The question of possibility to build additional entropy conservation law for the system of moment equations, which appear in mathematical modelling of the charge transport in semiconductors, is discussed

Blokhin, A. M.

Sokovikov, I. G.

Le Matematiche (Dipartimento di Matematica e Informatica, Università degli Studi di Catania)

LE MATEMATICHEVol. LX (2005) – Fasc. I, pp. 29–40ENTROPY APPROXIMATION FOR THE SYSTEM OFMOMENT EQUATIONS DESCRIBING CHARGETRANSPORT IN SEMICONDUCTORSA. M. BLOKHIN - I. G. SOKOVIKOVThe question of possibility to build additional entropy conservation lawfor the system of moment equations, which appear in mathematical modellingof the charge transport in semiconductors, is discussed.Introduction.It is well known that so called hydrodynamic models are widely usednowadays in mathematical modelling of the physical phenomena related tocharge transport in semiconductors. Last years, a lot of new mathematicalmodels of hydrodynamic type were proposed for describing similar phenomena.Notice, however, that mathematical basis for most models of this type is notideal.One of the last models of hydrodynamic type was proposed recently in thepapers [1], [8]. This model is a quasilinear system of equations written in theform of the conservation laws. These conservation laws were obtained from thesystem of moment equations for Boltzmann transport equation by the use of acertain closing procedure.However, additional entropy conservation law was not found for theequations system obtained in [1], [8]. In this work, we propose so called entropyEntrato in redazione il 4 Marzo 2004.30 A. M. BLOKHIN - I. G. SOKOVIKOVapproximation for this mathematical model. We discuss here one of the methodsof such approximation application — construction of calculation models forfinding of approximate solutions to the conservation laws system from [1], [8].1. Preliminary information.Remind that in [1], [8] the system of moment equations describing chargetransport in semiconductors was proposed. That system was obtained fromthe Boltzmann transport equation with use of entropy maximum principle.Application of this principle is, in essence, closing procedure (just diversityof closing procedures defines a large number of various mathematical models,which are used to describe charge transport in semiconductor devices).Following [3], write down a quasilinear nonstationary system of mentionedabove moment equations for two-dimensional case and in the dimensionlessform (the process of obtaining dimensionless form is described in detail in [3]):(1.1)⎧⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎩Rτ + divJ = 0,Jτ + ∇(R + P) = RQ+ c11J+ c22θ,32Pτ + div(J+ θ ) = (J,Q)+ cP,25θτ + ∇(P + P2R)= PQ+ c12J+ c22θ.Here R is an electron density;J = (J (x), J (y)) is a electron flow vector with components J (x) , J (y) (Cartesiancoordinates (x , y) are taken as a base);P = R(23E − 1);E is electron energy;θ = (θ (x), θ (y)) is a electron energy flow vector;Q = ∇ϕ; ϕ = ϕ(τ, x , y)) is an electric potential, satisfying Poisson equation:(1.2) 2ϕ = R − ρ, 2 = 1β;coefficients c, c11, . . ., c22 in (1.1) are smooth functions of E (exact form ofthem is presented in [1], [3], [8]);ρ = ρ(x , y) is an doping density;β > 0 is some constant (see [3]).ENTROPY APPROXIMATION FOR THE SYSTEM OF. . . 