Department of Physics, University of Sheffield, Sheffield S3 7RH, United ... same time, all other spectral features are the same from both sides of the structure.
PHYSICAL REVIEW B
VOLUME 58, NUMBER 23
15 DECEMBER 1998-I
Polariton-induced optical asymmetry in semiconductor microcavities A. Armitage and M. S. Skolnick Department of Physics, University of Sheffield, Sheffield S3 7RH, United Kingdom
A. V. Kavokin Dipartimento di Ingegneria Elettronica, Universita di Roma II, 00133 Roma, Italy
D. M. Whittaker Toshiba Cambridge Research Centre, Cambridge CB4 0HE, United Kingdom
V. N. Astratov and G. A. Gehring Department of Physics, University of Sheffield, Sheffield S3 7RH, United Kingdom
J. S. Roberts Department of Electronic and Electrical Engineering, University of Sheffield, Sheffield S1 3JD, United Kingdom ~Received 25 June 1998! Polariton-induced optical asymmetry in the reflectivity spectra of coupled semiconductor microcavity structures, containing quantum wells in one of the cavities, is reported. The central mode of the system is only observable when viewed from one side of the structure, but is dark and invisible from the other side. At the same time, all other spectral features are the same from both sides of the structure. Microscopically, interference between the exciton and cavity modes is shown to lead to the optical asymmetry. Furthermore, the asymmetric exciton-cavity mode is observable only for light incident on the cavity without excitons, a result shown to be fully consistent with a coupled oscillator model. @S0163-1829~98!02247-4#
The electronic states in atoms, or exciton states in semiconductors, can usually be characterized as ‘‘dark’’ or ‘‘bright,’’ that is, either optically inactive or active. A good example of a dark state is an exciton with spin 2, which corresponds to a transition that is forbidden by selection rules. In multilayer systems with n active regions, such as multiple quantum wells ~QW’s! or quantum microcavities ~QMC’s!, the exciton-photon interaction leads to m polariton states that have enhanced oscillator strength and are bright ~super-radiant!, and n-m states that have almost zero photon content and are dark, with m typically much less than n.1–3 We show here that the excitons in a semiconductor can be either bright or dark depending on the direction of study. We report the observation of optically asymmetric polariton modes in the reflectivity spectra of coupled QMC’s. The central modes of the system are only observable for one direction of observation, while at the same time all other spectral features are seen for both directions. Furthermore, we show that the central mode can be converted from bright to dark by tuning the optical mode energies relative to the exciton. Such directionally dependent reflectivity cannot arise in a system possessing time-reversal symmetry. The phenomena are permitted since time-reversal symmetry is broken by the presence of exciton absorption in only one of the cavities of the coupled QMC’s. The phenomena are explained within a three-oscillator model, taking into account exciton-resonance absorption and the two optical-cavity eigenmodes. The surprising observation that the exciton-cavity coupled modes are only visible for light incident on the cavity without excitons finds a natural explanation within the model. QMC’s represent one of the most rapidly developing areas of semiconductor physics. Important effects recently 0163-1829/98/58~23!/15367~4!/$15.00
found include vacuum Rabi coupling of excitons and photons,4 motional narrowing,5 and strong coupling between macroscopically separated excitons.6 QMC’s are Fabry-Pe´rot cavities whose optical length is a half-integer multiple of the wavelength of the excitonic excitations of QWs in the cavity.7 The high-reflectivity mirrors are formed from distributed Bragg reflectors ~DBR!.7 A coupled QMC as studied here ~Fig. 1!, contains two 1l cavities grown one on top of the other.8 The central DBR has sufficiently few repeats that significant overlap occurs between the optical fields of the cavities, leading to the formation of symmetric ~S! and antisymmetric ~AS! optical eigenmodes.8 The structures investigated, grown by metal-organic vapor-phase epitaxy on GaAs substrates, contain three In0.06Ga0.94As QW’s in either the lower or upper 1l GaAs cavities @samples A and B, respectively, in Figs. 1~a! and
FIG. 1. Schematic diagram of coupled-microcavity structures. Structure A in ~a! contains quantum wells in the lower cavity only, whereas in structure B they are in the upper cavity. 15 367
©1998 The American Physical Society
FIG. 2. Reflectivity spectra for sample A, with light incident on the top surface of the sample, as indicated by the arrow above the figure. At low and high angles, two reflectivity dips are observed from the symmetric and antisymmetric cavity modes. Three dips are observed on resonance at 29.4°, due to mixing of the exciton states with the cavity modes. The inset shows the angular dependence of the dip positions, with the full line fits to a transfer matrix model.
