- Email: [email protected]
Relaxation of polar excitons in a semiconductor
LUMINESCENCE
Journal of Luminescence 53 (1992) 313—316
JOURNALOF
Relaxation of polar excitons in a semiconductor F. Vallée, F. Bogani
1
and C. Flytzanis
Laboratoire d’Optique Quantique du Centre National de Ia Recherche Scientifique Ecole Polytechnique, 91128 Palaiseau Cédex, France
The dephasing time of the Z
3 exciton—polariton in CuCI is investigated using a new time resolved coherent technique. The measurements performed both as a function of polariton frequency and crystal temperature allow a precise determination of the redistribution channels of the initial energy between the various degrees of freedom of the crystal. The results show that relaxation is mediated by the acoustic phonons at low temperature and by the longitudinal optic phonons at high temperature (T 40 K). The dephasing time of the longitudinal exciton is also measured and the results are compared to those obtained for the transverse mode.
1. Introduction The exciton—polaritons govern the propagation of the electromagnetic energy near the band gap of a polar semiconductor and were consequently extensively investigated with a special emphasis on their spectral properties [1]. Their dynamics which is directly related to the absorption of light in the medium was also addressed but, except in one case [2], with incoherent techniques which only provide global information on the energy redistribution inside the crystal since both the initial and final states are simultaneously probed. We have taken advantage of the selectivity of the coherent methods to obtain qualitative and quantitative information on the elementary interactions of the exciton—polariton and, for the first time, of the associated longitudinal exciton with their environment. The technique we have developed reposes on picosecond coherent two-photon
Correspondence to: Dr. F. Vallée, Laboratoire d’Optique Quantique du Centre National de Ia Recherche Scientifique Ecole Polytechnique, 91128 Palaiseau Cédex, France This Permanent address: Dipartimento di Fisica, Universita di Firenze and Unità del Gruppo Nazionale di Struttura della Materia, Largo E. Fermi 2, 50125 Firenze, Italy. 0022-2313/92/$05.00 © 1992
—
excitation and detection of an exciton—polariton or longitudinal exciton wave packet [3]. Coherent excitation of an exciton mode of frequence We.~r and wave vector ke,r(We~) is realized in the bulk of the crystal by two-photon absorption of two synchronized picosecond pulses with frequencies w1 and w2 and wave vectors k1 and k2 such that w~. ~ + w2 and ke~ k1 + k2. Wave vector conservation can only be satisfied for the longitudinal exciton and for the upper branch polariton restricting the applicability of the technique to these modes [41.The evolution of the coherence of the initial non-equilibrium population is then selectively followed by phase matched parametric emission at wd We~ w~stimulated by a third picosecond pulse, w~,delayed with respect to the excitation. Using a Lorentzian oscillator model for the material excitation, it can be shown that the intensity of the signal decreases exponentially as the probe pulse is delayed with a characteristic time T2/2, directly measuring the dephasing time T2 of the excitation [3]. =
=
=
—
2. Exciton—polariton dephasing .
.
technique was first used to investigate two upper-branch polaritons in CuCl with energies hw~ 3.208 eV and hw~ 3.217 eV corre-
Elsevier Science Publishers B.V. All rights reserved
—
14
L I a/lee
it
a!. / Relaxation
0/
polar en/Ion
in a se,nieo,idiiitor
and up—Conversion into a lower branch polariton ~en
0.6
I /1
(ps-i)
0.4
,.‘
with, respectively, emission or absorption of a phonon [5]. At low temperature the relaxation is expected to be dominated by exciton—polariton
h
re~
e~
interaction with acoustic phonons. mediated mainly by the deformation potential (DP) for
~
A A 02
LA-phonons and by the piezoelectric effect (FE) for TA-phonons [61. The low temperature cxcidimping i itc I = 2/T ~ in thus
— —
0
25
where ~
15
TEMPERATURE (K)
Fig. I. Temperature dependence of the w~ and w~ polariton dephasing rate l~ (upper and lower curse respeciivelr). The dashed and full lines~:ire calculated for interaction respectivels with only the acoustic phonons and with the acoustic and LO—phonons.
