A relaxation scheme for the hydrodynamic equations for semiconductors

Transcription

1 Applied Numerical Mathematics 43 (00) A relaxation scheme for the hydrodynamic equations for semiconductors Ansgar Jüngel a,, Shaoqiang Tang a,b a Fachbereich Mathematik und Statistik, Universität Konstanz, Konstanz, Germany b Department of Mechanics and Engineering Science, Peking University, Beijing , People s 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 boundary conditions 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. 00 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 [5]. 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 hydrodynamic equations [9,36]. 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. addresses: juengel@fmi.uni-konstanz.de (A. Jüngel), tangs@fmi.uni-konstanz.de (S. Tang) /0/$ see front matter 00 IMACS. Published by Elsevier Science B.V. All rights reserved. PII: S (01)0018-9

2 30 A. Jüngel, S. Tang / Applied Numerical Mathematics 43 (00) 9 5 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 Bløtekjær [9] and subsequently thoroughly investigated 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 the heat 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 Bløtekjær, Baccarani and Wordeman read as follows: n t 1 div J = 0, q J t 1 ( ) J J q div n ( m E t div J J q 3 ε s V = q(n C). qk B q (nt ) + m m n V = C J, () + 5 ) k B n q TJ + κ T (1) = J V + C E, (3) 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 k B, the effective electron mass m, and the permittivity constant ε s. The doping concentration characterizing the 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 J q n + 3 k BTn= 1 m n u + 3 k BTn, where the electron velocity u is defined by J = qnu. The momentum and energy relaxation terms, respectively, are C J = J τ p, C E = 1 ( m J τ w q n + 3 ) k B(T T 0 )n, where T 0 is the lattice temperature, ( ) T r T τ p = τ po, τ w = τ wo + 1 T 0 T + T 0 τ p are the momentum and energy relaxation times, respectively, with τ po = m µ n q, τ wo = 3µ nk B T 0, (5) qvs (4)

3 A. Jüngel, S. Tang / Applied Numerical Mathematics 43 (00) µ n is the low-field mobility and v s the saturating velocity. Finally, the heat conductivity is assumed to be ( ) 5 k ( ) κ = + c B µ n T r q nt, T 0 where c,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 for n, u and T at the domain boundary (see Section 4. for Dirichlet boundary conditions for n and 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 τ po is much smaller than the energy relaxation time constant τ wo, the hydrodynamic equations reduce in the relaxation-time limit formally to the energytransport equations (see Section ). This limit has been proved rigorously in [17] under the assumption of uniform L bounds. If the momentum and energy relaxation times are of the same order, but the mean free path is much smaller than the typical device length, we obtain formally the drift-diffusion equations from the hydrodynamic model. This limit has been shown rigorously for constant temperature in [3]. The asymptotic limit in the full model has been studied under some conditions in [1,1]. For an overview of these limits, see [4]. The main objectives of this paper are first to adopt the relaxation scheme [] 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 [8], and a relaxation scheme was proposed in []. 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 [] is a special case when we take only two velocities ±λ as moments. To solve this relaxation model, we may apply a splitting method, i.e., first solving an ODE step and then solving a linear convection 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 hyperbolic elliptic 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.

4 3 A. Jüngel, S. Tang / Applied Numerical Mathematics 43 (00) 9 5 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 [6]. 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 [0], finite element methods of the Runge Kutta discontinuous Galerkin scheme [13], finite difference methods of the ENO (essentially non-oscillatory) scheme [19,37], and UNO (uniformly non-oscillatory) schemes with the Nessyahu Tadmor method [3, 3] have been developed. This paper is organized as follows. In Section 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 a benchmark problem are presented in Section 4.. Scaling of the equations and asymptotic models In this section we scale Eqs. (1) (4) appropriately and derive the energy-transport and drift-diffusion models by means of formal asymptotic analysis..1. Scaling We introduce the thermal voltage U T = k B T 0 /q, the mean free path kb T 0 ι = τ po m, and the scaled Debye length λ = ε s U T qc m L, where C m is a typical doping concentration and L is a typical device length. Furthermore, we define the dimensionless parameters ε = ι L, δ = τpo. τ wo Then, with the scaling n C m n, t τ po t, C C m C, x Lx, V U T V, T T 0 T, J ( q U T C m τ po /Lm ) J, we obtain the scaled equations n t div J = 0, (6) (7)

