- Email: [email protected]
Spatial and spectral features of polariton fluorescence
Journal of Luminescence 18/19 (1979) 27—31 North-Holland Publishing Company
©
SPATIAL AND SPECTRAL FEATURES OF POLARITON FLUORESCENCE C. WEISBUCH and R.G. ULBRICH Laboratoire de Physique de la Matière Condensée, Ecole Polytechnique, 91128 Palaiseau, France and Inst ituf für Physik, Universität Dortmund, 46-Dortmund 50, Fed. Rep. Germany
The spatial properties of polariton fluorescence are shown to be of crucial importance to determine polariton fluorescence lineshapes. Comparison in GaAs of front and back-side fluorescence under resonant and non-resonant optical excitation of the no-phonon line and its LO-replica gives evidence of this spatial dependence.
1. Introduction The use of the polariton concept [1] leads to substantial modifications in the description of direct-exciton fluorescence phenomena [2, 3]. Exciton fluorescence is then the propagation of polariton wavepackets to the crystal surface and their subsequent transmission outside the crystal as photons rather than the transition from exciton to photon states inside the crystal and subsequent photon propagation to the surface. In this description two essential questions have to be answered: (i) in which way is polariton damping taken into account and (ii) what is the explicit influence of crystal surface properties like transmission coefficients and non-radiative surface recombination? There are several previous experimental and theoretical studies on polariton fluorescence [4—10]. We present here an investigation in GaAs using various geometrical configurations and monochromatic cw dye laser excitations. At variance with previous works [4, 6—9] our analysis points out the fundamental importance of the spatial inhomogeneity in the polariton energy distribution function near the crystal surface.
2. Polariton fluorescence Ilneshape analysis The fluorescence of the n = 1 exciton-polariton in GaAs and its 1-LO replica are shown in fig. 1. The simple spectrum of the replica can be described along the same lines as in the work of Gross [11] in CdS: given an energy distribution function f(E) of polaritons (irrespective of their branch), the creation rate of 27
28
C. Weisbuch, R.G. Ulbrichf Features of polariton fluorescence
LO-RepI.
—
°
X20
~
10 2
~.
L
1.514
1.516 Photon
t
t
L
~g
1.518
1.520
Energy (eV)
Fig. Ia. Polariton fluorescence spectrum of GaAs under non-resonant front excitation at hwo. b. One LO-phonon replica under the same excitation conditions (abscissa shifted by one LO-phonon energy for clarity). Circles give the lineshape of a Maxwell—Boltzmann distribution at temperature T = 22K: 1— \/~exp(—i~/kT);~ = hw E 1~— hwLo.
polaritons at energy hw = E LO is the product of f(E) with the density of states p(E) and the LO-phonon emission probability. Assuming the latter to have a smooth dependence on E, the replica only reproduces the energy distribution of polaritons. It probes the whole excited volume because polaritons with energies well below the longitudinal exciton energy E[ can readily escape the crystal (small absorption coefficient at E ~ EL). The fluorescence intensity of the no-phonon line originates from both upper (upb) and lower (lpb) polariton branches and is given by [51: —
1(E) = ~
f1(E, z = 0)p~(E)v51~(E)Ti(E),
(I)
i=1,2
where f1(E, z = 0) is the value of the distribution function in space and energy f1(E, r) at the crystal surface, Vgj~ is the component of polariton group velocity perpendicular to the surface, T1(E) is the transmission coefficient of polaritons in the vacuum as photons. Now the index 1, 2 labels upb and lpb, respectively. In a spherical approximation p(E)vgj(E) = (2i~h)~k~ and is approximately constant, except in region of strong variations of k, (lpb above E1, upb near EL). The transmission coefficient T,(E) can be derived from the phenomenological Maxwell theory and requires appropriate abc’s above EL [12]. Independent of the abc’s is the fact that sufficiently away from the resonance, i.e. Ihw E1j ~ ELT (longitudinal—transverse splitting), the transmission coefficients of the lpb below EL and upb above EL are constant and equal to 1 R, R being the reflectivity in that range. In the resonance region hw ELI ~ ELT, luminescence originates from the two branches as their T1(E)’s are of the same order of magnitude. Due to the large K’s of the lpb above EL, T2(E) is very small there (large refractive index n) and the fluorescence in this range is mainly determined by the transmission of upb polaritons. The most difficult task in eq. (1) is the computation of f(E, r). Assuming spatial homogeneity (but finite crystal thickness), Sumi made an explicit cal—
—
—
C. Weisbuch, R.G. Ulbrich/ Features of polariton fluorescence
29
