Microscopic Statistical Theory of InhomogeneousBroadening in InGaN/GaN Quantum Wells

Mykhailo Klymenko, Oleksiy Shulika

Kharkov National University of Radio Electronics

Lenin ave. 14, Kharkov, 61166, Ukraine

Email: klymenko@daad-alumni.de

Abstract— Microscopic statistical theory of inhomogeneousbroadening in InGaN/GaN quantum-well structures is developed.It is shown that inhomogeneous broadening can be describedas energy-dependent daphasing time. Explicit relation betweeninhomogeneous broadening and spectral power density of thelateral interface potential fluctuations is derived.

I. INTRODUCTION

Light-emitting diodes and semiconductor lasers based on

InGaN/GaN quantum well (QW) structure are of actual interest

due to lasing in the spectral range from the visible to the

UV [1]. On the other hand, a widespread application and

designing of these devices are limited by the difficulties in

quantitative prediction of their optical spectral characteristics.

Indeed, all published theories of the optical response for

InGaN/GaN structures contain at least one fitting parameter.

The parameter which is encountered most frequently as fitting

one is the linewidth of the inhomogeneous broadening [2], [3].

This kind of broadening is related to 3D random variations

of the potential relief caused by the indium composition

fluctuations together with fluctuations of the QW width.

There are several physical reasons caused such fluctuations.

Most relevant are the indium surface segregation [4], [5] and

clusterization [6]. The theory relating quantum well width

fluctuations and inhomogeneous broadening is developed in

[10]. In this work, we develop microscopic statistical theory

connecting parameters of lateral potential fluctuations with the

linewidth of the inhomogeneous broadening. Obtained results

shows clear relationship between the spectral power density

of the potential fluctuations at interfaces and inhomogeneous

broadening which is represented as a wave-vector (energy)-

dependent dephasing time.

II. HAMILTONIAN

Here we consider 2 nm InxGa1−xN/GaN single-

quantum-well structure [2]. However, as we will see later, re-

sults of our derivations are applicable to any QW heterostruc-

ture, even such exotic one as QWs with kinetic confinement

[7], [8], [9]. The Hamiltonian of this structure can be presented

as a sum of the deterministic part H0 and time-independent

random potential fluctuations V (r||, z).

H = H0 + V (r||, z) (1)

As a basis functions for quantum fields expansion, we chose

the eigensates of the Hamiltonian H0 in the envelope function

approximation. In this case, quantum fields ψ(r) and ψ∗(r)can be expended as follows:

ψ(r) =∑j,k||

aj,k||φj(z)exp(−ik||r||) (2)

ψ∗(r) =∑j,k||

a†j,k||φj(z)exp(ik||r||) (3)

here: k|| is the in-plane wave vector, φj(z) is the envelope

function, r|| is the in-plane position vector, a†j,k||and aj,k|| are

creation and annihilation operators, j is the subband number.

In this representation, Hamiltonian reads:

H =∑

j=c,v,k||

εj,k||a†j,k||

aj,k|| +

∑k||

E(t)[dcvk||a

†c,k||

av,k|| + c.c.]+

∑k||,q

V cqa

†c,k||+qac,k|| +

∑k||,q

V vq a

†v,k||+qav,k|| (4)

here εj,k|| is the total energy of charge carriers, E(t) is the

electric field, dcvk|| is the interband dipole matrix element, V jq

is the matrix element for elastic scattering.

Hereafter, we omit electron-electron and electron-phonon

scattering terms and focus only on elastic scattering terms at

the potential fluctuations which are described by last two terms

in (4).

III. LATERAL POTENTIAL FLUCTUATIONS AND RANDOM

FIELDS

Here, the quantum well interfaces are treated as random

fields [11], [12]. We assume that these fields are uniform

and ergodic. In most general case related to the considered

structure, the matrix element of elastic scattering on the

potential reads:

V jq =

∫∫dr||dzV (r||, z)φ∗

j (z)φj(z)exp(iqr||

)(5)

Hereafter, we assume that 3D potential profile of the het-

erostructure can be factorized as V (r||, z) = V (r||)V (z).

