Polytypism of GaAs, InP, InAs, and InSb: An ab initio study

PHYSICAL REVIEW B 84, 075217 (2011)
Polytypism of GaAs, InP, InAs, and InSb: An ab initio study
Christian Panse, 1,* Dominik Kriegner, 2 and Friedhelm Bechstedt 1
1 Institut für Festkörpertheorie und -optik, Friedrich-Schiller-Universität Jena and European Theoretical Spectroscopy Facility (ETSF),
Max-Wien-Platz 1, D-07743 Jena, Germany
2 Institute of Semiconductor and Solid State Physics, Johannes Kepler University Linz, Altenbergerstrasse 69 A-4040 Linz, Austria
(Received 6 April 2011; published 17 August 2011)
A systematic study of ground state properties of cubic (3C) and hexagonal (6H ,4H ,2H ) polytypes of
normal (nonnitride) III-V compounds is reported using well-converged density-functional calculations within
local density approximation, ab initio pseudopotentials, and projector-augmented wave method. Equilibrium
results are obtained for lattice parameters, cohesive energies, and bulk moduli. Internal degrees of freedom,
i.e., atomic relaxations, are taken into account. Trends with the hexagonality are discussed. Within the axial
next-nearest-neighbor Ising (ANNNI) model driving forces of the polytypism as well as the stacking fault
formation are derived. The results are compared with available experimental data.
DOI: 10.1103/PhysRevB.84.075217
PACS number(s): 61.46.Km, 61.66.Fn, 61.50.Ah, 61.50.Ks
I. INTRODUCTION
Highly anisotropic needlelike crystals such as nanowires
and whiskers have recently become a strong interest because
of their potential as building blocks of nanoscale electronic
and photonic devices. 1–3 The synthesis of such quasi-onedimensional
structures is based on a wide range of material
systems. However, III-V semiconductors, e.g., arsenides, phosphides,
and antimonides, are of particular interest. Because of
their almost direct character and small electron masses they are
especially suitable for active optoelectronics and high-speed
electronics. 1,2
Under ambient conditions the arsenides, phosphides, and
antimonides of Al, Ga, and In crystallize in the cubic zincblende
(zb, 3C) geometry. However, studies of the atomic
structure of III-V nanowires and nanorods grown in [111]
direction show significant deviations interpreted first as the
occurrence of a large number of twin-plane defects. 4,5 Earlier
growth experiments of free-standing GaAs wires combined
with transmission electron microscope (TEM) analysis revealed
that the variation of the growth temperature could lead
to the formation of the hexagonal wurtzite (wz,2H ) structure. 6
This picture has been confirmed in principle later. 7,8 However,
the coexistence of wurtzite and zinc-blende regions and their
intermixing through stacking faults on the (111) planes have
also been pointed out. In addition to the temperature influence,
recent experimental and theoretical studies of the atomic
morphology of III-V nanowire surfaces showed the influence
of surface effects on the energetic stability of wurtzite- and
zinc-blende-type nanowires. 9,10 Growth kinetics and nucleation
also influence the atomic geometry of freestanding
nanowires. 11 Besides TEM experiments also phase-sensitive
coherent X-ray diffraction (XRD) with a microfocused beam 12
demonstrated that the wurtzite contribution is a result of
stacking faults distributed along the wire. Recent TEM
and XRD investigations showed that in whiskers of indium
compounds, e.g., InAs, InSb, 13 and GaAs, 14,15 the fluctuating
stacking in [111] direction can lead to ordered structures which
were identified as the hexagonal 4H 13–15 and even 6H 16
polytype of these materials. Even structural pure wurtzite
nanowires as well as polytypic and twin-plane superlattices
can be grown. 8,17–20
The discovery of 2H and 4H polytypes in addition to the 3C
equilibrium structure opens the way to novel physics and ideas
for nanowire applications, especially due to the variation of
the electronic structure with the crystal structure. 21 In the past
several one-dimensional heterostructures have been realized
in semiconductor nanowhiskers 22 or even heterostructures
based on core-shell nanowires. 23 From the basic-physics point
of view heterocrystalline but homomaterial junctions are of
special interest. Such systems have been first predicted 24 and
realized 25 on the base of SiC polytypes. A layer of 3C-SiC
with a small energy gap embedded in 4H -SiC with a much
larger fundamental gap may form quantum wells for both
electrons and holes. In InP and GaAs nanowires a type-II
band alignment 21,26 has been observed between regions of 3C
and 2H , which at least may lead to quantum well structures
for one carrier type, the electrons.
