A mathematical model of charge transport in semiconductors is considered. The model is a quasilinear system of differential equations. A problem of finding an additional entropy conservation law and system symmetrization are solved

Blokhin, Alexander M.

Sokovikov, I. G.

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

LE MATEMATICHEVol. LVII (2002)  Fasc. II, pp. 241253SYMMETRIZATION OF AMATHEMATICAL MODELOF CHARGE TRANSPORT IN SEMICONDUCTORSA. M. BLOKHIN - I. G. SOKOVIKOVA mathematical model of charge transport in semiconductors is consid-ered. The model is a quasilinear system of differential equations. A problemsof �nding an additional entropy conservation law and system symmetrizationare solved.Introduction.It is well known that hydrodynamic models are widely used in mathemati-cal modelling of various physical phenomena. This is typical, for example, forsemiconductor physics. In the last years a lot of new mathematical models ofhydrodynamic type for description of charge transport in semiconductors wereproposed. It is necessary to note, however, that mathematical justi�cation ofsuch models is far from perfect in most cases.Below we consider the mathematical model of charge transport in semi-conductors, which has recently been proposed in the article [1]. The model is aquasilinear system of rather complex equations (see section 1). We simplify thissystem using reliable reasons; next, for this simpler system, we formulate andsolve the problem of �nding the additional entropy conservation law. Moreover,we discuss symmetrization of this simpler equations system.Entrato in redazione il 23 Maggio 2003.242 A. M. BLOKHIN - I. G. SOKOVIKOV1. Preliminary information.Now, we brie�y describe the mathematical model suggested in [1]. Somerelated mathematical and calculational aspects are discussed in [2]. Follow-ing [3], we write down a dimensionless quasilinear system of hydrodynamicequations, which describe this model, in one-dimensional non-stationary case(reduction of the system to a dimensionless form is detailed in [7]):(1.1) Rτ + J (1)s = 0,(1.2) J (1)τ + J (2)s = F (2),(1.3) eτ + J (3)s = F (3),(1.4) στ + J (4)s = F (4),(1.5) J (3)τ + J (5)s = F (5).Here τ is the time; s , the spacial coordinate;R, the electron gas density;J (1) = Ru, the electron �ow, i.e. a variable which characterizes the electriccurrent in a semiconductor;u, the electron gas velocity;J (2) = Ru2 + P , the �ow in the momentum balance equation (1.2);P = P + � , the single component of the stress tensor in the one-dimensionalcase;P = Rϑ , the pressure in the electron gas;ϑ , the temperature;e = R�E = R�u22 + 32ϑ�, the volume density of the total energy;J (3) = J (1)(�E + ϑ)+�+ �u, the energy �ow in the energy conservation law(1.3);�, the heat �ow;σ = 23 Ru2 + � ;J (4) = uσ + 43uP + 815�;J (5) = u2e + 52u2P + 52 Pϑ + 72�ϑ + 165 �u, the �ow of the energy �ow;F(2) = RQ − J (1)τP ;SYMMETRIZATION OF A MATHEMATICAL MODEL. . . 