VOLUME 86, NUMBER 12PHYSICAL REVIEW LETTERS19 MARCH 2001
Spatial Correlations in GaInAsN Alloys and their Effects on Band-Gap Enhancement
and Electron Localization
Kwiseon Kim and Alex Zunger
National Renewable Energy Laboratory, Golden, Colorado 80401
(Received 14 August 2000)
In contrast to pseudobinary alloys, the relative number of bonds in quaternary alloys cannot be deter-
mined uniquely from the composition. Indeed, we do not know if the Ga0.5In0.5As0.5N0.5alloy should be
thought of as InAs 1 GaN or as InN 1 GaAs. We study the distribution of bonds using Monte Carlo
simulation and find that the number of In-N and Ga-As bonds increases relative to random alloys. This
quaternary-unique short range order affects the band structure: we calculate a blueshift of the band gap
and predict the emergence of a broadband tail of localized states around the conduction band minimum.
DOI: 10.1103/PhysRevLett.86.2609PACS numbers: 71.20.Nr, 71.55.Eq, 78.20.Bh
The need to simultaneously control both the band
gap and the lattice constant of semiconductor alloys
prompted interest in not only ternary (e.g., Ga12xInxAs)
but also in quaternary (e.g., Ga12xInxAs12yPy ,
Ga12xInxAs12yNy ) alloys. The latter class of alloys
exhibits a fundamental topological difference with re-
spect to the former class:
macroscopic composition x uniquely defines the number
of bonds of each type, this is not the case in quaternaries
.Indeed, the ratio between the number of In-As
and Ga-As bonds in Ga12xInxAs is x:?1 2 x?, but in
Ga12xInxAs12yNy the chemical formula itself does not
reveal this information, for the number of In-As bonds is
not necessarily x?1 2 y?M nor is the number of Ga-N
bonds ?1 2 x?yM, where M is the total number of bonds.
This means that we do not know if the equimolar system
Ga0.5In0.5As0.5N0.5is to be thought of as GaAs 1 InAs or
as GaN 1 InAs. Since the statistical distribution of bonds
of different types controls the optical properties of the
alloy (e.g., Ref. ), this inherent topological ambiguity
in the bond distribution of quaternary alloys poses an
One can formulate the problem by noting that, in ad-
dition to the macroscopic compositions x and y, another
parameter j must be known in order to specify the near-
est-neighbor bond count ni2j. This “short range order”
(SRO) parameter can be defined as
While in ternary alloys the
j ? nIn-N?M 2 xInyN,(1)
where nIn-As?M ? xIn?1 2 yN? 2 j, etc. In a perfectly
random arrangement, one has j ? 0, whereas j . 0
means that InN 1 GaAs is the correct description of the
alloy, and j , 0 means that InAs 1 GaN is correct. In
general, j must be determined by minimizing the alloy
free energy, yielding the equilibrium function j?x,y,T?.
The following consideration clarifies the main physical
factors at play: Since Ga and N are smaller atoms than In
and As, respectively, the “small atom–large atom” bond
configuration Ga-As 1 In-N will be better lattice-matched
(thus, possess less strain) than the “small atom–small
atom” plus “large atom–large atom” bond configuration
Ga-N 1 In-As, respectively. On the other hand, the cohe-
sive energies of the respective binary zinc blende solids
follow the sequence  GaN . InN . GaAs . InAs
(being, respectively, 2.24, 1.93, 1.63, and 1.55 eV per
bond), so the (highly strained) Ga-N 1 In-As configu-
ration is preferred in terms of bond energy.
find the equilibrium configuration, one must search for
a configuration that minimizes the sum of strain plus
“chemical” (bond) energies, mediated by configurational
entropy effects at finite T.
