Vibrational modes and phase transition of Si-Ge solid solution

Solid State Communications, Vol. 50, No. 11, pp. 1007-1010, 1984. Printed in Great Britain.

0038/1098-84 $3.00 + .00 Pergamon Press Ltd.

VIBRATIONAL MODES AND PHASE TRANSITION OF Si--Ge SOLID SOLUTION T. Soma, Y. Kitani and H.-Matsuo Kagaya Department of Applied Physics and Mathematics, Mining College, Akita University, Akita 010, Japan (Received 5 September 1983 by C. W. McCombie) We present the simplified treatment where the lattice vibrations of Si or Ge atoms in the Si---Ge solid solution are replaced with that of pure Si or Ge crystal at lattice constants of the alloy. Considering the volume effect on the force constants of the pure constituent, we obtain the phonon dispersion curves of the local and band modes for Si0.91Geo.o9 and Si0.~l Geo.a9 systems and the concentration x-dependence of the local and band modes frequencies in the Sil-xGex solid solutions. Then, from the calculation of the effective mode GrOneisen parameter ~i for the average phonon modes in the Sil_~Ge~ systems, we obtain the predominant correlation between TA mode Griineisen parameter 3'~A at the point X and the phase transition pressure Pt, and the softening of TA modes is related to the pressure-induced phase transition of the Si-Ge solid solution. 1. INTRODUCTION THEORETICAL STUDIES about the lattice dynamics of the alloy system have been devoted to Rb~_xKx systems, because both constituents were alkali metals and the lattice dynamics of the pure constituent has been investigated in detail. Roughly classifying, there are two theoretical treatments. One is the coherentpotential approximation, which is a mean-field type of approach employed to describe electronic density of states in random alloys. Along this line, some works [ 1--4] considering a mass defect and the differences of force constants etc. have been reported, and neutron scattering cross section for various momentum transfers have been calculated in comparison with experimental data. The other is the computer-simulation calculations [5, 6] in molecular dynamics, which consists of many particles involving randomly chosen solute atoms. The latter obtains the dynamical structure factor, but the effort in computer-simulation procedures is enormous. Previously, Wakabayashi et al. [7, 8] measured impurity phonon modes of Sio.o92Geo.9oasystem by neutron scattering investigation. Then, Lannion [9] obtained first- and second-order Raman spectra of Sil-xGex systems with x = 0.09,0.23, 0.34, 0.54, 0.65, 0.77, 0.89 and 0.935. Recently, Shen and Cardona [10] reported local and impurity-induced band modes of Sio.HGe0.s9 and Sio.16Geo.a4 systems from far infrared absorption spectra. Theoretically, Agrawal [11 ] calculated phonon density of states in crystalline and amorphous Si-Ge alloys using a five-atom cluster Bethe lattice method treating the short-range order. There were no theoretical works to estimate the local and band mode frequencies and the thermal properties

of the Si-Ge solid solution in the electronic theory of solids from first principle. In the present work, we are not bound to the rigid description of the lattice vibrations for the alloy system and present a simplified treatment to estimate quantitatively the phonon mode frequencies of the Si-Ge system. 2. LATTICE DYNAMICS OF Si-Ge SOLID SOLUTION The Sil-xGex system forms the substitutional solid solution over all region of x, and the crystal binding of the solid solution is unchanged compared with that of pure Si or Ge. The difference in atomic sizes and lattice constants of Si-Ge is smaller than that of R b - K system. Recently, we [12] have presented the electronic theory of Si-Ge solid solutions using the virtual crystal approximation and the pseudo alloy atom model, and calculated the static properties such as the bulk modulus [ 13], phase diagram [ 14], equation of state [ 15 ], and pressure4nduced phase transition [ 16]. The bulk properties of the heat of solution, phase diagram and the bulk modulus were reproduced, and the idea of the average atomic potential was suitable for Sil-xGex solid solution. Then, the remarkable characteristics such as equation of state [15] and the phase transition under pressure [16] of Si-Ge systems were obtained by considering the volume effect on the atomic pseudopotential of Si and Ge. In dynamical treatment, we can not treat the lattice dynamics of the hypothetical pseudo alloy atom directly from the idea of the average atomic potential, because the constituent Si or Ge atom thermally vibrates. Actually, it is well-known that local and band vibrational modes corresponding to the lattice vibration

