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Lattice dynamics of CuBr and CuI by a modified rigid ion model
Solid State Communications, Vol. 15, pp. 1667—1669, 1974.
Pergamon Press.
Printed in Great Britain
LATTICE DYNAMICS OF CuBr AND Cu! BY A MODIFIED RIGID ION MODEL B.P. Pandey and B. Dayal Physics Department, Institute of Advanced Studies, Meerut University, Meerut 250001, India (Received 12 July 1974 byA.R. Verma)
Phonon dispersion relations of CuBr and Cu! have been studied on the basis of a modified rigid ion model in which a non-central force system is used for the short-range interactions. The results of the calculations show a satisfactory agreement with the experimental data.
liiiI
ZINC BLENDE structure solids form an interesting class of partially ionic semiconductors. Lattice dynamical studies of these solids have been greatly
Laai
=
(4a~+ 8a2 + l4Ui + 1402) (‘Ia 2 + Oi)C2a(Ci(3 + C27) —
facilitated due to the advent of the recent neutronscattermg techmques which are capable of giving the exact values of the phonon frequencies. Phonon dispersion relations for CuBr and CuI have been j 1 and Hennion reported quite recently by Prevot et al. et al.2 These workers fitted their measured dispersion
+(401+2c72)C2(3C2 1111 [ai3j 1121 = (4a2 + 0i)S2aS2p + 2(a~+ 02)S21(C2a C2ii) = (4a 1 + 1 6a~+ 1 602)(CaC(3C7 iSaS0S7) a [121 = —8 ~ S C (~C S —
—
—
curves with some rigid ion model but have not reported details of the model used. In the present paper, we report the theoretical calculations of the dispersion curves of CuBr and Cu! by a modified rigid ion model containing five force-parameters.
—
La~3i
~
1
~1)’~ a (3 7
a (3 ~
1221
L~]
=
(4a~+ 8a3 + l4u~+ 1402) (4O~+ Oi)C2a(C2(3 + C21) —
The total interaction is supposed to consist of the two parts (i) the short-range interactions and the (ii) long range interactions between the ions. The short range term is supposed to comprise of the central as 3 type effective welltoassecond the angular force ofThe thelong-range C.G.W. interactions up neighbours. are the well known coulombic forces. We have used
+ (4o~+ 2a2)C2(3C27 1221
L ~]
= (4a3 + Ol)S2aS2(3 + 2(a~ + (72)S27(C2a C2~). Other elements can be obtained by the cyclic permutation of the indices. ai, a 2 and a3 define the central force constants between Cu—Br, Br—Br ions respectively while a~and 02 Cu—Cu are the and angular force constants. The notations of our earlier paper6 have been adopted.
the coupling coefficients of the long range 4 for ZnS structure. Theterm shortderived by Bergsma range coupling coefficients are derived exactly in the same way as described by Pandey and Dayal.5’6 The typical elements of the dynamical matrix are given below:
—
The dynamical matrix contains six parameters including the effective charge Z. There have been various methods used7’8 for deriving the parameter Z in the lattice dynamical studies. It is found that there 1667
1668
LATTICE DYNAMICS OF CuBr AND Cul
Vol. 15, No. 10
Table 1. The force constants and the input data for CuBr and Cul CuBr Ref. Force constants
Input data vLo(F) i’LO(X) ~LA(X)
= = =
4.62 THz 4.59 THz 3.69 THz
1 1 1
Input data VL0(F) = 4.80 THz = 4.65 THz = 3.72 THz
2 2 2
a~=
dynX cm~2 u~= 0.950 X iO~dyn/cm C12= 3.07 lO~ 02=—l.451 X lO4dyn/cm a=6.0427A Z = 0.692
13 14 9
01=
a~= 02 =
03=
3.077 X lO4dyn/cm —0.153 X lO4dyn/cm —0.051 X lO4dyn/cm
CuI Ref. Force constants
~LO(X) VLA(X)
C 12=3.44X lOll dyn cm2 a=5.6905A Z = 0.735
13 14 9
02 = 03=
1.915 X lO4dyn/cm 0.234 X lO4dyn/cm 0.078 X lO4dyn/cm
4dyn/cm 0.450X lO u 4dyn/cm 2=—0.680X lO
oL 50 4.0
4.~
3.0
~
_~__
oL oT~
oo
ao ~
I
~ 20 1.0~v~ 0
__ 0
d8 X
A
c~z~o.oc~ ct4 r A L
[100]
0.8 X
[iii]
oh
A [100]
0.0 E
0.2 Q4 A L [111]
FIG.!.
