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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. Numerical schemes for
semiconductors energy- transport models. Finite Volumes for Complex Applications IX, Jun 2020,
Bergen, Norway. pp. 75-90. �hal-02563093�
Numerical schemes for semiconductors
energy-transport models
Marianne Bessemoulin-Chatard, Claire Chainais-Hillairet and H´el`ene Mathis
Abstract We introduce some finite volume schemes for unipolar energy-transport models. Using a reformulation in dual entropy variables, we can show the decay of a discrete entropy with control of the discrete entropy dissipation.
Key words: energy-tranport model, finite volumes, entropy method. MSC (2010): 65M08, 65M12, 35K20.
1 Energy-transport models
Presentation
In this article, we are interested in the discretization 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 ef-fects. As they have a drift-diffusion form, they remain simpler than hydrodynamic equations or semiconductor Boltzmann equations. As explained for example in [17] (and the references therein), these energy-transport models can be derived from the Boltzmann equation by the moment method.
The unipolar energy-transport system consists in two continuity equations for the electron density ρ1and the internal energy density ρ2, coupled with a Poisson equation describing the electrical potential V . Following the framework adopted in
Marianne Bessemoulin-Chatard and H´el`ene Mathis
Laboratoire de Math´ematiques Jean Leray, Universit´e de Nantes & CNRS UMR 6629, BP 92208, F-44322 Nantes Cedex 3, France
e-mail: marianne.bessemoulin@univ-nantes.fr, helene.mathis@univ-nantes.fr Claire Chainais-Hillairet
Univ. Lille, CNRS,UMR 8524-Laboratoire Paul Painlev´e. F-59000 Lille, France e-mail: claire.chainais@univ-lille.fr
[6], we consider that the electron and energy densities are defined as functions of the entropy variables u1= µ/T and u2= −1/T where µ is the chemical potential and T the temperature. We set u = (u1, u2).
Let Ω be an open bounded subset of Rd (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= 0, (1a) ∂tρ2(u) + divJ2= ∇V · J1+W (u), (1b)
−λ2
∆V = C(x) − ρ1(u), (1c) where J1and J2are respectively the electron and energy current densities, ∇V · J1 corresponds to a Joule heating term and W (u) is an energy relaxation term. The doping profile C(x) describes the fixed charged background and λ is the rescaled Debye length. The electron and energy current densities are given by:
J1= −L11(u)(∇u1+ u2∇V ) − L12(u)∇u2, (2a) J2= −L21(u)(∇u1+ u2∇V ) − L22(u)∇u2, (2b) where L(u) = (Li j(u))1≤i, j≤2is a symmetric uniformly positive definite matrix.
The system (1)-(2) 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, we assume that Ω is an open bounded polygonal (or polyhedral) subset of Rd, such that its boundary ∂ Ω is split into ∂ Ω = ΓD∪ ΓN, with ΓD∪ ΓN= /0 and m
d−1(ΓD) > 0. We denote by n the normal to ∂ Ω outward Ω . The boundary conditions write
u1= uD1, u2= u2D, V = VDon ΓD× [0, Tmax], (3a) J1· n = J2· n = ∇V · n on ΓN× [0, Tmax]. (3b) We assume that the Dirichlet boundary conditions uD1, uD2 and VD do not depend on time and are the traces of some functions defined on the whole domain Ω , still denoted by uD1, uD2 and VD. Moreover, we assume that uD2 < 0 is constant on ΓDand that the energy relaxation term W (u) verifies, for all u ∈ R2and uD2 < 0,
W(u)(u2− uD2) ≤ 0. (4) The main results on the energy-transport model (1)-(2)-(3) are presented in [15]: existence of solutions to the transient system, regularity, uniqueness and existence and uniqueness of steady-states. The main assumptions needed on the function u 7→ ρ (u) = (ρ1(u), ρ2(u)) for the existence result are the following:
ρ ∈ W1,∞(R2; R2), (5a) ∃c0> 0 such that (ρ(u) − ρ(v)) · (u − v) ≥ c0|u − v|2 for u, v ∈ R2, (5b) ∃χ ∈ C1
(R2; R) strictly convex such that ρ = ∇uχ . (5c) These hypotheses are rather hard to satisfy in the applications (see Section 4), as well as the hypothesis on uniform positive definiteness of the diffusion matrix L. Exis-tence results for physically more realistic diffusion matrices (only positive semi-definite) are established in [10, 12] for the stationary model and in [4, 5] for the transient system, but only in the case of data close to thermal equilibrium. More recently, existence of solutions has been proved in a simplified degenerate case, namely for a model with a simplified temperature equation in [16] and for vanishing electric fields (avoiding the coupling with Poisson equation) in [20].
