Extrinsic point defects in aluminum antimonide

Paul Erhart,* Daniel Åberg, and Vincenzo Lordi†

Physical and Life Science Directorate, Lawrence Livermore National Laboratory, Livermore, California 94550, USA�Received 20 October 2009; revised manuscript received 9 April 2010; published 26 May 2010�

We investigate thermodynamic and electronic properties of group IV �C, Si, Ge, Sn� and group VI �O, S, Se,Te� impurities as well as P and H in aluminum antimonide �AlSb� using first-principles calculations. To thisend, we compute the formation energies of a broad range of possible defect configurations including defectcomplexes with the most important intrinsic defects. We also obtain relative carrier scattering strengths forthese defects to determine their impact on charge carrier mobility. Furthermore, we employ a self-consistentcharge equilibration scheme to determine the net charge carrier concentrations for different temperatures andimpurity concentrations. Thereby, we are able to study the effect of impurities incorporated during growth andidentify optimal processing conditions for achieving compensated material. The key findings are summarizedas follows. Among the group IV elements, C, Si, and Ge substitute for Sb and act as shallow acceptors, whileSn can substitute for either Sb or Al and displays amphoteric character. Among the group VI elements, S, Se,and Te substitute for Sb and act as deep donors. In contrast, O is most likely to be incorporated as an interstitialand predominantly acts as an acceptor. As a group V element, P substitutes for Sb and is electrically inactive.C and O are the most detrimental impurities to carrier transport, while Sn, Se, and Te have a modest to lowimpact. Therefore, Te can be used to compensate C and O impurities, which are unintentionally incorporatedduring the growth process, with minimal effect on the carrier mobilities.

DOI: 10.1103/PhysRevB.81.195216 PACS number�s�: 61.72.J�, 72.10.Fk, 71.55.Eq, 61.72.U�

I. INTRODUCTION

Applications in nuclear and medical imaging as well ashomeland security have recently led to a resurgence of re-search in the field of radiation detection. Specifically, this hasmotivated the search for new materials to improve the per-formance of room-temperature semiconductor radiationdetectors.1–3 Aluminum antimonide, which is a III-V semi-conductor with an indirect band gap of 1.69 eV, is a promis-ing candidates but its potential for radiation detection has notbeen conclusively established yet.4,5

The basic mechanism underlying radiation detection in asemiconductor can be summarized as follows: �1� an incom-ing quantum of radiation transfers energy to the electrons inthe material creating electron-hole pairs, �2� the charge car-riers are separated in an externally applied electric field, �3�the magnitude of the resulting electric current is measured.To achieve maximum energy resolution, the “loss” of chargecarriers due to scattering and recombination events must beminimized. This requires high carrier mobilities and verylow concentrations of intrinsic defects and impurities. In ad-dition, a very high resistivity is desirable to maintain a lowconcentration of background free carriers and thereby tominimize noise events. For instance, in germanium detectorsthat are used at cryogenic temperatures, defect and impurityconcentrations have to be reduced to less than 1010 cm−3 andfree carrier densities to less than 1014 cm−3 to reach optimalperformance.6 To meet such stringent bounds a highly opti-mized manufacturing process is needed, which typically im-plies years of research and development. This situation ren-ders the screening of candidate materials extremelychallenging. Predictive computations, however, offer a pos-sibility to circumvent this difficulty, since they allow study-ing the fundamental limits of materials in a much more timeeffective and systematic fashion. Specifically, quantum me-

chanical methods based on density functional theory can beused to compute the formation energies of intrinsic and ex-trinsic defects and their equilibrium concentrations. Theyalso allow one to obtain the defect-limited scattering rates,which determine the mobility and life time of free chargecarriers. Since these calculations can be done for idealizedsystems, it is possible to separate the effects of different in-trinsic and extrinsic defects and determine the ultimate per-formance limits.

The objective of the present paper is to elucidate the roleof extrinsic defects in aluminum antimonide on the basis offirst-principles calculations. Specifically, we seek to establishwhich elements are the most detrimental for the electronicproperties of AlSb and which elements can be introducedintentionally to compensate for intrinsic defects and impuri-ties. In short, it will be shown that �i� C, Si, and Ge substituteon Sb sites and act as acceptors with shallow transition lev-els, �ii� Sn can substitute on both Al and Sb sites and displaysamphoteric characteristics, �iii� S, Se, and Te substitute on Sband act as donors with deep equilibrium transition levels, �iv�O is most likely to be incorporated as an interstitial andpredominantly acts as an acceptor, �v� P substitutes for Sband is electrically inactive, �vi� C and O have the most det-rimental effect on carrier transport, while �vii� Te is a weakscatterer and can therefore be used to manipulate the netconcentration of charge carriers with a modest effect on theirmobilities. The present study represents a continuation of ourearlier work on intrinsic defects in AlSb7 and is connected toour recent fully quantum-mechanical calculations of defect-limited carrier mobilities.8

This paper is organized as follows. In Sec. II we give anoverview of the methods and computational parameters. Theformation energies of extrinsic defects and their binding en-ergies in complexes with intrinsic defects are summarized inSec. III. Then we discuss the effect of extrinsic defects oncharge carrier mobility using a measure that is based on their

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associated lattice distortions in Sec. IV. Finally, we analyzethe impact of selected extrinsic defects on the net chargecarrier and defect concentrations in the material and discussoptimal doping schemes in Sec. V.

II. METHODOLOGY

A. Defect thermodynamics

The equilibrium concentration, c, of a point defect de-pends on its free energy of formation, �Gf, via

c = c0 exp�− �Gf/kBT� , �1�

where c0 is the concentration of possible defect sites. Theformation free energy �Gf is usually approximated by theformation energy �Ef, which is legitimate if the vibrationalentropy and the pressure-volume term are small.7 In thepresent context we are interested in impurity-related defectsin a binary stoichiometric compound. In this case the forma-tion energy of a defect in charge state q can be written as9

�Ef = Edef −1

2�nAl + nSb��AlSb

bulk −1

2�nAl − nSb���Al

bulk − �Sbbulk�

−1

2�nAl − nSb���AlSb + q�EVBM + �e�

− �i

imp

�ni��ibulk + ��i� , �2�

where Edef is the total energy of the system containing thedefect, ni denotes the number of atoms of type i, and �i

bulk isthe chemical potential of component i in its reference state.Neglecting entropic contributions, the chemical potentials ofthe reference phases can be replaced by the cohesive ener-gies at zero Kelvin. The formation energy depends on thechemical environment via ��AlSb which describes the varia-tion of the chemical potentials under different conditions.The range is constrained by the formation energy of AlSb,���AlSb���Hf�AlSb�, where for the present convention��AlSb=−�Hf�AlSb� and ��AlSb=�Hf�AlSb� correspond toAl and Sb-rich conditions, respectively. The formation en-ergy also depends on the electron chemical potential, �e,which is measured with respect to the valence band maxi-mum, EVBM. The last sum in Eq. �2� runs over all the impu-rities in the system, where �ni denotes the difference in thenumber of atoms of impurity i between the ideal and thedefective system, and ��i is the change in chemical potentialof this component with respect to the bulk chemical poten-tial. For the latter we used the cohesive energy of the respec-tive most stable phase �e.g., O2 molecule, Si in the diamondstructure, etc.� as computed within the present computationalframework.