Unfortunately, less is known about the polytypism of
normal III-V compounds. Only the relationship of the 3C
and 2H structures has been studied. On the basis of ab
initio total-energy calculations it has been concluded that the
wurtzite phase is a metastable high-pressure modification of
InAs and GaAs compounds. 26 A theoretical investigation on
growth thermodynamics of nanowires by Dubrovskii et al. 27
assigns the lowest formation energy to the 4H polytype. In
the framework of a local-density approximation (LDA) of
the density functional theory (DFT) it has been demonstrated
that in the case of zinc-blende compounds with a direct
conduction-band minimum Ɣ 1c , i.e., with the L 1c state above
Ɣ 1c , the corresponding wurtzite crystal also has a direct gap
which is however slightly larger. 28 There is a prediction 29 that
3C/2H heterostructures made of a direct semiconductor are
of type II. Recently the relative stability between 2H and 3C
structures in III-V nanowires is systematically investigated
based on empirical approaches 30 or DFT-LDA calculations. 10
Also the first quasiparticle band structures have been published
for InAs and GaAs in wurtzite phase. 31 Homojunctions have
been studied theoretically for InAs and InP nanowires. 32–34
In the present paper we study the energetic, structural, and
elastic properties of the most important hexagonal polytypes
6H ,4H , and 2H in comparison to the well-known unstrained
zinc-blende 3C structure of GaAs, InAs, InP, and InSb. The
cell-internal degrees of freedom, i.e., the atomic relaxations,
1098-0121/2011/84(7)/075217(9) 075217-1
©2011 American Physical Society

PANSE, KRIEGNER, AND BECHSTEDT PHYSICAL REVIEW B 84, 075217 (2011)
are taken into account. We derive results for the lattice
constants, atomic positions, cohesive energies, and bulk
moduli. The results are discussed versus the hexagonality
of the polytype. The polytypism is discussed within the
axial next-nearest-neighbor Ising (ANNNI) model. 35,36 It also
allows the derivation of the formation energies of two types of
stacking faults in the zinc-blende polytype.
II. MODELING AND COMPUTATIONAL METHODS
A. Total-energy calculations
The parameter-free total-energy and force calculations are
performed in the framework of the DFT 37 within the LDA 38
as implemented in the Vienna ab initio simulation package
(VASP). 39 The exchange-correlation (XC) functional is used
as parametrized by Perdew and Zunger. 40 We do not take into
account gradients of the electron density within the generalized
gradient approximation (GGA), since LDA gives better
structural parameters for conventional III-V compounds. 41 The
outermost s, p, and (in the case of Ga and In) d electrons
are treated as valence electrons whose interactions with the
remaining ions is modeled by pseudopotentials generated
within the projector-augmented wave (PAW) method. 42 The
electronic wave functions between the cores are expanded in
a basis set of plane waves. Its energy cutoff is tested to be
sufficient with 450 eV for the four III-V compounds GaAs,
InP, InAs, and InSb under consideration. The Brillouin-zone
(BZ) integrations are carried out on Ɣ centered 10 × 10 × M
k-point meshes according to Monkhorst and Pack 43 to achieve
an overall energy convergence beneath 1 meV. The value of M
has been reduced according to the number of layers in stacking
direction of the III-V polytype and amounts to 10 for zb.
B. Polytype description
Under ambient conditions the four compounds crystallize
in zinc-blende (zb) structure with one free parameter, the cubic
lattice constant a 0 , and a primitive unit cell of the fcc lattice that
contains two atoms, cation and anion, in a distance √ 3a 0 /4
in a cubic [111] direction. The space group is Td
2 (F 43m).
The same structure can be represented in a hexagonal Bravais
lattice with a nonprimitive unit cell containing six atoms,
three cations and three anions. In this case the zb crystal is
described by the stacking of cation-anion bilayers in [111]
direction called A, B, and C. Taking the number of bilayers
(p = 3) and the cubic character into account the atomic
arrangement can be also called 3C geometry (see Fig. 1). The
corresponding hexagonal lattice constants a and c fulfill the
relation 2c/(pa) = √ 8/3 with a = a 0 / √ √
2, c = a 0 3 and the
number of bilayers per unit cell p. The stacking sequence ABC
can, however, be changed, e.g., to AB, ABCB, or ABCACB, as
indicated in Fig. 1. This leads to hexagonal polytypes pH with
p = 2, 4, or 6 bilayers in [0001] stacking direction to obtain
translation invariance. Thereby, the 2H crystal corresponds
to the well known wurtzite (wz) structure. After introduction
of hexagonal Cartesian coordinates x ‖ [110] ‖ [2110],
y ‖ [112] ‖ [0110], and z ‖ [111] ‖ [0001] using Miller or
Bravais indices 44 one can define primitive basis vectors of
FIG. 1. (Color online) Three-dimensional perspective view of
primitive hexagonal unit cells of 3C and pH (p = 2, 4, 6) polytypes
of binary III-V compounds. The chains of bonds in the (11¯20) plane
being characteristic for a certain polytype are represented by thick
solid lines. The stacking sequences are indicated. Red circles (blue
dots) are cations (anions).
the corresponding hexagonal Bravais lattice for pH and 3C
polytypes, according to
a 1 = a (1,0,0),
a 2 = a 2 (−1,√ 3,0), (1)
a 3 = c (0,0,1).
