Phase and energetic relaxation in resonant Raman scattering and hot luminescence

Journal of Luminescence 18/19 (1979) 673—677 © North-Holland Publishing Company

PHASE AND ENERGETIC RELAXATION IN RESONANT RAMAN SCATTERING AND HOT LUMINESCENCE V.V. HIZHNYAKOV and I.J. TEHVER Institute of Physics, Estonian SSR Academy of Sciences, 202400 Tartu, USSR

Classification of the resonant secondary radiation of vibronic systems with localized electrons is examined, using the criteria of different relaxation stages. The cases of one local mode (1), a strong electron phonon coupling (2), and random fluctuations (3) are taken under consideration.

The aim of the present paper is to consider the resonant scattering of light based on the theory of resonant secondary radiation [1—31.The term “resonant secondary radiation” (RSR) is used for the optical response of the system to excitation by light. The corresponding spectrum is described by the ordinary formula for the resonant scattering of light. Here we shall use the integral form of this formula [1,2] I(wo, 12) A(ji,

—

T, i-’) =

~-~--

J fJ d~t

dr di-’ e’~° “°~‘

(d~e’~’~de’~”dj~ 1 e

e

“~‘°~

‘~‘~

‘~A(j.~, i-, i-’),

(I)

(2)

‘~‘~),

where w0 and 12 are the frequencies of incident and secondary photons, d, and d~are the one-photon transition operators for absorption and emission, H H + i’~,H is the Hamiltonian of the electron-lattice system, ~ is the operator of radiative decay, and the angular brackets indicate an ensemble average. The variables i-, r’ in (1) and (2) are the delay times for transition amplitudes in the intermediate state. In the resonant case the main contribution in integral (I) is made by the region T -y ~ 10 8 s (y is the constant of the radiative decay for the intermediate state). Hence, the relaxation processes in the intermediate state have time to take part in the transformation of a primary photon, violating the coherence and changing the frequency and polarization. As the emission of a secondary photon at different stages of relaxation has different features, RSR can often be divided into ordinary luminescence (OL). hot luminescence (HL). and resonant Rayleigh and Raman scattering (RRS). In eq. (1) these components are determined by different integration regions on the plane r, r’, and the effect of relaxation processes is expressed by the dependence of A(~,‘i-, r’) on the variables. In particular, A(~i,i-, r’) as a function of s r r’ describes the phase —

—

673

—

674

V. V. Hizhnyakov, If. Tehver/ Phase relaxation in resonant Raman scattering

(transverse) relaxation in the intermediate state, and that of t (i- + T’)/2—the energetic (longitudinal) relaxation in this state. If the characteristic time of the energetic relaxation (te) essentially exceeds that of the phase relaxation (t~5)*, the integration region I < t~,hdescribes the part of RSR with correlation between primary and secondary photons (RRS), and I tph, RSR without correlation (HL + OL). Thus HL corresponds to the integration region I < t,~,and I ~ t,, to OL, i.e. HL is the radiation after phase relaxation but during energetic relaxation, and OL is the radiation after energetic (and phase) relaxation in the intermediate state. Naturally, the division of RSR into the components is to some extent conventional and has an approximate character. Fvidently the division makes no sense when relaxation processes are lacking or when the phase and energetic relaxations cannot be separated. To illustrate the general ideas, let us examine some concrete models for vibronic systems with localized electrons. ~‘

(1) A localized optical electron interacts with one local mode which changes the equilibrium position and frequency at electronic transition In this model the correlation function A(~.i-, r’) up to a constant multiplier is determined by [2] r,

r’)

e

y(r-+

)+~uI,~(~

(erH~eMHI e

rH,

e

‘~ +

,lH1~

(3)

where Q21 is the frequency of pure-electronic transition. H1 is the anharmonic Hamiltonian of the lattice when the localized electron is in the ground state, H2 e~’[H1 + (W2 w~)a~a]e V is the Hamiltonian for the excited state, e~’is the operator of shift, ~ 1(2) is the frequency of the mode in the ground (excited) state. a~(a)is the phonon creation (annihiliation) operator. The correlation function (3) and the corresponding RSR spectrum for the zero-temperature limit were calculated in [2]. For the case of monochromatic excitation the RSR spectrum is determined as following I(w .12)

