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electron/hole energy relaxation time constant electron ... submicron regime, electron velocity overshoot starts ... 20% in a 0.15 pm device with I',, - V, = IO V and.
Solid-Stare ElecrronicsVol. 41. No. 8. pp. I 119-I125, 1997

Pergamon PII: SOO3tJ-1101(97)00031-2

((0 1997ElsevierScienceLtd. All rights reserved Printed in &eat Britain $17.00+ 0.00 0038-IlOI/

VELOCITY OVERSHOOT OF ELECTRONS Si INVERSION LAYERS D. SINITSKY,

F. ASSADERAGHI,

M. ORSHANSKY,

AND HOLES IN

J. BOKOR and C. HU

Department of Electrical Engineering and Computer Sciences, University of California, Berkeley, 21 I- 179 Cory Hall #1772, Berkeley, CA 94720-1772, U.S.A. (Received 6 August 1996; in reoised form 25 November 1996)

Abstract-Velocity overshoot of inversion layer electrons and holes is studied experimentally and analytically in special test structures with nominally uniform electric field. The data were used to calibrate energy relaxation parameters in a commercial simulator MEDIC1 ver. 2.0. We propose an analytical model for velocity overshoot and show that it agrees well with experimental data. The amount of hole velocity overshoot is small. 0 1997 Elsevier Science Ltd.

NOTATION

gate voltage drain voltage source voltage threshold voltage ve - v,; Vdr= Vd- v,; vgt = V8- v, inversion layer charge distance x from source junction along the Si/SiO, interface gate oxide gate capacitance drain current source series resistance (=drain side series resistance) effective MOSFET channel length electron current density electron inversion layer concentration electron mobility electron low-field mobility electron diffusion coefficient Boltzman constant electron gas local temperature lattice temperature electron/hole energy relaxation time constant electron energy flux tangential component of electric field in the inversion layer electron mass electron drift velocity average electron velocity across the channel carrier saturation velocity electron transit time across the channel electron, hole velocity in the channel phenomenological electron, hole overshoot length 1.

INTRODUCTION

As the size of silicon devices shrinks into the deep submicron regime, electron velocity overshoot starts to play a significant role in determining current drive. Velocity overshoot was first predicted by Ruch[l] using Monte-Carlo simulations. N-MOSFET I. Iil, enhancement at 4.2 K was reported by Chou et a1.[2]. Similarly, room temperature velocity overshoot was observed experimentally by Shahidi et al. and

Sai-Halasz et al.[3a,b] by inference from large Id. A direct measurement of velocity overshoot as a function of field and device length in Si inversion layers in nominally uniform field was made in[4]. Teitel and Wilkins explained[S] that carrier velocity overshoot in a semiconductor is because the energy relaxation time is greater than the momentum relaxation time. While the momentum relaxation time is determined by the carrier temperature, that temperature itself may be lower than the steady-state value that corresponds to the local field in the channel. In other words, the carriers do not have time to “heat up”. Lower carrier temperature gives higher mobility and velocity. It is difficult to analytically model MOSFET behavior including velocity overshoot because of a complex spatially varying electric field. The same complexity makes it difficult to calibrate 2-D device simulators that include energy transport. For this reason there is disagreement in the literature over electron energy relaxation parameters. In this article, the test structures of[4] with nominally uniform electric field are used to quantitatively evaluate velocity overshoot of electrons and holes. An analytical model for a test structure is proposed. The results agree well with experimental data. Also, the commercial simulator, MEDIC1 v. 2.0[6], energy balance parameter was calibrated to our measurements. The organization of the article is the following; first, we describe the use of test structures used in[4] to calibrate transport parameters of the TMA MEDIC1 2-D device simulator. Then a quantitative analytical model for energy transport such test structures is proposed and fitted to experimental data as well. Finally, the amount of hole overshoot is estimated quantitatively. We first review the test structure with nominally

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D. Sinitsky et al.

