Resonant Raman scattering and luminescence due to excitonic molecule

~

Solid State Communications, Voi.32, pp.19-24. Pergamon Press Ltd. 1979. Printed in Great Britain.

RESONANT RAMAN SCATTERING AND LUMINESCENCE DUE TO EXCITONIC MOLECULE E. Hanamura and T. Takagahara Department of Applied Physics, Faculty of Engineering, University of Tokyo, Hongo, Bunkyo-ku, Tokyo 113 JAPAN

Competitive behavior of two channels of resonant Raman scattering and luminescence is theoretically discussed for giant two-photon excitation of excltonlc molecule. Time-integrated and tlme-resolved emission spectra of the excltonlc molecule excited resonantly by a short pulse are described in terms of the stochastic theory of intermediate state interaction. Particularly we propose the time-resolved spectrum in which the frequency-tlme uncertainty due to the finite observation time is incorporated. This theory is shown to explain the observed characteristics of the emission spectra, and the relaxation constants of the excltonic molecule in the intermediate state and the exclton in the final state are determined from comparison between the observed and the calculated spectra.

and is by an order of 106 larger than that of the ordinary two-photon absorption. This extremely strong absorption comes from the mutual enhancement due to the giant oscillator strength and the resonance effect. This giant oscillator strength is due to the fact that in the second photon absorption process which causes the transition from the intermediate state to the final state of an excitonic molecule created, one can excite any valence electron within the large molecular radius around the virtually created first exclton to make an excitonic molecule. The situation is quite different from that which occurs in ordinary two-photon absorption due to band-to-band transition. There, the same electron which is excited into the intermediate state has to interact again with the second photon. The further enhancement comes from the resonant effect. This is due to the smallness of the energy denominator in the expression of the second order perturbation with respect to the electron-radlatlon interaction V.. One year I 6 after this proposal, Gale and Mysyrowicz observed this giant two-photon absorption due to the excltonic molecule in the excitation spectrum and confirmed the expected large oscillator strength. This was observed also in the absorption spectrum7 as a sharp absorption llne at the expected position. The most rapid decay process of the excitonic molecule created by this giant two-photon absorption is the radiative process in which an electron-hole pair is radlatlvely annihilated and the other electron-hole pair remains as a transverse or a longitudinal exciton. The former process may be considered as two-polarlton scattering because two incident light as well as both the emitted photon and the remaining transverse exciton behave as two polarltons inside the crystal. 8 Here arises a serious problem whether this emitted light is considered as luminescence or Raman scattering. The excitonic molecule in CuCl is the most suitable subject to study this relationship because i) the crystal is very transparent to the radiation fields of

i. Introduction An excltonlc molecule is a bound state of two slngle excitons. This excltonlc molecule was for the first time observed in the emission spectrum of CuCI heavily excited through the I band-to-band transition by Mysyrowlcz et al. and successively by Ueta and his coworkers. 2 The electron to hole mass ratio in this material is very small such as 0.05 so that the internal motion of this excltonic molecule is understood by analogy with that of a hydrogen molecule. The mass ratio, however, is usually of an order of one in many semiconductors and there arises a question whether the excitonic molecule is stable or not depending upon the value of the electron to hole mass ratio. In 1972, we could prove by variational calculation that the excltonic molecule is stable for any values of the electron and hole masses.3 Triggered by this theoretical result, the excltonic molecules have been observed in many semiconductors, e.g., in CdS, CdSe and ZnO. 4 In these experiments, the excltonic molecule was formed from two electrons and two holes excited in band to band transitions by transferring extra energy to the lattice system. They observed the light emitted when the excltonlc molecule radiatlvely annihilated. The genuine study of the excitonic molecule started after proposal of direct and effective creation of the excitonlc molecule by giant two-photon absorptlon. 5 2. Giant Two-Photon Absorption due to Excitonic Molecule A crystal is usually transparent to radiation field with its frequency less than the exclton band. The excltonlc molecule can be directly and very effectively created by irradiating the crystal by radiation fieldwlth reasonable intensity and with tunable frequency by half the molecular binding energy lower than the exclton band. The absorption coefficient becomes as large as that of the single exclton 19

emitted lights and 2) the excitonic molecule has the strong and sharp absorption spectrum due to the giant two-photon excitation which depends on the excitation power. This makes it possible to use the excitation power as another freedom to study the competitive behavior of two channels.