The atoms in the hexagonal crystals are still fourfold coordinated.
Only the tetrahedrons could be slightly deformed in
c-axis direction as well known for SiC polytypes. 45
The resulting atomic geometries are depicted in Fig. 1. In
the hexagonal polytypes the atoms of the p cation-anion pairs
are all on trigonal axes in positions in agreement with the
C6v 4 (P 6 3mc) space-group symmetry. The general formation
laws are given in the textbook of Wyckhoff 46 or can be taken
from the characteristic chain structures within the unit cells in
Fig. 1. Using the three basis vectors a i (i = 1, 2, 3) in (1) and
displacing them eventually by a 1 or a 2 toward the coordinate
zero the 2p atoms are in the following positions. For the 2H
wurtzite structure one has
( )
1
(0,0,u),
2 ,2 3 ,1 2 + u (2)
with u(cation) = 0 and u(anion) = 3 + ε with a small dimensionless
cell-internal structural parameter ε describing
8
deviations
√
from ideal tetrahedrons together with 2c/(pa) ≠
8/3. In the 4H ,6H , and 3C cases the formation laws are
(0,0,u),
( 1
2 ,2 3 ,v )
,
( 2
3 ,1 3 ,w )
. (3)
For 4H it holds u(cation) = 0, u(anion) = 3
16 + ε(1),
v(cation) = 1 4 + δ(2) and 3 4 + δ(2), v(anion) = 7 + ε(2) and
16
15
16 + ε(2). w(cation) = 1 11
, w(anion) = + ε(1). For 6H one
2 16
has u(cation) = 0 and 1 2 , u(anion) = 1 8 + ε(1) and 5 8 + ε(1),
v(cation) = 1 6 + δ(2) and 5 6 + δ(3), v(anion) = 7 + ε(2) and
24
22
24 + ε(3), w(cation) = 1 3 + δ(3) and 2 + δ(2), w(anion) =
3
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POLYTYPISM OF GaAs, InP, InAs, AND InSb: AN ... PHYSICAL REVIEW B 84, 075217 (2011)
FIG. 2. (Color online) Isoenergy lines (in a distance of 1 meV) of the total energy E(c,a) per cation-anion pair of pH (p = 2, 4, 6)
polytypes of InSb without taking cell-internal relaxations into account. Their center allows to read the equilibrium values E, c, anda. The
optimized 2c/(pa) line is given in red. For the purpose of comparison the ideal ratio is also depicted in blue.
11
19
+ ε(3) and + ε(2). In the 3C case all positions are fixed
24 24
at u(cation) = 0, u(anion) = 1 4 , v(cation) = 1 3 , v(anion) = 7
12 ,
w(cation) = 2 11
, w(anion) = . The small parameters ε(i) and
3 12
δ(i) characterize the additional degrees of freedom, one for
2H , three for 4H , and five for 6H , which have to be optimized
in order to obtain the ground state structure.
Figure 1 suggests the inequivalence of the three bilayers
A, B, and C whose special arrangement leads to one of the
polytypes. In addition, comparing 3C with the hexagonal
polytypes the cubic (c) or hexagonal (h) character of the
cation-anion double layers can be defined according to
the parallel or nonparallel limiting bonds. In hexagonal layers
the bonds and hence the tetrahedra are rotated by 180 ◦ around
the c-axis. In other words, in a hexagonal layer the two adjacent
bilayers are equivalent in contrast to a cubic layer, where
the two adjacent bilayers differ. The ratio of the hexagonal
layers and the total number of bilayers per unit cell gives the
percentage hexagonality of the polytype, 0% (3C), 33% (6H ),
50% (4H ), and 100% (2H ). The denotation of the bilayers by
h and c leads to the earlier used polytype notation (h) 2 for 2H ,
(hc) 2 for 4H , and (hcc) 2 for 6H of Jagodzinski. 47
C. Atomic positions
In order to determine the equilibrium geometries we minimize
the total energy E with respect to the atomic coordinates.
