Lattice vibrational spectrum of diamond

I Pkys. Chrm. S&s Vol. 40, pp Ji%9.4 Pcrgamon Press Ltd.. 1979. Printed tn Grert Britain

LATTICE VIBRATIONAL SPECTRUM OF DIAMOND RAJENDRA GUPTA Department

of Mathematics.

University

of Calabar, Calabar, Nigeria

(Received 23 August 1978; accepted 6 October 1978)

Ah&net -The lattic dynamics of covalent crystals are discussed with reference to known models. The dispersion curves of diamond have been computed on the basis of the shell model of Cochran applicable to covalent crystals. Parameters have been determined using elastic constants and dispersion curves along the A and A directions from the neutron spectrometric data of Warren et al. In general there is good agreement between the calculated curves and experiment. The average error is about 4.8%. Although the calculated curves seem to be a definite improvement over the curves calculated by Smith there appears to be a certain discrepancy between theory and experiment. Plausible causes of this discrepancy are pointed out.

Covalent crystals are bounded by non-central forces and there is no accepted formula for the inter-atomic potential which would lead to the force constants. Models, used in the initial stages of development of lattice dynamics of covalent crystals, can be classified in two broad categories. The first involves models of “hard” atoms with adiabatic force constants and interaction between the nearest neighbours (1-41. Herman 141has made the corresponding reduction of force constants out to fifth neighbours in the diamond structure using a large number of adjustable parameters. In spite of the complicated calculations, results were not appreciably better than the first calculations made by SmithIf I for diamond. The results of all these models were later found to be in serious disagreement with experimental results for germanium, silicon and diamondI%71. A theoretical explanation of the long range of interatomic forces in diamond type structure has also been given by Lax[8], who observed that such long range forces may be of quadrupolequadrupole origin. It seems however that dipole-dipole forces which vary as I/r’ should be of greater significance. This has been the basis of the second category of models for covalent crystalsIP_161. CochranllO] proposed a shell model for germanium based on the simple model of Dick and Overhauser [ 171 for ionic crystals. This model takes into account the long range interaction as well as the quasi-elastic forces which connect the cores to the shells of zero mass. A displacement of the shell relative to the core gives rise to a dipole moment and an electrostatic interaction with atoms far away. With a moderate number of parameters (five) Cochran obtained fairly good agreement with experimentU1. This model also works for siliconll I]. Cochran’s formula for potential energy is ‘a natural generalisation of the expression used in 191which forms the basis of the long range force models using deformable atoms I IZ-161. 579

From the theoretical point of view, the calculation of the dispersion curves in diamond is of special interest, since it is evident that diamond, which is composed of one of the lightest of these elements (carbon), differs strongly from its homologues. The theory of lattic dynamics based on deformable ions was first applied to ionic crystals and only first order approximations were considered. In the case of covalent crystals the first order approximations diverged for germanium and silicon[ IO, 141, and the theory was extended to second and third order approximations. However, for diamond the first order approximation does not contain divergences[Nal and it can be calculated just as well as other approximations. In this sense diamond behaves more like ionic crystals than its homologues germanium and silicon. Thus diamond is the only crystal in which it is possible to compare the different theoretical approximations which are being used separately for alkali halides and homopolar crystals. Among the more recent modelsIl~231 perhaps the best one is that of Vasil’ev et af.[20] with four parameters, including the long range interaction in terms of rotation of valence bonds formed by the overlap of hybrid d~ection~ wave functions. Theoretical calculations have been made for germanium, silicon and diamond. Agreement with experiment is uniformly good, though best for silicon. Calculations have also been made for diamond and silicon with the valence force model[l8, 191. Bose et all231 have computed the dispersion curves of germanium and silicon on the basis of angular force models[21,221 DAF (De Launey angular force model) and CGW (Clark, Gazis and Wallis model), and have shown that the theoretical results of shell model [lo, I II are better than those of theirs. The most recent development in the lattice dynamics of convalent crystak is the pseudo-~tential

model[24-

271, in which the response of the electron system to the ion displacements is considered and only one parameter

