Jungel_2002, Relaxation Scheme for Hydrodynamic Eqs Semiconductors

8/13/2019 Jungel_2002, Relaxation Scheme for Hydrodynamic Eqs Semiconductors

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Applied Numerical Mathematics 43 (2002) 229252

www.elsevier.com/locate/apnum

A relaxation scheme for the hydrodynamic equationsfor semiconductors

Ansgar Jngel a,, Shaoqiang Tang a,b

a Fachbereich Mathematik und Statistik, Universitt Konstanz, 78457 Konstanz, Germanyb Department of Mechanics and Engineering Science, Peking University, Beijing 100871, Peoples Republic of China

Abstract

In this paper, we shall study numerically the hydrodynamic model for semiconductor devices, particularly in

a one-dimensional n+nn+ diode. By using a relaxation scheme, we explore the effects of various parameters,such as the low field mobility, device length, and lattice temperature. The effect of different types of boundaryconditions is discussed. We also establish numerically the asymptotic limits of the hydrodynamic model towards

the energy-transport and drift-diffusion models. This verifies the theoretical results in the literature.

2002 IMACS. Published by Elsevier Science B.V. All rights reserved.

1. Introduction

One of the main goals in semiconductor device modeling is to establish a hierarchy of models,

which allows for the choice of appropriate models for specific semiconductor applications [25].

Monte Carlo simulations of the kinetic Boltzmann equation provide a very accurate description of

charge transport in submicron devices [15]. However, their use is not practical for computer aided

design because of the large computer times needed. Macroscopic models derived from the Boltzmann

equation seem to be a compromise between physical accuracy and computational effort. The main

classes of macroscopic semiconductor models are the drift-diffusion, energy-transport and hydrodynamicequations [29,36].

The drift-diffusion models which are the most popular ones were first proposed in 1950 by Van

Roosbroeck [33]. The energy-transport equations include also the carrier energy (or temperature)which is constant in the drift-diffusion equations, and have been suggested one decade later by

Stratton [39]. The drift-diffusion and energy-transport models can be formally derived by a Chapman

* Corresponding author.E-mail addresses:[email protected] (A. Jngel), [email protected] (S. Tang).

0168-9274/02/$ see front matter 2002 IMACS. Published by Elsevier Science B.V. All rights reserved.

PII: S0168-9274(01)00182-9


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230 A. Jngel, S. Tang / Applied Numerical Mathematics 43 (2002) 229252

Enskog type expansion method from the Boltzmann equation [7,8]. The drift-diffusion equations give

satisfactory results for semiconductor devices with a typical size of a few microns and moderately

applied voltage [35], whereas energy-transport models can also be used for certain submicron

devices [14].

The hydrodynamic equations have been introduced by Bltekjr [9] and subsequently thoroughlyinvestigated by Baccarani and Wordeman [6]. They can be derived from the Boltzmann equation by

using a moment method. This yields usually a set of equations for the carrier density, momentum and

energy which is not in closed form. To obtain a closed set of equations, often the Fourier law for theheat flux is taken [9]. For different approaches of the derivation of the hydrodynamic equations and a

discussion of the closing problem, we refer to [3].The hydrodynamic equations derived by Bltekjr, Baccarani and Wordeman read as follows:

nt 1

qdiv J= 0, (1)

Jt 1q

div

J Jn

qkBm

(nT ) + q2

mnV= CJ, (2)

Et div

m

2q3J|J|2

n2 + 5

2

kB

qT J+ T

= J V+ CE , (3)

s V= q(n C). (4)Here, the physical variables are the electron density n, the current density J, the energy density E,

and the electrostatic potential V. The constants are the elementary charge q , the Boltzmann constant kB ,

the effective electron mass m, and the permittivity constant s .The doping concentration characterizingthe device under consideration is denoted by C = C(x). We assume the following constitutive relations.The energy density is given as the sum of kinetic and thermal energy

E = m

2q2|J|2

n+ 3

2kB T n =

1

2mn|u|2 + 3

2kB T n,

where the electron velocity u is defined by J= qnu. The momentum and energy relaxation terms,respectively, are

CJ= J

p,

CE

= 1

w m

2q2

|J|2

n +3

2

kB (T

T0)n,

whereT0is the lattice temperature,

p = po

T

T0

r, w = wo

T

T+ T0+ 1

2p

are the momentum and energy relaxation times, respectively, with

po =mn

q, wo =

3nkB T0

2qv2s, (5)


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A. Jngel, S. Tang / Applied Numerical Mathematics 43 (2002) 229252 231

n is the low-field mobility and vs the saturating velocity. Finally, the heat conductivity is assumed to be

=

5

2+ c

k2B n

qnT

T

T0

r,

wherec, r R are some phenomenological constants. In the numerical simulations we use c = r = 1.Eqs. (1)(4) have to be solved in a bounded domain. In this paper we will present numerical

simulations for the above equations in one space dimension. We use homogeneous Neumann boundary

conditions forn,u and T at the domain boundary (see Section 4.2 for Dirichlet boundary conditions for

nand T). Moreover, we impose initial conditions for n,u and T (see Section 3.1).

