Use of excitons in materials characterization of semiconductor system

Materials Science and Engineering R 34 (2001) 59±120

Use of excitons in materials characterization of semiconductor system K.K. Bajaj*

Physics Department, Emory University, Atlanta, GA 30322, USA Accepted 11 October 2000

Abstract The objective of this review article is to provide an overview of the use of excitons in the characterization of semiconductor alloys and quantum well structures. In particular, it is shown how the measurements of the excitonic linewidth at low temperatures using a variety of optical spectroscopic techniques such as photoluminescence (PL), cathodoluminescence (CL), and absorption can be used to provide information about the structural quality of alloys and quantum well structures. The results of several theoretical approaches that have been developed to calculate the excitonic linewidth in semiconductor alloys as a function of compositional disorder, which is primarily responsible for excitonic line broadening, are reviewed. The measurements of the excitonic linewidths in a variety of III±V and II±VI based semiconductor alloys are described and compared with the calculated values to obtain information about their quality. This is followed by a review of the results of a theoretical formalism, which has been used to calculate the excitonic linewidth due to interfacial disorder in quantum well structures. The combined effects of both the compositional and the interfacial disorders on the excitonic linewidth in quantum well structures are discussed. The results of the measurements of excitonic linewidth in several III±V and II±V semiconductor based quantum well structures are reviewed and compared with those calculated to gain insight into their quality. This article is intended to provide a balanced review of both the theoretical and experimental developments in this field. # 2001 Elsevier Science B.V. All rights reserved. Keywords: Excitons; Linewidth; Semiconductor alloys; Quantum wells; Photoluminescence

1. Introduction The last three decades have witnessed a remarkable progress in the development of epitaxial crystal growth techniques such as molecular beam epitaxy (MBE), metal±organic chemical vapor deposition (MOCVD) and their various variations such as chemical beam epitaxy (CBE), atomic layer epitaxy (ALE), etc. which have allowed the growth of very high quality semiconductors, their alloys and heterostructures. A variety of interesting structures such as quantum wells, quantum wires, and quantum dots have been fabricated with abrupt changes in composition and/or doping characteristics and their structural, electronic and optical properties have been investigated in considerable detail. Many of these structures, especially those based on quantum wells, have found important applications in a number of electronic and opto-electronic devices such as high electron mobility transistors (HEMTs), lasers, light emitting diodes (LEDs), photodetectors, spatial light modulators, etc. The use of these devices in opto-electronics for instance, has literally revolutionized this field during the past decade. *

Tel.: ‡1-404-727-0700; fax: ‡1-404-727-0873. E-mail address: [email protected] (K.K. Bajaj). 0927-796X/01/$ ± see front matter # 2001 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 7 - 7 9 6 X ( 0 1 ) 0 0 0 3 2 - 8

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All these structures involve semiconductor alloys and interfaces between two different semiconductors and, therefore, their quality has an important effect on the performance of these devices. It is therefore of crucial importance to determine their quality. Optical characterization techniques such as photoluminescence (PL), photoluminescence excitation (PLE), absorption and cathodoluminescence (CL) provide excellent means of determining the quality of alloys and of interfaces in the heterostructures. It is well known that the optical properties of semiconductors, their alloys and heterostructures, at low temperatures, are dominated by excitonic transitions. It was realized very early that the linewidth of the excitonic transitions in these systems can provide a great deal of useful information about their quality. During the past 20 years a great deal of effort has been devoted to both the theoretical and experimental investigations of the behavior of excitonic linewidth in these systems and important information concerning their quality has been obtained. The purpose of this article is provide a comprehensive review of these developments. In Section 2, we briefly review the results of the calculations of the eigenfunctions and eigenvalues of an exciton in bulk semiconductors. This is followed by a brief summary of the results of the calculations of the behavior of exciton binding energies in quantum well structures. In Section 3, we provide a critical review of the various calculations of the excitonic linewidth due to potential fluctuations in completely random semiconductor alloys. In Section 4, we describe the results of measurements of the excitonic linewidth in several ternary alloys and compare them with their calculated values. In Section 5, we describe a formalism to calculate the excitonic linewidth in quantum well structures due to interfacial disorder. In Section 6, we provide a brief review of the results of the measurements of excitonic linewidth in several different quantum well structures which have been fabricated with and without growth interruption at the interfaces, and in Section 7, we provide a summary. 2. Excitons in bulk semiconductors In this section, we shall briefly review some of the properties of excitons both in bulk semiconductors as well as in quantum well structures, which will be used later to study the behavior of excitonic linewidths in these systems. When a semiconductor is irradiated, say with a laser light whose photon energy is equal to or greater than the value of the energy band-gap, the electron from the valence band is excited to the conduction band. In this process an empty electron state in the valence band is created, which behaves like a particle with a positive charge, equal in magnitude to that of an electron and a positive mass, which is characteristic of the shape of the valence band. This particle called a hole exerts an attractive force on the electron, thus forming an entity analogous to a positron called an exciton. This then is the first excited state of a semiconductor in which an electron has been taken out of an otherwise filled valence band at temperature T ˆ 0 K. This system was first studied by Wannier [1] and later by Slater [2] who showed that if one assumes that the electron and the hole wave functions are extended over many lattice constants, the exciton can be described as consisting of an electron and a hole with effective band masses me and mh, respectively, coupled together with Coulomb potential screened by the static dielectric constant e0. The Hamiltonian H of such a system, therefore, could be written as Hˆ

~ ~ p2e p2 ‡ h 2me 2mh

e2 e0 …~ re ~ rh †

(1)

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where ~ pe and ~ ph are the electron and the hole momentum, respectively, and ~ re and ~ rh the respective position coordinates. We can write Eq. (1) in terms of center of mass (CM) and relative coordinates using the following well-known transformations: ~ ph P ˆ~ pe ‡ ~

(2a)

re ‡ mh~ rh me~ ~ Rˆ me ‡ mh

(2b)

~ p ˆ~ pe

~ ph

(2c)

~ r ˆ~ re

~ rh

(2d)

and M ˆ me ‡ mh

(2e)

The Hamiltonian can now be express as Hˆ

P2 p2 ‡ 2M 2m

e2 e0 r

(3)

where m ˆ …1=me ‡ 1=mh † 1 is the reduced mass. The corresponding SchroÈdinger equation is Hc ˆ Ec

(4)

where ~
~ c ˆ exp…i hK R†cn …~ r†

(5)

E ˆ EK ‡ En

(6)

and

Here EK ˆ

2 K 2 h 2M

describes the motion of the center of mass and En are the eigenvalues of  2  p e2 c ˆ En cn 2m e0 r n

(7)

(8)

This is the well-known equation for the hydrogen atom in which the electron mass is replaced by the reduced mass m and the eigenfunctions cn and eigenvalues En are [3] given as cnlm …r; y; f† ˆ Rnl …r†Ylm …y†Fm …f†

(9)

where 1 Fm …f† ˆ p exp…imf† 2p

(10)

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 …2` ‡ 1†…` jmj†! 1=2 jmj Ylm …y† ˆ pl cos y 2…` ‡ jmj†!

(11)

and  Rnl …r† ˆ

2 na0

3 

  r …n ` 1†! 1=2 rL2`‡1 exp n‡1 …r† 2nf…n ‡ 1†!g 2

(12)

in which rˆ

2 r na0

(13a)

2 h me2

(13b)

a0 ˆ jmj

Pl cos y are the associated Legendre functions and L2`‡1 n‡1 …r† are associated Laguerre polynomials. The quantities n, l and m represent the well-known total quantum member, angular momentum quantum number and z component of the angular momentum quantum number, respectively, and take the following values: n ˆ 1; 2; 3; . . . `ˆn mˆ

1; n

(14a) 2; . . .

`; ` ‡ 1; `

(14b) (14c)

The eigenvalues En are given as En ˆ

e4 m 2n2 e20  h2

For the ground state, for instance, the wave function c1s can be written as   1 r c1s ˆ q exp a 0 pa30

(15)

(16)

The wave functions for the excited states can be obtained using the above mentioned expressions. It should be pointed out that in the above discussion of the exciton, we have assumed that the electron and the hole motion can be described in terms of parabolic conduction and valence bands with isotropic masses. This is an approximation as in crystals with a zinc blende structure such as most III±V compounds, the valence band is four-fold degenerate at the G point (Fig. 1). The expression for the kinetic energy of the hole is more complicated and is given in terms of 4  4 matrix as described in the next section. 2.1. Excitons in quantum wells In this section, we shall briefly review the properties such as eigenfunctions and eigenvalues of excitons confined in quantum well structures. Before we do that, however, we shall briefly describe the behavior of electrons and holes confined in such structures.

K.K. Bajaj / Materials Science and Engineering R 34 (2001) 59±120

Fig. 1. Typical band structure of a direct-gap semiconductor.

As mentioned earlier, due to enormous advances in the epitaxial crystal growth techniques such as MBE and MOCVD during the past 30 years, it has become possible to grow structures with alternate layers of different semiconductors with very controlled thicknesses and composition. For instance, in the case of GaAs and AlAs whose lattices constants differ very little from each other, it is possible to grow these structures where the change from GaAs layer to AlAs layer takes place over 1 Ê ). Such ordered structures are commonly referred to as ML (one-half the lattice constant 2.8 A superlattices. The direct energy band-gap of GaAs is much smaller than that of AlAs. Thus, when a GaAs layer is sandwiched between two layers of AlAs, the two band-gaps of these two different semiconductors completely overlap with each other. A part of the band-gap difference (DEc) is accommodated by the conduction band and the rest by the valence band (DEv) such that DEc ‡ DEv ˆ Eg …AlAs† Eg …GaAs† ˆ DEg . The electrons in GaAs layer, therefore, find themselves confined in a one-dimensional (along the direction of growth usually referred to as z-axis) potential well whose height is determined by DEc. Similarly, the holes are also confined in a potential well whose height is given by DEv. In the case of holes, however, the degeneracy of the band at the G point is lifted due to reduction in symmetry thus leading to the presence of heavy- and light-holes (Fig. 2). As is well-known from the very early studies of a particle in a one-dimensional potential well in quantum mechanics, the energy

Fig. 2. Electron (Ee), heavy-hole (Ehh) and light-hole (Elh) subbands in a GaAs/AlGaAs quantum well structure.

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levels of electrons and holes in the z-direction are quantized. In the case of a well with infinitely high potential barriers, for instance, the energy levels are given as En ˆ

2 p2 n2 h mL2

(17)

where n ˆ 1; 2; 3; . . ., m the mass of the particle (electron or hole) along the z-axis and L the size of the well. In the case of a quantum well with finite potential barriers (actual situation), the energy levels are still strong functions of L though the dependence is somewhat weaker. This dependence of the energy levels En on the size of the quantum well is commonly referred to as quantum size effect (QSE) and is essential, as we shall see later, for the understanding of the behavior of excitonic linewidth in these structures caused by interface roughness. To understand the properties of electrons and holes confined in quantum well structures with finite potential barriers, we need to determine their eigenfunctions and eigenvalues. Toward that end effective mass approximation (EMA) has been used extensively which is essentially based on the assumption that the particle wave functions extend over a large number of units cells, an assumption the use of which is justified in most cases. In addition, energy levels of interest of both electrons and holes often lie near their respective band edges and, therefore, are well described by EMA. As mentioned earlier, both GaAs and AlAs crystallize in zinc blende crystal structure. The band edges relevant to the optical properties of quantum wells consisting of these materials have G6 (conduction band), G8 (valence bands for lightand heavy-holes) and G7 (split-off band) symmetries. A general expression of the wave function describing the physical state of the particles in a QW or in the barrier can be written as [4] X WB WB r† ˆ cWB r† exp‰i…k? r? ‡ kzWB z†Š (18) cWB …~ n …k? ; kz †un0 …~ n

where W and B stand for the well or the barrier, ~ k? …kx ; ky † and~ r? …x; y† are the two-dimensional wave and position vectors, respectively, and n is the bulk band index which runs over the host band edges r†. It should be mentioned that k? is real as described by the periodic part of the Bloch functions un0 …~ it describes the motion of free particles in the x±y plane where as kz can be either real or imaginary depending on whether the particle energy corresponds to a propagating or an evanescent state within the W or B layers, respectively. In order to solve for the eigenfunctions and eigenvalues of electrons and holes in quantum well structures, we make two simplifying approximations. 1. In the expansion of the wave function in Eq. (18), we confine ourselves to only eight terms in n, which pertain to G6, G7 and G8 bands. r† are the same both in the well and 2. We assume that the periodic parts of the Bloch functions un0 …~ in the barrier regions. This is a valid approximation as long as the band structures of the two materials are not too different from each other. The Hamiltonian of a particle (an electron or a hole) confined in a rectangular potential well is separable in x-, y-, and z-coordinates. The envelope eigenfunctions can, therefore, be expressed as products of eigenfunctions in and x-, y-, and z-directions. In the x±y plane both in the well and in the barrier regions the eigenfunctions are plane waves, whereas in the z-direction they are solutions of the following equation:   2 @ h 1 @f …z† ‡ V…z† ˆ Ef …z† 2 @z m…z† @z

(19)

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65

where we have considered the case in which the effective mass of the particle along the z-axis is different in the well and in the barrier regions. This equation is then solved using the boundary conditions in which f(z) and …1=m…z††…@f …z†=@z† are continuous throughout the quantum well structure including the interfaces at z ˆ L=2 (origin is assumed to be at the center of the well). The eigenfunctions and eigenvalues of this equation are well known and are given in literature (see, for example [3]). As stated earlier, the energy levels along the z-axis are quantized but are continuous functions of k? in the x±y plane. The conduction band mass is isotropic, however, the masses of the heavy- and light-holes are quite anisotropic due to the p-like character of the valence band. These masses are generally expressed in terms of the well-known Luttinger [5] parameters g1, g2, and g3, which describe the nature of the valence bands. For instance, the values of the masses of the heavyhole mhh and the light-hole mlh along (0 0 1) direction are given as [5] 2g2 Š

1

(20a)

mlh ˆ m0 ‰g1 ‡ 2g2 Š

1

(20b)

mhh ˆ m0 ‰g1 and

Similar expressions for mhh and mlh are also obtained in [1 1 0] and [1 1 1] directions and are given in [6]. The photon energy required to excite an electron from the lowest heavy-hole subband, for instance, to the first conduction subband, without exciton effects in a GaAs/AlAs quantum well structure is given as hn ˆ Eg ‡ Ee ‡ Eh

(21)

where Eg is the band-gap of GaAs (1.519 eV at 2 K) and Ee and Eh are the energies of the first electron and heavy-hole subbands, respectively. As described earlier, the presence of Coulomb interaction between an electron and a hole creates an excitonic state and, therefore, the photon energy required to create a heavy-hole exciton, for instance, is hnex ˆ Eg ‡ Ee ‡ Eh

EB

(22)

where EB is the binding energy of a heavy-hole exciton. Light-hole excitons are created in a similar fashion. Thus, to determine the value of the hnex , one needs to know the values of Ee, Eh and EB, which are dependent on the well size for a given material system. In bulk GaAs, for instance, the value of EB ˆ 4:2 meV [7]. We have already discussed how the dependence of the values of Ee and Eh on the well size L is determined. We shall now briefly review the results of calculations of Ee as a function of L. For the sake of illustration, we shall consider the case of GaAs/AlGaAs quantum wells though the results, we discuss here are equally well applicable to similar type 1 quantum well structures such as InGaAs/InP, etc. It is clear from the above brief description that the confinement of excitons in quantum well structures leads to dramatic modifications in their eigenfunctions and eigenvalues, which in turn affect their oscillator strength. This quantity is related to the degree of overlap of the electron and the hole-wave functions and is a measure of the magnitude of the excitonic absorption. Several calculations of oscillator strength [8] have been reported in literature and the reader is referred to this work. The Hamiltonian of an exciton in a structure consisting of a layer of GaAs sandwiched between two semi-infinite layers of Alx Ga1 x As grown along the (0 0 1) direction can be expressed,

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using an EMA as 2 r2e h ‡ Th … irh † 2me

Hˆ

e2 ‡ Ve …ze † ‡ Vh …zh † e0 j~ re ~ rh j

(23)

Here the first term denotes the kinetic energy of the conduction electron with effective mass me and the second term is the kinetic energy of the hole as first described by Luttinger [5]. In the present case, we can ignore the split-off valence band and express this as 2 3 L‡ M N 0 6 M L 0 N 7 7 (24) Th …k† ˆ 6 4 N 0 L M5 M  L‡ 0 N where L ˆ

2 h ‰…g  g2 †…kx2 ‡ ky2 † ‡ …g1  2g2 †kz2 Š 2m0 1

(25)

Mˆ

2 p h 3g3 kz …kx 2m0

(26)

Nˆ

2 p h 3‰g2 …kx2 2m0

iky †

and ky2 †

2ig3 kx ky Š

(27)

Here m0 is the free electron mass g1, g2, and g3 the Luttinger parameters and e0 the static dielectric constant of system assumed to be the same for GaAs and Alx Ga1 x As. The potential wells for the conduction electron Ve(ze) and for the holes Vh(zh) are assumed to be square wells of width L ( 0; jze j < 12 L (28a) Ve …ze † ˆ Ve ; jze j > 12 L and

( Vh …zh † ˆ

0; jzh j < 12 L Vh ; jzh j > 12 L

(28b)

The values of the potential heights Ve and Vh are determined from the Al concentration in Alx Ga1 x As, using the following expression [9] for the total band-gap discontinuity: DEg ˆ 1:155x ‡ 0:37x2

(29)

in units of electron volts. The values of Ve and Vh are assumed to be 60 and 40% of DEg, respectively. The first attempt to calculate the binding energies of the ground state (1s-like, hereafter referred to as 1s) and of the first excited state (2s-like, referred to as 2s) of excitons associated with the lowest hole subbands was made by Miller et al. [10]. They assumed that the heavy- and light-hole subbands were completely decoupled, namely, they ignored the contributions of the off-diagonal terms of

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Luttinger Hamiltonian (Eq. (24)). This leads to the formation of two types of excitons, one associated with say the lowest heavy-hole subband and the other with the lowest light-hole subband. With this approximation, the Hamiltonian of a heavy (light) hole exciton in a quantum well reduces to   2 1 @ @ h 1 @2 r ‡ Hˆ 2m r @r @r r2 @f2

2 @ 2 h 2me @z2e

h2 @ 2 2m @z2h

e2 ‡ Ve …ze † ‡ Vh …zh † e0 j~ re ~ rh j

(30)

where we have used cylindrical coordinates and have used the following definitions: 1 1 1 ˆ ‡ …g  g2 † m  me m0 1

