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Motional narrowing in semiconductor microcavities
Superlattices and Microstructures, Vol. 22, No. 1, 1997
Motional narrowing in semiconductor microcavities D. M. Whittaker Toshiba Cambridge Research Centre, 260 Cambridge Science Park, Milton Road, Cambridge, CB4 4WE, U.K.
P. Kinsler, T. A. Fisher, M. S. Skolnick, A. Armitage A. M. Afshar Department of Physics, University of Sheffield, Sheffield, S3 7RH, U.K.
J. S. Roberts, G. Hill, M. A. Pate Department of Electronic and Electrical Engineering, University of Sheffield, Sheffield, S1 3JD, U.K. (Received 15 July 1996) We describe experiments on a semiconductor microcavity which provide the first demonstration of motional narrowing in semiconductor inter-subband optical transitions. Significant narrowing occurs because of the small mass of the polaritons in a microcavity. The demonstration is made possible by the control provided in a microcavity of the mixing between photon and exciton states, and hence the dispersion of the polariton. c 1997 Academic Press Limited
Key words: motional narrowing, polaritons, microcavities.
1. Introduction Semiconductor microcavities are excellent systems in which to study the effects of strong coupling between excitons and photons [1–3]. The photon confinement produces a discrete optical mode which can be brought into resonance with the exciton state in a quantum well embedded in the cavity. If the coupling is sufficiently strong, the exciton and photon states anti-cross, giving a finite on-resonance energy separation, known as the vacuum Rabi splitting. In this regime, the excitations of the system are polaritons consisting of mixed photon and exciton states. One important difference between these polaritons and the ordinary excitons which are found in quantum wells is the much smaller effective mass: M p , the polariton mass is ∼ 10−5 m 0 , whereas Me , the quantum well exciton mass is ∼ 0.25m 0 . As a consequence, we are able to observe the phenomenon of motional narrowing in the region round the resonance; there is a reduction in the inhomogeneous contribution to the observed line widths, due to quantum mechanical averaging over the quantum well disorder. .
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2. Polaritons in microcavities Our semiconductor microcavity is a Fabry–P´erot structure with a λ GaAs cavity between two Bragg mirrors each consisting of 20 repeats of a λ/4 AlAs-Al0.13 Ga0.87 As bilayer. For a particular in-plane wave-vector, the 0749–6036/97/050091 + 06 $25.00/0
sm960275
c 1997 Academic Press Limited
92
Superlattices and Microstructures, Vol. 22, No. 1, 1997 T (K) 1.415
15
1.425
60
Reflectivity
85
95
105
110
1.40
1.41 1.42 Energy (eV)
1.43
Fig. 1. Microcavity reflectivity spectra at a field of 14 T (σ + polarisation) and a range of temperatures, showing tuning through the polariton resonance. The inset shows, for comparison, the exciton feature with the top mirror of the cavity removed by etching. .
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cavity supports only a single electromagnetic mode. This mode has a finite lifetime due to the finite finesse of the cavity. The structure is designed so that the cavity mode is close to resonance with the excitonic transition ˚ positioned close to the antinode of the cavity in three In0.13 Ga0.87 As quantum wells of nominal width 100 A, photon in order to maximize the coupling. The structure and experimental procedures are described in more detail in reference [4]. Figures 1 and 2 show experimental results demonstrating a typical anti-crossing when the excitons in the quantum well are tuned through resonance with the cavity photon. The exciton energy is varied by exploiting the reduction in the bandgap of a semiconductor which occurs when its temperature is raised. We chose a piece of the sample for which the exciton is well above the photon energy at low temperatures, and then increase the temperature to tune through the resonance. Figure 1 shows a series of reflectivity spectra at a range of temperatures spanning the anti-crossing: at 15 K, the weak exciton feature is well above the cavity, at 90 K, the two are close to resonance, and at 110 K the exciton is well below the cavity. The exciton feature is only strong close to resonance where it mixes significantly with the cavity photon—the exciton on its own does not couple to the electromagnetic field outside the cavity. Figure 2 shows the energies of the two features in Fig. 1 plotted as a funcion of temperature. The behaviour is that of a very typical anti-crossing, with the two branches approaching each other then parting, the minimum energy separation of 7.3 meV occuring at a temperature of ∼ 90 K. The mixed exciton–photon states near resonance are the microcavity polariton modes. They differ from ordinary quantum well polaritons in that, as our results show, the resonance occurs for optically active states with small in-plane wavevectors. The figure also shows, as dashed lines, the energies of the uncoupled exciton and cavity photon—the cavity photon variation was obtained from another piece of the sample, where the exciton is far from resonance, and the exciton variation was then deduced by fitting a simple two level anti-crossing model to the experimental data. .
