Damping of a two-phonon spectrum with cubic anharmonicity

Volume 122, number 1

PHYSICS LETFERS A

25 May 1987

DAMPING OF A TWO-PHONON SPECTRUM WITH CUBIC ANHARMONICITY J.M. WESSELINOWA Faculty ofPhysics, University ofSofia, 1126 Sofia, Bulgaria Received 9 March 1987; accepted for publication 24 March 1987 Communicated by V.M. Agranovich

A Green function technique is used to study a two-phonon spectrum. For the first time the damping y~ due to the cubic anharmonicity including interaction with acoustic phonons is calculated beyond the RPA. In contrast to the damping due to the fourth- and fifth-order anharmonicity y( ~ is finite at T= 0, then it increases strongly with increasing T.

Much attention has been focused in the last years on the temperature and anharmonicity dependence of the phonon spectrum (see for example refs. [1—3]).The influence ofa quartic anharmonicity on the one- and twophonon lines for different temperatures and anharmonic constants is discussed in our previous papers [4,5]; ref. [5] includes the interaction with the acoustic phonons. The damping ofone- and two-phonon states depends not only on the quartic anharmonicity, but on all anharmonic interactions between the phonon modes. The obtained large temperature broadening ofthe biphonon line in H103 [6] can be explainedby taking into account a cubic anharmonicity. The purpose of the present paper is to extend the treatment of refs. [4,5] by including the cubic anharmonicity. We consider the following vibrational hamiltonian: H ~1lCOqbbq+ b~+ ~

~

Cq + ~

hw~a~ ag

t4~+Ht” (1) q)b,~+qbp+F~(p,q)c,~÷qcp](aq +at~)+H where c~,b and Cq, bq are the optical phonon creation and annihilation operators, and a~and aq the acoustic phonon ones. The last two terms in (1) due to the fourth- and fifth-order anharmonicity are considered in refs. [4] and [5], respectively. In order to study the one- and two-phonon spectra we evaluate the following Green functions: ,

+~ [F~,(p,

Gc(k,t)=<>

(2)

,

Gs(k,

t)=~ <> ,

G’~(k,t)=~

<>.

(3)

Using the same method as in ref. [4] we obtain the energies of the one-phonon states in the generalized Hartree—Fock approximation: ~b,C ...hWb,C

—

~ F~,~(q,p)i~,,d, 0/w~,

with 2

(wbC)2

(4)

+4( W~3~c)2(3_cos k~a—cosIç,a—cos k~a)

(cob~C)

64

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Volume 122, number 1

PHYSICS LETTERS A

25 May 1987

W~is the width of the one-phonon bands, w~= vq. The last term in (4) appears due to the cubic anharmonicity. Calculations yield the following expression for the damping of the one-phonon modes: Yb,c(k),ci(k)+Y~,c~(k)+Yf,,c~(k)

y(k)

LeN ~

=

+ (fl~’

fl~q)d(

(5)

,

b,c F~~(k, q)[(1 +flkq +n~)ô((O~’Wa_qWq

W~ _W~q+W~”)]

ph

(6)

.

The terms y~jand yf,~9were evaluated and discussed in refs. [4] and [5], respectively. In contrast to the case of a quartic anharmonicity the damping y~(k) is finite at zero temperature, y~](T=0)=

Wb,CN ~

~

q)d(w~_w~b~Cq_w~h).

(7)

For the damping of the two-phonon states which enters in the Green functions (3) we obtain: 1(k,q) +y~4d~(k, q) +y~~(k, q) yS,d(k, q) =y~ ,

(8)

y~3~(k, q)

=

(w~+wg)N ~ [i~,

+

(1 +~)(l +n~) [d(WLq l+Pi~g+fl~

~,P)((l

1~q~p)d(W~~q

_W~_W~q,)

d(O)~q

W~_W~qp)

+ff~qpö(WLq

_W~ +W~qp)

_W~

+F~(k,q,p)((l+fl,)o(w~_w_w,)+n~po(w~_a+w~ q—p)

+ (1-l-cf;)(l+n~_q)~

(9)

,

l+fl~q+fl~

y~3~(k, q)

=

(w~—w8)N ~ ~

q,p)((l +~)d(co~±,_k—w~_k _~h) +n~d(a~+,_~

+ n~+,_~(l +n~)[ô(o4+p_a_w~_~~h)

+F~(k,q,p)((1 +,i~’)d(&~~,

~

—d(w~±,_~ W~_k

+w;h)

+w~)])

h) +n~d(co~ _w~, +w~’)

+ ~ nb +n~_,,)[d(w~—w~_, —wv) —d(w~ ~ ~

(10) 65

Volume 122, number 1

PHYSICS LETTERS A

40

Fig. 1. Temperature dependence of the two-phonón damping

4),~~5) (refs. [4,5]) due to the 3rd, 4th y~ paper), y~ and (present 5th order anharmonicity for w1=1400cm’, Wc= 60cm’ co~i=300 cm’ W’=lOcm’ for different interac-

~

0

25 May 1987

fCO

200

300

TTKJ

cm~(4)KI=2Ocm~(5)~Kj=30cm-’;(6)lFbi=3lcm~ iF~I=42cm’;(7) IFbi =35cm’, iF~i=53cm~(8) Fbi =41 cm’, iF~i=54cm’.

where ~ flC and ~ are the occupation numbers of the phonons. The terms y~and y~were calculated and discussed in refs. [4] and [5], respectively. In ref. [4] we have shown that the biphonon peak width corresponding to half-maxima intensity Sm~,,I2coincides with 2Ys,d. The damping y~was calculated numerically using the same model parameters as in ref. [4]. We suppose that the quantities Fb and F~in (1) do not depend on wave vectors. The results are plotted in fig. 1. It can be seen that at T= 0 we obtain a finite value of the damping, Ys,d(T0)7t(Fb+1~)/(WO±WO)

(11)

,

which is in agreement with the experimental data of Polivanov and Shiryaeva [6] for H10 3. The damping due to the quartic anharmonicity

[41was zero for T= 0. At

low temperatures we have

(12) 3)can be smaller, equal or larger than y~ depending on the relation of the whereas at higher temperatures anharmonic constants Fb, F~andy~K [5]. At room temperatures we have

~ ( T= 300

K) = 60—80 cm~ which is of the right order of magnitude compared with the experimental data for HIO ,

(13)

3 [6]. It can be concluded that if we want to obtain correct results, all interaction terms which contribute to the broadening of the biphonon lines must be taken into account.

References [1] [2] [3] [4] [5] [6]

66

V.M. Agranovich and I.J. Lalov, Usp. Fiz. Nauk 146 (1985) 267. M.V. Belousov, in: Excitons, eds. Ed. Rashba and M.D. Sturge (North-Holland, Amsterdam, 1982) ch. 18, p. 771. J.A. Reissland, The physics of phonons (Wiley, New York, 1973). J.M. Wesselinowa and I.J. Lalov, Physica B 141 (1986) 277. J.M. Wesselinowa and I.J. Lalov, Solid State Commun. 61(1987) 47. J.N. Polivanov and A.V. Shiryaeva, Kratk. Soobshch. Fiz. 11(1982) 37.