Analysis of numerical schemes for semiconductors energy-transport models

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Analysis of numerical schemes for semiconductors energy-transport models

Marianne Bessemoulin-Chatard, Claire Chainais-Hillairet, Hélène Mathis

To cite this version:

Marianne Bessemoulin-Chatard, Claire Chainais-Hillairet, Hélène Mathis. Analysis of numerical

schemes for semiconductors energy-transport models. Applied Numerical Mathematics, Elsevier, 2021,

168, pp.143-169. �10.1016/j.apnum.2021.05.030�. �hal-02940224v2�

ENERGY-TRANSPORT MODELS

MARIANNE BESSEMOULIN-CHATARD, CLAIRE CHAINAIS-HILLAIRET, AND H ´EL `ENE MATHIS

Abstract. Some finite volume schemes for unipolar energy-transport models are introduced. Using a reformulation in dual entropy variables, we can show the decay of a discrete entropy with control of the discrete entropy dissipation.

We establisha priori estimates which lead to the existence of a solution to the scheme. Similarly to the continuous framework, we prove the exponen- tial decay of the discrete relative entropy towards the thermal equilibrium.

Numerical results assess the good behaviour of the whole numerical scheme.

Key-words. Energy-transport model, finite volume schemes, entropy method.

2020 MCS.65M08, 65M12, 35K20

Contents

1. Introduction 2

1.1. The energy-transport models 2

1.2. Entropy structure 4

1.3. Goal and outline of the paper 5

2. Numerical schemes 5

2.1. Mesh notations 5

2.2. Schemes in primal and dual entropy variables 6 2.3. Equivalence of the schemes in the primal and dual entropy variables 8 3. Discrete entropy inequality anda priori estimates 10

3.1. Discrete entropy inequality 10

3.2. A prioriestimates on the approximate solutions 12

4. Existence of a solution to the schemes 14

5. Long time behavior 16

5.1. Definition of the thermal equilibrium 16

5.2. Exponential decay towards equilibrium 17

6. Numerical experiments 19

6.1. Physical models 19

6.2. Implementation of the whole method 20

6.3. Test cases 20

7. Conclusion 27

References 28

MBC and HM were partially funded by the Centre Henri Lebesgue (ANR-11-LABX-0020- 01) and ANR Project MoHyCon (ANR-17-CE40-0027-01). CCH was partially funded by ANR Project MoHyCon (ANR-17-CE40-0027-01) and was supported by the ANR Labex CEMPI grant ANR-11-LABX-0007-01.

1

1. Introduction

In this article, we are interested in the numerical analysis of unipolar energy- transport models for semiconductor devices. Such models describe the flow of electrons through a semiconductor crystal, influenced by diffusive, electrical and thermal effects. Contrary to classical drift-diffusion systems (see [34,21,22]), they take into account the temperature and permit to observe hot electrons effects.

As recalled in [29], these models can be derived either from hydrodynamic mod- els by neglecting some convection terms [23] or from the Boltzmann equation by the moment method [2]. There exists several other models which allow to take into account thermal effects. For instance non-isothermal drift-diffusion system, considered in [32], couples the drift-diffusion semiconductor equations to the heat flow equation for the temperature distribution in the device. We refer to [32] and references therein for details about this approach. In the sequel we focus on the energy-transport model with cross-diffusion structure, obtained while considering another moment of the Boltzmann equation [29].

1.1. The energy-transport models. The primal energy-transport system con- sists in two continuity equations for the electron densityρ₁and the internal energy densityρ₂, coupled with a Poisson equation for the electrical potential V. Follow- ing the framework adopted in [14], we consider that the electron and the energy densities are defined as functions of the entropy variablesu= (u₁, u₂) with

(1) u1= µ

T, u2=−1 T,

whereµdenotes the chemical potential andT the temperature. Then the electron densityρ1 and the internal energy densityρ2 are assumed to depend onu. Let Ω be an open bounded subset ofR^d,d≥1, describing the geometry of the considered semiconductor device and let Tmax > 0 be a finite time horizon. The energy- transport model writes in Ω×(0, Tmax)

∂_tρ₁(u) + divJ₁(u) = 0, (2)

∂tρ2(u) + div J2(u) =∇xV ·J1(u) +W(u), (3)

−λ²∆V =C(x)−ρ1(u), (4)

where the electron and energy current densities are given by:

(5) J1(u) =−L11(u)(∇u1+u2∇V)−L12(u)∇u2, J2(u) =−L21(u)(∇u1+u2∇V)−L22(u)∇u2,

whereL(u) = (L_ij(u))_1≤i,j≤2 is a symmetric uniformly positive definite matrix in the sense that there exists a constantα >0 such that

(6) (Lu,u)≥α|u|², ∀u∈R².

The term ∇V ·J1 corresponds to a Joule heating term and W(u) is an en- ergy relaxation term. The doping profile C ∈ L^∞(Ω) describes the fixed charged background andλis the rescaled Debye length.

The system (2)-(5) is supplemented with an initial condition u0 = (u1,0, u2,0) and with mixed boundary conditions. There are Dirichlet boundary conditions on the ohmic contacts and homogeneous Neumann boundary conditions on insulating

segments. More precisely the domain Ω is assumed to be an open bounded polyg- onal (or polyhedral) subset ofR^d (d≥1) and∂Ω is split into∂Ω = Γ^D∪Γ^N, with Γ^D∩Γ^N =∅ and md−1(Γ^D)>0. We denote bynthe normal to∂Ω outward Ω.

The boundary conditions write

u1=u^D₁, u2=u^D₂, V =V^D, on Γ^D×(0, Tmax), (7)

J1·n=J2·n=∇V ·n= 0, on Γ^N×(0, Tmax).

