The localised modes due to P defects in cadmium telluride

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The localised modes due to P defects in cadmium telluride

K R A M A C H A N D R A N and T M H A R I D A S A N

School of Physics, Madurai Kamaraj University, Madurai 625 021 MS received 25 July 1980; revised 3 November 1980

Abstract. Dutt and Spitzer experimentally observed the localised vibrational modes related to the phosphorus defects in CdTe andr eported that P would go either to substitutional or interstitial site. In this paper we have theoretically investigated the P defect behaviour in CdTe for those two possible sites by Green's function technique and we believe from our calculations that P goes interstitially rather than substitutionally.

Keywords. loealised vibrational modes; substitutional modes; interstitial modes;

Green's function technique; cadmium telluride.

1. Introduction

Although a n u m b e r o f papers appeared on the vibrational studies o f disordered solids and defect crystals, the II-VI c o m p o u n d s did not receive adequate attention until 1974. After the review o f Barker a n d Sievers (1975), these c o m p o u n d s were studied both experimentally a n d theoretically and most o f the studies todate involved iso- electronic substitutions in these compounds. Hayes and Spray (1969) measured the infrared absorption o f CdTe d o p e d with Be and in this case the fundamental local m o d e was at 391 cm -a at 4°K. Meanwhile Sennet et al (1969), in addition, f o u n d a resonant m o d e at 61 cm -1 which they explained using the Green's function method.

Recently D u t t and Spitzer (1977a) measured the infrared localised vibrational modes ( L V M ) in CdTe when defects like P, A1, In and G a are substituted. They reported two major P defect centres, an acceptor with a T a symmetry and an electri- cally inactive P complex having trigonal symmetry. Their measurements o f double d o p e d CdTe showed L V M absorption suggesting the presence o f the isolated acceptors and P impurity pairs.

It is generally k n o w n that the group V impurities behave as acceptors in II-VI c o m p o u n d s although the nature o f the defects and the self c o m p e n s a t i o n mechanism are not that clear. But phosphorus is known to exhibit b o t h electropositive and electronegative character in forming the c o m p o u n d s (Apple 1959). It is thus possible that P is an amphoteric impurity in CdTe. Dutt and Spitzer (1977a) predicted that the observed self compensation could arise in a number o f alternate ways (i) by different P centres compensating themselves; a priori one expects P o n a Te site PTe to be a single acceptor, P on a n interstitial site P~; a multiple d o n o r (ii) by some o f the defects forming neutral or chemically inactive associates (iii) by native defects with or without complexing with P acceptor centres.

P.--2

17

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18 K Ramachandran and T M Haridasan

From symmetry considerations, an isolated substitutional or an interstitial P atom in II-VI compounds, both have the same Td point symmetry and hence from LVM observations alone it is rather difficult to isolate these two possibilities since in either case we expect a triply degenerate localised modes. Further P, being an amphoteric impurity, can replace either Cd or Te substitutionally and also can occupy interstitial site surrounded by either Cd or Te as first neighbours. Thus to explore all the possibilities, some theoretical investigations are also simultaneously desirable apart from experimental measurements, and with this in mind we have investigated theoretically all these possibilities using Green's function technique. Further, we have also investigated the impurity pair problem associated with the additive impurities of Ga, In and A1.

2. Theory

Cadmium telluride has the zinc blende structure. Therefore, the Cd, Te as well as the two common types of interstitial sites are all tetrahedrally coordinated with a Td point group symmetry. If P is located at any of these sites one would expect each type of defect to give rise to a triply degenerate vibrational absorption band in the infrared.

We employ the general theory of impurities to work out the impurity modes both at substitutional and interstitial sites.

2.1 Substitutional studies

When a substitutional impurity atom is introduced into the crystal, the equation of motion becomes

( L - - ~ L ) U = 0 , (1)

where L is the force constant matrix of perfect crystal and t9 L is the perturbation brought about by the impurity. The elements of the matrix ~ L are

A 1 l'

where E is the mass defect parameter ~ --- (Mn--Ma)/Mh and Mk is the mass of the kth atom; Mh is the mass of the host atom and Ma is that of the defect atom.

(') (')

represents the force in the ~-direction on the atom k when the atom k' undergoes unit displacement along the fl direction.

OL has the dimensionality 3m where m is the number of atoms interacting with the impurity including it. G is the inverse of L and it can be shown to be equal to

Z [

1 e a ( k l q j ) e#* (k]qj) x exp 2~ri(t'r k k']J' ( n ~ M~,)I/~ o~ - - o~ (q)

qJ

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where e~ (k [q j) is the ath component o f the eigen vector corresponding to the kth atom for oJj (q), the eigen frequency. ~ is the phonon vector and j is the branch index (G is known as the Green's functions o f the perfect lattice).

