- Email: [email protected]
Organic superlattices with interface Fermi resonance: optical nonlinearity
SYRTIHIITIIr IIfl|TRLS S y n t h e t i c M e t a l s 64 (1994) 1 4 7 - 1 5 3
ELSEVIER
Organic superlattices with interface Fermi resonance: optical nonlinearity V.M. Agranovich a, P. Reineker b, V.I. Yudson a'b alnstitute of Spectroscopy, Russian Academy of Sciences, Troitsk, 142092 Moscow Region, Russian Federation bAbteilung far Theoretische Physik, Universitiit Ulm, 89069 Ulm, Germany Received 16 D e c e m b e r 1993
Abstract
A mechanism of optical nonlinearity of organic superlattices is considered which is based on the Fermi resonance between molecular excitations of alternating layers. Using Green's function formalism we have obtained a closed equation for the energy states in the vicinity of the Fermi resonance and found a general expression for the second-order nonlinear polarizability which determines the second-harmonic generation of light caused by the interface Fermi resonance coupling.
1. Introduction
Recent progress in engineering of organic superlattices [1-12] makes necessary a search for new efficient mechanisms of optical nonlinearity connected with their multilayer structure. An interesting and perspective mechanism of optical nonlinearity arises in connection with the phenomenon of the Fermi resonance between excitations of neighbouring layers [13]. This interface Fermi resonance takes place when the energy of several excitations in the layer A is close to some excitation energy in the neighbouring layer B. Due to this intermolecular anharmonicity a new type of excitation may arise in the vicinity of the interface of two contacting molecular crystals [14,15]. In the present work we consider not a single interface of two crystals but a periodic superlattice consisting of many alternating molecular layers of type A and B. As in Ref. [14] we assume the presence of the Fermi resonance 2ro,4= wB at each interface. When the thickness of each of these two layers is relatively small (say, of the order of several intermolecular spacings) the presence of many interfaces may result in new physical
properties of the superlattice on a macroscopic scale. Study of superlattices is especially important due to accumulation of nonlinear optical effects with an increase of sample length. In the present paper we use Green's function formalism to obtain the excitation spectrum in the energy range E ~2¢0A ~ ~OB. We have found a closed characteristic equation for the eigenvalues corresponding to states with wave vector K, determining their translational properties. The excitation spectrum can be straightforwardly calculated for the concrete model of intermolecular transfer integrals. The studied energy range is of particular interest as it determines a resonant second-harmonic generation of light in the superlattice. Using the developed formalism we calculate the nonlinear optical polarizability X (2) describing the second-harmonic generation in a superlattice with Fermi resonance interfaces. Within the considered model, a non-zero value of X (2) is completely caused by the Fermi resonance nature of nonlinear coupling. A dependence of )((2) on coupling constant, frequency detuning and excitation band structure is analysed. 2. The model and basic equations
* W e are happy to contribute this paper to the Festschrift for Professor H. Inokuchi, great scientist and friend. W e wish him many years of happiness and successful activity.
0379-6779/94/$07.00 © 1994 Elsevier Science S.A. All rights reserved
SSDI 0 3 7 9 - 6 7 7 9 ( 9 4 ) 0 2 0 9 9 - K
We consider a model described by the Hamiltonian Z = Z O -~- ~/~int "1- aCC~anh, where X o = Y.4 + ,'~8 is a sum
148
V.M. Agranovich et aL / Synthetic Metals 64 (1994) 147-153
of Hamiltonians of noninteracting excitations inside layers A and B, respectively. Here
= E'o4A N,I
. ,AN,+ , X V'u,AN,/IN,,, *
(1)
N,l,l'
and the Hamiltonian ,Tg'B is given by the replacement
A-*B. In Eq. (1) the lower indexN=O, +1, ... counts the periods of the superlattice, and l=1, ...n4 the molecular monolayers of type A(B) in one period of the superlattice. The matrix element Vu, describes excitation transfer between sites l, l'. The Hamiltonian ~ i . t describing the interface interaction corresponding to the Fermi resonance condition 2o4 = ~on is given by t
'~int = E[FBA(AN.I) N
2
t
2
O u - 1 , r i b -1- I ~ A B ( A N , nA) ON,1 "J~h . c . ]
(2) Here the coupling constants F4B and FBA correspond to the interfaces AB and BA, respectively. In a general case of molecules without inversion symmetry these constants are typically non-zero and may differ considerably. In the case of molecules with an inversion centre, the quantities /'AB and /"BA are not independent but connected to each other: F=F4B = +_FB4, if the states '2/1' and 'B' are of the same or of the opposite parity, respectively. In its turn, the constant F may vanish for sufficiently symmetric A and B molecules and their relative orientation. Hamiltonian (2) describes transformation of two A excitations into a single B excitation and vice versa. In its turn, the two 'A' interactions involved in this process may interact with each other. This intralayer interaction can be described by the following Hamiltonian (see, e.g., [14,16]): t 2 x : . . . --- u4 E(AN.,)
N,I
2
(3)
For the case when the state A is a phonon one, Hamiltonian (3) corresponds to the intralayer anharmonicity. The model (Eqs. (1)-(3)) is an extension of that considered in Ref. [14] to the case of a periodic superlattice. For simplicity and similar to Ref. [14], we treat the system as quasi-one-dimensional, i.e. we neglect excitation transfer in the transverse direction. A generalization to the three-dimensional situation can be done straightforwardly. Note that up to now we did not specify the nature of the excitations which could be, for example, optical phonons or Frenkel excitons. The operators entering Eqs. (1) and (2) will be boson and paulion ones, respectively. The following formalism allows us to treat both cases on equal footing. However, from the physical point of view one should consider only the cases of
exciton--exciton and phonon-phonon Fermi resonances, while the exciton-phonon resonance would correspond to an exotic situation of comparable energies of electron and phonon subsystems. The spectrum of lowest energy excitations of the model is very simple: it corresponds to single-particle excitation of the type A. This excitation is confined to a certain period of the superlattice. Thus, one cannot expect a manifestation of the interaction considered in the linear response of the system to an external field of frequency ~0= w4. A higher band of the energy spectrum of the model corresponds to a mixture of the two excitations of type A and a single excitation of type B. There is still a trivial case when two A excitations are created in different periods, then coupling to B excitations is obviously absent. An interesting situation arises when two A excitations are created at the same period, then mutual transformation of these excitations to the B excitation and vice versa takes place. The spectrum of such coupled states is determined by the poles of the time-energy Fourier transform of the single-particle Green's function:
GB(N,I,t;N',I',t') = -i(O[T{BN,t(t)B~,.t,(t')}[O)
(4)
or the two-particle Green's function:
GA(Nl,l~,tl;N2,12,t2;N'~,l'~,t'~ ;N'z,l'z,t'z ) -" ---t(OlT{AN,
t ~ t , a,(tl)AN2,t2(t2)Aul.r~(tm)A m.,~(tz)}lO) (5)
of the system with interaction. The operators entering Eqs. (4) and (5) are taken in the Heisenberg representation with respect to Hamiltonian Z . For a perturbative analysis it is more convenient to use the interaction representation [17]:
GB(N,I,t;N',I',t ') =-i(O[T{BN,,(t)B~,.,,(t') 00
where all the operators are taken in the Heisenberg representation with respect to Hamiltonian ~¢'o of noninteracting system, and the subindex 'c' denotes a part corresponding to connected Feynman diagrams. Eq. (6) is the starting point of the following consideration.
