Electronic-structure calculation of point defects in silicon

297

Computer Physics Communications 44 (1987) 297—305 North-Holland, Amsterdam

ELECI’RONIC-STRUCFURE CALCULATION OF POINT HEFEI F. BEELER, O.K. ANDERSEN, 0. GUNNARSSON, 0. JEPSEN Max-Planck-I nstitut für Festkorperforschung D-7000 Stuttgart-80, Fed. Rep. Germany

and M. SCHEFFLER Physikalisch-Technische Bundesanstalt, D-3300 Braunschweig; Fed~Rep. Germany

states, total energies, donor- and acceptor levels, and spi open questions and controversies, we present results chalcogens and 3d transition-metal impurities in silico positions in the silicon lattice as the substitutional site. and acceptor levels in good agreement with available exr ions are predicted to have low-spin ground-states whli Woodbury (LW). A large number of predictions will be

1. Introduction Impurities are known to affect significantly the electronic properties of semiconductor materials (such as silicon) which are relevant to device applications. There are two categories of impurities with completely different electronic properties, denoted as shallow and deep. Shallow impurities in silicon such as P or B induce bound states in the optical gap very close (~kT) to the conduction or valence band edge. Their electronic properties are described by a bydrogenic model, i.e. the effective-mass theory (EMT) [1]. In the EMT it is assumed that an electron (or hole) in a shallow state can be treated by a hydrogen-atom like theory, i.e. as independent of all other electrons of the system. Thus, an electron (or hole) occupying a shallow state in the gap is bound by the screened long-range Coulombic tail of the impurity potential, e2/cr, where e is the static dielectric constant of the crystal (Si: e 12). The resulting binding energy measured from the nearest band edge is less than —

isities of point defects in semiconductors. Focussing on existing ent theoretical studies of single substitutional and interstitial ir the chalcogen point defects S, Se and Te we identify their d impurities in silicon we obtain spin-multiplicities and donor ntal data. The early 3d interstitial and the late 3d substitutional in conflict with the generally accepted model of Ludwig and ~d to be tested by future experimental studies.

0.1 eV and the effective Bohr radius is large (Si: 30 A) compared with the interatomic distance (Si: 2.3 A). Deep impurities induce bound states somewhere within the band gap which are bound essentially by the short-range part of the defect potential. Contrary to shallow impurities, the wave functions of bound states associated with deep defects are well localized in r-space (delocalized in k-space). Deep-level defects can act as recombination (or killer) centers or as traps [2], and consequently they often limit the lifetime of carriers [3,4]. The theoretical description of these impurities has been a major challenge in the theory of semiconductor physics and is still an active field in semiconductor research. Since deep impurities are controlled significantly by the short-range potential, extensions of EMT have not proved successful. Recognizing the importance of the short-range part of the deep defect potential, earlier theoretical investigations have been based on a cluster approach in which the perfect crystal with the impurity is replaced (or simulated) by a

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finite cluster and taking into account only a number of atoms [5—9].The dangling bonds atoms at the cluster surface are usually term by hydrogen atoms. Unfortunately, theo ~tndies based on the cluster approach are ~ y considerable uncertainties due to the small :er-size and the termination of the cluster. In :r to eliminate both of these shortcomings ciated with the cluster approach theoretical tods based on a Green-function treatment have developed [10—14]. Here the impurity is ed as a localized perturbation in an otherwise ~ctinfinite crystal. All results presented in this ~r are based on a Green-function technique

.

-

2.1. Density-functio A major step towards a better descr the physical properties of atoms, molec solids was achieved by the density-functL.~ ory (DFT) [15,16]. Here we shall only discuss some features of the DFT which are important for understanding our calculated results later on.

dLUiIU~ Lw..yUucIg UIIILS

theory (DFT) [15,16], will discuss the local density and local spin-density approximation [17—19]which allows calculations of the many-body ground-state properties from first principles. We then continue with a schematic description of the Green-function method for deep-defect calculations. In section 3 and 4 we discuss two recent examples of applications of the self-consistent Green-function method and demonstrate the importance and power of the theory to get a comprehensive understanding of defect properties. In section 3 we report total-energy and electronic-structure calculations for the chalcogen point defects S, Se and Te in silicon [13]. We show that total-energy calculations clearly identify the stable position of these impurities as the substitutional site. In section 4 we discuss results of electronic-structure calculations for interstitial and substitutional 3d transition-metal impurities in silicon [20]. We find theoretical evidence for lowspin ground-states of the early interstitial and late substitutional 3d ions. This is in conflict with the generally accepted model of Ludwig and Woodbury (LW) [22], but not with existing experimental data. In fact, it will be shown that the calculated donor and acceptor levels obtained by our spinunrestricted theory reproduce all experimentally observed transitions and trends. A microscopic justification for the success of the LW model for all Electron Paramagnetic Resonance (EPR)-iden-