243F(3) = J (1)Q − R�E−3/2τW ;F(4) = 43 J (1)Q − �τσ ;F(5) =� 32 Ru2 + 52 P + ��Q − J (3)τq are the source terms in (1.2)(1.5);Q = ϕs , ϕ is the electric potential;τP = τP (�E), the relaxation time in (1.2);τW = τW (�E), the relaxation time in (1.3);τσ = τσ (�E), the relaxation time in the equation (1.4) for the stress � ;τq = τq(�E), the relaxation time in the equation (1.5) for the heat �ow Q .The system (1.1)(1.5) has to be completed with one-dimensional Poissonequation for the electric potential ϕ(τ, s) (see [7]), but since it is unrelated tothe problem of symmetrization discussed in this article it is not placed here.Remark 1.1. Now we brie�y describe the method to obtain system of thehydrodynamic type in the theory of charge transport in semiconductors. Mostly,the Boltzmann transport equation is basic (see [1]), for which it is possible towrite down, generally speaking, an in�nite system of the so called momentumcorrelations. In order to obtain a systemwith a �nite number of equations, somerealistic reasoning (the so called closing procedures) is used. While analyzingthe system of moment correlations, it is easy to understand that only the �rstthree equations (in the one-dimensional case) are physically clear. They arethe mass conservation law, the momentum conservation law, and the energyconservation law. The other moment correlations have no such a physicalmeaning; essentially they are differential equations for various additions in theimpulse and energy �ows. So, in the system (1.1)(1.5), the equations (1.4),(1.5) are differential equations in additions � , � which appear in the �owsJ (2), J (3).After simple but cumbersome calculations, the system (1.1)(1.5) can berewritten in a non-divergent form:(1.6) Uτ + BUs = F,where B = uI5 + B0,U =RuP�� , B0 =0 R 0 0 00 0 R−1 R−1 00 λ 0 0 2/30 µ 0 0 8/15b51 b52 b53 b54 0 , F =0F2F3F4F5λ =5P + 2�3 , µ =4P + 7�3244 A. M. BLOKHIN - I. G. SOKOVIKOVb51 = −5P + 7�2R2 P, b52 =165 �, b53 =5P2R , b54 =P −�R ,F2 = Q − uτP , F3 =23 Rg, g = u2� 1τP −12τW�+3(1− ϑ)2τW ,F4 = 43Ru2τP −�τσ,F5 = −�τq + �u� 1τσ+1τP −1τq�+52 Pu� 1τP −1τq�++Ru32� 53τW −1τq −1τP�−52 J (1)1− ϑτW ,I5 is an unitary matrix of order 5. We note that hyperbolicity of the system(1.6) has been discussed in [2]. As shown, it becomes hyperbolic underessential restrictions on components of the vector U . In our opinion, this makesdif�culties for using numerical methods to �nd solutions to (1.6).At the same time in [6, 7] the so called gas-dynamical model of chargetransport in semiconductors has been considered. This model has been derivedfrom (1.1)(1.5) by turning � and � into zero and truncating the last twoequations. The gas-dynamical model in a non-divergent form looks like thefollowing:(1.7) Uτ + BUs = F.HereU =� RuP�, B =� 0 R 00 0 R−10 5P/3 0�+ uI3, F =� 0F2F3�;the aggregates F2, F3 are described above. The system (1.7) is indeed agas-dynamical equations system (with right parts) for a polytropic gas withthe adiabatic exponent γ = 5/3. Results of numerical calculations (see [6])have shown that this model adequately depict peculiarities of test tasks insemiconductor physics.Remark 1.2. Actually, turning � and � into zero and truncating the differen-tial equations for them is an element of the closing procedure (see. Remark 1.1.)while obtaining the gas-dynamical model of charge transport in semiconductors.SYMMETRIZATION OF A MATHEMATICAL MODEL. . . 