Giant two-photon absorption due to excitonic molecule

~

0038-1098/9356.00+.00 Pergamon Press Ltd

Solid State Communications, Vol. 88, Nos. 11/12, pp. 1073-1075, 1993. Printed in Great Britain.

GIANT TWO-PHOTON ABSORPTION DUE TO EXCITONIC MOLECULE Eiichi Hanamura The Institute for Solid State Physics, The University of Tokyo, Roppongi, Minato-ku, Tokyo 106, Japan

(Received 12 February 1973 by .Y. Toyozawa)

The direct excitation of excitonic molecule due to the two-photon absorption process is shown to be strongly enhanced because of two effects of the resonance and the giant oscillator strength as known for the bound exciton. Then it is pointed out that the existence of the excitonic molecule can be confirmed also by this two-photon absorption spectroscopy. We discuss also the property of the excitonic molecules highly excited by this method.

SINCE Akimoto and the author z pointed out that the excitonic molecule should exist as stable complex excitation for any value of the electron-hole mass ratio. its existence has been confirmed in CdS and CdSe, 2 in addition to the classical case of CuCI and CuBr, a by the emission spectrum under strong excitation. The Bose condensation of these excitonic molecules4 was also observed on the emission spectrum under the pico-second laser pulse excitation, in agreement with the theortical prediction of the author. 5 In this short note, we point out theoretically that the two-photon absorption due to excitonic molecule is extremely enhanced due to the same effect of giant oscillator strength as in bound exciton ~ and the resonance effect, in contrast to the ordinary two-photon absorption. As a result, the existence of excitonic molecules will be confirmed also by the two-photon absorption spectroscopy. The high density excitonic molecules created by this direct excitation share the same translational momentum 2Ko and the same excitation phase, where Ko is the wave number vector of light. As a result, after collisional time of an order of 10-1:sec for the density 10-16cm -a of excitonic molecules under the pico-second pulse excitation, some part of excitonic molecules are expected to condense to K = 2Ko state. The property of these coherent excitations is also discussed. We give the expression 7 of the transition probability for the two ~o photon excitation of an

insulating crystal from its ground state Ig) to the excited state 1era) with an excitonic molecule as follows:

2n

wm= [i)(i[

i Eig--~

( g,ovl N 2 l ( e m l l ' ~ P/ • ~

1'

Pf[g)[" 6(E¢1--2~,o), (1)

where N is the number of co photons in the incident beam with polarization ~, g is a dielectric constant for frequency co, V is the volume of the crystal, Pi = exp(/Ko • rl)p~ and pj is the momentum of the/th electron. As the intermediate state ]i), we only need take the following one exciton state by paying attention to the resonance effect described below;

li,)=~r,p v/-V 8g'l?-pfn ( : ) a~l'avplg)'

wherefj.(:)=SV~o)/{l

+ a~(od~ + 0p)= }', ao _=

go h2//ze2 is the exciton Bohr radius, ot = ms/(me + mh) and 0 = 1 - a. Here go is the static dielectric constant of the crystal and/z - memh/(m¢ + mn) is the reduced mass of an exciton. The final state with one excitonic molecule is described as follows; F,l,',p,p'

p, a~]ea~r' a~p, avplg).

Previously published in: Solid State Commun. Vol. 12, No. 9, pp. 951-953 (1973)

1074

GIANTTWO-PHOTONABSORPTlONDUETOEXCITONICMOLECULE

Here we take the following formr *’ for the wave function of an excitonic molecule;

4r

8 single exciton

( 1 P

p’

P

P’

= ~GK.~+~‘-p-p~g(P-P-P)+P’)

f 18 0 ;

fi, 0 g

9

where g(K) is the Fourier component of the wave function describing the relative motion of two excitons in a molecule and is approximated by um (1 + t&K')*with the average distance aE given in reference (1). When we use the following familiar approximation in semiconductors; ~P,exp(&

lrj)lt

= (P,~~)~4p+~04rp.

the matrix elements in equation (1) are evaluated as follows;

where

4*,(r) = E f,,(k) exp(ikr). l

putting these matrix elements into equation (I), we obtain the transition probability of two photon absorp tion due to an excitonic molecule as follows;

l

Id,,(O)r’g2@) W2EI,

-E”, - 2hL

(2)

where J?$~is the binding energy’ of an excitonic molecule and I#,,(O)g(O)l* = 64(aB/ao)3. When this transition probability is compared with that of one photon absorption due to an exciton;

l

lc 1’ (PC”

w,l,w2

w,-

Vol. 88, Nos. 1l/l2

hw),

Wh2)/Wz' is estimated to be of an order of 3 X lo-” a)*/m 23 eV,(an/ue)2:3,1 hw WI v), where (P, l ‘v 3 eV and Et, - hw = Lz/2 2 20 meV (CUC~)~or 3 meV (CdS)2 are estimated, corresponding to the case of CuCl or CdS. Therefore when we use the dye laser as the source of tunable frequency above the photon density N/V = 1O”/cm-“ , we can expect the two-photon absorption coefficient as strong as

