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Title:
Native defects and self-diffusion in GaSb
Year: 2002
Version: Final published version
Please cite the original version:
Hakala, M. & Puska, M. J. & Nieminen, Risto M. 2002. Native defects and self-diffusion
in GaSb. Journal of Applied Physics. Volume 91, Issue 8. 4988-4994. ISSN 0021-8979
(printed). DOI: 10.1063/1.1462844.
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Native defects and self-diffusion in GaSb
M. Hakala, M. J. Puska, and R. M. Nieminen
Citation: Journal of Applied Physics 91, 4988 (2002); doi: 10.1063/1.1462844
View online: http://dx.doi.org/10.1063/1.1462844
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Native defects and self-diffusion in GaSb
M. Hakala,
a)
M. J. Puska, and R. M. Nieminen
Laboratory of Physics, Helsinki University of Technology, P.O. Box 1100, FIN-02015 HUT, Finland
共Received 19 July 2001; accepted for publication 29 January 2002兲
The native defects in GaSb have been studied with first-principles total-energy calculations. We
report the structures and the formation energies of the stable defects and estimate the defect
concentrations under different growth conditions. The most important native defect is the Ga
Sb
antisite, which acts as an acceptor. The other important defects are the acceptor-type Ga vacancy and
the donor-type Ga interstitial. The Sb vacancies and interstitials are found to have much higher
formation energies. A metastable state is observed for the Sb
Ga
antisite. The significantly larger
concentrations of the Ga vacancies and interstitials compared to the corresponding Sb defects is in
accordance with the asymmetric self-diffusion behavior in GaSb. The data supports the
next-nearest-neighbor model for the self-diffusion, in which the migration occurs independently in
the different sublattices. Self-diffusion is dominated by moving Ga atoms. © 2002 American
Institute of Physics. 关DOI: 10.1063/1.1462844兴
I. INTRODUCTION
Gallium antimonide is a technologically interesting ma-
terial due to its applications to high speed electronic and long
wavelength photonic devices 共for a review, see Ref. 1兲. The
narrow, direct band gap of 0.73 eV 共at 300 K兲 of GaSb en-
ables its use in the fabrication of infrared detectors and
sources. GaSb can be lattice matched with ternary and qua-
ternary III-V compounds to form a substrate material for
various optical communication devices, which operate in the
2–4
m regime. Furthermore, GaSb has a high electron mo-
bility and saturation velocity, which can be utilized in high
electron mobility transistors. In these devices band offsets of
GaSb with similar compounds can be used to create the con-
ducting two-dimensional electron gas. Interesting basic prob-
lems in GaSb are the origin of its residual acceptor concen-
tration and the recently observed strongly asymmetric
self-diffusion.
2,3
GaSb is always p type irrespective of
growth techniques and conditions.
1
The typical residual ac-
ceptor concentration is ⬃10
17
cm
⫺ 3
, and the acceptors are
associated with native Ga vacancies and antisite defects.
1
Nonstoichiometric growth conditions have been used to re-
duce the acceptor concentration and increase the hole
mobility.
4,5
Self-diffusion is related to material degradation,
and it is strongly correlated to lattice defects and the diffu-
sion of impurities.
The recent experiments by Bracht et al.
2,3
have demon-
strated that Ga diffuses several orders of magnitude faster
than Sb. The authors have proposed a model according to
which the diffusion takes place independently in the different
sublattices, which points to a next-nearest-neighbor diffusion
mechanism. According to the model Ga diffuses via vacan-
cies, whereas Sb diffuses via interstitials. The large differ-
ence in the diffusion coefficients derives from a significant
concentration difference between the Ga vacancies and the
Sb interstitials. The concentrations are explained to be
strongly affected by amphoteric transformations, which in-
crease the number of Ga vacancies and reduce the number of
Sb interstitials. Since the defects may possess several charge
states, the energetics of amphoteric reactions is proposed to
depend on the Fermi level position.
Electronic structure calculations have been extensively
used in the analysis of properties of native defects in III-V
materials. GaAs is the most studied material, and to a large
extent the results can be generalized to other III-V materials.