In the zinc-blende case this is easy since only the equilibrium
lattice constant a 0 has to be computed with an accuracy of the
total energy per cation-anion pair below 1 meV. Total energy
calculations for various values of a 0 gave the equilibrium
geometry through a fitting to the Murnaghan’s equation of
state. This leads to bulk lattice constants a 0 = 5.611 Å (GaAs),
5.830 Å (InP), 6.032 Å (InAs), and 6.454 Å (InSb) as well
as to cohesive energies per pair (without spin-polarization
corrections for the free atoms) of E coh = 9.607 eV (GaAs),
9.643 eV (InP), 9.002 eV (InAs), and 8.189 eV (InSb).
The typical overbinding effect of the LDA is obvious. The
calculated lattice constants are smaller by 0.8% (GaAs),
0.7% (InP), 0.4% (InAs), and 0.4% (InSb) in comparison to
experimental (room-temperature) values. 48 This fact has to be
taken in mind when predicting equilibrium lattice parameters
for the hexagonal polytypes.
In the case of the hexagonal polytypes we apply a three-step
procedure. For vanishing parameters ε(i) and δ(i) the total
energy E = E(c,a) is determined on a regular (c,a) grid.
Then the cell-internal geometry against the total energy is
optimized. In the last step E(c,a) is minimized, but now
under the constraint of the obtained cell-internal parameters.
Within the last step we resolve the interplay of unit cell shape
(c,a) and the cell-internal relaxation, at least in a first-order
manner. Compared to a simultaneous optimization of all
lattice parameters our method is sufficient, as shown later in
Fig. 4. The resulting energy surface is illustrated in Fig. 2 for
the example of InSb. Our experience with the cell-internal
optimization is that the parameters ε(i) (i = 1, 2, 3) and
δ(i) (i = 2, 3) for 6H are so small that an influence by the
numerical accuracy cannot be excluded in agreement with
the low hexagonality. For that reason we decided to set these
parameters to zero and only present the results for 2H (ε) and
4H [ε(1), δ(2), and ε(2)]. This is somewhat different to silicon
carbide polytypes 49 where also the small relative cell-internal
displacements play a role for 6H .
The resulting (equilibrium) lattice constants c and a define
the (equilibrium) volume per III-V pair V pair = √ 3
2 a2 c p .This
formula allows us to define a volume dependence of the
total energy E = E(V ). From the E(c,a) data sets we obtain
the bulk moduli of the hexagonal polytypes following the
procedure proposed in Ref. 50. A fit to Murnaghan’s equation
of state 51 then yields the isothermal bulk modulus B 0 .
III. EQUILIBRIUM GEOMETRY AND STABILITY
A. General trends
The results of the minimization of the total energy as well
as the fit procedure to Murnaghan’s equation of state are
listed in Table I. Here the cell-internal parameters of 2H ,
4H , and 6H have been fixed at the ideal values. Clear trends
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PANSE, KRIEGNER, AND BECHSTEDT PHYSICAL REVIEW B 84, 075217 (2011)
TABLE I. Structural (c, a, V pair ), energetic (E with respect to the 3C value), and average elastic properties (B 0 ) of III-V polytypes. The
3C lattice constants are given referring to the corresponding hexagonal unit cell, i.e., a = a 0 / √ 2andc = a 0
√
3.
Compound Polytype 2c/p a 2c/(pa) V pair E B 0
(Å) (Å) (Å 3 ) (meV) (GPa)
GaAs 2H 6.5095 3.9556 1.6456 44.103 23.1 74.7
4H 6.4919 3.9606 1.6396 44.096 10.4 74.4
6H 6.4881 3.9626 1.6374 44.114 6.3 74.4
3C 6.4790 3.9676 1.6330 44.164 0.0 73.9
InP 2H 6.7515 4.1148 1.6408 49.499 10.6 71.2
4H 6.7409 4.1182 1.6369 49.503 4.5 71.1
6H 6.7379 4.1194 1.6356 49.510 2.5 69.2
3C 6.7319 4.1224 1.6330 49.539 0.0 71.0
InAs 2H 6.9894 4.2570 1.6419 54.846 19.1 61.9
4H 6.9764 4.2605 1.6375 54.834 8.1 59.9
6H 6.9727 4.2620 1.6360 54.844 5.0 58.5
3C 6.9653 4.2653 1.6330 54.869 0.0 60.3
InSb 2H 7.4806 4.5547 1.6424 67.198 20.7 46.9
4H 7.4654 4.5586 1.6377 67.176 9.6 46.9
6H 7.4613 4.5606 1.6360 67.198 5.8 47.0
3C 7.4527 4.5638 1.6330 67.281 0.0 46.8
are observed along the row 2H ,4H ,6H , and 3C. The lattice
constant in c-axis direction normalized to two cation-anion
bilayers as in 2H ,2c/p, increases while the lateral lattice
constant a decreases along the row 3C→2H . This happens
for all considered III-V compounds. In contrast, the volume
per cation-anion pair V pair only shows a very weak dependence
on the polytype. The volume is almost conserved. The weak
dependence on the hexagonality is monotonous only for InP.
