TIC AM REPORT 97-07April 1997

Simulation of Hydrodynamic SemiconductorDevice Models Using a Modified

Scharfetter-Gummel Strategy

Anand L. Pardhanani and Graham F. Carey

Simulation of Hydrodynamic Semiconductor Device ModelsUsing A Modified Scharfetter-Gummel Strategy

Anand L. Pardhanani & Graham F. Carey

Abstract

An efficient numerical solution scheme based on a new generalized finite differencediscretization and iterative strategies is developed for submicron semiconductor devices.As a representative model we consider a non-parabolic hydrodynamic system. The dis-cretization is formulated in a mapped reference domain, and incorporates a transformedScharfetter-Gummel treatment for the current density and energy flux. This permitsthe use of graded, nonuniform curvilinear grids in the physical domain of interest, whichhas advantages when gridding irregular domain shapes or solution profiles. The solutionof the discrete system is carried out in a fully-coupled, implicit form, and non-symmetricgradient-type iterative strategies are investigated. Numerical results demonstrating theperformance and reliability of the scheme are presented for ID and 2D test problems.

1 Introd uction

A key requirement in the analysis and design of submicron semiconductor devices in two orthree dimensions is the need to model drift-dominated transport effects reliably, accuratelyand efficiently. The traditional drift-diffusion equations have been very effective for largerdevices, but more advanced carrier transport models such as augmented drift-diffusion, en-ergy balance and variants of the hydrodynamic model [2, 8, 18] have been recently developedfor submicron devices. These models are more complex, and both gridding and solutionconvergence are more sensitive problems. Efficient numerical simulation of these modelsis quite challenging, since the presence of steep gradients requires graded meshes, and thestrong nonlinear coupling frequently leads to numerical oscillations, inaccuracy, instabilityand poor computational efficiency. These difficulties become more pronounced as devicesizes shrink into the deep submicron regime.

Our primary objective in this work is to investigate new numerical methodology basedon iterative and multigrid strategies on general structured grids in two-dimensions. Thisimplies the numerical methods must be capable of handling nonuniform and curvilinearmeshes, which impose the least restrictions on structured mesh design. In the present workwe introduce a mapping strategy and generalize the Scharfetter-Gummel approach withinthis framework [13]. The ideas may also be directly extended to 3D and are generallyapplicable to a broad spectrum of advanced transport models. For clarity of exposition wefocus on a representative non-parabolic variant of the hydrodynamic model which has beenshown to predict velocity and energy effects more accurately in submicron devices [3, 20].Our emphasis in the present work is on developing accurate, efficient and robust methods,and demonstrating their potential using relatively simple MOSFET-type model problems.

1

This enables us to explore the numerical formulations and solution behavior more readily,and to examine several key issues including gridding, scaling of the variables, discretization,and solution strategies.

We consider finite difference schemes for structured (but strongly nonuniform) grids, anduse a mapping strategy to facilitate discretization. Fully-coupled, time-marching solutionschemes are used in conjunction with iterative strategies to enhance stability and efficiency.Our use of the mapping approach for discretization is motivated by the need for curvilineargrids to accommodate a wider range of domain shapes and solution profiles in the finitedifference formulation. The map introduces a mathematical transformation to a referencedomain, which permits the use of more general grids in the original domain. We develop amodified Scharfetter-Gummel approach that uses the mapping strategy for discretizing thecurrent density and energy flux in the non-parabolic hydrodynamic model. We remark thatthis approach can also be readily applied to the drift-diffusion analog of the hydrodynamicmodel (with constant energy) which we, in fact, routinely do to compute good startingiterates for the hydrodynamic model. The resulting sparse discrete systems are solvedusing the conjugate gradient squared (CGS) algorithm.

The rest of this paper is organized as follows: Section 2 introduces the hydrodynamicmodel, Section 3 outlines our numerical formulation and discretization method, Section 4discusses the time-marching and iterative solution approach, Section 5 presents numericalresults, and Section 6 offers some concluding remarks.

