NUMERICAL METHODS FOR THE SEMICONDUCTOR BOLTZMANN EQUATION BASED ON SPHERICAL HARMONICS EXPANSIONS AND ENTROPY DISCRETIZATIONS

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1 TRANSPORT THEORY AND STATISTICAL PHYSICS Vol. 31, Nos. 4 6, pp , 2002 NUMERICAL METHODS FOR THE SEMICONDUCTOR BOLTMANN EQUATION BASED ON SPHERICAL HARMONICS EXPANSIONS AND ENTROPY DISCRETIATIONS Christian Ringhofer Department of Mathematics, Arizona State University, Tempe, A ringhofer@asu.edu ABSTRACT We present a class of numerical methods for the semiconductor Boltzmann Poisson problem in the case of spherical band energies. The methods are based on spherical harmonics expansions in the wave vector and difference discretizations in space time. The resulting class of approximate solutions dissipate a certain type of entropy. Key Words: differences AMS Classification: Boltzmann equation; Galerkin methods; Finite 65N35; 65N DOI: /TT Copyright & 2002 by Marcel Dekker, Inc (Print); (Online) + [ :05am] [ ] [Page No. 431] i:/mdi/tt/31(4-6)/ _tt_031_4-6_r1.3d Transport Theory and Statistical Physics (TT)

2 432 RINGHOFER INTRODUCTION This paper is concerned with the space time discretization of a class of conservation laws which arise from the approximation of the semiconductor Boltzmann Poisson problem via an spherical harmonics expansion in the wave vector direction. We consider the Boltzmann Poisson problem t f þ Lð f Þþ 1 Qð f Þ¼0 ðbþ Lð f Þ :¼ r x ½r k "ðkþ f Š qr k ½r x VðxÞ f Š ðcþ x V þ q½d dop ðxþ Š ¼0, ðx, tþ :¼ f ðx, k, tþ dk where the kinetic density function f ðx, k, tþ depends on the spatial variable x 2 R d with d ¼ 1, 2, or 3, the wave vector k 2 R 3 and the time t. The acceleration of particles is given by the electric field qr x V which, using a mean field approximation, is determined by the Poisson Eq. (1c). Here, q denotes the charge of one particle, so q < 0 holds for electrons and q > 0 holds if the Boltzmann equation describes the transport of holes. The function D dop denotes a charge background, produced by the implantation of different species atoms into the crystal. is the dielectricity constant of the material. The dimensionless constant in Eq. (1a) denotes the scaled mean free path (the Knudsen number). Since, we are modeling transport in a crystal, as opposed to a vacuum, the velocity of a particle is given in terms of its wave vector through of a dispersion relation of the form vðkþ ¼r k "ðkþ where "ðkþ denotes the band energy. In vacuum "ðkþ ¼constjkj 2 =2 holds and the velocity vector and the wave vector are, up to a multiplicative constant, identical. For the purposes of this paper, we will take the band energy function to be spherically symmetric, so we will assume that "ðkþ ¼"ðjkjÞ holds. One of the most popular approximate choices for the band energy is the Kane dispersion model of the form jkj 2 "ðkþ ¼ p 1 þ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi, 1 þ jkj 2 but the approach outlined in this paper is certainly applicable to more complicated band structures. The collision operator Q in Eq. (1a) models the interaction of particles with a uniform phonon background, and can ð1þ + [ :05am] [ ] [Page No. 432] i:/mdi/tt/31(4-6)/ _tt_031_4-6_r1.3d Transport Theory and Statistical Physics (TT)

3 SEMICONDUCTOR BOLTMANN EQUATION therefore be taken to be linear. In the scaled and dimensionless form (1), the collision operator is of the form Qð f Þðx, k, tþ ¼ X1 c ¼ 1 R 3 k ( f ðx, k, tþ R 3 k dk 0 ð"ðkþ "ðk 0 Þþ!Þ dk 0 ð"ðk 0 Þ "ðkþþ!þ f ðk 0 ) Þ : ð2þ So a particle can gain or loose an amount! of energy, with a probability c 1, c 1, in the collision event. Because of the principle of detailed balance, the constants c have to be such that local Maxwellians of the form f ðx, k, tþ ¼ðx, tþe "ðkþ are in the kernel of the collision operator. Thus the constants c has to satisfy the relation c ¼ c e!. To express the symmetry properties and integral invariants of the phonon collision operator it will be more convenient to rewrite Q as h Qð f Þ¼ dk 0 S ", " 0 i fe " f 0 e "0, ð3þ R 3 k where the scattering cross section Sð", " 0 Þ is a symmetric function, so Sð", " 0 Þ¼Sð" 0, "Þ holds. We use the short hand f 0 for f ðx, k 0, tþ in Eq. (3) and from now on. S is then of the form ðaþ Sð", " 0 Þ¼ X1 ¼ 1 s ð", " 0 Þð" " 0 þ!þ ðbþ s ð", " 0! " "0 Þ¼c exp : ð4þ 2 The standard way to solve the Boltzmann Eq. (1) numerically is of course the Monte Carlo (MC) method. This paper is concerned with an alternative, deterministic approach, namely to expand the kinetic density function f into a series of basis functions in the wave vector direction, obtaining a system of hyperbolic conservation laws with source terms for the coefficients. While such an approach cannot hope to reproduce MC results in all physical detail, it has the advantage of being deterministic and computationally considerably less expensive, particularly in fewer space dimensions (d ¼ 1 or 2), since it has the ability to take advantage of symmetries. Usually these approaches, and the closely related moment closure methods, have to use sometimes quite crude approximations to the collision operators to be effective (see Ref. [3]). Alternatively, they + [ :05am] [ ] [Page No. 433] i:/mdi/tt/31(4-6)/ _tt_031_4-6_r1.3d Transport Theory and Statistical Physics (TT)

