VOLUME 41, NUMBER 1 P H Y S I C A L R E V I E W L E T T E R S 3 JULY 1978

Determination of the Excitonic Polariton Dispersion in CuCl by Resonant Two-Photon Raman Scattering

B. Honer lage , ( a ) A. B ivas , and Vu Duy Phach Laboratoire de Spectroscopic et d'Optique du Corps Solide, Associe au Centre National

de la Recherche Scientifique, Universite Louis Pasteur, 67000 Strasbourg, France (Received 14 April 1978)

Starting from Hopfield's theory of the excitonic polariton, the energies and momenta of the particles involved in a two-photon Raman scattering process may be calculated self-consistently from the direction of the exciting photons, their energy, and the direc­tion of the scattered light. Attributing to the calculated momenta the measured energies of the emitted photons, one obtains the dispersion relation of the excitonic polaritons with a high accuracy in a semiexperimental way.

Because of the strong coupling between the elec­tromagnetic radiation field and the polarization in the crystal, eigenstates of the coupled exciton-photon system have been shown to be mixed states of excitons and photons.1'2 In CuCl, the eigenval­ues of these states—so-called polaritons—give r ise to a nonlinear dispersion relation in the r e ­gion where the photon dispersion crosses the r 5

exciton band. In addition, the threefold degener­acy of the T5 exciton band is partly lifted by this interaction and one obtains one longitudinal exci­ton band and two twofold degenerate—upper and lower—polariton branches. The usual equation describing the polariton dispersion relation E(p) reads1

E4(p)-AE2(p)+B = 0,

where

A = (H?c2p2/cJ +ET2(p)(l + W°),

B = (H2c2p2/e00)ET2(p),

2TIP0^(PL-ET)/ET,

ET(p)=ET + (H2/2m*)p2.

(1)

(2)

(3)

(4)

(5)

For the transverse exciton band, in the effective-mass approximation, this equation depends on four parameters: the high-energy dielectric con­stant €oo, the effective exciton mass m*, and the energies EL of the longitudinal exciton and ET of the transverse exciton at a momentum p = 0.

The longitudinal exciton band, given by

EL(p)=EL + (H2/2m*)p2, (6)

and the upper polariton branch E2(p) [following from Eq. (1)] were first directly observed by non-resonant two-photon absorption.3 In this study, the momentum p of the excited quasiparticles was tuned by changing the angle between the two photon beams. Information on the lower polariton branch Ex(p) [following from Eq. (1)], however,

could not be obtained because of the conservation of energy and momentum in the two-photon ab­sorption process.

We will show in this work that a combined ab­sorption and emission process, like resonant two-photon Raman scattering via biexcitons, gives the possibility of performing spectroscopy in mo­mentum space of the lower polariton branch, too.

Biexcitons in their ground state can be excited resonantly by two-photon absorption in CuCl when the energy of the exciting photons is equal to half the biexciton energy EB/2, as shown unambigu­ously.4 When the energy of the exciting photons is detuned from this resonance, biexcitons are created only virtually, with a momentum equal to two times the momentum q, of the exciting po­laritons. These virtual biexcitons recombine, emitting a lower polariton of energy Ex{q), and leaving in the crystal a lower polariton El(Jz)> an upper polariton E2(k), or a longitudinal exciton EL(k)5

These processes conserve energy and momen­tum of the different particles involved; namely,

and

2Hu>l=E1(q)+E1(k),

2H(Jol=E1(q)+E2(k),

2Hul=E1(q)+EL(k),

I

(7)

(8)

(9)

(10)

In the experiment, the crystal is excited with a laser beam making an angle of incidence a on the sample. The photon energy of the laser is fooj. The photons emitted by the crystal are observed along a direction making an angle 0 with the nor­mal to the surface of the sample.

If we assume that the dispersion of the polar­iton is known, the momentum |q,| of the exciting polaritons is obtained from their energy Hix>l.