sponding, respectively, to a backward 0 18(1°. and forward, I) 00, excitation geometry (0 is the angle between k1 and k2). The experiments were =
and ~ arc the relaxation rates associated respectively with the DP and Ph scattering mechanisms. The efficiency of these mechanisms increases with w~ in agreement with our experimental results. Discrimination between the DP and FE processes can he realized because of
their different temperature behaviors which are governed by the Bose factors of the involved phonons. These arc different for the DP and FE processes and the temperature dependence of i~ is given approximately by:
=
conducted usingdelivering a passively locked 3~/glasslaser a singlemode 5 ps pulse in Nd infrared ~ the 1.054 ji.m) which is frequency converted to create the three independent pulses w-, and wi.,. The probe beam frequency w~is adapted to phase matching requirements and is either at the same frequency as w~ for 0 180° or frequency shifted to 1.107 ~m for U 0°.After polarization adjustment, the three beams are focused into a CuC1 sample with (110) surfaces. The signal is detected by a photomultiplier after spatial, spectral and polarization selection. In the 180°geometry, in order to discriminate the signal against the frequency degenerate diffused beam at 0)0 a temporal selection of the signal is added using a Kerr gate synchronized with the probe pulse. The results of the measurements are shown in fig. 1 where the measured dephasing rates l~. 2/T, of the upper and lower energy polaritons were plotted as a function of crystal temperature. The most probable processes for the relaxation of upper branch polaritons are extrahand down=
=
=
=
I~(T)
=
Yt)t’H
+
i,(
~
+
ti( wi\)}
11 + ii( to IA ) + n( w IA )1i (~) it~i where Y~i’ and Ypt are frequency dependent coupling parameters and ii(w) the occupation number for the to phonon. The frequencies w~ and w of the acoustic phonons involved. respectively, in the up- and down-conversion processes are imposed by energy and wave vector conservation. Using Y~i’and YPE as fitting parameters, we obtain the dashed line of fig. 1 which correctly reproduces the low temperature dependence of I~. The frequency dependence of the coupling parameters are in agreement with the theoretical estimations with, in particular, a faster increase of Yi’t~ with w~[3]. At higher temperature (1> 40 K) a strong deviation from the above lit is observed br the w~ polariton (the investigated temperature range is not sufficient to observe this deviation for the w~ polariton). This can he ascribed to LO-phonon assisted mechanisms mediated by the FrOhlich interaction [71.Here, only the up-conversion process needs to he taken into account + ~
--
‘
—
F. Vallée et al.
/ Relaxation of polar excitons in
because of the very low density of accessible states for the down-conversion mechanism and hence the contribution of the LO-phonons to the relaxation rate can be written: FLO(T) YLO~(~LO), (3) =
with 0)LO 26 meV for CuCI and YLO is a frequency dependent coupling constant. Including this mechanism we obtain the full line in fig. 1 in good agreement with our experimental results using YLO 26 meV for the ~ polariton and a related value for the w~ polariton. =
=
3. Longitudinal exciton dephasing Similar measurements were performed on the longitudinal mode in a CuC1 crystal with (111) surfaces. The experimental system corresponds to the forward (U 0) geometry as previously described for the investigation of the higher energy exciton—polariton. Here the crystal orientation allows the two-photon excitation and probing of the longitudinal-exciton whose energy is quasiidentical with hw~ (h(w~ WL) 0.3 meV). An exponential decay of the signal is measured for all the investigated temperatures between 20 and 70 K allowing the first determination of a =
a semiconductor
315
longitudinal exciton dephasing time. For lower temperatures, an exponential decay is also ohserved over four orders of magnitude but is followed by a non-exponential tail over more than one order of magnitude. The formation of this tail can be attributed to the spatial dispersion of the longitudinal exciton and to crystal defects and will be discussed in a forthcoming paper. The measured longitudinal exciton dephasing rate, TLE’ is essentially identical with the w~ polariton one, as expected from the quasi-degeneracy of these two modes. Its temperature dependence is depicted in fig. 2 and can be reproduced taking into account interactions with both the acoustic and the longitudinal optic phonons (eqs. (2) and (3)) with the same parameters as determined for the co~polariton. The higher time resolution of the forward configuration (about 1 ps) allows a clear demonstration of the role of the LO-phonons for temperatures higher than 40 K with in particular a very fast increase of TLE between 50 and 70 K which is perfectly correlated with the fast increase of the efficiency of the LO-phonon induced mechanism with temperature.