In this Letter we apply finite temperature Monte Carlo
(MC) simulation to an empirical energy functional that de-
scribes the strain plus chemical energy of GaInAsN for
any configuration. We find that (i) j . 0, so the proper
description of this disordered alloy is InN 1 GaAs. Thus,
N prefers to be surrounded by In atoms (“local In enrich-
ment”), whereas As prefers to be surrounded by Ga. This
type of SRO is a novel feature of quaternary isovalent sys-
tems. (ii) This SRO has important consequences on the op-
tical properties of the alloy, leading to significant blueshift
of the band gap (i.e., reduced bowing) with respect to ran-
dom alloys. By applying a plane-wave pseudopotential
description to large alloy supercells with atomic SRO ob-
tained from the Monte Carlo simulation, we find that the
GaInAsN alloy band gap increases by ?100 MeV relative
to the random alloy. (iii) The band gap increase is due
to the fact that an alloy with SRO has a larger statistical
presence of N-centered In3Ga1clusters (local In enrich-
ment), and a smaller presence of N-In0Ga4clusters than in
random alloys of the same overall composition. Since the
energy level of the N-In3Ga1cluster is higher than that of
N-In0Ga4cluster, the statistical depletion of Ga-rich clus-
ters in favor of In-rich clusters leads to a diminished down-
ward level repulsion of the conduction band, relative to the
random alloy, hence to a larger band gap. This discov-
ery of SRO-induced band-gap shift is important since one
might hope to control j?x,y,T?, and thus the band gap, via
control of growth conditions. (iv) While it was previously
0031-9007?01?86(12)?2609(4)$15.00 © 2001 The American Physical Society2609
VOLUME 86, NUMBER 12PHYSICAL REVIEW LETTERS19 MARCH 2001
thought (e.g., Ref. ) that, in quaternary semiconductor
alloys, one can simultaneously tune band gaps and the lat-
tice constant by altering only the composition ?x,y?, we
find that the ensuing band gap is, in fact, not unique and
an additional thermodynamic variable ?jSRO? controls it.
(v) The level repulsion in the conduction band has an im-
portant consequence on localization:
cal distribution of As-centered As-GamIn42m tetrahedra
in ternary InGaAs causes a small ??1 meV? broadening
of the conduction band minimum (CBM), the distribution
of N-centered N-GamIn42mtetrahedra in GaInAsN leads
to a far broader range ??60 meV? of quasi-N-localized
band tail states near the CBM. This wide range is due
to the stronger perturbations by the higher energy cluster
(L1c-derived) levels in the nitride alloy. Such strong alloy
band edge fluctuations can locally capture carriers, affect-
ing carrier dynamics .
Our model energy functional depends both on site oc-
cupancy variables Sc, Saand on atomic positions Ri:
E??Sc;Sa?;?Ri,i ? 1,...,M?? ? Echem1 Estrain. (2)
Here Sc? 1 ?21? if Ga (In) is on cation site c, and Sa?
1 ?21? if As (N) is on the anion site a. Also, ?Ri,i ?
1,...,M? are the atomic positions of all M atoms in the
cell. Since we are not aiming at an accurate calculation of
the absolute energy E but rather the atomic configuration
?Sc;Sa? that has the best balance of strain vs bond energy,
a simple energy functional is sufficient. We use
where ni2jis the number of bonds of type ij, and eijis the
respective bond energy for which we use the experimental
cohesive energy  of binary compound ij. For a ternary
such as Ga12xInxAs, the chemical energy Echemis a con-
figuration independent constant xEInAs1 ?1 2 x?EGaAs
and hence does not decide the thermodynamic energy bal-
ance. This is so because, in a ternary, any cation (Ga or
In) atom is always bonded to As atoms, regardless of the
configuration. For the strain energy we use the valence
force field (VFF) model  and bond-stretching and bond-
bending force constants of Refs. [8,9].