5 ( J J J t ε div n A. Jüngel, S. Tang / Applied Numerical Mathematics 43 (00) ) (nt ) + n V = J T r, (8) J J div (ε + 5 ) TJ + κ T n ) 1 ( J ( J n + 3 ) nt ε t ( T = J V δ T δ T r n + 3 λ V = n C. We used the same notation for the scaled and unscaled variables. In Eq. (9) we have set ( ) 5 κ = + c T 1+r n... Asymptotic models ) n(t 1), (9) ε In order to obtain the energy-transport and drift-diffusion models, we rescale Eqs. (7) (9) by t t/ε : n t div J = 0, ( J J ε J t + ε div n J (ε n + 3 ) nt t (10) (11) ) (nt ) + n V = J T, (1) r J J div (ε n + 5 ) JT + κ T ( ) T 1 = J V (δ T δ T r J n + 3 δ ) n(t 1). (13) ε The energy-transport equations are obtained by assuming that ε 1, δ 1. The relation ε 1 holds if the kinetic energy associated with the velocity needed to cross the device in time τ po is very large compared with the thermal energy. The relation ε 1 also means that we study the system at large times of the order of 1/ε. Furthermore, it holds δ 1 if the kinetic energy associated with the saturating velocity is much smaller than the thermal energy. We formally perform the limit ε 0, δ 0 such that ε/δ τ 0, where τ 0 > 0 is some constant. The limit ε/δ τ 0 means that the velocity needed to cross the device in time τ po is assumed to be of the same order as the saturating velocity. We obtain the equations: n t div J = 0, J = T r( (nt ) n V ), ( ) ( 3 5 nt div TJ + κ T t λ V = n C, ) = J V 3 (14) (15) n(t 1), (16) τ(t) where τ(t)= τ 0 T/(T + 1). Notice that this energy-transport model is not of the general form derived in [14] expect for c = r = 0. (17)

6 34 A. Jüngel, S. Tang / Applied Numerical Mathematics 43 (00) 9 5 For the derivation of the drift-diffusion model, we fix the parameter δ>0 and let formally ε 0in Eqs. (11) (13) to obtain T = 1and n t div J = 0, (18) J = n n V, (19) λ V = n C. (0) 3. Numerical scheme In this section, we shall put the model into a more concise form. Then we shall describe our numerical scheme in three parts, namely, the overall second-order splitting method, the relaxation scheme for the convection, and the treatment of the boundaries and diffusion Reformulation of the model We recast the hydrodynamic model in one space dimension as follows. U t + A(U) x = ( B(U,U x ) ) + S(U), (1) x coupled with the Poisson equation ε s V xx = q ( n C(x) ). () The vector quantities are [ ] ( σ n σ U = σ, A(U )= 3 n + E ) m, E 5σE 3n m σ 3 3n 0 [ ] 0 σ + nqv x B(U,U x ) = 0, S(U )= τ p m κt x qσv x E E. 0 Here σ = nu = J/q, E = 1 m nu + 3 nk BT, and E 0 = 3 nk BT 0 is the rest energy density. We recall the relaxation coefficients and heat diffusion coefficient ( ) τ p = m µ n T r, (3) q T 0 τ w = 1 τ p + 3µ nk B T 0 T qvs (T + T 0), (4) ( ) 5 nk ( ) κ = + c B µ n T T r, (5) q T 0 where we use c = r = 1. τ w

7 A. Jüngel, S. Tang / Applied Numerical Mathematics 43 (00) Table 1 Physical parameters Parameter Physical meaning Numerical value q elementary charge C m effective electron mass kg ε s permittivity constant F/m µ n low field mobility constant 0.1m /Vs k B Boltzmann constant J/K T 0 lattice temperature 300 K n i intrinsic electron concentration m 3 v s saturating velocity 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), (6) u(x, 0) = 0, (7) T(x,0) = T 0, (8) n x (0,t)= n x (L, t) = 0, (9) u x (0,t)= u x (L, t) = 0, (30) T x (0,t)= T x (L, t) = 0. (31) Taking into account the conservation of electrons, we notice that the reflecting boundary condition on n implies the fixed boundary condition n(0,t)= C(0), n(l, t) = C(L). For the electric field with an applied voltage V b, the boundary conditions are V(0) = T ( ) 0 n(0,t) q ln, (3) n i V(L)= T ( ) 0 n(l, t) q ln + V b. (33) n i 3.. Second-order Runge Kutta splitting scheme The system (1) can be solved by a splitting method. Given some data at the kth time step t = t k, one first solves an ODE step, namely the initial-value problem with electric field Vx k corresponding to the electron concentration n(x, t k ), U t = ( B(U,U x ) ) x + S(U), U( x,t k) = U k (x). (34) Denote the solution at t = t k + t as U k+1/ = R(U k,v k x, t). As we shall explain in a later subsection, the diffusion term (B(U, U x )) x is expressed explicitly by quantities at neighboring grid points. This makes (34) an ODE system.

