APPLIED PHYSICS LETTERS 97, 092119 共2010兲
Band offsets of semiconductor heterostructures: A hybrid density
functional study
Amita Wadehra,a兲 Jeremy W. Nicklas, and John W. Wilkins
Department of Physics, The Ohio State University, Columbus, Ohio 43210, USA
共Received 15 July 2010; accepted 17 August 2010; published online 3 September 2010兲
We demonstrate the accuracy of the Heyd–Scuseria–Ernzerhof 共HSE06兲 hybrid functional for
computing the band offsets of semiconductor alloy heterostructures. The highlight of this study is
the computation of conduction band offsets with a reliability that has eluded standard density
functional theory. A special quasirandom structure models an infinite random pseudobinary alloy for
constructing heterostructures along the 共001兲 growth direction. Our results for a variety of
heterostructures establish HSE06’s relevance to band engineering of high-performance electrical
and optoelectronic devices. © 2010 American Institute of Physics. 关doi:10.1063/1.3487776兴
Heterostructures are ubiquitous in semiconductor technology. For instance, AlInAs/GaInAs is used for quantum
cascade lasers 共QCLs兲,1 infrared photodetectors, and high
electron mobility transistors 共HEMTs兲;2 GaInP/AlGaAs for
HEMTs, heterojunction bipolar transistors 共HBTs兲, and
phototransistors;3 AlInAs/InP for HEMTs;4 GaInP/GaAs
for HBTs;5 GaInAs/InP for single-photon avalanche
photodiodes;6 and AlInP/GaInP and disordered/ordered
GaInP for solar cells.7 Among the most important properties
that determine the feasibility and performance of heterostructure devices are the band offsets. These are the discontinuities between the valence band maxima 共VBM兲 or conduction
band minima of each semiconductor at their common interface, and act as barriers to electrical transport across the
interface. Band engineering of devices with desired properties, particularly QCLs and quantum dot-based devices, critically require a precise knowledge of band offsets. However,
reliable measurements and predictions of band offsets continue to be challenging despite extensive theoretical and experimental efforts.8–10
Density functional theory 共DFT兲 is an efficient method
for calculating electronic structure. The accuracy of DFT calculations is controlled by the exchange-correlation 共XC兲
functional. Local and semilocal functionals such as local
density approximation and Perdew–Burke–Ernzerhof 共PBE兲
共Ref. 11兲 underestimate band gaps, and in extreme cases predict small gap semiconductors as metals. Hybrid XC functionals, that include a fraction of Hartree–Fock 共HF兲 exchange, provide a promising alternative. In this letter, we
demonstrate the success of a hybrid functional Heyd–
Scuseria–Ernzerhof 共HSE06兲 共Ref. 12兲 in computing band
offsets of several technologically important heterostructures.
HSE06 includes a fraction, ␣, of screened, short-range HF
exchange to improve the derivative discontinuity of the
Kohn–Sham potential for integer electron numbers 共default
HSE06 uses ␣ = 0.25兲. This functional was recently used to
predict the band alignments throughout the composition
range of InGaN.13
Figure 1 highlights the success of HSE06 in computing
valence and conduction band offsets of the classic
Al0.5Ga0.5As/ GaAs heterostructure in close agreement with
a兲
Author to whom correspondence should be addressed. Electronic mail:
amita@mps.ohio-state.edu.
0003-6951/2010/97共9兲/092119/3/$30.00
experiment. HSE06 also shows significant improvement over
PBE for computing accurate lattice constants, band gaps, and
cation outermost d-orbital binding energies for the bulk
III–V compound semiconductors.14 These results signal advantages of HSE06 over traditional functionals for computing electronic properties.
Since the percentage of HF exchange in a hybrid functional is not a universal constant and the optimal value may
be system-dependent, it is worthwhile to study the variation
in band gaps as a function of ␣ in HSE06. Figure 2 demonstrates close agreement between the computed and experimental direct band gaps for III–V phosphides and arsenides
using the default HSE06 functional. Since ␣ = 0.30 describes
the band gaps of both AlAs and GaAs so well, a reasonable
choice for ␣ in the AlGaAs alloy would be 0.30, which we
refer to as mod-HSE06. For all other systems we use the
default ␣ = 0.25 as no other value will work for both components of the alloy.
We employ the average electrostatic potential
technique21 to compute the band offsets of the heterostructures S1/S2, where S1 and S2 are the semiconductors constructing a heterostructure. The bulk valence band edges are
aligned through a reference potential calculated across the
interface of the heterostructure. The difference in VBM of S2
and S1 is ⌬EVBM. The discontinuity in this reference potential across the heterostructure interface is defined as ⌬Vstep.