2 Hydrodynamic Model

In this work we use the hydrodynamic models developed at the Microelectronics ResearchCenter at the University of Texas [1, 3]. Our emphasis is on the non-parabolic model, whichis considered more physically appropriate since it is based on a more realistic energy bandstructure. For single-carrier devices it is mathematically described by the following set ofcoupled equations

\J2¢

8n+V.(nv)8t

8(A(w)nv) + ~V(B(w)nw) - ~nV</J8t 3m* m

8(nw) + V. (Dnwv + Q) - qnv . V</J8t

onv

Tp

n(w - wo)

(1)

where </Jis the electrostatic potential, n is the electron density, v is the drift velocity, and wis the total energy. The momentum and energy relaxation times, Tp and Tw, are empiricalfunctions, which are chosen from the work of Bordelon et al. [3] as

0.007 q 10-12Tp = X S

W

Tw = 0.46 X 10-12 s

2

(2)

The system is closed by assuming a Fourier type constitutive relation for the heat flux, Q,of the form

2 ('YR)Q = -- - nVw (3)3 I<B

The other quantities in equations (1) - (3) are defined as follows: ND(X, y) = givendoping density, A(w) = 1 + 2DQ'!£, B(w) = (1 + Q'!£)/(1 + 2Q'!£), q = electron charge =q q q

1.602x 10-19 C, m* = effective mass = 2.367x 10-31 kg, E = permittivity of silicon = 11.9 Eo,

EO = 8.854 X 10-12 C2/(joule m), I<B = Boltzmann constant = 1.381 X 10-23 joule/kelvin,wo = ~I<BTL joule, TL = lattice temperature in kelvin, and Q' = 0.5eV-1, D = 1.3,1 = 4.2 X 10-26 (watt m2)/kelvin, R = 0 to 0.5 are empirical constants.

Note that in (1) it is also assumed that the drift kinetic energy is negligible, and thesystem reduces to the corresponding parabolic hydrodynamic model if we set Q' = 0 andD = 3/2.

3 Numerical Formulation

The first step prior to discretization is to scale the hydrodynamic system. We use thenotation ts, Xs, <Ps, ns, vs, Ws and Qs to denote normalizing constants for time, length,electrostatic potential, electron density, drift velocity, energy and heat flux respectively.The equations can then be written in the following non-dimensional form

where

>..2\12 <p8nFt+C1 V·(nv)

8(A(w)nv) + M1 ~V(B(w)nw) - M2 nV<p8t 3

8(nw)---at + E1 V· (Dnvw) + E2 V· Q - E3 nv . V<p

Q

(n - ND)

onv

Tp

n(w - wo)

(4)

Note that the functions A(w) and B(w) are already non-dimensional by the way they aredefined.

Although the scaling constants can be chosen in a variety of ways, our experience in-dicates that their choice has an im pact on the convergence behavior and efficiency of thenumerical algorithm [14]. Selberherr [16] and Markowich [9] have discussed some standardmethods for choosing these constants, which we combine with numerical tests to determinetheir values in the present work. Table 1 summarizes typical values for the scaling con-stants. The main difference from the values given in [9] and [16] is in the choice of <Ps,

3

Samplevalue

Selectionmethod.. , .. rJ,III, 111~1,'IIl".1 , Vn,llJr.

Xs maximum domain width 1 micron

ns maximum doping density 1.0 X 1024 m-3

<Ps maximum applied voltage 5 volts

Ws q <Ps 8 X 10-19 joule

Vs (ws/m*)1/2 1.8 X 106 m/s

ts xs/vs 5.5 X 10-13 s

Qs (ns Ws xs)/ts 1.45 X 1012 watt/m3

[--Scalingconstant

Table 1: Method of choosing scaling constants for the hydrodynamic system.

and all the quantities that depend upon it. We found that choosing the maximum appliedvoltage instead of I<BTL/q gave better convergence behavior in our test cases. The generalguideline we followed for scaling was to ensure that each scaled variable was bounded aboveby unity. However, we believe the best choice for the scaling parameters remains an openquestion that merits further research. The scaling issue is also closely connected to griddingquestions, as the numerical impact of poor scaling appears similar to that of inadequategrid density or insufficient smoothness.