3. Green's function formalism
In the absence of the coupling (in the zeroth order in X:'i.t) the single-particle Green's functions are given by
V.M. Agranovich et al. / Synthetic Metals 64 (1994) 147-153
G(a°(~)(N,l,t;N',l ',t') - 8N,N'GBc4)(I,t,I (o) . , ,t , )
(7)
where the Green's function on the right-hand side relates to a single period. The term linear in ,,~.~ does not contribute to Eq. (6) as it contains an odd number of B and B* operators. The contribution to Eq. (6) of the second order in J~int is determined by
Z J dr1 +
2
2
.
s,s'= +
(8)
A correlation function (OtT{B(t)B*(t')B*(tz)B(t,)}[O)~, which arises in Eq. (8), reduces to iZG(°)(t;tz)G(B°)(h;t ') for both boson and paulion operators B due to the particular time ordering (tz>q). Also we introduce a notation:
Note, in passing, that in a simple case when one neglects a mutual interaction of two A excitations, the correlation function (Eq. (11)) reduces to a convolution of two single-A-particle Green's functions:
M(E) ,s'= 2i
fd, ~
G(°)(K=O,E/2 + e;s,s') (15)
G(BZ)(K,E;I,I ') = G(B°),(E)Mss,(K,E)G(ff,),r(E)
M(12,t2;I,,tl)SN2.N,
= [IrBAI2M+ + ( E )
exp(-iKL) FsA F~BM + _ (E) exp(iKL)]
(9)
for another correlation function arising in Eq. (8); obviously, M ( l j 2 ; l , q ) ~ O(t2-q). All the states of the periodic superlattices can be classified by the wave vector K taking values in the interval ( - w / L , ¢r/L), where L is the period of the superlattice. It is natural to consider the Fourier transformation:
(16)
where the summation over repeated indices s, s ' = _+ is implied, and the matrix M(K,E) is connected with the matrix M(E) (Eq. (12)):
M(K,E)
i(OIT{(AN~,,~(t2))2(A~,.,,(q))2}IO)~
(14)
Taking into account Eqs. (9)-(14), the expression for the Fourier component of the second-order correction, Eq. (8), to Green's function can be represented in the following compact form:
1(,2)
+ I~AB(A*N,,,A)ZBu,.I](,I)}IO>¢
(o)
X G(A°)(K= O,E/2 - e;s,s')
X [I-~BA(AtN,,I)2BN, _ a,,,s
-
_
O(t -td
× (O]T{BN, z(t)B*u,.r(t' )[FBAB* t
a.( o )
149
IFA.I2M__(E)
]
(17)
One can easily check that the higher order corrections to Green's function are described by terms with a higher number of matrix 'self-energy' Iql insertions. Thus, Green's function Gn(K,E;I,I') may be represented as
G.(K,e;t,t') =
+
(18)
G(N,I,t;N',I',t')=
-~
-~
X e x p { i [ K L ( N - N ' ) - E ( t - t')]}G(K,E;l,l ') dE M(l,t;l ',t') = f - ~ e x p [ - i E ( t - t ' ) ] M ( E ; l , l ' )
where the T matrix is determined by (10)
"I'(K,E) = M(K,E) + M(K,E)G(n°)(E)M(K,E) +...
(11)
Summation of this geometrical series results in the following expression:
where N is a total number of superlattice periods. To shorten expressions it is convenient to introduce a matrix notation: M(E) = {Mss,(E)}; s,s'= +_
(12)
with M+ + (E) =M(E; 1,1);
M _ + (E) =M(E;nA,1);
M+ _ (E) =M(E; 1,hA)
M_ _ (E) =M(E;nA,nA) (13)
We will also write Green's function G~°)(E;l,l ') in the form Gsm(E) (o) replacing the indices l, l' by s, s' = + when 1, l' denote one of the end points: nB ~ + ; 1 ~ - . We introduce the following 2 × 2 matrix:
r(K,E)
(19)
= B -
× [G~O)(E)]-I _ [G~O)(E)]--1
(20)
where I denotes the unit 2 × 2 matrix. Eqs. (18)-(20) give a formal solution to the problem of Green's function calculation for the system (Eqs. (1) and (2)). The spectrum of excitations coupled by the Fermi resonance interaction (Eq. (2)) in the energy range of interest (E = tos--- 2WA) is determined by the poles of the "1" matrix or, equivalently, by the characteristic equation: det[I - M(K,E)G~°)(E)] = 0
(21)
This equation is one of the main results of our analysis. The quantities entering Eq. (21) are expressed
150
V.M. Agranovich et al. / Synthetic Metals 64 (1994) 147-153
through Green's functions of the system without interaction. Note that up to now all the obtained expressions are valid for both boson and paulion operators. The difference between them would arise only in the calculation of the two-particle Green's function (Eq. (9)). For noninteracting bosons this function reduces to the product of the single-particle Green's functions, while for paulions such factorization does not hold. Using Eq. (21) the excitation spectrum of interest can be straightforwardly calculated for concrete models of intermolecular transfer matrix elements Vtr. To get an idea of the Fermi resonance phenomenon we consider first a trivial case of very small transfer matrix elements V~t,. In this limit we have EA(B) --~ O)A(B)and the matrices G(B°)(E) (Eq. (14)) and M(K,E) (Eq. (17)) reduce to the diagonal ones with G(O) + (E) = ~(o) _ (E) = [ E - w B ] - ' B+
(K,E)
M+ + = 2IrBAIZ[E-M _ _ (X,E) = 21rAnl2[E-- 2WA]-'
(22)
where the dipole moment operator for the sublattice A is given by PA = ~AXN.,[A~,, +AN.,]
with that for the sublattice B being determined by the replacement A ~ B ; I~A denotes the transition dipole moment corresponding to these excitations. The nonlinear polarization of interest arises as the second-order response to the perturbation (Eq. (24)): e(2)(t) = ~
1
Tr{P(t)p(2)(t)}
0)A--
+2/~
(23)
Here F should be substituted by FAn o r Xg'nA for the interfaces AB or BA, respectively. Of course, in the absence of excitation transfer there is no K dependence in the right-hand side of Eq. (23); the solutions (Eq. (23)) correspond to the modes localized just at interfaces AB or BA and, naturally, coincide with those found in Ref. [14] for a single interface. Concerning now the case of finite transfer matrix elements Vtr we point out another interesting and important situation when they are still smaller than the interface coupling constants. In this case the energies (Eq. (23)) spread out into a rather narrow band with a width exponentially decreasing with an increase of parameters In IF/I~ and the superlattice period L. An elaborate analysis and numerical study of Eq. (21) will be given in a separate paper [18]. Now we proceed with nonlinear optical properties caused by the interface Fermi resonance coupling.