The existence result due to Degond, G´enieys and J¨ungel [6, 15] is based on a reformulation of the system in terms of dual entropy variables. This reformulation symmetrizes the system and allows to apply an entropy method. Since we are going to adapt the results of [6] to the discrete framework, let us now introduce the system reformulated in terms of dual entropy variables and give the outline of the entropy structure.
The system in dual entropy variables
The key point of the analysis of the primal model (1)-(2) is to use another set of variables which symmetrizes the problem, see [6]. Let us define the so-called dual entropy variables w = (w1, w2) (w1is an electrochemical potential):
w1= u1+ u2V, (6a)
w2= u2. (6b)
Through this change of variables, the problem (1)-(2) is equivalent to
∂tb1(w,V ) + divI1(w,V ) = 0, (7a) ∂tb2(w,V ) + divI2(w,V ) = eW(w) − ∂tV b1(w,V ), (7b)
−λ2
∆V = C − b1(w,V ), (7c) where the function b(w,V ) = (b1(w,V ), b2(w,V )) is related to ρ and V by
b1(w,V ) = ρ1(u), b2(w,V ) = ρ2(u) −V ρ1(u), (8) and the new energy relaxation term is defined by eW(w) = W (u). Moreover, the symmetrized currents are given by I1= J1and I2= J2−V J1, which leads to
I1(w,V ) = −D11(w,V )∇w1− D12(w,V )∇w2, (9a) I2(w,V ) = −D21(w,V )∇w1− D22(w,V )∇w2, (9b)
where the new diffusion matrix D(w,V ) = (Di j(w,V ))1≤i, j≤2is defined by
D(w,V ) = P(V )TL(u)P(V ), with P(V ) =1 −V0 1
. (10)
It is therefore clear that the new diffusion matrix D is also symmetric and uniformly positive definite.
Entropy structure
We recall in this section the entropy/entropy-dissipation property satisfied by the energy-transport model (1)-(3) established in [6]. The entropy function is defined by
S(t) =
Z
Ω
ρ (u) · (u − uD) − (χ(u) − χ(uD)) dx−λ 2 2 u D 2 Z Ω |∇(V −VD)|2dx. (11)
Since uD2 < 0 and χ is a convex function such that ρ = ∇uχ , S(t) is nonnegative for all t ≥ 0.
In addition to the hypotheses already given above, we assume that the Dirichlet boundary conditions are at thermal equilibrium, namely
∇wD1 = ∇wD2 = 0. (12) Then the entropy function satisfies the following identity:
d dtS(t) = − Z Ω (∇w)TD∇w + Z Ω W(u)(u2− uD2) ≤ 0. (13) The proof of (13) is given in [6], even for more general boundary conditions.
2 Numerical schemes
Different kind of numerical schemes have already been designed for the energy-transport systems, essentially for the stationary systems: finite difference schemes in [11, 19], finite element schemes in [7, 14]. We also refer to [3] for DDFV (Dis-crete Duality Finite Volume) schemes for the evolutive case. Up to our knowledge, there exists no convergence analysis of these numerical schemes. In this paper, we are interested in the design and the analysis of some finite volume schemes for the system (1)–(3), with two-point flux approximations (TPFA) of the numerical fluxes. We pay attention, while building the scheme, on the possibility of adapting the entropy method to the discrete setting. This will be crucial in order to fulfill the convergence analysis of the scheme.
