The ionic character and elastic moduli of zinc blende lattices

J. Phys. Chem. Solids

Pergamon Press 1957. Vol. 3. pp. 223-228.

THE IONIC

CHARACTER

AND ELASTIC MODULI OF

ZINC BLENDE

LATTICES*

ROY F. POTTERT National

Bureau

of Standards,

(Received

7 February

Washington,

D.C.

1957)

Abstract-The elastic moduli for a number of materials with the diamond or zinc blende (sphalerite) crystal structure are available in the literature. It has been noticed that the ratio C&u has values between 2.6 and 2.7 for Si and Ge, respectively, and 1.65 or 1.50 in the case of ZnS. InSb and GaSb have elastic moduli that give intermediate values for this ratio. A scale which gives the ionic character as a function of the ratio Cn/Cn has been constructed based on the Born diamond lattice theory and the relationship between the longitudinal optical mode and transverse optical modes when the lattice has some ionic character. This scale indicates that InSb has the ionic character that corresponds to an ionic charge of 0.65 electron. GaSb has an ionic character approximately two-thirds of that for InSb. Conversely, the elastic moduli may be estimated for crystals of this type if the optical data are available. INTRODUCTION

THE steadily increasing interest in the physical properties of the intermetallic semiconductors having the zinc blende (sphalerite) structure makes it desirable to have some estimate of the ionic character of the lattice. The effect that any polar character of the lattice may have on electron scattering mechanisms in these compounds is currently being considered by several investigators. If one considers the covalent diamond crystals Si and Ge as special cases of the zinc blende lattice having a higher order of symmetry, he will find that the elastic moduli of a number of these compounds have been measured, many by modern techniques. Table 1 comprises a list of such values and the investigators who reported them. The ratio C,,/C,, shows a definite variation between such covalent crystals as Si and Ge (2.65) and the partially ionic zinc blende (l-65, 1.49). The materials InSb and GaSb have intermediate values. On the other hand, the ratio C,,/C, shows no large deviation from the value 2.

The manner in which the value of C&‘,, varies from the covalent compounds to the partially ionic compounds suggests that the ratio could be related to the ionic character of the bond. SPITZER and FAN@) have given what they believe to be an upper limit for the value of the ionic charge of InSb. Based on an estimate of the difference between the static and high frequency dielectric constants the value given is 0.34 electron charge. LATTICE VIBRATION

* This research was supported by the United States Air Force through the Air Force Office of Scientific Research of the Air Research and Development Command under Contract No. CSO 670-53-12. t Present Address-U.S. tory, Corona, California.

Naval

Ordnance

THEORIES

Lattices with tetrahedral symmetry about each site have been considered by MOTT and FR~~HLICH,@)SZIGETI(~)and others.c4) This work has been summarized by BORN and HUANGWand their notation will be used here in presenting the resulting equations. It is shown that the transverse optical frequency (corresponding to the reststrahlen frequency) is related to the longitudinal mode frequency, the ionic charge, Ze, and the static and high-frequency dielectric constants, E,, and eco by the following expression : 6x? = +J(~)-

Labora-

223

(1)

ROY

224

F. POTTER

where iii is the reduced mass of the primitive cell and o, is its volume, i.e. as3/4 for zinc blende structures. By considering ionic overlap potentials SZIGETI(~) derived a relation between the bulk modulus [K = (Cu+2C,,)/3] and the optical modes which can be written

BORN@) and SMITH(‘) have investigated the diamond lattice. Only the nearest neighbor lattice theory will be considered here. The results of this theory predict that only two independent elastic constants for the diamond lattice exist, [4c11(c11--44)1/(c11+c12)2

-

1

(4)

and that the fundamental frequency w for the optical mode of vibration with lattice wave vector q = 0 is given by The bulk moduli of InSb and ZnS give values for frequency that are too low compared to the values obtained from measured reststrahlen frequencies, although in the case of ZnS the agreement is somewhat better. This discrepancy is not surprising in that a large amount of covalent bonding exists in these lattices. One may eliminate wr from equation (1) by writing UJ,,~= w~~(E~/E,J(Q. This gives the equivalent expression 4+e)s fJ$(WJ-~co) = ___ EV,

em+2 s 3 > (

69 = (4C~~as)/5.