1120

uniform electric field. The schematic of the device is shown in Fig. 1. The device channel lengths are between 0.1 and 0.5 pm. Thick oxide allows us to apply very high voltages (up to 50 V) without oxide breakdown. Let us limit the drain-to-source voltage Vdrto be less than 2 V. Then ( Vgs- I’#)>>Vdr. Here Vgs is the gate-to-source voltage and V, is the threshold voltage. The charge at point .Yin the channel is given by Q(X) = C,,( IJ’~- Vch(.y) - V,(x)). where Vch(_x)is the channel potential at a point Y. Variation of V, with .Y is minimized by applying a 10 V reverse bias on the body, and V(s)cc l~‘~~, since V(s) is between P’, and Vd. Hence. Q(.x) is approximately uniform along the channel. That also implies the electric field is nominally uniform as well, though velocity saturation makes the E-field less uniform than channel charge. Simulations show, however, that E is uniform within 20% in a 0.15 pm device with I’,, - V, = IO V and Vd5= 2 V, which is the worst case condition in our experiments (Fig. 2). Usually E-uniformity is much better. Carrier velocity I’ across the channel can be derived from the formula;

Id

I’ = ?@)=

Id WC,,( Vgs- I',

- IdR,)'

(1)

Note that DIBL correction for V, needs to be accounted for to get a correct result. The DIBL effect on V, can be extracted from the shift of the subthreshold curves. Since the field is uniform, it can be calculated easily as; E

=

I’d%- Id& L eR ’

where Leff is the effective device channel

(2) length.

2. 2-D ENERGY BALANCE TEST STRUCTURE SIMULATIONS

The thick gate oxide test structure is an excellent candidate for calibration of transport parameters of 2-D device simulations because of a nominally uniform field profile in the inversion layer. This is illustrated in Fig. 3. As explained above, the test structures have almost uniform field in the channel

‘t V&-revd

biased

Fig. 1. The thick gate oxide device schematic.



vp,=w 5v IW

250 -

isv -2W

V,, cases.

since (V,, - V,)>> Vds. Conventional (thin gate oxide) MOSFETs exhibit a highly non-uniform channel field profile. The gradient of electric field is very sensitive to doping profile in the channel, which is very hard to determine precisely. More importantly, the field profile is sensitive to charge density, i.e. velocity. This makes it difficult to calibrate the velocity or energy relaxation parameters. Devices studied had effective channel lengths ranging from 0.13 to 0.43 pm. The oxide thickness was measured to be 500 A, the doping in the channel was 2 x lO”/cm’. The source series resistance R, was measured to be 250 Rpm. The results of MEDIC1 fitting are presented in Fig. 4. The best fit was obtained with electron energy relaxation time rE,e equal to 0.32 ps. This agrees well with Monte-Carlo simulations[l I]. Note that there is little difference among measurement, energy balance and drift diffusion simulations in case of LeR= 0.43 pm. At Len = 0.13 pm, energy balance simulations give the same current as the measurement, but the difference between drift-diffusion simulation and experiment is large. This is because the velocity overshoot effect gets larger as the channel length gets smaller.

Fig. 3. Field profiles in a thin and thick gate oxide MOSFETs; the thin oxide MOSFET field profile is very sensitive to doping profile in the channel and source/drain and to the energy relaxation phenomena.

Velocity overshoot of electrons and holes in Si inversion layers

0.6

A

cllpulmantaldata

0

eneraYbdmcenl

1121

X

_..

-0. Drain Voltage (V)

0.5 1.0 1.5 Drain Voltage (V)

0

Drain Voltage (V)

)

Drain Voltage (V)

Fig. 4. MEDIC1 fits to experimental data for thick gate oxide devices. Vg-V, = 5, IO, 15 V.

The mobility models used were the gate field mobility reproducing the universe mobility model, srfmob2, the doping concentration mobility, conmob, tangential field dependent mobility model, fldmob, and SRH and auger scattering models consrh and auger. The three Id-Vdr curves were for V,, I’, = 5, 10, 15 V respectively. Note that the same value of 7E.e (0.32 ps) was used for all the three gate voltages and all channel lengths. That indicates relative insensitivity of electron energy relaxation time to the vertical field in the channel. 3. ANALYTICALMODEL

In this section we derive an analytical model to predict u-E curves incorporating electron velocity overshoot effects. We assume a step function field profile, which is a good approximation for the thick gate oxide test structures. A similar problem was solved in[9], which however predicted a very slow heating up of electrons on the scale of about 1 pm. In reality it happens very fast, within 0.1 pm from the electrical field step. First we write the current equation for electrons in the channel[ lo]; J. = nqyE + qD,Vn + pknVT,.

