Accepted Manuscript

Effects of Dilute Alloying on the Quality of Ultrathin InGaN/GaN Single-Quantum Wells

Nacir Tit, Abdullah A. Al-Shezawi

PII: S0925-8388(14)02805-9DOI: http://dx.doi.org/10.1016/j.jallcom.2014.11.146Reference: JALCOM 32700

To appear in: Journal of Alloys and Compounds

Received Date: 26 August 2014Revised Date: 11 November 2014Accepted Date: 24 November 2014

Please cite this article as: N. Tit, A.A. Al-Shezawi, Effects of Dilute Alloying on the Quality of Ultrathin InGaN/GaN Single-Quantum Wells, Journal of Alloys and Compounds (2014), doi: http://dx.doi.org/10.1016/j.jallcom.2014.11.146

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Effects of Dilute Alloying on the Quality ofUltrathin InGaN/GaN Single-

Quantum Wells

Nacir Tit* and AbdullahA. Al-Shezawi

Department of Physics, UAE University, P.O. Box 15551,

Al-Ain, United Arab Emirates

Abstract

We present a theoretical investigation on the electronic properties of alloyed InxGa1-xN ultrathin single-quantum

wells (SQWs) embedded in GaN matrix. The empirical tight-binding method with sp3s* basis set, including spin-

orbit interaction and nearest-neighbor two-center overlap integrals, is used to study the number of bound

states, quantum confinement (QC) energy and the band-gap energy of (InxGa1-xN)Nw/GaN SQWs versus the well

composition and parameters; namely width (Nw) and depth (via valence band offset, VBO). The results show

strong correlation between the bound states (number and QC energy) and the well’s composition and

parameters. Furthermore, the results were used to model experimental photoluminescence (PL) data of three

samples containing Nw= 1, 3 and 5 monolayers (MLs), which were fabricated by A. Yoshikawa and coworkers

using rate-flow plasma molecular-beam epitaxy (rf-MBE). The results have revealed that in all these three

samples, the indium mole fraction would not exceed 25% and, consequently, the three wells are shown to

contain at maximum 1, 2 and 3 electronic bound states, respectively. It is deduced that the maintaining of low

indium content (x < 0.25) is the secret for the achievement of high structural and optical qualities of the

produced samples with free of misfit dislocations.

Key words: Alloyed quantum wells,Nanostructures, Electronic structure, Nitrides,

Photoluminescence

(*) Corresponding author, email: ntit@uaeu.ac.ae

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1. INTRODUCTION

In the last two decades, the development and applications of low dimensional

semiconductors have been rapid and spectacular. Ever improving epitaxial growth and

device fabrication techniques have allowed access to some remarkable new physics in

quantum-confined nanostructures while a plethora of new devices has emerged. Among

these nanostructures, single and double quantum wells (SQW and DQW) have been

attractive because of both their fundamental physics properties [1] and their tunable

coherent-light sources for optical communications [2], besides having a good control on the

charge distribution and the strain morphology.

The compound semiconductors of group III-V usually possess relatively small bandgaps,

which limit their applications to electronics and telecommunications. On the other hand,

the semiconductors of group II-VI and GaN-based compounds possess higher bandgaps,

making them suitable for photonic devices. A technological breakthrough was achieved by

the incorporation of nitrogen in III-V compounds either as alloys (such as Ga1-xInxN [3,4] or

low-dimensional hetero-structures, such as multiple quantum wells (MQWs) [5-10] and

quantum dots (QDs) [11,12]. Furthermore, the nitrides offer a broad spectrum of bandgap

energies that span the entire visible-light energy spectrum, ranging from the infrared (IR) to

the ultraviolet (UV). This has made nitrides a strong competitor against II-VI materials in the

market of optoelectronic industry. In fact, their rapid rise has allowed them to dominate the

market especially as far as the abundance of the III-V substrates is concerned.