The formation energies that are shown in Figs. 2 and 5–7have been obtained in the impurity-rich limit, i.e., ��i=0 eV. We have chosen this limit because it provides a goodindication for how easy a certain impurity is incorporatedduring the growth process. In some cases, e.g., silicon andoxygen, this leads to negative formation energies. If one isinterested in formation energies under different chemical

conditions, the relevant formation energies can be easily ob-tained by shifting all formation upwards by the value of ��ichosen. In this way, one can, e.g., determine the formationenergies that are applicable if the AlSb sample is in contactwith say a block of Al2O3.10

Given the formation energies for two different chargestates, q1 and q2, at the valence band maximum ��e=0 in Eq.�2��, we can also obtain the corresponding equilibrium �ther-mal� transition level,

Etrth�q1 → q2� =

�Ef�q2� − �Ef�q1�q1 − q2

. �3�

Note that equilibrium transition levels Etrth are distinct from

electronic transition levels Etropt. While the former determine

the equilibrium electron chemical potential at which achange in the charge state occurs, the latter are related tooptical transitions. In the present work, we obtained elec-tronic transition levels from band structure calculations ofdefect cells. Examples are shown in Fig. 4.

Binding energies for defect complexes are calculated ac-cording to

�Eb = �Efcomplex − �

i

�Ef�i�, �4�

where the sum runs over the individual defects from whichthe complex is constructed. Note that �Eb changes with theelectron chemical potential, since �Ef for both the complexand the individual defects depends on �e. According to Eq.�4� negative binding energies indicate a driving force forcomplex formation.

B. Defect structures

We consider C, Si, O, S, P, and H as potential impuritiesincorporated during the growth and include Ge, Sn, Se, andTe as possible compensating dopants. Figure 1 provides anoverview of the defect configurations included in this work.To be specific, we took into account substitutional configu-rations with both tetrahedral �Td� and trigonal �C3v� symme-try as well as four variations of cation-cation bonded�CCBDX� configurations that have been discussed in con-nection with DX-centers in III-V and II-VI semiconductors.11

�Additional tests were carried out for several distorted sub-stitutional configurations, including molecular dynamicssimulations in order to check for the existence of other rel-evant local minima, but did not reveal any further configu-rations of importance�. With regard to interstitial configura-tions, we included tetrahedral �Xi,tet,Al ,Xi,tet,Sb� and hexagonalsites �Xi,hex� as well as several split-interstitial configurationsoriented along �100� and �110�. The latter configurationswere always observed to relax into a structure in which thebond between the impurity atom and its nearest neighbor isoriented slightly away from the �110� direction �compareFig. 1�h��. In the literature, this distorted structure has beenpreviously referred to as “bridge” interstitial, and for sim-plicity in the following we adopt this convention. We typi-cally calculated formation energies for charge states betweenq=−2 and +2, and where necessary also included chargestates −3 and +3.

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We also investigated the formation of defect complexes ofgroup IV and VI elements with aluminum vacancies VAl andantimony antisites SbAl which are the two most importantintrinsic defects.7

C. Computational details

Calculations were performed within density-functionaltheory �DFT� using the local density approximation �LDA�and the projector augmented wave method12,13 as imple-mented in the Vienna ab initio simulation package.14–17

In our earlier study7 of intrinsic defects in AlSb we em-ployed a variety of supercell sizes in combination with afinite-size scaling procedure in order to mitigate the limita-tions of the supercell method. Given the large number ofdifferent elements, defect structures, and charge states con-sidered in the present work, such a comprehensive approachis, however, beyond our present computational capabilities.We therefore carried out the majority of the calculations for64-atom supercells, which based on the results of Ref. 7implies that the error in our calculated formation energiesdue to the supercell approximation is still only on the orderof 0.1 eV. For Brillouin-zone integrations we used a 6�6

�6 Monkhorst-Pack mesh.18 For a small set of configura-tions, we performed additional calculations using 216-atomcells �see Fig. 4� and a 4�4�4 Monkhorst-Pack k-pointmesh to verify the viability of our 64-atom cell calculations.We used a plane-wave cutoff energy of 500 eV, which isimposed by the pseudopotential for oxygen, but for the sakeof consistency was used for all calculations. A Gaussiansmearing with a width of �=0.1 eV was used to determinethe occupation numbers. The ionic positions were optimizeduntil the maximum force was below 20 meV /Å. For thecharged defect calculations a homogeneous backgroundcharge was added to ensure charge neutrality of the entirecell.

For bulk AlSb we obtain a lattice constant of 6.12 Å ingood agreement with the experimental room temperaturevalue of 6.13 Å. The band gap is 1.12 eV, which is lowerthan the experimental room temperature value of 1.69 eV asexpected for a LDA calculation. A more detailed overview ofthe bulk properties of Al, Sb, and AlSb that are obtainedusing the present computational parameters can be found inRef. 7.

The underestimation of the band gap is a well-known de-ficiency of DFT within the LDA. In order to overcome thisdeficiency in Ref. 7 we have employed a post-mortem cor-

(a)XSb (C3v) (b)Xi,hex

(c)Xi,tet,Al (d)Xi,tet,Sb (e)ideal

(f)(X − Al)〈100〉Al (g)(X − Sb)

〈100〉Sb

(h)Xi,bb (i)Hi,bb

(j)XAl (α-CCBDX) (k)XAl (β-CCBDX)

(l)XSb (α-CCBDX) (m)XSb (β-CCBDX)

FIG. 1. �Color online� Overview of defect configurations considered in this study. Large violet, small gray, small orange and very smallgreen spheres represent Sb, Al, impurity atoms, and H, respectively.