2(L ROIM)11 2IL,)V ii-(TL2+y)[(wo Q2i w2M)+(TM+y)] 8L

2M 2+(TL+~ w0+w~L~) + [I’(L 3L 2+[F(L+L 2 1+L2)+ yI(l 7M) (4 (12 1221+w1L1 w2L7) 2)+y1 where (OjM) is the Franck Condon factor for the resonant absorption transition 0—sM, and (L 2IL) is that for the radiative transition L2—*L1 F denotes the constant of anharmonic decay of the mode (F~-y). The first term in brackets in eq. (4) describes the lines corresponding to transition M—s L1, i.e. the lines of RRS with maximum at 12 w~ w1L1, and half-width FL~(L~corresponds to the order of scattering). The second term takes into account the transitions from

tffl

I’Li

In vibronic systems tph 0 . t~~ w ‘, hereby always tph 1, (u is the dispersion of the resonant hand of absorption, w is the average frequency of vibrations).

*

V. V. Hizhnyakov, I.J. Tehver/ Phase relaxation in resonant Raman scattering

675

L2 ~ M to L1, i.e. HL (if L2 0) and OL (if L2 0) lines at (2 1221 w1L1 + w2L2 with half-widths [‘(L1 + L2) + y and FL1 + y, respectively. The intensity of RRS and HL lines is determined by the life-time of the vibronic levels (FL2+y) ~, and that of OL lines—by the life-time of the 0th vibrational level y of the excited electronic state. Consequently, as demonstrated by this simple model, OL is due to the transitions after vibrational relaxation, HL during vibrational relaxation in the excited state, and RRS before that. A single relaxation transition (M —s M 1) is necessary and sufficient to violate the phase correlation between the primary and secondary photons, leading to HL. ..>

—

—

—

—

(2) A strong electron—phonon coupling 2~‘ case the thereStokes is a broad resonant band absorption: ~ óiS~ ~ In (Sthis denotes losses; S ~ 1).vibronic Therefore, theoftime of phase0’relaxation for the vibrations in the excited electronic state is far shorter than that of the the energetic relaxation: tph 0’ ~ te ~‘ ~ Consequently, following the classification criteria given above, the RSR spectrum can be divided into RRS, HL and OL. To be convinced of that, let us examine A(~,i-, i-’) obtained in Ill: ~.

const.

1’, T’)

Here

F(T)

F(T)F*(T’)

~ [K(~r~

i-P)]fl

denotes the Fourier-transform of the absorption spectrum,

K(~,r, r’)

—

J

dw

J

dx e

e~~M(e0~r

l)(e

l)q,(co),

~

(6)

where ~(w)

-

~

l~x((v

(V))V(x)),

is the phonon correlation function, V

=

(7)

H

2 H1. Substituting eqs. (5) and (6) into eq. (1) and evaluating the integral, we obtain —

I(w0, 12)

—

~ I~(w0,(2),

(8)

where the nth order inelastic resonant scattering equals const I I I~(o~,12) n! dx1... j dx~q~(x~)...q2(x~)

J

x

~ pI...p,,

(_l)Pi+~~(Q+ p1x~+

+

p~x~)~3(flwo+ x1 —

+

‘~‘

+

x~).

0

(9) Here cP(coo)=ijdre

J

l~oorF(T)_ii,~K(~o)+

dxk(x) (dO X —

676

V. V. Hizhnvakov, If. Tehverf Phase relaxation in resonant Raman scattering

is the amplitude of the Rayleigh scattering. In the case of a broad absorption band K(wo) (and, consequently, a broad function I(w~)),the spectrum given by eq. (9) has a clear vibrational structure and the spectral features of Raman scattering for small n, which corresponds to the evaluation of integral (I) at small i-, i-’ a- However, increasing the scattering order (which corresponds to S r r’ a- ‘, t (r + r’)/2 a~ ), the spectrum becomes smooth and transforms into a quasi-continuous background. In the region 1221 Sii this background grows into intensive OL. Between OL and RRS the background corresponds to HL. The lack of the vibrational structure in HL + OL allows one to calculate them as an envelope of RSR, i.e. in the approximation of small jx and s ‘r T’. The features of HL depend on the function q~(w)reflecting the relaxation processes in the excited state. In particular, if there is no narrow maxima in the phonon spectrum 92(w) (no local or quasilocal mode), the vibrational relaxation is fast (t~ i~5 ‘). In this case the main contribution to HL is given at I i~i~ I/4 by the radiation from the first classical turning point of the configurational coordinate in the excited electronic state, and the HL spectrum at &
~‘

