Introduction

Semiconductor microcavities with engineered photonic states are a valuable platform to observe fundamental and emergent quantum electrodynamic phenomena1. Polaritons are formed as a quantum superposition of semiconductor-excitons and microcavity-photons when the exciton-photon interaction is much larger than their decay rates. The ability to tailor the photonic modes, light-matter interaction and polariton lifetime in semiconductor microcavities enables versatile control of polaritons and opens the possibility of novel quantum effects such as Bose-Einstein condensation (BEC) at elevated temperatures2,3,4,5,6. Picosecond time-scale quasiequilibrium room-temperature polariton BEC has been claimed in ZnO7,8, GaN9,10,11 and polymers12 in Fabry-Pérot microcavities. This is possible when the thermalization time is even shorter than the polariton lifetime. However, for both scientific studies and applications, it is more favorable to achieve room-temperature polariton BEC with polariton lifetimes on the scale of 100 ps – 1 ns. Realization of long-lived, room-temperature, equilibrium polariton BEC remains an important target for fundamental research and practical application13.

In this article, we demonstrate theoretically a route to simultaneously achieve very strong exciton-photon coupling and long polariton lifetime using a simple TiO2 based photonic band gap (PBG)14,15 material sandwiching a planar quantum-well slab containing as few as three monolayers of the transition-metal dichalcogenide MoSe216. This architecture provides a realistic, technologically accessible route toward equilibrium polariton BEC above room temperature. In contrast, the corresponding Fabry-Pérot microcavity has picosecond polariton lifetime due to radiative recombination into unwanted leaky (air) modes that are degenerate with the microcavity mode. These leaky modes couple strongly to any finite area polariton condensate in the quantum-well region. They are eliminated by replacing the 1D periodic structure with a 3D PBG material. Recently WSe2 and MoS2 monolayers in 2D photonic crystals have been experimentally realized17,18. However, a complete 3D PBG is absent in those cavities and radiative decay of polaritons is not suppressed effectively. Another realization of a MoS2 monolayer in a Fabry-Pérot microcavity19 reveals exciton-photon coupling of 31.5 meV but very rapid (sub-picosecond) polariton decay and significant line-broadening of 38.5 meV.

In a MoSe2 monolayer the exciton binding energy is as large as 0.55 eV and the exciton Bohr radius is as small as 1.2 nm16,20. Theoretical calculation16 predicts that 6% of incident light is absorbed during transmission through a MoSe2 layer16, indicating very strong light-matter interaction. As we show, this enables high-temperature BEC with only three quantum-well monolayers of MoSe2. Previous considerations of high-temperature, equilibrium BEC have involved on the order of 100 quantum-wells21. Another special property of MoSe2 is that due to correlation between wave-vector valley and spin polarizations at the electronic band edge of MoSe2, left- and right-circularly polarized light are coupled to different exciton spin and valley states22. In other words, optical, valley and spin polarizations are correlated with each other. Information encoded in the spin and valley channels of the condensate can be uncovered by measuring the polarization of emitted light. This may provide a new degree of freedom to store and manipulate coherent quantum information in the predicted polariton condensate23,24,25.

Optical Microcavity Architectures

A fundamental challenge to the realization of above-room-temperature BEC is the accurate fabrication of complex photonic nanostructures. Almost all previous experimental studies have focused on 1D dielectric multi-layers which provide strong exciton-photon coupling at the expense of concomitant strong coupling to low-quality-factor leaky modes that render picosecond time-scale polariton lifetimes. 3D PBG materials enable very strong coupling without the occurrence of leaky modes. However, structures made of CdTe21 require a very large number of embedded quantum wells in the active region to provide sufficiently strong coupling to approach room temperature BEC. Here, we propose a simple route to above-room-temperature BEC using TiO2-based photonic crystal architectures. As a reference, we provide comparison with a corresponding TiO2-SiO2 1D multi-layer photonic cavity.

In Fig. 1 we depict the Fabry-Pérot (FP), woodpile photonic crystal (PC) and slanted-pore (SP) PC microcavities, as well as their cavity-mode field intensities along the growth z-direction. There are 3 MoSe2 monolayers in the middle of each microcavity separated by 7 nm TiO2 layers (SiO2 for the FP). Inter-layer Coulomb correlation energy is suppressed due to the large static dielectric constant (~100) of TiO2 at room temperature26. The Coulomb interaction energy for two electrons (holes) separated by ≥7 nm with such high dielectric constant is ≲2 meV, which is much smaller than other energy scales such as the exciton-photon coupling ħΩ = 40 meV. Consequently, we treat excitons in each monolayer as independent. The structures of 3 microcavities are illustrated in Fig. 1a. In the woodpile PC, the height of each rod is h = 0.3 a and the width is w = 0.25 a where is the periodicity in the x-y plane (see Fig. 1a). The SP PC cavity is a SP2 structure of the S/[1, 1] ⊕ [−1, −1](0.5,0) family27 with the periodicity along the z direction of c = 1.4 a where is the in-plane lattice constant. The diameter of each cylindrical air pore is d = 0.69 a. For both types of PC microcavity, the thickness of the slab is 0.05 a. The SiO2-TiO2 FP cavity has two Bragg mirrors with periodicity of and a half-wave SiO2 slab of thickness in the middle, such that the lowest band edge cavity mode is close to the exciton recombination energy of 1.55 eV.

