semiconductor device simulation using a viscous-hydrodynamic model

Semiconductor device simulation using aviscous-hydrodynamic model q

Luca Ballestra a, Stefano Micheletti b, Riccardo Sacco b,*

a Dipartimento di Matematica ‘‘F. Enriques’’, Universit�aa degli Studi di Milano, Via Saldini 50, 20133 Milano, Italyb MOX–Modeling and Scientific Computing, Dipartimento di Matematica ‘‘F. Brioschi’’, Politecnico di Milano,Via Bonardi 9,

20133 Milano, Italy

Received 28 February 2001; received in revised form 1 June 2002

Abstract

In this article we deal with a hydrodynamic model of Navier–Stokes (NS) type for semiconductors including a

physical viscosity in the momentum and energy equations. A stabilized finite difference scheme with upwinding based

on the characteristic variables is used for the discretization of the NS equations, while a mixed finite element scheme is

employed for the approximation of the Poisson equation. A consistency result for the method is established showing

that the scheme is first-order accurate in both space and time. We also perform a stability analysis of the numerical

method applied to a linearized incompletely parabolic system in two space dimensions with vanishing viscosity. A

thorough numerical parametric study as a function of the heat conductivity and of the momentum viscosity is carried

out in order to investigate their effect on the development of shocks in both one and two space dimensional devices.

� 2002 Elsevier Science B.V. All rights reserved.

1. Introduction

The classical hydrodynamic model for semiconductors, introduced by Bløtekjær (see [8]), consists of a

hyperbolic system of conservation laws for describing charge transport, coupled with the Poisson equation

for the electric field. Compared with a parabolic model, such as the drift-diffusion or the energy-balance

models, the hydrodynamic model may exhibit more complex solutions. In particular, depending on thegeometry of the device and the physical working conditions, the solutions computed by the hydrodynamic

model may experience strong shocks or overshoot phenomena for the electron velocity, which are not seen

in the parabolic case.

Comput. Methods Appl. Mech. Engrg. 191 (2002) 5447–5466

www.elsevier.com/locate/cma

0045-7825/02/$ - see front matter � 2002 Elsevier Science B.V. All rights reserved.

PII: S0045-7825 (02 )00441-3

qThis research was supported by MURST Cofin�99 ‘‘Approssimazione di Problemi Non Coercivi con Applicazioni alla Meccanica

dei Continui e all�Elettromagnetismo’’, and project ‘‘Giovani Ricercatori-Es. Fin. 1999’’.* Corresponding author.

E-mail address: [email protected] (R. Sacco).

mail to: [email protected]

A numerical evidence of all of these effects can be found in the work of Gardner (see [12]) where a one-dimensional hydrodynamic model is solved by a finite difference (FD) method upwinded along the physical

velocity of the carriers, and in the work of Jiang (see [15]) where a SUPG finite element formulation is

applied to the simulation of a one- and a quasi-two-dimensional ballistic diode. From a numerical point of

view in both these works the use of extra artificial viscosity or the adding of first-order shock capturing

operators are needed to avoid the oscillations near the discontinuities. The dramatic loss of accuracy due to

a first-order shock capturing operator has been experienced in [2]. Among other numerical works present in

the literature, the hydrodynamic model has been investigated in [11,14] by using ENO schemes. These

methods are high-order accurate but the computed solutions are still affected by spurious oscillations nearthe discontinuities, in particular the electron current profile.

In the present article we adopt a slight variation of the standard hydrodynamic model, that we call

viscous-hydrodynamic, taking also into account viscous terms in the momentum and energy equations. Our

transport model is based on the hydrodynamic model of Navier–Stokes (NS) type (i.e., including viscous

stresses in the momentum and energy equations) derived in [4,5]. Electric effects are properly taken into

account by self-consistently coupling the Poisson equation with the NS system.

One of the aim of this work is to investigate the effect of the viscosity on the supersonic shocks that occur

in both one- and two-dimensional devices operating at low temperatures. The numerical approximation ofthe NS system is carried out using an explicit first-order FD method based on a centered discretization of

the derivatives of the fluxes plus an upwinding correction along the characteristic variables. FD schemes

can be written in conservation form and are robust in presence of discontinuous solutions; moreover, these

schemes have been widely employed in semiconductor device analysis (see, e.g., [14] for two-dimensional

simulations, and [3,6,12] for one-dimensional simulations). The discretization of the Poisson equation is

carried out using a dual-mixed finite element (MFE) scheme with hybridization (see [9, Chapter V]). This

formulation ensures flux conservation and yields equal-order accuracy for both electric potential and

electric field. The resulting FD/MFE discretization scheme can be proved to be first-order accurate in spaceand time and on nonuniform grids. Moreover, a stability analysis of the numerical method is carried out for

the classical linearized incompletely parabolic hydrodynamic model in two space dimensions. Despite

several simplifying assumptions, the theoretical estimate obtained agrees perfectly with the numerical ex-

periments.

The choice of working on nonuniform grids of rectangles not only improves the accuracy of the method

but allows also for reaching convergence in two-dimensional simulations where the number of nodes must

be limited to keep the computational effort as low as possible. In particular, we have numerically inves-

tigated a one-dimensional nþ–n–nþ ballistic diode, and a pair of two-dimensional devices, i.e., a submicronMESFET and a BJT.

In all cases, the monotonicity of the scheme seems to be crucial to compute solutions that exhibit strong

irregularities or shocks, since no further artificial viscosity is needed to reach convergence for a large

variety of geometries and nonsmooth initial data. Moreover, for all the simulated devices sharp discon-

tinuous solutions are obtained along with a satisfactory conservation of the electron current, despite the

first-order overall accuracy of the numerical scheme. We point out that, to our knowledge, this is the

first time that shock waves are experienced even in two-dimensional submicron devices at low tempera

ture.This paper is organized as follows. In Section 2, we present the system of PDEs constituting the hy-

drodynamic model. Only electron flow is considered for the sake of simplicity since we are modeling

unipolar devices. The numerical method is briefly described in Section 3 where a consistency result is given

establishing that the scheme is first-order accurate in both space and time and on nonuniform grids. In

Section 4 a discrete stability analysis for the linearized one and two-dimensional NS system is carried out.