580

R. GUPTA

(the electron static permittivity eO) is used. The dispersion curves for diamond, germanium and grey tin have been calculated by Al’tshuler er al. [28]. In this case also, agreement with experiment is inferior to that of Cochran’s shell model. Now in the second decade the shell model continues to be the principal phenomenological model used in lattic dynamical calculations involving non-metals. One of the reasons for its continued use is its ability to provide a simple description of long wave length electric and elastic phenomena while simultaneously accounting for measured phonon dispersion curves. The shell model has been justified from rather fundamental points of view and its parameters are realistic [9,29,30]. In recent years, extensions of the shell model have been made. These are the three-body force shell model (TSM) and extended three-body force shell model (ETSM)[31,32]. The dispersion curves calculated with the help of these models[31-351 compare excellently with experimental results. The theoretical results are superior compared with the results of the simple shell model, not only in the case of alkali halides but also in the case of other ionic crystals like silver bromide and silver chloride[M, 371which exhibit partial covalent behaviour. On the basis of the phenomenal success of TSM and ETSM for lattice dynamics of ionic crystals it is reasonable to expect that a suitably modified shell model can be used to describe the lattice behaviour of covalent crystals-especially diamond, which in some sense behaves like ionic crystals. Thus it seems desirable first to test the applicability of Cochran’s shell model to the case of diamond also, by comparing the theoretical results with the neutron spectroscopic experimental observations of Warren et al. 171.

constants and corresponding coefficients are introduced as follows #l”,‘(l3,1’4)

between the shells

&7)(/l, 1’2)

between the cores

dCF)(I1 *Y 91’4)= #F’(/3 w P1’2) between the core of one atom to the shell of the other. The dynamical matrix for the shell model of diamond contains 12 x 12 coefficients. It is well known that for wave vectors along [ IO01or [ 1111the usual sixth order equation factors into three equations each of order two, of which one gives the dispersion relation for the longitudinal modes and the others give two identical solutions for the two transverse modes. Therefore, attention is restricted to 4 x 4 matrices which refer to a particular mode. The equation of motion, therefore becomes m,02u(k) = $,

M(kk’)u(k’)

where M(kk’) is a linear combination of M,(kk’), &(kk’), etc. depending upon the mode and its polarisation (see Table 1). For covalent crystals m, = m2 = m and aI = a2 = a. Expanding the four equations of motion keeping ml = m2 = m and m3 = m4 = 0 and eliminating u(3) and u(4) we get mu*uW = A,u(l) + AU(~)

mw*u(2)=A*u(1)t A42)

(2.3)

which give the dispersion relation as mo2=Ao? {Al

2.THJ?.ORYANDCAWULM'ION

The basic features of the theory are the same as those for the shell model for ionic crystals[lO, 17,29, 381. The energy perturbation in the harmonic ap proximation is written as

(2.2)

(2.4

the t ue sign referring to optic modes and the - ue sign to acoustic modes. Using ~=~+~+2~=B(l2)tB(~)+B(32)

42 = - ; z 8 cb,,(lk, f’k’)U,(lk)UJl’k’).

(2.1)

The summation extends over four values of k instead of two, k = 3,4 denoting the shells while 1 and 2 refer to the co~espondi~ cores. Thus the potential depends not only on the nuclear coordinates but also on coordinates of the outer electrons. The equivalent of the adiabatic approximation is achieved by taking the mass of each shell to be negligible. Separate symbols for short range force

T=S+F=B(34)tB(l4) (The bonding coefficients S, D and F have been defined earlier.) and

fi=;

B,=k

T2

&=-

so k + Fo

the final forms of A, and A arefound to be

A,=%tB,

A=RtB,

+

(2.5)

581

Lattice vibrational spectrum of diamond Tabk

I. Expnssions for M(W)

dependingon modesand their polarization

Polarisation

M(kk')

1100 1

I100 1

Mxx(kk')

T I100 I

1010 1

Myy(kk')

L I111

1

I111 I

Mxx(kk')+2Mxy(kk')

T 1111

I

1li2 I

Mxx(kk')-Mxy(kk')

Mode (q) L

Table 2. Input data for diamond

Here for the purpose of calculation RI& S/S, and 7lT, are each taken to depend upon the same parameter y through the following expressions (a) S=-S

0 (cosfi+iysin@

2

=11 =

10.76 x 101*

dynes /cm2

Cl2

=

1.25 x 1012

dynes /cm2

=

5.76 x lO1*

dynes /cm*

2ro

=

3.5672 x 10-S cm.

m

=

1.995 x 1o-23 gm.