31It was shown in works [3], [7] that hyperbolicity condition of the system(1.1) is formulated as:E > 0.This inequality is physically defensible.Remind again that system of moment equations (1.1) (see. [3]), writtenin the form of conservation laws, was obtained in [1, 8] with use of entropymaximum principle. However, additional entropy conservation law(1.3) ητ + div = G,where η = η(U),  = ((x), (y)), (x),(y) = (x),(y)(U), G = G(U,Q), andwhich would be consequence of all equations of the system (1.1) and fulfilledon any its smooth solution, was not obtained in works mentioned above.Here η is entropy function; is entropy flow vector;G is entropy production;U =⎛⎜⎝RJPθ⎞⎟⎠.We failed to obtain additional entropy conservation law for conservationlaws system (1.1) as well. Therefore, in this work, we propose to consider thefollowing system instead of the system (1.1):(1.4)⎧⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎩Rτ + divJ = 0,Jτ + ∇(R + P) = RQ+ c11J+ c22θ,32Pτ + div(J+ θ ) = (J,Q) + cP,25θτ + ∇(P + F ′(D, x , y)) = PQ+ c12J+ c22θ,whereD = 32P − R + D0(x , y),F(D, x , y), D0(x , y) are some functions,F ′(D, x , y) = ∂F∂D.We call this system entropy approximation of initial system (1.1).32 A. M. BLOKHIN - I. G. SOKOVIKOVRemark 1.1. With use of simple but bulky computations, it could be shownthat hyperbolic condition of the system (1.4) is formulated as:(1.5) F ′′ = ∂2F∂D2(D, x , y) > −25.2. Obtaining of additional entropy conservation law for the system (1.4).Rewrite the system (1.4) as:(1.4) Pτ +Mx + Ny = F .HereP =⎛⎜⎜⎜⎜⎝RJ32P25θ⎞⎟⎟⎟⎟⎠ , M =⎛⎜⎜⎜⎜⎜⎝J (x)R + P0J (x) + θ (x)P + F ′0⎞⎟⎟⎟⎟⎟⎠ , N =⎛⎜⎜⎜⎜⎜⎝J (y)0R + PJ (y) + θ (y)0P + F ′⎞⎟⎟⎟⎟⎟⎠ ,F =⎛⎜⎝0RQ+ c11J+ c12θ(J,Q)+ cPPQ+ c12J+ c22θ⎞⎟⎠ .Multiply now the system (1.4) by the vector 2U scalar. We obtain as a result:(2.1){R2 + |J|2 + 32P2 + 25|θ |2}τ++ div{2(R + P)J + 2Pθ} + 2(θ,∇F ′) == ([2RJ + 2PJ + 2Pθ ],Q)+ 2cP2 + 2{c11|J|2 + μ12(J, θ )+ c22|θ |2},μ12 = c12 + c21.Extracting the first equation of the system (1.4) from the fourth one, we obtain:(2.2) divθ = (J,Q) + cP − Dτ .ENTROPY APPROXIMATION FOR THE SYSTEM OF. . . 33Since(2.3) (θ,∇F ′) = div(F ′θ )− F ′divθ,then in view of (2.2), (2.3) we rewrite (2.1) in the form (1.3) with the followingexpressions for η, , G :(2.4)⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩η = R2 + |J|2 + 32P2 + 25|θ |2 + 2F, = 2(R + P)J+ 2(P + F ′)θ,G = ([2(R + P + F ′)J+ 2Pθ ],Q)+ 2(P + F ′)cP ++ 2{c11|J|2 + μ12(J, θ )+ c22|θ |2}.3. Symmetrization of the system (1.4).It is known (see [4], [5]), that existence of additional conservation law(1.3) allows