—
J
The investigation of exciton—polaritons and longitudinal excitons by coherent techniques constitutes a new source of information on the ele-
___________________________
r (ps.1)
0 0
25
50
4. Conclusion
I
with the other collective modes of the crystal. As examplified in CuCl, the dominant relaxation channels for the polariton mode can be clearly identified and their importance quantitatively estimated. mentary interactions of the excitonic excitation Such experiments are readily applicable to other semiconductors allowing to gain a precise map of the different exciton—polariton decay channels in the low polariton density regime. Measurements could also be performed at higher densities where exciton—exciton scattering becomes dominant. We have already obtained pre-
75
TEMPERATURE (K)
Fig. 2. Measured temperature dependence of the longitudinal exciton dephasing rate TLE• The full line takes into account interaction with the acoustic and LO-phonons.
liminary results demonstrating the influence of
exciton—exciton interactions in the high excitation regime on the longitudinal Z 3 exciton dephasing in CuCI.
316
F. Vallée i’I a!.
/
Relaxation o/ i,i,lar ave/boa in a ie,nuondueior
Acknowledgements
[2) Y. Masurnoto, S. Shionova and T. Takagahara. Phys. Rev. Lctt. 51(1983)923.
We wish to thank Professor D. Fröhlich of the University of Dortmund, Germany, for providing samples and for valuable information; we also thank 5. Godard of the Laboratoire de Physique des Solides, Université de Paris-Sud, France, for preparing the samples.
[3] F. Vallée, F. Bogani and C. Flytzunis, Phys. Rev. Lcit. 66 1991) 1509 . .. -. . [4) 1). l-rohlich, F. Mohlcr and P. Wicsner, Phys. Re’. Felt. 26(1971) 554. 5] V.V. Travnikov and V.V. Krivolapchuk, Zh. [ksp. Tcor. F1L. 85 (1983) 2087 (Sos’. Phys. JETP 58(1983) 1210).
References [1] ES. Kotelcs. in: Exciton. eds. El. Sturge (North-Holland, 1982) p. 83.
Rashha and M.D.
[6) iD. Zonk, Phys. Rev. 136 (1964) A869. [7] C. Weishuch and R.G. Ulhrich, in: Light Scattering in Solids Ill. eds. M. (‘ardona and Ci. Guntherodi (Springer Verlag. 1982) p. 207.
Journal of Luminescence 53 (1992) 313—316
JOURNALOF
Relaxation of polar excitons in a semiconductor F. Vallée, F. Bogani
1
and C. Flytzanis
Laboratoire d’Optique Quantique du Centre National de Ia Recherche Scientifique Ecole Polytechnique, 91128 Palaiseau Cédex, France
The dephasing time of the Z
3 exciton—polariton in CuCI is investigated using a new time resolved coherent technique. The measurements performed both as a function of polariton frequency and crystal temperature allow a precise determination of the redistribution channels of the initial energy between the various degrees of freedom of the crystal. The results show that relaxation is mediated by the acoustic phonons at low temperature and by the longitudinal optic phonons at high temperature (T 40 K). The dephasing time of the longitudinal exciton is also measured and the results are compared to those obtained for the transverse mode.
1. Introduction The exciton—polaritons govern the propagation of the electromagnetic energy near the band gap of a polar semiconductor and were consequently extensively investigated with a special emphasis on their spectral properties [1]. Their dynamics which is directly related to the absorption of light in the medium was also addressed but, except in one case [2], with incoherent techniques which only provide global information on the energy redistribution inside the crystal since both the initial and final states are simultaneously probed. We have taken advantage of the selectivity of the coherent methods to obtain qualitative and quantitative information on the elementary interactions of the exciton—polariton and, for the first time, of the associated longitudinal exciton with their environment. The technique we have developed reposes on picosecond coherent two-photon
Correspondence to: Dr. F. Vallée, Laboratoire d’Optique Quantique du Centre National de Ia Recherche Scientifique Ecole Polytechnique, 91128 Palaiseau Cédex, France This Permanent address: Dipartimento di Fisica, Universita di Firenze and Unità del Gruppo Nazionale di Struttura della Materia, Largo E. Fermi 2, 50125 Firenze, Italy. 0022-2313/92/$05.00 © 1992
—
excitation and detection of an exciton—polariton or longitudinal exciton wave packet [3]. Coherent excitation of an exciton mode of frequence We.~r and wave vector ke,r(We~) is realized in the bulk of the crystal by two-photon absorption of two synchronized picosecond pulses with frequencies w1 and w2 and wave vectors k1 and k2 such that w~. ~ + w2 and ke~ k1 + k2. Wave vector conservation can only be satisfied for the longitudinal exciton and for the upper branch polariton restricting the applicability of the technique to these modes [41.The evolution of the coherence of the initial non-equilibrium population is then selectively followed by phase matched parametric emission at wd We~ w~stimulated by a third picosecond pulse, w~,delayed with respect to the excitation. Using a Lorentzian oscillator model for the material excitation, it can be shown that the intensity of the signal decreases exponentially as the probe pulse is delayed with a characteristic time T2/2, directly measuring the dephasing time T2 of the excitation [3]. =