culation of f(E) [8]. His main results are: for long enough lifetimes, a quasiequilibrium distribution exists above EL. This distribution is truncated somewhat below EL because of the decreasing escape time in that region (due to the increase in polariton group velocity). Fig. lb proves that the space-averaged polariton distribution deduced from the 1-LO phonon replica lineshape resembles indeed an equilibrium one. The nophonon lineshape (fig. la) is different: thermalization occurs in the lpb because of its larger density of states. The upb emission above EL is therefore due to the build-up of f1(E, 0) by scattering-in of polaritons thermalized on the lpb. 1(E) in this spectral region is thus the product of an equilibrium f2(E) with the interbranch transition probability integrated over space (T1(E) can be taken constant in this range). The spectral variation of the scattering probability explains well the difference between the shapes of upb emission and 1-LO emission. Below EL, the emission reflects the cutoff of the distribution function. A peculiar feature is the dip existing at EL, which is not observed on the replica. Such dips were previously attributed to the spectral variation of T(E) [6,7]. We will show that they are determined by the shapes of f1,2(E, 0). The spatial dependence of f(E, r) plays a fundamental role. Fig. 2 shows the fluorescence observed on the two sides of a 22 ~m thick sample: clearly the f(E, 0) is not the same on the excited and back-sides! Considering a simple diffusion model for polariton propagation, we deduce the following in analogy with the problem of carrier diffusion with surface recombination [13]: for volume excitation with a sink at the surface, when spatial diffusion is more important than direct creation, the surface density decreases with decreasing diffusion constant. In other words, only the polaritons within one diffusion length are able to reach the surface. As polariton scattering and trapping processes are resonantly increased in the vicinity of EL (as evidenced through the maximum of the absorption coefficient), f(E, 0) will exhibit a minimum at such energies. Thus, for a non-resonant (i.e. deep in volume) excitation, f(E, 0)
1.514
1.516 Photon
1.518 1.520 Energy 1eV)
Fig. 2a. Resonantly excited polariton fluorescence of a thin GaAs sample of 22 ~m thickness (front excitation at fho0 = EL). b. Spectrum observed from the back-side under the same conditions.
30
C. Weisbuch, R.G. Ulbrichl Features of polariton fluorescence
displays a dip at the maximum of scattering probability on an overall smooth shape with a cut-off below EL. This dip in f(E, r) exists only near the surface which acts as a sink through radiative and/or surface recombination. Inside the volume, thermalization processes have the time to fill-in this dip. This is clearly shown by the LO-replica in fig. lb. Under resonant excitation, the surface density f(E, 0) is not only due to the diffusion of thermalized polaritons from the bulk. The major contribution comes now from direct optical creation of non-thermalized polaritons. As shown in fig. 2a, the no-phonon line is thus much less thermalized and has no dip. Its intensity is also largely increased due to the increase in quantum efficiency, because of the shorter escape time of polaritons created near the surface [10]. Under such conditions, the LO-replica is observed to be still well thermalized: it represents the space average of the polariton distribution function, and still most polaritons have a chance to diffuse in the bulk and thermalize there.
3. Discussion In eq. (1), the three factors p. 0g~ T are fixed “geometrical” factors, whereas f(E, 0) is the quantity which can be externally varied as follows: (i) Varying excitation energy hw 0: as described above, resonant excitation leads to non-thermalized distribution function at the surface, and large radiative efficiency of the no-phonon line. Non-resonant (i.e. bulk) excitation leads to a dip in the distribution function (i.e. in the emission intensity) at the energy where the polariton scattering rate is maximum. (ii) Back-side luminescence in a thin sample exhibits a behaviour almost independent of exciting light energy: the value of f(E, 0) is non-negligible only at energies where the diffusion length is comparable with the crystal thickness, i.e. well away from EL. Created resonantly or not, the polaritons with an energy = EL do not propagate to the other surface. (iii) The LO-phonon replica lineshape is a direct evidence that the no-phonon polariton fitiorescence probes only part of a spatially inhomogeneous distribution. The replica is always well-thermalized, whereas the no-phonon line displays a variety of shapes under varying exciting energy, back-side observation. Compared with previous works, we point out here the fundamental importance of the spatial distribution of polaritons, which explains both the variation of lineshapes with hw0 and the very different behaviour of the nophonon line and 1-LO replica. Previous descriptions cannot describe the effects reported here. A first attempt to take into account the spatial aspect of polariton fluorescence was made by Bonnot and Benoit a Ia Guillaume [51,but the over-simplification of the model led to a conclusion where emission should be maximum at the energy of maximum polariton scattering rate.