The valence band offset, ⌬Ev, is calculated as
mod−HSE06
0.42
2.19
(ind)
1.52
0.31
2.10
(ind)
1.52
1.52
(ind)
GaAs
AlGaAs
GaAs
1.12
0.18
0.21
0.27
0.26
AlGaAs
PBE
Experiment
AlGaAs
GaAs
FIG. 1. Band alignments for Al0.5Ga0.5As/ GaAs heterostructure computed
with mod-HSE06 共␣ = 0.30, see text兲 and PBE, in comparison with experiment 共see Ref. 9兲 The direct and indirect band gaps are shown for GaAs and
Al0.5Ga0.5As, respectively. The hybrid functional shows significant improvement over PBE for bulk band gaps and both valence and conduction band
offsets of the heterostructure.
97, 092119-1
© 2010 American Institute of Physics
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092119-2
Appl. Phys. Lett. 97, 092119 共2010兲
Wadehra, Nicklas, and Wilkins
4
TABLE I. Comparison of HSE06 and PBE calculated valence band offsets
⌬Ev 共eV兲 and conduction band offsets ⌬Ec 共eV兲 of III–V binary and pseudobinary alloy heterostructures with experiment values. All the pseudobinary
alloys in the present study are of the form A0.5B0.5C. The band offsets
computed with HSE06 show much better agreement with experimental values than PBE. Asterisks indicate that mod-HSE06 共␣ = 0.30兲 is used instead
of the default HSE06 functional 共␣ = 0.25兲. A positive value of ⌬Ev共⌬Ec兲 for
a heterostructure S1/S2 implies that the valence 共conduction兲 band edge of
semiconductor S2 lies higher than that of the semiconductor S1. The valence
and conduction band offsets have opposite signs for type-I 共straddling兲 heterostructures and the same sign for type-II 共staggered兲 heterostructures.
(a)
AlAs
Bandgap (eV)
3
AlAs(i)
2
GaAs
1
InAs
0
0
PBE
0.1
0.2
α
0.3
HSE06
0.4
Heterostructure
S1/S2
5
(b)
AlAs/GaAs
AlPb/GaPb
AlSbb/GaSb
AlGaAsb/GaAs
GaInP/GaAs
InP/GaInAs
InP/AlInAs
AlInPb/GaInP
AlInAs/GaInAs
GaInP/AlGaAs
d/o GaInPk
AlP
Bandgap (eV)
4
GaP
3
GaP(i) AlP(i)
2
InP
1
0
⌬Ec
共eV兲
⌬Ev
共eV兲
HSE06
PBE
Expt.
HSE06
PBE
Expt.
0.52ⴱ
0.54
0.38
0.26ⴱ
0.32
0.36
0.16
0.22
0.23
0.11
0.01
0.45
0.49
0.35
0.21
0.26
0.27
0.14
0.19
0.18
0.08
0.02
0.53a
0.55c
0.38a
0.27a
0.31a
0.34e
0.17f
0.24g
0.22h
0.09i
−1.02ⴱ
0.58
⫺0.67
−0.42ⴱ
⫺0.24
⫺0.38
0.22
⫺0.23
⫺0.57
0.25
⫺0.24
⫺1.08
0.57
⫺1.21
⫺1.12
⫺0.34
⫺0.42
0.17
⫺0.74
⫺0.54
0.19
⫺0.18
⫺1.05a
0.38c
⫺0.51d
⫺0.31a
⫺0.18a
⫺0.27e
0.25f
⫺0.26g
⫺0.51h
0.28j
⫺0.15l
a
0
PBE
0.1
0.2
α
0.3
HSE06
0.4
FIG. 2. 共Color online兲 Calculated direct band gaps 共open symbols兲 vs fraction of HF mixing 共␣兲 in HSE06 functional for III–V 共a兲 arsenides and 共b兲
phosphides. The vertical line passes through ␣ = 0.25, the default fraction for
HSE06. PBE results are shown at ␣ = 0. Filled 共partially filled兲 red symbols
indicate direct 共indirect兲 experimental band gaps. The experimental band
gaps do not vary with ␣ but are positioned according to the ␣ needed in
HSE06 to reproduce those values. Default HSE06 gives band gaps close to
experimental values for InAs, GaP, and InP. An optimal value of ␣ = 0.3 is
required for both AlAs and GaAs.
共S2-S1兲
共S2-S1兲
⌬E共S2-S1兲
= ⌬EVBM
+ ⌬Vstep
.
v
共1兲
The conduction band offset is determined from ⌬Ev and the
difference in bulk band gaps, ⌬Eg, as
.
= ⌬E共S2-S1兲
+ ⌬E共S2-S1兲
⌬E共S2-S1兲
c
g
v
From Ref. 9 and references therein.