4 434 RINGHOFER have to use a difference method in the radial component of the wave vector to deal with the functions in the integral kernel (see Refs. [2,5,12]). This limits the use of these methods in more than one spatial dimension because of the large number of necessary unknowns. In this paper, we present a complete spectral approximation to the Boltzmann equation in the wave vector direction using the complete phonon collision operator Q as given in Eq. (3). The resulting system of hyperbolic conservation laws can be extremely stiff in the spatial direction, due to very large potential gradients and in the time direction due to small values of the Knudsen number. In Sec. 3 of this paper, we present several approaches to deal with these stiffness problems. In Sec. 4, we present a simple numerical test example which serves to demonstrate that the resulting method is capable to simulate transport quite far away from the equilibrium regime. 2. THE MOMENTUM DISCRETIATION The general idea of the approach outlined in this paper is to discretize the Boltzmann Eq. (1) in the wave vector direction by a spectral Galerkin method, thus obtaining a first order system of hyperbolic conservation laws in position and time, which will contain source terms due to the acceleration of particles by the electric field and the collision operator Q in Eq. (1). This system will then be discretized by methods suitable for hyperbolic conservation laws, which are described in the next section. An important concept which is used in making the choice of basis functions is that of an entropy, i.e., a function hð f, x, kþ which satisfies ðaþ ðbþ R 3 k R 3 k dk dk R d x R d x dx½hð f, x, kþlð f ÞŠ ¼ 0, dx½hð f, x, kþqð f ÞŠ 0: ð5þ A straight forward calculation yields, that for any function h satisfying Eq. (5) R d x dx dk½hð f, x, kþ@ t f Š¼ d dx dkhð f, x, kþ 0 R 3 dt R d k x R 3 k holds, where H denotes the antiderivative of h with respect to f. If the function H, the entropy, is a convex function of the variable f this guarantees that the Boltzmann Eq. (1) is well posed, and the basis functions for the ð6þ + [ :05am] [ ] [Page No. 434] i:/mdi/tt/31(4-6)/ _tt_031_4-6_r1.3d Transport Theory and Statistical Physics (TT)

5 SEMICONDUCTOR BOLTMANN EQUATION wave vector discretization will be chosen such that the property (6) is preserved for the semi-discrete system. To this end we rewrite the Boltzmann Eq. (1) in the variable gðx, k, tþ ¼hð f ðx, k, tþ, x, kþ, t ðg, x, kþþlððg, x, kþþ þ 1 Qððg, x, kþþ ¼ 0, ð7þ where denotes the functional inverse of h; so g ¼ hð f, x, kþ ()f ¼ ðg, x, kþ holds. Equation (7) is now discretized by a standard Galerkin procedure in the k-direction. So we express g as a linear combination of basis functions in k, whose coefficients depend on x and t, and integrate Eq. (7) against each basis function with respect to k. This gives the hyperbolic system of conservation laws for the expansion coefficients, which then automatically satisfies the entropy estimate (6), and is thus well posed. The physical entropy is given by logarithmic function (hð f, x, kþ ¼ln f, Hð f, x, kþ ¼f ðln f 1Þ). However, since we are using only the linearized collision operator, modeling the collision of electrons with phonons, there is a much wider choice of possible entropy functions. As a matter of fact, the equality (see Ref. [6]) dk½hqð f ÞŠ ¼ 1 h i dk dk 0 S fe " f 0 e "0 ðh h 0 Þ ð8þ R 3 2 R 3 R 3 k k k implies that any function hð f, x, kþ which is a monotonically increasing function of fe "ðkþ produces an entropy. The more complicated the chosen entropy function, the more difficult it will be to compute the discretization of the collision operator Q. Galerkin expansions which use the physical entropy (the logarithm) usually have to replace the collision operator by some BGK-type approximation to yield practical schemes (see Ref. [3]). In the approach outlined in this paper we will instead choose the simplest possible entropy, namely the linear function hð f, x, kþ ¼e "þqv f, Hð f, x, kþ ¼ 1 2 e"þqv f 2, ðg, x, kþ ¼e " qv g: Note that this choice of entropy seems to implicitly imply a time independent potential V. In the case, considered in this paper, when the potential is computed self consistently via the Poisson equation, the above choice of entropy can still be used. However the estimate (6), together with the proof that the resulting entropy function H yields a convex functional, becomes quite a bit more complicated. We refer the reader to Ref. [8] for the details. + [ :05am] [ ] [Page No. 435] i:/mdi/tt/31(4-6)/ _tt_031_4-6_r1.3d Transport Theory and Statistical Physics (TT)