Mesh and notations
Let ∆t > 0 be the time step and set tn= n∆t for all n ≥ 0. We now define the mesh of the domain Ω . It is given by a family T of open polygonal (or polyhedral in 3D) control volumes, a familyE of edges (or faces), and a family P = (xK)K∈T of points. The schemes we will consider are based on two-points flux approximations, so that we assume that the mesh is admissible in the sense of [9, Definition 9.1].
In the set of edgesE , we distinguish the interior edges σ = K|L ∈ Eint and the boundary edges σ ∈Eext. Due to the mixed boundary conditions, we have to distin-guish the edges included in ΓDfrom the edges included in ΓN:Eext=ED∪EN. For a control volume K ∈T , we define EK the set of its edges, which is also split into EK=EK,int∪EKD∪EKN.
In the sequel, we denote by d the distance in Rd and m the measure in Rd or Rd−1. For all σ ∈E , we define dσ= d(xK, xL) if σ = K|L ∈Eint and dσ= d(xK, σ ) if σ ∈Eext, with σ ∈EK. Then the transmissibility coefficient is defined by τσ = m(σ )/dσ, for all σ ∈E .
A finite volume scheme with two-point flux approximation provides, for an un-known v, a vector v = (vK)K∈T ∈ Rθ (with θ = Card(T )) of approximate values on each cells. We can associate to v a piecewise constant function, still denoted v. For all K ∈T and all σ ∈ EK, we define
vK,σ= vL if σ = K|L ∈Eint, vDσ if σ ∈ED, vK if σ ∈EN, and DK,σv = vK,σ− vK, Dσv = |DK,σv|.
Schemes in primal and dual entropy variables
Our aim is to design a scheme for the energy transport model in the primal entropy variables (1)-(3). This scheme must lead to an equivalent scheme for the system written in the dual entropy variables (7)-(9). Indeed, in this case, it will be possible to apply the entropy method at the discrete level. This step is crucial as it brings a prioriestimates on the sequences of approximate solutions, leading to compactness results. Moreover, it also permits to prove existence of a solution to the scheme.
One main difficulty in writing a TPFA scheme for the energy-transport model (1)-(3) comes from the approximation of the Joule heating term ∇V · J1. One possibility would be to apply the technique developed in [1], and further used in [18, 8], to discretize de Joule heating term. However, with such discretization, the rewriting of the scheme in dual entropy variables is not straightforward. Therefore, following [2], we propose an approximation of the Joule heating term which is based on its following reformulation:
Let us now turn to the definition of the scheme for the model (1)-(3). Initial and Dirichlet boundary conditions are discretized as usually: u0i,K is the mean value of ui,0over K for all K ∈T and i = 1,2, uDi,σ and VσD are the mean values of u
D i for i= 1, 2 and VDfor σ ∈EDand we define:
un1,σ= uD1,σ, un2,σ = uD2,σ, Vσn= VσD, ∀σ ∈ED, ∀n ≥ 0. (14) The scheme is backward Euler in time and finite volume in space with a two-point flux approximation. It writes, for all n ≥ 0, for all K ∈T :
m(K)ρ n+1 1,K − ρ1,Kn ∆ t +σ ∈E
∑
K Fn+1 1,K,σ= 0, (15a) m(K)ρ n+1 2,K − ρ2,Kn ∆ t +σ ∈∑
E K Fn+1 2,K,σ= m(K)WKn+1 +∑
σ ∈EK Vσn+1F1,K,σn+1 −VKn+1∑
σ ∈EK Fn+1 1,K,σ, (15b) − λ2∑
σ ∈EK τσDK,σV n+1 = m(K)(CK− ρ1,Kn+1), (15c) whereρi,Kn+1= ρi(un+1K ), i= 1, 2 and WKn+1= W (un+1K ) for all K ∈T . The numerical fluxes are given by
Fn+1 1,K,σ= −τσ Ln11,σ(DK,σu1n+1+ un+12,σDK,σVn+1) + Ln12,σDK,σu2n+1 , (16a) Fn+1 2,K,σ= −τσ Ln12,σ(DK,σu1n+1+ un+12,σDK,σVn+1) + Ln22,σDK,σu2n+1 , (16b) where the matrix Ln
σ= (L n i j,σ)1≤i, j≤nis defined as Lnσ= L un K+ unK,σ 2 for all K ∈T ,σ ∈ EK. (17)
At this point, it remains to define Vσn+1involved in (15b) and un+12,σ involved in (16) for all σ ∈E . We will do it later. The choice will be driven by the expected equiva-lence with a scheme for (7)–(10).