After scaling, a mathematical transformation of variables is introd uced from the phys-ical (x, y) domain to a reference (~, TJ) domain [4, 14]. This strategy yields a transformedhydrodynamic system in the mapped reference domain on which a rectangular grid can bedefined. This transformation relates a curvilinear, nonuniform grid in the physical domainto a corresponding orthogonal grid in the reference domain. The transformed PDE's arethen discretized in the reference domain using symmetric differencing in conjunction witha modified Scharfetter-Gummel type of scheme for the hydrodynamic model [12]. Similarschemes for finite volume discretization of the the current continuity equation in the stan-dard hydrodynamic model have been proposed by Fukuma and Uebbing [5], Rudan andOdeh [15] and others. We have generalized this approach to allow its use in conjunctionwith 2D finite difference mapping strategies. The mapping approach allows greater flexibil-ity in grid design when the domain shape is irregular or the solution contours are curvilinear.Although the mathematical formulation assumes the existence of a map, in practice it onlyappears in the form of transformation metric-coefficients that modify the original PDEs.These are discretized, along with the original PDEs, using finite difference approximationswith respect to a uniform, rectangular reference grid. Thus the map is never explicitlyconstructed.

The Scharfetter-Gummel formulation is derived from the steady-state momentum con-servation equations, which in the (~, TJ) domain may be written as

J =

4

(5)

(6)

where J = nv = current density (scaled), k1 and k2 are unit vectors in the ~- and TJ-direction respectively, and the other variables are defined previously. We make the usualassumptions about local behavior of the variables over a grid block: constant J and B(w),and linear <p and w. We also use the empirical relationship Tp ex: ~, as described in Section2. Equation (5) is projected along the respective ~- and TJ-directions, and each component isintegrated locally in the corresponding coordinate direction. The integration is performedalong the sides of the local grid block using the nodal values of n, wand <p for boundaryconditions. This yields the following expressions for the ~- and TJ-components of J

~ = -~M1CTB(wiV) [ni+1 {3(-X) _ ni {3(X)] (Wi+1 - Wi)IV~I 3 Wi+1 Wi In(wi+dwi)~~

~ = -~M1CTB(wav) [nj+1 {3(-Y) _ nj {3(Y)] (Wj+1 - Wj)IVTJI 3 J Wj+1 Wj In(wj+dwj)~TJ

where

[3M2 (<Pi+1- <Pi)]

X = 2 - "2 M1B(w'r) (Wi+1 _ Wi) In (Wi+dWi)

[3M2 (<Pj+1-<Pj)] ( /)

Y = 2 - "2 M1B(wr)(wj+1 _ Wj) In Wj+1 Wj

and {3(x) = x/(eX- 1) denotes the Bernoulli function. The other quantities introduced in

(6) are

W~vJ

d ffi· 0.007q 12scale coe Clent of Tp = -10- ,wsts

(~i+1 - ~i), ~TJ = (TJj+l - TJj),1

average energy along ~~ = -(Wi+1 + Wi),21

average energy along ~TJ = -(WJ+1 + Wj)2

(7)

In practice it is also necessary to consider the limiting case where W is locally constant(Wi = Wi+1 or Wj = Wj+d and the expressions in (6) break down. For this case it can beshown that equations (6) reduce to

J1 2 M1CTB(Wi)IV~I= -3 ~~ [ni+1{3(-XI) - ni{3(XI)]

h _ 2M1CTB(wj) [. f.l( Y) ·f.l(Y)]- -- n +11.1 - I - n I-' IIVTJI 3 ~TJ J J

with

5

A Scharfetter-Gummel type approach may also be applied for discretizing the trans-formed energy equation, for instance, by generalizing to the transformed case the schemesoutlined by Meinerzhagen and Engl [10] and Tang [17]. In this case the energy flux is inte-grated analytically over local grid blocks, and the resulting expression is used in discretizingthe energy equation. From equation (4), the energy flux in scaled form is