EXTRINSIC POINT DEFECTS IN ALUMINUM ANTIMONIDE PHYSICAL REVIEW B 81, 195216 �2010�

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rection scheme, which relies on a separation of valence andconduction band states. Analysis of the electronic structureof the defects in this work reveals that in many cases thedefect induces electronic levels in the band gap, which ren-ders the separation into valence, conduction or localizedstates highly unreliable. Since as a result the application ofthe correction scheme employed in Ref. 7 becomes ambigu-ous, we have decided to abstain from any band-gap correc-tions in the present work. Instead all results are presentedand discussed with respect to the calculated band gap, asdone previously by Du.10 For reference we have, however,included the experimental band gap in Figs. 2, 5, and 6. Tocompute the binding energies of defect complexes involvingaluminum vacancies and antimony antisites, we used the de-

fect formation energies obtained in Ref. 7 with for 64-atomcells and without band gap corrections

D. Lattice distortion and effect on charge carrier mobilities

Defects break the translational symmetry of an ideal crys-tal and therefore act as scattering centers for electrons andholes traveling through the crystal. The two most importantcontributions to the scattering cross section of a defect arerelated to the amount of charge localized at the defect �Cou-lomb scattering� and the lattice distortions that accompanydefect relaxation. Since we are primarily interested in com-paring defects that have identical absolute charge states �e.g.,CSb

−1 vs TeSb+1�, in the following we focus on the second con-

0

1

2

3

4

5

6

7

For

mat

ion

ener

gy(e

V) (a) C CAl (Td)

CSb (Td)CSb−SbAlCSb−VAlCi, tet, AlCi, tet, SbCi, hexAl-richSb-rich

-2

-1

0

1

2

3

4

5

For

mat

ion

ener

gy(e

V) (b) Si SiAl (Td)

SiSb (Td)SiSb-SbAlSiSb-VAlSii, tet, AlSii, tet, Sb

-2

-1

0

1

2

3

4

5

For

mat

ion

ener

gy(e

V) (c) Ge GeAl (Td)

GeSb (Td)GeSb-SbAlGeSb-VAlGei, tet, AlGei, tet, Sb

-2

-1

0

1

2

3

4

5

0.0 0.4 0.8 1.2 1.6

For

mat

ion

ener

gy(e

V)

Electron chemical potential (eV)

(d) Sn SnAl (Td)SnSb (Td)SnSb-SbAlSnSb-VAlSni, tet, AlSni, tet, Sb

-2

-1

0

1

2

3

4

5

EGLDA

EGexp

(e) P PAl (Td)PAl (C3)PSb (Td)Pi, tet, AlPi, tet, SbPi, hex

-3

-2

-1

0

1

2

3

4(f) S SAl (Td)

SAl (C3)SSb (Td)SSb-SbAlSSb-VAlSi, tet, AlSi, tet, SbSi, hex

-2

-1

0

1

2

3

4 (g) Se SeAl (Td)SeAl (C3)SeSb (Td)SeSb-SbAlSeSb-VAlSei, tet, AlSei, tet, SbSei, hex

-1

0

1

2

3

4

5

6

0.0 0.4 0.8 1.2 1.6

Electron chemical potential (eV)

(h) Te TeAl (Td)TeAl (C3)TeSb (Td)TeSb-SbAlTeSb-VAlTei, tet, AlTei, tet, SbTei, hex

FIG. 2. �Color online� Formation energies of impurity-related defects for �a-d� group IV �e� P, and �f-h� group VI elements calculated asa function of the electron chemical potential with ��imp=0 eV in Eq. �2�. Symbols indicate equilibrium transition levels �see Eq. �3��. Theslope of the lines corresponds to the charge state. The white region indicates the calculated band gap while the lighter gray region indicatesthe difference between the experimental and the calculated band gap.

ERHART, ÅBERG, AND LORDI PHYSICAL REVIEW B 81, 195216 �2010�

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tribution only. There are many different ways in which onecan quantify lattice distortions. We have empirically foundthat a particularly useful measure is given by

M2 = dr��r��V���2

, �5�

where �V denotes the potential difference between the de-fective and ideal cells. This quantity includes not only ionicrelaxations but also changes in the electronic charge distri-bution.

In Eq. �5� we use the potential and not, for example, thecharge density difference. This choice is motivated by a fullfledged computation of the scattering rates8 based on Fermi’sgolden rule for which one needs to evaluate matrix elementsof the form �� f��V��i�.19,20 In that study,8 we found that M2

reproduces very well the trends obtained from the scatteringrate calculations based on Fermi’s golden rule across a widerange of extrinsic defects. Since the computation of M2 is,however, orders of magnitude more efficient, in the follow-ing we use the relative carrier scattering strength M2 toassess the impact of defects on carrier transport. We note thatthis analysis does not differentiate between electron and holescattering. It is, however, unusual that a given defect has asignificantly different impact on electron and holescattering.8

E. Self-consistent charge equilibration

In Refs. 7 and 21 we have described in detail a self-consistent scheme to determine the concentrations of defectsand charge carriers as a function of temperature and chemi-cal environment. Here, we only provide a brief summary ofthe most important equations. The concentrations of elec-trons, holes, and charged defects in a material are coupled toeach other via the charge neutrality condition

ne + nA = nh + nD, �6�

where ne and nh are the electron and hole concentrations,respectively. They are given by

ne = CBM

�

D�E�f�E,�e�dE �7�

nh = −�

VBM

D�E��1 − f�E,�e��dE , �8�

where D�E� denotes the density of states, f�E ,�e�= �1+exp��E−�e� /kBT� −1 is the Fermi-Dirac distribution, �e isthe electron chemical potential, VBM and CBM indicate thevalence band maximum and the conduction band minimum,respectively. The concentrations of acceptors and donors, nAand nD, can be split into contributions due to intrinsic andextrinsic defects

nA = nAint + nA

ext and nD = nDint + nD

ext �9�

The concentration of intrinsic acceptors �donors� is obtainedby summing over all intrinsic defects with charge statesq0 �q0�,

nAint = �

i

acceptors

eqici,q and nDint = �

i

donors

eqici,q �10�

where e is the unit of charge and the defect concentration isgiven by Eq. �1�.

