- Email: [email protected]
Excitonic-biexcitonic polariton interference in thin platelet of CuCl
26 April
1999 PHYSICS
Physics
ELSEYIER
Letters
Excitonic-biexcitonic
A 254 (1999)
LETTERS
A
351-354
polariton interference in thin platelet of CUCl Z.G. Koinov ’
Department Received
17 July
1998;
of Physics, revised
Higher
manuscript
of Transport
Institute
Engineering,
1574 &$a.
received 22 December 1998; accepted Communicated by A. Lagendijk
Bulgaria
for publication
26 January
1999
K)stract position in Q-space of the resonance region, measured
The spectral in Zi-excitonic
in:erference
transmission
effect between two propagating
PI CS: 71.36; 71.35 Keywords: Polaritons;
Excitons;
Biexcitons;
by Mita
maxima of a 0.15 ,um thick CuCl single crystal with parallel plates and
Nagasawa,
excitonic-biexcitonic
has been
polariton
interpreted
as an indication
for
the
mutual
modes. @ 1999 Elsevier Science B.V.
CuCl
-
-
I. Introduction In recent years there has been a great deal of interest ir the exciton spectra and polariton interference in the
layers of CuCl [ l-51 and GaAs [ 6-91, It was pointed o 4t that depending on the thickness of layers the transn ission and reflection spectra of those materials can b-, explained either by treating the exciton system as a confined one or regarding it as a bulk system. In what follows we analyze the transmission spectrum of a 0.15 pm thick CuCl single crystal with parallel plates ir Z3-excitonic resonance region at low excitation intensities (N 300 W/cm2), measured by Mita and Nagasawa [ 11. This is not the case of a confined exciton s!~tem, and therefore, the usual type of Fabry-Perot polariton interference takes place. Assuming that the excitonic-polariton modes have been produced in the s.ab by the incident beam, Cho and coworkers [2,3]
’ E-mail:
have already analyzed the results by Mita and Nagasawa, primarily with respect to determining which of the various forms of the additional boundary conditions (ABC) are valid for the propagation of excitonic polaritons in CuCl thick crystal with parallel plates. The purpose of this paper is to explain the perioddoubling of the polariton interference in the energy region just below the longitudinal exciton energy, which has been observed in CuCl [ 11. The period-doubling means that for the case of double polariton modes interference the transmission maxima and reflection minima are equally spaced in Q-space with a period which is twice the period for the case of single mode interference. This long-period interference contradicts to the excitonic-polariton point of view because according to the excitonic-polariton model only one polariton mode is allowed in the above energy ration. We explain this period-doubling of the polariton interference by employing the so-called excitonic-biexcitonic polariton concept [ IO,1 I], which predicts two prop-
[email protected]
037%9601/99/$ - see front FII SO375-9601(99)00094-8
matter
@ 1999 Elsevier
Science
B.V. All rights
reserved
352
Z.G.
Koinov/Pl~ysicsics
Letters
agating polariton modes allowed in the energy region just below the longitudinal exciton energy. The dispersion curves of bulk excitonic polaritons w~.pO1.(Q) in this m aterial can be calculated by solving the following equation,
The solutions of the above equations are as follows, e;‘.p”.