In physical situations where the mean free path of the particles is much smaller than the typical device

length and the momentum relaxation time constant po is much smaller than the energy relaxation time

constant wo , the hydrodynamic equations reduce in the relaxation-time limit formally to the energy-

transport equations (see Section 2). This limit has been proved rigorously in [17] under the assumption

of uniformLbounds. If the momentum and energy relaxation times are of the same order, but the meanfree path is much smaller than the typical device length, we obtain formally the drift-diffusion equationsfrom the hydrodynamic model. This limit has been shown rigorously for constant temperature in [23].

The asymptotic limit in the full model has been studied under some conditions in [1,12]. For an overview

of these limits, see [24].

The main objectives of this paper are first to adopt the relaxation scheme [22] in order to solve

numerically the hydrodynamic model in one space dimension and secondly, to perform the above

asymptotic limits numerically. This allows to determine the numerical values for which the solution

to the hydrodynamic model behaves like the solution of the drift-diffusion or energy-transport equations.

Moreover, we will illustrate the effects of the mobility constant, lattice temperature and channel length

of a simulated n+nn+diode.

A relaxation model was first rigorously studied in [28], and a relaxation schemewas proposed in [22].Various generalizations have then been made, e.g., discrete BGK schemes [4,30]. The basic idea is as

follows. In general, a set of conservation laws, usually quasilinear, may be derived as a macroscopic

model from a Boltzmann type equation with certain equilibrium states (e.g., local Maxwellians). This

Boltzmann type equation is semilinear, yet contains an additional variable, namely the momentum.

It is therefore much more expensive to simulate numerically. However, we may design a discrete

BGK equation instead, i.e., an artificial Boltzmann equation with finite discrete moments and suitable

Maxwellians, which are constructed in such a way to give the desired set of hyperbolic conservation laws

when performing the limiting process. A relaxation model in [22] is a special case when we take only

two velocities as moments.

To solve this relaxation model, we may apply a splitting method, i.e., first solving an ODE step andthen solving alinearconvection step. As the relaxation parameter tends to zero, this solution tends to the

solution of the original problem (see Section 3). The resulting scheme has the advantage that it possesses

a modular structure, which is particularly good for coding and for higher space dimensions. An extensive

exploration on the stability, efficiency, as well as accuracy has been made for general discrete BGK

models in [4]. It is also shown to be robust when applied to hyperbolicelliptic systems, and strongly

degenerate parabolic systems in one dimension and multi-dimensions [5,31]. We thus deem it suitable

to treat semiconductor devices, where various complexities are present, such as being of hyperbolic

parabolic type, having stiff source terms, etc.


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The numerical discretization of the transient full hydrodynamic model in two space dimensions using

a discrete BGK method will be presented in a forthcoming publication [26].There are several other techniques to discretize the hydrodynamic equations. First the Scharfetter

Gummel method has been generalized for these equations, in particular for subsonic flow [34]. Later,

second-order upwind shock-capturing methods have been used for transonic flow [16]. In recent years,numerical techniques like streamline-diffusion schemes [20], finite element methods of the RungeKutta

discontinuous Galerkin scheme [13], finite difference methods of the ENO (essentially non-oscillatory)

scheme [19,37], and UNO (uniformly non-oscillatory) schemes with the NessyahuTadmor method [3,32] have been developed.

This paper is organized as follows. In Section 2 we scale Eqs. (1)(4) appropriately and explain

the relaxation-time limits towards the energy-transport and the drift-diffusion equations in more detail.

Section 3 is concerned with the numerical discretization of Eqs. (1)(4) in one space dimension.

Numerical simulations for a stationary one-dimensional n+nn+ diode which can be considered as abenchmarkproblem are presented in Section 4.

2. Scaling of the equations and asymptotic models

In this section we scale Eqs. (1)(4) appropriately and derive the energy-transport and drift-diffusionmodels by means of formal asymptotic analysis.

2.1. Scaling

We introduce the thermal voltageUT= kB T0/q, the mean free path

= po

kBT0

m ,

and the scaled Debye length

=

s UT

qCmL2

,

whereCmis a typical doping concentration and L is a typical device length. Furthermore, we define thedimensionless parameters

=

L,

=po

wo. (6)

Then, with the scaling

n Cmn, t po t, C CmC, x Lx,V UTV , T T0T , J

q2UTCmpo /Lm

J,we obtain the scaled equations

nt div J= 0, (7)


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Jt 2 div

J Jn

(nT) + nV= J

Tr, (8)

|J|22n

+ 32

nT

2 t div2 J|J|2

2n2 + 5

2T J+ T

= J V 2

T

T+ 1 +2

2Tr

1 |J|22n

+ 32

n(T 1)2

, (9)

2V= n C. (10)We used the same notation for the scaled and unscaled variables. In Eq. (9) we have set

=

5

2+ c

T1+r n.