For the other compounds the volume exhibits a minimum
for 4H , i.e., for 50% hexagonality. As a consequence of the
almost volume conservation the effective lattice constant ratio
2c/(pa) increases from its ideal value √ 8/3 = 1.6330 in 3C
to 2H . This behavior is in accordance with the empirical rule
of Lawaetz, 52 which states that compounds with zinc-blende
ground state should have c/a ratios larger than the ideal value
for their (unstable) wurtzite phase. This rule is supported
by first-principles calculations on III-V compounds 53 and
predictions of the ANNNI model. 54 The increase of 2c/(pa)
is almost linear in the polytype hexagonality as demonstrated
in Fig. 3(a). The structural trends along the polytypes are in
qualitative agreement with those for the prototypical material
SiC. 45
Along the variation of the anion X = P,As,Sbinthe
InX compounds, one observes a clear chemical trend. For
the deformation of the bonding tetrahedra c/(pa) is a clear
measure. It is the largest for the InX compound with the
largest bond length InSb. The trend with the size of the anion
and therefore with the average bond length is clearly visible.
A smaller anion covalent radius will reduce the deformation,
hence decrease c/(pa), smaller cations (In→Ga) will increase
it. Comparing only zinc-blende and wurtzite structures similar
findings have been made by Yeh et al. 55
FIG. 3. (Color online) Structural and energetic properties as function of the hexagonality of the polytype. (a) Renormalized lattice constant
ratio 2c/(pa). (b) Cohesive energy per cation-anion pair relative to the 3C value. Lines are only given to guide the eye.
075217-4

POLYTYPISM OF GaAs, InP, InAs, AND InSb: AN ... PHYSICAL REVIEW B 84, 075217 (2011)
The stability of the polytypes also follows a monotonous
decrease along the row 3C, 6H ,4H , and 2H (see Table I).
The variation of the relative total energy per pair is however
nonlinear as shown in Fig. 3(b). Among the considered
semiconductors the one with the lowest tendency to crystallize
in hexagonal polytypes is GaAs. The energy difference per
pair between wz and zb amounts to 23.1 meV and is almost
identical with previously published values of 25.0 28 and
23.4 meV. 29 Among the InX compounds a chemical trend
is visible. However, whereas the energy deviations from the
3C value are more or less identical for InAs and InSb with
the bigger anions, the energy differences for InP (see Table I)
are the smallest ones. That means that in thermal equilibrium
stacking fluctuations with respect to the ABC stacking of the
zinc-blende geometry should be easier possible for InP in
comparison with other III-V compounds. One reason may be
that among the covalent radii of 1.26 (Ga), 1.44 (In), 1.06 (P),
1.20 (As), and 1.40 Å (Sb) 56 the cation-anion radii difference
of InP is the biggest. To discuss chemical trends in a more
sophisticated way we compare the compounds via their charge
asymmetry coefficient (i.e., their ionicity) 0.294 (InSb), 0.316
(GaAs), 0.450 (InAs), and 0.506 (InP). 57 The largest value
for InP indicates its stronger ionic character which yields the
higher stability of the hexagonal phases among the considered
compounds.
The bulk modulus B 0 only weakly varies with the polytype.
The weak variations do also not show clear trends with the
hexagonality, similar to the case of SiC polytypes. 45 The
deviations to the experimental zinc-blende values are smaller
than 2%. 58
B. Influence of atomic relaxations
The relaxation of the atomic positions within the unit cells
of the hexagonal crystals 2H and 4H , i.e., the deviations of
the parameters ε (2H ) and ε(1), δ(2), ε(2) (4H ) are listed in
Table II. They are small but of the same order of magnitude as
in the case of the SiC and group-IV polytypes. 45,49,59 The most
important differences are the signs of ε (2H ) and δ(2) (4H ).
In contrast to SiC, in the III-V compounds there is a tendency
for a small increase of the cation-anion bond lengths parallel
to the c-axis relative to the bonds forming angles of about 70 ◦
with the c-axis (see Fig. 1). Our findings are not in agreement
with the conservation of the bond length in the average. Such
a conservation law leads to the relation ε = 1 8 [ 8 3 ( a c )2 − 1] in
the 2H case. 45 Considering a and c from Table I one obtains
the values ε =−22.4 (GaAs), −14.1 (InP), −16.1 (InAs),
−18.0 × 10 −4 (InSb). These displacements have the correct
sign but their absolute values are by a factor of 3 larger than
the optimized ones in Table II.
TABLE II. Cell-internal parameters ε (2H )andε(1), δ(2), ε(2)
(4H ). For definition see text.