(3)

Here p is electron mobility, D. is diffusion coefficient, T, is electron temperature, E is the electric field in the

channel, n is electron concentration in the channel. The energy balance equation is[lO]; $,kTe - To 2 -=n 7c

J . E_VF

W.

(4)

Here F, is the energy flux equal to[lO]; Fw = --

5 kT, -. 2 4

k2 J--npT,VT,. 4

(5)

Note that eqns (3H5) have a few assumptions[lO]. (a) The ballistic tail is small, and most electrons fall within a Maxwellian distribution, which is true for devices with LIP > 0.1 pm. (b) Temperature tensor is scalar, i.e. equipartition of energy between degrees of freedom. (c) Thermal energy is much bigger than drift energy (closely related to (b)). (d) Parabolic bands. (e) Wiedeman-Franz law of heat conduction[lO]. Although the temperature gradient terms are appreciable in short channel devices, the ratio of a temperature gradient to the term that has E in as a multiplying factor is on the order of

_~-..E

Vd

Assuming that T, is 10 times higher than room temperature, and Vd = 1 V, we get the ratio to be

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D. Sinitsky et al.

0.25. Thus, in solving (3x5) we will neglect all the temperature gradient terms that are of the order lower than E. We also neglect diffusion current in (3) because our test structures operate under a condition I’,, - I’,>>Vdarwhich guarantees good channel charge uniformity and E-field uniformity. Substituting (3) and (5) into (4) using the relation between mobility and electron temperature 11= +,

gradient

A’

0.10

,,o_o-o-o--o-~o

0.08

0.0

terms of

0.1

0.2

0.3

x (I@

To) _____E dT,

2q

(6)

Note that mobility model used is an approximation which works for hot electrons with energy approximately 60 meV, at which optical phonon scattering becomes significantj IO], and might give spurious velocity overshoot in some cases[l3]. However, a simple mobility model is essential for a closed form solution of transport equations. Since velocity overshoot occurs at high fields, this mobility approximation works well when velocity overshoot is significant, which is the case of our device. Equation (6) has a solution; + A

.A’

r

neglecting the temperature order less than E. we obtain;

T,(s) = +[I

A

A-A---A-A

0.12

Fig. 5. Temperature vs. position along the channel for a step electric field; E = 0 for x < 0, E = 0. I. 0.07. 0.04 MV cm-’ for .Y z 0.

field is already high, the electron gas is still cold, and therefore mobility is high. From (7) we can calculate the average velocity across the channel;

ttmt =

tanh(Bx+C)],

where L et7 V,” = -. t,rm,,

1 IO

B=

-&+)+($&~)1” C = tanh-If

.

(7)

The r, vs. .Yprofile is plotted in Fig. 5 for different E-fields. Note that the initial carrier thermal voltage in the channel is equal to 26 mV, and it sharply rises to the steady state hot value. It takes less than 0.1 pm for 7’, to saturate to the final value. Corresponding velocity with pc,= 250cm* V-’ s-’ is plotted on Fig. 6, using

(8)

Equation (8) was compared with experimentally extracted velocity. The result of analytical model fit is shown in Fig. 7. A good fit was obtained for a range of channel lengths, with the energy relaxation parameter T, equal to 0.27 ps, which is close to the value of MEDICI-extracted ~~~~ of 0.32 ps. Figure 8 shows the average electron velocity vs. channel length for different tangential fields as

Very close to the injection point, the velocity is determined by ballistic energy transfer from field to electron: 0.00

+nv’ = qEx. Note a significant velocity overshoot right after the electric field step. This is because although the electric

0.05

0.10

0.15

x (pm) Fig. 6. Velocity vs. distance for a step function field; velocity can transiently overshoot the final saturation value quite significantly.

