Existence of Bounded Steady State Solutions to Spin-Polarized Drift-Diffusion Systems

The large-time asymptotics of the density matrix solving a drift-diffusion-Poisson model for the spin-polarized electron transport in semiconductors is proved. The equations are analyzed in a bounded domain with initial and Dirichlet boundary conditions. If the relaxation time is sufficiently small and the boundary data is close to the equilibrium state, the density matrix converges exponentially fast to the spinless nearequilibrium steady state. The proof is based on a reformulation of the matrix-valued cross-diffusion equations using spin-up and spin-down densities as well as the perpendicular component of the spin-vector density, which removes the cross-diffusion terms. Key elements of the proof are time-uniform positive lower and upper bounds for the spin-up and spin-down densities, derived from the De Giorgi-Moser iteration method, and estimates of the relative free energy for the spin-up and spin-down densities.1.1. Model equations. We assume that the dynamics of the (Hermitian) density matrix N(x, t) ∈ C 2×2 , the current density matrix J(x, t) ∈ C 2×2 , and the electric potential V (x, t)

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“…for any ε > 0. Choosing ε = K 0 /(2q α K 1 ), which is equivalent to K 1 q α ε = K 0 /2, (22) becomes…”

Section: Appendix a A Boundedness Resultsmentioning

confidence: 99%

Large-time asymptotics for a matrix spin drift-diffusion model

Holzinger

Jüngel

2020

Journal of Mathematical Analysis and Applications

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“…The equations are weakly coupled through the last term, which expresses the well-known D'yakonov-Perel' spin relaxation. The spin drift-diffusion model with = 0 was suggested in [21] and mathematically analyzed in [12,13]. Model (8) in one space dimension and with nonlinear diffusion corresponds to the bipolar quantum drift-diffusion equations that were analyzed in [5].…”

Section: =1mentioning

confidence: 99%

“…Since The final product (h 1 • σ)# 0 (1) is just a multiplication. We apply rule (13) with 0 = 0 = 0 to obtain…”

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confidence: 99%

Formal derivation of quantum drift-diffusion equations with spin-orbit interaction

Barletti¹,

Holzinger²,

Jüngel³

2021

Preprint

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A. Quantum drift-diffusion equations for a two-dimensional electron gas with spin-orbit interactions of Rashba type are formally derived from a collisional Wigner equation. The collisions are modeled by a Bhatnagar-Gross-Krook-type operator describing the relaxation of the electron gas to a local equilibrium that is given by the quantum maximum entropy principle. Because of non-commutativity properties of the operators, the standard diffusion scaling cannot be used in this context, and a hydrodynamic time scaling is required. A Chapman-Enskog procedure leads, up to first order in the relaxation time, to a system of nonlocal quantum drift-diffusion equations for the charge density and spin vector densities. Local equations including the Bohm potential are obtained in the semiclassical expansion up to second order in the scaled Planck constant. The main novelty of this work is that all spin components are considered, while previous models only consider special spin directions.

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“…In case the free energy functional is seen as a complete description, D provides the time scale, else it may include density dependent corrections accounting for the missing parts in the free energy functional. For the general setting (not restricted to Fermi-Dirac integrals only) and a rigorous treatment including phase separation problems compare, e.g., [6][7][8][9][10][11][12][13] and cited literature. For relations with gradient structures, cf.…”

Section: Short Summary Of the Variational Formulationmentioning

confidence: 99%

“…All potential-like quantities are 'measured' in units of the thermal voltage U T := k B T /q, is the k B Boltzmann constant, T is the temperature, and q is the elementary charge. Thus (10) in the Lipschitzian domain Ω ⊂ R d . For reasons of simplicity the boundary conditions are homogeneous Neumann for insulating parts and Dirichlet for contacts (Γ D = ∪ i Γ D i with positive surface measure), where w applied i is added to the contact potentials (w i = w * i + w applied i , ϕ p,i = w applied i , ϕ n,i = w applied i , x i ∈ Γ D i ).…”

Section: Short Summary Of the Variational Formulationmentioning

confidence: 99%

Existence of bounded discrete steady state solutions of the van Roosbroeck system with monotone Fermi–Dirac statistic functions

Gärtner

2015

J Comput Electron

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If in the classic van Roosbroeck system (Bell Syst Tech J 29:560-607, 1950) the statistic function is modified, the equations can be derived by a variational formulation or just using a generalized Einstein relation. In both cases a dissipative generalization of the Scharfetter-Gummel scheme (IEEE Trans Electr Dev 16, 64-77, 1969), understood as a one-dimensional constant current approximation, is derived for strictly monotone coefficient functions in the elliptic operator ∇ • f (v)∇, v chemical potential, while the hole density is defined by p = F(v) ≤ e v . A closed form integration of the governing equation would simplify the practical use, but mean value theorem based results are sufficient to prove existence of bounded discrete steady state solutions on any boundary conforming Delaunay grid. These results hold for any piecewise, continuous, and monotone approximation of f (v) and F(v). Hence an implementation based on this discretization will inherit the same stability properties as the Boltzmann case based on the Scharfetter-Gummel scheme. Large chemical potentials and and related degeneracy effects in semiconductors can be approximated. A proven, stability focused blueprint for the discretization of a fairly general, steady state Fermi-Dirac like drift-diffusion setting for semiconductors using mainly new results to extend classic ideas is the main goal.

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Existence of Bounded Steady State Solutions to Spin-Polarized Drift-Diffusion Systems

Cited by 14 publications

References 34 publications

Large-time asymptotics for a matrix spin drift-diffusion model

Large-time asymptotics for a matrix spin drift-diffusion model

Formal derivation of quantum drift-diffusion equations with spin-orbit interaction

Existence of bounded discrete steady state solutions of the van Roosbroeck system with monotone Fermi–Dirac statistic functions

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