Optical spin hall effect in exciton–polariton condensates in lead halide perovskite microcavities

The spin Hall effect (SHE)^1,2 generates transverse spin current by an electric current, a process that fundamentally originates from spin–orbit coupling (SOC) in the material.³ The optical spin Hall effect (OSHE)^4,5 originates from the photonic SOC, which can be represented by the action of an effective magnetic field on the photon pseudo-spin, that is, the Stokes vector of light.⁶ For light reflecting off an interface, the effective magnetic field Ω_T arises from the transverse electric/transverse magnetic (TE/TM) mode splitting and scales with $k‖2e−2iφ$⁠, where k_|| and φ are the in-plane momentum vector and azimuthal angle, respectively.^7–9 The OSHE can be understood as a result of the precession motion of the photon pseudo-spin induced by Ω_T, which gives rise to a signature in the form of a quadrupolar distribution of opposite spin polarizations (σ⁺/σ⁻) in momentum space.^4,5 Because Ω_T scales with k_||,² the OSHE vanishes at k_|| ∼ 0 and its manifestation in the quantum fluid region has only been observed for propagating exciton–polariton condensates at sufficiently large k_||.^10,11 In contrast to the OSHE, Ohadi et al. discovered that spin polarization can form spontaneously in quasi-equilibrium exciton–polariton condensates in microcavities at k_|| ∼ 0.¹² While the appearance of +σ or −σ is stochastic, the stable condensate can be pinned to a particular spin state by a slight imbalance in the spin polarization of the non-resonant pump pulse. These authors attributed the findings to parity breaking bifurcation induced by different loss rates of the linear polarizations.

To realize exciton–polariton condensates with spin polarization from the OSHE at k_|| ∼ 0, we need a large photonic SOC, which can result from optical anisotropy in crystals with birefringence, as represented by a constant effective magnetic field Ω_XY.^13,14 We choose lead halide perovskite (LHP) single crystals because they possess large optical anisotropy in the orthorhombic phase^15,16 and strong light–matter interaction,^17–19 resulting in SOC exciton–polaritons in broad momentum spaces^16,20,21 We grow methylammonium lead bromide (CH₃NH₃PbBr₃) single crystal plates directly in a microcavity defined by two distributed Bragg reflectors (DBRs) and carry out Fourier space photoluminescence (FS-PL) imaging [Fig. 1(a)] with complete determination of the Stokes vector S(S₁, S₂, S₃) in a Poincaré sphere. Here, S₁, S₂, and S₃ represent the linear (TE/TM), diagonal, and circular (σ⁺/σ⁻) polarizations, respectively. We apply a motorized rotational quarter waveplate and a linear polarizer before the detector and use the corresponding Mueller matrices²² to obtain the Stokes vector; see Sec. S1 and Fig. S1 in the supplementary material.

All chemicals and regents were purchased from Sigma-Aldrich and used as received, unless noted otherwise. CH₃NH₃PbBr₃ precursor solution with a concentration of 1.0 M was prepared by dissolving stoichiometric 1:1 CH₃NH₃Br and PbBr₂ in dimethylformamide. The DBR mirrors were custom-produced by Thorlabs (subtractive magenta dichroic filter), and the center of the stop band was measured at 545 nm with a bandwidth of 160 nm. The perovskite precursor solution was filled into the cavity composed of two opposing DBRs through capillary action. The microcavity was then placed in a vacuum oven with the temperature set at ∼80 °C. The solvent in the cavity slowly evaporated to form single-crystal perovskites within 48 h. A subsequent epoxy seal was applied to bond the two opposing DBRs. Multiple batches of CH₃NH₃PbBr₃ single crystals in such a DBR-based cavity were prepared for measurements and their Stokes vectors under the same conditions were similar, indicating the microstructure and the quality of the samples are highly reproducible. Note that the direct crystal growth in the cavity defined by two DBR mirrors avoids damage to the crystal surfaces. The minimization of damage to the lead halide perovskite crystals has allowed us to carry out quantitative measurements over broad excitation densities, as highlighted by the unambiguous determination of two thresholds: stimulated scattering and stimulated emission. The downside of this sample preparation procedure is the potential introduction of disorder and gradients in cavity length due to spatial variations in crystal thickness.