6
The defects with intrinsic metastabilities have attracted par-
ticular interest in III-V materials.
7
For example, the detection
of a defect may depend on whether it is in the stable or the
metastable configuration. It is interesting to see whether this
structural property is found in GaSb. For GaSb, there are few
electronic structure calculations available. The existing data
is usually calculated as a part of a series where the chemical
trends of III-V materials have been analyzed. Ku
¨
hn et al.
8
have studied vacancies and antisite defects using charge-self-
consistent empirical tight-binding methods. Xu
9
has studied
vacancies with semiempirical tight-binding methods, while
Talwar and Ting
10
have studied vacancies and antisites using
Green’s function tight-binding methods. Puska
11
has studied
vacancies and antisites with linear-muffin-tin-orbital Green’s
function methods. All the above-mentioned studies are for
ideal atomic structures.
In this work we have studied the electronic structures
and the formation energies of native defects in GaSb with ab
initio methods. The calculations have been performed for
fully relaxed vacancies, interstitials, and antisite defects. We
show that the defects behave to a large extent in a similar
way to those in GaAs. We calculate the defect concentrations
at typical growth temperatures. We show that the residual
acceptor is the gallium antisite defect and estimate the effect
of growth conditions on the residual hole concentration. We
study the model by Bracht et al.
2,3
for the asymmetric self-
diffusion, the proposed amphoteric transformations, and
comment on the probability of the nearest-neighbor diffusion
mechanism.
a兲
Electronic mail: moh@fyslab.hut.fi
JOURNAL OF APPLIED PHYSICS VOLUME 91, NUMBER 8 15 APRIL 2002
49880021-8979/2002/91(8)/4988/7/$19.00 © 2002 American Institute of Physics
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The paper is organized as follows. In Sec. II we describe
the theoretical methods and the calculation of the defect for-
mation energies and concentrations. In Sec. III we present
the results for the atomic and electronic structure of the
stable native defects and study the concentrations of the de-
fects, especially those of residual acceptors. In Sec. IV the
self-diffusion is discussed in the light of our data. Finally,
Sec. V presents the conclusions.
II. METHODS
The structures and the total energies of the native defects
are calculated within the density-functional theory
12
and the
local-density approximation 共LDA兲.
13
The plane-wave
pseudopotential method
14
is used with nonlocal norm-
conserving pseudopotentials: for Ga a Hamann type
15
and for
Sb a Troullier-Martins type.
16
The 3d and 4d electrons of Ga
and Sb, respectively, are included in the core. The nonlinear
core-valence exchange-correlation scheme
17
is used for both
elements. With these pseudopotentials the theoretical
共experimental
1
兲 lattice constant is 6.01 Å 共6.10 Å兲, the bulk
modulus 0.548 Mbar 共0.563 Mbar兲 and the band gap is 0.31
eV 共0.82 eV at 0 K兲. The band gap underestimation is typical
for the LDA calculations.
We have used a 64-atom supercell with the optimized
lattice constant and a k-point sampling proposed by Makov
et al.
18
The sampling consists of points 共0, 0, 0兲 and 共
1
2
,
1
2
,
1
2
兲
in units of (2
/a), where a is the lattice constant of the
supercell. This sampling and supercell size have been found
to be a good approximation for the formation energies for
point defects in semiconductors,
19
tending to minimize the
interactions between a defect and its periodic replica.
18
The
approximation may lead to a slight indefiniteness in the point
symmetry group of the defects.
19
We use a 22-Ry kinetic
energy cutoff, which leads to well-converged atomic struc-
tures. Our starting configurations are the ideal structures for
vacancies, antisites, and interstitials. For interstitials we use
as starting configurations the two inequivalent tetrahedral
sites and the hexagonal site. A small random component is
added to the initial atomic coordinates to break the high sym-
metry of the initial configuration. The ions are allowed to
relax with no symmetry restrictions following the
Hellmann–Feynman forces until the largest force component
on any atom is less than 5 meV/Å.
The formation energies of the defects are calculated fol-
lowing the formalism used by Zhang and Northrup.
