the molecule mass, n is the number density such that ρf = mm n. Here, v = vm is the mean flow velocity. Thus, the Reynolds stress
113 can be formulated for non-equilibrium situations as (Chen et al., 2004) σij =
ρf (vim
− vi )
¡
vjm
¢
− vj =
Z
¡ ¢ dvm (vim − vi ) vjm − vj f (x, vm , t) ,
(3.32)
¡ ¢ where i 6= j. If Eq. 3.30 is multiplied by (vim − vi ) vjm − vj and integrated over vm , one can obtain (Yakhot and Colosqui, 2007) Z ¡ ¢ ∂σij σij 1 X ∂f , + =− dvm (vim − vi ) vjm − vj vαm ∂t τ ρf α ∂xα
(3.33)
where σij is as defined by Eq. 3.32. It is analytically challenging to obtain a general solution of Eq. 3.33 for any geometry. However, the solution has been recently formulated for the oscillating flow problem in the half-space (Yakhot and Colosqui, 2007). The Stokes’ second flow problem can be described in its simplest geometry as a flow formation in the y-direction for y > 0 with velocity amplitude v(y, t), generated by an infinite plate oscillating in the x-axis. The geometry of the problem is depicted in Fig. 3·2. Due to the infinite plate assumption, flow in all directions other than v(y, t) would be zero. Assuming the validity of the no-slip boundary condition, one can express the flow speed at the surface to be equal to that of the oscillator, expressed by Eq. 3.11, v(0, t) = v0 sin(ωt). The validity of the no-slip boundary condition has been a topic of numerical study (Colosqui and Yakhot, 2007), and it has been determined that the slip factor ² is relatively low for the ωτ > 1 regime, ²(ωτ ) ∼ 0.2 − 0.3 (Yakhot and Colosqui, 2007). Hence, for the solution presented here the no-slip boundary condition will be applied, while acknowledging that ²(ωτ ) can be included in the solution for enhanced accuracy, albeit with increased complexity in solution. Based on the above flow assumptions, the component of the stress tensor of
114
y
v(y,t ) h
k ω
Figure 3·2: Schematic of the Stokes’ second flow problem; an infinite plate oscillating with angular frequency ω sets up waves in the fluid in the normal (y) direction with amplitude v(y, t). interest to the Stokes problem would be σx,y = σ. Due to the symmetries in the problem, Eq. 3.33 can be reduced to (Yakhot and Colosqui, 2007) ∂v ∂σ = − , ∂t ∂y ∂σ ∂v τ + σ = −ν . ∂t ∂y
(3.34)
Here, ν = µ/ρf is the kinematic viscosity. The above set of equations can be combined to obtain τ
∂ 2 v ∂v ∂ 2v + = ν . ∂t2 ∂t ∂y 2
(3.35)
Based on the boundary condition v(0, t) = v0 sin(ωt), the solution to Eq. 3.35 can be found from v (y, t) = v0 e
− δy
−
µ
y sin ωt − δ+
¶ ,
(3.36)
with two length scales; δ− is the penetration length and δ+ is the wavelength of
115 the generated wave. These can be obtained from Eq. 3.35 as ¡
¢ 2 2 1/4
r
1/δ± = 1 + ω τ
¶ µ ¶¸ · µ 1 ω 1 −1 −1 tan (ωτ ) ± sin tan (ωτ ) . cos 2ν 2 2
(3.37)
It is important to note at this stage that, at the Newtonian limit of ωτ → 0, these two length scales converge to the Stokes’ penetration depth δ r δ− → δ+ → δ =
2ν . ω
(3.38)
At higher frequency levels, it is observed that the penetration depth approaches large values while the wavelength is reduced, such that δ− > δ > δ+ . Eq. 3.36 can be combined back into to Eq. 3.34 to extract the drag force acting on unit area of the structure as a function of ωτ , · ¸ v0 1 1 σ (0, t) = µ − (sin (ωt) − ωτ cos (ωt)) + (−ωτ sin (ωt) − cos (ωt)) . 1 + ω2τ 2 δ− δ+ (3.39) By defining the surface area of the rectangular beam as S = 2l(w + h), the total fluidic force on the structure can now be formulated as · µ ¶ µ ¶ ¸ µS 1 ωτ ωτ 1 Ff = −u0 ω sin (ωt) + + − u0 ω cos (ωt) . 1 + ω2τ 2 δ− δ+ δ− δ+
(3.40)
Using the definition in Eq. 3.10, the above fluidic force can be separated into its inertial fi and dissipative fd components as fi fd
¶ µ ωτ µS 1 − = ω (1 + ω 2 τ 2 ) δ+ δ µ ¶− µS 1 ωτ = + . 2 2 1+ω τ δ− δ+
(3.41)
Hence, the ratio of effective fluidic mass to effective structural mass can be fur-
116 ther clarified using βf = fi /ms and by inserting the appropriate definitions of penetration depth and wavelength from Eq. 3.37, βf
r · µ ¶ S 1 µρf 1 −1 = (1 − ωτ ) cos tan (ωτ ) ms (1 + ω 2 τ 2 )3/4 2ω 2 µ ¶¸ 1 −1 + (1 + ωτ ) sin tan (ωτ ) . 2
(3.42)
The fluidic damping coefficient γf can be similarly obtained as γf
3.2
r · µ ¶ S 1 1 µρf ω −1 (1 + ωτ ) cos = tan (ωτ ) ms (1 + ω 2 τ 2 )3/4 2 2 µ ¶¸ 1 −1 − (1 − ωτ ) sin tan (ωτ ) . 2
(3.43)
Measurement of Nanomechanical Resonator Characteristics
3.2.1 Experimental Setup In order to explore the fluidic dissipation effects in nanomechanical systems and analyze the validity of the flow theories outlined in Sec. 3.1 at the high-frequency limit, a wide range of nanoscale doubly-clamped beams were fabricated using the procedures outlined in Sec. 1.3. To extend the frequency range, two commercial AFM cantilevers were utilized.1 The devices were individually characterized using an optical NEMS characterization setup. The setup, whose schematic is provided in Fig. 3·3, was designed with both Michelson and Fabry-Perot interferometry capabilities. The nanomechanical resonators were measured using Fabry-Perot interferometry while the micro-cantilever characterization required the operation of Michelson interferometry, due to the reasons outlined in Sec. 2.4. During the Fabry-Perot operation mode, the reference reflection arm is simply blocked. 1
PointProbe Silicon-SPM-Sensor, NanoWorld AG
117 The samples were placed in a custom designed characterization chamber with pressure measurement and control capabilities. This measurement chamber was designed to provide optical access through a viewport and electrical control capabilities by two vacuum feedthrough connectors. Furthermore, as demonstrated in the setup schematic in Fig. 3·3, the chamber contained two flow connections. One channel was connected to the roughing pump and high-purity nitrogen tank for control of pressure level. Second flow line is designed as an outlet for pressure monitoring through two gauges of 1-1000 mTorr and 1-1000 Torr ranges,2 connected to dual channel digital readout unit.3 The chamber was sealed with an O-ring beneath the cover plate, held down by screws, and was demonstrated to reach pressure levels as low as P < 30 mTorr solely through the use of roughing pumps. With no positioning capabilities inside the chamber, optical alignment requirements were fulfilled by placing the sealed chamber on top of a three-axis flexural translation stage, with piezoelectric feedback control.4 The stage is equipped with stepper motors of 4 mm range and piezoelectric actuators of 20 µm range in each axis. This dual configuration allows for both large distance translation and precise alignment in short ranges. Furthermore, once aligned, the position of the stage can be fixed through the close-loop operation capabilities of the piezoelectric controller. A custom user interface was employed through the serial port connection to the multi-unit controller rack5 during the remote operation of the stage. The load limitations of the stage at ∼ 1 kg, was one of the major design constraints in the design and assembly of the chamber. 2
722 Baratron Absolute Capacitance Manometer, MKS Instruments, Inc. PDR2000 Digital Power Supply and Readout, MKS Instruments, Inc. 4 17 MAX 301 Three-Axis Parallel-Flexure Translation Stage, Melles-Griot, Inc. 5 17 MMR 001 Nanopositioning Main Rack, Melles-Griot, Inc. 3
118 Feedback Circuit
Beam Splitter (50/50)
Piezoelectric Actuator Reference Mirror
Optical Fiber, Single Mode λ=632 nm Preamplifier Preamplifier w/ w/ low low pass pass filter Filter filter
Collimation Lens
Photodetector For Signal Detection
High Frequency Amplifier
