An Asymptotic-Preserving Scheme for the Semiconductor Boltzmann Equation toward the Energy-Transport Limit

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1 An Asymptotic-Preserving Scheme for the Semiconductor Boltzmann Equation toward the Energy-Transport Limit Jingwei Hu Li Wang January, 4 Abstract We design an asymptotic-preserving scheme for the semiconductor Boltzmann equation which leads to an energy-transport system for ectron mass and internal energy as mean fr path goes to zero. As opposed to the classical drift-diffusion limit where the stiff collisions are all in one scale, new difficulties arise in the two-scale stiff collision terms because the simple BGK penalization [5] fails to drive the solution to the correct limit. We propose to set up a spatially dependent threshold on the penalization of the stiffer collision operator such that the evolution of the solution resembles a Hilbert epansion at the continuous lev. Formal asymptotic analysis and numerical results confirm the efficiency and accuracy of our scheme. Keywords: semiconductor Boltzmann equation, energy-transport system, asymptotic-preserving scheme, fast spectral method. Introduction The semiconductor Boltzmann equation describes the transport of charge carriers in semiconductor devices. By incorporating the quantum mechanical effects semiclassically, it provides accurate description of the physics at the kinetic lev [3, 8, 7]. A non-dimensional form of this equation usually contains small parameters such as the mean fr path or time, which pose tremendous computational challenge since one has to numerically resolve the small scales. To save the computational cost, in the past decades various macroscopic mods were derived from the Boltzmann equation based on different dominating effects. One of the widy-used mod is the drift-diffusion equation a mass continuity equation for ectrons or holes [34, 7]. Ideally, if the parameters in the kinetic equation are uniformly small in the entire domain of interest, then it suffices to solve the macroscopic mods [9, 9, 35]. In practical applications, however, the validity of these mods may break down in part of the domain and one has to resort to kinetic equations [5, 6]. To handle such multiscale phenomena, a typical approach is to use domain decomposition [4, 3], but the task of setting up a good interface condition coupling the two mods at different scales can be highly non-trivial. An alternative approach sks a unified scheme for the kinetic equation such that when the parameter goes to zero it automatically becomes a macroscopic solver. This design concept leads to the asymptotic-preserving AP scheme [], first proposed by S. Jin for transport equations in diffusive regimes []. In the semiconductor framework, the first AP scheme was introduced in [3] for the Boltzmann equation with an anisotropic collision term. The scheme was further improved in []. Recently a high-order scheme was constructed in [3], which raed the strict parabolic stability condition to a hyperbolic one. All these works deal with a single, linear collision operator with smoothed kern which uniquy defines an equilibrium state. As a result, the corresponding macroscopic equation is in the form of a drift-diffusion equation. Although this equation gives satisfactory simulation results for semiconductor devices on the micrometer scale, it is not able to capture the hot-ectron effects in submicron devices [8]. Some works in The Institute for Computational Enginring and Sciences ICES, The University of Teas at Austin, East 4th St, Stop C, Austin, TX 787 hu@ices.uteas.edu. Department of Mathematics, University of California, Los Anges, 5 Portola Plaza, Los Anges, CA 995 liwang@math.ucla.edu.

2 the high fid scaling considered this effect, but it only makes sense for the situation where the fid effect is strong enough to balance the collision [5, 7]. In this work, we consider a more realistic semiconductor Boltzmann equation [, ]. By assuming the astic collision as dominant, and ectron-ectron corration as sub-dominant effects, one can pass on the asymptotic limit to obtain an energy-transport ET mod, which is a system of conservation laws for ectron mass and internal energy with flues defined through a constitutive ration [6]. To design an AP scheme for such kinetic equation, we face two-fold challenge:. the convection terms are stiff;. two stiff collision terms live on different scales. The convection terms can be treated by an even-odd decomposition as in [4, 3]. For the collision terms, due to their complicated forms, we choose to penalize them with a suitable BGK operator [5]. However, unlike the usual collision operator with smoothed kern, the leading astic operator has non-unique null space. Only when the ectron-ectron operator at net lev takes into effect, the solution can be eventually driven to a fied equilibrium a Fermi-Dirac distribution. A closer eamination of the asymptotic behavior of the numerical solution reveals that the penalization should be performed wisy, otherwise it won t capture the correct limit. To this end, we propose a thresholded penalization scheme. Simply speaking, when a certain threshold is satisfied, we turn off the leading order mechanism and move to the net order, which in some sense resembles the Hilbert epansion at the continuous lev. We will show that this new scheme, under certain assumptions, satisfies the following four properties required in a standard AP scheme: let denote the small parameter;,, and k be the time step and mesh size in spatial and wave vector momentum domain, then. for fied, it is consistent to the kinetic equation when,, k ;. for fied,, k, it becomes a discretization to the limiting ET system when ; 3. it is uniformly stable for a wide range of, from = O to ; 4. implicit terms can be implemented eplicitly fr of Newton-type solvers. The rest of the paper is organized as follows. In the net section we give a brief review of the semiconductor Boltzmann equation and the derivation of its ET limit in diffusive regimes. Section 3 is devoted to a detailed description of our AP schemes. We first consider the spatially homogeneous case with an emphasis on the two-scale collision terms, and then include the spatial dependence to treat the full problem. In either case, the asymptotic property of the numerical solution is carefully analyzed. In Section 4 we present several numerical eamples to illustrate the efficiency, accuracy, and AP properties of the schemes. Finally, the paper is concluded in Section 5. The semiconductor Boltzmann equation and its energy-transport limit The Boltzmann transport equation that describes the evolution of ectrons in the conduction band of a semiconductor reads [3, 8, 7] t f + kk f + q V, t k f = Qf, Ω R d, k B R d, d =, 3,. where f, k, t is the ectron distribution function of position, wave vector k, and time t. is the reduced Planck constant, and q is the positive ementary charge. The first Brillouin zone B is the primitive cl in the reciprocal lattice of the crystal. For simplicity, we will restrict to the parabolic band approimation, where B can be etended to the whole space B = R d, and the energy-band diagram k is given by k = m k, with m being the effective mass of ectrons. In principle, the ectrostatic potential V, t is produced sf-consistently by the ectron density with a fied ion background of doping profile h through the Poisson equation: ɛ ɛ r V, t = qρ, t h,.