(5)

The last column of Table 1 gives values for the left side of equation (4). It can be seen that they become markedly less than unity for the more ionic crystals. The ratio C,,/C,, decreases in value as the lattices become more ionic (see Table l), however, this ratio must have unity as a lower bound. This limit results from the critical conditions for lattice stability,(8)

(3) c11-Cl2

which was used by SPITZER and FAN(~) in estimating the effective charge for InSb. Instead of using the estimated dielectric constant difference, it is the purpose here to investigate the possibilities of getting values for wr from the elastic moduli. Once this is obtained, a value for 2 can be determined from equation (1).

>

0,

c44 >

c11+2c12

>

0,

(6)

0.

This would indicate that lattices for which C,, -C,, = 0 are unstable in that they approach what BORNrefers to as a gel state. Furthermore, the bulk modulus for such a lattice would be equal to either C,, or C,,.

Table 1. Room temperature values of the elastic constants (units are 1O1l dyn/cm2) = =

I

4Cn(C11-Cd Crystal Si(a)

Ge(@ GaSb(a) I&b(b) I&b(c) Z&W) Zdw

CII

CI,

16.57 12.89 8.849 6.717 6.472 9.45 10.7

6.392 4.832 4.037 3.665 3.265 5.70 7.22

.-

G

_-

7.957 6.712 4.325 3.018 3.071 4.36 4.12

=

.2.6 2.7 2.2 1.83 1.98 I.65 1.49

=

cldc,, ~~

CUIC,,

0.8 0.72 0.93 I.21 1.06 1.76

2.08 1.93 2.04 2.22 2.11 2.16 2.50

(c,,+C,)P 1.07 I.015 0.965 0.90 0.91 0.85 0.89

=

(a) MCSKIMIN H. J. j? Appl. Phys. 24, 988 (1953). (b) MCSKIMIN H. J., BOND W. L., PEARSONG. L. and HROST~W~KI H. J. Bull

(c) POTTERR. F. Phys. Rev. 103, 42 (1956). (d) VOIGHTW. see HEARMANR. F. S. Rev. Mod. Phys. 18, 409 (1946). (e) BHAGAVANTAM S. Proc. Indian Acad. Sci. 41, 72 (1955).

Amer. Phyf. Sm. 1, (1956).

THE

IONIC

CHARACTER

AND

ELASTIC

MODULI

OF ZINC

BLENDE

LATTICES

225

Table 2. Data usedfor determining the values of Z in Fig. 1 = crystal

f;

a,(A)

Si Ge GaSb InSb InSb zns ZnS

2.32 6.02 7.36 9.81

.5*43(U) 5.66(b) 6-10(c) 6*48fa) 6.48 5.40(“) 5.40

3.57

,[W

1.55 x10*6 48.6 29.3 17.75 17.10 59-5 64.8

W’W

4(P)

15 27 35 45 46 24-5 23-5

16*5(f) 29(f) 41(k) 52(r) 52(h) 33(‘) 26(h)

Z(5)

-16 14 16 16 5 5

0 0 o-71 0.65 1.29 0.9

= (a) SWANSONH. and J?UYATR. NBS Circular 539 II, 6 (1953). (b) SWANSONH. and TATCB E. NBS Circular 539 I, 19 (1953). (c) MCSKIMIN H., BONDW., PEARSONG., and HROSTOWSKIH. Bull. Aw. Phys. Sot. II 1,111(1956). (d) SWANSONH., FUYAT R., and UCRINIC G. NBS Circular 539 IV, 73 (1955). (e) Calculated from equation (2a). (f) COLLINS R. J. and FAN H. Y. Phys. Rtw. 93, 676 (1954). (g) SPITZER W. and FAN H. Phys. Rm 99, 1893 (1955). (h) YOSHINADAH. Plays. Rev. 100,753 (1955). (i) PARADIM. C.R. Acad. Sci., Paris 205, 1224 (1937). (j) Calculated from equation (1). (R) See ref. 9. APPLICATION

TO MIXED IONIC AND COVALEBONDS

It is assumed here that the case for C,/C,, = 1 corresponds to the completely ionic zinc blende lattice. This does not appear unreasonable from the characteristics of the compounds in Table 1. This assumption further implies that any stable sphalerite form must have some covalent bonding. Equation (2) based on ionic overlap potentials is now written as

6~1~= 4Cllai-Jm.