Figure 1
figure 1

Strong coupling in microcavities (a) Schematics of the FP, woodpile PC and SP PC microcavities. Green regions in the middle represent the active layer containing 3 MoSe2 monolayers embedded in TiO2 for the PC cavities and SiO2 for the FP cavity. For the FP cavity, the dark-blue regions are TiO2 while the light-blue regions are SiO2. For woodpile PC cavity the white regions denote the air regions while the blue regions are TiO2. For SP PC cavity the light-blue regions denote slanted air pores while the dark-blue regions are TiO2. Top view of the active layer for the SP structure depicts breaking of x-y symmetry due to adjacent air pores, leading to the lifting of polariton ground state degeneracy. (b) The distributions of the averaged in-plane photonic field intensity, at , along z direction for the woodpile PC and SP PC cavities and that at for the FP cavity.

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Sandwiched PC–quantum-well–PC structures28,29 are fabricated in a layer-by-layer process30,31. The PCs and the quantum-well structure are fabricated independently, the quantum-well structure is then fused onto the lower PC and finally the upper PC is fused onto the quantum-well structure30,31. Corresponding technologies for TiO2 PC growth have become mature in the last decade32,33,34,35,36. In addition, high quality MoSe2 thin films can be grown by molecular beam epitaxy16 and chemical vapor deposition37.

The 3D PBG's in the woodpile and the SP PC's are δω/ωc = 8.7% and 14%, respectively (δω and ωc are the bandwidth and the central frequency of the PBG, respectively). Photonic band structures near the 3D-PBG for the woodpile and SP PC cavities reveal several confined 2D guided bands in the PBG, among which only the lowest one (depicted in Fig. 2) is relevant for polariton BEC21.

Figure 2
figure 2

Photonic and polaritonic band structures.

Spectra of photon, exciton and polariton for (a) woodpile and (b) SP PC microcavities. Light-blue regions are 3D photonic bands. The red curves are the lowest 2D planar guided modes, the dotted curves denote the exciton recombination energy and the blue curves represent the upper and lower polariton branches. The lowest band edges of the 2D guided (cavity) modes are in resonance with the exciton recombination energy, 1.55 eV. This is realized when a = 380 nm for the woodpile and a = 370 nm for the SP PC. Spectra of the lowest 2D planar guided photonic band in 2D wave-vector space for woodpile (c) and SP (d) PC microcavities. The polariton ground state is doubly degenerate for the woodpile but nondegenerate for the SP microcavity.

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Strong Exciton-Photon Coupling

The Hamiltonian of the coupled exciton-photon system can be written as21

where creates an exciton in the l-th monolayer with total angular momentum ħIz = ±ħ along z direction. The energy of the 1s-exciton is where EX0 = 1.55 eV and mX are exciton emission energy and effective mass, respectively. In this optical transition from valence to conduction band, the electron spin is preserved but its orbital angular momentum changes by ħ. creates a photon with wave-vector and frequency in the lowest guided 2D photonic band. The exciton-photon coupling is

where dcv = 3.6 × 10−29 Cm (Supplementary Information) is the absolute value of the interband dipole matrix element, φ(0) is the exciton wavefunction of the 1s-excitonic state at zero electron-hole distance, is the vacuum permittivity, Su.c. = a2 is the area of the unit cell of PC in the x-y plane, zl is the z coordinate of the l-th MoSe2 monolayer plane. where is the polarization vector of the exciton state with Iz = ±1 and is the periodic part of the photonic electric field . Here S is the area of the system, with and . The special electronic band structure (see Fig. 3a) leads to optical selection rules for the lowest excitonic transitions: the σ+ photon excites only spin-up hole and spin-down electron in the K valley, while σ− photon excites only spin-down hole and spin-up electron in the −K valley22,38,39. Here Jz is the total angular momentum along z direction.