Following the method o f Maradudin et al (197 l) we find that the condition to give all the perturbed modes is

t l - g O l l =o,

where g and tgl are the partitioned parts of the matrices G and OL in the defect space and have the order o f (3m x 3m).

In all determination o f impurity modes, both the real and imaginary parts o f the function G M k ; °J~ are found to be useful. For LVM calculations, however, we encounter only the real part of the Green's functions.

2.2 Application to CdTe

As is known, CdTe is tetrahedratly bonded crystal and the position o f the atoms is given in figure 1.

As a first approximation, the contribution from the second neighbours is neglected.

It is found when Te is substituted with phosphorous the symmetry did not get altered.

i.e. it retained the Td symmetry.

When Te is taken as the origin, we have four Cd neighbours and so the defect space will be o f dimension (15 x 15) and the Green's function matrix for the host lattice in the defect space will also be (15 x 15).

One can make use o f group theoretical arguments to block diagonalise the g and 01 matrices. T o facilitate block diagonalisation of the matrix, we have to find a matrix U which partitions gOl by the process (U + [ gOl I U). The U matrix can be worked out from the symmetry coordinates o f the normal modes. With that U matrix, block diagonalisation of g and 0 l matrices was separately carried out.

The symmetry coordinates for this symmetry are available from the work of Brice (1965).

( )¢4

Cd

Figure 1. Isolated substitutional case with Te at origin and Cd as first neighbours ; phosphorus is substituted at Te site.

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The (15 x 15) matrix will reduce into blocks in the following representations.

F15 = A1 + E + F 1 -~- 3F 2, A1 -> (1 X 1) matrix E -> (2 x 2) matrix F1 -+ (3 x 3) matrix F2 ~ (3 x 3) matrix The blocks are given in Appendix 1.

The position of the atoms and the definition of various Green's functions are given below

No. Atom

1 Te

2 Cd

3 Cd

4 Cd

5 Cd

The Green's functions are (0 0 ) _ B ; g~x Te Te

(0 g ~ Cd

(0 g~x Cd

(0 g~r Cd

Position 0 0 0

2 2 2

1 1

½ ½ ½

(o o)

gxx Cd Cd = A1;

) (0

½½½ = Y; gx, Cd Te

Te

110 (0 110

C d ] --- Pa; gzz Cd C d ] = Q1;

110~ ( 0 110'~

Cd ] --- R1; gxz Cd Cd ] : Sx"

The Green's functions are evaluated for CdTe in a modified rigid ion approximations, using the parameters given by Plumblle and Vandevyuer (1976). This modified rigid ion model though simple indeed reproduces the dispersion relations fairly well and hence can be used as a good model for interpolating frequencies at the different mesh points of the Brillouin zone. More refined lattice dynamical model when available in future would not give widely different values for these Green's functions since these functions are obtained as average over the number of wavevector points in the entire Brillouin zone. Hence, the Green's functions are now available from X = 0"01 to 2"99 where X = o~]OJma x and oJ is the vibrational frequency and Wma x is the maximum allowed frequency for this host crystal.

Since our main interest is on the LVM, we concentrate on the Green's functions after X : 1.00.

To start with, let us consider P at Te site, which is at the origin. The proper Green's functions are fed in the computer along with a prescribed value of change in force constant (i.e. AA and AB which are change in force constants in terms of

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Kellerman's constants A a n d B, between the host lattice a n d defect lattice). T h e force constant is actually varied from 10% to 100~o and corresponding to each value, 11--g ~ I I is w o r k e d o u t for nearly 50 values of X. T h e value o f X which gives

[1--gOl[

zero was picked corresponding to each AA value a n d a graph is plotted with this X versus AA. T h e X value corresponding to the experimentally suggested L V M value is taken a n d was used to fit the corresponding AA value in the F 1 representation.

The value o f AA o b t a i n e d f r o m the graph is 4.4451 x 104 dynes/cm. The AA obtained corresponds to increase in force constant when a P a t o m is substituted in the place o f Te.

The molecular m o d e l was also worked out ( R a m a c h a n d r a n a n d Haridasan 1977) to study this unusual behaviour. We found that there was an increase in force constant for the case o f PTe" The actual change obtained was AA = 4-6066 x 104 dynes/cm which is very close to our presently calculated value. D u t t and Spitzer (1977a) also predicted an increase in force constant in their experimental results.