(8)

We assume that the Dirichlet boundary conditionsu^D₁,u^D₂ andV^Ddo not depend on time and are the traces of some functions defined on the whole domain Ω, still denotedu^D₁,u^D₂ andV^D, and such that

(9) u^D₁, u^D₂ ∈H¹(Ω), V^D∈H¹(Ω)∩L^∞(Ω).

Moreover we assume that

(10) u^D₂ is a given negative constant on Γ^D

and that the energy relaxation termW(u) verifies, for allu∈R²andu^D₂ <0, that

(11) W(u)(u2−u^D₂)≤0.

The main results on the energy-transport model (2)-(5) are presented in [14,28]:

existence of weak solutions to the system, regularity, uniqueness and long-time con- vergence towards thermal equilibrium. They are based on a reformulation of the sys- tem in terms of dual entropy variables (electrochemical potentials). This change of variables symmetrizes the equations and allows to derive a crucial entropy–entropy dissipation estimate.

The proofs established in [14] rely on some assumptions on ρ recalled below (Assumptions 1) and on the assumption of uniform definiteness of the diffusion matrixL. This is the framework we will adopt here. However, these assumptions are hardly satisfied by practical test cases, since they require that the electron density ρ₁ and the temperature T have positive lower bounds, which is not the case in general. Existence results for physically more realistic diffusion matrices (only positive semi-definite) are established in [18,24] for the stationary model and in [12, 13] for the transient system, but only in the case of data close to thermal equilibrium. More recently, existence of solutions has been proved in simplified degenerate cases, namely for a model with a simplified temperature equation in [30] and for vanishing electric fields (avoiding the coupling with Poisson equation) in [35].

We consider the following assumptions on the functionu7→ρ(u) = (ρ1(u), ρ2(u)).

Assumptions 1. The functionρis such that

(i) ρ∈W^1,∞(R²;R²), in particular there existsCρ>0 such thatkρ(u)k_∞≤Cρ

for allu∈R²,

(ii) there exists a constantC0 such that

(12) (ρ(u)−ρ(v))·(u−v)≥C₀|u−v|², ∀u,v∈R², (iii) there existsχ∈ C¹(R²;R) convex such thatρ(u) =∇uχ(u).

The key point of the analysis of the primal model (2)-(5) is to use another set of variables which symmetrizes the problem, see [14]. Let us define the so-called dual

entropy variablesw= (w1, w2) (w1is actually an eletrochemical potential):

(13)

(w1=u1+u2V, w2=u2.

Through this change of variables, the problem (2)-(5) is equivalent to

∂tb1(w, V) + div I1(w, V) = 0, (14)

∂tb2(w, V) + divI2(w, V) =Wf(w)−∂tV b1(w, V), (15)

−λ²∆V =C−b1(w, V), (16)

where the functionb(w, V) = (b₁(w, V), b₂(w, V)) is related toρ andV by (17)

(b1(w, V) =ρ1(u),

b₂(w, V) =ρ₂(u)−V ρ₁(u),

and the new energy relaxation term is defined byWf(w) =W(u). The symmetrized currents are given by

(18) I1=J1,

I2=J2−V J1, which leads to

(19) I1(w, V) =−D11(w, V)∇w1−D12(w, V)∇w2, I2(w, V) =−D21(w, V)∇w1−D22(w, V)∇w2. The new diffusion matrixD(w, V) = (Dij(w, V))_1≤i,j≤2 is defined by (20) D(w, V) =P(V)^TL(u)P(V), withP(V) =

1 −V

0 1

, so that

(21)

D11(w, V) =L11(u),

D₁₂(w, V) =D₂₁(w, V) =L₁₂(u)−V L₁₁(u), D₂₂(w, V) =L₂₂(u)−2V L₁₂(u) +V²L₁₁(u).

It is clear that the diffusion matrix D is also symmetric and uniformly positive definite.

1.2. Entropy structure. The existence results for model (2)-(5) given in [14] are mainly based on entropy estimates obtained thanks to the dual model (14)-(19).

We recall in this section the entropy/entropy-dissipation property. The entropy function is defined by

(22) S(t) =

Z

Ω

ρ(u)·(u−u^D)−(χ(u)−χ(u^D))

dx−λ² 2 u^D₂

Z

Ω

|∇(V −V^D)|²dx.

Sinceu^D₂ <0 andχis a convex function, see Assumption1-(iii),S(t) is nonnegative for allt≥0.

Moreover, if the boundary conditions are at thermal equilibrium, namely

(23) ∇w^D₁ =∇w₂^D= 0,

one can prove, under Assumptions1, that the entropy satisfies the following iden- tity:

(24) d

dtS(t) =− Z

Ω

∇w^TD∇w+ Z

Ω

W(u)(u2−u^D₂).

Thanks to the uniform positivity of the matrixDand the dissipation of the relax- ation termW, see hypothesis (11), one deduces that the entropy function decreases along time. The identity (24) gives alsoa priori estimates onw.

More general boundary conditions are considered in [14], but they involve the control of several additional terms.

1.3. Goal and outline of the paper. Different kind of numerical schemes have already been designed for the energy–tranport systems, mainly for the stationary systems: finite difference schemes in [19, 33], finite element schemes in [15, 20], mixed finite element schemes in [25,26]. We also refer to [9] for DDFV (Discrete Duality Finite Volume) schemes for the evolutive case. Up to our knowledge, there exists no analysis of these numerical schemes.

The purpose of this paper is to design some finite volume schemes for the sys- tems (2)–(4) (and (14)–(16)) with a two-point flux approximation (TPFA) of the numerical fluxes. We pay attention while building the scheme on the possibility of adapting the entropy method (briefly described in Section 1.2) to the discrete setting. This will be crucial in order to adapt the proof of [14] to the discrete level.

Thea prioriestimates obtained for the approximate solution thanks to the entropy method are sufficient to establish the existence of a solution to the scheme via a fixed point theorem.