(31a)

1 1 ˆ …g  2g2 † m m0 1

(31b)

and

In Eqs. (31a) and (31b), the upper sign refers to the Jz ˆ 3=2 (heavy-hole band) and the lower sign to the Jz ˆ 1=2 (light-hole band). Fairly accurate values of the Luttinger parameters g1, g2 and g3 for GaAs are now available [11]. Miller et al. [10] followed a variational approach and used the following form of the trial wave function: c ˆ fe …ze †fh …zh †g…r; z; f†

(32)

where z ˆ ze zh ; f e …ze † and fh(zh) are the ground-state solutions of the electron and the hole, respectively, in the quantum well and g(r, z, f) describes the internal motion of the exciton. In order to further simplify their calculations, they assumed infinite potential barriers and used for g wave Ê ). Using the known values of the dielectric functions which are appropriate for thin wells (<200 A constant and the various mass parameters, they minimized the expectation values (E) of the Hamiltonian (Eq. (30)) with respect to the variational parameters of the trial wave function (Eq. (32)). The binding energy of the exciton say for the 1s state (E1s) was then obtained by subtracting E from the total ground-state-energy of the electron and the hole in the wells, namely, the sum of the electron and hole subband energies. They found that the binding energies of the 1s and the 2s states of both the heavy- and light-hole excitons increased as the well size (L) was reduced and reached their respective two-dimensional values as the well size went to zero. Bastard et al. [12] and Shinozuka and Matsuura [13] also calculated the binding energies of the ground state of these excitons using infinite potential barriers and obtained results similar to those derived by Miller et al. [10]. Greene and Bajaj [14] were the first to calculate the binding energies of the 1s state of both the heavy- and light-hole excitons as a function of the well size using a more realistic case of finite potential barriers for electrons and holes. They solved the Hamiltonian (Eq. (30)) following a variational approach using a trial wave function of the form given by Eq. (32) where fe(ze) and fh(zh) were now the well-known solutions of the particle in a one-dimensional box problem with finite barriers. They used the following expression for the function g: g…r; z; f† ˆ …1 ‡ az2 † exp… s…r2 ‡ z2 †1=2 †

(33)

where a and s were the variational parameters. In order to improve the accuracy of their results, Greene et al. [15] later used a more general expression for g, which consisted of a linear combination of three basis functions and calculated the

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values of the binding energies of 1s, 2s and 2p states of a heavy-hole exciton and a light-hole as a function of L for values of Al concentrations x ˆ 0:15 and 0.3. In order to compare their results with those of Miller et al. [10] and Bastard et al. [12], they also calculated the binding energies of these levels for the case of infinite potential barriers. The values of the various physical parameters used in this calculation are given in [15]. The variation of the binding energies of the 1s state of a heavy-hole exciton E1s(h) (solid lines) and light-hole exciton E1s(l) (dashed lines) as a function of the well-size L for three different values of the potential-barrier heights are displayed in Fig. 3. One finds that for a given value of x, the value of E1s(h) increases as L is reduced until it reaches a maximum and then drops quire rapidly. The value of L at which E1s(h) reaches a maximum is smaller for larger x. Essentially, the same behavior is exhibited by E1s(l). The reason for this behavior is quite simple. As the value of L is reduced the exciton wave function is compressed in the quantum well, leading to enhanced binding. However, below a certain value of L, which depends on the Al concentration, the spread of the exciton wave function into the surrounding Alx Ga1 x As layers becomes more important. This forces the binding energy of the exciton to go over to the value characteristic of bulk Alx Ga1 x As as L is reduced further. In case of infinite potential barriers, the values of E1s(h) and E1s(l) increase monotonically as L is reduced and go over to their respective two-dimensional values, i.e. four times their bulk values, as L goes to 0. Similar behavior was found for 2s and 2p states. In the foregoing calculations, the effect of the off-diagonal terms in the exciton Hamiltonian (Eq. (23)) on its binding energy has been ignored. It is known that in quantum wells, the inclusion of the off-diagonal terms results in strong mixing between the heavy- and light-hole states, which leads

Fig. 3. Variation of the binding energy of the ground state, E1s, of a heavy-hole exciton (solid lines) and of a light-hole exciton (dashed lines) as a function of GaAs well size (L) in a GaAs/Alx Ga1 x As quantum well structure for Al concentration x ˆ 0:15 and 0.3, and for an infinite potential barrier.

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Fig. 4. Variation of the calculated binding energy of the ground state of a heavy-hole exciton in a GaAs/Al0.4Ga0.6As quantum well as a function of well width. Thin solid line: assuming equal effective masses and dielectric constants for the well and the barrier materials and a parabolic conduction band. Dotted line: including the effective mass mismatch. Dashed line: including the effective-mass mismatch and the conduction band non-parabolicity. Dashed±dotted line: including the effective-mass and dielectric constant mismatches but a parabolic conduction band. Thick solid line: full calculation, including all the effects mentioned above. The inset shows how the binding energy tends to its barrier value for very narrow wells.

to a hole subband structure which is highly anisotropic and non-parabolic in the transverse direction. These features, when included in the calculations of the exciton binding energies, lead to results different from those obtained by using decoupled parabolic valence subbands. In addition, we have also not included the Coulomb coupling between excitons belonging to different subbands, the nonparabolicity of the bulk conduction band, and differences between the mass parameters and the dielectric constants between the well and the barrier materials. Each of these effects has been studied, mostly separately, and has been found to enhance the values of the exciton binding energies by almost comparable amount (typically 1 2 meV for thin wells). Andreani and Pasquarello [16] and more recently, Iotti and Andreani [17] have calculated the values of the binding energies of the 1s and 2s states of both the heavy- and light-hole excitons in GaAs/Alx Ga1 x As quantum wells including the above mentioned affects using a variational approach. Their results for the 1s state for the heavy-hole exciton are shown in Fig. 4 for GaAs/Al0.4Ga0.6 quantum well structures. It is clear that the inclusion of the effects originally ignored by Greene et al. [15] leads to considerably larger values of the exciton binding energies especially for narrow wells. They have also calculated the variation of the binding energy of the 1s state a heavy-hole exciton as a function of the well width L for several values of the Al concentration in the barrier layers. As expected, for a given value of the well width the value of the binding energy increases with Al concentration. As a matter of fact the value of the binding energy exceeds the two-dimensional limit, namely, four times the bulk Rydberg (4 meV) for narrow wells with AlAs barriers. This enhancement is largely due to the contribution of the electrostatic potential due to the image charges to the Coulomb potential. For future details, the reader is referred to this work [16,17]. 3. Excitonic line broadening in semiconductor alloys: theoretical formulation In this section, we provide a brief review of the various theoretical attempts that have been made to calculate the excitonic linewidth in completely random semiconductor alloys at low temperatures.

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As pointed out earlier, the excitonic transitions in semiconductor alloys as observed in optical measurements such as photoluminescence, absorption, photo-reflectance, etc. are considerably broader than those observed in their components. This broadening has been attributed to the compositional disorder, which is always present in these systems. In high quality alloys, this disorder is expected to be completely random. The physical origin of the excitonic line broadening due to this disorder lies in the fact that the average alloy composition inside the region occupied by an exciton is different from that inside the region of another exciton. Even though the global value of the alloy composition is fixed, excitons in different regions of the alloy experience different values of the alloy composition. As the values of the conduction and valence band edges experienced by an exciton are determined by the local alloy composition, excitons in different regions of the alloy have different transition energies thus leading to an inhomogeneously broadened transition. As far as we know, the first attempt to calculate the effect of the compositional disorder on the excitonic linewidth was made by Goede et al. [18] who used their results to explain the reflectivity data in CdSeS systems. They calculate the probability of an exciton with a volume Vex experiencing an alloy composition xexc about the mean value x and relate it to the transition energy using virtual crystal approximation (VCA). In their calculation, Goede et al. [18] use for the volume of the exciton, Vex ˆ …4p=3†hr3 i, where hr 3 i is an expectation value of r3 and is equal to 7:5R3ex . Here Rex is the Bohr radius of an exciton. Singh and Bajaj [19] not being aware of the work of Goede et al. [18] calculated the variation of the excitonic linewidth as a function of alloy composition using a somewhat different but equivalent approach. In the following, we briefly outline their method as it is somewhat more transparent and can be generalized to more complex systems such as confined quantum structures. Consider a binary alloy AB where the constituent atoms A and B each form a semiconductor. Our treatment also applies to ternary alloys such as ABC where AC and BC each have semiconducting properties. A well-known example of the binary semiconductor alloy is SiGe, whereas there are many III±V and II±VI ternary alloys such as AlGaAs, InGaAs, GaInP, ZnCdSe, ZnSeS, ZnCdTe, etc. which have been studied extensively in recent years. We assume that there is no disorder in C. We shall therefore use the symbols A and B to denote the components of the alloy and suppress C. We denote the alloy by AxBy, where x and y represent the mean concentration of atoms A and B, respectively. The conduction and valence band edges of the two components are assumed to be given by EcA, EvA and EcB and EvB , respectively. We further assume that the electronic structure of the alloy is represented by the VCA according to which EcVCA ˆ xEcA ‡ yEcB

(34a)

EvVCA ˆ xEvA ‡ yEbB

(34b)

and the direct band-gap EgVCA ˆ EcVCA

EvVCA

(34c)

If an exciton is created at ~ r, its emission energy is given as r† ˆ Ec …~ r† Eex …~

Ev …~ r†

EB …~ r†

(35)

where EB …~ r† is the binding energy of the exciton, Ec …~ r† and Ev …~ r† are the conduction and the valence band edges as experienced by the exciton in its volume and need not be the same as the VCA values. These values are derived in the following.

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First, we need to determine the probability of a concentration x 0 to occur over the exciton volume Vex. This, according to Lifshitz [20] is given as    0  0  Vex x y 0 0 P…x; Vex † ˆ exp x ln ‡ y ln (36) Vc x y where Vc is the smallest volume over which the fluctuation can occur and x ‡ y ˆ 1. The shift in the exciton transition energy from the value corresponding to the VCA is given as DE…x 0 ; x; Vex † ˆ ‰Ec …x 0 †

Ev …x 0 †Š

‰Ec …x†

Ev …x†Š ˆ ‰x 0

xŠD1

(37)

as the value of EB does not change significantly over small compositional variations, and D1 is the difference in the band-gaps of the two components of the alloy. Since V ex =V c @ l, …x 0 x† has to be small if P(x 0 x, Vex) is to have any significant value. Therefore, for small values of (x 0 x), the logarithms in Eq. (36) are expanded to give " # Vex …x 0 x†2 0 (38) P…x ; x; Vex † ˆ exp Vc 2x…1 x† This is a Gaussian distribution and the full width at half maximum s of this distribution is given as s ˆ 2…2 ln 2†

1=2



x…1

x†Vc Vex

1=2

D1

(39)

This expression for s is identical to the one derived by Goede et al. [18]. We note that Eq. (38) has two different volumes: smallest volume over which the composition fluctuation takes place Vc and the volume of the exciton. For alloys with zinc blende structure, V c ˆ a3 =4 where a is the lattice constant. The volume of an exciton, however, is not a well defined quantity. As mentioned earlier, Goede et al. [18] use for Vex the expression …4p=3†hr3 i ˆ 7:5pR3ex . Singh and Bajaj [19], however, use …4p=3†R3ex as the exciton volume. The value of s in [19], therefore, is about 2.7 times larger than that calculated by Goede et al. [18]. Schubert et al. [21] have also calculated the variation of s as a function of alloy composition using binomial statistics and have obtained the same expression as Eq. (39). They use, for the exciton volume, the same expression as that used by Singh and Bajaj [19]. A serious drawback of the calculations described above lies in their use of the exciton volume, which is a classical entity but not a well-defined quantum mechanical quantity. The ambiguity in defining the exciton volume has thus led to large discrepancies in the calculated linewidths. In an effort to resolve this situation, Singh and Bajaj [22] followed a quantum mechanical approach that does not explicitly depend on the exciton volume. In their quantum mechanical formulation, they follow essentially the same approach as described in [19] except they now write the shift in energy DE as DE ˆ …x 0

x†D1 Pin

(40)

jc…~ r;~ r0 †j2 d~ r

(41)

where Z Pin ˆ

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Here c…~ r;~ r0 † is the exciton wave function centered at ~ r0 and Pin is the fraction of the localized probability function inside an arbitrary volume V. Substituting the value of (x 0 x) from Eq. (40) in Eq. (39), they get " # V DE2 (42) P…X; V† ˆ exp Vc x…1 x†D21 P2in They now use an extremal principal for P(x, V ) and select configurations which will contribute most to a given shift P(x, V ) and hence to the linewidth. Thus, one needs to evaluate V0 for which the quantity   @ V ˆ0 (43) @V Pin VˆV0 Using the following expression for the excitonic wave function   1 …r r0 † c…~ r ~ r0 † ˆ exp Rex HpR3ex

(44)

they calculate Pin and the corresponding value of V0 and substitute these in Eq. (42) and obtain 1=2

s ˆ 2…2 ln 2†

 x…1

x†Vc

1=2

Vex

D1 …0:327†

(45)

where Vex ˆ 43 pR3ex This expression of s can be obtained from Eq. (39) by using an effective exciton volume ˆ 12:42pR3ex . It should be pointed out that they do not use any particular definition of the exciton volume but obtain it by using an extremal principle. Another attempt to determine the variation of s as a function of alloy composition was made by Zimmerman [23] who used statistical theory of exciton line shape which results from the superposition of delta functions. Using first-order perturbation theory on the fluctuating Hamiltonian, he was able to cast the expression for the lineshape function in the form of a Gaussian with the following expression for the excitonic linewidth: 1=2

s ˆ 2…2 ln 2†

 x…1

x†Vc 0 Vex

1=2

D1

(46)

0 is now defined as where the statistically relevant exciton volume Vex 0 Vex ˆR

1 d~ rjc…~ r†j4

(47)

Since c…~ r† is normalized to unity, Eq. (47) has the dimension of a volume. Substituting the expression for c…~ r† from Eq. (44) in Eq. (47), we get Vex ˆ 8pR3ex

(48)

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Thus, the value of s calculated by Zimmerman is larger than those obtained by Goede et al. [18], and by Singh and Bajaj [22] and smaller than those obtained earlier by Singh and Bajaj [19] and Schubert et al. [21]. In the following, we summarize the results we have discussed so far. Let us define s0 as   …2 ln 2†x…1 x†Vc 1=2 D1 s0 ˆ 2 …4p=3†R3ex

(49)

the expression obtained by Singh and Bajaj [19] and Schubert et al. [21]. The various expressions calculated for s can be written as follows: 8 …Goede et al: ‰18Š† < 0:36s0 s ˆ 0:319s0 …Singh and Bajaj ‰22Š† (50) : …Zimmerman ‰23Š† 0:41s0 A few years ago, Lee and Bajaj [24] not being aware of Zimmerman's work derived an expression for s following an approach somewhat similar to that of [23]. In their formalism, they use a quantum mechanical description for the excitonic system and calculate the mean deviation of its transition energy (and hence the linewidth) due to statistical potential fluctuations using first-order perturbation theory. They treat the fluctuation potential at each lattice site as an independent disturbance on the excitonic wave function and then sum over the contributions from all lattice sites. In other words, the excitonic wave function is perturbed by the fluctuating potential from each lattice site independently. This method therefore does not property account for the closely spaced potential fluctuations in the random alloy. This approach, therefore, is valid only for very small alloy compositions. The values of s they obtain are considerably smaller than those calculated by Zimmerman [23]. The reason for this discrepancy is not clear. Later Lee and Bajaj [25] generalized their earlier work by calculating the effect of the closely spaced potential fluctuations directly. In addition, they also calculate the effect of the applied magnetic field on the value of s. This will be discussed in the latter part of this section. The details of the calculations are given in [25] and will not be reviewed here. Lee and Bajaj [25] obtain an expression for s which is evaluated numerically. However, with a couple of simplifying but valid assumptions, the expression for s can be cast in the following simple analytic form: x†SŠ1=2

(51)

1 X 1 X …jcjk j2 Vc D1 †2

(52)

s ˆ 2‰…2 ln 2†x…1 where Sˆ

kˆ 1 jˆ1

jcjk j2 ˆ

1 X iˆ 1

jcijk j2

(53)

where cijk is the value of the exciton wave function at the site of the unit cell denoted as~ r ˆ~ rijk . The ~ exciton is assumed to be localized at r ˆ 0. Thus, the value of s is proportional to the square root of R jc…~ r†j4 d~ r; a result also found by Zimmerman [23]. Thus, in the absence of a magnetic field, our expression for s is identical to that of Zimmerman [23]. It should be pointed out that the expression for s as given in Eq. (51) reduces to the well-known expressions involving exciton volume [18,19,21]

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if we treat the exciton as a sphere of volume Vex, i.e. if we express the exciton wave function as follows:  c…~ r† ˆ

1 Vex

1=2

reVex ˆ 0; otherwise

(54)