We start with the simplest case of T ? 0.
ergies of some limiting thermodynamic states, e.g., the
random alloy or phase separation into constituents that
are coherent with the substrate, are depicted in the upper
half of Fig. 1(a). We consider an alloy that is lattice-
matched to a given substrate. This produces a relation-
ship between xInand yNfor each substrate. The energy
of the random j ? 0 alloy is calculated in Fig. 1(a) by
relaxing atomic positions of the random configuration us-
ing the VFF method while maintaining coherence. Fig-
ure 1(a) shows that for a GaAs substrate, at T ? 0, phase
separation into the strain-minimizing (but chemically un-
favorable) InN 1 GaAs configuration is much lower in
energy than phase separation into the chemical-energy-
minimizing (but highly strained) InAs 1 GaN configura-
While the statisti-
0.0 0.2 0.4 0.6 0.8 1.0
In concentration (x)
In concentration (x)
N concentration (y)
0.0 0.07 0.14 0.22 0.29 0.37
to InN + GaAs
to InAs + GaN
to InN + GaAs
phase sep. to InAs + GaN
N concentration (y)
Correlation parameter (ξ)
Random Alloy, T=
Phase separation to InN + GaAs
to InAs + GaN
states of GaInAsN lattice-matched to GaAs at T ? 0.
lowest energy for coherent phase separation (upper part) is
for InN 1 GaAs while, for incoherent phase separation (lower
part), the lowest energy is for InAs 1 GaN.
parameter j [Eq. (1)] expected for the limiting cases of phase
separation into InN 1 GaAs and InAs 1 GaN lattice-matched
to GaAs.For the random alloy, j ? 0, whereas for the
disordered alloy with SRO at T ? 360 K, j . 0.
(a) The energies of some limiting thermodynamic
(b) The SRO
tion. If one removes the coherence condition, permitting
each material to attain its own lattice parameter [so that
Estrain? 0, see bottom half of Fig. 1(a)], then InAs 1
GaN is the preferred decomposition. Note that, near the
free surface, atoms can relax freely so that Estrain? 0 and
the preferred configuration is GaN 1 InAs, just as in the
Figure 1(b) shows the j expected for the limiting cases
of phase separation at T ? 0 for the Ga12xInxAs12yNy
lattice-matched to GaAs. For phase separation into InN 1
GaAs, the proportion of InN bonds is the lesser of x
and y (in this case, y), thus j ? y 2 xy ? ?1 2 2.7y?y,
which is positive. For the InAs 1 GaN case, there are
two cases: (i) when y , 1 2 x, the number of InN bonds
is 0, thus j ? 2xy ? 22.7y2, which is negative. (ii),
when y . 1 2 x, the proportion of InN bonds is x 1 y 2
1, thus j ? 2?1 2 x??1 2 y? ? 2?1 2 2.7y??1 2 y?,
which is also negative.
The next question is how these T ? 0 behaviors of
Figs. 1(a) and 1(b) are modified at finite temperatures.
To obtain finite-temperature results we subject our energy
functional of Eq. (2) to a MC treatment, randomly flipping
spins and simultaneously relaxing atomic positions, so as
to minimize the energy of each spin configuration. This
“spin flip 1 atomic relaxation MC”  is executed us-
ing a cell of 512 atoms with 100000 spin flips per tem-
perature (clearly, a first-principles energy functional for
Eq. (2) is prohibitively costly). The simulation tempera-
ture is exponentially lowered with 12 steps, to equilibrate
the configuration at each temperature as it reaches the
ground state at the low temperature. Figure 1(b) shows
the resulting j?x,y,T? at T ? 360 K. Our result shows
that, for GaInAsN, j . 0, i.e., the InN 1 GaAs configu-
ration is preferred. The SRO parameter j is positive, be-
ing about 0.013 and 0.004 for Ga0.94In0.06As0.98N0.02 at
360 and 1200 K, respectively, and 0.042 and 0.031 for
Ga0.75In0.25As0.91N0.09at 360 and 1200 K, respectively.
VOLUME 86, NUMBER 12 PHYSICAL REVIEW LETTERS19 MARCH 2001
loy Ga0.89In0.11As0.96N0.04 (a) for a random configuration and
(b) with SRO determined by MC simulation at T ? 360 K.
Note the preponderance of N-In3Ga1and N-In4clusters. Solid
dots denote the positions of Ga and As atoms.
