The Quantum Well in an Electric Field A Density of States Approach

Abstract

A quantum well (QW) in the presence of a homogeneous electric field forms a system with continuous energy spectrum. Its density of states (DOS) is calculated. A general DOS formula is applied which avoids the well-known difficulties for systems with continuous spectra. The Schrodinger equation is solved exactly, and the 1D DOS is expressed in closed analytical form by means of Airy functions of first and second kind. Quasistationary energy levels are defined as zeros of a certain 4 x 4 matrix. Their positions are calculated in dependence on the field strength over a wide range. The 1D DOS peaks arising from quasistationary levels are studied for different field strengths. The same is done for the 1D DOS in a larger energy interval ranging from far below the QW bottom up to far above the QW barriers. Franz-Keldysh oscillations as well as QW interferences are observed above the barriers.Ein Quantum-Well (QW) im homogenen elektrischen Feld ist ein System mit kontinuierlichem Energiespektrum. Seine Zustandsdichte (DOS) wird berechnet. Dabei wird eine allgemeine DOS Formel verwendet, die die bekannten Schwierigkeiten vermeidet, die bei Systemen mit kontinuierlichem Energiespektrum auftreten. Die Schrodinger-Gleichung wird exakt gelost, und die 1D DOS wird in geschlossener analytischer Form durch Airy-Funktionen erster und zweiter Art ausgedruckt. Quasistationiire Energieniveaus werden als Nullstellen einer gewissen 4 x 4-Matrix definiert. Ihre Positionen in Abhangigkeit von der Feldstiirke werden in einem weiten Bereich berechnet. Die den quasistationiiren Niveaus entsprechenden Maxima der 1D DOS werden fur verschiedene Feldstlrken untersucht. Das gleiche wird durchgefuhrt fur die 1D DOS in einem grol3eren Energieintervall, das von weit unterhalb des QW-Bodens bis weit oberhalb der QW-Barrieren reicht. Fur Energien oberhalb der Barrieren werden Franz-Keldysh-Oszillationen und QW-Interferenzen beobachtet.

Analytic methods for field induced tunneling in quantum wells with arbitrary potential profiles

“…There are two different but similar numerical techniques [7,8,19] to obtain Ö and in the real Airy function methods. The technique followed by Endelein et al [7] is described first.…”

“…This method has been used in quantum well problems by several groups [7,8,26]. The time-dependent wave function is defined as (17) and (18) and using eq.…”

“…Austin and Jaros [6] have calculated Ö and in the phase shift method. Enderlein et al [7] have used the density of states method for calculating Ö and . Ghatak et al [8] have presented this method in the spirit of wave mechanics.…”

“…The Stark shifted energy levels in a single rectangular quantum well (RQW) are thoroughly studied using both analytic Airy function methods [6][7][8][9] and numerical methods [10][11][12][13][14]. The Airy function methods are also efficient in calculating the mean tunneling lifetimes´ µ.…”

“…In the single RQW there are three different Airy function methods available to obtain Stark shifted energies and mean tunneling lifetimes [6][7][8][9]. Under the uniform electric field the Stark shifted energy levels and mean tunneling lifetimes are calculated as the resonance position´ Ö µ and width´ µ in the Breit-Wigner formalism [16].…”