8 36 A. Jüngel, S. Tang / Applied Numerical Mathematics 43 (00) 9 5 Secondly, one solves a convection step, namely the initial-boundary-value problem for the homogeneous system U t + A(U) x = 0, U ( x,t k) = U k+1/, U x (0,t)= U x (L, t) = 0, (35) by a hyperbolic problem solver. The solution after one time step is denoted as U k+1 = W(U k+1/,vx k, t) (see Section 3.3 for an explicit formula). Finally the electric field is updated from the Poisson equation () with electron concentration n k+1 = first component of U k+1, denoted as Vx k+1 = P(n k+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 data U 0 = U(x, 0) and update the electric field Vx 0 = P(n0 ). () For k = 0,...,k T 1, compute U k+1,vx k+1 as follows: (a) Set (U [0],V x [0]) = (U k,vx k). (b) Compute (U [1],V x [1]) and U [] as follows: U [1] = ( n [1],σ [1],E [1]) = U [0] + [ W ( R ( U [0],V x [0], t),v x [0], t) U [0]] /, V x [1] = P ( n [1]), U [] = W ( R ( U [1],V x [1], t),v x [1], t). (c) Set U k+1 = (U [1] + U [] )/, Vx k+1 = P(n k+1 ). We note that this is an explicit splitting, hence one should take t ( x) for stability. It can be improved by applying implicit techniques 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 ε + Yx ε = 0, (36) Yt ε + λ Ux ε = A(U ε ) Y ε, (37) ε where ε is a small parameter and λ is a constant, larger than the maximum wave speed of the original system (sub-characteristic condition) for stability. As ε approaches towards 0, formally Y ε approaches towards A(U ε ). In turn the first equation approximates (35). Rigorous results are obtained, e.g., in [8, 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 may use 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., [7]). In the ODE step

9 A. Jüngel, S. Tang / Applied Numerical Mathematics 43 (00) (for (36), (37) without the terms Yx ε and λ Ux ε), on the other hand, it is observed that U ε keeps unchanged, therefore the Y ε -equation can be solved explicitly as ( Y ε (t) = Y ε (0) exp t ) + A ( U ε)[ ( 1 exp t )]. ε ε A relaxed scheme is obtained, at this step, when putting ε to zero, or projecting Y ε to the flux function (Maxwellian) A(U ε ). Note that λ then becomes the only numerical parameter introduced by the relaxation method. We introduce Ui k = U k (x i ) and ζ = λ t/ x. The explicit expression of the relaxed scheme for one-step marching U k+1 = W(U k+1/,vx k, t)is with U k+1 i M ± i = (1 ζ)u k+1/ i + ζ ( M + i 1 + ) ζ(1 ζ) M i+1 [ minmod ( M + i+1 M+ i,m + i M + ) ( i 1 minmod M + i M + i 1,M+ i 1 ) M+ i + minmod ( M i+ M i+1,m i+1 ) ( M i minmod M i+1 M i,m i M )] i 1 = U k+1/ i ± A(U k+1/ i 3.4. Source terms and boundaries )/λ. In the ODE step, the first equation in (34) implies that n remains unchanged, and consequently so for the electric field. However, the relaxation times τ p, τ w and the heat diffusion parameter κ are expressed in terms of the temperature T, and therefore of the energy E. The ODE system cannot be solved explicitly in general. It takes the following form: σ t = σ τ p + nqv x m, (38) E t = qσv x E E 0 + (κt x ) x. (39) τ w We may apply a Runge Kutta method to integrate this system. As a remark, since t ( x),even the forward Euler method is enough to maintain the accuracy. If an implicit second order splitting is employed, one may use a second-order integrator for this ODE step to match the accuracy. It is worth mentioning the treatment of the heat diffusion term. In our code, it is approximated in conservative form with second-order accuracy by 1 [ (κt x ) x (κi+1 + κ ( x) i )(T i+1 T i ) (κ i + κ i 1 )(T i T i 1 ) ], (40) where κ i is the local heat diffusion parameter in (5) with n i, T i plugged in. The Poisson equation () is discretized by the standard central difference scheme. The resulting tridiagonal algebraic system can be readily solved. In all the discretizations, reflecting boundary conditions are used for n, u, T,andV x, i.e., with reflectingly-valued ghost-points once needed. This is consistent with the boundary conditions in the continuous form (6) (31).