=
=
—
2. Exciton—polariton dephasing .
.
technique was first used to investigate two upper-branch polaritons in CuCl with energies hw~ 3.208 eV and hw~ 3.217 eV corre-
Elsevier Science Publishers B.V. All rights reserved
—
14
L I a/lee
it
a!. / Relaxation
0/
polar en/Ion
in a se,nieo,idiiitor
and up—Conversion into a lower branch polariton ~en
0.6
I /1
(ps-i)
0.4
,.‘
with, respectively, emission or absorption of a phonon [5]. At low temperature the relaxation is expected to be dominated by exciton—polariton
h
re~
e~
interaction with acoustic phonons. mediated mainly by the deformation potential (DP) for
~
A A 02
LA-phonons and by the piezoelectric effect (FE) for TA-phonons [61. The low temperature cxcidimping i itc I = 2/T ~ in thus
— —
0
25
where ~
15
TEMPERATURE (K)
Fig. I. Temperature dependence of the w~ and w~ polariton dephasing rate l~ (upper and lower curse respeciivelr). The dashed and full lines~:ire calculated for interaction respectivels with only the acoustic phonons and with the acoustic and LO—phonons.
sponding, respectively, to a backward 0 18(1°. and forward, I) 00, excitation geometry (0 is the angle between k1 and k2). The experiments were =
and ~ arc the relaxation rates associated respectively with the DP and Ph scattering mechanisms. The efficiency of these mechanisms increases with w~ in agreement with our experimental results. Discrimination between the DP and FE processes can he realized because of
their different temperature behaviors which are governed by the Bose factors of the involved phonons. These arc different for the DP and FE processes and the temperature dependence of i~ is given approximately by:
=
conducted usingdelivering a passively locked 3~/glasslaser a singlemode 5 ps pulse in Nd infrared ~ the 1.054 ji.m) which is frequency converted to create the three independent pulses w-, and wi.,. The probe beam frequency w~is adapted to phase matching requirements and is either at the same frequency as w~ for 0 180° or frequency shifted to 1.107 ~m for U 0°.After polarization adjustment, the three beams are focused into a CuC1 sample with (110) surfaces. The signal is detected by a photomultiplier after spatial, spectral and polarization selection. In the 180°geometry, in order to discriminate the signal against the frequency degenerate diffused beam at 0)0 a temporal selection of the signal is added using a Kerr gate synchronized with the probe pulse. The results of the measurements are shown in fig. 1 where the measured dephasing rates l~. 2/T, of the upper and lower energy polaritons were plotted as a function of crystal temperature. The most probable processes for the relaxation of upper branch polaritons are extrahand down=
=
=
=
I~(T)
=
Yt)t’H
+
i,(
~
+
ti( wi\)}
11 + ii( to IA ) + n( w IA )1i (~) it~i where Y~i’ and Ypt are frequency dependent coupling parameters and ii(w) the occupation number for the to phonon. The frequencies w~ and w of the acoustic phonons involved. respectively, in the up- and down-conversion processes are imposed by energy and wave vector conservation. Using Y~i’and YPE as fitting parameters, we obtain the dashed line of fig. 1 which correctly reproduces the low temperature dependence of I~. The frequency dependence of the coupling parameters are in agreement with the theoretical estimations with, in particular, a faster increase of Yi’t~ with w~[3]. At higher temperature (1> 40 K) a strong deviation from the above lit is observed br the w~ polariton (the investigated temperature range is not sufficient to observe this deviation for the w~ polariton). This can he ascribed to LO-phonon assisted mechanisms mediated by the FrOhlich interaction [71.Here, only the up-conversion process needs to he taken into account + ~
