The Over-Barrier Resonant States and Multi-Channel Scattering in Multiple Quantum Wells

Alexander Polupanov,Alexis Kruglov

doi:10.1260/1750-9548.6.2.99

Abstract

We demonstrate an explicit numerical method for accurate calculation of the scattering matrix and its poles, and apply this method to describe the multi-channel scattering in the multiple quantum-wells structures. The S-matrix is continued analytically to the unphysical region of complex energy values. Results of calculations show that there exist one or more S-matrix poles, corresponding to the over-barrier resonant states critical for the effect of the absolute reflection of holes in the energy range where only the heavy ones may propagate over barriers in a structure. Light- and heavy-hole states are described by the Luttinger Hamiltonian matrix. In contrast to the single quantum-well case, at some parameters of a multiple quantum-wells structure the number of S-matrix poles may exceed that of the absolute reflection peaks, and at different values of parameters the absolute reflection peak corresponds to different resonant states. The imaginary parts of the S-matrix poles and hence the lifetimes of resonant states as well as the widths of resonant peaks of absolute reflection depend drastically on the quantum-well potential depth. In the case of shallow quantum wells there is in fact a long-living over-barrier resonant hole state.

Highlights

INTRODUCTIONThe multi-channel scattering by quantum-well structures was studied in [1] for particle states obeying the system of ordinary differential equations a
The multi-channel scattering by quantum-well structures was studied in [1] for particle states obeying the system of ordinary differential equations a d2 dz2 +b d dz + c + V (z) Ψ(z) = EΨ(z) (1)where V(z) is a bounded piecewise analytic potential function with a finite number of “pieces”, and V (z) ~ V1± / z + V2± / z2 + ..., z → ±∞; a, ib, and c are piecewise constant hermitian n×n matrices; Ψ(z) is the n-component wave function; E is the energy
Results of calculations show that at all realistic values of parameters there exist one or more S-matrix poles corresponding to the over-barrier resonant states critical for the effect of absolute reflection of the heavy hole in the energy range where only heavy holes may propagate over barriers in the structure

Summary

INTRODUCTION

The multi-channel scattering by quantum-well structures was studied in [1] for particle states obeying the system of ordinary differential equations a. The Over-Barrier Resonant States and Multi-Channel Scattering in Multiple Quantum Wells where group vveelcotcoirtsy:uvααa=reiudα*et(e2rimkαinae+d from b)uα. The S-matrix component for the channel β → α has the form: Sαβ = Xαβ vα / vβ 1/2. The nature of the states related to such pattern of scattering can be clarified by examining the analytic properties of the S-matrix [3]. In this work we generalize our study to the case of multiple quantum-wells structures

FORMULATION OF THE METHOD

RESULTS AND DISCUSSION

CONCLUSION