statistical terminal current monte carlo device...

VLSIDESIGN1998, Vol. 6, Nos. (1-4), pp. 303-306Reprints available directly from the publisherPhotocopying permitted by license only

(C) 1998 OPA (Overseas Publishers Association) N.V.Published by license under

the Gordon and Breach Science

Publishers imprint.Printed in India.

Statistical Enhancement of Terminal Current EstimationMonte Carlo Device Simulation

P.D. YODERa, U. KRUMBEINa, K. G,RTNERa, N. SASAKIb and W. FICHTNERa

aSwiss Federal Institute ofTechnology, Integrated Systems Laboratory, Gloriastrasse 35, CH-8092 Ziirich, Switzerland,bFujitsu Laboratories, Ltd., ULSI Research Division Atsugi 243-01, Japan

We present a new generalized Ramo-Shockley theorem (GRST) to evaluate contact currents,applicable to classical moment-based simulation techniques, as well as semiclassical MonteCarlo and quantum mechanical transport simulation, which remains valid for inhomogeneousmedia, explicitly accounts for generation/recombination processes, and clearly distinguishesbetween electron, hole, and displacement current contributions to contact current. We thenshow how this formalism may be applied to Monte Carlo simulation to obtain equations forminimum-variance estimators of steady-state contact current, making use of informationgathered from all particles within the device. Finally, by means of an example, we demon-strate this technique’s performance in acceleration of convergence time.

Keywords: Monte Carlo simulation, Terminal current calculation, Generalized Ramo-Shockley theorem

1 INTRODUCTION: THE RAMO-SHOCKLEYTHEOREM

such that all contacts are grounded with the exceptionof contact k, which is set to Volt.

Through application of the method of Green functionsin the quasi-electrostatic approximation, Shockleyand Ramo [2] introduced the original domain integra-tion formula relating the currents induced on an arbi-trary number of contacts to the motion of charges inmultiple dimensions

I(k) -EqjE)’vj, (1)J

where the index k indicates the contact at which thecurrent is to be evaluated, qj and vj represent particlecharge and velocity, respectively, and the index j runsover all particles within the volume. The symbol E)k)

denotes the electric field at the position of particle jwhich would result if all charges were removed fromthe volume, and boundary conditions were imposed

303

2 A GENERALIZED RAMO-SHOCKLEYTHEOREM

One of the most fundamental properties of all types oftransport are the continuity equations, which may bederived by taking moments of the appropriate trans-

port equations. For classical and semi-classical sys-tems, this is most often the Boltzmann Transportequation [3], and either the Wigner-Boltzmann equa-tion [4] or the quantum Liouville equation [5] forquantum systems. The 0th moments take on the fol-lowing general form:

-V.jn eGn(r) eh, (2)

--V’jp eGp(r) + ep, (3)

304 ED. YODER et al.

Furthermore, Maxwell’s equations may be used to

derive equally familiar identities for the displacementcurrent Jd and total current Jr"

-V. Jd V. (in + jp) -e(p + h) (4)

V. Jt 0, (5)

where the variables n and p refer to the electron andhole free carrier densities, respectively. Let hkl be a

suitable set of test functions satisfying

hkl IFl 1, hkl IFj-- 0, j 1, k n,p,d,t, (6)

where F denotes the boundary, consisting of bothNeumann (FN) and Dirichlet (FD) parts, andFD L)iFi. Upon multiplication of an equation of theform-V. Jk Rk with a test function hkl, integrationby parts yields [6,7]

(-V.j,h) (R,h:t), (7)

fahklV "jkdV frh,ljkdS + faVhkl "jkdV.

The definition of hkl requires that the surface integralis equivalent to the contact current, because

0. Equations (2), (3),hkllr 51m and (Jk h)[rN(4), (5), and (7), lead to the following formulas for theterminal currents at an arbitrary contact

Inl f vh.( .a.ev + efahnt(r)dV

e ]Gn(r)hm(r)dV, (8)

lpl f Vhpl(r).jpdV ef 1)hpl(r)dV

eJ Gp(r)hpl(r)dV,+ (9)

Idl [ hdl(r) .jddV / hdl(r) (in +jp)dV

fVhdl(r).jddV +ef(p-i,)hdl(r)dV, (10)

Itl- f Vh,l(r).(j,+jp+jd)dV. (11)

This method is also applicable to terminal currents ofquantities derived from higher-order moments of thecarrier distribution functions, inhomogeneous mediaand high frequencies [8]. The steady-state form ofthese equations is obtained by setting all time deriva-tives equal to zero.

3 OPTIMIZED STEADY-STATE FORMOF THE GRST

Convergence time for steady-state particle-based sim-ulation is limited by estimator variance. Minimizingthe functional form of the steady-state terminal cur-

2rent estimator variances ((It-Ikl) with respect to

the test functions hkl, under the assumption that fluc-tuations in particle trajectories about the spatially-dependent mean trajectory have only weak interparti-cle correlation, results in the following set of Euler-Lagrange equations

--’kt (r) 0 (12)

Opt Irm- 5lm, h E H k- n,p t. (13)kl

The quantity 5 (R) 5) is the position-dependent cur-rent density variance tensor. In steady-state simula-tion, it has the following numerically convenienttensor and scalar approximations

(Jk @ jk)ii (13k)2 2(’OT,k, )2k,i)T,c,k/(Nkt), (14)

where the symbols Pk, agT,k and )k are the charge den-sity, thermal velocity and mean velocity, respectively,for particles of type k. Nk and Xc,k are the grid-depend-ent mean type-k particle number and momentumrelaxation time, and is the simulation time. The cor-responding terminal current estimators then become:

opt optnl- f Vhnl (r).jndV-ef Gn(r)h,1 (r)dV, (16)

opt /Vhpl (r).jpdV + e ap(r)h;f (r)dV, (17)

tl f Vhlpt (r). (in + jp)dV. (8)