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Superlattices and Microstructures, Vol. 22, No. 1, 1997
93
1425
Energy (meV)
1420
1415
1410
1405
0
20
40 60 80 Temperature (K)
100
120
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Fig. 2. The peak positions of Fig. 1 plotted as a function of temperature. The dots are the experimental points. The dashed lines indicate the behaviour of the uncoupled states. .
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The separation between the two polaritons on resonance is known as the vacuum Rabi splitting. Its size is determined by the strength of the coupling between the cavity photon and the exciton, which in turn depends on the magnitude of the photon field at the position of the quantum wells and on the intrinsic oscillator strength of the excitons. The vacuum Rabi splitting of 7.3 meV which we obtain is fairly large, a consequence of the enhancement of the exciton oscillator strength due to the additional confinement which occurs in a magnetic field. The cavity only confines the photon in the direction perpendicular to the mirrors, with no confinement in the parallel plane. Like the excitons in the quantum wells, the cavity photon has an in-plane dispersion [2]. The perpendicular component of the photon wavevector is fixed by the mirrors to be 2π/L, where L is the cavity width. Hence a photon with in-plane wave-vector k has energy s 2 1 h¯ c L 2 h¯ c h¯ c 2π 2π + k El (k) = + k2 ≈ n L n L 2 n 2π .
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with n the cavity refractive index. For small k, the in-plane dispersion takes a parabolic form, so we can describe it using a cavity photon ‘effective mass’, Ml = nh/(Lc) ∼ 10−5 m e . The in-plane dispersion of the polariton depends on the dispersion of the photon and the extent of the mixing with the exciton. For sufficiently small k, the in-plane kinetic energy can be treated as a perturbation on the polariton states. First order perturbation theory then gives dispersions which are parabolic, with an effective mass M p = (|ce |2 /Me + |cl |2 /Ml )−1 , where |ce |2 is the exciton fraction in the polariton state, |cl |2 the photon fraction. For larger k, the dispersions become non-parabolic, and this result is no longer applicable. The dependence of M p on the coefficients ce , cl means that the polariton effective mass varies as the system
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is tuned through the resonance. Since Ml is very small compared to Me , we can, unless |cl |2 is close to zero, neglect Me and take M p ≈ Ml /|cl |2 ∼ 10−5 m 0 .
3. Motional narrowing Motional narrowing occurs when a quantum mechanical particle moves in a disordered potential. Quantum mechanics requires that the particle is in a state with a finite spatial extent, characterised by its De Broglie wavelength, λqm . Hence the particle does not ‘see’ fine scale structure in the potential, only an average over a length ∼ λqm . Since the variance of such an average is always smaller than the variance of the underlying potential, this leads to a reduction in the linewidths of transitions involving the particle [5]. Whether significant motional narrowing is likely to occur in a particular system can be readily determined by comparing the De Broglie wavelength of the particle with the length scale, λc , of typical fluctuations in the potential. If λc λqm , quantum mechanical effects are not important and motional narrowing should be negligible. On the other hand, if λc λqm , significant motional narrowing should occur. λc is a property of ˚ [6]. If we take the linewidth 0 as a characteristic energy for the the disorder potential, and is typically ∼ 100 A √ particle, the De Broglie wavelength, λqm ∼ h¯ / 2M0. For a typical quantum well exciton, M = Me ∼ 0.25m 0 ˚ which is less than λc so motional narrowing is not very important [6]. and 0 ∼ 5 meV, giving λqm ∼ 50 A, By contrast, for a polariton in a microcavity, M = M p ∼ 10−5 m 0 , and from our data 0 ∼ 1 meV, which ˚ much greater than λc . Hence motional narrowing should be significant in microcavities. gives λqm ∼ 104 A, Furthermore, since we can change M p by tuning through the resonance, we can vary the amount of the motional narrowing. We obtain a quantitative description of line width variation as the system is tuned through resonance by using a scaling argument, valid in the limit when the line widths are small compared with the vacuum Rabi splitting. In this limit, scattering between the two branches can be neglected, and each branch can be treated as an independent particle with mass M p = Ml /|cl |2 as described above. The potential seen by each of these branches has the same spatial variation as the underlying disorder potential Ve (R), but the amplitude is reduced by a factor |ce |2 , since only the exciton component of the polariton couples to the quantum well disorder. The effective Hamiltonian for the branch is then .