A visualization of the real space supercell of the al-
Figure 2 provides a visualization of the real space atomic
positions in the supercell of Ga0.89In0.11As0.96N0.04 with
SRO obtained with MC simulation at T ? 360 K. Tak-
ing statistics over many final configurations such as that
shown in Fig. 2 reveals that the main effect of SRO is
that the concentration of the N-centered In3Ga1clusters is
statistically enhanced significantly (local In enrichment),
whereas the In0Ga4clusters are statistically depleted rela-
tive to the random alloy. Also, the first nearest-neighbor
N-N pairs are depleted and the third nearest-neighbor N-N
pairs (“NN3”) are enhanced.
To understand the consequence of this special qua-
ternary SRO on the electronic properties, we calculated
the energy levels of supercells such as Fig. 2 by us-
ing the empirical pseudopotential method (EPM) as
described in Ref. .We use the pseudopotential
parameters of Bellaiche . Comparing the band gap of
a Ga0.94In0.06As0.98N0.02 supercell with MC-determined
SRO to that of an equivalent random alloy, we see
that SRO increases the band gap by ?100 meV, thus
reducing the optical bowing. This effect is similar to the
case of Zn12xMgxS12ySey alloys, where clustering of
MgS 1 ZnSe also increases the band gap , but the
effect here is considerably larger. Figure 3 depicts the
most important energy levels calculated for (i) isolated N
in GaAs, (ii) isolated N-InmGa42mclusters ?0 # m # 4?,
each embedded in the ternary In0.06Ga0.94As, (iii) con-
centrated (rather than isolated) but random 2% N alloy
of Ga0.94In0.06As0.98N0.02, and (iv) correlated (rather than
random) 2% N alloy of Ga0.94In0.06As0.98N0.02. To identify
the nature of the levels we project the corresponding alloy
wave functions onto the G, L, and X Bloch wave functions
, and indicate in Fig. 3 the percentage character. We
note the following features.
(i) Isolated N-InpGa42p clusters in InGaAs.—Much
like an isolated N impurity in GaAs [Fig. 3(a)] intro-
duces an a1?N? N-localized level above the CBM (at
Ec1 180 meV) and an L1c-derived level a1?L1c? at 
Ec1 286 meV, so does isolated N in GaInAs [Fig. 3(b)].
Energy levels (meV)
Γ / L / X (%)
percentages calculated for (a) isolated N in GaAs, (b) iso-
lated N-InmGa42m clusters ?0 # m # 4? each embedded in
a 4096-atom supercell of In0.06Ga0.94As, (c) random 2% N
alloy of Ga0.94In0.06As0.98N0.02, and (d) SRO 2% N alloy of
Ga0.94In0.06As0.98N0.02. Shaded areas denote band tails. The
zero of energy is the CBM of GaAs.
The most important energy levels with ?G?L?X?
However, here one can distinguish five different types of
isolated nitrogens, depending on their nearest-neighbor
coordination N-InmGa42m, where 0 # m # 4. We can
determine the energies of such clusters in the ultradi-
lute limit by placing them, one at a time, in a large
GaInAs supercell [Fig. 3(b)].
of the N-centered In0Ga4 level is the lowest, occurring
at Ec1 215 meV, while the In3Ga1 level is highest,
occurring at Ec1 229 meV, where Ecis the CBM energy
of GaAs. The N-cluster levels are strongly localized
around N and their wave functions are composed mainly
of the L1calloy states. The alloy CBM at Ec2 90 meV
is a delocalized G1c-like state. Note in Fig. 3 that the N
level in GaAs:N is closer to the CBM of GaAs than the N
levels in InGaAs:N. Thus, the addition of N to InGaAs
causes weaker coupling to the CBM (less bowing) than
the addition of N to GaAs. Specifically, 1% N reduces the
GaAs gap by 90 meV more than it reduces the InGaAs
gap.We will next see that SRO in GaInAsN further
reduces the bowing relative to random GaInAsN.