The calculated dispersion curves for CuBr. The experimental points are those of Prevot et al.1
FIG. 2. The dispersion curves for Cul. The experimental points are those of Hennion et a!.2
is no unique value of Z which has been determined from the above fitting procedures. It is possible to get approximately the same dispersion curves by using a wide range of the values of Z. Very recently, Phillips9 has calculated the fractional ionicity of the ZnS type compounds from the fundamental principles. We have used accordingly, Phillips’ values ofZ in the present calculations. The remaining
m2w~A(X)= ~
parametersby have been determined fromused physical quantities following the procedure by Banerjee and VarshnL10 The linear expressions for the zone centre optic mode, the elastic constant C 12 and the longitudinal zone boundary modes for [001] direction have been used for this purpose. The relevant expression for them are pw~(F)= 4a~+
16a~+ 1602 +
3
~2
V
+ 1603 + 20o~+ 1602 + 4.3366z2
mi~,Lo(x)zr~ 4a~+
1602 +
20u~+
aC
1602 +
4.3366z2 e2
2 12 = ~ + 202 + 203— 3o~ 202— 2.648z V where m 1 and m2 are the masses of the ions and 1/ji3/4 = (1/rn1 + I/rn2).ofethe is the and V = a is the volume unitelectronic cell. Somecharge workers11’12 have taken the second neighbour force constants 02 ~—
—
and 03 to be equal for both the ions which is physically unjustified. We have arrived at the value 02/03 = 3 in our calculations which is more reasonable. The values of the force parameters and the input data to derive them are listed in Table 1.
Vol. 15, No. 10
LATTICE DYNAMICS OF CuBr AND Cu!
The calculated dispersion curves for CuBr and Cu! are shown in Figs. 1 and 2, respectively along with the experimental points. It is observed that the agreements between the theoretical and the experimental curves are quite good. The general agreements for the acoustic modes are better than for the optic modes. The experimental data for optic modes are quite sparse and there are relatively larger uncertainties in their measurements. Therefore, any rigorous comparison is not possible. There are some minor deviations in the agreements of the acoustic modes. According to Martin,’5 the numerical values of the phonon frequencies of the acoustic modes are very
I.
1669
small which indicates a large amount of cancellation among the forces involved. Therefore a~yapproximations present in the model are liable to give exaggerated effects in the acoustic modes. The agreements may be taken as satisfactory in view of the simplicity of the model and small number of parameters used. Acknowledgements Authors are highly grateful to Dr. B. Prevot for supplying his measured phonon frequencies and to Prof. B.B. Srivastava for providing the facilities of work. One of us (B.P.P.) acknowledges the financial support of C.S.I.R. Govt. of India. —
REFERENCES PREVOT B., CARABATOS C., SCHAWB C., HENNION B. and MOUSSA F., Solid State Commun. 13, 1725 (1973).
2. 3.
HENNION B., MOUSSA F., PREVOT B., CARABATOS C. and SCHAWB C.,Phys. Rev. Lett. 28, 964 (1972). CLARK B.C., GAZIS D.C. and WALLIS R.F., Phys. Rev. 134, 1486 (1964).
4. S.
BERGSMA J., Reactor Centre Netherlands Report, RCN- 121(1970). PANDEY B.P. and DAYAL B., Indian J. Pure and Appl. Phys. 10, 777 (1972).
6.
PANDEY B.P. and DAYAL B., Solid State Commun. 11, 775 (1972).
7. 8.
KAPLAN H. and SULLIVAN J.J.,Phys. Rev. 130, 120 (1963). KUNC K., BALKANSKI M. and NUSIMOVICI M., Phys. Status Solidi 41, 491 (1970).
9.
PHILLIPS J.C., Rev. Mod. Phys. 42, 317 (1970).
10. 11.
BANERJEE R. and VARSHNI Y.P., Can. J. Phys. 47,451(1969). RAJGOPAL A.K. and SRINIVASAN R., Z. Phys. 158, 471 (1960).
12. 13.
VETELINO J.F., MITRA S.S. and BRAFMAN O.,Solid State Commun. 7, 1809 (1969). HANSON R.C., TAYLOR J. and SCHAWB C., Bull. Am. Phys. Soc. 17, 144 (1972).
14.
WYCKOFF R.W.G., Crystal Structures Vol. 1, p.1 10, Interscience, New York.
15.