rewriting the system (1.4) in the form of symmetric t-hyperbolic(by Friedrichs) system. Symmetrization formalism is described in detail in [4],[5], and we present here just some essential remarks. Let’s define componentsof the vector q:q =⎛⎜⎜⎜⎜⎜⎝q1q2q3q4q5q6⎞⎟⎟⎟⎟⎟⎠ ,such as(q,Pτ +Mx + Ny −F ) = ητ + div − G = 0.To gain this we present differential dη in the following form (see (2.4)):dη = 2(R − F ′)dR + 2(J, dJ) ++ 2(P + F ′)d(32P)+ 2(θ, d(25θ))= (q, dP).Consequently, components of the vector q (so caleed canonic variables) couldbe defined as:(3.1)q1 = 2(R − F ′), q2 = 2J (x), q3 = 2J (y),q4 = 2(P + F ′), q5 = 2θ (x), q6 = 2θ (y).34 A. M. BLOKHIN - I. G. SOKOVIKOVNow define so called generating functionsL,M, N :(3.2){L = (q,P)− η,M = (q,M)− (x),N = (q,N)− (y).Then, in view of (3.2), we obtain:(3.3) Lq =⎛⎝Lq1...Lq6⎞⎠ = P, Mq = M, Nq = N.Taking into account (3.3), we rewrite the system (1.4) as:(Lq)τ + (Mq)x + (Nq)y = F ,and then in the following form:(3.4) Aqτ +Bqx + Cq y = F ,where A = (Lqiqj ), B = (Mqiqj ), C = (Nqiqj ), i, j = 1, 6 are symmetricmatrices.Let’s obtain matrices A, B, C. For this, we use the method described in[4], [5]. We find at first matrices I , I0, I (x) , I (y) , such as:dq =⎛⎝ dq1...dq6⎞⎠ = IdU (see (3.1)),dLq =⎛⎝ dLq1...dLq6⎞⎠ = I0dU,dMq = I (x)dU,dNq = I (y)dU.ThenA = I0 I−1 = 12⎛⎜⎜⎜⎜⎜⎜⎜⎜⎝2+ 3F ′′2+ 5F ′′ 0 03F ′′2+ 5F ′′ 0 00 1 0 0 0 00 0 1 0 0 03F ′′2+ 5F ′′ 0 03(1+ F ′′)2+ 5F ′′ 0 00 0 0 0 25 00 0 0 0 0 25⎞⎟⎟⎟⎟⎟⎟⎟⎟⎠,ENTROPY APPROXIMATION FOR THE SYSTEM OF. . . 35B = I (x) I−1 = 12⎛⎜⎜⎜⎜⎜⎝0 1 0 0 0 01 0 0 1 0 00 0 0 0 0 00 1 0 0 1 00 0 0 1 0 00 0 0 0 0 0⎞⎟⎟⎟⎟⎟⎠ ,C = I (y) I−1 = 12⎛⎜⎜⎜⎜⎜⎝0 0 1 0 0 00 0 0 0 0 01 0 0 1 0 00 0 1 0 0 10 0 0 0 0 00 0 0 1 0 0⎞⎟⎟⎟⎟⎟⎠ .Notice, that matrix A > 0 when hyperbolicity condition (1.5) is fulfilled (itcould be verified directly). Hence, the system (3.4) is symmetric t -hyperbolic(by Friedrichs).4. Another form of the system (1.4) when function F specially selected.Symmetric system (3.4) is one of the forms of the system (1.4). One caneasy find with simple calculations that:(4.1)Aq = P +,Bq =M,Cq = N,where =⎛⎜⎜⎜⎜⎜⎜⎜⎜⎝2[F ′′(D − D0)− F ′]2+ 5F ′′00−2[F′′(D − D0)− F ′]2+ 5F ′′00⎞⎟⎟⎟⎟⎟⎟⎟⎟⎠.Let(4.2)F ′′(D − D0) − F ′2+ 5F ′′ =15D1.Here D1 = D1(x , y) is a certain function.36 A. M. BLOKHIN - I. G. SOKOVIKOVConsidering (4.2) as equation to find function F , we take such its solution:(4.3) F(D, x , y) = − 2D15(D0 + D1) ·D22,i.e.F ′ = − 2D15(D0 + D1)D, F′′ = − 2D15(D0 + D1) .Introduce into consideration functions R∗(x , y) > 0, E∗(x , y) > 0 andP∗(x , y) = R∗(x , y)(23E∗(x , y) − 1). Assume that P(τ ∗, x , y) = P∗(x , y),R(τ ∗, x , y) = R∗(x , y) are valid for solution to the system (1.1) under certainτ = τ ∗ .Choose functions D0, D1 such as:(4.4) − 2D15(D0 + D1)(32P∗ − R∗ + D0)= (P∗)2R∗.If D1 = −P∗ , D0 = R∗ + P∗ , then equality (4.4) is fulfilled. Hence, if functionF ′ in the system (1.4) is selected as:(4.5) F ′ = 25P∗R∗(32P − R + R∗ + P∗),then this