245We note that for the system (1.7) it is easy to write down an additionalentropy law (see [6, 7]):(1.8) (RS)τ + (RuS)s = g˜,where S is the mass entropy connected with the density R and the pressure Pby the state equation(1.9) P = R5/3 exp{(2/3)S},g˜ = Rϑg (the aggregate g is described above).In this article, we consider a simpler system which is obtained from (1.1)(1.5)by turning � into zero and truncating the last equation (1.5). The system in anon-divergent form looks as follows:(1.10) Uτ + BUs = F.HereU =RuP� , B =0 R 0 00 0 R−1 R−10 λ 0 00 µ 0 0+ uI4, F =0F2F3F4 ;the aggregates F2, F3, F4 are described above. Under some natural physicalrestrictions(1.11) R > 0, P > 0,the system (1.10) is hyperbolic. The aim of this article is to �nd an additionalentropy conservation law for the system (1.10)(1.12) ητ +�s = G,(where η, �, G are some functions in components of the vector U ), whichwould hold at any smooth solutions to the system (1.1)(1.4). Here η is theentropy function; �, the entropy �ow; G , the entropy production.246 A. M. BLOKHIN - I. G. SOKOVIKOV2. On �nding the entropy law for the system (1.10).As well know (see [5]), existence of the additional entropy conservationlaw (1.12) means existence of functionsqi = qi (U ), i = 1, 4,called the Lagrange multipliers (or canonical variables (see [3])) such that thefollowing relations are valid:(2.1) dη = q1dR + q2d J (1) + q3de + q4dσ,(2.2) d� = 4�i=1qid J (i),(2.3) G = 4�i=2qiF (i).In what follows we assume that the entropy function η is of the form(2.4) η = RS + f (R, u, ϑ, �).Moreover, the addition f must satisfy the following condition:(2.5) f ���=0 = 0.Taking into account the relation (1.9), we easily obtain from (2.4):(2.6) dη = (S − 1+ fR )dR + fu du + �3R2ϑ + fϑ�dϑ + f�d�.On the other hand, we deduce from (2.1):(2.7) dη = q˜1dR + q˜2d J (1) + q3de + q4d� == (q˜1 + uq˜2 + q3�E)dR + (q˜2 + uq3)Rdu + 32 Rq3dϑ + q4d�,whereq˜1 = q1 − 23u2q4,SYMMETRIZATION OF A MATHEMATICAL MODEL. . . 247q˜2 = q2 + 43uq4.Comparing (2.6) and (2.7), we �nally have:(2.8)q1 = S − 1+ fR − (u/R) fu + (u2 − �E)�+ 23u2 f�,q2 = R−1 fu − u�− 43u f�,q3 = �,q4 = f�.Here � = 1ϑ+ 23R fϑ .Now, we compose the aggregate(2.9) L = q1R + q2 J (1) + q3e + q4σ − RS − f = −R + R fR + � f� − f.Following the classic article [9], we call the aggregate L by the generatingfunction (see also [3, 4]).We rewrite (2.2) as follows:(2.10) d� = u(q1dR + q2d J (1) + q3de + q4dσ ) ++ (q1R + q2 J (1) + q3e + q4σ )du ++�q2 + u�q3 + 43q4��dP + P�q3 + 43q4�du.In a view of (2.1), (2.8), (2.9), the expression (2.10) transforms into:(2.10�) d� = d(uη) + 1R fudP +�R fR + 2P3R fϑ + µ f� + �ϑ − f �du.The aggregate µ is described above. The right hand expression in (2.10�) is atotal differential if fu = 0, i.e. f = f (R, ϑ, �), and(2.11) R fR + 2P3R fϑ + µ f� = f −�ϑ.We consider (2.11) as a differential equation for �nding the function f . Wenote, however, that the solution to (2.11) can be found in the form(2.12) f = f (R, r), r = �ϑ, f ��r=0 = 0.248 A. M. BLOKHIN - I. G. SOKOVIKOVIn this connection, (2.11) can be rewritten as:(2.11�) R fR + �53r − 2r23R + 43 R� fr = f − r.We �nd the general solution to (2.11�). For this purpose we form the so calledconcomitant system of ordinary differential equations (see [9]):(2.13)drdR =43 +53 y −23 y2, y =rR ,d fd R =fR − y,.The �rst integrals of this system can be taken in the form:(2.14) y + 1R2(2− y) = C1,fR +12ln� 27(y + 1)(2− y)2�= C2 .Therefore, the general