time 0 and reaches the final stateJe>at tlme t on t h e ~ b 0 t h ~ . We consider the excitonic molecule in the intermediate state and the single exciton in the final state as the relevant system, and other excitons and excltonlc molecules created already as well as phonons and imperfections as reservoirs. The interaction between the system and the reservoirs is assumed to be Gausslan and Markoffian process. The giant two-photon absorption process due to formation of an excitonlc molecule is completed in much shorter time than any involved relaxation times. Therefore we may re~lace V~exp{-i(t~-t~)

3. Third Order Optical Process In order to answer the problem of two channels, we study the third order optical process in which two incident photons of ml are absorbed and an excltonlc molecule is created, and subsequently a photon of w2 is emitted leaving behind a single exciton in the final state el This latter matrix element is denoted by V~. The probabillty P(t) of observing the emitted light w9 at time t is given by taking trace of the product of this probability amplitude $(3)(t) and its conjugate ~(3)(t)* with respect to the reservoir coordinate R; P(t) = TrR[~(3)(t) PR ~(3)(t)*],

,(3) (t)

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20

it

=

/tI

H}VT i n eq.(2) by Nm~ ~ - i ( V 1 ) m i ( V T ) t ~ / ( ~ i . - ~ l ) " tim~s 6(t2-t3). In ~valnation of'P~), w8 e~ploy the stochastic theory of intermediate state interaction. ~ The product of propagators to the left and to the right for a common interval T, which appears in the expression for the third order optlcal process, is simplified in the fast modulation limit as TrRPX{} - 6ac6dbexp{-i~abZ-YabZ}. Here the effect of the reservoirs can b~ described by a relaxation constant Yah = Y~b +w(ra+rb), which is composed of elastic ana inelastic collisions Y~b and decay rates ra and rb of levels a and b, respectively. For the diagonal component a - b, Yaa (=Y~+ ra) is a sum of inelastic collision rate ¥~ and decay rate Fa. The diagrsms inflg.l are useful not only in calculating the emission spectrum but also in understanding the physics involved. The upper llne describes propagation of the state to the

(i)

(t2

J0dtl)0dt2J° dt 3

x
v~exp{-i(t2-t3)H}Viexp{-it3H}lg>.

(2)

The s t a t e propagates t o the l e f t and t o the right starting from the ground state Ig>pR
2cut ~

/

~--g

°J2

BX

P

m ..

g Ymg

m 7"eg ~ /

Yme

.i

me

--.ex

ex

"~'---

g

Ymg

~ ~-.- m ~ . ~ - e x

"j// F i g . l . Decomposed diagrams of the t h i r d order o p t i c a l process v i a an e x c i t o n i c molecule. Three types of c o n t r i b u t i o n s depend on time orderings of the i n t e r a c t i o n with the r a d i a t i o n f i e l d , g, m and e denote the i n i t i a l ground state, the intermediate state with an e x c i t o n i c molecule and the fSnal e x c i t o n s t a t e , r e s p e c t i v e l y . The wavy l i n e s r e p r e s e n t photon propagation.

-

\o.

LUMINESCENCE DUE TO EXCITONIC MOLCULE

Vol. 32, No. 1

left-hand side and the lower does that to the right-hand side in the density matrix expression. Then we have three kinds of diagrams according to the chronological order among times of twophoton absorption and subsequent emission in the upper llne and those in the lower one. For example, in the time interval T 1 in the i-diagram, an excitonic molecule is crea[ed in the upper line and it is in the ground state in the lower llne, so that the effect of its environment is taken into account by the relaxation constant 7mg as a result of frequency modulation on that energy difference as well as radiative decay. Corresponding to three energy differences, we have three relaxation constants 7m~, 7me and 7eg. In the time interval T2 in il- and iiidiagrams, the excitonic molecule is really created so that the radiative decay and inelastic collision contribute to the decay constant Tm. Therefore these two diagrams describe the luminescence process while i-diagram contains the R-m-n scattering component because this represents the correlation between two polarization operators. i. Under stationary irradiation by a monochromatic laser light with ~I' the emission spectrum F(2~I, ~2 ) is calcuIated as t ÷ = limit of P ( t ) / t as F ( 2 ~ l , ~ 2) = ~_~ P ( t ) / t ffi