(8)

which can be expanded and written in the (~, TJ) domain as

8w 8wS = [E1DJ1w+ IV~IE2H1n 8~]k1 + [E1Dhw+ IVTJIE2H1n 8TJ]k2 (9)

As before, we assume locally constant J. In addition, S is assumed locally constant, andn is assumed to have an exponential behavior over a local grid block. Each component ofequation (9) is then integrated analytically in the corresponding coordinate direction alongthe local grid block, and the nodal values of ware used for boundary conditions. This yieldsthe following expressions for the ~- and TJ- components of the energy flux on the sides of thegrid block

where

and

Ddwi+1{3(-Xs) - wi{3(Xs)]D1j[WJ+1{3(-Ys) - Wj{3(Ys)] (10)

(11)

Note that J1 and h in the above expressions correspond to those given in equations (6) -(7). In the degenerate case corresponding to the identity map, the metric coefficients areconstant and we recover the standard Scharfetter-Gummel approximations.

Discretization of the current density and energy flux using equations (6) - (7) and (10)yields piecewise constant values for these quantities along the sides of each mesh elementin the reference domain. These values may then be used in a standard central differencingscheme to discretize the V . J and V . S terms in the carrier continuity and energy equa-tions. Central differencing is also used for discretizing the electrostatic potential equation.After completing the spatial discretization, we obtain a large, nonlinearly coupled systemof ordinary differential equations (ODE's) in time.

We remark that this Scharfetter-Gummel treatment can also be extended to certainother more complex forms of the momentum and energy relaxation times. In particular, ifthe relaxation times can be written as some power of the energy, integration factors can befound to derive expressions analogous to those in (6).

6

4 Solution Approach

The ODE system that arises from semi-discretizing the hydrodynamic PDE system isstrongly coupled and nonlinear, and we integrate it to steady-state in fully-coupled form us-ing implicit methods. This makes it necessary to solve large systems of algebraic equationsat each integration step, which can be computationally very expensive. The need for goodpreconditioning and iterative methods is crucial here, since they can significantly enhancethe efficiency of algebraic system solution.

We consider the backward Euler method and a second-order semi-implicit Runge-Kuttamethod for performing the numerical integration. For convenience, we write the ODEsystem for U = [<p, n, nv, nw]T as

dd~ = F(U)

Then, a backward Euler scheme applied to (12) yields

(12)

(13)

where I is the identity matrix, An = oOOn is the Jacobian matrix, and the subscripts denotetime steps. A two-stage semi-implicit Runge-Kutta method applied to (12) can be writtenin the form (see, for example, Lapidus and Seinfeld [7])

where

[F(U n) + ~t al A(U n) k1]

[F(U n + ~t b1 kd + ~t a2 A(U n + ~t C1k1) k2]

(14)

(15)

Here Q'1, Q'2, ai, a2, b1 and C1 are constants that depend upon the specific integrationscheme under consideration. For example, the second-order Rosenbrock version of the semi-implicit scheme used in the present work corresponds to the following choice of constants

.;2al = a2 = 1- 2'

b1.;2-1

(16)= ,2

C1 - 0, Q'1 = 0, Q'2 = 1

Note that both integration schemes require solution of linear algebraic systems duringeach step which, in the case of the hydrodynamic operator, are sparse, banded and nonsym-metric. The choice of iterative method depends upon the properties of the Jacobian matrixof the hydrodynamic system. Our experience indicates that the Jacobian matrix is verysensitive to the choice of grid and scaling. We have observed that minor perturbations inthe grid can affect the success or failure of a given iterative scheme. In the present work theJacobian systems are nonsymmetric, so we use the Conjugate Gradient Squared (CGS) andBi-Conjugate Gradient (BCG) methods to solve the algebraic systems [6, 19]. The solutionalgorithm is set up to maximize efficiency by first trying iterative methods for solving each

7

algebraic system. Convergence is monitored continuously, and if it becomes unsatisfactory,the algorithm automatically switches to a band solver. Numerical tests indicate that withproper choice of grid most of the algebraic systems can be solved with the iterative methods(see Section 5 for more details).

