VLSI DESIGN 2001, Vol. 13, Nos. 1-4, pp. 329-334 Reprints available directly from the publisher Photocopying permitted by license only (C) 2001 OPA (Overseas Publishers Association) N.V. Published by license under the Gordon and Breach Science Publishers imprint, member of the Taylor & Francis Group. An Upstream Flux Splitting Method for Hydrodynamic Modeling of Deep Submicron Devices MIN SHENa, WAI-KAY YIPb, MING-C. CHENG c’* and J. J. LIOU a aAdvanced Materials Research Institute; bDepartment of Electrical Engineering, University of New Orleans, New Orleans, LA 70148, USA; CDept. of Electrical and Computer Engineering, Clarkson University, Box 5720, Potsdam, New York 13699-5720; aDepartment of Electrical and Computer Engineering, University of Central Florida, Orlando, FL 32816 The advective upstream splitting method (AUSM) developed for fluid dynamics problems has been applied to solving hydrodynamic semiconductor equations coupled with the Poisson’s equation. In the AUSM, the flux vectors of a fluid system are split into a convective component and a diffusive pressure component. Discretization of these two physically distinct fluxes is thus performed separately in AUSM. Application of the developed hydrodynamic AUSM to a GaAs MESFET with a gate length of 0.1 gm has demonstrated its simplicity, efficiency and effectiveness in dealing with the highly nonlinear hydrodynamic device system. Keywords: Device simulation; Hydrodynamic model; Upwind difference; Convective flux 1. INTRODUCTION Many upwind flux splitting schemes have been developed to solve Euler and Navier-Stokes eq- uations [1-4]. These approaches are usually classi- fied as flux vector splitting (FVS) or flux difference splitting (FDS). Among these schemes, the FVS approaches, such as Steger-Warming splitting (SWS) [1] and Van Leer splitting (VLS) [2], appear to be simpler and more efficient but the accuracy is limited due to numerical diffusion. On the other hand, the FDS approaches, such as Roe splitting (RS) [3] and Osher splitting (OS) [4] are more accurate. Formulation using FDS is however more complicated, and the computational cost for FDS is considerably higher. Recently, the advective up- stream splitting method (AUSM) [5], which com- bines the features of FYS and FDS approaches, has been developed to achieve the efficiency and simplicity of FVS and the accuracy of FDS. The AUSM has been demonstrated to be robust and efficient for a wide variety of flow problems [5-7]. In this study, we apply the AUSM to the hydro- dynamic semiconductor equations coupled with the Poisson’s equation to examine its capability for simulation of deep submicron devices. The * Corresponding author: Tel.: 1-315-268-7735, Fax: 315-268-7600, e-mail: [email protected]329
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VLSI DESIGN2001, Vol. 13, Nos. 1-4, pp. 329-334Reprints available directly from the publisherPhotocopying permitted by license only
(C) 2001 OPA (Overseas Publishers Association) N.V.Published by license under
the Gordon and Breach Science Publishers imprint,member of the Taylor & Francis Group.
An Upstream Flux Splitting Methodfor Hydrodynamic Modelingof Deep Submicron Devices
MIN SHENa, WAI-KAY YIPb, MING-C. CHENGc’* and J. J. LIOUa
aAdvanced Materials Research Institute; bDepartment of Electrical Engineering,University ofNew Orleans, New Orleans, LA 70148, USA; CDept. of Electrical
and Computer Engineering, Clarkson University, Box 5720, Potsdam, New York 13699-5720;aDepartment of Electrical and Computer Engineering, University of Central Florida,
Orlando, FL 32816
The advective upstream splitting method (AUSM) developed for fluid dynamicsproblems has been applied to solving hydrodynamic semiconductor equations coupledwith the Poisson’s equation. In the AUSM, the flux vectors of a fluid system are splitinto a convective component and a diffusive pressure component. Discretization of thesetwo physically distinct fluxes is thus performed separately in AUSM. Application of thedeveloped hydrodynamic AUSM to a GaAs MESFET with a gate length of 0.1 gm hasdemonstrated its simplicity, efficiency and effectiveness in dealing with the highlynonlinear hydrodynamic device system.
Many upwind flux splitting schemes have beendeveloped to solve Euler and Navier-Stokes eq-uations [1-4]. These approaches are usually classi-fied as flux vector splitting (FVS) or flux differencesplitting (FDS). Among these schemes, the FVSapproaches, such as Steger-Warming splitting(SWS) [1] and Van Leer splitting (VLS) [2], appearto be simpler and more efficient but the accuracyis limited due to numerical diffusion. On the otherhand, the FDS approaches, such as Roe splitting(RS) [3] and Osher splitting (OS) [4] are more
accurate. Formulation using FDS is however morecomplicated, and the computational cost for FDSis considerably higher. Recently, the advective up-stream splitting method (AUSM) [5], which com-
bines the features of FYS and FDS approaches,has been developed to achieve the efficiency andsimplicity of FVS and the accuracy of FDS. TheAUSM has been demonstrated to be robust andefficient for a wide variety of flow problems [5-7].