6 436 RINGHOFER This leaves us with the choice of basis functions. The structure of the collision operator (i.e., the averaging over surfaces of equal energy) suggests to rewrite the Boltzmann Eq. (1) in polar coordinates and to discretize by spherical harmonic functions in the k-direction. The idea to expand solutions of the Boltzmann equation into spherical harmonic functions goes back a long time (see Ref. [1] for an overview). This set of basis functions was first used numerically and in the context of semiconductors in Ref. [12] and later on by Ref. [2], and more recently in Refs. [4,5]. All of these approaches use a difference approximation of the density function in the radial direction. Because of the number of gridpoints necessary in the radial direction to resolve the collision operator, they are, given current computer resources, restricted to the case of one space dimension (d ¼ 1). In this paper, we will instead use a spectral method in the radial direction as well, which enables us to dramatically reduce the number of variables. According to the entropy principles used, it is natural to use a polynomial basis in the radial direction for the entropy variable g, and correspondingly, polynomials time the equilibrium Maxwellian for the density function f. Thus we set gðx, k, tþ ¼ X m2m g m ðx, tþ m ðr,, Þ, mðr,, Þ ¼P m1 ðrþ m2 m 3 ð, Þ, m ¼ðm 1, m 2, m 3 Þ where the triple index m varies in some finite index set M. r,, denote polar coordinates and is the spherical harmonics given by 0 1 r cos m2 m 3 ð, Þ ¼ m 3 m 2 ðcos Þðsin Þ m 3 expðim 3 Þ, k r sin sin A: r sin cos Here m 3 m 2 denote the associated Legendre polynomials and P m1 are some suitable basis polynomials in the radial direction. Correspondingly the Boltzmann density function f is approximated by f ðx, k, tþ ¼ðg, x, kþ ¼ X f m ðx, tþ m ðr,, Þe "ðrþ, m2m where we have absorbed the factor e qv into the coefficients f m for simplicity. The expansion (9) is inserted into the Boltzmann equation and integrated against the basis functions for g of the form P m1 ðrþ m2 m 3 ð, Þ with respect to the wave vector k. Thus the whole procedure reduces, because of the linear choice for h, to a Galerkin approximation for the Boltzmann density f with a scalar product given by the weight function e "ðjkjþ. Correspondingly, the polynomials P m1 in Eq. (9) should be chosen orthogonal with respect to ð9þ + [ :05am] [ ] [Page No. 436] i:/mdi/tt/31(4-6)/ _tt_031_4-6_r1.3d Transport Theory and Statistical Physics (TT)

7 SEMICONDUCTOR BOLTMANN EQUATION this weight function, so as to diagonalize the mass matrix. So the P m1 will satisfy 1 0 P m1 ðrþp n1 ðrþe "ðrþ r 2 dr ¼ m1 n 1, P 0 ðrþ ¼const With this choice the resulting system of conservation laws becomes ðaþ ðbþ ðcþ t F þ Xd ¼1 A ðm, nþ ¼ B ðm, nþ ¼ Cðm, nþ ¼ x F qð@ x VÞB FŠþ 1 CF ¼ 0, F R 3 k R 3 k R 3 k dk½ m ð@ k "Þe " nš, dk k ½e " nš, dk m Q½e " nš, ¼ðf 111,...Þ ðeþ m ¼ P m1 ðrþ m2 m 3 ð, Þ ð10þ Once the matrices A, B,andC are computed, Eq. (10) constitutes a system of hyperbolic conservation laws whose discretization in space and time will be the subject of the next section. In the case of non-parabolic bands, i.e., "ðkþ 6¼ ðjkj 2 =2Þ these coefficient matrices will have to be computed numerically. A considerable computational effort can be spent on this computation since it only has to be performed once for a given band structure and a given set of basis functions. However, the numerical evaluation of the coefficient matrices should be done in such a way that the entropy property (6) translates exactly to the semi-discrete system (10). We will therefore briefly outline how to do this. First, integrating the k ðe " m n Þ¼ m ð@ k "Þe " n þ k ½e " nšþ k ½e " mš with respect to the wave vector k implies that A ¼ B B T, ¼ 1,..., d ð11þ holds. This relation should hold exactly and therefore, we compute the matrices B by some numerical integration rule and define the matrices A via Eq. (11). Second, using the identity (8), we write the collision matrix C as Cðm, nþ ¼ 1 dk dk 0 ½Sð 2 m 0 mþð n 0 nþš, ð12þ R 3 k R 3 k + [ :05am] [ ] [Page No. 437] i:/mdi/tt/31(4-6)/ _tt_031_4-6_r1.3d Transport Theory and Statistical Physics (TT)