Angle-resolved PL measurements were carried out on a home-built confocal setup (see Fig. S1 in the supplementary material). Low-temperature PL measurements were performed on a close-cycle liquid helium cryostat (Montana). The second harmonic of a Clark-MXR Impulse laser (0.5 MHz repetition rate, 250-fs pulses, and 1030 nm wavelength) was used to pump a home-built noncollinear optical parametric amplifier to generate 800-nm pulses, which was subsequently used to generate 400-nm pulses via second harmonic generation. The beam size is expanded to ensure large-area illumination with a spot size of ∼25 µm through a 100× vacuum objective with a numerical aperture of 0.85. The cavity exciton–polaritons are formed from non-resonant excitation by linearly polarized light at 400 nm. The emission spectra were collected with a liquid nitrogen-cooled charge-coupled device (Princeton Instruments, PyLoN 400B) coupled to a spectrograph (Princeton Instruments, Acton SP 2300i). We used the Lightfield software suite (Princeton Instruments) and LabVIEW (National Instruments) for data collection. A sample temperature of 77 K is used in all experiments.

The first effect of the Ω_XY field is the persistence of energetic splitting of the exciton–polaritons^16,18 as shown by dispersions below the condensation threshold (P_th) along k_x for the two modes, A and B [Fig. 1(b)]. Both polariton branches undergo condensation above P_thr, as shown by the flat dispersions k_x ∼ 0 at E_A = 2.262 ± 0.001 eV and E_B = 2.272 ± 0.001 eV with a full-width-at-half-maximum (FWHM) of ∼0.4 meV [Fig. 1(c); see Fig. S2 in the supplementary material for analysis of peak widths]. Note that, in a perfect planar cavity, condensation occurs at k_|| (k_x or k_y) = 0. The unavoidable presence of cavity disorder leads to some propagation of the condensates,^16,23 as reflected in the flat dispersion near k_x ∼ 0 in Fig. 1(c). The FWHM corresponds to a total dephasing time of t ∼ℏ/FWHM ∼ 1.6 ps and is dominated by cavity decay and corresponds to a quality factor of the cavity of Q ∼ 5000 in the condensation region. As detailed before,¹⁶ excitation power-dependent measurements clearly establish a condensation threshold P_thr due to stimulated scattering and a second lasing threshold attributed to stimulated emission at an order of magnitude higher power density. For the CH₃NH₃PbBr₃ cavity probed here, we focus on condensation and find the threshold of P_th = 8.0  µJ  cm⁻² at the cavity detuning (cavity photon energy – exciton energy) of δ = −23.8 meV. This threshold is clearly shown in the excitation power (P) dependence of the integrated PL intensity of both modes (A, blue; B, red) as a function of excitation power (P), in agreement with a more detailed study on exciton–polariton condensation in DBR cavities containing the birefringent CH₃NH₃PbBr₃ or CsPbBr₃ crystals.¹⁶ 

While our previous study focused on the two eigenstates of linear polarizations,¹⁶ we find that the treatment in the Hamiltonian equation (1) is insufficient, as there are also substantial circular polarization components. Figures 2(a) and 2(b) show the two-dimensional momentum distributions of S₃ for modes A and B, respectively; see the supplementary material, Fig. S3, for the 2D momentum distributions of S₁ and S₂ and Fig. S4 for sample fittings to extract the Stokes vector parameters. Each image is characterized by a center region around (k_x, k_y) = (0, 0) of the condensate with a particular spin polarization (σ⁻ or σ⁺) and a ring at higher k_|| of a cavity mode weakly coupled to excitons. Note that the deviation from (k_x, k_y) = (0, 0) of the exciton–polariton in each image is a result of the variation in sample thickness within the probe area; the extents of variation vary from spot-to-spot and from sample-to-sample, but the magnitudes of circular polarization are highly reproducible and depend on cavity detuning (see below). The momentum integrated Stokes vectors of the two condensates, S_A(S₁, S₂, S₃) = (0.65, −0.36, −0.68) and S_B = (−0.73, 0.55, 0.40), are shown on the Poincaré sphere in Fig. 2(c) with the two condensates exhibiting opposite spin polarizations. The degrees of spin-polarization ρ₃ = |S₃|/(S₁² + S₂² + S₃²)^1/2 are 0.70 ± 0.05 and 0.50 ± 0.05 for condensates A and B, respectively.

In the OSHE effect,^4,5 the pseudo-spin vector S is subject to a torque $S×Ωk‖$⁠, which rotates a nominally in-plane pseudo-spin vector toward the two poles on the Poincaré sphere. The resulting precession-like motion is coupled with cavity decay to give net polarizations of opposite signs for two opposite linear polarizations, as shown in Fig. 2(d) for the TE (blue) and TM (red) modes. In a microcavity without birefringence, $Ωk‖$ = Ω_T, and its variation in direction with φ leads to a distinct quadrupolar pattern of S₃ in momentum space for microcavity excitons,^4,5 including propagating condensates,^11,24 at |k_||| > 0. This OSHE effect is confirmed by the quadrupolar patterns of S₃ for the cavity modes in Figs. 2(a) and 2(b) (outer rings). Without birefringence, $Ωk‖$ and the OSHE effect vanish when k_|| approaches zero. In the present case, the large birefringence of the CH₃NH₃PbBr₃ perovskite crystal in the orthorhombic phase^15,16 gives a constant effective magnetic field $ΩXY=Δph±/ℏ$⁠, which is 1.5 × 10¹³ ps⁻¹ for the experimental energy splitting of Δ_ph± = 10 meV. This rate is of the order of or higher than most scattering rates in an exciton–polariton condensate.^25,26 Thus, for Bose scattering of exciton–polaritons of orthogonal linear polarizations (TE/TM) into the condensate states, the OSHE from the Ω_XY field biases the condensates into σ⁺ and σ⁻ polarizations, respectively. This mechanism differs from that of Ohadi et al.¹² in terms of the source for polarization pinning. In their approach, the pinning of the condensate in a σ⁺ or σ⁻ state originates from the external seeding of spin polarization in the excitation light. In our case, this seeding is internal, from the OSHE effect in the presence of Ω_XY.