In Section2, we introduce the finite volume schemes for the system written in the primal entropy variables (2)–(5) and the system written in the dual entropy vari- ables (14)–(19). We also establish the equivalence between the different schemes.

Section3is devoted to the proof of a discrete entropy-entropy dissipation estimate, which yieldsa priori estimates on the approximate solutions. In Section4, we es- tablish the existence of a solution to the schemes and in Section5, we study the long time behavior of the numerical solutions. Finally, Section6is devoted to numerical experiments.

2. Numerical schemes

2.1. Mesh notations. The mesh of the domain Ω is given by a familyT of open polygonal (or polyhedral in 3D) control volumes, a familyEof edges (or faces), and a familyP = (xK)K∈T of points. As it is classical in the finite volume discretization of diffusive terms with two-point flux approximations, we assume that the mesh is admissible in the sense of [17, Definition 9.1]. It implies that the straight line between two neighboring centers of cells (xK, xL) is orthogonal to the edgeσ=K|L.

In the set of edges E, we distinguish the interior edges σ = K|L ∈ Eint and the boundary edges σ ∈ Eext. Due to the mixed boundary conditions, we have to distinguish the Dirichlet boundary edges included in Γ^D from the Neumann boundary edges included in Γ^N: E_ext =E_ext^D ∪ E_ext^N . For a control volumeK ∈ T, we defineE_K the set of its edges, which is also split intoE_K=E_K,int∪E_K,ext^D ∪E_K,ext^N . In the sequel, we denote byd the distance in R^d and m the measure in R^d or R^d−1. For allσ∈ E, we defined_σ=d(x_K, x_L) ifσ=K|L∈ Eintandd_σ=d(x_K, σ)

ifσ ∈ Eext, with σ∈ EK. Then the transmissibility coefficient is defined by τσ = m(σ)/dσ, for allσ∈ E.

We assume that the mesh fulfills the following regularity constraint: there exists ξ >0 such that

(25) d(xK, σ)≥ξdσ, ∀K∈ M,∀σ∈ EK.

Concerning the time discretization, let ∆t >0 be the time step. ForTmax>0, we set NT the integer part of Tmax/∆t and tⁿ =n∆t for all n = 0, . . . , NT. We denote byδ= max(∆t,size(T)) the size of the space–time discretization.

A TPFA finite volume scheme provides, for an unknown v, a vector vT = (vK)K∈T ∈ R^θ (with θ = Card(T)) of approximate values on each cells. Then, a piecewise constant function, still denotedv_T, can be defined by

v_T = X

K∈T

vK1K,

where 1K denotes the characteristic function of the cell K. However, since there are Dirichlet boundary conditions on a part of the boundary, we also need to define approximate values forvat the corresponding boundary edges: v_ED = (vσ)_σ∈ED

ext ∈ R^θ

D (with θ^D = Card(E_ext^D )). Therefore, the vector containing the approximate values both in the control volumes and at the Dirichlet boundary edges is denoted byv_M= (v_T, v_ED). We denote byX(M) the set of the discrete functionsv_M. For any vectorv_M= (v_T, v_ED), we define for allK∈ T and allσ∈ EK:

(26) vK,σ=







vL ifσ=K|L∈ EK,int, vσ ifσ∈ E_K,ext^D , vK ifσ∈ E_K,ext^N , and

DK,σv_M=vK,σ−vK, Dσv_M=|DK,σv_M|.

Then, we can define the discreteH¹-seminorm| · |_1,MonX(M) by

|vM|²_1,M=X

σ∈E

τσ(DσvM)², ∀vM∈X(M).

We also define the discreteH¹-normk · k1,M onX(M) by kvMk²_1,M=kvTk²_L2(Ω)+|vM|²_1,M.

2.2. Schemes in primal and dual entropy variables. Our aim is to design a scheme for the energy-transport model in the primal entropy variables (2)-(5). This scheme must lead to an equivalent scheme for the system written in the dual entropy variables (14)-(19). Indeed, in this case, it will be possible to apply the entropy method at the discrete level. This step is crucial since it bringsa priori estimates on the sequences of approximate solutions, which allows to prove existence of a solution to the scheme.

One main difficulty in writing a TPFA scheme for the energy-transport model (2)-(5) comes from the approximation of the Joule heating term∇V ·J1, see equa- tion (3). One possibility would be to apply the technique developed in [5], and further used in [32, 16], to discretize de Joule heating term. However, with such

discretization, the rewriting of the scheme in dual entropy variables is not straight- forward. Therefore, following [7], we propose an approximation of the Joule heating term which is based on its following reformulation:

(27) ∇V ·J₁= div (V J₁)−VdivJ₁.

Let us now turn to the definition of the scheme for the model (2)-(5). Initial and Dirichlet boundary conditions are discretized as usual by taking the mean values of the data on the cells or on the Dirichlet boundary edges:

u⁰_1,K, u⁰_2,K

= 1

m(K) Z

K

(u_1,0(x), u_2,0(x))dx, ∀K∈ T, (28)

u^D_1,σ, u^D_2,σ, V_σ^D

= 1

m(σ) Z

σ

u^D₁(γ), u^D₂(γ), V^D(γ)

dγ, ∀σ∈ E_ext^D , (29)

and we set

(30) uⁿ_1,σ=u^D_1,σ, uⁿ_2,σ =u^D_2,σ, V_σⁿ=V_σ^D, ∀σ∈ E_ext^D , ∀n≥0.