MARTIN R.M.,Phys. Rev. 186, 871 (1969).
Pergamon Press.
Printed in Great Britain
LATTICE DYNAMICS OF CuBr AND Cu! BY A MODIFIED RIGID ION MODEL B.P. Pandey and B. Dayal Physics Department, Institute of Advanced Studies, Meerut University, Meerut 250001, India (Received 12 July 1974 byA.R. Verma)
Phonon dispersion relations of CuBr and Cu! have been studied on the basis of a modified rigid ion model in which a non-central force system is used for the short-range interactions. The results of the calculations show a satisfactory agreement with the experimental data.
liiiI
ZINC BLENDE structure solids form an interesting class of partially ionic semiconductors. Lattice dynamical studies of these solids have been greatly
Laai
=
(4a~+ 8a2 + l4Ui + 1402) (‘Ia 2 + Oi)C2a(Ci(3 + C27) —
facilitated due to the advent of the recent neutronscattermg techmques which are capable of giving the exact values of the phonon frequencies. Phonon dispersion relations for CuBr and CuI have been j 1 and Hennion reported quite recently by Prevot et al. et al.2 These workers fitted their measured dispersion
+(401+2c72)C2(3C2 1111 [ai3j 1121 = (4a2 + 0i)S2aS2p + 2(a~+ 02)S21(C2a C2ii) = (4a 1 + 1 6a~+ 1 602)(CaC(3C7 iSaS0S7) a [121 = —8 ~ S C (~C S —
—
—
curves with some rigid ion model but have not reported details of the model used. In the present paper, we report the theoretical calculations of the dispersion curves of CuBr and Cu! by a modified rigid ion model containing five force-parameters.
—
La~3i
~
1
~1)’~ a (3 7
a (3 ~
1221
L~]
=
(4a~+ 8a3 + l4u~+ 1402) (4O~+ Oi)C2a(C2(3 + C21) —
The total interaction is supposed to consist of the two parts (i) the short-range interactions and the (ii) long range interactions between the ions. The short range term is supposed to comprise of the central as 3 type effective welltoassecond the angular force ofThe thelong-range C.G.W. interactions up neighbours. are the well known coulombic forces. We have used
+ (4o~+ 2a2)C2(3C27 1221
L ~]
= (4a3 + Ol)S2aS2(3 + 2(a~ + (72)S27(C2a C2~). Other elements can be obtained by the cyclic permutation of the indices. ai, a 2 and a3 define the central force constants between Cu—Br, Br—Br ions respectively while a~and 02 Cu—Cu are the and angular force constants. The notations of our earlier paper6 have been adopted.
the coupling coefficients of the long range 4 for ZnS structure. Theterm shortderived by Bergsma range coupling coefficients are derived exactly in the same way as described by Pandey and Dayal.5’6 The typical elements of the dynamical matrix are given below:
—
The dynamical matrix contains six parameters including the effective charge Z. There have been various methods used7’8 for deriving the parameter Z in the lattice dynamical studies. It is found that there 1667
1668
LATTICE DYNAMICS OF CuBr AND Cul
Vol. 15, No. 10
Table 1. The force constants and the input data for CuBr and Cul CuBr Ref. Force constants
Input data vLo(F) i’LO(X) ~LA(X)
= = =
4.62 THz 4.59 THz 3.69 THz
1 1 1
Input data VL0(F) = 4.80 THz = 4.65 THz = 3.72 THz
2 2 2
a~=
dynX cm~2 u~= 0.950 X iO~dyn/cm C12= 3.07 lO~ 02=—l.451 X lO4dyn/cm a=6.0427A Z = 0.692
13 14 9
01=
a~= 02 =
03=
3.077 X lO4dyn/cm —0.153 X lO4dyn/cm —0.051 X lO4dyn/cm
CuI Ref. Force constants
~LO(X) VLA(X)
C 12=3.44X lOll dyn cm2 a=5.6905A Z = 0.735
13 14 9
02 = 03=
1.915 X lO4dyn/cm 0.234 X lO4dyn/cm 0.078 X lO4dyn/cm
4dyn/cm 0.450X lO u 4dyn/cm 2=—0.680X lO
oL 50 4.0
4.~
3.0
~
_~__
oL oT~
oo
ao ~
I
~ 20 1.0~v~ 0
__ 0
d8 X
A
c~z~o.oc~ ct4 r A L
[100]
0.8 X
[iii]
oh
A [100]
0.0 E
0.2 Q4 A L [111]
FIG.!.