In Section 5 we show and discuss some numerical results on several benchmark problems of relevance in

microelectronic applications, while some conclusions are drawn in Section 6.

5448 L. Ballestra et al. / Comput. Methods Appl. Mech. Engrg. 191 (2002) 5447–5466

2. The hydrodynamic model

The hydrodynamic model for charge transport in semiconductor, derived by Anile and Pennisi in [4],

comprises the conservation laws for electron mass, momentum and energy, i.e.

@tUþ ›xðFcðUÞ þ FdðU;rUÞÞ þ BðUÞU ¼ GðUÞ; ð1Þwhere U ¼ ðn; nv; neÞT denotes the set of conservative variables, n being the density of electrons, v ¼ðu1; u2; u3ÞT the electron velocity and e the total energy per unit mass, defined as e ¼ 3=2RT þ 1=2jvj2 whereT is the absolute electron temperature, R ¼ kB=m, m is the electron effective mass, kB is the Boltzmann

constant and jvj is the magnitude of the velocity vector. The advective flux is

Fc ¼ ðnv; nv� vþ nRT d; nevþ nRT vÞT;where d is the identity tensor and ðv� vÞij ¼ uiuj. The diffusive flux is

Fd ¼ ð0;�s;�svþ qÞT;where q is the heat flux and s is the viscous stress tensor, which represents the main difference with respect to

the standard hydrodynamic model derived in [8].

The quantities

BðUÞ ¼

0 0 0

0d

sp0

0 01

sw

266664

377775; GðUÞ ¼

0

� qnnmE

� qnmnE vþ ne0

sw

266664

377775

model the reaction and source terms in the semiconductor, qn being the electron charge (>0). Finally,

e0 ¼ 3=2RT0 is the internal energy of the lattice, T0 being the temperature corresponding to the thermo-dynamic equilibrium, and E is the electric field. As made by Gardner in [12], we take for the relaxation

times, sp and sw, the Baccarani–Wordeman relations (see [7])

sp ¼ ml0

qn

T0T; sw ¼ m

l0

2qn

T0Tþ 3

2

l0

qnv2skBT0; ð2Þ

where the mobility l0 obeys the following law (see [7,18])

l0 ¼Dl

1þ NDþNA

Nref

� �a ; ð3Þ

ND and NA being the donor and acceptor doping concentrations, respectively, and Dl, Nref , a suitable

parameters depending on T0. For the heat flux the following constitutive law is derived in [4]

q ¼ � 5

2sqnR2TrT þ 5

2sq

1

sp

� 1

sq

nRT v: ð4Þ

However, since no expression in closed form is available for sq, we prefer to adopt the following standard

Fourier-type relation in order to compare our results with those of the literature

q ¼ �krT ; ð5Þwhere the semiconductor heat conductivity k is given by the Wiedemann–Franz law

k ¼ kW–F

l0

qnk2BnT0: ð6Þ

Notice that (4) and (5) coincide provided that the convective term in (4) is neglected and assuming the

particular choice

L. Ballestra et al. / Comput. Methods Appl. Mech. Engrg. 191 (2002) 5447–5466 5449

sq ¼2

5

kW–Fl0mqn

T0T: ð7Þ

As in [4], we use the following relation for the viscous stress tensor in the two-dimensional case

sijðUÞ ¼ nRT sroujoxi

þ ouioxj

� dijoxv

while in the one-dimensional case we set

sðUÞ ¼ nRT srouox

;

sr being a characteristic time for the viscous interactions. The NS system (1) must be coupled with thePoisson equation for the electric field

oxð�EÞ ¼ qnðN � nÞ; ð8Þwhere N ¼ ND � NA is the net doping profile in the semiconductor. The Poisson equation is supported by

the constitutive relation

E ¼ �r/

between the electric field and the electrostatic potential /. In (8) � is the dielectric permittivity of the

semiconductor medium.

3. The numerical method

We describe in this section the FD method for the discretization of system (1) starting from the one-dimensional case.

Let us consider the following spatial and time grids

0 ¼ x0 < x1 < < xNx ¼ Lx; 0 ¼ t0 < t1 < < TM�1 < tMand define the intervals

DXr ¼ xiþ1 � xi; i ¼ 1; . . . ;Nx � 1;

DXl ¼ xi � xi�1; i ¼ 1; . . . ;Nx � 1;

Dtk ¼ tkþ1 � tk; k ¼ 0; . . . ;M � 1:

We denote henceforth by DX ¼ maxðDXl;DXrÞ and by Dt ¼ maxðDtkÞ. We use the forward Euler method

for time discretization while for the discretization of the convective fluxes at time tk we use the following

weighted-centered formula

oFc

ox

��xi

¼:DXl

DXrðFkciþ1

� FkciÞ þDXr

DXlðFkci � F

kci�1

ÞDXl þ DXr

: ð9Þ

This approximation is second-order accurate in DX and reduces to the standard centered scheme on uni-

form grids, but is unstable for strongly discontinuous solutions. To achieve stability we add an upwinding

term of the form

Qkiþð1=2Þ

Ukiþ1 �Uk

i

DXr

�Qki�ð1=2Þ

Uki �Uk

i�1

DXl

and we choose the matricesQkiþð1=2Þ andQ

ki�ð1=2Þ in order to upwind the derivative (9) along the characteristic

variables of the hyperbolic part of system (1). To do this, let us consider the Jacobian of the vector of the

convective fluxes


AðUÞ ¼ dFc

dUðUÞ

whose eigenvector matrix is PðUÞ and eigenvalues are ðk1ðUÞ; k2ðUÞ; k3ðUÞÞ. Let us define the matrices

K ¼ 1

2

k1 jk1j 0 0

0 k2 jk2j 0

0 0 k3 jk3j

24

35:

Then, in order to introduce the correct upwinding of the derivative (9) along the characteristic variables

P�1ðUÞU one must choose

Qkið1=2Þ ¼

ðPkið1=2ÞÞ

�1

DXl þ DXr

ðDXlKþkið1=2Þ � DXrK

�kið1=2ÞÞP

kið1=2Þ;

where the subscript iþ ð1=2Þ and i� ð1=2Þ refer to the average states

Uiþð1=2Þ ¼Uiþ1 þUi

2; Ui�ð1=2Þ ¼

Ui þUi�1

2:

