Bound states of two Frenkel excitons in layer antiferrodielectrics

Volume 46A, number 1

PHYSICS LETTERS

19 November 1973

BOUND STATES OF TWO FRENKEL EXCITONS IN LAYER ANTIFERRODIELECTRICS Yu.B. GAIDIDAY and V.M. LOKTEV Institute of Theor. Phys., Kiev, USSR Received 13 August 1973 Rigorous conditions for the appearance of two-exciton bound states in antiferrodielectrics (AFD) with a point symmetry group D2h are obtained and the energy of these states is found.

The problem of finding the energy of the bound states of quasiparticles was considered by Bethe 11] and Wortis [2] for two magnons, by Freeman and Hopfield [3] for an exciton and magnon, by the authors [4] for two excitons. Exact solutions were obtained for one-dimensional crystals [1,4] and for some symmetrical directions in three-dimensional and plane (with a point group D4h) crystals [2, 3]. Below we give the exact solution of the problem for a plane crystal with a point group D2h (or D4h) using the example of two excitons. In refs. [4, 5] it is shown that the two-exciton states may be ofand twoodd kinds: even with respect to inversion ones. Theones energies of two-exciton eigenstates are derived from from the solution of the Dyson equation which is the case of a plane crystal splits into four independent equations [5] for the Green functions G~= V1 {l—V(I

00~I10tJ01~i11)}~— v,

G~1

~1

~

V

_____

~L2 L a

Lb _______

and assume that 1~> y> 0. Then the range of twoexciton continuous spectrum is situated in the interval ~ ~i3 +y.(Ifi3 and ‘y have different signs, then this interval is j3—y~.)Outside this interval, in the range xl > f3+7, the poles of the even functions ~ (1) are defined from the equations 1p(O ; k) tanO = K—E + K ~~n~+cos~ x~ ~ ~ ~3±~ (13) —

‘

and those of the odd functions GS~(2)are defined from the equations 2K~~o(O;k) IT x~E—k’ X k k ~ respectively. In (3) and (4) we use the notations

~1V(J~JJJ)}’V(2)

-

(4)

1TIT

‘nm

—

1ff w—~(L cosnt cosmz dt dz cost+Lbcosz)_iO’

(n,m

=

0,1). ~c(O;k)

Here ~ are the Green functions of even and odd states, I~is the interaction between the nearest excited ions in AFD [4] ;La and Lb are the two-exciton band widths in the principal directions of a layer cyrstal [5], w is the frequency. The poles (1) and (2) define the presence of bound levels in the system. For the sake of convenience, we introduce the dimensionless parameters

=~—

k

KE(U;k’)—(E—K)F~O;k’);

= ~

2./~ cosO

x2—(~—7)2 tan = ~

where F(O ; k) and E(O ; k) are the incomplete elliptic integrals of the first and second kind, respectively;

67

Volume 46A, number 1

PHYSICS LETTERS

19 November 1973

K F(ir/2 ; k), E E(x/2, k). These equations define the spectrum of bound states for arbitrary ~ and y. Here the crystals with the lattice symmetry D4h and

The solutions of eqs. (3) and (4) can be found both for small and large ~ [5~. For example, when

D2h are considered simultaneously. Wortis [2] has so far succeeded in treating only the case ~3= y. The analysis of eqs. (3) and (4) shows that in the interval x < ._(j3+y), to the left of the range of the continuous spectrum, bound states occur subject to

and their energies are equal to

~

2

—

IT

l~iCOt~i1 ‘ for tan~/2

~< 0

for ~ ~

IT

~{2

~<

+ 51fl~

~

~°~‘~‘

1’

i~l 3+’y, all the branches of the bound states exist ~‘

+ ~ =

+

72

2

—

~~:P

~

To these roots there correspond the stationa~states

for ~

of AFD belonging to the exciton type. It is to be noted that the bound states are subject to a splitting which has no analog in the one-particle spectrum. This splitting can be observed, for example, in double absorption. And here the polarization of the lines

for ~

along the principal axes of the AFD lattices will take place.

~

and in the interval x > $+ 7, to the right of the continuous spectrum, when

At present a great number of layer (two-dimen. sional) AFD are known: Rb2MnF4, NiCl2, K2NiF4, (C~H2~~1NH4)2MnX4 (n = 1,2..., X = Cl, Br) and

~> 0

for

others. The calculations presented will be useful in

for

identifying the observable lines at double absorption and their polarization properties.

for ~

References

~>

~

2 1 i~icot ipl—l ~ tan~,j~J~ —-

~ 1

lIT ~>

~

l—cos~1i l~

11] HA. Bethe, Z. Phys. 71(1931) 205.

2 IT

~i

1

— —

sin

cos~iif

for ~

At ~ y these conditions change to the conditions for a one-dimensional AFD [4], and at ,3 = 7 to the conditions given in [2] ,but only for even states (odd states were not considered at all in [2]).

68

[2] M. Wortis, Phys. Rev. 132 (1963) 85.

[3] S. Freeman and J. Hopfield, Phys. Rev. Lett. 21(1968) 910. [4] E.G. Petrov, V.M. Loktev and Yu.B. Gaididay, Phys. St.

So!. 41(1970)117.

[51 Yu.B. Gaididay and V.M. Loktev, preprint !TP-73-115E, Kiev.