A
=
B2 =
( Q)
= 5
[A
+
d-1
‘/’
)
~‘12’ El, + E+(Q) + Et - E$ > c2Q2E;(Q,
(lb)
where ~g accounts for all other oscillators not explicitly considered in this one-oscillator model. ET(Q) = ET + Q’/~MT is the dispersion of the transverse excitons (MT is the transverse exciton mass) and EL = ET + ALL is the longitudinal exciton energy at the point Q = 0 (Aw is the longitudinal-transverse splitting, caused by the Elliott exchange interaction). The corresponding parameters for CuCl are [ 121 ET = 3.2022 eV, EL = 3.2079 eV, MT = 2.3mo, Et) = 5.59. Due to the reflectivity of the sample surfaces the incident beam is splitted into many partial waves which interfere with one another. According to the excitonic-polariton model, for the photon energy below the longitudinal exciton energy EL at the point Q = 0 (w < EL), only one mode can be produced in the sample. The dispersion of the partial waves is identical to the dispersion of the lower excitonicpolariton branch (LBP) w:‘.~“. (Q) . In this case we have a usual type of Fabry-Perot interference which is determined by the constructive and deconstructive condition for the phase differences arising from a single round trip of the lower excitonic-polariton mode across the slab. For normal incidence a constructive interference of partial waves occurs if the phase shift for one round trip in the sample (the way is 24 equals an integer multiply by 27r. As a result, a standing wave with a large amplitude builds up in
A 254 (1999)
351-354
the crystal, resulting in transmission maxima and reflection minima, which are equally spaced in Q-space with jAQ,,,/ = n-/d. When the energy of the photon is above the longitudinal exciton energy, the situation becomes complicated because there are two modes of excitonic polariton allowed in a slab: the lower polariton branch w~c’pO”(Q) and the upper polariton branch WY.“‘- (Q) (UBP) The corresponding microscopic calculations of the transmission and reflection amplitudes of the electromagnetic waves [2,3] show the period-doubling of the polariton interference. This result can be understood from macroscopic point of view in the following way. The incident beam produces two waves with equal frequency but considerably different wave vectors ( ]QuuPI < /QLspi). The amplitude of the LBP decreases with increasing energy but in the range closed to EL, both modes have comparable amplitudes. Due to the reflection at the back side of the sample, the UBP and LBP waves produce both UBP and LBP waves, so the EBP wave can find a partner for constructive interference after one half round trip in the sample (the way is d). On the way from the front to the backside of the sample, both waves get phase differences Ap~np = lQLBPId and Apunp = /QusPId. In the above energy region Ap~ar >> Apusr, and as a result, we can go energetically from one transmission maximum to the next in Q-space by equal steps ~AQLsp~zz 2rr/d. It should be noted that the period-doubling of the polariton interference for the photon energy above the longitudinal exciton energy does not depend on the corresponding ABC. The ABC are required for solving the problem of deciding how the intensity of the incident radiation beam impinging on a surface is partitioned among two excitonic-polariton modes. The thickness of the slab is closely related with the positions of the transmission maxima in Q-space, while ABC are needed to evaluate the sharpness and relative amplitude of each transmission maximum. Since the geometrical thickness of the sample is d = 0.15 pm, from the excitonic-polariton point of view follows that the equal step of IAQLBPI M 2s-/d = 4.2 x lo5 cm-’ above the energy 3.2079 eV should be observed. But, according to the transmission data, measured by Mita and Nagasawa, the following maxima are clearly seen:
Z.G. Koinov/Physics
Letters
El RZ3.2067 eV,
E2 M 3.2074 eV, E3 zz 3.2082 ev , . . , .
Tke corresponding wave vectors IQLBpI, i = 1,2,3, ca.culated by using the dispersion w-exc’po’.(Q) , are as fo- lows. jQ~,,l = 52.314 x lo5 cm-’ , If&I = 56.498 x IO5 cm-’ , I~~,,~ = 60.681 x 10’ cm-‘. The spectral positions in Q-space of the above transm ssion maxima are in a very good agreement with th:: interference condition lAQLsPI = 2rr/d. From th.:se calculations follows that below the longitudinal exciton energy EL there exist two propagating polariton modes and the constructive interference after one half round trip in the slap is observed. This statement cannot be understood by employing the excitonicpolariton point of view.