© 1978 The American Physical Society 49

VOLUME 41, NUMBER 1 P H Y S I C A L R E V I E W L E T T E R S 3 JULY 1978

The direction of qz inside the crystal is deduced from the index of refraction

and

n(ql)=kc\ql\/E1(ql)9 (ID and the angle of incidence a.

The momenta q and £ of the two polaritons created in the crystal and the scattering angle 0 between incoming and observed polaritons can be computed self-consistently, as suggested previ­ously,6*7 by fulfilling the two laws of energy and momentum conservation [Eqs. (7) to (10)] and the index of refraction [expression (11)]. Thus the energies Ex{q) of the emitted polariton and Ex(k) of the quasiparticle left in the crystal are ob­tained.

The first process, where the two particles created in the crystal are two lower polaritons, has been studied in two different experimental configurations.

In the backward configuration, the scattering angle is large (0^ 180°). As shown in Fig. 1, the polariton left in the crystal is excitonlike, and its energy is almost equal to the energy of the transverse exciton ET. The photonlike polariton gives r ise to an emission line RT which is detect­ed experimentally. For different energies of the exciting photons, we have fitted the experimental positions of the RT line with the computed solu­tions. Thus we have been able to deduce the fol­lowing parameters of the dispersion curve of the polariton:

3.2025± 10"4 eV, (12)

e=i80:

225

FIG. 10 Resonant two-photon Raman scattering pro­cesses involving the lower polariton branch.

5.0±0.2. (13)

In a forward-scattering configuration, the angle 0 is small (0 <30°). As shown in Fig. 1, the ener­gies of the two polaritons created in the crystal are in the bottleneck region of the polariton dis­persion curve. By changing the scattering angle, the frequency of the observed Raman lines can be tuned. Furthermore, at these scattering angles, three simultaneous solutions of Eqs. (7) and (10) of conservation of energy and momentum exist for each given direction of observation. Two of these solutions correspond to the RT~ a n d # r

+

Raman lines, shown in the typical Raman spectra drawn in Fig. 2. The third one corresponds to an energy inside the exciton absorption band and was observed, too, in another set of experiments.

The second process, where a lower and an up­per polariton are created in the crystal, is not observed in our experimental conditions.

In a third process, a lower polariton and a lon-

3.140 3.160 3.180 3.190 3.200 eV

FIGo 20 Biexciton emission spectra in CuCl for dif­ferent angles of incidence a and for a fixed angle of observation (3 = - 57°. The exciting photon energy is chosen to be Kul = 3.1865 eV.

50

VOLUME 41, NUMBER 1 PHYSICAL REVIEW LETTERS 3 JULY 1978

gitudinal exciton are created in the crystal. This gives r ise to theRL emission line observed in the backward- and forward-scattering configurations. As shown in Fig. 3, a small energy difference (0.16± 0.02 meV) of this line is observed between the two configurations, for each given exciting photon energy. It is due to the curvature of the longitudinal exciton band. It is remarkable that the exciting photon energy being at half the biex-citon resonance H(jot = 3.1860 eV, the observed bi-exciton luminescence does not fit into the straight lines obtained for the Raman process. This is a consequence of the biexciton distribution due to the relaxation processes, as discussed elsewhere.

Knowing from our calculation the momentum of the final exciton state, we have been able to de­duce from this energy difference the exciton ef­fective mass:

3.1670

m* = (2.5±0.3)m0. (14)

This i s , to our knowledge, the first time that the exciton mass has been directly determined in CuCl. This value is in close agreement with the one given by Honerlage, Klingshirn, and Grun8

and deduced from the electron-exciton scattering. We have also deduced, from the spectral posi­

tion of the RL line, the value of the energy EL of the longitudinal exciton at the center of the Bril-louin zone:

E £ = 3.2080±10"4 eV. (15)

3.16104-3.1850

This value is in close agreement with the one ob-

FIGo 3. Energy of the two-photon Raman line KuR

as a function of the exciting photon energy Hool in two different scattering configurations. Crosses, back­ward-scattering configuration: a = 58°, /3 = - 32°; circles, forward-scattering configuration: a = 45°, jS = - 45°; full lines, calculated curves using the pa­rameters given in the text»

tained by Frohlich, Mohler, and Wiesner.3

Thus, all parameters describing the polariton dispersion relation in Eqs. (2) to (5) are known

E(eV>

3.220

3.200

3.180

(

-

D

«»«»«««»

/ ^

|

5

. -*>*—

I 10

i „ 15|p|(105cm-1)'

FIG. 4. Polariton dispersion relation in CuCl. Full line, theoretical; crosses, direct observation; solid circles, indirect, using energy conservation; open circles, Ref. 3.

