First-principles study of As interstitials in GaAs: Convergence, relaxation, and formation energy

0.25 MB·English

by  J T Schick

First-principles study of As interstitials in GaAs: Convergence, relaxation, and
formation energy
∗
J. T. Schick
Physics Department, Villanova University, Villanova, PA 19085, USA
C. G. Morgan and P. Papoulias
Department of Physics and Astronomy, Wayne State University, Detroit, MI 48202, USA
(Dated: August 19, 2002)
1 Convergenceofdensity-functionalsupercellcalculationsfordefectformationenergies,chargetran-
1 sition levels, localized defect state properties, and defect atomic structure and relaxation is inves-
0 tigated using the arsenic split interstitial in GaAs as an example. Supercells containing up to 217
2 atomsandavarietyofk-spacesamplingschemesareconsidered. Itisshownthatagooddescription
ofthelocalizeddefectstatedispersionandchargestatetransitionlevelsrequiresatleasta217-atom
n
supercell, although the defect structureand atomic relaxations can be well converged in a65-atom
a
J cell. Formation energies are calculated for the As split interstitial, Ga vacancy, and As antisite
defects in GaAs, taking into account the dependence upon chemical potential and Fermi energy.
7
It is found that equilibrium concentrations of As interstitials will be much lower than equilibrium

concentrations of Asantisites in As-rich, n-typeor semi-insulating GaAs.
]
i
c
PACSnumbers: 61.72.Bb,61.72.Ji,71.55.-i
s
-
l
r
t I. INTRODUCTION or diffuse ‘clouds’ of arsenic interstitials have been re-
m
ported in GaAs grown by the horizontal Bridgman and
. liquid-encapsulatedCzochralskimethods,basedonX-ray
t Interstitials are the most complicated of the simple
a diffusescattering7–9 andquasi-forbiddenX-rayreflection
point defects, and the most elusive. For example, even
m
intensity measurements.10 However, the atomic compo-
thougharsenicinterstitialsmustbecreatedbyirradiation
- sition and microscopic structure of these defects cannot
d ofGaAswithsufficientlyenergeticparticles,andtheycan
be unambiguously determined from these experiments.
n subsequently be observed to recombine with arsenic va-
o cancieswhenthesampleisheatedabove220◦ C,isolated Gallium arsenide grown by arsenic-rich molecular
c arsenic interstitials have not been observed directly in beam epitaxy at low temperature (LT GaAs) is a semi-
[ EPR, electrical, or optical experiments.1 insulating material with a host of potentially useful
1 It has been argued based on a thorough analysis2–4 applications.11 This material contains up to 1.5% excess
v of a variety of experimental data including titration As,12 which is accommodated by high concentrations of
3 experiments5 and measurements of density and lattice point defects in UN-annealed samples, and arsenic pre-
1 parameter6 that melt-grown GaAs is always As-rich un- cipitatesplussomewhatlowerconcentrationsofpointde-
4
less the concentration of Ga in the melt is substantially fectsinannealedsamples. ConcentrationsofAsantisites
.1 greaterthan50%,andthatthisdeviationfromstoichiom- (AsGa) up to 1020 cm−3 are observed in LT GaAs, as
1 etry is due primarily to the creation of large numbers of measured by electron paramagnetic resonance (EPR),13
0 As interstitials(As )duringgrowth. Inparticular,Hurle near-infrared absorption (NIRA) and magnetic circular
1 i
hasargued2 thatthe measureddeviationofthe massper dichroismofabsorption(MCDA),14andscanningtunnel-
1
: unitcellasafunctionofarsenicconcentrationinthemelt ingmicroscopy(STM).15 ConcentrationsofGavacancies
v must be explained by arsenic interstitials and/or arsenic (VGa)upto1018 cm−3 aremeasuredinLTGaAsbyslow
Xi vacancies, since the number of arsenic antisites which positronannihilation.16 Ionchannelingexperimentshave
would be required to fit the data is unrealistically large been interpreted as providing evidence for large concen-
r
a (up to several percent), due to the small difference be- trationsofAsinterstitialsinLTGaAs.12 However,itwas
tween the atomic masses of arsenic and gallium. Hurle’s laterpointedoutthattheobservedhighconcentrationof
workalsocontainsanextensivethermodynamicanalysis, atomsinthechannelnearthenormalarseniclatticesites
including estimates of the mass action constants for the could also be due to outward relaxation of the nearest
formation of all the neutral native point defects. These neighbors of the As antisites.14,17
estimates arederivedby fitting to a largequantity ofex- Withincertainwelldefinedlimitsofthegrowthparam-
perimental data on both doped and undoped GaAs, un- eters for LT GaAs, a linear correlationbetween the neu-
dertheassumptionthatnativedefectanddopantconcen- tralAs concentrationandthe latticedilationhasbeen
Ga
trations are near equilibrium close to the melting point found.14,16 It was therefore proposed that As are the
Ga
and during high temperature growth from the melt or dominant defects which determine the lattice expansion
from solution. for growth within this regime. Staab et al.17 used a self-
In the high temperature growth regime, observations consistentdensity-functional-basedtight-bindingmethod
of defects tentatively described as high concentrations to study the lattice distortion induced by point defects
2
in As-rich GaAs, and concluded that only As are nec- dralsites.20 Theeffectsoflatticerelaxationwereignored.