Ym~ (2~l_~mg) 2 + 7 2

7me

(~2-%e)2+~me2

x .2__+ Yeg{l+(2~l-~m8 ) (~2-~me)/Tm67me} ~%

( 2 ~ l - ~ 2 - e e g ) 2 + ye2g

- (2~l-~2-Ueg) { (2~l-~m~)/Ym$- (~2-~me)/Tme} ] . 2 2 ( 2 ~ l - ~ 2 - ~ e g ) + Tag (3) The first term in the angular parentheses came from ii- and ill-diagrams in fig.l and describes the luminescence spectrum. This is also realized from the fact that the peah frequency ~2 due to this term is independent of ~i" On the other hand, the second and third terms originated from i-dlagram and contain the Haman component as read from that the emission peak ~2 due to these two terms shifts with change of 2~ I. The absorption spectrum due to giant two-photon excitation of an excitonic molecule is represented by the first factor of eq. (3) beside unimportant constant. Therefore Tmg is determined by full width at half the maximum in the absorption spectrum. Teg is fixed by half width at half the maximum of the Raman component and 7me is determined by half width at half the maximum of the luminescence component in the emission spectrum. The last remaining decay constant y_ will be fixed from the ratio of Raman peak to l~minescence peak as {l+(2~l-~mg)2/YmgYme}/Yeg versus 2/ym. 2. In real experiments, they use laser pulse with some distribution of carrier frequency as the excitation source. I~hen the intensity distribution of two-photon excitation frequency 2 ~ has, e.g., theGausslan form f(2~l)= I / 2 ~ x ± p [- (2~i-2~ I) 0 2 /2o 2 olex 1] , the observed emission spectrum is re~resented by the following superposition:

21

3. Furthermore w h m the incident pulse has the Gausslan envelope in time: e~[-(to2/2)2] and t h e e m i s s i o n s p e c t r u m i s i n t e g r a t e d i n t i m e , t h e e m i s s i o n s p e c t r u m i s g i v e n by P(-) - Tr[~ (3) (t=-)pR~(3) (tffi~)*], where "~--~ 2~1_2~ 2 ~(3) (t) ®d (2~i) exp [- (--~--~) ] ~t (tI -2/~it2+i~2tl ,iH(t-tl ) x I dt./ dt.e . Then the spectrum is shown to have the same form as e q . ( 4 ) w i t h t h e d e v i a t i o n o = ~ . When a v a l u e o f o i s l e s s t h a ~ t h ~ relaxat i o n c o n s t a n t s 7 _ and Te, , t h e F-men and l u m i nescence lines Re clearly resolved. On t h e o t h e r hand, t h e 9 - - ~ u l i n e i s smeared o u t and only the luminescence line is observable for the much l a r g e r v a l u e o f o t h a n ~mg and 7e=. These f a c t s were a l r e a d y o b s e r v e d . / Ueta an~ Mita 10 o b s e r v e d c o e x i s t e n c e o f t h e 9-m-n and l u m i n e s c e n c e l i n e s due t o t h e e x c i t o n i c m o l e c u l e i n t h e t i m e - i n t e g r a t e d emission spectrum under the g i a n t t w o - p h o t o n e x c i t a t i o n by t h e n a r r o w band laser pulse. The r e l a x a t i o n and d e c a y c o n s t a n t s a r e u n i q u e l y d e t e r m i n e d and e q . ( 4 ) g i v e s t h e e m i s s i o n s p e c t r u m which d e s c r i b e s whole o f t h e obs e r v e d c h a r a c t e r i s t i c s a s shown i n f i g . 2 . In the second experiment, the s p e c t r a l width of exc i t a t i o n l i g h t was i n c r e a s e d by an o r d e r o f magnitude. Then t h e Raman l i n e b r o a d e n s by t h e same o r d e r o f m a g n i t u d e w h i l e t h e l u m i n e s c e n c e line is still sharp because it is independent of the spectral width of the incident light and is determined by Yme" These observed characteristics can he represented as shown in fig.3 by the calculated spectra in terms of the same relaxation and decay constants in fig.2. It is noted that another broad emission llne is observed in the experiment and that this is assigned to the secondary (and higher) step emission of the hot excitonic molecules formed from two single excitons left behind in the first (and higher) step emission processes. 4. Transient Emission Spectrum The competitive behavior of the Raman and luminescence channels on time coordinate is studied by the time-resolved emission spectrum after the pulse excitation. Let us assume the observation system open between T and T + AT. The probability amplitude that the system falls into the final state in this time interval is g i v e n by A~ (3) (T) ~T+AT

