- Email: [email protected]
Stimulated scattering of excitons into microcavity polaritons
Microelectronic Engineering 47 (1999) 325-327
Stimulated Scattering of Excitons into Microcavity Polaritons R. Huanga, aEdward
F. Tassonea,
L. Ginzton
and Y. Yamamotoalb
Laboratory,
Stanford
bNTT Basic Research Laboratories, 3-1, Morinosato-Wakamiya, Atsugi-shi,
University,
Kanagawa
Stanford,
CA 94305, USA
243-0198, Japan
We have observed the bosonic enhancement of the elastic exciton-exciton scattering rate in a semiconductor microcavity. The bosonic signature is in the enhancement of the scattering rate that is proportional to the final state occupation number. This gain process for exciton-polaritons is distinctly different from that in a conventional photon laser, because of the fermionic nature of the exciton’s constituents, the electron and hole. This distinction becomes prominent at large final state occupation numbers, where we observe a saturation and reduction in the gain. We discuss the origins of this saturation behavior.
1. INTRODUCTION Although electrons and holes in a semiconductor are fermions, their bound state, the exciton, behaves approximately as a boson at low exciton density [l]. The polariton, a hybrid mode of the exciton and a photon, is also a boson. It is well known that the bosonic nature of a particle leads directly to quantum statistical effects such as BoseEinstein condensation and stimulated emission [2]. We can expect to achieve final state stimulation for excitons via a scattering process involving lower and upper polaritons [3]. The scattering process starts with (1) two nearly exciton-like lower polaritons created at large, and opposite, in-plane momenta, and results in (2) an upper and a lower polariton created, both at zero in-plane momentum (in order to satisfy conservation of momentum and energy). From the rate equations of this scattering process, we expect that the final state upper polariton intensity should exhibit a quadratic dependence on the pump which initially created the two exciton-like lower polaritons. Furthermore, the upper polariton intensity should scale linearly with an additional probe beam, which serves to create population at the lower polariton at zero in-plane momentum. In this paper, we describe experimental evidence for this bosonic stimulation of the elastic exciton-exciton scattering rate. 0167-9317/99/S - see front matter PII: SO167-9317(99)00225-7
Q 1999 Elsevier Science B.V. All rights IWXV~~.
326
R.Huang et al. I Microelectronic Engineering 47 (1999) 325-327
2. STIMULATED
SCATTERING
IN THE LOW DENSITY
REGIME
2.1. Rate-Equations for Exciton-Exciton Scattering To model this scattering process, we set up a rate equation
d&p -= dt
clNesc+
CS(NLP)N&,
(LP), and where Nup, Nhp, and N,,, are the upper polariton (UP), lower polariton exciton populations, respectively. The coefficients cl and cz (NLP) = cze + c2r x (1 + NLP) are scattering rates for the physical scattering processes. The first term represents the scattering from the exciton into the upper polariton due to acoustic phonon emission and absorption. The second term represents the spontaneous and stimulated scattering processes, which have two contributions: (1) a process of an exciton scattering with another exciton to produce a third exciton and an UP (i.e., the czo term) and (2) two excitons at large in-plane momentum which scatter into an UP and a LP (i.e., the ~21 term). The objective of our experiment is to observe this stimulated scattering due to LP population, which is the signature of bosonic final state stimulation. 2.2. Experimental Setup In the experiment, two CW pump beams are incident on the sample at an angle of 4~51”from the normal direction to create the large in-plane momentum excitons, and a CW probe beam is incident on the sample in the normal direction to inject LPs with zero in-plane momentum. The emission is collected by a lens and is detected with a 0.75 meter spectrometer, with an 1800 grooves/mm grating, and a high quantum efficiency, liquid nitrogen cooled CCD detector. 2.3. Results for Low Density We observe the UP luminescence as a function of the pump and probe. The fact that this luminescence has a typical polariton linewidth means that the process is not a fourwave mixing, since in that case, we would expect to see a very narrow linewidth (defined by the nearly monochromatic pump and probe beams), rather than the full linewidth of the polariton. The measured constancy of the center wavelength of the UP throughout this experiment indicates that the polariton has not collapsed (i.e., we have not exceeded the exciton saturation density). We confirmed this by making simultaneous reflectivity measurements and found that the positions of the upper and lower polaritons did not shift over the range of pump intensities used in the experiment. We observe a quadratic dependence of the intensity of the luminescence of the UP state, as a function of pump power. We extracted the quadratic coefficients in these experimental curves, and plot the ratio of the quadratic component to the linear component in figure 1. This ratio is given by 2, from equation 1. The rates for acoustic phonon scattering were calculated via the deformation potential scattering, and the elastic exciton-exciton scattering rates were calculated by the exciton exchange interaction. The theoretically determined rates give ac1 = 3.5 x lO_llN~p + 3.8 x lo-‘; we compare this with the experimental value, z = (1.5 f 0.2) x lo-l1 NLP + (1.3 f 0.8) x lo-‘. The agreement is reasonable considering the theoretical and experimental uncertainties.