4. Nonlinear optical response
For not too thick samples and for normal incidence (for simplicity) of an external electromagnetic (EM) field, both the field and induced polarization inside the sample may be considered as uniform and the Hamiltonian of interaction of the superlattice with the EM field E(t) can be represented in the form:
V(t) = - [PA + PB]E(t~ - - PE(/)
(24)
(26)
where L is the superlattice period and N is a total number of the periods; S is an area of the elementary cell in the plane transversal to the growth direction. The second-order correction p(2) to the density operator is given by
o(2'(tl=(-i)2f
dt' dt" O(t > t"> t')[V(t"l,[V(t'),O(°)]]
The characteristic Eq. (21) reduces in this limit to a couple of quadratic ones with the following roots: E±(K)=tOA+ ~- _+
(25)
(27) Operators entering Eqs. (26) and (27) are taken in the interaction representation with respect to the interaction Eq. (24) or, in other words, in the Heisenberg representation with respect to the Hamiltonian YU of the system without an external EM field. The operator p(0) in Eq. (27) is the density operator of the unperturbed system. With use of Eqs. (24), (26) and (27) one obtains
1 ~ -2 dt' dt" O(t>t">t') e ( z ) ( t ) = - NL--S xTr{[[P(t),P(t")],P(t')]p(°)}E(t")E(t ')
(28)
We assume the exciting EM field:
E(t) = ~o, exp( - itot) + c.c.
(29)
to be in an approximate resonance with the type-A transition, i.e. tO-----tOA. In resonance (rotating wave) approximation one takes into account only the processes where the EM field (Eq. (29)) interacts with the A excitations, thus only the A part (Eq. (25)) of the operators P(t"), P(t') should be retained in Eq. (28). Similarly, only the part B of the operator P(t) contributes to the second-order polarization (Eq. (28)). Assuming that no excitations are present in the unperturbed system, i.e., p(O)= 10( >0l, where 10> is the ground state, we come to the following expression for the Tr{..-} in the right-hand side of Eq. (28): Tr{.--} =
(30)
Representing the nonlinear polarization P~2~(t) in the form: P(2)(t) = X(2)(2w;to, co)~ exp( - 2i~) + c.c.
(31)
151
V.M. Agranovich et al. / Synthetic Metals 64 (1994) 147-153
and using Eqs. (25), (28) and (30), we obtain the following expression for the second-order nonlinear optical susceptibility X(2): X(2)(2oJ;to, o~)=
tz~B L-----~V(2to;to, to)
(32)
kK~)(t,t",t ') = -~ ~
d~l
× (FBAG(z°)(N,I,t;N~ - 1,nB,q) × GA(o)(NI,I,ta;N ,l ,t ) ,,
where
× G(A°)(N,,1,q;N',I
v(2;to,o,)=f f
,,
',t')
+ FABG(8°)(N,I,t;N~,I,q)
dt'dt"V(t,t",t')
× G~°)(N~,nA,l,t~;N",l",t ")
× exp[ - 2/to(t' + t" - 2t)]
(33)
The 'vertex' function V(t,t",t') in Eq. (33) is determined
as
V(t,t",t') = - 2---N 1 ~(OlT{BN't(t)A~"'r(t")A*u"r(t')}lO) (34) where the sum goes over all lower indices (molecular sites). It is easily seen that the vertex function (Eq. (34)) is the only term of all the possible arrangements of the operators in Eq. (30) making non-zero contribution to ,)((2). The obtained expressions (Eqs. (32)-(34)) reduce calculation of the nonlinear susceptibility X~2) to the problem of calculation of the vertex function V(t,t",t'). The latter problem allows a regular field-theoretical analysis carried out in the next section.
5. Calculation of the vertex function and estimation of X (2)
It is convenient to rewrite the expression for the vertex function (Eq. (34)) in the interaction representation with respect to the Hamiltonian of the interface Fermi resonance coupling ,'Y:i,, (Eq. (2)):
lz(I {
V(t,t",t')= - - ~
,,
0 T BN, t(t)A~.r(t")A~,r(t')
× G~°)(NI,nA,I,t,;N ',l',t')}
(36)
where the summation goes over all the sites Nl, N'l', N'T', N1 and ll; G~°) and G~°) are the single-particle Green's functions of noninteracting subsystems A and B. For simplicity, the operators A, A* have been assumed to be boson ones, while operators B, B* may be both boson or paulion ones. Using the Fourier transformation we obtain the following expression for the vertex function V°)(2w;to, o~) (Eq. (33)): V~a)(Zto;w,~o)=
~
l,l',l";s
FsG~°)(K=O,Zw;l,s)
× G
=/'BA, F_ =
GB(K=O,E;I,s) = ~ G~°)(K=O,E;I,s ') S=-t-
Xexp[--i f®,,~int(tl)dt,]} O>c
(35)
where the lower index c denotes a connected component of the average. In the zeroth order in X/i,t we have for the vertex function (Eq. (35)): V~°)=0, as in this case only correlation functions with equal numbers of creation and annihilation operators could differ from zero. In the first order in Y~,t the vertex function (Eq. (35)) is given by
× [I - M(E)G(B°)(E)];:
(38)
Note that the matrix [. -- ]- 1 entering Eq. (38) is closely connected with the T matrix (Eq. (20)) for the general case K:~0; its poles determine the excitation spectrum of the system in the Fermi resonance range. The vertex function V(2to;to, to) (Eq. (33)) is obtained from Eq. (37) by replacement of G
V.M. Agranovich et aL / Synthetic Metals 64 0994) 147-153
152
X(z)(2oJ; o~,oJ) =
tz2 lzB ~ F,G(B°)(K=O,20;I,s) L S t. r, t'; s, s"
X (2)(2o~;oJ,oJ) = (/An + FBA)
× [ I - M(20)G~°)(Zo)];, ' × G~°)(K = O,~o;s',l')G~°)(K = O, o4s',l")
XZ
(40)
1
we obtain eventually a simpler expression for X (z) (Eq. (39)): X
(41)
To get an idea of the manifestation of the Fermi resonance phenomenon, we consider first the simple case of narrow bandwidths for bothA and B excitations, i.e., small transfer matrix elements Vn,. In this limit
art,
G (A°0B)(K= O,E;l,l') - E - OJA(m+ iO
2[r'lz
a.,
• 1 s= ± ( 2 ~ o - ) ( ~ o - WA)--Ir, I=
(39)
This equation is the main result of our analysis. The quantities entering Eq. (39) are expressed through Green's functions of the system without the interface Fermi resonance interaction. However, this expression holds also for the case when there is an interaction between excitations of the same type, e.g., intralayer anharmonism, etc. The formalism developed is an extension of an earlier one [16] applied to a uniform medium to the case of interface Fermi resonance in superlattices. The 2 × 2 matrix structure of Eq. (39) results from the presence of two alternating constituents (A and B) of the considered superlattice. Note that during the derivation we did not draw a distinction between macroscopic and local electromagnetic fields. Thus, strictly speaking, expression (39) relates to the case of optically ratified media. An extension of Eq. (39) to the case of optically dense media is achieved in quite a standard way (see, e.g. [19]) introducing in Eq. (39) local field correction factors. The presence of these factors would result in a shift of resonance frequencies. This shift depends on the density of the molecular superlattice and its period. Using Eq. (39) the second-order nonlinear optical susceptibility can be straightforwardly calculated for concrete models. If the system is symmetric with respect to the interchange of the arguments s , ~ , - s in the following quantity: E G ~A°?m(K= O,E;l,s) = GAW)(E)
/Z2 IZB
2LS(o~- o~A)
(42)
M(E)=, = E - 2¢0A + iO
and neglecting the local field correction we obtain the following expression for X(2) (Eq. (39)):
(43)
This expression possesses the following resonances: the first, at w=~oA, corresponds to single-particle excitations in the sublattice A, while the second, o~= E/2, determined by zeros of the square bracket in the denominator of Eq. (43), just corresponds to excitation of Fermi resonance interface modes with eigenenergies E (Eq. (23)). In the case of finite transfer matrix elements V~r the degenerated resonances in Eq. (43) are spread out into a band. An important feature pointed out in section 3 is that for transfer integrals smaller than the interface coupling constants the bandwidth of the Fermi resonance coupled states can be rather small. It can result in relatively sharp resonances of X(2) which is favourable for an enhancement of optical nonlinearity. Numerical results for various values of the parameters will be given in Ref. [18]. Note that the non-zero value of X(2) completely originates from the Fermi resonance nature of nonlinear coupling (Eq. (2)). As is seen from Eq. (41) (see also Eq. (43)), the quantity X(2) increases with a decrease of the superlattice period L, which is a consequence of an increase of the 'concentration of interfaces'.