Supporting our interpretation of polarization pinning, we find that the degree of spin polarization is higher as the condensate is closer to a quasi-equilibrium condition and the Bose scattering becomes more efficient. The first experimental evidence comes from the consistently higher degree of spin-polarization in condensate A than that in condensate B [Fig. 2(c)]. For the negative cavity-detuning that is probed here, the lower energy mode A is closer to a quasi-equilibrium, while mode B is kinetically favored.¹⁶ The second experimental evidence comes from the detuning dependence. Figures 3(a) and 3(b) show the degrees of polarization (DOP) for linear (S₁² + S₂²) and circular (S₃²) polarizations of the two polariton condensates (at 1.4 P_thr) as a function of cavity detuning, δ. The corresponding momentum–energy maps of polariton condensates at the four δ values can be found in Fig. S5 in the supplementary material and the complete Stokes vector parameters, including the degree of unpolarized light expected from photonic defects and heterogeneities, are shown in Fig. S6 in the supplementary material. The detuning-dependent polarization state is nearly independent of the excitation power above P_th (see Fig. S7 in the supplementary material). For the thermodynamically favored mode (A), the DOP for circular polarization in Fig. 3(a) increases from 17% ats δ = −38.0 meV to ∼50% as detuning decreases to δ = −22.5 meV. A smaller detuning value means that the condensate is closer to a quasi-equilibrium condition and the polarization state approaches that of the spin-pinned eigenstate dictated by Bose scattering. For the higher energy condensate (B), it is far from quasi-equilibrium in the whole range of δ_c values, and scattering processes in the kinetically favored condensate may reduce the degree of spin polarization predicted for the eigenstates.

The results presented here reveal highly spin-polarized emission from exciton–polariton condensates in optical cavities containing CH₃NH₃PbBr₃ single crystals. This is attributed to strong photonic SOC at k_|| ∼ 0 due to the birefringence of the CH₃NH₃PbBr₃ single crystal in the orthorhombic phase. The resulting OSHE leads to the pinning of condensates in spin-polarized states from linearly polarized excitation. Such spin-polarized condensates may serve as model systems for the exploration of many-body SOC interactions in the quantum fluid phases and as high-temperature and solid-state alternatives to Bose–Einstein condensates in ultracold atomic gases.^27–30 

See the supplementary material for additional data and analysis.

This work was solely supported by the Center of Programmable Quantum Materials (Pro-QM), an Energy Frontier Research Center of the U.S. Department of Energy, under Grant No. DE-SC0019443. We thank Yongping Fu for help with crystal growth.

The authors have no conflicts to disclose.

X.-Y.Z. and B.X. conceived this work. B.X. and Y.L. performed all optical experiments presented here. B.X. and M.S.S. built the experimental setup, with assistance from Y.D. and Y.B. B.X. carried out sample fabrication. B.X. and X.-Y.Z. wrote the manuscript, with input from all coauthors. X.-Y.Z. and D.N.B. supervised the project. All authors participated in the discussion and interpretation of the results.

Bo Xiang: Conceptualization (equal); Data curation (lead); Formal analysis (equal); Investigation (lead); Methodology (lead); Visualization (equal); Writing – original draft (equal). Yiliu Li: Data curation (supporting); Investigation (supporting). M. S. Spencer: Formal analysis (equal); Methodology (equal). Yanan Dai: Data curation (supporting); Investigation (supporting); Methodology (supporting). Yusong Bai: Data curation (supporting); Investigation (supporting). Dmitri N. Basov: Conceptualization (supporting); Funding acquisition (equal); Methodology (supporting); Resources (equal). X.-Y. Zhu: Conceptualization (equal); Formal analysis (equal); Funding acquisition (equal); Project administration (equal); Resources (equal); Supervision (equal); Writing – original draft (equal).

The data that support the findings of this study are available within the article and its supplementary material.