We propose a backward Euler in time and finite volume in space scheme with a two-point flux approximation. It reads for alln≥0, for allK∈ T:

m(K)ρⁿ⁺¹_1,K −ρⁿ_1,K

∆t + X

σ∈E_K

F_1,K,σⁿ⁺¹ = 0, (31)

m(K)ρⁿ⁺¹_2,K −ρⁿ_2,K

∆t + X

σ∈EK

F_2,K,σⁿ⁺¹ = X

σ∈EK

V_σⁿ⁺¹F_1,K,σⁿ⁺¹ (32)

− V_Kⁿ⁺¹ X

σ∈EK

F_1,K,σⁿ⁺¹ + m(K)W_Kⁿ⁺¹,

−λ²X

σ∈EK

τ_σD_K,σV_Mⁿ⁺¹ = m(K)(C_K−ρⁿ⁺¹_1,K), (33)

where

ρⁿ⁺¹_i,K =ρ_i(uⁿ⁺¹_K ), i= 1,2, and W_Kⁿ⁺¹=W(uⁿ⁺¹_K ),for allK∈ T. The numerical fluxes are given by

(34) F_1,K,σⁿ⁺¹ =−τσ(Lⁿ_11,σ(DK,σuⁿ⁺¹_1,M+uⁿ⁺¹_2,σ DK,σV_Mⁿ⁺¹) +Lⁿ_12,σDK,σuⁿ⁺¹_2,M), F_2,K,σⁿ⁺¹ =−τ_σ(Lⁿ_12,σ(D_K,σuⁿ⁺¹_1,M+uⁿ⁺¹_2,σ D_K,σV_Mⁿ⁺¹) +Lⁿ_22,σD_K,σuⁿ⁺¹_2,M).

The coefficients Lⁿ_ij,σ are approximations of the coefficients of the matrixLat the interfaceσ. The discrete matrixLⁿσ= (Lⁿ_ij,σ)_1≤i,j≤2 is defined as

Lⁿσ=L

uⁿ_K+uⁿ_K,σ 2

, for allK∈ T, σ∈ EK.

To complete the definition of the scheme (31)–(34), it remains to define V_σⁿ⁺¹ involved in (32) anduⁿ⁺¹_2,σ involved in (34). These definitions will be given later on, our choice being driven by the expected equivalence with a scheme for (14)–(19).

In order to obtain an equivalent scheme for the energy-transport system in the dual entropy variables (14)–(19), we apply the change of variables (13), associated with the definitions of the functions b_i(w, V) given in (17) and the definitions of the currentsI_i(w, V) given in (19), into the numerical scheme (31)–(34).

For allK∈ T and for alln≥0, we set

w_1,Kⁿ =uⁿ_1,K+uⁿ_2,KV_Kⁿ, w_2,Kⁿ =uⁿ_2,K, (35)

bⁿ_1,K =ρⁿ_1,K=b1(w_Kⁿ, V_Kⁿ), bⁿ_2,K =ρⁿ_2,K−ρⁿ_1,KV_Kⁿ=b2(wⁿ_K, V_Kⁿ).

(36)

The boundary conditions are similarly defined:

(37) wⁿ_1,σ =w^D_1,σ, wⁿ_2,σ =w^D_2,σ, ∀σ∈ E_ext^D , ∀n≥0.

From the numerical scheme (31)–(32), we deduce m(K)bⁿ⁺¹_1,K −bⁿ_1,K

∆t + X

σ∈EK

F_1,K,σⁿ⁺¹ = 0,

m(K)bⁿ⁺¹_2,K −bⁿ_2,K

∆t + X

σ∈EK

F_2,K,σⁿ⁺¹ −V_σⁿ⁺¹F_1,K,σⁿ⁺¹

= m(K)fW_Kⁿ⁺¹−m(K)V_Kⁿ⁺¹−V_Kⁿ

∆t bⁿ_1,K.

It leads to the following scheme for the system written in the dual entropy variables (14)–(16):

m(K)bⁿ⁺¹_1,K −bⁿ_1,K

∆t + X

σ∈EK

G_1,K,σⁿ⁺¹ = 0, (38)

m(K)bⁿ⁺¹_2,K −bⁿ_2,K

∆t + X

σ∈E_K

G_2,K,σⁿ⁺¹ = m(K)fW_Kⁿ⁺¹−m(K)V_Kⁿ⁺¹−V_Kⁿ

∆t bⁿ_1,K, (39)

−λ²X

σ∈EK

τσDK,σV_Mⁿ⁺¹= m(K)(CK−bⁿ⁺¹_1,K), (40)

whereWf_Kⁿ⁺¹=W_Kⁿ⁺¹=Wf(w_Kⁿ⁺¹, V_Kⁿ⁺¹) and numerical fluxes are defined by:

(41) G_1,K,σⁿ⁺¹ =F_1,K,σⁿ⁺¹ ,

G_2,K,σⁿ⁺¹ =F_2,K,σⁿ⁺¹ −V_σⁿ⁺¹F_1,K,σⁿ⁺¹ .

The crucial point now is to ensure that the new numerical fluxesG_i,K,σⁿ⁺¹ ,i= 1,2, can be seen as approximations of the currentsI1andI2defined by (19). This means that we want the numerical fluxes to be written as

(42) G_1,K,σⁿ⁺¹ =−τσ(D^∗_11,σDK,σwⁿ⁺¹_1,M+ D^∗_12,σDK,σw_2,Mⁿ⁺¹), G_2,K,σⁿ⁺¹ =−τσ(D^∗_21,σD_K,σwⁿ⁺¹_1,M+ D^∗_22,σD_K,σw_2,Mⁿ⁺¹).

The coefficients (D^∗_ij,σ)_1≤i,j≤2are approximations of the coefficients of the matrix D(w, V) given by (21) at the interfaceσand on the time interval (tⁿ, tⁿ⁺¹]. They compose the associate matrixD^∗σwhich has to be symmetric and uniformly positive definite. This property depend on the definition of the quantitiesV_σⁿ⁺¹ anduⁿ⁺¹_2,σ , respectively involved in the equations (32) and (34), for each σ∈ E.

2.3. Equivalence of the schemes in the primal and dual entropy variables.

Observe that the scheme is not a fully backward Euler discretization in time since the last term in the equation (39) involves the quantitybⁿ_1,K. This is explained by the proof of the following proposition.