ε × 10 4 ε(1) × 10 4 δ(2) × 10 4 ε(2) × 10 4
GaAs −8.5 4.4 5.3 −3.5
InP −4.2 2.5 1.6 −3.0
InAs −5.3 3.0 2.5 −3.5
InSb −6.2 3.9 3.9 −3.0
The additional degrees of freedom in the atomic positions
and hence the higher flexibility of the bonds lead to larger
relative changes a/a and c/c of the lattice constants with
respect to the zinc-blende values (see Table III). However,
these changes remain small. The absolute values for all
compounds are below 0.5%. As a consequence of the different
signs the largest changes happen for the effective c/a ratio,
more precisely 2c/(pa). The additional relaxation of the
atomic positions further amplifies this increase of the aspect
ratio for the hexagonal polytypes.
Additionally the cell-internal relaxation has significant
impact on the bond length along the c-axis inside the cubic
(d cub ) and hexagonal (d hex ) bilayers. While cubic bilayers
are compressed along the c-direction, the hexagonal ones
are stretched. This selective (inhomogeneous) deformation
leads to a relative change in the characteristic bond lengths
along the c-axis of about d cub =−0.15% for the cubic and
d hex =+0.42% for the hexagonal bilayers of the 4H -InSb
polytype. The different signs of the relative variations can
easily be explained by the different stacking sequences of a
hexagonal (ABA) and cubic (ABC) layer. In the hexagonal
case the neighboring layers are on top of each other which
yields a third next neighbor repulsive force as described by
Ito et al. 60 In contrast to that the adjacent bilayers of a cubic
layer are twisted by 60 ◦ which facilitates a compression. With
the calculated signs of the internal parameters (Table II) the
following relation is valid for the 4H phase of the calculated
III-V polytypes:
d hex − d cub = [ε(1) − ε(2) + δ(2)] c
= [|ε(1)|+|ε(2)|+|δ(2)|] c. (4)
In other words, the directions of the displacements through
the cell-internal relaxation, i.e., the signs of the internal
parameters, increase d hex − d cub . Therefore, the role of the
cell-internal parameters for the ground state geometry lies
within some kind of partial compensation of the inequality
of cubic and hexagonal bilayers. This bond-length difference
between cubic and hexagonal stackings explains the linear
scaling with hexagonality for the different properties. We point
out that, even if the internal relaxation is neglected, the cell
shape will change in a way that an average bilayer stretching
is realized. In the case of 4H -InSb both characteristic bond
lengths are stretched by about +0.17% with respect to the
value in 3C.
Our results on the cell-internal relaxation differ significantly
from former first-principles calculations by Wang
et al., 61 which reach comparable accuracy concerning internal
parameters. Wang et al. report c/a ratios beneath 1.6330 for
InAs and InSb with nonnegative ɛ of about +5 × 10 −4 (InAs)
and 0 (InSb). These findings are in contrast to the empirical
rule of Lawaetz 52 confirmed by Yeh et al. 53 and our work, after
which compounds with zb ground state should have c/a ratios
above the ideal 1.6330 in their wz phase. Also these findings
do not agree with the most recent XRD measurements. 20
The differences may come from an inappropriate treatment
of the cell-internal relaxation and the analytically described
pseudopotentials where the shallow d electrons are frozen into
the core.
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PANSE, KRIEGNER, AND BECHSTEDT PHYSICAL REVIEW B 84, 075217 (2011)
TABLE III. Relative deviations of lattice constants from those of
most stable 3C polytype (in percent). The deviations without (w/o)
and with (w) inclusion of the finite cell-internal parameters from
Table II are given separately.
a/a c/c 2c
pa / 2c
pa
w/o w w/o w w/o w
GaAs 2H −0.30 −0.35 0.48 0.55 0.77 0.91
4H −0.17 −0.19 0.24 0.28 0.40 0.47
6H −0.13 – 0.14 – 0.26 –
InP 2H −0.18 −0.21 0.30 0.36 0.48 0.57
4H −0.10 −0.11 0.14 0.17 0.24 0.28
6H −0.07 – 0.09 – 0.16 –
InAs 2H −0.20 −0.23 0.35 0.42 0.55 0.65
4H −0.11 −0.13 0.16 0.20 0.27 0.33
6H −0.08 – 0.11 – 0.18 –
InSb 2H −0.20 −0.25 0.38 0.47 0.58 0.73
4H −0.11 −0.13 0.17 0.22 0.28 0.35
6H −0.07 – 0.12 – 0.18 –
C. Comparison with experiment
Experimental data of the structural properties for hexagonal
polytypes of bulk III-V compounds are generally not available.
However, very recently lattice constants of the 2H and 4H
phases of InAs and InSb 20 have been obtained from XRD in
grazing incidence.
