Lineshape of Light Absorption by Thermalized ...

N. I. GRIGORCIIUKand L. G. GRECHKO:Light Absorption by Thermalized Excitons

633

phys. stat. sol. (b) 153,633 (1989) Subject classification : 71.35 and 78.20

Institute for Chemistry of %rfaces, Academy of Sciences of the Ukrainian SSR, Kiev1)

Lineshape of Light Absorption by Thermalized Frenkel Excitons under Band-to-Band Transitions N. I. GRIGORCHUK and L. G. GRECHKO It is assumed that sublevels of the lowest exciton band in consequence of pumping are filled by excitons, which thermalize during their lifetime. The probability of throwing of such excitons to the next more higher lying band by means of another radiation source is investigated. By simple assumptions, when the distance between the exciton bands is much greater than the bandwidth difference and the damping parameter is independent of the frequency, the lineshape of light absorption under band-to-band transitions is calculated. For small damping in the case of broad bandwidth it is shown that the absorption is determined rather by temperature of the crystal than by bandwidth difference. I n the case of narrow bands only that difference plays the essential role.

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1. Introduction Traditional investigations of the radiation absorption are based on a n assumption according t o which the transitions to higher-lying excited states come from the ground state. An analysis of the experimental data [1,2], however, shows that a number of molecular crystals give a luminescence with quantum yield close to unity and that the radiation in such crystals comes from the statistical equilibrium state. I n other words, this means that in such crystals during a time smaller than the radiation lifetime of excitons their statistical distribution on sublevels of the lowest band occurs, and its number may be considered as given. Therefore, the question about the photoexcitation of such excitons from the lowest band to the next higher one arises, and the spectral peculiarities of the light absorption a t such transitions need investigation. Earlier the problem of interband electron spectra and their structure were considered [3], and theoretically the transitions from the phonon band of a ground vibronic state to an exciton band were investigated [a].I n [3]the energies and the intensity of the exciting states were calculated. Assuming for benzene, anthracene, and naphthacene crystals narrow phonon bands it was shown in [4]that the absorption contour under such transitions can repeat the density of band exciton states. l)

Prospekt Nauki 31, SU-252028 Kiev, USSR.

N. I. GRICORCHUK and L. G . GRECHKO

634

Those investigations, however, do not allow t o explain a number of experimentally observable facts. Particularly, as is well known [5], the absorption spectra in Hparaphine remain invariable from nitrogen temperature till room temperature, antl a t low temperatures the contour does not split into tens and hundreds single lines. It is not yet clear what are the reasons for such lineshape formations if inultiphonon processes do not play any role. We shall t r y a n answer t o those and some other questions in the present, paper.

2. General Expressions for the Transition Probability Let 11sconsider a molecular crystal with simple nondegenerate bands whose extremum lies in the centre of the Brillouin zone. Let every elementary cell contaiii one molecule. We assume t h a t the exciton bands are sufficiently separated antl configuration mixing of excitoii states in each band is absent. The state of an exciton in the i - t h band will be described by the function If,k,), where f t denotes a set of quantum numbers associated with the i-th excited state of a molecular crystal, k, is the exciton quasimomentum in the i-th band. The energy which corresponds to a such state is

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E j " ( k )= Ef, F t ( k 2 ) . (1) We shall also suppose that in the initial state the excitons are distributed on the lowest band sublevels with probability W(le,).Under a photon flus of concentration no and polarization a ( = 11, I) the excitons are excited from the i-th to the i 1st state. Let us introduce the phenoinenological parameter which will take into account the exciton damping due t o their scattering on imperfections of the crystal lattice and other collisions. Further, for convenience, we p u t i = 1 for the lowest band and i = 2 for a n upper band. Then the transition probability Iflk,) -+ ),&fI by means of the above-mentioned parameters may be written in the form [G]

r

+

m

where H' represents an interaction operator of the radiation field of v frequency with the crystal and Nl is the total number of excitons in the first band. The matrix element between the ~ a v functions e of Frenkel excitons in the dipole approximations is

where clj2f1and cuf2f, are the dipole momentum matrix element and the transition frequency from the state (flkl) to the state lf2k2),respectively, Q the phot.on wave vector with polarization CJ ( 1 01 = 1). The 8-symbol reflects the momentum conservation law of interacting particles. Using (3) one can rewrite ( 2 ) as follows:

where the dimensionless frequency function m

Lineshape of Light Absorption by Thermalized Frenkel Excitons

635

defines in general form the lineshape of light absorption by excitons under band-toband transitions. Below we will calculate this function for particular cases in more detail. 3. Band-to-Band Exeiton Light Absorption in the Case of Broad Bands