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For the III-V semiconductors, the incorporation of nitrides has paved the way for a

great domain of applications in photonics [13]. Among them and perhaps is the most

popular is gallium nitride (GaN), which crystallizes into two phases zinc-blende and wurtzite

that can easily be grown on sapphire. Besides,GaN possesses well distinct and unique

characteristics, such as a direct bandgap of 3.4 eV with relatively high exciton’s binding

energy (26 meV). Its flexible properties have made its applications extend beyond

photonics. For instance: (i) In photonics, it is commonly used in bright light-emitting diodes.

Its substrate makes violet (405 nm) laser diodes possible without use of nonlinear optical

frequency-doubling. Its low sensitivity to ionizing radiation has made it suitable material for

solar cell arrays for satellites [14]. (ii) In electronics, the achievement of p-doping with Mg

[15] has paved the way for fabrication of GaN-based MOSFET devices. Because GaN

transistors can operate at much higher temperatures and work at much higher voltages

than GaAs transistors, they make ideal power amplifiers at microwave frequencies [16]; (iii)

In telecommunication, the Ga1-xInxNyAs1-y ternary and quaternary alloys have been of great

interest as desirable bandgaps below infra-red (IR) can easily be achieved (0 <Eg<< 1 eV) [17-

18].

In the field of photonics, nanostructures such as quantum wells have been

predominantly used in laser and LED devices. Dealing with GaN-based layered

nanostructures for quite a long time, Yoshikawa has been studying the fabrication and

characterization of InN-GaN-based nanostructures including quantum dots (QDs),

nanowires (NWs), and conventional two-dimensional multiple quantum wells (MQWs) [7,9-

10]. Using sophisticated growth technique, such as rate-flow plasma molecular-beam

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epitaxy (rf-MBE), the group achieved the growth of high-quality semiconductor hetero-

structures even originating from highly lattice mismatched materials such as InN/GaN

(whose lattice mismatch is about 11%). As a result the strained slabs being kept thin (in

order to avoid the formation of misfit dislocations). The good control in this latter growth

technique has further permitted the exploration of the strain effects in the bandgap

engineering of the hetero-structures. The optical qualities of the produced structures are

normally tested using photoluminescence (PL) experiments.

The huge discrepancy in growth temperatures betweenGaN and InN are actually due the

fact that In-N bond being much weaker than the Ga-N bond. The rf-MBE method effectively

rose to this challenge and made it possible to achieve the growth of InN/GaN MQWs

consisting of ultrathin InN wells of thickness around 1-5 monolayers (MLs) coherently

embedded in a GaN matrix [7,8-9]. The qualities of such structures have been assessed by

several methods including X-ray diffraction (XRD), high-resolution transmission electron

microscopy (TEM), atomic force microscopy (AFM) and PL measurements.

On the computational side, various methods have been applied for the calculations of the

band structures and the optical properties of nitrides. Many methods were limited by the

system size and their ability to only deal with ground-state properties (with an

underestimation of bandgap energy), e.g. the first-principle methods. Other limitations

include the complete neglect of band-mixing effects, e.g. the Hȕckel method and the

effective-mass approach (based on the Kronig-Penney model). To overcome such

difficulties, we have used the sp3s* tight-binding (TB) method with the inclusion of spin-

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orbit interaction [19,20]. The TB method succeeded in simulating the experimental data

while incorporating the microscopic description of the material and including the point-

group symmetry of the system. Within the Slater-Koster scheme [21], the TB method uses

minimal basis set of atomic orbitals, enabling the method to deal with large systems

consisting of thousands of atoms. In addition, the TB method takes into account the band-

mixing effects that are essential in the band structures of systems, such as alloys and

quantum wells. Besides, the inclusion of spin-orbit interaction is not only necessary for II-VI

materials and nitrides but also for the calculation of optical properties.

For the present work on the alloyed InGaN/GaN quantum wells, two theoretical

assumptions are considered valid: (i) the virtual crystal approximation (VCA) is assumed to

provide a valid description of the InxGa1-xN alloy consisting the wells. The VCA assumes that

the atoms remain in ideal positions and completely ignores the lattice relaxation effects.

The VCA commonly provides qualitative explanations for most of the important features in

the band structures. A critical analysis of the VCA was recently addressed by Dragam et al.