Thus, one sees an unusual increase o f about 74% in the nearest neighbour force constant, even when a lighter impurity like P is substituted in Te site.

Past experience in o t h e r solids indicate that when a lighter a t o m is substituted, only a reduction in force constant is expected (Fritz et al 1965), unlike in the present situation. One is therefore tempted to eliminate the possibility o f P going into T e site substitutionally.

The second configuration, i.e. Cd at origin and Te as neighbours, can also be tried, because b o t h sites are possible and we have worked out the AA for this case also;

i.e. when Cd is replaced by phosphorus. Here again, an increase in force constant was seen and for the same reason mentioned with regard to P at Te site, it is felt that one should eliminate this possibility as well.

3. Isolated interstitial--Theory

Brice (1965) gave a basic theory when an atom occupies an interstitial site with particular reference to Si and Ge.

It is to be u n d e r s t o o d that by the introduction o f an interstitial the n u m b e r o f degrees o f freedom o f the system is increased by three. Brice introduced a new Green's function o f the host crystal, combined with that o f the interstitial, as

g = g (interstitial) ~ g (lattice)

where g ~ (interstitial) is a (3 x 3) matrix given by (m ~o~)-1 ~ and g (lattice) is defined as usual (equation (1)).

The interstitial is t a k e n to be at the origin and if the interstitial interacts with m host atoms around, the g and ~ l matrix would be o f dimension 3 ( m ÷ 1) x 3 (m-}- 1).

So, it is now b r o u g h t in the line o f regular dynamics o f crystals with substitutional defects.

Brice (1965) assumed that the coupling between the interstitial and the host a t o m s is weak and here when g matrix is written, it has two blocks, a (3 x 3) block f o r the interstitial and a (3m x 3m) block for the host a t o m s surrounding it. T h e r e is no matrix element in g, coupling the interstitial and any o f the host atoms. In 8l,

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the first three rows and first three columns contain the actual interaction o f the interstitial with the surrounding atoms. The remaining part o f the cO I matrix however would represent the force constant changes at the host a t o m sites due to the interstitial defect.

F o r the CdTe lattice, the interstitial site is at (½ ½ ½) for the configuration des- cribed earlier. A phosphorus a t o m occupying the interstitial site is surrounded either by 4 Cd atoms as immediate neighbours or 4 Te atoms as first neighbours. It is assumed that charge compensation is achieved by some other foreign atoms situated much f a r t h e r away from the P site.

I f the interstitial P is taken as the origin then the positions o f other atoms are given below.

(for the first configuration with Cd as neighbours)

No. Atom Position

1 P 0 0 0

2 Cd l- 1 1 _2- _P.- _2-

3 Cd 2

4 Cd ½ ½ ~

5 Cd ½ ½ e- 1

6 Te ~ 0 0

7 Te 0 i 0

8 Te 0 0

9 Te 1 0 0

10 Te 0 1 0

11 Te 0 0 1

(in units o f a = 2r0)

We can look f o r both localised modes as well as r e s o n a n t modes, but as we explained earlier, we do n o t go into detail for the resonant modes, as there are no experimental results yet available.

I f we consider the first neighbours alone g a n d t91 matrix will have dimension (15 x 15) whereas when second neighbours are included it will be (33 x 33). T o begin with, we consider the second neighbours also a n d construct the g a n d O l matrix.

When an a t o m occupies an interstitial site in a crystal, then one has to consider the lattice relaxation also i.e. the nearest neighbour Cd a t o m s and the next nearest neighbour Te a t o m are displaced from their n o r m a l positions and due to this mecha- nical relaxation, assume new equilibrium positions. N o r g e t t (1974) developed a p r o g r a m m e to work out the new position, to estimate the relaxation, by a selfconsis- tent calculations with lattice potential as input. H a r i d a s a n et al (1979) w o r k e d out this part when they studied the impurity modes due to interstitial in CaF~. Such relaxation calculations are possible only when the interatomic pair potential is well known. But in CdTe the analytic form o f such a potential is less clear at the m o m e n t and hence relaxation calculation for these crystals would have to wait till such potentials b e c o m e available. However the effect o f relaxation is to alter the nearest neighbour force constants and in our present calculations we try to fit this force

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constant f r o m the experimental L V M frequencies. Thus the effect o f relaxation is indirectly b r o u g h t out in o u r theory through these force constant changes.