In order to obtain an equivalent scheme for the energy transport system in the dual entropy variables (7)–(10), we apply the change of variables (6), associated with the new functions defined in (8), (9) and (10), to (15)-(16). Let us define for all K∈T , for all n ≥ 0,
w1,Kn = un1,K+ un2,KVKn, wn2,K= un2,K, (18a) bn1,K= ρ1,Kn = b1(wnK,VKn), bn2,K= ρ2,Kn − ρ1,Kn VKn= b2(wnK,VKn). (18b)
We similarly define wD1,σand wD2,σ for σ ∈ED. From (15a) and (15b), we deduce m(K)b n+1 1,K − bn1,K ∆ t +
∑
σ ∈EK Fn+1 1,K,σ= 0, m(K)b n+1 2,K − bn2,K ∆ t +σ ∈E∑
K Fn+1 2,K,σ−V n+1 σ F n+1 1,K,σ = m(K)WKn+1− m(K)V n+1 K −VKn ∆ t b n 1,K. It leads to the following scheme for the system written in the dual variables (7):m(K)b n+1 1,K − bn1,K ∆ t +σ ∈E
∑
K Gn+1 1,K,σ= 0, (19a) m(K)b n+1 2,K − bn2,K ∆ t +σ ∈E∑
K Gn+1 2,K,σ= m(K) ˜W n+1 K − m(K) VKn+1−Vn K ∆ t b n 1,K, (19b) − λ2∑
σ ∈EK τσDK,σVn+1= m(K)(CK− bn+11,K), (19c) with Gn+1 1,K,σ =F n+1 1,K,σ, ∀K ∈T ,∀σ ∈ EK, (20a) Gn+1 2,K,σ =F n+1 2,K,σ−V n+1 σ F n+1 1,K,σ, ∀K ∈T ,∀σ ∈ EK, (20b) and ˜WKn+1= WKn+1= ˜W(wn+1K ).The crucial point now is to ensure that the new numerical fluxesG1,K,σn+1 ,G2,K,σn+1 can be seen as approximations of the currents I1and I2defined by (9). This means that we want to rewrite the numerical fluxes as
Gn+1 1,K,σ= −τσ(D ∗ 11,σDK,σwn+11 + D∗12,σDK,σwn+12 ), (21a) Gn+1 2,K,σ= −τσ(D ∗ 21,σDK,σwn+11 + D ∗ 22,σDK,σwn+12 ), (21b) with the coefficients (D∗i j,σ)1≤i, j≤2defined such that the associate matrix D∗σis sym-metric and uniformly positive definite. This property will now depend on the def-inition of Vσn+1 and un+12,σ, respectively involved in (15b) and (16), for each edge σ ∈E .