The concentrations of extrinsic acceptors and donors aregiven by expressions that are analogous to Eq. �10� but sub-ject to the constraint that the total concentration of a givenimpurity or dopant is constant, a condition that can be easilyachieved by proper normalization of the defect concentra-tions. The set of Eqs. �6�–�10� are solved self-consistentlyusing an iterative algorithm.

III. FORMATION ENERGIES AND DEFECT STRUCTURES

A. Group IV elements: C, Si, Ge, and Sn

The formation energies of defects involving the first fourgroup IV elements �C, Si, Ge, and Sn� are shown in Figs.2�a�–2�d�. In all cases the most important configuration is thesubstitutional defect on an Sb-site, XSb �X=C, Si, Ge, or Sn�.Regardless of the initial condition, we find the defects torelax into a configuration with Td symmetry. The XSb defectsoccur predominantly in charge state q=−1 and thus act assingle acceptors. This observation is in agreement with Halland resistivity measurements.22–24

Within the error bars of our calculations the 0 /−1 equi-librium transition levels for C, Si, and Ge coincide with theVBM whereas for Sn the equilibrium transition level is lo-cated 0.1 eV above the VBM �compare Fig. 3�. In agreementwith our calculations, shallow acceptor levels have been ob-served for Si by Bennett et al.23 while C has been found toact an acceptor by McCluskey et al.24 For XSb defects, ourcalculations do not reveal any electronic states inside theband gap that could give rise to strong optical signatures.

For C the substitutional defect on an Al-site XAl, whichalways adopts a local Td symmetry, is very high in energyand therefore should not occur in thermodynamic equilib-rium. For Si, Ge, and Sn, however, this defect becomes in-creasingly more important. In fact in the case of Sn, theformation energies for substitutional defects on Al and Sb-sites are very similar. These results are consistent with theexperiments by Shaw22 who observed that Sn can occupyboth Al and Sb sites. For Si, Ge, and Sn the calculationspredict XAl to be a donor with a deep +1 /0 equilibrium tran-sition level �compare Fig. 3�. For Si, Ge, and Sn we also findelectronic levels located inside the band gap. This is explic-itly demonstrated for the case of GeAl in Fig. 4�b�, whichreveals a localized defect state approximately 0.24 eV belowthe CBM.

The similar formation energies of SnAl donors and SnSbacceptors gives rise to an interesting type of amphoteric be-havior for Sn-doped AlSb. Usually this property is associatedwith a single defect type which undergoes electronic �andusually also structural� changes as a function of charge state.In the case of AlSb:Sn, however, the amphoteric behavior isdue to two different defects occupying two different latticesites that possess opposite electronic characteristics. Thisleads to an intriguing opportunity for compensation dopingthat will be explored in Sec. V D.

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Regarding interstitial defects, both tetrahedral configura-tions Xi,tet,Al and Xi,tet,Sb �Figs. 1�c� and 1�d�� are found to belocal minima on the energy landscape with Xi,tet,Al beingslightly more stable. The hexagonal interstitial Xi,hex is a lo-cal minimum for C only, whereas it relaxes to the Xi,tet,Alconfiguration for Si, Ge, and Sn. All interstitials are, how-ever, significantly higher in energy than the most stable sub-stitutional defects, therefore in thermodynamic equilibriumtheir concentrations should be negligible. Since the migra-tion barriers for substitutional defects are, however, typicallyvery large, it is possible that interstitial configurations play arole with respect to the diffusion of group IV elements.

We also considered defect complexes of group IV impu-rities with the intrinsic defects VAl and SbAl. The formationenergies for XSb−VAl defects which act as acceptors arerather high due to the like charges of the two isolated defects.The XSb−SbAl complexes on the other hand display rathersmall formation energies that are typically less than 1 eVhigher than the formation energy of the most stable configu-ration. These defects exhibit amphoteric behavior with thedonor state dominating over the wider range of the band gap�also see Fig. 7�a��.

B. Group V elements: phosphorus

Phosphorus is included in the present work as a prototypi-cal example for group V elements which are isoelectronic to

Sb. The calculated formation energies are shown in Fig. 2�e�which shows that P prefers substitution on a Sb-site main-taining Td symmetry. PSb is neutral throughout the band gapand thus does not introduce any transition levels inside thegap. All other possible configurations are much higher inenergy.

C. Group VI elements: S, Se, Te

Since oxygen behaves very differently from the othergroup VI elements, we first consider the less complex casesof S, Se, and Te. A detailed discussion of oxygen is given inthe next section. For S, Se, and Te the most stable defectconfiguration for electron chemical potentials over the widestrange of the band gap is XSb which possesses Td symmetryand acts as a donor �Figs. 2�f�–2�h��. The +1 /0 equilibriumtransition levels for these defect are deep for all three ele-ments �Fig. 3�. While all interstitial configurations consid-ered are local minima, they are always much higher inenergy than the respective XSb defect. Although there is sup-porting experimental evidence of the existence of shallowdonor levels,25,26 it is firmly established that the ground stateof these donors for electron chemical potentials near the con-duction band minimum is actually a DX-center27,28 associ-ated with a deep level.29–31 As indicated in Sec. II B, we haveinvestigated a variety of possible DX-center structures.While we find several of these configurations to be localenergy minima for charge state q=−1, their formation ener-gies are always higher than the respective Td configurationsin charge state 0 as long as the electron chemical potentialremains within the band gap. This behavior can be attributedto the LDA, which in this instance fails to properly capturethe energetics of localized vs delocalized defect states. Adetailed discussion is, however, beyond the scope of thisarticle and will be published elsewhere.

