Generalized valence-force-field model of (Ga,In)(N,P) ternary alloys
Koushik Biswas, Alberto Franceschetti, and Stephan Lany
National Renewable Energy Laboratory, Golden, Colorado 80401, USA
Received 17 June 2008; published 22 August 2008
We present a generalized valence-force-field VFF model for the ternary III–V alloys III=Ga, In and
V=N, P to predict the formation energies and atomic structures of ordered and disordered alloy configura-
tions. For each alloy GaInN, GaInP, GaNP, and InNP the VFF parameters, which include bond-angle/bond-
length interactions, are fitted to the first-principles calculated formation energies of 30 ternary structures.
Compared to standard approaches where the VFF parameters are transferred from the individual binary III–V
compounds, our generalized VFF approach predicts alloy formation energies and atomic structures with con-
siderably improved accuracy. Using this generalized approach and random realizations in large supercells
4096 atoms, we determine the temperature-composition phase diagram, i.e., the binodal and spinodal decom-
position curves, of the Ga, InN, P ternary alloys.
DOI: 10.1103/PhysRevB.78.085212 PACS numbers: 81.05.Ea, 61.66.Dk, 64.75.Qr
I. INTRODUCTION
Alloys of group-III nitride semiconductors such as GaN
and InN and conventional III–V binary semiconductors
such as GaAs and InP are being extensively studied for
their potential applications in different fields. For example,
the Ga
1-x
In
x
N ternary pseudobinary alloy has widespread
applications in blue-green light emitting diodes and other
optoelectronic devices.
1,2
Recently, there have been attempts
to use III–V ternary and quaternary alloys for photoelectro-
chemical water-splitting applications.
3,4
In this respect, such
alloys offer several advantages over the III–V binaries. The
band edges of III–V alloys can be tuned over a wide range of
values to match the redox potentials for the water-splitting
reactions. As an example, it has been shown that the band
gap of GaN can be significantly reduced by alloying with
other elements from groups III or V, e.g., In and P.
5–7
Fur-
thermore, nitride alloys are relatively stable under photoelec-
trochemical operating conditions, which is not the case for
other III–V semiconductor alloys.
4,8–11
Accurate calculations of the formation energy and the
phase diagram of III–V and III-N ternary and quaternary
alloys often require the use of very large supercells that con-
tain hundreds if not thousands of atoms. Such calculations
are not feasible using standard first-principles methods.
Therefore, the idea is to develop an energy functional that
can be evaluated inexpensively and that can reliably predict
the formation energies of large and/or complex structures.
Many recent computational studies of III–V and III-N alloys
have relied on the valence-force-field VFF method
12–15
or
the cluster-expansion method.
16,17
In the VFF approach
originally developed by Keating
18
and later refined by
Martin
19
, the ground-state atomic positions and lattice vec-
tors are obtained by minimizing the strain energy, which is
described by a set of bond-stretching and bond-bending pa-
rameters. The ground-state strain energy can then be used to
compute the formation energy of ordered and disordered
structures
20–27
as well as the phase diagram of ternary and
quaternary alloys.
20–26,28–31
A major advantage of the VFF
method over first-principles calculations is the relatively
small computational cost required to relax large structures
and calculate the formation energy of systems including
thousands of atoms. At the same time, the VFF approach has
the advantage over “discrete” methods such as the cluster
expansion
32
of being able to accurately predict the atomic
positions of lattice-mismatched semiconductor alloys and
superlattices.
33,34
The input parameters of the VFF energy functional are
usually obtained from the experimentally measured or theo-
retically calculated elastic constants of the binary
constituents
35–39
and are then used to calculate the formation
energy of ternary and quaternary alloy systems.
20–24,28–31
In
the original VFF scheme developed by Keating
18,19
Keating
valence force field KVFF, several constraints are imposed
on the bond-bending parameter. For example, the bond-
bending parameter is assumed to be identical for the cation
or the anion-centered bonds e.g., the bond-bending param-
eter for N-Ga-N equals that for Ga-N-Ga. In addition, a
ternary or a quaternary alloy has mixed bonds, where three
different atomic species form the bond configuration e.g.,
Ga-N-In. In such cases, conventional KVFF defines the
bond-bending parameter as the arithmetic average of the
bond-bending parameters of the binary constituents that form
the mixed bond e.g., Ga-N-In is the “average” of Ga-N-Ga
and In-N-In. As we will show below see Sec. II and Table
III, such parametrization of the KVFF functional often pro-
duces rather large deviations in the predicted alloy formation
energies compared to density-functional calculations. In or-
der to improve on this model, Silverman et al.
25,26
proposed
an approach where these restrictions are lifted by the intro-
duction of individual bond-bending parameters for the differ-
ent atomic combinations. After fitting all VFF parameters to
first-principles calculations of the formation energy of or-
dered structures, they found for the specific case of the
Ga
1-x
In
x
P alloy that the description of the alloy formation
energies was improved over KVFF.
Using the VFF formulation without the restrictions of
KVFF, and additionally considering the bond-angle/bond-
length interaction parameters,
40
we have developed a gener-
alized ternary valence-force-field TVFF model for the four
ternary alloys in the Ga, InN, P zinc-blende system. For
each ternary alloy, i.e., GaInN, GaInP, GaNP, and InNP, the
TVFF parameters are obtained from a fit to the formation
PHYSICAL REVIEW B 78, 085212 2008
1098-0121/2008/788/08521210 ©2008 The American Physical Society 085212-1