Ga
essary to understand the observed lattice expansion in BaraffandSchlu¨terconcludedthatsimpletetrahedralar-
the regime where the linear correlation is observed, and senic interstitials were less likely to occur than vacancy
that if concentrations of isolated As comparable to the and antisite defects under all equilibrium conditions, al-
i
measured concentrations of As were also present, the thoughtheycouldnotruleoutthepossibilitythatother,
Ga
lattice expansion would be three times greater than is more complicated interstitial configurations might have
experimentally observed. However, Luysberg et al. have a lower energy.20
reportedthat whenthe As/Gaflux ratiois increasedbe- Jansen and Sankey calculated the formation energies
yondabeamequivalentpressure(BEP)ratioof20,there for unrelaxed native defects with tetrahedral symme-
is adeparturefromthe linear correlationbetweenlattice try inGaAs, including arsenicinterstitials intetrahedral
dilation and antisite concentration.16 It was pointed out sites, using a density-functional pseudopotential method
thatother defects mustbe presentto accountforthe de- with a basis set of pseudo-atomic orbitals and a single
viation from stoichiometry and the lattice expansion at specialk-pointinsupercellscontainingabout32atoms.21
high As/Ga flux ratios.16 Inordertocalculatetheformationenergiesforindividual
Nonequilibriumprocessessuchasdiffusionandcompo- defects instead of reaction energies for defect reactions
sitional intermixing at interfaces can also be strongly af- whichconservethenumberofatomsofeachspecies,they
fectedbypointdefectsthatarepresentinhighconcentra- were required to choose a value for the arsenic chemical
tions. Sincethepointdefectswhichhavebeenunambigu- potential (or equivalently, for the gallium chemical po-
ouslydocumentedaspresentinhighconcentrationsinLT tential). Anarbitraryvaluewaschosen,correspondingto
GaAs, As and V , occupy sites on the gallium sub- theconditionthattheformationenergiesforneutralgal-
Ga Ga
lattice, they cannot contribute directly to interdiffusion lium vacancies and for neutral arsenic vacancies should
on the arsenic sublattice. However, substantial concen- be equal. Jansen and Sankey concluded that arsenic in-
trations of arsenic interstitials may affect interdiffusion terstitials in tetrahedral sites should be less numerous
on the arsenic sublattice. For example, an experimental than vacancies and antisites in GaAs under equilibrium
study showing a positive dependence of GaAsP/GaAs conditions,21 in agreement with Baraff and Schlu¨ter.
andGaAsSb/GaAsinterdiffusiononarsenicpressurehas Zhang and Northrup used density functional theory
indicated that a kickoutmechanisminvolvingarsenicin- (DFT) within the local density approximation (LDA)
terstitialsisthedominantprocessfortheAs-PandAs-Sb and supercells of about 32 atoms to calculate the for-
interdiffusion in the material studied.18 mation energies for vacancies, antisites, and tetrahedral
Similarly, annealing LT-GaAs δ-doped with Sb was interstitials in GaAs as a function of arsenic chemical
found to produce substantially greater compositional potential, over the physically allowable range of chemi-
intermixing than annealing conventional stoichiometric calpotentials, fromGa-richto As-rich.22 This physically
GaAs similarly δ-doped.19 This enhancement of As-Sb allowable range is set by the heat of formation of bulk
interdiffusion was attributed to an oversaturation of ar- GaAs and by the requirement that the arsenic chemical
senicinterstitialsintheLTGaAssamples,resultingfrom potential may not exceed the chemical potential of bulk
the balance of arsenic interstitials with arsenic clusters arsenic, since the material is in equilibrium with arsenic
and all the other excess-arsenic-containingdefects in the precipitates in the arsenic-richlimit. The atomic coordi-
material. TheeffectiveactivationenergyforAs-Sbinter- nates were allowed to relax in these calculations, within
diffusioninLTGaAs deducedfromthis work,0.6±0.15 theconstraintsimposedbythetetrahedralsymmetry. In
eV,19 is reasonably close to the migration energy of 0.5 agreement with the previous work, Zhang and Northrup
eV for arsenic interstitials deduced from annealing ex- foundthatantisitesand/orvacanciesshouldbemorenu-
periments on defects produced by electron irradiation,1 merous than arsenic interstitials in tetrahedral sites un-
as well as to migration energies subsequently ascribed der all equilibrium conditions.22
to arsenic interstitial defects produced in GaAs by other ChadiusedDFT-LDAcalculationsand33-atomsuper-
means. Theconcentrationofarsenicinterstitialsrequired cells to investigate many different types of bonding con-
to produce sufficient oversaturation to eliminate com- figurations for self-interstitials in GaAs, including vari-
pletely any contribution of the interstitial formation en- ous split interstitials, as well as hexagonal, two-fold co-
ergytothe activationenergyforAs-Sbintermixingmea- ordinated, and tetrahedral interstitials, all fully relaxed
suredintheLTGaAssamplewasestimatedtoberoughly within the constraints of the chosen symmetry.23 He
1018cm−3,usingHurle’sthermodynamicanalysisincon- found that the lowest energy configuration for arsenic
junction with the experimental data.19 interstitials in the neutral or −1 charge state is a split
Theoreticalattempts to obtain a picture of the micro- interstitialconsisting oftwo As atomssharinganarsenic
scopic structure and properties of the lowest energy ar- latticesite,displacedfromthissiteinoppositedirections
senic interstitial configuration(s)beganwith the workof along a <110>-like axis, while the lowest energy config-
BaraffandSchlu¨ter,whouseddensityfunctionalGreen’s uration for positively charged arsenic interstitials in the
function calculations to investigate the energies of re- +1 or +2 charge state is a split interstitial consisting of
actions creating native defects with T symmetry in an As atom and a Ga atom sharing an gallium lattice
d
GaAs, including arsenic interstitials in the two tetrahe- site, displaced from this site in opposite directions along
3
a <100>-likeaxis. Since we will be interested below pri- Since the theoretical investigations described above
