Magnetoresistance characterization of nanometer Si metal-oxide

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Nov 15, 2004 - (down to 75-nm gate length) n-type Si metal-oxide-semiconductor ... electron magnetoresistance mobility of these nanometer devices was ...
JOURNAL OF APPLIED PHYSICS

VOLUME 96, NUMBER 10

15 NOVEMBER 2004

Magnetoresistance characterization of nanometer Si metal-oxide-semiconductor transistors Y. M. Meziani,a) J. Łusakowski,b) W. Knap, N. Dyakonova, and F. Teppe Groupe d’Etude des Semiconducteurs, Université Montpellier II, Unité Mixte de Recherche CNRS 5650, 34095 Montpellier, France

K. Romanjek, M. Ferrier, R. Clerc, and G. Ghibaudo Institut de Microélectronique, Electromagnétisme et Photonique, Ecole Nationale Supérieupe d’Electronique et de Radioélectricité de Grenoble, BP 257, 38016 Grenoble, France

F. Boeuf and T. Skotnicki STMicroelectronics, BP 16, 38921 Crolles, France

(Received 21 April 2004; accepted 25 August 2004) We report on the high-field (up to 10 T) magnetoresistance measurements performed on the short (down to 75-nm gate length) n-type Si metal-oxide-semiconductor field-effect transistors. The electron magnetoresistance mobility of these nanometer devices was determined for a wide range of the electron concentration (107 – 1013 cm−2, i.e., from a weak to a strong inversion) and gate length 共10 ␮m – 75 nm兲. In the case of long samples, the magnetoresistance mobility was compared to the effective mobility obtained by the standard parameter extraction and the split C – V techniques. The results are discussed in terms of the scattering power-law two-dimensional transport analysis. The data clearly indicate a significant decrease of the mobility with the gate length reduction below 100 nm. © 2004 American Institute of Physics. [DOI: 10.1063/1.1806991] method allows us to extract the ␮MR from weak to strong inversion conditions without knowing the device length. In the case of the long samples, the MR mobility is compared to the effective mobility, ␮eff, obtained by the conventional parameter extraction and the split C – V techniques. The difference between the ␮MR and the ␮eff is interpreted by an analysis of the two-dimensional transport taking into account the different scattering mechanisms and kinetic-energy dispersion relations (the scattering power-law transport analysis).

I. INTRODUCTION

The mobility is one of the key parameters that determine the transport properties in the semiconductor devices. In the metal-oxide-semiconductor field-effect transistors (MOSFETs), it can be measured by various techniques based on static parameter extraction1,2 or split capacitance measurements.3,4 Generally, these methods are restricted to the long channel devices and are very difficult to apply to the short channel ones. In the latter case, the static parameter extraction breaks when the mobility becomes gate length dependent, whereas the split C – V method is perturbed by parasitic capacitances for small area devices. Alternatively, the transport measurements in a magnetic field perpendicular to the plane of the current flow provide an alternative way to measure the carrier mobility independently of the carrier concentration.5 The results of such measurements strongly depend on the geometry of the current flow. In the case of the long and narrow devices (L Ⰷ W, where L is the length in the direction of the current flow and W is the width of the device), the Hall voltage builds up, which allows to determine the Hall mobility, ␮H and the Hall carrier concentration. On the other hand, for standard MOSFET structures, L Ⰶ W and the Hall voltage is shortened by a metallic source and drain electrodes. This Corbino-like geometry enables to measure the magnetoresistance (MR) mobility, ␮MR.5 In this paper, we present the MR mobility measurements performed on the n-type Si MOSFETs as a function of the gate length down to 75 nm. The applied experimental

II. MAGNETOTRANSPORT THEORY

The conductivity of the two-dimensional electron gas in the presence of the magnetic field, B, perpendicular to the current flow plane is described by5 jx = ␴xxEx + ␴xyEy , j y = − ␴xyEx + ␴xxEy ,

where Ex and Ey are the components of the electric field in the 共xy兲 plane and ␴xx and ␴xy are the components of the conductivity tensor. In the case of the degenerate electron gas, the Drude-Boltzmann theory gives

␴xx = ␴0/共1 + ␮2B2兲, 共2兲

␴xy = ␴0␮B/共1 + ␮ B 兲, 2 2

where ␴0 = nse␮, with ns and e being the sheet carrier concentration and charge, respectively. These equations should be supplied with boundary conditions consistent with the geometry of the considered sample. In the case of the Hall bar geometry 共L Ⰷ W兲 with the applied voltage in the x direction,

a)

