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Tuning moiré excitons in Janus heterobilayers for high-temperature Bose-Einstein condensation

Using first-principles calculations, we predict that moiré excitons in twisted Janus heterobilayers could realize tunable and high-temperature Bose-Einstein condensation (BEC). The electric dipole in the Janus heterobilayers leads to charge-transfer interlayer and intralayer moiré excitons with exceptionally long lifetimes, in the absence of spacer layers. The electric dipole is also expected to enhance exciton-exciton repulsions at high exciton densities and can modulate moiré potentials that trap excitons for their condensation. The key parameters for exciton condensation, including exciton Bohr radius, binding energy, effective mass, and critical Mott density, are examined as a function of the twist angle. Last, exciton phase diagrams for the Janus heterobilayers are constructed from which one can estimate the BEC (>100 K) and superfluid (~30 K) transition temperatures. In addition to indirect interlayer excitons, we find that direct intralayer excitons can also condense at high temperatures, consistent with experiments.


INTRODUCTION
Two-dimensional (2D) moiré superlattices formed in van der Waals (vdW) heterostructures with a small angular or lattice mismatch have emerged as a fascinating and versatile platform to explore and engineer quantum states of matter. Driven by the interplay between flat energy bands and strong electron-electron interactions, fascinating quantum phenomena, such as unconventional superconductivity (1), charge density waves (2), and correlated insulating states [e.g., striped phases (3), Mott states (4,5), and Wigner crystals (6)] have been observed in the moiré superlattices. Recently, moiré superlattices in semiconducting transition metal dichalcogenides (TMDs) have garnered tremendous interest because they could host localized, long-lived, and valley-polarized moiré excitons (7)(8)(9)(10), which are envisioned as single-photon emitters (11) in quantum information applications and as an essential element in excitonic devices (12).
Bose-Einstein condensation (BEC) is an exotic quantum state of matter where bosons, such as atoms, in a macroscopic system spontaneously condense into a single quantum state, leading to marked physical consequences, such as superfluidity (13). Since the first achievement of atomic BEC at nanokelvin temperatures (14), the quest to realize even higher BEC temperatures has fascinated physicists. Compared to atoms, excitons have much smaller masses and thus are expected to condense at much higher temperatures. Fogler et al. (15) are the first to predict that indirect or interlayer excitons in TMD bilayers could form BEC at ~100 K, and experimental evidence for high-temperature (~100 K) exciton condensation in MoSe 2 /WSe 2 heterobilayers has recently been reported (16). In both cases, the BEC is formed by interlayer or indirect excitons whose electrons and holes are separated by hexagonal boron nitride spacer layers. The spatial separation of the electrons and the holes is deemed critical to achieve long-lived excitons necessary for BEC. Recent first-principles calculations (17) suggested that high-temperature BEC could also be realized in organic-TMD heterostructures in the absence of a spacer layer. This raises an intriguing question of whether moiré excitons in coupled TMD bilayers (i.e., without a spacer layer) could also condense at high temperatures and whether direct or intralayer moiré excitons with electrons and holes at the same TMD layer could realize high-temperature BEC. Note that localized moiré excitons have distinct optical selection rules and stronger correlations than the delocalized excitons examined in the previous works (15,16). In particular, moiré excitons can be readily tuned by the twist angle of the heterostructures. How the condensation of localized moiré excitons depends on the moiré superlattice or the twist angle is completely unknown. These are fascinating questions and, if answered, could have substantial impacts on the burgeoning field of excitonic BEC.
The excitonic BEC transition temperature T c depends on several key factors: (i) the excitons should have longer lifetimes (>10 to 100 ns) than their thermalization times (~ps) (18); (ii) the exciton binding energy should be large enough, typically >10k B T c (k B is Boltzmann constant) (15,16); (iii) the presence of repulsive interactions between the excitons at high densities, which can stabilize the exciton condensate against the formation of competing exciton complexes; and (iv) the presence of potential traps that confine the exciton condensate. The last factor is of particular importance; breakthroughs in BEC were made possible by the realization of magneto-optical traps for atomic BEC (14,19) and in-plane potential traps for excitonic BEC in coupled semiconductor quantum wells (20,21).