20
The
formation energy of a defect is written as
E
f
⫽ E
D
⫹ q
共
e
⫹ E
v
兲
⫺ n
Ga
Ga
⫺ n
Sb
Sb
, 共1兲
where E
D
is the total energy of the supercell containing the
defect in question, q the charge 共electrons or holes兲 trans-
ferred to the defect from a reservoir,
e
the chemical poten-
tial of the reservoir, and E
v
the valence band maximum
共VBM兲.
e
corresponds to the Fermi level calculated from
the VBM. To account for the Coulomb interaction between a
charged defect and its periodic replicas, we have added a
Madelung-type monopole correction term to E
D
.
21
n
Ga
and
n
Sb
are the number of each element in the supercell. The
atomic chemical potentials obtain values below their bulk
precipitates:
Ga
⭐
Ga(bulk)
and
Sb
⭐
Sb(bulk)
. It is also re-
quired that the sum of the chemical potentials equals the
chemical potential per atom pair in the bulk compound:
Ga
⫹
Sb
⫽
GaSb
.
Equation 共1兲 can be rewritten as
E
f
⫽ E
D
⬘
⫹ q
e
⫺
1
2
共
n
Ga
⫺ n
Sb
兲
⌬
, 共2兲
where
E
D
⬘
⫽ E
D
⫺
1
2
共
n
Ga
⫹ n
Sb
兲
GaAs
⫺
1
2
共
n
Ga
⫺ n
Sb
兲
共
Ga
共
bulk
兲
⫺
Sb
共
bulk
兲
兲
⫹ qE
v
共3兲
and
⌬
⫽
共
Ga
⫺
Sb
兲
⫺
共
Ga
共
bulk
兲
⫺
Sb
共
bulk
兲
兲
. 共4兲
Ga(bulk)
and
Sb(bulk)
are calculated from their elemental
bulk phases. The electron chemical potential
e
is allowed to
vary between zero and the experimental band gap value, 0
⭐
e
⭐E
g
. The limits for the chemical potential ⌬
are
given by the heat of formation ⌬H of GaSb, ⫺ ⌬H⭐⌬
⭐⌬H. The values closer to ⫺ ⌬H correspond to Sb-rich
growth conditions and values closer to ⌬H to Ga-rich
growth conditions. If ⌬
were defined using the chemical
potentials of other than the elemental solid phases, i.e., those
of the elemental molecular gases, the range of variation of
⌬
would increase. We use the experimental value of 0.43
eV for ⌬H.
22
Our calculated theoretical value is (⫺ 0.1
⫾ 0.3) eV; the discrepancy may be explained by the short-
comings of the LDA calculation, since ⌬H requires a very
accurate calculation of the energy difference between the
compound and the elements in bulk phase.
The concentration of a given defect is
C
D
⫽ z
D
N
s
exp
共
⫺ F
f
/k
B
T
兲
, 共5兲
where z
D
is the number of equivalent configurations for a
particular defect per sublattice site, N
s
the density of sublat-
tice sites (1.7⫻10
22
cm
⫺ 3
), F
f
the free energy of defect
formation, k
B
the Boltzmann constant, and T the equilibrium
sample temperature. In our calculations we have omitted the
entropy part in F
f
and use only the formation energies when
estimating the concentrations. The entropy differences can be
of the order of 3k
B
,
6
which corresponds to internal energies
⬃0.2 eV in the temperature range of 400 to 500 °C. The
charge neutrality condition is used to define the Fermi level
position.
20
Thermally created charge carriers in the valence
and the conduction bands are included in the self-consistent
determination of the Fermi level. The effective densities of
states of the valence and conduction bands are calculated
from the effective masses of electrons and holes.
1
In showing
the relative defect concentrations we use the temperature
450 °C, which corresponds to the growth temperature of the
molecular beam epitaxy 共MBE兲 method used in the experi-
ments by Bracht et al.
