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

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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