[22,23]. The effects of relaxations will be treated as perturbation added posteriori to the

results as it will be discussed in section 3. (ii) The macroscopic theory of elasticity (MTE) [24]

is also assumed to be valid for the atomic structure of the computational super-cell (SCell),

in which pseudomorphic growth is considered. Strain can place severe restrictions on the

hetero-structure that must be grown in order to avoid misfit dislocations. The strained layer

may relax toward its unstrained lattice parameter when it exceeds a critical thickness dc,

corresponding to the misfit dislocations to be nucleated at the neighborhood of the

interface. For instance in case of InN/GaN, dc is expected to be about 4 monolayers (MLs).

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As consequence of this latter restriction, experimental growth are limited to deal with

ultrathin wells composed of Ga1-xInxN alloys while keeping indium content sufficiently low

to ensure the coherent growth [10]. Within this latter experimental restriction, the above

two theoretical assumptions appear reasonable. Nonetheless, any needed theoretical

correction will be posteriori included in the calculation as a perturbation to the TB

Hamiltonian.

The aim of the present work is to investigate the electronic structure of (Ga1-

xInxN)NwSQW embedded in GaN matrix versus the well’s thickness, composition and

valence-band offset (VBO). One important aim shall be under scope is to determine the

correlation between the indium content and the well thickness (i.e., Lw not dc) to ensure the

coherent growth by modeling the PL data of three wells Lw = 1, 3 and 5 MLs, that were

reported by Yoshikawa and coworkers [10]. The paper is organized as follows: Section 2

gives details of the TB method; Section 3 illustrates a discussion of the results. The last

section summarizes our main conclusions.

2. COMPUTATIONAL METHOD

The present work utilizes the sp3s*-TB models, with the inclusion of spin-orbit coupling,

developed by Hernandez-Cocoletzi et al. [25]. Great efforts were focused by the authors on

the fitting of the valence bands (VBs) and low-energy lying conduction bands (CBs) [25],

while the band-gap energy and carrier effective masses were fit to the experimental data

[13]. For sake of completeness, the TB parameters are presented in Table 1. It is worth

mentioning that bit modifications were done on two parameters (and ) of the original

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parameters in order to adjust the bulk bandgaps to the experimental ones: = 3.60 eV

and = 0.676 eV, reported by Yoshikawa and coworkers [10]; It is as our aim in this

present investigation to model the PL data of the wells reported in the same paper [10].

Then the on-site energies were shifted in order to make the VB-edge as an energy reference

for both GaN and InN bulks. In the supercell calculations, besides the two assumptions

discussed in the introduction about the ideal atomic structure of SCell (i.e., VCA and MTE

approximations), the problem of energy reference between the two constituents is sorted

out by taking the VBO into account [26] (for instance, in our present case, InN on-site

energies are shifted up by VBO since the VB-edge of InN is always higher than that of GaN

as an interface of type-I is usually formed between them; namely, VBO = Ev(InN) – Ev(GaN) ≥

0). We emphasize that in case of lattice matched common-anion hetero-junction, the VBO is

usually vanishingly small. Nonetheless, due to the biaxial strain existing in the case of highly

lattice mismatched InN/GaN hetero-junction, first principle all-electron calculation by Wei

and Zunger [27] estimated VBO to be about 0.26 eV. In our present work effects of VBO

variation will be also discussed.

With the inclusion of spin-orbit coupling, the sp3s*-TB Hamiltonian is expressed in the

Löwdin basis set [28] as follows:

= ∑ ,, |, , | + ∑ |, , ("#$"%), , | (1)

where i and j refer to atoms at the respective positions ri and rj; µ and ν refer to one of the

ten spin-orbit orbitals on the atom i and j, respectively; Ei,µ is an on-site (diagonal) energy

element of orbital µ on site i; and Uiµ,jν is the overlap integral between the respective

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orbitals µ of atom i and ν of atom j. The sum just runs over the nearest neighbors. Moreover,

we emphasize that the TB parameters are shown in Table 1 for sake of completeness.

