Annegret Glitzky Energy Models for Semiconductor Devices Where The

Weierstraß-Institut für Angewandte Analysis und Stochastik

GAMM Annual Meeting 2006, Section 13 Annegret Glitzky

Energy models for semiconductor devices where the various equations are de®ned on different domains

Mohrenstr. 3910117Berlin glitzky@wias-berlin.de www.wias-berlin.de March2006 Energy models for heterogeneous semiconductor devices mass and charge and energy transport described by

Γ • continuity equations for densities n, p D ΓN1 − + of electrons e and holes h on 0 Ω1 • balance equation for the Γ density of the total energy e on ΓN01 Ω S

• Poisson equation for the ΓN0 Ω0 electrostatic potential ϕ on ΓD strongly coupled PDEs, different domains of de®nition, heterogeneous materials, mixed boundary conditions restrict us to the stationary energy model

2 Stationary energy model

∇· jn = R, ∇· jp = R in 0 ∇· je = 0 in

f − n + p in −∇ · (ε∇ϕ) = 0 ( f in 1

R reaction rate of the direct electron-hole recombination-generation e− + h+ ⇋ 0 jn, jp particle ¯ux densities of electrons and holes je ¯ux density of the total energy ε dielectric permittivity f prescribed charge density system has to be completed by state equations, kinetic relations, mixed boundary conditions

3 Stationary energy model: state equations, kinetic relations

ζn, ζp ± electrochemical potentials of electrons and holes, T ± lattice temperature use variables ζn ζp 1 z , z de®ned on z = , , − ,ϕ , 1 2 0 T T T z3, z4 de®ned on state equations:

n = Hn(·, z), p = Hp(·, z) n − p = h(·, z) reaction:

R = r(·, z) ez1+z2 − 1

4 Stationary energy model:kinetic relations

¯uxes:

jn σnnT σnpT τ˜n ∇z1 jp = − σpnT σppT τ˜p ∇z2 on 0    2    je τ˜n τ˜p κT +τ ˜0 ∇z3      

τ˜i = σikT (ζk + PkT ), i = n, p, τ˜0 = σikT (ζi + Pi T )(ζk + PkT ) k=n,p i,k=n,p X X

2 je = −˜κT ∇z3 on 1 conductivities κ˜, κ, σnn, σpp > 0, σnp = σpn ≥ 0 depend nonsmoothly on x and smoothly on the state variables, Pn, Pp - transported entropies Onsager relations are ful®lled

5 Stationary energy model: strongly coupled system (1)

a11(z) a12(z) a13(z) 0 ∇z1 −R(z) a21(z) a22(z) a23(z) 0 ∇z2 −R(z) −∇·     =   on 0 a31(z) a32(z) a33(z) 0 ∇z3 0  0 0 0 ε   ∇z4   f − h(z)             

a˜33(z3) 0 ∇z3 0 −∇· = on 1 0 ε ∇z4 f

6 Stationary energy model: boundary conditions (2)

D zi = zi , i = 1,..., 4, on ŴD,

N0 ν · k=1,2,3 aik(x, z)∇zk = gi , i = 1, 2, 3, on ŴN0, N0 ν · (ε∇z4) = g on ŴN0, P 4 Γ ν · k=1,2,3 aik(x, z)∇zk = 0, i = 1, 2, on ŴN01, D ΓN1 νP·a ˜ (z ) = gN1, ν · (ε∇z ) = gN1 on Ŵ 33 3 3 4 4 N1 Ω1

Γ ΓN01 Ω N0

ΓN0 Ω0

ΓD

7 Notation

G0 = 0 ∪ ŴN0 ∪ ŴN01, G = ∪ ŴN ,ŴN = ŴN0 ∪ ŴN1

1,s 2 1,s 2 Xs = (W0 (G0)) × (W0 (G)) , 1,s 2 1,s 2 Ws = (W (0)) × (W ()) de®ne vectors of data