=

JT

[tI

dt

lJ_®dt 2 ¢ ( t 2

-2~0~t2+i~2t 1 )e


-iH(T+AT-t I) + -iH(tl-t 2) -iHt 2 x e V2e We Ig>,

22

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LUMINESCENCE DUE TO EXCITONIC MOLECULE

where

Retative ML Y'mg=O.086 mev Excitation "6"me---0.152 Energ~ Yeg =0.066 f (1632 me~ "~J Ym =0.05 g0.885 FWHM of incident hl.839 light =0.4 mev

¢ ( t 2) - I d ( 2 = l ) e x p [ - i ( 2 = l - 2 w ~ ) t 2 ]

x 1//~'~oexp[-(2wl-2~)2/o

2] ,

Relative ML "Y~mg=O.086mev Excitation "Y'me=0.152 E n e r g y ( 2 u ~ Y e g =0.066 f 0.157 mev "'# Yrn =0.05 • g 0215 FWHM of incident .-. h 0,314 light =0.053 mev ® i 0.430 f//

izgo8

u

,114 °

m k 0.711

~

m

I IRol=

10

2.0

1.0

0.0

Relative Photon Energy(mev) F i g . 3. C a l c u l a t e d e m i s s i o n s p e c t r a v i a e x c i t o n ic molecule with recoil of a longitudinal e x c i t o n u n d e r wide band e x c i t a t i o n (FWHM0.4 meV).

1.0 0.5 0.0 Relative Photon Energy (mev) Fig.2. Calculated emission spectra under the giant fifo-photon excitation of an excitonic molecule with recoil of a longitudinal exc i t o n u n d e r n a r r o w bend e x c i t a t i o n (FWHM= 0.053 meV). The origin of the abscissa corresponds to ~-Imelneacence llne. and t h i s d e s c r i b e s t h e i n c i d e n t p u l s e c o r r e s ponding to the giant two-photon excitation of an e x c i t o n i c m o l e c u l e . Here b o t h t h e e f f e c t s o f c a r r i e r f r e q u e n c y d i s t r i b u t i o n and o f p u l s e e x c i t a t i o n a r e t a k e n i n t o a c c o u n t by t h e d e v i a t i o n u. Under t h e a s s u m p t i o n t h a t t h e a m p l i t u d e o f the emitted radiation is proportional to the probability amplitude of finding the final state o f t h e t o t a l s y s t e m i n t h e t i m e i n t e r v a l T and T+AT, we d e f i n e t h e e m i t t e d l i g h t i n t e n s i t y by ~(Z) - ~ z T r

[A* (3) (T)%A* (3) (T)*].

(5)

In our previous paper, we defined provisionally t~. time, resolved emission spectrum in terms of ,~J) (t) in eq. (2) as T(t) = d ~ T r [,(3) (t)PR,!3) ( t ) * ] .

(6) 'b

We may say that the definition of I(t) and Z(t) come from the particle and the wave pictures for the emitted light, respectively. Unfortunately, we can not guarantee the po~itlveuess of I(t). As to the new definition ~(t), this is free from that drawback of l(t) and the fre-

quency-time uncertainty due to the finite observation t i m e i s s u i t a b l y i n c o r p o r a t e d . F~ the d e t a i l e d e x p r e s s i o n ~ ( T ) , see t h e p a p e r ÷ Under t h e e x c i t a t i o n by r e c t a n g u l a r p u l s e 0 < t 2 < z, t h e ~Rm~n component d i s a p p e a r s s u d d e n l y after the pulse is switched off (T
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LUMINESCENCE DUE TO EXCITONIC MOLECULE

Vol. 32, No. I

YmgffiG15 "Y'me=0.25 mev Yeg fiat "fro =0.1 FWHM of incident iight=O.Smev off resonance energy of

/

/.

,

incident lioht = - 0 . 6 mev

Pulse Dur~ion Time=25ps ,'2._ Duration Time of / ~ ~ Observ=tion=251~ ....

"

,

~ -

/ r

~

, .......

