Spatial and orientational ordering of charge-transfer excitons at intense pumping. Absorption and fluorescence spectra

Volume 123, number 2

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20 July 1987

SPATIAL AND ORIENTATIONAL ORDERING OF CHARGE-TRANSFER EXCITONS AT INTENSE PUMPING. ABSORPTION AND FLUORESCENCE SPECTRA B.P. ANTONYUK Institutefor Spectroscopy, USSR Academy ofSciences, Troitsk, Moscow Region 142092, USSR

and B.E. STERN Institutefor Nuclear Research, USSR Academy ofSciences, Troitsk, Moscow Region 142092, USSR Received I May 1987; accepted for publication 7 May 1987 Communicated by V.M. Agranovich

The non-linear properties of charge-transfer exciton dynamics at intense pumping are investigated. The shift of an absorption band ofan arbitrary cell, induced by the electric field of the available excitons, results in spatial and orientational ordering of the excitons. The side bands appear in the absorption and fluorescence spectra.

In a charge-transfer exciton the electron and the hole are spatially separated. It results in two important peculiarities of the above-mentioned excitons: every exciton has a static electric dipole moment and is localized in space (self-trapped). The origin of self-trapping is a strong Coulomb interaction between the separated electron and hole which provides a strong lattice distortion and, as a result, leads to a large effectivemass of the chargetransfer exciton. The dipole moment of an exciton is of the order ofea (e is the electron charge, a is the lattice constant). It means that each exciton generates a strong long-range electric field and every new exciton is created in the field varying in space. The cross section of photon absorption with creation of a single charge-transfer exciton is [1,21 a=a0 exp{— [(w—~—A)I4]2}, where cv is the frequency of the photon, h~is the excitation energy, A is the Stokes shift, 4 10_2 eV is the band width, o~—~10~19cm2. Assume that two exciton dipole moment orientations are possible. The excitation energy in the electric field, generated by the available excitons, is he + h V, where V, is the dipole—dipole energy ‘~

~ =

~11~1,

~

d~(1—3cos29,~)

±1 corresponds to two possible dipole moment orientations of created excitons, n

3 = ±1 corresponds to orientations of available excitons, n~=0 means that site j is nonexcited. In the case discussed the cross section becomes dependent on the number of the site i and on the sign of n~, (1) Our previous investigations [3,4] within the framework of a model with one orientation have shown that 0375-960 11871$ 03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)

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d~ d~’ ‘S

a Fig. 1. Possible homogeneous states: (a) ~/a

0 <0; (b) ~/a0>0.

the sensitivity of the cross section of exciton creation in an arbitrary cell to the positions of excitons already existing results in spatial ordering of excitations. The ordering discussed is possible in spite of the fact that each exciton annihilates at the same place where it has been created and a motion of excitons is absent. In the present paper for a model with two orientations it is shown that not only spatial ordering, but also orientational ordering is possible. One, two, or many orientations are realized in various charge-transfer molecular crystals [5]. Let ps,, be the probability of occupation of site i by the exciton with dipole moment orientation v= ±1. In the self-consistent approximation Pva obey the equations ~

(2a)

where I is the photon flux and y is the damping constant. The sum L~p~’,v’ d1 is the average dipole moment of the site i. For the variables d, eqs. (1) give =2exp

[_ (~_~

—2exp

[_ (~+~ a~cij)~]—di,

(2b)

with 2—J~o/Y, r=yt,

~=(cv—e—A)I4,

a,~=V~I4.

The static homogeneous solution d 1 = d of the equation 2]—2 exp[ (~+a 2] d=2 exp[— (~—a0d) 0d) can be easily obtained from fig. 1. We consider the lattices with >.~a~, ct —

0 ~ 0. In the case ~Ict~, <0 there exists only one solution cii’ = 0 for all pumping 2 (fig. 1 a). If c~/a0>Otwo more solutions appear at high enough 2 (fig. lb). With the increase of 2, di’—~d?and the system jumps from the unpolarized state d~’to the polarized state d’~or di’. The jump back occurs at a smaller intensity 2, when di’ = di’ the system reveals bistability. The states di’ and di’ may coexist and be separated by a diffusible domain wall. For a perturbation d1—’exp( —iwt+ik.R~)of the homogeneous states we find from (2) 2]+2a~A(~—a 2]—1, —icvk=2cEA2(4~—aOd)exp[—(~—a0d) 0d) exp[—(c~+ot0d) —

—

—

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~

_______

_______

20 July 1987

_______

~‘j

i~i;’

Fig. 2. Possible states: (a) ~/a <0, antiferroelectric-type ordering; (b) c/a> 0, ferroelectric-type ordering.