10 38 A. Jüngel, S. Tang / Applied Numerical Mathematics 43 (00) 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 Neumann boundary conditions. Finally we shall explore the numerical limits of the hydrodynamic model towards the dimensionalized energy-transport model and drift-diffusion model, respectively Benchmarks on the numerical scheme We make numerical tests on an n + nn + ballistic silicon diode with an applied voltage of 1.5 V. The semiconductor domain is described by the interval [0,L] with L = 0.6 µm. The channel length is 0.4 µm. The doping profile is: { 10 C(x) = 1 m 3, x (0.1 µm, 0.5 µm), m 3 (41), elsewhere. We make a convergence study with different meshes. In all these tests, the numerical relaxation parameter in the relaxation scheme is set as λ =.5. A computation shows that this value satisfies the subcharacteristic condition. We perform simulations with successively double space grid numbers, i.e., with 100 grid points ( x = µm), 00 grid points ( x = µm), 400 grid points ( x = µm), 800 grid points ( x = µm), and 1600 grid points ( x = µm), respectively. The numerical solution with 00 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. They are 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 10 1 µm for the space variable x, 10 1 m 3 for the concentration n, 10 5 m/s for the velocity u, 10 1 Jm 3 for the energy density E, 10 1 ev = Jforthetotal energy w = E/n, 10 3 K for the temperature T, and 10 6 V/m for the electric field V x. This scale of units will be used throughout the figures hereafter. We depict the stationary solution at t = 15 ps in Fig.. In the velocity profile, it is observed that besides 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 may give different profiles [19]. The low field mobility µ n also makes big differences, e.g., as reported for a GaAs diode [11]. We shall describe the numerical simulations with different constants µ n in the next subsection. We also observe a cooling zone around the first junction, and a heating zone around the second with highest temperature about 5 times the lattice temperature. We now describe the numerical convergence study. The solutions with different meshes are shown in 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. For finer 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

11 A. Jüngel, S. Tang / Applied Numerical Mathematics 43 (00) Fig. 1. Numerical solution with 00 grid points ( x = µm), (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 in n 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 L 1 errors in Table. The numerical convergence rates for the L 1 error are also computed. The units of the L errors are the same as the

12 40 A. Jüngel, S. Tang / Applied Numerical Mathematics 43 (00) 9 5 Fig.. Numerical solution with 1600 grid points ( x = µm). corresponding quantities, respectively, whereas the units for L 1 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 smearing effects, 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 [1]. The losing of accuracy around the discontinuity makes the L 1 convergence rate between 1 and. For a more comprehensive description, see Fig 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) µ n in Fig. 5. When decreasing

13 A. Jüngel, S. Tang / Applied Numerical Mathematics 43 (00) Fig. 3. Numerical solutions with successive double grid points. Dotted: 100 grid points; dashed: 00 grid points; solid: 400 grid points; heavy-dotted: 800 grid points. the mobility constant µ n, it turns out that there are not much change in the concentration n, nor in the electric field V x. A distinct feature lies in the gradual diminishing of the hump in the velocity profile. A flat interval appears within the channel for mobility µ n 0.04 m /Vs. Moreover, the temperature profile gets flatter, yet keeps slightly higher than the lattice temperature T 0 = 300 K on the right boundary. 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. µm channel. For µ n = 0.1m /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 right boundary almost cools down to the lattice temperature. The electric field intensifies and gets sharper 1 As the mobility µ n becomes smaller, also the parameter ε is becoming smaller (see (5) and (6)), whereas δ is fixed. This will therefore correspond to the drift-diffusion limit as explained in Section.. 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.

