Qualitative Physics of Defects in Quantum Wells: Interface Roughness
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QUALITATIVE PHYSICS OF DEFECTS IN QUANTUM WELLS: INTERFACE ROUGHNESS
HAROLD P. HJAIMARSON Sandia National Laboratories,
Albuquerque,
New Mexico 87185.
ABSTRACT The electronic properties of interface roughness in a quantum well are described. Interface roughness is shown to always produce localized bound states. Thus intrinsic roughness can explain the giant oscillator strength observed for "free" excitons in quantum wells. INTRODUCTION Interface roughness can affect both the optical and electrical properties of semiconductor quantum wells. Typically such roughness has a height of 3-10 A (1-3 monolayers) and a lateral size of 50 - 250 A. This roughness causes local fluctuation in the quantum sub-band energy which leads to photoluminescence line broadening [1,2] and increased carrier scattering [3,4]. In earlier work, Gaussian type wavefunctions were used to compute exciton binding energies; no binding occured if the lateral size of the roughness was smaller than a critical size of approximately 40 A [1]. In other work, the Born approximation was used to compute carrier mobility in quantum wells [3]. In this paper the qualitative properties of interface roughness are described. In an idealized model, interface roughness consists of islands with a distribution of radii a and thicknesses b. A multi-band variational calculation and a single band two-dimensional (2D) square well approximation are used to compute the electronic properties. By analogy to bulk defect problems, the wavefunction is expressed in terms of quantum well eigenfunctions X(p,z). Assuming a single quantum well of depth A for 0 : z < d, the wavefunction can be written as
with In
f(r) - i cXgp,z)
(1)
Xkn(p,z)
(2)
- #k(p) on(z).
the limit of an infinitely deep well (A-a=), 1
- d" /
-(z)
2
sin(r n/d);
for an isotropic effective mass, #(p)
-
(3) 0(p) is
a Bloch function of the form
A-I/2exp(ikp)
(4)
in which A is the unit cell area. For this defect problem with cyclindrical symmetry, it is replace the plane waves with Bessel functions. Thus we use x(r) - Nmi Jm(kip) O•z)
T18)
convenient to
(5)
as a orthonormal complete set ofates in which J (kp) is a Bessel function of the first kind and T_(0) - r exp(im8) [5,6]. Boundary conditions on a cylinder of radius R >> »R a) quantize the radial wavevector; assuming that Me wavefunction vanishes at p - R defines ki - Z mi/R in which Zmi is the i zero of the Bessel function [5,6]. mm
Mat. Res. Soc. Symp. Proc. Vol. 163. 'c 1990 Materials Research Society
362
The energies are obtained by diagonalizing the matrix 6
n',n Si,,i
[(En + ci - E)
+
] Cni - 0
(6)
2m in whi~h ci - Y2m k? is the Bessel function "kinetic energy" and En : (nif/d) are the bounA state energies of the quantum well. The potential V(p,z) - -A iff d S z : d+b and 0 : p : a. The matrix elements are separable; - V(n',n) S(i',i) with V(n',n) -
On,(z)*O(z) dz
-A d1'
(7)
and S(i',i)
- NMi,Nmi of ap J m(kip) JM (kip) dp,
(8)
an overlap function. It is important to use accurate bound state wavefunctions. For examp
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