Defect complexes of XSb with SbAl antisites for the groupVI impurities show similar trends as in the case of the groupIV impurities being about 1 eV higher than the isolated XSbdefects �see Fig. 7�c��. In all cases, these complexes displaydonor character. In contrast to the group IV elements, how-ever, the XSb−VAl complexes have much lower formationenergies resulting from the Coulombic interaction of the op-positely charged isolated defects. In fact, for electron chemi-cal potentials in the upper half of the band gap, this mutual

+0.0

+0.4

+0.8

+1.2C

Sb

Si S

b

Si A

l

Ge S

b

Ge A

l

Sn S

b

Sn A

l

Oi,A

l

OS

b,1

OS

b,2

SS

b

Se S

b

Te S

b

−1.2

−0.8

−0.4

+0.0

Ene

rgy

w.r

.t.V

BM

(eV

)

Ene

rgy

w.r

.t.C

BM

(eV

)

Conduction band

Valence band

group IV group VI

0

−1

0

−1

+1

0

0

−1

+1

−1

0

0

−1

+1

0

0

−2

−1+1

−10

0

−1

+1

0+1

0 +1

0

donoracceptorsamphoteric

+0.0

+0.4

+0.8

+1.2

Si A

l

Ge A

l

Sn A

l

OS

b,1

OS

b,2

SS

b

−1.2

−0.8

−0.4

+0.0

Ene

rgy

w.r

.t.V

BM

(eV

)

Ene

rgy

w.r

.t.C

BM

(eV

)

Conduction band

Valence band

group IV group VI

(a) (b)

FIG. 3. �Color online� Equilibrium �left� and electronic �right� transition levels of the most relevant impurity related defects for group IVand VI elements. The equilibrium transition levels have been extracted from the formation energies using Eq. �3� and correspond to theposition of the symbols in Figs. 2 and 5. The electronic transition levels have been identified using band structure calculations of the typeshown in Fig. 4. OSb,1 and OSb,2 stand for OSb�C3v� and OSb��−CCBDX��, respectively. All other substitutional defects have Td symmetry.The oxygen interstitial has tetrahedral symmetry.

−2

−1

0

1

2

M X Γ R X

Ene

rgy

w.r

.t.V

BM

(eV

)

(a) ideal

M X Γ R X

(b) GeAl

M X Γ R X

(c) OSb (C3v)

FIG. 4. �Color online� Band diagrams for �a� an ideal system aswell as for systems containing �b� a GeAl defect and �c� a OSb ��−CCBDX� defect. Conduction and valence bands are indicated byred and blue lines, respectively. Defect states are shown in thickyellow lines. All calculations were performed with 216-atom super-cells in charge state q=0.

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attraction renders the XSb−VAl acceptor complex the moststable defect. The donor character of XSb combined with theacceptor character of XSb−VAl could allow for an interestingself-compensation effect. Since this would require that alu-minum vacancies remain mobile down to very low tempera-tures ��400 K� in order to achieve full equilibrium, such ascenario is, however, unlikely to occur in reality.

D. Group VI elements: oxygen

Oxygen behaves distinctly differently from all the otherelements considered so far, in that its interstitial configura-tions have very low formation energies and there is a multi-tude of different defect configurations which are all veryclose in energy �Fig. 5�. Over the widest range of the elec-tron chemical potential, the most stable defect is the tetrahe-dral interstitial surrounded by Al atoms Oi,tet,Al in chargestate q=−2 �Fig. 1�c��. This acceptor defect displays transi-tion levels 0.1 eV �0 /−1� and 0.5 eV �−1 /−2� above theVBM �see Fig. 3�.

For Al-rich conditions and electron chemical potentials inthe lower half of the band gap, the lowest energy defect is aninterstitial which assumes a bridgelike structure �Fig. 1�h��and is indicated as Oi,bb in Fig. 5. It maintains the neutralcharge state throughout the band gap. Very close to the VBMthe OSb�C3v� defect emerges as the ground-state configura-tion. This defect adopts a configuration with C3v symmetryand is associated with a pronounced lattice distortion �Fig.1�a��. It displays amphoteric characteristics with equilibriumtransition levels 0.2 eV �+1 /0� and 0.4 eV �0 /−1� above theVBM. As shown in Figs. 3 and 4�c�, it also introduces twoelectronic levels into the band gap that are located 0.45 eVabove the VBM and right at the CBM, respectively. For Sb-rich conditions and electron chemical potentials in the lowerhalf of the band gap the OSb-SbAl defect complex whichexhibits donor character prevails.

Finally, there are several other configurations with forma-tion energies within about 0.3 eV of the respective ground-state configurations. These include the �O-Sb�Sb

�100� split inter-stitial �Fig. 1�g��, which is neutral throughout the band gapand electrically inactive, as well as the OSb��−CCBDX� de-fect �Fig. 1�m��.

We observe a large number of oxygen configurations withvery similar formation energies, and most of these defects

lead to one or more deep equilibrium transition levels. Deepstates induce defect-mediated recombination and/or trappingand thereby limit the lifetime of charge carriers. Oxygen in-corporation is therefore very detrimental to the operation andenergy resolution of a detector device.

E. Hydrogen

For hydrogen, we focused on interstitial configurationsand considered both tetrahedral sites and the bridge bondconfiguration �Fig. 1�i��. The results are shown in Fig. 6. Forelectron chemical potentials in the lower half of the bandgap, the bridge bond configuration is the energetically mostfavorable one. It acts as a deep donor with an equilibriumtransition level 0.3 eV below the CBM. For electron chemi-cal potentials in the upper half of the band gap, the Hi,tet,Aldefect prevails which behaves as an acceptor with an equi-librium transition level 0.1 eV above the VBM. We also notethat although the formation energies for Hi,bb and Hi,tet,Al inthe neutral charge state are virtually identical, the structuresare nonetheless distinct.

F. Defect complexes

In this section, we consider the interaction of the twomost important intrinsic defects7 SbAl and VAl with group IVand VI impurities. In the preceding sections, we have already

−4

−3

−2

−1

0

1

0.0 0.4 0.8 1.2 1.6

For

mat

ion

ener

gy(e

V)

Electron chemical potential (eV)

EGLDA

(a) Al-rich

0.0 0.4 0.8 1.2 1.6

Electron chemical potential (eV)

EGexp

(b) Sb-rich OAl (Td)

OAl (C3v)

OAl (β-CCBDX)

OSb (Td)

OSb (C3v)

OSb (β-CCBDX)

OSb-VAl

OSb-SbAl

Oi, tet, Al

Oi, tet, Sb

Oi, bb

(O-Al)Al100

(O-Sb)Sb100

FIG. 5. �Color online� Formation energies of oxygen-related defects under Al-rich and Sb-rich conditions with ��O=0 eV in Eq. �2�.Symbols indicate equilibrium transition levels. The slope of the lines corresponds to the charge state.

0

1

2

3

0.0 0.4 0.8 1.2 1.6

For

mat

ion

ener

gy(e

V)

Electron chemical potential (eV)

EGLDA

EGexp

Hi, tet, AlHi, tet, SbHi, hexHi, bbHAl (C3v)

H

FIG. 6. �Color online� Formation energies of hydrogen-relateddefects with ��H=0 eV in Eq. �2�. Symbols indicate equilibriumtransition levels. The slope of the lines corresponds to the chargestate.