14 4 A. Jüngel, S. Tang / Applied Numerical Mathematics 43 (00) 9 5 Table Numerical errors for different number of grid points Grid number Concentration n Momentum σ = nu L error L 1 error rate L error L 1 error rate Grid number Energy E Electric field V x L error L 1 error rate L error L 1 error rate Grid number Velocity u Temperature T L error L 1 error rate L error L 1 error rate around the junctions. When the low field mobility decreases, the velocity hump again decreases in the amplitude, and almost disappears at around µ n 0.05 m /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 ε and δ. Thirdly, we simulate a Si-diode with 50 nm-channel. Under different model equations, this kind of device has been studied in []. We consider a 50 nm long device with doping profile { C(x) = 10 1 m 3, x (0.1 µm, 0.15 µm), m 3, elsewhere. (4) 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 = 5psinFig.7. The concentration differs much from the doping profile, particularly by a shift to the right. This can be explained 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. Though the right n + region is twice the channel length, it is This means that the total energy goes completely in the kinetic energy.

15 A. Jüngel, S. Tang / Applied Numerical Mathematics 43 (00) Fig. 4. Convergence study of the scheme. Left: L errors; right: L 1 errors. Legend: +: concentration n; : momentumσ ; : energy density E; : electric field V x ; o: velocity u; : temperature T. 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 T 0 = 300 K, 00 K, 100 K, 80 K, 60 K are displayed in Fig. 8. Although this is by no means a limit study for T 0 0, the numerical results suggest that n, u, E, w, and V x might converge to certain profiles, yet not clear for the scaled temperature T/T 0. We observe a kink developing around x 0.35 µm in the profiles of u, E, w, and a hump in n. A second temperature peak lies at this point. For T 0 00 K, this temperature peak surpasses the hot point near the second junction. 4 3 Notice that the parameter ε becomes larger as the channel length decreases, i.e., the system is far away from the energytransport regime and convective effects are important. 4 As T 0 decreases, the parameter ε becomes smaller. However, we are not in the drift-diffusion regime, since δ becomes larger as T 0 0.

16 44 A. Jüngel, S. Tang / Applied Numerical Mathematics 43 (00) 9 5 Fig. 5. Numerical solutions for different low field mobility. Dotted: µ n = 0.1 m /Vs, dash-dot: µ n = 0.08 m /Vs, dashed: µ n = 0.06 m /Vs, solid: µ n = 0.05 m /Vs, heavy-dotted: µ n = 0.04 m /Vs. Finally, instead of the Neumann boundary conditions (9) and (31) on n and T, we impose fixed boundary conditions n(0,t)= C(0), n(l, t) = C(L), T(0,t)= T 0, T(L,t)= T 0. As explained before, the Neumann boundary conditions on n are equivalent to the Dirichlet boundary conditions. However, this is not the case for T. In fact, we observed before an elevation of temperature on the right boundary from the lattice temperature T 0. Quite naturally, the Dirichlet boundary conditions lays a stronger effect when the solution under Neumann boundary conditions differs more from T 0 at the boundary. In Fig. 9, we compare the solutions with the same parameters as those in Section 4.1. The deviation is barely observable. However, for the 50 nm-channel, the difference is fairly big. See Fig. 10. (43) (44) (45) (46)

17 A. Jüngel, S. Tang / Applied Numerical Mathematics 43 (00) Fig. 6. Numerical solutions for different low field mobility for channel length 0. µm. Dotted: µ n = 0.1 m /Vs, dash-dot: µ n = 0.06 m /Vs, dashed: µ n = 0.03 m /Vs, solid: µ n = 0.05 m /Vs, heavy-dotted: µ n = 0.0 m /Vs Approximation to the energy-transport model and the drift-diffusion model As discussed in Section, the hydrodynamic model approaches to the energy-transport model in the limit ε 0, δ 0 with the ratio ε/δ fixed. In our numerical study, we fix all other parameters as in Section 4.1 but µ n = αµ n0,v s = αv s0, with µ n0 = 0.1 m /Vs and v s0 = m/s, and α is a parameter. Correspondingly, we have ε = αε 0,δ = αδ 0 with ε 0 = µ n0 m k B T 0 /ql = , δ 0 = m vs0 /3k BT 0 = The limiting system as α 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 (1) numerically for α = 1, 0.5, 0., 0.1, and 0.05, respectively. The solutions are plotted at time T = 15 ps/α. The numerical solutions seem to converge. A spike appears again in the profile of the rescaled velocity u/α. We note that the time step size and the discrete BGK parameter λ 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. 1, the numerical limit of the hydrodynamic