Here, by means of first-principles calculations, we predict that moiré excitons in Janus TMD heterobilayers could realize tunable and high-temperature BEC. A Janus TMD monolayer consists of two distinct chalcogen atoms above and below the metal atom, producing an out-of-plane electric dipole. The electric dipole can yield charge-transfer intralayer or interlayer excitons, leading to exceptionally long exciton lifetimes (microseconds and milliseconds); the electric dipole can also enhance the repulsive interactions among the excitons. Both the intralayer and interlayer moiré excitons are predicted to have large binding energies (~0.40 eV), and they are trapped by deep moiré potentials, which are tunable by twist angle, strain, pressure, electric field, etc. (11,(22)(23)(24) This is in sharp contrast to the coupled semiconductor quantum wells whose trap potentials are formed by random fluctuations of the quantum well thickness and the alloy disorder (20,25), which are difficult to control. The phase diagrams of the moiré excitons are constructed from which one can estimate the BEC (>100 K) and superfluid (~30 K) temperatures. We find that in addition to indirect interlayer excitons, direct intralayer excitons can also condense at high temperatures (123 to 193 K), in support of a recent experiment (26). Fig. 1A, the top (or bottom) Te atoms in MoTe 2 monolayer are replaced by Se atoms to form a Janus monolayer MoSeTe (or MoTeSe). The Janus monolayers are then stacked with a WSe 2 monolayer to form vdW heterobilayers. Here, we focus on three twisted heterobilayers, including WSe 2 /MoTe 2 , WSe 2 /MoTeSe, and WSe 2 /MoSeTe, as shown in Fig. 2A. We then examine how excitonic properties may be tuned by the chemical modifications (MoTe 2 ➔ MoTeSe ➔ MoSeTe). Compared to other means of tuning [e.g., electric field, pressure, strain, and defects (11,(22)(23)(24)], chemical tuning could have drastic effects yet received less attention.

Electric dipole and charge-transfer excitons in Janus monolayers
Owing to the asymmetry in the out-of-plane (or z) direction, Janus monolayer MoSeTe develops an electric dipole P as indicated by the purple arrow in Fig. 1A. The effect of the dipole is illustrated in Fig. 1B where the average electrostatic potentials along the z direction are plotted for both MoTe 2 and MoSeTe monolayers. The dipole yields an asymmetric electrostatic potential, manifested by the potential energy difference E = 0.7 eV between the top and bottom vacuum of the Janus monolayer, as shown in Fig. 1C. In contrast, no such energy difference (E = 0) exists in the MoTe 2 monolayer.
The electric dipole has important consequences on the excitonic properties. In Fig. 1D, we present the charge density of the lowest energy excitons in both monolayers. In MoTe 2 , both the electron and the hole reside at the Mo layer because the conduction/valence edge states are contributed by the Mo d-orbitals. In MoSeTe, on the other hand, the dipole P drives the electron and the hole apart and the hole extends to the Se sublayer, forming a charge-transfer exciton. Thus, the wave function overlap between the electron and the hole is much smaller in MoSeTe, resulting in a dark exciton whose optical dipole is four orders of magnitude smaller than the corresponding bright exciton in MoTe 2 . Therefore, the lowest exciton in Janus monolayer MoSeTe is expected to have a long lifetime, which is beneficial for BEC. Note that recent first-principles calculations predicted that charge-transfer interlayer excitons in organic/ TMD heterostructures could condense at high temperatures (17). In this paper (and consistent with convention), we regard the excitons in the Janus monolayers as intralayer excitons despite their charge-transfer character. In contrast to conventional charge-transfer excitons that are formed between two distinct materials (i.e., donor and acceptor), the charge-transfer excitons in the Janus monolayers are in the same material.

Moiré potentials and flat bands in twisted Janus heterobilayers
We next examine the twisted heterobilayers. In Fig. 2 (A and B), we display the atomic structures and the band alignments of the three heterobilayers with a twist angle  = 30°. The first two heterobilayers exhibit the type I band alignment, while the third one has the type II band alignment. Note that the last two heterobilayers have the identical chemical composition but with the bottom Janus monolayer flipped upside down. Thus, the electric dipole in the Janus monolayer is reversed between the two (fig. S1), switching the band alignment from type I in WSe 2 /MoTeSe to type II in WSe 2 /MoSeTe.