D D D D N0 N0 N1 N1 w = (z , g, f ), z = (z1 ,..., z4 ), g = (g1 ,..., g4 , g3 , g4 ) ∞ 4 ∞ 2 ∞ H = Wp × L (ŴN0) × L (ŴN1) × L () look for solutions in the form z = Z + z D, where z D corresponds to a function ful®lling the Dirichlet boundary conditions and Z represents the homogeneous part of the solution

D D Mq,τ = (Z, z ) ∈ Xq × Wp : |Zi + zi | < τ, i = 1, 2, on 0, n 1 − τ < Z + z D < − , |Z + z D| <τ on , q ∈ (2, p], τ > 1 3 3 τ 4 4 o 8 Weak formulation of the stationary energy model

∞ 4 ∞ 2 ∞ ∗ Fq,τ : Mq,τ × L (ŴN0) × L (ŴN1) × L () → Xq′

hFq,τ (Z, w), ψiXq′ = aik(·, z)∇zk ·∇ψi + ε∇z4 ·∇ψ4 dx 0 i,k=1 Z n X o + R(·, z)(ψ1 + ψ2) + h(·, z)ψ4 dx − f ψ4 dx Z 0 n o Z + a˜33(·, z3)∇z3 ·∇ψ3 + ε∇z4 ·∇ψ4 dx 1 Z n 4 o N0 N1 N1 − gi ψi dŴ − (g3 ψ3 + g4 ψ4) dŴ, ψ ∈ Xq′ ŴN0 i=1 ŴN1 Z X Z Problem (P):

Find (q,τ, Z, w) such that q ∈ (2, p], τ > 1, (Z, w) ∈ Xq × H,

D Fq,τ (Z, w) = 0, (Z, z ) ∈ Mq,τ .

9 Existence and uniqueness of thermodynamic equilibria

data compatible with thermodynamic equilibrium

D H D N0 Q := w = (z , g, f ) ∈ : zi = const, gi = 0, i = 1, 2, 3, n N1 D D D g3 = 0, z1 + z2 = 0, z3 < 0 o

Theorem 1. Let w∗ = (z D∗, g∗, f ∗) ∈ Q be given.

∗ 1,q0 i) Then there exist q0 ∈ (2, p], τ > 1 and Z4 ∈ W0 (G) such that ∗ D∗ ∗ D∗ ∗ ∗ (Z , z ) = ((0, 0, 0, Z4), z ) ∈ Mq0,τ and Fq0,τ (Z ,w ) = 0. ∗ ∗ In other words, (q0,τ, Z ,w ) is a solution to (P). ii) z∗ = Z ∗ + z D∗ is a thermodynamic equilibrium of (1), (2). ∗ ∗ iii) If (q, τ, Z,w ) is solution to (P) then Z = Z in Xq with q = min{q0, q} holds.

b e e e e b e

10 Results

Lemma 1. (Differentiability) For all parameters τ > 1, all exponents q ∈ (2, p] the operator

∞ 4 ∞ 2 ∞ ∗ Fq,τ : Mq,τ × L (ŴN0) × L (ŴN1) × L () → Xq′ is continuously differentiable.

Proof: Differentiability properties of Nemyzki operators (Recke'95)

Lemma 2. (Fredholm property and injectivity of the linearization) ∗ D∗ ∗ ∗ ∗ ∗ Let w = (z , g , f ) ∈ Q be given. Let (q0,τ, Z ,w ) be the equilibrium solution to Problem (P) (see Theorem 1). Then there exists a q1 ∈ (2, q0] such that the operator ∗ ∗ ∂Z Fq1,τ (Z ,w ) is an injective Fredholm operator of index zero.

11 Sketch of the proof ∗ ∗ ∂Z Fq,τ (Z ,w ) = Lq + Kq 3 4 ∗ hLq Z, ψiXq′ = aik(·, z )∇ Zk ·∇ψi + ε ∇ Z4 ·∇ψ4 + Zi ψi dx Z0 i,k=1 i=1 X 4 X ∗ + a˜33(·, z3)∇ Z3 ·∇ψ3 + ε ∇ Z4 ·∇ψ4 + Zi ψi dx, Z1 i=3 X 4 ∗ ∗ hKq Z, ψiXq′ = r(·, z )(Z1 + Z2)(ψ1 + ψ2) + ∂zh(·, z ) · Z ψ4 − Zi ψi dx Z0 i=1 n4 X o − Zi ψi dx , ψ ∈ Xq′. 1 i=3 Z X Kq compact Lq injective, surjective for q ∈ (2, q1] (surjectivity result in Theorem 3)