C. Weisbuch, R.G. Ulbrich/ Features of polariton fluorescence
31
References [1] J.J. Hopfleld, Phys. Rev. 112 (1958) 1555; SI. Pekar, Zh. Eksp. Teor. Fiz. 33 (1957) 1022 [Soy. Phys.-JETP 6 (1958) 785]. [2] Y. Toyozawa, Prog. Theor. Phys. Suppl. 12 (1959) 111. [3] J.J. Hopfield, J. Phys. Soc. Japan Suppl. 21 (1966) 77. [4] W.C. Tait and R.L. Weiher, Phys. Rev. 178 (1969) 1404. [51A. Bonnot and C. Benoit a Ia Guillaume in Polaritons, Proc. Taormina Res. Conf. Structure of Matter, Taormina, 1972, eds. E. Burstein and F. de Martini (Pergamon, New York, 1974) p. 197. [6] E. Gross, S. Permogorov, V. Travnikov and A. Selkin, Solid State Commun. 10 (1972) 1071. [71D.D. Sell, S.E. Stokowski, R. Dingle and J.V. DiLorenzo, Phys. Rev. B7 (1973) 4568. [8] H. Sumi, J. Phys. Soc. Japan 41 (1976) 526. [9] I. Broser and R. Broser, J. Luminescence 12/13 (1976) 201. [10] C. Weisbuch and R.G. Ulbrich, Phys. Rev. Letters 39 (1977) 654. [11] E.F. Gross, S.A. Permogorov and B.S. Razbirin, Usp. Fiz. Nauk 103 (1971) 431 [Soy. Phys. usp. 14 (1971) 104]. [12] See e.g. C.S. Ting, Mi. Frankel and J.L. Birman, Solid State Commun. 17 (1975)1285 and ref s. therein. [13] See e.g. SM. Sze, Physics of semiconductor devices (Wiley, New York, 1969) ch. 2.
©
SPATIAL AND SPECTRAL FEATURES OF POLARITON FLUORESCENCE C. WEISBUCH and R.G. ULBRICH Laboratoire de Physique de la Matière Condensée, Ecole Polytechnique, 91128 Palaiseau, France and Inst ituf für Physik, Universität Dortmund, 46-Dortmund 50, Fed. Rep. Germany
The spatial properties of polariton fluorescence are shown to be of crucial importance to determine polariton fluorescence lineshapes. Comparison in GaAs of front and back-side fluorescence under resonant and non-resonant optical excitation of the no-phonon line and its LO-replica gives evidence of this spatial dependence.
1. Introduction The use of the polariton concept [1] leads to substantial modifications in the description of direct-exciton fluorescence phenomena [2, 3]. Exciton fluorescence is then the propagation of polariton wavepackets to the crystal surface and their subsequent transmission outside the crystal as photons rather than the transition from exciton to photon states inside the crystal and subsequent photon propagation to the surface. In this description two essential questions have to be answered: (i) in which way is polariton damping taken into account and (ii) what is the explicit influence of crystal surface properties like transmission coefficients and non-radiative surface recombination? There are several previous experimental and theoretical studies on polariton fluorescence [4—10]. We present here an investigation in GaAs using various geometrical configurations and monochromatic cw dye laser excitations. At variance with previous works [4, 6—9] our analysis points out the fundamental importance of the spatial inhomogeneity in the polariton energy distribution function near the crystal surface.
2. Polariton fluorescence Ilneshape analysis The fluorescence of the n = 1 exciton-polariton in GaAs and its 1-LO replica are shown in fig. 1. The simple spectrum of the replica can be described along the same lines as in the work of Gross [11] in CdS: given an energy distribution function f(E) of polaritons (irrespective of their branch), the creation rate of 27
28
C. Weisbuch, R.G. Ulbrichf Features of polariton fluorescence
LO-RepI.
—
°
X20
~
10 2
~.
