Superlattices and Microstructures 36 (2004) 799–806
www.elsevier.com/locate/superlattices
Band offset calculations applied to III–V nitride
quantum well device engineering
A. Bhouria,∗, A. Ben Fredja, J.-L. Lazzarib, M. Saida
a Unité de Physique des Solides 99/UR/13-19, Département de Physique, Faculté des Sciences de Monastir,
Boulevard de l’Environnement, 5019 Monastir, Tunisia
b Centre de Recherche en Matière Condensée et Nanosciences, CRMC-N, UPR-CNRS 7251,
Laboratoire associé aux Universités Aix-Marseille II et III, Campus de Luminy, Case 913,
13288 Marseille cedex 9, France
Available online 2 November 2004
Abstract
Band offset calculations for zinc-blende pseudomorphically strained Al1−x Gax N/Al1−y Gay N
and Inx Ga1−x N/Iny Ga1−y N interfaces have been performed on the basis of the model solid theory
combined with ab initio calculations. From the results obtained, we have calculated, separately,
the valence and conduction band discontinuities of Inx Ga1−x N/GaN and GaN/Al1−x Gax N as
a function of the indium and gallium contents respectively. Using the latter results, we have
extended our study to simulate band discontinuities for strained Ga1−x Inx N/relaxed Al1−y Gay N
heterointerfaces. Information derived from this investigation will be useful for the design of lattice
mismatched heterostructures in modeling optoelectronic devices emitting at ultraviolet to near
infrared wavelengths.
© 2004 Elsevier Ltd. All rights reserved.
1. Introduction
Wide band gap (AlGaIn)N-based heterostructures are of considerable current interest
for realization of optoelectronic devices operating in the UV to near infrared wavelength
∗ Corresponding author. Tel.: +216 73500276; fax: +216 73500278.
E-mail address: bhouri_amel@yahoo.fr (A. Bhouri).
0749-6036/$ - see front matter © 2004 Elsevier Ltd. All rights reserved.
doi:10.1016/j.spmi.2004.09.036
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A. Bhouri et al. / Superlattices and Microstructures 36 (2004) 799–806
spectral range. The commercialization of high brightness blue and green light emitting
diodes has established (AlGaIn)N as an important material system for optoelectronic
devices [1,2]. In fact, the Alx Ga1−x N alloy system covers a wide ultraviolet (UV) spectral
range between the direct band gap of 3.4 eV for GaN and 6.2 eV for AlN at room
temperature [3]. The ternary alloy, Inx Ga1−x N, has the advantage of growing coherently on
GaN and it appears to be useful as quantum well (QW) layers in blue laser diodes [4]. In
addition, Al1−x Gax N is used to form strained heterostructures with GaN and Inx Ga1−x N in
light emitting diodes and in GaN/Al1−x Gax N field-effect transistors [3]. The vast majority
of research on III–V nitrides has been focused on the wurtzite crystal phase. The reason is
that most of the III–V nitrides have been grown on sapphire substrates which generally
transfer their hexagonal symmetry to the nitride film. Nevertheless, interest in zincblende nitrides has been growing recently. According to the general trends of the material
properties of III–V compounds semiconductors, zinc-blende GaN should be better suited
for controlled n-type and p-type doping than wurtzite. This latter system usually exhibits
a high n-type background carrier concentration, mostly originating from native defects
and/or from residual impurities [4]. Moreover, cubic GaN has a higher drift velocity and
a somewhat lower band gap than the wurtzite structure [5]. AlN can be grown in both
wurtzite and zinc-blende forms as well [5]. The existence of GaN and AlN in two crystal
phases with different electron band structures adds further interest to the Al1−x Gax N alloy
system.