2. Excitonic-biexcitonic of the transmission CuCl slab
polariton interpretation spectrum of 0.15 pm thick
In principle, the more accurately consideration of the transmission spectrum can be based on the so-called excitonic-biexcitonic polariton concept [ 10,111, because this model provides the calculai.ion of the elementary excitation spectrum of the whole photon-exciton-biexciton system. In Fig. 1 we have shown a schematic representation of the dispersion curves for the interacting system of photons, excitons and biexcitons in the crystal to illustrate ttz main differences between the excitonic-polariton rr od.el and the excitonic-biexcitonic polariton conccpt. According to the exitonic-polariton point of VI&W, the elementary excitation spectrum in the system under consideration consists of three branches: tire first two are the upper and lower excitonic polaritc.n branches and the third represents the biexciton g.,-ound state (the direct interaction between biexcitons and photons is forbidden due to the selection rules). If we take into consideration the interaction b-tween the biexcitons and the exciton components or the corresponding excitonic polaritons, then the e.>:citonic-biexcitonic polariton concept has to be
353
A 254 (1999) 351-354
used. According to this concept, the elementary excitation spectrum in the above system consists of three branches L!;(Q), i = 1,2,3. The dispersion of those quasiparticles can be determined by solving the following equation [ 111, fJ - 00(Q)
-
IA+(Q>l” IA-(Q)l” fJ _ we+xc+O’.- 0 - w~c+O’. (Q) (Q)
=o.
(2)
Here 00 (Q) is the biexciton ground state energy and A+(Q) are the matrix elements of the interaction between the biexcitons and the exciton components of the corresponding excitonic polaritons. For small wave vectors IQ1 << ( &/c) ET, the lowest excitonicbiexcitonic polariton branch is almost identical to the photon-like lower excitonic-polariton branch, f& (Q) M a~~-~~‘.(Q) ,
(3a)
while the other two branches are given by the following relations, 02,3(Q)
= ; [~e:c~p”‘~(Q) + G(Q)
i &("y.POl.
(Q>- f20(Q>>~ + W+(Q>12]. (3b)
According to (2)-( 3)) the elementary excitation spectrum at the point Q = 0 is defined as follows, l-2, zz w-exc.pol. (Q = 0) = 0 , 03 = @,+A,,
fit2 KSEL - A0 , (ha)
where L$ is the biexciton ground energy at the point Q = 0 and
lA+(Q= WI2 A’ = 0, - ET(Q = 0) ’
(4b)
The quantity A0 determines both the biexciton groundstate energy shift and the longitudinal exciton energy shift. For the photon energy w just below the longitudinal exciton energy EL (EL - A0 c w < EL), there exist two excitonic-biexcitonic-polariton modes (01 (Q) and L$( Q)). We note that according to the excitonic-polariton model, in the energy region w < EL only one mode (the lower excitonicpolariton mode) can be produced in the sample. Therefore, the excitonic-biexcitonic polariton concept and
Z.G.
354
V
Koinov/Physics
Letters
Wavevector
Fig. I. Dispersion curves for the interacting system of photons, excitons and biexcitons; - - - - -, the dispersion curves of photons w(Q) = c*[Q[*/eo, excitons Er(Q) = Er+Q2/2Mr and biexcitons (la(Q) = & + Q2/2Mt,tcxc; p, the dispersion curves , of upper and lower excitonic polaritons wi=.PO’. ( Q) ; the dispersion curves of the three excitonic-biexcitonic polariton branches q(Q), i= 1,2,3.
the excitonic-polariton model give a different number of propagating modes in the energy region just below EL, and this fact can be used for the experimental verification of the above two models. The results by Mita and Nagasawa confirm the prediction of the excitonic-biexcitonic polariton concept, that even in the range just below the energy EL (EL - A0 < o) there exist two propagating excitonic-biexcitonicpolariton modes (0, (Q) and L$( Q)) and the constructive interference after one half round trip in the sample has to be observed. According to the measured position of the transmission maxima, the value of the quantity Aa in CuCl should be about 1 meV.