3 where g ρ, t := f, k, t R π d dk d is the ectron density the spin degeneracy g = s +, with s = / being the spin of ectrons. ɛ and ɛ r are the vacuum and the rative material permittivities. The doping profile h takes into account the impurities due to acceptor and donor ions in the semiconductor device. The collision operator Q mods thr different effects: Q = Q imp + Q ph + Q, where Q imp and Q ph account for the interactions betwn ectrons and the lattice defects caused by ionized impurities and crystal vibrations also called phonons; Q describes the corrations betwn ectrons themsves. We will specify their forms later.. The nondimensionalized semiconductor Boltzmann equation In this paper, we are interested in a high energy scale [, ] at which the rative energy gain or loss of ectron energy during a phonon collision is very small, then it is valid to consider an astic approimation of the ectron-phonon interactions Q ph. Combining astic collisions Q imp, which is astic by nature, and the astic part of Q ph and performing careful nondimensionalization on., one arrives at the following scaled semiconductor Boltzmann equation in diffusive regime []: t f + k f + V k f = Q f + Q f + Q in ph f,.3 where the parameter can be interpreted as the scaled mean-fr path for astic collisions. The energy-band diagram k is now simply k = k..4 The astic collision Q and the ectron-ectron collision Q are given by the eact form of Q in ph will not be nded in the following discussion and is thus omitted: Q fk = Φ k, k δ f f dk,.5 R d Q fk = Φ k, k, k, k δ + δk + k k k R [ 3d ] f f ηf ηf ff ηf ηf dk dk dk,.6 where δ is the Dirac measure,,, f, f, are short notations for k, k, f, k, t, f, k, t,.the scattering matrices Φ k, k is symmetric in k and k : Φ k, k = Φ k, k; Φ k, k, k, k is pairwise symmetric in four variables: Φ k, k, k, k = Φ k, k, k, k = Φ k, k, k, k. They are determined by the underlying interaction laws. The Poisson equation. becomes where C is the square of the scaled Debye length, and ρ, t = f, k, t dk. R d. The energy-transport limit C ɛ r V, t = ρ, t h,.7 This subsection is devoted to a formal derivation of the asymptotic limit of.3 as. Our approach, following that of [], is a combination of the Hilbert epansion and the moment method which mainly ry on the properties of the collision operators Q and Q. To this end, we list them at first. They will also be useful in designing numerical schemes. 3

4 Proposition. []. For any regular test function gk, R d Q fg dk = In particular, for any gk, R d Φ k, k δ f f g g dkdk. R d Q fg dk =.. Q f is a sf-adjoint, non-positive operator on L R d. 3. The null space of Q f is given by 4. The orthogonal complement of N Q is N Q = {gk N Q = {fk, f}. S gk dn k =, R}, where the integral is defined through the coarea formula [4]: for any regular functions f and k : R d R, it holds fk dk = fk dn k d..8 R d R S Here S = {k R d, k = } denotes the surface of constant energy, dn k is the coarea measure, and N := S dn k is the energy density-of-states. Under the parabolic band approimation.4,.8 is just a spherical transformation: fk dk = f k σ k d dσ d, R d S d and N = S d k d dσ = ms d k d. 5. The range of Q is RQ = N Q. The operator is invertible as an operator from N Q onto RQ = N Q. Its inverse is denoted by Q. 6. For any ψk, we have Q fψ = ψq f and Q fψ = ψq f. Proposition. []. For any regular test function gk, Q fg dk = Φ k, k, k, k R 4 δ + δk + k k k d R [ 4d ] f f ηf ηf ff ηf ηf g + g g g dkdk dk dk. In particular, we have the conservation of mass and energy Q f dk = Q f dk =. R d R d. H-theorem: let Hf = lnf/ ηf, then R d Q fhf dk, and if f = fk, R d Q fhf dk = f = Mk Q f =, where Mk =.9 η z e k T + is the Fermi-Dirac distribution function [33]. The variables z and T are the fugacity and temperature. Alternativy, M can be defined in terms of the chemical potential µ = T ln z and T. 4