In this study, we apply the AUSM to the hydro-dynamic semiconductor equations coupled withthe Poisson’s equation to examine its capabilityfor simulation of deep submicron devices. The
numerical flux in the AUSM is separated into theconvective and pressure-like contributions at thecell interface. The convective flux is carried bythe carrier average velocity but the pressure-likeflux is governed by the carrier random velocity.Discretization of these two physically distinctfluxes is thus performed separately in AUSM.
The hydrodynamic equations [8-12] for chargecarriers in semiconductor devices include conser-vation equations for carrier density n, momentump and energy w, which are
OnOt--+ V. (nv) 0, (la)
Onp np+ V- npv + VnkBTe -enE , (1b)Ot -p
Ot
(w-wo)-enE. v (lc)7-w
where p m’v, v is the average velocity of carriers,m* effective mass, kB the Boltzmann constant, qthe elementary charge, E the applied electric field,-p, the momentum relaxation time, -w the energyrelaxation time, and w0 the thermal energy atlattice temperature. The coefficients -m, -m, and m*are assumed energy dependent and extracted fromMonte Carlo simulation. Q is the heat flux anddescribed by the gradient of electron temperature
Q--VTe and -2 m*(2)
where is the thermal conductivity. The meanenergy w is related to Te and v by
3 re,v2w - kBTe -+- - (3)
The electric field E is obtained from the potentialb which is solved from Poisson’s equation"
E--Vb, and vzb--q(Na-n), (4)gs
where es is the dielectric constant of the semi-conductor and Nd is the n-type impurity concen-tration. In this study, only the n-type impurity.The hydrodynamic model described by Eqs.
(1)- (3) and the Poisson’s equation in Eq. (4) aresolved self-consistently. The AUSM is used toobtain the solution of the hydrodynamic equa-tions, and central difference is employed to deter-mine the potential from Poisson’s equation. Itshould be noted that the hydrodynamic modelused to demonstrate the capability of the upstreamflux splitting method is a semiclassic approach.For ultra- small devices where quantum phenom-ena have strong influences on device character-istics, quantum corrections are required.
3. FORMULATIONFOR THE ADVECTIVE UPSTREAMSPLITTING METHOD
The hydrodynamic equations given in Eqs. (la)-(1 c) can be expressed by a single equation in vectorform. In a 2D device, this vector equation iswritten as
0U OF 0G0---- + xx +-y H, (5)
where U is the vector of independent variables andchosen to be
U- [n npx t’lpy nw] r, (6a)
the flux vector in the x direction is given as
F [nVx n(pxVx + ksTe) npylx l"llx(W + kBZe)] T,(6b)
DEEP SUBMICRON MODELING 331
the flux vector in the y direction is expressed as
c. ,,px y , (py y + k re) +(6c)
and the source vector becomes
0-nqEx npx
.rp
-nqEy P’-rp
O(nQx) O(nQy) n(w-Wo)-qnExVx qEyly Ox Oy Tw
(6d)
The fluxes given in Eqs. (6b) and (6c) can be de-composed into convective and pressure-like fluxes:
n
F F +Fp npxVx +npy
n(w-+ kBTe)
0
(7a)
n 0
G_Gc+Gp_ npxVy + k’Senpy
n(W + kBTe) 0
(7b)
The convective fluxes, F and Gc, are carried bythe carrier average velocities, Vx and Vy, respec-tively. The pressure-like fluxes, Fp and Gp, are
governed by the carrier random velocity. Therandom velocity can be show to be c=
v/5k,Te/3m (analogy to the sound velocity influid dynamics). As proposed in [5], these twofluxes are discretized differently and weighted bythe Mach numbers. Evaluation of the fluxes at theinterface of the adjacent cells is described below.
3.1. Convective Fluxes
The modified Mach numbers, Mxl/2 and myl/2, atthe interface are given as
M/e M]+ / M-s and
My 2 M?D -+- M;u(8)
where subscripts L and R denote the parametervalues taken at the grid point to the left and to theright of the interface, respectively. The split Machnumbers, Mx+ and M-, are defined as
where Mach numbers are Mx Vx/C and My Vy/C.Using the modified Mach numbers, the convectiveflux F at the interface of two adjacent grid pointsin the x direction is given as
F/2 Vxl/2
Mxl/2
n
n(w + k,Te) L/R
cn
cnpx
cnpy
cn(w + k,Te) L/R
(lOa)
where the flux at the grid point left to the interfaceis taken for Mxl/2 >_ 0 and right to the interface for
Mx/2 <0. Similarly G at the interface of twoadjacent grid points in they direction is given as
G/2 Yyl/2
Mxl/2
n
n(w + k,Te) D/U
cn
cnpx
cnpy
cn(w + k,Te) D/U
(10b)
where the flux at the grid point below the interfaceis taken for Myl/2 >_ 0 and above the interface for
myl/2 % O.