The resulting approximation for the convective fluxes can be written in conservation form as

oFc

ox

��xi

¼:Fkciþð1=2Þ

� Fkci�ð1=2Þ

DXl þ DXr

; ð10Þ

where

Fkciþð1=2Þ¼ Fkciþ1

DXl

DXr

þ Fkci

DXr

DXl

� DXl þ DXr

DXr

Qkiþð1=2ÞDU

kiþð1=2Þ

Fkci�ð1=2Þ¼ Fkci

DXl

DXr

þ Fkci�1

DXr

DXl

� DXl þ DXr

DXl

Qki�ð1=2ÞDU

ki�ð1=2Þ

and

DUkiþð1=2Þ ¼ Uk

iþ1 �Uki ; DUk

i�ð1=2Þ ¼ Uki �Uk

i�1:

The approximation of the viscous fluxes in the NS system is carried out using a similar weighted-centered

formula as done in (9) for the centered contribution to the convective fluxes. As for the discretization of the

Poisson equation (8) a dual-hybrid MFE method using the lowest-order Raviart–Thomas finite element

space is employed (cf. [9, Chapter V]). This method guarantees that both the Lagrange multiplier (nodal

electric potential) and the electric field are approximated with the same first-order accuracy with respect to

the maximum norm (see [9, Section V.4]).

A proper use of Taylor�s expansions yields the following consistency result for the numerical methodresulting from the combined use of the FD upwinded (FDUP) scheme and of the MFE method.

Proposition 1. Assuming that the exact solution of the coupled system (1) and (8) is smooth enough, theFDUP/MFE discretization scheme is first-order consistent in both nonuniform space and time discretizations.

We point out that the choice of working on cartesian grids aims at minimizing the computational effort

without degrading the order of accuracy. As a matter of fact, a finite volume formulation using piecewise

constant functions on triangles would suffer a loss of accuracy by a factor one half (see [13]), a behavior thatwe have experienced in several numerical experiments.


Remark 1 (The two-dimensional case). The approximation of system (1) in two space dimensions using

structured nonuniform grids along x and y is a straightforward extension of the one-dimensional procedure

illustrated above. In particular, the two-dimensional convective fluxes can be treated using (10) in both the xand y directions. The approximation of the viscous fluxes is again carried out using the weighted-centered

formula (9) along the x and y directions. As done in the one-dimensional case, a dual-hybrid MFE method

is used to deal with the Poisson problem. The discretization is implemented on a staggered grid in order to

have the numerical unknowns defined at the nodes of the FD grid.

4. Stability analysis of the numerical scheme

In this section we carry out a stability analysis for a simplified version of the discretized hydrodynamic

system in two space dimensions and assuming to neglect the momentum viscosity. For the sake of sim-

plicity, we will consider in the original system of equations (1), which is incompletely parabolic, only the

differential contributions, since the linearized reaction terms can be easily controlled by the Gronwall�slemma without introducing a further restriction on the time step. Despite all the simplifying assumptions,we will find a limitation on the spatial and time intervals which reveals, by numerical experiments, to be

quite close to the actual one (see Section 5).

The discretized two-dimensional system (1) on a uniform mesh with no reaction terms reads

Ukþ1i;j �Uk

i;j

DtþFXk

iþ1;j � FXki�1;j

2DXþFYk

i;jþ1 � FYki;j�1

2DY� jAXjiþ1=2;j

Ukiþ1;j �Uk

i;j

2DX

þ jAXji�1=2;j

Uki;j �Uk

i�1;j

2DX� jAYji;jþ1=2

Uki;jþ1 �Uk

i;j

2DYþ jAYji;j�1=2

Uki;j �Uk

i;j�1

2DY� KXiþ1=2;j

Wkiþ1;j �Wk

i;j

DX 2

þ KXi�1=2;j

Wki;j �Wk

i�1;j

DX 2� KYi;jþ1=2

Wki;jþ1 �Wk

i;j

DY 2þ KYi;j�1=2

Wki;j �Wk

i;j�1

DY 2¼ 0; ð11Þ

where ui;j and vi;j are the velocity components at each grid node and

Uki;j ¼ ½nki;j; nki;juki;j; nki;jvki;j; nki;jeki;j�

T; Wk

i;j ¼ ½nki;j; uki;j; vki;j; T ki;j�

T

are the conservative and primitive variables, respectively. Moreover, we denote by FX and FY the hori-

zontal and vertical components of the convective flux Fc

FXki;j ¼

nki;juki;j

nki;juki;ju

ki;j þ nki;jRT

ki;j

nki;juki;jv

ki;j

nki;jeki;ju

ki;j þ nki;jRT

ki;ju

ki;j

26664

37775; FYk

i;j ¼

nki;jvki;j

nki;juki;jv

ki;j

nki;jvki;jv

ki;j þ nki;jRT

ki;j

nki;jeki;jv

ki;j þ nki;jRT

ki;jv

ki;j

26664

37775

with

eki;j ¼ 32RT k

i;j þ 12ððuki;jÞ

2 þ ðvki;jÞ2Þ

while AX and AY are the Jacobian matrices of the convective fluxes, i.e.

AXkiþð1=2Þ;j ¼

dFX

dU

Uki;j þUk

iþ1;j

2

!; AYk

i;jþð1=2Þ ¼dFY

dU

Uki;j þUk

i;jþ1

2

!:

Let A be any square matrix and denote by K and P its associated eigenvalue/eigenvector matrices. Then, we

have


jAj ¼ PjKjP�1;

where jKj is the diagonal matrix of the modules of the eigenvalues of A. Finally, KX and KY are square

matrices of rank one representing the parabolic contribution to system (1) due to the heat dissipation

KXkiþð1=2Þ;j ¼

0 0 0

0 0 0

0 0 kkiþð1=2Þ;j

24

35; KYk

i;jþð1=2Þ ¼0 0 0

0 0 0

0 0 kki;jþð1=2Þ

24

35:

Let Z ¼ Zðt; x; yÞ be a sufficiently smooth function of space and time and Zki;j a space-time grid function.