8 438 RINGHOFER which implies that the matrix C is symmetric and positive semi-definite. Moreover, the first row and column of C vanish identically since 000 ¼ const holds. These two properties should also hold exactly for the numerical approximation of the collision matrix C. Inserting the spherical harmonic basis functions into Eq. (12) gives Cðm, nþ ¼ 1 dk dk 0 ½Sð 2 m 0 mþð n 0 nþš R 3 R 3 k k ¼ dr d d dr 0 d 0 d 0 ½r 2 sinðþðr 0 Þ sinð 0 ÞSð", " 0 ÞðP m1 m2 m 3 P 0 m 1 0 m 2 m 3 ÞðP n1 n2 n 3 P 0 n 1 0 n 2 n 3 ÞŠ: Since the band energy function " depends only on the modulus r of the wave vector, this sixfold integral can be simplified considerably by integrating out the angular variables. A direct calculation gives 1 1 Cðm, nþ ¼2 m2 0 m3 0 n2 0 n3 0 dr dr 0 r 2 ðr 0 Þ 2 Sð", " 0 Þ P m1 P 0 m P n1 P 0 n 1 þ 4 m2 n 2 m3 n 3 m2 0 m3 0 n2 0 n dr dr 0 r 2 ðr 0 Þ 2 Sð", " 0 ÞP m1 P n1, ð13þ 0 0 where we have used the fact that the pspherical harmonic functions m2 m 3 are mutually orthogonal and 00 ¼ 1= ffiffiffiffiffi 4 holds. For the remaining integrals in the radial direction one integral can be eliminated by using the delta functions in the kernel Sð", " 0 Þ. For this we need the inverse of the band energy function, which we denote by, so u ¼ "ðrþ ()r ¼ ðuþ holds. For a general function Gðr, r 0 Þ 1 dr ¼ X1 1 ¼ 1 0 dr 0 ½Sð", " 0 ÞGðr, r 0 ÞŠ dr c e " Hð" þ!þgðr, r 0 ðrþþ d du ð" þ!þ ð14þ holds, where the variable r 0 is evaluated at r 0 ðrþ ¼ð"ðrÞþ!Þ and H denotes the Heaviside function. The one dimensional integral on the right hand side of Eq. (14) has to be evaluated numerically by some integration formula. However, inserting Eq. (14) into Eq. (13) now yields a symmetric + [ :05am] [ ] [Page No. 438] i:/mdi/tt/31(4-6)/ _tt_031_4-6_r1.3d Transport Theory and Statistical Physics (TT)

9 SEMICONDUCTOR BOLTMANN EQUATION positive definite matrix C, independent of what integration formula is used for this purpose. Because of the Kronecker symbols in Eq. (13), and since the polynomial P 0 is a constant, the first row and the first column of the collision matrix C will vanish as well. The semi-discrete version of the entropy relations is now given through Eq. (11) and the fact that the matrix C is positive semi-definite. Because of Eq. (11) we have e qv F T x F qð@ x V ÞB F x ½e qv F T B FŠ, R d x ¼ 1,..., d Thus, multiplying the semi-discrete Boltzmann Eq. (10) from the left with e qv F T and integrating with respect to x gives dx e qv F T t F ¼ 2 dx e qv F T CF 0: ð15þ R d x In the case of a time-independent potential, this immediately gives the estimate that R R d dx½eqv jfj 2 Š is non-increasing in time. In the case of a x self consistent potential V, the potential is computed from the Poisson equation x V þ q D dop ðxþ 1 F 000 ¼ 0, F 000 ¼ ðxþ ¼ f ðx, kþ dk, ð16þ 000 (remember that 000 is a constant,) and Eq. (15) only yields the t dx e qv jfj 2 q dx ð@ t VÞe qv jfj 2 R d x R d x which, using the Poisson equation again, can be used to show that some appropriate convex functional of the solution F decays (see Ref. [8]). 3. THE SPACE TIME DISCRETIATION We now turn to the discretization of the hyperbolic system of conservation laws (10) in space and time. In principle, any numerical method, suitable for hyperbolic systems would be applicable. However, we will make use of the special structure of the system which is due to the entropy based construction in the previous section. Moreover, there are some features of the system (10) which are special to the discretization of the Boltzmann equation. Standard methods for hyperbolic conservation laws + [ :05am] [ ] [Page No. 439] i:/mdi/tt/31(4-6)/ _tt_031_4-6_r1.3d Transport Theory and Statistical Physics (TT)

10 440 RINGHOFER would treat the field term and the associated matrices B, the approximations of the wave vector derivative, as a source term, when in fact, it is a transport term and part of the free streaming operator which as a whole is skew self adjoint under the weight function e qv. Moreover, we will also derive a spatial discretizaton, which is suitable for the direct solution of the steady state problem, while in the usual methods for hyperbolic conservation laws steady states are computed by letting the transient problem converge to a steady state, which is a terribly expensive approach. We will here only outline the basic idea of the approach and refer the reader to the papers of Refs. [6,7] for the details. The first idea is based on the observation that the free streaming operator L in Eq. (1) maps even functions of k into odd functions and vice versa as long as the band energy function " is even, which can always be assumed. So from now on, we assume that "ðkþ ¼"ð kþ holds. Accordingly, we split the kinetic density function f in its even and odd parts f ¼ f e þ f o, f e ðx, k, tþ ¼f e ðx, k, tþ, f o ðx, k, tþ ¼ f o ðx, k, tþ, and obtain the system of Boltzmann t f e þ Lð f o Þþ 1 Qe ð f e, f o t f o þ Lð f e Þþ 1 Qo ð f e, f o Þ¼0, where Q e, Q o denote the even and odd parts of the collision operator, Q. Moreover, a closer inspection of the collision operator Q in Eq. (2) yields that, since the scattering cross sections are independent of the angular variables, Q maps even functions into even functions of k and odd functions into odd functions; so Q e ð f e, f o Þ¼Qðf e Þ, Q o ð f e, f o Þ¼Q o ð f o Þ¼ð"Þf o, ð"þ ¼e " Sð", " 0 Þ dk 0 holds. If we split the basis function space of Maxwellians times spherical harmonics into even and odd basis functions, and reorder the coefficient vector F such that the first components correspond to even basis functions and the last component correspond to odd basis functions, we obtain that the matrices in the semi-discrete Boltzmann Eq. (10) are of the form F ¼ F e, F o A ¼ 0 Aeo, A oe B ¼ 0 Beo B oe 0 0, C ¼ Cee 0 0 C oo, + [ :05am] [ ] [Page No. 440] i:/mdi/tt/31(4-6)/ _tt_031_4-6_r1.3d Transport Theory and Statistical Physics (TT)