Asymmetric reciprocal space maps have been recorded in
order to obtain the lattice parameters of the 3C, 2H , and 4H
polytypes independently. The nanowires were grown using
chemical beam epitaxy (InAs) and metal-organic vapor phase
epitaxy (InAs and InSb). The cubic reference values used in
this study were a 0 = 6.058 Å (InAs) and a 0 = 6.478 Å (InSb)
taken from Ref. 62.
The experimental studies 20 yielded the relative changes
a/a =−0.23%, c/c = 0.42% for 2H -InAs, a/a =
−0.23%, c/c = 0.54% for 2H -InSb, a/a =−0.14%,
c/c = 0.19% for 4H -InAs, and a/a =−0.14%,c/c=
0.28% for 4H -InSb. The signs and the absolute values
are in excellent agreement with the computed deviations in
Table III. This holds especially after inclusion of the atomic
relaxation, which improves the agreement. Only for c/c of
InSb the calculated results seem to slightly underestimate
the measured ones. However, taking the inaccuracies of
the computational approaches and uncertainties due to the
nanowire measurements into account, we can state a excellent
prediction of the atomic geometries of the hexagonal III-V
polytypes by the DFT studies. Only the absolute distances have
to be rescaled by the small deviations (

POLYTYPISM OF GaAs, InP, InAs, AND InSb: AN ... PHYSICAL REVIEW B 84, 075217 (2011)
FIG. 4. (Color online) Influence of atomic relaxation on the relative lattice constant deviations from their ideal (i.e., 3C) values for InAs
and InSb. Constant effective 2c/(pa) ratios are indicated by blue dotted lines, while black dotted lines represent constant volumes V pair per
cation-anion pair for the 3C value and small deviations. Theoretical results (a) without and (b) with internal relaxation are connected by arrows.
For comparison also the result (c) for one full optimization, i.e., a, c,andε are optimized independently, is shown for 2H -InAs. Experimental
data (c/c, a/a) 20 are given by squares.
and J 1 /J 2 as variables. In the interesting parameter region one
multiphase degeneracy appears. For J 3 = 0 and J 1 =−2J 2
the phases of low hexagonality 3C, 6H , and 4H degenerate.
The III-V compounds are far away from the phase boundary
between 3C and 6H . Nevertheless, there are differences. While
3C-InSb is clearly most stable, the 3C polytype of InAs and
InP are much closer to the phase boundary. Therefore, these
compounds should show the strongest stacking fluctuations in
the nanorods under equilibrium conditions. The comparison
with the energy differences in Table I makes obvious that the
energy differences between wurtzite and zinc-blende phases
not only solely determine such a tendency. The phase diagram
directly reflects the chemical trend with respect to the charge
asymmetry coefficient g given in Table IV. As the ionicity
(respectively, the charge asymmetry coefficient g and the
covalent radii difference) increases the distance to the 3C-6H
phase boundary decreases. In a simple picture one can explain
this fact with an increasing intrinsic inhomogenity due to the
bigger difference of the covalent radii, which makes easier the
incorporation of stacking faults or twist of atomic layers.
C. Stacking faults
Further information can be extracted from the ANNNI
model. Most important is the energetics of the formation
of stacking faults in the zinc-blende polytype. 64 Since the
normal stacking in the zinc-blende polytype is ABCABC... the
intrinsic stacking fault (ISF) represents a removed bilayer from
the infinite stacking sequence. It may result in a ABCA/CAB...
stacking if a B layer is taken away. A possible formation
process could be related to condensation of vacancies. Another
process generating an ISF is a plastic glide caused by
TABLE IV. Interaction parameters J i (in meV per cation-anion
pair) of the ANNNI model (5). For comparison SiC values 64 are
also given. To illustrate the chemical trend the charge asymmetry
coefficients 57,65 g as a measure of the ionicity are given.
g J 1 J 2 J 3 J 3 /J 2 J 1 /J 2
InSb 0.294 10.80 −0.38 −0.45 1.200 −28.8
GaAs 0.316 12.03 −0.58 −0.48 0.828 −20.9
InAs 0.450 9.85 −0.73 −0.30 0.414 −13.6
InP 0.506 5.68 −0.40 −0.38 0.938 −14.2
SiC 0.476 1.18 −2.34 −0.32 0.137 −0.5
FIG. 5. (Color online) Phase diagram for the polytypes within
the ANNNI model. The 3C equilibrium structures are indicated by
black dot (GaAs), blue square (InP), red triangle (InAs), and green
diamond (InSb). For comparison the 4H equilibrium structure of SiC
is indicated by an open circle.
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PANSE, KRIEGNER, AND BECHSTEDT PHYSICAL REVIEW B 84, 075217 (2011)
TABLE V. Stacking fault formation energies E (meV/atom) and
γ (mJ/m 2 )forthe3C structure. The experimental values are taken
from Refs. 67 and 68.