Recently, Lyo [26] has calculated the dependence of s on alloy composition and on applied magnetic field. His results for the zero magnetic fields are discussed here. He points out that the theoretical approaches described so far are characterized by two limitations. First, the previous authors have only considered the relative motion of the electron and the hole and second, the local exciton * transition energy was determined by the local random band-gap energy fluctuation Eg … r †. He suggests that in a disordered alloy, the exciton transition energy is determined by the absolute positions of the electron and the hole and not by their relative coordinates. The electron and the hole coordinates should, therefore, be treated independently right from the beginning. The electron and the hole are not at the same position most of the time and, therefore, the appropriate parameter to use is the non-local band-gap energy fluctuation which is the sum of the separate local band-edge * * * fluctuations of the electron (Ee … r †) and the hole (Eh … r †) rather than Eg … r †. In his approach, the total exciton wave function is expressed as a product of the wave function of the center of mass coordinates and the relative coordinates. The true exciton volume is determined both by the wave function of the relative coordinates and of the center of mass coordinates. He then considers two different cases of exciton localization, namely, pinning by defect centers and localization by the alloy disorder. The localization of excitons by the defect centers leads to a Gaussian lineshape and the value of s depends on the localization radius R0 and the values of the conduction and valence band-edge fluctuations. The values of R0 are not known and even the values of the individual bandedge fluctuations are often not accurately determined. One, however, knows much more accurately, from experimental measurements, the values of the energy band-gap fluctuations. In addition, in the case of an exciton bound to a shallow donor or an acceptor (which is often the case), the complex has a spherical symmetry with the center of mass of the whole system located at the ion. As the two electrons or the two holes are indistinguishable, it is not clear as to what the parameter R0 really means. As a matter of fact, R0 should be 0 in these situations. In this case their results are similar to those of Lee and Bajaj [25] except the electron band-edge fluctuation (which is not known accurately anyway), replaces the energy band-gap fluctuation. Lyo [26] has also considered the effect of the exciton localization due to alloy disorder on the excitonic linewidth. Localization of the noninteracting electrons and holes through band-edge fluctuations caused by the composition fluctuations has been studied in considerable detail [27]. Following Baranovski and Efros [27] he calculates the effect of the localization of the center of mass of the exciton by the fluctuating bandedge potentials on the excitonic linewidth and finds that the value of s depends on the localization radius R0 and the values of the individual electron and hole fluctuating potentials. The value of R0, in this case, can be estimated, however, the values of the other two parameters are again not accurately known. The excitonic lineshape turns out to be highly asymmetric with a long tail toward the lowphoton energy side, in strong disagreement with the observed Gaussian lineshape in alloys based on III±V semiconductors. It appears that in these alloy systems the localization of excitons by the pinning defect centers plays a major role in determining the value of s. For further details, the reader is referred to Lyo's work [26]. An interesting but somewhat different approach to calculate the dependence of excitonic linewidth s on the compositional disorder was followed by Baranovski and Efros [27]. In the following, we briefly review the main ideas on which their calculations are based. Let us consider the

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behavior of a conduction electron in an alloy Ax B1 x, where x again will be referred as the ``composition''. Let Ec(x) be the conduction band minimum for the crystal with composition x. The value of x varies from point to point with the mean value of x0. Then for a give volume with a large number of substitutional atoms, one can write x ˆ x0 ‡ Dx. Assuming Dx ! x0 (small fluctuations), the energy at the bottom of the conduction band can be written as Ec …x† ˆ Ec …x0 † ‡ ae Dx dEc ae ˆ dx xˆx0

(55a) (55b)

Now the deviation of the fractional concentration of A atoms from its mean value can be written as Dx ˆ

D…~ r† N

(56)

where D…~ r† is the deviation of the (absolute) concentration of A atoms from the mean value and N the total concentration of sublattice sites occupied by either A or B atoms. D…~ r† is assumed to be averaged over a large number of A and B atoms, centered at point ~ r. The potential energy of an electron at the bottom of the conduction band can be written as V…~ r† ˆ ae

D…~ r† N

(57)

Assuming a completely random distribution of atoms A and B, the two point correlation can be approximated as hV…~ r†V…~ r 0 †i ˆ

a2e x…1 N

x†d…~ r ~ r 0 †;

(58)

where h


i denotes averaging over different configurations of atoms. The valence bands are affected in the same fashion except ae is replaced by ah. The effect of the composition fluctuations on the conduction band, for instance, is reflected in the density of states in that at energies below the bottom of the conduction band corresponding to x0, the density of states does not vanish but forms an exponentially decaying tail. These states are the lowest bound states confined in the potential wells produced by the composition fluctuations. For large values of energies E, measured from the bottom of the conduction band Ec(x0), the density of states g(E) is assumed to be proportional to exp[ F(E)], where F(E) is some function of B. In the low energy part, g(E) can be expressed as [27]   jEj (59) g…E† ˆ exp E0 where E0, the characteristic decay energy for the density of states in the band-gap, is given as [27] E0 ˆ

x2 …1

  x†2 a4e m4e 1 178 h6 N 2 

(60)

This expression is valid as long as jEj ! E0 or jEj ! ae x. These states are discrete levels in the potential wells induced by the composition fluctuations. A similar result applies to the tail of

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the valence band. One can regard E0 as a measure of the band-edge smearing in semiconductor alloys. Its value in most III±V based alloys is estimated to be quite small (<1 meV) for the conduction band-edge but considerably larger for the valence band-edge due to the much larger value of the hole mass. Baranovski and Efros [27] show that E0 also characterizes the exciton line broadening when the exciton generated by a photon is localized by the composition fluctuation. It is further assumed that the exciton kinetic energy is small and the localization distance is much larger than the exciton Bohr radius Rex. The broadening of the exciton occurs because the exciton localization energy is different at different regions in the sample. Under these conditions, the exciton broadening is of the order given by Eq. (60) except ae is replaced by a …a ˆ …dEg =dx†jxˆx0 † and the electron mass me is replaced by the total exciton mass M ˆ me ‡ mh . Suslina et al. [28] were able to explain their data on excitonic linewidth in Znx Cd1 x S and Znx Cd1 x Te rather well using Eq. (60). The presence of the finite density of states in the energy band-gap of a semiconductor alloy allows the creation of excitons with different energies determined by the energy of the absorbed photon since the momentum of these low energy excitons is not a good quantum number. The wave functions of states lying immediately above E0 are also strongly modulated by the potentials produced by the composition fluctuations and excitons with such energies can be created by the absorption of light. However, as the exciton energy increases, the nature of its wave function changes from being localized to a plane wave like and, therefore, the conservation of momentum starts to play its role and the absorption tends to 0. This accounts for the short wavelength absorption wing of the exciton line. The long wavelength part of the absorption spectrum is simply due to the exponential reduction in the density of states. It is not possible to develop an analytic theory for a three-dimensional system which would describe the above mentioned absorption profile in the energy range of interest. Ablyazov et al. [29] have, however, calculated exactly the short wavelength and the long wavelength asymptotic values of the absorption coefficient. They then propose an interpolation scheme, which connects these two types of asymptotic behavior in such a way that the area under the corresponding curve is equal to the required value. Following the procedure outlined above they obtain an expression for the width of the excitonic line D as [29]  4 2  a x …1 x2 †M 2 D ˆ 0:08 h6 N 6 

(61)

As pointed out earlier this expression for D is valid when the length characterizing the interaction of an exciton with composition fluctuations RD ˆ h=2MD is large compared to the exciton Bohr radius Rex. When me and mh are comparable in value then the condition RD ! Rex implies that D ! Eex where Eex is the exciton binding energy. For mh @ me, RD ! Rex and Eq. (61) is still valid with a replaced by ah. Alyazov et al. [29] have also considered the situation in which ah is so small compared to ae, that the interaction of the hole with the composition fluctuations can be ignored. In addition, they assume that RD ! Rex (i.e. mh @ me ). They then evaluate the mean square fluctuations of such a potential and derive the following expression for D, i.e.     x…1 x† 1=2 x…1 x†Vc 1=2 ˆ 0:5ac D ˆ 0:5ac NR3ex R3ex

(62)

For semiconductor alloys with zinc blende structure N ˆ 4=a3 ˆ 1=V c where a is the lattice constant.

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Fig. 5. Conduction band energy minima in eV at G, X and L, relative to the top of the valence band as a function of alloy composition x in Alx Ga1 x As.

In this section, we have reviewed the results of various calculations of the variation of s as a function of alloy composition in semiconductor alloys. To illustrate these results, we now consider the case of Alx Ga1 x As. In Fig. 5, we display the variation of the energies of the G, L and x band edges as measured from the top of the valence band as a function of alloy composition x. This alloy system has a direct band-gap for values of x upto about 0.45. In Fig. 6, we show the variation of s as a function of alloy composition x upto x ˆ 0:4 as calculated by several groups.

Fig. 6. Variation of the excitonic linewidth (s) as a function of Al concentration in Alx Ga1 x As. Dotted line: Goede et al. [18]. Dashed line: Singh and Bajaj [19], Schubert et al. [21]. Solid line: Zimmerman [23] and Lee and Bajaj [25].

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3.1. Effect of external perturbations In this section, we shall briefly review the results of various calculations of the effect of the external perturbations such as magnetic and electric fields on the excitonic linewidth in semiconductor alloys. The application of such external perturbations affect the excitonic wave function and hence the excitonic linewidth. 3.1.1. Magnetic field As far as we know, the first calculation of the effect of the magnetic field on the value of s was reported by Raikh and Efros [30]. They showed that the value of a increases with the increasing magnetic field and they attributed this broadening to two mechanisms. The first mechanism involves the relative motion of the election and the hole which averages the fluctuations on a scale smaller than the exciton Bohr radius Rex, and, therefore, a relatively smoothed out potential acts on the exciton as a whole. The application of a strong magnetic field reduces the overall volume of the exciton thus reducing the importance of the averaging of the random potential. The second mechanism leading to the enhancement of the value of s with increasing magnetic field depends on the fact that the center of mass of exciton in a plane perpendicular to the magnetic filed increases with increasing magnetic field. This facilitates the localization of the exciton by the compositional fluctuations. In the following, we briefly review their calculations of the effects of the magnetic field on the excitonic linewidth due to these two mechanisms. We first consider the second mechanism. In the presence of a strong magnetic field such that b ˆ … heB=mc†=Eex @ 1, the effective exciton mass in the plane perpendicular to the direction of the magnetic field is considerably enhanced over its value at zero field. With an accuracy upto logarithmic terms ln b @ 1, the mass in the transverse direction is given by [30]   Rb MB ˆ m ln b l

(63)

where l ˆ …c h=eB†1=2 is the magnetic length and Rb the longitudinal size of the exciton and is equal to Rex/ln b. Eq. (63) is valid only when …heB=mh c† @ Eex . Using the magnetic field dependent total mass in the SchroÈdinger equation of the exciton in the presence of a magnetic field and fluctuating electron and hole potentials, Raikh and Efros [30] derive an expression for E0 and hence D, following a procedure essentially similar to that described in [27]. The expression for D they obtain is D ˆ 14E0 ˆ

0:08…ae ‡ ah †4 MMB2 x2 …1 h6 N 2 

x2 †

(64)

Again this expression for D is valid provided the linewidth is much smaller than the binding energy of the exciton and the transverse and the longitudinal dimensions of the exciton are much less than the length characterizing the interaction of the exciton with composition fluctuations. In this approach, the effect of the magnetic field on the excitonic linewidth is estimated by studying the behavior of an exciton in compositional fluctuations, in which the center of mass of the exciton is enhanced by the magnetic field. They next consider the case when Rb ˆ Rex =ln b is considerably larger than the dimension of a typical well of localization. As in the case of B ˆ 0, they also calculate excitonic linewidth assuming that mh @ me but ah is much smaller than ae, in the presence

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of a magnetic field following a procedure essentially similar to that described in [27] and obtain the following expression: D2 ˆ

0:3a2e x…1 x†MB Nh2 Rex

(65)

They also consider the situation in which the diameter of the region of motion of an exciton as a whole in a plane perpendicular to the magnetic field becomes comparable with the magnetic length l. In this case they find that the profile of the absorption line is Gaussian  2 1 E (66) A…E† ˆ p exp 2y2 2py where y2 ˆ

a2e x…1 x†Vc 4Rb l2

(67)

They also discuss, in considerable detail, the conditions of validity of their results. For more details, the reader is referred to their work. Singh and Bajaj [19], not being aware of the work of Raikh and Efros [30], suggested that for a given alloy composition, the application of a magnetic field would increase the value of the excitonic linewidth as it shrinks the excitonic wave function and hence reducing its effective size. Following this suggestion, Mena et al. [31] have calculated the variation of s in perfectly random alloys as a function of magnetic field. Following Coede et al. [18], they calculate the expectation value of the exciton volume as a function of magnetic field using the following exciton wave function [32]: c ˆ A exp‰ ar

b2 r2

c2 z2 Š

(68)

where a, b, and c are variational parameters which are determined by minimizing the exciton binding energy in a magnetic field and A is a normalization constant. Variations in the local concentrations of alloy components from their global values are accounted for as described in [19] by using statistical mechanical arguments developed by Lifshitz [20] and then relating them to the excitonic linewidth. One obtains an expression for s identical to that given by Eq. (39) except now the exciton volume Vex is field dependent. In Fig. 7, we display the variation of s as a function of magnetic field for several values of x …x  0:4† in Alx Ga1 x As for which this alloy system has a direct band-gap. As expected the value of s, for a given value of x, increases as a function of magnetic field. Mena et al. [31] do not consider the effect of the motion of the center of mass of an exciton in a magnetic field on the excitonic linewidth. It is implicitly assumed in their calculation that the exciton is localized at an impurity center. As noted earlier, Lee and Bajaj, in a series of two papers [24,25], have developed a theory of excitonic linewidth in semiconductor alloys in which they use a quantum mechanical description of the excitonic system and calculate the mean deviation of its transition energy (and hence the linewidth) due to statistical fluctuations using a first-order perturbation theory. Again the dependence of s on the magnetic field arises from the dependence of the wave function on the magnetic field used in the first-order perturbation calculation. In the first version of their theory [24], they treat the fluctuation potential at each lattice site as an independent disturbance on the exciton wave function and find that the rate of increase of s with magnetic field is considerably smaller than that reported

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Fig. 7. Variation of the excitonic linewidth (s) as a function of a magnetic field (B) for several Al concentrations x in Alx Ga1 x As.

by Mena et al. [31]. In the second version, they generalized their approach as described in [25] and find that the rate of increase of s with magnetic field is close to that calculated by Mena et al. [31]. In Fig. 8, we display the variation of the calculated values of s as a function of the applied magnetic field for a few values of the Al composition in Alx Ga1 x As. The solid lines are obtained using calculations of Lee and Bajaj [25] and dashed lines represent the results of Mena et al. [31]. For low

Fig. 8. Variation of the excitonic linewidth (s) as a function of magnetic field (B) at a few typical values of Al composition x. Solid curve: Lee and Bajaj [25]. Dashed curve: Mena et al. [31].

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values of the magnetic field, the values of s as calculated by Lee and Bajaj [25] are somewhat larger than those obtained by Mena et al. [31]. At higher magnetic fields, the two results are almost identical. As noted earlier, Lyo [26] has calculated the dependence of s on alloy composition and on magnetic field. His calculations for the case of a zero magnetic field have been reviewed earlier. The excitonic linewidth s has two major sources of contributions, namely, one that arises from the alloy fluctuations experienced by the center of mass and is independent of the magnetic field and second which originates from the alloy fluctuations inside the region where the exciton probability is significant. The latter contribution is a function of the magnetic field as the exciton probability depends on the strength of the field. The SchroÈdinger equation for an exciton in a magnetic field is solved numerically and these results are then used to calculate the excitonic linewidth due to potential fluctuations following a procedure described earlier. As in the case of a zero magnetic field, the excitonic linewidth depends on the values of R0 and the individual electron and hole band-edge fluctuations and increases with magnetic field. For further details, the reader is referred to his work [26]. Application of an electric field on the excitonic system increases the average separation between the electron and the hole and thus considerably modifies the excitonic wave function and eventually leads to its disassociation. This is the well-known Stark effect which has been studied in considerable detail in atomic physics. The excitonic transition is expected to broaden in the presence of an electric field. As far as we know, no systematic study of the behavior excitonic linewidth in alloys in the presence of an electric field has been reported. The effects of the application of the hydrostatic pressure on the electronic band structure of semiconductors has been studied in great detail during the past several decades. In a direct band-gap semiconductor the application of hydrostatic pressure increases the value of the energy band-gap, which in turn enhances the value of the conduction band mass. This leads to a larger value of the reduced mass and hence the excitonic linewidth increases as a function of pressure as long as the band-gap remains direct. Such an effect has recently been observed in several semiconductor alloys and will be discussed later. 4. Experimental studies In this section, we shall briefly review the results of the experimental measurements of the excitonic linewidths in several III±V and II±VI alloy systems and compare them with their theoretical values as calculated by several authors, described in the previous section. Due to extensive technological applications of these alloys in a variety of electronic devices a great deal of effort has been devoted to study their optical properties using several spectroscopic techniques. A number of groups have studied the behavior of the excitonic linewidth in these alloys using primarily photoluminescence spectroscopy. We shall review only a few illustrative cases as a comprehensive review of this work is beyond the scope of this article. 4.1. Alx Ga1 x As Due to the close lattice matching of AlAs and GaAs, this alloy system has been grown by several growth techniques such as MBE, MOCVD and LPE over the complete range of alloy composition with excellent electronic and optical properties. The electronic band structures of this system have been studied, both theoretically and experimentally, by a number of groups during the

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past two decades. In Fig. 5, we display the variation of the energies of the G, L and X band edges as measured from the top of the valence band, as a function of alloy composition x. We find that this system has a direct band-gap for values of x up to about 0.45 and, therefore, a number of photoluminescence (PL) measurements of excitonic linewidth (s) have been made in this system in this particular range of alloy compositions. Wicks et al. [33] were among the first to grow high quality AlGaAs using MBE with As4 as the arsenic source. The best PL spectra as determined from the widths of the bound excitonic transitions was obtained from layers grown under low arsenic flux at substrate temperature of 7008C. In Al0.19Ga0.81As doped with tin (5  1015 cm3), the width of the bound exciton transition was determined to be 3.0 meV, whereas in undoped Al0.24Ga0.76As, its value was measured to be 4.0 meV, at 40 K. These were the lowest values of the excitonic linewidths in AlGaAs reported at that time. These values are somewhat larger than those calculated by Zimmerman [23] and Lee and Bajaj [25] due to the presence of other broadening mechanisms in this alloy system. As mentioned in the previous section, Schubert et al. [21] calculated the variation of s as a function of alloy composition using binomial statistics. They also measured the values of s for several different alloy compositions x in Alx Ga1 x As and compared these with their theoretical values. Their samples, typically 2±3 mm thick, were grown by MBE on (1 0 0) Ð oriented undoped semi-insulating GaAs substrates at temperatures ranging from 590 to 6708C in different growth runs using As4 as the arsenic source. For a given value of x, they optimized their growth conditions to obtain the most intense and the sharpest excitonic transitions. Their samples were nominally undoped and were p-type with hole concentrations in the high 1014 cm 3 range at 300 K. The total impurity content was estimated to be in the range of 1015 cm 3. The PL measurements were carried out at 2 K and the Al concentration along the surface of the samples as determined by the position of the excitonic transition was found to be extremely uniform. The values of s were measured at three different alloy compositions in the range x ˆ 0:2±0.45 and were found to increase from 5 to 8 meV. These values are considerably larger than those calculated by Zimmerman [23] and Lee and Bajaj [25]. It should be pointed out that in semiconducting alloys, the free exciton transitions are observed generally for small values of x (i.e. x  0:15) [34]; the transitions associated with excitons bound to shallow impurities are usually the most intense. The values of s, measured by Schubert et al. [21], are most likely those of excitons bound to shallow acceptors, presumably carbon, the most commonly found impurity in MBE or MOCVD grown Alx Ga1 x As. Reynolds et al. [35] have also measured the variation of s as a function of alloy composition in Alx Ga1 x As using high resolution PL spectroscopy at 2 K. Their samples were also grown by MBE on (1 0 0) oriented semi-insulating GaAs substrates and varied in thickness from 1 to 2 mm. The highest quality samples, both in terms of transport and optical properties, were grown at substrate temperatures in the range of 680±7008C with V/III flux ratios in the range of 6±10. The substrates were azimuthally rotated during growth and arsenic in the form of As2 flux was used. The alloy composition x was varied from a few percent to about 45%. In Fig. 9 is plotted the variation of their measured values of s as a function Al concentration x. There is a considerable scatter in their data however, a general trend is clear; s increases as a function of x. The transitions observed in their work are among the narrowest reported in literature; for example, s ˆ 2:1 meV for x ˆ 0:36, thus indicating a high quality cluster-free material. Reynolds et al. [35], in their paper, attribute the observed transitions to the radiative decay of free excitons. We suggest that these transitions may be due to the bound excitons rather than free excitons, especially at larger values of x. Cunningham et al. [36] have investigated the growth of AlGaAs in the temperature range of 800±8708C, using MBE. This range of temperature is considerably larger than that commonly used (<7008C) in MBE growth of this material. They find that growing AlGaAs at these elevated temperatures has a beneficial effect