Velocity overshoot of electrons and holes in Si inversion layers -extracted

0

t-

from measurement modeI %; r,=O.27ps

1123

6~10~

10

6

0 0.0

4.0x104

8.0X104

1.2x16

E tmgential

Electric Field (kV/cm)

Wm)

(a)

Fig. 7. An analytical model fit of (5) and (6) to the experimental data taken from test structures with nominally uniform field.

calculated from (8). Note that velocity overshoot can be seen only at short channel lengths and high tangential fields in the uniform-field test structures used in the study. 4. HOLE VELOCITY OVERSHOOT

hole velocity-tangential field curves for devices with .Leffranging from 0.36 to 0.16 pm. All the hole velocity curves overlap perfectly, in contrast to the electron velocity-field curves which show higher average velocity for shorter devices (Fig. 9(b)). Thus no hole velocity overshoot is observed at these structure lengths, though electron velocity overshoot clearly occurs. Note that the hole velocity shown reaches 5.5 x IO6cm s-’ and only just begins to saturate even for tangential field as high as 0.1 MV cm-‘. This indicates a rather high value of saturation velocity for holes, in fact, comparable to v,, of electrons. Figure 10(a) shows hole velocity-tangential field Figure 9(a) shows experimental

D

012

0:4

016

L,

(elm)

Oil

Fig. 8. Average velocity of electrons vs. channel length in a uniform field device; the field is a parameter. Data calculated from (6).

u.u

2.5110’

5.0X10’

J

75104

Fig. 9. (a) Hole v-E characteristics extracted from device measurements at room temperature. (b) Electron v-E characteristics extracted from device measurements at room temperature.

curves for the same devices at 77 K. The average hole velocity in the high tangential field regime is higher for .& = 0.16 pm than for L.~ff= 0.36 pm. We attribute this to a slight velocity overshoot of holes at 77 K. As can be seen from Fig. 10(b), electrons exhibit much larger velocity overshoot at 85 K, than holes at 77 K. The amount of velocity overshoot of electrons was found[l2] to be correlated with carrier mobility. At 77 K the hole mobility is comparable to the electron mobility at room temperature, but the amount of velocity overshoot is much smaller. The reason is that the origin of carrier velocity overshoot is dependent upon the interrelation between momentum and energy relaxation rates[ lo]. The difference in overshoot behavior of holes and electrons is not only because of differences in mobilities (i.e. momentum relaxation rates), but because of differences in energy relaxation rates as well. We will now estimate the hole energy relaxation time. The phenomenological overshoot length P‘ of holes and electrons can be expressed roughly as a product of the carrier velocity in the channel and the energy relaxation time; Pa) (9b)

D. Sinitsky e/ al.

1124

Here u is carrier velocity, TV is carrier energy relaxation time, and subscript e or h stands for electrons or holes. We now use u = pE which holds in the pre-saturation region, to get the ratio of overshoot lengths of electrons and holes; P _ h p’-e

Ph7E.h pL7E.e

The electron energy relaxation time is about 0.3 ps, as verified by Monte-Carlo simulations and experiments[ 111, [ 121. Hole energy relaxation parameters are not available, but we can estimate the ratio of the two energy relaxation times by comparing low-temperature electron and hole velocity measurements. We measure about 10% of hole velocity overshoot at 0.16 pm; that corresponds to the behavior of electrons in roughly 0.37 pm channel length devices at low temperature, as can be seen from Fig. 8. So, the ratio of the two overshoot lengths is approximated as 0.16/0.37 z 0.43. Since the ratio of electron and hole mobilities is roughly 2.5, we get that; z Assume

x (0.43) . (2.5) 2 I, at low temperature. that the ratio of energy relaxation

(11)

times is

6x1$ s P 5x106 I s

4X106 P v 3x106 s 3 2x106 4

1x106 0

0

2x104 4xld bd

6x104 fixlO4

lxld

Or/cm)

also 1 at 300 K. Since 10% overshoot was observed for 0.23 pm NMOS test structures at 300 K (see Fig. 7), we expect the same amount of overshoot for l/2.5 of that channel length, i.e. 0.09 pm PMOS test structures. As discussed in[l I], the maximum amount of velocity overshoot can be obtained if cold carriers are injected into a high-field region right at the source of the device. This makes our test structure ideal for velocity overshoot observation. In practical MOSFETs. the high-field peak occurs closer to the drain, towards which carriers are already somewhat heated up. That is why in practical MOSFETs velocity overshoot is suppressed, when compared with our test structure. Thus, a roughly 10% increase in current at room temperature should be observed for thin gate oxide P-MOSFETs with channel lengths even shorter than 0.09 pm. The opportunity of using velocity overshoot to enhance current drive in Si p-channel MOS devices is limited. 5. CONCLUSION

