VIBRATIONAL RELAXATION IN THE EXCITED ELECTRONIC STATE AND THE DEPOLARIZATION OF EMISSION

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VIBRATIONAL RELAXATION IN THE EXCITED

ELECTRONIC STATE AND THE DEPOLARIZATION

OF EMISSION

I. Tehver

To cite this version:

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VIBRATIONAL RELAXATION IN T H E EXCITED ELECTRONIC STATE AND T H E DEPOLARIZAT ION OF EM1 SS ION

I . Tehver

I n s t i t u t e o f P h y s i c s , Estonian SSR Acad. S c i . , 202400 T a r t u , U. S.S. R.

Abstract

-

The spectral and polarization characteristics of the resonance

secondary emission, such as the first order Raman scattering and hot luni-

nescence (mltiphonon Raman tail), are examined for mltimode systems with a

broad absorption band (i.e. strong vibronic coupling) on the example of cubic impurity centers. The transform method is generalized for nontotally symmet- ric modes.

I

-

INTRODUCTION

The emission during vibrational relaxation in an excited electronic state represents resonance Raman scattering (at the very beginning of the relaxation) and hot luminescence (in further stages of relaxation). When 'the electronic state is split, e.g. by Jahn-Teller effect, the transitions between the sheets of the adiabatic poten- tials surface will cause the depolarization of emission during vibrational relaxation (hot depolarization). Below the emission in the initial stages of relaxation mentioned will be examined, with depolarization processes taken into account for such systems as impurity centers in crystals or molecules in solutions with broad absorption bands.

I1 - GENERAL FORMULAF.

Resonance Raman scattering is described by

where the polarizability operator P is determined as

a B d u b 4 d g

In (1) and (2) subscripts a , ~ indicate the direction of the polarization vector for excitation (a,a' ) and emission (B, B')

,

1

i>, ID and 10are the initial, final and intermediate states with energies Ei, Ef and Em, ym is the radiative damping con- stant,

<...>

denotes the average over initial states, the 6 function describes

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C7-430 JOURNAL DE PHYSIQUE

energy conservation law, wo and n denote the frequencies of exciting and scattered

light,

d

is the electronic dipole moment, h= 1. The calculation of spectral intensities (-polarizations) for the case ofmultimode systems cannot be correctly carried

out directly on the basis (1)

-

(2) without limiting oneself to only a few vibra-

tional states in sum over m. Such kind of mathematical obstacles can be surmounted

by time-representation of 6-function and resolvents, and operator technique which

enables to sum up effectively over all the states m. Now the tensor of secondary

radiation has the following form /I, 2/

with the correlator

In (3)

-

(4) the Condon and adiabatic approximations are used, Ho is the vibrational

Hamiltonian in the nondegenerate ground electronic state, H = HoI +V, the vibronic

Hamiltonian of the degenerate excited electronic state, for n-fold degenerate state H is a n x n matrix, the angular brackets

<...>

denote a thermal average.

The integral over v represents the energy conservation law; the integrals over r and

T I arise from resolvents in (3) and have the meaning of averaging over the time

spent in the intermediate electronic state in two transition amplitudes. Correspond-

ingly, (.r+r11/2 has the meaning of the time spent by the system in this state, while

T'-T describes the difference between the phases of two amplitudes.

I11

- STRONG VIBRONIC COUPLING. MAMPLE OF CUBIC CENTERS

This case is realized e.g. for F-centers or ~l+-type centers in-alkali halides. They

are characterized by a broad absorption band with the width cr>>w, the average vibrational frequency in the ground el ctro ic state. In this case, actual values of

rl-r in (3) are small,

1

rl-r

1

< i.e. the fast phase relaxation occurs. In the centers, due to phonon dephasing, during the first period of vibrations of the configurational coordinates their amplitudes are reduced by an approximate factor 22 /2/. Now, taking into account the Jahn-Teller effect, .which removes the degene- racy of the excited electronic state during the relaxation following absorption, it is obvious that hot depolarization transitions can efficiently occur only near the crossing point of the potential surface (Fig. 1).

F ! g

. 1 - Diagram of the potential energies and the optical excitation followed by

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may be considered without taking into account the hot depolarization processes. Here we are interested in the short-time emission accompanying hot depolarization processes. This emission corresponds to the small delay time (T+T1)/2 in the excited electronic state.-Therefore one may use the short-time approximation for the times

T and ~ l :T ,TI << W-1. In this approximation /I ,2/,

where

V(u) = exp (iuH01V exp(-iuH01.

We consider the emission in the initial stage of relaxation in cubic systems for

the case of excitation in resonance with A electronic transition. The vib-

ronic coupling is described by a 3 x 3 matr&)CT&&h takes into account the linear coupling to the vibrations of a

-,

e

-

and T -representations:

lg g 2g

v

=

va

+

ve

/1,2/.

tg

The polarization characteristics of the resonance secondary radiation of cubic sys-

tems are determined by three independent components of the tensor Wa BB

a, :

w1

= W _{xxxx' 2 xyyx'}W = W W3=W + W

=YY XPY'

For the case of excitation with polarization of (100) direction, W1 = I,, and W2 = IL, W3 = 0. The polarization of emission is determined by p = I

,,

/I1 = W1 / W ,

To find the intensities I I and I, on the basis of (3) and (S), we use tie correlator technique, expanding the exponents in (5) in a series of pair correlators (W(u)>. The zero-order term describes the Rayleigh line, the first, second, etc.

order terms, the RRS of the first, second, etc. order.