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Heff = −|cl |2
h¯ 2 ∇ 2 + |ce |2 Ve (R). 2Ml
In order to proceed further, we approximate the disorder potential by a Gaussian white noise potential, which has zero correlation length scale and is characterised by a two point correlation function of the form hVl (R1 )Vl (R2 )i = αδ(R1 − R2 ). This is a reasonable approximation to make, bearing in mind that the actual correlation length scale of the potential is less than 1% of λqm . The important property of the Gaussian white noise potential is that it has no intrinsic scale: changes in length scale lead to a similar potential, with only a different value of α in the correlator. We exploit this property by rescaling both the lengths and energies in the problem: E 0 = E|cl |2 /|ce |4 , R 0 = R|ce |2 /|cl |2 . Then 2 h¯ 2 ∇ 2 |cl |2 |cl | 0 h¯ 2 ∇ 2 0 + V R + Ve0 (R 0 ). = − Heff = − e 2Ml |ce |2 |ce |2 2Ml The scaling is chosen such that the α value in the correlator of the scaled potential Ve0 (R 0 ) is exactly the same (in two dimensions) as for the original Ve (R). Hence, whatever the values of ce , cl , the scaled Hamiltonian has identical properties, including the linewidth of the spectral features it predicts. This means that in real units, the linewidth must vary as the energy scaling, that is 0 = |ce |4 /|cl |2 0e , where 0e is the value of the scaled width.
Superlattices and Microstructures, Vol. 22, No. 1, 1997
95
2.50 2.25
Line width (meV)
2.00 1.75 1.50 1.25 1.00 0.75 0.50 0.0
0.2
1.0
0.4 0.6 0.8 Exciton fraction, |ce|2
Fig. 3. Fit to experimental data for the lower (solid symbols) and upper branch (open symbols) of the polariton. The solid line is the motional narrowing fit, while the dashed line is a ‘fit’ to a model without motional narrowing. .
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4. Evidence for motional narrowing In the previous section we obtained a quantitative prediction for the variation of the inhomogeneous linewidth of a polariton feature as the system is tuned through resonance. In order to compare this with our experimental linewidths, it is necessary to account for the homogeneous contribution to the linewidth due to the finite finesse of the microcavity. This is treated in detail in reference [7]. If the cavity photon has a line width 0l , a polariton will have a smaller homogeneous width |cl |2 0l , the reduction occuring because the polariton can only escape in the fraction |cl |2 of the time it spends as a photon. The total line width is obtained by convolving the lineshapes due to the homogeneous and inhomogeneous processes [7]. The homogeneous line shape is clearly Lorentzian, and we take the inhomogeneous shape to be Gaussian. This is not strictly correct, but the central portion is known to be Gaussian [6], and anyway the effects of the convolution are small for our data. Figure 3 shows the fit we obtain to our experimentally measured widths using our motional narrowing theory. The widths are plotted as a function of the exciton fraction in the polariton, |ce |2 , which was obtained by using the fit to the two level system discussed above. The most significant feature of the data is the decrease in the line width which occurs close to resonance, at least for the lower polariton branch (solid squares). This is a clear signature of the occurence of motional narrowing. The figure also shows (triangles) a restricted set of data points obtained using electric field tuning at low temperatures [3], which demonstrates that the narrowing is not an effect of the magnetic field or an artefact of the temperature tuning. The experimental widths at small and large exciton fractions were fitted to obtain the values of the constants 0e and 0l . The motional narrowing model (solid line) then gives a very good fit to the behaviour of the lower branch over the .