H= T+ U+ V 2 ~.,

I~, ‘I

i~—~’

~

(1)

where T is the operator for the kinetic energy, U is the electron—electron interaction and V is the interaction with the external potential, Hohenberg and Kohn showed that the ground-state cT.’ is a unique functional of the ground-state electron density n(r) only fi(r1s1

rNsN)=cI~[n(r)].

(2)

Further, they have shown that the energy functional, E[n(r)J

=

<~T+

UJ ~>+ fd3r n(r)vext(r) (3)

has its minimum for the proper ground-state density n(r) which is the ground-state energy. Kohn and Sham showed that the variational problem of the energy functional (3) with respect to the density can be solved by self-consistently solving

299

F Beeler et at. / L

the following set of single-particle equationt [_v2+v(r)+fd3r’

2n(r’)F

Ir—r

I

+ &E~ [n~

&n(r

(4)

N

I 4.(r)

=

(5)

2

s within the DFT the N-electron problem (1) ie ground-state is transformed into the probof solving a system of non-interacting quasi-

mally exact, self-consistent one-particle scheme (the so-called Kohn—Sham equations), as defined by (4) and (5), is designed to yield ground-state properties and that the single-particle energies e, and wave-functions 4.(r) have no direct physical meaning. The general form of the exchange correlation functional ~ is not known and in order to convert the Kohn—Sham scheme (defined by (4) and (5)) into a practical method, the functional ~ must be specified. Most calculations are based on the local density approximation (LDA), where the exchange-correlation functional ~ is calculated for a homogeneous electron gas, i.e. ~ En] is approximated by E~~[n(r)]~fd3rn(r)~~(n(r)),

(6)

where ~ is the exchange-correlation energy per electron for a homogeneous electron gas of the density n(r). If magnetic properties are important (as is in the case of 3d transition-metal impurities in silicon, see section 4), an approximation more sophisticated than the LDA (6) is required. It is, however, a rather simple matter to generalize the formalism to a spin-density scheme with two independent variables, the electron density n(r) and the spin-density m(r), n(r)

=

n ~(r)

m(r)=n

+

n

(r)—n

(r),

E,~(n,m), wnicn I ergy per electron I with density n(r) the LSDA and neglecung spin—orou two decoupled Kohn—Sham equations direction of spin have to be solved. The local approximation (LDA and L~JJrt) i~ formally justified for such systems in which tht electron density variesand slowly [16]. However; iti i~’i-.e~ f.-~rmti1p~’ii1p~ Qr,lide indic~ateQthat itc

LSDA during the last years has shown that for perfect crystals most ground-state properties can be calculated with errors of typically 1—10%: The lattice constant usually being accurate to within 1% and the cohesive energy being the most inaccurate. However, for trend-studies of electronic properties between similar physical systems (as it is the case for the chosen examples discussed in section 3 and 4), the LDA and LSDA scheme is a particularly appropriate tool. 2.2. Green-function method The electronic properties associated with chalcogen (section 3) and transition-metal impurities (section 4) have been calculated using a selfconsistent ab-initio Green-function technique which is based on the linear-muffin-tin-orbital (LMTO) method in the atomic spheres approximation (ASA). A detailed discussion of this method is given in refs. [12,23]. Here we present only a brief scenario on how the calculations were performed and we shall comment only on those aspects which are relevant for understanding the results presented later. Within the Green-function method the impurity electronic-structure problem separates into two parts: First, after having solved self-consistently the band-structure problem for the infinite crystal, the Green-function G° of the perfect crystal is calculated. Secondly, the Green-function G for the

JU’J

I.

DtCtC~ Ci Lit.