Equivalence of the schemes in the primal and dual entropy variables
Proposition 1. Let us supplement the scheme (15)-(16) with the definition of the (Vσn+1)σ ∈E , n≥0and(un+12,σ )σ ∈E , n≥0. We distinguish two cases:
un+12,σ =u n+1 2,K + un+12,K,σ 2 and V n+1 σ = VKn+1+VK,σn+1 2 . (22) • Case 2: upwind scheme. For all σ ∈E and n ≥ 0, we set:
un+12,σ = (
un+12,K,σ, if DK,σVn+1> 0, un+12,K, if DK,σVn+1≤ 0,
and Vσn+1= min(VKn+1,VK,σn+1). (23)
Then, in both cases, the scheme(15)-(16) written in the primal entropy variables is equivalent with the scheme(19)-(21) written in the dual entropy variables, provided that D∗σ= (P n+1 σ ) T LnPn+1σ with P n+1 σ = 1 −Vn+1 σ 0 1 . (24)
Proof. Starting from the definition (20) of the numerical fluxesG1,K,σn+1 andG2,K,σn+1 , we want to establish (21) with D∗σdefined by (24).
Let us first notice that, due to the change of variables (18a), we can rewrite DK,σu1n+1and DK,σu2n+1for all K ∈T and σ ∈ EK. It is clear that DK,σu2n+1= DK,σw2n+1. Moreover, we have
DK,σu1n+1= DK,σw1n+1−VKn+1DK,σw2n+1− wn+12,K,σDK,σVn+1, = DK,σw1n+1−VK,σn+1DK,σw2n+1− wn+12,K DK,σVn+1. This yields, for Case 1 as well as for Case 2,
DK,σu1n+1= DK,σw1n+1−Vσn+1DK,σw2
n+1− wn+1 2,σ DK,σV
n+1,
with wn+12,σ = un+12,σ. Therefore, from (16) and (20), we deduce that Gn+1 1,K,σ= −τσ L n 11,σDK,σw1n+1+ (Ln12,σ−Vσn+1L n 11,σ)DK,σw2n+1 , Gn+1 2,K,σ= −τσ (Ln12,σ−Vσn+1L n 11,σ)DK,σw1n+1 +(Ln22,σ− 2Vσn+1Ln12,σ+ (Vσn+1)2Ln11,σ)DK,σw2n+1 . This corresponds to (21) with D∗σ defined by (24). We have shown that the scheme (15)-(16), supplemented either with (22) or (23), implies (19)-(21)-(24). Starting from (19)-(21)-(24), we similarly get (15)-(16).
3 Discrete entropy inequality
In this Section, we establish the discrete counterpart of the decay of the entropy, with the control of its dissipation, (13). The result is stated in Proposition 2.
Main result
First of all, since the functions uD1, uD2, VDare assumed to be defined on the whole domain Ω , we can set
(uD1,K, uD2,K,VKD) = 1 m(K)
Z
K
(uD1(x), uD2(x),VD(x))dx, ∀K ∈T .
Moreover, we remember that uD2 is a constant function, such that
uDK,2= uD2 < 0, ∀K ∈T . (25) Let (unK= (un1,K, un2,K)T,VKn)K∈T ,n≥0be a solution to the scheme (14)–(17), sup-plemented with either (22) or (23). For all n ≥ 0, we define the discrete entropy functional as follows: Sn=
∑
K∈T m(K)ρKn· (unK− uKD) − (χ(unK) − χ(uDK)) (26) −λ 2 2 u D 2∑
σ ∈E τσ(Dσ(Vn− VD))2.We recall that ρKn= ρ(unK) = (ρ1(unK), ρ2(unK))T and that ρ is related to χ by (5c). Therefore, Snis nonnegative for all n ≥ 0.