Banach's open mapping theorem, Nikolsky's criterion =⇒ ∗ ∗ ∂Z Fq1,τ (Z ,w ) Fredholm operator of index zero, ∗ ∗ ∂Z Fq1,τ (Z ,w ) injective

12 Local existence and uniqueness of steady states

∗ D∗ ∗ ∗ ∗ ∗ Theorem 2. Let w = (z , g , f ) ∈ Q, and let (q0,τ, Z ,w ) be the equilibrium solution to Problem (P) according to Theorem 1.

Then there exists q1 ∈ (2, q0] such that the following assertion holds: There exist neigh- ∗ H ∗ D∗ ∗ ∗ 1 bourhoods U ⊂ Xq1 of Z and W ⊂ of w = (z , g , f ) and a C -map 8: W → U such that Z = 8(w) iff

D D Fq1,τ (Z, w) = 0, (Z, z ) ∈ Mq1,τ , Z ∈ U, w = (z , g, f ) ∈ W.

For data w = (z D, g, f ) near w∗ = (z D∗, g∗, f ∗) ∈ Q there exists a locally unique solution z = Z + z D of the stationary energy model (1), (2).

13 Surjectivity result for Lq1

∗ consider A : Xq → Xq′,

4 4

hAZ, ψiXq′ = dik∇ Zk ·∇ψi + di Zi ψi dx Z0 i,k=1 i=1 X4 X4 + d˜ik∇ Zk ·∇ψi + d˜i Zi ψi dx, Z ∈ Xq, ψ ∈ Xq′ Z1 i,k=3 i=3 X X assume: ∞ ∞ dik, di ∈ L (0), i, k = 1,..., 4, d˜ik, d˜i ∈ L (1), i, k = 3, 4, ∃ M, m > 0 such that

4 4 4 2 2 2 2 R4 diktkti ≥ m|t|R4, diktk ≤ M |t|R4 ∀t ∈ , m ≤ di ≤ M a.e. in 0 i,k=1 i=1 k=1 X X X 4 4 4 ˜ 2 ˜ 2 2 2 R2 ˜ diktkti ≥ m|t|R2, diktk ≤ M |t|R2 ∀t ∈ , m ≤ di ≤ M a.e. in 1 i,k=3 i=3 k=3 X X X

14 Surjectivity result for Lq1

1,2 −1,2 J2,G : W0 (G) → W (G), hJ2,Gu,vi = (uv +∇u ·∇v)dx ZG 1,q −1,q q > 2: Jq,G := J2,G| 1,q : W (G) → W (G) continuous, W0 (G) 0 exists r(G) > 2 such that Jq,G is onto for q ∈ [2, r(G)] (GroÈger'89)

1,q Mq,G := sup{kuk 1,q : u ∈ W (G), kJq,GukW −1,q(G) ≤ 1}, M2,G = 1 W0 (G) 0 analogously for G0

Jq := (Jq,G0, Jq,G0, Jq,G, Jq,G) ∗ Jq : Xq → Xq′ is onto for q ∈ [2, rˆ], rˆ = min{r(G0), r(G)}

M := sup{kuk : u ∈ X , kJ uk ∗ ≤ 1} q Xq q q Xq′ ≤ θ θ 1 = θ + 1−θ Mq max (Mrˆ,G0) , (Mrˆ,G) , q rˆ 2

15 Surjectivity result for Lq1

∗ Theorem 3. The operator A maps Xq onto Xq′ provided that q ∈ [2, rˆ] and m2 1/2 41/2−1/q M 1 − < 1. q M 2

∗ Proof. Let h ∈ Xq′, q ∈ [2, rˆ], de®ne Qh : Xq → Xq by m Q Z := Z − J −1(AZ − h), Z ∈ X h M 2 q q m2 1/2 kQ Z − Q Z¯ k ≤ 41/2−1/q M 1 − kZ − Z¯ k h h Xq q M 2 Xq m2 1/2 m2 1/2 41/2−1/q M 1 − → 1 − < 1 for q → 2 q M 2 M 2 Banachs Fixed Point Theorem

16 References

G. Albinus, H. Gajewski, and R. HuÈnlich, Thermodynamic design of energy models of semiconductor devices, Nonlinearity 15 (2002), 367±383.