marilyinarsenicinterstitialsinsemi-insulatingorn-type have been carried out over a long period of time, it has
GaAs,wewillusethe notationAs -Asforthe interstitial gradually become possible not only to include lattice re-
i
withtwoatomssharinganarsenicsite andalignedalong laxation and to investigate more complicated interstitial
a <110>-likeaxis,whichshouldbe the lowestenergyin- configurations,butalsotodomoreaccuratecalculations,
terstitial configuration in semi-insulating or n-type ma- using larger unit cells and better sets of k-points for the
terial. summationsoverk-space. Po¨ykkoet al.showedhowsen-
sitive calculated defect properties can be to the k-space
Chadi also showed that neutral arsenic interstitials,
samplingmethodandsupercellsizeintheirinvestigation
which have unpaired spins, are unstable relative to for-
mation of a pair of +1 and −1 charged interstitials — ofthe VAs-SiGa complex inGaAs.28 They found thatthe
i.e. arsenic interstitials form a negative-U system. This use of the Γ point can produce misleading results even
whensupercellsare64-atomsinsize,reinforcingthecon-
suggested that arsenic interstitials may not be observ-
able in EPR experiments.23 Chadi reported the relative clusionsofMakovthatthe Γ pointproducesparticularly
slowlyconvergingresultswithrespectto cellsize.29 Soit
energies for the most energetically favorable arsenic in-
isessentialtouseaspecialpointmeshinthistypeofcal-
terstitial configurations in each of these charge states,
culation. Furthermore, Puska et al. concluded that cell
including ineachcaseanumberofmetastableconfigura-
sizes of 128 to 216 atoms are needed to properly assess
tions somewhat higher in energy than the lowest energy
the physical properties of the silicon vacancy in bulk sil-
configurations, all of which were more complicated than
the simple tetrahedral configurations.23 However, since icon, because of the dispersion of energy levels and long
range ionic relationships.30
Chadi did not report absolute interstitial formation en-
ergiesasafunctionofarsenicchemicalpotential,nocom- In this paper, we investigate the combined effects of
parisonwiththeformationenergiesofdefectsinvolvinga cell size and k-space sampling on the formation energy,
different number of excess arsenic atoms,suchas arsenic charge state transitions, atomic relaxations, and char-
antisites, was possible from this work. acterization of localized defect states for arsenic self-
interstitialsinGaAs. Becauseofthemoreionicnatureof
Landmanet al. investigatedthe relative formationen-
the material and the complicated split interstitial defect
ergies of the point defects containing excess As, As ,
Ga structure, comparison of these results for interstitials in
V , and the lowest energy As configuration in semi-
Ga i GaAstothepreviousresultsforvacanciesinsilicon30can
insulatingorn-type,As-richGaAs,As -As,24usingDFT-
i enhance our understanding of the range of behavior for
LDA-based calculations with the Harris-Foulkes func-
different defects in different materials. We compare the
tional and a basis of pseudo-atomic orbitals.25,26 They
formationenergy of the lowestenergy arsenic interstitial
placed the defects in 64-atom supercells, and estimated
in n-type or semi-insulating GaAs, As -As, with the for-
summations in k-space by using a single Chadi and Co- i
mation energies of As and V at the arsenic-richend
hen special point.27 Since Harris-Foulkes,pseudoatomic- Ga Ga
of the range of physically allowed chemical potentials,
orbital calculations do not give as accurate results for
all calculated by state-of-the-artDFT pseudopotential31
semiconductor heats of formation or for the relative
calculations, using the larger supercells and sets of spe-
energies of compound semiconductor and pure, metal-
cial k-points which we have determined to be necessary.
lic phases as fully self-consistent DFT-LDA calculations
We conclude our study by discussing the relative con-
withasufficientlylargebasisofplanewaves,theydidnot
centrations of these defects in equilibrium in As-rich, n-
use this method to calculate the arsenic chemical poten-
type or semi-insulating GaAs at growth temperatures,
tial in the arsenic-richlimit. Instead, the relative forma-
and reporting the computed charge transition levels and
tion energies of the tetrahedral As with arsenic nearest
i expected electrical behavior of As -As as a function of
neighbors,As ,andV inthearsenic-richlimitforthe i
Ga Ga Fermi level.
chemicalpotentialwere takenfrominformationgivenby
ZhangandNorthrup,22andtherelativeformationenergy
ofAs -AswasdeterminedbyLandmanetal.’sresultthat
i
II. COMPUTATIONAL METHOD
theneutralAs -Asis4eVlowerinenergythantheunre-
i
laxed,neutraltetrahedralAs witharsenicnearestneigh-
i
bors. Since the tetrahedral interstitial was found to be We have used the molecular dynamics code developed
unstable, relaxing to another configuration in Landman attheFritzHaberInstitut(FHIMD)31 forthisinvestiga-
et al.’s calculation, they were obliged to compare their tion,usingdensity-functionaltheory(DFT)32 withinthe
results for the ideal, unrelaxed tetrahedral interstitial to local density approximation (LDA), with the Ceperley-
Zhang and Northrup’s results for a tetrahedral intersti- Alder33 form for the exchange and correlation poten-
tial which had been relaxed while constrained to keep tials as parameterized by Perdew and Zunger.34 The
its tetrahedral symmetry. This led to an additional un- core electrons are treated in the frozen-core approxima-
certainty in the relative formation energies between zero tion and the ion cores are replaced by fully-separable35
and 0.8 eV.24 However, Landman et al. concluded that norm-conservingpseudopotentials.36 Planewavesarein-
the lowestenergysplitAs mayhavea concentrationap- cluded up to the energy cutoff of 10 Ry. The atoms
i
proaching that of As for certain Fermi levels.24 are allowed to relax until the force components are are
Ga
4
less than 5 × 10−4 hartrees per bohr radius and the conductionbandstatesthemselves, while leavingthe de-
zero temperature formationenergies change by less than fect states with predominantly valence band character
5×10−6 hartrees per step for at least 100 steps. fixed relative to the valence band edge.