Electronic mail: [email protected] On leave from Institute of Experimental Physics, Warsaw University, Warsaw, Poland.

b)

0021-8979/2004/96(10)/5761/5/$22.00

共1兲

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J. Appl. Phys., Vol. 96, No. 10, 15 November 2004

one requires no current in the y direction, i.e., j y = 0. This leads to Ex = jx / ␴0, which is independent on B. In other words, no magnetoresistance appears at this geometry of the current flow. On the other hand, in the case of a very short but wide device 共L Ⰶ W兲, the Hall voltage is short circuited by long current-supplying contacts. Then, Ey = 0 and jx = ␴xxEx. In such a case, the measured Ex = jx共1 + ␮2B2兲 / ␴0, and one expects a parabolic increase of the sample resistance as 共1 + ␮2B2兲. The mobility determined in this way is the magnetoresistance mobility, ␮MR. In the case of the small magnetic fields, i.e., for ␮2B2 Ⰶ 1, the MR mobility can be deduced from Eq. (2) within the Kubo-Greenwood formalism, after an integration over the carrier kinetic energy, ⑀, as6

␮MR =



冕 冕



冉 冊 冉 冊

⑀ · N共⑀兲 · ␮共⑀兲3 −

0



0

⳵f d␧ ⳵⑀

⳵f ⑀ · N共⑀兲 · ␮共⑀兲 − d⑀ ⳵⑀

,

共3兲

where f共⑀兲 is the Fermi-Dirac distribution, N共⑀兲 is the density of states, and ␮共⑀兲 = q␶共⑀兲 / m* is the energy-dependent mobility. ␶共⑀兲 and m* is the scattering relaxation time and the electron effective mass, respectively. For comparison, we give also the expressions for the effective mobility, ␮0 = ␴ / Qi, Qi being the inversion charge6

␮0 =





冉 冊

⑀ · N共⑀兲 · ␮共⑀兲 −

0





⳵f d⑀ ⳵⑀

共4兲

,

N共⑀兲 · f共⑀兲d⑀

0

and the Hall mobility

冕 冕



␮H =

0 ⬁ 0

冉 冊 冉 冊

⑀ · N共⑀兲 · ␮共⑀兲2 −

⳵f d⑀ ⳵⑀

⳵f ␧ · N共⑀兲 · ␮共⑀兲 − d⑀ ⳵␧

·

共5兲

It can be shown from Eqs. (3) and (4) that ␮MR reduces to ␮0 only if ␮共⑀兲 = const., i.e., for a degenerate (metallic) electron gas or for scattering mechanisms with no kineticenergy dispersion.

FIG. 1. Transfer characteristics Id共Vg兲 of bulk n MOSFETs with various gate lengths 共Vd = 10 mV兲.

measurements at 300 K, a variable-temperature insert was introduced into the coil, which enabled a vacuum thermal isolation of the sample chamber from the helium bath. The samples were placed in the microwave-ceramic supports and bonded using gold and/or aluminum wires. IV. RESULTS AND DISCUSSION A. Static parameter extraction

Figure 1 shows a typical drain current 共Id兲 versus the gate voltage 共Vg兲 transfer characteristics for various gate lengths ranging from 75 nm up to 5 ␮m, showing the excellent subthreshold behavior of these devices. The threshold voltage, Vt, and the low electric-field mobility, ␮0, were extracted using the Y共Vg兲 = Id / 冑gm共Vg兲 function, allowing an elimination of the first-order mobility attenuation factor and source-drain series resistance effects.2 Note that in the calculations, the electrical length, i.e., the effective device gate length, was taken a priori to be equal to the physical gate length L in order to avoid any assumptions concerning the gate length dependence of the mobility. Figure 2 displays the variation of the reduced threshold voltage Vt共L兲 − Vt共L = 10 ␮m兲 with the gate length L for the two values of the bulk substrate bias, Vb. Note the very good control of Vt共L兲 even for Vb = −3 V due to the presence of highly doped halos, i.e., implanted channel regions closed to the source/drain with a higher doping level.7 Figure 3 shows the corresponding relative variations of the low-field mobil-

III. EXPERIMENTAL DETAILS

The n-type MOSFETs studied here feature a 1.2 nm gate oxide with a 1500 Å N+ polygate. The test structures consisted of transistor batteries (with common source, bulk substrate, and gate contacts) with variable gate lengths L down to 75 nm and a fixed gate width W = 10 ␮m. The static parameter extraction was carried out with the HP 4155 semiconductor analyzer. The capacitance effective gate oxide thickness 共⬇2 nm兲 was determined from the gate-to-channel capacitance measurements performed at the strong inversion conditions on isolated gate test structures available on the same chip. The high magnetic field up to 10 T was provided by a superconducting coil working at 4 K. In order to make the

FIG. 2. Variations of Vt共L兲 − Vt共L = 10 ␮m兲 with gate length L for two bulk voltages Vb [Vt共L = 10 ␮m , Vb = 0兲 = 0.52 V and Vt共L = 10 ␮m , Vb = −3 V兲 = 0.83 V].