5 We also nd the properties of the operator Q, an energy space counterpart of Q : for any F k, Q F := Q F k dn k. S Proposition 3. []. Conservation of mass and energy R Q F d = R Q F d =.. H-theorem: let HF = lnf/ ηf, then R Q F HF d, and Q F HF d = F = M Q F =. R Now that the mathematical priminaries are set up, we are ready to derive the macroscopic limit. The main result is summarized in the following theorem. Theorem 4. [] In equation.3, when, the solution f formally tends to a Fermi-Dirac distribution function.9, with the position and time dependent fugacity z, t and temperature T, t satisfying the so-called Energy-Transport ET mod: t ρ ρe + j ρ j E where the density ρ and energy E are defined as ρz, T = f dk = R d M dk, R d the flues j ρ and j E are given by jρ z, T j E z, T with the diffusion matrices and the energy raation operator W in ph = V j ρ Ez, T = ρ D D = D D z W in ph R d f dk = ρ z V T T T,. R d M dk;.. D ij = k Q k M ηm i+j dk; R d.3 is W in ph z, T = Q in ph M dk..4 R d Proof. Inserting the Hilbert epansion f = f + f + f +... into equation.3 and collecting equal powers of leads to O : Q f =,.5 O : Q f = k f + V k f Q f..6 From.5 and Proposition 3, we know there eists some function F such that f, k, t = F, k, t. Plugging f into.6: Q f = k F + V F Q F..7 5

6 The solvability condition for f Proposition 4 5 implies k F + V F dn k = Q F dn k. S S Clearly the left hand side of above equation is equal to zero k is odd in k, so S Q F dn k = Q F =. By Proposition 3, we know that F is a Fermi-Dirac distribution M with position and time dependent z, t and T, t. Therefore, Q F itsf is equal to zero by Proposition, and.7 reduces to Then using Proposition 5 6, we have Q f = k M + V M. f = Q k M + V M..8 Going back to the original equation.3, multiplying both sides by, k T and integrating w.r.t. k gives: [ t f + ] [ R k f + V k f dk = d R Q f + Q f + Q in ph f] dk. d.9 Terms involving Q f and Q f vanish due to Propositions and. Since Q in ph [8],.9 simplifies to ρ t + ρe R k f + V k f dk = W in d ph f where W in ph f = Q in R d ph f dk. conserves mass,. From the previous discussion, we know f = f + f +... with f being the Fermi-Dirac distribution.9, and f given by.8. Plugging f into., to the leading order we have O term drops out since f is an even function in k: t ρ ρe + k f + V k f R d Utilizing the special form of M, one can rewrite f as f = Q z k V z T + T T Then a simple manipulation of. yids the ET system.. dk = W in ph M ηm. M.. The ET mod. is widy used in practical and industrial applications s [8] for a review and references therein. It can also be derived from the Boltzmann equation through the so-called SHE Spherical Harmonics Epansion mod [6]. In either case, the rigorous theory behind the formal limit is still an open question. Remark 5. If the ectron-ectron interaction Q is assumed as one of the dominant terms in.3, i.e., the same order as astic collision Q which could be the physically revant situation for very dense ectrons, one can still derive the ET mod. via a similar procedure [], but the diffusion coefficients D ij are different. A rigorous result is available in this case [3]. 6

7 Remark 6. If we consider even longer time scale such that the energy raation term Wph in equilibrates the ectron temperature T to lattice temperature T L, and further assume that M is the classical Mawlian, then. reduces to a single equation [ ρ t ρ + D ρ ] V =. T L This is the classical drift-diffusion mod [36, 34]. Remark 7. Unlike the classical statistics, given macroscopic variables ρ and E., finding the corresponding Fermi-Dirac distribution.9 is not a trivial issue. Under the parabolic approimation.4, ρ and E are rated to z and T via [8] F d z, ρ = πt d η E = d T F d +z F d z,. where F ν z is the Fermi-Dirac function of order ν F ν z = ν Γν z e d, + < z <,.3 and Γν is the Gamma function. For small z < z <, the integrand in.3 can be epanded in powers of z: n+ zn F ν z = n ν = z z ν + z3 3 ν.... n= Thus, when z, F ν z behaves like z itsf and one recovers the classical limit. 3 Asymptotic-preserving AP schemes for the semiconductor Boltzmann equation Equation.3 contains thr different scales. To overcome the stiffness induced by the O and O terms, a fully implicit scheme would be desirable. However, neither the collision operators nor the convection terms are easy to solve implicitly. Our goal is to design an appropriate numerical scheme that is uniformly stable in both kinetic and diffusive regimes, i.e., works for all values of ranging from O to, while the implicit terms can be treated eplicitly. We will first consider a spatially homogeneous case with an emphasis on the collision operators, and then include the spatial dependence to treat the convection terms. To facilitate the presentation, we always make the following assumptions without further notice:. The inastic collision operator Q in ph in.3 is assumed to be zero, since it is the weakest effect and its appearance won t bring etra difficulties to numerical schemes.. The scattering matrices Φ and Φ are rotationally invariant: Φ k, k = Φ k, k, Φ k, k, k, k = Φ k, k, k, k. Then it is not difficult to verify that s Proposition 4 Q fk = λ [f] fk, 3. where λ k := Φ k, k δ dk = Φ k, k N, R d 7

