Asymptotic behavior of weak solutions of the drift diffusion equations for semiconductors coupled with Maxwell’s equations

F. Jochmann Mathematisch Naturwissenschaftliche Fakultät II Institut für angewandte Mathematik Humboldt Universität Berlin Unter den Linden 6, 10099 Berlin Germany

                         jochmann@mathematik.hu.berlin.de

In this talk the transient drift-diffusion model describing the charge transport in semiconductors is considered.

∂_{t}ρ_{k}(t, x) = −∇ · j_{k}(t, x) − R(x, ρ_{1}(t, x), ρ_{2}(t, x)),  (1)
j_{k}(t, x) = −D_{k}(x)∇ρ_{k}(t, x) − (−1)^{k}μ_{k}(x)ρ_{k}(t, x)E(t, x)  (2)

for k ∈ {1, 2}

ε(x)∂_{t}E(t, x) = curl H(t, x) + j_{2}(t, x) − j_{1}(t, x)  (3)
μ(x)∂_{t}H(t, x) = − curl E(t, x)  (4)
div (ε(x)E(t, x)) = ρ_{1}(t, x) − ρ_{2}(t, x) + C(x), div (μ(x)H(t, x)) = 0  (5)

Here ρ_{1}, ρ_{2} denote the charge densities and j_{1}, j_{2} denote the current densities of the holes and electrons respectively. The self-consistent electromagnetic field (E, H) obeys Maxwell’s equations 3, 4 and 5. The unknown functions ρ_{1}, ρ_{2}, E, H depend on the time t ∈ IR and space variable x ∈ Ω Ω ⊂ IR^{3} is a bounded Lipschitz-domain with ∂Ω = Γ_{D} ∪ Γ_{N}, where Γ_{D}, Γ_{N} are disjoint subsets of ∂Ω. Γ_{D} represents the perfectly conducting Ohmic contacts and Γ_{N} represents the insulating boundary of the semiconductor device. The mobilities μ_{1}, μ_{2} of the holes and electrons resp. are assumed to be positive constants. The diffusion coefficients D_{1}, D_{2}and the recombination generation rate R are functions of the densities ρ_{1}, ρ_{2} and the space variable x. The system 2 - 5 is supplemented by suitable physically motivated initial boundary conditions. Analysis of the drift diffusion model for semiconductors has been presented in [1] - [9] and [11]-[14] in the case that Maxwell’s equations 3 - 5 are replaced by Poisson’s equation − div (ε∇V) = ρ_{1} − ρ_{2} − C for an electrostatic field E = −∇V. However, at very high frequencies the time dependent magnetic field generates a non curl-free electric field, which cannot be written as the gradient of an electrostatic potential. Therefore, Poisson’s equation has to be replaced by Maxwell’s equations 3, 4 - 5 for the electromagnetic field. ¿From the mathematical point of view this means that the elliptic Poisson equation is replaced by a hyperbolic system, which complicates in particular the problem of uniqueness and regularity. Global existence of weak solutions (ρ, E, H) to problem 1 - 5, with

ρ ∈ L_{loc}^{2}([0, ∞), H^{1}(Ω)) ∩ L_{loc}^{∞}([0, ∞), L^{∞}(Ω)) ∩ C([0, ∞), L^{2}(Ω))

and (E, H) ∈ C([0, ∞), L^{2}(Ω)) is proved, see [6] and [9]. Uniqueness and L^{p}-regularity (p > 2) of weak solutions in the two-dimensional case is proved in [7]. The main question of this talk is the asymptotic behavior of weak solutions for t → ∞ . For this purpose assume that φ ∈ H^{1}(Ω) and r_{1}, r_{2} ∈ L^{∞}(Ω) ∩ H^{1}(Ω) is a solution of the stationary drift-diffusion-equations. Then sufficient conditions for the convergence of ρ, E, h to the stationary state r, φ, i.e. lim_{t→∞} ||E(t) + ∇φ||_{Ł^{2}(Ω)} = 0 and lim_{t→∞} ||ρ(t) − r||_{Ł^{p}(Ω)} = 0 for all p ∈ [1, ∞) will be given.

References

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[6] F. Jochmann, Existence of weak solutions to the drift diffusion model for semiconductors copled with Maxwell’s equations, J. Math. Anal. Appl.204, (1996), 655-676

[7] F. Jochmann, Uniqueness and regularity for the two-dimensional drift diffusion model for semiconductors coupled with Maxwell’s equations, Preprint, submitted for publcation

[8] F. Jochmann, A compactness result for vector-fields with divergence and curl in L^{q} satisfying mixed boundary conditions, Appl. Anal. vol. 66, pp. 189-203

[9] F.Jochmann On the drift-diffusion model for semiconductors with prescribed currents, submitted for publication.

[10] F. Jochmann, A singular limit in the drift diffusion model for semiconductors coupled with Maxwell’s equations, to appear in Appl. Anal.

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