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Superlattices and Microstructures, Vol. 22, No. 1, 1997
whole range. This contrasts strongly with the poor ‘fit’ obtained using a model without motional narrowing, but including the convolution effects discussed above (dashed lines). The comparison with theory for the upper branch is less satisfactory, with the experimental points lying above the predicted values. Numerical simulations [8] show that when the coupling between the two branches is included, the effect is to broaden the lower branch relative to the upper branch. Hence the poor fit for the upper branch suggests that there is another source of broadening in operation. Previously, additional broadening of the upper branch has been attributed to the effects of absorption into higher energy exciton and continuum states [9], but this cannot apply at high magnetic fields, where the nearest optically allowed transitions occur at much higher energies. We speculate that the difference may be due to coupling with higher energy optically forbidden states, mixed into the wavefunction by the disorder potential. .
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References .
[1] }C. Weisbuch, M. Nishioka, A. Ishikawa, and Y. Arakawa, Phys. Rev. Lett. 69, 3314 (1992). } Houdr´e, R. P. Stanley, U. Oesterle, M. Ilegems, and C. Weisbuch, Phys. Rev. B 49, 16761 (1994). [2] R. } A. Fisher, A. M. Afshar, D. M. Whittaker, M. S. Skolnick, J. S. Roberts, G. Hill, and M. A. Pate, Phys. [3] T. Rev. B 51, 2600 (1995). } A. Fisher, A. M. Afshar, M. S. Skolnick, D. M. Whittaker, J. S. Roberts, G. Hill, and M. A. Pate, [4] T. Proceedings of 7th International Conference on Modulated Semiconductor Structures Madrid, 1995. } I. Halperin, Phys. Rev. 139, A104 (1965). [5] B. } F. Schnabel, R. Zimmermann, D. Bimberg, H. Nickel, R. Losch, and W. Schlapp, Phys. Rev. B 46 9873 [6] R. (1992); R. Zimmermann and E. Runge, J. Luminesc. 60-1, 320 (1994). } Kinsler, D. M. Whittaker, Phys. Rev. B 54, 4998 (1996). [7] P. } M. Whittaker, unpublished results. [8] D. [9] J. } Tignon, P. Voisin, C. Delalande, M. Voos, R. Houdr´e, U. Oesterle, and R. P. Stanley, Phys. Rev. Lett. 74, 3967 (1995). .
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Motional narrowing in semiconductor microcavities D. M. Whittaker Toshiba Cambridge Research Centre, 260 Cambridge Science Park, Milton Road, Cambridge, CB4 4WE, U.K.
P. Kinsler, T. A. Fisher, M. S. Skolnick, A. Armitage A. M. Afshar Department of Physics, University of Sheffield, Sheffield, S3 7RH, U.K.
J. S. Roberts, G. Hill, M. A. Pate Department of Electronic and Electrical Engineering, University of Sheffield, Sheffield, S1 3JD, U.K. (Received 15 July 1996) We describe experiments on a semiconductor microcavity which provide the first demonstration of motional narrowing in semiconductor inter-subband optical transitions. Significant narrowing occurs because of the small mass of the polaritons in a microcavity. The demonstration is made possible by the control provided in a microcavity of the mixing between photon and exciton states, and hence the dispersion of the polariton. c 1997 Academic Press Limited
Key words: motional narrowing, polaritons, microcavities.
1. Introduction Semiconductor microcavities are excellent systems in which to study the effects of strong coupling between excitons and photons [1–3]. The photon confinement produces a discrete optical mode which can be brought into resonance with the exciton state in a quantum well embedded in the cavity. If the coupling is sufficiently strong, the exciton and photon states anti-cross, giving a finite on-resonance energy separation, known as the vacuum Rabi splitting. In this regime, the excitations of the system are polaritons consisting of mixed photon and exciton states. One important difference between these polaritons and the ordinary excitons which are found in quantum wells is the much smaller effective mass: M p , the polariton mass is ∼ 10−5 m 0 , whereas Me , the quantum well exciton mass is ∼ 0.25m 0 . As a consequence, we are able to observe the phenomenon of motional narrowing in the region round the resonance; there is a reduction in the inhomogeneous contribution to the observed line widths, due to quantum mechanical averaging over the quantum well disorder. .
.
.
.