1~b!#. The Bragg mirrors are composed of 17.5, 14.5, and 12 GaAs/AlAs layers, respectively, in the lower, intermediate, and upper mirrors. Spectra were measured at 10 K using white-light techniques. Following reflection ~or transmission! by the sample, the light was dispersed, and detected by a Ge photodiode. A key feature is the control of the optical mode energies relative to those of the excitons. We take advantage of the 1/cos uint dependence of the Fabry-Pe´rot mode energy on internal angle u int . Variation of the external angle of incidence u ext allows modes to be selected in resonance with the excitonic states.9 However, other tuning techniques, such as temperature or electric field10 could just as easily be used. Reflectivity spectra for sample A, which contains QW’s only in the lower cavity @see Fig. 1~a!#, are shown in Fig. 2 as a function of u ext . The spectra are taken with light incident on the top surface of the sample. At 20° two dips are observed arising from the S and AS ~C S , C AS! optical modes of the coupled cavities.11 The weak feature ~X! to higher energy arises from the excitons of the QW’s. With increasing u ext the cavity modes shift to higher energy and the excitonic feature becomes stronger, until at resonance at ;29° the three dips have very nearly equal intensities. The energies of the dips as a function of u ext are shown in the inset to Fig. 2; the characteristic anticrossing for a coupled mode system is observed. When u ext is increased further, the cavity modes shift further to higher energy, until at 44.4°, the excitonic feature is no longer visible.12 Spectra corresponding to Fig. 2, again measured from the top surface, but for sample B @Fig. 1~b!# with QW’s only in
FIG. 3. Reflectivity spectra for sample B, observed from the top surface. By contrast with sample A in Fig. 2, only two dips are observed on resonance at 39.8°. The inset shows a fit to the intensity of the central mode as a function of detuning, using the model of Eqs. ~1!–~4!.
the upper cavity, are shown in Fig. 3.13 Between 25° and 33.6° the excitonlike feature increases in intensity as it is approached by the C AS mode, but then most surprisingly as u ext is increased further, the central feature loses intensity, until at resonance at 39.8° it is unobservable and a two-dip spectrum is seen. For further increase of u the central dip reappears and the three-feature spectrum is restored. The central mode for sample B is thus dark on resonance. Detuning to either lower or higher energy restores the third mode, which becomes bright again. To verify that this result does not arise from peculiarities of sample B, we show spectra in Fig. 4~a! from sample A, but measured with light incident from the substrate side. Sample A is then equivalent to sample B in the geometry of Fig. 3 since the light is incident on the cavity containing the QW’s. It is seen that on resonance at 32.8° the central mode is unobservable, even though when measured from the surface ~Fig. 2!, three modes were seen.14 Thus, the central mode is unobservable when light is incident on the cavity containing the QW’s. By contrast, when light is incident on the empty cavity ~Fig. 2!, the central mode is strong. These unexpected results can be understood by consideration of a three-coupled oscillator model composed of the optical modes from the two cavities and the excitonic states of the QW. The interaction Hamiltonian for the system can be expressed as
DS D S D
c1 c1 c 2 5E c 2 , cx cx
FIG. 4. ~a! Reflectivity spectra for sample A from the substrate side of the structure, as indicated by the arrow. The central mode is dark onresonance. ~b! On-resonance transmission spectra for sample A. The central mode is absent for both directions of observation.