From a fundamental viewpoint, to provide a basis for future wide energy gap
device concepts and applications based on zinc-blende III–V semiconductors, particularly
Al1−x Gax N/GaN lattice matched heterostructure devices, we have computed the band gap
energy and the electron effective masses of unstrained cubic Al1−x Gax N. Such calculations
are done using the empirical pseudopotential (EPM) method under the virtual crystal
approximation (VCA) [6] which is simple and expected to give quick and reasonably
reliable results. Recently, taking for the intrinsic InN band gap the commonly accepted
value of 0.78 eV [7–9], we have performed band offset calculations for lattice matched and
pseudomorphically strained Inx Ga1−x N/Iny Ga1−y N [10] heterointerfaces using a model
solid theory combined with ab initio calculations [6,10]. Using the electronic band
parameters obtained, a simulation of the band edges of an Inx Ga1−x N/GaN heterostructure
has revealed that the Inx Ga1−x N channel can emit photon energies in the spectral range
0.680–3.299 eV with relatively high oscillator strengths [10].
In the following, we present first the modeling of an Inx Ga1−x N/GaN and GaN/
Al1−x Gax N heterostructure. Then, we turn to considering the calculation of the conduction
and valence band alignment that results when the materials Al1−x Gax N and Inx Ga1−x N are
joined to form heterojunctions in various combinations.
2. Results and discussion
The band offsets E v and E c between the valence band (VB) maxima and the
conduction band (CB) minima, respectively, of two semiconductor compounds A and
B forming a heterostructure are among the most important parameters of interfacial
structures, governing both transport and quantum confinement.
A. Bhouri et al. / Superlattices and Microstructures 36 (2004) 799–806
801
2.1. Calculation of band offsets for strained Al1−x Ga x N/ Al1−y Ga y N
Using the empirical pseudopotential method under the virtual crystal approximation, the
band gap of zinc-blende Al1−x Gax N alloy is found to exhibit a bowing parameter of 1.0 eV
and to change from being indirect (Γ → X) for x = 0 to direct at Γ at a concentration of
x = 0.43 [6].
For an Al1−x Gax N/Al1−y Gay N heterostructure with the growth direction along 001,
the lattice mismatch gives rise to a biaxial strain in the (001) plane. The effect of this strain
on the energy band edges can be decomposed into hydrostatic and uniaxial contributions.
The hydrostatic strain shifts the overall energetic positions of the bands. The shearstrain component can split degenerate bands and also leads to an additional splitting of
the valence band energies when it is coupled to the spin–orbit interaction. For the ∆
conduction band under [001] uniaxial strain, the two valleys along [001] (denoted as
∆2 ) and the four valleys along [100] and [010] (denoted as ∆4 ) shift with respect to
their zero-strain positions [11]; however, the Γ conduction band is not affected by this
strain component. The strain also influences the band discontinuity at the strained layer
interfaces. To calculate the band offsets for the Al1−x Gax N/Al1−y Gay N heterointerfaces
we have adopted the procedure outlined in references [11]. The valence and conduction
band offsets E v and E c are obtained from
E vstrhh,lh = E vstrhh,lh (Al1−x Gax N) − E vuns
(Al1−y Ga y N)
hh,lh
= E vuns
+ δ E vhy + δ E vshhh,lh
hh,lh
(1a)
and
E cstr = E cstr (Al1−x Gax N) − E cuns (Al1−y Ga y N)
hy
= E vuns + E guns + δ E csh + δ E c
(1b)
where E vuns is the natural valence band discontinuity and E guns is the band gap
hy
sh are the band energy shifts under hydrostatic and uniaxial stains
difference. δ E v,c and δ E v,c
for the valence and conduction bands respectively. The subscripts hh and lh refer to heavy
holes and light holes respectively.
Using the above set of equations, we have calculated the band discontinuities for
heterointerfaces between strained Al1−x Gax N and relaxed Al1−y Gay N [6].