It is worth to mention that before diagonalizing the Hamiltonian (1), it is extremely important

to take care about some crystal point-group symmetry considerations. Namely, the existence

of an inversion symmetry with respect to the planar-spin-averaged valence-electron charge

density, and this symmetry should be properly implemented into the TB Hamiltonian [29]. In

the present work, the inversion center is taken to be the atom located at the middle of the

InGaN well.

The Bloch wavefunction|' , of course, should diagonalize the TB Hamiltonian and can

be written as:

|' = (|' (2)

where n is a band index; k is a wave-vector, usually taken either from within the irreducible

wedge (IW) of the Brillouin zone (BZ) if the aim is to calculate the density of states or along

the high-symmetry lines if the aim is to calculate the bands; ( is the eigen-energy

corresponding to the eigen-function (Bloch wave-function). In our particular case, both

constituents possess direct band gaps at Γ-point. Thus with exception of band structure

calculations, the band-gap energy (Eg) and the quantum-confinement energy (EQ) are

calculated at Γ-point.

Moreover, the obtained eigen-functions at Γ-point are used to calculate the spin-averaged

wavefunction-squared amplitudes, for each eigen-energy, and its expression is given by:

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|)()|* = ∑ +|),()(↑)|* + |),()(↓)|*./0

1/ (3)

wheren is a band index and µ is an orbital on the site i. The sum is carried over all the 10 TB

orbitals belonging to the same atom; ),()(↑)and ),

()(↓) are eigen-functions’ components

on the TB-basis orbital µ of atom icorresponding to the spin-degenerate band (with spin up

and down, respectively). The quantity in Eq.(3) is related to the valence-electron local

charge density (at site i) as follows:

2 = *34 ∑ |)

()|*51/ (4)

where e is the electron charge; Ω is the volume of the supercell and 2 stands for spin

degeneracy. The sum is carried out over the eigen-states up the Fermi level (here, Nf = 4Nat,

and Nat is the total number of atoms in the supercell).

The hetero-junction made of either InN/GaN or InxGa1-xN/GaN is considered to be of

type-I [27]. Computationally, we have found that it is sufficient to achieve the complete

separation of the well from its mirror (due to periodic boundary condition along z-direction)

by just making 67 → 30 MLs, which is easily affordable in our TB method.

Finally, we mention that in the computation, we have used a supercell of structure

(GaN)N1(Ga1-xInxN)Nw(GaN)N2–GaN (001) to simulated the isolated alloyed InGaN well

embedded in GaN matrix. Each monatomic layer contains 4 atoms. We took N1+N2 to be

about 30 MLsand with a constraint (N1+N2+Nw) must be an even number in order to fulfill

the periodic boundary condition in the Z-direction.The total number of atoms used in most

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of our calculations is ranging from to 240 to 272 atomsto correspond to the range of Nw = 1

to 5 MLs, respectively. The results will be discussed next.

3. RESULTS AND DISCUSSIONS

3.1. Effects of alloying:

Figure 1 shows the eigen-energies of Hamiltoniancalculated at Gamma-point of the BZ for

(InxGa1-xN)Nw/GaN SQWs versus the well widths (Lw) for four different alloying

concentration: (a) x = 0.25, (b) x = 0.50, (c) x = 0.75 and (d) x = 1.0. The range of well width is

kept within the experimental realizations (i.e, 1-5 monolayers). The energy reference is

taken to be the VB-edge of GaN bulk and VBO = 0. Ten top valence-band (VB) and twenty

conduction band (CB) states are shown in each panel. The eigen-states, with energies

ranging in 0 < E < 3.6 eV, are considered to be localized within the InxGa1-xN well. We

emphasize the existence of two wells corresponding to the two types of charge carriers: (i)

The electronic well (e-Well) whose depth is the conduction band offset (CBO ≈ 2.924 eV)

and extends in the energy range 0.676 ≤ E ≤ 3.6 eV; and (ii) The hole well (h-Well) whose

depth is about 150 meV formed by bi-axial strain at the interfaces and it extends in the

energy range 0 ≤ E ≤ 0.15 eV. This h-Well consists of two small potential dips located at the

interfaces and each has capacity to localize just one bound state (see below for more

details).