Proposition 1 (Schemes equivalence). Let us supplement the scheme (31)–(34) with the definitions of the quantities (V_σⁿ⁺¹)_{σ∈E, n≥0} and (uⁿ⁺¹_2,σ )_{σ∈E, n≥0}. We dis- tinguish two cases:

• Case 1: centered scheme. For allσ∈ E and n≥0, we set:

(43) uⁿ⁺¹_2,σ = uⁿ⁺¹_2,K +uⁿ⁺¹_2,K,σ

2 and V_σⁿ⁺¹= V_Kⁿ⁺¹+V_K,σⁿ⁺¹

2 .

• Case 2: upwind scheme. For allσ∈ E andn≥0, we set:

(44) uⁿ⁺¹_2,σ =

(uⁿ⁺¹_2,K,σ, ifDK,σV_Mⁿ⁺¹>0,

uⁿ⁺¹_2,K, ifD_K,σV_Mⁿ⁺¹≤0, and V_σⁿ⁺¹= min(V_Kⁿ⁺¹, V_K,σⁿ⁺¹).

Then, in both cases, the scheme (31)–(34) written in the primal entropy variables is equivalent to the scheme (38)–(42) written in the dual entropy variables, provided that

(45) D^∗σ= (Pⁿ⁺¹σ )^TLⁿσPⁿ⁺¹σ with Pⁿ⁺¹σ =

1 −V_σⁿ⁺¹

0 1

.

Proof. Starting from the definition (41) of the numerical fluxesG_1,K,σⁿ⁺¹ and Gⁿ⁺¹_2,K,σ, we want to establish the relations (42) with D^∗σ defined by (45). Observe that, from (45), one deduces the following relations, which are the discrete counterparts of (21):

D^∗11,σ =Lⁿ_11,σ, (46)

D^∗12,σ =D^∗21,σ =−V_σⁿ⁺¹Lⁿ_11,σ+Lⁿ_12,σ, (47)

D^∗22,σ = V_σⁿ⁺¹2

Lⁿ_11,σ−2V_σⁿ⁺¹Lⁿ_12,σ+Lⁿ_22,σ. (48)

Then, due to the change of variables (35), we can rewriteDK,σuⁿ⁺¹_1,MandDK,σuⁿ⁺¹_2,M for all K ∈ T and σ ∈ EK. It is straightforward that DK,σuⁿ⁺¹_2,M = DK,σwⁿ⁺¹_2,M. ConcerningDK,σuⁿ⁺¹_1,M, two formulations are possible, either

DK,σuⁿ⁺¹_1,M=DK,σw_1,Mⁿ⁺¹−V_Kⁿ⁺¹DK,σw_2,Mⁿ⁺¹−w_2,K,σⁿ⁺¹ DK,σV_Mⁿ⁺¹, or

D_K,σuⁿ⁺¹_1,M=D_K,σw_1,Mⁿ⁺¹−V_K,σⁿ⁺¹D_K,σwⁿ⁺¹_2,M−w_2,Kⁿ⁺¹D_K,σV_Mⁿ⁺¹.

The definitions of the interfacial quantities given by (43) in the centered case and by (44) in the upwind case lead to the same definition ofD_K,σuⁿ⁺¹_1,M, namely

DK,σuⁿ⁺¹_1,M=DK,σw_1,Mⁿ⁺¹−V_σⁿ⁺¹DK,σwⁿ⁺¹_2,M−w_2,σⁿ⁺¹DK,σV_Mⁿ⁺¹,

wherewⁿ⁺¹_2,σ =uⁿ⁺¹_2,σ . Finally, plugging the expressions (34) of the numerical fluxes F_i,K,σⁿ⁺¹ and the latter expressionsD_K,σuⁿ⁺¹_i,M, i= 1,2, into the relations (41) of the numerical fluxesG_i,K,σⁿ⁺¹ gives

Gⁿ⁺¹_1,K,σ=−τσ

Lⁿ_11,σDK,σw_1,Mⁿ⁺¹+ Lⁿ_12,σ−V_σⁿ⁺¹Lⁿ_11,σ

DK,σwⁿ⁺¹_2,M , Gⁿ⁺¹_2,K,σ=−τσ

Lⁿ_12,σ−V_σⁿ⁺¹Lⁿ⁺¹_11,σ

DK,σwⁿ⁺¹_1,M + Lⁿ_22,σ−2V_σⁿ⁺¹Lⁿ_12,σ+ (V_σⁿ⁺¹)²Lⁿ_11,σ

DK,σwⁿ⁺¹_2,M .

This corresponds to the expected relations (42) with the coefficients (46)–(48) of the matrix D^∗σ. Hence we have shown that the scheme in primal variables (31)–

(34), supplemented by either the centered or the upwind definitions of the interfacial quantitiesV_σⁿ⁺¹anduⁿ⁺¹_2,σ , is equivalent to the numerical scheme in the dual entropy variables (38)–(42). Conversely, starting from the dual scheme (38)–(42) with the matrixD^∗σ given in (45), one deduces the primal scheme (31)–(34).

3. Discrete entropy inequality and a priori estimates

In this section, we establish the discrete counterpart of the entropy inequal- ity (24) and then deduce some a prioriestimates on the approximate solution.

First of all, recall that the functionsu^D₁,u^D₂,V^D are assumed to be defined on the whole domain Ω. Hence we can set

(u^D_1,K, u^D_2,K, V_K^D) = 1 m(K)

Z

K

(u^D₁(x), u^D₂(x), V^D(x))dx, ∀K∈ T. Moreover the assumption (10) on the boundary condition implies that (49) u^D_K,2=u^D₂ <0, ∀K∈ T.