5 Results and Discussion

Numerical studies using our approach have been carried out on a variety of graded meshes inone- and two-dimensions. The results presented in this section are computed using the non-parabolic form of the hydrodynamic model. The systems are integrated to steady state,and their corresponding solution profiles displayed. In all cases the Scharfetter-Gummeltreatment is used for the current density and the energy flux. The drain voltage is ap-plied directly, without using an incremental continuation strategy. However, our numericalstudies indicate that the logarithmic expressions in the Scharfetter-Gummel form require agood starting iterate to ensure that their arguments remain positive. We provide this byfirst computing the solution to a constant-energy form of the hydrodynamic model, which isanalogous to a drift-diffusion model. This approach is quite efficient, as the constant-energysolution typically requires less than 10% of the total computational time. When plotting1- Y characteristics it is used only for computing the starting iterate for the first bias point,after which the subsequent points needed for the plot constitute a natural bias continuationstrategy. Our experience indicates that the constant-energy solution is a more efficient androbust starting iterate than bias continuation if the simulation is to be performed for asingle point alone.

A series of 1D tests have been performed using n+ - n - n+ diode structures withchannel-lengths ranging from 0.6µm down to O.08µm, and applied drain bias up to 5 volts.Figure 1 shows simulated results for the 0.08µm structure at 3 volts bias [12]. In thesecalculations we use the value R = 0.05 in equation (3). For this case the doping densitywas 5 X 1020 per cm3 in the n+ regions (0 ~ x ~ 0.025 and 0.125 ~ x ~ 0.15), and 2 X 1015

per cm3 in the n region (0.035 ~ x ~ 0.115), with a linear transition at the junctions.These computations were performed using a band solver, which is preferable in 1D sincethe matrices have small, non-sparse band-widths. In all our 1D test cases the simulation isefficient and robust, typically requiring less than one CPU minute on a Sun Sparc 1.

Two-dimensional studies have been carried out using MOSFET-type structures withgate-lengths from 0.5 - 0.6µm. In the first test case we use a simple doping profile withrectangular source and drain regions as shown in Figure 2. The grid used in this problemconsists of 61 X 41 nodes, graded strongly toward the junction regions in the device. Thenon-dimensional time step (with reference to the scaling in Table 1) in this simulationwas 0.01 units, and integration was carried out to steady-state, which was assumed to bereached when the absoiute maximum change in the computed variables between consecutivetime steps had dropped below 10-6 (non-dimensional) units. The total time required toreach steady-state depends on the initial conditions. The initial constant-energy solutionis typically integrated through about 0.5 units of time, while the steady-state solution forthe full hydrodynamic model generally requires 2 to 3 units of time per bias point. We useTL = 300I< and R = 0.05 in equation (3) for these calculations. The numerical algorithm isimplemented to optimize efficiency by using iterative methods for solving the linear systems

8

E630

25

200-Q)

~ 15~.~ 10ga;> 5

a

2.0

1.5

>1.0~>-e>~ 0.5w

0.0

-50.00 0.b3 0.b5 0.b8 0.10 0.12 0.15

X (microns)

-0.50.00 0.b3 0.b5 0.b8 0.10 0.12 0.15

X (microns)

Figure 1: Hydrodynamic simulation of carrier velocity and energy at 3 volts bias for 0.08µmn+ - n - n+ diode with 129 grid points; doping concentration varies 5 orders of magnitudeat junctions.

whenever possible, and automatically reverting to the band solver if divergence is detected.For the 2D test problem, the ratio of CPU time for band solution to iterative solution isabout 150 to 1. In practice we found that over 90 % of the linear systems were successfullysolved using the iterative method. However, we must emphasize that this is very muchdependent on the grid - algebraic systems arising out of smoother grids are much moreamenable to iterative solution. In fact, grid density and smoothness have a critical impacton the stability and efficiency of the numerical algorithms. Our experience underscores theimportance of maintaining smooth grid transitions from low- to high-density regions overthe domain.