3. Conclusion As was mentioned above, the thickness d of the sample is closely related with the positions of the transmission maxima in Q-space. A more
A 254 (1999)
351-354
quantitative analysis, however, needs to be carried out to evaluate not only the spectral positions, but also the sharpness and relative amplitude of each transmission maximum. The solution of this problem is primarily connected to the problem of ABC which are needed for a microscopic calculation of the dielectric constant of the sample, assuming slab geometry. Unfortunately, at the present time we do not know the explicit form of A&(Q) . For this reason the excitonic-biexcitonic polariton dispersion cannot be exactly calculated even in a bulk case, and therefore, the final understanding of the transmission spectra of a CuCl slab from excitonic-biexcitonic polariton point of view is still an open problem. References [II
T. Mita,
N. Nagasawa,
Solid
State
Commun.
44
( 1982)
1003.
[21 K.
Cho, M. Kawnta, J. Phys. Sot. Jpn 54 (1985) 4431. K. Cho, A. D’Andrea, R. Del Sole, H. Ishihara, J. Phys. Sot. Jpn 59 (1990) 1853. [41 Z.K. Tang, A. Yanase. T. Yasui, Y. Segawa, Phys. Rev. Lett.
r31
71 (1993)
1431.
r51 Z.K.
Tang, A. Yanase, Y. Segawa, N. Matsuura, Phys. Rev. B 52 (199.5) 2640. [61 Y. Chen, F. Bassani, J. Massies, C. Deparis, Europhys. Lett. 14 ( 1991) 483. Y. Chen, G. Czajkowski, F. Bassani, I71 A. Tredicucci, 3 (1993)
K. Cho, G.
Neu,
J. Phys.
389.
[sl
F. Bassani. Y. Chen, G. Czajkowski, A. Tredicucci. Phys. State Sol. (b) 180 (1993) 115. Y. Chen, F. Bassani, J. Massies, C. Deparis, 191 A. Tredicucci, G. Neu, Phys. Rev. B 47 (1993) 10348. IlO1 Z.G. Koinov, J. Phys. Condens. Matter 8 (1996) L391. rtt1 Z.G. Koinov, J. Phys. Condens. Matter 10 (1998) 2389. rt21 T. Mita, K. Salome, M. Ueta, Solid State Commun. 33 (1980)
1135.
1999 PHYSICS
Physics
ELSEYIER
Letters
Excitonic-biexcitonic
A 254 (1999)
LETTERS
A
351-354
polariton interference in thin platelet of CUCl Z.G. Koinov ’
Department Received
17 July
1998;
of Physics, revised
Higher
manuscript
of Transport
Institute
Engineering,
1574 &$a.
received 22 December 1998; accepted Communicated by A. Lagendijk
Bulgaria
for publication
26 January
1999
K)stract position in Q-space of the resonance region, measured
The spectral in Zi-excitonic
in:erference
transmission
effect between two propagating
PI CS: 71.36; 71.35 Keywords: Polaritons;
Excitons;
Biexcitons;
by Mita
maxima of a 0.15 ,um thick CuCl single crystal with parallel plates and
Nagasawa,
excitonic-biexcitonic
has been
polariton
interpreted
as an indication
for
the
mutual
modes. @ 1999 Elsevier Science B.V.
CuCl
-
-
I. Introduction In recent years there has been a great deal of interest ir the exciton spectra and polariton interference in the
layers of CuCl [ l-51 and GaAs [ 6-91, It was pointed o 4t that depending on the thickness of layers the transn ission and reflection spectra of those materials can b-, explained either by treating the exciton system as a confined one or regarding it as a bulk system. In what follows we analyze the transmission spectrum of a 0.15 pm thick CuCl single crystal with parallel plates ir Z3-excitonic resonance region at low excitation intensities (N 300 W/cm2), measured by Mita and Nagasawa [ 11. This is not the case of a confined exciton s!~tem, and therefore, the usual type of Fabry-Perot polariton interference takes place. Assuming that the excitonic-polariton modes have been produced in the s.ab by the incident beam, Cho and coworkers [2,3]
’ E-mail:
have already analyzed the results by Mita and Nagasawa, primarily with respect to determining which of the various forms of the additional boundary conditions (ABC) are valid for the propagation of excitonic polaritons in CuCl thick crystal with parallel plates. The purpose of this paper is to explain the perioddoubling of the polariton interference in the energy region just below the longitudinal exciton energy, which has been observed in CuCl [ 11. The period-doubling means that for the case of double polariton modes interference the transmission maxima and reflection minima are equally spaced in Q-space with a period which is twice the period for the case of single mode interference. This long-period interference contradicts to the excitonic-polariton point of view because according to the excitonic-polariton model only one polariton mode is allowed in the above energy ration. We explain this period-doubling of the polariton interference by employing the so-called excitonic-biexcitonic polariton concept [ IO,1 I], which predicts two prop-
[email protected]
037%9601/99/$ - see front FII SO375-9601(99)00094-8
matter
@ 1999 Elsevier
Science
B.V. All rights
reserved
352
Z.G.