8 and [f]k is the mean value of f over sphere S : In particular, for any odd function fk, [f]k := N S fk dn k. Q fk = λ fk, Q fk = λ fk. This observation is crucial in designing our AP schemes. 3. The spatially homogeneous case In the spatially homogeneous case, equation.3 reduces to t f = Q f + Q f, 3. where f only depends on k and t. An eplicit discretization of 3., e.g., the forward Euler scheme, suffers from severe stability constraints: has to be smaller than O. Implicit schemes do not have such a restriction, but require some sort of iteration solvers for Q and Q which can be quite complicated. To tackle these two stiff terms, we adopt the penalization idea in [5], i.e., penalize both Q and Q by their corresponding BGK operators: t f = Q f β M f + β M f + Q f β M f + β M f. 3.3 Using the properties of Q and Q from the last section, M can be naturally chosen as the Fermi-Dirac distribution M in the homogeneous case ρ and E are conserved, so M is an absolute Mawlian and can be obtained from the initial condition, whereas M can in principle be any function of such that it shares the same density and energy as f the choice of M is not essential as we shall s, and we will get back to this when we consider the spatially inhomogeneous case. As the goal of penalization is to make the residue Q f β M f as small as possible so that it is non-stiff or less stiff, and Q, Q can be epressed symbolically as Q fk = Q + f λ fk; Q fk = Q + fk Q fkfk, we choose β ma λ ; β ma Q fk. 3.4 k Other choices are also possible [5]. Generally speaking, we only nd the coefficient to be a rough estimate of the Frechet derivative of the collision operator around equilibrium. Therefore, a first-order IMEX scheme for 3.3 is written as f n+ f n = Q f n β M f n + β M f n+ + Q f n β M f n + β M f n Asymptotic properties of the numerical solution To better understand the asymptotic behavior of the numerical solution, in this subsection we assume that Q f = M f. Then scheme 3.5 becomes f n+ f n = Q f n β M f n + β M f n+ + β M f n + β M f n+. 8

9 This is equivalent to f n+ = β λ + β β + β f n + = β λ + β β + β β + β f n + some function of. Q + f n + Iterativy, it yids + f n β = λ + β n β + β f + some function of. β + β So when is small, for arbitrary initial data f and any integer m, there eists some integer N, s.t. f n O m + some function of, for n > N, 3.6 which means that f n can be arbitrarily close to the null space of Q : so Q f n O m, for n > N. 3.7 At this stage, if we eamine the distance betwn f and M, we found that where f n+ M = β + β β + β f n M + β + β f n+ M r f n M + O m, for n > N, r = Then for proper β, we have < r < and β + β. β + β M Q f n, 3.8 f n+n M r n f n M + O m, for n > N. 3.9 This implies that no matter what the initial condition is, f will eventually be driven to the desired Fermi- Dirac distribution, but the convergence rate can be rather slow for small. What even worse is, when, r, we s from 3.8, 3.6 that f will stay around some function of, but nothing guarants it is M! This violates the property mentioned in the Introduction. On the other hand, we know from 3.7 that Q f n can be arbitrarily small after some time. What if we just set it equal to zero afterwards? Dropping this term as wl as its penalization leads to which is f n+ f n = β M f n + β M f n+, f n+ M = β β f n M. 3. Hence as long as β > /, f n will converge to M regardless of the initial condition. Compared with 3.8, this one has much faster convergence rate. 9