Figure 3
figure 3

Optical selection rules and polariton BEC transition temperatures in MoSe2 cavities (a) Electronic band structure and optical selection rules for MoSe2. The spin-degeneracy in the valence band is lifted by spin-orbit interaction. For an excitonic optical transition, the σ+ photon (angular momentum parallel to its wave-vector) is coupled to an exciton in the K valley consisting of a spin-up hole and a spin-down electron. The σ− photon is coupled to an exciton in the −K valley consisting of a spin-down hole and a spin-up electron. Polariton BEC transition temperature Tc vs. detuning Δ and polariton density for (b) SP PC, (c) woodpile PC and (d) FP microcavities. The side length of the square-shaped exciton-trap is D = 5 μm.

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The resulting dispersion of the lower polariton branch is

Here the collective exciton-photon coupling21 is given by . Strong light trapping leads to exciton-photon interaction as large as 40 meV and 38 meV in the SP PC and woodpile PC microcavities, respectively.

Above Room Temperature Polariton BEC

The polariton BEC transition temperature, Tc, is calculated by the criterion40N0/Ntot = 10% where N0 and Ntot are the population on the ground state and the total polariton number respectively40. We consider a quantum box trap for polaritons with box area defined by the finite area of embedded MoSe2 thin films. As shown below, the trap size, characterized by the side length D of the square box, does not alter the transition temperature considerably over a range from a few microns to one centimeter. The quantum box confinement leads to quantization of wave-vector and energy. At thermal equilibrium, , , where E0 is the ground state energy, j runs over all discrete energy levels and μ is the chemical potential. As shown previously21, Tc is essentially defined by the polariton dispersion depth, , where Δ = EX0 − ħω0 is the exciton-photon detuning and is the exciton-photon coupling at photonic band edge. As kBT approaches Vlp, the population on exciton-like states with very large effective mass becomes significant21. The pure exciton BEC transition temperature is calculated to be less than 4 K for density less than (10aB)−2 and trap size D of 1 micron in a MoSe2 monolayer. On the other hand for positive detuning Δ = 30 meV and ħΩ = 40 meV, the dispersion depth is 58 meV. This can support polariton BEC up to .

Figs. 3(b)–(d) give the polariton BEC transition temperatures for different detuning Δ and polariton densities for the SP PC, woodpile PC and FP cavities. We assume that the exciton gas is initially created by incoherent pumping that (after thermal relaxation) equally populates all degenerate dispersion minima (-points) in the photonic band structure. As seen in Fig. 2, the woodpile PC has a doubly degenerate polariton minimum whereas the polariton ground state is nondegenerate for the SP PC. The trap size is D = 5 μm. The Tc is enhanced with increasing detuning as the polariton dispersion depth increases. The highest Tc is reached at the largest detuning Δ = 40 meV which is 590 K, 450 K and 430 K for the SP PC, woodpile PC and FP cavities, respectively. The BEC transition temperature Tc is highest in the SP PC cavity since it contains a nondegenerate polariton ground state. This is provided by the placement of the active layer at a plane that breaks the x-y symmetry of the criss-crossing pores above and below (see Fig. 1a). In contrast, there are two degenerate polariton minima in the woodpile that reduces effective polariton density for BEC by a factor of two. In the case of the FP cavity, the polariton density available for BEC is divided by the two degenerate optical polarization states at . We do not consider larger detuning because a too large Δ will reduce the exciton fraction of polariton as well as the polariton-phonon and polariton-polartion scattering, which leads to much longer equilibration time.

The polariton BEC transition temperatures at Δ = 30 meV for different trap sizes D and polariton densities for each cavity are shown in Fig. 4. The trap size dependence is prominent for low polariton densities, but very weak for high polariton densities21. At low densities, and polariton equilibrium is governed by the low-lying excited states of very small effective mass particles in the box trap that are very sensitive to D. At high polariton densities, kBTc≲Vlp and a significant fraction of polaritons acquire the bare exciton mass for which the level spacing is less sensitive to D. Eventually Tc tends to zero for very large trap size D (when the low-lying excited states are extremely close to the ground state) according to the Mermin-Wagner theorem41. In all these calculations the largest polariton density, 5 × 104 μm−2, corresponds to an exciton density in each MoSe2 monolayer less than (11aB)−2, well below the exciton saturation density of (5aB)−242.

Figure 4
figure 4

High-temperature polariton BEC Polariton BEC transition temperature Tc vs. trap size and polariton density for (a) SP PC, (b) woodpile PC, (c) FP microcavities.

Detuning is Δ = 40 meV, corresponding to λ = 820 nm for the FP cavity, a = 390 nm for woodpile PC cavity and a = 380 nm for SP PC cavity.