~

/

/

/

=-10(mev-1) -1.O

-OS

0.0

0.5

1.0 Relative Photon Energy (mev)

Fig.4. Time-resolved emission spectra under excitation by a G a u s s i a n p u l s e . The t~me coordinate is drawn in units of (meV)-*~ 4.136 ps.

/'~

ML

Yma=0.3Ymef$.Smev

\

T,.,,. o,

// .,~

~U

b.I

scence

/ / E ,///,.._.,Raman,,/,~ " ,'.," ",

-10

0.0

,\

',. 10

~

~ 20

• -1 . T,i m e ( m e v ) 30

Fig. 5. Decay profile of M L llne under Just resonant excitation of excltonlc molecule. The total emission intensity is decomposed into R-m-n and luminescence components as shown by dashed line.

with the tion and from the Then the observed

radiative life time I/ym. The relaxadecay constants have been determined t/me-integrated emission spectrum. time dependence of flg.5 represents the features very well. 5. Discussions

The competitive aspects of the Raman and luminescence channels are studied both in the emissionspectrum as a function of frequency and in the time dependence of emitted light intensity. The re-

laxation constants used in figs. 2 and 3 have been determined in the weak excitation case but they increase as the excitation power increases. This is because the colllslonsof the excitonlc molecule in the intermediate state and the single exclton in the final state with other highly excited molecules contribute to the relaxation constants Tm~, Yme and y _ . The increase of Ym~ with increas~ of excltat~n power was observed as broadening of the giant two-photon absorption spectrmn due to the excitonic molecule.7 The ratio of the peak value of the Raman llne to

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LUMINESCENCE DUE TO EXCITONIC MOLECULE

that of the luminescence llne is given approximately by i/Ve~{l+(2~l-~nK)2/ymgYme } versus 2/7m and this ratio is expected-to decrease as the excitation power increases. We could determine the values of relaxation and decay constants by comparison between the observed and calculated emission spectra. These values look to be of an order of reasonable magnitud e . The calculated spectra catch also the characteristic features observed by Segawa et al. 12 both for the M L and M T processes leaving behind the longitudinal and transverse excitons, respectively. It is noted that the Raman and luminescence lines have the same order of magnitude for the M L process, while the Raman llne in the M T process is much

Vol. 32, No. 1

stronger than the luminescence llne. This is clear because the peak value of the Raman line is approximately proportional to the reciprocal of Yeg and the value of Veg is much smaller for the M T process than for the M L process. This is understood as follows: In t h e m L process, the inelastic scattering of the longitudinal exciton into the transverse one plays an important role in determining the value of Xeg, while in the M T process the reverse process is quenched at low temperature. At present, almost all experimental facts have been analyzed in terms of our stochastic theory in the fast modulation limit. It will be more interesting to look for effects beyond the damping theory.

References

i. A.Mysyrowicz, J.B.Grun, R.Levy, A.Bivas and S.Nikitine: Physics Letter 26A (1968)615. 2. H.Souma, T.Goto, T.Ohta and M.Ueta: J. Phys. Soc. Japan 29 (1970)697, 3. O.AklmoCo and E.Hanamura: J. Phys. Soc. Japan 33 (1972)1537. 4. See, e.g., E.Hanamura, in Optical Properties of Solid, ed. by B.O.Seraphin (North-Holland Amsterdam, 1976) Ch.3. 5. E.Hanamura: Solid State Commun. 12 (1973) 951. 6. G.M.Gale and A.Mysyrowlcz: Phys. Rev. Letters 32 (1974)727.

7. M.Ueta and N.Nagasawa: Lecture Notes in Physics 5_~7 (Springer-Verlag, 1976)p.i. 8. T.Itoh and T.Suzuki: J. Phys. Soc. Japan 45 (1978)1939. 9. T.Takagahara, E.Hanamura and R.Kubo: J. Phys. Soc. Japan 43 (1977)802, 811, 1522; ibid. 44 (1978)728, 742. i0. T.Mita and M.Ueta: Solid State Commun. 27 (1978)1463. Ii. E.Hanamura and T.Takagahara: to be published in J. Phys. Soc. Japan. 12. Y.Segawa, Y.Aoyagi, O.Nakagawa, K.Azuma and S.Namba: Solid State Commun. 27 (1978)785.