11 SEMICONDUCTOR BOLTMANN EQUATION and we obtain the system (10) in the form t F e þ L eo F o þ 1 Cee F e ¼ 0, t F o þ L oe F e þ 1 Coo F o ¼ 0, ðcþ ðdþ L eo F o ¼ Xd L oe F e ¼ Xd A x F o qð@ x VÞB eo F o, ¼1 ¼1 A x F e qð@ x VÞB oe F e The symmetry property (11) becomes, expressed in this split form A eo ¼ðA oe Þ T ¼ B eo ðb oe Þ T ð17þ ð18þ There are of course many approaches one can take for the spatial discretization of the system (17), which will in general depend on the grid structure used and the desired order of the space time discretization. Rather than going into the details of a specific discretization, we will outline the basic underlying idea. Generally speaking, we would like the discretization to exhibit the following features:. The spatial discretization should be suitable for a direct solution of the steady state problem without having to iterate the transient method to a steady state.. The discretization should reproduce the entropy estimate (15) on the discrete level.. The discretization should yield the correct diffusion limit in the Hilbert limit! 0.. In regimes other than the Hilbert limit, where the behavior of the solution is dominated by transient waves, we would like to minimize the amount of artificial diffusion necessary to solve the system. Of course, the third and the fourth items are somewhat contradictory. We will start with the spatial discretization. The basic idea is to use a conservative difference discretization for the even moment Eq. (17a) and to use a discretization for the odd moments which guarantees a discrete version of the entropy estimate (15). To this end, it is essential to discretize the even and odd coefficient vectors F e and F o on adjoint meshes, i.e., we will use a staggered grid approach. Difference operators approximating the partial derivatives in x direction will map one mesh into the other. One problem arises in this approach, since acceleration terms B F will connect these meshes as well and therefore, some interpolation would be needed. The easiest way to + [ :05am] [ ] [Page No. 441] i:/mdi/tt/31(4-6)/ _tt_031_4-6_r1.3d Transport Theory and Statistical Physics (TT)

12 442 RINGHOFER circumvent this problem by rewriting the operator L eo in Eq. (17a) as L eo F o ¼ Xd x ða eo F o qvb eo F o ÞþqVB x F o : ð19þ This way only spatial derivatives of the odd coefficient vector F o appear in the equation for F e, and, using a staggered grid, these spatial derivatives are defined on the same mesh as F e. For the spatial discretization of L oe in Eq. (17b), we use the fact that L is skew self adjoint under the L 2 scalar product using the weight function e qv. Therefore, L oe is the negative adjoint of L eo under this weight function and is discretized as L oe F e ¼ Xd ¼1 ða eo qvb eo Þ T e x ðe qv F e ÞþðB eo Þ T e x ðqve qv F e Þ : ð20þ Using the identity (18), a direct calculation shows that this form of L oe is equivalent to Eq. (17d). Note that, when written in this form the skew self adjoint property dx e qv ðf e Þ T L eo F o þ e qv ðf o Þ T L oe F e ¼ 0 R d x is immediate. It is in this form that the spatial discretization is carried out. Given difference approximations D eo, D oe of the partial derivatives on the even and odd meshes which satisfy a discrete integration by parts formula of the form I e ½g e ðd eo g o ÞŠ ¼ I o ½g o ðd oe g e ÞŠ ð21þ for any grid functions g e and g o defined on the corresponding meshes. Here, I e, I o denote some suitable integration rules on the even and odd meshes. The semi-discrete system (17) is then replaced by t F e þ L eo F o þ 1 Cee F e ¼ 0, t F o þ L oe F e þ 1 Coo F o ¼ 0, ðcþ ðdþ L eo F o ¼ Xd ¼1 L oe F e ¼ Xd ¼1 ½D eo ða eo F o qvb eo F o ÞþqVB eo D eo F o Š, h ð A eo qvb eo Þ T e qv D oe e qv F e þðb eo Þ T e qv D oe qve qv F e i, ð22þ + [ :05am] [ ] [Page No. 442] i:/mdi/tt/31(4-6)/ _tt_031_4-6_r1.3d Transport Theory and Statistical Physics (TT)