L
1.514
1.516 Photon
t
t
L
~g
1.518
1.520
Energy (eV)
Fig. Ia. Polariton fluorescence spectrum of GaAs under non-resonant front excitation at hwo. b. One LO-phonon replica under the same excitation conditions (abscissa shifted by one LO-phonon energy for clarity). Circles give the lineshape of a Maxwell—Boltzmann distribution at temperature T = 22K: 1— \/~exp(—i~/kT);~ = hw E 1~— hwLo.
polaritons at energy hw = E LO is the product of f(E) with the density of states p(E) and the LO-phonon emission probability. Assuming the latter to have a smooth dependence on E, the replica only reproduces the energy distribution of polaritons. It probes the whole excited volume because polaritons with energies well below the longitudinal exciton energy E[ can readily escape the crystal (small absorption coefficient at E ~ EL). The fluorescence intensity of the no-phonon line originates from both upper (upb) and lower (lpb) polariton branches and is given by [51: —
1(E) = ~
f1(E, z = 0)p~(E)v51~(E)Ti(E),
(I)
i=1,2
where f1(E, z = 0) is the value of the distribution function in space and energy f1(E, r) at the crystal surface, Vgj~ is the component of polariton group velocity perpendicular to the surface, T1(E) is the transmission coefficient of polaritons in the vacuum as photons. Now the index 1, 2 labels upb and lpb, respectively. In a spherical approximation p(E)vgj(E) = (2i~h)~k~ and is approximately constant, except in region of strong variations of k, (lpb above E1, upb near EL). The transmission coefficient T,(E) can be derived from the phenomenological Maxwell theory and requires appropriate abc’s above EL [12]. Independent of the abc’s is the fact that sufficiently away from the resonance, i.e. Ihw E1j ~ ELT (longitudinal—transverse splitting), the transmission coefficients of the lpb below EL and upb above EL are constant and equal to 1 R, R being the reflectivity in that range. In the resonance region hw ELI ~ ELT, luminescence originates from the two branches as their T1(E)’s are of the same order of magnitude. Due to the large K’s of the lpb above EL, T2(E) is very small there (large refractive index n) and the fluorescence in this range is mainly determined by the transmission of upb polaritons. The most difficult task in eq. (1) is the computation of f(E, r). Assuming spatial homogeneity (but finite crystal thickness), Sumi made an explicit cal—
—
—
C. Weisbuch, R.G. Ulbrich/ Features of polariton fluorescence
29
culation of f(E) [8]. His main results are: for long enough lifetimes, a quasiequilibrium distribution exists above EL. This distribution is truncated somewhat below EL because of the decreasing escape time in that region (due to the increase in polariton group velocity). Fig. lb proves that the space-averaged polariton distribution deduced from the 1-LO phonon replica lineshape resembles indeed an equilibrium one. The nophonon lineshape (fig. la) is different: thermalization occurs in the lpb because of its larger density of states. The upb emission above EL is therefore due to the build-up of f1(E, 0) by scattering-in of polaritons thermalized on the lpb. 1(E) in this spectral region is thus the product of an equilibrium f2(E) with the interbranch transition probability integrated over space (T1(E) can be taken constant in this range). The spectral variation of the scattering probability explains well the difference between the shapes of upb emission and 1-LO emission. Below EL, the emission reflects the cutoff of the distribution function. A peculiar feature is the dip existing at EL, which is not observed on the replica. Such dips were previously attributed to the spectral variation of T(E) [6,7]. We will show that they are determined by the shapes of f1,2(E, 0). The spatial dependence of f(E, r) plays a fundamental role. Fig. 2 shows the fluorescence observed on the two sides of a 22 ~m thick sample: clearly the f(E, 0) is not the same on the excited and back-sides! Considering a simple diffusion model for polariton propagation, we deduce the following in analogy with the problem of carrier diffusion with surface recombination [13]: for volume excitation with a sink at the surface, when spatial diffusion is more important than direct creation, the surface density decreases with decreasing diffusion constant. In other words, only the polaritons within one diffusion length are able to reach the surface. As polariton scattering and trapping processes are resonantly increased in the vicinity of EL (as evidenced through the maximum of the absorption coefficient), f(E, 0) will exhibit a minimum at such energies. Thus, for a non-resonant (i.e. deep in volume) excitation, f(E, 0)
1.514
1.516 Photon
1.518 1.520 Energy 1eV)
Fig. 2a. Resonantly excited polariton fluorescence of a thin GaAs sample of 22 ~m thickness (front excitation at fho0 = EL). b. Spectrum observed from the back-side under the same conditions.