From these, we have deduced the valence and the conduction band offsets of
Al1−x Gax N/GaN and GaN/Al1−x Gax N heterostructures as a function of x over the whole
range of compositions. In fact the two latter systems are inequivalent and must be specified
separately. For the Al1−x Gax N/GaN interface, the valence and conduction band offsets
E v and E c are fitted by the following laws:
str
E v,hh
(Al1−x Gax N/GaN) = −0.928x + 0.928
(2a)
str
(Al1−x Gax N/GaN)
E v,lh
(2b)
= −0.926x + 0.920
1.341x 2 − 0.789x + 0.794
str
E c (Al1−x Gax N/GaN) =
2.756x 2 − 5.198x + 2.442
x < 0.43
x > 0.43.
(2c)
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A. Bhouri et al. / Superlattices and Microstructures 36 (2004) 799–806
However, E v and E c for the GaN/Al1−x Gax N heterostructure are approximated by
str
(GaN/Al1−x Gax N) = −0.928x + 0.928
E v,hh
(2d)
str
E v,lh
(GaN/Al1−x Gax N)
(2e)
= −0.926x + 0.920
−1.293x 2 + 0.694x − 0.747
str
E c (GaN/Al1−x Gax N) =
−2.709x 2 + 5.105x − 2.396
x < 0.43
x > 0.43.
(2f)
As reported in recent review papers by Vurgaftman et al. [12], the valence band offsets
(VBO) at both zinc-blende and wurtzite GaN/AlN interfaces range from 0.7 to 0.85 eV.
According to the above equations, we have obtained for cubic AlN/GaN and GaN/AlN a
VBO value of 0.92 eV. Unlike the valence band case, there is no report on the conduction
band discontinuity.
As an application, we present the modeling of an undoped heterostructure GaN quantum
well of 4 nm thickness, embedded between relatively thick Al1−x Gax N barriers. The
gallium composition is treated as a parameter. In this case, the band alignment for
GaN/Al1−x Gax N is of type I. This means that both electrons and holes are confined in
the GaN well layer.
In the one-band version of the envelope wave function approximation, the subband
energies for the conduction and valence bands can be computed from the effective
Hamiltonian:
Heff (z) = −
2 d 1 d
+ V (z)
2 dz m ∗ (z) dz
(3)
where z is the growth direction, m ∗ (z) is the effective mass of free carriers, V (z) is the total
potential energy. The values of the electron and hole effective masses are taken from [6,
10,13]. A linear interpolation is used to calculate these electronic band parameters for
the Al1−x Gax N layers. Using Eq. (2), we have calculated the conduction and valence
band edges as well as the energy levels for the GaN/Al1−x Gax N heterostructure. We
report in Fig. 1 the x-dependent band gap diagram of GaN/Al1−x Gax N (plot (a)) and
Inx Ga1−x N/GaN [10] (plot (b)) heterostructures, calculated for 4 nm well thickness. The
subscripts e, hh and lh refer to electrons, heavy holes and light holes. Plot 1(a) shows
Al1−x Gax N band edges and GaN subband energies with the emission energy as the
separation between the subbands. It can be seen that the transitions e1 → hh1 and
e1 → lh1 vary slightly in energy emission as the gallium composition increases and they
are around 3.275 eV (378.473 nm). This is due substantially to the strain in the GaN well
layer as the GaN energy band gap varies with the gallium composition in the Al1−x Gax N
substrate. Plot 1(b) shows that the transitions e1 → hh1 and e1 → lh1 shift down in
energy emission as the indium composition increases. In addition, the energy emission of
these interband transitions ranges from 0.851 (1456.5) to 3.299 eV (375.7 nm). This means
that the Inx Ga1−x N/GaN QW heterostructure can be made to emit at both visible and near
infrared wavelengths by simply adjusting the strain [10].
The idea that arises is to combine the two systems Al1−x Gax N and Inx Ga1−x N in
the hope of achieving emissions ranging from ultraviolet to near infrared wavelengths.
This has inspired us to carry out band offset calculations for Inx Ga1−x N/Al1−x Gax N
heterointerfaces.