2.2. Asymptotic models

In order to obtain the energy-transport and drift-diffusion models, we rescale Eqs. (7)(9) by t t/2:nt div J= 0, (11)2Jt + 2 div

J J

n

(nT ) + nV= J

Tr, (12)

2|J|22n

+ 32

nT

t

div

2J|J|2

2n2 + 5

2J T+ T

= J V

T

T+ 1 +2

2Tr

12

|J|22n

+ 32

2

2n(T 1)

. (13)

The energy-transport equations are obtained by assuming that

1, 1.The relation 1 holds if the kinetic energy associated with the velocity needed to cross the device intimepo is very large compared with the thermal energy. The relation 1 also means that we study thesystem at large times of the order of 1/2. Furthermore, it holds 1 if the kinetic energy associatedwith the saturating velocity is much smaller than the thermal energy.

We formally perform the limit 0, 0 such that / 0,where 0>0 is some constant.The limit/ 0 means that the velocity needed to cross the device in time po is assumed to be ofthe same order as the saturating velocity. We obtain the equations:

nt div J= 0, (14)J= Tr(nT ) nV, (15)

3

2nT

t

div

5

2T J+ T

= J V 3

2

n(T 1) ( T )

, (16)

2V= n C, (17)where (T ) = 0T/(T+ 1).Notice that this energy-transport model is notof the general form derivedin [14] expect forc = r = 0.


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For the derivation of the drift-diffusion model, we fix the parameter >0 and let formally 0 inEqs. (11)(13) to obtain T= 1 and

nt div J= 0, (18)J= n nV , (19)2V= n C. (20)

3. Numerical scheme

In this section, we shall put the model into a more concise form. Then we shall describe our numericalscheme in three parts, namely, the overall second-order splitting method, the relaxation scheme for the

convection, and the treatment of the boundaries and diffusion.

3.1. Reformulation of the model

We recast the hydrodynamic model in one space dimension as follows.

Ut + A(U)x =

B(U,Ux)

x+ S(U), (21)

coupled with the Poisson equation

s Vxx = q

n C(x). (22)The vector quantities are

U= n

E

, A(U ) =

2

3

2

n +E

m

5 E

3n m

3

3n2

,

B(U,Ux) =

00

Tx

, S(U ) =

0

p

+ nqVxm

q Vx E E0

w

.

Here = nu = J /q, E = 12

mnu2 + 32

nkB T , and E0 = 32 nkB T0 is the rest energy density. We recallthe relaxation coefficients and heat diffusion coefficient

p =mn

q

T

T0

r, (23)

w =1

2p +

3nkB T0T

2qv2s (T+ T0), (24)

=

5

2+ c

nk2B nT

q

T

T0

r, (25)

where we usec = r = 1.


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Table 1

Physical parameters

Parameter Physical meaning Numerical value

q elementary charge 1.6

1019 C

m effective electron mass 0.26 9.11 1031 kgs permittivity constant 11.7 8.85 1012 F/mn low field mobility constant 0.1 m

2/Vs

kB Boltzmann constant 1.38 1023 J/KT0 lattice temperature 300 K

ni intrinsic electron concentration 1.4 1016 m3vs saturating velocity 1.03 105 m/s

We list the physical parameters used in our simulations in Table 1.

The initial and boundary conditions are assigned as:

n(x, 0) = C(x), (26)u(x, 0) = 0, (27)T(x, 0) = T0, (28)nx (0, t) = nx (L,t) = 0, (29)ux (0, t) = ux (L,t) = 0, (30)Tx (0, t) = Tx (L,t) = 0. (31)

Taking into account the conservation of electrons, we notice that the reflecting boundary condition onnimplies the fixed boundary condition

n(0, t) = C(0), n(L, t) = C(L).For the electric field with an applied voltage Vb, the boundary conditions are

V (0) = T0q

ln

n(0, t)

ni

, (32)

V (L) = T0q

ln

n(L,t)

ni

+ Vb. (33)

3.2. Second-order RungeKutta splitting scheme

The system (21) can be solved by a splitting method. Given some data at the k th time step t = tk , onefirst solves an ODE step, namely the initial-value problem with electric field Vkx corresponding to theelectron concentration n(x, tk ),

Ut=

B(U,Ux)

x+ S(U ), U x, tk= Uk(x). (34)

Denote the solution att = tk + tasUk+1/2 =R(Uk, Vkx, t). As we shall explain in a later subsection,the diffusion term (B(U,Ux))x is expressed explicitly by quantities at neighboring grid points. This

makes (34) an ODE system.