N o w let us construct the g matrix for the present case. Again at the interstitial site, we have the Ta symmetry.

The symmetry coordinates have been obtained for the t e t r a h e d r o n plus interstitial by Brice (1965). The set o f coordinates spans the representation Fe o f the group Ta given by

I'¢ --- 2 A 1 + 2 E + 3F1+6F2.

The block diagonalisation o f the (33 × 33) matrix will now give (2 X 2) matrix for A 1 block,

(2 × 2) matrix f o r E block, (3 × 3) matrix for F 1 block, (6 × 6) matrix for Fz block.

Looking at the (33 × 33) defect space, one finds that g matrix contains the following independent lattice G r e e n ' s functions, which are labelled below for convenience o f discussion.

(

₀ ₀

)

₌ _AI' g~* Cd Te

(0 ~ )

g ~ Cd Cd = Y'

"$ "~ = Z,

0 0 -- B, g*Y Cd Te

g;'~' Te Te

g~' Cd C d ] : P~ &,x Cd Te =

= ~ "g½ D1 '

g:'r Cd C d ] RI' gxr Cd T e =

(0 110~ (0

^{a T}

= ] ½ ~ ' ~ E1 '

g = Cd C d ]

SI'

gyx Cd T e ] =

(

⁰ ^{110~ =--}

( ~ - )

⁰ ^{~ ½ ½} ^O~,

g= Cd Cd ] 01, g'Y Cd T e ---

(

^o ^{1 1 o ~} ^_-

(~:~)

^o ^{= =} ^j~,

gxx Te T e ] P' gY= Cd

Co 1,o~ (o ~oo~

g~Y Te Te ] = R, gxx T e Te ] = A2'

(o ,lO~ (o ,~)= ~.

gx~ Te T e ] = S, g'Y Te Te

(0 ll0h

g= Te T e ] = Q'

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24 K Ramachandran attd T M Haridasan

The form of the (33 × 33) g matrix, in terms of the above labelled Green's functions, is shown in appendix 2.

Now let us see the 0 l matrix. It has to naturally contain two parts (i) arising from the Coulombian interaction which is essentially a long range interaction (ii) and the other, short range in nature.

The a I matrix will have elements representing the various interactions and changes of interactions, in terms of the parameters a, b, c, d, AE1 and AF v etc.

Interactions between k and k ' atoms are represented as (k - - k'); then 1--2 for example (1 is the interstitial) can be written as

I a --b --b ]

--b a b ,

--b b a

1-3, 1-4,

example can be written as

1-5 can be obtained from this 1-2 by symmetry considerations; 1-6 for

c 0 O J

0 d 0 ,

0 0 d

1-7, 1-8, 1-9, 1-10, 1-11 are all derived from this.

The 2-6 matrix (now is Cd-Te interaction) can be given in terms of change of interaction, as

aF1 AE1 aFx ,

AF1 AF1 AE1

where AE1 and AF 1 represent the change in force constant between Cd and Te as the result o f the P at interstitial site. All the force constants are defined in terms of the Kellerman's parameter, i.e.

a ---- (ACdTe + 2BCdTe)/3, b = (ACdTe -- BCdTe)/3.

The other possible interactions are derived from the above from symmetry considerations. The whole structure of the (0/)short range is given in appendix 2.

The long range contribution can be straightaway obtained from the Coulombic potential.

4. Group theoretical simplifications

Symmetry coordinates have already been worked out for this configuration (Brice 1965) and are therefore not repeated here. With the U matrix constructed from the

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symmetry coordinates and properly orthonorrnalised, we did U+(/9 I)SR U, U+(/9 I)LR U, U+gU separately in the 33 × 33 space. (LR denotes long range; SR denotes short range). However long range contribution is neglected for our present calculations.

The block diagonalised form of the g and 19 / matrices obtained using the symmetry coordinates are given in appendix 3.

5. Results

To start with, we took just first neighbours alone and neglected the second neighbour contributions. Group theory considerations reduce g and 0 l to blocks. For this case, LVM comes under F 2 representation which is only (3 × 3) matrix. The reduced blocks of g and 01 are given in appendix 4.

Just as we have done in the substitutional case, we calculated the I I--g 01 [ for nearly 50 values of X, for various values of force constants. Actually we tried both the configurations (i) Cd as neighbours and (ii) Te as neighbours. A graph was drawn connecting the force constant and x, where the sign of the determinant changes. The force constant is fixed corresponding to the localised mode frequency of 322 cm -1 (phosphorus mode), we found

a : 3"8468 × 104 dynes/cm.