Then, for any time level tk we define the centered second-order FD approximations of oxxZ and oyyZ at the

grid node ðxi; yjÞ

D2X ki;jZ ¼

Zkiþ1;j � 2Zk

i;j þ Zki�1;j

DX 2; D2Y k

i;jZ ¼Zk

i;jþ1 � 2Zki;j þ Zk

i;j�1

DY 2:

To carry out the stability analysis we introduce the following simplifying assumptions:

1. System (11) is linearized about a given equilibrium state (denoted by a bar) which will be assumed for

ease of presentation to be constant in both space and time.2. The boundary terms appearing in the discrete summations will be neglected.

Under these assumptions, a linearized version of (11) is

Ukþ1i;j �Uk

i;j

Dtþ AX

Ukiþ1;j �Uk

i�1;j

2DXþ AY

Ukiþ1;j �Uk

i;j�1

2DY� jAXjDX

2D2X k

i;jU

� jAYjDY2

D2Y ki;jU� KXD2X k

i;jU� KYD2Y ki;jU ¼ 0 ð12Þ

with

AX ¼

0 1 0 0

� 2

3�uu2 þ 1

3�vv2

4

3�uu � 2

3�vv

2

3��uu�vv �vv �uu 0

� �uu3

6� �uu�vv2

6� 5

2�uuRT � �uu2

6þ �vv2

2þ 5

2RT � 2

3�uu�vv

5

3�uu

2666664

3777775;

AY ¼

0 0 1 0

��uu�vv �vv �uu 0

� 2

3�vv2 þ 1

3�uu2 � 2

3�uu

4

3�vv

2

3

��vv3

6� �vv�uu2

6� 5

2�vvRT � 2

3�uu�vv ��vv2

6þ �uu2

2þ 5

2RT

5

3�vv

26666664

37777775;

KX ¼

0 0 0 0

0 0 0 0

0 0 0 0

0 0 0 �kk

2664

3775; KY ¼

0 0 0 0

0 0 0 0

0 0 0 0

0 0 0 �kk

2664

3775:

The linear transformation between conservative and primitive variables is W ¼ TU with


T ¼ oW

oU¼

1 0 0 0

� �uu�nn

1

�nn0 0

� �vv�nn

01

�nn0

� T�nnþ �uu2 þ �vv2

3�nnR� 2�uu3�nnR

� 2�vv3�nnR

2

3�nnR

26666664

37777775:

Multiplying (12) by T we get

Wkþ1i;j �Wk

i;j

Dtþ AXnc

Wkiþ1;j �Wk

i�1;j

2DXþ AYnc

Wki;jþ1 �Wk

i;j�1

2DY� jAXncjDX

2D2X k

i;jW

� jAYncjDY2

D2Y ki;jW� KXncD2Xk

i;jW� KYncD2Y ki;jW ¼ 0; ð13Þ

where

AXnc ¼ T�1AXT; AYnc ¼ T

�1AYT:

As a consequence, we have

AXnc ¼

u n 0 0

RTn

u 0 R

0 0 u 0

02

3T 0 u

266664

377775; AYnc ¼

v 0 n 00 v 0 0

RTn

0 v R

0 02

3T v

2666664

3777775

and

KXnc ¼ T�1KX; KYnc ¼ T

�1KY;

so that

KXnc ¼

0 0 0 0

0 0 0 00 0 0 0

0 0 02

3nRk

26664

37775; KYnc ¼

0 0 0 0

0 0 0 00 0 0 0

0 0 02

3nRk

26664

37775:

Our next step is to cast system (13) in symmetric form. With this aim, let us introduce the following positive

diagonal matrix

M ¼

RTn2

0 0 0

0 1 0 0

0 0 1 0

0 0 03R

2T

2666664

3777775:

Multiplying (13) by ðWkþ1i;j ÞTM, we get

ðWkþ1i;j ÞTM

Wkþ1i;j �Wk

i;j

Dtþ ðWkþ1

i;j ÞTMAXnc

Wkiþ1;j �Wk

i�1;j

2DXþ ðWkþ1

i;j ÞTMAYnc

Wki;jþ1 �Wk

i;j�1

2DY

�ðWkþ1

i;j ÞTMjAXncj2DX

D2X ki;jW�

ðWkþ1i;j ÞTMjAYncj

2DYD2Y k

i;jW� ðWkþ1i;j ÞTMKXncD2X k

i;jW

� ðWkþ1i;j ÞTMKYncD2Y k

i;jW ¼ 0: ð14Þ


It is easily checked that the matricesMAXnc,MAYnc are symmetric and thatMKXnc,MKYnc are symmetric

positive semidefinite. The next lemma enables us to prove that this latter property holds also for the ma-

trices MjAXncj and MjAYncj.

Lemma 1. Let M and A be two matrices such that M is symmetric positive definite and the product MA issymmetric. Then, the product MjAj is symmetric and the matrices MðjAj þ AÞ and MðjAj � AÞ are bothsymmetric positive semidefinite.

Proof. Let PKP�1 ¼ A be the eigenvalue/eigenvector decomposition of A. Then, by assumption the matrix

A1 ¼MA ¼MPKP�1 is symmetric and this holds also for the following matrix

PTA1P ¼ PTMPK:

Let us now suppose that the matrix K is partitioned into diagonal blocks of size equal to the algebraic

multiplicity of the corresponding eigenvalue. Then, the product of the symmetric matrix PTMP with the

diagonal matrix K is symmetric provided that PTMP is block diagonal and the dimensions of the (sym-

metric) blocks of PTMP coincide with the corresponding blocks of K. In particular, if A has distinct eigen

values then PTMP is diagonal. Let us now prove that the matrix

A2 ¼MjAj ¼MPjKjP�1

is symmetric. Indeed, we have

PTA2P ¼ PTMPjKj

and since PTMP is block diagonal then it follows that the matrix PTMPjKj is symmetric and, in turn, that

the matrix A2 is symmetric, which proves the first part of the lemma.To prove that the symmetric matrices A3 ¼MðjAj þ AÞ and A4 ¼MðjAj � AÞ are also positive semi-

definite we proceed as follows: we have

PTA3P ¼ PTMPðjKj þ KÞ: ð15Þ

The symmetric matrix above is the product of two block diagonal matrices, PTMP and ðjKj þ KÞ, so that

the eigenvalues of the resulting matrix are those of the product blocks. Now, in this operation the blocks of

PTMP of dimension 1 (corresponding to distinct eigenvalues) are multiplied by nonnegative numbers, while

the blocks of PTMP of dimension greater than one (corresponding to coinciding eigenvalues) are multiplied

by the corresponding diagonal blocks of ðjKj þ KÞ which are formed by the same nonnegative number. It

turns out that in the matrix product in (15) the eigenvalues of the blocks of PTMP are simply scaled by anonnegative number. By that, the eigenvalues of PTMPðjKj þ KÞ may be null or positive since M and

PTMP are symmetric positive definite. �

We are now in a position to prove the following stability result.