3.1. Discrete entropy inequality. Let uⁿ_K = (uⁿ_1,K, uⁿ_2,K)^T, V_Kⁿ

K∈T,n≥0 be a solution to the scheme in primal variables (31)–(34), supplemented with either the definition (43) or (44) of the interfacial quantities.

For alln≥0, we define the discrete entropy functional as follows:

Sⁿ = X

K∈T

m(K)

ρⁿ_K·(uⁿ_K−u^D_K)−(χ(uⁿ_K)−χ(u^D_K)) (50)

−λ² 2 u^D₂ X

σ∈E

τσ(Dσ(V_Mⁿ −V_M^D))²,

whereρⁿ_K =ρ(uⁿ_K) = (ρ1(uⁿ_K), ρ2(uⁿ_K))^T. Sinceρis related to the convex potential χby Assumptions 1-(iii), one deduces that Sⁿ is nonnegative for alln≥0.

We now establish the discrete counterpart of the entropy inequality (24).

Proposition 2 (Discrete entropy dissipation). Let assume that Assumptions 1, (10) and (11) hold. Let us also assume that the boundary conditions are at thermal equilibrium, namely that

(51) D_K,σw^D_1,M=D_K,σw^D_2,M= 0.

Let (uⁿ_K, V_Kⁿ)_K∈T,n≥0 be a solution to the scheme in primal variables (31)–(34), supplemented with either (43) or (44). The discrete entropy (50) satisfies the following inequality: for alln≥0,

(52) Sⁿ⁺¹−Sⁿ

∆t ≤ −X

σ∈E

τσ(DK,σwⁿ⁺¹_M )^TD^∗σDK,σwⁿ⁺¹_M

+ X

K∈T

m(K)W_Kⁿ⁺¹(wⁿ⁺¹_2,K −w^D_2,K)≤0, whereDK,σwⁿ⁺¹_M = (DK,σwⁿ⁺¹_1,M, DK,σwⁿ⁺¹_2,M)^T.

Proof. Using the definition (50) of the discrete entropy, one has Sⁿ⁺¹−Sⁿ=A+B,

where

A= X

K∈T

m(K)

ρⁿ⁺¹_K ·(uⁿ⁺¹_K −u^D_K)−(χ(uⁿ⁺¹_K )−χ(u^D_K))

−ρⁿ_K·(uⁿ_K−u^D_K) +χ(uⁿ_K)−χ(u^D_K) , B=−λ²

2 u^D₂ X

σ∈E

τσ

(Dσ(V_Mⁿ⁺¹−V_M^D))²−(Dσ(V_Mⁿ −V_M^D))² .

We first consider the term A, where the quantities χ(u^D_K) cancel each others.

Adding and subtractinguⁿ⁺¹_K ·ρⁿ_K, the termAnow reads A= X

K∈T

m(K)(ρⁿ⁺¹_K −ρⁿ_K)·(uⁿ⁺¹_K −u^D_K)−(χ(uⁿ⁺¹_K )−χ(uⁿ_K))−ρⁿ_K·(uⁿ_K−uⁿ⁺¹_K ).

Since χ is a convex function such that ρ =∇uχ, leading to ρⁿ_K =∇uχ(uⁿ_K), we have:

χ(uⁿ⁺¹_K )−χ(uⁿ_K)−ρⁿ_K·(uⁿ⁺¹_K −uⁿ_K)≥0.

It yields

(53) A≤ X

K∈T

m(K)(ρⁿ⁺¹_K −ρⁿ_K)·(uⁿ⁺¹_K −u^D_K).

We now address the termB. Since (a²−b²)≤2a(a−b) for alla, b∈R, one has B≤ −λ²u^D₂ X

σ∈E

τσDK,σ(V_Mⁿ⁺¹−V_M^D)DK,σ(V_Mⁿ⁺¹−V_Mⁿ).

A discrete integration by part leads to B ≤λ²u^D₂ X

K∈T

(V_Kⁿ⁺¹−V_K^D) X

σ∈E

τσDK,σ(V_Mⁿ⁺¹−V_Mⁿ)

! . Using the scheme for the Poisson equation (33), one gets

(54) B≤u^D₂ X

K∈T

m(K)(V_Kⁿ⁺¹−V_K^D)(ρⁿ⁺¹_1,K −ρⁿ_1,K).

Combining (53) and (54) gives (55) Sⁿ⁺¹−Sⁿ≤ X

K∈T

m(K)(ρⁿ⁺¹_1,K −ρⁿ_1,K)h

(uⁿ⁺¹_1,K −u^D_1,K) +u^D₂(V_Kⁿ⁺¹−V_K^D)i

+ X

K∈T

m(K)(ρⁿ⁺¹_2,K −ρⁿ_2,K)(uⁿ⁺¹_2,K −u^D_2,K).

Using the primal scheme (31)-(32), the inequality (55) becomes Sⁿ⁺¹−Sⁿ

∆t ≤X

K

m(K)W_Kⁿ⁺¹(uⁿ⁺¹_2,K −u^D_2,K) +C+D,

with

C=− X

K∈T

X

σ∈E

F_1,K,σⁿ⁺¹

!

(uⁿ⁺¹_1,K −u^D_1,K) +V_Kⁿ⁺¹(uⁿ⁺¹_2,K −u^D_2,K)

−u^D₂ X

K∈T

X

σ∈E

F_1,K,σⁿ⁺¹

!

(V_Kⁿ⁺¹−V_K^D),

D=− X

K∈T

X

σ∈E

F_2,K,σⁿ⁺¹ −V_σⁿ⁺¹F_1,K,σⁿ⁺¹

!

(uⁿ⁺¹_2,K −u^D_2,K).

Now using the relation (49), the change of variables (35) and the relations (41) between the primal and dual numerical fluxes, it holds

C=−X

K∈T

X

σ∈E

G_1,K,σⁿ⁺¹

!