In each panel of Figure 1, the quantum confinement effects are clearly responsible for

the reduction of energy confinement (in the e-Well) and gap with the increase of well width

Lw. The number of bound states in the e-Well is also demonstrated to be dependent on both

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well width, Lw, and composition, x. For instance, in Figure 1a, the number of bound states

increases from one to three as the well width increases from 1 to 5 MLs, respectively.

Within the same range of Lw,the number of bound states raises from 1 to 6, and from 2 to

11, and last from 3 to 18 corresponding to Figures 1b, 1c and 1d, respectively.

3.2. Effects of well parameters:

Of course, as it is well known that both the well’s depth and width do control the number

and confinement energy of bound states. In our present computational model, the h-Well

depth is about VBO and the e-Well depth is about CBO (which is correlated to VBO by the

relationship: VBO + CBO + Eg(alloy) = Eg(GaN) = 3.6 eV). We further emphasize that VBO is

usually and reliably calculated using ab-initio methods as it is a ground state property and

should abide for interface specific effects. Besides, its inclusion within the TB scheme should

take care of the problem of energy reference between the two sides of hetero-junction.

Nonetheless, it is well know that in common-anion lattice-matched hetero-junction, VBO is

vanishingly small. In case of highly strained hetero-junction such as GaN/InN, VBO is

evaluated by ab-initio methods to be about 0.26 eV.

Figure 2 shows the effect of varying VBO (within a hypothetical range of 0-2 eV) on the

bound states and band-gap energy. In Figure 2a, we have fixed the well’s width and

composition to be: Lw = 1 ML and x = 0.25 and let VBO vary. As VBO increases, the number

of bound states in h-Well increases on the expense of the reduction of number of electronic

bound states in the e-Well. The compromised total confinement energy of hole and

electron reach its optimum value at about VBO ≈ 1 eV (i.e., where the band gap reaches its

maximum value). Figure 2b shows the variation of band-gap energy versus VBO for three

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different well’s widths Lw = 1, 3 and 5 MLs with same indium content x = 0.25. The optimum

value of VBO is basically the same for all the three wells and equals to VBO ≈ 1 eV.

3.3. Bands and eigen-functions:

In order to study the nature of the eigen-states whose energies are around the band-gap

energy, Figure 3 shows both the bands along the ΓZ high-symmetry line (lower panel VBs

and upper panel CBs) and the squared amplitude of eigen-functions along the Z-direction

calculated at Γ-point of the BZ. The indium content in Figure 3 is kept constant x = 0.25 and

the VB-edge of bulk GaN is taken as an energy reference with VBO = 0. Three different well

widths are considered: (a and d) Lw = 1 ML; (b and e) Lw = 3 MLs; and (c and f) Lw = 5 MLs

corresponding the same well widths of experimental samples [10]. All the bound states in the

e-Well are shown by red curves.

Particularly, in case of well width of 1 ML (see Figures 3a and 3d), the e-Well contains just

one bound state (denoted C1). This bound state is displayed by a flat band in Figure 3a and

well localized wave-function in Figure 3d. The next upper band, which is close to be flat,

correspond to a delocalized state exhibiting some resonance quantum mechanical effect as its

energy is just above the top of the e-Well as can be depicted from the profile of its

corresponding wave-function in Figure 3d. Concerning the states in the VB, it seems that the

bi-axial strain at the two interfaces of the well has caused a formation of two potential dips of

depth of about 20 meV. Each of these dips can accommodate one quantum bound state.

Figure 3a clearly shows two flat bands (denoted V1 and V2). Figure 3d shows that these

states (V1 and V2) to be somewhat localized within the well region. Note that the state V3 is

almost localized within the well just because the well is extremely thin and the separation

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between the two interface potential dips is very small; so that the combined/overlapped dips

can accommodate about three states. As the well’s thickness increases, the two potential dips

will get more separated and retain justV1 and V2 as bound states. All other VB states clearly

belong to the continuum as their bands are dispersive.