I n media with high dielectric constant, where the interaction is weak, the description of a n exciton may be carried out in the effective-mass approximation. Hence, for the exciton states of lowest energy one can write the following expression:

where m,,2 means the exciton mass of the first and second band, respectively. From here follows that the distribution function of excitons of the first band, obviously, will be a Boltzmann function,

where V is the volume of crystal. Substituting (6) and ( 7 ) into ( 5 ) , after integration in spherical coordinates over all quasimomenta of the first exciton band, one gets m

P ( v ) = D3/2QRe

f

&t(v-R)

dt ( D

- (Pt/Z)

-

+ it 4 L ) 3 / 2-

0

=2

D fz Q AL R e [ z ecZzerfc (iz)]. ~

Here

Deriving (8) we have neglected by photon momentum as an infinitesimal quantity in comparison with others in the system, and supposed that the energy distance between first and second exciton bands is much greater than the bandwidth, so outside the exponent it was put L u f z f i Z $2. For satisfactory convergence of the integral over dk, it was enough that the parameter D 1. I n the case of damping, when r j 2 -+ 0, (8) takes one a simple form,

>

From (10) one can see already that the absorption lineshape under band-to-band transitions depends t o the same extent on crystal temperature and on the difference of bandwidths of two bands. Moreover, both factors act in the same direction. For 41 physica (b) 155/2

N. I. GRIGORCHUK and L. G. GRECHKO Big. 1. The light absorption curves by Frenkel excitons a t band-to-band transitions in the broad-band case. (1) D = 2, A L = 517; (2) D = 1, A L = 52"; (3) D = 2, AL = (4) D = I, AL =

zr;

0

2

4

V-R

ri2

zr

6 c

instance, the decrease of the difference AL gives the same effect as temperature lowering : the absorption intensity decreases. A more particular analysis, with accounting of the damping, can be carried out with the help of numerical calculations according t o (9). As a n illustration, in Fig. 1 the form of the absorption contour derived for the considered case is graphically represented. Curves 1 and 2 (as 3 and 4)are related t o the same crystal (AL,m, = const), while curves 1 and 3 correspond t o a temperature twice higher than curves 2 and 4. For a definite exciton number of the first band the area under the absorption curve, as follows from (8), does not depend on temperature, but with decreasing temperature the lifetime of excitons increases, what a t the same generation rate leads t o a n area enlargement. Comparing curves 1 and 3 or 2 and 4 one can state that the difference of bandwidth increase a t a definite temperature (D= const) leads to a decrease of the absorption maximum and to the displacement t o the short-wave spectrum range. A similar result can also be received by increasing the crystal temperature under constant AL (compare curves 1 and 2 or 3 and 4).Physically such a n equivalence is caused by the fact that due t o the momentum conservation law a n exciton with definite momentum k, can relax a t a phonon or another exciton with momentum k,,with an equal probability. Numerical estimations adopting the values specified under Fig. 1 have shown that a twofold increase of both AL and T ought the intensity of absorption approximately to such a time decrease. The account of damping, as is obvious from the above, qualitatively does not change the conclusions in comparison with the ones a t deriving an analysis of (10). Thus, in the case of broad exciton bands, the difference of bandwidth and crystal temperature are the two main comparable factors defining the contour shape and the absorption intensity. The increase of this difference gives the same result as a temperature increases: the absorption intensity decreases and its maximum shifts in the short-wave spectrum range. I n the extreme case, when the bandwidths are equal, i.e., when AL = 0, the absorption curve is a symmetric Lorentzian with halfwidth T12.