13 SEMICONDUCTOR BOLTMANN EQUATION where, for simplicity, we have used the same symbols L eo, L oe for the difference and differential operators. Because of reformulating the free streaming operators L eo and L oe in the semi-discrete Boltzmann equation in the form (19), (20) they now map the mesh for the even expansion coefficients into the mesh for the odd coefficients and vice versa. However, the potential V is now needed on the even as well as on the odd meshes. More precisely, Eq. (22c,d) should really read ðaþ ðbþ L eo F o ¼ Xd ¼1 L oe F e ¼ Xd ¼1 ½D eo ða eo F o qv o B eo F o ÞþqV e B eo D eo F o Š, ½ðA eo qvb eo Þ T e qvo D oe ðe qv e F e Þ þðb eo Þ T e qvo D oe ðqv e e qve F e ÞŠ, ð23þ where V e is given on the even mesh by the solution of the discrete Poisson equation Xd ¼1 D eo D oe V e þ q D dop 1 F000 e ¼ 0: 000 Thus, we still need a suitable interpolation formula to compute V o in Eq. (23) from V e. This formula is usually chosen in such a way that the resulting scheme reduces to the well known Scharfetter-Gummel scheme for the Drift-Diffusion equations in the Hilbert limit (see Refs. [6,7] for the details of the interpolation formula and c.f. Ref. [11] for an overview of the properties of the Scharfetter Gummel scheme). Note that, by virtue of construction, the entropy estimate I e e qv ðf e Þ t F e þ Io e qv ðf o Þ t F o ¼ 1 2 I e e qv ðf e Þ T C ee F e þ Io e qv ðf o Þ T C oo F o 0 holds for the resulting system of ordinary differential equations. The discretization Eq. (22) is particularly convenient for the solution of the steady state problem t ¼ 0 in Eq. (22a,b)). In this case the odd coefficient vector F o can be eliminated altogether by locally inverting C oo and inserting into Eq. (22a). C oo is invertible since the kernel of the collision operator only exists of even functions. So, in the steady state case the equation 2 L eo ðc oo Þ 1 L oe F e þ C ee F e ¼ 0 ð24þ + [ :05am] [ ] [Page No. 443] i:/mdi/tt/31(4-6)/ _tt_031_4-6_r1.3d Transport Theory and Statistical Physics (TT)

14 444 RINGHOFER is solved. By virtue of construction, the operator L eo ðc oo Þ 1 L oe F e is a self adjoint positive semi-definite second order difference operator. If only one expansion term (the charge density) for F e is used, Eq. (24) reduces to a standard discretization for the resulting steady state drift-diffusion equations. So, although the momentum as well as the space time discretization have been derived on the basis of entropy principles, which are inherently transient concepts, the resulting structure of self adjoint and skew self adjoint operators are essential for the numerical solution of the steady state problem as well. It can be shown (see Ref. [7]) that the linearization of the resulting scheme in the steady state case is stable precisely because of this structure. When discretizing the conservation laws (17) in the temporal direction, a distinction has to be made as to the purpose of solving the time dependent problem. Either the time dependent problem is solved mainly to compute a steady state. In this case the focus of the design will be on dealing with the collision terms and with the resulting temporal stiffness close to the Maxwellian regime efficiently. This is the idea pursued in Refs. [9,10]. The resulting methods will necessarily exhibit a large amount of diffusion since they are designed to solve the system close to the diffusive regime. We are, on the other hand interested in approximate solutions to the Boltzmann equation quite far away from the Maxwellian regime, i.e., for values of the scaled Knudsen number close to ¼ 1. In this case the behavior of the solution will be dominated by dispersive waves, due to the acceleration induced by the electric field (see Ref. [6]). In order to compute accurately the transient behavior of the system, it is necessary to resolve these waves with as little artificial diffusion as possible. Splitting the system into even and odd parts provides a natural way to do this, namely by effectively discretizing a second order wave equation with central differences in space and time. To compute a transient solution, we discretize Eq. (22) as ðaþ ðbþ t ½F e ðt þ tþ F e ðtþš þ L eo F o ðtþþ 1 Cee F e ðt þ tþ ¼0, t ½F o ðt þ tþ F o ðtþš þ L oe F e ðt þ tþ þ 1 Coo F o ðt þ tþ ¼0: The Scheme (25) is explicit in time (except, of course, for the solution of the Poisson equation) in the following sense. Given the solution at time tf e at the next time step t þ t is computed from Eq. (25a). Once F e ðt þ tþ is known F o at the next time step can be computed from Eq. (25b). The collision terms are always taken implicitly to avoid any resulting stiffness problems for <<1. A rather straight forward analysis (see Ref. [6]) ð25þ + [ :05am] [ ] [Page No. 444] i:/mdi/tt/31(4-6)/ _tt_031_4-6_r1.3d Transport Theory and Statistical Physics (TT)