R.Huang et al. I Microelectronic Engineering 47 (1999) 325-327
Figure 1. The bosonic enhancement rate as a function of relatively small lower polariton population (points: experiment, solid line: fit to experiment).
3. SATURATION
OF SCATTERING
321
Figure 2. The integrated upper polariton luminescence at a fixed exciton density as a function of relatively large lower polariton number (points: experiment, solid line: theory, dotted line: linear rate).
IN THE HIGH
DENSITY
REGIME
As we increase the injected lower polariton population to a level close to the exciton saturation density, we observe a striking change in the upper polariton luminescence. At fixed exciton density, the UP intensity dependence on the number of LPs, which is proportional to the stimulated scattering rate, becomes sublinear. This behavior may be partly due to the phase space filling and exciton exchange effects, which become important near the exciton saturation density [4]. In figure 2, we have plotted this characteristic along with a first-order theoretical estimation of the curve which is valid only in the regime below the polariton saturation. The depleted exciton population due to strong stimulated scattering might also be responsible for this saturation behavior [5]. REFERENCES 1. E. Hanamura and H. Haug, Phys. Rep., 33C, 209-284 (1977). 2. A. Griffin, D. W. Snoke, and S. Stringari, (eds)., Bose-Einstein Condensation, Cambridge University Press, Cambridge, 1995. 3. F. Tassone, R. Huang, and Y. Yamamoto, submitted for publication. 4. S. Schmitt-Rink, D.S. Chemla, and D.A.B. Miller, Adv. in Phys., 38, 89-188 (1989). 5. F. Tassone and Y. Yamamoto, submitted for publication.
Stimulated Scattering of Excitons into Microcavity Polaritons R. Huanga, aEdward
F. Tassonea,
L. Ginzton
and Y. Yamamotoalb
Laboratory,
Stanford
bNTT Basic Research Laboratories, 3-1, Morinosato-Wakamiya, Atsugi-shi,
University,
Kanagawa
Stanford,
CA 94305, USA
243-0198, Japan
We have observed the bosonic enhancement of the elastic exciton-exciton scattering rate in a semiconductor microcavity. The bosonic signature is in the enhancement of the scattering rate that is proportional to the final state occupation number. This gain process for exciton-polaritons is distinctly different from that in a conventional photon laser, because of the fermionic nature of the exciton’s constituents, the electron and hole. This distinction becomes prominent at large final state occupation numbers, where we observe a saturation and reduction in the gain. We discuss the origins of this saturation behavior.
1. INTRODUCTION Although electrons and holes in a semiconductor are fermions, their bound state, the exciton, behaves approximately as a boson at low exciton density [l]. The polariton, a hybrid mode of the exciton and a photon, is also a boson. It is well known that the bosonic nature of a particle leads directly to quantum statistical effects such as BoseEinstein condensation and stimulated emission [2]. We can expect to achieve final state stimulation for excitons via a scattering process involving lower and upper polaritons [3]. The scattering process starts with (1) two nearly exciton-like lower polaritons created at large, and opposite, in-plane momenta, and results in (2) an upper and a lower polariton created, both at zero in-plane momentum (in order to satisfy conservation of momentum and energy). From the rate equations of this scattering process, we expect that the final state upper polariton intensity should exhibit a quadratic dependence on the pump which initially created the two exciton-like lower polaritons. Furthermore, the upper polariton intensity should scale linearly with an additional probe beam, which serves to create population at the lower polariton at zero in-plane momentum. In this paper, we describe experimental evidence for this bosonic stimulation of the elastic exciton-exciton scattering rate. 0167-9317/99/S - see front matter PII: SO167-9317(99)00225-7
Q 1999 Elsevier Science B.V. All rights IWXV~~.