EXTRINSIC POINT DEFECTS IN ALUMINUM ANTIMONIDE PHYSICAL REVIEW B 81, 195216 �2010�

195216-7

seen that these defect complexes can have very low forma-tion energies, in particular in the case of group VI elements,where for certain electron chemical potentials they are themost stable defect configuration.

In Fig. 7, we show the binding energies and the respectiveformation energies of Sn and Te complexes that are proto-typical for group IV and VI elements, respectively. To illus-trate the trends we show in Fig. 8 the most negative bindingenergies �indicating the strongest driving force for defectcomplex formation� for all group IV and VI impurities con-sidered.

In the case of Sn, both types of complexes display strongbinding for electron chemical potentials in the lower 2/3 ofthe band gap with binding energies as negative as −2.2 eVfor SnSb−SbAl at the valence band maximum. On the otherhand, for electron chemical potentials in the upper third ofthe band gap, the interaction is strongly repulsive. Thesetrends hold for the other group IV elements as well with theexception of the CSb−VAl complex, which is never bound. Ingeneral, the vacancy complexes have high formation ener-

gies and act as acceptors, while the antisite complexes haverather low formation energies near the valence band edgeand display mostly donor characteristics.

As exemplified by Te, the trends described above forXSb−SbAl prevail also for group VI elements. With respect toXSb−VAl complexes with group VI impurities, we obtainbinding energies which are consistently negative throughoutthe band gap. As discussed in Sec. III C, the vacancy com-plexes act as acceptors, have relatively low formation ener-gies, and are actually the most stable defect for electronchemical potentials near the CBM.

IV. RELATIVE CARRIER SCATTERING STRENGTHS

The relative carrier scattering strengths calculated usingEq. �5� for the most important intrinsic and extrinsic defectsvary over several orders of magnitude as shown in Fig. 9.While the absolute values of the relative scattering strengthbear no meaning, they still allow us to compare differentextrinsic and intrinsic defects with respect to their impact oncharge carrier mobility.8

Figure 9 clearly shows that oxygen induces by far thestrongest carrier scattering and thus has the most detrimentalimpact on charge carrier mobilities, with carbon also show-ing a very high-scattering strength. The relative scatteringstrengths for these impurities are about one order of magni-tude larger than for the aluminum vacancy, which is the mostdetrimental intrinsic defect.

For the elements within the same group of the periodictable, the scattering strength generally decreases with in-creasing atomic number. This effect is owed to the diminish-ing size mismatch between the impurity atom and Sb, forwhich most impurities substitute, leading to decreased latticedistortion upon substitution. As a result, the elements thathave the least negative effect on carrier mobility are Sn, Se,and particularly Te. In fact, many of the substitutional impu-rities scatter carriers less strongly than the intrinsic defects.

−2

−1

0

0.0 0.4 0.8 1.2

Bin

ding

ener

gy(e

V)

Electron chemical potential (eV)

SnSb-VAlSnSb-SbAl

(b)

0

1

2

3

For

mat

ion

ener

gy(e

V)

EGLDA

SnSb (Td)SnSb-SbAlSnSb-VAlAl-richSb-rich

(a)

−1

0

1

0.0 0.4 0.8 1.2

Bin

ding

ener

gy(e

V)

Electron chemical potential (eV)

TeSb-VAlTeSb-SbAl

(d)

0

1

2

3

4

For

mat

ion

ener

gy(e

V)

EGLDA

TeSb (Td)TeSb-SbAlTeSb-VAlAl-richSb-rich

(c)

FIG. 7. �Color online� �a,c� Formation and �b,d� binding energies of defect complexes of intrinsic defects with �a,b� Sn and �c,d� Te. Thetwo cases are exemplifying the situation for group IV and VI elements, respectively.

−3

−2

−1

0

1

2

C O Si S

Ge

Se

Sn

TeM

ostn

egat

ive

bind

ing

ener

gy(e

V)

Atomic number

group IV, XSb-VAl

group IV, XSb-SbAl

group VI, XSb-VAl

group VI, XSb-SbAl

FIG. 8. �Color online� Most negative binding energies forXSb−VAl and XSb−SbAl complexes where X is a group IV or VIelement. According to Eq. �4�, negative binding energies indicate adriving force for defect complex formation.

ERHART, ÅBERG, AND LORDI PHYSICAL REVIEW B 81, 195216 �2010�

195216-8

These elements are therefore excellent candidates for dopingthe material in order to achieve carrier compensation withminimal impact on mobility, an opportunity that is exploredin the following section.

V. CHARGE CARRIER CONCENTRATIONS

In the previous sections we have determined the forma-tion energies of a wide range of extrinsic defects and dis-cussed, which of them act as donors or acceptors. Further-more, we have obtained their relative scattering rates, whichenabled us to identify the defects which are, respectively, theleast and most detrimental to the electron and hole mobilitiesin the material. Now, we are in a position to combine thesedata to predict net charge carrier concentrations and to deter-mine conditions for obtaining optimal material. For radiationdetection, the objective is to obtain a material which featuresmaximal charge carrier mobilities, which requires small scat-tering rates, and very high resistivity, which requires low netcharge carrier concentrations.

In the analysis that follows, we consider four importantcases. First, we analyze the pure material �containing onlyintrinsic defects�. Then, we include in our model uninten-tional impurities from the growth process, followed byanalysis of carrier compensation with intentional Te doping.Finally, we consider the case of Sn doping, which exhibitsunique behavior due to its amphoteric character.