4. Band-to-Band Exciton Light Absorption in the Case of Narrow Bands As is well known, narrow bands characterize triplet excitons in molecular crystals. Their widths are = 10 em-' for the lowest states and enhance with increasing state number.

Lineshape of Light Absorption by Thermalizcd Frenkel Excitons

637

From the formal point of view, transitions between triplet states are not distinguished from singlet ones. Therefore, for the calculation of the absorption lineshape we make me of the same equation (5) as in the previous case. Let 3

&i(ki) =

-Li

C cos kija;

i

= 1,2

,

j=1

define the law of exciton dispersion on the sublevels of the first and second bands. The exciton distribution function over the first band in this case takes the form [7] 3

W(/c,) = N ; ~ I G ~ ( ~exp L , ) [BL,

c cos kip] ,

j=1

where Io(x)is the modified Bessel function of zeroth order. Then, substituting (11), (12) into ( 5 ) ,after integration over dk,, dk12dk,,, we obtain m

where

Qo = Q k2j %

kij

+3AL>

6 AL,

.

If now the multiplication theorem of Bessel functions is used and we confine ourselves t o the first three terms in the sum (that is verified by the infinitesimal arguments of Bessel functions for the temperatures and bandwidths discussed here), so after simple transformations one derives for the function P ( Y )the expression dt eiat [J:(t)

AL

+ QJg(t) Jl(t)]

0

in which AL

& = SII(BL1) i p IO(BL1) The integration in (14) can be carried out if one takes into account that

Then for the function F ( v ) one finds

Formula (17) describes the absorption lineshape in the rather general case, when the ratio A L l F is arbitrary. If A L / r 1, then the series (17) converges quickly and the


AL) and T = 4 K. (1) A L = 0 and (2) 0.W Fig. 3. The light absorption curves by Frenkel excitons at band-to-bandtransitions in the narrowband case. (1) T = 100 K, 2AL = r; ( 2 ) T = 20 K, 2 A L = r; ( 3 ) T = 100 K, 2AL = 5I'; ( 4 ) T = 20 K, 2AL = 5I'

Lineshape of Light Absorption by Thermalized Frenkel Excitons

639

one can answer the question which of the two bands is wider, and from the magnitude of these parameters one can judge about the quantitative meaning of AL. The change of temperature only slightly affects the absorption band form. Now let us consider the case A L / F > 1 in more detail. I n this case the series (17) converges badly, therefore it is more convenient t o carry out numerical integration in (14). Taking into account that the parameter AL is purely imaginary, one finds 00

F ( Y )= % AL

dx e

_ _r

X

J:(z) cos

2AL

v ~

0, AL x -

-

0 m

As a n illustration in Pig. 3 are plotted curves of the absorption lineshape by (20). Analysing them one can find some main peculiarities of light absorption a t exciton transitions between narrow bands. First of all one can see that the absorption curves are characterized by a weak asymmetry: a t small difference AL they are close t o Lorentzians and a t large one they are bell-like curves. Absorption bands are now more sensitive t o the band-width difference AL change and practically do not respond t o a temperature change. Comparing curves 1 to 3 or 2 to 4 (Fig. 3) one can check to see that a fivefold enhancement of the band-width difference AL, for instance, a t given temperature leads to a fourfold decrease of absorption intensity. The band maximum can be shifted both in the short-wave range of the spectrum (at high T )and in the long-wave one (at low T). Absorption bands become wide (bell-like) with approximately the same intensity all over the width. The temperature change mentioned above (fivefold) only insignificantly affects the lineshape and the value of absorption intensity. I t s decrease leads to a weak enhancement of absorption intensity in the long-wave wing and a t the cost of this t o a weak asymmetry (compare curves 1 with 2 and 3 with 4). The above-mentioned absorption peculiarities in the case of narrow bands agree with known experimental results [5, S] and qualitatively explain them. First of all, as estimations show, the Boltzmann factor for narrow bands (Ll = 3 cm-l, for example) a t kl = 0, within the temperature range 50 t o 250 K, changes from 0.63 t o 0.75. Thus, the temperature becomes a n inessential factor and the main role in contour formation will play the distinction between band state densities. I n other words, due t o the conservation law, for the same momentum the higher band excitation energy is necessary the greater the difference AL is. But, on the other hand, as the number of states with great energies is small, as a result one can obtain a contour of finite width with a maximum. It is clear from here that the width of such contour is proportional to the difference AL of the two band-widths, independently of the crystal temperature.

