Numerical discretization of energy-transport model for semiconductors using high-order compact schemes

Applied Mathematics PERGAMON

Letters

Applied MathematicsLetters15 (2002) 721-726

www.elsevier.com/locate/aml

Numerical Discretization of Energy-Transport Model for Semiconductors Using High-Order Compact Schemes M. FOURNII? Mathematiques pour 1’Industrieet la Physique UMR CNRS 5640, UFR MIG, Universite Paul Sabatier 118 route de Narbonne, 31062 Toulouse Cedex, Prance (Received

and accepted

June

2001)

Communicated by P. Markowich Abstract-This paper deals with numerical discretization of energy-transport model for nondegenerate semiconductors with a parabolic structure. The scheme is based on high-order computations using compact stencil. Numerical simulations of a ballistic diode in 1D are performed for different energy relaxation time and are compared with the results obtained by a drift-diffusion model. @ 2002 Elsevier Science Ltd. All rights reserved.

Keywords-High-order

compact

schemes, Semiconductors,

Parabolic

band structure.

1. INTRODUCTION This work describes one fourth-order compact scheme used to solve the stationary energy transport model (ET model) [l] formulated in terms of entropic variables (21. We refer to [3-51 for the derivation of such models from the Boltzmann equation incorporating the electron-electron and the elastic collisions. This is an intermediate approach between the drift diffusion model the hydrodynamic model (DD model) [6], which is an equation for the electronic density-and (HD model) [7], w h’ICh is an hyperbolic system of equations for the electronic density, momentum and energy. The ET model is advocated for a submicrometer model device. Indeed, the DD model which assumes a local equilibrium of the mean carrier energy is not accurate enough owing to the rapidly changing electric fields, and HD model yields unphysical solutions in some cases, coupled with difficult numerical stability due to the hyperbolic nature of model equations [8]. This paper is organized as follows. In Section 2, we present an efficient conservative numerical method based on an accurate computation of the electron and energy currents using high-order compact scheme (HOC scheme). Finite difference schemes of this type only require three consecutive points in any direction (three points in lD, 3 x 3 points in 2D, 3 x 3 x 3 points in 3D), the order still being greater than 2. Such good candidates as these schemes are to discretize The author thanks S. Gdnieys and A. Jiingel for their assistance and discussions and the reviewers for their helpful suggestions and comments. 0893-9659/02/$ - see front matter @ 2002 Elsevier Science Ltd. All rights reserved. PII: SOS9S9659(02)00033-2

Typeset

by &S-T&$

M. FOURNI~

722

advection-dominated equations may be viewed as an extension of the standard central differencing scheme in which the lowest-order terms of the truncation error are locally approximated using the differential

equation

to an extension grids

[lO,ll].

formed.

analogously

of a previous In Section

3, numerical

The new scheme preserves

oscillations)

to the Lax-Wendroff

one already

and the efficiency

studied

simulations

the excellent

already

observed

2. HIGHER-ORDER We summarized

in Table

idea [9]. The new scheme corresponds

on DD model

with uniform

of a one-dimensional numerical

stability

retained

(without

and nonuniform diode are per-

producing

artificial

for the DD model.

DIFFERENCE

1, the scaling

ballistic

SCHEME

for the ET model.

Table 1. Scaling factors.

r

Quantity _

1 Symbol

Scaling Parameter

1

Doping profile

c

Gn

the maximal value of C

Distance

r

L

the diameter of the device

Electron temperature

T

To

the lattice temperature

Electric potential

li,

UT = y

the thermal voltage

Energy relaxation time

70

L2

Electron density

n

mo

Current density

Jl

Energy flux density

J2

Diffusion coefficients

Di?j

With

constant) charge)

(q is the elementary

(7%‘)3’2 c,

(r%* is the scaled effective mass)

L2q2

Effective electron mass

Source term

(k,y is the Boltzmann

UTPO

(po

i&To

is the carrier electron mobility)

SpOuTcnx L

W

such a scaling,

the ET model in 1D is given by, -- aJ1

ax x2 a2+ arc2-

n+C=O,

J=(;)=DV(;).