4 APPLICATION TO MOSFET SIMULATION

To extract current-voltage characteristics of an 0.5 gmn-Mosfet device, several simulations were performedwith the Monte Carlo simulator Degas. Test functionsfor the terminal current estimators were computed

STATISTICAL ENHANCEMENT OF TERMINAL CURRENT ESTIMATION 305

FIGURE Simulated electron density and velocity profiles for a 0.5 micron MOSFET. Vgs 2 V, Vds 4.875 V

12

10

8

1

:.,_, e,. - 1.51.0 2.0 3.0 4.0 5.0

Vd IV]

H ExllentDrifl-DillusionHydrodamicMonte Carlo

0.0 1.0 2.0 3.0 4.0 5.0Time [PSI

sourcedrain

e---esubstrate 25

FIGURE 2 Simulated and experimental Id Vcl characteristics, along with convergence times for source, drain and substrate currents at

Vel 4.875 V and Vg 2 V

from equations (12), (15) and (16), and used alongwith simulated carrier density and velocity profiles,such as shown in Fig. 1, to obtain the cumulativelytime-averaged steady-state drain currents shown inFig. 2. All simulations were run with 20000 electronsand 10000 holes. In comparison to the particle count-

ing technique, convergence is reached with the opti-mzed GRST method between 10 and 40 times fasterfor these examples.

AcknowledgementsThe authors are grateful to H. Gotoh, Fujitsu Limited,Kawasaki, for providing experimental data.

References[1] W. Shockley, Journal ofApplied Physics, 9, 635 (1938).[2] S. Ramo, Proceedings ofthe IRE, 27, 584 (1939)[3] K. Hess, Advanced Theory of Semiconductor Devices, Pren-

tice Hall, Englewood Cliffs, New Jersey (1988).

[4] C. L. Gardner, SIAM Journal on Applied Mathematics 54,409 (1994).

[5] H. L. Grubin, T. R. Govindan, J. P. Kreskovsky, and M. A.Stroscio, Solid-State Electronics, 36, 1697 (1993).

[6] M. S. Mock, Analysis of Mathematical Models of Semicon-ductor Devices, Boole Press Limited, Dublin, Ireland 1983.

[7] A. Gajewski, private communications, ca. 1985.[8] P. D. Yoder, K. Girtner and W. Fichtner, Journal ofApplied

Physics, 79, 1951 (1996).

Biographies

P. D. Yoder was born in Washington, D.C., on Febru-ary 15, 1968. He received the B.S.E.E. degree fromCornell University, Ithaca, NY, in 1989, and the M.S.and Ph.D. degrees in electrical engineering from the

University of Illinois, Urbana, IL, in 1991 and 1993,respectively. Since graduation he has been employedat the Integrated Systems Laboratory of the SwissFederal Institute of Technology (ETH) in Zurich,Switzerland, working in the area of silicon devices.

306 ED. YODER et al.

His professional interests include semiconductordevice physics, applied mathematics, and MonteCarlo simulation.

Ulrich Krumbein was born in Braunschweig, Ger-many in 1964. In 1990 he recieved the Diploma inphysics from the University of Kaiserslautern, Ger-many. In 1991 he joined the Integrated Systems Labo-ratory at the ETH in Zurich, Switzerland, where he is

currcntly working towards his Ph.D. in semiconduc-tor device simulation. His research interests are inphysical models for silicon device simulation, with

special emphasis on EEPROM devices.K. Girtner was born in Zittau, Germany, on

March 4, 1950. He studied theoretical physics inDresden (Germany), and received a PhD in nuclearreactor physics. After working ten years in the field ofneutron transport, he joined the Karl-WeierstrafS-Insti-tute for Mathematics in Berlin in 1982. Since thistime, his main interest has been in numerical prob-lems connected to PDE’s and semiconductor devicemodels. In 1992, he joined the InterdisciplinaryCenter for Supercomputering at the ETH, and since1994 has been working with Prof. Fichtner at the ETHIntegrated Systems Lab.Nobuo Sasaki was born in Matsuyama, Japan, in

1949. He received the B.S. degree in Physics andPh.D. degree in Electrical Engineering in 1971 and

1984, respectively, both from the University of

Tokyo, Japan. He joined Fujitsu Limited, Kawasaki,in 1971. He moved to Fujitsu Laboratories, Atsugi, in1993, where he is currently a director of ULSI Tech-nology Laboratory. He has studied device physics ofMOSFET’s, 3-dimensional integrations, and electri-

cally conducting polymers. His main interest is nowin Gbit-DRAM’s. In 1985, he received the award ofMinister of Science and Technology Agency for his

contributions to the SOI technology and 3-dimen-

sional integrations.Woifgang Fichtner received the Dipl. Ing. degree

in physics and the Ph.D. degree in electrical engineer-ing from the Technical University of Vienna, Austria,in 1974 and 1978, respectively. From 1975 to 1978,he was an Assistant Professor in the Department ofElectrical Engineering, Technical University ofVienna. From 1979 through 1985, he worked at

AT&T Bell Laboratories, Murray Hill, NJ. Since 1985he is Professor and Head of the Integrated SystemsLaboratory at the Swiss Federal Institute of Technol-

ogy (ETH). In 1993, he founded ISE Integrated Sys-tems Engineering AG, a company in the field oftechnology CAD. Wolfgang Fichtner is also a fellowof the IEEE and a member of the Swiss National

Academy of Engineering.

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