To facilitate exciton condensation, it is highly desirable to create in-plane potentials to trap excitons; this can be achieved by moiré potentials in twisted heterobilayers. Moreover, one can modulate the magnitude of the moiré potential by changing the twist angle. In the following, we consider twisted heterobilayers with a small twist angle  = 0.22°, which is expected to result in deeper moiré potentials. In Fig. 2G, we show the relaxed moiré supercell for WSe 2 / MoTeSe heterobilayer ( = 0.22°) with 1356 atoms. On the basis of the relaxed structure, we can estimate the moiré potential as the bandgap variation E g = E g − 〈E g 〉 of the supercell; 〈E g 〉 is the average value of the bandgap E g (the computation detail for the moiré potential can be found in the Supplementary Materials). As shown in Fig. 2 (C and E), a much deeper moiré potential (0.61 eV) is formed in WSe 2 /MoTeSe as compared to that in WSe 2 /MoTe 2 (0.02 eV). The deep and tunable moiré potentials in Janus heterobilayers could provide more robust and controllable traps for exciton condensation than those in the coupled semiconductor quantum wells.
To shed light on the contrasting moiré potentials between WSe 2 / MoTeSe and WSe 2 /MoTe 2 , we determine the spatial distribution of E, which represents the dipole energy distribution in WSe 2 /MoTeSe superlattice. As shown in Fig. 2D, there is a strong resemblance between the distribution of E and the distribution of E g in WSe 2 /MoTeSe, suggesting a connection between the dipole and the moiré potential. The connection is also apparent in WSe 2 /MoTe 2 , where the negligible moiré potential is consistent with the absence of the dipole (E = 0). Hence, the deep moiré potential in WSe 2 /MoTeSe can be attributed to the dipole in the Janus monolayer. Another important consequence of the deep moiré potential is the formation of an isolated flat valence band (2.7 meV) in WSe 2 /MoTeSe, as shown in Fig. 2F. The flat band implies small kinetic energy of the hole (and exciton), which could benefit the trapping and condensation of the excitons. A less flat valence band is also observed in twisted WSe 2 / MoTe 2 and WSe 2 /MoSeTe ( fig. S3). The moiré bands in WSe 2 /MoTe 2 heterobilayers have recently been shown to yield finite Chern numbers due to the lattice relaxation (27).

Tuning moiré exciton lifetimes
Exciton lifetime is one of the most important factors for BEC. In the following, we first explore chemical tuning of exciton lifetime. To this end, we compare the lifetimes of the lowest moiré excitons in WSe 2 /MoTe 2 , WSe 2 /MoTeSe, and WSe 2 /MoSeTe heterobilayers with the same twist angle ( = 30°). As shown in Fig. 3A, because WSe 2 / MoTe 2 and WSe 2 /MoTeSe have a type I band alignment, their lowest excitations are intralayer or direct excitons with the electrons and holes residing at the same monolayer (MoTe 2 or MoTeSe); no electron or hole is at WSe 2 layer because of its wider bandgap (Fig. 2B). The intralayer exciton in WSe 2 /MoTeSe has the charge-transfer character due to the electric dipole in MoTeSe as mentioned before. WSe 2 /MoSeTe, on the other hand, has a type II band alignment, and its lowest energy exciton has an interlayer character with the electron and hole separated at different monolayers. The spatial separation of the electron and the hole renders the interlayer exciton dark, whose optical dipole is three (or two) orders of magnitude smaller than that of the intralayer exciton in WSe 2 /MoTe 2 (or WSe 2 / MoTeSe). The charge-transfer character yields a smaller optical dipole for the intralayer exciton in WSe 2 /MoTeSe as compared to the intralayer exciton in WSe 2 /MoTe 2 . The radiative lifetimes of the excitons exhibit the opposite trends as the optical dipoles. The intralayer exciton in WSe 2 /MoTe 2 has the shortest lifetime of  0 = 145 fs, while the intralayer exciton in WSe 2 /MoTeSe has a much longer lifetime of  0 = 23.7 ns. The interlayer exciton in WSe 2 /MoSeTe is predicted to have an exceptionally long lifetime  0 ~ 2 s. Previous first-principles GW-Bethe-Salpeter equation calculations have also reported similarly long lifetimes for dark excitons in TMD monolayers and heterobilayers, ranging from a few microseconds to milliseconds (28). It is generally believed that BEC should be possible if the exciton lifetime is longer than 10 to 100 ns (18); thus, both the chargetransfer intralayer exciton in WSe 2 /MoTeSe and the interlayer exciton in WSe 2 /MoSeTe could realize BEC, and we will estimate their transition temperatures in the next section.
