[Paper Review] Semiclassical hydrodynamics of a quantum Kane model for semiconductors

Luigi Barletti, G. Borgioli+1 more|arXiv (Cornell University)|Feb 14, 2014

Advanced Thermodynamics and Statistical Mechanics13 references3 citations

TL;DR

This paper derives a semiclassical hydrodynamic model for electrons in semiconductors with two energy bands using the $k\cdot p$ Hamiltonian and the Wigner-BGK formalism. By applying the Maximum Entropy Principle to close the moment equations, it uniquely parameterizes local equilibrium states via density and velocity fields, enabling a consistent fluid description that captures non-parabolic band dynamics and interband transitions through BGK-like relaxation terms.

ABSTRACT

In this paper we derive a semiclassical hydrodynamic system for electron densities and currents in the two energy bands of a semiconductor. We use the semiclassical Wigner equation with a k.p Hamiltonian and a BGK dissipative term to construct the first two moment equations. The closure of the moment system is obtained using the Maximum Entropy Principle, by minimizing a Gibbs free-energy functional under suitable constraints. We prove that the constraint equations can be uniquely solved, i.e. that the local equilibrium state can be parametrized by the density and velocity field. Some BGK-like models are proposed to mimic the quantum interband migration.

Motivation & Objective

• To develop a hydrodynamic model for electron transport in two-band semiconductors that balances physical rigor with numerical tractability.
• To incorporate non-parabolic band effects and interband transitions into a hydrodynamic framework while preserving consistency with quantum mechanics.
• To close the moment system using the Maximum Entropy Principle, ensuring thermodynamically consistent fluid variables.
• To propose physically motivated BGK-like collision terms that model interband scattering and relaxation processes.
• To prove the unique solvability of the Lagrange multipliers in the MEP closure in terms of macroscopic densities and velocities.

Proposed method

• Formulate the $k\cdot p$ Hamiltonian as a $2\times2$ matrix operator to describe valence and conduction bands using pseudo-spinor formalism.
• Use matrix-valued Wigner functions to represent electron states in phase space, decomposed into scalar and pseudo-spinorial parts.
• Derive zeroth- and first-order moment equations for band densities $n_{\pm}$ and currents $n_{\pm}\mathbf{u}_{\pm}$ from the Wigner-BGK equation.
• Apply the Maximum Entropy Principle to close the moment system by minimizing a Gibbs free-energy functional under moment constraints.
• Introduce three BGK-like collision terms to model band relaxation, isotropic interband scattering, and polarization relaxation.
• Prove that the Lagrange multipliers $A_{\pm}$ and $\mathbf{B}_{\pm}$ are globally invertible functions of the macroscopic densities and velocities, ensuring unique closure.

Experimental results

Research questions

• RQ1 Can a consistent hydrodynamic model be derived for two-band semiconductors that retains non-parabolic band effects from the $k\cdot p$ model?
• RQ2 How can the Maximum Entropy Principle be applied to close the moment system of a matrix-valued Wigner function in a two-band system?
• RQ3 What is the mathematical structure of the pressure and effective-mass tensors in the hydrodynamic model, and how are they related to the macroscopic variables?
• RQ4 How can interband transitions be modeled in a semiclassical hydrodynamic framework without full quantum kinetic treatment?
• RQ5 Is the local equilibrium state uniquely determined by the density and velocity fields in this two-band system?

Key findings

• The Lagrange multipliers $A_{\pm}$ and $\mathbf{B}_{\pm}$ in the Maximum Entropy closure are globally invertible functions of the macroscopic densities $n_{\pm}$ and velocity fields $\mathbf{u}_{\pm}$, ensuring a unique and smooth parametrization of local equilibrium.
• The pressure tensor $\mathbb{T}_{\pm}$ is expressed as the Hessian of the logarithmic moment generating function of the energy and velocity distribution, derived from the MEP.
• The effective-mass tensor $\mathbb{Q}_{\pm}$ is given by the integral of the inverse of the band mass tensor weighted by the exponential of the energy and momentum terms.
• Three distinct BGK-like collision terms are proposed: one for band relaxation, one for isotropic interband scattering, and one for polarization relaxation, each conserving total density and momentum.
• The band relaxation term $C^{\mathit{br}}$ depletes the upper band and populates the lower band with a characteristic time $\tau_{\mathit{br}}$, while the isotropic scattering term $C^{\mathit{is}}$ redistributes momentum isotropically and relaxes density and current polarization.
• The resulting hydrodynamic system (53) is Euler-Poisson-like, with source terms $N_{\pm}$ and $\mathbf{U}_{\pm}$ that depend on the band densities and velocities, and includes self-consistent electrostatic potential via Poisson's equation.