Proposition 2. Assume that the matrix

E2d ¼M d

� jAXncj

DtDX

� jAYncjDtDY

� 2KXncDt

DX 2� 2KYnc

DtDY 2

ð16Þ

is positive semidefinite. Then, the following stability inequality holdsXi;j

ðWkþ1i;j ÞTMðWkþ1

i;j Þ6Xi;j

ðWki;jÞ

TMðWk

i;jÞ 8kP 0: ð17Þ


Proof. Rearranging terms in (13), we get

ðWkþ1i;j ÞTMWkþ1

i;j ¼ ðWkþ1i;j ÞTE2dW

ki;j þ ðWkþ1

i;j ÞTðMðjAXncj þ AXncÞÞDt2DX

Wki�1;j

þ ðWkþ1i;j ÞTðMðjAXncj � AXncÞÞ

Dt2DX

Wkiþ1;j þ ðWkþ1

i;j ÞTðMðjAYncj þ AYncÞÞDt2DY

Wki;j�1

þ ðWkþ1i;j ÞTðMðjAYncj � AYncÞÞ

Dt2DY

Wki;jþ1 þ ðWkþ1

i;j ÞTMKXncDt

DX 2Wk

iþ1;j

þ ðWkþ1i;j ÞTMKXnc

DtDX 2

Wki�1;j þ ðWkþ1

i;j ÞTMKYncDt

DY 2Wk

i;jþ1

þ ðWkþ1i;j ÞTMKYnc

DtDY 2

Wki;j�1: ð18Þ

From the assumption on (16) and Lemma 1 it follows that all the square matrices in the previous expression

are symmetric and positive semidefinite. Therefore, we can apply the discrete Young inequality to obtain

ðWkþ1i;j ÞTE2dW

ki;j 6

12ðWkþ1

i;j ÞTE2dWkþ1i;j þ 1

2ðWk

i;jÞTE2dW

ki;j: ð19Þ

By an analogous estimate in (18) and summing over i and j (neglecting the contributions at the boundary),we finally get the stability result (17). �

Requiring that the matrix E2d be positive semidefinite means that for every fixed spatial interval the time

step Dt cannot become greater than a precise positive value Dt2d , in correspondence of which one of the

eigenvalues of the matrix E2d becomes negative. This statement can be interpreted as a CFL condition for

the numerical method in both one and two space dimensions.

The CFL condition will be verified on several numerical experiments in the one-dimensional case (see

Section 5). In such a case it is also possible to better characterize the restriction on Dt by requiring that thematrix

E1d ¼M d

� jAXncj

DtDX

� 2KXncDt

DX 2

is positive semidefinite. We have

PTE1dP ¼ PTMP d

� jKj Dt

DX� 2P�1KXncP

DtDX 2

:

Simple calculations show that

P�1KXncP ¼ 2�kk3�nnR

2

10

2

10� 3

102

10

2

10� 3

10�4

10

�4

10

6

10

26666664

37777775:

This matrix can be diagonalized by premultiplication with the diagonal matrix

D ¼1 0 0

0 1 0

0 0 3=4

24

35


while

PTMP ¼ RT�nn2

10=3 0 0

0 10=3 0

0 0 5=2

24

35:

This yields

D�1PTMP ¼ 10RT3�nn2

d

from which it turns out that the symmetric matrix PTE1dP is a scaled version (up to a positive factor) of the

matrix

E�1d ¼ D d

� jKj Dt

DX� 2P�1KXncP

DtDX 2

:

Since E1d and E�1d are symmetric matrices it follows that to have stability in one dimension we must require

that the matrix E�1d is positive semidefinite.

Letting g ¼ ð�kk=ð30�nnRÞÞDt=DX 2 we have

E�1d ¼

1� jk1jDtDX

� 8g �8g 12g

�8g 1� jk2jDtDX

� 8g 12g

12g 12g3

4� 3

4jk3j

DtDX

� 18g

2666664

3777775:

Denoting by Dt1d the limiting value of the time step in correspondence of which the E1d is no longer positive

definite, a lower bound Dtest1d for Dt1d can be computed by requiring that the matrix E�1d is diagonally

dominant and with positive diagonal entries. This yields the following CFL condition

Dtest1d ¼ mink¼1;2;3

DX

jkkj þ bk�kk

�nnRDX

; ð20Þ

where

b1 ¼ b2 ¼ 14=15; b3 ¼ 28=15:

5. Numerical results

In this section we carry out a numerical validation of the discretization scheme proposed in this article on

both one- and two-dimensional device structures, that are very well-known benchmark problems in the

mathematical semiconductor community. We point out that the discrete formulation has also been ex-

tended and applied to the numerical simulation of a two-dimensional bipolar transistor, which is a widely

employed structure in semiconductor device technology.

5.1. Test case 1

We deal with the simulation of a nþ–n–nþ diode, with a structure similar to the one studied in [12].

Namely, we take a 0.6 lm silicon diode with 0.1 lm source, 0.4 lm channel and 0.1 lm drain. The lattice

temperature is T0 ¼ 77 K and the applied potential is /bias ¼ 1 V. Moreover, we assume the following

doping profile:


NðxÞ ¼ 5� 1023 m�3 if 06 x6 0:1 lm or 0:56 x6 0:6 lm;

5� 1021 m�3 if 0:126 x6 0:48 lm:

�

The above piecewise constant values are connected by a C1 smooth cosine law. In our simulation we use the

following values of the physical quantities (which are similar to those in [12]):

m ¼ 0:26me; � ¼ 11:7�0; Nref ¼ 1:44� 1021 m�3;

a ¼ 0:659; vs ¼ 1:2� 105 ms�1; Dl ¼ 1:8 m2Vs�1:

A nonuniform mesh of 600 nodes is used. The stepsize is smaller where strong variations of the doping

occur since high gradients of the solution are expected here, while a coarser grid size is used elsewhere.