6. Conclusions
We have considered a periodic superlattice consisting of alternating molecular layers A and B with the Fermi resonance coupling between excitations of neighbouring layers at each interface. By means of the Green's function formalism we have found a closed equation (Eq. (21)) for the excitation spectrum in the energy range E ~ 20~A~ WB. For a given wavevector g related to the translational symmetry of the superlattice, this equation is a characteristic equation for a definite 2 × 2 matrix. The quantities entering Eq. (31) are expressed through the Green's functions of the system without interaction. For simplicity, we have considered only a quasi-onedimensional problem; the formalism and main results can be straightforwardly extended to a more general situation when excitations can propagate not only along the superlattice growth direction but also in transverse directions. Another possible generalization of the considered model is worth mentioning. It is connected to the case when the Fermi resonance takes place between an exciton in the sublattice B and a state 'exciton A +
V.M. Agranovich et al. / Synthetic Metals 64 (1994) 147-153
phonon C' in the sublattice A. The interface coupling of these three modes can also be modelled by Eq. (2) with operator products AA, A *A* replaced by A C , A ' C * with C, C* being phonon operators. The correlation function (Eq. (9)) would factorize into the product of two single-particle Green's functions for the A excitons and phonons, respectively. The studied energy range is of particular interest as it determines the second-order response to the external electromagnetic field. Using the developed formalism we have studied the nonlinear polarizabily X~2) which determines a resonant second-harmonic generation of light in the superlattice. We have found a closed expression (Eq. (39)) for X~z). The quantities entering Eq. (39) are expressed through the Green's functions of the system without Fermi resonance interaction. The value of X °) is completely caused by the Fermi resonance nature of nonlinear coupling (Eq. (2)). To prevent possible misunderstanding it is worth discussing the particular case when the molecules and superlattice possess an inversion symmetry. One expects that the second-order nonlinear polarizability, being the thirdrank tensor, vanishes in this case. This is indeed the case. In the presence of inversion symmetry the coupling constants at the interfaces A B and B A differ in sign: F A B = - - I'BA , SO the quantity X(2) (Eq. (41)) equals zero. Here in the simple estimations of our general expressions we have not clearly taken into account the intralayer interaction (Eq. (4)) between the excitation. Such an interaction (for phonons it would be an anharmonicity) may itself considerably change the twoparticle excitation spectrum resulting, for instance, in the appearance of bound states: biphonons, etc. (see [16] and references therein). This interaction would manifest itself also in nonlinear optics. The present approach can be straightforwardly extended to this case along the following line. The matrix correlation function M (Eq. (9)) could not be expressed as a simple convolution (Eq. (15)) of two single-particle Green's functions, but should be calculated as an exact two-particle propagator in the presence of the interaction between particles [16]. Analysis of this situation will be carried out in a subsequent paper. In the absence of an intralayer anharmonicity, for the considered model with the nearest-neighbour transfer of site excitations along the superlattice growth direction, we have estimated the dependence of X~2) on frequency detuning, Fermi resonance coupling constant and band structures of the constituent sublattices. The most interesting case corresponds to the situation when the states caused by the interface Fermi resonance coupling lie well apart from the quasi-continuum of
153
'bulk' excitations. In this case the bandwidth of these Fermi resonance states is very small. As a consequence, they provide a sharp resonance peak of the secondorder nonlinear polarizability lying outside the range of the main absorption due to 'bulk' excitations. Thus, the mechanism of the optical nonlinearity considered is rather perspective for attaining of relatively large nonlinear optical response at a relatively small damping.
Acknowledgements This work was supported by an Alexander von Humboldt Award (V.M.A.), NSF Grant DMR-8918894 (V.M.A.), Grant 1-044, Physics of Solid State Nanostructures, Russian Ministry of Sciences (V.M.A.), and by the Volkswagen Foundation (V.I.Y.).
References [1] F.F. So, S.R. Forrest, Y.Q. Shi and W.H. Steier, Appl. Phys. Lett., 56 (1990) 674. [2] D.-Y. Zang, Y.Q. Shi, F.F. So, S.R. Forrest and W.H. Steier, Appl. Phys. Lett., 58 (1991) 562. [3] F.F. So and S.R. Forrest, Phys. Phys. Lett., 66 (1991) 2649. [4] M. MObus, M. Schreck and N. Karl, Thin Solid Films, 175 (1989) 89. [5] U. Zimmermann, G. Schnitzler, N. Karl and E. Umbach, Thin Solid Films, 175 (1989) 85. [6] U. Zimmermann and N. Karl, Surf. Sci., 268 (1992) 296. [7] M. M6bus and N. Karl, Thin Solid Films, 215 (1992) 213. [8] N. Karl and N. Sato, Mol. Cryst. Liq. Cryst., 218 (1992) 79. [9] C. Ludwig, B. Grompf, W. Glatz, J. Peterson, W. Eisenmenger, M. M6bus, U. Zimmermann and N. Karl, Z. Phys. B, 86 (1992) 397. [10] A.J. Dann, H. Hoshi and Y. Maruyama, J. Appl. Phys., 67 (1990) 1371. [11] N.R. Armstrong, K.W. Nebesny, G.E. Collins, L.-K. Chau, P.A. Lee, C. England, D. Diel, M. Douskey and B.A. Parkinson, Thin Solid Films, 216 (1992) 90. [12] G.E. Collins, K.W. Nebesny, C. England, L.-K. Chau, P.A. Lee, B.A. Parkinson and N.R. Armstrong, J. Vac. Sci. Technol. A, 10 (1992) 2902. [13] V.M. Agranovich, Phys. Scr., T49 (1993) 699. [14] V.M. Agranovich and O.A. Dubovsky, Chem. Phys. Lett., 210 (1993) 458. [15] V.M. Agranovich and J.B. Page, Phys. Lett. A, 183 (5/6) (1993). [16] V.M. Agranovich, in V.M. Agranovich and R. Hochstrasser (eds.), Spectroscopy and Excitation Dynamics of Condensed Molecular Systems, North-Holland, Amsterdam, 1983, p. 83. [17] A.A. Abrikosov, L.P. Gorkov and I.E. Dzyaloshinski, Methods of Quantum Field Theory in Statistical Physics, Prentice-Hall, Englewood Cliffs, NJ, 1963. [18] Th. Neidlinger, P. Reineker, V.M. Agranovich and V.I. Yudson, to be published. [19] R.W. Boyd, Nonlinear Optics, Academic Press, Orlando, FL, 1992, Sect. 3.8.