1007

1008

Vol. 50, No. 11

VIBRATIONAL MODES OF Si-Ge SOLID SOLUTION

of the solute and solvent atoms are observed. But, the lattice site of the solute and solvent atoms in the substitutional solid solution is not determined and the theoretical treatment of the lattice dynamics has many difficulties. Some works [1--4] stated in Section 1 have a future subject in calculating the macroscopic thermal properties obtained by summing up the contributions from the individual vibrational modes over/-branches and q-space. We introduce the drastic approximation as follows. When Siz-xGex solid solution is formed, Si atoms in the solid solution are in a state of volume expansion compared with those in pure Si, and Ge atoms in a state of volume compression compared with those in pure Ge. Because the difference in atomic sizes of Si-Ge is small, we consider apparently the lattice vibration of Si or Ge atoms in the solid solution as that in pure Si or Ge crystal at lattice constants of the solid solution. The calculated results of thus simplified treatment have been already reported for local and band mode frequencies of Rbl-xKx alloy (for example, see [6]), and did not deviate fatally from those by the coherent potential approximation and by the molecular dynamics simulation.

I" [tOO]

[ I I0 ] Sio,,Geoo, [

T

-z

,o

;..,_-~./ LA L~ / T •

I 7., / LA,

I

.__._'T~,"..

/i///

0.4

TA2--'TW"~',\ \

0.8

0.8

0.4

Reduced

(*)

,,

0

Wove

4. MODE GRONEISEN PARAMETER AND PRESSURE-INDUCED PHASE TRANSITION The mode Griineisen parameter 7i(q) is the measure for the volume I2-dependence o f i t h mode phonon

0.2

0.4

Number

X

Sio.,,Geo.,,

[100] LO

TOI

TO~-"-.,-.- . . . . . . /

._

LA// LO /

I0

[111]

L 1101

TO

....

~o-~.~ ; ' ~ ' "

LOs .# LA '~%, *~ TOI

LAI/t

TO /

TAZ~

"% \

0.8

0.4

/

3. LOCAL AND BAND MODE FREQUENCIES Previously, we [17] have calculated phonon dispersion curves of pure Si and Ge crystal using the higherorder perturbation and the local Heine-Abarenkov model potential. We consider the force constants up to sixth neighbours in Herman's formulation [18] and the volume effect on the force constants due to the alloying. The obtained phonon curves for local and band modes of Sio.91Ge0.o9 and Sio.ll Geo.s9 solid solutions are shown in Fig. l(a) and (b). In Fig. 1 and what follows, the results with Hubbard's dielectric function are given, because the best agreement with the observed data was obtained [17] for pure Si and Ge crystal. Then, we show the calculated results for the concentration x-dependence of the local and band mode frequencies in the Sil_xGe= solid solutions in Figs. 2(a), (b) and (c), representatively at F, X and L points. Although the concentration x-dependence of the local and band mode frequencies is weak, the volume effect on the force constants in the solid solution is appreciable. The obtained data in Figs. 1 and 2 are important in further research of the lattice dynamics of the Sil-x Gex solid solution.

L [fill

,'Y? 0.4

0.8

(b)

Reduced

Wove

0

0.2

0.4

Number

Fig. 1. Calculated phonon curves for the local (dashed) and band mode (solid curves) of (a) Sio.91Geo.o9 and (b) Sio.ll Geo.a9 solid solutions. The points are the observed data [10].

frequency pi(q) and defined by "ri(q) -

d [In vi(q)] d(ln ~2)

(1)

Considering the volume effect on the force constants, we obtain the corresponding mode Griineisen parameter 7}1)(q) and 3,}2)(q) for the local band modes, and show the calculated results for the concentration x-dependence of the local and band TA mode Gr0neisen parameter 7XA at the point X for the Sil-xGex solid solutions in Fig. 3, representatively. Other mode Griineisen parameters 3'To, 3%0 and 7LA for the local and band modes are also monotonous functions of the concentration x. Recently, we [16] have studied the pressureinduced covalent-metallic phase transition of the Sil-xGex solid solution using the electronic theory based

Vol. 50, No. 11

VIBRATIONAL MODES OF Si-Ge SOLID SOLUTION

1009

1.0 15

t

I0~- . . . . . . . . . . . . . . . . .