A. Pure material

In Ref. 7, we employed the self-consistent charge equili-bration scheme described in Sec. II E to determine the con-centrations of intrinsic defects and charge carriers in pureAlSb �no impurities� using defect formation energies calcu-lated from DFT, which had been corrected for the band gaperror. As discussed in Sec. II C, in the present work we didnot employ any such corrections. Before we can apply theself-consistent charge equilibration scheme to study the be-havior of extrinsically doped AlSb, we therefore have to dis-cuss to which extent the band-gap error influences the re-sults. According to Eq. �7� the intrinsic electron and holeconcentrations depend exponentially on the band gap. To be

consistent with the calculations discussed above, we nowhave to use the LDA band gap and the corresponding forma-tion energies for the intrinsic defects. The absolute carrierconcentrations are therefore no longer comparable to our cal-culations in Ref. 7. For the present purpose, we are, however,interested not in absolute carrier concentrations but in netcharge carrier concentrations, i.e., the difference between theconcentrations of electron and holes,

�n = ne − nh. �11�

The impact of the band-gap error on �n is not immediatelyobvious. In Fig. 10, we therefore compare the net electronconcentrations for pure AlSb using as-calculated formationenergies as well as formation energies, which have been cor-rected for the band gap error. We find that while the actualvalues deviate somewhat, the conduction type �n vs p�,chemical potential dependence, and the order of magnitudeof �n are very similar. This finding enables us to use theas-calculated formation energies to compute net charge car-rier concentrations for AlSb with extrinsic defects. Figure 10also shows that pure AlSb exhibits n-type behavior over the

10−1

100

101

+1 0

−1

−2

−3

+1

SbAl VAl Ali, Al

Rel

ativ

esc

atte

ring

stre

ngth

Sb surplus Al

surplus

10−1

100

101

−1

−1

−1

+1

−1 0

−1

−2

+1 0

−1

+1 0

+1 0

+1 0

CSb SiSb GeSb SnAl SnSb PSb Oi, tet, Al OSb (C3v) SSb SeSb TeSb

Rel

ativ

esc

atte

ring

stre

ngth

group IV group VI

(a) (b)

FIG. 9. �Color online� Relative carrier scattering strengths for the most prevalent intrinsic �left� and extrinsic �right� defects calculatedusing Eq. �5�. The substitutional defects all have Td symmetry with the exception of oxygen for which the results for the C3v configurationare shown. In the left panel, red arrows mark the most important charge states for the intrinsic defects �Ref. 7�.

1011

1012

1013

1014

1015

1016

1017

−0.4 −0.2 0.0 0.2 0.4

Sb−rich Al−rich

Net

elec

tron

conc

entr

atio

n(c

m−

3 )

Chemical potential difference (eV/atom)

corr.as calc.

1300 K1100 K

900 K

FIG. 10. �Color online� Net electron concentration for pure AlSbat three different temperatures using band-gap corrected �solidlines� and as-calculated �dashed lines� formation energies assumingfull equilibrium. The horizontal bar indicates the range of chemicalpotentials that are accessible based on the calculated compoundformation energy.

EXTRINSIC POINT DEFECTS IN ALUMINUM ANTIMONIDE PHYSICAL REVIEW B 81, 195216 �2010�

195216-9

entire range of the chemical potential difference. In contrast,as-grown AlSb has been found to exhibit p-type conductiv-ity. In the following section, we show that this apparent dis-crepancy is caused by the accidental incorporation of impu-rities during growth.

B. Unintentional background doping

In the experimental growth process, certain impurities areimpossible to avoid. In particular this includes oxygen andcarbon.24,32 Oxygen can be incorporated from the gas phaseor the components of the growth chamber and is also presentin the raw starting materials. Carbon enters because graphitesusceptors are used to heat the melt.33 Using a highly opti-mized growth process, it is possible to reduce the concentra-tion of oxygen to about 1015 cm−3 and the concentration ofcarbon to about 1017 cm−3.32

In Fig. 11, we show the net charge-carrier concentrationsfor AlSb with c�C�=1017 cm−3 and c�O�=1015 cm−3. Infact, these concentrations cause the conductivity to switchfrom n-type to p-type, which resolves the apparent discrep-ancy between theory and experiment mentioned above. Theinversion of the conductivity type is indeed expected since inSec. III both of these impurities have been identified as ac-ceptors.

C. Compensation with Te

Since the background doping by C and O renders thematerial p-type, compensation requires addition of donor

ions. To simultaneously achieve high carrier mobilities andlow net carrier concentrations, the dopant should possess ascattering potential which is as small as possible. Accordingto Figs. 2 and 9 Te is an ideal candidate based on thesecriteria. In Fig. 11 we show the effect of adding Te at aconcentration of 1017 cm−3 to AlSb containing C and O im-purities. We find that by using Te doping in this concentra-tion range the net charge carrier concentration can be tunedvery efficiently. The net carrier concentration changes as afunction of the chemical potential difference between Al andSb, with a minimum close to the Sb-rich limit �typical an-nealing condition�. Intentional doping with Te therefore en-ables us to compensate the background doping due to C andO impurities while minimizing the detrimental impact on thecarrier mobilities. In fact, using these data as a guidance forthe experimental growth process led to a dramatic improve-ment of the material for radiation detection, which is re-ported elsewhere.32

D. Ambipolar doping with Sn

According to Fig. 2, Sn can be incorporated both on theSb and the Al sublattice, which gives rise to acceptor anddonor-type defects, respectively. Also, Fig. 9 shows that bothof these defects are weak scatterers compared to the alumi-num vacancy. Figure 12�a� illustrates the effect of Sn on thenet charge carrier concentration. In thermodynamic equilib-rium and for a Sn concentration of about 1017 cm−3, thematerial is n-type if grown under Sb-rich conditions andp-type if grown under Al-rich conditions �case II in Fig.12�a��. In the presence of C and O impurities this behavior isretained although the transition point is shifted toward Sb-rich conditions �case C in Fig. 12�a��. By increasing the Snconcentration the minimum in �n can be shifted back towardthe impurity-free case �case IV in Fig. 12�a��. Figure 12�b�reveals that the origin of this behavior is the subtle equilib-rium between SnSb

−1 and SnAl+1 defects, which act as acceptors

and donors, respectively.We suggest that this intriguing property of Sn-doped AlSb

can be exploited to achieve self-compensation. In a typicalexperiment, an AlSb single crystal is grown under Al-richconditions close to the melting point of 1330 K. The growthprocess is sufficiently slow to assume that we are near ther-modynamic equilibrium, which implies that the vast majorityof Sn occupies Sb-sites. The growth step is usually followedby a high-temperature anneal at about 1200 K under Sb-richconditions. This inversion of chemical conditions leads to astrong driving force for redistributing Sn from the Sb to theAl sublattice. This conversion can be mediated by aluminumvacancies which at this temperature are both very mobile34

and exist in high concentrations.7 By carefully controllingthe annealing time it should thus be possible to achieve aSnSb /SnAl ratio close to one. At this point the material wouldbe perfectly compensated due to the opposing effects of SnSband SnAl defects. Due to the moderate scattering strength ofSn-related defects such a material would exhibit high carriermobilities in conjunction with high resistivity, as needed forthe radiation detection applications.