The upper curve, Fig. 1, shows a scale based on the assumptions of this section using equations

PI

Thus the relationship between the modulus C, and the longitudinal optical mode has nearly the identical form for the extreme covalent bond (equation 5) and the completely ionic bond (equation 2a). The crystals of interest, of course, are those with mixed ionic and covalent bonding. The above conclusions would indicate that a reasonable estimate for wz can be made for all zinc blende lattices by using equation (5). With wz, the reststrahlen frequency w,,, the high frequency dielectric constant, the reduced cell mass, and the cell volume, one may estimate the effective charge of the lattice from equation (1). In Table 2 are shown the results for the several zinc blende lattices for which data are available.

0

0.2

0.4

0.6

0.8

1.0

FIG. 1. The relationship between the ratio C,,/C, and the ionic charge of zinc blende lattices. The solid curve is based on equations (1) and (5) using data for A B C D E

Ge Si Insb InSb ZnS

MCSKIMIN MCSKIMIN MCSKIMIN et al. POTTER VOIGHT.

The broken curve is based on values for 2 determined from equation (3) with data from F InSb G ZnS

SPITZERand FAN SZIGET~.

226

ROY

F.

POTTER

Table 3. Data used for estimating elastic moduli of copper halide crystals

(a) SZICETIB. Proc. Roy. Sot. 204, 51 (1950). (6) PARADIM. C.R. Acad. Sci., Paris 205, 1224 (1937). (c) SWANSON H., FUYAT R., and UGRINIC G. NBS Circular 539 IV, 35 (1955).

(1) and (2a). Th’ IS curve is given approximately by Z/Z,, = [(1/1.65)(~r-265)]~, where u = C,,/C,,. When the elastic constants for GaSb are used with this curve the charge Zis estimated to be O-39. This value of 2 with the application of equations (1) and (2a) determines the reststrahlen wavelength to be h, = 37 p. PICUS(~) has recently determined the reststrahlen wavelength for GaSb to be 41~. The lower curve in Fig. 1 is based on values of Z/Z, based on the SPITZER and FAN estimate for AE < 1.5 in the case of InSb(l) and that AP = 3 for ZnS.(3) This gives an alternative scale for Z based on the elastic constants. The discrepancy between the two curves in the low Z/Z,, region may be due to the fact that equation (2a) is not rigorously true. TWO METHODS

FOR ESTIMATING CONSTANTS

THE ELASTIC

(a) If the bulk modulus, high frequency constant, and reststrahlen frequency

dielectric are known

In the case of the copper halides, the bulk modulus, the dielectric constants, and the reststrahlen frequencies have been determined experimentally.(s) The present scheme for relating the elastic constants and the optical modes then allows an estimate for the elastic constants. It follows from equations (1) and (5) that

In Table 3 are shown the resulting Cu and C,, for the copper halides. The values for C, were arbitrarily taken as one-half the values of Cu. This seems to be a reasonable procedure suggested by experimental values in Table 1. The coefficient of linear expansion of CuCl and CuBr have been measured by X-ray techniques.(lOJ1) The fact that two phase transformations are observed at elevated temperatures indicates the relative instability of these compounds. ‘The crystals exhibit transitions from the sphalerite form to the hexagonal wurtzite form and back to a final form with cubic symmetry. (b) If the characteristic Debye temperature is known Early in the century there were derivations for isotropic materials relating 0, to the bulk modulus K.(i2) However, a general expression for lattices with cubic symmetry is given by(13)

0 = (2h/k)(9/Y~~)1'3(l/ao)[(C11-C44)/p11'2.

Several schemes for evaluating Yn, which is a function of the elastic constants, have been developed. The method of QUIMBY and SUTTON, which has been simplified in application by SUTTON,(~@ seems to be the most facile. With the assumption that C,,jC,, = 2, then equation (8) becomes, for the case of zinc blende structure,

1.285 x 103Z2(
00 = (h/k)(9/ YR47T)1/3(2C,l/pao2)1/2

TZa03cd02

=

W/(a+2)1~4ao_ 1 WO%i

(8)

(8’)

and (7)

The values of Z/Z, and a = C,,/C,, which satisfy equation (7) are determined from the curve of Fig. 1.

3.0 < Yn < 8 0.7 > Yl+J3

> 0.5.