326
R.Huang et al. I Microelectronic Engineering 47 (1999) 325-327
2. STIMULATED
SCATTERING
IN THE LOW DENSITY
REGIME
2.1. Rate-Equations for Exciton-Exciton Scattering To model this scattering process, we set up a rate equation
d&p -= dt
clNesc+
CS(NLP)N&,
(LP), and where Nup, Nhp, and N,,, are the upper polariton (UP), lower polariton exciton populations, respectively. The coefficients cl and cz (NLP) = cze + c2r x (1 + NLP) are scattering rates for the physical scattering processes. The first term represents the scattering from the exciton into the upper polariton due to acoustic phonon emission and absorption. The second term represents the spontaneous and stimulated scattering processes, which have two contributions: (1) a process of an exciton scattering with another exciton to produce a third exciton and an UP (i.e., the czo term) and (2) two excitons at large in-plane momentum which scatter into an UP and a LP (i.e., the ~21 term). The objective of our experiment is to observe this stimulated scattering due to LP population, which is the signature of bosonic final state stimulation. 2.2. Experimental Setup In the experiment, two CW pump beams are incident on the sample at an angle of 4~51”from the normal direction to create the large in-plane momentum excitons, and a CW probe beam is incident on the sample in the normal direction to inject LPs with zero in-plane momentum. The emission is collected by a lens and is detected with a 0.75 meter spectrometer, with an 1800 grooves/mm grating, and a high quantum efficiency, liquid nitrogen cooled CCD detector. 2.3. Results for Low Density We observe the UP luminescence as a function of the pump and probe. The fact that this luminescence has a typical polariton linewidth means that the process is not a fourwave mixing, since in that case, we would expect to see a very narrow linewidth (defined by the nearly monochromatic pump and probe beams), rather than the full linewidth of the polariton. The measured constancy of the center wavelength of the UP throughout this experiment indicates that the polariton has not collapsed (i.e., we have not exceeded the exciton saturation density). We confirmed this by making simultaneous reflectivity measurements and found that the positions of the upper and lower polaritons did not shift over the range of pump intensities used in the experiment. We observe a quadratic dependence of the intensity of the luminescence of the UP state, as a function of pump power. We extracted the quadratic coefficients in these experimental curves, and plot the ratio of the quadratic component to the linear component in figure 1. This ratio is given by 2, from equation 1. The rates for acoustic phonon scattering were calculated via the deformation potential scattering, and the elastic exciton-exciton scattering rates were calculated by the exciton exchange interaction. The theoretically determined rates give ac1 = 3.5 x lO_llN~p + 3.8 x lo-‘; we compare this with the experimental value, z = (1.5 f 0.2) x lo-l1 NLP + (1.3 f 0.8) x lo-‘. The agreement is reasonable considering the theoretical and experimental uncertainties.
R.Huang et al. I Microelectronic Engineering 47 (1999) 325-327
Figure 1. The bosonic enhancement rate as a function of relatively small lower polariton population (points: experiment, solid line: fit to experiment).
3. SATURATION
OF SCATTERING
321
Figure 2. The integrated upper polariton luminescence at a fixed exciton density as a function of relatively large lower polariton number (points: experiment, solid line: theory, dotted line: linear rate).
IN THE HIGH
DENSITY
REGIME
As we increase the injected lower polariton population to a level close to the exciton saturation density, we observe a striking change in the upper polariton luminescence. At fixed exciton density, the UP intensity dependence on the number of LPs, which is proportional to the stimulated scattering rate, becomes sublinear. This behavior may be partly due to the phase space filling and exciton exchange effects, which become important near the exciton saturation density [4]. In figure 2, we have plotted this characteristic along with a first-order theoretical estimation of the curve which is valid only in the regime below the polariton saturation. The depleted exciton population due to strong stimulated scattering might also be responsible for this saturation behavior [5]. REFERENCES 1. E. Hanamura and H. Haug, Phys. Rep., 33C, 209-284 (1977). 2. A. Griffin, D. W. Snoke, and S. Stringari, (eds)., Bose-Einstein Condensation, Cambridge University Press, Cambridge, 1995. 3. F. Tassone, R. Huang, and Y. Yamamoto, submitted for publication. 4. S. Schmitt-Rink, D.S. Chemla, and D.A.B. Miller, Adv. in Phys., 38, 89-188 (1989). 5. F. Tassone and Y. Yamamoto, submitted for publication.