(1)

The electron density n = ( -27r/V)3/2 exp(U - V$), where U = (p - $)/T, V = -l/T are the entropic variables depending on p the chemical potential of the electrons [4] and D is the diffusion matrix. In the case of Boltzmann statistics, parabolic band structure and under additional hypothesis on the elastic collisions with transition matrix independent of the particle energy, D takes the form,

= $

exp(U - V$)

(2)

Numerical

The parameter (1 - T) where

723

Discretization

the scaled squared Debye length W = C~~LT-~/~ equal to 3/2(2n) 3/2~g*&% and 7; is the scaled energy relaxation

X2 = dJT/qCmL2

C’s is a constant

represents

time. investigated in many Model (1) (with various choices of the diffusion coefficients) is numerically based on high-order conservative and papers [8,12-141. H ere, we describe a new discretization compact finite difference schemes to compute the fluxes. We approximate the problem on uniform grid with step h and we denote

spatial

mesh point xi = hi. We present an accurate points

(i) and (i+l).

central

difference

the energy

computation

equation

First,

to determine

difference

formulas.

discretization

of the continuity

can be obtained

a fourth-order The terms

of a generic

of Ji at the grid point (i + l/2)

Thus, a final conservative

approximation

balance

by Ui an approximation

equation;

and (E)

only using the neighboring

can be obtained a similar

coefficients

(Dii,

ui+1/2

Thus,

%+l

=

of J1 = Dllg

are approximated

+ Dlzg,

we use

central

by,

,

(3)

+ Ui

+

0 (h4)

we obtain,

(2) of the diffusion

V, $f$,$$,

$J). Each derivative

order truncation two grid points. equations

coefficients,

(5) can be formulated

must be approximated

in terms of (a,

2,

to the second order to obtain

U, g, a fourth-

error. The difficulty lies in the approximation of such derivatives by only using To preserve the compactness characteristic of our discretization, the differential

(1) and the equations

approximation of (3). finite difference operator, relation

with

by

2

Using definition $$

Diz)

by using a classical

scheme for Jz coupled

+ 0 (h4)

and the diffusion

u at the

as well.

discretization

($$)

function

(4) truncated

deduced

by differentiation

are used.

The simplest

case is the

Instead of using three grid points to apply the classical second-order we reduce the number of grid points using the Poisson’s equation and

at order 2. Thus, we obtain

i+1/2

Ci+l/2)= & (%+I + % -

Ci+l

-

Ci)

+ 0 (h2) .

(6)

The treatment of the other terms is more delicate because of the coupled form of the problem and the technique used in [lo] for the the DD model can not be applied. Here, we must consider two systems deduced from (l), (&)

= {g

and (S2) = {

= 0,

$ =

0,

2 = -w} g

= -g>.

M. FOURNIJ?

724

We denote

by Fi and Fi, generic

functions

of (U, V, T/J) of order up to 1. System

where 1 indicates

the dependences

(Si) can be solved to determine

(g

on the derivatives

) and

(g

) in terms

of the lower derivatives,

The uniqueness

of system

a same way, we can deduce

Substituting

(7) is guaranteed (a)

by the positive-definiteness

and (g)

(7) into (8) gives the expressions

order up to 1. Using truncated

relations

of the matrix

D [3,4]. In

from (Sz),

of (g)

and (a)

in terms

of the derivatives

(3),(4) and (6) into (7),(8) second-order

of

approximations

of all derivatives appearing in (5) and the final approximation of the current Ji are obtained. The computation of the energy flux J2 is analogous and the discretization of the Poisson’s equation is performed using a fourth-order compact scheme [lo]. The boundary conditions are taken into account in the interior scheme (due to the compactness of the dicretization). In lD, no fictitious points outside the domain are defined in opposite to the multidimensional cases where particular treatments must be done [lo]. This discretization yields a system of nonlinear equations linearized by Newton iteration (requiring the definition of the Jacobian matrix) and inverted using the solver GMRES. The manipulations required for the discretization and the linearization of the problem are very intricate algebraic manipulations which are achieved using a symbolic computations system. The fortran code is automatically generated assuming reliable and optimized programming.

