Abstract
The spin Hall effect consists of the generation of a spin current perpendicular to the charge current flow. Thirty-five years after its prediction by Dyakonov and Perel’1, it is the focus of experimental and theoretical investigations and constitutes one of the most remarkable effects of spintronics. Owing to scattering and dephasing in electronic gases, it is difficult to observe and has only been demonstrated for the first time a few years ago2,3,4,5. Recently, three of us have predicted the optical spin Hall effect6, which consists of a separation in real space and momentum space of spin-polarized exciton–polaritons generated by a laser in a semiconductor microcavity7. The separation takes place owing to a combination of elastic scattering of exciton–polaritons by structural disorder and an effective magnetic field coming from polarization splitting of the polariton states. The excitonic spin current is controlled by the linear polarization of the laser pump. Here, we report the first experimental evidence for this effect and demonstrate propagation of polariton spin currents over 100 μm in a high-quality GaAs/AlGaAs quantum microcavity. By rotating the polarization plane of the exciting light, we were able to switch the directions of the spin currents.
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The spin carries a quantum bit of information, which makes spin-based devices, also called spintronic devices, extremely promising for quantum-information processing8. More specifically, the spin Hall effect (SHE) has recently raised an increasing interest owing to its potentialities in spintronics2,3. In a similar way to the Hall effect, in which a magnetic field causes a particle to deviate depending on its charge, the spin Hall effect results in a spin current owing to the spin-dependent scattering of electrons by charged impurities or other defects (extrinsic SHE1,9) or to the spin–orbit effects on the carrier energy dispersion (intrinsic SHE10,11). It leads to a separation in real space and momentum space between spin-up and spin-down electrons and thus enables control of electron spin in semiconductor spintronic devices12. Optical2,3 and electronic4 measurements of the extrinsic SHE have been reported. Although successfully generated, electronic spin currents are subject to rapid dephasing and decay, mostly due to electron scattering. Recent theoretical studies5,13 show that the intrinsic SHE is fully suppressed in macroscopic-sized samples because of electron scattering, which directs the system towards equilibrium. The use of neutral particles as carriers for spin currents is an alternative approach that may overcome the problem of dephasing. Here, we demonstrate the spin currents carried by (neutral) exciton–polaritons in a semiconductor microcavity and propagating coherently over 100 μm. We demonstrate for the first time experimentally that the spin currents can be controlled by an optical spin Hall effect (OSHE), which exhibits remarkable analogy to the spin Hall effect.
Exciton–polaritons (or polaritons for short) are the ‘mixed’ exciton–photon quasiparticles associated with the normal modes of semiconductor microcavities. They have been studied extensively since the discovery of strong light–matter coupling in these systems in 1992 (ref. 14). Many unique properties of polaritons are derived from their non-parabolic in-plane dispersion and composite bosonic nature7. Polaritons are particularly suitable as spin current carriers owing to their long coherence length, of the order of 100 μm (ref. 15), compared with that of electronic gases. Furthermore, the recent observation of Bose–Einstein condensation of polaritons in microcavities16 confirms their high potential for fascinating quantum coherent effects17,18.
Polaritons can be created by directly shining a laser onto the microcavity where the angle of incidence determines their in-plane wavevector. The polaritons scatter elastically with disorder in the system and redistribute around an elastic circle in reciprocal space (the modulus of the in-plane wavevector is conserved). A splitting in energy between polaritons with transverse electric and transverse magnetic linear polarizations causes different spin currents to develop in different quadrants of the elastic circle. This OSHE can be considered as the result of an effective magnetic field acting on a pseudospin.
The polariton spin (angular momentum) has ±1 projections on the growth axis of the microcavity. Therefore, the polariton can be treated as a two-level system with an effective pseudospin S=1/2. Furthermore, there is a strict correspondence between the polariton pseudospin vector in the Poincaré sphere (Fig. 1a) and the polarization of light emitted by the microcavity at a given angle. Similar to an angular momentum, the polariton spin state can be controlled by an appropriate polarized excitation and can be analysed by a polarization-resolved detection. The polarization of light can be fully characterized by measuring a set of values, the so-called Stokes parameters19 that correspond to the following polarization degrees:
where l and d denote the linear polarization degree measured in the x y and diagonal axes, respectively, and c denotes the circular polarization. and I−45∘ represent the intensities measured through linear polarizers oriented at 0∘, 90∘, 45∘ and − 45∘ to the x axis; Iσ+ and Iσ− represent the intensities measured in the circular polarization basis. The OSHE is the precession of the pseudospin of the polariton around an effective magnetic field, , owing to the longitudinal–transverse splitting, ΔLT (ref. 20), which is induced by the long-range part of the exciton exchange interaction21 and by the transverse electric–transverse magnetic splitting of the photon mode of the cavity22. The value of this magnetic field is proportional to ΔLT and its direction depends on the polar angle, θ, of the polariton in-plane wavevector, k=(kx,ky) in momentum space6:
The evolution of a pseudospin is critically dependent on the orientation of . Figure 1a,b shows the evolution of a pseudospin having an initial linear polarization into a left (σ+) or right (σ−) circular polarization depending on the orientation of . Figure 1c shows the dependence of the orientation on the orientation of k and its consequences on the pseudospin evolution. Similar to the role of the Rashba field in the intrinsic SHE10,11, the effective magnetic field causes different polarizations to develop in different directions in the OSHE. Recently, the existence of this field has been demonstrated in experiments on linear polarization conversion in the ballistic transport of exciton–polaritons23,24.