A. Bhouri et al. / Superlattices and Microstructures 36 (2004) 799–806
803
Fig. 1. The subbands e1, hh1 and lh1 versus x for GaN/Al1−x Gax N (a) and Inx Ga1−x N/GaN (b) heterostructures
with a well 4 nm thick. Symbols CB and VB refer to the Al1−x Gax N and GaN band edges respectively. The
e1 → hh1 and e1 → lh1 energy emissions are shown as separations between the subbands.
2.2. Calculation of band offsets for strained I n x Ga1−x N/ Al1−y Ga y N
Valence and conduction band offsets for lattice matched and pseudomorphically
strained Inx Ga1−x N/Al1−y Gay N are calculated using results from Section 2.1 above as well
as the band offset transitivity. The so-called transitivity rule is given by
E v (A/C) = E v (A/B) + E v (B/C).
(4)
A similar expression would hold for E c . From Eq. (4), the Inx Ga1−x N/Al1−y Gay N band
offsets can be expressed as summations of Inx Ga1−x N/GaN and GaN/Al1−y Gay N band
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A. Bhouri et al. / Superlattices and Microstructures 36 (2004) 799–806
Fig. 2. Conduction band offsets E c = E c (x)−E c (y) and (b) valence band offsets E v = E v (x)−E v (y) for the
heavy holes (solid lines) and light holes (dotted lines) at (001) oriented Inx Ga1−x N/Al1−y Gay N heterointerfaces.
The band offsets are in eV.
A. Bhouri et al. / Superlattices and Microstructures 36 (2004) 799–806
805
offsets:
E vhh,lh (Inx Ga1−x N/Al1−y Ga y N) = E vhh,lh (Inx Ga1−x N/GaN)
+ E vhh,lh (GaN/Al1−y Ga y N)
E c (Inx Ga1−x N/Al1−y Ga y N) = E c (Inx Ga1−x N/GaN)
+ E c (GaN/Al1−y Ga y N).
(5a)
(5b)
Furthermore, we have recently calculated valence and conduction band discontinuities
for the Ga1−x Inx N/GaN interface as a function of the gallium composition [10]. We fit the
band offset results obtained with the following analytical expressions:
str
(Ga1−x Inx N/GaN) = −0.0192x 2 + 0.766x
E v,hh
(6a)
str
(Ga1−x Inx N/GaN) = −0.185x 2 + 0.766 − 0.008
E v,lh
E cstr (Ga1−x Inx N/GaN) = 0.964x 2 − 3.064x + 0.005.
(6b)
(6c)
With the use of Eqs. (2) and (6), we simulate the valence band and the conduction band
offsets for lattice matched and pseudomorphically strained Inx Ga1−x N/Al1−y Gay N(001)
oriented heterostructures as a function of x and y over the whole range of compositions.
Fig. 2 shows the conduction band (plot (a)) and valence band (plot (b)) offsets. The
plot 2(b) distinguishes between hh (solid lines) and lh (dotted lines) band offsets.
The separation in the valence band offsets is of interest in separately evaluating the
confinements of the two kinds of holes in Inx Ga1−x N QW structures. As can be noted from
Fig. 2, the conduction band offsets are negative and the valence band offsets are positive
for all compositions, meaning that the line-up is of type I.
3. Conclusion
Using our band offset calculations [6,10], we have deduced the valence and conduction
band offsets for GaN/Al1−y Gay N, Al1−x Gax N/GaN and Ga1−x Inx N/GaN strained/relaxed
heterointerfaces. As an application, we simulated the band edges of a GaN/Al1−y Gay N
heterostructure. A peculiar feature was revealed: the GaN channel can emit photon
energies in the ultraviolet wavelengths at around 3.275 eV. Using the results obtained,
we have estimated the valence and conduction band offsets for lattice matched and
pseudomorphically strained Inx Ga1−x N/Al1−y Gay N heterostructures. The line-up at
Inx Ga1−x N/Al1−y Gay N is found to be of “type I”. In view of the lack of theoretical and
experiment data, our results will be useful as a reference and more especially for designing
optoelectronic devices intended to operate over a large spectrum range.
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