—

If the remarkable contribution to q~(w)is made by a local mode, the relaxation will have fast and slow stages, and HL can be regarded as adenote radiation 4 (S~r the from Stokesa large number of turning points. Then at ü Q
(1)

const.(w

(3) Effect of random fluctuations Let us briefly consider the features of RSR for inhomogeneous systems. The effect of static inhomogeneities was examined earlier in [4]. Here we take into account the change of the inhomogeneous field in time leading to the fluctuations of the energy of the resonant electronic transition. These fluctuations become essential near the temperature of the structural phase transitions, and cause the central peak in the spectra of non-resonant scattering of photons and neutrons; besides, these fluctuations are essential in glasses at low temperatures. For RSR the influence of such fluctuations is also important as they lead to the phase break-down. We propose a simple model for fluctuations of a finite rate, multiplying the previous correlation function A(~x,r, i-’) by T,

i-)

exp{f(~x)+f(~x +

f(p.

+

T’)

f Cu.

r’

r)+f(i-’)+f( i-)},

‘r)

(12)

V. V. Hizhnyakov, If. Tehver/ Phase relaxation in resonant Raman scattering

677

where f(x)

=

[~s [ds’((/s)

—

())(s’))

—

T2

Fix!

-

(e

flX~

1)].

(13)

It was proposed that ((c(s) ())c(s’) = e~exp( FIs s’I), ~ denotes the amplitude of fluctuations of the electronic energy , and F their rate. Here we restrict ourselves to the examination of the effect of fluctuations on the resonant part of the total RSR spectrum when exciting at the zero-phonon line of absorption. In the case of fast fluctuations (F ~ ~), the second term in brackets in eq. (13) can be neglected. In this approximation, which corresponds to the case examined in [1,5,61, the resonant line of emission consists of two components: the scattering line of 6-shape and OL line of the Lorentzian shape with the width 2~/F. In accordance with the classification criteria the first line corresponds to the emission before the fluctuations occur, the second one originates during fluctuations. If the fluctuations are slow and other relaxation processes are lacking, there will be no luminescence, all RSR is elastic (6-shape) and quasielastic scattering. In the case of a large amplitude of fluctuations (Eo~’y), the latter is more intensive, having the following shape: —

‘2

1

I(w~,12)

expj

ill

\I/25

wI(~-~~) ~. I

(14)

The width of the line is ~o(F/2y)”2, whereas the zero-phonon absorption line has the Gaussian shape and the width e~,by far larger than that of the emission line. In glasses at T —sO F T4, and the width of the resonant emission line* T2. Hence, for glasses the mechanism proposed here for the broadening of the zero-phonon emission line at low temperatures is more important than the modulation mechanism of McCumber and Krivoglaz, which leads to the broadening T7. References [11 Ii. Tehver and V.V. Hizhnyakov. Eesti NSV Tead. Akad. Toimet.. Füüs. Mat. Tehn. teaduste seeria 15 (1966) 9; V. Hizhnyakov and I. Tehver. Phys. Stat. Sol. 21 (1967) 75S. [2] V. Hizhnyakov and I. Tehver, Phys. Stat. Sol. 39 (1970) 67; (b) 82 (1977) K89. [3] K.K. Rebane. t.J. Tehver and V.V. Hizhnyakov, in Theory of light scattering in condensed matter, Proc. of the First Joint USA—USSR Symp., eds. B. Bendow, J. Birman, V. Agranovich (Plenum Press. New York, 1976) p. 393. [41 Ii. Tehver, Preprint F-5, Estonian SSR Academy of Sciences, Tartu (1977) in Russian. [5] D.L. Huber, Phys. Rev. 170 (1968) 418. [6] T. Takagahara, E. Hanamura and R. Kubo, J. Phys. Soc. (Japan) 43 (1977) 811. [7] P.W. Anderson, B.l. Halperin and CM. Varma, Philos. Mag. 25 (1972)1. *

In glasses at T—e0 the rate of fluctuations is determined by the emission rate of phonons from

quasi-spin levels which depends on temperature as T4 for the model of Anderson. Halperin and Varma [7].