(ii) Concentrated N-InpGa42p clusters in InGaAs al-
loy.—As the N concentration increases from the ultradilute
limit [Fig. 3(b)] to 2% [Fig. 3(c)], all conduction levels
move down in energy, reflecting the alloy’s large bowing
parameter . Since we have only 6% In in our alloy,
in a random arrangement the In-poor clusters In0Ga4and
In1Ga3form statistically the majority clusters. But, in a
correlated arrangement, with SRO [Fig. 3(d)] the In-poor
In0Ga4and In1Ga3clusters are statistically nearly elimi-
nated in favor of In-rich In3Ga1clusters, (See also Fig. 2)
even though the total In content is conserved. Since the
L1c-like energy level of the In0Ga4cluster is closer to the
alloy G1c-like CBM [Fig. 3(b)], the L1c-like level would
interact more strongly with the G1c-like CBM than with the
higher energy level of the In3Ga1cluster. Consequently,
We find that the energy
VOLUME 86, NUMBER 12 PHYSICAL REVIEW LETTERS19 MARCH 2001 Download full-text
-0.4-0.3 -0.2 -0.10.0 0.10.20.3
(a) Strongly localized states
(b) Quasi-localized states
a1 (Γ1c )
Localization 1/R (A-1)
radius around N that contains 20% of the amplitude of the wave
function) vs the energy ei. (a) The distribution of localized
states for which there are at least 10% of N with the value of
R , 13 Å. (b) The distribution of the rest less-localized states.
The zero of energy is the average alloy CBM.
Average of inverse of the localization radius R?ei? (the
in the random alloy with its statistically abundant In0Ga4
clusters, the alloy CBM is pushed downwards more
??200 meV? than in the SRO alloy, which lacks In0Ga4
clusters.This explains why the correlated alloy has a
larger band gap (smaller bowing) than the random alloy.
(iii) Localized states near the CBM.—The mechanism
of the repulsion between the L1c-like N-InmGa42mcluster
states and the G1c-like CBM leads to another interesting
effect—strong alloy potential fluctuations near the CBM.
Because the N-InmGa42mclusters have an inherent statisti-
cal distribution of m values (unlike N in pure GaAs, having
a single N-Ga4local nearest-neighbor environment), and
because the energies of these L1c-like cluster levels
depend strongly on m (shifting by 14 meV for m ? 1 to
m ? 3, far more than the 0.1 meV shift in As-centered
clusters), we find a corresponding broad distribution of
G1c-like band tail levels near the CBM. By distributing
2% N (41 atoms) in a 4096-atom cell of In0.06Ga0.94As in
15 independent configurations s and calculating the elec-
tronic structure, we are able to examine the distribution
of localized states near the CBM. To see the localization,
we calculate for each N site a in alloy configuration s
the radius Ra?s,e? that localizes within it 20% of the
wave function c?e? at energy e. Figure 4(a) shows the
distribution of strongly localized states, while Fig. 4(b)
shows the distribution of the rest, less-localized states. We
see in Fig. 4(b) a ?60 meV (FWHM) wide distribution
of G1c-like band tail states (compared with ?1 meV in
InGaAs). The G1c broadening results from differ-
ent degrees of repulsion of G1c by the higher energy
N-InmGa42mcluster states of L1ccharacter . Below
the a1?G1c? CBM we see strongly localized states inside
the gap: the nearest-neighbor pairs N-N and triplets N3
and N30. These gap states and the quasilocal G1c-like band
tail states could explain the very different carrier dynamics
in quaternary GaInAsN relative to the ternary InGaAs
(Ref. ) including the longer lifetime associated with
stronger alloy fluctuations (e.g., Ref. ) and weaker
temperature dependence of the band gap (Ref. ).
In summary, we find that the strong perturbation of the
N-centered clusters leads in the quaternary GaInAsN to
a special form of SRO, a blueshift of the gap, reduced
bowing, and broad, localized band tail states as well as
This work was supported by the DOE/OS/BES/MSD
under Contract No. DE-AC36-99-GO10337.
L. Bellaiche for providing us with his EPM parameters.
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