To evaluate the defect formation energy, we used the
formalism of Zhang and Northrup,22 which gives for the
formationenergyintheAs-richlimitatzerotemperature III. RESULTS AND DISCUSSION
∆E =E(N ,N ,q)−N µ
f Ga As Ga GaAs
A. Defect formation energies, charge transition
−(N −N )µ +qǫ . (1)
As Ga As(bulk) F levels, and localized defect states
Here q electrons have been transferred to a reservoir at
the Fermi energy ǫ in order to produce a defect in We used cubic supercells with dimensions of both two
F
the desired charge state. E(N ,N ,q) is the zero- andthreetimesthecomputationallydeterminedbulklat-
Ga As
temperature total energy produced by the ab initio code tice constant, corresponding to bulk cells of 64 and 216
for a supercell containing the desired defect, the chemi- atoms, along with three different Brillouin zone (BZ)
cal potential µ is the energy per atomic pair of bulk sampling schemes to examine the effects cell size and
GaAs
GaAs, and the arsenic chemical potential in the As-rich sampling scheme have on the formation energies and
limit, µAs(bulk), is the energy per atom of pure bulk As transition levels for the Asi-As. In order to investigate
computed using the same ab initio code and pseudopo- the most efficient choice of k-points to obtain good re-
tentials. N and N are the numbers of atoms of each sults,wehaveuseda13 Monkhorst-Pack(MP)mesh,46 a
Ga As
speciesinthesupercellcontainingthedefect. Wewilldis- 23 MP mesh, and the Γ and L points, which was recom-
cuss the effect of temperature, which can be important mendedasagoodminimalsetofk-pointsforcubicsuper-
for defect concentrations, in Section III. cellswithnoparticularsymmetrywithinthesupercell,29
Becausethezerooftheenergylevelsfloatsfreely,37 re- and which has subsequently been used in defect calcula-
sults from different DFT supercell calculations must be tions,e.g.tostudythestructuresassociatedwithdopants
aligned in order to obtain the correct charge transition in highly n-doped Si.47 Calculations comparing different
levels. We apply the procedure outlined by Kohan et summation schemes for vacancies in Si show that use of
al.38 in which we first compute the difference between the Γ+L points produces a reasonably well-converged
the electrical potential in the supercell with the neutral formation energy in a 64-atom supercell.30
defect and the electrical potential for the corresponding This earlier work on the vacancy in silicon has shown
bulk crystal supercell, averaged over parallel planes, as that the neutral vacancy formation energy computed
a function of position along a line normal to the planes. withdifferentk-spacesamplingschemesconvergesatdif-
Far from the defect within the supercell, this difference ferent rates with respect to supercell size.30 However,an
becomes a constant. In order to make the potential far acceptably converged value for the neutral vacancy for-
from the neutral defect equal to the corresponding po- mation energy can be attained more easily than an ac-
tential in the bulk cell, a uniform shift is applied to the ceptably converged description of the charge transition
potentialandtheenergylevels,yieldingtheproperalign- levels and the atomic relaxations and defect symmetry
ment of the energy levels of the defect with the energy for different charge states.30
levels of ideal bulk crystal supercell. The same shift is To augment our understanding of the effects of cell
applied for all charge states of a given defect. size and sampling scheme, we computed the formation
Awell-knownshortcomingoftheLDAisthatitunder- energiesofthedifferentchargestatesforthefullyrelaxed
estimatesthebandgapsofmaterials. Thetypicalmethod split interstitial Asi-As in GaAs. Charge was balanced
for dealing with this problem is to simply shift the con- by a uniform background to avoid long range Coulomb
duction band states up uniformly by the amount needed interactions between the supercells.
to reproduce the experimental gap, using the so-called Table I lists the formationenergies we obtained for all
‘scissorsoperator’.39Morerecently,ananalytically-based the excess-arsenic-containingelementarypointdefects in
model justifying the rigid shifting upward of all conduc- thearsenic-richlimit,withGaAsinequilibriumwithbulk
tion band states by a scissors-type correction has been arsenic,includingcompleteresultsforthevariousk-space
shown to produce good results for a large number of sumsforAs -As. Forcomparison,the formationenergies
i
semiconductors.40 Since the LDA can produce similarly of the unrelaxed, ideal tetrahedral As interstitials with
large errors in the energies of the deep defect states, it As neighbors (As ) and with Ga neighbors (As ) are
i1 i2
is also important to correct for these errors when deter- also shown. These tetrahedral interstitials are unstable,
mining where the charge transition levels corresponding and will relax to other configurations if allowedto break
to deep defect states lie in the experimental gap. Unfor- their tetrahedral symmetry.
tunately,afullGWcalculation,41–45 whichwouldcorrect Table I displays formation energies evaluated for the
these errors,isnotcurrentlypossible forthe largesuper- Fermi level at the valence band maximum (VBM), and
cells needed for studies of defects. Therefore, we apply alsofor the Fermilevelpinned atthe calculatedposition
the same upward shift to the defect states with predom- ofthe(+1/0)chargestatetransitionoftheAs ,whichis
Ga
inantly conduction band character as is applied to the atVBM+0.54eVforthe23MPmeshandVBM+0.45eV
5
TABLE I. Formation energies for excess-arsenic-containing
4.0
defects computed in theAs-richlimit. Thesewere calculated
with supercellscorresponding tothebulk216 atom cell. The
3.8
values in the last column are computed with the Fermi level
pinnedatthecalculated(+1/0)transitionleveloftheAsan- )
V 3.6
tisite defect.