Koinov/Pl~ysicsics
Letters
agating polariton modes allowed in the energy region just below the longitudinal exciton energy. The dispersion curves of bulk excitonic polaritons w~.pO1.(Q) in this m aterial can be calculated by solving the following equation,
The solutions of the above equations are as follows, e;‘.p”.
A
=
B2 =
( Q)
= 5
[A
+
d-1
‘/’
)
~‘12’ El, + E+(Q) + Et - E$ > c2Q2E;(Q,
(lb)
where ~g accounts for all other oscillators not explicitly considered in this one-oscillator model. ET(Q) = ET + Q’/~MT is the dispersion of the transverse excitons (MT is the transverse exciton mass) and EL = ET + ALL is the longitudinal exciton energy at the point Q = 0 (Aw is the longitudinal-transverse splitting, caused by the Elliott exchange interaction). The corresponding parameters for CuCl are [ 121 ET = 3.2022 eV, EL = 3.2079 eV, MT = 2.3mo, Et) = 5.59. Due to the reflectivity of the sample surfaces the incident beam is splitted into many partial waves which interfere with one another. According to the excitonic-polariton model, for the photon energy below the longitudinal exciton energy EL at the point Q = 0 (w < EL), only one mode can be produced in the sample. The dispersion of the partial waves is identical to the dispersion of the lower excitonicpolariton branch (LBP) w:‘.~“. (Q) . In this case we have a usual type of Fabry-Perot interference which is determined by the constructive and deconstructive condition for the phase differences arising from a single round trip of the lower excitonic-polariton mode across the slab. For normal incidence a constructive interference of partial waves occurs if the phase shift for one round trip in the sample (the way is 24 equals an integer multiply by 27r. As a result, a standing wave with a large amplitude builds up in
A 254 (1999)
351-354
the crystal, resulting in transmission maxima and reflection minima, which are equally spaced in Q-space with jAQ,,,/ = n-/d. When the energy of the photon is above the longitudinal exciton energy, the situation becomes complicated because there are two modes of excitonic polariton allowed in a slab: the lower polariton branch w~c’pO”(Q) and the upper polariton branch WY.“‘- (Q) (UBP) The corresponding microscopic calculations of the transmission and reflection amplitudes of the electromagnetic waves [2,3] show the period-doubling of the polariton interference. This result can be understood from macroscopic point of view in the following way. The incident beam produces two waves with equal frequency but considerably different wave vectors ( ]QuuPI < /QLspi). The amplitude of the LBP decreases with increasing energy but in the range closed to EL, both modes have comparable amplitudes. Due to the reflection at the back side of the sample, the UBP and LBP waves produce both UBP and LBP waves, so the EBP wave can find a partner for constructive interference after one half round trip in the sample (the way is d). On the way from the front to the backside of the sample, both waves get phase differences Ap~np = lQLBPId and Apunp = /QusPId. In the above energy region Ap~ar >> Apusr, and as a result, we can go energetically from one transmission maximum to the next in Q-space by equal steps ~AQLsp~zz 2rr/d. It should be noted that the period-doubling of the polariton interference for the photon energy above the longitudinal exciton energy does not depend on the corresponding ABC. The ABC are required for solving the problem of deciding how the intensity of the incident radiation beam impinging on a surface is partitioned among two excitonic-polariton modes. The thickness of the slab is closely related with the positions of the transmission maxima in Q-space, while ABC are needed to evaluate the sharpness and relative amplitude of each transmission maximum. Since the geometrical thickness of the sample is d = 0.15 pm, from the excitonic-polariton point of view follows that the equal step of IAQLBPI M 2s-/d = 4.2 x lo5 cm-’ above the energy 3.2079 eV should be observed. But, according to the transmission data, measured by Mita and Nagasawa, the following maxima are clearly seen:
Z.G. Koinov/Physics
Letters
El RZ3.2067 eV,
E2 M 3.2074 eV, E3 zz 3.2082 ev , . . , .
