On the phonon self-energy

Solid State Communications.

Vol. 13. Pp. 1211—1214, 1973.

Pergamon Press.

Printed in Great Britain

ON THE PHONON SELF-ENERGY F. Gervais Centre de Recherches sur la Physique des Hautes Temperatures, C.N.R.S., 45045 OriCans, France (Received 16 May 1973 by A.A. Maradudin)

It is reported that aspecifIc phonon—phonon interaction process described by the sixth-order Hamiltonian can contribute to the real part of the phonon self-energy of an order which involves a quadratic dependence of the resulting frequency shift on temperature, in the high temperature limit. The possible connection between the occurrence of such a process and experimental results in a-quartz is discussed.

THE SELF-ENERGY of phonon mode in crystals is most easily evaluated by using Green’s function method with the help of diagrammatic perturbation theory.1 On the basis of an anharmonic Hamiltonian inc1udin~ cubic and quartic terms. Maradudin and Fein1 and later

from infrared reflectivity analysis). The linear dielectric susceptibility is the analytic continuation of a sum of phonon propagators over all polar vibration modes at k~ 13 x = lim ~ M~~M~ 1~G(Ojj ,w±,e) (4) 1’ e—o Oh’ .

2

—

Wallis eta!, gave expressions for the phonon selfenergy of order Q(~2) and Q(774) respectively, where 77 lS a dimensionless classification operator. The anharmonic Hamiltonian ofa crystal may indeed be expanded as —

“A

—

u(3) 77i~

In the notation of Wallis er tonian ~ —

2gj(4)

j..

-t

77

~

A’J,~and M~1.are the usual first-order dipolar moment tensor components,13= 1/kBTand Vis the volume of the crystal.

11

The phonon propagator G is the Fourier coefficient of the one-phonon thermodynamic Green’s function. The elements of the phonon propagator matrix G(Oj/’, iwo) may generally be known by solving the

‘.‘

2

the nth-order Hamil.

V V ‘5~ k3j~ k2j~ k~j~

.

1—3

Dyson equation. G(Oj/’,iw8)

V~(k1j1.k2j2. k~j~)A(k1i1)A(k2j2) ...A(k.,j~) ...

(2)

=

g(O/.ic.~)S~1, +

g(O/,iw5)I P(Ofj1,iw~)G(Of1/’,iw8)

(5)

where the operators A(k~j1)are defined in term of the usual phonon creation and annihilation operators by 4(—kJ A(k1f1) = a 1)+a(k1j1) (3)

where w~ 2irs/j3h, s an integer. g(Oj.iw~)= {[~.,(Oj)+iw~]’+

Before expliciting a specific effect derived from the term ~ we shall briefly review some fundamental expressions and useful results. Results concerning the self-energy of a polardielecphonon mode are commonly obtained through tric susceptibility measurements (for instance deduced 1211

(6) is the free-phonon propagator, and P(Ojj1, iw8) is an element of the self-energy matrix. The Dyson 4equation of may be solved withinP(0 the good approximation assuming the matrix 111. iw~)as diagonal over the branch indices. One gets

1212

ON THE PHONON SELF-ENERGY G(O//’, iw~)= 2w(O/) (7) J31i

~2(Of) +

—

2w(Oj)P(Of, iw8)/i’3h Oj

k~ii i ”

Vol. 13, No. 8

Experimental results~7suggest that the frequency shift ~~.~(O/) derived from the contribution p}4) and ~.~ 2(Oj,magnitudes w)..~=~0J) have opposite signs and comparable so that they balance each 3’8 confirm these observations. other. Calculations

in Figs. 2(a) and 2(b). The corresponding contribuhigher 2 are order, diagrammatically that isare Q(~4),represented the tionsAttothe thenext self-energy respectively purely quartic terms P~8~(0j,iw~) = 96132 ~ E V~4~(0/,k 1/1,k2/2,k3j3)

k 117

~1’2

ra )

(b ~

k1i1 k312k3j3 S~S~S~

FIG. 1. Q(~2) contributions to the phonon self.energy.