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Fig. 9. Variation of the excitonic linewidth (s) as a function of Al concentration x in Alx Ga1 x As: (&), experimental points. Dashed curve: Singh and Bajaj [19]. Solid curve: Singh and Bajaj [22].

on the quality of the material and also reduces significantly the incorporation of carbon impurities. For instance, in Al0.33Ga0.67As grown on (0 0 1) oriented GaAs substrates at 8708C with As4 as a source of arsenic, they find an excitonic linewidth of 3.6 meV at 2.2 K, which is comparable to that measured by Reynolds et al. [35]. More recently, Olsthorn et al. [37] have measured the values of s in Alx Ga1 x As grown by MOCVD at two different alloy compositions, namely, x ˆ 0:12 and 0.244 and have observed the narrowest excitonic transitions in a MOCVD grown material. Their samples were grown on (1 0 0) 28 off towards (1 1 0) semi-insulting GaAs substrates in two different MOCVD reactors using different growth conditions. Hall measurements at 77 K showed n-type carrier concentration of 1:7  1015 and 1:8  1015 cm3 in the two samples with Al concentration x of 0.12 and 0.244, respectively. Their PL measurements were performed at 4.5 K and the optical excitation was provided by 2.41 eV line from an Ar‡ laser with excitation densities ranging from 1:8  10 2 to 5.6 W/cm2. The Al concentration in the two samples was determined from the known positions of the exciton peaks. In Fig. 10, we display their excitonic spectra of a sample with x ˆ 0:244 recorded at various excitation densities. At the lowest excitation density, they observe three transitions which they attribute to excitons bound to neutral acceptors (A , X), neutral donors (D , X) and ionized donors (D‡, X). As far as we know, this is the first report of an observation of all these bound exciton transitions in a Alx Ga1 x As samples with such a large value of x. As the excitation density is increased, the concentration of bound exciton centers increases, the transitions merge into each other and finally a broad peak is observed. Olsthorn et al. [37] therefore suggest that in their samples, the broadening effect due to residual ionized impurity centers plays a significant role. However, they do not provide any physical explanation of the behavior of these bound exciton transitions as a function of excitation density. We suggest that a likely explanation of their observation lies in the fact that as

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Fig. 10. Exciton spectra of Al0.244Ga0.756As recorded at various excitation densities (P). The relative gain used to record each spectrum is given on the left. The linewidth of the separate (D , X) peak (FWHM) of each spectrum is given on the right.

the excitation density is increased, more ionized impurity centers are neutralized, thus leading to a larger concentration of bound exciton systems. The increased overlap between the wave functions of these complexes leads to larger broadening of the transitions and eventually a broad peak. At the lowest excitation densities used in their experiment, the values of the linewidths of say (D , X) transitions are at least one-half of those observed in other MOCVD grown samples and are comparable to those measured by Reynolds et al. [35] in MBE grown samples. Chin et al. [38] have grown high quality Alx Ga1 x As (0:28 < x < 0:33) on both (1 0 0) and (1 1 1) B oriented GaAs substrates using MBE. They optimize their growth conditions for both these orientations. For Al0.3Ga0.7As grown on (1 0 0) GaAs at 7008C they find a minimum value of the excitonic linewidth of 2.4 meV at 40 K using PL spectroscopy. Similar measurements on Al0.3Ga0.7As grown on (1 1 1) B oriented GaAs at 6508C yield a value of 2.9 meV for the excitonic linewidth. Recently, Zhuravlev et al. [39] have measured the variation of s as a function of Al concentration in Alx Ga1 x As using PL spectroscopy at 4.2 K. Their samples were grown by MBE on (0 0 1) oriented semi-insulating GaAs substrates at 6308C and the Al concentration as determined by Ê ) AlAs PL was varied from 0 to 0.295. The Alx Ga1 x As layers were sandwiched between thin (250 A layers to decrease the rates of surface and interface recombinations. They took great care to reduce the incorporation of residual impurities (mostly carbon) in their samples, which all showed p-type conductivity with hole concentrations in the range of …1 5†  1014 cm3 at room temperature. Their

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Fig. 11. Variation of the free excitonic linewidth (FWHM) as a function of Al concentration x in Alx Ga1 x As. The experimental points are: (1) current work, (2) from Reynolds et al. [35], (3) from Olsthoorn et al. [37]. The solid line is the theoretical curve calculated using the theory of Lee and Bajaj [25]. The inset shows high resolution PL spectrum in Al0.15Ga0.85As.

PL spectra were dominated by the free exciton emission with very weak impurity related transitions. In Fig. 11, we display the variation of their measured values of s as a function of Al concentration x along with that calculated using the theory of Lee and Bajaj [25]. There is an excellent agreement between theory and experiment except for a sample with x ˆ 0:295, thus suggesting a very high quality of the material without clustering and with low impurity concentration. They attribute a somewhat larger value of s for x ˆ 0:295 to inter-valley scattering. Also shown in Fig. 11, for comparison are the values of s reported by Reynolds et al. [35] and Olsthorn et al. [37]. The values of PL excitonic linewidths measured by Zhuravlev et al. [39] are among the lowest reported in Alx Ga1 x As grown by any technique and are very close to those calculated by Lee and Bajaj [25]. In the foregoing, we have reviewed briefly the results of only a few representative PL measurements of the excitonic linewidth in AlGaAs alloys grown by MBE and MOCVD. As suggested by the low values of the excitonic linewidths, it is now possible to grow almost cluster free AlGaAs alloys with low impurity concentration. A vast body of work on the optical properties of AlGaAs alloy system exists in literature. 4.2. InGaAs High quality Inx Ga1 x As alloys covering the whole range of energy band-gaps between GaAs and InAs have been grown both on GaAs and InP substrates using a variety of epitaxial techniques such as vapor phase epitaxy (VPE), LPE, MOCVD and MBE with excellent electronic and optical properties. Because of its widespread applications in electronic and opto-electronic devices, its materials properties have been studied both theoretically and experimentally by a number of groups.

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InGaAs is miscible and has a direct energy band-gap over the whole range of alloy composition. InGaAs grown on GaAs substrates is always in a state of compression. However, when grown on InP substrates, it can be either in the compressive or in the tensile state depending on the alloy composition. In0.53Ga0.47As is grown lattice matched to InP. Excitonic linewidths have been measured Inx Ga1 x As over a wide range of alloy compositions. The narrowest PL linewidths, however, have been observed in In0.53Ga0.47As grown on InP. We shall therefore review briefly the behavior of excitonic linewidths in this particular system. Schubert and Tsang [40] have studied the PL spectra of In0.53Ga0.47As grown on InP by CBE, at 2 K as a function of excitation intensity. Their samples were grown on InP substrates using triethylgalium (TEGa), trimethylindium (TMIn), cracked arsine (AsH3) and phosphine (PH3). The growth temperature was 550±6008C and the growth rate was typically 3.65 mm/h. Excellent structural quality of their layers was attested by very sharp (0 0 4) Bragg reflections obtained in double-crystal X-ray diffraction measurements. In one of their samples with a thickness of 0.2 m, they observe an excitonic linewidth of 1.2 meV at the low excitation intensity of 5 mW/cm2 (Fig. 12). As far as we know, this is the narrowest linewidth ever reported in this alloy system. This is to be compared with its calculated value of about 0.6 meV. They also find that with decreasing excitation intensity the excitonic line narrows; the rate of narrowing being proportional to the linewidth. In addition, the transition shifts to lower energies and gets increasingly asymmetric. They also derive an expression of the PL lineshape of excitons in semiconductor alloys using a semi-classical model, based on the migration of carriers to the low-energy sites of the alloy and use it to explain their experimental data. Several other groups have grown InGaAs lattice matched to InP using mostly MOCVD. The values of the excitonic linewidths they measure are larger than those reported by Schubert and Tsang [40] and will not be discussed here.

Fig. 12. Low temperature photoluminescence spectrum of an undoped 0.2 mm thick Ga0.47In0.53As layer clad by InP barriers.

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4.3. InGaP The direct band-gap ternary alloy In0.48Ga0.52P lattice matched to GaAs substrates and from here on referred to as InGaP, is important for a variety of technological and fundamental reasons. Depending on the growth conditions, this alloy system can be grown both in ordered and completely disordered phases. The dominant radiative process in ordered InGaP is rather complicated and is generally considered not to be excitonic in nature, whereas the dominant high-energy emission processes at low temperature in disordered InGaP are primarily excitonic in character. We shall therefore confine ourselves to the study of excitonic linewidth in completely disordered InGaP system. Jones et al. [41] have measured the variation of the PL excitonic linewidth as a function of magnetic field in InGaP grown on GaAs substrates by low-pressure MOCVD. Substrates used were n‡ type GaAs, oriented (1 0 0) with a mis-orientation of 28 to the nearest h1 1 0i or 58 to the nearest h1 1 0i A. The measurements were made at 1.4 K and the magnetic field ranged between 0 and 13.6 T. They observe a value of s of 4.3 meV at zero field which increases to 6.0 meV when the field is increased to 13.6 T. In Fig. 13, we display the variation of s with magnetic field which agrees rather well with that calculated by Mena et al. [31] as well as with that calculated by Lee and Bajaj [25]. The latter formalism, of course, does not involve the use of an exciton volume. The values of the physical parameters used in these calculations are given in [25]. The difference of about 1 meV observed between the measured and the calculated values is attributed to other broadening mechanisms as discussed earlier. Recently, Zeman et al. [42] have measured both the diamagnetic shift and the linewidth of an excitonic transition in InGaP as a function of magnetic field up to 22 T at 4.2 K. Their sample was grown on a GaAs substrate using low pressure MOCVD at 7008C. The substrate was mis-oriented by 158 from [0 0 1] towards [0 1 1] direction. This value of misorientation is considerably larger than those used by most groups. The value of s at zero field was measured to be 5.0 meV, a value close to that measured by Jones et al. [41]. However, they find that the variations of the values of both the diamagnetic shift and s with magnetic field are about one-half of those reported by Jones et al. [41] and also those calculated by Lee and Bajaj [25] using a free or a weakly bound exciton model. To explain this rather unexpected behavior, they propose that the hole,

Fig. 13. Variation of the FWHM for the excitonic transition in In0.52Ga0.48P as a function of applied magnetic field. Circles represent experimental data. The line drawn through the data is a smooth curve to provide a visual aid. Solid curve without symbols is obtained from the theory of Lee and Bajaj [24]. The dashed curve with squares is based on the theory of Mena et al. [31]. The dashed curve with triangles is obtained from the earlier formalism of Lee and Bajaj [25].

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Fig. 14. Variation of excitonic linewidth in In0.48Ga0.52P as a function of magnetic field. Dotted curve is calculated using a free exciton model. Solid curve is calculated using a localized exciton model. And solid circles represent experimental data.

for reasons which are not completely understood but may be related to the fact that the sample was grown 150 mis-oriented from [0 0 1] direction towards [0 1 1], is completely localized either by the potential fluctuations or by some defect. They therefore treat the hole mass as infinite and use the formalism developed by Lee and Bajaj [25] to calculate both the variation of s and of the diamagnetic shift as a function of magnetic field. The exciton wave function is now much more localized and, therefore, is much less responsive to the magnetic field. The calculated variations of s and of diamagnetic shift with magnetic field agree very well with their measured values as shown in Fig. 14. 4.4. III±V nitrides There been an enormous interest in studying the electronic and optical properties of group III nitrides: AlN, GaN and InN and their ternary alloys in recent years. This interest has largely been motivated by their applications in a variety of opto-electronic devices covering the visible to ultraviolet spectral range. Optical properties of both AlGaN and InGaN alloy systems have been studied in considerable detail in the past few years. In the following, we briefly review the behavior of excitonic linewidths in these alloys. 4.4.1. AlGaN Recently, Steude et al. [43] have studied the optical properties of coherently strained Alx Ga1 x N layers (x < 0:22) grown on GaN which in turn was grown on sapphire substrates using MOCVD. They employ several characterization techniques such as PL, CL, reflection, transmission and absorption over a large temperature range (4±500 K). In particular, they measure the value of s as a function of alloy composition (x) at 4 K and find, as expected, that it increases as a function of x. They compare their experimental values with those calculated by Schubert et al. [21] and find, in general, a fair agreement, with the calculated values being somewhat larger for each value of the alloy composition. Based on this comparison between theory and experiment they suggest that their samples have negligible clustering.

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Fig. 15. Variation of excitonic linewidth (s) as a function of Al composition x in Alx Ga1 x N. The solid circles are data taken from Steude et al. [43] and solid curve is determined from the theory of Lee and Bajaj [25]. The dashed curve is due to Schubert et al. [21].

As mentioned earlier the calculations of Schubert et al. [21] give values of s which are more than twice as large as those calculated by other groups. In Fig. 15, we display the variation of s as a function of alloy composition in Alx Ga1 x N. The solid circles are data taken from [43] (Fig. 4) and solid line is calculated by Bajaj and Coli [44] using the theory of Lee and Bajaj [25] with the same set of physical parameters as given in [21]. The dashed curve is due to Schubert et al. [21]. The difference between the measured values of s and those calculated using the theory of Lee and Bajaj [25] may be attributed to the effects of the presence of random electric fields due to ionized impurities and clustering in this alloy system. This system is expected to be highly compensated and may have ionized impurity concentration in the range of 1016 to 1017 cm3 depending on the value of x. This effect therefore can contribute a few meVs to the values of the excitonic linewidth. 4.4.2. InGaN During the past few years, Inx Ga1 x N alloys have been used extensively in the fabrication of a variety of emitters such as bright LEDs and laser diodes (LDs) with excellent results. However, the quality of this material system is still far from satisfactory. In particular, the growth of good quality Inx Ga1 x N alloys with x larger than 0.2 is still considered to be a challenging task. The large difference between the interplane distance between InN and GaN and the high volatility of In at elevated temperatures required to grow this material system pose major problems in growing high quality layers. A number of groups have studied the structural, electronic and optical properties of InGaN alloys. Radiative recombination at low temperatures is found to be due to localized excitons [45,46] and the excitonic linewidths are rather large (>50 meV). These alloys are generally inhomogeneous, have considerable clustering, and a high density of threading dislocations. Recently, O'Donnell et al. [47] have studied the variation of PL linewidth as a function In composition in Inx Ga1 x N from 15 K to room temperature. Their samples were grown by MOCVD on (0 0 0 1) oriented sapphire substrates and the In composition varied from 9 to 50%. At 15 K they find that the value of PL linewidth increases from 70 to 200 meV as the In composition is varied from 9 to 50%.

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The contribution to the excitonic linewidth due to compositional disorder in a perfectly random alloy is estimated to be less than 5 meV. To account for such large values of PL linewidths, they propose a model in which the strain induced varying piezoelectric fields separate the electrons and holes to sites of different energies, leading to an inhomogeneously broadened PL transition. The effect of these strain induced piezoelectric fields increases as the In concentration is increased thus leading to broader transitions. It should be pointed out several different models to explain the broad emission lines in InGaN alloys have been proposed. Discussion of their relative merits and demerits is beyond the scope of this article. 4.4.3. GaAsN Since the early work of Weyers and coworkers [48,49] there has been a considerable interest in the study of the electronic and optical properties of GaAsN and InGaAsN semiconductor alloy systems. These early reports, which have been verified by several other groups, showed that an addition of a small amount of nitrogen, e.g. about 2% in GaAs could cause a band-gap reduction approaching 0.4 eV. This ability to tune the band-gap energy with small addition of nitrogen makes it an attractive material system for opto-electronic devices operating in the 1.0±1.3 mm region. This large value of dEg/dx in GaAs1 x Nx alloys is responsible for the large values of s observed in these systems using PL spectroscopy at low temperatures. For instance, Jones et al. [50] have observed PL linewidths of about 14 and 22 meV in In0.07Ga0.93As1 x Nx alloys for x ˆ 0:01 and 0.02, respectively (Fig. 16). A small amount of In is added to GaAs1 x Nx to make it lattice match to readily available high quality GaAs substrates. As the mass parameters have not been measured in these alloys, the values of the reduced masses of excitons, however, can be determined from the measured values of using the formalism developed by Lee and Bajaj [25]. Such a determination has been made by Jones et al. [50] who have also studied the variation of s as a function of hydrostatic pressure up to 110 kbar. They find that the value of s increases as a function of pressure until about 100 kbar and then saturates. For instance, in In0.07Ga0.93As0.99N0.01 the value of s increases from 14 to 130 meV when

Fig. 16. Ambient pressure PL spectra at 4 K for 1% nitrogen (right trace and 2% nitrogen (left trace) in InGaAsN. The values of s are 14 meV (1% nitrogen) and 22 meV (2% nitrogen).

