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Transcript of Effects of dilute alloying on the quality of ultrathin InGaN/GaN single-quantum wells
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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
This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customerswe are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, andreview of the resulting proof before it is published in its final form. Please note that during the production processerrors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
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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: [email protected]
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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
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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-
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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].
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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
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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).
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REFERENCES
[1] For a review, see for instance, B.R. Nag in: Physics of Quantum-Well Devices, Kluwer,
Academic Publishers, Dordrecht, 2002.
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[3] J. Wu et al., Appl. Phys. Lett. 80 (2002) 4741.
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[5] E. Dimakis et al., Phys. Stat. Solidi A 205 (2008) 1070.
[6] M. Zhu et al., Phys. Rev. B 81 (2010) 125325.
[7] A. Yoshikawa et al., J. Crystal Growth 311 (2009) 2073.
[8] S. Che et al., Appl. Phys. Express 2 (2009) 021001
[9] A. Yoshikawa, et al., Appl. Phys. Lett. 90 (2007) 073101.
[10] A. Yoshikawa, et al., J. Vac. Sci. Technol. B 26 (2008) 1551.
[11] S. Schulz, et al., Phys. Status Solidi C 3 (2006) 3827.
[12] A.F. Jarjour, et al., Phys. Rev. Lett. 99 (2007) 197403.
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[14] R.H. Horng, et al., IEEE Electron Device Letters 30 (2009) 724.
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[15] H. Amano, M. Kito, K. Hiramatsu, and I.Akasaki, Jap.J. Appl. Phys. 28 (1989) L2112.
[16] Y. Dora, A. Chakraborty, L. McCarthy, S. Keller, S.P. Denbaars, and U.K. Mishra, IEEE
Electron Device Letters 27 (2006) 713.
[17] N. Tit, M.W.C. Dharma-wardana, Appl. Phys. Lett. 76 (2000) 3576.
[18] M. Henini, “Dilute Nitride Semiconductors”, by M. Henini (Elsevier, Oxford, 2006).
[19] A. Kobayashi, O.F. Sankey, J.D. Dow, Phys. Rev. B 25 (1982) 6367.
[20] N. Tit, J. Alloys Compd 503 (2010) 529.
[21] J.C. Slater and G.F. Koster, Phys. Rev. 94 (1954) 1498.
[22] T.G. Dargam, R.B. Capaz, B. Koiller, Brazilian Journal of Physics 27/A (1997) 299.
[23] G. Geneste, J.-M.Kiat, C. Malibert, Phys. Rev. B 77 (2008) 052106.
[24] C.G. Van de Walle, Phys. Rev. B 39 (1989) 1871.
[25] H. Hernandez-Cocoletzi, D.A. Contreras-Solorio, S.-J.Vlaev, I. Rodriguez-Vergas, Physica
E 41 (2009) 1466.
[26] N. Tit, A. Al-Zarouni, J. Phys. Condens. Matter 14 (2002) 7835.
[27] S.-H Wei, A. Zunger, Appl. Phys. Lett. 72 (1998) 2011.
[28] P.O. Lӧwdin, Phys. Rev. 18 (1950) 365.
[29] N. Tit, I. Obaidat, J. Phys. Condens. Matter 20 (2008) 165205.
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[30] M. Ferhat, J. Furthmȕller, F. Bechstedt, Appl. Phys. Lett. 80 (2002) 1394.
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.
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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
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FIGURE-1
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FIGURE-2
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FIGURE-3
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FIGURE-3
FIGURE-4
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FIGURE-5