As the well thickness increases, more bound states appear in the e-Well. Namely, in

Figure 3b and 3e where Lw = 3 MLs, the e-Well contains 2 bound states (denoted C1 and

C2). The rest of CBs are among the continuum CB states. In Figure 3c and 3f, where Lw = 5

MLs, the e-Well contains 3 bound states (C1, C2 and C3). The rest of CB states, whose

energy are above Eg(GaN) = 3.6 eV, might be considered among CB continuum. One further

remark about Figure 3 is that the two bound states in the h-Well (i.e., V1 and V2) raise their

energies as getting more confined in the potential dips. This reveals that the potential dips are

enlarging their depths from 20 meV to 100 meV to more than 100 meV as the well width

increases from 1 to 3 to 5 MLs, respectively. The increase of depth of potential dips can be

justified by the augmentation of bi-axial interface strain as the well gets broader.

3.4. Number of electronic bound states:

From theoretical point of view, the largest compromised confinement energy is obtained

when VBO = 1.0 eV (as been indicated in sub-section 3.2). When VBO = 1.0, then CBO =

1.924 eV when the well is purely InN (i.e., x=1.0). We have considered two extreme VBO

values (VBO = 0 and 1 eV) in counting the number of bound states in the e-Well to

correspond to maximal and minimal numbers (Nmax and Nmin), respectively. Figure 4 shows

the results of Nmax and Nmin versus well’s width (Lw) and composition (x).

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The number of bound states increases with the increasing well’s width as the

quantization of both energy and wave-vector permits in similar fashion to finite 1

dimensional quantum well. Furthermore, the number of bound states also enlarges with the

increase of indium mole fraction. This latter fact reveals that the bound states in the e-Well

is mainly supported by In atoms. The increase of Nmax appears to be varying close to linear

as function of either Lw or x.

3.5. Modeling of photoluminescence data:

It is worth to emphasize that the InN/GaN is a highly lattice mismatched hetero-junction (of

lattice mismatch of about 11%). The dilute doping of the well is necessary for two reasons:

(1) To pave the way for coherent (pseudomorphic) growth of thin-layered nanostructure

composed of InGaN well in GaN matrix; and (2) To gain control on bandgap engineering for

desirable photonic applications. In the last two decades, dealing with InN- and GaN-based

nanostructures for a long period of time, Yoshikawa and coworkers [10] were able to

fabricate high quality InN/GaN nanostructures. Their successful growth was experimentally

confirmed by the X-ray diffraction (XRD), the transmission electron microscopy (TEM), the

atomic force microscopy (AFM) and the photoluminescence (PL) measurements. In their

work, the state-of-the-art rf-MBE method has been used to achieve the growth of 1-ML and

fractional monolayer single-quantum wells;while the barriers’ thicknesses of GaN are kept

constant equal to about 150 Å. It is worth to mention that the rf-MBE is considered the

most suitable for InN epitaxy process compared to the metal-organic-vapor-phase epitaxy

(MOVPE) method, because very large epitaxy temperature diversity between InN and other

15

nitrides should be better under control. Furthermore, to overcome difficulty of the high-

lattice mismatchwas achieved by lowering the indium-gallium content ratio during the

growth of the wells. We emphasize that with such 11% lattice mismatch, it has been

reported that the critical thickness can be up to about 4 MLs unless the mismatch is

reduced through for instance alloying. Successful growth of 1-ML, 3-ML and 5-ML SQWs

have been reported by Yoshikawa’s group [10] with PL emissions of wavelengths 363 nm

(3.4 eV), 398 nm (3.11 eV) and 437 nm (2.84 eV), respectively. While, in principle, the

experimental evidence of defect-free samples is confirmed, the fractional or full filling of

the well with indium should remain an open question [8,10] to be addressed in the agenda

of further inspection; as it will be discussed here below.

