The electronic states of oxygen in gallium phosphide an example of weak bonding

Physica 116B (1983) 131-138 North-HoUandPublishingCompany

Paper presented at ICDS-12 Amsterdam, August31 - September3, 1982

T H E E L E C T R O N I C S T A T E S O F O X Y G E N IN G A L L I U M P H O S P H I D E A N E X A M P L E O F WEAK B O N D I N G

T.N. M O R G A N

IBM Thomas J. Watson Research Center, Yorktown Heights, N.Y. 10598, U.S.A.

A n u m b e r of i m p o r t a n t electronic properties of oxygen in GaP, which c a n n o t b e a c c o u n t e d for by existing theories, can be explained by a simple model b a s e d on the a 1 and t 2 spin orbitals formed from the four n e i g h b o r i n g gallium valence bonds. I analyze this model and show that it provides a more realistic description of the physics of such a weakly coupled system t h a n do c o n v e n t i o n a l defect theories. I show f u r t h e r t h a t in this model, for which two b o n d i n g electrons have b e e n p r o m o t e d to a n t i b o n d i n g orbitals, the spin multiplet states are stabilized by the C o u l o m b and exchange energies of the G a orbitals. 1. I N T R O D U C T I O N S u b s t i t u t i o n a l oxygen in G a P exhibits n u m e r o u s a n o m a l o u s properties, which have r e m a i n e d unexplained for several years. L2 In this paper I analyze a model 3 which a c c o u n t s in detail for these unusual optical properties and predicts others, m a n y of which h a v e b e e n observed. The most striking a n o m a l y in the G a P : O system is the a p p e a r a n c e of two types of transitions, as indicated by their p h o n o n sidebands, b e t w e e n the neutral D O a n d positive D + defect charge states. O n e involves hole capture, as in D - A pair r e c o m b i n a tion, a n d the o t h e r e l e c t r o n capture or release. W h e n this fact was n o t e d previously 2 it was recognized t h a t the t r a n s i t i o n s were different, a n d a proposal was made that one of t h e m might involve the ' 2 - e l e c t r o n ' D - charge state. F r o m more recent work, 4 h o w e v e r , it is a p p a r e n t t h a t b o t h t r a n s i t i o n s c o n n e c t D O a n d D +. H e n c e , one of these latter charge states must be c o m p o s e d of two nearly degenerate, but different, electronic states. A clue as to which was double came from Samuelson a n d M o n e m a r 5 w h o p r e s e n t e d e v i d e n c e t h a t D + consisted of two very different states separated by only 0.4meV. T h e s e p r o p e r t i e s i n d i c a t e d clearly t h a t a m o r e complex model t h a n t h a t n o r m a l l y devised for a deep substitutional defect was needed. A likely source of this complexity was f o u n d in the G a b o n d s s u r r o u n d i n g the O atom, b o n d s which differ from those in most o t h e r defects because of the d e p t h of the oxygen atomic levels a n d the resulting w e a k n e s s in G a P of the G a - O b o n d . I have 0378-4363/83/0000-0000/$03.00 © 1983 North-Holland

d e m o n s t r a t e d 3 t h a t the properties described a b o v e can be explained by the energy level diagrams in Fig. 1, which are a fit to e x p e r i m e n t of an approximate molecular orbital ( M O ) analysis of a weaklyb o n d e d defect o n a P site. 3 This model differs from the c o n v e n t i o n a l o n e b y p r o m o t i n g two b o n d i n g electrons into a n t i b o n d i n g levels a n d optimizing the C o u l o m b a n d e x c h a n g e energies of the resulting multiplets. In this p a p e r I analyze and c o m p a r e these two models.

D0/D* I (+) 3_(+)

D~D 0 e~cb

A:, T2F'-.p.~'~e~

('9 .(+)

4_(0)

::

I 15~8o '.~.?,+°,

2..(o) 84, 1 [ I

~2 4T(I°' ~[ .

-" ~

I

3 ~

.7

T!

I(-)

E

%+%

I I

".. " " h~+b)

1452 17 1765" 3T~+) 1. (+)

(..) ~ e vb

2T(0)

1740

--2 TI

4 (o)

A1

Fig. 1. Oxygen transitions and energy levels in GaP: 'lst electron' transitions in the left half, '2nd' in the fight. The states consist of the deep MO configurations on the left plus, where shown, the EM electrons e- or holes h +. Energies are in meV, cb = conduction band, vb = valence hand edge, and * marks transitions exciting Ga breathing mode phonons. All transitions shown, except the strongly forbidden one XE(-) have been measured. (See Ref. 6). The 1765 meV threshold is obscured by satelites of the 1740meV line.