Proposition 2 (Discrete entropy dissipation). Assume (4), (5), (25) and let (unK= (un1,K, un2,K)T,VKn)K∈T ,n≥0be a solution to the scheme(14)–(17), supplemented with either(22) or (23). The discrete entropy satisfies the following inequality: for all n≥ 0, Sn+1− Sn ∆ t ≤ −σ ∈E
∑
τσ(DK,σw n+1)T D∗σDK,σw n+1 +∑
K∈T m(K)WKn+1(wn+12,K − wD 2,K) ≤ 0, (27) where DK,σwn+1= (DK,σwn+11 , DK,σwn+12 ) T .Proof. Using the definition (26) of the discrete entropy, one has
Sn+1− Sn= A + B, (28) where A=
∑
K∈T m(K)ρKn+1· (un+1K − uDK) − (χ(un+1K ) − χ(uDK)) − ρKn· (unK− uDK) + (χ(unK) − χ(uDK)) , (29) B= −λ 2 2 u D 2∑
σ ∈E τσ h (Dσ(Vn+1− VD))2− (Dσ(Vn− VD))2 i . (30)We first consider the term A. As χ is a convex function such that ρ = ∇uχ , leading to ρKn= ∇uχ (unK), we have: χ (un+1K ) − χ(unK) − ρKn· (un+1K − unK) ≥ 0. This yields A≤
∑
K∈T m(K)(ρKn+1− ρn K) · (un+1K − uDK). (31) We now address the term B. Since (a2− b2)/2 ≤ a(a − b), for all a, b ∈ R, and uD2 ≤ 0, we get:
B≤ −λ2uD 2
∑
σ ∈E
τσDK,σ(Vn+1− VD) DK,σ(Vn+1− Vn).
A discrete integration by part leads to
B≤ λ2uD2
∑
K∈T (VKn+1−VKD)∑
σ ∈EK τσDK,σ(Vn+1− Vn) ! .Using the scheme for the Poisson equation (15c), we obtain B≤ uD
2
∑
K∈Tm(K)(VKn+1−VD
K)(ρ1,Kn+1− ρ1,Kn ). (32) From (28), (31) and (32), we deduce:
Sn+1− Sn≤
∑
K∈T m(K)(ρ1,Kn+1− ρn 1,K) (un+11,K − uD1,K) + uD2(VKn+1−VKD) +∑
K∈T m(K)(ρ2,Kn+1− ρn 2,K)(un+12,K − u D 2,K). (33)Using the primal scheme (15a), (15b), the inequality (33) becomes Sn+1− Sn ∆ t ≤ C + D +K∈
∑
Tm(K)W n+1 K (un+12,K − uD2,K), (34) with C= −∑
K∈T σ ∈E∑
K Fn+1 1,K,σ ! h (un+11,K − uD 1,K) +VKn+1(un+12,K − u D 2,K) i − uD2∑
K∈T σ ∈E∑
K Fn+1 1,K,σ ! (VKn+1−VKD), D= −∑
K∈T σ ∈E∑
K Fn+1 2,K,σ−V n+1 σ F n+1 1,K,σ ! (un+12,K − uD 2,K).Using the change of variables (18a), the relations (20) on the numerical fluxes written in the primal and dual entropy variables and the hypothesis (25), we get
C= −
∑
K∈T σ ∈E∑
K Gn+1 1,K,σ ! (wn+11,K − wD 1,K), D= −∑
K∈T σ ∈E∑
K Gn+1 2,K,σ ! (wn+12,K − wD2,K). (35)Accounting for the boundary conditions, we conclude by a discrete integration by parts which gives (27):
Sn+1− Sn ∆ t ≤σ ∈
∑
EG n+1 1,K,σDK,σwn+11 +∑
σ ∈E Gn+1 2,K,σDK,σwn+12 +∑
K∈T m(K)WKn+1(wn+12,K − wD 2,K). (36)The formulation (21) of the numerical fluxesGi,K,σn+1 permits to rewrite
∑
σ ∈EG n+1 1,K,σDK,σwn+11 +∑
σ ∈EG n+1 2,K,σDK,σwn+12 = −∑
σ ∈E τσ DK,σwn+11 DK,σwn+12 T D∗σ DK,σwn+11 DK,σwn+12 . (37)From (36) and (37), we deduce (27). The hypothesis (4) on the energy relaxation term and the positive definiteness of the matrices Dσensure the nonpositivity of the right-hand-side in (27) and the decay of the discrete entropy.