2. Polaritons in microcavities Our semiconductor microcavity is a Fabry–P´erot structure with a λ GaAs cavity between two Bragg mirrors each consisting of 20 repeats of a λ/4 AlAs-Al0.13 Ga0.87 As bilayer. For a particular in-plane wave-vector, the 0749–6036/97/050091 + 06 $25.00/0
sm960275
c 1997 Academic Press Limited
92
Superlattices and Microstructures, Vol. 22, No. 1, 1997 T (K) 1.415
15
1.425
60
Reflectivity
85
95
105
110
1.40
1.41 1.42 Energy (eV)
1.43
Fig. 1. Microcavity reflectivity spectra at a field of 14 T (σ + polarisation) and a range of temperatures, showing tuning through the polariton resonance. The inset shows, for comparison, the exciton feature with the top mirror of the cavity removed by etching. .
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cavity supports only a single electromagnetic mode. This mode has a finite lifetime due to the finite finesse of the cavity. The structure is designed so that the cavity mode is close to resonance with the excitonic transition ˚ positioned close to the antinode of the cavity in three In0.13 Ga0.87 As quantum wells of nominal width 100 A, photon in order to maximize the coupling. The structure and experimental procedures are described in more detail in reference [4]. Figures 1 and 2 show experimental results demonstrating a typical anti-crossing when the excitons in the quantum well are tuned through resonance with the cavity photon. The exciton energy is varied by exploiting the reduction in the bandgap of a semiconductor which occurs when its temperature is raised. We chose a piece of the sample for which the exciton is well above the photon energy at low temperatures, and then increase the temperature to tune through the resonance. Figure 1 shows a series of reflectivity spectra at a range of temperatures spanning the anti-crossing: at 15 K, the weak exciton feature is well above the cavity, at 90 K, the two are close to resonance, and at 110 K the exciton is well below the cavity. The exciton feature is only strong close to resonance where it mixes significantly with the cavity photon—the exciton on its own does not couple to the electromagnetic field outside the cavity. Figure 2 shows the energies of the two features in Fig. 1 plotted as a funcion of temperature. The behaviour is that of a very typical anti-crossing, with the two branches approaching each other then parting, the minimum energy separation of 7.3 meV occuring at a temperature of ∼ 90 K. The mixed exciton–photon states near resonance are the microcavity polariton modes. They differ from ordinary quantum well polaritons in that, as our results show, the resonance occurs for optically active states with small in-plane wavevectors. The figure also shows, as dashed lines, the energies of the uncoupled exciton and cavity photon—the cavity photon variation was obtained from another piece of the sample, where the exciton is far from resonance, and the exciton variation was then deduced by fitting a simple two level anti-crossing model to the experimental data. .
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Superlattices and Microstructures, Vol. 22, No. 1, 1997
93
1425
Energy (meV)
1420
1415
1410
1405
0
20
40 60 80 Temperature (K)
100
120
.
Fig. 2. The peak positions of Fig. 1 plotted as a function of temperature. The dots are the experimental points. The dashed lines indicate the behaviour of the uncoupled states. .
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The separation between the two polaritons on resonance is known as the vacuum Rabi splitting. Its size is determined by the strength of the coupling between the cavity photon and the exciton, which in turn depends on the magnitude of the photon field at the position of the quantum wells and on the intrinsic oscillator strength of the excitons. The vacuum Rabi splitting of 7.3 meV which we obtain is fairly large, a consequence of the enhancement of the exciton oscillator strength due to the additional confinement which occurs in a magnetic field. The cavity only confines the photon in the direction perpendicular to the mirrors, with no confinement in the parallel plane. Like the excitons in the quantum wells, the cavity photon has an in-plane dispersion [2]. The perpendicular component of the photon wavevector is fixed by the mirrors to be 2π/L, where L is the cavity width. Hence a photon with in-plane wave-vector k has energy s 2 1 h¯ c L 2 h¯ c h¯ c 2π 2π + k El (k) = + k2 ≈ n L n L 2 n 2π .