5. Discussion of Results The study of transitions between high-excited states in molecular crystals also can be regarded as important from the point of view of the investigation of octupole interactions and so-called configuration interactions between molecular excited states and charge transfer states. For ionic crystals the study of exciton transitions between different bands and between two states inside the same band (n, 1 + n, 1 1) can give important information about IR and mm-wave spectra. It was shown in [9] that

+

640

N. I. GRIGORCHUKand L. G. GRECHKO: Light Absorption by Thermalized Excitons

such transitions for Cu,O crystals have appreciable, as compared with atomic transitions, oscillator strengths and can be considered as sources of IR or mm-waves. For their observation low temperatures and experimental refinements are required. Lowest exciton states can be filled by intensive pulse light irradiation or by electron beams of corresponding energy. After that transitions to higher-lying states are stimulated by light from another source. If transitions between triplet states of different bands are studied, the lower states are usually filled from higher-lying singlet states. Excitation methods for naphtalene crystals, for example, are presented in [S] in more detail. An essential condition for the observation of such transitions is t o produce sufficient concentrations of triplet excitons in the first band. I n [S], in one electron pulse a concentration of triplet excitons above lo1' ~ m was - ~achieved. The typical impurity concentration is much smaller, therefore, it cannot influence the observation results. Under band-to-band transitions the only selection rule, independently of the value of k,, is Ak = 0. That is why, if states of the lower band, for example, are evenly filled by excitons, one can study the states density of higher-lying bands. Then, in accordance with the obtainable density, one can also find the exciton dispersion law in the band. An even filling may be achieved if the band is narrow enough. Our assumption about the absence of a dependence of T on frequency in the area of the fundamental exciton absorption holds true, but it can strongly change a t the resonant frequencies in the system. If in (6) simultaneously the signs of the effective mass are reversed, it is not complicated to demonstrate that the absorption lineshape does not change and will be defined by the same equation (S), as in the above-considered case. In the case when the signs of the exciton effective mass in the first and second bands are opposite, for example, a E ~ ( J C ~ )= L, C cos klja , j=l

L, C cos k2ja , j=1

+

then, if one uses AL, for the magnitude h-l(Ll L2),it is not difficult to show that the absorption curve Fl(v)will be mirror-symmetric about the curve F ( v ) with AL = AL, and SZ, = SZ 3 AL,. The sign change in (21) does not change this conclusion.

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References [I] V. Kn. BRIKENSHTEIN, V. A. BENDERSKII,and P. G. FILIPPOY, phys. stat. sol. (b) lli, 9 (1983). [2] V. L. BROUDE,G. V. KLIMUSHEVA, A. F. PRIKCHODKO, E. F. SHEKA, and L. P. YAZENKO, Spektry pogloshcheniya molekulyarnykh kristallov, Izd. Naukova Dumka, Kiev 1972. [3] R. PARISER,5. chem. Phys. 24, 250 (1956). [4] V. L.BROUDE, E. I. RASHBA,and E. F. SHEKA,Spektroskopiya molekulyarnykh eksitonov, Izd. Energoizdat, Moskva 1981. [ 5 ] A. F. LUBCHENXO and N. I. GRIGORCHUK, Optika i Spektroskopiya 41, 82 (1976). [6] N. I. GRIGORCHUK, Fiz. tverd. Tela 25, 387 (1983). 171 N. I. GRIGORCHUK, phys. stat. sol. (b) 128, 599 (1985). HIGUCHI, I. T. NAKAYAMA, and N. ITIN,J. Phys. Soc. Japan 40, 250 (1976). [8] & [9] 9. NIKITINE,J. Phys. Chem. Solids 46,949 (1984). (Received August 10, 1988)