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Fabrication induced disorder and exciton line broadening tend to degrade high-temperature polariton BEC. In a PBG material, light in the cavity mode is well-protected from scattering into unwanted modes. Consequently, the role of disorder is primarily to shift the cavity resonance frequency. For photonic (dielectric) disorder, our calculation indicates that in the woodpile and SP PC cavities, photonic structure deviations less than 6 nm in the active slab layer, less than 12 nm in the logs close to the active slab layer, or less than 20 nm in the logs two periods away, do not affect the band edge of the lowest planar guided mode by more than 2%. This fluctuation is equivalent to alteration of the detuning by ±30 meV, which can reduce the Tc to 360 K (480 K) if the detuning decreases to 10 meV for the woodpile (SP) PC cavity. Polariton BEC is also found to be robust to exciton homogeneous and inhomogeneous broadening although the Rabi-splitting and polariton dispersion depth are slightly reduced (e.g., Rabi splitting in SP cavity is reduced from 80 meV to 62 meV) (see Supplementary Information).

In contrast there is no 3D PBG for the FP cavity and light-trapping is not robust to disorder. Excitons in the FP with wave-vector q ≳ 0.3q0 with q0 = EX0/(ħc) are strongly coupled to leaky (air) modes, leading to rapid radiative decay43. This decay into leaky modes is not alleviated by increasing the quality of the FP cavity mode and is exacerbated by imperfect fabrication accuracy. In real FP cavities, dielectric disorder scatters polaritons with wave-vector q~ 0 into leaky modes with the same energy. Such scattering significantly limits the quality factor of the (nonleaky) cavity mode (~103). The overall polariton lifetime due to radiative decay is very short (≲) in SiO2/TiO2 multilayers44 and in other8 FP cavities. The resulting sub-picosecond polariton lifetime (see also calculation in the Supplementary Information) is still comparable with the thermalization time, although calculations have shown that phonon scattering is very efficient (about 0.1 ps scattering time) in MoSe2 at room temperature45. In contrast, PC cavities protected by a 3D PBG enable light-trapping in all directions, leading to polaritons with lifetime limited only by nonradiative decay (see Supplementary Information). This nonradiative decay is further suppressed by our detuning (Δ > 0) that renders our polariton more photon-like than exciton-like. The quantum coherence of polaritons in the cavity can be probed and manipulated using light propagating through engineered waveguide channels in the 3D PBG cladding31,46.

In the absence of any microcavity, optically excited valley polarization of excitons can last only about 9 ps in MoSe2 at room temperature47. In high-quality PC microcavities strong light trapping over all directions can enhance polariton lifetime beyond 130 ps (the exciton lifetime in MoSe2 monolayer without any cavity at room temperature)47. When the polaritons form a BEC, optical, valley and spin polarization coherences maintained on such long time-scales may be useful for quantum information processing based on manipulation of the spin and valley degrees of freedom24,25.

Conclusions

We have theoretically demonstrated photonic crystal architectures based on TiO2 with very strong light-matter interaction and long polariton lifetime for above-room-temperature, equilibrium exciton-polariton BEC. The 3D photonic band gap robustly supports strongly confined guided modes in a slab cavity containing three separated excitonic MoSe2 monolayers. Unlike 1D periodic Fabry-Pérot structures that suffer from substantial radiative decay through directions other than the stacking direction, the 3D PBG structures suppress radiative decay in all directions enabling equilibration and long-time polariton coherence. An important feature of the slanted pore PC architecture is that it provides a non-degenerate lower polariton dispersion minimum, without recourse to a symmetry-breaking external field23. This enables a remarkable equilibrium high-temperature Bose-Einstein condensate, above 500 K, with only three quantum well layers. Both TiO2 based photonic crystals and MoSe2 thin films have been fabricated with mature technologies providing a very accessible route to high-temperature BEC. This BEC is robust to disorder, fabrication imperfections, exciton homogeneous and inhomogeneous line-broadening. The unique coupling among the optical, valley and spin degrees of freedom in MoSe2 provides a condensate with novel internal symmetries that can be manipulated by external fields. Polariton BEC with long-lived optical, valley and spin coherence persisting above room-temperature offers new opportunities for macroscopic quantum physics.

Methods

The photonic band structures and electric field distributions for PC microcavities are calculated using the plane wave expansion method, while those for the FP microcavity are calculated using the transfer matrix method. The refractive indices of TiO2 and SiO2 are 2.7 and 1.5, respectively48. In MoSe2 the effective mass for conduction band electron and valence band hole are, me = 0.7 m0 and mh = 0.55 m0, respectively49. The effective mass of exciton is mX = me + mh. The exciton Bohr radius aB and interband band dipole matrix element dcv are obtained from existing studies on electronic band structure in MoSe2 and excitonic optical absorption16,22 (see Supplementary Information). The calculation of Tc involves summation over numerous polariton states which is broken into two parts: the summation over the first 1600 states and the integration over other higher states. Polariton energy in a square (hard-wall) quantum box is calculated according to wave-vector quantization.