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with n the cavity refractive index. For small k, the in-plane dispersion takes a parabolic form, so we can describe it using a cavity photon ‘effective mass’, Ml = nh/(Lc) ∼ 10−5 m e . The in-plane dispersion of the polariton depends on the dispersion of the photon and the extent of the mixing with the exciton. For sufficiently small k, the in-plane kinetic energy can be treated as a perturbation on the polariton states. First order perturbation theory then gives dispersions which are parabolic, with an effective mass M p = (|ce |2 /Me + |cl |2 /Ml )−1 , where |ce |2 is the exciton fraction in the polariton state, |cl |2 the photon fraction. For larger k, the dispersions become non-parabolic, and this result is no longer applicable. The dependence of M p on the coefficients ce , cl means that the polariton effective mass varies as the system
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Superlattices and Microstructures, Vol. 22, No. 1, 1997
is tuned through the resonance. Since Ml is very small compared to Me , we can, unless |cl |2 is close to zero, neglect Me and take M p ≈ Ml /|cl |2 ∼ 10−5 m 0 .
3. Motional narrowing Motional narrowing occurs when a quantum mechanical particle moves in a disordered potential. Quantum mechanics requires that the particle is in a state with a finite spatial extent, characterised by its De Broglie wavelength, λqm . Hence the particle does not ‘see’ fine scale structure in the potential, only an average over a length ∼ λqm . Since the variance of such an average is always smaller than the variance of the underlying potential, this leads to a reduction in the linewidths of transitions involving the particle [5]. Whether significant motional narrowing is likely to occur in a particular system can be readily determined by comparing the De Broglie wavelength of the particle with the length scale, λc , of typical fluctuations in the potential. If λc λqm , quantum mechanical effects are not important and motional narrowing should be negligible. On the other hand, if λc λqm , significant motional narrowing should occur. λc is a property of ˚ [6]. If we take the linewidth 0 as a characteristic energy for the the disorder potential, and is typically ∼ 100 A √ particle, the De Broglie wavelength, λqm ∼ h¯ / 2M0. For a typical quantum well exciton, M = Me ∼ 0.25m 0 ˚ which is less than λc so motional narrowing is not very important [6]. and 0 ∼ 5 meV, giving λqm ∼ 50 A, By contrast, for a polariton in a microcavity, M = M p ∼ 10−5 m 0 , and from our data 0 ∼ 1 meV, which ˚ much greater than λc . Hence motional narrowing should be significant in microcavities. gives λqm ∼ 104 A, Furthermore, since we can change M p by tuning through the resonance, we can vary the amount of the motional narrowing. We obtain a quantitative description of line width variation as the system is tuned through resonance by using a scaling argument, valid in the limit when the line widths are small compared with the vacuum Rabi splitting. In this limit, scattering between the two branches can be neglected, and each branch can be treated as an independent particle with mass M p = Ml /|cl |2 as described above. The potential seen by each of these branches has the same spatial variation as the underlying disorder potential Ve (R), but the amplitude is reduced by a factor |ce |2 , since only the exciton component of the polariton couples to the quantum well disorder. The effective Hamiltonian for the branch is then .
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Heff = −|cl |2
h¯ 2 ∇ 2 + |ce |2 Ve (R). 2Ml
In order to proceed further, we approximate the disorder potential by a Gaussian white noise potential, which has zero correlation length scale and is characterised by a two point correlation function of the form hVl (R1 )Vl (R2 )i = αδ(R1 − R2 ). This is a reasonable approximation to make, bearing in mind that the actual correlation length scale of the potential is less than 1% of λqm . The important property of the Gaussian white noise potential is that it has no intrinsic scale: changes in length scale lead to a similar potential, with only a different value of α in the correlator. We exploit this property by rescaling both the lengths and energies in the problem: E 0 = E|cl |2 /|ce |4 , R 0 = R|ce |2 /|cl |2 . Then 2 h¯ 2 ∇ 2 |cl |2 |cl | 0 h¯ 2 ∇ 2 0 + V R + Ve0 (R 0 ). = − Heff = − e 2Ml |ce |2 |ce |2 2Ml The scaling is chosen such that the α value in the correlator of the scaled potential Ve0 (R 0 ) is exactly the same (in two dimensions) as for the original Ve (R). Hence, whatever the values of ce , cl , the scaled Hamiltonian has identical properties, including the linewidth of the spectral features it predicts. This means that in real units, the linewidth must vary as the energy scaling, that is 0 = |ce |4 /|cl |2 0e , where 0e is the value of the scaled width.