N

"I" I--

0.0

-!.0

i

0

i

i

0.2

i

i

0.4

(Si)

0:6

'

o18

/ S

2.0

I

(Ge)

X

/f

(o}

/

/ f

d

s

s i

0

0.2

0.4

(SI) TO

............

i

0.6

X

0.8

ID

"

(Gel

Fig. 3. The concentration x-dependence of the local (dashed) and band mode (solid curves) Grtineisen parameter TrxA in the Sil-xGex solid solutions.

¸-O.8

TO

A N

LO= LA

3: I--

"rA___

r

. . . . . . . . . . . . . . .

5 TA

o

i

I

0.2

I

i

0.4

(si)

i

i

0.6

i

i

0.8

X ~

(Ge)

(b)

-I.2 15 ~

\

TO

"~----'---'~-"~L°~~ZZZTZ::::__._==~,.,., 1(3 N

Z I'-

...........

TO

........

LO

IOC

-I.4

LA i

0 (Si)

TA

.i

02 Si)

i'

i

i

i

0.4

0.6

0.8

X

1.0 (Ge)

Fig. 4. The correlation between the effective mode Griineisen parameter 7~rA and the phase transition pressure Pt [16] as function o f x .

TA

h

0.2

i

i

0.4

i

0.6 X

~

i

i

i

0.8 (Ge)

(c)

Fig. 2. The concentration x-dependence of the local (dashed) and band mode (solid curves) frequencies in the Sil-xGex solid solutions. (a) D, (b) X and (c) L. on pseudopotentials. The calculated data for the transition pressure Pt had a maximum near the atomic

fraction x = 0.55, and then, this prediction was experimentally observed by Werner et al. [19]. Previously, Weinstein [20] has proposed the correlation between 3~rA and Pt for the tetrahedrally-bonded covalent crystals. In order to demonstrate Weinstein's correlation, we define the effective mode Griineisen parameter 7i(q) for the average phonon modes in the Si~-xGex solid solutions given by

1010

VIBRATIONAL MODES OF Si~Ge SOLID SOLUTION

7i(q) = (I --x). v/~-7}l)(q) v~1)(q) + x v!2)(q) v/--~-~q) 7!2)(q), (2)

4. 5.

where 6. vi(q) = (1 --x) v~l)(q) + xv~Z)(q).

(3)

We plot both of T~rAobtained in the equation (2) and Pt in our work [16] as function of the atomic fraction x in Fig. 4. From Fig. 4, we see that the good correspondence between 3~rA and Pt is obtained, and that the softening of TA modes is related to the pressure-induced phase transition of the Sil-xGex solid solutions.

REFERENCES 1.

2. 3.

G. Griinewald & K. Scharnberg, Proc. Int. Conf. on Lattice Dynamics, Paris, September 1977 (Edited by M. Balkanski) p. 443. Flammarison Science. M. MostaUek & T. Kaplan, Phys. Rev. B16, 2350 (1977). W.A.Kamitakahara & J.R.D. Copley, Phys. Rev. BI8, 3772 (1978).

7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20.

Vol. 50, No. 11

G. Griinewald & R. Schopohl, J. Phys. F: Metal Phys. 9, 1047 (1979). G. Jacucci, M.L. Klein & R. Taylor, Solid State Commun. 24, 685 (1977). G. Jacucci, M.L. Klein & R. Taylor, Phys. Rev. BI8, 3782 (1978). N. Wakabayashi, R.M. Nicklow & H.G. Smith, Phys. Rev. B4, 2558 (1971). N. Wakabayashi, Phys. Rev. ii8, 6015 (1973). J.S. Lannin, Phys. Rev. BI6, 1510 (1977). S.C.Shen & M. Cardona, Solid State Commun. 36, 327 (1980). B.K.Agrawal, SolidState Commun. 37,271 (1981). T. Soma, Phys. Status Solidi (b) 95,427 (1979). T. Soma, Phys. Status Solidi (b) 95, Kl17 (1979). T. Soma, Phys. Status Solidi (b) 98,637 (1980). T. Soma, H. Iwanami & H. Matuso, Phys. Status Solidi (b) 110, 75 (1982). T. Soma, H. Iwanami & H. Matsuo, Solid State Commun. 42,469 (1982). T. Soma, Phys. Status Solidi (b) 87,345 (1978). F. Herman, J. Phys. Chem. Solids 8,405 (1959). A. Werner, J. Sanjurjo & M. Cardona, Solid State Commun. 44, 155 (1982). B.A. Weinstein, Solid State Commun. 24, 595 (1977).