where aa = ~ a,~exp [iF (R, —Ri)]. One can easily see that the solutions ±d~are unstable at all parameters (the derivative from curve 2 at d= ± di’ is greater than 1, see fig. lb ). The unpolarized state di’ (fig. 1 b) becomes unstable if 4a0~Ae~2 > 1. The polarized states ±di’ are stable against homogeneous perturbations, but become unstable against perturbations with large enough k when a,~ has sign opposite to a0. The solution 4ak~Aedi’~2>in the 1 is case fulI a) is unstablesolutions against inhomogeneous perturbations when the condition — filled.<0 So(fig. all homogeneous are unstable at high enough pumping. We shall take only into account the interaction of neighbouring sites a,~ = a and shall analyze the competition between the homogeneous state and the simplest inhomogeneous state with the wave number q = (it Ia)(1, 1, 1), (aq= — a 0). There are two sublattices of dipole moments d, and d2 in the second state. For the values d1 and d2 the system (2) gives 2]—2exp[—(~+2pad 2], d1=2exp[—(~—2pad2) 2) d 2 = 2 exp [ — (~ — 2pad, ) 2] —2 exp [— (~ + 2pad, ) 2], (3) -

where p is the dimensionality of the crystal. The graphic solutions of eqs. (3) are shown in fig. 2. If ~/a>0 and 2 is large enough one can see the above-mentioned homogeneous solutions di’, ± di’, ± di’. In addition there appear inhomogeneous solutions ±d~,± di’, for which d1 # d2. In the case ~Ia>0 the dipole moments d1 and d2 have the same sign (ferroelectric-type ordering). If ~/a <0 the sign of d, is opposite to d2 (antiferroelectrictype ordering). For small perturbations dk exp ( ik~R1)+ ôk+q exp [i (k +q) .R,J of the states found we have ô1 =k2d2

~

~2

k,öI

—d2,

where 5 exp(ik’R,) and d2 exp(ik~R,)are the deviations in the first and second sublattices, ki=—~-{2exp[_(~_aod,)2]_2exp[_(,~+aod,)2]}~, a0 k2=_~_{2exp[_(~_aod2)2]_2exp[_(~+aod2)2]}~ a2 a0

t is are (3). The frequency of oscillation 5~, ö2—~e~’° — icy the = — appropriate 1 ±~ In derivatives the states ±of di’, the ± di’,functions k, k 2 <0; these solutions are stable at all parameters. For the solutions di’, ± di’, ± di’, k, k2> 0; the homogeneous states, as mentioned above, become unstable at high enough pumping 2. So, in the case ~/a <0 and high enough 2 the system is in the state in which both the values and

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orientations of the dipole moments are modulated. In the case ~/a>O the orientation is constant, but the magnitude of the dipole moment varies in space. To prove the above statements we have performed computer simulations of the excitation process for the two-orientational model. The approach is similar to that in refs. [3,4]. The one-dimensional chain with interaction a,~=a li—il 3n,n~(n,, n~=(0, ±1)) has been considered. The computer simulation is done in the following way. If at a moment a series of sites is excited, then during the next time interval dt each of the excitons could randomly annihilate with probability ydt. The non-excited site also randomly, with probability 1a 1( ±1), changes into the excited state ±1. The excitation probability per unit time Ia,( ±1) has been calculated with regard to the distribution n,( t) ofexcitons at the moment according to (1). At the moment t= 0 all values n, = 0. The positions and orientations ofall excitons are displayed. We have directly observed spatial and orientational ordering of charge-transfer excitons at 2 ~ 1. In the case ~/a>0 excitons form a superlattice with the ferroelectric-type ordering of orientations, in the case ~/a <0 the ordering is of the antiferroelectric type. For parameters ~T=—2, a= —10, 2=5 and for time ‘r ~ all excitons form the crystal and have the same orientation. In the case ~T=—1, a= —8,2=5 we have r0= 12. If the mismatch ~ and interaction a have opposite sign the system behaves differently. The excitations with different orientations coexist for all time, the total polarization is equal to zero. The following functions have been calculated: the average dipole moments d, and densities p1, the correlation functions of dipole moments D~and densities K~,the absorption and fluorescence spectra .Ea( cv’), E~(cv’); —

di=~Jnj(~)dt, pe=~J~ni(r)l th, 2T

2T

D~=~Jnjer)nj(r)_d2,

K1j=~JInj~r)IInj(T)Jdr_p2,

2] Ea(~a)rzz

~cro(l_IniI){exP[_

(~a_

+ex~[_(~a+

~a~ini)2]},

~a~ini)

~

+J_~exp[_(~r+~atini)2]}, 1

100

1

100

d_~j~.>,di~ PJöö~Pi~

~a(cv’eA)/4,

~fZZ(w_e+A)IA.