132

T.N. Morgan / The electronic states o f oxygen in GaP

2. T H E G a - O B O N D In the oxygen defect the G a - O b o n d s are formed from the O atomic orbitals a n d the four G a sp 3 bond orbitals extending into the P vacancy occupied by O. The covalent nature and the orientation of these b o n d orbitals are fixed by o r t h o g o nality to the G a - P ' b a c k ' b o n d s s u r r o u n d i n g the defect. This G a - O b o n d , however, is o b s e r v e d to be very weak. There are three principal reasons for this: (1) The atomic 2p level in O ° lies a b o u t 7 eV below 3p in p0, with 2s a n o t h e r 15 eV lower, b o t h far below the a] and t 2 c o m b i n a t i o n s of the G a orbitals. (2) The overlap of the O and G a orbitals, already small in the unrelaxed lattice, is further reduced by relaxation of the weakly b o n d ed G a away from O. (3) The atomic 2p levels in O - and O = lie close to the v a c u u m level, far above the c o n d u c t i o n b a n d edge, so that configurations containing these ions are not an i m p o r t a n t part of the defect wave function. One cannot, therefore, make the usual assumption that the ( b o n d i n g ) oxygen 2p shell must be filled (with 6 electrons) before the ( a n t i b o n d i n g ) gallium levels begin to fill. O n e must, instead, c o m p a r e the energies of the various configurations. ~1

n

r

0

1A1

0

1

2AI

I

2

IA~

2

n

r

n(a~)

D÷

2

1A~

3/2

DO

3

0-

4

D*

) WITH

f

D-

(a)

~2s

2 2p'

%

wrrH -

n(a1)

41-1

1

%2

5/4

IE

1

~r,

1

Fig. 2. Weak-bonding (a) and Conventional (b) models for O in GaP. n is the number of electrons in antibonding orbitals and n(a 1) the number of a I electrons. In (a) only the two multiplets observed for each charge state are listed.

The M O model described in Ref. 3, s h o w n in Fig. 2a, assumes t h a t six valence electrons go onto the oxygen and remain in a nearly atomic O ° configuration, while the 2, 3 or 4 defect electrons (for the D +, D O or D - charge states, respectively) go into the s u r r o u n d i n g G a orbitals (as in the P vacancy). (This is the c o n f i g u r a t i o n e x p e c t e d intuitively in the limit of no G a - O bonding.) Hence, these defect states allow a wider choice of spin multiplets t h a n the usual ' b o n d - o r b i t a l ' models 7 in which eight electrons fill the O 2s a n d 2p shells, and only 0, 1 or 2 are assigned to the G a bonds, Fig. 2b. A l t h o u g h b o t h m e t h o d s find nearly the same radial distribution of charge, with O approximately neutral and the remaining electrons on the bonds, the latter is unable to take m a x i m u m a d v a n t a g e of the C o u l o m b and e x c h a n g e energies associated with the G a spin orbitals. O n the o t h e r h a n d , in Ref. 3 the exchange i n t e r a c t i o n b e t w e e n the O electrons and the G a orbitals has b e e n neglected. The effects of this latter i n t e r a c t i o n will be considered in future work. S t a n d a r d defect theory uses delocalized basis functions which are the 1-electron b o n d i n g and antib o n d i n g c o m b i n a t i o n s of the oxygen s a n d p atomic-like orbitals and the a I a n d t 2 orbitals formed from the G a dangling bonds. W h e n the b o n d i n g is strong e n o u g h to c o n c e n t r a t e most of the electron probability in the b o n d region, these functions can be adjusted to describe the b o n d e d configurations r a t h e r well (although they provide a r a t h e r poor description of the a n t i b o n d i n g ones, as in the c o n d u c t i o n b a n d ) . W h e n the b o n d i n g is weak, h o w e v e r , as in the p r e s e n t example, such f u n c t i o n s give a very misleading picture for all states. To u n d e r s t a n d b e t t e r what the p r o b l e m s are, we consider the h y d r o g e n molecule, which is u n d o u b tedly the best u n d e r s t o o d example of b o n d i n g . E i t h e r the delocalized molecular orbital ( M O ) or the localized H e i t l e r - L o n d o n ( H - L ) approximation can be used to describe the i n t e r a c t i o n of two H atoms to form H2.8 The b o n d i n g M O provides a r a t h e r accurate d e s c r i p t i o n near the equilibrium nuclear separation but fails completely w h e n the nuclei are widely separated, unless a nearly equal a m o u n t of the a n t i b o n d i n g M O is mixed in. O n the other hand, the H - L a p p r o a c h is exact in the limit of infinite s e p a r a t i o n b u t b e c o m e s p o o r e r near equilibrium. Thus, the best way of describing two weakly i n t e r a c t i n g H a t o m s is by using the