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Secondly, one solves a convection step, namely the initial-boundary-value problem for the homoge-

neous system

Ut + A(U)x = 0, Ux, tk

= Uk+1/2, Ux (0, t) = Ux (L,t) = 0, (35)

by a hyperbolic problem solver. The solution after one time step is denoted as Uk+1 =W(Uk+1/2, Vkx, t)(see Section 3.3 for an explicit formula).

Finally the electric field is updated from the Poisson equation (22) with electron concentration

nk+1 = first component ofUk+1, denoted asVk+1x =P(nk+1).This is just a first-order splitting. There are different ways to make it of high-order accuracy, e.g., [10,

38]. We simply apply the mid-point method to combine two Euler-forward steps for second-order

accuracy. More precisely, the scheme is:

(1) Set the initial dataU0 = U(x, 0)and update the electric fieldV0x= P(n0).(2) Fork = 0, . . . , kT 1, computeUk+1, Vk+1x as follows:

(a) Set(U[0

], V[0

]x ) = (Uk

, V

k

x).(b) Compute(U[1], V[1]x )and U[2]as follows:

U[1] = n[1], [1], E[1]= U[0] + WRU[0], V[0]x , t, V[0]x , t U[0]/2,V[1]x = P

n[1]

,

U[2] =WRU[1], V[1]x , t, V[1]x , t.(c) SetUk+1 = (U[1] + U[2])/2, Vk+1x = P(nk+1).

We note that this is an explicit splitting, hence one should take t (x)2 for stability. It can beimproved by applying implicit techniques.

3.3. Relaxation scheme for the convection step

The numerical resolution for homogeneous hyperbolic systems has been the main advance in

computing science during the last two decades. We adopt a relaxation scheme in the simulations. That is,we approximate the quasi-linear system (35) by a semi-linear one,

Ut+ Yx= 0, (36)Yt+ 2Ux=

A(U) Y

, (37)

where is a small parameter and is a constant, larger than the maximum wave speed of the originalsystem (sub-characteristic condition) for stability. As approaches towards 0, formally Y approaches

towards A(U). In turn the first equation approximates (35). Rigorous results are obtained, e.g., in [28,30] and references therein.

At the numerical level, this relaxation scheme bears many nice features, such as high accuracy,modular structure, easy to code, ready for high-dimensional generalization, etc. Numerically one mayuse the same splitting method in the previous section for this system. Since (37) is semilinear, the linear

convection part (for (36), (37) without the term on the right-hand side) is readily solved by some existing

scheme, e.g., a second-order MUSCL type scheme with minmod limiter (see, e.g., [27]). In the ODE step


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4. Numerical simulations

In this section, we shall first describe the numerical tests with our scheme. Then we shall illustrate

the effects of the mobility constant, channel length, and lattice temperature. We also demonstrate the

effects of taking Dirichlet boundary conditions for n and T, instead of the aforementioned Neumannboundary conditions. Finally we shall explore the numerical limits of the hydrodynamic model towards

the dimensionalized energy-transport model and drift-diffusion model, respectively.

4.1. Benchmarks on the numerical scheme

We make numerical tests on an n+nn+ ballistic silicon diode with an applied voltage of 1.5 V. Thesemiconductor domain is described by the interval [0, L] withL = 0.6 m. The channel length is 0.4 m.The doping profile is:

C(x) = 2 1021 m3, x

(0.1 m, 0.5 m),

5 1023 m3, elsewhere. (41)We make a convergence study with different meshes. In all these tests, the numerical relaxation

parameter in the relaxation scheme is set as = 2.5. A computation shows that this value satisfies thesubcharacteristic condition. We perform simulations with successively double space grid numbers, i.e.,with 100 grid points (x = 6 103 m), 200 grid points (x = 3 103 m), 400 grid points (x =1.5 103 m), 800 grid points (x = 7.5 104 m), and 1600 grid points (x = 3.75 104 m),respectively.

The numerical solution with 200 gridpoints is displayed in Fig. 1. In the first few picoseconds,

oscillations occur due to the sudden application of the electric field and the initial conditions. Theyare damped out gradually, and the solution tends to a steady state. In fact, it is fairly stationary-like at

t= 5 ps. The units are taken as 101 m for the space variable x, 1021 m3 for the concentration n,105 m/s for the velocity u, 101 Jm3 for the energy density E, 101 eV = 1.6 1020 J for the totalenergy w = E/ n, 103 K for the temperature T, and 106 V/m for the electric fieldVx . This scale ofunits will be used throughout the figures hereafter.

We depict the stationary solution at t= 15 ps in Fig. 2. In the velocity profile, it is observed thatbesides an overshoot in the second junction, an even bigger hump appears in a fairly wide region around

the first junction. This hump has already been observed in the simulation of the hydrodynamic model [11].

It results from our choice of the parameters. The change of the heat diffusion constant exponent r maygive different profiles [19]. The low field mobility n also makes big differences, e.g., as reported for

a GaAs diode [11]. We shall describe the numerical simulations with different constants n in the next

subsection. We also observe a cooling zone around the first junction, and a heating zone around thesecond with highest temperature about 5 times the lattice temperature.