18 46 A. Jüngel, S. Tang / Applied Numerical Mathematics 43 (00) 9 5 Fig. 7. Numerical solutions for different low field mobility for channel length 50 nm. Dotted: µ n = 0.1 m /Vs, dash-dot: µ n = 0.08 m /Vs, dashed: µ n = 0.06 m /Vs, solid: µ n = 0.04 m /Vs, heavy-dotted: µ n = 0.0 m /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 µ n = αµ n0 with µ n0 = 0.1 m /Vs, and α is again a parameter. Correspondingly, we have ε = αε 0 and a fixed δ = δ 0,where ε 0 and δ 0 are defined above. The limiting system is the drift-diffusion model (18) (0). 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.). At ε = , the maximal difference is about 1 K. The rescaled velocity converges to a profile. 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) (0) 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.

19 A. Jüngel, S. Tang / Applied Numerical Mathematics 43 (00) Fig. 8. Numerical solutions for different lattice temperatures. Dotted: T 0 = 300 K, dash-dot: T 0 = 00 K, dashed: T 0 = 100 K, solid: T 0 = 80 K, heavy-dotted: T 0 = 60 K. Fig. 9. Numerical solutions for different type of boundary conditions. Dotted: Dirichlet boundary conditions, solid: Neumann boundary conditions.

20 48 A. Jüngel, S. Tang / Applied Numerical Mathematics 43 (00) 9 5 Fig. 10. Numerical solutions for different type of boundary conditions (50 nm-channel). Dotted: Dirichlet boundary conditions, solid: Neumann boundary conditions. Fig. 11. Approximation to the energy-transport model. Dotted: (ε, δ) = ( ,.0116), dash-dot: (ε, δ) = ( , ), dashed: (ε, δ) = (0.0063, 0.403), solid: (ε, δ) = (0.0103, 0.01), heavy-dotted: (ε, δ) = ( , ).

21 A. Jüngel, S. Tang / Applied Numerical Mathematics 43 (00) Fig. 1. Comparison between the stationary energy-transport solution and the hydrodynamic model with (ε, δ) = ( , ). 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 semiconductor 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 [6].

22 50 A. Jüngel, S. Tang / Applied Numerical Mathematics 43 (00) 9 5 Fig. 13. Approximation to the drift-diffusion model with δ = Dotted: ε = , dash-dot: ε = , dashed: ε = , solid: ε = , heavy-dotted: ε = Acknowledgements The authors acknowledge partial support from the Gerhard-Hess Program of the Deutsche Forschungsgemeinschaft, 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 The second author has been partially supported by Chinese Special Funds for Major State Basic Research Project, NSFC under grant , and a DAAD grant.