10 3.. The thresholded AP scheme The above simple analysis illustrates that the penalization idea [5] should be applied wisy, especially when there are stiff terms on different scales. Back to the original equation 3., we propose to solve it as follows. At time step t n+, check the norm of Q f n in k: if Q f n > δ, apply scheme 3.5; otherwise, apply 3.5 with Q f n = β =. As eplained previously, the threshold can be chosen based on the property we epect in 3.7, say, δ = m, some m >. However, in practice, as any numerical solver of Q has certain accuracy, we therefore set where l denotes the order of accuracy of the numerical solver for Q. δ = O k l, 3. Remark 8. The choice of threshold 3. does not violate the consistency of our method, since when k, we have δ, and we are back to the original scheme 3.5 whose consistency to 3. is not a problem i.e. the scheme satisfies the property in the Introduction. 3. The spatially inhomogeneous case We now include the spatial dependence to treat the full problem.3. To handle the newly added stiff convection terms, we follow the idea of [3] to form it into a set of parity equations. Denote f + = f, k, t, f = f, k, t, then they solve t f + + k f + + V k f + = Q f + + Q f +, t f k f + V k f = Q f + Q f. Now write we have r, k, t = f + + f, j, k, t = f + f, t r + k j + V k j = Q r + Q f + + Q f, 3. t j + k r + V k r = λ j + Q f + Q f, 3.3 where we used the fact that j is an odd function in k, thus Q j = λ j. For , the same penalization as in the homogenous case suggests t r + k j + V k j = Q r β M r + β M r + Q f + + Q f β M r + β M r, t j + k r + V k r = λ j + Q f + Q f + β j β j. Note here M = M,, t is the local Fermi-Dirac distribution, and M = M,, t is some function of whose form will be specified later. The coefficients β and β are chosen the same as in 3.4, ecept that β can also be made space dependent. The above equations can be formed into a diffusive raation system [4]: t r + k j + V k j = G r, j + β M r + β M r, 3.4 t j + θ k r + V k r = G r, j β j [ λ j + θ k r + V k r ], 3.5

11 where G r, j = Q r β M r + Q f + + Q f β M r, 3.6 G r, j = Q f + Q f + β j, 3.7 and θ / is a control parameter simply chosen as θ = min {, / }. A first-order IMEX scheme for the system thus reads r n+ r n j n+ j n + k j n + V n k j n = G r n, j n + β M n+ r n+ + θ k r n + V n k r n = G r n, j n β jn+ + β M n+ r n+, 3.8 [λ j n+ + θ k r n+ + V n+ k r n+ ] Asymptotic properties of the numerical solution Leaving aside other issues such as the spatial discretization, let us for the moment again assume that Q f = M f, then simplify to r n+ r n + k j n + V n k j n = Q r n β M n rn + β M n+ r n+ + β M n r n + β M n+ r n+, j n+ j n + θ k r n + V n k r n = β j n β jn+ The scheme for j is actually j n+ = + β λ + β j n θ λ + β [λ j n+ + θ k r n+ + V n+ k r n+ ]. k r n+ + V n+ k r n+ λ + β [ k r n+ + V n+ k r n+ k r n + V n k r n ]. When so θ =, suppose all functions are smooth, we have The scheme for r results in j n+ = λ k r n+ + V n+ k r n+ + O. r n+ = β λ + β β + β r n β + β k j n + V n k j n + β + β Q + rn + β β + β M n+ M n + β + β M n + β β + β M n+ M n = β λ + β β + β r n + O + some function of.

12 Iterativy, this gives + r n β = λ + β n β + β r + O + some function of. Clearly the first term on the right hand side will be damped down as time goes by no matter what the initial condition is. After several steps, we have which implies r n = O + some function of, for n > N, Q r n = O, for n > N. At this stage, if we continue to penalize Q r n, similarly as before, r n+ M n+ = β + β β + β r n M n + β + β Q r n [ β + β k j n + V n k j n + M n+ M n ] β M n+ M n β + β M n+ M n = β + β β + β r n M n + O +, for n > N. 3. From 3. we s that r will converge to M with a dominant O error. Therefore, a good choice of M is M = M, which will not only simplify the scheme, but also improve the asymptotic error from O to O. Moreover, like the spatially homogeneous case, we suffer from the same problem that the convergence rate is too slow for small but finite. To accerate the convergence, we again choose some threshold δ as in 3. such that once Q r n < δ check it for every, we set Q r n = and turn off its penalization. Then 3. becomes β r n+ M n+ = β β r n M n = β β r n M n + O, and we gain much faster convergence rate. 3.. The thresholded semi-discrete AP scheme [ k j n + V n k j n + M n+ M n ] Based on the discussion above, we integrate the thresholding idea into to propose the following semi-discrete AP scheme. At time step t n+, given f n = f + n, f n, r n, j n, ρ n, E n, and V n : Step : Compute M n+ used in 3.8 and V n+ in 3.9. Although 3.8 appears implicit recall M = M, M n+ can be computed eplicitly similarly as in [5]. Specifically, we multiply both sides of 3.8 by, k T and integrate w.r.t. k. Utilizing the conservation properties of Q, Q, and the BGK operator, we get r n+ dk = r n dk R d R k j n + V n k j n dk. d R d