aggregate is linear approximation of the function P2/R in neighbor ofτ = τ ∗ (see (4.4)). We call system (1.4) entropy approximation of the system(1.1) by this reason namely. Notice, that F ′′ = 25 P∗R∗ = − 25(1− 23 E∗)> − 25in this case in view of natural physical restriction: E∗ > 0.So, let aggregate F ′ in the system (1.4) is found with use the formula (4.5).Then, in view of (4.1), (4.2), the system (1.4) can be rewritten as follows:(4.6) (Aq)τ + (Bq)x + (Cq)y = F .System (4.6) is another form of the system (1.4) (when function F ′ is speciallyselected).ENTROPY APPROXIMATION FOR THE SYSTEM OF. . . 375. Predictor-corrector type difference scheme for the system (1.1).The fact that the system (1.4) can be written in the form (4.6), suggestsan idea to use this fact when constructing finite-difference schemes to initialsystem (1.1). Let’s describe, for example, possible construction of the finite-difference scheme of the predictor-corrector type for the system (1.1). Weplot difference grid with steps  = τ , hx = x , hy = y in the domainR3+ = {τ > 0, (x , y)∈ R2}. Introduce new designations:Uni j = U(n, ihx , j hy) = Un,x , y, −1x , −1y are shift operators:±1x Un = Uni±1, j , ±1y Un = Uni, j±1;ξx , ξ x , ξy, ξ y, Lx , Ly, ξ0, η0 are difference operators:ξx = x − 1, ξ x = 1− −1x , ξy = y − 1, ξ y = 1− −1y ,Lx = x + −1x2, Ly =y + −1y2,ξ0 = x − −1x2, η0 =y − −1y2,rx = hx , ry =hy.Besides, we assume, that(5.1) B = B+ − B−, C = C+ − C−,where B± , C± are symmetric positive-definite matrices. It is known, thatexpansion (5.1) for arbitrary symmetric matrix is always fulfilled.We can takeB± = 12⎛⎜⎜⎜⎜⎜⎝1 ±1/2 0 0 0 0±1/2 1 0 ±1/2 0 00 0 1 0 0 00 ±1/2 0 1 ±1/2 00 0 0 ±1/2 1 00 0 0 0 0 1⎞⎟⎟⎟⎟⎟⎠ ,38 A. M. BLOKHIN - I. G. SOKOVIKOVC± = 12⎛⎜⎜⎜⎜⎜⎝1 0 ±1/2 0 0 00 1 0 0 0 0±1/2 0 1 ±1/2 0 00 0 ±1/2 1 0 ±1/20 0 0 0 1 00 0 0 ±1/2 0 1⎞⎟⎟⎟⎟⎟⎠ .for matrices B, C. Notice, thatB+ + B− = I6, C+ + C− = I6 , where I6 isunitary matrix of the order 6. Eigen values of matrices B± , C± can be easyfound:lk (B±,C±) = 12(1− cos(kl)), l = π5, k = 1, 4;l5,6(B±,C±) = 12 .Calculational model, which we will use to find approximate solution to initialsystem (1.1), is build as follows. At the predictor stage we use the system (1.4),written in the form (4.6). Using difference scheme from the work [6] (see also[2]), we find auxiliary quantities q∗1,...,6:Aq∗ − Aqn + r∗x ξ x{B+qn} + r∗y ξ y{C+qn} −− r∗x ξx {B−qn} − r∗y ξ y{C−qn} = ∗F ni jor(5.2) Aq∗ = (A − (r∗x + r∗y )I6)qn + r∗x (B+qni−1, j + B−qni+1, j )++ r∗y (C+qni, j−1 + C−qni, j+1)+ ∗F ni j .Here (see sec. 3 and sec. 4):A = A(F ′′), F ′′ = 25Pni jRni j,q∗ =⎛⎝ q∗1...q∗6⎞⎠ is the vector of auxiliary quantites,0 < ∗ ≤ , r∗x = ∗hx , r∗y = ∗hy .ENTROPY APPROXIMATION FOR THE SYSTEM OF. . . 