The influence of critical points on polariton dispersion in the band of two-particle states

Solid State Communications,Vol. 19, pp. 503—505, 1976.

Pergamon Press.

Printed in Great Britain

THE INFLUENCE OF CRITICAL POINTS ON POLARITON DISPERSION IN THE BAND OF TWO-PARTICLE STATES V.M. Agranovich and I.!. Lalov Institute of Spectroscopy, Academy of Sciences U.S.S.R., Academgorodok, Podolskii r-n., Moscow, 142092, U.S.S.R. (Received 25 November 1975 by E. Burstein) The presence of critical points in the two-phonon band in conditions of the Fermi resonance on a polariton may result in a considerable polariton dispersion inside the band of two-particle states. Moreover, the polariton linewidth, conditioned by the decay into two particles, is shown to become zero near critical points; this fact holds certain promise for the determination of the positions of critical points in side the band of two-particle states. IT WAS SHOWN in previous papers”2 that on a polariton crossing the band of two-particle states (i.e. the twophonon band), a gap may be formed in its spectrum (above or below this band). The gap in this case appears only in the case when the interaction of quasiparticles leads to the bound states of quasiparticles (biphonons) being formed. An effect of this kind was actually observed in experiments on Raman scattering by polaritons (see, e.g. references 3—5). In this case in [references4 and 5] a rather strong polariton dispersion was also noticed inside the very band of two-particle states. In this spectrum region there opens a channel for a polariton decay into two quasiparticles. Besides, the character of the distribution of state densities, also influenced by anharmonicity, may also turn out to be an important factor affecting polariton dispersion inside the band of two-particle states. Considering the above mentioned experiments,4’5 in the present paper we shall follow in more detail, as compared to references I and 2, the effect of the above factors on the peculiarities of polariton dispersion inside the band of two-particle states. In so doing we shall base it on the model of an anharmonic crystal, described in references 1 and 2; in this model the cross-section of Raman scattering by a polariton in the considered spectrum region proves to be proportional to the following spectral density* 2~~EK~ JtE, K) = ~2 (E, K) + 4ir 22 g (E, K) [A(E, K)] 2 (1) ‘

*

~

Here, for simplicity, only that situation is discussed when the main part of Raman scattering intensity by a polariton results from Raman scattering intensity by

where E and K are the energy and the wave vector, respectively, transmitted to the crystal and where ~(E K) = ~ + ~ ‘(E K)I(E K) ‘

‘

A’ E K

—

A

‘

I(E, K)

“

‘

E

—

F2(K)

—

(2) 3

K) de (4) E In (3) the value E~(K)is the polariton energy with a wave vector K, determined without due regard for the interaction with the band of two-particle states. Values A and 1’, according to references I and 2, are anharmonicity constants. The value A here gives half of the energy decrease for two optical phonons which occurs at their confluence in the lattice site n (a corresponding Hamiltonian part has the form —A ~ whereas the value F appears in the Hamiltonian of interaction 1 ~ r(K) b~_qb~q + c.c ~

—

—

k,q

relating the polariton with two optical phonons (~ and b are decay operators for the polariton and the phonon respectively). The function g(E, K) appearing in (4) is equal to the density of states of two free quasiparticles (e.g. phonons) with the total momentum K; the dependence g(E, K) on K is weak, however, at K ~ lid (d is the lattice constant) and is neglected hereinafter (it is assumed = 0). Quasiparticles with the opposite in direction Kmomenta correspond to the states with K = 0 in the band of two-particle states. Therefore, for the band two-particle states, corresponding, forthe example, to theofpresence of quasiparticles of one kind, den.

sity of states g(E, 0)

=

~gi(E/2), where g

1(E) is the density of states in the band with a single quasiparticle. It means that in the given case peculiarities of the

a band of two-particle states. The general theory is given in reference 1. 503

504

CRITICAL POINTS ON POLARITON DISPERSION

functionsg(E, 0) and g1(Ei2) are of the same character; this fact will be used in later considerations. If the energy E, transmitted to the crystal, lies outside the band of two-particle states, then g(E) -÷ 0. In this case the value f(E, K) becomes proportional to the s-function [E E0(K)], where E0(K) is determined from the equation ~

~TH

(5)

Namely from the analysis of this equation (see references 1 and 2), giving a polariton spectrum (for the model considered) outside the band of two-particle states, it follows that in the given energy region E a gap in the polariton spectrum appears only in the case, when anharmonicity leads to the formation of bound states of quasiparticles (biphonons). If the energy E lies within the band of two-particle states, then for determining the polariton dispersion equation (5) may be also used; in this case the integral 1(E) should be taken already as the principal value [seeequation (4)]. Inside the band, however, as has already been mentioned, a new decay channel opens for the polariton, It follows from (1) that the corresponding level with

y(K)

______

—

~(E,K) = 0.

=

2irg[E0(K)] A’(E, K)

dE)E=E0(K)

—

—

S~

£rnin.

~S2 £ mar.

Fig. 1. The distribution of state phonon band.densities in a two-

( I

/~+Reg(e),

2b~~arctg

.v id

I b~s,/~in 11 + ~~

e~0

+ Re g(e),

—

0

~

where Re g(e) is the function e, being regular at Similarly, near the upper edge of the band (i.e. at = E Em~ 0) we have

0.