--
‘
—
F. Vallée et al.
/ Relaxation of polar excitons in
because of the very low density of accessible states for the down-conversion mechanism and hence the contribution of the LO-phonons to the relaxation rate can be written: FLO(T) YLO~(~LO), (3) =
with 0)LO 26 meV for CuCI and YLO is a frequency dependent coupling constant. Including this mechanism we obtain the full line in fig. 1 in good agreement with our experimental results using YLO 26 meV for the ~ polariton and a related value for the w~ polariton. =
=
3. Longitudinal exciton dephasing Similar measurements were performed on the longitudinal mode in a CuC1 crystal with (111) surfaces. The experimental system corresponds to the forward (U 0) geometry as previously described for the investigation of the higher energy exciton—polariton. Here the crystal orientation allows the two-photon excitation and probing of the longitudinal-exciton whose energy is quasiidentical with hw~ (h(w~ WL) 0.3 meV). An exponential decay of the signal is measured for all the investigated temperatures between 20 and 70 K allowing the first determination of a =
a semiconductor
315
longitudinal exciton dephasing time. For lower temperatures, an exponential decay is also ohserved over four orders of magnitude but is followed by a non-exponential tail over more than one order of magnitude. The formation of this tail can be attributed to the spatial dispersion of the longitudinal exciton and to crystal defects and will be discussed in a forthcoming paper. The measured longitudinal exciton dephasing rate, TLE’ is essentially identical with the w~ polariton one, as expected from the quasi-degeneracy of these two modes. Its temperature dependence is depicted in fig. 2 and can be reproduced taking into account interactions with both the acoustic and the longitudinal optic phonons (eqs. (2) and (3)) with the same parameters as determined for the co~polariton. The higher time resolution of the forward configuration (about 1 ps) allows a clear demonstration of the role of the LO-phonons for temperatures higher than 40 K with in particular a very fast increase of TLE between 50 and 70 K which is perfectly correlated with the fast increase of the efficiency of the LO-phonon induced mechanism with temperature.
—
J
The investigation of exciton—polaritons and longitudinal excitons by coherent techniques constitutes a new source of information on the ele-
___________________________
r (ps.1)
0 0
25
50
4. Conclusion
I
with the other collective modes of the crystal. As examplified in CuCl, the dominant relaxation channels for the polariton mode can be clearly identified and their importance quantitatively estimated. mentary interactions of the excitonic excitation Such experiments are readily applicable to other semiconductors allowing to gain a precise map of the different exciton—polariton decay channels in the low polariton density regime. Measurements could also be performed at higher densities where exciton—exciton scattering becomes dominant. We have already obtained pre-
75
TEMPERATURE (K)
Fig. 2. Measured temperature dependence of the longitudinal exciton dephasing rate TLE• The full line takes into account interaction with the acoustic and LO-phonons.
liminary results demonstrating the influence of
exciton—exciton interactions in the high excitation regime on the longitudinal Z 3 exciton dephasing in CuCI.
316
F. Vallée i’I a!.
/
Relaxation o/ i,i,lar ave/boa in a ie,nuondueior
Acknowledgements
[2) Y. Masurnoto, S. Shionova and T. Takagahara. Phys. Rev. Lctt. 51(1983)923.
We wish to thank Professor D. Fröhlich of the University of Dortmund, Germany, for providing samples and for valuable information; we also thank 5. Godard of the Laboratoire de Physique des Solides, Université de Paris-Sud, France, for preparing the samples.
[3] F. Vallée, F. Bogani and C. Flytzunis, Phys. Rev. Lcit. 66 1991) 1509 . .. -. . [4) 1). l-rohlich, F. Mohlcr and P. Wicsner, Phys. Re’. Felt. 26(1971) 554. 5] V.V. Travnikov and V.V. Krivolapchuk, Zh. [ksp. Tcor. F1L. 85 (1983) 2087 (Sos’. Phys. JETP 58(1983) 1210).
References [1] ES. Kotelcs. in: Exciton. eds. El. Sturge (North-Holland, 1982) p. 83.
Rashha and M.D.
[6) iD. Zonk, Phys. Rev. 136 (1964) A869. [7] C. Weishuch and R.G. Ulhrich, in: Light Scattering in Solids Ill. eds. M. (‘ardona and Ci. Guntherodi (Springer Verlag. 1982) p. 207.