We now describe the numerical convergence study. The solutions with different meshes are shownin Fig. 3. For a better presentation of the differences in the concentration and energy density profiles,

logarithmic plots are used. With finer grids, the velocity overshoot in the second junction clearly becomes

sharper. It is likewise in the total energy profile. These are known effects of the numerical viscosity. Forfiner mesh, numerical viscosity is smaller, thus the numerical solution has less smearing around the

discontinuities or the place where large gradients occur. Differences in other quantities are relatively

small. For the electric field, the difference is even negligible. This can be explained by analyzing


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Fig. 1. Numerical solution with 200 grid points (x = 3 103 m), (t,x) [0, 10] [0, 6].

the steady states. As the numerical current should keep constant, the difference in u is reflected in n

only reciprocally. Because n ranges from around 10 to 5000, this difference is barely observable. The

difference inn interferes the electric field through the Poisson equation. The twice integration therefore

further diminishes the difference.

Let us make a quantitative analysis of the differences. Taking the solution with the finest grid

(1600 grid points) as an exact solution, we list the L and L1 errors in Table 2. The numericalconvergence rates for the L1 error are also computed. The units of the L errors are the same as the


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Fig. 2. Numerical solution with 1600 grid points (x = 3.75 104 m).

corresponding quantities, respectively, whereas the units for L1 errors should take the unit of space into

account.

As explained before, large gradients (discontinuities) occurs in the solution around the second junction.

Around this point, our scheme only maintains first order accuracy, which is the case for most existing

hyperbolic solvers. Moreover, the L error is reached here. Different meshes yield different smearingeffects, and right at the spike, the L difference is not negligible. Away from this point, the L

difference is indeed very small. We also remark that the spike may be involved with a numerical

artifact similar to that in a slowly moving discontinuity [21]. The losing of accuracy around thediscontinuity makes the L1 convergence rate between 1 and 2. For a more comprehensive description,

see Fig. 4.

4.2. The effects of the mobility constant, channel length, lattice temperature, and Dirichlet boundary

conditions

It is known that the solution of the hydrodynamic model changes along with the physical parameters.

In particular, we display the numerical results for different (constant) n in Fig. 5. When decreasing


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Fig. 3. Numerical solutions with successive double grid points. Dotted: 100 grid points; dashed: 200 grid points; solid: 400 gridpoints; heavy-dotted: 800 grid points.

the mobility constant n, it turns out that there are not much change in the concentration n, nor in theelectric field Vx . A distinct feature lies in the gradual diminishing of the hump in the velocity profile.A flat interval appears within the channel for mobility n 0.04 m2/Vs. Moreover, the temperatureprofile gets flatter, yet keeps slightly higher than the lattice temperature T0= 300 K on the rightboundary.1

Secondly, if we simulate a device with shorter channel length, the basic picture is quite similar. For

instance, Fig. 6 depicts the solution at t=

15 ps for a device with a 0.2 m channel. Forn =

0.1 m2/Vs,

the velocity hump is more profound than in the previous case. An energy density peak appears after the

second junction. The heating effect in the second junction is even stronger, with highest temperature

about eight times the lattice temperature. As there is a longer n+ range, the temperature on the rightboundary almost cools down to the lattice temperature. The electric field intensifies and gets sharper

1 As the mobility n becomes smaller, also the parameter is becoming smaller (see (5) and (6)), whereas is fixed. This

will therefore correspond to the drift-diffusion limit as explained in Section 2.2. By Eq. (13), we expect that the temperature T

becomes closer to the lattice temperature. A numerical limit study will be presented in Section 4.3.


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Table 2

Numerical errors for different number of grid points

Grid number Concentrationn Momentum= nuL

error L1 error rate L

error L1 error rate

100 52.398 21.566 1.67 32.636 21.411 1.92

200 46.019 6.7560 1.31 14.902 5.6418 1.79

400 20.522 2.7227 0.50 7.7074 1.6296 2.02

800 19.797 1.9311 6.8576 0.4030

Grid number EnergyE Electric fieldVx

Lerror L1 error rate Lerror L1 error rate100 64.681 52.995 1.32 0.9787 0.5162 2.14

200 46.911 21.203 2.05 0.4713 0.1173 0.88

400 19.840 5.1181 0.89 0.1583 0.0637 0.76

800 16.906 2.7647 0.1044 0.0377

Grid number Velocityu TemperatureT

Lerror L1 error rate Lerror L1 error rate100 1.2121 0.2987 1.22 0.2420 0.2224 1.51

200 1.2257 0.1278 1.93 0.1675 0.0782 1.66

400 0.8105 0.0336 1.07 0.0614 0.0248 1.58

800 0.6609 0.0160 0.0376 0.0083

around the junctions. When the low field mobility decreases, the velocity hump again decreases in the

amplitude, and almost disappears at around n

0.025 m2/Vs. The energy density, total energy, and

temperature within the channel decrease considerably. This may be explained as above from the change

of the parameters and .