where E 1 and E 2 are the unperturbed energies of the optical modes in cavities 1 and 2, respectively, E x is the exciton energy, D/2 is the coupling constant between the optical modes ~;3.5 meV in sample A!, and V/2 is the excitonphoton coupling potential @2.5 meV for In0.06Ga0.94As QW’s ~Ref. 5!#. Equation ~1! corresponds to the QW’s positioned in cavity 1, so there is only interaction between cavity 1 and the excitons. c 1 , c 2 , and c x are the coefficients of the basis functions for cavities 1 and 2 and the excitons, respectively. Solution of the secular determinant for resonance where E 1 5E 2 5E x ~taken equal to zero! leads to the eigenvalues E 50, 6(V 2 1D 2 ) 1/2/2. Thus, three modes are expected, the center mode at the unperturbed energy and the outer two modes shifted by 6(V 2 1D 2 ) 1/2/2 to higher and lower energy. The eigenvectors for the upper, central, and lower modes are
D & ~ V 1D ! 2
& ~ V 1D ! 2
V & ~ V 1D 2 ! 1/2 2
& ~ V 1D !
fx , ~2!
& ~ V 1D 2 ! 1/2
V & ~ V 1D 2 ! 1/2 2
fx , ~4!
where f 1 , f 2 , and f x are the unperturbed basis functions. It is seen from Eq. ~3! that the eigenvector of the central mode contains no component from the basis function f 1 , of cavity 1; the optical-field amplitude in the cavity containing the QW’s is zero for the central mode. The full weight of f 1 resides equally instead in the outer two components. In reflectivity the optical field is probed in the cavity on which the light is incident. If light is incident on the cavity containing the QW’s, Eq. ~3! predicts that the central mode will be dark, thus explaining the most important result of Figs. 3 and 4~a!. By contrast, if light is incident on the empty cavity, then the central mode will be bright @since the f 2 component of Eq. ~3! is nonzero# as in Fig. 2. These surprising results arise from the nature of the (333) matrix Hamiltonian of Eq. ~1!, which has one basis function ( f 1 ) that is coupled to the other two basis functions ( f 2 , f x ). f 2 and f x by contrast are not coupled directly. The eigenvector C c for the central mode contains no component from f 1 since its admixtures with f 2 and f x are of opposite sign and cancel. At the same time the two outer modes contain components from both f 1 and f 2 , and are thus observable for both directions. This situation for the coupled two-cavity one-exciton case is closely analogous to that found for the normal modes of three masses connected linearly by springs15 or of linear triatomic molecules.16 In these cases there is one mode corresponding to zero displacement of the central mass, with the outer two masses undergoing stretching mode vibrations. This is in accordance with Eq. ~3! where the oscillator ( f 1 ) connected to the other two has zero weight in the central mode. In the QMC case, if only a single cavity containing QW’s is studied, two bright polaritons are found.1–4 Growth of a second empty cavity on top of the first leads to a threemode system, but with the central mode dark when reflectivity is studied with light incident on the cavity containing the QW’s. In the time domain this corresponds to a photon emitted from the first cavity on a subpicosecond time scale (h/D;0.3 ps). This photon is reflected by the second cavity with a phase that exactly cancels the amplitude in the first cavity, leading to a dark feature when the first cavity is probed. As a result, the exciton-cavity mixed mode is only observable when probed from the cavity without excitons. The principal features of Figs. 2, 3, and 4~a! are thus understood. The dark nature of the central mode is, however, only expected for exact resonance when the couplings of f 1 to f 2 and f x cancel. Away from resonance, the central mode is expected to be visible, as found in Fig. 3. The intensity of the central dip for sample B is plotted as a function of detuning in the Fig. 3 inset. The results are compared with the variation expected from Eqs. ~2!–~4! for the cavity 1 fraction in the central mode. Good agreement is found, further supporting the model. Figures 2 and 4~a! show that the visibility of the central mode depends on the direction of observation. Transmission spectra do, however, exhibit mirror symmetry as shown in Fig. 4~b!, for sample A, where transmission spectra are found to be the same for the two directions of observation, with the central mode absent. This is expected, since for either direction of propagation the light must pass through cavity 1 where the optical field is zero. Transmission ~T! must be independent of direction for any optical system.17 Since
T1R1A must equal 1, where R and A are reflected and absorbed intensities, R will also be independent of direction, if A50. However, the presence of exciton absorption, which breaks time-reversal symmetry, permits R to be dependent on the direction of observation. To summarize, we have shown that coupled microcavities containing QW’s in only one of the cavities exhibit polariton modes only visible in reflectivity for one direction of observation. The coupled exciton-photon mode is only visible
when light is incident on the cavity without excitons, a result shown to arise from interference between the two-cavity and one-exciton mode of the system. The radiative properties of the excitons have a highly unusual angular dependence, exhibiting a zero for particular angles. Device applications, for example as optical switches, may result by applying an electric field to shift the exciton states, and thus to convert the dark modes to bright.