Appropriate boundary conditions are (see [12]):

• at the source (x ¼ 0): n ¼ Nð0Þ, T ¼ T0, / ¼ /built-in,• at the drain (x ¼ LTot): n ¼ NðLTotÞ, T ¼ T0, / ¼ /built-in þ /bias,

where /built-in ¼ ððkBT0Þ=qnÞ logðN=niÞ is the built-in voltage arising between the semiconductor and the

metal contact and ni is the intrinsic concentration in the semiconductor. As in [12] we have used

ni ¼ 2:84� 10�14 m�3. At t ¼ 0 we set n ¼ N , u ¼ 0 and T ¼ T0.Fig. 1(a) and (b) shows the steady-electron velocity and the electron temperature for different values of

kW–F and without the momentum viscosity (i.e., sr ¼ 0); the results are essentially the same as in [12].

Precisely, in absence of heat flux (kW–F ¼ 0) a shock wave is generated at approximatively 0.25 lm; as theheat dissipation increases, the shock is spread out and an overshoot of the electron velocity is produced at

0.45 lm.

We have also numerically checked the validity of the CFL condition (20) by computing at each iteration

the limiting value Dt1d . In order to construct the matrix E�1d at each node of the grid we have considered all

the quantities in E�1d to be constant on each spatial cell. Moreover, a uniform mesh of only 250 nodes has

been chosen in order to reduce the computational cost of matrix eigenvalues computation. Finally, we

denote by CFL the value of the CFL number computed according to the procedure described above.

Numerical experiments show that using the value Dtest1d for computing CFL would typically underestimatethe actual CFL number by a factor 1.2.

In order to check the validity of the stability analysis, we set kW–F ¼ 0:1 which aims at balancing the

hyperbolic and the parabolic characters of system (1) and (8). In this regard, see Fig. 1(a) where the heights

of the two overshoots are almost the same in the case kW–F ¼ 0:1.

Fig. 1. (a) Electron velocity for different values of kW–F; (b) electron temperature for different values of kW–F.


Figs. 2(a,b) and 3(a) show the electron velocity, at t ¼ 0:159 ps, for CFL ¼ 0:95, CFL ¼ 1:05 andCFL ¼ 1:1 respectively. Notice that in the first case the velocity profile is smooth, in the second case some

small oscillations appear at approximatively 0.12 lm, and in the latter case the oscillations are severe.

Notice that, due to the nonlinearity of the equations, in the case with CFL ¼ 1:1, the spurious oscillationsproduced at t ¼ 0:159 ps are spread out as time increases and a steady smooth solution can thus be

achieved. On the contrary, if CFL is approximatively greater than 1.2, wild spurious oscillations arise in the

velocity profile, the temperature becomes negative and an overall instability occurs.

In Fig. 3(b) the steady-electron current profile obtained with the nonuniform grid is compared to the one

obtained using a uniform grid with the same number of nodes: as we can see, in this latter case the error onthe conservation of the current is more than three times greater.

Finally, we have investigated the effect of the momentum viscosity taking values for sr close to those

reported in [16]. In Fig. 4(a) and (b) the contribution of the momentum viscosity to the electron velocity

and to the electron temperature is considered assuming zero heat conductivity (kW–F ¼ 0): the shock wave is

spread out for both velocity and temperature, as in the case of heat dissipation, but no velocity overshoot is

produced. Notice also that no cooling effect is present at the source end of the channel unlike what happens

when sr ¼ 0.

Fig. 5(a) and (b) compares the velocity and temperature profiles obtained for different values of themomentum viscosity at kW–F ¼ 0:5: as we can see, the velocity overshoot effect tends to disappear as sr

Fig. 2. (a) Electron velocity at t ¼ 0:159 ps for CFL ¼ 0:95; (b) electron velocity at t ¼ 0:159 ps for CFL ¼ 1:05.

Fig. 3. (a) Electron velocity at t ¼ 0:159 ps for CFL ¼ 1:1; (b) electron current on a uniform and a nonuniform grid.


increases. Notice that the influence of the momentum viscosity is less evident than in the case with no heat

conductivity since the problem is already dissipative.

5.2. Test case 2

We have considered a two-dimensional submicron MESFET device with the same structure as in [14].The geometry is shown in Fig. 6(a) where LTot ¼ 0:6 lm, LCh ¼ 0:4 lm and d ¼ 0:6 lm. The doping profile

is defined by Nþ ¼ 5� 1023 m�3 in ½0; 0:08� � ½0; 0:08� and in ½0:52; 0:6� � ½0; 0:08� lm, N ¼ 1� 1022 m�3

elsewhere.

The above piecewise constant values are connected with a suitable two-dimensional cosine law on a

radial interval of 0.02 lm. The physical parameters and the constitutive laws are the same as in the previous

test case, the lattice temperature is 77 K, and an external voltage /bias ¼ 2 V is applied at the drain contact.

A voltage bias of /Gate ¼ �0:8 V is applied at the gate, which is considered to be a Schottky contact. As

in [14], a very small concentration value

Fig. 4. (a) The momentum viscosity reduces the shock (at 0.25 lm) and does not give rise to the velocity overshoot (case where

kW–F ¼ 0); (b) electron temperature for different values of sr (case where kW–F ¼ 0).