13 Note that by definition r dk = f dk = R d R d ρ ρe so the preceding scheme just gives an evolution of the macroscopic variables ρ n+ ρ n ρ n+ E n+ = ρ n E n k j n + V n k j n R d, dk. Once ρ n+ and E n+ are computed, we can invert the system. to get z n+ and T n+ details s [8]. Plugging them into.9 then defines M n+. Given ρ n+, V n+ can be easily obtained by solving the Poisson equation.7. Step : Compute r n+. At every spatial point, check the norm of Q r n in k: if Q r n > δ, apply scheme 3.8; otherwise, apply 3.8 with Q r n = β =. Step 3: Employ scheme 3.9 to get j n+. Step 4: Reconstruct f n+ = f + n+ = r n+ + j n+ and f n+ = r n+ j n Space discretization We finally include the spatial discretization to the previous semi-discrete scheme to construct a fully-discrete scheme. We will show that when, it automatically becomes a discretization for the limiting ET mod., thus is Asymptotic Preserving satisfies the property in the Introduction. For the sake of brevity, we will present the method in a splitting framework, namy, separating the eplicit and implicit parts in This is equivalent to an unsplit version since our scheme is of an IMEX type. We also assume a slab geometry: Ω R. Etension to higher dimensions is straightforward. Let r n l,m, jn l,m denote the numerical approimation of r l, k m, t n and j l, k m, t n, where < l N, < m = m,..., m d N d k, N and N k are the number of points in and k directions respectivy. We have at time step t n+ : Step : Solve the eplicit part of to get rl,m and j l,m. The thresholding idea is embedded in this step when computing G rl,m n, jn l,m : if Q r n < δ at l, set Q r n l =. For convection terms, we use the upwind scheme with a slope limiter [3] on the Riemann invariants. Step : Solve for the macroscopic quantities ρ l and E l and thus define M l,m and V l,m. ρ l = m rl,m k d, El = r l,m k m k d /ρ l. Finding Ml,m is then eactly the same as in the semi-discrete scheme. V finite-difference discretization of the Poisson equation. m l,m is solved from a simple Step 3: Solve the implicit part of to get r n+ l,m and jn+ l,m if the threshold is satisfied in Step, set β = in r s equation as wl: r n+ l,m j n+ l,m r l,m j l,m = β M n+ l,m rn+ l,m + β M n+ l,m = [ λ j n+ l,m + θ rn+ l,m k m r n+ l+,m rn+ l,m, 3. + V n+ r n+ l,m+ ] rn+ l,m l k β jn+ l,m. 3. 3

14 First, it is easy to s that macroscopic quantities ρ l and El remain unchanged during this step the right hand side of 3. is conservative. Therefore, the previously obtained Ml,m and V l,m are in fact n+ and V. From 3. one can easily obtain rn+, and then 3. directly gives rise to jn+ M n+ l,m l,m 3..4 Asymptotic properties of the fully discrete scheme As already shown in Section 3.., sending to zero in for Q = M f leads to which in -D fully discrete form read r n+ l,m r n+ = M n+, j n+ = λ k r n+ + V n+ k r n+, l,m = M n+ l,m, j n+ l,m = λ k m r n+ l+,m rn+ l,m + V n+ r n+ l,m+ rn+ l,m l. k Plugging these rations into the discrete scheme of r resulted from the last subsection, we get after multiplication by, k T and integration w.r.t. k: ρ n+ ρ n ρ n+ E n+ ρ n E n + V n l+ k m +k m+ V n l k M n l+,m+ M n l+,m k { [ km Ml+,m n k M l,m n + M l,m n m V n Ml,m+ n M n ] l,m l k [ V n Ml,m+ n M l,m n + M l,m n l k k ] R d λ M n l+,m Ml,m n + Ml,m n V n + l Ml+,m+ n M l,m+ n M n l,m+ Ml,m n + Ml,m } n Ml+,m n k M l,m n m l,m. dk =, 3.3 which is a kinetic scheme [] for the ET mod. compare with. and.8. Here for notation simplicity, we only consider the upwind scheme, the slope limiter can be added in the same manner. 4 Numerical eamples In this section, we present several numerical eamples using our AP schemes. In order to perform the tests, we nd accurate numerical solvers for the collision operators. Since the ectron-ectron interaction Q falls into a special case of the quantum Boltzmann operator, we adopt the fast spectral method devoped in []. For the astic collision Q, it is desirable to compute it in the same spectral framework but the direct evaluation would be very epensive. Here we introduce a fast approach by eploring the low-rank structure in the coefficient matri. Details are gathered in the Appendi. In the following, we always assume k [ L k, L k ] in -D and [, L ] in -D. N k is the number of points in each k direction and N is the number of spatial discretization. 4. The spatially homogeneous case We first check the behavior of the solution in the spatially homogeneous case. Consider the non-equilibrium initial data f k, k = [ k + k.5 ] e [k +k.5 ]. π 4