Beam Splitter (90/10) Photodetector For Power Level Monitor Dry Nitrogen
CCD
Illumination Beam Splitter (50/50)
Objective Lens (NA=0.5) NEMS Sample
Vacuum Pump
Pressure Gauges
Three Axis Translation Stage
Network Analyzer DC Voltage Source
Figure 3·3: Optical measurement setup for characterization of micromechanical and nanomechanical structures under controlled pressure environments. In the inset, a typical CCD image of a family of doubly-clamped nanomechanical beams with the focused laser spot is displayed. The chamber loaded stage is placed under a specifically constructed optical microscope, designed for operation at λ = 633 nm. In this setup, a remote HeliumNeon laser6 output is transferred to the setup by single-mode fiber,7 and collimated to a diameter of ∼ 4 mm, slightly overfilling the back-aperture of the objective lens.8 The reflection from the sample, collected through the same objective lens is directed onto a high-speed silicon photodetector.9 In the Michelson inter6
1144P, 15 mW Helium-Neon Laser, JDS Uniphase Corporation P1-630A-FC-5, Single Mode Fiber Patch Cable, Thorlabs, Inc. 8 M Plan APO SL 80x, Mitutoyo Corporation 9 DET210, High-Speed Silicon Detector, Thorlabs, Inc. 7
119 ferometry configuration, this reflection signal is combined with the reflection from the reference mirror that is placed on a piezoelectric actuator. The piezoelectric actuator is connected to a feedback circuit for stabilization purposes, as explained at length in Sec. 2.1.1. Furthermore, the complete mirror-actuator setup is placed on a tilt platform to compensate for misalignments. Finally, for alignment purposes, an observation arm was constructed with white-light illumination of the sample and imaging capabilities using a CCD camera.10 A typical sample image with the incident focussed laser spot is provided in the inset of Fig. 3·3. Due to the low saturation limit of the camera, the imaging had to be performed at low laser power and reflections to the camera needed to be blocked during signal acquisition. The stray reflection of laser beam from the 90/10 splitter was utilized for power monitoring purposes. As will be explained in Sec. 3.2.4, this precaution was required to minimize fluctuations due to the thermal effects induced by laser absorption by the nanomechanical resonator. 3.2.2
Device Actuation
For nanomechanical beams, data acquisition was performed by electrostatic actuation of the beams. In this technique, the beams and the substrate are coated with metal, essentially forming a parallel plate capacitor. The potential energy in the parallel plate capacitor with separation U0 can be expressed as 1 Ecap = CV 2 , 2
(3.44)
where C ∝ U0−1 is the capacitance between the capacitor plates and V is the applied voltage. From here, the force F0 on one of the plates can be approximated 10
XC-ST50 Monochrome Camera, Sony Corporation
120 as F0 = −
∂Ecap C 2 ∼ V . ∂z 2U0
(3.45)
Hence, harmonic oscillations can be induced in the above described geometry through an application of a time variant voltage difference between the substrate and the beam. The applied voltage difference can be defined in general terms as a combination of time variant and constant terms, V = Re{Vdc + Vac eiωt },
(3.46)
with an oscillation frequency ω. As a result, the actuation force on the structure becomes F0 ∝ Vdc2 + 2Vdc Vac eiωt + Vac2 ei2ωt .
(3.47)
In the above equation, the first term will deform the beam statically, while the second term (∝ Vdc Vac ) excites the beam at the frequency of the applied signal, ω. It is important to note here that solely applying an oscillating voltage at frequency ω, with Vdc = 0, will excite the beam only at frequency 2ω. Experimentally, the use of the network analyzer is often a practical method in characterizing the beams. In the detection scheme used in this study, the output of the network analyzer11 at ω, was biased with a constant voltage Vdc and applied across the beam-substrate connections. The induced electrostatic force (Eq. 3.45) on the suspended beam creates an oscillation in the optical intensity incident on the photodetector, through effective path length change mechanisms explained in Chapter 2. The high frequency components of the photodetector current, proportional to the device displacement were amplified and input into the network 11
8712ET RF Network Analyzer, Agilent Technologies, Inc.
121 analyzer operated in the transmission measurement mode. The network analyzer allows for the measurements of the transmission signal only at ω, within a specified bandwidth. Thus, the network analyzer provides the capability of performing sweeps of ω for frequency domain analysis of the nanomechanical beams. It is clear from Eq. 3.45 that strong actuation forces can be obtained at small separation distances U0 . However, there exists fabrication challenges in preparation of suspended nanomechanical structures with high aspect ratios at small separations due to surface tension effects. Additionally, the thin-film deposition process to fabricate the parallel metal plates of the capacitor can create undesired electrical contact in devices with small U0 values. Such short circuited connections are detrimental to the electrostatic actuation method, as undesired current would flow across the electrical contacts creating permanent damage in the devices. The challenges in preparation of samples with high beam-substrate resistivity can significantly decrease the fabrication yield. In a typical successful sample, actuation power level of ∼ 1 mW was employed to drive the beams. The voltage offset Vdc was adjusted to keep the resonant response within linear limits of individual beams while optimizing the signal-to-noise ratio. Typically, voltage levels of approximately Vdc ∼ 20 V were applied. The AFM cantilevers characterized in this study were not actuated since their thermal motion was within the detection capabilities of the Michelson interferometer. Hence, their response was recorded simply by feeding the amplified photodetector signal to a spectrum analyzer, as demonstrated in Sec. 2.1.1.
122 3.2.3
Analysis of Resonance Data in the Frequency Domain
As discussed in detail in Sections 1.2, the linear frequency response of the damped simple harmonic oscillator can be expressed as u=
F0 1 h i. ms (ω 2 − ω 2 ) + i ωωd d
(3.48)
Q
Here, two device parameters are of critical importance; the damped resonance frequency ωd and the quality factor Q, as defined in Eqs. 3.16 and 3.17, respectively. These parameters can be extracted through curve fitting to the spectral response of the resonator. The transmission power measurement of the network analyzer allows for the separate measurements of amplitude urms
v F0 u 1 u =√ ³ ´ , t 2ms (ω 2 − ω 2 )2 + ωd ω 2 d
and phase
· θ = tan
−1
(3.49)
Q
¸ ωd ω/Q − 2 . ωd − ω 2
(3.50)
Typical |u|2 and θ measurements of a nanomechanical beam are presented in Fig. 3·4(a), for a resonator of dimensions (w × h × l) 500 nm ×280 nm ×11.2 µm, with a resonance frequency of ω0 /2π = 18.1 MHz in a vacuum of P = 49 mTorr. Extraction of device characteristics is usually performed through Lorentzian curve fitting to the measured power data, proportional to |u|2 . A custom code was developed for automated extraction of ωd and Q from experimental data using least mean squares approach.12 In the fitting function, a constant offset was included to compensate for a spectrally flat background signal. A typical Lorentzian 12
fminsearch algorithm of Matlab, Mathworks, Inc.
123 30
70
|u|
30
-90
20
(a.u.)
-60
(degrees)
40
2
(a.u.)
-30
50
-2
-4
-6 -120
10 0 18.06
0
0
60
18.09
18.12
18.15
-150
-8 -2
(MHz)
(a)
0
2
4
6
(a.u.)