Figure 3 shows steady-state surface plots for the carrier concentration, velocities andenergy with 3 volts applied at the drain. The solution detail can be seen more clearly in across-sectional view of these results near the MOSFET surface, which is shown in Figure 4for a section parallel to the oxide interface and about 40A below it. The actual simulationwas carried out over a range of drain voltages, and a plot of current versus voltage at thedrain is shown in Figure 5. The calculations were performed on a Cray Y-MP, and the CPUtime for computing a typical bias point on the ID - VD curve is about 2 minutes.

Another MOSFET structure we consider has the doping profile and geometry shown inFigure 6. Here the peak donor doping in the n-type source and drain regions is 1018 cm-3,

and the acceptor doping in the p region is 1016 cm -3, with abrupt transition (over one meshcell) along the junctions. The gate-length is 0.5µm. We use a numerical simulation strategysimilar to the one outlined previously, using a 49 X 25 grid. Figures 7 show steady-statesurface plots for electrostatic potential, carrier concentration and velocities for the constant-energy model, with applied bias VDS = 4 volts and Vas = 1 volt. We observe an overshoot inthe x-velocity profile near the drain junction, which persists even as the solution converges tosteady-state. We also observed that as the drain voltage is increased past about 3 volts, westart seeing a deterioration in computational efficiency, as the conditioning of the Jacobian

9

5c+174.5c+17

4e+173.5c+17

3c+172.5e+17

2e+171.5e+17

le+175e+16

o

o

Figure 2: Doping profile for 2D MOSFET test case (61 X 41 grid).

5c+17

4c+17

3c+17

2e+17

Ic+17

o

(a) Electron density (per cm3)

Figure 3: (a)-(e) Surface plots of steady-state results for 2D MOSFET test with VD = 3volts.

10

o

o

(b) X-velocity (cm/sec)

Figure 3: (continued)(c) V-velocity (cm/sec)

11

0.5

0.4

0.3

0.2

0.1

o

o

32.5

21.5

I

n.5n

-0.5

o

Figure 3: (continued)

(d) Energy (eV)

(e) Electrostatic potential (volts)

12

5e+17 5.5e+17~4.5e+17 c;;- 5e+17~ 4e+17 E 4.5e+17E~ 3.5e+17

(,) 4e+17a;CI> -3: 3.5e+17-3: 3e+17Z- Z- 3e+17'iii 2.5e+17 'iiic

2e+17~ 2.5e+17

CI>0 Cl 2e+17g> 1.5e+17 ce 1.5e+17'0.. 1e+17 t)

1e+170 CI>0 5e+16 iii 5e+16

0 00 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8

X (microns) X (microns)

1.2e+07 i i i i i i 2.5e+06

1e+07 2e+06

U -g- 1.5e+06CI>8e+06~ ~

E E 1e+06~ ~Z- 6e+06 Z-'0 'g 5000000Qi 4e+06 Qi> > 0X ~

2e+06 -500000

0 -1e+060 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8

X (microns) X (microns)

0.55 3.50.5 3

0.450.4 ~

2.5;;- 0.35 (5 2CI> ~~ 0.3>- (ij 1.5en 0.25a; ~c 0.2 CI>w 0

0.15 c-0.5

0.10.05 0

0 -0.50 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8

X (microns) X (microns)

Figure 4: Projection of 2D results of Figure 3 on ID cross-section 40..4.below interface.

13

0.06

........0.05c

0....uE 0.04::cE'-" 0.03.....cQ)........:::l 0.02uc

"(ij0.01....