X V~ ~(Of, —k1/1 ,—k2/2 ,—k313)g(k1 /1,iw~)g(k2/2 ,iw~) X g(k3/3, iw8) ~ (11) +S2~$3,0

The diagrammatic representations of the Q(772) lowest order contributions to the phonon self-energy are shown in Figs. 1(a) and 1(b). The corresponding analytic1 expressions through the usual set respectivelyobtained are of rules P~4>(0f,i~ 5) = —1213 ~ k1J

4~(0f, 0/,—k

(8)

P~6~(0j,i~

8) = 18132 V — k,111, s,s, 3~(0j. k k1j2) V~ 1j1 k1j2) k1j2, iw9 )g(k1j1, !~s )ó~+~ +s~.0 (9)

v~3~(of, k 1/1, X g(

—

—

—

,

The frequency shift ~w(0/, w) and the damping r(oi, theimaginary phonon mode correspond to the w) realofand parts respectively of the self-energy limP(0/,w±ie) = —13h{~o.(Oj)~iF(0j,w)}

(10) The summation ofg(k 1j1, iw3)over 2n 1 + I, where =

{ exp

[j3hw(k~j1)]

k1k3i,J2j~s1s2s~

XV ~ ~(k1/1,—k1j2 ,k3j3 ~ ,—k3j3)g(k1j1,iw~)g(—k1 /2(12) ,i(~.)83) X g(k~f~,

notice that, neglecting 89 the two-phonon contributions statestobounded the dampvia anharmonicity, 6)and ing derive from the imaginary part of terms p~ p~s)since other terms are real. Inspection of the right side of equations (11) and (12) shows that the product of three g functions and 6 restriction gives rise to a quadratic temperature dependence of terms 8~/13hand —F~8~/13h —P~ in the high temperature limit. Their order indeed is 0(n2).1° The temperature dependence of damping of a 12

S1

is equal to

—

is the mean number of phonons. Thus both relations (8) and (9) are of order 0(n) which give rise to a linear temperature in condition the high temperature liniit1 takingdependence the restrictive s 1 = S S2 into account in equation (9). Moreover it may be shown that the real part ~o.2(0/, w)wof±i)113h 6~(o/ tim P~ e-0 is negative in the vicinity of the normal mode frequency. —

—

P~~(Of, iw~) = 0/, —k 4~(o/, ~ v~ 1/1, k1j2)

Explicit results for the phonon self-energy are given in reference 2.We shall restrict ourselves to

Si

1j1,k1j1)g(k1/3,iw~)550

V~

~

144132

large number of phonon modes has been observed~~’ over a range including elevated temperatures, and the quartic interaction processes represented in Fig. 2(a) have been evidenced to occur in some cases. These results are obtained in situations in which the implicit temperature dependence of damping due13toremains lattice thermal expansion pointed out by Mooij negligible. Moreover, even if the observed dependences of damping on temperature are corrected to take thermal expansion relation effects into (in addition to the equivalence 3.14account of reference 14), it seems that four-phonon interactions contribute to limit the phonon lifetimes to a same order of magnitude

Vol. 13, No.8

ON THE PHONON SELF-ENERGY

1213

k

~2~2

2j~

4)contribution to the phonon self-energy. FIG.3. Additional Q(~

0;

Oj (a

~w

(b

2Iin the vicinity of the normal mode, may be FIG.2. Q(774) purely quartic contributions to the phonon self-energy. as three-phonon interactions, in alkali halide crystals’5 at intermediate and high temperatures. These observations confirm the results of Ipatova et a!.14 and that of Bruce8 obtained through ab initlo calculations, The purpose of this paper is then to point out thatorder another contribution to the phonon self-energy of 0(~~) is to be considered. This contribution arises from the sixth-order Hamiltonian and is represented by the diagram shown in Fig. 3. Its value is P~6~~0’~ = ~ ‘ ~‘ 1 V~6~(0/,0j,k I 1/1,—k1f1.k2J2.—k2/2) ~‘

72013

~ I

2 2

X g(k 1/1, iw5)g(k2/2, iw82)ó, 1.s2 .0 (13) The frequency-independent result is real and thus contributes to the frequency shift

characteristic of a soft-mode behavior. A limiting case of this feature is the possibility of stabilization 4 (0/), quartic as predicted first in of a paraelectnc phase by strong anharmonicity expressed P1~ 1963.16 Up to through date, experimental results have mainly been obtained in the vicinity of room temperature or lower. In the range of higher temperatures, 0(77k) terms can become important as observed for the imaginary part ofPP~.Whether or not ~o~(’0j) should counterbalance the real part ~ w w) of 8~+3(0/, \Ia~ ~ ,D~ e—.o depends on the sign of V~6~ potentials in equations (14). Calculations3’8 and experimental resuIts~’7 —

~

‘~ ~

—

indicate a positive sign for quartic same situation prevails for the all cases, thepotential. frequencyIf the 8~(0/)/j3hisnecessarily negative. Moreover potentials are second derivatives of shift ~w4 V~6~ = —P4 4>potentials and are thus expected to have the V~ same positive sign. The expectations may be compared to the observed