e
(
Defect k-space Charge Formation Formation y 3.4
sum state energy (eV) energy (eV) g
r
ǫF at VBM semi-insulating ǫF ne 3.2
AsGa MP 23 +2 0.9 2.0 e
AsGa MP 23 +1 1.3 1.8 on 3.0
AsGa MP 23 0 1.8 1.8 ati 217 & 2x2x2 mesh
AsGa MP 13 +2 0.9 1.8 m 2.8 217 & 1x1x1 mesh
AsGa MP 13 +1 1.1 1.6 or 217 & G + L
AsGa MP 13 0 1.6 1.6 f 2.6 65 & 2x2x2 mesh
Asi1 MP 23 0 6.9 6.9 65 & 1x1x1 mesh
Asi2 MP 23 0 6.2 6.2 2.4 65 & G + L
Asi-As MP 23 +1 3.5 4.1
Asi-As MP 23 0 3.8 3.8 0.0 0.2 0.4 0.6 0.8
Asi-As MP 23 −1 4.4 3.8 Fermi level (eV)
Asi-As MP 13 +1 3.4 3.8
Asi-As MP 13 0 3.7 3.7
Asi-As MP 13 −1 4.1 3.7 FIG. 1. Defect formation energies for theAssplit interstitial
Asi-As Γ+L +1 3.6 computed in the As-rich limit. Comparisons are presented
Asi-As Γ+L 0 3.6 for different k-space sums and two supercell sizes. Dashed
Asi-As Γ+L −1 4.1 linesareusedforcellscontaining65atoms,andsolidlinesfor
VGa MP 23 0 2.9 2.9 s2t1a7teastoamrecmelalsr.keTdrawnsitithiocnirlcelveselsfobretthweee2n3tMhePdimffeersehn,tsqchuaarrgees
VGa MP 23 −1 3.0 2.5 forthe13 MPmesh,andtrianglesfortheΓ+Lk-spacesum.
VGa MP 23 −2 3.1 2.1
VGa MP 23 −3 3.4 1.8
lessfinelyspacedsetofk-pointsthanthe23 MPmeshto
covertheBrillouinzone. Itisclearthatthe217atomcell
forthe13MPmesh. ThischoiceofFermilevelwasbased with the 23 MP mesh is well converged. We see that the
on the experimental finding that there can be high con- two less finely spaced sampling schemes produce some-
centrationsofAsGa inAs-richGaAs,andthatthesehigh what converged results in the 217 atom cell, as does the
concentrations of AsGa can pin the Fermi energy near 23 MP mesh in the 65 atom cell. In agreementwith pre-
thistransitionlevel. UsingeitherchoiceofFermilevelas vious results for the vacancy in bulk Si,30 we find that
a reference, the formation energies and equilibrium con- differentsamplingschemescanbeusedtoproduceeither
centrationsofthedefectscanbedeterminedasafunction an attraction between defects (i.e. a lowering of energy
of Fermi energy (or doping level). in the smaller supercell), as in the 13 MP or the Γ+L
In Fig. 1, we present the results of the formation en- cases,oracharge-statedependentrepulsionorattraction
ergy calculations for As -As in which both cell size and between defects, as in the 23 MP case.
i
k-space sampling methods have been varied. This figure When comparing the results for the 65-atom and the
shows the formation energy for the Asi-As in its pre- 217-atom cells, using the 23 MP mesh, we see that the
ferred charge state. For Fermi levels in the range where charge transition levels converge less rapidly than the
the neutral Asi-As is preferred, the formation energy is neutral defect formation energy with increasing cell size,
independentofFermilevel. IftheFermilevelisdecreased due to long-range interactions of the electrons in the lo-
past the (0/+1) charge transition level, so that the +1 calizeddefectstatewiththechargeddefectsinneighbor-
chargestate is preferred,the formationenergyvs. Fermi ing cells. The neutral As -As formation energy changes
i
levelcurvehasaslopeof+1,ascanbeseenfromEq.(1), bylessthan0.1eVwhenthecellsizeisreducedfrom217
andiftheFermilevelisincreasedpastthe(0/−1)charge to 65 atoms, while the charge transition levels move by
transition level, so that the −1 charge state is preferred, about0.4eV,whenusingthe 23 MPmesh. Thistrendis
the formationenergyvs. Fermilevelcurvehasa slopeof notseenintheless-well-convergedresultsobtainedusing
−1. the 13 MP mesh or the Γ+L points, where the neutral
The most obvious feature in Fig. 1 is the wider varia- As -As formation energies move by 0.5 to 0.6 eV when
i
tionintheenergiescomputedusingthesmallersupercell. the cell size is reduced from 217 to 65 atoms, perhaps
Convergencewith cellsize is also visibly slowerwhen us- duetointeractionsofthedeepdefectlevelwiththeband
ing the 13 MP mesh or the Γ+L points, which use a edges.
6
is also found to be localized in the p and p orbitals of
y z
the two As atoms of the neutral defect when the Γ+L
sampling is used.
Insmallersupercells,thedefectstateinteractswithits
images and becomes less localized. One way to observe
this is to examine the variation in energy of the defect
state across k-space, i.e. the dispersion of the state. We
characterize this variation by computing the difference
between the highest and lowest energy obtained from a
k-spacesurveyalongthe∆,Σ,Λ,andT linesinthecubic
Brillouin zone. The dispersion measured in this way is
0.1 eV in the 217 atom supercell, and 0.5 eV in the 65
atom cell. This interaction of the defects in neighboring
supercells contributes to the variation in the positions
of the charge transition levels between the different cell
sizes and sampling schemes, noted above, and supports
the conclusion that the larger 217 atom supercell should
be used for accuracy in describing the charge transition
levels and deep defect states.
FIG. 2. This contour plot shows levels of constant charge
density for the localized defect state of the neutral Asi-As
B. Defect atomic structure and relaxation
in the 217 atom supercell, evaluated in the plane containing
thetwo defect Asatoms, whichis 0.48˚A abovetheAslattice
siteassociatedwiththedefect. Thedarkcirclesrepresentthe The detailed structure of the atomic positions is also
positionsoftheAsatoms,includingthedefectatomsandthe expected to be dependent on the cell size and k-space
As atoms of theoriginal lattice plane. The light circles show sampling approach. However, in contrast to the behav-
thelocationsoftheneighboringGaatomsprojectedontothe iorobservedforthevacancyinsilicon,30 wherethedefect
plane.