332 M. SHEN et al.
3.2. Pressure-like Fluxes
The pressure-like fluxes are first split as follows:
FpT(1 :k_Mx) if Mx <_ 1,FP
FP (Mx 4--IMxl)/Mx if Mx >-2-
Gp
Gp +/- T(1 -+-My) if My <_ 1,
T (My +/- IMyI)/My if My >
and
(11)
The fluxes at the interface of the adjacent cells aretaken as
F/2-F++Fp and G/2:Gp++Gpe-.(12)
Using Eqs. (10a) and (10b), upwind difference isweighted by the modified Mach numbers, Mxl/2and Myl/2, for the convective fluxes at the interfaceof adjacent cells in the discretized central-differ-ence equation for Eq. (5). On the other hand,central difference weighted by Mx and My appearsin the pressure-like fluxes.
4. NUMERICAL SIMULATIONAND DISCUSSION
Simulations using the AUSM have been per-formed on an n-channel GaAs MESFET with agate length of 0.1 tm at room temperature. Thedetailed device structure is given in Figure l(a).The current density vectors in this 2D structureat Vgs:-0.8V and Vds= 1.5V are also shownin Figure l(b) where the current flow and thedepletion region in the device can be clearlyobserved.The electron concentration and temperature are
illustrated in Figures 2(a) and 2(b). Due to thereverse-biased Schottky barrier under the gate, thechannel is form on the bottom of the device asshown in Figure l(b) and Figure 2(a). The
Source Gate Drain
(a)
Source Gate Drain
0.15 ti ;1
...x..,.... ,
O0 0.1 0.2 0.3 0.4 0.5
x-Position rn]
FIGURE (a) The 2D depletion-model GaAs n-channelMESFET structure used in the hydrodynamic simulation.Source. Gate, and drain metal contacts are shown on the topof the device. Nd--1017cm -3 in the n region, and Nd--3 1017 cm -3 in the n + source/drain. (b) The current densityvectors at Vds 1.5 V and Vgs 0.8 V, where the vector lengthis proportional to the magnitude of the current density vector.
reverse-biased Schottky barrier and applied drain-source voltage are also responsible for the ex-tremely large density gradient near the drain metaledge at (x, y) (0.45 tm, 0.15 tm). Similar discon-tinuity for electron temperature at some metaledges is also observed in Figure 2(b), particularlynear the gate metal edges. This is because thelattice temperature is taken on the metal boundaryand zero Ex is set on the metal boundary whichcauses the discontinuity of the tangential compo-nent of the electric field (Ex) near the gate metaledges. As shown in Figure 3(a), Ex 0 along themetal boundary, and extremely strong electricfields (as high as 8 107 V/m) are found at metaledges near x 0.2 and 0.3 tm. Strong field normalto the gate (Ey) under the gate is also shown due tothe reverse-biased Schottky barrier. Because of thezero normal component of electric field on thenon-metal boundary, discontinuity for 14 is alsoobserved at the metal near x 0.2 and 0.3 tm.The AUSM appears to be very efficient for
hydrodynamic simulation of the GaAs MESFET.
DEEP SUBMICRON MODELING 333
3E+23
2E+23
1E+23
0.250.125’POsition
" 3000Q.
2000
01000
0.375;,,, 0.25
FIGURE 2 (a) Electron density and (b) electron temperature in the MESFET at Vas 1.5 V and Vg -0.8.
8E+07
4E+07
-4E+07
-8E+07
0.250.125
FIGURE 3 (a) x-component and (b) y-component electric fields in the MESFET at Vd 1.5 V and Vgs= -0.8.
It requires approximately 35 minutes on a DigitalAlpha 533 MHz workstation. This study showsthat regardless of high gradients ofmany variables,stable and smooth numerical results can be reach-ed. It demonstrates the effectiveness and efficiencyof the AUSM in handling the convective flux andthe extremely high-gradient situations.
5. CONCLUSION
A hydrodynamic AUSM has been developed andapplied to obtain self-consistent solution from
hydrodynamic equations and Poisson’s equationfor electrons in a 2D GaAs MESFET with a gatelength of 0.1 gm. The flux splitting schemesproposed in the AUSM [5] are easy to formulate,and the application to the hydrodynamic devicesimulation appears to be very efficient. Regardlessof high gradients of the variables, the AUSM isable to provide stable and smooth numericalsolution. The method has been shown to be veryefficient and effective in handling convective andpressure-like fluxes in the highly nonlinear hydro-dynamic semiconductor equations in a very smallGaAs device.
334 M. SHEN et al.
Acknowledgement
The authors Cheng and Liou are partiallysupported by SRC under Grant Number 2000-RJ-873G.
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[3] Roe, P. L. (1981). J. Comput. Phys., 43, 357.
[4] Osjer, S. (1981). Math. Studies, 47, North-Holland,Amsterdam, 179.
[5] Meng-Sing Liou and Christopher J. Steffen, Jr. (1993).J. Comput. Phys., 107, 23.
[6] Meng-Sing Liou (1996). J. Comput. Phys., 129, 364.[7] Richard W. Smith (1999). J. Comput. Phys., 150, 268.[8] Ming-C. Cheng Liangying Guo, Robert Fithen and
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[10] Benvenuti, A., Coughran, W. M. Jr. and Pinto, M. R.(1997). IEEE Trans. Elect. Dev., 44, 1349.
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