~w 1440

~

17

5(0j) V”~(0i.Of, k

behavior of the lowest-frequency A2 mode of a-quartz. The anharmonic frequency shift may be known by

=

1!1, —k1/1 k212. -

13

—

k2f2)

k,j1k2j2

+ + ~2+ ~2 I

x ~1~l

—

~‘-‘i —(-‘-~2

(14)

The frequency dependence of such a term is quadratic in the high-temperature limit, It is of interest to notice a same 6) correspondance (0/) and between ~ the± 0(77~) contributions p~ F~8~(0j, Ic), and the 0(772) contributions P 4~(0j) and P~”(0/,~ ±ic), keeping in mind that both 1last terms may have comparable magnitudes. Authors6’7 have suggested that the predominance of~w 1on

deducing the effect of thermal expansion from the observed shift. To the author’s knowledge mode 5 thatnot a rough estimation of Grtineisen parameters available for a-quartz. the frequency shift of are TO modesA It has been reported’ 2 andE due to thermal expansion, based on the Gruneisen’s approximation reveals positive anharmonic frequency shifts for most of the modes. Moreover the increase in most of the frequencies with increasing temperature seems to obey a of lawthe more than linearofparticularly at the approach temperature the a 13 phase transition (845 K). —~

Consequently, of the contributions to the selfenergy considered here and under the hypothesis

1214

ON THE PHONON SELF-ENERGY

Vol. 13, No. 8

possible contribution of the term ~w

concerning their sign, only the contribution p~6) represented in Fig. 3 might explain the mode frequency behavior of most of the A2 and £ modes in a-quartz although this can not be stated with certainty in view of the present state of experimental data. Anyhow, as long as anharmonic contributions are retained up to 0(~~) (but neglecting orders higher than the eighth in the notation of Wallis eta!. 2), the

5(O/) to the anharmonic frequency shift should be considered at intermediate and high temperatures. Acknowledgements I am grateful to Professor F. Cabannes and Dr. B. Piriou for their interest in this work, and to Professor S.S. Mitra for reading the manuscript and valuable discussions. —

REFERENCES

1. MARADUDIN AA. and FEIN A.E.,Phys. Rev. 128, 2589 (1962). 2.

WALL1S R.F., IPATOVA I.P. and MARADUDIN A.A., Soy. Ph;’s. Solid State 8,850(1966).

3.

COWLEY R.A.,Adv. Phys. 12,421 (1963).

4. 5. 6.

GUREVITCH L.E. and IPATOVA I.P., Soviet Phys. JETF 18, 162 (1964). POSTMUS C., FERRARO J.R. and MITRA S.S.,Phys. Rev. 174, 983 (1968). LOWNDES R.P.,Phys. Rev. B6, 1490 (1972).

7.

SAMAR.AG.A.andPEERCYP.S.,Phys.Rev. B7, 1131 (1973).

8. 9.

BRUCE AD., J. Phys. C., Solid State P/U’s. 6, 174 (1973). Two-phonon bound states are essentially characterized by resonance effects in the high-frequency region (RUWALDS J. and ZAWADOWSKI A., Phys. Rev.8 Anyhow B2, 1172)their and lowest-order their contribution contribution in the vicinity to the self-energy of the one-phonon mode frequency appears to be small. j~p(lO)~ A factor (2n 3 + l)seems to be omitted in equation 2.9(a) of reference 2.

10. 11.

GERVAIS F., PIRIOU B. and CABANNES F., Ph vs. Status Solidi (b) 51, 701 (1972).

12. 14.

GERVAIS F., PIRIOU B. and CABAN1~ESF., Phys. Status Solidi (b) 55. 143 (1973). MOOIJ J.E.,Phys. Lett. 29A, 111(1969). IPATOVA I.P., MARADUDIN A.A. and WALLIS R.F.,Phvs. Rev. 155, 882 (1967).

15. 16.

GERVAIS F., Thesis, Orleans (1973). SILVERMAN B.D. arid JOSEPH R.L,Phj.’s. Rev. 129, 2062 (1963).

17.

GERVAIS F., PIRIOU B. and CABANNES F.,Phys. Letr. 4lA, 107 (1972).

13.

On signale l’éxistence d’un type d’interaction phonon—phonon décrit par l’Hamiltonien du sixième ordre, dont Ia contribution a I’Cnergie du phonon provoque, dans la limite des hautes temperatures, une evolution quadra. tique du glissement de frCquence avec la temperature. L’Cventuaiité d’un tel processus est discutCe en liaison avec certains rCsultats expCrirnentaux sur Ic quartz-a.