absorption. Under this condition, it is guaranteed that two photon absorption coefficient due to excitonic molecule overcomes very much the absorption coefficient of an exciton at the exciton absorption tail E,, - E./Z.Furthermore in comparison with the case of an ordinary two-photon absorption due to a single electron excitation, we have two strong enhancement factors in our case; the first enhancement factor 64(0n/ue)~ comes from the fact that our case corresponds to the giant oscillator strength6 and the second one comes from the resonance effect ((f$ - hw)/(E,, - LJ))*, where k$, is the effective energy difference between the intermediate state and the state in the valence band. The first enhancement is explained as follows: in the process of transition from Ii,) to le,,, ), we only need excite another exciton in the range within the molecular radius aB around the first exciton in order to make an excitonic molecule coherently, in contrast with the ordinary case where the electron excited to the intermediate state should interact again with the second photon. As a result, we have the factor WlrB /aa )’ z 10-j. The second enhancement factor of the resonance effect is estimated to be of an order of 10’ atw=ol,pm /2R if we assume I!?&2: 5 eV and Eb ,,, 2: 40 meV (CuCl). Because it is difficult to consider the other factors of two-photon absorption process work 1O6 times unfavourably on the case of the excitonic molecules, we can expect the two-photon absorption peak due to excitonic molecules to be observed dominantly. As a result, in the two-photon spectroscopy, the excitonic molecule will be confirmed as the absorption peak at E;,- t”,/2 embedded by the rather weak background of one-photon absorption tail of an exciton and ordinary two-photon absorption. Next we discuss the coherent property of these excitonic molecules highly excited by this method. If we have the mode-locked picosecond pulse with the frequency (E,, -pm/Z) and the enough power with (N/v) > 10”cm-3, the high density excitonic molecules directly created by this pulse share the common translational momentum 2Ke. As a result, after collisional time of an order of 1O-“set for the density 1Or6cmm3of excitonic molecules, some fraction of excitonic molecules are expected to condense to K = 2K,, state. The group velocity 2hKe/M of these Bosecondensed excitonic molecules is less than the Landan critical velocity l@Ne W&f) in our case,

Vol. 88, Nos. 11/12 GIANTTWO-PHOTON ABSORPTION DUE TO EXCITONIC MOLECULE where Wo is the scattering matrix element between molecules. Therefore the high density excitonic molecules, created at the surface of an order of (!/g~)), are expected to show superfluidity of band gap energy, where gm ~2) is the absorption coefficient of two-photon absorption due to excitonic molecules and is estimated to be 104/cm. When we deposit the traps of excitonic molecules, for example, the hetero-junction with narrower band gap, at the back of the sample with the depth l, we can observe the emission signal from the

1075

back surface after the time/M/2hKo. These Bose condensed excitonic molecules have not only long longitu. dinal relaxation time but also long transverse relaxation time. Therefore these objects are considered as a candidate of study of nonlinear optical phenomena.

Acknowledgements - The author is grateful to Professors Y. Toyozawa and S. Shionoya for fruitful discussions.

REFERENCES I. 2.

AK]MOTOO. and HANAMURA E., Solid State Comraun. lO, 253 0972); ibid. I l, No. 9, p. xiii 0972); J. Phys. Soc. Japan 33, 1537 (1972). SHIONOYAS., SAITO H., HANAMURA E. and AKIMOTO O., Solid State Comraun. 12, 223 (1973).

3.

MYSYROWICZ A., GRUN J.B., LEVY R., BIVAS A. and NIKITINE S.,Phys. Lett. 26A, 615 (1968); SOUMA H., GOTO T., OHTA T. and UETA M.,J. Phys~ Soc. Japan 29,697 0970).

4.

KURODAH., SHIONOYA S., SAITO H. and HANAMURA E., SolidSmte Commun. 12, 533 0973).

5.

HANAMURAE. and INOUE M., Proc. 11th Int. Conf. on the Physics o f Semiconductors, ~/arsaw, 711 0972); HANAMURA E., l~oc. Int. Con/. Luminescence, Leningrad 0972), to appear in J. Luminescence.

6.

HENRYC.H. and NASSAU K.,J. Luminescence 1,2,299 0970).

7.

INOUEM. and TOYOZAWAY., J. Phys. Soc. Japan 20, 363 (1965).

8.

HANAMURAE.,J. Phys. So¢. Japan 28, 120 0970).

Die direkte Anregung des Exzitonenmolekuls durch den 2-Photonen-Absorption Process wird dutch zwei Effekte erheblich verst~'kt, durch die resonante Wechselwirkung und die gigantische Oszillator-St~rke. Es wird gezeigt dass die Existenz des Ezxitonenmolekt~Is auch durch das 2-Photonen-Absorption Spektrum bestatigt werden kann. Wir diskutieren den Charakter yon ExzitonenmolekUlen hoher Dichte die bei dieser Methode angeregt werden.