solution to (2.11�) can be written as:(2.15) f = R2 ln� (r + R)(2R − r)227R3�+ RF0� r + RR2(2R − r)�,where F0 is an arbitrary function in a single variable.In order to satisfy (2.12),we take the constant (1/2)ln(27/4) as the functionF0. Finally, we have(2.16) f = R2 ln� (r + R)(2R − r)24R3�.The function (2.16) satis�es all the desired conditions. Additionally, we �nd thederivatives fR , fr :(2.17)fR = 12ln� (r + R)(2R − r)24R3�+3r22(r + R)(2R − r) ,fr = − 3Rr2(r + R)(2R − r) .So, the system (1.10) admits existence of the additional entropy conservationlaw (1.12) which we write as follows:(2.18) (R�)τ + (Ru�)s = G.SYMMETRIZATION OF A MATHEMATICAL MODEL. . . 249Here(2.19) � = S + 12ln�Pω24P3�, ω = 2P −� (�= 0),(2.20) G = (Ru)2PτP+3R�22Pωτσ −R2(2P + �)Pω�E − 3/2τW.Now the formulae (2.8), (2.9) for the Lagrange multipliers qi , i = 1, 4 of thegenerating function L look as follows:(2.8�)q1 = 12 ln�Pω24R5�+Ru2 − 5P2P ,q2 = − RuP,q3 = R(2P +�)Pω,q4 = −3R�2Pω ,(2.9�) L = −R.In a view of (1.1), the relation (2.18) can be rewritten as:(2.18�) �τ + u�s = GR .Consequently, in a view of (2.18�), the system (1.10) can be rewritten as asymmetric t -hyperbolic system:(2.21)d�dτ =GR ,R dudτ + Ps = RF2,1λdPdτ + us =1λF3,1µd�dτ + us =1µF4,provided that(2.22)λ =5P + 2�3 > 0,µ =4P + 7�3 > 0, i.e. � > −47 P.Here ddτ = ∂∂τ + u ∂∂s .250 A. M. BLOKHIN - I. G. SOKOVIKOVRemark 2.1. The additional entropy conservation law (2.18�) is a consequenceof equations in (1.10). Therefore, we can replace any one of these equationsby the relation (2.18�). In some cases, such a substitution leads to a symmetricsystem immediately. In our case, substituting the �rst equation in (1.10) for(2.18�), we come to the symmetric t -hyperbolic (by Friedrichs) system (2.21).Concluding this section, we note that the additional entropy law for thesystem (1.10) is not unique. Indeed, taking a smooth function h = h(�), wehave(2.18��) (Rh)τ + (Ruh)s = h�G.Such an ambiguity can be used while numerically studying the system (1.10)(see, e.g., [8]).3. Symmetrization of the system (1.10).In the previous section we have proposed a variant of symmetric form ofthe system (1.10) (see (2.21)). To obtain another variant, we rewrite the system(1.10) in a slightly different form. Summing up the third and forth equations in(1.10), �nally we obtain:(3.1) dPdτ + 3Pus = F3 + F4.Next, we multiply the third equation by 2 and subtract the forth equation:(3.2) dωdτ + ωus = 2F3 − F4and, using (1.9), rewrite the expression for � (see (2.19)) in the following form:(3.3) � = 12ln�Pω24�−52lnR.In a view of (3.1), (3.2), (3.3), we derive another symmetric form of the system(1.10)(3.4)d�dτ =GR ,dXdτ = 2F3 + F43P − 22F3 − F4ω,Re−π dudτ + πs = Re−π F2,13dπdτ + us =F3 + F43P .SYMMETRIZATION OF A MATHEMATICAL MODEL. . . 251Hereπ = lnP, X = 23π − lnω2.While obtaining (3.4), we have assumed that inequalities (1.11) are ful�lled. Wehave also assumed that � �= 2P (ω �= 0) (see (2.19))If (1.11) are valid the system (3.4) is symmetric t -hyperbolic (byFriedrichs).Finally, one more variant of symmetric form of the system (1.1)(1.4) isconnected with the obtained in section 2 canonical variables qi , i = 1, 4 andthe generating function L . We need one more generating function M :(3.5) M = 4�i=1qi J (i) −� = −Ru = uL.Further symmetrization formalism is detailed in [3, 4, 5]. It follows from (2.9),(3.5) that R = Lq1 , J (1) = Lq2 , e = Lq3 , σ = Lq4 ,J (1) = Mq1 , J (2) = Mq2 , J (3) = Mq3, J (4) = Mq4 .Next, the system (1.1)(1.4) can be rewritten as follows:(3.6) (Lqi )τ + (Mqi )s = F (i), i = 1, 4 (F (1) = 0).We obtain from (3.6):(3.6�) A0qτ + A1qs = F ,whereq =q1q2q3q4 , F =0F(2)F(3)F(4) ,A0 = (Lqi qj ), A1 = (Mqiqj ), i, j = 1, 4 are symmetric matrices. Aftercumbersome calculations we �nd the matrices A0 , A1 :A0 = −R Ru R�E σR uRu2 + P Ru32 + 32λu 23 Ru3 + µuR�E Ru32 + 32λu a033 a034σ 23 Ru3 + µu a043 a044 ,252 A. M. BLOKHIN - I. G. SOKOVIKOVwherea033 = Ru44 +32λu2 +3(5P2 + �2)4Ra034 = a043 = Ru43 +14P + 11�6 u2 +�(7P +�)2Ra044 = 4Ru49 +16P + 28�9 u2 +2(2P2 + 2P� + 3�2)3R ;A1 = uA0 −P�A1,�A1 =0 1 u 34u1 2u 32u2 + 3λ2R 2u2 + µRu 32u2 + 3λ2R u3 + 3λR u 43u3 + 14P+11�3R43u 2u2 + µR 43u3 + 14P+11�3R u 169 u3 + 5uµ3R ,and prove thematrix B0 = −A0 is positive de�nite provided that P > 0. Takinginto account this fact, �nally we rewrite the system (1.1)(1.4) as a symmetrict -hyperbolic system (by Friedrichs):(3.6��) B0qτ + B1qs = −F .Here B1 = uB0 + P�A1.Authors are thankful to V. Romano for useful discussions of this article.The work is supported by RFFI(grants N. 01-01-00781 and N. 02-01-00641)and INTAS, grant N. 868.REFERENCES[1] A.M. Anile - O. Muscato, Improved hydrodynamical model for carrier transportin semiconductors, Physical Rev. B., 51 - 23 (1995), pp. 1672816740.[2] A.M. Anile - V. Romano - G. Russo, Extended hydrodynamical model of carriertransport in semiconductors, SIAM J. on Appl. Math., 61 (2000), pp. 74101.[3] A.M. Blokhin, Energy integrals and their application to gas-dynamical prob-lems, Novosibirsk: Nauka, (1986), p. 240.[4] A.M. Blokhin - R.D. Alaev, Energy integrals and their application to studydifference schemes stability, Novosibirsk: NSU Publishing house, (1993), p. 213.[5] A.M. Blokhin - V.N. Dorovsky, Problems of mathematical modelling in the theoryof multispeed continuum, Novosibirsk, (1994), p. 183.SYMMETRIZATION OF A MATHEMATICAL MODEL. . . 253[6] A.M. Blokhin - A.A. Iordanidy, Numerical investigation of a gas dynamical modelfor charge transport in semiconductors, COMPEL., 18 - 1 (1999), pp. 637.[7] A.M. Blokhin - A.A. Iordanidy - I.Z. Merazhov, Numerical investigation of thegas-dynamic model for charge transport in semiconductors, Novosibirsk, (1996),p. 51 (Preprint / Sobolev Inst. of Math. Siberian Branch. of the Russian Acad. ofSci, N. 33).[8] A.M. Blokhin - I.G. Sokovikov, On the approach to designing of differenceschemes for quasilinear equations of gas-dynamics, Siberian Math. Journal, 40- 6 (1999), pp. 12361243.[9] S.K. Godunov, On an interesting class of quasilinear systems, RAS. 139 - 3(1961), pp. 521523.[10] I.G. Petrovsky, Lectures on the theory of ordinary differential equations, Mos-cow: Nauka, 1970.Sobolev Institute of Mathematics,Russian Academy of Sciences, Koptyuga pr., 4,630090, Novosibirsk, (RUSSIA)e-mail: blokhin@math.nsc.ru

Symmetrization of mathematical model of charge transport in semiconductors

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Symmetrization of mathematical model of charge transport in semiconductors

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