symmetry and atomic relaxations are very sensitive to
the supercellsize andk-space sampling,we findthat the
atomicstructureoftheAs -Asdefectisremarkablysimi-
i
In the neutral charge state, the topmost filled elec- larforthe65-atomand217-atomsupercells,andalsode-
troniclevelishalffilled. Thislevelcorrespondstoalocal- pends little on the k-space sampling approach. In Fig. 3
izedstatecenteredonthe splitinterstitialorientedalong we showthe structure ofthe neutralsplit interstitialori-
a <110>-like direction. In Fig. 2, the charge density as- ented along the <011> direction, viewed from the [100]
sociatedwiththisdefectstateisshowninaplaneparallel direction. The two Ga atoms labeled Ga(1) and Ga(3)
to an arsenic lattice plane, but slightly above this plane, are bonded to both As atoms of the defect, while those
so that it includes the two arsenic atoms of the split in- labeled Ga(2) and Ga(4) are bonded to only one of the
terstitial,whichhaverelaxedslightlyawayfromtheideal defect atoms. The structure exhibits C symmetry in
2v
lattice site. This plot clearly shows that the defect state all charge states and returns to this symmetry when the
is localized in p-like orbitals which appear to be forming atomic coordinates are perturbed from their equilibrium
a π antibonding state, as evident from the node in the positions and allowed to relax.
charge density midway between the two As atoms form- Althoughthelocallatticeexpansionintroducedbythe
ing the split interstitial. This result is corroborated by additional atom of the interstitial might be expected to
anexaminationofthecharacterizationofthestate,using converge slowly with supercell size, we find that reason-
the217-atomcellandthe23 MPmesh,whichshowsthat able convergence is more easily reached for the atomic
in contrast to the very extended character of the bulk- positions than for the electronic properties discussed in
like states, 20% of the defect state is localized in the py the previous section. The As-As and As-Ga distances
and pz orbitals of the two As atoms of the neutral de- found in the 65 atom supercell differ by less than 0.01 ˚A
fect. Since the charge density in this defect state points from those found in the 217atom supercell, using the 23
roughly along the direction of the As-Ga bonds between MPmesh. ThedistancebetweenthetwoAsatomsofthe
the defect As atoms and the two Ga atoms bonded to As -As defect is 2.39˚A. (For comparison,the GaAs bulk
i
both of these As atoms, we note that this state makes a nearest neighbor distance in this calculation is 2.41 ˚A.)
bonding contributionto these As-Ga interactionsas well The distance between Ga(1) or Ga(3) and either of the
asmakinganantibondingcontributiontotheinteraction As atoms of the defect is 2.60 ˚A. The distance between
between the As atoms of the defect. Ga(2)orGa(4)andthenearestAsofthesplitinterstitial
The characterization of the deep defect state in the is 2.32 ˚A.
217-atom supercell is not very sensitive to the k-space The two different supercell sizes produce slightly dif-
sampling method used. For example, 20% of the state ferent results in the position of the center of mass of
7
The effect of these small variations in bond lengths is
to reduce the changes in bond length seen in the 23 MP
mesh calculations when the defect becomes charged, if
one of the other k-space sampling methods is used in-
stead. In particular, the Γ+L sampling is observed to
produce a somewhat smaller dependence of these bond
lengthsonchargestate. Thisdependenceofbondlengths
on the charge state of the defect is seen below to result
from the changes in occupation of the deep defect state
when the charge state is changed, and from the bond-
ing or antibonding character of this state for particular
bonds.
Focusing on results for the 217 atom cell with the
23 MP mesh, we observe that the distance between the
FIG. 3. The neutral Asi-As defect from the 217-atom su- atoms of the defect has a significant dependence on the
percell is shown (viewed from a direction slightly displaced charge state of the system. The As-As bond expands to
from the [100] direction) with larger dark spheres represent- 2.47 ˚A (about 3.5% compared to the neutral state bond
ing the As atoms and smaller light spheres representing the length), when the system is allowed to relax in the −1
Ga atoms. charge state. This can be easily understood, since the
antibonding defect state on the two As atoms is doubly
occupiedandcancauseastrongerrepulsivecontribution
the two As atoms of the split interstitial. The center of to the interaction between these two atoms for the −1
mass of the pair of As atoms is shifted away from the chargestateofthedefect,whileitisonlysinglyoccupied
bulk lattice site by 0.48 ˚A in the [100] direction toward for the neutral state of the defect.
the plane containing Ga(1) and Ga(3) in the case of the When the system is allowed to relax in the +1 state
217 atom supercell, using the 23 MP mesh. This shift is theAs-Asbondshrinksto2.31˚A,acontractionofabout
0.50 ˚A in the smaller supercell. Since the nearest neigh- 3%. This can also be easily understood, since the anti-
bor distances are very similar in the two supercells, this bondingdefectstateonthetwoAsatomsofthedefectis
difference is accomplished through variation of bond an- completely unoccupied for the +1 charge state, so it no
gles. The angle between the bonds from one of the As longer makes a repulsive contribution to the interaction
atoms to the Ga(1) and Ga(3) atoms is 107.7◦ in the between the two As atoms. The bond length between
small supercell and 106.8◦ in the large supercell. Ga(1) or Ga(3) and either As atom of the defect is 3.5%
We findthatthe atomic structureofthe Asi-As defect longer and the bond length between Ga(2) or Ga(4) and
depends very little on the k-space summation method, thenearestdefectAsatomis1.9%longerinthe+1state
ascanbe shownbyexamining the bond lengthsbetween thanintheneutralstate. Sincethedeepdefectstateacts
the As atoms of the defect and between those As atoms as a bonding state for the As-Ga bonds between the de-
and the four neighboring Ga atoms in the 217 atom su- fect As atoms and Ga(1) and Ga(3), as discussed above
percell. FortheneutralAsi-Asdefect,thesebondlengths inSectionIIIA,itiseasytounderstandwhythesebonds
change by less than 0.1% when the summation method are longer in the +1 state, when the defect state is fully
is changed. unoccupied andcanno longercontribute to the strength
For the +1 charge state of the defect, we find a small of these bonds.