In this article velocity overshoot was investigated using special thick gate oxide transistors with nominally uniform electric field. The devices are ideal for simulator calibration. Calibration was performed successfully using TMA simulator MEDIC1 ver. 2.0. An analytical model describing overshoot in uniform field was presented. The model agrees closely with experimental measurements. Although electron velocity overshoot can be seen clearly, hole velocity overshoot could not be observed at room temperature down to LeR= 0.16 pm, and could be barely seen at 77 K. A simple qualitative estimate of the amount of hole velocity overshoot yielded to be about 2.5 times smaller than electron. We estimate that hole energy relaxation time is about the same as electron energy relaxation time ( _ 0.32 ps), and that comparable velocity overshoot effect will be found in P-MOSFET with 2.5 times shorter length than in N-MOSFET.

Ackno~ledgemenrs-This

work was supported by a grant from Intel Corp., Joint Services Electronics Program, under grant number -F49620-94-C-0038, and Air Force Office of Scientific Research. Air Force Material Command, USAF. under grant number F49620-94-l-0464. We acknowledge TMA Inc. for providing a device simulator MEDICI. Device fabrication was done in the UCBerkeley Microlab. Useful discussions with Dr Henry Claw from Intel and Mr Paul Scrobohaci from TMA Corp. are greatly appreciated. REFERENCES

(b) Fig. IO. (a) Hole c-E characteristics extracted from device measurements at low temperature. (b) Electron v-E characteristics extracted from device measurements at low temperature.

I. Ruth, J. G. IEEE Trans. Electron. Deu., 1972. ED-19, 652. 2. Chou, S. Y., Antoniadis, D. A. and Smith, H. I. IEEE Electron Device Left., 1985, EDL-6( 12). 665667. 3. (a) Shahidi, G. G., Antoniadis, D. A. and Smith, H. I. IEEE Electron Device Lett., 1988, 9(2), 94-96.

Velocity overshoot of electrons and holes in Si inversion layers 3. (b) Sai-Halasz, G. A., Wordeman, M. R., Kern, D. P., Rishton, S. and Ganin, E. IEEE Electron Device Lett., 1988, 9(9), 464466. 4. Assaderaehi. F.. Sinitskv. D.. Gaw. H.. Bokor. J.. Ko, P. k. ‘and Hu, 6.’ IEbM Tech. ‘Dig., 1994; 419482. 5. Teitel, S. L. and Wilkins, J. W. IEEE Trans. Electron Deu., 1983, ED-30 (2).

6. MEDIC1 version 2.0, TMA Inc., 1994. 7. Choi, W. S., Assaderaghi, F., Park, Y. J., Min, H. S., Hu, C. and Dutton, R. W. IEEE Electron Device Left.. 1995, 16, 333-335. 8. Bagwell, P. F., Antoniadis, D. A. and Orlando, T. P. in Advanced MOS Device Physics, ed. N. G.

9. IO. Il. 12. 13.

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Einspruch and G. Gildenblat, Vol. 18, Academic Press, 1989, Ch. 8. Huang, R. S. and Ladbrooke, P. H. J. Appl. Phys., 48,(l I) 1977. Lundstrom, M. Fundamentals of Carrier Transport, X Modular Series on Solid Stare Devices, ed. G. W. Neudeck and R. F. Pierret, Addison Wesley Co., 1990. Baccarani, G. and Wordeman, M. R. Solid-St. Electron., 1985, 28,(4), 407416. Assaderaghi, F. PHD Thesis, University of California, Berkeley, 1993. Chen, D., Kan, E., Ravaioli, U., Shu, C.-W. and Dutton, R. W. IEEE Electron Device Left., 1992. 13,(l), 2628.