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the pressure is increased from its atmospheric value to 100 kbar. They, therefore, determine the variation of the exciton reduced mass as a function of pressure which agrees reasonably well with that obtained from first-principle band structure calculation using local density approximation. As far as we know, Jones et al. [50] are the first to determine the values of reduced masses of excitons from measured values of s in any semiconductor alloy system. It should be pointed out that the values of the exciton reduced mass thus determined are the upper limits. 4.5. II±VI semiconductors There has been a remarkable revival of interest in studying the electronic and optical properties of II±VI semiconductors and their alloys during the past decade. In addition to the growth of bulk systems based on these semiconductors, a variety of heterostructures, such as quantum wells, quantum wires and quantum dots have also been fabricated using MOCVD and MBE. The interest in studying these material systems is largely been motivated by their potential for applications in optoelectronics in the blue±green region of the spectrum. Remarkable progress has been made in growing high quality alloys and quantum well structures based on a variety of II±VI semiconductors. In the following, we briefly review the behavior of excitonic linewidth as a function of alloy composition in a few II±VI alloy systems. To the best of our knowledge the first observation of the variation of s as a function of composition in a semiconductor alloy was made in Znx Cd1 x S and Znx Cd1 x Te by Suslina et al. [28] using reflection spectroscopy. They found that at 4.2 K the value of s in Znx Cd1 x S increased as the composition x was increased and reached a maximum of 8.5 meV at x ˆ 0:5. Similar behavior was also found in Znx Cd1 x Te alloys where the maximum value of 3.5 meV was determined for x ˆ 0:5. The values of s in both alloy systems decrease as the value of x is increased beyond x ˆ 0:5. The value of s as a function of x was calculated using the theory developed by Baranovskii and Efros [27], described earlier, and a good agreement with the experimental data was found (Fig. 17a and b). As far as we know, a similar study of the variation of s with compositional disorder has not been carried out in other II±VI alloy systems such as Zn1 x Cdx Se, which has recently found applications in LEDs and LDs operating in the blue±green region of the spectrum. However, several groups have reported the values of s for specific values of x. For instance, Snoeks et al. [51] have measured the PL linewidth in Zn0.5Cd0.5Se layers grown on (0 0 1) InP substrates by MBE using an InP buffer

Fig. 17. Variation of excitonic linewidth E, measured in reflection, as a function of alloy composition x in (a) Znx Cd1 x S and (b) Znx Cd1 x Te at 4.2 K. Points are experimental data, solid curves are calculated using the theory of Baranovski and Efros [27].

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layer at 19 K. They find that the value of s is 6.7 meV when the sample was grown without the buffer layer and reduces to 5.2 meV when the buffer layer was used. Pelekanos et al. [52] have measured Ê wide Zn0.75Cd0.25Se/ZnSe quantum well at 10 K using absorption the value of s in a 200 A Ê ) is much spectroscopy and find it to be 5.0 meV. Since the value of the exciton Bohr radius (40 A smaller than the width of the well, this value of the exciton linewidth is almost entirely due to alloy compositional fluctuations. Similar results have also been reported by several other groups [53]. These values of s are comparable to those calculated using the theory of Lee and Bajaj [25]. 5. Excitonic line broadening in quantum wells There has been a remarkable progress in the development of crystal growth techniques such as MBE and MOCVD and their various variations during the past three decades, which has allowed us to grow high quality semiconductors and their heterostructures with abrupt changes in composition and/or doping characteristics. A variety of structures based on quantum wells, quantum wires and quantum dots have been fabricated and their structural, electronic and optical properties have been investigated in considerable detail. Many of these structures especially those based on quantum wells have found important applications in a variety of electronic and optical devices, such as HEMTs, lasers, spatial light modulators, photodetectors, LDEs, etc. The use of these devices in optoelectronics, for instance, has literally revolutionized this field during the past decade. All these structures involve interfaces between two different semiconductors and, thus the quality of the interfaces has clearly an important effect on the performance of these devices. Under appropriate set of conditions, which are often achieved in MBE and MOCVD, growth of a semiconductor takes place in a layer-by-layer growth mode. However, before the growth of a particular layer on the surface is completed, another layer starts growing on the previously partially filled layer. Now if a layer of a different semiconductor is grown, the interface will not be completely abrupt but will be characterized by a structural disorder on the atomic scale, which is often referred to as ``interface roughness''. Thus, the presence of interface roughness in semiconductor heterostructures such as quantum wells and quantum wires is inevitable. The challenge facing the crystal grower is to minimize this interface roughness and thus reduce its deleterious effect on the performance of electronic devices. To obtain a better insight into the nature of interface roughness let us consider the growth of GaAs/AlAs and GaAs/AlGaAs heterostructures along (0 0 1) direction by MBE. In this orientation cation and anion layers grow alternately. In the case of GaAs/AlAs heterostructure, an ideal interface will be one in which the Ga layer is full, followed by a full As layer on which a full Al layer is grown. This interface is therefore completely free of any disorder. In the case of GaAs/AlGaAs, however, the situation is somewhat different. An ideal interface will now consist of a full Ga layer followed by a full As layer on which a full AlGaAs layer is grown in which Ga and Al are distributed completely randomly. This interface is therefore not free of disorder but has minimum interface roughness. Any deviation from complete randomness will make it worse. A perfect interface in the case of a GaAs/AlGaAs heterostructure which is free of any disorder will consist of AlGaAs grown in the form of a digital alloy, i.e. a short period superlattice consisting of alternate layers GaAs and AlAs with perfect interfaces. It should be pointed out that ideal interfaces are rarely, if ever, realized in actual practice due to the inherent random nature of the growth process in MBE and MOCVD. Almost 20 years ago, Weisbuch et al. [54] were the first to point out that in GaAs/AlGaAs quantum well structures, the topological disorder present at the interfaces has a significant effect on their optical properties and that the optical techniques such as PL and PLE provide useful means to

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characterize this disorder, complementary to the more microscopic methods of transmission electron microscopy (TEM) and X-ray diffraction. In high quality multiple quantum well structures where the low temperature PL is primarily due to intrinsic excitonic transitions, the width of these transitions is controlled mainly by two effects. 1. Layer-to-layer thickness variation in the sample which leads to different confinement energies of electrons and holes in different layers. 2. Thickness fluctuations within each layer, all the layers having the same average thickness. These fluctuations arise, as mentioned earlier, due to the fact that after a layer of GaAs or AlGaAs has been grown, there exists a number of islands at the free surface because the number of atomic planes in a layer is never an integer. The next layer grown will freeze those islands so that the microscopic interface position cannot be defined to better than 1 ML (half the lattice constant), although the microscopic average can be defined more accurately. 5.1. Theoretical formalism In the following, we will consider the effect of the thickness fluctuations within a given layer on the excitonic line broadening in quantum well structures. Though we will focus our attention on GaAs/AlGaAs structures for the purposes of illustration, the results discussed in this section are equally applicable to other similar III±V and II±VI systems. We will first review briefly the results of various calculations of excitonic linewidths in quantum wells. We will then examine the results of experimental measurements in the light of insights gained from theoretical investigations. As discussed earlier, the energies of electrons and holes, confined in quantum wells, are quantized in the direction of growth and depend strongly on the well width. For instance, in the case of a quantum well with infinite potential barriers the energies of the electrons (holes) are given as (Eq. (17)) En ˆ

2 p2 n2 h 2me mh L2

A small change in the well size dL produces a change in energy as dEn ˆ

2 p2 n2 dL h 2me mh L3

(69)

In an actual quantum well, the positions of the heterointerfaces in the z-direction vary statistically along x and y, and give rise to a distribution of well widths which can be described by a Gaussian function. This distribution of the well widths is reflected in the distribution of the emission energy and the full width at half maximum (s) of the recombination line is related to the full width at half maximum (DL) as sˆ

2 p2 DL h mL3

(70)

where m is the reduced mass of the exciton. An important objective of any theoretical effort, therefore, is to calculate DL and relate it to s using a topological model of the heterointerfaces. After the early work of Weisbuch et al. [54] and Singh et al. [55] developed a theory correlating the linewidths of excitonic luminescence spectra in quantum wells with the microscopic details of

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the structure of the interfaces. This theory was based on the realization that the optical probe, Ê in GaAs, for instance, and the energy of the namely, an exciton has effectively a finite extent 250 A emitted radiation reflects the average well width of the region seen by the exciton. Using arguments similar to those invoked by Lifshitz [20] to understand the spectra of disordered alloys, Singh et al. [55] calculated the probability distribution of the compositional fluctuations at the interface. In other words they maximize the entropy of mixing of the structural components of a rough interface. This procedure is very similar to that followed by Singh and Bajaj [19] to calculate the excitonic linewidth in random alloys. In the following, we outline briefly the essential features of the formalism developed by Singh et al. [55]. To discuss the model for the interface, let us consider the most studied quantum well structure, i.e. GaAs/AlGaAs. The nature of the compositional disorder present at the interfaces depends on the growth conditions and the crystal growth technique used as well as the physical and chemical properties of the two semiconductors involved. In a given crystal growth technique conditions vary from run to run and often during the same run as all the parameters that characterize the growth conditions are not accurately measurable. Strictly speaking there are no such structures as those grown under ``identical conditions,'' due to inherent random nature of the growth processes in MBE and MOCVD. In principle, every growth run for a quantum well structure produces a unique structural disorder at each interface. To gain some physical insight and to make the problem tractable, we shall assume a relatively simple model for the structural disorder at the interfaces, which we describe in the following. Depending on the growth conditions there will be localized fluctuations in the well size around a mean value L0. These fluctuations arise for example, if the growth is not perfectly two-dimensional. In such a case an interface is not a step function but has a certain amount of diffusiveness. If z ˆ 0 represents an ideal abrupt interface between GaAs and AlGaAs, in the real interface one may find that in the region z < 0 there are small islands of AlGaAs and in the region z > 0, small islands of GaAs. In addition, there can be fluctuations in the average composition of Al from one part of the interface to another, which also leads to non-ideal interfaces. We shall assume for the sake of simplicity that such is not the case. We now assume that a non-ideal interface can be represented by an average interface at z ˆ 0 with the fluctuations extending to a distance  d1 from the interface. These fluctuations, of course, arise from the presence of islands of AlGaAs in the well and the islands GaAs in the barrier region. We further suppose that the correlated lateral radius of these islands (assumed circular again for the sake of simplicity) is d2, i.e. the smallest island has a radius d2, larger islands have radii which are integral multiples of d2. This is an approximation to the structure of a real interface which is much more complicated and consists of islands of different shapes and sizes which are determined by the growth conditions. However, it is hoped that this model provides a reasonable approximation to the actual disorder present at the interface. Based on this model, we view the interface as being described on a global scale by parameters Ca0 ; Cb0 and Cc0 representing the mean concentration of islands protruding into the well, islands projecting out of the well (valleys) and regions that are flat, respectively. As mentioned earlier, the luminescence spectrum at low temperature arises as a result of radiative recombination of an exciton and the emission energy depends on the mean value of the width of the quantum well as seen by the exciton namely. *

*

*

Eex … r ; L† ˆ Ee … r ; L† ‡ Eh … r ; L† ‡ Eg *

*

Eb … r ; L†

(71)

where r denotes the center of mass of the exciton and Eh refers to either the heavy- or light-hole exciton. The value of Eb is quite insensitive to small fluctuations in well size. The values of Ee and Eh

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as pointed out earlier, are quite sensitive to the changes in well size, particularly for narrow wells. The exciton wave function extends over an effective region of radius say, R (to be defined later) so that the relevant information regarding the quantum well corresponds to that part of the well with lateral extent R and centered around the position f of the exciton. The average width of the well * around the point r will be given by the average microscopic nature of the interface. If it is assumed that the excitons are created randomly in the well, the emission line shape is determined by the probability distribution P(R, L) of finding fluctuations in the well size extending over the effective exciton size. To determine this probability distribution required to calculate the excitonic line shape, Singh et al. [55] as mentioned earlier, used the statistical thermodynamic arguments due to Lifshitz [20]. They assumed that the islands, valleys and flat regions occur randomly at the interfaces and that the radii of the smallest clusters associated with islands, valleys and flat regions are d2a, d2b and d2c, respectively. At a given interface the probability of finding fluctuation concentration Ca, Cb and Cc over a region of effective radius R when the global concentrations are Ca0 ; Cb0 and Cc0 is given as [55] ( " P…Ca ; Cb ; Cc ; R† ˆ exp

     # ) R2 Ca R2 Cb R2 Cc Ca ln 0 ‡ 2 Cb ln 0 ‡ 2 Cc ln 0 2 Ca Cc Cb d2a d2b d2c

(72)

A similar expression is obtained for the other interface of a quantum well, where the values of average global concentrations Ca0 , etc. and radii of islands d2a, etc. may be quire different. If we assume that the two interfaces of a quantum well are not correlated in terms of their growth, which is likely to be the case if the quantum well is not too narrow (well size > 10 MLs), then the total probability of the well size fluctuations will be given by the product of the probability distributions of the two interfaces. In general, the quality of the two interfaces is often quite different. However, to simplify our discussion, we can consider two extreme cases for the nature of the two interfaces. In the first case, we assume that the quality of well is determined primarily by the quality of only one interface. This is often the case for GaAs/AlGaAs quantum wells where the inverted interface (GaAs grown on AlGaAs) is of inferior quality than the normal interface (AlGaAs grown on GaAs). In the second case, we assume that the two interfaces are of comparable quality. For the first situation, the probability distribution is determined using Eq. (72) with the effective well size defined as W ˆ W0 ‡ d1 ‰…Ca

Ca0 †

…Cb

Cb0 †Š

(73)

In the second case, R2 and d1 in Eq. (72) are replaced by 2R2 and 2d1, respectively. To evaluate probability distribution, Eq. (72), R2 is replaced by the expectation value of 2 r ˆ x2 ‡ y2 using the variational wave function of Greene and Bajaj [14] who have calculated the binding energies of heavy- and light-hole excitons as a function of well width as discussed earlier. Combining Eqs. (72) and (73) one can determine the excitonic linewidth as a function of well size. The probability distribution of the excitonic transition energy is related to probability distribution of the effective well size as P…EgQW …L0 †; EgQW …L†† dEgQW …L0 † ˆ P…L0 ; L† dL0

(74)

where EgQW …L0 † ˆ Eg ‡ Ee …L0 † ‡ Eh …L0 †

(75)

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This leads to a Gaussian function for the excitonic broadening " # EgQW …L0 † EgQW …L† 1 QW QW P…Eg …L0 †; Eg …L†† ˆ p exp 2s2E 2psE with the standard deviation @EQW …L0 † g sE ˆ dL @…L0 †

(76)

(77)

L; 0

where dL is the standard deviation of the probability distribution P(L0, L) of the effective well width. The excitonic linewidth is related to s ˆ 2:36sE

(78)

Using the above described formalism Singh et al. [55] have calculated the variation of s as a function of well width L0 for a heavy-hole exciton in GaAs/Al0.3Ga0.7As quantum well structures. They choose, for the sake of illustration, a simple model in which it is assumed that the interface is made up of valleys and hills only with Ca0 ˆ Cb0 ˆ 0:5 and d2a ˆ d2b ˆ d2 and take d1 equal to Ê ). In Fig. 18a, we display the variation s of a heavy-hole thickness of 1 ML of GaAs (i.e. 2.83 A exciton as a function of well width L0 for three different values of d2. We note that for a given value of L0 the value of s increase as a function of d2. In addition, for a given value of d2 the value of s increases as the well size is reduced due to larger changes in the values of Ee and Eh. These results, of

Ê (solid line), Fig. 18. Variation of the linewidth s of a heavy-hole exciton as a function of well width (L0) for d2 ˆ 20 A Ê (dotted line), and d2 ˆ 160 A Ê (dashed line) in GaAs/Al0.3Ga0.7As quantum wells for (a) B ˆ 0 and (b) d2 ˆ 80 A Ê ). B ˆ 75 kG. The value of d1 is assumed to be 1 ML (2.83 A

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course correspond to the situation where the excitons are produced uniformly in the quantum well and do not migrate to region of lower energies (i.e. wider regions of the well). It should be pointed out that as the value of d2 becomes significantly larger than the value of hr2i, the excitons experience flat surfaces and, therefore, the emission lines originating from different regions of the quantum well become sharp again. Thus, in the limit of very small (micro roughness) and very large values of d2 compared with the effective size of the exciton, the emission lines are sharp. In the case of large values of d2, there are more than one emission lines associated with a free exciton. The latter situation is often realized when the quantum wells are grown with growth interruption either at one or at both interfaces and will be discussed in the latter part of this article. 5.1.1. Excitonic linewidth in electric and magnetic fields In this section, we shall discuss the effects of the applied electric and magnetic fields on the values of the excitonic linewidth in quantum well structures. We shall see that the application of these fields affects the excitonic wave function in such a way as to enhance the values of the excitonic linewidth. It is well known that in a quantum well structure the application of an electric field parallel to the direction of growth reduces the effective band-gap and thus the value of the exciton absorption or emission energy. This effect is commonly referred to as quantum confined Stark effect (QCSE) and is exploited in an opto-electronic device called the spatial light modulator in which one modifies the transmission intensity of a signal beam coming in at a fixed wavelength by applying an appropriate varying electric field thus producing a high speed optical modulation. One finds that the depth of the modulation, an important parameter characterizing this device, depends on the excitonic linewidth and its variation as a function of the electric field. It is, therefore, of interest to calculate the effect of the electric field on the excitonic linewidth in quantum well structures. Hong and Singh [56] in fact performed such a calculation using a model for the interface described in Section 5.1. The application of the transverse electric field forces the electron and the hole of the exciton to move away from each other and closer to the two interfaces of the quantum well. The exciton, therefore, experiences the effect of the roughness at the two interfaces more strongly which leads to an enhancement of the excitonic linewidth. Using the formalism for the calculation of s Ê described in Section 5.1, they find that in a GaAs/Al0.3Ga0.7As quantum well structure for a 100 A thick well, the linewidth increases from 1.2 to 1.9 meV when the electric field is increased from 0 to Ê and d2 ˆ 150 A Ê . However, for a very small island size, namely, 100 kV/cm; for d1 ˆ 2:83 A Ê d2 ˆ 20 A, there is practically no change in linewidth as a function of electric field as expected. For further details the reader is referred to their work [56]. As discussed in the Section 5.1, changing the size of the well modifies the effective size of the exciton. It was pointed out by Singh et al. [55] that a more convenient approach to changing the effective exciton size would be the application of a magnetic field, say along the growth direction. Application of a magnetic field significantly modifies the excitonic wave function so that the study of the linewidth as a function of magnetic field for a given quantum well size may yield useful information concerning the microscopic structural details of the interface. Some time ago Lee et al. [57] calculated the excitonic linewidth in quantum wells in the presence of a magnetic field applied parallel to the growth direction using the model for the interface as described earlier. The probability function in Eq. (72) is now obtained by replacing R2 by hr2i calculated using exciton wave function in a magnetic field [57]. The value of hr2i decreases as a function of magnetic field and, therefore, the excitonic linewidth increases as the magnetic field is increased. In Fig. 18b, we display the variation of s for a heavy-hole exciton as a function of well Ê as the value of d1. We find that for a given value of size for a magnetic field B ˆ 75 kG using 2.83 A L0 and d2 the value of s increases as a function of the magnetic field as expected. For instance, for

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Fig. 19. Variation of the linewidth (s) of a heavy-hole exciton as a function of well width (Lz) in a GaAs/Al0.3Ga0.7As quantum well for several values of the electric (F ) and magnetic fields (B).