Thirdly, we simulate a Si-diode with 50 nm-channel. Under different model equations, this kind of

device has been studied in [2]. We consider a 250 nm long device with doping profile

C(x) =

2 1021 m3, x (0.1 m, 0.15 m),5 1023 m3, elsewhere. (42)

A voltage of 0.6 V is applied. The other physical constants remain the same as before. As the system

tends to equilibrium quicker at this length scale, we illustrate the solution at time t= 5 ps in Fig. 7.The concentration differs much from the doping profile, particularly by a shift to the right. This can beexplained by the positive mean velocity field. For all the mobility constants we have tried, no flat velocity

interval is observed. Moreover, the hottest position is located quite far away from the second junction,

moving gradually to the left when the mobility constant decreases. It is similar for the maximal velocity

point. Quite interestingly, we observe that the cooling zone moves to the right to such an extent that

it is relatively cool inside the channel.2 Though the right n+ region is twice the channel length, it is

2 This means that the total energy goes completely in the kinetic energy.


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Fig. 4. Convergence study of the scheme. Left: L errors; right: L1 errors. Legend: +: concentration n; : momentum ;: energy densityE ; : electric field Vx ; o: velocityu; : temperatureT.

not enough to cool the device down to the lattice temperature. These facts together yield energy density

profiles lifted on the right part.3

Now we investigate the effect of the lattice temperature. Solutions at T0 = 300 K, 200 K, 100 K, 80 K,60 K are displayed in Fig. 8. Although this is by no means a limit study for T0 0, the numericalresults suggest that n,u,E,w, and

Vx might converge to certain profiles, yet not clear for the scaled

temperatureT /T0. We observe a kink developing around x 0.35 m in the profiles ofu,E,w,and ahump inn. A second temperature peak lies at this point. ForT0 200 K, this temperature peak surpasses

the hot point near the second junction.4

3 Notice that the parameter becomes larger as the channel length decreases, i.e., the system is far away from the energy-

transport regime and convective effects are important.4 As T0 decreases, the parameter becomes smaller. However, we are not in the drift-diffusion regime, since becomes

larger asT0 0.


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Fig. 6. Numerical solutions for different low field mobility for channel length 0.2 m. Dotted: n= 0.1 m2/Vs, dash-dot:n = 0.06 m2/Vs, dashed: n = 0.03 m2/Vs, solid:n = 0.025 m2/Vs, heavy-dotted:n = 0.02 m2/Vs.

4.3. Approximation to the energy-transport model and the drift-diffusion model

As discussed in Section 2, the hydrodynamic model approaches to the energy-transport model inthe limit 0, 0 with the ratio / fixed. In our numerical study, we fix all other parametersas in Section 4.1 but n= n0, vs= vs0, with n0= 0.1 m2/Vs and vs0= 1.03 105 m/s, and is a parameter. Correspondingly, we have

=0,

=0 with 0

=n0

mkB T0/qL

=0.10315,

0 =

2mv2s0/3kB T0 = 2.0116.The limiting system as 0 is the energy-transport model (14)(17) in non-dimensionalised form.

In Fig. 11 we illustrate the approximation. Taking the same doping profile as before, we solve (21)

numerically for = 1, 0.5, 0.2, 0.1, and 0.05, respectively. The solutions are plotted at time T= 15 ps/.The numerical solutions seem to converge. A spike appears again in the profile of the rescaled velocityu/. We note that the time step size and the discrete BGK parameter are adjusted in the simulations, for

the sake of stability and computing load. Compared with the stationary solution of the energy-transport

model (solved with the numerical method of [18]) in Fig. 12, the numerical limit of the hydrodynamic


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Fig. 7. Numerical solutions for different low field mobility for channel length 50 nm. Dotted: n= 0.1 m2/Vs, dash-dot:n = 0.08 m2/Vs, dashed: n = 0.06 m2/Vs, solid:n = 0.04 m2/Vs, heavy-dotted:n = 0.02 m2/Vs.

model agrees reasonably well, particularly the density profile and the electric field. The deviations in the

rescaled velocity and temperature probably result from the numerical viscosity, which is not negligible

around the sharp gradient near the second junction.

On the other hand, if we keep all the parameters as in Section 4.1 but n= n0 with n0=0.1 m2/Vs, and is again a parameter. Correspondingly, we have = 0 and a fixed = 0, where

0and

0are defined above. The limiting system is the drift-diffusion model (18)(20).

The numerical results are displayed in Fig. 13. The temperature differs less from the lattice

temperature, when decreasing the mobility constant. This is clear from the asymptotic limit (see

Section 2.2). At = 0.00258, the maximal difference is about 1 K. The rescaled velocity converges to aprofile. As a result, the total energy density w converges to constant, as the main contribution comes from

the thermal energy. The energy density therefore converges to a profile similar to that of the concentration.