Tke corresponding wave vectors IQLBpI, i = 1,2,3, ca.culated by using the dispersion w-exc’po’.(Q) , are as fo- lows. jQ~,,l = 52.314 x lo5 cm-’ , If&I = 56.498 x IO5 cm-’ , I~~,,~ = 60.681 x 10’ cm-‘. The spectral positions in Q-space of the above transm ssion maxima are in a very good agreement with th:: interference condition lAQLsPI = 2rr/d. From th.:se calculations follows that below the longitudinal exciton energy EL there exist two propagating polariton modes and the constructive interference after one half round trip in the slap is observed. This statement cannot be understood by employing the excitonicpolariton point of view.
2. Excitonic-biexcitonic of the transmission CuCl slab
polariton interpretation spectrum of 0.15 pm thick
In principle, the more accurately consideration of the transmission spectrum can be based on the so-called excitonic-biexcitonic polariton concept [ 10,111, because this model provides the calculai.ion of the elementary excitation spectrum of the whole photon-exciton-biexciton system. In Fig. 1 we have shown a schematic representation of the dispersion curves for the interacting system of photons, excitons and biexcitons in the crystal to illustrate ttz main differences between the excitonic-polariton rr od.el and the excitonic-biexcitonic polariton conccpt. According to the exitonic-polariton point of VI&W, the elementary excitation spectrum in the system under consideration consists of three branches: tire first two are the upper and lower excitonic polaritc.n branches and the third represents the biexciton g.,-ound state (the direct interaction between biexcitons and photons is forbidden due to the selection rules). If we take into consideration the interaction b-tween the biexcitons and the exciton components or the corresponding excitonic polaritons, then the e.>:citonic-biexcitonic polariton concept has to be
353
A 254 (1999) 351-354
used. According to this concept, the elementary excitation spectrum in the above system consists of three branches L!;(Q), i = 1,2,3. The dispersion of those quasiparticles can be determined by solving the following equation [ 111, fJ - 00(Q)
-
IA+(Q>l” IA-(Q)l” fJ _ we+xc+O’.- 0 - w~c+O’. (Q) (Q)
=o.
(2)
Here 00 (Q) is the biexciton ground state energy and A+(Q) are the matrix elements of the interaction between the biexcitons and the exciton components of the corresponding excitonic polaritons. For small wave vectors IQ1 << ( &/c) ET, the lowest excitonicbiexcitonic polariton branch is almost identical to the photon-like lower excitonic-polariton branch, f& (Q) M a~~-~~‘.(Q) ,
(3a)
while the other two branches are given by the following relations, 02,3(Q)
= ; [~e:c~p”‘~(Q) + G(Q)
i &("y.POl.