(b)
Figure 3·4: Resonance of a silicon doubly-clamped beam of width w = 500 nm, thickness h = 280 nm and length l = 11.2 µm at vacuum pressure level of P = 0.049 Torr. (a) Displacement |u|2 and phase θ in the frequency domain, along with the Lorentzian fit to |u|2 (solid line). (b) Quadrature plot of the real κ and imaginary χ components of the response, as defined by Eq. 3.52. The device quality factor was determined to be Qs ≈ 1530. fit to |u|2 is plotted as a solid line in Fig. 3·4(a). Using this fit, the structural quality factor of this resonator was determined to be Qs ≈ 1530. The Lorentzian fits to the signal amplitude are highly accurate for strong detection signals with flat background noise. However, as the device dimensions are reduced, the frequency response of the resonator diminishes into a small signal superposed onto a large, frequency dependent background. This frequency dependent background, with amplitude B and phase φ, is created due to the many components of the detection circuit, and is often inevitable. In a first order approximation, the amplitude of the background signal can be expressed with a linear function B = B0 + B1 (ω − ωd ). Accounting for this background, the detected
124 displacement becomes u˜ =
eiφ F ³ ´ + B0 + B1 (ω − ωd ) . m (ω 2 − ω 2 ) + i ωd ω d
(3.51)
Q
However, the above equation contains many unknowns which complicate the fitting process. Furthermore, fitting solely to the amplitude of Eq. 3.51 can create significant errors as the algorithms make an effort to describe the insignificant tail in the data. In such situations, quadrature fitting to the real (κ) and imaginary (χ) components of the measured transmission signal (Petersan and Anlage, 1998) can provide more accurate prediction of ωd and Q. Therefore, to explore possible improvements to the results of the Lorentzian fitting algorithm, quadrature fitting analysis using Eq. 3.51 was also performed. From Eq. 3.48, the in-phase and out-of-phase components of the displacement U (ω) can be expressed as κ = Re{u} = urms cos (θ) , χ = Im{u} = urms sin (θ) .
(3.52)
As demonstrated in Fig. 3·4(b), a parametric plot of κ − χ will form a circle, with a diameter equal to the amplitude of urms at resonance. As ω → ±∞, urms → 0 and the circle will pass through the origin. In the absence of the background circuit signal (an ideal situation), at resonance frequency ω = ωd , θ = −π/2 and κ = 0, χ = −urms . Hence, in such an ideal case, the frequency of the crossing point of (0, −urms ) is essentially the resonance frequency. However, as can be deduced from Eq. 3.51 and observed in Fig. 3·4(b), the background phase φ rotates the circle and its amplitude B shifts the center, making the fitting algorithm dependent on
125 too many parameters, requiring better initial guesses. To overcome this difficulty, the initial fitting can instead be performed on the derivative of the phase, 2
dθ Q2 (ωd2 − ω 2 ) = 2 dω Q2 (ωd2 − ω 2 ) + ω 2 ωd2
(3.53)
Clearly, Eq. 3.53 is free of all background information, and is dependent only on the resonance frequency and phase. However, the numerical derivative of experimental data often amplifies the noise and fitting to the derivative can lead to errors. Therefore, the fitting to Eq. 3.53 is only appropriate in obtaining rough initial guesses on Q and ωd . The results from Eq. 3.53 can be used as initial conditions in performing a simultaneous fitting of κ = Re{˜ u} and χ = Im{˜ u}. In this parallel approach, the errors in both real and imaginary components can be minimized through an optimization algorithm using non-linear least squares approach.13 The success of quadrature fitting algorithms were determined to be very sensitive to the initial predictions provided, especially in low-Q (high pressure) situations. The above described quadrant fitting algorithm was tested using experimental data, and it was observed that approach fails in high pressure resonance data with Q < 300. A significant improvement was observed when the results of Lorentzian fitting algorithm were supplied into the Quadrant fitting code as initial guess values. Then, the Quadrant fitting outcome was determined to independently verify the Lorentzian results, with an agreement between the two results by ±%5 across a wide range of Q factors. However, the significantly longer computation time of the multi-step quadrant fitting algorithm with negligible benefits did not justify its use for all data points. The quadrant approach was only used as a verification of the Lorentzian fitting, the primary approach employed in this 13
lsqnonlin algorithm of Matlab, Mathworks, Inc.
126 study. 3.2.4
Thermal Effects due to Laser Absorption
Optical detection methods are often regarded as weakly perturbing. However, no observation scheme can be perfectly decoupled from the system it is measuring. In optical setups, the encountered perturbation is often through thermal effects created by the absorption of the high intensity probe laser. Thermal effects due to focused laser beams in nanomechanical structures is a widely observed phenomena, especially at visible wavelengths where silicon absorption is strong. While this effect has been previously exploited for photothermal excitation of devices through laser modulation (Ilic et al., 2005; Sampathkumar et al., 2006), thermal effects often lead to an undesired change in system characteristics. The laser-induced temperature increase in the resonator creates a decrease in the structural stiffness k, resulting in a decrease in the resonance √ frequency ωd ∝ 1/ k. The effect has been demonstrated to be a significantly nonlinear phenomena at high intensity laser power (Sampathkumar et al., 2006), and can cause errors in determining the natural resonance frequency of the device. In an effort to characterize the extent of the thermal effects, the resonance frequency was monitored as the power level of the optical probe beam was varied at an optical wavelength λ = 633 nm, using the setup displayed in Fig. 3·3. The observed thermal effect for a typical nanomechanical resonator of dimensions (w × h × l) 230 nm ×200 nm ×9.6 µm with a chromium coating of ∼ 20 nm is plotted in Fig. 3·5. A linear frequency decrease up to 1.2% was determined as the incident optical power level was increased up to P0 ∼ 0.5 mW. As indicated by the results of Fig. 3·5, thermal effects on the device resonance
127
0.996
(
d
/
d,0 )
1.000
0.05 Torr
0.992
0.9 Torr 20 Torr 150 Torr 0.988 0
100
200
P
0
300
400
( W)
Figure 3·5: Thermally-induced frequency shift due to laser absorption, as the incident laser power is increased, measured using a doubly-clamped resonator of dimensions (w × h × l) 230 nm ×200 nm ×9.6 µm with an undamped resonance frequency of 24.2 MHz. The measured resonance frequency ωd at each pressure level was normalized by the damped resonance frequency ωd,0 at the lowest power level, to remove inertial loading effect of the pressure increase from the comparison. can significantly hinder reliable measurements of the quality factor Q. Instabilities in the resonance frequency during network analyzer frequency sweeps, of approximately 12 seconds, can cause an artificial broadening of the recorded spectral response. Such an effect can lead to significant levels of overestimation in determining the quality factor, especially in high-Q resonance curves. Hence, power level incident on the structure must be maintained to be constant. Incoming beam power and the relative position of the optical spot are the two parameters that determine the incident optical power. Therefore precautions were taken in the setup design for both situations; (a) the alignment of the optical spot was stabilized to a large extent through the use of close-loop feedback on all axis of the translation stage, and (b) the incoming laser power was constantly monitored in real-time through a secondary photodetector, shown in Fig. 3·3.