/

crystal with the impurity is calculated by sel sistently solving Dyson’s equation. Let’s start with the band-structure problt a semiconductor crystal such as silicon. Rec the relatively open diamond structure fo1

~n-

in the LMTO—ASA method we pack the ~al with spheres in such a way that a close:ed structure is obtained. Therefore we divide Si crystal into space-filling spheres which are ered at all Si and interstitial sites. The basis onsists of s, p and d orbitals in all spheres. In kSA the crystal potential is spherically averwithin each sphere. This approach has proved ield very accurate results [241. The resulting

.

where the potenti lattice and, consequently, the wave-functi Bloch symmetry. The Green-function G° of the perfec may be defined by G°(Z: r, r’)

Whi~.1C

(8)

V(r)=~ø(rR/sR)vR(rR), R

where the local coordinates are rR = r R and l(rR/sR) is a step function being unity inside a sphere of radius SR and zero outside. The potential VR in the sphere at site R is within the local density approximation (LDA) 2ZR vR(r)=I.L(nR(r))-I-2J d,3r’ —

-.

Z.. — A.

=

Y~~

V iy

LA.,

r)~b~(k, r’)

y i~.ai, y ~r

~J)

io

ct

p~.JuiLILL

the complex energy plane, and E~Pand are the single-particle energies and wave-functions of the perfect crystal obtained from solving the bandstructure problem (11). The Green-function G for the crystal with an impurity is obtained by self-consistently solving Dyson’s equation G=G°+G°~VG,

(13)

—~

‘~R(~)

I-

where ~V = V V° is the perturbation due to the impurity. The Dyson equation (13) is turned into a matrix equation in which the range of the perturbation ~V determines the size of the matrices. The major advantage of the Green-function method arises from the fact that the impurity perturbation AV is large only in a small region of space, i.e. i~Vis localized. In our calculations, the perturbation was included inside 9 atomic spheres centered at the impurity site, the 4 nearest Si and the 4 nearest interstitial sites. We would like to emphasize that the localization of the impurity perturbation i~Vdoes not require that the wavefunctions are localized. The Green-function treatment is very convenient for studies of deep impurities because it separates the problem into two parts and takes advantage of the properties of both parts, the periodicity of the perfect crystal as well as the localization of the defect potential. For a start potential V 1~ for the perturbed system, eq. (13) is solved for a set of energy points —

+

~

R-R’

R’,LR’

I

k-ri

r

The first term is the exchange-correlation potential in the LDA [17—19]and ~R = 11v;R + ~c•R ~ the density of the valence and core electrons. The valence-electron density n v;R is obtained after solving the band-structure problem (11) and the core-electron density n C; R is taken from a freeatom calculation, that is, we use the frozen-core approximation. The second term is the electrostatic potential from the total electron density n and the third term is the electrostatic potential from the nucleus. The last term is the electrostatic Madelung potential from the net charges qR’ in all other spheres, qR=

_ZR+fd3rnR(r)~

where

ZR

(10)

is the atomic number. The band-struc-

1.

AJCCtC~ CI Lit.

/

Z along a contour y in the lower-half c complex energy plane, starting at E (belo lower valence band edge) and ending at the: level EF (somewhere in the gap). The va —‘--~rondensity of the perturbed crystal is Lned from the imaginary part of the GreenLion G by

onance (EPR) and onance (ENDOR) intensive expenme~~ ble to identify whether isolated chalcoger ties in silicon occupy substitutional or Td tial sites. The difficulty of this site iden~ was largely due to the fact that both sites have the same local environment (i.e. 4 Si nearest neighbors). ..

)

=

—

~

(dz Im G(Z;

r, r),

(14)

ITT

y is the energy path defined above. From ‘alence-electron density (14) and the impurity •electron density a new impurity potential ~ut ~

~

I..’

~

fQ\

-r~

,~,

V~”=(1 —f)V

(15)

.~

-

-

-

We SL~UIC SILC UI ~ OJ.IU silicon as substitutional [13]. ~,

0”t.

--

Ic

-

~UUIL

1~+fV1

Self-consistency is achieved, when V” is equal to V0~~t.From the calculated G, all electronic and magnetic ground-state properties of the perturbed system such as the defect-induced density of states below and in the valence band as well as in the gap, charge states, total electron spin, total energies and donor and acceptor levels, etc., can be obtained. 3. Chalcogen point defects in silicon 3.1. Introduction Chalcogens in silicon are well known to form impurities consisting of singly incorporated atoms [25]. The behaviour of oxygen is quite different from the other chalcogens which is probably due to its high electronegativity and small atomic radius. Therefore oxygen will be excluded from the following discussion. The stable configuration of the chalcogen point defects 5, Se and Te in silicon has for a long time been unclear. Extensive experimental work was devoted to infrared absorption studies [25,26], which allow the determination of the energetic positions and symmetries of the ground-state and various excited states. The spectra could be interpreted in a one-electron picture. From an analysis