15 SEMICONDUCTOR BOLTMANN EQUATION gives a CFL condition of the form t Oðx= max Þ, where x denotes the spatial mesh-size and max denotes the maximal spectral radius of the matrix A in Eq. (10). A condition of this form has to be expected, since the largest wave speeds in Eq. (17) will be of order Oð1=Þ. Moreover, the Scheme (25) also reproduce the correct Hilbert limit for small Knudsen numbers. For <<t, Eq. (25b) becomes approximately F o ðtþ ¼ ðc oo Þ 1 L oe F e ðtþ, and Eq. (25b) then becomes t ½F e ðt þ tþ F e ðtþš L eo ðc oo Þ 1 L oe F e ðtþþ 1 Cee F e ðt þ tþ ¼0, which yields the right diffusive limit. More importantly, however, the Scheme (25) minimizes diffusion. This can be seen by neglecting the collision terms in Eq. (25). In the absence of collisions, differencing Eq. (25a) in time and inserting from Eq. (25b) gives 2 t 2 ½F e ðt þ tþ 2F e ðtþþf e ðt tþš L eo L oe F e ðtþ ¼0, which is the centered difference in space and time approximation to the second order wave equation 2 t f e ½ðr k "Þr x ðkþ qðr x VÞr k Š 2 f e ¼ 0: This scheme has the property that it exhibits no artificial diffusion, i.e., all the eigenvalues of the resulting linear operator lie on the unit circle as long as the CFL condition is satisfied. It is usually considered as mildly unstable, since round-off errors and other perturbations will grow linearly in time. However, the damping terms introduced by the collision operators usually take care of this linear growth in practice. Thus the only damping in the Scheme (25) arises from the presence of the collision operators, which is precisely the same behavior as for the analytical solution. This property of minimizing the artificial diffusion is extremely important when computing the transient behavior of the problem, since, c.f. rise times (the time it takes to get from one steady state to another approximate steady state) will be extremely sensitive to the propagation of electron waves and artificial diffusion will result in underestimating these rise times. 4. A NUMERICAL TEST EXAMPLE In this section, we present as a numerical test example the simulation of a standard n þ n n þ silicon diode in one spatial dimension and in steady state. The purpose of this example is to demonstrate that the + [ :05am] [ ] [Page No. 445] i:/mdi/tt/31(4-6)/ _tt_031_4-6_r1.3d Transport Theory and Statistical Physics (TT)

16 RINGHOFER presented method is capable to simulate situations which are quite far from equilibrium. In one spatial dimension the choice of mesh difference operators D eo 1, D oe 1 in Eq. (21) is quite obvious. We define the grids for the even and odd coefficient vectors F e, F o by M e ¼fx 0 < < x J g, M o ¼ y j : y j ¼ 1 2 ðx j þ x jþ1 Þ, j ¼ 0,..., J 1 and the difference operators by ðd oe u e Þðy j Þ¼ ue ðx jþ1 Þ u e ðx j Þ x jþ1 x j, ðd eo u o Þðx j Þ¼ uo ðy j Þ u o ðy j 1 Þ y j y j 1 : If we define the discrete integral operators I e, I o in the usual way D eo and D oe satisfy a discrete integration by parts formula of the form (21). The n þ n n þ silicon diode has a channel length of 50 nm. This means that the doping concentration D dop in Eq. (1c) is given by a step function of the form m 3 0 < x < 50 nm D dop ðxþ@ m 3 50 nm < x < 100 nm A m nm < x < 150 nm The results below were obtained by varying the applied bias (the difference of the potential V between the right and left endpoint) from 0:1V to 0:5V. The phonon emission/absorption energy! in Eq. (4) was chosen as ev, and computations were performed using a parabolic band structure ("ðkþ ¼ðjkj 2 =2Þ). The plots in Figs. 1 8 were obtained using a uniform mesh with 150 gridpoints in the x-direction (x ¼ 1 nm), and with 16 expansion terms in the energy direction and eight terms in the angular direction. (For the one dimensional problem the solution can be expected to be cylindrically symmetric around the k 1 direction.) So a system of 128 conservation laws has been solved. It was verified that doubling the number of expansion terms or the number of gridpoints does not significantly change the obtained pictures. Figures 1 4depict the potential as well as the physical densities for electron, electron velocity and electron energy as functions of the spatial variable x. Figure 1 shows the potential distribution VðxÞ. Figure 2 shows the electron density h1iðxþ ¼ R f ðx, kþ dk. Figure 3 shows R the velocity distribution uðxþ ¼ðh@ k1 "iðxþ=h1iðxþþ with h@ k1 "ðkþ f ðx, kþ dk. Figure 4shows the corresponding energy densities given by wðxþ ¼ðh"iðxÞ=h1iðxÞÞ. Figures 5 8 depict the kinetic density f ðx, kþ for the bias value of 0.4V for various values of x. In one spatial dimension the kinetic density f will be F1 F8 + [ :05am] [ ] [Page No. 446] i:/mdi/tt/31(4-6)/ _tt_031_4-6_r1.3d Transport Theory and Statistical Physics (TT)

17 SEMICONDUCTOR BOLTMANN EQUATION Figure 1. Potential. Figure 2. Particle Density. + [ :05am] [ ] [Page No. 447] i:/mdi/tt/31(4-6)/ _tt_031_4-6_r1.3d Transport Theory and Statistical Physics (TT)

18 448 RINGHOFER Figure 3. Velocity. Figure 4. Energy Density. + [ :05am] [ ] [Page No. 448] i:/mdi/tt/31(4-6)/ _tt_031_4-6_r1.3d Transport Theory and Statistical Physics (TT)