Fig. 5. (a) The momentum viscosity reduces the velocity overshoot produced by the heat dissipation; (b) electron temperature for

different values of sr (case where kW–F ¼ 0:5).


n ¼ 1:051� 105 m�3

obtained from Eq. (5.1-19) of [17] is fixed there. Finally, we set kW–F ¼ 0:01.The boundary conditions are chosen as follows:

• at the source (06 x6 0:1 lm, y ¼ 0):

/ ¼ /built-in; n ¼ 5:0� 1023 m�3; T ¼ 77 K; u ¼ 0;

• at the drain (0:56 x6 0:6 lm, y ¼ 0):

/ ¼ /bias þ /built-in; n ¼ 5:0� 1023 m�3; T ¼ 77 K; u ¼ 0;

• at the gate (0:2256 x6 0:375 lm, y ¼ 0):

/ ¼ /Gate þ /built-in; n ¼ 1:051� 105 m�3;

and homogeneous Neumann boundary condition for the temperature.

• On all the other parts of the boundary we impose the velocity to be tangential and Neumann boundary

conditions for the temperature and the electrostatic potential.

We use a relatively coarse grid of 160� 80 nodes: the horizontal nodes are concentrated near the drain and

source contacts (including the junctions which are the curves separating the Nþ and N doping regions),since numerical experiments show that this choice is optimal if we are interested in preserving the con-

servation between incoming and outcoming electron current. As for the vertical direction, we again dis-

tribute the most of the nodes near the junctions since there the jump Nþ–N is very high, much more than

the one considered in [14] and large oscillations in time (which are no numerical products) occur. These

oscillations must be captured, even with the small number of vertical nodes, with sufficient accuracy, to

prevent from nonphysical negative temperatures which inevitably lead to instability.

The following figures refer to steady state. Fig. 6(b) shows the electron concentration in semilogarithmic

scale: notice the very sharp layer near the gate contact. Fig. 7(a) shows the electron current vector field:notice the correct flow of electrons that are injected from the source contact into the drain contact. Fig. 7(b)

shows the electric vector field: notice the high values at the gate contact.

Fig. 8(a) shows the Mach number: the sharp transition from supersonic to subsonic regime reveals the

presence of a shock wave, which is an almost horizontal line at approximatively 0.13 lm.

Fig. 8(b) shows the electron temperature which exhibits a strong peak at the drain–bulk junction.

Fig. 9 refers to a vertical section at x ¼ 0:094 lm of the two-dimensional MESFET and shows, for the

vertical velocity, the diffusion of the shock wave due to the momentum viscosity.

Fig. 6. (a) Two-dimensional MESFET; (b) electron concentration.


5.3. Test case 3

As a final test case, we have considered a two-dimensional BJT device (see, e.g., [10] for a similar

structure). Due to the bipolar nature of charge transport, the numerical simulation of the device is carried

Fig. 7. (a) Electron current; (b) electric field.

Fig. 8. (a) Mach number; (b) electron temperature.

Fig. 9. The momentum viscosity spreads out the shock wave (vertical cut at x ¼ 0:094 lm).


out taking into account a hydrodynamic model for both electron and hole carriers as described in Section 2.

In the following, we shall specify the complete set of physical parameters which define the bipolar model,and for more details about the exact formulation of the equations we refer, e.g., to [1].

The geometry of the device is shown in Fig. 10(a), where LTot ¼ 1:2 lm, L1 ¼ 0:4 lm, L2 ¼ 0:2 lm,

L3 ¼ 0:4 lm, LB ¼ LE ¼ 0:3 lm, with the origin of the x, y coordinate system at the bottom left corner of

the figure. The doping profile is defined by the following piecewise constant function

N ¼ ND � NA ¼5� 1023 m�3 in ½0:8; LTot� � ½0; L2� lm;

1� 1023 m�3 in ½0; LTot� � ½0:8; LTot� lm;

�1� 1022 m�3 elsewhere

8><>:

and is shown in figure Fig. 10(b).

The above piecewise constant values are connected with a suitable two-dimensional cosine law on a

radial interval of 0.08 lm. The physical parameters and the constitutive laws are almost the same as in theprevious test case. Precisely, we set the lattice temperature equal to 300 K, the electron and hole effective

masses equal to m ¼ mn ¼ 0:26me, m ¼ mp ¼ 0:38me, respectively. Relation (3) is assumed to hold for

electrons and holes with Dl ¼ 2 and 1.2 m2 V s�1, respectively, and take for both carriers Nref and a with the

same values as in the previous test cases. The same convention holds for the momentum and energy re-

laxation times (2), with m ¼ 0:2mn, m ¼ 0:1mn, respectively, for electrons, while we take m ¼ mp for holes.

For both carriers vs ¼ 1� 105 ms�1. The heat flux relaxation time in (7) is treated analogously as above,

with m ¼ mn for electrons, and m ¼ mp for holes. We assume kW–F ¼ 0:5 throughout and use in (6) the

bipolar version of (3) with the values of the parameters as defined above for electrons and holes, respec-tively. A generation/recombination mechanism is also included in the model, namely, the classical

Shockley–Read–Hall recombination rate, that is defined as (see [1])

U ¼ pn� n2itnðp þ niÞ þ tpðnþ niÞ

;

where p is the hole concentration, tn ¼ tp ¼ 10�7 s and ni ¼ 1:1� 1016 m�3.

The boundary conditions are chosen as follows:

• at the base (06 x6 LB, y ¼ 0 lm):

/ ¼ /built-in þ 0:67 V; n ¼ n2i =1022 m�3; p ¼ 1022 m�3; T ¼ 300 K; u ¼ 0;

Fig. 10. (a) Geometry of the BJT; (b) Doping profile: log10 ðjND � NAjÞ.


• at the emitter (0:86 x6 LTot, y ¼ 0, lm):

/ ¼ /built-in; n ¼ 5:0� 1023 m�3; p ¼ n2i =ð5� 1023Þ m�3; T ¼ 300 K; u ¼ 0;

• at the collector (06 x6 LTot, y ¼ LTot lm):

/ ¼ /built-in þ 1:5 V; n ¼ 1023 m�3; p ¼ n2i =1023 m�3; T ¼ 300 K; u ¼ 0:

• On all the other parts of the boundary we impose zero normal component for the velocity, and homo-

geneous Neumann boundary conditions for the temperature and the electrostatic potential.

In the numerical computations, we use a relatively coarse grid of 80� 80 subdivisions.

The following figures refer to steady state.

Fig. 11(a) and (b) shows the electron and hole concentrations in logarithmic scale. Notice thesmoothness of the profiles and the presence of sharp boundary layers.

