Approximation of Thermal Equilibrium for Quantum Gases with Discontinuous Potentials and Application to Semiconductor Devices

Abstract

We derive an approximate solution valid to all orders ofh to the Bloch equation for quantum mechanical thermal equilibrium distribution functions via asymptotic analysis for high temperatures and small external potentials. This approximation can be used as initial data for transient solutions of the quantum Liouville equation, to derive quantum mechanical correction terms to the classical hydrodynamic model, or to construct an effective partition function in statistical mechanics. The validity of the asymptotic solution is investigated analytically and numerically and compared with Wigner's O(h 2 ) solution. Since the asymptotic analysis results in replacing second derivatives of the potential in the correction to the stress tensor in the original O(h 2 ) quantum hydrodynamic model by second derivatives of a smoothed potential, this approach represents a definite improvement for the technologically important case of piecewise continuous potentials in quantum semiconductor devices.

An effective potential approach to modeling 25 nm MOSFET devices

“…Equation (4) is usually referred to as the Bloch equation and has been used in a similar context to derive quantum fluid models for semiconductor transport [7,8]. In the Wigner picture, the Bloch equation reads, using the Weyl quantization formula (2),…”

Smooth Quantum Hydrodynamic Model vs. NEMO Simulation of Resonant Tunneling Diodes

“…In Refs. [1][2][3] Gardner and Ringhofer present an extension of the classical hydrodynamic model which can handle in a mathematically rigorous way the discontinuities in the classical potential energy which occur at heterojunction barriers in quantum semiconductor devices. This smooth quantum hydrodynamic model is valid to all orders of h 2 /(mT 0 l 2 ) (where m is the effective mass of electrons or holes, and l is a typical length scale for the problem) and to first order in the classical potential energy.…”

Effective Potentials and Quantum Fluid Models: A Thermodynamic Approach

“…Although (13) completely defines the thermodynamic equilibrium in terms of the Hamiltonian, and therefore in terms of the potential V , we will need an explicit expression for f eq in order to obtain a formula for the effective quantum potential V Q in (14) and the closure moments P and q in (15). Of course, it is not possible to find an exact expression for f eq , and we will need some kind of approximate formula.…”

“…So we formally set V = εV ε and expand the solution of (18) in powers of the parameter ε. It should be pointed out that ε is a purely formal parameter and that the procedure below corresponds to the Born approximation of e −βH [13], [14]. So we expand the solution of…”