T.N. Morgan / The electronic states o f oxygen in GaP

H-L picture and adding, if necessary, coupling to excited configurations. When the states of the two weakly coupled atoms differ in energy, the bonding and antibonding combinations are little different from the H - L functions, because the mixing is small, and either basis could be used. In this case, however, it is most important to use those wave functions which leave O neutral in the weak bonding limit--and, thus, to consider configurations built from both antibonding and bonding functions. As a consequence, the O defect is qualitatively similar to a widely separated "diatomic molecule" in which the two weakly coupled parts are the central O ° atom and the surrounding Ga bonds. This is the rationale behind the approach followed in Ref. 3. 3. T H E M O L E C U L A R O R B I T A L M O D E L The basis states used in the MO model are the 2s and 2p atomic orbitals of neutral O and the even a 1 and odd t 2 combinations of the four dangling bond orbitals of the surrounding Ga. 9 The goal of the calculation is to find the n-electron spin multiplets ( n = 8 - 1 0 ) constructed from these functions, which have the lowest energies. In order to reduce the complexity of such a many-electron system, the model of Ref. 3 made two simplifying assumptions. (1) The O ° atom does not participate in the electronic transitions and, hence, can, with its 6 valence electrons, be eliminated from the calculation. Its effect is approximated by a spherical potential. (2) The Coulomb repulsion between two electrons on the same Ga bond is large enough to make the 'ionic' configurations--those having more than one electron per bond--an unimportant part of the stable solutions. With these two assumptions the defect becomes equivalent to the P vacancy with 2, 3 or 4 electrons distributed among the a I and t 2 orbitals of the Ga bonds in such a way that no two electrons occupy any one bond. This model leads to the spin multiplets with the properties shown in Fig. 2(a). 3 The most important properties of the multiplets are their energies, their spins and the composition of the states. The energies identify the stable levels, the spins determine the allowed transitions, while the composition, which specifies the number of a 1 and t 2 electrons in the wave function, determines both the k-conservation selections rules and the phonon coupling observed in optical transitions. 3 Note that the a~ and t 2 M O ' s are formed from nearly unbonded Ga orbitals. Thus,

133

this model explains in a natural way why the bound electron state is composed of nearly equal parts of conduction and valence bands. The wave functions and properties of the important multiplets are described in Appendix A. These have been compared with the experimental data to obtain the level structure shown in Fig. 1. The ' t w o - e l e c t r o n ' structure shown on the right side of the figure is somewhat less certain than that on the left for the 'one-electron' configurations because of the paucity of experimental data. It is significant to note that the exchange and Coulomb energies are important in determining the ordering of the energy levels. Thus, the lower energy a 1 orbitals are not necessarily filled before electrons begin to occupy the t 2 orbitals. For example, in D + the level order is not: al 2 below alt 2 below t22. Rather, the lowest energy 1AI(+) level is found to be that linear combination of both the al 2 and the t2 2 configurations which avoids double occupancy of the bonds, while the triplet level formed from the configuration alt2, which also avoids double occupancy, occurs at nearly the same energy. It is6also instructive to determine the energy of the (t2) configuration, because this is the one which generates the filled shell configuration 1A 1(+) of conventional theory. This is shown in Appendix A to be raised partly by the energy difference between the a 1 and t 2 orbitals and partly by the Coulomb energy associated with double occupancy of the Ga bonds. This result agrees with the argument that: the bonding p (or t2) shell, when filled, approximately neutralizes the charge on the oxygen with four electrons, this being 2 / 3 of 2p 6, an~d puts the remaining two electrons, as 1/3 of (t2) v, in the t 2 Ga bonds. Hence, it includes multiple occupancy of the bonds and, further, cannot take advantage of the lower energy of the a 1 symmetry orbitals. The fit to experiment fixes certain conditions on the parameters of the problem. Although these are both approximate and insufficient to determine all of the energies which enter the calculations, they do convince one that the results are reasonable. In summary I have shown that the weak bonding model of the oxygen defect in GaP provides an explanation of the complex optical properties observed and is in remarkably good agreement with experiment. Further, the success of this model,