ELSEVIER
Organic superlattices with interface Fermi resonance: optical nonlinearity V.M. Agranovich a, P. Reineker b, V.I. Yudson a'b alnstitute of Spectroscopy, Russian Academy of Sciences, Troitsk, 142092 Moscow Region, Russian Federation bAbteilung far Theoretische Physik, Universitiit Ulm, 89069 Ulm, Germany Received 16 D e c e m b e r 1993
Abstract
A mechanism of optical nonlinearity of organic superlattices is considered which is based on the Fermi resonance between molecular excitations of alternating layers. Using Green's function formalism we have obtained a closed equation for the energy states in the vicinity of the Fermi resonance and found a general expression for the second-order nonlinear polarizability which determines the second-harmonic generation of light caused by the interface Fermi resonance coupling.
1. Introduction
Recent progress in engineering of organic superlattices [1-12] makes necessary a search for new efficient mechanisms of optical nonlinearity connected with their multilayer structure. An interesting and perspective mechanism of optical nonlinearity arises in connection with the phenomenon of the Fermi resonance between excitations of neighbouring layers [13]. This interface Fermi resonance takes place when the energy of several excitations in the layer A is close to some excitation energy in the neighbouring layer B. Due to this intermolecular anharmonicity a new type of excitation may arise in the vicinity of the interface of two contacting molecular crystals [14,15]. In the present work we consider not a single interface of two crystals but a periodic superlattice consisting of many alternating molecular layers of type A and B. As in Ref. [14] we assume the presence of the Fermi resonance 2ro,4= wB at each interface. When the thickness of each of these two layers is relatively small (say, of the order of several intermolecular spacings) the presence of many interfaces may result in new physical
properties of the superlattice on a macroscopic scale. Study of superlattices is especially important due to accumulation of nonlinear optical effects with an increase of sample length. In the present paper we use Green's function formalism to obtain the excitation spectrum in the energy range E ~2¢0A ~ ~OB. We have found a closed characteristic equation for the eigenvalues corresponding to states with wave vector K, determining their translational properties. The excitation spectrum can be straightforwardly calculated for the concrete model of intermolecular transfer integrals. The studied energy range is of particular interest as it determines a resonant second-harmonic generation of light in the superlattice. Using the developed formalism we calculate the nonlinear optical polarizability X (2) describing the second-harmonic generation in a superlattice with Fermi resonance interfaces. Within the considered model, a non-zero value of X (2) is completely caused by the Fermi resonance nature of nonlinear coupling. A dependence of )((2) on coupling constant, frequency detuning and excitation band structure is analysed. 2. The model and basic equations
* W e are happy to contribute this paper to the Festschrift for Professor H. Inokuchi, great scientist and friend. W e wish him many years of happiness and successful activity.
0379-6779/94/$07.00 © 1994 Elsevier Science S.A. All rights reserved
SSDI 0 3 7 9 - 6 7 7 9 ( 9 4 ) 0 2 0 9 9 - K
We consider a model described by the Hamiltonian Z = Z O -~- ~/~int "1- aCC~anh, where X o = Y.4 + ,'~8 is a sum
148
V.M. Agranovich et aL / Synthetic Metals 64 (1994) 147-153
of Hamiltonians of noninteracting excitations inside layers A and B, respectively. Here
= E'o4A N,I
. ,AN,+ , X V'u,AN,/IN,,, *
(1)
N,l,l'
and the Hamiltonian ,Tg'B is given by the replacement
A-*B. In Eq. (1) the lower indexN=O, +1, ... counts the periods of the superlattice, and l=1, ...n4 the molecular monolayers of type A(B) in one period of the superlattice. The matrix element Vu, describes excitation transfer between sites l, l'. The Hamiltonian ~ i . t describing the interface interaction corresponding to the Fermi resonance condition 2o4 = ~on is given by t
'~int = E[FBA(AN.I) N
2
t
2
O u - 1 , r i b -1- I ~ A B ( A N , nA) ON,1 "J~h . c . ]
(2) Here the coupling constants F4B and FBA correspond to the interfaces AB and BA, respectively. In a general case of molecules without inversion symmetry these constants are typically non-zero and may differ considerably. In the case of molecules with an inversion centre, the quantities /'AB and /"BA are not independent but connected to each other: F=F4B = +_FB4, if the states '2/1' and 'B' are of the same or of the opposite parity, respectively. In its turn, the constant F may vanish for sufficiently symmetric A and B molecules and their relative orientation. Hamiltonian (2) describes transformation of two A excitations into a single B excitation and vice versa. In its turn, the two 'A' interactions involved in this process may interact with each other. This intralayer interaction can be described by the following Hamiltonian (see, e.g., [14,16]): t 2 x : . . . --- u4 E(AN.,)
N,I
2
(3)
For the case when the state A is a phonon one, Hamiltonian (3) corresponds to the intralayer anharmonicity. The model (Eqs. (1)-(3)) is an extension of that considered in Ref. [14] to the case of a periodic superlattice. For simplicity and similar to Ref. [14], we treat the system as quasi-one-dimensional, i.e. we neglect excitation transfer in the transverse direction. A generalization to the three-dimensional situation can be done straightforwardly. Note that up to now we did not specify the nature of the excitations which could be, for example, optical phonons or Frenkel excitons. The operators entering Eqs. (1) and (2) will be boson and paulion ones, respectively. The following formalism allows us to treat both cases on equal footing. However, from the physical point of view one should consider only the cases of
exciton--exciton and phonon-phonon Fermi resonances, while the exciton-phonon resonance would correspond to an exotic situation of comparable energies of electron and phonon subsystems. The spectrum of lowest energy excitations of the model is very simple: it corresponds to single-particle excitation of the type A. This excitation is confined to a certain period of the superlattice. Thus, one cannot expect a manifestation of the interaction considered in the linear response of the system to an external field of frequency ~0= w4. A higher band of the energy spectrum of the model corresponds to a mixture of the two excitations of type A and a single excitation of type B. There is still a trivial case when two A excitations are created in different periods, then coupling to B excitations is obviously absent. An interesting situation arises when two A excitations are created at the same period, then mutual transformation of these excitations to the B excitation and vice versa takes place. The spectrum of such coupled states is determined by the poles of the time-energy Fourier transform of the single-particle Green's function:
GB(N,I,t;N',I',t') = -i(O[T{BN,t(t)B~,.t,(t')}[O)
(4)
or the two-particle Green's function:
GA(Nl,l~,tl;N2,12,t2;N'~,l'~,t'~ ;N'z,l'z,t'z ) -" ---t(OlT{AN,
t ~ t , a,(tl)AN2,t2(t2)Aul.r~(tm)A m.,~(tz)}lO) (5)
of the system with interaction. The operators entering Eqs. (4) and (5) are taken in the Heisenberg representation with respect to Hamiltonian Z . For a perturbative analysis it is more convenient to use the interaction representation [17]:
GB(N,I,t;N',I',t ') =-i(O[T{BN,,(t)B~,.,,(t') 00
where all the operators are taken in the Heisenberg representation with respect to Hamiltonian ~¢'o of noninteracting system, and the subindex 'c' denotes a part corresponding to connected Feynman diagrams. Eq. (6) is the starting point of the following consideration.