butperceptibledependenceofthebondlengthsonthek- We note that because of the contributionof the defect
space summation method. The As-As distance obtained state to the As-Ga bonds between the defect As atoms
using the 23 MP mesh is about 1% smaller than that and Ga(1) and Ga(3), these two Ga atoms move signif-
obtained using the Γ+L sampling. The bond between icantly when the charge state is changed, changing the
either of the As atoms and the Ga(1) or Ga(3) atom occupationofthedefectstate. TheseGaatomsareabout
is about 1% longer when using the 23 MP mesh than 4% closer to each other in the −1 state and about 5.5%
when using the Γ+L summation method. The distance farther apart in the +1 state than in the neutral state.
between the Ga(2) or Ga(4) atom and the nearest As TheothertwoGaatoms,eachbondedtoonlyonedefect
atom of the split interstitial is about 0.5% longer for the Asatom,donotmoveinresponsetothechangeincharge
23 MP mesh calculation than for the Γ+L calculation. state.
There is a similar but weak effect (under 0.5%) in the In performing these calculations, we fixed the lattice
defect bond lengths observed in the −1 state, with the constant at the value determined through minimization
rolesofthe23 MPmeshandtheΓ+Lpointsreversed— oftheenergyofthebulkcrystal. Whiletheidealcalcula-
i.e. the 23 MP mesh now gives a larger As-As distance tion should include a full lattice constant determination
andsmallerAs-Gadistances. The 13 MP meshproduces witheachchangeofdefectconfiguration,forsimplicitywe
resultsbetweenthoseoftheothertwok-spacesummation did not perform this relaxation. This may be deemed a
methods. reasonablechoiceinlightofevidencepresentedbyPuska,
8
et al.30 for ab initio supercell calculations in the LDA,
usingsupercellsofsizescomparableto ours,inwhichva- 5
cancies in bulk Si are found to alter the lattice constant
by around 0.2%, while artificially introduced distortions
in the lattice constant of up to 1% are seen not to affect
4 As - As
their reported results significantly. V) i
0
e +1 -1
(
y
g 3
C. Relative defect concentrations in equilibrium er 0
en 1- 2- VGa

We now compare our well-converged results for the on 2 -3 AsGa
i
formation energies of the elementary excess-arsenic- t
a 0
containing point defects, As , V , and As -As (the m +1
Ga Ga i
r
mostfavorableAsi configurationin semi-insulatingorn- fo 1 +2
type GaAs), computed using the large supercell and the
23 MP mesh. These formation energies in the As-rich
limit, corresponding to GaAs in equilibrium with bulk
arsenic, are presented as a function of Fermi energy in 0
0.0 0.2 0.4 0.6 0.8
Fig. 4. The formation energies for two specific choices
of Fermi energy have also been listed in Table I in Sec- Fermi level (eV)
tion IIIA.
In Fig. 4, we can see that the V and As defects
Ga Ga
possess significantly lower formation energies than the
Asi-As for all Fermi energies. For example, the forma- FIG. 4. Defect formation energies for selected defects over
tion energy for AsGa is seen to be at least 2 eV lower thecalculatedbandgapintheAs-richlimit. ZeroFermilevel
than the formation energy for As -As for all Fermi ener- corresponds to thevalence band maximum.
i
gies. Small uncertainties in the formation energy should
notalterthisstrongqualitativeorderingoftheformation
energies or the prediction based on this ordering that cancel out any net charge resulting from the concentra-
equilibrium concentrations of As -As should be signifi- tions of allpositively andnegatively chargeddefects and
i
cantly lowerthan equilibrium concentrationsofAs , as impurities. This equation can then be solved to deter-
Ga
discussed below. mine the Fermi level. Once the Fermi level is known,
To estimate equilibrium concentrations of the excess- it may be used to determine the formation energies and
arsenic-containing defects we begin with the usual ex- theresultingequilibriumconcentrationsofallthedefects
pression present.
If the defect formation energy E for the most ener-
f
C =Ne−∆Ef/kBTeSf/kBe−P∆Vf/kBT , (2) getically favorable charge state of Asi-As in a particu-
lar sample is within k T of the formation energy of the
B
where N is the number of sites at which the defect can mostfavorablechargestate ofAs ,wemayexpectthat
Ga
form in the crystal per unit volume, ∆E is the total the equilibrium concentrations of these two defects are
f
energy of formation of the defect, k is the Boltzmann comparable, assuming that the effects of the entropy of
B
constant, and T is the temperature. The formation en- formation S and the change in the crystal volume ∆V
f f
tropy of the defect is S , P is the pressure, and ∆V associated with the defect formation can be neglected.
f f
is the change in the crystal volume associated with the Wewillnowconcentrateontherelativedefectconcentra-
defect formation. tions at 1500 K (near the melting point of GaAs), since
We note that the defect formation energy ∆Ef for defects with a higher formation energy such as Asi-As
chargeddefects,asgivenbytheformulainEq.(1),hasan have their greatest chance to attain equilibrium concen-
explicit dependence on the Fermi energy, in addition to trationscomparableto thoseofmoreenergeticallyfavor-
its dependence on the calculated energies for defect for- able defects at high temperature. We will estimate the
mationatzeroFermienergy. Thereforewemustcompute effectivecorrectionstothe formationenergywhichoccur
the Fermi energy self-consistently, in order to determine at this temperature due to the entropy of formation and
the native defect concentrations present in a particular change in volume associated with the defects.
sample. If any electrically active impurities or dopants First, to estimate the effect of the change in volume,
are present in the material, the concentrations of these we let P be atmospheric pressure and overestimate ∆V
f
impurities or dopants in all charge states must also be tobethevolumeperbulkatominthecell,whichgivesan
taken into account. We must set up the charge balance effectivecorrectiontothe defectformationenergyP∆V
f
equation, requiring that the free electron and hole con- of 9×10−5 eV. We may safely neglect this correction.
centrations (which also depend on the Fermi level) must Previouscalculations48 on defects in Si found that the
9
formation entropy S is dominated by vibrational con-
f
tributions,andthatthe formationentropiesare6kB and 1.4
5k forthe self-interstitialandthe vacancy,respectively.