(w_1,Kⁿ⁺¹−w^D_1,K),

D=−X

K∈T

X

σ∈E

G_2,K,σⁿ⁺¹

!

(w_2,Kⁿ⁺¹−w^D_2,K).

Accounting for the boundary conditions at thermal equilibrium (51), we conclude by a discrete integration by parts which gives:

(56)

Sⁿ⁺¹−Sⁿ

∆t ≤X

σ∈E

G_1,K,σⁿ⁺¹ DK,σwⁿ⁺¹_1,M+X

σ∈E

G_2,K,σⁿ⁺¹ DK,σw_2,Mⁿ⁺¹

+ X

K∈T

m(K)W_Kⁿ⁺¹(wⁿ⁺¹_2,K −w_2,K^D ).

The formulation (42) of the numerical fluxesG_i,K,σⁿ⁺¹ permits to rewrite

(57)

X

σ∈E

G_1,K,σⁿ⁺¹ DK,σwⁿ⁺¹_1,M+X

σ∈E

G_2,K,σⁿ⁺¹ DK,σwⁿ⁺¹_2,M

=−X

σ∈E

τ_σ(D_K,σw_Mⁿ⁺¹)^TD^∗σD_K,σwⁿ⁺¹_M

From equations (56) and (57), one deduces the entropy inequality (52). The as- sumption (11) on the energy relaxation term and the positive definiteness of the matricesD^∗σ ensure the nonpositivity of the right-hand-side of (52) and the decay

of the discrete entropy.

3.2. A prioriestimates on the approximate solutions. We start with a uni- formL^∞bound on (V_Kⁿ)_{K∈T, n≥0}, which will be useful to derive other estimates.

Lemma 1 (L^∞ bound onV_T). Let (uⁿ_K, V_Kⁿ)_K∈T,n≥0 be a solution to the scheme (31)–(34). There exists a constant CV > 0 only depending on Ω, kCk_∞, Cρ and kV^Dk_∞such that

(58) |V_Kⁿ| ≤CV, ∀K∈ T,∀n≥0.

Proof. The assumptions stated in Section1.1ensure that the doping profileC, the electron density ρ₁(u) and the potential boundary condition V^D are in L^∞(Ω).

Hence, one can apply [8, Proposition A.1] to get the expected bound.

From the discrete entropy estimate (52), one may now deduce the following uniform discrete bounds.

Proposition 3 (Discrete a priori estimates). Assume Assumptions 1 hold. Let T be an admissible mesh of Ω and ∆t > 0. Assume the initial and boundary conditions satisfy the hypotheses (10) and (11). Let (uⁿ_K, V_Kⁿ)_K∈T,n≥0be a solution to the scheme in primal variables (31)–(34), supplemented with either (43) or (44).

Then there exist a constant C1 > 0, depending only on the initial and boundary conditions and on the Debye lengthλ², and a constantC2>0, depending only on the initial and boundary conditions, onαdefined in (6) and onCV defined in (58) such that

sup

n=0,...,NT

kuⁿ_1,T −u^D_1,Tk_L2+kuⁿ_2,T −u^D_2,Tk_L2+|V_Mⁿ⁺¹−V_M^D|²_1,M

≤C1, (59)

NT−1

X

n=0

∆t

w_1,Mⁿ⁺¹

2 1,M+

wⁿ⁺¹_2,M

2 1,M

≤C₂. (60)

Proof. Proposition 2 ensures the dissipation of the discrete entropy. Since χ ∈ C¹(R²,R) satisfies∇χ=ρ, the Taylor’s theorem gives, for allK∈ T

χ(uⁿ_K)−χ(u^D_K) = Z 1

0

ρ(u^D_K+s(uⁿ_K−u^D_K))·(uⁿ_K−u^D_K)ds.

Then one can rewrite the first term of the discrete entropySⁿ, defined in equation (50), as follow:

X

K∈T

m(K) Z 1

0

ρ(uⁿ_K)−ρ u^D_K+s uⁿ_K−u^D_K

· uⁿ_K−u^D_K ds= X

K∈T

m(K) Z 1

0

ρ(uⁿ_K)−ρ u^D_K+s(uⁿ_K−u^D_K)

· uⁿ_K−(suⁿ_K+ (1−s)u^D_K) ds 1−s. Using the monotony assumption (12) ofρ, we finally get

S⁰≥Sⁿ≥ C₀ 2

X

K∈T

m(K)|uⁿ_K−u^D_K|²−λ² 2 u^D₂ X

σ∈E

τ_σD_σ(V_Mⁿ −V_M^D)², which yields the a priori estimate (59). The estimate (60) is obtained by sum- ming the discrete entropy inequality (52) forn= 0, . . . , N_T −1. Since the energy relaxation satisfies (11), the control of the entropy dissipation writes

(61)

N_T−1

X

n=0

∆tX

σ∈E

τσ(DK,σwⁿ⁺¹_M )^TD^∗σDK,σwⁿ⁺¹_M ≤S⁰.

Using the definition (45) of D^∗σ and assumption (6) on Lⁿσ, we have, for all σ∈ E and allv∈R²

v^TD^∗σv≥α|Pⁿ⁺¹σ v|²≥ α

k(P^n+1σ )⁻¹k²|v|².

Now, using the definition ofPⁿ⁺¹σ , we observe thatk(Pⁿ⁺¹σ )⁻¹k is a nondecreasing function of|V_σⁿ⁺¹|, whereV_σⁿ⁺¹ is defined either by (43) or by (44). Using theL^∞

bound (58), we finally obtain that there exists a constantβ >0, depending onCV

andα, such that for allσ∈ E, for allv∈R² v^TD^∗σv≥β|v|².

Applying this estimate in (61) yields thea priori estimate (60).