3.2. Identification of chalcogen point defect sites in silicon To discuss the stability of the two possible chalcogen impurity configurations of Td symmetry, we first consider the two processes in which chalcogens diffuse from the gas phase into a Si crystal and occupy a substitutional (sub) or Td interstitial (int) site. In both cases the initial state is an isolated chalcogen atom and a perfect Si crystal. The final state for the process “int” is a Si crystal with an interstitial chalcogen point defect, and for the process “sub” a Si crystal where a chalcogen atom replaces a Si crystal atom which has been moved to the surface. The differences i~Ebetween the calculated total energies of the final states for the process “sub” and “int” are given in table 1 for intrinsic, n-type and p-type silicon. Before we discuss the results in table 1 we emphasize that the charge state of a defect is determined by the position of the Fermi level. At the Fermi level we assume a reservoir of non-interacting electrons. For a given position of the Fermi level, which is determined by the doping of the crystal, the electrons of the system will distribute themselves between the Fermi-level reservoir and the defect-induced states in order to attain the

r.

Detle~

ci Ut.

/

fld

lowest total electronic energy. Substitutional and Te point defects are calculated to h localized state of A1 symmetry at the bott the conduction band. In the case of p-type state will be empty and thus the defects CAISI te double-positive charge state. For n-type ~rial the a1 state is filled with two electrons the defects are neutral. For the Td interstitial ~ogens we found a bound state of T2 symmen the gap containing four electrons in the ral charge state. For p-type silicon we conhere the double positive charge state where electrons are removed from the t2 state in the For n-tvj,e silicon the t., state is comoletelv

~

uuIIa.L pl(fi.css

is cIiCI~cLI(.duy

I4VUICU

uy

scvcrai

eV compared to the interstitial incorporation for intrinsic, n-type as well as p-type silicon. Further we see that the stability of the substitutional site with respect to the Td interstitial site increases significantly from S to Se to Te. To comment on the diffusion of chalcogens in silicon, we further consider the process in which a distant defect pair consisting of a substitutional chalcogen atom and a Si self-interstitial is transformed into an interstitial chalcogen. Here we only consider the Td interstitial site for the Si self-interstitial, assuming that other interstitial sites are roughly of the same energy. This assumption is supported by recent self-consistent pseudopotential calculations [30]. The interstitial interchange (iic) process where a Si self-interstitial and a substitutional chalcogen atom interchange their sites is calculated to be exothermic only for S in n-type silicon (see table 2).

Table 1 A E denotes 1’d interstitial the total-energy chalcogen point difference defects between for intrinsic, substitutional n-type and p-type silicon AE (eV) Intrinsic n-type p-type

S

Se

Te

—3.7 —3.2 —4.6

—6.4 —6.1 —7.1

—11.0 —10.9 —11.4

_________________

Intnnsic n-type

S +0.3 —0.2

-s- i.u +2.7

+ I.

p-type

+ 2.2

+ 4.7

+9.

+7.

This theoretical result suggests that for n-type silicon self-interstitials may kick-out sulohur atoms

con appears to be partly determined by an interstitial component, if Si self-interstitials are present. From the calculated energies ~ in table 2 it furthermore follows that the process lic is quite unlikely for Se and Te. In the total-energy calculations in tables 1 and 2 the distortions of the silicon lattice were not taken into account. This effect was calculated independently by the self-consistent pseudopotential method [31,32]. It is found that the lattice relaxation would lower the total-energies by less than 0.3 eV for substitutional and by less than 0.8 eV for Td interstitial chalcogen point defects. As a consequence, the effect of lattice distortions is not relevant for the above arguments.

4. 3d transition-metal impurities in silicon 4.1.

Introduction

Qualitative understanding of the electronic properties of single transition-metal (TM) impurities in silicon comes from the pioneering work of Ludwig and Woodbury (LW) more than 20 years ago [22]. Based on their Electron Paramagnetic Resonance (EPR) studies they developed a phenomenological model for the electronic structure of TM ions in silicon: A TM ion with n valence electrons is in a d” configuration if incorporated at the tetrahedral (Td) interstitial site and in a d”4

p. neeter er al. /

~tIJj

P.

configuration if it is substitutional. In the case 4 of the n free ion valence electror supposed to substitute for the 4 Si electro moved from the valence band and therefon i electrons are in localized 3d-like statcs. iii .~Wmodel it is assumed that by incorporation TM ion at a site of Td symmetry the atomic d ~alsplits into e and t2 states by the tetrahedral

--

al field. For interstitials the triplet t2 states Id have lower energy than the doublet e states Seas for substitutionals the level ordering Lld be inverse. A further assumption in the model is that in the ground-state the t2 and e s are oooulated according to Hund’s rule. i.e.