The lower limit of Yn is evaluated

(9) from the

THE

IONIC

CHARACTER

AND

ELASTIC

constants for Ge and Si while the upper limit is based on the estimate for the copper halides. The elastic constants for a Sn (gray tin) have been worked out from the 0n (0, = 260’K) given by HILL and PARKINSON.@~)The values for p and a, are the same as those for InSb. The elastic constants evaluated from equation (8’) are: C,, = 9.6, C,, = C,,/2*7 = 3.5, and C, = 4.8, all in units of 101r dynes/cm2. From equation (5) the optical mode wavelength for a Sn is given as X = 38 p.

This hybrid method for determining the relationship between the optical modes of the zinc blende lattice and the elastic constants is not as satisfactory as a more sophisticated approach might be. It does, however, offer a better approximation than existing theories in their application to the Table 4. ca) Electronegativities

=

Element

Compound

_-

X

(X.4 -XB)

I.5 2.5

Zn S

1 .o

ZnS 1.5 I.8

In Sb InSb

0.3 I.5 I.8

Ga Sb GaSb

0.3 I.8 3.0 2.8 2.5 I.8

cu Cl Br I A8

OF ZINC

BLENDE

LATTICES

227

and the reststrahlen frequency if the elastic moduli are known. Conversely, the elastic moduli and ionic character can be determined under this scheme if the bulk modulus and reststrahlen frequency are known. Examples of both applications have been given. The ionic character estimated by this method can be compared to that given from the electronegativity scale of PAULING.~‘) GORDY and THOMA@) have published a list of electronegativities (X) for the elements. The electronegativities of the elements of particular interest are shown in Table 4. It can be seen that the ionic character of a molecule AB (a monotonic function of (XA -XB) generally follows in the order shown in the last column of Table 1. The exceptions are that in Table 4 GaSb and InSb have the same electronegativity character as do ZnS and CuBr The scheme outlined here predicts a difference. PAULIN&‘) shows a scale of ionic character based on the ionic character and electronegativity of hydrogen halide molecules. The values for Z/Z, based on his scale are lower than those shown in Fig. 1 except in the case of GaSb. GOODMAN has discussed the electronegativity scale of these semiconducting compounds and the ionic character to be expected. The scheme outlined above for determining the optical modes of zinc blende type crystals from a knowledge of elastic moduli undoubtedly has serious shortcomings; nevertheless, in view of the present state of dynamic lattice theory it may provide a rough tool where none existed. REFERENCES

CuCl CuBr GUI AgI

-

MODULI

I.2 I.0 0.7 0.7

=

(a) GORDY W. and THOMAS W. J. 0. r. Chem. Phys. 24, 439 (1956).

intermetallic compounds. The entire structure rests on three assumptions: (1) that equation (5) holds, (2) that the 100 per cent ionic zinc blende lattice is non-existent because C,, = C’i2, and (3) that the theory for polar lattices holds. It does provide an estimate for ionic character

1. SPITZER W. G. and FAN H. Y. Phys. Rev. 99, 1893 (1955). 2. FR~HLICH H. and Mow N. F. Proc. Roy. Sot. A 171, 496 (1939). 3. SZIGETI B. Proc. Roy. Sot. A204, 51 (1950). 4. BORN M. and HUANG K. Dynamical Theory of Crystal Lattices. Oxford University Press, Oxford (1954). 5. LYDDANE R. H., SACHS R. G. and TELLER E. Phys. Rw. 59, 673 (1941). 6. BORN M. Ann. Physik 44, 605 (1914). 7. SMITH H. M. J. Trans. Roy. Sot. 241, 105 (1948). 8. See reference 4, p. 142. 9. Prcus G. Private correspondence. 10. MIYAKE S., HOSHINA S., and TAKENAKAT. J. Phys. Sot. Japan 7, 19 (1952).

228

ROY

F. POTTER

11. HOSHINA S. J. Phys. Sot. Japan 7, 560 (1952). 12. BLACKMAN M. Crystal Physics I Encyclopedia of Physics p. 376. Springer-Verlag, Berlin (1955). 13. LEIBFRIED G. Ibid. p. 247 ff. 14. QUIMBY S. L. and SUTTONP. M. Phys. Rev. 91,1122

16. HILL R. W. and PARKINSOND. H. Phil. Mug. 43,309 (1952). 17. PAULING L. Nature of the Chemical Bond. Cornell

(1953). 15. SUTTONP. M. Phys. Rev. 99, 1826 (1955).

439 (1956). 19. GOODMANC. H L. Proc. Phys. Sot. B 67,258 (1954).

University Press, Ithaca (1948). 18. GORDY W. and THOMASW. J. 0. r. Chews. Phys. 24,