128 Finally, to explore the possibility of a convective cooling effect that can also corrupt Q-measurements during pressure sweeps, the frequency shift measurements were repeated at a wide range of pressure levels 0.05 < P < 150 Torr, as indicated in Fig. 3·5. Each set of measurements was normalized by the respective resonance frequency at the lowest optical power level, P0 ∼ 20 µW. Through this normalization, frequency shift due to inertial fluidic effect was removed from the analysis. The amount of thermally-induced frequency shift at each pressure level overlapped, indicating that convective cooling effects were not significant. Quantifying the thermal effect on Q is significantly more challenging than monitoring the shift in resonance frequency. As the optical power level is lowered, the signal level decreases substantially with respect to the background, making accurate curve fitting using any of the methods in Sec. 3.2.3 difficult. Despite these fitting uncertainties, efforts to quantify the change in Q indicated that a maximum of 10% overestimation may be present in measurements at P0 = 0.5 mW when compared with data obtained at P0 = 100 µW, for resonances of Q ∼ 700. This value is expected to be significantly lower for low-Q data, as thermally induced frequency shifts become less important. However, efforts to extract this information at both low laser power and low-Q yielded signals too low for accurate fitting. Additionally, it was difficult to assess the origin of the observed Q decrease using the data obtained from the frequency domain sweep technique. The cause could be attributed to the above described “artificial” broadening, as well as possible “real” thermally-induced damping mechanisms. In summary, the lack of convective cooling effect and the relatively small thermal perturbations observed provides the confidence to characterize fluidic interactions in nanomechanical beams using optical methods, provided the optical
129 spot is maintained at a specific power level and location during individual pressure sweeps. 3.2.5
Pressure Effects on Resonator Performance
The experimental configuration described in Sec. 3.2.1 and the data analysis methods outlined in Sec. 3.2.3 provide the tools for studying the nanomechanical resonator response characteristics as a function of pressure. By capturing frequency domain response as the pressure is increased, the nanofluidic interactions between the resonator and the surrounding gas can be examined. In Fig. 3·6(a), the response of a single resonator of 18.1 MHz frequency at various pressure levels is displayed. Two significant effects are immediately apparent: a decrease in resonance frequency ωd and a broadening of the linewidth, indicating a decrease in quality factor Q. These effects are more clearly quantified in the pressure sweep results plotted in Fig. 3·6(b). Both parameters saturate at their intrinsic values, ω0 and Qs , in vacuum. From Eq. 3.14, the observed changes in ωd and Q allow for the direct extraction of the inertial and dissipative effects. These forces, as demonstrated in Sec. 3.1, are functions of device geometry, frequency as well as pressure. As a result, only a collective analysis of multiple beams can clarify the nature of the interaction. Comparative results from such a detailed study is presented in the following section. All of the above mentioned theories predict amplitude independent γf in the linear regime. To verify this, the quality factor measurements were quantified as a function of displacement amplitude through the use of Michelson interferometry. The amplitude of oscillation urms was varied by changing the drive voltage Vdc , as
130 -3
0.049 Torr
10 (a. u.)
5.4 Torr
1 3
10
32 Torr
d
/
Q
urms
-4
10
0
0.99 100 Torr
302 Torr 942 Torr
0.98
Q d
0
2
0.97
10
17.6
17.8 / 2
18.0
(MHz)
(a)
0.1
1
10
100
1000
P (Torr)
(b)
Figure 3·6: (a) Resonance of a silicon doubly-clamped beam of width w = 500 nm, thickness h = 280 nm and length l = 11.2 µm at various N2 pressures in the chamber: P = 0.049, 5.4, 32, 100, 302 and 942 Torr. (b) The extracted quality factor Q and normalized resonance frequency ω/ω0 of the same device as a function of pressure. Here, Qs ≈ 1530 and ω0 /2π = 18.1 MHz. discussed in Eq. 3.46. The measurements were performed with a 58.6 MHz beam of dimensions (w × h × l) 0.73 × 0.28 × 5.6 µm at pressure P = 500 Torr. The results, presented in Fig. 3·7, demonstrate the linear relation between Vdc and urms . In Fig. 3·7(b), fluidic dissipation is observed to be uncorrelated to the amplitude of motion, and the quality factor is measured to be Q ∼ 250 ± 10% regardless of drive amplitude. This result, in well-agreement with expectations, gives us confidence to compare dissipation measurements from various devices, regardless of the drive amplitude.
3.3
Analysis of Fluidic Interaction
In order to understand the fluidic effects, a large parameter space was explored by varying the resonator frequency and geometry as well as pressure levels. A set of doubly-clamped beam resonators with various dimensions was fabricated, as
131
16 V
120
urms
(pm)
8 V
100
4 V
80 60
140 100 60 40
20
20 0.996
1.000
/
0
(a)
1.004
250
80
40
0
300
120
Q
23 V
urms (pm)
140
0
200
urms Q
0
5
10
15
Vdc (V)
20
150 25
(b)
Figure 3·7: (a) NEMS displacement urms at drive voltages Vdc = 4, 8, 16, 23 V, measured with Michelson interferometry at P = 500 Torr. The device dimensions are (w × h × l) 0.73 × 0.28 × 5.6 µm and the undamped resonance frequency is ω0 /2π = 58.6 MHz. (b) The effect of drive voltage Vdc on displacement urms and quality factor Q for the same nanomechanical beam. listed in Table 3.1, using the techniques outlined in Sec. 1.3. To further extend the frequency range of the measurements, two commercial silicon AFM cantilevers were also characterized at their fundamental and 1st harmonic modes, as indicated in Table 3.1. All dissipation experiments were performed using ultra-high purity nitrogen gas. However, a comparative study of these significantly different devices requires consideration of the variations in geometry and structural damping, as discussed below. As listed in Table 3.1, each device demonstrated a different intrinsic quality factor Qs . Intrinsic dissipation in nanomechanical resonators is often attributed to structural impurities, thermoelastic damping mechanisms (Lifshitz and Roukes, 2000; Pelesko and Bernstein, 2003), clamping losses, and more dominantly to surface effects (Mohanty et al., 2002). Despite the use of perfect crystals with minimal number of impurities, the miniaturization trend has been damaging to structural
132
Table 3.1: Device parameters, transition pressure P (Wi = 1) and the approximate lower pressure limit Pmin for accurate measurements for the devices used in the study. 1st harmonic mode was also employed for some AFM cantilevers. (w × h × l) (µm) 53 × 2 × 460 (1st Harmonic) 36 × 3.6 × 125 (Fundamental) 36 × 3.6 × 125 (1st Harmonic) 0.50 × 0.28 × 17.1 0.50 × 0.28 × 11.2 0.93 × 0.22 × 9.9 0.76 × 0.22 × 9.9 0.23 × 0.20 × 9.6 0.50 × 0.28 × 9.1 0.32 × 0.20 × 7.7 0.50 × 0.28 × 5.9 0.25 × 0.20 × 5.6 0.73 × 0.28 × 5.6 0.24 × 0.20 × 3.6
ω0 /2π (MHz) 0.078 0.31 1.97 10.4 18.1 22.8 22.9 24.2 27.1 33.2 45.7 53.2 58.6 102.5
Qs
P (Wi = 1) (Torr) 1.0 3.0 17.5 110 200 176 216 280 290 320 310 400 490 −
8321 8861 3522 1840 1530 1335 1200 415 909 780 1066 571 525 495
Pmin (Torr) 0.05 0.05 0.06 1.9 2.9 0.8 1.2 2.77 2.6 15.8 2.3 1.2 19.0 11.9
quality of mechanical resonators. In short, surface losses due to defects and oxide formations is believed to be the dominant factor in determining the quality factor for devices of increasing surface-to-volume ratios (Ekinci and Roukes, 2005). This trend is generally reproduced in the results listed in Table 3.1. Differences in intrinsic dissipation are harmful to this comparative study fluidic interactions, and need to be removed. Since the dissipative forces are additive in their nature, the fluidic dissipation γf can simply be extracted as µ γf = γ − γs = ω d
1 1 − Q Qs
¶ .
(3.54)
While Eq. 3.54 removes the effect of individual intrinsic quality factors from the dissipation calculations, it does not reduce the benefit of high quality factor
133 devices in the measurements. At low pressure levels, the dissipation becomes dominated by structural losses, and 1/Qs along with the uncertainties in our measurements (of approximately 5%) determine the lower pressure limit of accurate fluidic dissipation measurements. The minimum pressure of accurate measurements Pmin for each device is reported in Table 3.1. A clear correlation between Qs and Pmin is apparent. The upper limit of the measurements in this study were set by capabilities of the gauges, at 1000 Torr. Structural differences also need to be accounted for in this comparison of devices with varying geometry. Fluidic energy dissipation, whether through viscous or molecular collision mechanism, is a surface effect and the amount of energy loss is proportional to the surface area of the structure. However, the stored energy in the device scales with the effective mass of the structure ms . Hence, a natural scaling of the quality arises here, Q ∝ ms /S. Furthermore, the factor S/ms clearly appears in all the theoretical fluidic dissipation expressions presented in Sec. 3.1, specifically Eqs. 3.26, 3.29 and 3.43. It is less apparent in the viscous flow derivation around an oscillating cylinder of (Eq. 3.24), where the S/ms effect is burried inside the Reynolds number. To remove this geometry dependent factor from the comparison of fluidic dissipation in various devices, the analysis will be based on the normalized fluidic dissipation γn = γf
ms . S
(3.55)
It is important to note that the structural mass ms used in Eq. 3.55 includes the appropriate effective mass constant from Table 1.1 or 1.2, as well as any additional mass due to the metal coating.