The situation is not very different when configuration (ii) is taken (i.e. Te as neighbours).

This shows that in the interstitial configuration (either (i) or (ii)) we get a smaller force constant between the interstitial and its first neighbours as we ordinarily expect.

The effect of second neighbours is also worked out in the same fashion as we had done in the first neighbour case. Here F 2 mode is a (6 × 6) matrix. The block diagonalised form of g and t91 are given in appendix 3. The case of Cd being neighbours is tried first. All the 19 independent Green's functions are used in this second neighbours case. The force constant we estimated from the graph (force constant versus X) fitted to the experimental frequency of 322 cm -1 is

a = 3"9314 × 10 i dynes/cm.

The contribution from the second neighbours is less than 5% (phosphorus- tellurium interaction was estimated by making proper scaling). It appears therefore more plausible that P goes rather interstitially and not substitutionally.

In order to substantiate the conclusions reached from isolated substitutional and isolated interstitial cases we extended the study to the LVM calculations for these P impurities paired with substitutional impurities like Ga, In and A1 at the nearest neighbour sites of the P center. Apart from charge compensations these additional defects reduce the site symmetry of the P center giving more LVMs.

6. Substitutional pair modes

Dutt and Spitzer (1977a) studied such pair modes and hence a comparison with their experimental results is possible, if theoretical investigations are made. The configu-

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ration o f atoms to indicate the above situation in the case o f substitutional P center is as follows (figure 2). We have experimentally measured LVMs only for this configuration.

L e t us try to explain the LVMs for the situation as in figure 2. The original dimension o f the g and tgl matrix are o f the o r d e r 15 × 15 and group theoretical considerations would not reduce the order to a convenient one so easily, as we have done in the isolated substitutional impurity case, since the site symmetry is now f u r t h e r reduced to Car.

T h e r e are therefore two impurities namely p h o s p h o r u s at Te site and Ga (In or A1) at one o f the Cd sites. The interaction matrix will have now two ~ factors

(Te)m - - Pm (Te)m and e~ = (Cd)m - - ( , G a , m, ~

(Cd)

where suffix m denotes the mass o f the corresponding atoms.

N o w AA in the interaction matrix will have two p a r t s , AA 1 arising f r o m Te replaced by P and A z from Cd by Ga. The f o r m e r is actually evaluated separately as in the isolated Substitutional case and the latter is evaluated as follows.

A A 2 is the change in force constant between ,4CdTe and AGaP; so,

A A 2 = ACdTe q- 2BCdTc A G a e -q- 2BGa P

3 3

ACdTe - - BCdTe AGaP - - BGa P

3 3

T o a very good approximation we treat in almost all the cases A A = A B which is quite valid for negligible relaxation.

,4Ga P, ,4CdTe, BGa P and BCdTe are taken f r o m the respective crystal data o f the corresponding host lattice.

()ca

C~r. Te C~OG °

F i g u r e 2 . Substitutional pair ease w i t h P defect at origin and one of the Cd n e i g h -

b o u t s is replaced by Ga or In or AI.

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The Green's function matrix g is unaltered whereas p e r t u r b a t i o n matrix Ol gets slightly modified as mentioned above. The modified Ol matrix for the present situation is given in appendix 5.

Due to the presence o f two defects, we get 2 LVM frequencies corresponding to the reduced symmetry o f C3v for the P center. Thus the determinant ] I--g 011 gives zero at two X values and the corresponding LVM frequencies obtained are given in table 1.

The cases like P T e - G a c d and PTe - - I n c d could reproduce the experimental results to a reasonable accuracy by our present calculations a n d PTe - - AIcd could n o t now be c o m p a r e d with as there is no experimental observations o f LVMs in the pair mode case.

The reason why D u t t and Spitzer could not find L V M in the case o f PTe - - AIcd is perhaps due to the fact that they did experiment only in the range u p t o 400 cm -1 whereas o u r calculation shows the range to be more than 400 cm -a for at least one o f the modes. They predicted that there should be a very high increase in the force constant for the cases CdTe :P and CdTe :AI. This conclusion derived from the fact that the recent experimental observations showed the wLV M o f A1Cd to be 299 cm -1 ( D u t t and Spitzer 1977b) and to explain this, an increase in force constant was required even in o u r calculations. Here again is an a n a m o l o u s case o f a lighter impurity demanding simultaneously an increase in force constant to reproduce the LVM.

We calculated the change in force constant in CdTe:A1 as 5 0 ~ increase which is not very great c o m p a r e d to the case o f CdTe :P, where it is more t h a n 74 ~o.