134

7~N. Morgan / The electronic states o f oxygen in GaP

where others have failed, indicates that great care must be taken in future calculations of the properties of weakly bonded defects and vacancies.

tential V0(r), having the matrix elements 0 •

V1=<11V0(r) l i>

(A4c)

and APPENDIX A

0

-

V2--=-

We express the wave functions in terms of two orthogonal basis sets constructed from the Ga dangling b o n d s - - t h e a 1 and t 2 symmetry orbitals and the orthogonalized bond orbitals centered along each bond. The former,

Hence, the total one-electron hamiltonian becomes, H = H 1 + V0(r). The (D +) Charge State

l a ! > = ' t ' a (r)

In the D + charge state the two electrons are to be distributed among the four bonds so that the states have definite spin and spatial symmetry, satisfy the Pauli principle and make no bond doubly occupied. This can be done for three singlets (lA,(+), 1E(-) and IT2(+)), and two triplets (3Tl(-I and 3T2(+)). The spatial parts of the wave functions in the bond basis for the three lowest energy states are:

and [ g>=q't~2(r), g = x, y, z, satisfy the one-electron equations, H 1 l a ! > = E 1 la~> and

= H llg>

= E2ltx>,

(A4d)

~IAI

(A1)

where HI, the effective one-electron hamiltonian for the vacancy, is the sum of the kinetic energy T 1 and the potential energies for the four Ga atoms, i= 1-4, (excluding the central O atom, i=0), 4 HI = T1 + 2 Vi(r)" (A2) i=l

1 ~ [ij>, ¢ - ~ - i:¢j 1

(A5)

~T2 = V78 x [1--(1,2)][I 13> + 124> +1 14> + 1 2 3 > 1 and ff~T2 = 1 [ 1 + ( 1 , 2 ) ] [ 1 1 2 >

-- 134>]

We denote each single bond orbital along bond i, which can be formed from the I a l > and I tL> by

where the operator (1,2) interchanges the two indices, and the convention is used that

I i>=Bi(r).

l i j > - l i> I j > - Bi(rl)Bj(r2 ).

Thus,

I 1> = ( 1 / 2 ) [ I a l > + I x > + l Y > + I z > l 12>=(1/2)[lal>+lx>-lY>-Iz>] (A3) I 3> = (1/2)[ I al>- I x> + I Y>- I z>l 14> = ( 1 / 2 ) [ l a ! > - I x > lY>+ Iz>]

= I ( E 1 + 3E2),

(A4a)

= I(E1-E2),i¢j.

4'3T2 = ~ ! 2_ [ l a , x > -

I xai>]

consists of one each a! and t 2 electron.

with the interbond energy V2_-__

~PtAlCC[3 I a l a l > - - ( I xx> + I YY> + I z z > ) ] consists of ( 3 / 2 ) a I and ( 1 / 2 ) t 2 electrons, while

and the bond energies become, EB=--

The numbers of a I electrons in the state, shown in Fig. (2a), are easily derived from the wave functions when expressed in the symmetry basis (al, x, y, z). Thus, for example,

(A4b)

In the approximation of this work the central oxygen atom is considered to contribute only the po-

It is important to note that, although these two states, IAI(+) and 3T2(+), are the ones which experiment shows to lie lowest, neither is considered in the Green's function formalism normally applied to defects and vacancies. 7 Instead, the conventional

7~N. Morgan / The electronic states o f oxygen in GaP

approach is, after the bonding levels are filled, to fill the one-electron a I antibonding level with two electrons before the t 2 levels begin to contribute. Hence, the'two-electron" J A l ( - ) state usually calculated contains an additional part, namely 25°/'0, composed of doubly occupied bonds, l ii>, which introduce a significant increase in the calculated energy. The energies are derived from the two-electron hamiltonian, H 2 = H l ( r l) + H l ( r 2) + V0(r I ) + V0(r 2) + HI2,

number of a 1 electrons in the state. ÷

TABLE I, The energies of the D states, given by the sums of the products of the coefficients Isted and the integrals shown.

1A 1

3T2

1T2

3T 1

~E



1

I

I

1

I



4

2

0

-2

-2



1

-1

1

-1

1

4

-2

0

2

-2

0

-2

o

2



where H12

----

e2/r12 .