978-1-4244-6997-0/10/$26.00 ©2010 IEEE

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Fig. 1. Model surface structure

Also, we use approximation that only the lateral disorder exists

[13], and the potential profile V (z) is caused by deterministic

effects such as indium surface segregation. This is good

approximation if the interwall cross-correlation function [14]

of the interface roughness is constant. As follows from the

TEM experiments, it is the case when the width of the

quantum well is several nanometers. Applying mentioned

above approximations, one gets following expression for the

matrix element:

V jq = δi,jCj

∫ ∞

−∞dr||V (r||)exp

(iqr||

)(6)

where: Cj =∫dzV (z)φ∗

j (z)φj(z)The ergodic property of the random field allows us to pass

from random quantity in the integral (7) to the ensemble aver-

age. This is realized propagating lateral sizes in all directions

as is illustrated in Fig. 1. The surface area Sj is transformed

to NSj . Due to ergodicity, the integration over expanded area

leads to the averaged Fourier transform Vq,j :

V jq = δi,jCj

N

N

N∑j=1

∫Sj

dr||V (r||)exp(iqr||

)= NVq,j (7)

Complex function Vq,j can be represented in exponential

form:

V jq =

√Wq,jexp(iφq,j)

here Wq,j is the spectral density function, and φq,j is the

averaged phase.

The averaged phase is equal to zero if the random process

is uniform one.

IV. KINETIC EQUATIONS AND INHOMOGENEOUS

BROADENING

With Hamiltonian (4), the polarization equation reads:

Fig. 2. Elastic scattering in k-space with energy conservation

pk|| = iωk||pk||+

i(nc,k|| − nv,k||

)dcvk||E(t)/�+∑

q

NV cq 〈a†c,k||+qav,k||〉−

NV vq 〈a†c,k||

av,k||−q〉 (8)

here pk|| = 〈a†c,k||av,k||〉 is the microscopic polariza-

tion. Eq. 8 is resulted from Heisenberg equation written

with respect to operator a†c,k||av,k|| . This equation states

that the microscopic polarization is dependent on correlations

〈a†c,k||+qav,k||〉 and 〈a†c,k||av,k||−q〉. This term appears due to

states mixing within a single energy band caused by potential’s

fluctuations. We approximate these correlations by following

expressions:

〈a†c,k||+qav,k||〉 = 〈a†c,k||+qac,k||〉〈a†c,k||av,k||〉 (9)

〈a†c,k||+qav,k||〉 = 〈a†v,k||av,k||−q〉〈a†c,k||

av,k||〉 (10)

The second factor in the right hand side is microscopic

polarization. The first term is intraband correlation describing

elastic scattering process. We calculate this correlation using

Fermi’s Golden Rule:

〈a†c,k||+qac,k||〉 = V cq δ(εc,k||+q − εc,k||)nc,k||(1− nc,k||+q)

(11)

〈a†v,k||+qav,k||〉 = V vq δ(εv,k||+q − εv,k||)nv,k||(1− nv,k||+q)

(12)

In this case, the polarization equation reads.

pk|| = iωk||pk|| + i(nc,k|| − nv,k||

)dcvk||E(t)/�+Ω(k||)pk||

(13)

Here, Ω(k||) is the inhomogeneous broadening defined by

the expression:

Ω(k||) = N∑q

Wq,cδ(εc,k||+q − εc,k||)nc,k||(1− nc,k||+q)−

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−Wq,vδ(εv,k||+q − εv,k||)nv,k||(1− nv,k||+q) (14)

Calsulation of expressions (13) and (14) can be simplified

using axial approximation for

εj,k|| and nj,k|| :

Ω(k) =NL2

4π2nc,k(1− nc,k)

∮L

Wkx,ky,cdl−

−NL2

4π2nv,k(1− nv,k)

∮L

Wkx,ky,vdl (15)

here k = |k|||, L : E(kx − k, ky) = constThe formula (15) contains curvilinear integral over isoener-

getic contour in k-space. In Fig. 2, such a contour is denote by

the red curve. The reference point for the expression (15) is

the point where q = 0. Therefore, the isoenergetic contour is

shifted in kx direction so coordinates of the center of Brilloun

zone is [−k, 0]. It is necessary to note that the formula (15) is

obtained without any restrictions on the spectral density func-

tion. Thus, the inhomogeneous broadening can be computed

for any kind of random surfaces including anisotropic ones.

V. CONCLUSION

In this work, we have developed microscopic statistical the-

ory of inhomogeneous broadening in InGaN/GaN quantum

well structures. The inhomogeneous broadening is derived as a

wave-vector (energy) dependent dephasing time. Resulted ex-

pression contains spectral power density of 2D random surface

describing potential fluctuations as quantum well interfaces.

We expect that results obtained here will have an impact

on calculation of gain spectra of either asymmetric MQW

heterostructures [15], [16] or conventional QWs with digital

alloy barriers [17], because starting point in that case is band

diagram [18]. It should also influence results on nonlinear gain

[19], and, possibly, on Auger recombination rate in QWs [20].

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