The C d T e :P, In spectrum is the simplest one to consider. Substitution o f In for Cd results in no significant size or mass changes. We may therefore expect the L V M as a first approximation, gtill to involve primarily the m o t i o n o f P, b u t now in a potential having a point group symmetry o f C3v with the trigonal axis being along the In-P pair direction. Thus for the pairs, the 322 cm -1 band should split into two bands.

But C d T e : P , G a is a bit complicated; here again G a c d - P T e pairs have a point group symmetry o f Czv. But G a is lighter than Cd and the pair bands might involve the localised vibrational m o t i o n o f b o t h G a and P.

Table 1. Localised frequencies in the case of substitutional pair modes (Cd as neigh- bours)

Calculated LVMs

No. System Molecular Green's Experimental

function LVMs

model technique 1 P T e - - G a c d 319"654 318"97

365"613 349"21

2 PTe -- InCd 308"93 315.59

320.32 345.97

3 PTe --A1cd 322"51 399.98

368"51 430.36

301.5 352-5 357.5 305 331.5

The LVM frequencies are in units of cm -x

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T h e r e are two configurations in this pair m o d e s case (i) p h o s p h o r u s paired with G a as given in figure 2 (ii) G a paired with p h o s p h o r u s .

Since b o t h the a b o v e configurations are possible for infrared a b s o r p t i o n measure m e n t s f o r the pair m o d e case, the intermixing o f the two cases m a y give rise to three bands.

W h e n we try one configuration (figure 2) separately we could get only two b a n d s in a c c o r d a n c e with g r o u p theoretical principles.

7. Interstitial pair modes

A l t h o u g h we could explain the pair modes reasonably well, it d e m a n d s a n increase in force c o n s t a n t between the g r o u p I I I substitutional defects a n d their neighbours.

F u r t h e r we could n o t rule out whether the 322 c m -1 b a n d c o r r e s p o n d s to PTe or P~

a n d so, we proceeded to work out in detail the localised m o d e s when p h o s p h o r u s occupies interstitial site, surrounded with Ga, In or AI in one o f the neighbouring C d site.

T h e C d - P interaction is already c o m p u t e d in the previous case o f isolated interstitial. Since one G a a t o m presents at the G a site, the new i n t e r a c t i o n o f G a - P is taken f r o m the interaction o f G a P crystal. With the presence o f two defects the T d sym- m e t r y o f the host lattice is again reduced to Car as in the substitutional case.

T h e modified f o r m o f the t9 l matrix (second neighbours neglected) is shown in a p p e n d i x 6.

T h e p a i r m o d e s are now calculated for three cases (AI, G a , In) a n d they are c o m - p a r e d with the experimental m e a s u r e m e n t s o f D u t t a n d Spitzer as can be seen in table 2.

8. Conclusions

T h e L V M study o f CdTe with particular reference to the p h o s p h o r u s i m p u r i t y is carried o u t as m o s t o f the relevant experimental infrared results are n o w available.

Table 2. Localised frequencies when P at interstitial site and one of the Cd neighbours is replaced by Ga or AI or In.

Calculated LVMs

Experimental

No. System Cd neigh- Te neigh- LVMs

bours bours

1 (Ga)cd--Te 318.92 339.22

P interstitial 345.97 346.43

2 (In)cd--Te 322"4 322

3 (AI)c d - Te 352.72 322

301-5 352.5 357.5 305 331.5

The LVM frequencies are in units of cm -~

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Dutt and Spitzer in their principal experimental investigations pointed out that phosphorus can go into any one of the possible two sites (i) substitutional site (ii) interstitial site. We have analysed the two possibilities by using Green's function formalism.

Apart from looking into the localised modes due to isolated substitutional and interstitial P centers we also investigated the effect on LVM due to pairing when Ga, In or A1 is substituted in the nearest Cd site for both configurations. The study reveals an unusual increase in force constants even for lighter impurities when the substitutional P center is assumed. On the other hand for the interstitial configurations one gets considerable reduction in force constants between the impurity (Pi) and its nearest neighbours. This is the behaviour one expects for lighter impurity.

This fact as also the fair agreement for the pair modes for interstitial case with experiment gives us a feeling that the P defect have more affinity to occupy the interstitial site rather than substitutional site in CdTe. Since the site symmetries are identical in both cases, it would be advisable to do more experiments especially by the application of P defects to throw more light on the nature of the LVM associated with these P centers.

