Impurity-Band Tails in the High-Density Limit. I. Minimum Counting Methods

Abstract

A new approximate method is presented for calculating the density of states and one-electron Green's function in the low-energy tail of a high-density impurity band. Such states occur in regions of large attractive potential produced by unusually high (random) concentrations of attractive centers and/or unusually low concentrations of repulsive centers. These well-separated deep wells each have one bound state of lowest energy. The distribution of these lowest levels yields the low-energy tail. For any one well, a bound-state energy E(x 0 ) is estimated variationally using ^(x) = /(x -x 0 ), where /(x) has any fixed form. The best energy for any well is obtained by minimizing £(x 0 ) with respect to x 0 in the vicinity of that well. The number of wells that contribute to the density of states at energy E is given by the number of local minima in E(xo) that occur at the level E. In the high-density limit, Gaussian statistics are adequate for treating the potential fluctuations and the expectation value for the density of states is easily calculated. At low energies the best choice for the function / is that which maximizes the expected density of states. Application to an exactly soluble one-dimensional model, a Gaussian "white noise" potential, yields the correct asymptotic form p(E) = const \E | exp[-(f) | IE | 3 / 2 ]. In three dimensions, the density of states in the Gaussian approximation is found to have the form p(E) = [_A CE)/£ 2 ]exp[ -23(£)/(2£)], where £ is proportional to the concentration of impurities and to the square of the strength of the impurity potential. For screened, charged impurities,where E Q = h 2 Q 2 /(2m*), (1/Q) is the screening radius, and v= (EQ-E)/EQ is the energy below the mean potential Eo in units of Eg. Computer-calculated curves are provided for the "universal" dimensionless functions a(v) and b(v). The exponent b(v) behaves like v m when v is small (strong screening) and behaves like v 1 when v is large (weak screening).

Mean-field phase diagram of the one-dimensional Bose gas in a disorder potential

“…The numerical results give two power-law dependencies of the boundary in these limiting cases /E c = C(U/E c ) γ , with γ equal to 3/4 and 1, respectively. The lower part of the phase diagram represents the WN limit: in this regime a single energy scale characterizes the disordered potential [27] (see the appendix)…”

Evaluation of optical quality and properties of Ga0.64In0.36N0.006As0.994 lattice matched to GaAs by using photoluminescence spectroscopy

“…1(a)] which is attributed to localized excitons emission [8,[18][19][20]. Also similarly to GaNAs, the large localization potential in GaInNAs is at least partly attributed to the composition and strain non−uni− formity of the alloy which originates the N existence.…”

Effect of random well-width fluctuations on the exciton optical absorption spectrum in single quantum wells

“…It has been previously shown 32 that in a smalltime approximation the expression for the propagator ͓equivalent to retaining only high-exciton-energy states in 1 (E) and physically understood by considering the ''pseudo'' Heisenberg uncertainty relation Etуប͔ leads to Thomas-Fermi semiclassical results while a large-t approximation reproduces the results of Halperin and Lax 29 deep in the band tail.…”

“…The quantity L , having the dimension of energy squared, was first introduced by Halperin and Lax 29 and represents a measure for the depth of the typical potential well. In the 2D case L is given by…”

“…This procedure is analogous to the use of a ''universal wave function'' by Halperin and Lax. 29 The harmonic oscillator potential is equivalent to modeling the real wave function by a quadratic function ͑Gaussian͒. The use of a harmonic trial action is equivalent to assuming that all the fluctuating potentials have the same quadratic shape.…”