Since the mesh satisfies the regularity constraint (25), and as a consequence of a discrete Poincar´e inequality [4], it holds

NT−1

X

n=0

∆t

kw_1,Mⁿ⁺¹k²_1,M+kwⁿ⁺¹_2,Mk²_1,M

≤c, (62)

sup

n=0,...,N_T

kV_Mⁿ⁺¹−V_M^Dk1,M

≤c.

(63)

4. Existence of a solution to the schemes

In this section, we state the existence of a solution to the schemes (both primal and dual, due to the equivalence property, see Proposition1). The proof is based on a Leray-Schauder fixed point theorem, and follows the same guidelines as the existence proof given in [14, Lemma 3.2].

Theorem 1 (Existence of a solution). Let (T,E,P) be an admissible mesh of Ω satisfying the regularity constraint (25). Assume Assumptions 1, (10) and (11) hold. Let (w_1,T⁰ , w_2,T⁰ ) be defined by (35) and (28). Then, for all 0 ≤n ≤NT, the primal scheme (31)–(34) has a solution (uⁿ_1,T, uⁿ_2,T, V_Tⁿ). As a consequence, by equivalence, the dual scheme (38)–(42) has a solution (w_1,Tⁿ , wⁿ_2,T, V_Tⁿ) for all 0≤n≤N_T.

Proof. The result is proved by induction on 0≤n≤NT. The principle of Leray- Schauder fixed point theorem is to transform continuously the nonlinear system into a linear one while ensuring that a priori estimates controlling the solution remain all along the homotopy. We sketch the main idea of the proof, following [14, Lemma 3.2], parametrizing the homotopy byκ∈[0,1].

Letn≥0 andw_1,Tⁿ , wⁿ_2,T, V_Tⁿbe given, as well as the Dirichlet boundary condi- tionsw_1,σ^D ,w^D_2,σ andV_σ^D,∀σ∈ E_ext^D . We apply the Leray-Schauder theorem to the application

Lⁿ:R^2θ×[0,1]→R^2θ (ueT, κ)7→uT

defined in four steps as follows. Hereθ denotes the cardinal ofT, see Section2.1.

Step 1. LetV_T ∈R^θ be the solution to the following linear system (64)







−λ²X

σ∈E

τσDK,σV_M= m(K) (CK−ρ1(ueK)), ∀K∈ T,

Vσ=V_σ^D, ∀σ∈ E_ext^D .

Step 2. We make use of the change of variables (13) to definewe_T = (we_1,T,we_2,T)∈ R^2θ as

we_1,K=eu_1,K+V_Keu_2,K, we_2,K =ue_2,K, ∀K∈ T.

Step 3. We define wT = (w1,T, w2,T) ∈ R^2θ as the solution to the following linear problem: for allK∈ T,

(65)

−X

σ∈E

τ_σ(D_11,σ(w, Ve )D_K,σw_1,M +D_12,σ(w, Ve )D_K,σw_2,M)

=−κm(K)b1(weK, VK)−b1(w_Kⁿ, V_Kⁿ)

∆t ,

−X

σ∈E

τσ(D21,σ(w, Ve )DK,σw_1,M +D22,σ(w, Ve )DK,σw_2,M)

=κ

−m(K)b₂(we_K, V_K)−b₂(wⁿ_K, V_Kⁿ)

∆t

+m(K)fW(we_K, V_K)−m(K)V_K−V_Kⁿ

∆t b₁(wⁿ_K, V_Kⁿ)

, associated with the following boundary conditions

(66)

w1,σ=κw_1,σ^D , w2,σ =κw^D_2,σ, ∀σ∈ E_ext^D ,

D11,σ(w, Ve )DK,σw1,M+D12,σ(w, Ve )DK,σw2,M= 0, σ∈ E_ext^N , D21,σ(w, Ve )DK,σw_1,M+D22,σ(w, Ve )DK,σw_2,M= 0, σ∈ E_ext^N .

Step 4. Making use of the entropy-primal change of variables (13), we define u_T = (u_1,T, u_2,T)∈R^2θas

u1,K =w1,K−w2,KVK, u2,K =w2,K, ∀K∈ T.

We remark that a fixed pointu_T ofLⁿ(·, κ= 1), together with the corresponding V_T, gives a solution (uⁿ⁺¹_1,T , uⁿ⁺¹_2,T , V_Tⁿ⁺¹) to the scheme (31)–(34). First of all, we have to check that the applicationLⁿ is well-defined. Existence and uniqueness of VT solution to the linear system (64) is obvious. Moreover, Lemma1ensures anL^∞ bound onV_T. Then, we prove the existence and uniqueness ofw_T = (w_1,T, w_2,T), solution to (65)–(66). Since it is a finite dimensional linear system, it is sufficient to prove uniqueness ofw_T, and next that ifκ= 0, then w_T =0. Let us multiply the first equation of (65) byw_1,Kand the second byw_2,K. Adding both equations, summing over all the control volumesK∈ T and performing a discrete integration by parts leads to

X

σ∈E

τ_σ(D_K,σw_M)^TDeσD_K,σw_M= 0,

where (Deσ)ij =Dij,σ(w, Ve ),i= 1,2. Since the matricesDeσare symmetric positive definite, DK,σw_i,M = 0 for allσ∈ E, i= 1,2. Due to the homogeneous Dirichlet boundary conditions (66) (withκ= 0), the discrete Poincar´e inequality [4] ensures thatw1,K=w2,K= 0, for allK∈ T. Hence the system is invertible.

We now have to verify the assumptions of the Leray-Schauder theorem. First observe that every step of construction ofLⁿ is continuous onR^2θ×[0,1]. Since the problem is set in finite dimension, it ensures the required compactness. We have already established thatLⁿ(ueT,0) =0. Now, it remains to verify that there exists a constantM >0 such that, for allκ∈[0,1], for allu_T satisfyingLⁿ(u_T, κ) =u_T, it holds kuTkL² ≤ M. Let us remark that if Lⁿ(u_T, κ) = u_T then we_T = w_T.