Acknowledgements

One of the authors (KR) wishes to thank the University Grants Commission, New Delhi, for financial assistance.

Appendix 1. Isolated substitutional case with first neighbours alone.

Block diagonalised Ol matrix A 1 . . . . ( A A + 2 AB),

--(AA-- AS) 0

)]

E ~

--(AA-- AB

I --(AA--AB) 0

F~ = 0 --(AA--AB)

0 0

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- 1 (16emo~ ~-- 100 AA) -- -4 - cmoJ 2 - - 40 AB

l0 V'160

l em°~Z 0

5

_ - - ( a a + AB)

Block diagonalised g matrix:

A1 = (A1--4S1--2Pl + QI--R1) ,

E ^{- - - -} -(A1-2PI+ Q1--RI+2S1) (AI-[- Q1--RI+2S1--2P1) 1]

[ (A~--2Px+ Q~ + Ra +2S0 0 0 ~

J

F~ = / 0 (A~--2P 1 + QI + RI + 2S1) 0

/

k 0 0 (A1--2PI-? Q I + R 1 +2S1)

( _ 1 6 ( 2 Y _ B ) + 4 A 1 + 4 ( - - ( B + 3 y ) +

8Px+4Qi)

A~+2PI+ Q1)

I ( B + 8 Y+.

5

a(A,+P~+ Qi))

V'160 (--Rl+4Z)

--(R~+Z)

X/40

(AI--Qi--Ri)

-

In all aplxndices only the upper half of the matrices are given, as all of them are symmetric.

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C~

-- I

,.¢: --

~ + I I

. . I I

I I

I

.~.

<

~ . . o , ~ . - ~ : , ~ ~ . - o o o o o o o o o o o o o o o o . ~ o o ~

I< ~ "~ <~ <] < 3 I

<~ I<~

~ ~ I

<

-- < ~ < ] < i <] <] <~

I I

I

~ l l ~ I I I

(16)

32 K Ramachandran and T M Haridasan

0#

~: II

-

~

t ~ p.,+,

~ O m O 0 0 ~ , ~

.~.

- ~ ~ ® ~ o - ~ ~ ~ ~ ~ ~

(17)

Appendix 3

Isolated interstitial case with second neighbours included

Block diagonalised g matrix

A 1

12 j

~/72 ( Y - G - 2 E ~ - 2 Z ) 6 (Ax+ Qi_2Pi_2Ri+4Si )

12 Ei = [ (B--Az+2R) 24 (y_G+E_j_Z) ] V288

(A1--2Pi--2Si + Ri + Qi)

--(B--,B~+.2Q)

i A B G H I C D E F i J K

F~= L M N O

T

--2S -- V'32 (D1 --J~--2Z) 8 (y+Z+Di_G1) --(--B+B2--2R)

- -

(Ai+ Qi--Ri--2Si) _

where

A =½(2a+B+Az); B=8(~--B--Az+3P)/x/504 C= [8a+ 28( Y+ C~)--8(B+Az)--32P]/~/1848 D = [--2~+4( Y+ C1)-4-2(B-+-A2)+ 8P]/v/66;

E=--8S/v'24 F--8(Z--Ei)/x/48

G =(16a + 68B+ 32A z -- 192P+36Bz+ 72 Q)/84

H=(16a+56( Y+ 3G1--2Ci)--(2B--4A~ +6Bs--4P +12 Q) )/ X/ 25872 I=(--4a+ 8( Yq-3G1--2Ci)q--20B+8A~--56P + 12B2+ 24Q)/ v/924 J= 56S/a/336

K=8(4E~--4Z--3Di--3Ji)/X/672

L=(16a+ 196(A1+ Q~+2P1)--224(3 Y+2Gi+Ci)+64(2Q+Bs+4P)+32(A~+3B)) / 308

M = (--4a + 28(A1+ 2/°1+ ax) + 12(2G 1 + 3 Y+ C~ -- 2B)-- 8(A ~q- 8P+ 2B 2 + 4 Q))/

v'3388

P.--3

(18)

34 K Ramachandran and T M Haridasan N= 56(Z--E~)/,v/1232

O = --8(7Rlq-4(Z-- D~--J~--E~))/~/2464

P=(a-k4( A~ @ Q~ q-Bq-B~)-k 8(Pl-k Qq- c1+ 3 Y-k 2al q-2P) q-2B-k 2A~))/11 a =8(Z--E1--Jl-~E1)/x/44

R=8(Z--R~-- D~--]~--e1)/ V88 S=(B--B~--2R)