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Meziani et al.

J. Appl. Phys., Vol. 96, No. 10, 15 November 2004

FIG. 3. Variations of ␮0共L兲 / ␮0共L = 10 ␮m兲 with gate length L for two bulk voltages Vb 关␮0共L = 10 ␮m , Vb = 0兲 = 794 cm2 / V s , ␮0共L = 10 ␮m , Vb = −3 V兲 = 386 cm2 / V s兴.

ity with the gate length. Degradation of the mobility by 40% should be noticed at small gate lengths for a zero bulk bias, whereas ␮0 remains almost constant but two times smaller for Vb = −3 V. This behavior could be attributed to the attenuation of the halo’s influence and the increase of the vertical field at a high 兩Vb兩. B. Magnetoresistance measurements

Magnetoresistance measurements were carried out at room temperature on the devices with gate lengths ranging from 75 nm to 5 ␮m. Typical variations of the channel resistance, R, with the magnetic field squared, B2, are shown in Fig. 4 for the gate voltages below and above the threshold voltage and at Vd = 5 mV. The excellent linearity of R versus B2 confirms the validity of the magnetoresistance analysis carried out in this study. Despite the use of the high B values, the channel resistance changes typically by a few percent only due to the small electron mobility in the investigated transistors. An influence of the Landau quantization on the electron motion can be observable for the magnetic fields satisfying ␻c␶ ⬃ 1, where ␻c is the cyclotron frequency and ␶ is the scattering time. Since ␻c␶ ⬃ ␮B Ⰶ 1 in the case of the present experiment, one can safely neglect the influence of the Landau quantization on the electron conductivity. The MR mobility was extracted from the slope of ⌬R / R共B2兲 ⬅ ␮MR2B2 plots for several gate voltages covering the range from the weak to the strong inversion and for all the gate lengths. Figure 5 gives the variation of thus obtained ␮MR mobility as a function of L for several values of Vg. It

FIG. 4. Variations of channel resistance with magnetic field squared at different values of gate voltage (below and above the threshold).

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FIG. 5. MR mobility vs gate length L for two Vg ranges below and above the threshold 共Vb = 0兲.

should be stressed that this analysis does not require any assumption or even knowledge of the gate length. There are clearly two sets of curves in Fig. 5 corresponding to Vg above or below Vt. In order to analyze the weak inversion case in more detail, we calculated the inversion charge for each gate voltage from the equation Qi共Vg兲 =

L · Id共Vg兲 , W · ␮MR · Vd

共6兲

assuming that the MR mobility equals the effective one to the first order. An example of ␮MR on Qi dependence is illustrated in Fig. 6 for L = 185 nm. Several remarks should be given in connection with the data presented in Fig. 6. First, it should be emphasized that no other technique is capable nowadays to provide mobility values in a so deep weak inversion regime. For example, the split C – V technique becomes unreliable below 1011 q / cm2.8 The static parameter extraction methods give only access to the low-field mobility around the threshold. Second, for the deep weak inversion, a nice plateau in the ␮MR共Qi兲 below the threshold voltage is present, corresponding to the charge concentration from 107 up to 1011 q / cm2. This particular behavior was already found in the mid’70s for the Hall mobility in long channel MOS devices (with a thick oxide and a lightly doped substrate).9 It was further interpreted as a manifestation of the weak localization transport related to the surface gate potential fluctuations originating from the randomly distributed oxide charges.10 In such a situation, for gate voltages below the threshold value, there is a mobility

FIG. 6. MR Mobility vs inversion charge for device 6, L = 185 nm 共Vb = 0兲.

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J. Appl. Phys., Vol. 96, No. 10, 15 November 2004

FIG. 8. Relative difference between the MR (Hall) and effective mobility as a function of the scattering mechanism exponent n. X denotes MR or H.