134 3.3.1
Fluidic Dissipation Measurements and Theoretical Fits
With the above considerations, the experimental results can now be compared to the theoretical predictions outlined in Sec. 3.1. For this purpose, pressure dependent γn will be theoretically obtained for all beam geometries from molecular collision prediction (Eq. 3.29), Sader formulation of viscous flow around oscillating rectangular beam (Eq. 3.24), and Yakhot-Colosqui model of Stokes 2nd problem at high-frequencies (Eq. 3.43). The results of these comparisons are plotted for each beam as a function of nitrogen pressure, in Figs. 3·8-3·10. Eq. 3.43 is multiplied by a fitting factor in each plot as indicated in the captions. The nanomechanical doubly-clamped structures tested in this study were suspended at a height of U0 ≈ 400 nm above the silicon substrate, as displayed in Fig. 1·5. Hence, the possibility of squeeze-film damping of Eq. 3.26 was also considered for these devices. As the commercially acquired AFM cantilevers were freely suspended with no near-by surface, the squeeze-film damping analysis was irrelevant for these devices. The calculation of dissipation through Yakhot-Colosqui model (Eq. 3.43) requires a prediction of relaxation time τ . In order to obtain the fits, it was assumed that τ satisfies the empirical form τ ∝ 1/P (Rodahl et al., 1995; Watts et al., 1990). The key prediction of the Yakhot-Colosqui (Yakhot and Colosqui, 2007) theory is that the dissipative effects of the fluid changes slope when plotted against pressure. This turning point in γn occurs when Wi = ωτ ≈ 1. This is in fact clearly visible in the experimental results presented in Figs. 3·8-3·10. The turning point in each plot is marked with an arrow, and the pressure level of the observed transition for each device is listed in Table 3.1. No such transition is observed for the 102 MHz beam, as it falls the beyond the upper pressure limit of the measurement
135 capabilities in our setup.
Figure 3·8: Normalized fluidic dissipation γn as a function of pressure obtained from beams at indicated frequencies, as listed in Table 1.2. The lines are from models of molecular collision (Eq. 3.29), Sader (Eq. 3.24) and Yakhot (Eq. 3.43) using τ ≈ 1850/P . For results obtained from nanomechanical structures, squeeze film damping (Eq. 3.26) was also included. The turning point of Wi = 1 is marked with an arrow in each plot. Yakhot-Colosqui formulation of Eq. 3.43 was multiplied with a fitting constant of (a) 2.8 (b) 2.8 (c) 3.5 (d) 5.0. Thus, the specific transition in each resonator of frequency ω can be correlated to the relaxation time at the given pressure level, through the formula τ = 1/ω. The relaxation time extracted from the transition of each resonator is plotted in Fig. 3·11. The trend agrees significantly with the predictions of τ ∝ 1/P , and
136
Figure 3·9: As in Fig. 3·8, for the beams at frequencies as indicated on the plots. Yakhot-Colosqui formulation of Eq. 3.43 was multiplied with a fitting constant of (a) 3.8 (b) 3.0 (c) 3.8 (d) 2.5 (e) 3.6 (f) 2.9.
137
Figure 3·10: As in Fig. 3·8, for the beams at frequencies as indicated on the plots. The turning point of Wi = 1 is marked with an arrow in each plot. Yakhot-Colosqui formulation of Eq. 3.43 was multiplied with a fitting constant of (a) 3.2 (b) 2.0 (c) 3.7 (d) 3.2.
138
3
10
(ns)
2
10
1
10
0
10
1
10
PWi=1 (Torr)
100
Figure 3·11: Relaxation time τ as a function of pressure. The points were extracted from fluidic dissipation measurements of 13 resonators, displayed in Figs. 3·8-3·10. The solid line is a least-mean-squares fit and indicates that τ ≈ 1850/P . The dashed line represents the theoretical calculation of bulk relaxation time from τ ∼ lmf p /cs . fitting to the data reveals the correlation of τ = 1850/P , where τ is in units of nanoseconds and P is expressed in Torr. This correlation was successful in obtaining the Yakhot-Colosqui fits to all plots in Figs. 3·8-3·10. Thus our experiments provided a direct and unique way to extract τ as a function of pressure P . The obtained relaxation time results are of the same order of magnitude as the approximate values obtained from τ ∼ lmf p /cs , plotted in Fig. 3·11 as a dashed line. Furthermore, it is important to note that this formulation for estimating τ is considered to be valid for bulk flow whereas the experimental dissipation measurements presented here are due to surface effects, where τ can be significantly different than bulk behavior. Thus, the discrepancy by a factor of ∼ 8.8 in the absolute values of relaxation time between the experimental results presented here and analytic estimations can be regarded as acceptable.
139 Comparison of Theoretical Descriptions The dissipation plots of Figs. 3·8-3·10 provide significant information regarding the applicability of the various fluidic damping theories to the high-frequency nanofluidics domain. An initial, clear observation is that the squeeze-film damping predictions (Eq. 3.26) for the geometries studied here remained significantly low when compared to the observed energy losses. The squeeze-film damping effect is more strongly observed in structures with lateral dimensions greater than the separation distance, w À U0 (Hutcherson, 2004). For the nanomechanical resonators explored in this study w ∼ U0 , the effect is determined to be negligible. Hence, the presence of the “near-by” substrate can be ruled out as a significant source of dissipation. The results obtained from micromechanical cantilevers (Fig. 3·8(a)-(c)), display a strong agreement with the Newtonian fluid dynamics solution of oscillating flow around a rectangular beam Eq. 3.24 (Sader, 1998), down to the transition pressure level. Below Wi = 1, the assumptions underlying the Newtonian descriptions fail and the continuum model diverges from the results as the pressure is decreased. For nanomechanical beams which are at higher frequencies, the continuum model converges with the data only at high pressure levels, at best. For pressure levels significantly below the transition level, the molecular collision model of Eq. 3.29 is observed to be successful in its description to the data, with dissipation γf being proportional to the pressure level P . In the past, the molecular collision model and the continuum model have been employed in combination to describe the fluidic dissipation across wide range of pressure levels (Blom et al., 1992; Bhiladvala and Wang, 2004). However, this approach fails to provide a smooth transition, as can be observed from the results in Figs. 3·8-3·10.