15 The parameters are chosen as = e 3 diffusive regime, η = strong quantum effect; the equilibrium is very different from the classical Mawlian, L k =.5, N k = 64. Under this condition, a stable eplicit scheme would require = Oe 6, while our scheme gives fairly good results with much coarser time step =. Figure shows the AP error in L -norm errorap n L = ma k,k f n M n 4. with time. Here we can s that during the initial period of time, this error decreases very slowly as eplained in 3.6: f is only driven to some function of, not M, because of the dominating mechanism Q. Once the threshold comes into play as shown in 3., f will start to converge to M at a reasonable spd, which appears as a sharp transition in dashed curve in Figure. As a comparison, the solid curve is obtained by the regular AP scheme without threshold, the error decreases very slowly as in 3.9. Figure displays the evolution of f at different times, where we clearly s that f first transits from non-radially symmetric to radially symmetric and then moves toward the desired Fermi-Dirac distribution M. =e 3 3 errorap in L without threshold errorap in L with threshold time Figure : Plots of the asymptotic error 4. versus time. Here η =, =, L k =.5, and N k = The spatially inhomogeneous case In the rest of simulation, we always take L =, L k = 9., and assume periodic boundary condition in and zero boundary in k. The time step is chosen to only satisfy the parabolic CFL condition: = O independent of. 4.. AP property Consider equation.3 with non-equilibrium initial data f, k, k = L 8 e + e [k +k] + e [k + +k]. π The ectric fid V is set to be one. We check the asymptotic property by looking at the distance betwn r and M, i.e., errorap n L =,k,k r n M n k, errorap n L = ma,k,k r n M n. 4. The results are gathered in Figure 3, where we observe a similar trend as in the space homogeneous case that r converges to M in two stages: first to a radially symmetric function some function of and then the local Mawlian M. This in some sense mimics the Hilbert epansion in the derivation of ET mod in Section. 5

16 initial distribution f at t= k 5 k 5 k 5 k 5 f at t=75 f at t= k 5 k 5 k 5 k 5 Figure : Evolution of f at times t =, 5, 75 and 5. Here η =, =, L k =.5, and N k = 64. = e 3 = e 3 errorap in L errorap in L errorap in L errorap in L time time Figure 3: Plots of the asymptotic error 4. versus time. Here N = 4, =.. Left: η =. classical regime, N k = 3. Right: η = 3 quantum regime, N k = D n + n n + ballistic silicon diode We finally simulate a -D n + n n + ballistic silicon diode, which is a simple mod of the chann of a MOS transistor. The initial data is taken to be f, k, k =. + tanh 4 5L 8 tanh 4 3L 8 e [k +k] + e [k + +k]. For Poisson equation.7, we choose h = ρ = f dk, ɛ r, C = /, with boundary condition V =, V L =. The doping profile h is shown in Figure 4. We consider two regimes: one is the kinetic regime with =, where we compare our solution with the 6

17 one obtained by the eplicit scheme forward Euler; the other is the diffusive regime with = e 3, where our solution is compared with that of the ET system using the kinetic solver 3.3. Good agrments are obtained in Figures 5, 6. Here the macroscopic quantities plotted are mass density ρ, internal energy E defined in., ectron temperature T and fugacity z obtained through., ectric fid E = V, and mean vocity u defined as u = j ρ /ρ..8 h Figure 4: Doping profile h for -D n + n n + ballistic silicon diode. 5 Conclusion We constructed an asymptotic preserving scheme for a multiscale semiconductor Boltzmann equation coupled with Poisson equation that in the diffusive regime captures the energy-transport limit. Besides the stiff convection terms, the two-scale collision operators pose new difficulties since the simple BGK penalization is unable to capture the correct limit. The key ingredient in our scheme is to set a suitable threshold for the leading astic collision such that once this threshold is crossed, the less stiff collision begins to dominate. In this way, the convergence of the numerical solution to the local equilibrium resembles the Hilbert epansion at the continuous lev. We analyzed this asymptotic behavior using a simplified BGK mod. Several numerical results confirmed the asymptotic-preserving property for any non-equilibrium initial data, as wl as the uniform stability of the scheme with respect to the mean fr path, from kinetic regime to diffusive regime. As a variant of the scheme in this paper, we are currently investigating another formulation using a splitting approach. The results will be reported in a companion paper [9]. It is also interesting to include the inastic ectron-phonon interaction into the equation and we leave it to the future work. Acknowledgments This project was initiated during the authors participation at the KI-Net Conference Quantum Systems hd by CSCAMM, University of Maryland, May 3. Both authors acknowledge the generous support from the institute. J.H. thanks Prof. Shi Jin for initially pointing out the ET mod, and Prof. Irene Gamba and Prof. Leing Ying for hpful discussion and support. L.W. thanks Prof. Robert Krasny for fruitful discussion on moding of semiconductors. 7

18 .4..8 AP eplicit 5 5 ρ.6.4. u T z. E Figure 5: Density ρ, vocity u, internal energy E, temperature T, fugacity z, and ectric fid E at time t =.5. is the AP scheme, is the forward Euler scheme. Here =, η =, N = 4, N k = 64, =.. Appendi: Fast spectral methods for collision operators Q Q and In this appendi, we briefly outline the numerical methods for computing the collision operators Q and Q. For numerical purpose, we assume the scattering matrices Φ k, k = Φ k, k, k, k, and the wave vector k R. 8

19 ρ AP ET u T z. E Figure 6: Density ρ, vocity u, internal energy E, temperature T, fugacity z, and ectric fid E at time t =.5. is the AP scheme, is the kinetic scheme for the ET system. Here = e 3, η =, N = 4, N k = 64, =.. Computation of Q Under the parabolic band assumption,.6 reads k Q fk = δk + k k k δ R + k k k [ 6 ] f f ηf ηf ff ηf ηf dk dk dk. By a change of variables, it is not difficult to rewrite the above integral in the center of mass reference system: Q fk = [ ] f f ηf ηf ff ηf ηf dσ dk, 5. R S 9