FIG. 7. Normalized variations of ␮MR with gate length L for strong (a) and weak (b) inversion regions 共Vb = 0兲.

reduction by exp共−⌫ / kT兲, where ⌫ is the standard deviation of the surface-potential fluctuations and kT is the thermal energy. In the present case with the ultrathin gate oxide and high substrate doping, it is believed that such fluctuations could mostly originate from the depletion charge 共Qd兲 fluctuations due to the random dopant impurity distribution. Following an analysis similar to that for the oxide charge,10 we get ⌫=

␴Qd , Cox + Cd + Ci

共7兲

where ␴Qd is the standard deviation of the depletion charge fluctuations (proportional to Qd), Cox is the gate oxide capacitance, Cd is the depletion capacitance, and Ci is the inversion charge capacitance. Applying this formula to the mobility data of Fig. 6, one gets ␴Qd to be equal to about 1.5 ⫻ 1011 q / cm2, which represents only 5% of the depletion charge. Indeed, Eq. (7) also clearly indicates how ⌫ diminishes for gate voltages above the threshold, when Ci becomes much larger than Cox + Cd due to the onset of a strong screening with entering the inversion region.10 An alternative explanation of this subthreshold mobility behavior could also be simply based on an assumption of a dominant Coulomb scattering progressively screened with entering the strong inversion.5,6 Both of these hypotheses look reasonable, and further investigation, especially as a function of temperature, is necessary before drawing the final conclusions.

For the sake of comparison of the MR mobility to the effective mobility data of Fig. 3, we plotted in Fig. 7 the normalized MR mobility versus the gate length. A good agreement between ␮0 and ␮MR versus L dependence can be noticed for Vb = 0, especially for gate voltages below and near the threshold value. This clearly underlines an overall consistency of the two methods of the mobility extraction and confirms the 50%–60% mobility degradation at small gate lengths. The discrepancy between the effective and MR mobility was further studied by the transport analysis presented in Sec. II. To this end, we calculated the relative difference between ␮eff, ␮H and, ␮MR, as given by Eqs. (3)–(5), in the case of nondegenerate (deep weak inversion conditions) 2D gas with one sub-band. The calculation results are plotted in Fig. 8 as a function of the power-law exponent of the mobility on energy dependence, ␮共␧兲 ⬃ ␧n. A value of n = −0.5 corresponds to the acoustic phonon scattering, whereas n = 1 is associated to the 2D Coulomb scattering.5,6 Near the threshold voltage, the relative difference between ␮0 and ␮MR experimental values is about 65% for Vb = 0. According to the horizontal axis of Fig. 8, this should correspond either to n = −0.5 or n ⬇ 1. Since our bulk MOSFET devices are characterized by a relatively high channel doping, the Coulomb scattering should dominate the collision interactions, and therefore it explains reasonably well the observed difference between the effective and MR mobility values. V. CONCLUSIONS

In conclusion, the MR mobility measurements were performed on sub-100 nm Si MOSFETs. The applied experimental method enabled the electron mobility to be measured from the weak to the strong inversion conditions without knowing the device channel length. In the case of the long samples, the MR mobility results are with a good agreement with the effective mobility data obtained by the standard parameter extraction techniques. The MR data have clearly indicated a significant decrease of the mobility with the gate length reduction due to the increased halo’s influence. The difference between the MR and the effective mobility values was well interpreted by a two-dimensional transport analysis. The applied method of mobility extraction from the magnetoresistance measurements is a promising tool for the mobil-

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Meziani et al.

J. Appl. Phys., Vol. 96, No. 10, 15 November 2004

ity assessment in the case of deep sub-100 nm MOSFET devices, including the ultrathin silicon-on-insulator and the double gate structures. 1

Y. Taur et al., IEEE Electron Device Lett. 13, 267 (1992). G. Ghibaudo, Electron. Lett. 24, 543 (1988). 3 J. Koomen, Solid-State Electron. 16, 801 (1973). 4 C. Sodini, T. Ekstedt, and J. Moll, Solid-State Electron. 25, 833 (1982). 2

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A. C. Beer, Galvanomagnetic Effects in Semiconductors (Academic, New York, 1963). 6 G. Ghibaudo, in Encyclopedia of Electrical and Electronics Engineering, edited by J. Webster (Wiley, New York, 1998). 7 M. Jurczak et al., IEEE Trans. Electron Devices 48, 1770 (2001). 8 S. Takagi and M. Takayanagi, Jpn. J. Appl. Phys., Part 1 41, 2348 (2002). 9 J. T. Chen and R. S. Muller, J. Appl. Phys. 45, 828 (1974). 10 G. Ghibaudo, J. Phys. C 17, 3067 (1984).

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