Thus, the energy of the 1Al(+) state becomes EiAi =
I ij> +4 I i k > + I ji> +4 I k i > +2 I k l > > , where no two of the indices i, j, k, and 1 are equal. Similar expressions are easily derived for the other states. The coefficients of the five integrals in these energy expressions are collected in Table I for the five D + states. These integrals can, in turn, be expressed as the sum of the appropriate Coulomb (and exchange) integrals of H12, denoted by U, and the one-electron energies listed in Eqs. (A4). Thus, = 2(E B + V 0) + U 2 0 = V 2 + V 2 + U 3 = U " 2

(A7)

= U ' 3

= U 4

The relative energies of the states are determined by their Coulomb and exchange energies U and b~( U the term nee, where the energy e is e = V 2 + V 2 which is found to be negative, e , ~ - l e V , The coefficients ne, which appear in the second row of Table I are related, through Eq. (A8) below, to the

135

2

Thus, the Coulomb and exchange energies are responsible for bringing the 3T2(+) and 1Al(+) states, which contain both a I and the higher energy t 2 electrons, below the state composed of two a 1 electrons. Table I shows also that the nearly equal energies found experimentally for the 1AI(+) and 3T2(+) states is a consequence of a cancellation between the Coulomb energy difference in nee, shown in the second row, and the exchange energies in the third and fourth rows. The composition C = n ( a l ) a l + n ( t 2 ) t 2 of any of these n-electron states can be calculated from the relation C = (n/4)(a l+3t2)+(nj4)(al-t2).

(A8)

The D o and D - Charge States The wave functions of the three- and four-electron configurations D O and D - , although too complex to be written here, can be obtained by standard means, l° The D O states are listed, together with the coefficients which determine their energies in Table II. TABLE 9. The energies of the D- states, as in Table I.



I

I

I

1

1



1

2

0

-2

-3

-5

1

-3

1

3

136

T.N. Morgan / The electronic states o f oxygen in GaP

Thus,

This work was supported in part by ONR grant, No. N00014-80-C-0376.

= 3(E a + V °) + 3U2, < i j k l H 3 I ijl> = V 2 + V 0 + 2U 3

(A9)

= U' 3, For D - the energies of the states 5A2(-) , 3Tl(-) and 1E(-) are found to be < i j k l l H 4 l i j k l > minus 6, 2 and 0, respectively, times the exchange integral , where the integrals are = 4(E B + V °) + 3U 2 (A10) = U2". These values are useful only in estimating the energies, because of the simplicity of the calculation. In fact, the order of these last three levels if found to be 1E(-) below 3T1(-) below 5A2(-!1° Because of this uncertainty in the calculated energy values, the results listed in Tables I and II can only be relied on to show that the results are reasonable. Finally, it is important to note that the filled bonding t 2 shell of conventional theory includes contributions from both the filled 2p of oxygen and the filled t26 shell of the P vacancy. The latter, with six electrons distributed over the four Ga bonds, contains a large contribution from multiply occupied bonds with its correspondingly large Coulomb energy. This cost in energy, together with the avoidance of the lower energy a t antibonding orbitals, makes this configuration inappropriate for weakly-bonded defects.

References: 1. See the forthcoming reviews by P.J. Dean: (to be published) and at the Montpellier conference. 2. T.N. Morgan, Phys. Rev. Letters 40, 190 (1978). 3. T.N. Morgan, Phys Rev Lett 49, 173 (1982) and at Montpellier. 4. M. Gal, B.C. Cavenett and P.J. Dean, J.Phys. C, 14, 1507 (1981). 5. L. Samuelson and B. Monemar, Phys. Rev. B18, 830 (1978). 6. The IE(--)level is hypothetical, having been suggested by the results in Ref. 10 that I E should lie lowest. 7. G.A. Baraff and M. Sehltiter, Phys. Rev. 25, 548 (1982). 8. See, for example, J.C. Slater, " Q u a n t u m Theory of Matter" (McGraw-Hill, New York, 1951). 9. The coupling between the 2p orbitals of O ° and the t 2 Ga orbitals may be eliminated first by forming their bonding and antibonding combinations. The weak bond formed stabilizes the O position but leaves the wave functions only slightly altered. 10. C.A. Coutson and Mary J. Kearsley, Proc. Roy. Soc. A, 241, 433 (1957) treated three (+, 0 and - ) charge states of the diamond vacancy; G.T. Surratt and W.A. Goddard, III, Phys. Rev. BI8, 2831 (1978), have calculated the neutral silicon vacancy. Both papers find for four electrons that ~E lies lowest.