"k

=

2.51 x 1014

2 > for [MO]transverse

(b) S = -S 0 cos qr, 2 for [lOOIlongitudinal (c) S = -S,[cos3 0 + y sin* 8 cos B + i(sin3 B t y cos’ B sin O)]

C44

Rad./Sec.

(Raman Frequency).

J

where e =% 2v3 >

13.68x lo’ dyne cm-’ and y = 0.573 the elastic constants calculated from (2.7) dilfer from the experimental values uniformly by nearly 11%. In order to determine the parameters f3, B, and & we (d) S= -S,[cos3e-2ysin2ecose t i(sin’ e - 2y cos’ e sin e)] have to search for other relations between the constants and known frequencies like Raman frequency and from other frequencies at the points lying on the zone boun(e same as in c) daries, taken from the experimental data. Making q = 0, for (1111longitudinal. (2.6) R = -& and c, = c2, the Raman frequency is given by for [ 111)transverse

The unknown parameters involved in (2.5) are taken to be the constants III, El, 8, R, and y; of these the latter two are fixed with the help of the elastic constants through the relations. CII = p:

0

mo2=2R

mw2=R,kR-

The modulii of elasticity for diamond known since 1946[39] were measured again in 1957 by McSkimin and Bond[40]. The results differ greatly, and the latest data are assumed to be correct because they appear to be in agreement with measurements of Desnoyers and Morrison [41] on the specific heat of diamond. This was further confirmed by Kucher and Nechiporuk[lS]. The elastic constants[40] and other data are given in Table 2. The set of elastic constants for diamond does not satisfy Born’s identity: ~CI,(CI~-CM) = (cl1 + c,# (Huntington(421). Further, it is impossible to satisfy the relations (2.7) and also to arrive at the above values of elastic constants, for any value of R,, and y. With & =

--

4B, I+282

(2.8)

At the zone boundaries the frequencies are given by

Cl2= c,,(2y - 1); cu = c,,(l - J). (2.7)

o

2 B, 12; ( 0>

(2.9)

I + /3(c, k ~2)t Bz l? ; ( 0> where the tue sign applies lo optic modes, and the -ue sign to acoustic modes. Considering the longitudinal and transverse modes of [MO] and [I 11) directions each having optic and acoustic modes, the relation (2.9) gives seven distinct frequencies as two of them Y, and vfOin the [MO] direction are degenerate. Thus two relations, (2.8) and (2.9) are actually equivalent to eight relations, of which (2.8) and any two of (2.9) are sufficient to fix the parameters p, BI and &. But it is seen that the values of the parameters thus obtained do not satisfy the other relations closely. In order to have uniform agreement the parameters are so chosen that the Raman frequency is fixed and the agreement with the other frequencies is as

582

R. GUPTA Table 3. Comparison of experimental and calculated frequencies (Shell model and Smith model) at the zone boundaries Mode and direction

w x lo-l4

Shell

model

Rad/Sec

Experimental

Smith

model

TA 1100

1.54

1.51

1.40

TO I100

2.13

2.03

2.23

LA [ 100

2.10

2.25

2.23

2.10

2.25

2.40

1.11

1.04

0.99

2.49

2.28

2.34

1.95

1.95

2.11

2.25

2.34

2.50

LO

[loo 1

L

L

Table 4. Model parameters for diamond Ro

=

13.68

V

=

0.573

B

=

0.026

=

8.87 x lo5

Sl

x lo5

dynes

dynes

1

cm-'

cm-l

0

02

04

08

IO

o/&.%1 Fig. 2. Dispersion curves in [Ill] direction in diamond. Theoretical curves: solid lines, shell model; dashed lines, Smith model. Experimental points: shaded circles, longitudinal branches; circles, transverse branches.