The calculated dispersion curves for CuBr. The experimental points are those of Prevot et al.1
FIG. 2. The dispersion curves for Cul. The experimental points are those of Hennion et a!.2
is no unique value of Z which has been determined from the above fitting procedures. It is possible to get approximately the same dispersion curves by using a wide range of the values of Z. Very recently, Phillips9 has calculated the fractional ionicity of the ZnS type compounds from the fundamental principles. We have used accordingly, Phillips’ values ofZ in the present calculations. The remaining
m2w~A(X)= ~
parametersby have been determined fromused physical quantities following the procedure by Banerjee and VarshnL10 The linear expressions for the zone centre optic mode, the elastic constant C 12 and the longitudinal zone boundary modes for [001] direction have been used for this purpose. The relevant expression for them are pw~(F)= 4a~+
16a~+ 1602 +
3
~2
V
+ 1603 + 20o~+ 1602 + 4.3366z2
mi~,Lo(x)zr~ 4a~+
1602 +
20u~+
aC
1602 +
4.3366z2 e2
2 12 = ~ + 202 + 203— 3o~ 202— 2.648z V where m 1 and m2 are the masses of the ions and 1/ji3/4 = (1/rn1 + I/rn2).ofethe is the and V = a is the volume unitelectronic cell. Somecharge workers11’12 have taken the second neighbour force constants 02 ~—
—
and 03 to be equal for both the ions which is physically unjustified. We have arrived at the value 02/03 = 3 in our calculations which is more reasonable. The values of the force parameters and the input data to derive them are listed in Table 1.
Vol. 15, No. 10
LATTICE DYNAMICS OF CuBr AND Cu!
The calculated dispersion curves for CuBr and Cu! are shown in Figs. 1 and 2, respectively along with the experimental points. It is observed that the agreements between the theoretical and the experimental curves are quite good. The general agreements for the acoustic modes are better than for the optic modes. The experimental data for optic modes are quite sparse and there are relatively larger uncertainties in their measurements. Therefore, any rigorous comparison is not possible. There are some minor deviations in the agreements of the acoustic modes. According to Martin,’5 the numerical values of the phonon frequencies of the acoustic modes are very
I.
1669
small which indicates a large amount of cancellation among the forces involved. Therefore a~yapproximations present in the model are liable to give exaggerated effects in the acoustic modes. The agreements may be taken as satisfactory in view of the simplicity of the model and small number of parameters used. Acknowledgements Authors are highly grateful to Dr. B. Prevot for supplying his measured phonon frequencies and to Prof. B.B. Srivastava for providing the facilities of work. One of us (B.P.P.) acknowledges the financial support of C.S.I.R. Govt. of India. —
REFERENCES PREVOT B., CARABATOS C., SCHAWB C., HENNION B. and MOUSSA F., Solid State Commun. 13, 1725 (1973).
2. 3.
HENNION B., MOUSSA F., PREVOT B., CARABATOS C. and SCHAWB C.,Phys. Rev. Lett. 28, 964 (1972). CLARK B.C., GAZIS D.C. and WALLIS R.F., Phys. Rev. 134, 1486 (1964).
4. S.
BERGSMA J., Reactor Centre Netherlands Report, RCN- 121(1970). PANDEY B.P. and DAYAL B., Indian J. Pure and Appl. Phys. 10, 777 (1972).
6.
PANDEY B.P. and DAYAL B., Solid State Commun. 11, 775 (1972).
7. 8.
KAPLAN H. and SULLIVAN J.J.,Phys. Rev. 130, 120 (1963). KUNC K., BALKANSKI M. and NUSIMOVICI M., Phys. Status Solidi 41, 491 (1970).
9.
PHILLIPS J.C., Rev. Mod. Phys. 42, 317 (1970).
10. 11.
BANERJEE R. and VARSHNI Y.P., Can. J. Phys. 47,451(1969). RAJGOPAL A.K. and SRINIVASAN R., Z. Phys. 158, 471 (1960).
12. 13.
VETELINO J.F., MITRA S.S. and BRAFMAN O.,Solid State Commun. 7, 1809 (1969). HANSON R.C., TAYLOR J. and SCHAWB C., Bull. Am. Phys. Soc. 17, 144 (1972).
14.
WYCKOFF R.W.G., Crystal Structures Vol. 1, p.1 10, Interscience, New York.
15.
MARTIN R.M.,Phys. Rev. 186, 871 (1969).