23 A. Jüngel, S. Tang / Applied Numerical Mathematics 43 (00) Fig. 14. Comparison between the drift-diffusion solution and the hydrodynamic model with (ε, δ) = (0.0058,.0116). Solid: drift-diffusion model solution, heavy dotted: hydrodynamic model solution. (a) Electron density, (b) rescaled velocity. We would like to thank Professor José 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-transport model solution for comparison. References [1] G. Ali, D. Bini, S. Rionero, Global existence and relaxation limit for smooth solutions to the Euler Poisson model for semiconductors, SIAM J. Appl. Math. 3 (000) [] 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 (001) [3] M. Anile, V. Romano, G. Russo, Extended hydrodynamic model of carrier transport in semiconductors, SIAM J. Appl. Math. 61 (000) [4] D. Aregba-Driollet, R. Natalini, Discrete kinetic schemes for multidimensional system of conservation laws, SIAM J. Numer. Anal. 37 (000) [5] D. Aregba-Driollet, R. Natalini, S. Tang, Numerical study of diffusive BGK approximations for nonlinear systems of degenerate parabolic equations, Preprint, 000. [6] G. Baccarani, M. Wordeman, An investigation on steady-state velocity overshoot in silicon, Solid-State Electr. 9 (198) [7] N. Ben Abdallah, P. Degond, On a hierarchy of macroscopic models for semiconductors, J. Math. Phys. 37 (1996) [8] N. Ben Abdallah, P. Degond, S. Génieys, An energy-transport model for semiconductors derived from the Boltzmann equation, J. Statist. Phys. 84 (1996) [9] K. Bløtekjær, Transport equations for electrons in two-valley semiconductors, IEEE Trans. Electr. Dev. 17 (1970) [10] R. Caflisch, S. Jin, G. Russo, Uniformly accurate schemes for hyperbolic systems with relaxation, SIAM J. Numer. Anal. 34 (1997) [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 (000) [1] 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 5 A. Jüngel, S. Tang / Applied Numerical Mathematics 43 (00) 9 5 [13] Z. Chen, B. Cockburn, J. Jerome, C.-W. Shu, Mixed-RKDG finite element methods for the -D hydrodynamic model for semiconductor device simulation, VLSI Des. 3 (1995) [14] P. Degond, A. Jüngel, P. Pietra, Numerical discretization of energy-transport models for semiconductors with non-parabolic band structure, SIAM J. Sci. Comput. (000) [15] M. Fischetti, S. Laux, Monte Carlo study of electron transport in silicon inversion layers, Phys. Rev. B 48 (1993) [16] C. Gardner, Numerical simulation of a steady-state electron shock wave in a submicron semiconductor device, IEEE Trans. Electr. Dev. 38 (1991) [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) [18] S. Holst, A. Jüngel, 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 [0] X. Jiang, A streamline-upwinding/petrov Galerkin method for the hydrodynamic semiconductor device model, Math. Models Meth. Appl. Sci. 5 (1995) [1] S. Jin, J. Liu, The effects of numerical viscosities I: slowly moving shocks, J. Comput. Phys. 16 (1996) [] S. Jin, Z.P. Xin, The relaxation schemes for systems of conservation laws in arbitrary space dimensions, Comm. Pure Appl. Math. 48 (1995) [3] S. Junca, M. Rascle, Relaxation of the isothermal Euler Poisson system to the drift-diffusion equations, Quart. Appl. Math. 58 (000) [4] A. Jüngel, Asymptotic limits in macroscopic plasma models, in: Proceedings of the IMA Workshop, Minneapolis, 001, to appear. [5] A. Jüngel, Quasi-Hydrodynamic Semiconductor Equations, in: Progress in Nonlinear Differential Equations, Birkhäuser, Basel, 001. [6] A. Jüngel, S. Tang, Article in preparation, 001. [7] R. Levesque, Numerical Methods for Conservation Laws, Birkhäuser, Basel, [8] T.P. Liu, Hyperbolic conservation laws with relaxation, Comm. Math. Phys. 108 (1987) [9] P.A. Markowich, C.A. Ringhofer, C. Schmeiser, Semiconductor Equations, Springer, Berlin, [30] R. Natalini, Recent mathematical results on hyperbolic relaxation problems, in: H. Freistühler (Ed.), Analysis of Systems of Conservation Laws, Chapman and Hall, London, 1999, pp [31] R. Natalini, S. Tang, Discrete kinetic models for dynamic phase transitions, Comm. Appl. Nonlinear Anal. 7 (000) 1 3. [3] V. Romano, G. Russo, Numerical solution for hydrodynamic models of semiconductors, Preprint, Università dell Aquila, Italy, [33] W. van Roosbroeck, Theory of flow of electrons and holes in germanium and other semiconductors, Bell Syst. Techn. J. 9 (1950) [34] M. Rudan, F. Odeh, Multi-dimensional discretization scheme for the hydrodynamic model of semiconductor devices, COMPEL 5 (1986) [35] D. Scharfetter, H. Gummel, Large signal analysis of a Silicon Read diode oscillator, IEEE Trans. Electr. Dev. ED-16 (1969) [36] S. Selberherr, Analysis and Simulation of Semiconductor Devices, Springer, Berlin, [37] C.-W. Shu, Essentially non-oscillatory and weighted essentially non-oscillatory schemes for hyperbolic conservation laws, ICASE Report No , NASA Langley Research Center, Hampton, VA, [38] C.-W. Shu, S. Osher, Efficient implementation of essentially non-oscillatory shock capturing schemes II, J. Comput. Phys. 83 (1989) [39] R. Stratton, Diffusion of hot and cold electrons in semiconductor barriers, Phys. Rev. 16 (196)