3. Green's function formalism
In the absence of the coupling (in the zeroth order in X:'i.t) the single-particle Green's functions are given by
V.M. Agranovich et al. / Synthetic Metals 64 (1994) 147-153
G(a°(~)(N,l,t;N',l ',t') - 8N,N'GBc4)(I,t,I (o) . , ,t , )
(7)
where the Green's function on the right-hand side relates to a single period. The term linear in ,,~.~ does not contribute to Eq. (6) as it contains an odd number of B and B* operators. The contribution to Eq. (6) of the second order in J~int is determined by
Z J dr1 +
2
2
.
s,s'= +
(8)
A correlation function (OtT{B(t)B*(t')B*(tz)B(t,)}[O)~, which arises in Eq. (8), reduces to iZG(°)(t;tz)G(B°)(h;t ') for both boson and paulion operators B due to the particular time ordering (tz>q). Also we introduce a notation:
Note, in passing, that in a simple case when one neglects a mutual interaction of two A excitations, the correlation function (Eq. (11)) reduces to a convolution of two single-A-particle Green's functions:
M(E) ,s'= 2i
fd, ~
G(°)(K=O,E/2 + e;s,s') (15)
G(BZ)(K,E;I,I ') = G(B°),(E)Mss,(K,E)G(ff,),r(E)
M(12,t2;I,,tl)SN2.N,
= [IrBAI2M+ + ( E )
exp(-iKL) FsA F~BM + _ (E) exp(iKL)]
(9)
for another correlation function arising in Eq. (8); obviously, M ( l j 2 ; l , q ) ~ O(t2-q). All the states of the periodic superlattices can be classified by the wave vector K taking values in the interval ( - w / L , ¢r/L), where L is the period of the superlattice. It is natural to consider the Fourier transformation:
(16)
where the summation over repeated indices s, s ' = _+ is implied, and the matrix M(K,E) is connected with the matrix M(E) (Eq. (12)):
M(K,E)
i(OIT{(AN~,,~(t2))2(A~,.,,(q))2}IO)~
(14)
Taking into account Eqs. (9)-(14), the expression for the Fourier component of the second-order correction, Eq. (8), to Green's function can be represented in the following compact form:
1(,2)
+ I~AB(A*N,,,A)ZBu,.I](,I)}IO>¢
(o)
X G(A°)(K= O,E/2 - e;s,s')
X [I-~BA(AtN,,I)2BN, _ a,,,s
-
_
O(t -td
× (O]T{BN, z(t)B*u,.r(t' )[FBAB* t
a.( o )
149
IFA.I2M__(E)
]
(17)
One can easily check that the higher order corrections to Green's function are described by terms with a higher number of matrix 'self-energy' Iql insertions. Thus, Green's function Gn(K,E;I,I') may be represented as
G.(K,e;t,t') =
+
(18)
G(N,I,t;N',I',t')=
-~
-~
X e x p { i [ K L ( N - N ' ) - E ( t - t')]}G(K,E;l,l ') dE M(l,t;l ',t') = f - ~ e x p [ - i E ( t - t ' ) ] M ( E ; l , l ' )
where the T matrix is determined by (10)
"I'(K,E) = M(K,E) + M(K,E)G(n°)(E)M(K,E) +...
(11)
Summation of this geometrical series results in the following expression:
where N is a total number of superlattice periods. To shorten expressions it is convenient to introduce a matrix notation: M(E) = {Mss,(E)}; s,s'= +_
(12)
with M+ + (E) =M(E; 1,1);
M _ + (E) =M(E;nA,1);
M+ _ (E) =M(E; 1,hA)
M_ _ (E) =M(E;nA,nA) (13)
We will also write Green's function G~°)(E;l,l ') in the form Gsm(E) (o) replacing the indices l, l' by s, s' = + when 1, l' denote one of the end points: nB ~ + ; 1 ~ - . We introduce the following 2 × 2 matrix:
r(K,E)
(19)
= B -
× [G~O)(E)]-I _ [G~O)(E)]--1
(20)
where I denotes the unit 2 × 2 matrix. Eqs. (18)-(20) give a formal solution to the problem of Green's function calculation for the system (Eqs. (1) and (2)). The spectrum of excitations coupled by the Fermi resonance interaction (Eq. (2)) in the energy range of interest (E = tos--- 2WA) is determined by the poles of the "1" matrix or, equivalently, by the characteristic equation: det[I - M(K,E)G~°)(E)] = 0
(21)
This equation is one of the main results of our analysis. The quantities entering Eq. (21) are expressed
150
V.M. Agranovich et al. / Synthetic Metals 64 (1994) 147-153
through Green's functions of the system without interaction. Note that up to now all the obtained expressions are valid for both boson and paulion operators. The difference between them would arise only in the calculation of the two-particle Green's function (Eq. (9)). For noninteracting bosons this function reduces to the product of the single-particle Green's functions, while for paulions such factorization does not hold. Using Eq. (21) the excitation spectrum of interest can be straightforwardly calculated for concrete models of intermolecular transfer matrix elements Vtr. To get an idea of the Fermi resonance phenomenon we consider first a trivial case of very small transfer matrix elements V~t,. In this limit we have EA(B) --~ O)A(B)and the matrices G(B°)(E) (Eq. (14)) and M(K,E) (Eq. (17)) reduce to the diagonal ones with G(O) + (E) = ~(o) _ (E) = [ E - w B ] - ' B+
(K,E)
M+ + = 2IrBAIZ[E-M _ _ (X,E) = 21rAnl2[E-- 2WA]-'
(22)
where the dipole moment operator for the sublattice A is given by PA = ~AXN.,[A~,, +AN.,]
with that for the sublattice B being determined by the replacement A ~ B ; I~A denotes the transition dipole moment corresponding to these excitations. The nonlinear polarization of interest arises as the second-order response to the perturbation (Eq. (24)): e(2)(t) = ~
1
Tr{P(t)p(2)(t)}
0)A--
+2/~
(23)
Here F should be substituted by FAn o r Xg'nA for the interfaces AB or BA, respectively. Of course, in the absence of excitation transfer there is no K dependence in the right-hand side of Eq. (23); the solutions (Eq. (23)) correspond to the modes localized just at interfaces AB or BA and, naturally, coincide with those found in Ref. [14] for a single interface. Concerning now the case of finite transfer matrix elements Vtr we point out another interesting and important situation when they are still smaller than the interface coupling constants. In this case the energies (Eq. (23)) spread out into a rather narrow band with a width exponentially decreasing with an increase of parameters In IF/I~ and the superlattice period L. An elaborate analysis and numerical study of Eq. (21) will be given in a separate paper [18]. Now we proceed with nonlinear optical properties caused by the interface Fermi resonance coupling.