VI. CONCLUSIONS

In this work we have investigated the thermodynamic andelectronic properties of group IV and group VI impurities as

1015

1016

1017

1018

−0.4 −0.2 0.0 0.2 0.4

Net

char

geca

rrie

rco

ncen

trat

ion

(cm

−3 )

Chemical potential difference (eV/atom)

1200 K

(I) intrinsic

(II) w/ C and O

(III) w/ C, O, Te

1015

1016

1017

1018Sb−rich Al−rich

1300 K

n-type p-type

FIG. 11. �Color online� Net charge carrier concentration for �i�pure AlSb, �II� AlSb with c�C�=1017 cm−3 and c�O�=1015 cm−3,and �III� AlSb containing c�C�=1017 cm−3, c�O�=1015 cm−3, andc�Te�=1017 cm−3 for two different temperatures assuming fullequilibration and using as-calculated formation energies. If �n inEq. �11� is positive �negative� as shown by the solid �dashed� linesthe material exhibits n-type �p-type� conductivity. The horizontalbar indicates the range of chemical potentials that are accessiblebased on the calculated compound formation energy.

ERHART, ÅBERG, AND LORDI PHYSICAL REVIEW B 81, 195216 �2010�

195216-10

well as P and H in AlSb using density functional theorycalculations. First, we computed the formation energies for awide range of possible defect configurations and chargestates to identify the most important defect configurations,their electronic characteristics, and their equilibrium andelectronic transition levels. We found that C, Si, and Ge pref-erentially substitute for Sb, act as acceptors and exhibit shal-low transition levels. Sn, however, can substitute for Sb andAl and displays amphoteric behavior.

With regard to group VI elements, S, Se, and Te substitutefor Sb and have deep donor levels. Oxygen in contrast can beincorporated in several different configurations that display avariety of electronic characteristics and transition levels, pre-dominantly of the acceptor type. For the widest range of

electron chemical potentials a tetrahedral interstitial configu-ration is the lowest in energy.

Phosphorus substitutes for Sb and is electrically inactivewhile H is incorporated on interstitial sites and can act asacceptor or donor depending on conditions.

With respect to defect complexes, our calculations showthat group IV elements bind to both aluminum vacancies�VAl� and Sb antisites �SbAl� for electron chemical potentialsin the lower two-thirds of the band gap. �The CSb−VAl com-plex is an exception and is never bound�. The vacancy com-plexes have high formation energies and act as acceptorswhile the antisite complexes have rather low formation en-ergies near the valence band edge and display mostly donorcharacteristics. Similar trends hold for group VI elements butin this case the vacancy complexes show negative bindingenergies over the entire range of the band gap. In addition,the formation energies of antisite complexes are very lownear the conduction band edge, which for S, Se, and Te ren-ders them even more stable than the individual XSb defects ina small range of electron chemical potentials in the upperthird of the band gap.

To characterize the effect of different impurities on thecharge carrier mobilities, we calculated relative carrier scat-tering strengths. This analysis revealed that C and O have byfar the most detrimental impact on carrier transport. Thescattering strengths for Sn, Se, and Te, however, are similaror even smaller than for the most important intrinsic defects.Tellurium, in particular, is therefore an excellent candidatefor compensation doping the material.

The interplay of electronic characteristics and impurityconcentrations was investigated using a self-consistentcharge equilibration scheme that allowed us to compute netcharge carrier concentrations. While pure material was foundto exhibit n-type conductivity, the incorporation of C and Oleads to p-type material. This observation is consistent withexperimental measurements which show p-type conductivityin as-grown samples which contain an unintentional back-ground concentration of C and O impurities. Our calculationsfurthermore revealed how Te can be used to compensatethese impurities in order to achieve minimal net charge car-rier concentrations. Finally, we discussed how the amphot-eric character of Sn could be used to achieve self-compensated material via controlled annealing.

ACKNOWLEDGMENTS

This work performed under the auspices of the U.S.Department of Energy by Lawrence Livermore NationalLaboratory under Contract No. DE-AC52-07NA27344 withsupport from the Laboratory Directed Research and Devel-opment Program and from the National Nuclear Security Ad-ministration Office of Nonproliferation Research and Devel-opment �NA-22�.

1015

1016

1017

−0.4 −0.2 0.0 0.2 0.4

Sb−rich Al−richN

etch

arge

carr

ier

conc

entr

atio

n(c

m−

3 )

Chemical potential difference (eV/atom)

(I) intrinsic

(II) w/ Sn

(III) w/ C, O, Sn (1)

(IV) w/ C, O, Sn (2)

(a)

Def

ectc

once

ntra

tion

(cm

−3 )

1014

1015

1016

1017

−0.4 −0.2 0.0 0.2 0.4

Sb−rich Al−rich

Chemical potential difference (eV/atom)

SnSb (0)

SnSb (−1)

SnAl (0)

SnAl (+1)

(b)

FIG. 12. �Color online� �a� Net charge carrier concentration for�I� pure AlSb, �II� AlSb with c�Sn�=1017 cm−3, �III� AlSb withc�C�=1017 cm−3, c�O�=1015 cm−3, and c�Sn�=1017 cm−3, and�IV� AlSb with c�C�=1017 cm−3, c�O�=1015 cm−3, and c�Sn�=2�1017 cm−3 at a temperature of 1200 K assuming full equilibrationand using as-calculated formation energies. Note that in cases �II–IV� going from Sb-rich to Al-rich conditions changes the conduc-tivity type from n to p. �b� Defect concentrations for case �II� in �a�revealing that the inversion of the conductivity type observed inSn-doped AlSb is caused by a shift from SnAl

+1 to SnSb−1. The horizon-

tal bars indicates the range of chemical potentials which are acces-sible based on the calculated compound formation energy.

EXTRINSIC POINT DEFECTS IN ALUMINUM ANTIMONIDE PHYSICAL REVIEW B 81, 195216 �2010�

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*erhart1@llnl.gov†lordi2@llnl.gov

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published�.33 Additional impurities have been measured in high-purity

samples using secondary ion mass spectroscopy including S, Si,and P. These impurities can, however, be fairly well controlledexperimentally and according to our calculations in Sec. IV arealso not as detrimental for carrier mobility as C and O.

34 P. Erhart, D. Åberg, and V. Lordi �unpublished�.

ERHART, ÅBERG, AND LORDI PHYSICAL REVIEW B 81, 195216 �2010�

195216-12