140 800 Torr
10
200 Torr
2
100 Torr
n
(kg/m 2-s)
400 Torr
10
1
1
10
6
100
1000
(10 /s)
Figure 3·12: Normalized fluidic dissipation γn as a function of the resonator frequency ω0 for several resonators at four different pressures. From top to bottom, τ ≈ 1850/P =2.3, 4.6, 9.2, 18.5 ns. The lines are fits calculated using Eq. 3.43. Wi ≈ 1 points are marked with an arrow for each pressure. In fact, this transition regime which is not described by either formulation can be a significantly large flow regime, i.e., 40 < P < 1000 Torr for the 10.4 MHz resonator as demonstrated in Fig. 3·8(d). The Yakhot-Colosqui formulation of Eq. 3.43 was successful in describing the transition regime observed in all the nanomechanical resonator, with a fitting factor ∼ 2.8. This multiplicative constant is suspected to arise from adapting the theoretical expressions of (Yakhot and Colosqui, 2007) for an infinite plate oscillating in-plane to the finite and rectangular resonators oscillating out-of-plane. It is important to note here that the results of the Yakhot-Colosqui model essentially acts as the missing link in connecting the continuum conditions and the rarefied environments, for all frequency levels. The transition at ωτ ≈ 1 is clearly observed by varying the relaxation time through pressure sweeps, as demonstrated in Figs. 3·8-3·10. A study in the fre-
141 4 3.5 1 /2
3
)
2.5
1.5
n
/(
0
f
2
1
0.5 0.01
0.1
1
10
√ Figure 3·13: The scaling of γn / ωµρf for all resonators with ωτ . Each symbol corresponds to an individual resonator and the solid line is Eq. 3.56. All the predictions were multiplied by the same fitting factor of 2.8. quency domain further supports this result. The normalized dissipation at pressure levels 100, 200, 400 and 800 Torr was extracted from each measured device, and plotted as a function of the device resonator frequency ω0 in Fig. 3·12. The transition point is marked with an arrow for each pressure level and the solid lines as fits to Eq. 3.43 with τ ≈ 1850/P , using the identical fitting coefficient of 2.8 for all resonators. Fig. 3·12 provides a very significant insight to the YakhotColosqui model; in a fluid of fixed relaxation time, increasing the resonator above 1/τ saturates the fluidic dissipation of the structure. In a final effort to compare all obtained data on a single scale, the plot of dissipation as a function of ωτ is presented in Fig. 3·13. Here, γn is normalized once
142 more to remove all factors except ωτ , ¶ · µ 1 γn 1 −1 = √ tan (ωτ ) (1 + ωτ ) cos √ µρf ω 2 2(1 + ω 2 τ 2 )3/4 µ ¶¸ 1 −1 − (1 − ωτ ) sin tan (ωτ ) . 2
(3.56)
In the resultant plot of Fig. 3·13, each symbol corresponds to an individual resonator with the fitting factor of 2.8, and the solid line is Eq. 3.56. A general agreement between all the data points and Yakhot-Colosqui model is obtained in Fig. 3·13. Once again, a clear interpretation of Fig. 3·13 is that increasing the Weissenberg number of the flow beyond Wi > 1 leads to the significant benefits in terms of the fluidic dissipation in the resonators. 3.3.2 Inertial Load Measurements Also apparent in Fig. 3·6 is a small decrease in the resonance frequency as the pressure is increased. The observed decrease ∆ω = ω0 − ωd is primarily due to the inertial mass of the fluid mf that is being displaced in-phase with the resonator (Ekinci and Roukes, 2005), and can be obtained from Eq. 3.14 as βf mf ∆ω ≈ = . ω0 2 2m
(3.57)
The normalized frequency shift in the above equation corresponds to the approximate fluid mass per unit beam length. Both the geometry and the frequency of the moving surface is expected to play a role in determining mf and thus the frequency shift ∆ω. In an effort to rule out the geometry effects, the frequency shift was compared for four beams of identical widths (w = 500 nm) and thicknesses (h = 280 nm) but varying lengths and resonance frequencies as a function of pressure. The result of ∆ω/ω0 is plotted in Fig. 3·14. The arrows mark Wi ≈ 1 for each
143
10.4 MHz 18.1 MHz 27.1 MHz 0
45.7 MHz
10
10
-2
-3
10
100
P (Torr)
1000
Figure 3·14: Normalized frequency ∆ω for beams of identical widths w = 500 ω0 nm and thicknesses h = 280 nm, but varying lengths and resonance frequencies. The pressure for Wi ≈ 1 is marked with an arrow. Dashed lines are least-mean-squares fits to ∆ω ∝ P 1/3 . beam and the dashed lines represent best line fits corresponding to ∆ω ∝ P 1/3 . The molecular flow model valid at the low pressure limit predicts no inertial load on the structure as the density of the fluid is negligible with respect to the structure. At high pressure, continuum limit, the Stokes’ expression for an oscillating plate (Landau and Lifshitz, 1987) can be used to obtain an approximation for the boundary layer thickness of s δ=
2µ . ρf ω
(3.58)
This formulation results in boundary layer thicknesses of δ ∼ 1 − 5 µm for the experimental case plotted in Fig. 3·14. Calculating the inertial fluidic load from
144 the boundary layer in Eq. 3.58, Eq. 3.57 becomes µ ∆ω ρf δ 2 = . ≈ ω0 2ρs wh ρs whω
(3.59)
Hence, the frequency shift ∆ω/ω0 in Eq. 3.59 becomes independent of the pressure, clearly not the result observed in Fig. 3·14. The penetration depth δ− derived by the Yakhot-Colosqui theory (Yakhot and Colosqui, 2007) in Eq. 3.37, is essentially the same as the classical Stokes prediction with corrections for high Weissenberg number flows. Nonetheless, the inertial load predictions of Eq. 3.42 from the Yakhot-Colosqui model underestimate the frequency shift significantly. The deficiency of Yakhot-Colosqui model in describing the frequency shift can be attributed to important geometric considerations. First, the model was developed for in-plane oscillations and does not account for the potential drag effects of the structure pushing against the flow. Secondly, the presence of the near-by substrate which is well within the penetration depth of the flow, is unaccounted for in Eq. 3.42. Although it has been demonstrated in the above results that the presence of the substrate does not result in creating additional dissipative forces, its inertial interaction can not be ruled out at this stage. A comprehensive understanding of the inertial effects is a topic of future numerical studies.
3.4 Implications on Device Design In the experimental data presented throughout this chapter, a clear transition in flow behavior has been observed at Wi = ωτ ≈ 1. It has been observed that the simple linear relation between stress and rate-of-strain in the Newtonian fluid breaks down as the flow time scale is approaches the relaxation time of the fluid. The recently developed Boltzmannian theory from Yakhot and Colosqui has been
145 rather accurate in its description of this transition. This transition at Wi = ωτ ≈ 1 was described (Yakhot and Colosqui, 2007) as a “viscoelastic to elastic” transition owing to the fact that the waves generated in the fluid by the resonator motion do not decay as ω → ∞. There also appears to be some universality with respect to device geometry as both cantilevers and doubly-clamped beams demonstrate consistent results when analyzed through the same geometric normalization. 7
10
800 Torr 400 Torr
100 Torr
6
10
Qfn
(m 2/kg)
200 Torr
1
10
6
100
1000
(10 /s)
Figure 3·15: Normalized quality factor in fluid Qf n = ω0 /γn as a function of the resonator frequency ω0 for several resonators at four different pressures. From top to bottom, τ ≈ 1850/P = 18.5, 9.2, 4.6, 2.3 ns. The lines are fits calculated using Eq. 3.43. Wi ≈ 1 points are marked with an arrow for each pressure. There is a relentless effort to develop nanomechanical resonators operating in gaseous (Li et al., 2007) and liquid environments (Verbridge et al., 2006a). The above presented results are expected to impact the design of next-generation nanomechanical resonators. Figure 3·12 suggests that fluidic dissipation saturates at high frequencies. For a comparison independent of geometry, normalized resonator quality in fluidic environment can be defined as Qf n = ω0 /γn . A plot of Qf n is presented in Fig. 3·15 for the measured devices at several pressure levels.
146 As before, the solid line is the result of Yakhot-Colosqui model. The transitions √ from Qf n ∝ ω to Qf n ∝ ω at Wi = 1 are marked with arrows. From Fig. 3·15, it is clear that there is a significant benefit to increasing the resonator frequency. However, the extent of the benefit can be discussed further. Taking into comparison two doubly-clamped beam resonators with identical widths and thicknesses but different lengths, i.e., identical
S ms
≈
1 w
+
1 h
but different fre-
quencies such that ω1 < ω2 , the following statements can be made: ω1 < ω2 < 1/τ ⇒ ω1 < 1/τ < ω2 ⇒ 1/τ < ω1 < ω2 ⇒
r Q2f ω2 ∼ , Q1f ω1 r Q2f τ ∼ ω2 , Q1f ω1 Q2f ω2 ∼ . Q1f ω1
(3.60)
From the above arguments, it is clear that the shorter, higher-frequency resonator will always be more resilient in a given fluid but the degree of resilience depends upon the fluid τ . However, from Eq. 3.55, it is clear that for two devices with identical frequencies, the smaller one with the larger S/ms will demonstrate higher fluidic dissipation and lower quality factor Qf . For the case where both S/ms and device frequency increase, the scaling factors need to be examined in detail to determine the end result.