4. Nonlinear optical response
For not too thick samples and for normal incidence (for simplicity) of an external electromagnetic (EM) field, both the field and induced polarization inside the sample may be considered as uniform and the Hamiltonian of interaction of the superlattice with the EM field E(t) can be represented in the form:
V(t) = - [PA + PB]E(t~ - - PE(/)
(24)
(26)
where L is the superlattice period and N is a total number of the periods; S is an area of the elementary cell in the plane transversal to the growth direction. The second-order correction p(2) to the density operator is given by
o(2'(tl=(-i)2f
dt' dt" O(t > t"> t')[V(t"l,[V(t'),O(°)]]
The characteristic Eq. (21) reduces in this limit to a couple of quadratic ones with the following roots: E±(K)=tOA+ ~- _+
(25)
(27) Operators entering Eqs. (26) and (27) are taken in the interaction representation with respect to the interaction Eq. (24) or, in other words, in the Heisenberg representation with respect to the Hamiltonian YU of the system without an external EM field. The operator p(0) in Eq. (27) is the density operator of the unperturbed system. With use of Eqs. (24), (26) and (27) one obtains
1 ~ -2 dt' dt" O(t>t">t') e ( z ) ( t ) = - NL--S xTr{[[P(t),P(t")],P(t')]p(°)}E(t")E(t ')
(28)
We assume the exciting EM field:
E(t) = ~o, exp( - itot) + c.c.
(29)
to be in an approximate resonance with the type-A transition, i.e. tO-----tOA. In resonance (rotating wave) approximation one takes into account only the processes where the EM field (Eq. (29)) interacts with the A excitations, thus only the A part (Eq. (25)) of the operators P(t"), P(t') should be retained in Eq. (28). Similarly, only the part B of the operator P(t) contributes to the second-order polarization (Eq. (28)). Assuming that no excitations are present in the unperturbed system, i.e., p(O)= 10( >0l, where 10> is the ground state, we come to the following expression for the Tr{..-} in the right-hand side of Eq. (28): Tr{.--} =
(30)
Representing the nonlinear polarization P~2~(t) in the form: P(2)(t) = X(2)(2w;to, co)~ exp( - 2i~) + c.c.
(31)
151
V.M. Agranovich et al. / Synthetic Metals 64 (1994) 147-153
and using Eqs. (25), (28) and (30), we obtain the following expression for the second-order nonlinear optical susceptibility X(2): X(2)(2oJ;to, o~)=
tz~B L-----~V(2to;to, to)
(32)
kK~)(t,t",t ') = -~ ~
d~l
× (FBAG(z°)(N,I,t;N~ - 1,nB,q) × GA(o)(NI,I,ta;N ,l ,t ) ,,
where
× G(A°)(N,,1,q;N',I
v(2;to,o,)=f f
,,
',t')
+ FABG(8°)(N,I,t;N~,I,q)
dt'dt"V(t,t",t')
× G~°)(N~,nA,l,t~;N",l",t ")
× exp[ - 2/to(t' + t" - 2t)]
(33)
The 'vertex' function V(t,t",t') in Eq. (33) is determined
as
V(t,t",t') = - 2---N 1 ~(OlT{BN't(t)A~"'r(t")A*u"r(t')}lO) (34) where the sum goes over all lower indices (molecular sites). It is easily seen that the vertex function (Eq. (34)) is the only term of all the possible arrangements of the operators in Eq. (30) making non-zero contribution to ,)((2). The obtained expressions (Eqs. (32)-(34)) reduce calculation of the nonlinear susceptibility X~2) to the problem of calculation of the vertex function V(t,t",t'). The latter problem allows a regular field-theoretical analysis carried out in the next section.
5. Calculation of the vertex function and estimation of X (2)
It is convenient to rewrite the expression for the vertex function (Eq. (34)) in the interaction representation with respect to the Hamiltonian of the interface Fermi resonance coupling ,'Y:i,, (Eq. (2)):
lz(I {
V(t,t",t')= - - ~
,,
0 T BN, t(t)A~.r(t")A~,r(t')
× G~°)(NI,nA,I,t,;N ',l',t')}
(36)
where the summation goes over all the sites Nl, N'l', N'T', N1 and ll; G~°) and G~°) are the single-particle Green's functions of noninteracting subsystems A and B. For simplicity, the operators A, A* have been assumed to be boson ones, while operators B, B* may be both boson or paulion ones. Using the Fourier transformation we obtain the following expression for the vertex function V°)(2w;to, o~) (Eq. (33)): V~a)(Zto;w,~o)=
~
l,l',l";s
FsG~°)(K=O,Zw;l,s)
× G
=/'BA, F_ =
GB(K=O,E;I,s) = ~ G~°)(K=O,E;I,s ') S=-t-
Xexp[--i f®,,~int(tl)dt,]} O>c
(35)
where the lower index c denotes a connected component of the average. In the zeroth order in X/i,t we have for the vertex function (Eq. (35)): V~°)=0, as in this case only correlation functions with equal numbers of creation and annihilation operators could differ from zero. In the first order in Y~,t the vertex function (Eq. (35)) is given by
× [I - M(E)G(B°)(E)];:
(38)
Note that the matrix [. -- ]- 1 entering Eq. (38) is closely connected with the T matrix (Eq. (20)) for the general case K:~0; its poles determine the excitation spectrum of the system in the Fermi resonance range. The vertex function V(2to;to, to) (Eq. (33)) is obtained from Eq. (37) by replacement of G
V.M. Agranovich et aL / Synthetic Metals 64 0994) 147-153
152
X(z)(2oJ; o~,oJ) =
tz2 lzB ~ F,G(B°)(K=O,20;I,s) L S t. r, t'; s, s"
X (2)(2o~;oJ,oJ) = (/An + FBA)
× [ I - M(20)G~°)(Zo)];, ' × G~°)(K = O,~o;s',l')G~°)(K = O, o4s',l")
XZ
(40)
1
we obtain eventually a simpler expression for X (z) (Eq. (39)): X
(41)
To get an idea of the manifestation of the Fermi resonance phenomenon, we consider first the simple case of narrow bandwidths for bothA and B excitations, i.e., small transfer matrix elements Vn,. In this limit
art,
G (A°0B)(K= O,E;l,l') - E - OJA(m+ iO
2[r'lz
a.,
• 1 s= ± ( 2 ~ o - ) ( ~ o - WA)--Ir, I=
(39)
This equation is the main result of our analysis. The quantities entering Eq. (39) are expressed through Green's functions of the system without the interface Fermi resonance interaction. However, this expression holds also for the case when there is an interaction between excitations of the same type, e.g., intralayer anharmonism, etc. The formalism developed is an extension of an earlier one [16] applied to a uniform medium to the case of interface Fermi resonance in superlattices. The 2 × 2 matrix structure of Eq. (39) results from the presence of two alternating constituents (A and B) of the considered superlattice. Note that during the derivation we did not draw a distinction between macroscopic and local electromagnetic fields. Thus, strictly speaking, expression (39) relates to the case of optically ratified media. An extension of Eq. (39) to the case of optically dense media is achieved in quite a standard way (see, e.g. [19]) introducing in Eq. (39) local field correction factors. The presence of these factors would result in a shift of resonance frequencies. This shift depends on the density of the molecular superlattice and its period. Using Eq. (39) the second-order nonlinear optical susceptibility can be straightforwardly calculated for concrete models. If the system is symmetric with respect to the interchange of the arguments s , ~ , - s in the following quantity: E G ~A°?m(K= O,E;l,s) = GAW)(E)
/Z2 IZB
2LS(o~- o~A)
(42)
M(E)=, = E - 2¢0A + iO
and neglecting the local field correction we obtain the following expression for X(2) (Eq. (39)):
(43)
This expression possesses the following resonances: the first, at w=~oA, corresponds to single-particle excitations in the sublattice A, while the second, o~= E/2, determined by zeros of the square bracket in the denominator of Eq. (43), just corresponds to excitation of Fermi resonance interface modes with eigenenergies E (Eq. (23)). In the case of finite transfer matrix elements V~r the degenerated resonances in Eq. (43) are spread out into a band. An important feature pointed out in section 3 is that for transfer integrals smaller than the interface coupling constants the bandwidth of the Fermi resonance coupled states can be rather small. It can result in relatively sharp resonances of X(2) which is favourable for an enhancement of optical nonlinearity. Numerical results for various values of the parameters will be given in Ref. [18]. Note that the non-zero value of X(2) completely originates from the Fermi resonance nature of nonlinear coupling (Eq. (2)). As is seen from Eq. (41) (see also Eq. (43)), the quantity X(2) increases with a decrease of the superlattice period L, which is a consequence of an increase of the 'concentration of interfaces'.