30
C. Weisbuch, R.G. Ulbrichl Features of polariton fluorescence
displays a dip at the maximum of scattering probability on an overall smooth shape with a cut-off below EL. This dip in f(E, r) exists only near the surface which acts as a sink through radiative and/or surface recombination. Inside the volume, thermalization processes have the time to fill-in this dip. This is clearly shown by the LO-replica in fig. lb. Under resonant excitation, the surface density f(E, 0) is not only due to the diffusion of thermalized polaritons from the bulk. The major contribution comes now from direct optical creation of non-thermalized polaritons. As shown in fig. 2a, the no-phonon line is thus much less thermalized and has no dip. Its intensity is also largely increased due to the increase in quantum efficiency, because of the shorter escape time of polaritons created near the surface [10]. Under such conditions, the LO-replica is observed to be still well thermalized: it represents the space average of the polariton distribution function, and still most polaritons have a chance to diffuse in the bulk and thermalize there.
3. Discussion In eq. (1), the three factors p. 0g~ T are fixed “geometrical” factors, whereas f(E, 0) is the quantity which can be externally varied as follows: (i) Varying excitation energy hw 0: as described above, resonant excitation leads to non-thermalized distribution function at the surface, and large radiative efficiency of the no-phonon line. Non-resonant (i.e. bulk) excitation leads to a dip in the distribution function (i.e. in the emission intensity) at the energy where the polariton scattering rate is maximum. (ii) Back-side luminescence in a thin sample exhibits a behaviour almost independent of exciting light energy: the value of f(E, 0) is non-negligible only at energies where the diffusion length is comparable with the crystal thickness, i.e. well away from EL. Created resonantly or not, the polaritons with an energy = EL do not propagate to the other surface. (iii) The LO-phonon replica lineshape is a direct evidence that the no-phonon polariton fitiorescence probes only part of a spatially inhomogeneous distribution. The replica is always well-thermalized, whereas the no-phonon line displays a variety of shapes under varying exciting energy, back-side observation. Compared with previous works, we point out here the fundamental importance of the spatial distribution of polaritons, which explains both the variation of lineshapes with hw0 and the very different behaviour of the nophonon line and 1-LO replica. Previous descriptions cannot describe the effects reported here. A first attempt to take into account the spatial aspect of polariton fluorescence was made by Bonnot and Benoit a Ia Guillaume [51,but the over-simplification of the model led to a conclusion where emission should be maximum at the energy of maximum polariton scattering rate.
C. Weisbuch, R.G. Ulbrich/ Features of polariton fluorescence
31
References [1] J.J. Hopfleld, Phys. Rev. 112 (1958) 1555; SI. Pekar, Zh. Eksp. Teor. Fiz. 33 (1957) 1022 [Soy. Phys.-JETP 6 (1958) 785]. [2] Y. Toyozawa, Prog. Theor. Phys. Suppl. 12 (1959) 111. [3] J.J. Hopfield, J. Phys. Soc. Japan Suppl. 21 (1966) 77. [4] W.C. Tait and R.L. Weiher, Phys. Rev. 178 (1969) 1404. [51A. Bonnot and C. Benoit a Ia Guillaume in Polaritons, Proc. Taormina Res. Conf. Structure of Matter, Taormina, 1972, eds. E. Burstein and F. de Martini (Pergamon, New York, 1974) p. 197. [6] E. Gross, S. Permogorov, V. Travnikov and A. Selkin, Solid State Commun. 10 (1972) 1071. [71D.D. Sell, S.E. Stokowski, R. Dingle and J.V. DiLorenzo, Phys. Rev. B7 (1973) 4568. [8] H. Sumi, J. Phys. Soc. Japan 41 (1976) 526. [9] I. Broser and R. Broser, J. Luminescence 12/13 (1976) 201. [10] C. Weisbuch and R.G. Ulbrich, Phys. Rev. Letters 39 (1977) 654. [11] E.F. Gross, S.A. Permogorov and B.S. Razbirin, Usp. Fiz. Nauk 103 (1971) 431 [Soy. Phys. usp. 14 (1971) 104]. [12] See e.g. C.S. Ting, Mi. Frankel and J.L. Birman, Solid State Commun. 17 (1975)1285 and ref s. therein. [13] See e.g. SM. Sze, Physics of semiconductor devices (Wiley, New York, 1969) ch. 2.