Ê and d2 ˆ 80 A Ê , the value of s at B ˆ 75 kG is almost 50% larger than its value at L0 ˆ 100 A zero field. Similar results are also obtained in lattice matched quantum well structures such as InGaAs/InP [57]. A few years ago, Bajaj and Lee [58] developed a formalism to calculate the linewidths of excitonic transitions in semiconductor quantum well structures with arbitrary potential profiles in the presence of electric an magnetic fields. They assume that at low temperatures the dominant mechanism responsible for the line broadening is the interface roughness, which is always present even in the structures with so-called perfect interfaces. As mentioned earlier, in the case of GaAs/ AlGaAs quantum well structures for instance, the AlGaAs layers at the interfaces have a completely random distribution of Al and Ga atoms in the ideal material and will always lead to excitonic line broadening. They generalized the formalism which they had developed earlier to calculate the excitonic linewidths in semiconductor alloys due to compositional disorder [25] and applied it to the calculation of the excitonic linewidth in quantum well structures. In particular, they have calculated the variation of s of a heavy-hole exciton as a function of well size and electric and magnetic fields in GaAs/Al0.3Ga0.7As quantum wells structures with three different potential well profiles, namely, rectangular, parabolic and asymmetric triangular. In rectangular quantum wells, the effects of broadening arise from the compositional disorder present at the interfaces and in the barrier alloy. They assume abrupt interfaces and calculate the line broadening using exciton wave functions appropriate for this quantum well structure in the presence of electric or/and magnetic fields. In Fig. 19, we display the variation of s of a heavy-hole exciton as a function of well size in rectangular (square), GaAs/Al0.3Ga0.7As quantum well structures. The value of s increases as the well size is reduced, as expected. It should be pointed out that these are the lowest values of s possible in these structures. 5.1.2. Excitonic line broadening due to combined effects of alloy and interfacial disorders In the first part of this review, we have described the theoretical and experimental investigations of the behavior of excitonic linewidth in semiconductor alloys. In Section 5.1.1, we have reviewed briefly the results of the calculations of the effect of interface roughness on the values of the excitonic linewidths in quantum wells in which the well is composed of a binary semiconductor. This approach describes the situation for wide wells and those with large conduction and valence band offsets, such that the excitonic wave function is completely confined in the well. In the case of

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narrow wells, however, a significant part of the excitonic wave function resides in the barrier. Therefore, both the compositional disorder in the barrier alloy and the interface roughness affect the values of the excitonic linewidth in the narrow quantum wells of GaAs/AlGaAs or similar structures where the well is composed of a binary semiconductor and the barrier is a semiconductor alloy. In addition, in quantum well structures such as InGaAs/InP and InGaAs/InAlAs where the alloy constitutes the well material and the binary or an alloy forms a barrier the values of the excitonic linewidths are determined both by the interface roughness and the compositional disorder present in the alloy. In the following, we briefly describe a formalism which takes into account both the effects of the compositional disorder in alloys and the interfacial quality on the excitonic linewidth in quantum well structures as developed by Singh and Bajaj [59]. We first consider the contribution of the effect of the compositional disorder in the alloy on the excitonic linewidth in three different varieties of quantum well structures. Case 1. As a ®rst case, we consider the situation where the barrier region is formed by an alloy and the well region is a single component material. This case would include quantum wells such as GaAs/AlGaAs, ZnSe/ZnSeS, etc. To calculate the contribution of the alloy broadening in this case, we note that in a quantum well only a certain fraction Pex 0 of a given heavy-hole or a light-hole exciton is outside the well. Thus, in Eq. (36) the volume over which the ¯uctuation x0 has to be calculated contains not R3ex but R3ex P ex 0 . Following the procedure used in Section 2.1, Singh and Bajaj [59] obtain the following expression of excitonic linewidth in a quantum well saw due to the effect of the alloy disorder: saw ˆ saB

Dw 1 ex D1 …P 0 †1=2

(79)

where saB is the excitonic linewidth in the bulk alloy forming the barrier, Dw is the variation of the excitonic emission energy as a function of barrier height and is defined as dE eh (80) Dw ˆ dCa x where Ehe ˆ Ee ‡ Eh ‡ E w g

Eb

(81)

For wider wells …L > Rex †; P ex 0 goes to 0 but Dw/D1 approaches 0 faster, leading to a negligible contribution to the linewidth form the barrier alloy. When L ! 0 P ex 0 ! 1 and Dw ! D1 so that saw ˆ saB

(82)

Case 2. Here an alloy forms the well region and the barrier consists of a single component semiconductor; for example, InGaAs/InP. In this case the situation is reverse of that in Case 1 Pex and the excitonic linewidth is obtained by replacing Pex 0 by 1 0 in Eq. (79) so that saw ˆ s aB

Dw 1 p D1 1 P ex 0

(83)

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Case 3. Here both the well and the barrier regions are formed by alloys and the linewidth is, therefore, determined by alloy disorder in two regions and can be expressed as q (84) saw ˆ s21 ‡ s22 where s1 ˆ saB1 ˆ

Dw 1 p D1 Pex 0

(85a)

s2 ˆ saB2 ˆ

Dw 1 p D1 1 Pex 0

(85b)

and

Here saB1 and saB2 are the values of the excitonic linewidths in the alloy forming the barrier and the quantum well, respectively. It is clear that in all three cases, the quality of the alloy plays an important role in determining the linewidth of the exciton spectra. To determine the value of the excitonic linewidth s in a quantum well due to alloy disorder and interface roughness, we combine these as sˆ

q s2a ‡ s2i

(86)

where sa is the value of the excitonic linewidth due to alloy disorder and si is that due to the interface roughness. We now use the results derived above to calculate the excitonic PL linewidths in three representative lattice matched quantum well structures, namely, GaAs/Al0.3Ga0.7As, In0.53Ga0.47As/ InP and In0.53Ga0.47As/In0.53Al0.47As. The results for the GaAs/Al0.3Ga0.7As quantum well are shown in Fig. 20, where we have plotted the contributions to the linewidth from interface roughness and alloy disorder. Curve a1 represents the shift in the exciton emission energy when the well size W 0 changes by 1 ML. This shift would correspond to the linewidth if the lateral dimensions of the island were comparable to the exciton size. Also shown in Fig. 20 is the linewidth when the lateral Ê (curve a2). The value of s is now much smaller as dimension of the island (d ˆ d2a ˆ d2b ) is 20 A the exciton is unable to see the individual islands at the interface. The effect of the alloy disorder is shown by curve b, which displays the variation of saw =saB as a function of well size. Here saB is the Ê ) the linewidth in Al0.3Ga0.7As. As clearly demonstrated in Fig. 20, for very narrow wells (W0  20 A effect of interface roughness on the linewidth starts to decrease, while the linewidth due to alloy disorder continues to grow. Also, if the interfaces are dominated by very small islands, the alloy disorder in the barrier becomes the dominant source of line broadening in the quantum well structure. Fig. 21 displays results for a In0.53Ga0.47As/InP quantum well structure. As in Fig. 20, curve a1 represents an excitonic linewidth when the well size changes by 1 ML and (d ˆ d2a ˆ d2b ) Ê . In this case, the is about Rex while curve a2 represents the case when d1 ˆ 1 ML and d2 ˆ 20 A effect of alloy disorder on the excitonic linewidth increases with increasing well width. Finally, in Fig. 22, we show results for In0.53Ga0.47As/In0.53Al0.47As quantum well structures. Again curves a1, Ê ), the alloy disorder in the a2 and b have the same meaning as before. For very narrow wells (<20 A barrier makes a dominant contribution to the line broadening and the linewidth approaches saB1 when Ê ) the linewidth approaches sa . W0 ! 0. On the other hand, for large well sizes (>100 A B2

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Fig. 20. Variation of the linewidth (s) of a heavy-hole exciton as a function of well size in GaAs/Al0.3Ga0.7As quantum Ê) well structures. Curve (a1) represents shift in the excitonic transition energy when the well size changes by 1 ML (2.83 A Ê and d2 ˆ 20 A Ê . Curve (b2) is the and d2 is exciton radius; curve (a2) is the shift in transition energy when d1 ˆ 2:83 A linewidth due to alloy disorder and is represented by saw =saB.

Fig. 21. Variation of the linewidth (s) of a heavy-hole exciton as a function of well size in InGaAs/InP quantum well structures. The various symbols have the same meanings as in Fig. 20.

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Fig. 22. Variation of the linewidth (s) of a heavy-hole exciton as a function of well size in InGaAs/InAlAs quantum well structures. The various symbols have the same meanings as in Fig. 20.

6. Experimental studies During the past 25 years, a large number of papers have been published on the optical properties of quantum well structures based on a variety of III±V and II±VI semiconductors. A significant portion of these papers deals with the behavior of excitonic linewidth as a function of several parameters, such as growth conditions, well width, etc. Due to the large amount of published material available in this area, we shall confine ourselves to reviewing only those papers which we believe illustrate the basic physical mechanisms responsible for the dependence of the excitonic linewidth on the quality of the interfaces as well as that of the constituent semiconductors and their alloys. An excellent review of the experimental work in this area till the middle of 1989 has been given in [60]. We shall, therefore, focus more on the discussion of the results of studies that have been performed during the past 10 years. 6.1. GaAs/AlGaAs quantum wells In this section, we shall review briefly the results of the studies of the behavior of excitonic linewidth s in the most extensively studied quantum well structure, i.e. GaAs/AlGaAs as a function of growth conditions and well width. In the first part, we shall consider the case in which no growth interruption is used and in the second, we shall discuss the effect of growth interruption on the behavior of excitonic linewidth. As we shall see a great deal of qualitative and sometimes quantitative information concerning the interfacial structure has been gleaned from these studies. As far as we know Weisbuch et al. [54] were the first to report the results of an extensive study of the influence of interfacial quality on interband absorption, PL and PLE spectra in a range of GaAs/AlGaAs multi-quantum wells grown by MBE on (0 0 1) oriented GaAs substrates. In order to investigate the effect of the substrate temperature Ts on the optical properties of these quantum wells,

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Ê thick wells. These structures were grown with Ts varying they grew MQW structures with 200 A from 570 to 7308C and their optical properties were studied using PL and PLE spectroscopies at 2 K. They find that the value of s as measured by PLE decreases as the substrate temperature is increased and reaches a minimum value of about 1 meV for T s ˆ 6908C (from 5 meV for T ˆ 5708C). In addition, the PL efficiency increases by a factor of about 100 when Ts is varied from 570 to 6908C. They, therefore, suggest that an ideal substrate temperature at which GaAs/AlGaAs MQW structures should be grown is about 6908C in their MBE set up. They then grew a series of seven samples with Ê sequentially without opening the MBE chamber at GaAs well thicknesses ranging from 327 to 51 A Ê . They study the optical properties of these 6908C. The AlGaAs barrier width was kept at 200 A samples using PL and PLE at 2 K. They find that the layer to layer average thickness varies only by about one tenth of a monolayer. In addition they find that the excitonic linewidth, as expected, increases with decreasing well width. Based on their observations they propose a model of the Ê in lateral dimension. They interface, which consists of islands of 1 ML in height and about 300 A also suggest that in samples grown at 5708C the island heights can be as much as 5 MLs. Following the pioneering work of Weisbuch et al. [54] several groups have measured the excitonic linewidth in GaAs/AlGaAs single and multiple quantum well structures grown by MBE and MOCVD using PL and PLE spectroscopies. For instance, Reynolds et al. [61] have reported the observation of very sharp PL spectra associated with both the intrinsic and the extrinsic excitonic transitions in GaAs/Al0.25Ga0.75As multi-quantum well structures at 1.6 K. Their samples were Ê . The width of the barrier was kept at grown by MBE and varied in well width from 100 to 400 A Ê . Transitions as sharp as 0.15 meV associated with free heavy-hole excitons were reported, thus 100 A indicating an extremely high quality of the material and interfaces. Reynolds et al., however, do not report a systematic study of the variation of excitonic linewidth as a function of well width. Later Reynolds et al. [62] reported low temperature (1.6 K) high resolution PL studies in six GaAs/Al0.25Ga0.75As multi-quantum well structures grown by MBE. In their samples the well width Ê while the barrier width was kept at 100 A Ê . Transitions associated with free varied from 87 to 192 A heavy-hole excitons and heavy-hole excitons bound to neutral donors were clearly observed and were very well resolved. Each of these transitions exhibited fine structure and were very sharp, some Ê well for instance. The observed fine structure namely, the energy as narrow as 0.1 meV in 192 A separation between the individual components was explained in terms of changes in the average well width which are integral multiples of an effective 0.5 ML. The sharpness of the individual peaks clearly indicates that most of the islands present at the interfaces have sizes, which are an order of magnitude smaller than the exciton diameter. They also demonstrate that the energy separation between the individual peaks scales with the well sizes, namely, the energy separation decreases as the well size is increased, as expected. They propose a tentative model for the interfacial structure in which the Al concentration varies from 25 to 0% over 2 MLs. It should be mentioned that no growth interruption was used in the fabrication of these structures. Juang et al. [63] have measured the values of s of heavy-hole excitons in single quantum wells Ê thick GaAs layers and Al0.3Ga0.7As barriers as well as those with GaAs/Al0.3Ga0.7As with 120 A superlattice barriers using PL spectroscopy at 2 K. Their samples were grown by MBE on (0 0 1) Ê GaAs substrates at 6308C. Two sets of samples were grown. One consisted of a single 126 A Ê GaAs and 43 A Ê Al0.3Ga0.7As as GaAs well with a 20 period undoped superlattice consisting of 30 A a barrier. In the second structure, the superlattice barrier was replaced by Al0.3Ga0.7As alloy. In the first set of samples, they measure 0.3 meV as the value of s where as in the second set the value of s was about 0.8 meV. These values represent almost the theoretically calculated limit for very sharp interfaces and are comparable to the best results published in samples without growth interruption.

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Table 1 Experimental FWHM data from a series of GaAs/Al0.38Ga0.72As quantum wells measured using PL (PLhh) and PLE (PLEhh and PLElh) spectroscopiesa Ê) Calculated QW width (A 5 15 20 48 53 68 93 a

FWHM (meV) PLhh

PLEhh

PLElh

4.4 11.2 10.9 6 6.4 4.9 2.8

7.6 18.1 15.5 5.8 9.7 4.9 3.6

8.8 19.5 23.6 6.6 7.6 4.8 3.4

Label hh and lh are heavy-hole and light-hole excitons.

Bertlot et al. [64] have studied the behavior of the excitonic linewidth as a function well size in two different quantum well systems, namely, GaAs/AlGaAs and InGaAs/GaAs at several temperatures (80, 8 and 2 K) using PL and PLE spectroscopies. Their samples were grown by atmospheric MOCVD and in the case of GaAs/Alx Ga1 x As system they study quantum well structures with two different Al concentrations x ˆ 0:17 and 0.38 in the barriers. In a GaAs/ Ê and in a GaAs/ Al0.17Ga0.83As system they grew four different well sizes, i.e. 7, 23, 56 and 83 A Ê Al0.38Ga0.62As system the four different well sizes were 5, 10, 15 and 20 A. They find that the value of the linewidth of the heavy-hole exciton, for instance, increases as the well size is reduced, reaches a maximum and then decreases in accordance with the theoretical predictions of Singh and Bajaj [59] (Table 1). Similar behavior is also found for the light-hole exciton as well as for excitons in InGaAs/GaAs quantum wells. Their results are shown in Fig. 23. As far as we know this is the first reported observation of the theoretically predicted PL linewidth narrowing in very thin quantum wells in these two material systems. Srinivas et al. [65] have investigated the variation of s of free excitons as a function of well width in a series of extremely high quality GaAs/Al0.3Ga0.7As quantum wells using PL spectroscopy. Their samples were grown by MBE and the well width was Ê . They find that their values of s measured at 8 K vary from 0.125 to 3 meV varied from 20 to 325 A Ê . These are among the lowest values of s ever reported in as the well size is varied from 325 to 20 A

Fig. 23. Variation of FWHM of a heavy-hole exciton transition as a function of well width in In0.12Ga0.8As/GaAs quantum Ê (0.5 ML) and d2 ˆ 100 A Ê. well. The solid line is obtained using the theory of Singh et al. [55] assuming d1 ˆ 1:42 A

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such a large range of quantum well widths. The interfaces in their quantum wells are obviously of very high quality and are clearly dominated by small island sizes. In an effort to improve the quality of hetero-interfaces in GaAs/AlGaAs quantum wells, several groups have grown these structures on GaAs substrates, which are differently oriented than commonly used (0 0 1) direction. The values of s in GaAs/AlGaAs quantum wells grown on GaAs(3 1 1) A, (3 1 1) B, (1 1 1) B and (1 1 0) substrates are generally larger than those measured in high quality structures grown on (0 0 1) substrates. During the past several years, Shimomura and his coworkers in a series of papers [66±70] have investigated the quality of interfaces in GaAs/Alx Ga1 x As quantum wells grown on (4 1 1) A oriented GaAs substrates by MBE using low temperature (4.2 K) PL spectroscopy. The Al composition x in their samples was varied from 0.3 to 1.0 and the well widths ranged from 23 to Ê . They grow their samples both on (4 1 1) A and (0 0 1) oriented GaAs substrates under 115 A nominally identical conditions. For instance, in the case of GaAs/Al0.3Ga6.7As quantum wells the Ê in samples values of s increase from 1.25 to 5.4 meV as the well width is decreased from 115 to 23 A grown on (4 1 1) A oriented substrates (Fig. 24). The values of s in samples grown on (0 0 1) oriented substrates however, are at least twice as large, thus indicating that the quality of the interfaces in samples grown on (4 1 1) A oriented substrates is considerably superior to that in samples grown on (0 0 1) substrates. Similar results are also obtained in quantum wells with x ˆ 0:7 and 1.0. They only observe a single sharp excitonic transition in each quantum well whose width does not change over a macroscopic area of about 1 cm  1 cm which is comparable to the contact area of tunnel transistors and optical modulators. The use of these quantum well structures in the design of these devices can lead to a greatly improved performance. They suggest that the reason for the sharp PL transitions may be mainly due to the enhanced two-dimensional growth because of larger migration length of Ga atoms and the layer growth by the step-flow mode on the (4 1 1) A GaAs surface. The interfaces in their samples are dominated by the presence of very small islands. Shimomura et al. do not report the impurity concentrations in their wells and hence their contributions to the excitonic broadening is not known. It is clear from the foregoing discussion that an enormous progress has been made in fabricating very high quality interfaces in GaAs/AlGaAs based heterostructures during the past three decades

Fig. 24. Photoluminescence spectra (4.2 K) from GaAs/AlGaAs quantum wells; (a) grown on (1 0 0) oriented substrates and (b) grown on (4 1 1) substrates.