The distribution of d, and correlation function D,~for ~ = —2, a = — 10, 2 = 5, T= 100 are shown in figs. 3 and 4 (p, = d,, K~=D,~).Figs. 3 and 4 demonstrate the orientational and spatial ordering with the correlation length 1—~1 Oa. The displayed distribution of n.(~)reveals the polycrystal structure. The perfect pieces ofcrystal 1010101 with length 1Oa are divided by the defects 1001. The defects move, can annihilate and can be created. The existence of defects results in a finite correlation length 1. In the case (~ = 2, a = — 10, 2 = 5) d is at the level of fluctuations d—~10 2, p 0.4. So, the computer simulation confirms the above self-consistent picture of orientational and spatial ordering (crystallization) of charge-transfer excitons at high pumping. The absorption and fluorescence spectra are shown in fig. 5. One can see that at high pumping new bands appear. The absorption spectrum is given for weak trial field of arbitrary frequency. Side bands appear due to absorption or fluorescence by the cells with neighbouring excitations. The unusual feature is the existence of fluorescence band with energy higher than the pumping photon energy. But it is easy to see that in the case 98

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:,~ 3031

Fig. 3. The distribution ofthe average dipole moments d

1= thffiin1(t) d~over the sites (~=—2, a=

— 10, A= 5, ~

when the exciton cluster is created by some ordered steps and annihilates in another order, the fluorescence photon may have both lower and higher energy than the pumping photon energy. Domain structure is possible if some homogeneous states It is easy to seea*; that width wall of the 1d in a one-dimensional chain is of the order ofare the stable. superlattice constant thethe domain is domain wall In the case of interaction of neighbouring sites we have microscopic. =2exp[—(~—ad,, —ad

2]—2exp[—(~+ad, 21. 1+,) 1 +ad,+1) Assume that at t=0 a macroscopic state with asymptotics lim~. ,,,d 1= — di’, lim,. ÷0,,d1=di’ is given. For such a state d, ± d and according to the equations the values d1 in every site increase if d,>0 or decrease if d <0, tending to stationary states di’; the domain wallwith sharpens and becomes microscopic (/d is equivalent a). In the case of 3 the± superlattice appears the constant a* = aa”3. The system to that interaction a,~=a/li—il

Fig. 4.

The

dipole

moments

correlation

Fig. 5. The absorption (a) and fluorescence (b) spectrafor

function

~=

— 1,

r=l00. 99 2 (~=—2, a= —10,

Fig. 4. The dipole moments correlation function D,~=thD~f~~n 1(t)n~(t) dr—d

~.=

5, K,~_=D,~).

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with neighbouring sites interaction on the superlattice a*; hence the domain wall width is 1d a*. So in general the crystal consists ofan arbitrary number of all possible domains (di’, ± di’), separated by microscopic domain walls. If an exciton annihilates in the boundary of the domain, the strong field may reconstruct the same domain or the neighbouring one, so domain walls are diffusible. Assume that the system is in the stable state di’ and a static electric field Elld 0 is applied. The energies of excitons with + 1 and — 1 orientations are changed by values 6 e = ~ d0E (d0E> 0). This shift results in the appearance of polarization in the field d=~E,where from (2) 2 4d~2e~ X — 4(1 _4çcao2e~2) (l4~ao2e~I <1), —

where ~ is the dielectric susceptibility per cell. As one can see x may be both positive or negative and in the case i~a 0>0it =1. 2 diverges with the increase of pumping near the threshold of ferroelectric-type ordering, 4i~a0Ae~ The strong shift of the absorption band by the neighbouring excitation permits one to construct various exciton clusters with the help of a few fields with proper frequencies. Let, for example, a weak resonant field be applied (A’ ~ 1, = 0). It generates independent random excitations of low concentration. If at the same time a strong field (2 ~ 1) with the mismatch ~ = a,,n 1n~is applied, near each exciton in the relative position I~,—1~ and orientations n•, n~a new exciton will be created with probability 1. The resonances for close cells are shifted more than the resonance width (I a~I ~ I). It permits one to generate only a given type of biexcitons. It is interesting that the lifetime of the biexcitons in the fields 2, 2’ may be much larger than the lifetime of a single exciton. It happens because after annihilation of the first exciton the biexciton may be reconstructed by an intensive resonant field A before the annihilation of the second one. The probability of biexciton annihilation without any reconstruction during the time I is ~‘

w(t)=2Jydti Jydt2exp[_yti_Ay(t2_ti)_yt2],

[

where the factor exp Ay( 12—ti)] is the probability of the absence of reconstruction between the annihilation time 11 of the first exciton and the second one 12. The calculation of the integral gives 2~’—e~~~’) + (1 —e2~’). w(t) = — 22 1 (e —

Let A~1, Ayt>> 1, ytic 1, then w(1) =2yt/A. For a short time interval yt<< 1 the calculated direct channel of the annihilation (without reconstruction) is dominant, hence Y2 = 2yIA is the full damping constant of the biexciton. It means that the lifetime of the biexciton in the field 2 exceeds the lifetime of single excitons by a factor 12>> 1. The authors are grateful to Professor V.M. Agranovich and Professor E.I. Rashba for stimulating discussion.

References [1] V.M. AgranovichandA.A. Zakhidov, Chem. Phys. Lett. 50(1977)278. [21 D.J. Haarer, J. Chem. Phys. 67 (1977) 4076. [3J B.P. Antonyuk and B.E. Stern, Solid State Commun. 61(1987) 675. [4] B.P. Antonyuk and B.E. Stern, Fiz. Tverd. Tela 29 (1987) 1206. [5] D.J. Haarer and M.L. Philpott, in: Spectroscopy and excitation dynamics of condensed molecular systems. I, eds. V.M. Agranovich and R.M. Hochstrasser (North-Holland, Amsterdam, 1983).

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