(Q>- f20(Q>>~ + W+(Q>12]. (3b)
According to (2)-( 3)) the elementary excitation spectrum at the point Q = 0 is defined as follows, l-2, zz w-exc.pol. (Q = 0) = 0 , 03 = @,+A,,
fit2 KSEL - A0 , (ha)
where L$ is the biexciton ground energy at the point Q = 0 and
lA+(Q= WI2 A’ = 0, - ET(Q = 0) ’
(4b)
The quantity A0 determines both the biexciton groundstate energy shift and the longitudinal exciton energy shift. For the photon energy w just below the longitudinal exciton energy EL (EL - A0 c w < EL), there exist two excitonic-biexcitonic-polariton modes (01 (Q) and L$( Q)). We note that according to the excitonic-polariton model, in the energy region w < EL only one mode (the lower excitonicpolariton mode) can be produced in the sample. Therefore, the excitonic-biexcitonic polariton concept and
Z.G.
354
V
Koinov/Physics
Letters
Wavevector
Fig. I. Dispersion curves for the interacting system of photons, excitons and biexcitons; - - - - -, the dispersion curves of photons w(Q) = c*[Q[*/eo, excitons Er(Q) = Er+Q2/2Mr and biexcitons (la(Q) = & + Q2/2Mt,tcxc; p, the dispersion curves , of upper and lower excitonic polaritons wi=.PO’. ( Q) ; the dispersion curves of the three excitonic-biexcitonic polariton branches q(Q), i= 1,2,3.
the excitonic-polariton model give a different number of propagating modes in the energy region just below EL, and this fact can be used for the experimental verification of the above two models. The results by Mita and Nagasawa confirm the prediction of the excitonic-biexcitonic polariton concept, that even in the range just below the energy EL (EL - A0 < o) there exist two propagating excitonic-biexcitonicpolariton modes (0, (Q) and L$( Q)) and the constructive interference after one half round trip in the sample has to be observed. According to the measured position of the transmission maxima, the value of the quantity Aa in CuCl should be about 1 meV.
3. Conclusion As was mentioned above, the thickness d of the sample is closely related with the positions of the transmission maxima in Q-space. A more
A 254 (1999)
351-354
quantitative analysis, however, needs to be carried out to evaluate not only the spectral positions, but also the sharpness and relative amplitude of each transmission maximum. The solution of this problem is primarily connected to the problem of ABC which are needed for a microscopic calculation of the dielectric constant of the sample, assuming slab geometry. Unfortunately, at the present time we do not know the explicit form of A&(Q) . For this reason the excitonic-biexcitonic polariton dispersion cannot be exactly calculated even in a bulk case, and therefore, the final understanding of the transmission spectra of a CuCl slab from excitonic-biexcitonic polariton point of view is still an open problem. References [II
T. Mita,
N. Nagasawa,
Solid
State
Commun.
44
( 1982)
1003.
[21 K.
Cho, M. Kawnta, J. Phys. Sot. Jpn 54 (1985) 4431. K. Cho, A. D’Andrea, R. Del Sole, H. Ishihara, J. Phys. Sot. Jpn 59 (1990) 1853. [41 Z.K. Tang, A. Yanase. T. Yasui, Y. Segawa, Phys. Rev. Lett.
r31
71 (1993)
1431.
r51 Z.K.
Tang, A. Yanase, Y. Segawa, N. Matsuura, Phys. Rev. B 52 (199.5) 2640. [61 Y. Chen, F. Bassani, J. Massies, C. Deparis, Europhys. Lett. 14 ( 1991) 483. Y. Chen, G. Czajkowski, F. Bassani, I71 A. Tredicucci, 3 (1993)
K. Cho, G.
Neu,
J. Phys.
389.
[sl
F. Bassani. Y. Chen, G. Czajkowski, A. Tredicucci. Phys. State Sol. (b) 180 (1993) 115. Y. Chen, F. Bassani, J. Massies, C. Deparis, 191 A. Tredicucci, G. Neu, Phys. Rev. B 47 (1993) 10348. IlO1 Z.G. Koinov, J. Phys. Condens. Matter 8 (1996) L391. rtt1 Z.G. Koinov, J. Phys. Condens. Matter 10 (1998) 2389. rt21 T. Mita, K. Salome, M. Ueta, Solid State Commun. 33 (1980)
1135.