A. Glitzky, Stationary energy models for semiconductor devices where the equations live on different domains, in preparation.

A. Glitzky, R. HuÈnlich, Stationary solutions of two-dimensional heterogeneous energy models with multiple species, Banach Center Publ. 66 (2004), 135-151.

A. Glitzky, R. HuÈnlich, Stationary energy models for semiconductor devices with incom- pletely ionized impurities, ZAMM 85 (2005), 778-792.

K. GroÈger, A W 1,p±estimate for solutions to mixed boundary value problems for second order elliptic differential equations, Math. Ann. 283 (1989), 679±687.

L. Recke, Applications of the Implicit Function Theorem to quasi-linear elliptic boundary value problems with non-smooth data, Comm. Partial Differential Equations 20 (1995), 1457±1479.

17 Assumptions

2 (A1) , 0 are bounded Lipschitzian domains in R , G, G0 are regular in the sense of GroÈger'89

∞ 0 (A2) ε ∈ L (), 0 <ε0 ≤ ε(x) ≤ ε < ∞ in .

De®nition. Let V = R2 × (−∞, 0) × R. Then b : 0 × W → R is of the class D0 iff b is a Caratheodory function, which is continuously differentiable with respect to the second argument and for which the function itself as well as its derivative with respect to the second argument are locally bounded and locally uniformly continuous.

b : 1 × V1 → R is of the class D1 if in the de®nition V and are substituted by V1 = (−∞, 0) and 1.

18 Assumptions

(A3) aik : 0 × V → R are of the class (D0), i, k = 1, 2, 3. For every compact subset K ⊂ V there exists an aK > 0 such that 3 2 3 aik(x, z)ξi ξk ≥ aK kξk for all z ∈ K , all ξ ∈ R and f.a.a. x ∈ 0. i,k=1 X a˜33 : 1 × V1 → R+ is of the class (D1), for every k > 1 there exists an a˜k > 0 such that a˜33(x, z) ≥a ˜k for all z ∈ [−k, −1/k] and a.a. x ∈ 1.

(A4) Hi : 0 × V → R+, i = n, p, are of the class (D0), h = Hn − Hp : 0 × V → R, h(x, z1, z2, z3, ·) is monotonic increasing 2 for all z ∈ R × (−∞, 0) × R and a.a. x ∈ 0. c|z | |h(x, z)|≤ cke 4 for all 2 (z1, z2, z3) ∈ [−k, k] × [−k, −1/k], z4 ∈ R and a.a x ∈ 0.

z +z (A5) R(x, z) = r(x, z) e 1 2 − 1 , r : 0 × V → R+ is of the class (D0).

19 Outline of the proof of Theorem 1

E 1 −1 1. : H0 (G) → H (G),

D∗ ∗ N0∗ N1∗ hE(φ), φ¯i 1 = ε ∇(φ + z ) ·∇φ¯ − f φ¯ dx − g φ¯ dŴ − g φ¯ dŴ H0 (G) 4 4 4 Ŵ Ŵ Z n o Z N0 Z N1 D∗ ¯ ¯ 1 + h(·, (0, 0, 0, φ) + z ) φ dx ∀φ ∈ H0 (G) Z0 1 E is strongly monotone, hemicontinuous, exists a unique solution φ ∈ H0 (G) of (φ) = 0.

1,q 2. higher regularity φ ∈ W 0(G), q0 > 2 (GroÈger'89), ∗ ∗ ∗ exists τ > 0 such that (q0,τ, Z ,w ) with Z = (0, 0, 0, φ) is a solution to (P) z∗ = Z ∗ + z D∗ is a thermodynamic equilibrium of (1), (2)

3. uniqueness (indirect proof)