Superlattices and Microstructures, Vol. 22, No. 1, 1997
95
2.50 2.25
Line width (meV)
2.00 1.75 1.50 1.25 1.00 0.75 0.50 0.0
0.2
1.0
0.4 0.6 0.8 Exciton fraction, |ce|2
Fig. 3. Fit to experimental data for the lower (solid symbols) and upper branch (open symbols) of the polariton. The solid line is the motional narrowing fit, while the dashed line is a ‘fit’ to a model without motional narrowing. .
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4. Evidence for motional narrowing In the previous section we obtained a quantitative prediction for the variation of the inhomogeneous linewidth of a polariton feature as the system is tuned through resonance. In order to compare this with our experimental linewidths, it is necessary to account for the homogeneous contribution to the linewidth due to the finite finesse of the microcavity. This is treated in detail in reference [7]. If the cavity photon has a line width 0l , a polariton will have a smaller homogeneous width |cl |2 0l , the reduction occuring because the polariton can only escape in the fraction |cl |2 of the time it spends as a photon. The total line width is obtained by convolving the lineshapes due to the homogeneous and inhomogeneous processes [7]. The homogeneous line shape is clearly Lorentzian, and we take the inhomogeneous shape to be Gaussian. This is not strictly correct, but the central portion is known to be Gaussian [6], and anyway the effects of the convolution are small for our data. Figure 3 shows the fit we obtain to our experimentally measured widths using our motional narrowing theory. The widths are plotted as a function of the exciton fraction in the polariton, |ce |2 , which was obtained by using the fit to the two level system discussed above. The most significant feature of the data is the decrease in the line width which occurs close to resonance, at least for the lower polariton branch (solid squares). This is a clear signature of the occurence of motional narrowing. The figure also shows (triangles) a restricted set of data points obtained using electric field tuning at low temperatures [3], which demonstrates that the narrowing is not an effect of the magnetic field or an artefact of the temperature tuning. The experimental widths at small and large exciton fractions were fitted to obtain the values of the constants 0e and 0l . The motional narrowing model (solid line) then gives a very good fit to the behaviour of the lower branch over the .
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Superlattices and Microstructures, Vol. 22, No. 1, 1997
whole range. This contrasts strongly with the poor ‘fit’ obtained using a model without motional narrowing, but including the convolution effects discussed above (dashed lines). The comparison with theory for the upper branch is less satisfactory, with the experimental points lying above the predicted values. Numerical simulations [8] show that when the coupling between the two branches is included, the effect is to broaden the lower branch relative to the upper branch. Hence the poor fit for the upper branch suggests that there is another source of broadening in operation. Previously, additional broadening of the upper branch has been attributed to the effects of absorption into higher energy exciton and continuum states [9], but this cannot apply at high magnetic fields, where the nearest optically allowed transitions occur at much higher energies. We speculate that the difference may be due to coupling with higher energy optically forbidden states, mixed into the wavefunction by the disorder potential. .
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References .
[1] }C. Weisbuch, M. Nishioka, A. Ishikawa, and Y. Arakawa, Phys. Rev. Lett. 69, 3314 (1992). } Houdr´e, R. P. Stanley, U. Oesterle, M. Ilegems, and C. Weisbuch, Phys. Rev. B 49, 16761 (1994). [2] R. } A. Fisher, A. M. Afshar, D. M. Whittaker, M. S. Skolnick, J. S. Roberts, G. Hill, and M. A. Pate, Phys. [3] T. Rev. B 51, 2600 (1995). } A. Fisher, A. M. Afshar, M. S. Skolnick, D. M. Whittaker, J. S. Roberts, G. Hill, and M. A. Pate, [4] T. Proceedings of 7th International Conference on Modulated Semiconductor Structures Madrid, 1995. } I. Halperin, Phys. Rev. 139, A104 (1965). [5] B. } F. Schnabel, R. Zimmermann, D. Bimberg, H. Nickel, R. Losch, and W. Schlapp, Phys. Rev. B 46 9873 [6] R. (1992); R. Zimmermann and E. Runge, J. Luminesc. 60-1, 320 (1994). } Kinsler, D. M. Whittaker, Phys. Rev. B 54, 4998 (1996). [7] P. } M. Whittaker, unpublished results. [8] D. [9] J. } Tignon, P. Voisin, C. Delalande, M. Voos, R. Houdr´e, U. Oesterle, and R. P. Stanley, Phys. Rev. Lett. 74, 3967 (1995). .
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