B
We note that the self-interstitial in silicon is a <110> As
1.2 Ga
split interstitial with the same basic structure as As -As (+1/0)
i
in GaAs. Therefore, in analogy to the results for de-
fects in silicon,48 we may assume that it is unlikely for V) 1.0 (+2/+1)
e
the split interstitial Asi-As to have a very different for- el ( 0.8
mation entropy when compared to defects such as AsGa, v
e
wSfhi=ch1o0nklyBc(oanntaoinvearteosmtimsoactceu)pfyoirnAgslai-tAtisc,etshitiess.giIvfewserlieset mi l 0.6 As i - As(0/1- )
r
to an effective reduction of the defect formation energy e
F 0.4 V
by S T,or 1.3eV at1500K.Evenif we apply no reduc- Ga
f (+1/0)
tion to the As formation energy due to entropy, this (2- /3- )
Ga
still leaves the effective formation energy about 0.7 eV 0.2 (1- /2- )
higher for As -As than for As , producing equilibrium (0/-1)
i Ga
0.0
concentrations of As -As which are about 0.4% those of
i
As at 1500 K.
Ga
We conclude that evenusing this extremely liberal es-
timate for the formation entropy of As -As and ignoring
i FIG. 5. Transition levels computed with a rigid shift applied
the formation entropy of As cannot lead to an As -As
Ga i to the conduction band and conduction band derived states
concentration approaching that of the antisites in ther-
to correct theLDA band gap underestimate.
mal equilibrium.
similar to the character of the valence band edge states.
D. Defect electrical behavior However,theAs doubledonordefectstatederivesfrom
Ga
an antibonding state of predominantly conduction band
Although the placement ofthe calculatedchargetran- character,whichhasbeenloweredinenergyduetothere-
sition levels in the experimental gap has an uncertainty placementof the originalanion-cationbonds ofthe ideal
farexceeding0.1eVduetotheshortcomingsoftheLDA crystalbyanion-anionbondsbetweentheantisiteandits
in calculating the gap, as discussed in Section II, our as- nearest neighbors. This donor state is occupied by the
calculated band gap and charge transition levels are re- twoextraelectronscontributedbythe arsenicatomthat
ported here to 0.1 eV (or 0.01 eV, for the closely spaced has been substituted for a gallium atom, which cannot
V levels),fortheconvenienceofthe readerwhoprefers be accommodated in the bonding states of the valence
Ga
not to read them off the picture in Fig. 4. For the As , band.
Ga
the(+2/+1)transitionlevelappearsat0.4eVabovethe We conclude that the charge transition levels of the
VBM,andthe (+1/0)levelisatEVBM+0.5eV. Forthe VGa and the Asi-As should remain fixed relative to the
Asi-As, the (+1/0) transition is at EVBM +0.3eV, and valence bandedge,while the donorlevels ofAsGa,which
the (0/−1)transitionis at E +0.5eV. The levels for possess a conduction band character, should be shifted
VBM
the V defect areat0.09eV, 0.13eV, and0.2eV above up together with the conduction band states. In Fig. 5,
Ga
the VBM for the (0/−1), (−1/−2),and (−2/−3)transi- weshowthechargetransitionlevelsforthesedefectscor-
tions, respectively. The calculated band gap of 0.8 eV is rected by the above procedure, using the room temper-
underestimated by 0.7 eV comparedto the experimental ature gap of 1.4 eV. The transition levels of AsGa are
zero-temperature gap of 1.5 eV. shifted to 1.0 eV and 1.1 eV, in fortuitously good agree-
As discussed previously in Section II, we can get a ment with the MCDA results identified with this defect
rough estimate of where the charge transition levels fall inLTGaAs,14,49 althoughthetransitionsarebothabout
within the experimental gap by shifting the conduction 0.4 eV higher than those associated with AsGa in melt-
bandderivedstates(includingthedeepdefectstateswith grown GaAs.50
primarily conduction band character) by the amount
neededtocorrectthe gap,while leavingthe defectstates
withpredominantlyvalencebandcharacterfixedrelative IV. SUMMARY
tothevalencebandedge. Sincetheacceptorlevelsofthe
V are derived from the dangling bonds on the arsenic In this work we have shown that larger supercells and
Ga
neighbors of the vacancy, which require three extra elec- a better k-space sampling than have been used in a
trons to fill them, they should have the predominantly large number of previous DFT defect calculations are
valencebandcharacterofaniondanglingbondstates. In required to give accurate results for the formation en-
Section IIIA, the deep defect state of the As -As was ergies, charge transition levels, defect state properties,
i
shown to have predominantly arsenic p-type character, and atomic structure and relaxation for the arsenic split
10
interstitialinGaAs. Inparticular,wefindthat217-atom or semi-insulating GaAs.
supercellsarenecessarytogetgoodresultsforthecharge
transition levels and the dispersion of the deep defect
state,eventhoughthearrangementofatomsinthestruc- ACKNOWLEDGMENTS
tureiswellconvergedina65-atomcell,particularlyifone
uses the finely spaced 23 MP k-space mesh. This work was supported in part by AFOSR Grants
We have calculated formation energies for As split in- No.F49620-96-1-0167andNo.F49620-97-1-0472,andby
terstitials,Gavacancies,andAsantisitesinAs-richGaAs grants of time on the Cray T3e computers at the DOD
using the larger supercells and better k-point sampling HPC Centers at NAVO and ERDC and at the NPACI
which we have determined to be necessary. Using these SanDiegoandUniversityofTexasSupercomputingCen-
results,wefindthattheequilibriumconcentrationsofar- ters,andontheNPACIAMDclusterattheUniversityof
senicinterstitialswillbesubstantiallylowerthanequilib- Michigan. WealsothankG.SchwartzandM.Bockstedte
riumconcentrationsofarsenicantisitesinAs-rich,n-type for highly useful conversations.
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