I I.0

Fig. I. Dispersion curves in [lOtI] direction in diamond. Theoretical curves: solid lines, shell model; dashed lines, Smith model. Experimental points: shaded circles, longitudinal branches: circles, transverse branches.

uniform as possible. For /3 =0.026 B, = 8.87 x lo5 dynes cm-’ and B1 = 0.7 the calculated frequencies at the zone boundaries are compared with the experimental values in Table 3. For the sake of comparison the corresponding values of o calculated by the Smith model are also Riven. Final values of the parameters fed into the calculations of the dispersion curves are Riven in Table 4. The method employed to determine the parameters and their values are diflerent from similar computations made by Warren et a[.[431 where it is claimed that shell model parameters for diamond can not be determined from the measured dispersion curves along A and A

583

Lattice vibrationalspectrumof diamond Table 5. Position of

q

lculatedand experimentalfrequenciesfor diamond w X lo-l4

Mode Shelf Acoustic 0

Model Optical

T

RaWSec Experimental

Acoustic

Optical

O.O,O

L

$o,o

L

$o,o

L

1.10

2.45

$o,o

L

1.53

2.37

1.68

&or0

L

1.86

2.26

2.00

2.42

l,O,O

L

2.10

2.10

2.25

2.25

O,O*O

T

.86

2.37

0.58

2.51 2.49

0

2.51 2.48

-$.o,o

T

0.46

$Lo

T

0.87

2.40

+o

T

1.22

2.29

1.16

2.25

+o,o

T

1.45

2.19

1.37

2.12

l.O,O

T

1.54

2.13

1.51

2.03

O,OtO 111 g's,x 111 3'3'7 111 T'I'Z

L

0

2.51

L

0.92

2.47

L

1.58

2.36

L

1.95

2.25

1.95

2.34

1.04

2.26

O,O,O 111 1#7;'8 111 5Tjt‘5 ill '1'1'Z

0

2.51

T

0.57

2.50

T

0.97

2.49

T

1.11

2.49

T

Frequencies have been calculated for wave vectors along the symmetry directions [IO01 and [ill] with the help of the general formulae (2.4) and (2.5). The results are tabulated in Table 5 and the dispersion curves plotted in Figs. 1 and 2. For comparison the computed dispersion curves for Smith’s model are also shown along with the experimental pointsl7J. fRWlLTsANDrrHX!BW Considering the limitations of the simple shell model the agreement between the theoretical computations and experiment is quite good. In general these computed values yield not only results better than Smith model but also better than most of the models discussed in the introduction. Moreover the number of parameters is quite small. Certain branches like LA along the fill] direction show exact agreement with the experiment, The average error for the calculated values from the shell model is neatly 4X%. The maximum error is for the LA frequency at the point &O,O) and the TO frequency at d. f. i). and is about 9%.

It is to be observed that the value of p = 0.026 does not satisfy the Claussius-Mossoti formula 4n 2a r-l -3 ( v ) =z

(3.1)

If we take e = 5.5 for Diamond this gives r!I= 0.072. This discrepancy cannot be explained on physical grounds. In our calculation t3 is treated as a parameter adjusted from the knowledge of frequencies-a procedure independent of relation (3.1). It is mere coincidence that for the cakulation of the dispersion curves of geranium by Cochran (1959) B fixed in this way comes out to be close to the value given by (3.1). On this basis it seems probable that the agreement might still be better by &eating R, and y aiso as adjustable parameters. A plausible explanation for this is that the limitations of the shell model are counter-balanced by the variation in the parameters. It is further noted that the number of relations available from the ex~~men~ frequencies at the points lying on zone boundaries are many more than the

584

R. GUP~A

parameters involved in the theory. Thus there is scope for introducing more parameters while modifying the shell model to suit covalent crystals as has been done in the case of ionic crystals. Acknowledgements-The author wishes to express his gratitude to Prof. S. Sampanthar for his helpful discussions, and to the University of Calabar for the award of a Senate Research Grant.

-Es

6. 7.

8. 9.

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