—

( —bsJIeIln 11 _\‘~U~ + v’~Ii~’ + Reg(),

(6)

is determined not only by the density of states, but also by a number of other factors and it will be considered below. Now, basing in equation (5), we shall discuss the peculiarities of polariton dispersion in the vicinity of critical points of the density of states in the band of two-particle states. It is known that in three-dimensional crystals the function of the density of states always remains continuous and its derivative is broken only at critical points (see, e.g. reference 6). In this case from different sides of the critical point E = E~the density of states in its vicinity (E~ 2 ö’, E~+ s”) follows differentand~”are laws: —E~i,a~0,b~0;6’ ~E)~a+b~.JiE positive values describing the size of the interval where the peculiarity is observed (the sign + before the root corresponds to the energy E~on the edge of the band of two-particle states, herewith a = 0; the sign before the root corresponds to saddle points S~and S 2 with b=0atE>E~forthepointsSj,whereasforS2, b = 0 at E
Vol. 19, No.6

1(E)

=

f

—

2b~ arctg

Re g(e),

C

~0

e~ 0

In the vicinity of the critical point S~(i.e. at e = E 0) we have: ( +e 2b a ln Re g(e), e ~ 0 1(E) = ~ + I + Re g(e), ln + 2b~’~ arctg —

~‘

~,,

—

~,,

—

—

—

~

0

~

0

in the vicinity of the critical point 52 (i.e. at e = E—E~2~0)wehave ~‘

a in 1(E)

=

I

~,,

~‘

a ln

~,,

+ e —

+ —

—

2b~~i arctg ~+ 1

Re me),

—

~ 2b~~ in 1 +

~

+ Re g(e), e~0

Thus, at critical points (E E~)the integral 1(E) has either its minimum (near the lower edge of the band and the points S1) or its maximum (near the upper edge of the band and the pointsS2); the absolute value ofits

Vol. 19, No.6

CRITICAL POINTS ON POLARITON DISPERSION 505 first derivatives from the right (at points Em~and E~2~) g(E) is small. It is important, however, that at the very and from the left (at the points Emin and E~)increases critical points the value ‘y(E) always becomes zero, indefinitely as the critical point is approached. Such because of the value i(d~/dE)~,E~I turning to zero. behaviour of the function 1(E) may lead in some cases to Actually, in accordance with (2) the appreciable polariton dispersion in the band of twoparticle states. This follows directly from equation (5). d~ 2r21(E) + In fact, if in the spectrum region considered we neglect dE [E—E~(K)]2 “ ‘dE the inessential here dispersion of polariton velocity in zero approximation and assume E~(K)= h(c/no)K, and the aforesaid is valid, since the derivative dudE on where c is light velocity in vacuum and n 0 = const. then the band edges and at critical points (from the right or from equation (5) [see also equation (2)] it follows that from the left) becomes ± as it has already been 21(E) stated. It is this very fact which provides certain possiK = E (7) biities for the determination of the positions of critical hc 2r no l+2A1(E) point by the Fermi-resonance on poiaritons. If at E = E~the inequality 12A1(E)I ~ 1 is followed, Generally speaking, polariton damping 7 between then hcK/n 21(E), so that maxima and critical points must be not too small. In these spectrum 0 ~ relationship E 2F minima of the K = K(E) correspond to regions the condition of synchronism should be conmaxima and minima of the integral 1(E). If the reciprosiderably disturbed. In this case the angular density of cal inequality is followed, then hcKJno = E 1721A, so the scattered radiation should drop which may lead, that at the critical point the peculiarity in the relationwith all other things being equal, to the illusory break of ship K = K(E) disappears. the polariton branch inside the zone of two-particle However, even in the case when the function K = states. K(E), determined by the relationship (7), has its peculiIn conclusion it should be noted that only the arity, strong dispersion caused by it may show itself, simplest pattern of the density of states in the band generally speaking, only in those polariton spectrum has been considered above. In many cases, however regions, to which a sufficiently weak damping y(E) (see, e.g. the analysis in reference 7) the density of corresponds. Since outside the band g(E) = 0, the aforestates in the band of two-particle states may differ said [seeequation (6)1 always holds true when moving considerably from the one used here (e.g. points S 1 and along the polariton branch to the edge of the band of S2 may approach one another, etc). Relationships two-particle state. Naturally, the damping 7(E) may (1—4) make it possible to also consider for these cases turn out to be weak also inside the band in the spectrum all peculiarities of polariton dispersion in the condition region where for some reasons the density of states of Fermi resonance. —

~

—

~

—

.

—

—

REFERENCES 1.

AGRANOVICH V.M. & LALOV I.I.,JETP61, 656 (1971); Fiz. Tverd. Tela 13, 1032 (1971).

2.

AGRANOVICH V.M., The Strong Anharmonicity Effects in Raman spectra. Supplement to Russian edition, Moscow, (1973) of the book by POULET H. & MATHIEN J.-P., Spectres de Vibration etSymetryc des Ciystaux. Paris (1970).

3.

(a) MAVRIN B.N. & STERIN KILE., JETPPisma v red. 16, 265 (1972); (b) WINTER F.X. & CLAUS R., Opt. Commun. 6, 22 (1972). MITIN G.G., GORELIK V.S., CULEVSKII L.A., POLIVANOV J.N. & SUSHCHINSKII M.M., JETP 68, 1757 (1975). AKUPPETROV K.A., GEORGIEV G.M., MIKHAILOVSKII A.G., PENIN A.N. & MITUSHEV I.V., Fiz. Tverd. Tela 17, 2027 (1975). MARADUDIN A.A., MONTROLL E.M. & WEISS G.H., Theory ofLattice Dynamics in the Harmonic Approximation. Academic Press, New York and London (1963).

4. 5. 6. 7.

BIRMAN J.L., TUBINO R., Phys. Rev. Lett. 35, 670 (1975).