ESF
ISF
E γ E γ
Calc. Calc. Calc. Expt. Calc. Expt.
GaAs 39.7 46.7 43.9 47 ± 5 51.6 55 ± 5
InP 16.5 18.0 19.6 17 ± 3 21.3 18 ± 3
InAs 31.2 31.7 35.3 30 ± 3 35.9 30 ± 3
InSb 36.6 32.5 39.9 43 ± 4 35.4 38 ± 4
shear stress applied to the 3C crystal. One speaks about an
extrinsic stacking fault (ESF) after adding a double layer to
the stacking sequence, for example, due to condensation of
eigeninterstitials. Adding a C bilayer the resulting stacking
sequence is ABCA/C/BCABC.... The occurrence of such
stacking faults can also be discussed in terms of a twist by
180 ◦ of the three equivalent bonds between two bilayers in
a bonding tetrahedron nonparallel to the [111] axis. Then,
besides staggered (cubic) layers also eclipsed (hexagonal)
bilayers appear. 64
Within the ANNNI model (4) the formation energies of
such a stacking fault E(ISF/ESF) with respect to the infinite
stacking of the 3C polytype can easily be derived. Per atom
in a two-dimensional cell perpendicular to the [111] axis one
finds
E ISF = 4J 1 + 4J 2 + 4J 3 ,
E ESF = 4J 1 + 8J 2 + 8J 3 .
With the sign of the interaction parameters in Table IV
it becomes obvious that the extrinsic stacking faults are
energetically favorable versus the intrinsic stacking faults in
agreement with experimental values, as known for silicon
crystals. 66 The stacking-fault formation energies γ (ISF/ESF)
per unit area follow from E(ISF/ESF) by division with the
area √ 3a0 2 /4 of one atom in a (111) plane.
The calculated energy results are listed in Table V and
compared with measured values. 67,68 The interesting point
is that the stacking fault formation energies are not directly
related to the bonding energy or the cohesive energy of
the compound. Within the In compounds there is a clear
correlation with the bond length or the lattice constant.
However, the direct proportionality is in contrast to the scaling
with the inverse lattice constant found for the stacking fault
formation energies of covalent group-IV semiconductors. 64,69
(7)
Experimentally it is rather difficult to distinguish between
extrinsic and intrinsic stacking faults. In the literature 67,68 only
formation energies of stacking faults are given. However, the
measured values given in Table V are in excellent agreement
within the error bars with the formation energies of intrinsic
stacking faults computed within the ANNNI model (see
interaction parameters in Table IV). The absolute values of
the formation energies for especially extrinsic stacking faults
E(ESF) of the order of the thermal energy k B T at room
temperature indicate that even in bulk zinc-blende crystals of
III-V compounds stacking faults are likely. In nanorods grown
in [111] direction stacking fluctuations and therefore the local
formation of hexagonal polytypes pH are probable.
V. SUMMARY AND CONCLUSIONS
In summary, we have performed first-principles calculations
for the structural, energetic, and elastic properties of
hexagonal polytypes 2H , 4H , and 6H of common III-V
semiconductors which crystallize in zinc-blende 3C geometry
under ambient conditions. We have observed clear tendencies
for the enlargement of the average cation-anion bilayer
thickness c/p in [0001] direction but a decrease of the lateral
(in-plane) lattice constant a with respect to their ideal values
derived from the 3C crystal. On the one hand, the volume
per cation-anion pair is almost conserved. On the other hand,
the effective ratio 2c/(pa) increases with the hexagonality of
the polytype. The agreement with measured lattice-constant
variations c/c and a/a for 2H -InAs, 4H -InAs, 2H -InSb,
and 4H -InSb is reasonably good. It is improved after inclusion
of the relaxation of the atomic positions within the hexagonal
unit cells.
The energy findings can easily be explained within an
ANNNI model taking the interaction up to third-nearestneighbor
bilayers into account. The phase diagram constructed
within the ANNNI model shows that InP and InAs are closest
to the phase boundary between 3C and 6H . A comparison of
the charge asymmetry coefficients of the compounds relates
the chemical trend for the energetics to the bond ionicity.
The ANNNI model also allows the computation of stacking
fault formation energies. Excellent agreement is found with
measured values.
ACKNOWLEDGMENTS
The work was financially supported by the Fond zur
Förderung der Wissenschaftlichen Forschung (Austria) in the
framework of SFB 25, Infrared Optical Nanostructures, and the
EU e-I3 ETSF project (GA No. 211956). Grants of computer
time from the HLRZ Stuttgart are gratefully acknowledged.
* christian.panse@uni-jena.de
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