20 where k = k + k k = k + k + k k σ, k k σ. This is just the usual form of the quantum Boltzmann collision operator for a Fermi gas of -D Mawlian molecules. Our way of computing Q follows the fast spectral method in []. The starting point is to further transform 5. to a Carleman form [ ] Q fk = δ y f f ηf ηf ff ηf ηf d dy, 5. R R where k = k + + y, k = k +, and k = k + y. If Suppfk B S a ball with radius S, we can truncate 5. as [ ] Q R fk = δ y f f ηf ηf ff ηf ηf d dy B R B R [ ] = δ y f f ff ηf f f + f f f f f f f f f d dy B R B R with R = S. We net choose a computational domain D L = [ L, L] for k, and etend the function fk periodically to the whole space R. L is chosen such that L 3 + S to avoid aliasing effect. We then approimate fk by a truncated Fourier series where fk = N/ j= N/ ˆf j = L fke i π L j k dk. D L ˆf j e i π L j k, 5.3 Inserting the Fourier epansion of f into Q R f, and performing a spectral-galerkin projection, we can get the governing equation for Q R f j. The computation is sped up by discovering a convolution structure and a separated epansion of the coefficient matri. The final cost is roughly OMN 3 log N, where N is the number of points in each k direction, and M is the number of angular discretization. More details can be found in []. Computation of Q Under the parabolic band assumption.4,.5 reads s also 3. Q fk = δ f f dk = R f k σ dσ πfk. S 5.4 Compared to Q, this one is much easier to compute. For instance, one can do a direct numerical quadrature plus interpolation to approimate the integral over S. To achieve better accuracy, we here present an efficient way to compute Q f based on the same spectral framework of Q f. Specifically, we still adopt the Fourier epansion 5.3. Inserting it into 5.4, we get Q fk = N/ j= N/ Bk, j ˆf j πfk, 5.5 where Bk, j = πj π L k j.

21 Here L has to be L + S to avoid aliasing. A direct computation of the above summation requires obviously ON 4 flops, which can be quite costly. But note that the coefficient matri Bk, j only depends on the magnitude of k and j, which means that its rank is roughly ON. Therefore, we can find a low rank decomposition of Bk, j as for -D problems this can be precomputed via a SVD Bk, j = Then computing the summation in 5.5 becomes N/ j= N/ Bk, j ˆf j = N/ ON j= N/ r= ON r= U r kv r j. U r kv r j ˆf j = ON r= U r k N/ j= N/ V r j ˆf j. The cost is reduced to ON 3. To summarize, we have two fast spectral solvers for the collision operators Q and Q the cost of Q is dominant due its intrinsic compleity. Both of them provide high-order accuracy and are thus suitable for testing asymptotic properties of our schemes. References [] N. B. Abdallah and P. Degond, On a hierarchy of macroscopic mods for semiconductors, J. Math. Phys., , pp [] N. B. Abdallah, P. Degond, and S. Genieys, An energy-transport mod for semiconductors derived from the Boltzmann equation, J. Stat. Phys., , pp [3] N. B. Abdallah, L. Desvillettes, and S. Genieys, On the convergence of the Boltzmann equation for semiconductors toward the energy transport mod, J. Stat. Phys., 98, pp [4] J. F. Bourgat, P. Le Tallec, B. Perthame, and Y. Qiu, Coupling Boltzmann and Euler equations without overlapping, in Domain decomposition methods in science and enginring, vol. 57, AMS, 994, pp [5] J. Carrillo, I. Gamba, A. Majorana, and C.-W. Shu, D semiconductor device simulations by WENO-Boltzmann schemes: efficiency, boundary conditions and comparison to Monte Carlo methods, J. Comput. Phys., 4 6, pp [6] Y. Cheng, I. Gamba, A. Majorana, and C.-W. Shu, A discontinous Galerkin solver for Boltzmann- Poisson systems in nano devices, Comput. Methods Appl. Mech. Engrg., 98 9, pp [7] N. Crouseilles and M. Lemou, An asymptotic preserving scheme based on a micro-macro decomposition for collisional Vlasov equations: diffusion and high fid scaling limits, Kinet. Rat. Mods, 4, pp [8] P. Degond, Mathematical modling of microectronics semiconductor devices, vol. 5 of AMS/IP Studies in Advanced Mathematics, AMS and International Press,, pp. 77. [9] P. Degond, A. Jüng, and P. Pietra, Numerical discretization of energy-transport mods for semiconductors with non-parabolic band structure, SIAM J. Sci. Comput.,, pp [] P. Degond, C. D. Levermore, and C. Schmeiser, A note on the Energy-Transport limit of the semiconductor Boltzmann equation, in Transport in Transition Regimes, N. B. Abdallah et al, ed., vol. 35 of The IMA Volumes in Mathematics and its Applications, Springer, New York, 4, pp

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