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using both MBE and MOCVD growth techniques. It is now possible to grow structures with 1 ML abruptness and with interfaces which are dominated by very small islands in this material system. As far as we know Aksenov et al. [71] were the first to investigate the behavior of s as a function of applied magnetic field in a single GaAs/Al0.23Ga0.77As quantum well using PL spectroscopy at 6 K. Their sample was grown on Cr doped (0 0 1) oriented GaAs substrate using Ê thick GaAs layer sandwiched between 500 A Ê thick Al0.23Ga0.77As MBE and consisted of 170 A barriers. The magnetic field was applied parallel to the growth direction and was varied from 0 to 6 T. The observed lineshape of the excitonic transition consists of convolution of two distribution functions: (1) Lorentzian contribution with full width at half maximum G0 arising from the final state interactions and (2) the Gaussian contribution s arising from the potential fluctuations (interface roughness) of the quantum well interfaces. They find that the value of G0 decreases with magnetic field, whereas the value of s first decrease slightly and then increases with magnetic field, varying from 0.24 to 0.39 meV as the magnetic field is increased from 0 to 6 T. They use the theoretical Ê) model developed by Singh and Bajaj [59] to analyze their data and find that for d1 ˆ 1 ML (2.83 A Ê the value of the lateral dimension of the interface island to be about 100 A. They find a small difference between the experimental data and the calculated values and they attributed this to the simplifying assumptions used in the model of Singh and Bajaj [59]. As far as we know such a study has not been reported in any other quantum well system. As discussed in Section 5.1.1 Bajaj and Lee [58] and Singh and Bajaj [59] have calculated the variation of s as a function of magnetic field in quantum well structures using two different approaches. Apparently Aksenov et al. [71] were not aware of this work. The variation of s with magnetic field they observed is very close to that calculated by those two groups. 6.2. InGaAs/InP quantum wells In this section, we shall briefly review the results of the studies of the behavior of excitonic linewidth s in Inx Ga1 x As/InP quantum well structures as a function of growth conditions and well width. We shall first consider the case where Inx Ga1 x As (x ˆ 0:53) is lattice matched to InP and then discuss the coherently strained structures where values of x differ from 0.53. A great deal of effort has been devoted to the study of the electronic and optical properties of lattice matched In0.53Ga0.47As/InP (from hereon referred to InGaAs/InP) quantum well structures during the past 25 years. Several groups have investigated the behavior of s as a function of well width and growth conditions in these structures, which were grown using a variety of epitaxial growth techniques. In the following, we shall only review those investigations which have reported sharp excitonic transitions in high quality quantum well structures. Tsang and Schubert [72] were the first to report very sharp excitonic transitions in InGaAs/InP quantum wells. Their samples were grown by CBE on InP substrates and the widths of the single Ê . The width of the InP wells grown in a multi-quantum well structure were varied from 6 to 150 A Ê barrier layer was kept at 700 A. Their PL measurements were made at 2 K and they observed a single sharp excitonic peak from each well (Fig. 25). The values of s varied from 2.7 to 8.3 meV as the well Ê . These are among the lowest values of s ever reported in this width was reduced from 150 to 10 A quantum well system. The values of s measured earlier by other groups in InGaAs/InP quantum wells were larger and are discussed in [72]. Kamei and Hayashi [73] have measured the variation of s as a function of well width in InGaAs/InP quantum well structures using PL spectroscopy at 2 K. Their samples were grown by Ê . The values of s they measure vary from 2.4 MOCVD and the well width was varied from 6 to 100 A Ê . These values of s are slightly less than to 10.0 meV as the well width is decreased from 100 to 6 A

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Ê thick InP Fig. 25. Typical PL spectrum from a stack of quantum wells with different thicknesses separated by 700 A barriers at 2 K. The pumping power is 1 mW and the pumping area is about 50 mm in diameter.

those reported by Tsang and Schubert [72] and are the lowest ever reported in this quantum well system. It is clear that the interfaces of the quantum wells grown by CBE by Tsang and Schubert [72] and those grown by Kamei and Hayashi [73] by MOCVD are dominated by very small islands Ê ) without any steps. (<20 A It should be mentioned that a number of groups have studied the optical properties of coherently strained InGaAs/InP quantum well structures using PL, PLE and absorption techniques. The values of the excitonic linewidths in these structures are larger than those measured in lattice matched systems and will not be reviewed here. 6.3. Inx Ga1 x As/GaAs quantum wells We shall now briefly review the results of a few experimental studies of the behavior of s in coherently strained Inx Ga1 x As/GaAs quantum well structures as a function well width. The lattice mismatch between InAs and GaAs is about 7% and, therefore, the maximum value of the width of coherently strained quantum wells is determined by the indium composition x, which therefore, limits the range of the values of the well widths that can be studied. As far as we know the first systematic study of the variation of s as a function of well width in Inx Ga1 x As/GaAs quantum wells using PL spectroscopy at 2 K was reported by Bertolet et al. [64]. Two sets of samples with x-values of 0.12 and 0.18 were grown by atmospheric MOCVD at 6508C and Ê , all in the same multi-layer consisted of single quantum wells of thicknesses varying from 15 to 160 A structure. The sharpest lines were observed in samples with x ˆ 0:12. They find that in a sample with x ˆ 0:12 the value of s increases as the well width is reduced, reaches a maximum and then decreases, Ê again as predicted by Singh and Bajaj [59]. For instance, the value of s is 1.0 meV for L ˆ 60 A Ê Ê increases to 3.2 meV for L ˆ 27 A and drops to 0.9 meV for L ˆ 11 A (Fig. 23). They calculate the variation of s as a function of well width due to interface roughness using the formalism developed by Ê (0.5 ML) Singh et al. [55] and Singh and Bajaj [59]. They choose for the values of d1 and d2 1.42 A Ê , respectively, and assume a simple case of equal coverage of hills and valleys. They find and 100 A that the calculated values of s agree very well with those measured for narrow wells. In their

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Fig. 26. Variation of linewidth (FWHM) of a heavy-hole exciton as a function of well width (L) in Inx Ga1 x As/GaAs quantum wells with x ˆ 0:09 and 0.18. The data were taken at 5 K.

calculations, they do not include the effect of alloy broadening and attribute the difference between the measured and the calculated values of s for wide wells to the neglect of this effect. Recently, Patane et al. [74] have investigated the behavior s as a function well width in Inx Ga1 x As/GaAs quantum wells using PL spectroscopy at 5 K. They studied three different sets of samples. The first set contained samples with In concentration of 0.09 and the well width was varied Ê . In the second set of samples, the value of x was 0.18 and the well width ranged from 15 to 250 A Ê from 10 to 140 A. The third set of samples consisted of five InAs/GaAs quantum well structures with well widths ranging from 0.8 to 1.8 ML in steps of 0.2 ML. All three sets of samples were grown by MBE at 5208C and the thickness of each sample was kept less than the critical value to prevent the formation of misfit dislocations. The PL measurements were carried out 5 K. As shown in Fig. 26, in the first set of samples with x ˆ 0:09, the value of s varied from 1.1 meV for the widest well to Ê . Similar behavior was 0.6 meV for the narrowest well reaching a maximum value of 1.4 meV at 50 A found for the second set of samples with s varying from 2.7 to 0.8 meV for the widest to the Ê . Patane et al. [55] analyze narrowest well with a maximum value of 3.1 meV reaching at about 40 A their experimental data in the first (Fig. 27) and the second set of samples using a formalism developed by Singh et al. [55] and Singh and Bajaj [59]. In the first set of samples, they assume a Ê (1 ML) and treat d2 as a free parameter. They determine the values of r2 and value of d1 to be 2.8 A the ``exciton volume'' from their experimental values of the exciton binding energy using a threedimensional spherical exciton model in the hydrogenic approximation. The total value of s is calculated using Eq. (86). q s ˆ s2i ‡ s2a where si and sa are the excitonic linewidths due to interface roughness and alloy fluctuations, respectively. From fitting their data in this set of samples, they determine the value of d2 to be Ê and the excitonic volume to be 6.5Vex where Vex ˆ …4p=3†R3 . This value of the excitonic 24  3 A ex volume is close to that calculated by Singh and Bajaj [22], Zimmerman [23] and Lee and Bajaj [25], thus lending credence to these theories. An application of a similar analysis of the data obtained on

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Fig. 27. Variation of linewidth (FWHM) of a heavy-hole exciton as a function of well width (L) in In0.09G0.91As/GaAs quantum well. Solid line: theory; dots: experimental data. The separate contributions of the interface roughness (gint, dotted Ê and b ˆ 6 were used in the calculations. line) and the alloy disorder (gall, dashed line). The values of d2 ˆ 24 A

Ê leads to different values of the second set of samples with x ˆ 0:18 assuming again that d1 ˆ 2:8 A d2 and exciton volume. Ê and the exciton volume ˆ …2  0:4†Vex. It is not clear They find that in this case d2 ˆ 13  2 A why the value of exciton volume obtained in this set of samples is considerably smaller, Patane et al. [74], however, do suggest some possible reasons, including the presence of clustering to account for this difference. A similar analysis of their experimental data in the third set of samples consisting of InAs/GaAs quantum wells using only broadening due to interface roughness yields a value of Ê assuming d1 ˆ 2:8 A Ê . The quantum wells in this set of samples are very narrow and d2 ˆ 29  3 A the volume of the exciton in bulk GaAs was used throughout their analysis as a reasonable approximation in this system. It should be pointed out that the choice of d1 ˆ 1 ML in all samples is reasonable, the values of d2, however, can vary considerably from sample to sample even when the growth conditions are nominally similar due to the inherently random nature of the growth process in MBE. Kirby et al. [75] have studied the behavior of linewidths of both the heavy-hole free excitons and the heavy-hole excitons bound to neutral donors in Inx Ga1 x As/GaAs single quantum well structures at about 5 K. Their samples were grown by MBE on (0 0 1) oriented GaAs substrates, with InGaAs layers grown at 5408C and GaAs layers grown at 5808C. The In composition x was varied Ê . In In0.11Ga0.89As/GaAs single quantum from 0.05 to 0.15 and the well width ranged from 6 to 160 A well structures, for instance, they observe a double peak in their PL spectra recorded at 4.8 K for all Ê ). For well widths less than 40 A Ê the two peaks are well values of the well widths studied (6±160 A resolved. However, for wider wells the positions and the linewidths are determined by deconvolution of the spectra as the two peaks could not be resolved. These two peaks are identified with the radiative decay of free and bound excitons and their linewidths are observed to decrease rapidly as Ê . Their results are displayed in Fig. 28. For well widths larger the well width is reduced from 40 A Ê than 40 A, the two excitonic linewidth is remain almost constant, i.e. about 3 meV. The value of s Ê ) is determined to be 0.23 meV. The above observations can (free exciton) for the narrowest well (6 A Ê , more and more of the exciton be explained by the fact that as the well width is decreased from 40 A wave function is located in the GaAs barrier, reducing the effect of the alloy fluctuations and the interface roughness on the linewidth. In wider wells the alloy fluctuations seem to play a dominant

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Fig. 28. Comparison of the PL linewidth (FWHM) of a heavy-hole exciton in GaAs/Al0.38Ga0.62As quantum wells (x) and in In0.11Ga0.89As/GaAs single quantum wells (* and * represent the FWHM of the free exciton and donor bound exciton emission, respectively). Data for GaAs/Al0.38Ga0.62As wells from Bertolet et al. [64].

role in determining the value of the excitonic linewidth. Kirby et al. [75], however, do not provide a quantitative analysis of the behavior of their data on excitonic linewidths. Reynolds et al. [76] have also measured the variation of s of heavy-hole free excitons as a function of well width in In0.10Ga0.90As/GaAs quantum wells using PL spectroscopy at 2 K. Their samples consisted of five single wells of In0.10Ga0.90As varying in width from 2 to 17 ML Ê ) and were sandwiched between 500 A Ê thick GaAs barriers. This multi-layer well (1 ML ˆ 2:985 A ‡ structure was grown on n GaAs(0 0 1) oriented substrates by MBE in which GaAs layers were grown at 5808C, whereas In0.1Ga0.90As layers were grown at 5408C. The linewidth was observed to decrease from 3.8 to 0.11 meV as the well width was reduced from 17 to 2 ML, a trend similar to that observed by Kirby et al. [75]. Reynolds et al. [76] unlike Bertolet et al. [64] and Patane et al. [74] do not provide a quantitative analysis of their data. Strained Inx Ga1 x As/GaAs quantum well structures grown on (1 1 1) oriented substrates have attracted some interest because of the presence of large piezoelectric fields and hence their possible applications in opto electronic devices such as blue shifting modulators, all optical self electrooptical devices (SEEDS) and integrated laser modulators. Recently, Khoo et al. [77] have studied the behavior of excitonic linewidth as a function of well size in strained Inx Ga1 x As/GaAs quantum wells grown on (1 1 1) B GaAs substrates and compared it with that observed in quantum well structures grown on (0 0 1) GaAs substrates. Their structures were grown by solid source MBE either on n‡ (1 1 1) B GaAs substrates mis-oriented towards (2 1 1) or on undoped (0 0 1) GaAs substrate. Ê separated by 100 A Ê thick GaAs They consisted of single QWs of sizes 120, 100, 80, 60 and 40 A barriers. They studied two structures with compositions of x ˆ 0:10 and 0.17 grown on (1 1 1) B substrates (samples P10  P17, respectively) and one structure with x ˆ 0:17 grown on (0 0 1)

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substrate (sample N17). PL measurements were carried out at 10 K and linewidths as narrow as 2.0, 2.4 and 2.2 meV were observed in samples P10, P17 and N17, respectively. The excitonic linewidths in samples P10 and P17 were the narrowest ever reported in such structures thus attesting to the very high quality of these structures. The behavior of s as a function of well width in samples P17 and N17 was observed to be quite different; the value of s decreased as the well width increased in N17, as expected. In sample P17, however, the value of s first decreased and then increased to almost Ê wide well. Similar trend was also observed sample P10. 5 meV for 120 A Khoo et al. [77] were able to explain the behavior s as a function of well size in sample N17 at least qualitatively, using the theory of Singh and Bajaj [59]. To explain the behavior of s in samples P10 and P17, however, they propose a model in which the lateral variation in excited carrier density across the illuminated spot produces corresponding changes in emission energy because of the difference in screening of the piezoelectric field. They, however, do not apply an external electric field, opposite in direction to the built-in piezoelectric field, to check the validity of their model. As mentioned in Section 5.1.1 Hong and Singh [56] have calculated the variation of s as a function of applied transverse electric field in quantum well structures. They find that for a given well size the value of s increases as a function of electric field. In addition, for a given electric field the value of s increases as a function of well width. Khoo et al. [77] do not consider the contribution of this effect to the value of s in their analysis. 6.4. GaInAs/AlInAs quantum wells There has been a great deal of interest in the study of the electronic and optical properties of GaInAs/AlInAs quantum well structures lattice matched to InP, during the past two decades. This material system offers excellent opportunities for applications in both ultra-high-speed devices and optical telecommunications. The band-gap of Ga0.47In0.53As at room temperature is 0.75 eV (1.65 mm) but can be enhanced by quantum confinement effect by incorporating this in Ga0.47In0.53As/Al0.48In0.52As quantum well structures which in turn can be used in light sources for applications in fiber optics. Several groups have studied various aspects of the optical properties of these quantum well structures. In the following, we shall briefly review some of these investigations which deal with the behavior of s with well width. Scott et al. [78] have measured the variation of s as a function of well width in GaInAs/AlInAs single quantum wells lattice matched to InP using PL spectroscopy at 4 K. Their samples were Ê . In Fig. 29, we display grown by MBE at 5308C and the well width was varied from 6.7 to 122 A their results of the variation of s as a function of well width L. The solid circles represent their data. Ê . In The value of s increases from 4 to about 18 meV as the well width is decreased from 122 to 6.7 A addition to the contributions of the variations of the well width and the alloy fluctuations to s, they also consider the effect of band filling by the transfer of electrons from the barriers to the wells. The broken curve a in Fig. 29 shows broadening due to sheet concentration ns ˆ 1011 cm 3. The contribution due to alloy fluctuations is shown by the solid curve which they obtain by appropriately scaling the theoretical results of Singh and Bajaj [59] to conform with the observed values of s in AlInAs and GaInAs reference layers of 18.9 and 4.0 meV, respectively. The dot±dash curve c is calculated using the theory of Singh and Bajaj [59] where the interface is assumed to have islands with 1 ML height and lateral extent comparable to the exciton radius. Their experimental results are described well by the solid curve and they, therefore, suggest that the dominant mechanism responsible for line broadening in their samples is due to alloy fluctuations. The interfaces are apparently dominated by island sizes much smaller than the Bohr radius and thus contribute very

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Fig. 29. Variation of the linewidth (FWHM) of a heavy-hole exciton as a function of well thickness (Lz) in InGaAs/InAlAs present work (‡) from [78] and (x) from Tsang and Schubert [72]. Solid curve shows broadening due to variations in alloy composition. Dot±dash and broken curves indicate broadening due to variation in well width by 1 ML and a sheet concentration of 1011 cm 3, respectively.

little to excitonic linewidth. It should be pointed out, however, that the calculated values of s in InGaAs and AlInAs due to completely random compositional disorder are 0.8 and 3 meV, respectively, thus indicating the presence of some clustering in their alloys. 6.5. III±V nitride based quantum wells During the past few years, III±V nitride based quantum well structures have attracted a great deal of attention due to their applications in opto-electronics in the blue±green region of the spectrum. A number of groups [79] have investigated the electronic and optical properties of GaN/ AlGaN, InGaN/GaN and InGaN/AlGaN quantum well structures grown by MBE and MOCVD. Fir instance Lerouc et al. [80] have studied the temperature dependent PL and reflectively in GaN/ AlGaN quantum wells grown on (0 0 0 1) sapphire substrates by MBE. The well widths and the Al concentration in the barriers were determined by using reflection high energy electron diffraction (RHEED) intensity oscillations. Three different samples were studied. Ê )/GaN quantum wells of width 5, 9 and 13 MLs The first contained single Al0.11Ga0.89N (50 A Ê (1 ML ˆ 2:59 A) and the second 3, 7, 11 and 15 MLs while the third sample consisted of 17 MLs wide GaN well embedded in Al0.09Ga0.91N. At 9 K, they were able to observe PL spectra originating from each well thus demonstrating a monolayer control of thickness. The excitonic linewidths varied from 20 to 30 meV, thus indicting a relatively poor quality of the interfaces. These values are comparable to those reported by other groups. As far as we know, no systematic study of the variation of excitonic linewidth as a function of well size has been reported in these systems. More work is needed to improve the quality of the interfaces in these structures. As mentioned earlier, the quality of InGaN alloys is still far from satisfactory. The values of the PL linewidths observed in InGaN/GaN and InGaN/AlGaN quantum well structures are larger (77 meV) due to the poor quality of the alloys and interfaces [81]. At this time the nature of the physical mechanisms responsible for radiative recombination in these systems are not understood. A great deal of effort is being devoted to improve the quality of these heterostructures.