On the simulation side, Figure 5 displays the TB results of Eg versus indium content, x,

for three single quantum wells: (a) Lw = 1 ML shown in full green stars, (b) Lw = 3 MLs shown

in full red triangles, and (c) Lw = 5 MLs shown in full bleu circles. By performing a non-linear

least-square fitting to the theoretical data using the bowing relation of alloys:

;<=; = > + (1 − >) − A>(1 − >)

where: ;<=; is the bandgap energy of the alloy; and are the bandgap energies

of bulk InN and GaN, respectively; and B is the bowing parameter, which is considered constant

independent of composition,as in the case of low indium content cases. The results of the

fittings are shown in dotted curves with the following obtained bowing parameters: B = 0.39,

1.17 and 1.58 eV for 1-ML, 3-ML and 5-ML SQWs respectively. Furthermore, based on the idea

of existence of Stokes shift to take account of atomic relaxation and other effects in the InxGa1-

16

xN ternary alloys (which can reach even an energy shift of about ΔE ≅ 200 meV in case of a

single crystal alloy with x = 0.5) [20,30], then we have accounted such effect by including error

bars into the bowing parameters. One has to make sure to exclude any anti-bowing effects (i.e.,

B cannot be negative as is never been experimentally observed in such alloys). Effectively, the

theoretically possible bandgap energies are shown in the regions bounded between two solid

curves of same color as the corresponding TB results (i.e., green curves for 1-ML SQW, red

curves for 3-ML SQW, and bleu curves for 5-ML SQW). The experimentally observed

fundamental transitions in these respective wells are presented by the energies: 0/BC = 3.4 eV,

0DBC = 3.11 eV, and 0EBC = 2.84 eV. Figure 5 brings into evidence that these transition energies

are relatively high and should rather correspond to low indium content cases. Theoretically,

one predicts them to occur for cases of indium contents of: 0.09 ≤ x ≤ 0.20 ⇒ 1-ML SQW; 0.13 ≤

x ≤ 0.20 ⇒ 3-ML SQW; and 0.17 ≤ x ≤ 0.25 ⇒ 5-ML SQW. The first estimate is consistent with

the experimental work of Che and coworkers [8], who reported x = 0.15-0.20 for 1-ML SQWs. As

a matter of fact, each region between pairs of solid lines are justified not only to take care of

the Stokes shift but also to take account of both the band offsets between the two constituents

to alter the interface-specific effects and the TB parametrization shortcomings.

The results shown in Figure 5 suggest the existence of a clear restriction in the indium

content to ensure coherent growth on InGaN/GaN SQWs. As thick-wells are beyond

experimental consideration, ultrathin wells can be experimentally realized free of misfit

dislocations in condition to keep indium content below 25%. This claim is indeed corroborated

with experimental evidence in the work by Che and coworkers [8].

17

4. CONCLUSIONS

The sp3s* tight-binding method, with inclusion of spin-orbit coupling, is employed to investigate

the effects of dilute alloying on the electronic band structures of (InxGa1-xN)Nw/GaN (001) single-

quantum wells. In the computational model, the GaN barrier is kept large enough to ensure the

complete isolation of the SQW by taking 30 MLs of GaN, and the band structures are studied

versus well width (Lw= Nwap/2, where apis the perpendicular lattice constant of the bi-axially

strained InGaN well, and Nw is the number of MLs in it), well composition (x) and VBO. The

results can be summarized as follows:

(i) The band-gap energy decreases with the increasesof well width,Lw, and indium

content, x. The former variation is due to the reduction in quantum confinement

whereas the latter one is due to the bowing effects in ternary alloys. Both Eg and

number of bound states are found to be dependent on well width and composition.

(ii) The PL data from Yoshikawa and coworkers [10] on three samples containing 1-ML,

3-ML and 5-ML SQWs have been modeled. Taking into account the error bars due to

the shortcomings originating from the TB parametrization in treating Stokes shift,

band offsets, lattice relaxation and interface-specific effects, the fitting of the TB

results to the PL data has led to the following two concluding remarks: (1) The

experimental samples are defect free and of high quality as being produced by a

successful pseudomorphicgrowth (rf-MBE). There is sufficient experimental evidence

18

supporting this point; (2) The high energy PL picks associated with the fundamental

emissions in the three studied wells must correspond to low indium contents with:

0.09 ≤ x ≤ 0.20; 0.13 ≤ x ≤ 0.20; and 0.17 ≤ x ≤ 0.25 to correspond to the three 1-ML, 3-

ML, and 5-ML SQWs, respectively. The first estimation of indium content is quite consistent

withx = 0.15-0.20 for 1-ML SQWs reported in the experimental work by Che and

coworkers [8]. Finally, this constraint of fractional filling is likely to remain not only a

challenge but a recipe inevitably to be used whenever coherent growth of highly-

lattice mismatched layered nanostructures is aimed.