19 _TT_031_4 6_R1.pdf SEMICONDUCTOR BOLTMANN EQUATION Figure 5. r2*f: Bias ¼ 0.4 V, x ¼ nm Figure 6. r2*f: Bias ¼ 0.4 V, x ¼ 50 nm. + [ :05am] [ ] [Page No. 449] i:/mdi/tt/31(4-6)/ _tt_031_4-6_r1.3d Transport Theory and Statistical Physics (TT)

20 _TT_031_4 6_R1.pdf 450 RINGHOFER Figure 7. r2*f: Bias ¼ 0.4 V, x ¼ 100 nm Figure 8. r2*f: Bias ¼ 0.4 V, x ¼ nm. + [ :05am] [ ] [Page No. 450] i:/mdi/tt/31(4-6)/ _tt_031_4-6_r1.3d Transport Theory and Statistical Physics (TT)

21 SEMICONDUCTOR BOLTMANN EQUATION cylindrically symmetric pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi around the k 1 axis. So Figs. 5 8 show the function f plotted over k 1 and k 2 2 þ k2 3 for fixed values of x. The purpose of these computations is to investigate how far away from the fluid dynamic regime the presented approach is applicable. (For such a short channel and the given applied bias, we expect to see some distinctly non-equilibrium phenomena). Figure 5 shows the distribution to the left of the channel. It is essentially given by a forward and a backward traveling Maxwellian of roughly the same amplitude. Figures 6 and 7 show the distribution at the beginning and the end of the channel. First, we notice that the wave has developed a second peak in the forward as well as in the backward traveling component. Moreover, it has become definitely asymmetric in the k 1 direction at the end of the channel (in Fig. 7). After the electron has left the channel in Fig. 8 remnants of the second peak are still visible, but the solution has become symmetric in k 1 again. Figure 5 could have been produced by a hydrodynamic model or even by drift-diffusion equations, considering the low values of the group velocity in Fig. 3 at this point. The second peak in Fig. 8 (after leaving the channel) could have been produced by a SHE-model. However, the strongly asymmetric distribution at the channel end (Fig. 7) is a truly kinetic phenomenon and could not have been produced by either of the above approximations. 5. CONCLUSIONS We have presented a methodology to discretize the Boltzmann equation for semiconductors based on entropy principles by a spectral approximation in the wave vector direction and by difference methods in the space time direction. The use of entropy principles results in certain self adjoint properties of the involved discrete operators which are essential for an efficient numerical solution of the resulting discrete equations. One of the key ideas in the presented approach was to split the kinetic density function in its even and odd parts, and to effectively solve a system of second order wave equations in the transient case and a second order elliptic problem in the steady state case. Although the wave vector discretization consists effectively of polynomial corrections to Maxwellians the resulting discrete equations are capable of handling situations quite far away from thermodynamic equilibrium very well. ACKNOWLEDGMENT This work was supported by NSF grants DMS and INT (Submitted to TTSP, July 2001). + [ :06am] [ ] [Page No. 451] i:/mdi/tt/31(4-6)/ _tt_031_4-6_r1.3d Transport Theory and Statistical Physics (TT)

22 452 RINGHOFER REFERENCES 1. Cercignani, C. The Boltzmann Equation and its Applications; Applied Mathematical Sciences, Springer Verlag: 1988; Vol Goldsman, N.; Henrickson, L.; Frey, J. A Physics Based Analytical Numerical Solution to the Boltzmann Equation for use in Semiconductor Device Simulation. Solid State Electr. 1991, 34, Levermore, D. Moment Closure Hierarchies for Kinetic Theories. J. Stat. Phys. 1996, 83, Majorana, A.; Pidatella, R. A Finite Difference Scheme for Solving the Boltzmann Poisson System for Semiconductor Devices, Manuscript Majorana, A. Spherical Harmonics Type Expansion for the Boltzmann Equation in Semiconductor Devices. Le Matematice LIII 1998, Ringhofer, C. Space Time Discretization Methods for Series Expansion Solutions of the Boltzmann Equation for Semiconductors. SIAM J. Numer. Anal. 2000, 38, Ringhofer, C. A Mixed Spectral-Difference Method for the Steady State Boltzmann Poisson System. Submitted, SIAM J. Numer. Anal Ringhofer, C. An Entropy-Based Finite Difference Method for the Energy Transport System. Math. Mod. Meth. in Appl. Sci. 2001, 11, Schmeiser, C.; wirchmayr, A. Galerkin Methods for the Semiconductor Boltzmann Equation. Proc. ICIAM 95, Hamburg, Schmeiser, C.; wirchmayr, A. Convergence of Moment Methods for the Semiconductor Boltzmann Equation. SIAM J. Num. Anal. 1998, 36, Selberherr, S. Analysis of Semiconductor Devices, 2nd Ed.; Wiley: New York, Ventura, D.; Gnudi, A.; Baccarani, G.; Odeh, F. Multidimensional Spherical Harmonics Expansions for the Boltzmann Equation for Transport in Semiconductors. Appl. Math. Let. 1992, 5, Received July 4, 2001 Revised May 19, 2002 Accepted May 30, [ :06am] [ ] [Page No. 452] i:/mdi/tt/31(4-6)/ _tt_031_4-6_r1.3d Transport Theory and Statistical Physics (TT)