3. NUMERICAL

RESULTS

diode In this section, we simulate an electronic flow in a n + -n-n+ submicrometric which is a simple model for the channel of a MOSFET. Both the n+ regions have to cl = 0.1 pm. The doping densities in the n+ and n region are equal respectively and q = 2.1021 rnm3, Tc = 300 K and ~_lc= lo3 cm2V-r s- ‘. These data enable us

L = 0.6 pm a width of 5.1023 rnT3 to compare

our results to numerical results Neumann boundary conditions

of the physical literature [8,13,14]. The classical mixed Dirichletare restricted in 1D case to ohmic contacts : T(0) = T(L) = To, boundary n(0) = n(L) = CI, +(O) = tiappl = 1.5V, and $J(L) = OV. The scaled corresponding conditions on the variables (U, V, $J) are given by,

where ?%iis the scaled intrinsic concentration (scaled by cl) and $&,,,r is the scaled applied voltage (see Table 1). The computations are made using successive simulations for different values of the applied voltage $appi. We first compute the solution for $&,,, = 0, then we use it as the initial data to compute the solution for $&,, = $,“,,, + LY$,~~~,and so on, to finish with $+,I = 1.5V. We present some numerical results (scaled) for a uniform mesh of 100 nodes. Figures 1 and 2

Numerical Discretization

80

1

1

I

f

I

1

1

I I DD _._._ ET, no relax. ET, with relax. .........

725

5.51

,

,

5

!

4.5

( -.,,

,

,

,

, D,,

.,

I

I I

4

, \

ET, no relax. ‘. \ ,, ET, with relax.

, _._._

, _

_ .-...-...

\ ‘. ‘. I;’ ‘.., ‘. I,. .‘.. ‘. ;; ‘...... .._.. ‘. .... \

3.5 3 2.5 2 30

-

20

-

1.5

1 I

10 0

0.1

I

I

0.2 0.3

I

I

I

I

0.4 0.5 0.8 0.7

I

0.5 0

I

0.8 0.9

1

I

I

0.1 0.2 0.3 0.4 0.5 0.8 0.7 0.8 0.9

Figure 2. Mean velocity position.

Figure 1. Potential $, versus position.

16

0

1

1

I

I

/

I

ET, no relax. ET, with relax.

14

of the electrons,

1

versus

I

..........

12 10 8 6 4 2 I

0 0

I

0.1 0.2

I

I

I

I

I

I

I

0.3 0.4 0.5 0.6 0.7 0.6 0.9

Figure 3. Electron temperature

1

T, versus position.

represent the potential and the mean velocity of the electrons (Jl/qn) for the DD model and the ET model with 70 = 0.4 x 10-12s and rc = 00 (W = 0), and the temperature is given in Figure 3. Numerical comparisons with a Scharfetter-Gummel (SG) type discretization [15] show that the HOC scheme admits a bigger step S$J,,,~ than the SG scheme: &+!J~~~~z ~S$J::~,. An overshoot spike appears near the anode junction with the maximal value of the (scaled) 14.99 when there is no relaxation term and T = 7.67 in the other case.

temperature

T =

4. CONCLUSION In this work, we have developed

one technique

energy transport model. We have used after solving algebraic coupled systems. puter systems and enables us to define The scheme gives a good treatment of gards to the applied bias. Extension of discretization of the 2D model (covering investigations.

to obtain

a new finite difference

scheme for the

a fourth-order compact scheme to compute the currents This approach can be automatized with algebraic coma more accurate scheme than a classical discretization. abrupt junctions (no oscillation) and is robust with rethe new scheme to nonuniform grids is possible and the the bipolar case, electrons and holes) is under advanced

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M. FOURNIB

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