T=8( DI-kG~ @Z-- Y)/~/32 U=8(A1--Qa--Ry--2S1)/8

Block diagonalised form of ~l matrix

A I - I N a/7---2 I2 (AEj--2AE~) l (L --2M)

~ = [ _{½ N} x/2S'8 (AE~-b AF 0 8 ]

½ (L-t-M)

I ^V ^U ^16AFx/~32 ¹

Fx = V 8(AE~-k AFx)/~32 (L-I-M)

l

^{- 1} ²

⁷ ¹²

³

⁸ ¹³

⁴

⁹ ¹⁰ ¹⁴

⁵

¹¹ ¹⁵

^{6 - i}

F1 = 16 17 18

19 20 21 where

1 =(4(k--2c)-k2N)/6 2=8(k-kc--N--3d)/ ~/ 504

3=(8(k +4d--N--7a+c)+ 28 AE1)/x/1848 4=2(--c--k+2 AE1--4a--4d+ N)/ v'66

5=0

6 =8(2bq-AE~)]~/48

7 =(16(k--6d+4c-k2N)+36 V)/84

8=(16(k +4c-~d--3 Vq-2N)--56(2a@ AEx))/X/25872

9=(--(k q-dq-4c--2 AEl +4a--3 V-k 2N)/ ~/924

10 = -- 12 U/a/336

(19)

11 =32(b-- AF~)/V'672

12=(16(k--14a@4e@8d+4V+ 2N)+ 28(7L--24

^AEI))/308

13 = - - 4 ( k - - 3 a @

8d@4c--7L--90

AE~+4

V+2N)/~v/3388

14=8(2U--7 AF1)/~/1232 15 = 8 ( 4 b + 7 M - - 4 AFx)/~/2464

16=(k@8a+4c@4L+24 AE~+4V+2N)/11

17=--4(U@2 AF1)/~v/44 18 = - - 8 ( b - - M - -

AFI)/V88

1 9 = V

20 = 8 ( A E ~ - - AFa)/,V/32 21

=L--M

Appendix 4

Isolated interstitial case with first neighbours alone Block diagonalised g matrix

A 1 = [(A1--2P1+

Qx--2Rlq-4S1)]

EI = [ ( A1-2PI-~ QI-I- R1- 2Sa) ( AI_}_ QI__ 2pI_+_ RI_~_ 2S1) ]

0

F 1 = (A l - Q]-I-RI-{-2S1)

j- 4 (_aq_Al_k2plq_ QI) --8R1 -

0 (4¢L+Ax-}-2PI@ Q1) -- 10 x/160

F,. = 1 ( ~ + 4 + ( 2 P ~ + QI+AO) --8R~ [

5 V40 [

(A1-- Q1---R1--2Sx) _1 Block diagonalised 0 l matrix

A a = [(a+2b)]

F~ = [

= F --5a

F, [

0 0

= [--(a--b) E1

o ]

--(a--b) 0 --(a--b)

V'160 0

(a+b)

(20)

I

I I I

t I

,J

o ~

i

I I

I

I I

1

I

I I

II II

(21)

O

I I

"d

0

I ~ ~ ,.~ ..~

" 0

"d I z

..=

II II

(22)

38 K Ramachandran and T M Haridasan References

Apple E F 1959 J. Electrochem. Soc. 106 271

Barker Jr and Sievers A J 1975 Rev. Mod. Phys. 47 suppl. 2 Brice D K 1965 Phys. Rev. AI40 1211

Dutt B V and Spitzer W G 1977a J. Appl. Phys. 48 954 Dutt B V and Spitzcr W G 1977b J. Appl. Phys. 47 565 Fritz B, Cross U and Bauerle 1965 Phys. Status Solidi 11 231

Haridasan T M, Govindarajan J, Neremberg M A and Jacobs P W M 1979 Phys. Rev. 1320 3462 Hayes W and Spray A R L 1969 J. Phys. C2 1129

Maradudin A A, Montroll E W, Weiss G H and Ipatova I P 1971 Theory o f lattice dynamics in the harmonic approximation, I I e d . (New York: Academic Press)

Norgett M J 1974 Report R7650 Atomic Energy Res. Estt., Harwell Plumblle P and Vandevyuer M 1976 Phys. Status Solidi B73 271

Ramachandrart K and Haridasan T M 1977 Prec. Nucl. Phys. Solid State Symposium (India) Sennet C T, Bosomworth D R, Hayes W and Spray A R L 1969 J. Phys. C2 1137

The localised modes due to P defects in cadmium telluride