147
Chapter 4
Conclusions and Future Directions 4.1
Summary of Achievements
Despite the increasing interest in NEMS applications in fluidic environments, an understanding of fluidic effects at this length and time scale has been lacking until now. In this thesis, in an attempt to clarify some of the nanofluidic issues that the NEMS community currently faces, I have demonstrated resonant operation of nanoelectromechanical systems in high pressure environments with an in-depth study of the pressure effects on device performance. Beyond the implications of this work in engineering future devices, these efforts have also contributed to the fundamental understanding of fluid dynamics. In the optical part of this thesis, I have presented the application of several detection schemes to subwavelength devices by both experimental demonstrations and theoretical analysis. In the case of interferometric detection, the results from both numerical and experimental studies that indicate a strong benefit of employing Fabry-Perot interferometry for characterizing the out-of-plane motion of nanomechanical resonators. However, the limitations of this technique to specific device geometries and the inability to extract exact device motion amplitude has been insufficient in some applications. Therefore, for the purposes of employing this microscope for detecting motion in cantilevers and even membrane structures, I eventually constructed a hybrid Michelson/Fabry-Perot interferom-
148 eter setup. I was able to demonstrate a Michelson interferometer displacement √ detection sensitivity of Su = 0.017 √pm . Hz Additionally, I designed and assembled a characterization setup for analysis of NEMS devices under various fluidic conditions. Feedback mechanisms were employed as necessary to ensure the long term stability of the samples and the optical components. I fabricated a wide range of device geometries, with resonance frequencies reaching 102 MHz, using the protocols that were presented in Section 1.3. Employing the above described optical setup, I initially examined the extent of the thermal perturbations on the NEMS samples. The thermal effects due to the incident laser power were determined to be minimal and controllable. Similarly, I demonstrated that the amplitude of motion was indeed an irrelevant parameter in analyzing fluidic dissipation, within the linear operation region. The nanomechanical resonators were then driven in their fundamental modes at pressure levels up to P ∼ 1.5 atm. Furthermore, in an attempt to extend the region of exploration, I performed Michelson interferometry on commercially obtained micro-cantilevers in various vibrational modes. In order to extract fluidic interactions from frequency domain resonance characteristics, I prepared automated fitting algorithms to analyze the spectral response of resonators. Consequently, by observing the changes of resonator response as a function of pressure variation, I was able to obtain sufficient data to discuss the applicability of various fluid models to the nano-scale oscillatory flow conditions that form around NEMS structures. With the ability to change pressure from 0.01 < P < 1000 Torr, I was essentially able to tune the relaxation time of the fluid by over four orders of magnitude. The obtained fluidic dissipation results confirmed the applicability of the con-
149 tinuum formulations to fluidic dissipation problems in relatively low-frequency micromechanical resonators. At the opposite end of the pressure spectrum, the molecular collision theory was demonstrated to be in agreement with the obtained experimental data. Undoubtedly, the most significant implication of this thesis arises from the fluidic effects observed in between these two flow conditions, namely the transition flow regime. I have demonstrated that manomechanical resonators are ideal structures to probe the transition between non-continuum and Newtonian flow, at the convenient pressure levels of 0.1 < P < 1000 Torr. The non-Newtonian conditions that emerged were well described by an extension of the Stokes’ oscillating plate problem to the high frequency limit. Furthermore, the observed transitions were employed in estimating the near-surface relaxation time of the fluid. Besides the obvious academic interest in the above results, there are significant implications to device design in developing the next generation NEMS for operation in fluidic environments. The results clearly demonstrate the benefits in pursuing high resonance frequency in order to decrease fluidic dissipation. However, an increase in surface-to-volume ratio, encountered as device sizes are decreased, creates a counter effect of lower Qf . Therefore, an optimization will be required for future devices. The use of materials with higher stiffness are expected to be beneficial to the device performance. Even at this early stage, it is evident that the NEMS devices can be used without the need for vacuum conditions, reducing the need for many expensive packaging processes. Furthermore, many application possibilities, like molecule sensing in gaseous environments, are now within reach.
150
4.2
Suggestions on Future Directions
Even though the analytical flow model of Yakhot and Colosqui was successful in describing the observed interactions, certain aspects of the analysis still requires clarification using numerical methods that fall beyond the scope of this study. Universality of the proposed analytical model is clearly observed in the data presented here. In fact, the applicability of the theory to both doubly-clamped structures and cantilever geometries is indeed impressive. The exact nature of the fitting factor employed in this study can perhaps be clarified through numerical modeling of various geometries. Furthermore, an improved understanding of the inertial effects created by high-frequency oscillatory flows could be beneficial in obtaining a more comprehensive understanding of the nanofluidic effects. Additionally, the surface roughness, especially for very small devices, is expected to have an important role in nanofluidics of nanomechanical resonators (Palasantzas, 2007) and should be explored in future work. Having established the applicability of NEMS in gaseous environments, efforts are already under way for similar explorations in liquids. Very recently, resonant operation of nanowire structures were reported in aqueous solutions (Verbridge et al., 2006a), albeit with rather low quality factors Qf ∼ 4. Evidently, improvements in the under-liquid quality factor is needed before applications can emerge. This requires a similar study of dissipation now in liquid phase. In the optimization discussions above, the emphasis was placed on increasing frequency to reach the elastic flow regime. In gas phase, the tuning of the onset of elastic response was achieved by changing the pressure level. Similar tunability in liquids could perhaps be obtained through various mixtures of liquids with different relaxation time (like glycerol-water) or by introduction of polymers into solutions.
151 Additional technical challenges lay ahead in developing a systematic analysis under liquid. Obviously, actuation and detection in liquids requires the development of some novel methods. One of the foremost technical challenges is the development and integration of a compact flow cell that allows for actuation and detection of NEMS resonators under controllable flow conditions. Recently, successful fabrication of nano-scale flow channels in PDMS1 by surface micromachining was demonstrated (Han et al., 2006). Additionally, its integration with microchip technology has been successful (Wood et al., 2005). At this stage, adaptation of PDMS technology to implement flow channels across suspended nanomechanical resonators seems to be a feasible option. Electrical conductivity of the fluid is also a major concern in liquid operation if electrostatic actuation is to be applied. This method can present a limitation to the types of fluids that can be studied, so implementation of alternative actuation approaches may prove to be more beneficial. Along these lines, I have performed preliminary studies into the possibility of resonant operation of NEMS by base excitation using piezoelectric actuators. In this technique, the device chip is bonded onto a piezoelectric ceramic and the combination is vibrated from the base by applying an AC voltage across the ceramic. This method, while being conceptually simple, presents some technical challenges of its own. Often, the piezoelectric actuators themselves have low resonance frequencies, and therefore demonstrate very weak mechanical responsivity at higher frequencies. Furthermore, attempts to drive the piezoelectric actuators harder are hindered by their high impedance values at the radio-frequencies, causing significant signal reflections. Nonetheless, I was recently able to achieve piezoelectric actuation using very thin (∼ 100 µm) 1
Polydimethylsiloxane
152
25
(arb. units)
(pm)
100
pd
zrms
10
15
10
V
1
20
5
0.1
0 0
5
10
15
/ 2
20
25
30
35
46.8
(MHz)
(a)
47.0
/ 2
47.2
(MHz)
(b)
Figure 4·1: (a) Displacement response of a 100 µm thick single crystal piezoelectric ceramic2 with 15 dBm actuation actuation power applied from the network analyzer. The displacement signal of the ceramic, measured using Michelson interferometry, was buried below the background signal beyond 35 MHz. (b) The resonance signal of a doubly-clamped nanomechanical resonator of dimensions (w × h × l) 590 nm ×220 nm ×4.6 µm, actuated with the same piezoelectric ceramic in P = 100 mTorr pressure, measured with Fabry-Perot interferometry with an incident optical power level P0 ∼ 1 mW. The quality factor was determined to be Q ∼ 500. ceramic pieces2 with actuation power level of 15 dBm (32 mW), limited by the network analyzer. The displacement of the piezoelectric actuator with the sample load, displayed in Fig. 4·1(a), was measured using Michelson interferometry with an incident laser power level P0 ∼ 1 mW. A strong decay of piezoelectric response is apparent at higher frequencies. It should also be noted that at these relatively high drive levels, increased precautions of cable shielding were necessary to minimize electromagnetic coupling between the actuation and detection signal lines. Despite the additional precautions, this coupling prevented the recording of the photodetector signal beyond 35 MHz. However, the benefit of high-Q in NEMS allowed for resonant displacement detection up to ∼ 47 MHz using the 2
TRS-X2B Single Crystal Piezoelectric, TRS Technologies, Inc.