6. Conclusions
We have considered a periodic superlattice consisting of alternating molecular layers A and B with the Fermi resonance coupling between excitations of neighbouring layers at each interface. By means of the Green's function formalism we have found a closed equation (Eq. (21)) for the excitation spectrum in the energy range E ~ 20~A~ WB. For a given wavevector g related to the translational symmetry of the superlattice, this equation is a characteristic equation for a definite 2 × 2 matrix. The quantities entering Eq. (31) are expressed through the Green's functions of the system without interaction. For simplicity, we have considered only a quasi-onedimensional problem; the formalism and main results can be straightforwardly extended to a more general situation when excitations can propagate not only along the superlattice growth direction but also in transverse directions. Another possible generalization of the considered model is worth mentioning. It is connected to the case when the Fermi resonance takes place between an exciton in the sublattice B and a state 'exciton A +
V.M. Agranovich et al. / Synthetic Metals 64 (1994) 147-153
phonon C' in the sublattice A. The interface coupling of these three modes can also be modelled by Eq. (2) with operator products AA, A *A* replaced by A C , A ' C * with C, C* being phonon operators. The correlation function (Eq. (9)) would factorize into the product of two single-particle Green's functions for the A excitons and phonons, respectively. The studied energy range is of particular interest as it determines the second-order response to the external electromagnetic field. Using the developed formalism we have studied the nonlinear polarizabily X~2) which determines a resonant second-harmonic generation of light in the superlattice. We have found a closed expression (Eq. (39)) for X~z). The quantities entering Eq. (39) are expressed through the Green's functions of the system without Fermi resonance interaction. The value of X °) is completely caused by the Fermi resonance nature of nonlinear coupling (Eq. (2)). To prevent possible misunderstanding it is worth discussing the particular case when the molecules and superlattice possess an inversion symmetry. One expects that the second-order nonlinear polarizability, being the thirdrank tensor, vanishes in this case. This is indeed the case. In the presence of inversion symmetry the coupling constants at the interfaces A B and B A differ in sign: F A B = - - I'BA , SO the quantity X(2) (Eq. (41)) equals zero. Here in the simple estimations of our general expressions we have not clearly taken into account the intralayer interaction (Eq. (4)) between the excitation. Such an interaction (for phonons it would be an anharmonicity) may itself considerably change the twoparticle excitation spectrum resulting, for instance, in the appearance of bound states: biphonons, etc. (see [16] and references therein). This interaction would manifest itself also in nonlinear optics. The present approach can be straightforwardly extended to this case along the following line. The matrix correlation function M (Eq. (9)) could not be expressed as a simple convolution (Eq. (15)) of two single-particle Green's functions, but should be calculated as an exact two-particle propagator in the presence of the interaction between particles [16]. Analysis of this situation will be carried out in a subsequent paper. In the absence of an intralayer anharmonicity, for the considered model with the nearest-neighbour transfer of site excitations along the superlattice growth direction, we have estimated the dependence of X~2) on frequency detuning, Fermi resonance coupling constant and band structures of the constituent sublattices. The most interesting case corresponds to the situation when the states caused by the interface Fermi resonance coupling lie well apart from the quasi-continuum of
153
'bulk' excitations. In this case the bandwidth of these Fermi resonance states is very small. As a consequence, they provide a sharp resonance peak of the secondorder nonlinear polarizability lying outside the range of the main absorption due to 'bulk' excitations. Thus, the mechanism of the optical nonlinearity considered is rather perspective for attaining of relatively large nonlinear optical response at a relatively small damping.
Acknowledgements This work was supported by an Alexander von Humboldt Award (V.M.A.), NSF Grant DMR-8918894 (V.M.A.), Grant 1-044, Physics of Solid State Nanostructures, Russian Ministry of Sciences (V.M.A.), and by the Volkswagen Foundation (V.I.Y.).
References [1] F.F. So, S.R. Forrest, Y.Q. Shi and W.H. Steier, Appl. Phys. Lett., 56 (1990) 674. [2] D.-Y. Zang, Y.Q. Shi, F.F. So, S.R. Forrest and W.H. Steier, Appl. Phys. Lett., 58 (1991) 562. [3] F.F. So and S.R. Forrest, Phys. Phys. Lett., 66 (1991) 2649. [4] M. MObus, M. Schreck and N. Karl, Thin Solid Films, 175 (1989) 89. [5] U. Zimmermann, G. Schnitzler, N. Karl and E. Umbach, Thin Solid Films, 175 (1989) 85. [6] U. Zimmermann and N. Karl, Surf. Sci., 268 (1992) 296. [7] M. M6bus and N. Karl, Thin Solid Films, 215 (1992) 213. [8] N. Karl and N. Sato, Mol. Cryst. Liq. Cryst., 218 (1992) 79. [9] C. Ludwig, B. Grompf, W. Glatz, J. Peterson, W. Eisenmenger, M. M6bus, U. Zimmermann and N. Karl, Z. Phys. B, 86 (1992) 397. [10] A.J. Dann, H. Hoshi and Y. Maruyama, J. Appl. Phys., 67 (1990) 1371. [11] N.R. Armstrong, K.W. Nebesny, G.E. Collins, L.-K. Chau, P.A. Lee, C. England, D. Diel, M. Douskey and B.A. Parkinson, Thin Solid Films, 216 (1992) 90. [12] G.E. Collins, K.W. Nebesny, C. England, L.-K. Chau, P.A. Lee, B.A. Parkinson and N.R. Armstrong, J. Vac. Sci. Technol. A, 10 (1992) 2902. [13] V.M. Agranovich, Phys. Scr., T49 (1993) 699. [14] V.M. Agranovich and O.A. Dubovsky, Chem. Phys. Lett., 210 (1993) 458. [15] V.M. Agranovich and J.B. Page, Phys. Lett. A, 183 (5/6) (1993). [16] V.M. Agranovich, in V.M. Agranovich and R. Hochstrasser (eds.), Spectroscopy and Excitation Dynamics of Condensed Molecular Systems, North-Holland, Amsterdam, 1983, p. 83. [17] A.A. Abrikosov, L.P. Gorkov and I.E. Dzyaloshinski, Methods of Quantum Field Theory in Statistical Physics, Prentice-Hall, Englewood Cliffs, NJ, 1963. [18] Th. Neidlinger, P. Reineker, V.M. Agranovich and V.I. Yudson, to be published. [19] R.W. Boyd, Nonlinear Optics, Academic Press, Orlando, FL, 1992, Sect. 3.8.