D. S. Citrin, Solid State Commun. 89, 139 ~1994!; L. C. Andreani, Phys. Status Solidi B 188, 29 ~1995!; G. Bjo¨rk, S. Pau, J. M. Jakobson, H. Cao, and Y. Yamomoto, Phys. Rev. B 52, 17 310 ~1995!. 2 H. Hu¨bner et al., Phys. Rev. Lett. 74, 2391 ~1995!. 3 V. M. Agranovich, G. C. La Rocca, and F. Bassani, Phys. Status Solidi A 164, 39 ~1997!. 4 C. Weisbuch, M. Nishioka, A. Ishikawa, and Y. Arakawa, Phys. Rev. Lett. 69, 3314 ~1992!. 5 D. M. Whittaker, P. Kinsler, T. A. Fisher, M. S. Skolnick, A. Armitage, A. M. Afshar, and J. S. Roberts, Phys. Rev. Lett. 77, 4792 ~1996!. 6 A. Armitage et al., Phys Rev B 57, 14 877 ~1998!. 7 For a review, see V. Savona, C. Piermarocchi, A. Quattropani, P. Schwendimann, and F. Tassone, New Aspects in Optical Properties of Materials ~Gordon and Breach, New York, 1997!. 8 Previous work on coupled cavities, but in different contexts can be found in e.g., R. P. Stanley et al., Appl. Phys. Lett. 65, 2093 ~1994!; P. Pellandini et al., ibid. 71, 864 ~1997!, and in Ref 6. 9 D. Baxter, M. S. Skolnick, A. Armitage, V. N. Astratov, D. M. Whittaker, T. A. Fisher, J. S. Roberts, D. J. Mowbray, and M. A. Kaliteevski, Phys. Rev. B 56, R10 032 ~1997!. 10 T. A. Fisher et al., Phys. Rev. B 51, 2600 ~1995!.
We thank J. J. Baumberg for very helpful discussions.
The background reflectivity level is very close to 100%, determined by the high reflectivity DBRs. 12 The extra width of the C s , C AS dips at 44.4° compared to 20° arises from absorption from excitonic continuum states. See e.g., Ref. 9. 13 The C S , C AS splitting for sample B is 10.3 meV, as compared to 7.1 meV in sample A, most probably due to a small variation in the growth parameters of the central mirror. 14 In Fig. 2, the C AS mode is slightly stronger than C S , whereas in Fig. 4~a!, C S is stronger than C AS . Transfer-matrix simulations show that this arises since the upper cavity is ;0.5% shorter than the lower cavity, resulting in the C AS (C S ) mode having slightly greater upper ~lower! cavity character. This leads to the cavity that is probed @upper in Fig. 2, lower in Fig. 4~a!# appearing slightly more strongly in the spectra. 15 See, e.g., H. Goldstein, Classical Mechanics ~Addison-Wesley, Reading, MA, 1951!, p. 336. 16 See, P. W. Atkins, Molecular Spectra ~Clarendon, Oxford, 1970!, p. 353. 17 This is a consequence of reciprocity. If a linear system contains absorbing and reflecting regions, T will be the same whether the light passes through the absorbing or nonabsorbing regions first.