Indirect band gaps of these semiconductors are used to get ⌬Ec.
c
From Ref. 15.
d
No expt. data; ⌬Ec calculated from expt. ⌬Ev and ⌬Eg.
e
From Ref. 16.
f
From Ref. 17.
g
From Ref. 18.
h
From Ref. 10 and references therein.
i
No expt. data; ⌬Ev calculated from expt. ⌬Ec and ⌬Eg.
j
From Ref. 19.
k
d = disordered and o = ordered with Cu–Pt 共L11兲 ordering.
l
From Ref. 20.
b
共2兲
Table I shows the success of HSE06 over PBE in computing
conduction and valence band offsets of 11 important semiconductor heterostructures in excellent agreement with experiments. All the pseudobinary alloys in this study are of the
form A0.5B0.5C. The error in ⌬Ec mainly stems from the
error in band gaps 关see Eq. 共2兲兴, the reason behind PBE’s
failure. HSE06 predicts the accurate nature and magnitude of
band gaps and hence band offsets. Sampling Table I, we start
with AlSb/GaSb. PBE underestimates the band gaps of both
constituents,14 predicts GaSb metallic, and produces a large
⌬Ec. HSE06 corrects the band gaps as well as offsets. A
major success of HSE06 is evident for the more complex
heterostructures such as AlInP/GaInP; HSE06 predicts AlInP
as indirect band gap material,14 in agreement with experiment, and computes accurate offsets. PBE predicts AlInP as a
direct gap semiconductor and gives incorrect ⌬Ec. Although
both HSE06 and PBE give similar band offsets for AlInAs/
GaInAs, the latter is misleading since PBE makes GaInAs a
metal;14 likewise for InP/GaInAs. The predictive power of
HSE06 for optoelectronic materials, demonstrated here for
heterostructure band offsets, holds also for the alloy concentration of direct-indirect band gap crossovers for pseudobinary alloys.22
We use the plane wave projector augmented-wave
method23 with PBE and HSE06 functionals in the VASP
code.24–26 We use a plane wave energy cut-off of 500 eV and
treat the outermost d electrons of cations as valence. The
Brillouin zone integration for the III–V binaries and their
heterostructures is performed on a ⌫-centered 8 ⫻ 8 ⫻ 8 and
8 ⫻ 8 ⫻ 1 k-point meshes, respectively. For the pseudobinary
alloys and their heterostructures, 4 ⫻ 8 ⫻ 4 and 4 ⫻ 8 ⫻ 2
⌫-centered k-point meshes are used, respectively. We relax
the interfacial atoms only when the anions are different in the
two semiconductors making a heterostructure.
The computation of an infinite random alloy requires a
large supercell and is arduous, particularly with the hybrid
functional. In order to simulate such an alloy using a finite
supercell with reasonable computational effort, we employ a
special quasirandom structure 共SQS兲,27 generated using the
alloy theoretic automated toolkit.28 Only the 16 cations on a
fcc sublattice were distributed according to the SQS construction, whereas the 16 anions are located on the separate
sublattice that makes up the 32 atom zinc blende supercell.14
We search all possible 16-atom fcc supercells with two
lattice-vectors orthogonal to 共001兲 for easy construction of a
heterostructure in a 共001兲 growth direction. The degree to
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092119-3
which this SQS matches an infinite perfect random alloy is
based on the behavior of the first few radial pair-correlation
functions. The SQS employed in this work has radial correlation functions that match the perfect random alloy up to the
fourth nearest neighbor pairs. The heterostructures of alloys
共binary III–V’s兲 are modeled as 4 + 4-layer thick supercells,
64 atoms 共16 atoms兲 with a 共001兲 interface. For either functional, the substrate lattice constant and average of the lattice
constants of the nearly lattice-matched bulk materials, computed with that particular functional, are used for the pseudobinary and binary alloy heterostructures, respectively.
To conclude, DFT originally created for ground state
properties has recently been extended to conduction states,
especially important for optoelectronic property predictions,
through the use of hybrid exchange correlation functionals.
Here HSE06 has been extensively exploited for semiconductor alloy heterostructures mimicked by appropriately constructed SQSs. For a broad selection of technologically important heterostructures, HSE06 succeeds in predicting 共i兲
the magnitude of optically direct band gap and 共ii兲 band offsets, especially the conduction band offset, so important to
optoelectronic properties. These achievements greatly extend
the utility of DFT for technological relevance. More computational resources may conquer strained and latticemismatched interfaces increasingly being used for advanced
devices.
This work was supported by DOE-BES-DMS 共Grant No.
DE-FG02-99ER45795兲. We used computational resources of
the NERSC, supported by the U.S. DOE 共Grant No. DEAC02-05CH11231兲, and the OSC. We thank Steven A. Ringel, Siddharth Rajan, and Richard G. Hennig for useful suggestions, and Georg Kresse for the beta version, VASP5.1.
1
Appl. Phys. Lett. 97, 092119 共2010兲
Wadehra, Nicklas, and Wilkins
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