Consequences
From Proposition 2, we deduce the uniform bound: Sn≤ S0for all n ≥ 0. The control of the dissipation writes
N
∑
n=0σ ∈E∑
τσ(DK,σwn+1)TD∗σDK,σw
n+1≤ S0.
This yields a discrete L2(0, Tmax, H1) estimates on w1 and w2. But, following the ideas of [6, 17], we may obtain other a priori estimates on the solution. They permit first to prove the existence of a solution to the scheme, thanks to a topological degree argument, and second to show the compactness of the sequence of approximate solutions leading to the convergence of the scheme. The existence result and the convergence analysis will be detailed in a forthcoming paper.
4 Numerical experiments
For the numerical experiments, we consider the unipolar energy-transport model under Boltzmann statistics, as in [17, 6]. It is based on the following definitions of the densities ρi(u), i = 1, 2:
ρ1(u) = −1 u2 3/2 exp(u1), ρ2(u) = 3 2 −1 u2 5/2 exp(u1). (38)
so that ρ(u) = ∇uχ (u) with χ (u) = (−u2)−3/2exp(u1).
The diffusion matrix L(u) = (Li j(u))1≤i, j≤2actually depends on u under the fol-lowing form [17]: L = coρ1(u)T1/2−β 1 (2 − β )T (2 − β )T (3 − β )(2 − β )T2 , (39)
where c0> 0 is a constant (and we recall that T = −1/u2). The usual values of β are 1/2, corresponding to the Chen model, and 0, corresponding to the Lyumkis model [17]. The matrix L(u) is symmetric positive definite.
Presentation of the test case
We consider a test case of a 2-D n+nn+silicon diode, uniform in one space direc-tion, already introduced in [7, 13, 3]. It is a simple model for the channel of a MOS transistor. The adopted model is the Chen model (β = 1/2 in (39)). Additional test cases will be given in a forthcoming paper.
The domain is Ω = (0, lx) × (0, ly) with lx= 0.6 µm and ly= 0.2 µm. The channel length is 0.4 µm, see Fig. 1.
xin µm 0 0.1 0.5 0.6 0 0.2 yin µm n+ n n+ Γ1D Γ2D
Fig. 1 Geometry of the n+nn+ballistic diode.
The numerical values of the physical parameters for a silicon diode are given in Table 1. The doping profile is
Table 1 Physical parameters.
Parameter Physical meaning Numerical value q elementary charge 10−19As ε permittivity constant 10−12AsV−1cm−1 µ0 low field mobility 1.5 × 103cm2V−1s−1
UT thermal voltage at T0= 300K 0.0259 V
τ0 energy relaxation time 0.4 × 10−12s
C= Cm= 5 × 1017cm−3 in the n+region, C= Cm= 2 × 1015cm−3 in the n region. The boundary conditions are
V= 1.5V on Γ1Dand V = 0 on Γ2D,
u2= −1/T0, with T0= 300K, on Γ1D∪ Γ2D, ρ1(u) = Cmon Γ1D∪ Γ2D,
the latest giving the boundary condition for u1according to (38). The initial condi-tions for u1and u2are constant and equal to the boundary conditions.
The function W reads
W(u) = c1ρ1(u) − c2ρ2(u), with c1= 3 2 l2x τ0µ0UT , c2= lx2 τ0µ0UT , and the scaling ensures that the Debye length is
λ2= εUT ql2
xCm .
Numerical results
We use an admissible mesh made of 896 triangles. Figure 2 presents the results ob-tained by the scheme (15)-(16) in the centered case (22). The results are plotted for the final time Tfinal= 1s, as the equilibrium state is reached. Although the discretiza-tion is fully implicit, it is necessary to use an adaptative time step during the first iterations, in order to allow the convergence of the Newton’s method. As expected, the computed quantities are almost uniform in one space direction. Moreover one observes the expected hot electron effect in the channel, which compares with the results given in [7, 13, 3].
Fig. 2 2-D n+nn+diode: temperature (above) and electrostatic potential (below).
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