For the first order RRS we obtain the following expressions: the Raman excitation

profile (REP) of a + e modes,

lg g

-2z02 2

I ! '

) =

&

[vzo2e + (I-~z~w(z~))

1 ,

the REP of T mode

2g

where z, = (wo-Vo)/cs fi is the dimensionless excitation frequency, Vo is the maximum

of the absorption band, a and 6 are the dimensionless interaction parameters for e

and T modes /I/, Erf x is the error function, W(zo)

,

the Dawson function. g

2g

(1

1

(1 )

Fig. 2

- Raman excitation profiles for a e modes

(Ill ) and r mode (Il ).

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C7-432 JOURNAL DE PHYSIQUE

Fig. 2 shavs the important contribution of e modes t o the f i r s t order scattering by

~2~ mode. We can see that with wo shifted off from VO, e modes begin t o counteract the depolarization transitions: the stronger the e coupfing is, the sharper is the

T,- Raman profile. g

4

The high-order Raman processes produce a smooth multiphonon Raman t a i l which can be interpreted as a hot luminescence spectrum /2/. I t reveals a remarkable polarization as known from the F-center experiments /3,4/. This polarization points t o the depolarization switch-off due t o the Jahn-Teller s p l i t t i n g a t the very beginning of vibrational relaxation. The depolarization of the emission t a i l has the approximate form

where z = (coo-n)/ad.

Fig. 3

-

Depolarization of hot luminescence (multiphonon Raman t a i l ) as a function of the frequency of emission (z) and excitation (zo).

Fig. 3 demonstrates the switch-off of depolarization transitions i n relaxation: with

n moving f a r f r o m the excitation frequency p - l obtains an asymptotic v a l e. One also notes that p- depends on the excitation frequency wo(zo), f o r wo > V O p-' passes through a maximum with n increasing, before it obtains the asymptotic value. The in- terpretation of t h i s maximum i s the following: directly a f t e r the transition t o ano- ther sheet of adiabatic potential surface the system moves down slowlier than before. Therefore IL is enhanced i n comparison with I,,

.

The maxinunn as a function of

wo shows that depolarization occurs with a maximum probability when the system re- laxes t o the crossing point.

Fig. 3 r e f l e c t s also the violation of the coherence of the electronic s t a t e i n the very beginning stage of relaxation, a s the linear dependence of on wo-n l a s t s only near wo = a.

Above we have considered that depolarization of RRS is caused only by .r2 coupling. As is known, there may be additional interactions, e .g

.

spin-orbit intergction which may lead t o spin-flip transitions. This mechanism may be considered analogously,

since the spin-orbit coupling can be formally taken into account by replacing

412 > by

>

+ h2 /2/, where A = <x

1

<+

l~~~

1 ->

l

p

(Hso is the Hamiltonian of

' 2 e 2n

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Let us regard once more the Raman excitation profiles (REPs) from the point of view of transform method (see, e.g. /5/). According t o t h i s method, REPs are determined by the absorption spectra ~ ( w ) owing t o certain transform laws, e.g. the REP of the f i r s t order Raman l i n e (of the mode of frequency w;) is determined

1 1 (W) %

s

Q(W0)

-

Q ( ~ 0 - u ~ )

l 2

3

J where

is the complex refraction index, and S - , the linear coupling strength of the scattered mode. The simple relation is obtiined for standard assumption such as the adiabatic and Condon approximations, a nondegenerate excited electronic s t a t e with linear vibronic coupling. The transform law w i l l be more complicated i f these assumptions are not f u l f i l l e d .

The practical importance of t h i s technique i s obvious. By modelling REPs, it allows one to take into account a l l the multimode information carried by the absorption band. When disagreement with observed REP arises, one can search f o r i t s origin i n the violation of the assumptions.

Now, a question arises what transform law may be used f o r the case of a degenerate electronic s t a t e . By applying for the absorption spectrum the short-time approxima- t i o n used above it can be shown t h a t f o r "diagonal" (a,-,e-) modes i n cubic centers

L B B

the transform law retains the form (8) when expanded i n a Taylor series,

while the REP of a "non-diagonal" mode, such a s .r

23'

-

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i s determined by the d i f f e r e n t i a l spectrum of absorption KC16

,

i.e. by the correc- tion of the absorption under application of a low symmetry perturbation. Hence, t o determine REPs f o r Jahn-Teller systems, i n addition the differential absorption spectrum should be known.

REFERENCES

/1/ Hizhnyakov, V. and Tehver, I . , i n "Light Scattering i n Solids", edited by M.Ba1- kanski (Flannnarion, Paris) 1971, p. 57.

/2/ Hizhnyakov, V., Phys. Rev. B 30 (1984) 3490.

/3/ Liity, F., Semicond. Insul. 5 m 8 3 ) 249.

/4/ Mori, Y., Hattori, R. and OT;kura, H., J. Phys. Soc. Jpn. 15 (1982) 2713.