153
Figure 4·2: An optical microscope image, obtained with the setup described in Fig. 3·3, of a family of suspended nanomechanical beams of identical length l = 5.6 µm, and thickness h = 230 nm and varying width 200 < w < 730 nm, obtained while being immersed in a glycerol:water (1:1) solution. The scanning electron microscope image of the same sample is presented in Fig. 1·4. The focused laser spot is visible in the center. same actuator. In Fig. 4·1(b), the Fabry-Perot measurement of the resonance of a doubly-clamped beam with dimensions (w × h × l) 590 nm ×220 nm ×4.6 µm at P = 100 mTorr pressure (Q ∼ 500) is plotted. Clearly, the signal levels as well as the Lorentzian lineshape are deteriorated in comparison to the electrostatic actuation. Improvements to the actuation can perhaps be made through the use of high-power amplifiers or impedance matching. The non-contact, remote sensing nature of optical displacement detection schemes can provide significant benefits in liquid experimentation. My initial attempts in imaging the nanomechanical resonators under water have been successful provided that a flat surface at the liquid-gas interface could be achieved to minimize image distortion. This problem was overcome in early attempts by trapping the liquid layer between the NEMS chip and a thin glass cover. A sample image of a family of nanomechanical resonators under 1:1 glycerol-water solution
154 is presented in Fig. 4·2. The clear visibility of suspended beams of width w ∼ 200 nm is very promising. Nonetheless, absorption spectra of the liquid and the cover material, as well as possible wave aberrations will need to be analyzed before optimized operation is achieved.
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CURRICULUM VITAE Devrez Mehmet Karabacak
Department of Aerospace and Mechanical Eng. Office: +1-617-358-2797 Boston University Fax: +1-617-353-5866 110 Cummington St.
[email protected] Boston, MA 02215 USA http://people.bu.edu/devrez/
EDUCATION Boston University, College of Engineering Ph.D. in Mechanical Engineering Thesis: “Resonant Operation of Nanoelectromechanical Systems in Fluidic Environments”
Boston, USA Sept. 2003 – Jan. 2008
Boston University, College of Engineering M.Sc. in Mechanical Engineering
Boston, USA Sept. 2003 – May 2005
Middle East Technical University, College of Engineering B.Sc. in Mechanical Engineering, Minor in Mechatronics Engineering
Ankara, Turkey Sept. 1998 – June 2002
ENGINEERING AND RESEARCH EXPERIENCE Boston University, College of Engineering Graduate Research Assistant in Nanoscale Engineering Laboratory
Boston, USA Sept. 2003 – Aug. 2007
Drexel University, College of Engineering Graduate Research Assistant in Combustion and Fuels/Fluids Group
Philadelphia, USA Sept. 2002 – Aug. 2003
168 Drexel University, College of Engineering Graduate Teaching Assistant in Mechanical Engineering and Mechanics Department
Philadelphia, USA Sept. 2002 – Aug. 2003
Robert Bosch Group, Diesel Fuel Injection Systems Intern Engineer in Process and Method Development Department
Bursa, Turkey July 2001 – Aug. 2001
Roketsan Missiles Industries Inc. Intern Engineer in Mechanical Production Department
Ankara, Turkey June 2000 – July 2001
JOURNAL PUBLICATIONS • D. M. Karabacak, V. Yakhot and K. L. Ekinci, “High-Frequency Nanofluidics: An Experimental Study using Nanomechanical Resonators,” Physical Review Letters 98, 254505 (2007) ¨ u, ¨ S. B. Ip• D. M. Karabacak, K. L. Ekinci, C. H. Gan, G. Gbur, M. S. Unl polito, B. B. Goldberg, and P. S. Carney, “Diffraction of evanescent waves and nanomechanical displacement detection,” Optics Letters 32, 1881 (2007) • D. Karabacak, T. Kouh, C. C. Huang, and K. L. Ekinci, “Optical knife-edge technique for nanomechanical displacement detection,” Applied Physics Letters 88, 193122 (2006) • D. Karabacak, T. Kouh, and K. L. Ekinci, “Analysis of Optical Interferometric Displacement Detection in Nanoelectromechanical Systems,” Journal of Applied Physics 98, 124309 (2005) • T. Kouh, D. Karabacak, D. H. Kim, and K. L. Ekinci, “Diffraction Effects in Optical Interferometric Displacement Detection in Nanoelectromechanical Systems,” Applied Physics Letters 86, 013106 (2005) CONFERENCE PROCEEDINGS ¨ u, ¨ Y. Meydbray, E. R. Behringer, J. I. Quesnel, D. M. Karabacak, • F. H. Kokl ¨ u, ¨ ”Subsurface Imaging with S. B. Ippolito, B. B. Goldberg, and M. S. Unl Widefield and Confocal Numerical Aperture Increasing Lens Microscopy,” Proceedings of IEEE Lasers and Electro-Optics Society Annual Meeting (2006) • K. L. Ekinci, D. Karabacak, T. Kouh and D. H. Kim, ”Optical Probing of NEMS,” Proceedings of the SPIE 6186, 61860M (2006)
169 • T. Kouh, D. Karabacak, D. H. Kim and K. L. Ekinci, “Ultimate Limits to Optical Displacement Detection in Nanoelectromechanical Systems,” Technical Proceedings of NSTI-Nanotech 2004, ISBN 0-9728422-9-2 Vol. 3, pg. 1-4 (2004) • H. Pearlman, M. Foster and D. Karabacak, “The Effect of Gravity on the Shape, Propagation, and Stability of Cool Flames,” Proceedings of Eastern States Section of the Combustion Institute, Fall Technical Meeting (2003) • H. Pearlman, M. Foster and D. Karabacak, “Cool Flames in Propane-Oxygen Premixtures at Low and Intermediate Temperatures at Reduced-Gravity,” Proceedings of 7th International Workshop on Microgravity Combustion and Chemically Reacting Systems, NASA/CP-2003-212376, Paper 49 (2003) CONFERENCE SEMINARS AND PRESENTATIONS • D. Karabacak and K. L. Ekinci, “Resonant Operation of Nanoelectromechanical Systems in a Viscous Fluid,” APS March Meeting, Baltimore, USA, March 13-17 (2006) ¨ u¨ and K. L. Ekinci, “Nano• P. S. Carney, D. Karabacak, S. B. Ippolito, M.S. Unl scale Motion Detection by Diffraction of Evanescent Waves,” APS March Meeting, Baltimore, USA, March 13-17 (2006) ¨ u, ¨ B. B. Goldberg, K. L. Ek• D. Karabacak, T. Kouh, S. B. Ippolito, M. S. Unl inci, ”Solid Immersion Lens Microscopy Techniques for Enhanced Optical Displacement Detection in Nanoelectromechanical Systems,” International Workshop on Nanophotonics and Nanobiotechnology, Istanbul, Turkey, June 28July 8 (2005) ¨ u, ¨ B. B. Goldberg, K. L. Ekinci, “Solid Im• D. Karabacak, T. Kouh, M. S. Unl mersion Lens Microscopy Techniques for Enhanced Optical Displacement Detection in Nanoelectromechanical Systems,” APS March Meeting, Los Angeles, USA, March 21-25 (2005)
170 HONORS AND AWARDS Boston University, College of Engineering Dean’s Research Fellowship Middle East Technical University, College of Engineering Dept. of Mechanical Engineering Third Best Design Award, Senior Design Project Competition Middle East Technical University, College of Engineering Dean’s High Honor and Honor Listings (multiple)
Boston, USA Sept. 2003 – May 2004 Ankara, Turkey May 2002
Ankara, Turkey Sept. 1998 – June 2002
PROFESSIONAL ORGANIZATION MEMBERSHIPS • American Physics Society (APS), 2004 – current • The American Society of Mechanical Engineers (ASME) membership, 1999 – current • ASME-Middle East Technical University Student Section founding member and vice-president, 1999 – 2001, president, 2001 – 2002