Fig. 12(a) displays the electric potential.

Figs. 12(b) and 13(a,b) show the streamlines together with a contour plot of the absolute value for the

total, electron and hole current densities, respectively. In particular, it can be appreciated that the total base

current density is dominated by the hole contribution, while elsewhere the electrons determine the most part

of the total current density. This is in accordance with the physics of the n–p–n type BJT (see, e.g., [19]).

Fig. 11. (a) Electron concentration; (b) hole concentration.

Fig. 12. (a) Electric potential; (b) streamlines and strength of the total current density.


Fig. 14(a) and (b) shows the electron and hole temperatures. Notice a strong peak at the base–collector

junction for both electrons and holes and that the latter heat up more than the former around both emitter

and collector junctions.

6. Conclusions

In this article we have investigated a hydrodynamic model for semiconductors with a physical viscosity

in the momentum/energy equations. A stabilized FD scheme on nonuniform grids is used for the discret-

ization of the mass, momentum and energy conservation laws in both one and two space dimensions, while

a dual-hybrid MFE method is used to deal with the Poisson equation. The proposed numerical formulation

has been validated on both unipolar and bipolar benchmark device structures, with applications including

the simulation of a one-dimensional nþ–n–nþ ballistic diode, and of a pair of two-dimensional devices, i.e.,a submicron MESFET and a BJT. We have established a consistency result for the difference scheme

showing that the method is first-order accurate in both space and time on nonuniform grids. We have also

performed a stability analysis of the numerical method applied to the classical linearized incompletely

parabolic hydrodynamical system in both one and two space dimensions. This analysis provides us with a

sharp estimate of the maximum time step allowed in practical computations. Numerical evidences of this

estimate have been proved in one-dimensional simulations. A parametric study of the device performance

Fig. 13. (a) Streamlines and strength of the electron current density; (b) streamlines and strength of the hole current density.

Fig. 14. (a) Electron temperature; (b) hole temperature.


as a function of the heat conductivity and of the momentum viscosity has been carried out to investigatetheir effect on shocks both in one and two space dimensions. To our knowledge, this is the first time that

shock waves are experienced even in two-dimensional submicron devices at low temperature. Finally, we

notice that the computed steady-state solutions are sharp and monotone and that the scheme yields a good

conservation of the electron current density.

References

[1] N.R. Aluru, K.H. Law, A. Raefsky, P.M. Pinsky, R.W. Dutton, Numerical solution of two-carrier hydrodynamic semiconductor

device equations employing a stabilized finite element method, Comput. Methods Appl. Mech. Engrg. 125 (1995) 187–220.

[2] N.R. Aluru, A. Raefsky, P.M. Pinsky, K.H. Law, R.J.G. Goossens, R.W. Dutton, A finite element formulation for the

hydrodynamic semiconductor device equations, Comput. Methods Appl. Mech. Engrg. 107 (2000) 269–298.

[3] A.M. Anile, C. Maccora, R.M. Pidatella, Simulation of nþ–n–nþ devices by a hydrodynamic model: subsonic and supersonic

flows, COMPEL 14 (1995) 1–18.

[4] A.M. Anile, S. Pennisi, Thermodynamic derivation of the hydrodynamical model for charge transport in semiconductors, Phys.

Rev. B 46 (1992) 13186–13193.

[5] A.M. Anile, O. Muscato, Improved hydrodynamical model for carrier transport in semiconductors, Phys. Rev. B 51 (1995) 16728–

16740.

[6] A.M. Anile, V. Romano, G. Russo, Extended hydrodynamical model of carrier transport in semiconductors, SIAM J. Appl.

Math. 61 (2000) 74–101.

[7] G. Baccarani, M.R. Wordeman, An investigation of steady-state velocity overshoot effects in Si and GaAs devices, Solid State

Electron. 28 (1985) 407–416.

[8] K. Bløtekjær, Transport equations for electrons in two-valley semiconductors, IEEE Trans. Electron Dev. ED-17 (1970) 38–47.

[9] F. Brezzi, M. Fortin, Mixed and Hybrid Finite Element Methods, Springer, New York, 1991.

[10] A. Forghieri, R. Guerrieri, P. Ciampolini, A. Gnudi, M. Rudan, G. Baccarani, A new discretization strategy of the semiconductor

equations comprising momentum and energy balance, IEEE Trans. Computer-Aided Des. 7 (2) (1988) 231–242.

[11] E. Fatemi, J. Jerome, S. Osher, Solution of the hydrodynamic model us-ing high-order nonoscillatory shock capturing algorithms,

IEEE Trans. Computer-Aided Des. 10 (1991) 501–507.

[12] C.L. Gardner, Shock waves in the hydrodynamic model for semiconductor devices, in: W.M. Coughran Jr., J. Cole, P. Lloyd, J.K.

White (Eds.), Semiconductors Part II, IMA Volumes in Mathematics and Its Applications, vol. 59, Springer, New York, 1994.

[13] E. Godlewsky, P.A. Raviart, Numerical Approximation of Hyperbolic Systems of Conservation Laws, Springer, New York, 1996.

[14] J.W. Jerome, C.W. Shu, Energy models for one-carrier transport in semiconductor devices, in: W.M. Coughran Jr., J. Cole, P.

Lloyd, J.K. White (Eds.), Semiconductors Part II, IMA Volumes in Mathematics and Its Applications, vol. 59, Springer, New

York, 1994.

[15] X.L. Jiang, A streamline-upwinding/Petrov–Galerkin method for the hydrodynamic semiconductor device model, Math. Models

Meth. Appl. Sci. 5 (1995) 659–681.

[16] O. Muscato, Monte Carlo evaluation of the transport coefficients in a nþ–n–nþ diode, COMPEL 19 (2000) 812–828.

[17] S. Selberherr, Analysis and Simulation of Semiconductor Devices, Springer, New York, 1984.

[18] S. Selberherr, MOS device modeling at 77 K, IEEE Trans. Electron Dev. 36 (1989) 1464–1474.

[19] S.M. Sze, Physics of Semiconductor Devices, 2nd Ed., Wiley–Interscience, New York, 1981.


semiconductor device simulation using a viscous-hydrodynamic model

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