IiLcIIuii U4Ld

ble charge states a and acceptor levels 4.2. Electronic-structure trend from Ti to ( The calculated single-particle energies ground states of the neutral 3d impurities are shown in fig. 1: The atomic 3d orbital splits into a t.,-branch (full curve~ and a e-branch (broken

snown in IUIC

IL lIdS UILCII UCCit ~ppncu

0.150

LU 5U~Ii I IVI

ions where no EPR data exist. The aim of our theoretical investigation is to give a microscopic justification of the LW model and a discussion of its validity range. In order to investigate the electronic and magnetic groundstate properties of TM impurities in silicon spinunrestricted Green-function calculations of the energetically lowest spin-configuration have been

~

ror interstitiais tue t

2-Dranclt lies below the e-branch, and for substitutionals e is below t2. This is in agreement with the model of LW. However, if we consider the occupation numbers of the defect-induced e and t2 states or what is the same the total electron spin S, we find: Going from interstitial Ni°,a characteristic 0 and Ti°to Cr° interrupts the otherjump between V wise monotonous decrease of the t 2 single-par~.

~MEEE

ISUBSTITUTIONAL 3d -IMPURITIE

10

rig. 1.

~°°

IINTERSTITIAL 3d -IMPURITIES I

: : flfl~o~ifl (2)

SvO

~

1

~

0

~

1

Sli

Fig. 1. Single-particle energies calculated for the ground sta occupancy of a localized gap state, or of a resonance in the val~ Cr° the high-spin state was used. Interstitiá

~

2

~

1

-~

0

neutral 3d impurities. Numbers in the parentheses give the r conduction band from which the spin S result. For interstitial tnly exists in the single-positive charge state.

LJCCtC~ Vi Lit.

/

i~i

tide energy which is caused by a change low-spin to high-spin ground-states. This c

originates from the fact that the crystal field ting between the e- and t 2-branch increases ~~stitial Ni° to Ti° and dominates ovt11 •splitting in the case of V°and Ti°.Thus, for ‘stitial Ti and V our theory predicts a breakof the LW model. For the substitutional 3d’s —t2 splitting increases from Ti°to Cu°(fig. 1 part) and therefore the switch from high- to spin occurs to the right of Mn. In summary ,redict low-spin ground states for the early stitials Ti°,V~,Ti, V°and V—2~, [35],and the Co°. Nit, substitutionals Fe°, Cot, Ni

5L0.Lll~

haviour of the sing] For substitution perimental level donor level (0/ +) of Mn [36] which ag with our calculation). We hope that our cally predicted donor- and acceptor enei~... refs. [20,23]) could be of help in future experimental investigations. ~

~

‘ 1~a.1..LIiaI.~&iIl’I

‘....l

(ILIU

~..1

W’...

0.LLIIUUI.%..

Ill

part to the neglected lattice distortions (magneticpressure effect) (see Refs. [20,23]). A detailed discussion of all predicted possible charge states, their wave-function localizations, spin-densities and spin-multiplicities is given in ref. [23]. The calculated donor- and acceptor levels for interstitial 3d impurities in silicon in comparison with experimental data are shown in fig. 2. The

1.0

0

—

t2s

1/0

-•-.-

‘.....

r-..ILLCI dull

fl.fl.

IVULCUCII,

FUyb.

IXCV.

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J.M. Luttinger and W. Kohn, Phys. Rev. 97 (1955) 869. W. Kohn, Solid State Phys. 5 (1957) 257. [2] W. Shockley and W.T. Read, Phys. Rev. 87 (1952) 835. H.J. Queisser, Solid State Electron. 21(1978) 1495. [3] A. Rohatgi, JR. Davis, RH. Hopkins, P. Rai-Chaudhuiy, P.G. McMullin and J.R. McCormick, Solid State Electron. 23 (1980) 415. R.H. Hopkins, R.G. Seidensticker, J.R. Davis, P. RaiChoudhury, PD. Blais and J.R. McCormick, J. Crystal Growth 42 (1977) 493.

eê2/1 (-/0)

—

t24 1/0

Ti~~t~

Cr~

Fig. 2. Calculated (full lines) and experimental (broken lines silicon. For each level the related single-particle state and its o Cr°and Cr~ hig

Fe1~t ~oflt

Ni1~t

Cu~

~7] acceptor- and donor levels for interstitial 3d impurities in Ltion before and after the ionization is indicated. In the case of i states were used.