2,3
The additional error sources in the formation energy are
as follows. The plane-wave convergence error was found to
be ⬃0.05 eV and the convergence error in the atomic chemi-
cal potentials ⬃0.25 eV. The used supercell size and the
k-point sampling may also induce errors in the formation
4989J. Appl. Phys., Vol. 91, No. 8, 15 April 2002 Hakala, Puska, and Nieminen
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energy. We assume that these deviations are not critical for
the formation energy and concentration differences.
Finally, the ionization level (q/q
⬘
) of a defect is defined
as the position of the Fermi level when two given charge
states have the same total energy:
E
D
q
⫹ Q
关
E
v
q
⫹
e
共
q/q
⬘
兲
兴
⫽ E
D
q
⬘
⫹ q
⬘
关
E
v
q
⬘
⫹
e
共
q/q
⬘
兲
兴
. 共6兲
One should note that the used monopole correction 共see
above兲 affects the level positions strongly for the higher
charge states. For example, the 共3⫺/2⫺兲 level is corrected by
0.6 eV. As the higher-order terms in the expansion for the
electrostatic energy decrease the magnitude of the correction,
the monopole shifts provide an upper bound.
III. NATIVE DEFECTS
A. Structures and energies
The charge states, the symmetries and the formation en-
ergies for the native defects in GaSb are presented in Table I.
The ionization levels are given in Table II. In general, the
electronic structure of the vacancies, antisites and interstitials
is in qualitative agreement with the other III-V compounds.
6
The formation energy is affected by the Fermi level position
due to the fact that the defects can exist in different charge
states. Among the vacancy defects, V
Ga
acts as a triple ac-
ceptor: 1–3 electrons can be added to the defect with a posi-
tive extra energy per each added electron. The electrons are
distributed to the four Sb dangling bonds. In the 共3⫺兲 charge
state the defect preserves the T
d
symmetry with an inward
relaxation 共12%兲 of the nearest neighbor atoms of the va-
cancy. For the 共2⫺兲 and 共1⫺兲 charge states there is an addi-
tional symmetry-breaking distortion, which indicates that the
partially occupied t
2
level is split. V
Sb
can be found with the
charge states 共1⫹兲, 共0兲 and 共1⫺兲.Inthe共1⫹兲 charge state the
symmetry group is ⬃T
d
共Ref. 23兲 with an inward relaxation
of 5%. The smaller relaxation indicates that the accompany-
ing energy release is smaller than for V
Ga
. The 共1⫹/0兲 and
共0/1⫺兲 levels were found close to each other at ⬃0.2 eV. In
the 共0兲 charge states the symmetry of V
Sb
is lowered to C
2
v
as a gap state is occupied. Double occupancy of the gap level
leads to a very strong inwards relaxation 共⬃20%兲 with the
symmetry of D
2d
. The strong relaxation is very similar to
that found for the negatively charged V
As
in GaAs.
24
The
systematic feature in the present calculation is that the relax-
ation is always towards the vacancy. In GaAs, an outward
relaxation for V
As
关for the 共1⫹兲 and 共0兲 charge states兴 has
been reported.
24
The antisite Ga
Sb
is stable in the 共0兲, 共1⫺兲, and 共2⫺兲
charge states and acts as a double acceptor. In the 共2⫺兲
charge states the bonding t
2
level is fully occupied and the
defect preserves the T
d
symmetry relaxing inwards 共9%兲.
The transition levels are in the lower half of the band gap.
The additional symmetry-breaking distortions for the 共1⫺兲
and 共0兲 charge states indicate that the partially occupied t
2
level is split. The double acceptor behavior is similar to the
Ga antisite in GaAs.
20,25
For the Sb
Ga
antisite in the 共0兲
charge state two electrons occupy the antibonding a
1
state.
The T
d
symmetry is preserved and the defect relaxes out-
wards 共11%兲. As electrons are removed from the single-
particle state the anion–anion distance decreases. The donor
levels 共2⫹/1⫹兲 and 共1⫹/0兲 as calculated from the total en-
ergies are, however, below the VBM. This is a significantly
different behavior from the anion antisite defects in GaAs. In
GaAs and InP the donor levels of the anion antisites are
located at midgap or above.