We solve the drift-diffusion model (18)(20) numerically employing a relaxation scheme, and observe

a nice agreement with the numerical limit of the hydrodynamic model (see Fig. 14) except some small

oscillations around the junctions, due to numerical viscosity.


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Fig. 8. Numerical solutions for different lattice temperatures. Dotted: T0 = 300 K, dash-dot: T0 = 200 K, dashed: T0 = 100 K,solid: T0 = 80 K, heavy-dotted: T0 = 60 K.

Fig. 9. Numerical solutions for different type of boundary conditions. Dotted: Dirichlet boundary conditions, solid: Neumann

boundary conditions.


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Fig. 12. Comparison between the stationary energy-transport solution and the hydrodynamic model with (,)=(0.00516, 0.1006). Solid: energy-transport model solution, heavy dotted: hydrodynamic model solution. (a) Electron density,

(b) rescaled velocity, (c) temperature, (d) electric field.

5. Conclusions

In this paper, we have applied a relaxation scheme to simulate the hydrodynamic model for semi-

conductor devices. We have demonstrated its accuracy and efficiency through numerical experiments.

With this scheme, we have further investigated interesting features of the system when varying different

parameters and the geometry. The dependence of the velocity overshoot on the low field mobility and

the channel length has been studied. It turns out that for sufficiently small mobilities, the hump near

the first junction disappears. Numerical limits, as well as a theoretical study by formal expansion, have

been performed, yielding the energy-transport model and the drift-diffusion model, for different limit

processes.

The relaxation approach has recently been applied in higher space dimension, and will be presented

in a forthcoming paper [26].


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Fig. 13. Approximation to the drift-diffusion model with = 2.0116. Dotted: = 0.10315, dash-dot: = 0.05158, dashed: = 0.01032, solid: = 0.00516, heavy-dotted: = 0.00258.

Acknowledgements

The authors acknowledge partial support from the Gerhard-Hess Program of the Deutsche Forschungs-

gemeinschaft, grant JU 359/3-1, and from the AFF Project of the University of Konstanz, grant 4/00. The

first author has been supported partially by the TMR Project Asymptotic Methods in Kinetic Theory,

grant ERB-FRMXCT-970157. The second author has been partially supported by Chinese Special Funds

for Major State Basic Research Project, NSFC under grant 10002002, and a DAAD grant.


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Fig. 14. Comparison between the drift-diffusion solution and the hydrodynamic model with (,) = (0.00258, 2.0116). Solid:drift-diffusion model solution, heavy dotted: hydrodynamic model solution. (a) Electron density, (b) rescaled velocity.

We would like to thank Professor Jos A. Carrillo for interesting discussions, as well as the help of

Professor Irene M. Gamba. We also thank Mr. Stefan Holst for providing the stationary energy-transportmodel solution for comparison.

References

[1] G. Ali, D. Bini, S. Rionero, Global existence and relaxation limit for smooth solutions to the EulerPoisson model for

semiconductors, SIAM J. Appl. Math. 32 (2000) 572587.

[2] M. Anile, J.A. Carrillo, I.M. Gamba, C.-W. Shu, Approximation of the BTE by a relaxation-time operator: Simulations for

a 50 nm-channel Si Diode, VLSI Des. 13 (2001) 349354.

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

Math. 61 (2000) 74101.

[4] D. Aregba-Driollet, R. Natalini, Discrete kinetic schemes for multidimensional system of conservation laws, SIAM J.

Numer. Anal. 37 (2000) 19732004.

[5] D. Aregba-Driollet, R. Natalini, S. Tang, Numerical study of diffusive BGK approximations for nonlinear systems of

degenerate parabolic equations, Preprint, 2000.

[6] G. Baccarani, M. Wordeman, An investigation on steady-state velocity overshoot in silicon, Solid-State Electr. 29 (1982)

970977.

[7] N. Ben Abdallah, P. Degond, On a hierarchy of macroscopic models for semiconductors, J. Math. Phys. 37 (1996) 3308

3333.

[8] N. Ben Abdallah, P. Degond, S. Gnieys, An energy-transport model for semiconductors derived from the Boltzmannequation, J. Statist. Phys. 84 (1996) 205231.

[9] K. Bltekjr, Transport equations for electrons in two-valley semiconductors, IEEE Trans. Electr. Dev. 17 (1970) 3847.

[10] R. Caflisch, S. Jin, G. Russo, Uniformly accurate schemes for hyperbolic systems with relaxation, SIAM J. Numer. Anal. 34

(1997) 246281.

[11] C. Cercignani, I. Gamba, J. Jerome, C.-W. Shu, Device Benchmark comparisons via kinetic, hydrodynamic, and high-field

models, Comput. Methods Appl. Mech. Engrg. 181 (2000) 381392.