(iii) In the three experimental samples, the maximum number of bound states in the e-

Well is 1, 2, and 3 to correspond to 1-ML, 2-ML, and 3-ML SQWs, respectively.

Conversely, The minimum number of bound states in the h-Well for all these

samples is 2 to be associated to the two interfaces. These bound states are

predicted for the least VBO value (if VBO = 0) and are due to the formation of two

potential dips at the interfaces asbeeninduced by the bi-axial strain. Likely VBO

should be greater than zero, and our prediction is in favor of enhancement of

oscillator strength to explain the observed sharp PL peaks.

ACKNOWLEDGEMENT

The authors are indebted to thank Drs. Bashar Issa and Thomas Fowler for critical reading of

the manuscript. This project is partially supported by a research grant from the College of

Science at UAE University (project number: COS/IRG-21/13).

19

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Figure Captions

Figure 1:Variations of eigen-energies (10 VBs and 20 CBs), calculated at Γ-point, versus InxGa1-

xN-well width Lw (in MLs units), for the cases of indium contents: (a) x = 0.25, (b) x = 0.50, (c) x =

0.75 and (d) x = 1.0. The VB-edge of bulk GaN is taken as an energy reference and VBO = 0. The

dashed lines are guide to the eye to show the energy limits of the e-Well.

Figure 2:(a) Same like in Figure 2 but Lw = 1 ML, x = 0.25 and the variation is for eigen-energies

versus VBO; (b) Variation of Eg versus VBO for three different well widths Lw = 1, 3 and 5 MLs.

Figure 3:(a-c)Energy bands along the ΓZ-high symmetry line of the BZ for three different well

widths Lw = 1, 3 and 5 MLs. The VBs are shown in lower panel with different energy scale for the

sake of clarity. The VB-edge of bulk GaN is taken as an energy reference and VBO = 0. (d-f) The

eigen-functions’ squared-amplitudes are monatomic-layer integrated and shown for the same

three preceding samples of SQWs. 6 VBs and 6 CBs are displayed. The bound states in the e-

Well are shown by red curves.

Figure 4:Statistics of bound states in the e-Well versus well width (Lw) and composition (x).

The maximum number (Nmax) is when VBO = 0, and the minimum number (Nmin) is when VBO =

1 eV are shown in red and black bars, respectively.

Figure 5:The TB results of three different samples of SQWs: Lw = 1, 3 and 5 MLs are shown in

the indicated symbols. The region between each pair of solid curves, which are similar in colors,

represent the theoretical estimation of Eg including the model and method error bars. The

experimental PL data for the fundamental transitions in the respective three SQWs are:0/BC =

3.4 eV, 0DBC = 3.11 eV, and 0EBC = 2.84 eV due to the work of Yoshikawa and coworkers [10]

are also shown.

22

Table Captions

Table 1:The sp3s* tight-binding parameters, with inclusion of spin-orbit interaction, are shown

for both zinc-blende GaN and InN. The original set was developed by the authors of Ref.[25]

and the ones shown have been a bit modified by us to fit the bulk experimental data of

Yoshikawa [10]. The on-site energies are shifted in order to make the VB-edge as an energy

reference.

Table 1

Compound F F F∗ F∗ H H

GaN -12.616 3.290 -1.284 9.050 14.300 14.300 0.003 0.015 InN -13.085 2.484 -0.623 9.528 14.776 14.776 0.003 0.002

Compound I0 4KF,F 4KL,L 4KL,M 4KF,, 4K,F, 4KF∗,, 4K,F∗,

GaN 4.50 -8.900 5.464 8.721 6.715 -7.352 7.844 -2.383 InN 4.98 -4.229 4.868 6.751 3.323 -5.609 8.976 -3.051

23

FIGURE-1

24

FIGURE-2

25

FIGURE-3

26

FIGURE-3

FIGURE-4

27

FIGURE-5