The evolution of the pseudospin can be well described by the following precession equation:
The first term describes the precession of the pseudospin. The second term describes the input term due to the Rayleigh scattering from the initial state (τ1 is the Rayleigh scattering time constant). The last term describes the radiative relaxation (τ is the polariton lifetime).
Resonant Rayleigh scattering is due to the static disorder in the microcavity25, which is inevitably present in any realistic microcavity owing to thickness variations in the cavity and Bragg mirror layers. Thus, when an initial polariton state with a given in-plane wavevector, k=(kx,ky), and a given polarization state, transverse magnetic or transverse electric, is created, it is efficiently scattered towards states with in-plane wavevectors different from the initial one and with the same polarization state. As the modulus of k is conserved, this scattering gives rise to the Rayleigh scattering ring in the far-field emission26,27. Then the anisotropy in the orientation of the effective magnetic field creates an anisotropy in the polarization: in two opposite quadrants of the elastic circle, σ+ polarization appears, whereas in the two others, σ− polarization appears (see Fig. 1c). The spin separation on the elastic circle leads to spin separation in real space owing to ballistic propagation of the polaritons, thus the polariton spin currents are formed.
To detect the OSHE, we use the experimental set-up shown schematically in Fig. 2. We use a continuous titanium-sapphire pump laser at 836 nm with an angle of incidence of 12∘. The polarization of the pump is linear and can be adjusted to transverse magnetic or transverse electric polarization. The far-field emission transmitted by the microcavity is polarization resolved and imaged on a 1,024×1,024 CCD (charge-coupled device) camera. This set-up allows a precise measurement of the Stokes parameters of the light in the far-field, thus providing information on the pseudospin of a polariton state with a given in-plane wavevector. The polariton spin currents in real space have been detected using a near-field polarization-resolved set-up. A complete description of the experiment is available in the Supplementary Information.
The microcavity was excited by a transverse magnetic polarized pump resonant with the lower polariton branch with a slight cavity–exciton detuning of 0.3 meV. The intensities Iσ+ and Iσ− of the light transmitted by the microcavity were imaged and measured successively using a CCD camera (Fig. 3a,b). The value of the circular Stokes parameter, ρc, was calculated for each pixel using equation (1). The result is shown in Fig. 3c. The dependence on θ of the circular parameter is in excellent agreement with the theoretical predictions shown in Fig. 1c. The maxima are located at ±π/4 and ±3π/4 and the sign of ρc is reversed between π/4 and 3π/4. For our experimental values of ΔLT (0.05 meV) and τ (∼10 ps), the maximal expected value of ρc is then close to 0.45 in excellent agreement with our experimental value of 0.48±0.03. Similar results are shown in Fig. 3d, where the experiment is carried out in the same conditions but with a transverse electric polarized excitation. The sign of ρc is then clearly reversed. We have also checked that ρc is very close to zero on the whole far-field if another spot with ΔLT=0 of the microcavity sample is excited. These experimental results are also in very good agreement with numerical simulations based on the solution of a two-component Schrödinger equation for the exciton polaritons that accounts for the disorder potential responsible for the Rayleigh scattering (Fig. 3e,f). The details of our formalism are given in the Supplementary Information.
Figure 4a shows the near-field images of the circular polarization degree of emission by the microcavity excited by transverse magnetic polarized light. It can be seen that the polaritons carrying the σ+ or σ− polarization propagate ballistically over about 100 μm, which means that macroscopic spin currents are generated. Their direction is controlled by the polarization of excitation light. This observation is in excellent agreement with the results of the simulation (Fig. 4b).
In the OSHE, the spin currents are carried by electrically neutral exciton–polaritons, which are subject to bosonic statistics. This represents two important advantages with respect to the electronic SHE: there is no dephasing owing to the Coulomb repulsion of the carriers and there is a possibility to create superfluid spin currents. Clearly, our microcavity acts as an optically controlled spin generator allowing for integration into photonic circuits. Future research is needed to investigate the OSHE in the nonlinear regime and superfluid spin currents.
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Acknowledgements
T.C.H.L. was supported by E.P.S.R.C. A.V.K. acknowledges support from the EU project STIMSCAT (contract 517769).
We are very grateful to R. Houdré (Ecole Polytechnique Fédérale de Lausanne, Switzerland) for providing the high-quality microcavity sample.
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C.L., M.R., J.P.K., E.G. and A.B. contributed to the experimental work, A.V.K. developed the theoretical concept and T.C.H.L., M.M.G. and G.M. contributed to the theoretical work.
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Leyder, C., Romanelli, M., Karr, J. et al. Observation of the optical spin Hall effect. Nature Phys 3, 628–631 (2007). https://doi.org/10.1038/nphys676
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DOI: https://doi.org/10.1038/nphys676
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