'0

00 0.5 1 1.5 2 2.5 3

drain voltage (volts)

Figure 5: Plot of drain current vs. drain voltage for MOSFET test.

systems gets worse. Since the CGS iterative solver is less effective for such algebraic systems,the algorithm resorts to the band solver more frequently, which is considerably less efficientthan the iterative solver.

Finally, we also performed some test runs using the parabolic form of our hydrodynamicmodel (Q' = 0 and D = 3/2). In general, for the same set of input conditions we saw nosignificant difference in convergence rate or performance between the two models.

6 Conclusion

We have developed an efficient finite-difference approach for simulating submicron semicon-ductor devices using a representative non-parabolic hydrodynamic model to illustrate theapproach. The discretization scheme is formulated in a general framework, which permitsthe use of curvilinear, graded, nonuniform finite difference grids. It extends the Scharfetter-Gummel treatment for the current density and energy flux terms in the hydrodynamic modelto the mapped system. It is important to note that the mapping ideas apply quite generallyto other transport models and can be directly extended to 3D. We are currently workingwith Bell Laboratories and Stanford University on implementing some of these ideas in thePROPHET simulation platform.

Our numerical results demonstrate the efficiency and robustness of the present schemefor both ID and 2D device structures. However, we have also shown that grid and scalingissues warrant attention, as they can have a significant impact on efficiency.

The primary challenge in grid design for submicron devices is the need to provide ad-equate resolution of very high gradients, while maintaining sufficient smoothness of thetransition between regions of disparate grid densities. Although grid smoothness is desir-able in most numerical applications, the semiconductor problem appears to be particularly

14

1.5 Y

-I II II I

1.0

0.5

00 I I I I I I0.0 0.5 1.0 I.S 2.0 2.5

Figure 6: Doping profile and geometry for 2nd MOSFET test structure with n-type sourceand drain, p-type substrate.

sensitive. Adaptive grid techniques that can address these requirements are needed to min-imize the frequent reliance on trial-and-error procedures. Adaptive schemes can be basedon either mesh refinement or redistribution. We are currently investigating adaptive redis-tribution and optimization methods to handle this problem [11]. We are also developingmultilevel preconditioning and continuation strategies to further improve the performanceof iterative schemes. These investigations will be carried out using PROPHET, and willalso expand PROPHET's device simulation capabilities.

Acknowledgements

This research is supported by DARPA contract # DABT 63-96-C-0067.

References

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[2] T. J. Bordelon, X.-L. Wang, C. M. Maziar, and A. F. Tasch. An evaluation of energytransport models for silicon device simulation. Solid-State Electronics, 34(6) :617-628,1991.

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[4] G. F. Carey. Computational Grids: Generation, Adaptive Refinement and SolutionStrategies. John Wiley, UK, 1996 (in preparation).

15

phi

5 -4.5

43.5

32.5

21.5

10.5o

ox

2.5

1.5

(a) Electrostatic potential (volts)

le+20

le+lO

o

1.5

(b) Electron density (per cm3)

Figure 7: (a)-(d) Surface plots of steady-state results for 2nd MOSFET test with VD = 4volts and VG = 1 volt.

16

u (em/sec)

6e+085e+084e+083e+082e+08le+08

o-le+08

o 0.5

2.5

(c) X-velocity (cm/sec)

1.5

v (em/sec)

2e+08 -le+08

o-le+08-2e+08-3e+08-4e+08

ox

2.5

1.5

Figure 7: (continued)

(d) V-velocity (cm/sec)

17

[5] M. Fukuma and R. Uebbing. Two-dimensional mosfet simulation with energy transportphenomena. IEDM Technical Digest, pages 621-624, 1984.

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[12] A. L. Pardhanani and G. F. Carey. Adaptive grid and iterative techniques for sub-micron device simulation. In Proceedings of the Third International Workshop onComputational Electronics, pages 248-251, Portland, OR, May 18-20 1994.

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