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6.6. II±VI semiconductor based quantum wells As mentioned earlier, there has been a remarkable resurgence of interest in the electronic and optical properties of II±VI semiconductors and their heterostructures during the past decade due to their applications in opto-electronic devices operating in the blue±green region of the spectrum. Both single and multiple quantum well structures based on Zn1 x Cdx Se/ZnSe have been used to fabricate LEDs and LDs with decent quantum efficiencies. As far as we know there has been no systematic experimental study of the behavior of s as a function of well width in these quantum well structures for different values of x. However, several groups have reported values of s in specific structures. Pelekanos et al. [52] have measured the values of s in Zn0.75Cd0.25Se/ZnSe multi-quantum well structures at 10 K using absorption spectroscopy. Their samples were grown by MBE on (0 0 1) oriented GaAs substrates and the well Ê with barriers kept at 500 A Ê . All samples were determined to be widths were varied from 30 to 200 A pseudo-morphically strained with the light-hole exciton transitions well separated from the heavyhole exciton transitions. They report the values of s to be 5.0, 9.0 and 13.0 meV for well widths of Ê , respectively. This behavior of s with well width is expected because the role of 200, 90 and 30 A interface roughness becomes more important as the well width is reduced. These authors do not report any measurements of s in single quantum wells where its value is expected to be lower. Pellegrini et al. [53] have also measured the values of s in Zn1 x Cdx Se/ZnSe quantum well structures using PL and absorption spectroscopies at 10 K. Their samples were also grown by MBE on (0 0 1) Ê . Two different values of x were oriented GaAs substrates with well width varying from 20 to 50 A Ê . Both single and chosen, namely, 0.11 and 0.31 and the barrier width in their samples was 300 A Ê multiple quantum well structures were investigated. They find that the value of PL linewidth in 40 A thick Zn0.69Cd0.31Se/ZnSe multi-quantum well structure is about 12 meV, whereas in a single quantum well, this value drops to 4.5 meV due to the absence of well width fluctuations. In addition, the value of s in absorption ranges between 5 meV (x ˆ 0:11) and 10 meV (x ˆ 0:31) in 10 different multi-quantum well samples examined. 6.7. Effect of growth interruption In this section, we shall discuss briefly the effect of interrupting the growth for certain duration, especially in MBE, on the nature of interfaces and hence on the behavior of excitonic linewidth in quantum well structures. Growth interruption has been used extensively in GaAs/AlGaAs, InGaAs/ GaAs and a variety of other heterostructures during the past 15 years and its effects on optical and transparent properties have been studied in considerable detail. In the following, we shall focus on the effect of growth interruption on interfacial disorder and hence on the excitonic linewidth in quantum well structures. The concept of growth interruption was introduced in the early 1980s in an effort to improve the smoothness of a surface of a growing material. The basic idea in the case of GaAs, for instance, was to close the shatter of the Ga oven for a certain duration in the presence of As flux. As the sticking coefficient of As in the absence of Ga flux is negligible, Ga atoms are able to migrate on the heated surface and get incorporated at appropriate sites thus smoothening out the surface. This effect was first observed by reflection RHEED and was called the ``recovery effect.'' A typical duration of growth interruption in most cases is about 100 s though shorter and longer intervals have also been used in some cases. In the case of quantum well structures, growth interruption at either one interface or on both interfaces has been used to improve their quality. In Fig. 30, we display a model for the interface disorder in the case of a AlGaAs/GaAs/AlGaAs quantum well structure grown under three

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Fig. 30. Model for the interface disorder of AlGaAs/GaAs/AlGaAs quantum well and of the energy state(s) of a heavyhole exciton X(e, hh) in the well of such a structure grown (a) without growth interruption and (b) with interruption of the growth at the interfaces for several minutes. Smoothening of both interfaces is assumed in this model. (c) Interfaces with interruption of the growth. The GaAs surface smoothens more rapidly than the AlGaAs one.

different conditions and energy states of a heavy-hole exciton in the well of such a structure grown: (a) without any interruption of growth, (b) with interruption of the growth at all the interfaces of several minutes. In this model both the interfaces are assumed to be smoothened, (c) interfaces with interruption of the growth. GaAs surface on which AlGaAs layer is grown (normal) smoothens more rapidly than the AlGaAs surface on which GaAs is grown (inverted). In case (a) one generally observes a single PL peak though multiple peaks have been observed in some cases as discussed earlier. In case (b) as many as three discreet peaks are observed and in case (c) only two discreet peaks are observed. We now briefly discuss the mechanism which is responsible for the smoothening of the interface during growth interruption, especially in MBE. In the case of epitaxial growth of GaAs for instance, the impinging fluxes of atomic Ga and molecular arsenic As4 or As2 (if As4 is cracked) on a heated GaAs substrate move along the surface till they find appropriate sites for incorporation. In the case of a (0 0 1) oriented GaAs substrate, the growth of GaAs films takes place in the form of alternate

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layers of As and Ga. Under ideal growth conditions a cation layer will be completely filled before the next cation layer will start growing. However, due to the random nature of the growth process in MBE, the subsequent cation layer starts growing before the preceding one is completely filled. When AlAs for instance, is grown on this surface, Al cations go to unoccupied Ga cation sites to fill that preceding layer. Thus, the transition from GaAs layer to AlAs layer is not abrupt and, therefore, leads to interfacial disorder at least over a range of 1 ML. If the growth is interrupted, namely, Ga flux is cut off for some time before the Al flux is allowed to impinge on the surface, Ga atoms have time to migrate to unfilled sites and get incorporated. Thus, without growth interruption the surface of GaAs film and, therefore, the interface between GaAs to AlAs consists of islands of various sizes with at least 1 ML in depth. It has been shown that after the growth is interrupted by a few seconds the micro-roughness of the interface is reduced. This ordering process seems to continue up to about 100 s after which the interface reaches a relatively stable condition. Much longer durations of growth interruption are undesirable as they allow the higher incorporation of background impurities and thus degrade the quality of the interface. Thus, an optimum value of the duration of the growth interruption needs to be determined for each material system under a given set of growth conditions. An interface soon after a growth interruption begins to get populated by islands Ê of lateral dimensions larger than the critical two-dimensional exciton diameter of about 250 A and the heights of these islands under ideal growth conditions are 1 ML. These islands in turn merge with larger ones with lateral dimensions much bigger than that of an exciton. Each of these large islands along with the similar large islands on the opposite interface of the GaAs layer constitutes a well. Thus, the width of the GaAs well varies from …n 1†a=2 to …n ‡ 1†a=2 with each well size having its own energy level scheme. As shown in Fig. 30b, similar smoothening of both the interfaces upon growth interruption is assumed. As mentioned earlier this leads to three discreet energy levels for excitons. At a low temperature the electron±hole pairs produced by the incident radiation thermalize in the lowest energy levels produced by the widest regions of the well where they eventually form excitons. The higher energy levels produced in the narrower regions of the well remain unoccupied. Therefore, at low temperatures one only expects to observe a single relatively sharp excitonic transition. As the temperature is increased the higher energy levels associated with narrower regions of the well become increasingly more populated and one observes three distinct transitions associated with three distinct regions of the quantum well. An analysis of the temperature dependence of the three emission lines can, therefore, provide us with useful information concerning the relative size of the sum of the surfaces of the three different quantum well islands as a function of growth interruption. The absence of the observation of multiple peaks as a function of temperature in the sample for instance, which has been subjected to growth interruption indicates that the growth interruption has not been successful in smoothening out the disorder on either of the two interfaces. At a given temperature, electron±hole pairs created in quantum well islands lose excess energy by emitting phonons and come to equilibrium with lattice temperature. There are then two possible paths to follow for the electrons and the holes created in narrower (higher energy) wells. Either they can diffuse to wider wells and form excitons there or they can form excitons in narrower wells, which then diffuse into wider wells. The third possibility is that both these processes take place simultaneously with certain probabilities. In the case of thermalization of the free carriers, the emission in wider well is determined by the diffusion length of the slower particles most probability the holes which have lower mobilities. However, in the case of thermalization by free excitons, the emission in wider well is determined by the diffusion length of free excitons. It is not clear if either or both of these different mechanisms of diffusion are responsible for thermalization. In any event the observation of thermalization also allows the derivation of the upper limit for the

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size of the islands as their average size should be smaller than the value of the diffusion length for thermalization to take place. As mentioned earlier the effect of growth interruption on smoothening out the interfaces is much more pronounced on the normal interface than on the inverted interface of a quantum well. This corresponds to the case of Fig. 30b and leads to two rather than three emission lines as illustrated in Fig. 30c. Thus, in the case of quantum well structures where the island sizes on the interfaces are considerably larger than the exciton diameter one observes sharp transitions in the PL spectra. In the opposite case, if the lateral size of the islands is much smaller than the exciton diameter the exciton really does not see the presence of islands and the transition again is sharp. In this case one observes a single peak in the PL spectrum and the interfaces are generally referred to as pseudo-smooth. It should be pointed out that even in the case of heterostructures which have been grown with interruptions at all interfaces large islands which appear flat actually consist of very small subislands (pseudo-smooth). Thus, most such interfaces are bimodial in nature. Bimberg et al. [82±84] have examined the effect of growth interruption on the PL spectra of a single QaAs/AlGaAs quantum well structure. Their sample was grown by MBE at 6208C at a rate of 1 ML s 1 with interruption of growth Dt at both interfaces of 1.0, 10.0 and 100.0 s, respectively. The Ê . They measure their PL spectra at 100 K and find that the excitonic transitions well width was 50 A become sharper as the value of Dt is increased. For example, with a growth interruption of 100 s their sample exhibits two excitonic transitions. As the temperature is increased no additional transition is observed. However, when the temperature is decreased, the higher energy transition disappears. This behavior of the PL spectra strongly suggests that the inverted interface is still rough, whereas the normal interface has become considerably smoother. The disappearance of the higher energy transition with decreasing temperature is indicative of thermalization in a two-level system. The higher energy transition corresponds to a well width of 18 MLs, whereas the lower energy transition arises from 19 MLs wide region of the quantum well. It is clear that the lateral size of the islands is smaller than the diffusion length of excitons to allow thermalization to take place. They estimate the value of diffusion length of excitons at T ˆ 2 K to be in the range of 0.9±1.3 mm and hence suggest that the lateral size of the very smooth islands to be larger than 1 mm. This conclusion is supported by their work using CL spectroscopy. To further check their conclusions concerning the behavior of thermalization, they grew a sample similar to the one discussed above but with a growth interruption of 120 s. The PL spectra obtained at several temperatures between 2 and 110 K show three distinct excitons peaks corresponding to three different well widths, i.e. 16, 17 and 18 MLs, respectively, thus suggesting the complete absence of thermalization. In addition, the relative intensities of the three peaks do not change significantly as the temperature is varied between 2 and 80 K. These observations strongly support the conclusion that the lateral extension of the growth island is larger than the diffusion length of excitons and that the majority of the excitons recombine before they are able to reach neighboring valleys at the interface where the well thickness is larger. Bimberg et al. [82±84] have also performed a lineshape fit of their spectra at T < 2 K taking into account both the lifetime and interface roughness broadening effects. They find that they need a very small contribution of the effects of interface roughness to explain their spectra, thus suggesting the presence of large very smooth growth islands at the interfaces. As mentioned earlier a number of groups have investigated the effects of growth interruption on the excitonic properties of quantum wells based on a variety of different material systems during the past 15 years. Their work has essentially led to conclusions similar to those arrived at by Bimberg et al. [82±84] and will not be reviewed here.

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6.8. Cathodoluminescence spectroscopy Another powerful technique that has proven very useful in studying the optical properties of quantum well structures especially the behavior of excitonic linewidth as it is affected by various growth conditions, is the CL spectroscopy. In this technique the excitation of the material is carried out using an energetic beam of electrons whose energy is varied from 200 eV to 40 keV. The projected range of such electrons in GaAs, for example, varies from only a few nm to almost 5.3 mm, respectively. Thus, the effects of both near surface and surface free excitations can be studied by varying the energy of the incident electron beam. In addition, the excitation is much more homogeneous in volume than after laser illumination. Following are some additional advantages of CL spectroscopy. 1. The exciting pluses can be varied in width by almost four orders of magnitude without altering the pulse rise or decay time: temporal resolutions of the experiment is independent of pulse width. This is of particular advantage in time resolved studies. 2. The electron beam can be easily scanned over the surface of the crystal and a two-dimensional pattern of luminescence intensity and lifetime can be taken in a few minutes. Perhaps a more interesting and appropriate luminescence technique especially for the study of quantum well structures is the cathodoluminescence imaging (CLI) technique where the electron beam is digitally scanned over a rectangular area (typical dimensions: 5:6 mm  10 mm to 200 mm  500 mm), which is equally divided into 12 pixels  100 pixels. The experimental details of this technique are described in an excellent review article by Herman et al. [60] and Bimberg and his coworkers [82±85] have used CLI technique extensively to study the effect of growth interruptions on GaAs/AlGaAs and InGaAs/GaAs quantum well structures. Some of the samples that were studied using PL spectroscopy were also investigated using CLI technique and similar results about the nature of interfaces were obtained. It should be mentioned that this technique has mostly been used to study the effects of growth interruption. For further details the reader is referred to a body of work on this subject by Bimberg and coworkers and other investigators mentioned in [60]. Bimberg et al. [82±84] have performed a detailed comparative study of lattice matched InGaAs/InAlAs quantum wells by advanced high-resolution electron transmission microscopy (HRTEM) and PL lineshape analysis developed by Singh and Bajaj [59]. They find that these two methods yield complimentary information. Their study shows that the well width variation induced by the inequivalent interface roughness is primarily responsible for the spectral broadening in wider wells, whereas very narrow wells where the exciton wave function is predominately in the barriers, alloy broadening plays a major role. The samples studied in this work were grown by atmospheric MOCVD and consisted of five Ê ) In0.53Ga0.47As/In0.52Al0.48As quantum wells. These structures were grown on InP thin (<30 A substrates at a relatively high temperature of 6408C and a 30 s growth interruption was used at each Ê in each sample but the well width was varied. interface. The barrier width was kept at 200 A HRETM images were recorded in each sample and based on their analysis it was suggested that InGaAs grown on InAlAs has an appreciably more abrupt interface extending from 1 to 2 MLs than InAlAs grown on InGaAs where the interface width varied from 2 to 3 MLs. Such a behavior was explained in terms of three-dimensional growth of InGaAs layers at 6408C, which causes greater interface roughness rather than two-dimensional growth which takes place at lower temperature. Ê from 1.5 They also performed PL measurements on one of their samples with nominal width of 29 A to 300 K. The value of s varied from 35 to 47 meV as the temperature was raised from 1.5 to 300 K. No additional lines were observed as the temperature was increased thus indicating the absence of

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flat islands with large lateral extension. They analyze their data using the theory of Singh and Bajaj [59] and suggest the presence of considerable clustering in InAlAs barriers which causes local variations in the band-gap and contributes to excitonic linewidth in narrow quantum wells. The Ê . They also calculate the effect of interface average size of the cluster they estimate is about 15 A roughness on the excitonic linewidth especially in wider wells, assuming 0.5 ML for the width of the interface. They find no clear agreement between theory and experiment, the measured values in wider wells being much larger than those calculated. They therefore suggest that the interface roughness extends beyond their assumed value of 0.5 ML to perhaps 1±2 MLs, which is consistent with their conclusions drawn from HRTEM measurements. It should be pointed out that their samples were not grown under optimum conditions. PL measurements on higher quality InGaAs/ InAlAs quantum wells have been reported and some of this work has been discussed earlier in this article. 7. Summary In this article, we have attempted to provide a comprehensive and a critical review of both the theoretical developments and the experimental investigations of the use of excitons in determining the materials quality of semiconductor alloys and quantum well structures using a variety of optical techniques such as PL, PLE and CL. The physical parameter that is directly related to the quality of these semiconductor systems is the width of the excitonic transition. We have reviewed the results of several calculations of the excitonic linewidth in semiconductor alloys where the statistical potential fluctuations caused by the components of the alloys are primarily responsible for the broadening. We have also discussed the effect of the applied magnetic field on the behavior of the excitonic linewidth in semiconductor alloys. We have described the results of measurements of excitonic linewidth in several III±V and II±VI semiconductor based ternary alloys and compared them with their calculated values. We have described a formalism to calculate the excitonic linewidth in quantum well structures due to interfacial disorder. This is followed by a review of the results of the measurements of excitonic linewidth in several different quantum well structures which have been fabricated with and without growth interruption at the interfaces. Acknowledgements Work on this review article was started while I was a visiting professor at the Universite Pierre et Marie Curie, Paris during 1997. I am very thankful to Professor M. Balkanski for his very kind and generous hospitality and for many stimulating discussions. I would also like to thank Ms. Connie Copeland for her help in preparing this manuscript. Lastly, I want to thank my wife, Swarnlata, for her patience with me during the time I was working on this review article. References [1] [2] [3] [4] [5]

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