7
For the interstitial defects we studied Ga and Sb atoms at
the hexagonal 共H兲 and the two tetrahedral sites (T
a
,T
c
). At
the T
a
(T
c
) site the defect has four anions 共cations兲 as near-
est neighbors. We find that the interstitials stabilize at posi-
tive charge states due to the removal of high-energy dangling
bond electrons. Ga
i
is found stable only at the T
c
site at the
共1⫹兲 charge state with a rather low formation energy. In fact,
the T
a
and H starting configurations relax to the T
c
configu-
ration. The absence of states in the band gap indicates that
the a
1
level is within the valence band and the unoccupied t
2
level in the conduction band. Sb
i
is found stable at the H site
and both the T sites. For the T
c
site the stable charge state is
共1⫹兲. 共In fact, the point symmetry is lowered to D
2d
.兲 This
is the energetically most favored configuration for all Fermi
level positions. The small symmetry-breaking distortion sug-
TABLE I. Charge states, point symmetry groups, and formation energies of
the native defects in GaSb. The point symmetries are deduced from the
converged positions of the ions neighboring the defect.
Defect Symmetry
a
E
f
共eV兲
V
Ga
3⫺
T
d
2.62⫹ (1/2)⌬
⫺ 3
e
V
Ga
2⫺
⬃T
d
2.07⫹ (1/2)⌬
⫺ 2
e
V
Ga
1⫺
⬃D
2d
1.76⫹ (1/2)⌬
⫺
e
V
Sb
1⫹
⬃T
d
1.93⫺ (1/2)⌬
⫹
e
V
Sb
0
⬃C
2
v
1.11⫺ (1/2)⌬
V
Sb
1⫺
⬃D
2d
2.31⫺ (1/2)⌬
⫺
e
Ga
Sb
2⫺
T
d
1.43⫺ ⌬
⫺ 2
e
Ga
Sb
1⫺
⬃T
d
1.17⫺ ⌬
⫺
e
Ga
Sb
0
⬃C
3
v
1.13⫺ ⌬
Sb
Ga
0
T
d
1.33⫹ ⌬
Ga
i
1⫹
(T
c
)
T
d
0.77⫺ (1/2)⌬
⫹
e
Sb
i
3⫹
(T
a
)
T
d
3.77⫹ (1/2)⌬
⫹ 3
e
Sb
i
1⫹
(H)
C
3
v
2.80⫹ (1/2)⌬
⫹
e
Sb
i
1⫹
(T
c
)
⬃D
2d
2.96⫹ (1/2)⌬
⫹
e
(V
Ga
Ga
Sb
)
3⫺
C
3
v
2.70⫺ (1/2)⌬
⫺ 3
e
(V
Ga
Ga
Sb
)
2⫺
⬃C
3
v
2.18⫺ (1/2)⌬
⫺ 2
e
(V
Ga
Ga
Sb
)
1⫺
⬃C
3
v
1.90⫺ (1/2)⌬
⫺ 1
e
(V
Sb
Sb
Ga
)
1⫹
C
1h
2.35⫹ (1/2)⌬
⫹
e
(V
Ga
Sb
i
)
0
C
3
v
1.56⫹ ⌬
a
Reference 23.
TABLE II. Ionization levels for the stable native defects.
Defect
Ionization
level
Energy
共eV兲
Ga
Sb
共0/1⫺兲 E
v
⫹ 0.04
共1⫺/2⫺兲 E
v
⫹ 0.26
V
Ga
共0/1⫺兲 E
v
⫹ 0.00
共1⫺/2⫺兲 E
v
⫹ 0.31
共2⫺/3⫺兲 E
v
⫹ 0.55
V
Sb
共1⫹/0兲 E
v
⫹ 0.18
共0/1⫺兲 E
v
⫹ 0.20
V
Ga
Ga
Sb
共1⫺/2⫺兲 E
v
⫹ 0.28
共2⫺/3⫺兲 E
v
⫹ 0.52
4990 J. Appl. Phys., Vol. 91, No. 8, 15 April 2002 Hakala, Puska, and Nieminen
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