I.

DCCtC~ Vt lit.

/

3U)

t.

[4) S.M. Sze, Physics of Semiconductor Devices (Wib York, 1969). [5] R.P. Messmer and G.D. Watkins, Phys. Rev. B 2568. [6] B.G. Cartling, J. Phys. C 8 (1975) 3171,3183. l.A. Hemstreet, Phys. Rev. B 15 (1977) 834. 3.G. DeLco, G.D. Watkins and W.B. Fowler, Phys. Rev. ~ 23 (1981) 1851, B 25 (1982) 4962, 4972. L..V.C. Assail, JR. Leite and A. Fazzio, Phys. Rev. B 32 1985) 8085. L.V.C. Assail and J.R. Leite, Phys. Rev. Lett. 55 (1985) ~80. I. Bernholc, NO. Lipari and S.T. Panteides, Phys. Rev. Lett. 41 (1978) 895; Phys. Rev. B 21 (1980) 3545. 3.A. Baraff and M. Schjllter, Phys. Rev. Lett. 41 (1978) 192; Phys. Rev. B 19 (1979) 1965.

XXIX, eds. F. Ba Holland, Ainsterds [25] P. Wagner, C. Hol in: Festkbrperprob vol. XXIV, ed. P. Grosse (Vieweg, Braunschwc 191. [26] R. Janzen, R. Stedmann, G. Grossmann and I meiss, Phys. Rev. B 29 (1984) 1907. [27] G.W. Ludwig, Phys. Rev. A 37 (1965) 1520. [28] J.R. Nildas and J.M. Spaeth, Solid State Commun. 46 (1983) 121. 1~k U7k..

in Semiconductors (1985) p. 117. [141 M. Scheffler, in: Festkorperprobleme: Advances in Solid State Physics, vol. XXII, ed. P. Grosse (Vieweg, Braunschweig, 1982) p. 115. [15] P. Hohenberg and W. Kohn, Phys. Rev. 136 (1964) B 864. [161 W. Kohn and U. Sham, Phys. Rev. 140 (1965) A 1133. [17] L. Hedin and B.I. Lundqvist, J. Phys. C 4 (1971) 2064. [18] U. von Barth and L. Hedin, J. Phys. C 5 (1972) 1629. [19] 0. Gunnarsson, M. Jonson and B.!. Lundqvist, Phys. Rev. B 20 (1979) 3136. [20] F. Beeler, O.K. Andersen and M. Scheffler, Phys. Rev. Lett. 55 (1985) 1498; Mater. Res. Soc. Symp. Proc. on Microscopic Identification of Electronic Defects in Semiconductors (1985) p. 129. [211 H. Katayama-Yoshida and A. Zunger, Phys. Rev. Lett. 53 (1984) 1256; Phys. Rev. B 31 (1985) 7877, 8317. [22] G.W. Ludwig and H.H. Woodbury, Solid State Phys. 13 (1962) 223.

V D

V.T:1~1.... ..._.1

V

..~.L

[30] R.Car, P.J. Kelly, A. Oshiyama and S.T. Panteides, Phys. Rev. Lett. 52 (1984) 1814. Y. Bar-Yam and J.D. Joannopoulos, Phys. Rev. Lett. 52 (1984) 1129. [31] M. Scheffler, J.P. Vigneron and G.B. Bachelet, Phys. Rev. Lett. 49 (1982) 1765; Phys. Rev. B 31 (1985) 6541. [32] M. Scheffler, F. Beeler, 0. Jepsen, 0. Gunnarsson, O.K. Andersen and G.B. Bachelet, J. Electron. Mater. 14A (1985) 45. [33] D.A. von Wezep and C.AJ. Ammerlaan, J. Electron. Mater. 14 (1985) 863. [34] E.R. Weber, Appl. Phys. A 30 (1983) 1. [35] Similar results for interstitial 3d ions in silicon were found by Katayama-Yoshida and Zunger [21]. [36] R. Czaputa, H. Feichtinger, J. Oswald, M. Haider and H. Sitter, Phys. Rev. Lett. 55 (1985) 758. [37] K. Graff and H. Piper, in: Semiconductor Silicon 81-5 (Electrochemical Society, Pennington, NJ, 1981) p. 331.