39Using formulae (3.1), we find components of auxiliary vector U∗ by compo-nents of the vector q∗ , and as F ′ we take the following expression:F ′ = 25Pni jRni j(32P∗i j − R∗i j + Rni j + Pni j).At the corrector stage we use initial system (1.1), which we rewrite as:(5.3) Uτ + Vx +Wy = Z,whereV =⎛⎜⎜⎜⎜⎜⎜⎜⎜⎝J (x)0R + P23(J (x) + θ (x))52(P + P2R)0⎞⎟⎟⎟⎟⎟⎟⎟⎟⎠, W =⎛⎜⎜⎜⎜⎜⎝J (y)R + P23(J (y) + θ (y))052(P + P2R)⎞⎟⎟⎟⎟⎟⎠ ,Z =⎛⎜⎜⎜⎜⎝0RQ+ c11J+ c12θ23(J,Q)+ 23cP52(PQ+ c21J+ c22θ )⎞⎟⎟⎟⎟⎠ .To find numeric solution to the system (5.3), we use, for example, modified Laxscheme (see [2]):(5.4) Un+1 = Lx LyUn − rx Lyξ0V∗ − ry Lxη0W∗ + Lx LyZ∗,where V∗ = V(U∗), etc.Described above difference model (5.2), (5.4) is one of the possible vari-ants of using entropy approximation of the system (1.1) when finding its approx-imate solution. In the future, authors plan to make the series of computationalexperiments on using entropy approximation to find numeric solutions to typicalproblems of semiconductors physics.Authors appreciate V. Romano for useful discussions of this work materi-als. The work is supported by RFFI (project N. 04-01-00900); INTAS, projectN. 868.40 A. M. BLOKHIN - I. G. SOKOVIKOVREFERENCES[1] A.M. Anile - V. Romano, Non parabolic band transport in semiconductors:closure of the moment equations, Cont. Mech. Thermodyn., 11 (1999), pp. 307–325.[2] A.M. Blokhin - R.D. Alaev, Energy integrals and its applications in investigationof difference schemes stability, Novosibirsk, NSU publishing, 1993.[3] A.M. Blokhin - R.S. Bushmanov - V. Romano, Asymptotic stability of the equilib-rium state for the macroscopic balance equations of charge transport in semicon-ductors, Computational technologies, 8 (3) (2003), pp. 7–22.[4] A.M. Blokhin - V.N. Dorovsky, Mathematical modelling problems in the multi-speed continuum theory, Novosibirsk, Nauka, 1994.[5] A.M. Blokhin - I.Yu. Druzhinin, Strong discontinuities in magnetohydrodyna-mic, Novosibirsk, Nauka, 1993.[6] A.M. Blokhin - I.G. Sokovikov, On a certain approach to construction of dif-ference schemes for quasilinear equations of gas dynamics, Sib. Math. Jour., 40(1999), pp. 1236–1243.[7] V. Romano, Non-parabolic band hydrodynamical model of silicon semiconduc-tors and simulation of electron devices, Math. Meth. Appl. Sci., 24 (2001), pp.439–471.[8] V. Romano, Non parabolic band transport in semiconductors: closure of theproduction terms in the moment equations, Cont. Mech. Thermodyn., 12 (2000),pp. 31–51.Sobolev Institute of Mathematics,Russian Academy of Sciences,Koptyuga pr., 4 - 630090, Novosibirsk (RUSSIA)e-mail: Blokhin@math.nsc.ru

Entropy approximation for the system of moment equations describing charge transport in semiconductors

The question of possibility to build additional entropy conservation law for the system of moment equations, which appear in mathematical modelling of the charge transport in semiconductors, is discussed.<br /

A. M. Blokhin

I. G. Sokovikov

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Entropy approximation for the system of moment equations describing charge transport in semiconductors

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