[12] G.-Q. Chen, J. Jerome, B. Zhang, Existence and the singular relaxation limit for the inviscid hydrodynamic energy model,

in: J. Jerome (Ed.), Modelling and Computation for Application in Mathematics, Science, and Engineering, Clarendon

Press, Oxford, 1998.


24/24


[13] Z. Chen, B. Cockburn, J. Jerome, C.-W. Shu, Mixed-RKDG finite element methods for the 2-D hydrodynamic model for

semiconductor device simulation, VLSI Des. 3 (1995) 145158.

[14] P. Degond, A. Jngel, P. Pietra, Numerical discretization of energy-transport models for semiconductors with non-parabolic

band structure, SIAM J. Sci. Comput. 22 (2000) 9861007.

[15] M. Fischetti, S. Laux, Monte Carlo study of electron transport in silicon inversion layers, Phys. Rev. B 48 (1993) 2244

2274.

[16] C. Gardner, Numerical simulation of a steady-state electron shock wave in a submicron semiconductor device, IEEE Trans.

Electr. Dev. 38 (1991) 392398.

[17] I. Gasser, R. Natalini, The energy-transport and the drift-diffusion equations as relaxation limits of the hydrodynamic

model for semiconductors, Quart. Appl. Math. 57 (1999) 269282.

[18] S. Holst, A. Jngel, P. Pietra, A mixed finite-element discretization of the energy-transport equations for semiconductors,

SIAM J. Sci. Comp., in press.

[19] J. Jerome, C.-W. Shu, Energy models for one-carrier transport in semiconductor devices, in: W. Coughran, J. Colde,

P. Lloyd, J. White (Eds.), Semiconductors, Part II, in: IMA Vol. in Math. Appl., Vol. 59, Springer, New York, 1994,

pp. 185207.

[20] X. Jiang, A streamline-upwinding/PetrovGalerkin method for the hydrodynamic semiconductor device model, Math.

Models Meth. Appl. Sci. 5 (1995) 659681.

[21] S. Jin, J. Liu, The effects of numerical viscosities I: slowly moving shocks, J. Comput. Phys. 126 (1996) 373389.[22] S. Jin, Z.P. Xin, The relaxation schemes for systems of conservation laws in arbitrary space dimensions, Comm. Pure Appl.

Math. 48 (1995) 235278.

[23] S. Junca, M. Rascle, Relaxation of the isothermal EulerPoisson system to the drift-diffusion equations, Quart. Appl.

Math. 58 (2000) 511521.

[24] A. Jngel, Asymptotic limits in macroscopic plasma models, in: Proceedings of the IMA Workshop, Minneapolis, 2001,

to appear.

[25] A. Jngel, Quasi-Hydrodynamic Semiconductor Equations, in: Progress in Nonlinear Differential Equations, Birkhuser,

Basel, 2001.

[26] A. Jngel, S. Tang, Article in preparation, 2001.

[27] R. Levesque, Numerical Methods for Conservation Laws, Birkhuser, Basel, 1990.

[28] T.P. Liu, Hyperbolic conservation laws with relaxation, Comm. Math. Phys. 108 (1987) 153175.

[29] P.A. Markowich, C.A. Ringhofer, C. Schmeiser, Semiconductor Equations, Springer, Berlin, 1990.[30] R. Natalini, Recent mathematical results on hyperbolic relaxation problems, in: H. Freisthler (Ed.), Analysis of Systems

of Conservation Laws, Chapman and Hall, London, 1999, pp. 128198.

[31] R. Natalini, S. Tang, Discrete kinetic models for dynamic phase transitions, Comm. Appl. Nonlinear Anal. 7 (2000) 132.

[32] V. Romano, G. Russo, Numerical solution for hydrodynamic models of semiconductors, Preprint, Universit dellAquila,

Italy, 1997.

[33] W. van Roosbroeck, Theory of flow of electrons and holes in germanium and other semiconductors, Bell Syst. Techn. J. 29

(1950) 560607.

[34] M. Rudan, F. Odeh, Multi-dimensional discretization scheme for the hydrodynamic model of semiconductor devices,

COMPEL 5 (1986) 149183.

[35] D. Scharfetter, H. Gummel, Large signal analysis of a Silicon Read diode oscillator, IEEE Trans. Electr. Dev. ED-16 (1969)

6477.

[36] S. Selberherr, Analysis and Simulation of Semiconductor Devices, Springer, Berlin, 1984.

[37] C.-W. Shu, Essentially non-oscillatory and weighted essentially non-oscillatory schemes for hyperbolic conservation laws,

ICASE Report No. 97-65, NASA Langley Research Center, Hampton, VA, 1997.

[38] C.-W. Shu, S. Osher, Efficient implementation of essentially non-oscillatory shock capturing schemes II, J. Comput.

Phys. 83 (1989) 3278.

[39] R. Stratton, Diffusion of hot and cold electrons in semiconductor barriers, Phys. Rev. 126 (1962) 20022014.

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