51

V O L U M E 41 , N U M B E R 1 P H Y S I C A L R E V I E W L E T T E R S 3 J U L Y 1978

with a rather high precision. Using these param­eters , we have drawn in full line, in Fig. 4, the dispersion curve of the polariton and of the lon­gitudinal exciton. We have also plotted the meas­ured energies of the Raman emission lines as functions of the wave vectors of the correspond­ing polaritons calculated as explained above. They are represented by crosses in Fig. 4. The values of the energies of the longitudinal exciton and of the lower polariton at large |E| values could be deduced from the energies of the ob­served Raman lines and from the law of energy conservation given by Eqs. (9) and (7). They are represented by the points marked by solid circles in Fig. 4.

Then, when we add the results obtained from the two-photon absorption measurements,3 marked by open circles in Fig. 4, the dispersion relation of the exciton polariton in the bottleneck region is completely known.

An enlarged portion of the lower polariton branch is given in Fig. 5. It shows how good the fit is between the experimental points and the the­oretical curve. All anomalies in the dispersion relation of the polariton should show up in this representation. If the parameters of the polar­iton dispersion had not been correctly chosen, a systematic deviation between the experimental points and the theoretical curve would have been observed.

The spectroscopy in momentum space described above should also work with more complicated exciton and polariton structures, if an intermedi­ate transverse exciton and biexciton states lead to a resonance enhancement of the two-photon Raman process. This spectroscopy is compar­able to the resonant Brillouin scattering which also gives information on the exciton and polar­iton dispersion curves.9 '10

We acknowledge an important contribution to this work by Dr. J. B. Grun and Professor Dr. U. Rossler. This work has been supported by a

M<*4

3.190^

3.180 f-

3.170 h

FIG, 50 P a r t of the lower polar i ton b ranch . C i r c l e s , exper imenta l ; full l ine , t heo re t i ca l .

contract Action Thematique Programmge of the Centre National de la Recherche Scientifique.

(a)On leave f rom Fachbere ich Phys ik , Univers i ta t Regensburg , D 8400 Regensburg , W. Germany.

i3. J . Hopfield, P h y s . Rev. _112, 1555 (1958). 2 S. I. P e k a r , Zh„ Eksp , Teor 0 F i z . 3 3 , 1022 (1957)

[Sov. P h y s , J E T P 6, 785 (1958)]. 3D. Frbhl ich , E . Mohler , and P . Wiesne r , P h y s .

Rev. Let t . 26 , 554 (1971). 4Vu Duy Phach , A. Bivas , B, Honer lage , and J . B.

Grun, P h y s . Status Solidi (b) _84, 731 (1977). 5E0 Os t e r t ag , t h e s i s , Univers i te de S t r a sbourg ,

F r a n c e , 1977 (unpublished). eVu Duy Phach , A. B ivas , B . Honer lage , and J„ B .

Grun, P h y s . Status Solidi (b) JB6, 159 (1978). 7 E. Os t e r t ag , A. Bivas , and J0 B. Grun, P h y s . Status

Solidi (b) 814, 673 (1977). 8 B . Honer lage , C. Kl ingshirn , and J . B . Grun, P h y s .

Status Solidi (b) 2 § , 599 (1976). 9R. Go Ulbrich and C„ Weisbuch, P h y s . Rev. Let t .

38 , 865 (1977). ™G. Winter l ing and E. Kote les , Solid State Commun.

253, 95 (1977).

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