Next, we examine exciton lifetime as a function of the twist angle. Specifically, we focus on the lowest intralayer exciton in WSe 2 /MoTe 2 by considering six twist angles ( = 0.22°, 1.93°, 10.58°, 30°, 48.07°, and 58.07°). Note that the moiré supercell with  = 1.93° (or 10.58°) has the same lattice parameters as that with  = 58.07° (or 48.07°). In Fig. 3B, we present the optical dipole and the radiative lifetime of the exciton as a function of the twist angle. The optical dipole increases with the twist angle until  = 30° and then decreases. The radiative lifetime follows the opposite trend. Thus, to increase the exciton lifetime in WSe 2 /MoTe 2 , one should tune the twist angle to be near either 0° or 60°. For example, for  = 0.22°, the lifetime of the intralayer exciton in WSe 2 /MoTe 2 could reach ~100 ns (this amounts to six orders of magnitude increase in the lifetime from  = 30°), rendering it possible for BEC. In Fig. 3C, we display the charge density of the intralayer exciton in WSe 2 /MoTe 2 with  = 0.22° and  = 30°. The electron and hole are much more separated in the former ( = 0.22°) than in the latter ( = 30°); hence, the exciton has a much longer lifetime in the former. The exciton lifetimes in WSe 2 / MoTeSe and WSe 2 /MoSeTe follow the similar twist angle dependence. In particular, for  = 0.22°, we calculate the lifetime of the intralayer exciton in WSe 2 /MoTeSe as ~1.87 s, and the lifetime of the interlayer exciton in WSe 2 /MoSeTe can reach ~3 ms. These are exceptionally long lifetimes, making these moiré excitons highly promising for BEC. We emphasize that these exceptionally long lifetimes are realized in the absence of a spacer layer in the heterostructures.

BEC temperature of moiré excitons
In this section, we examine the exciton phase diagram and estimate BEC and superfluid transition temperatures in the three heterostructures, following an approach proposed by Fogler et al. (15). As shown in the phase diagram (Fig. 4A), four competing phases, including classical exciton gas, classical electron-hole gas, degenerate exciton Bose gas, and degenerate electron-hole Fermi gas are possible as a function of temperature (T) and exciton density (n). The phases are divided by two lines: a blue vertical line that separates the exciton gases from the electron-hole gases at the critical Mott density n c and a red sloped line that separates the classical gases from the degenerate Fermi or Bose gases. BEC takes place below the critical Mott density and under the degenerate temperature (red line), which depends on the exciton density n. Hence, the BEC transition temperature (T c ) is where the two lines cross.
The critical Mott density n c can be determined by the exciton Bohr radius R via n c ≈ 1 _  R 2 (15,17,29,30). Here, we estimate the exciton Bohr radius R by the root mean square (RMS) radius of the exciton wave function defined as (17) Here, (r) is the square modulus for the electronic part of the excitonic wave function; r 0 denotes the center of the electron distribution with the hole fixed at the corner of the moiré supercell. The electronic part of the excitonic wave function for the lowest intralayer exciton in WSe 2 /MoTeSe ( = 0.22°) is shown in Fig. 4B, where the hole is indicated by the red cross. The Bohr radius R is estimated as 1.61 nm, yielding a critical Mott density n c = 1.23 × 10 13 cm −2 , which sets the high-density limit for BEC in WSe 2 /MoTeSe ( = 0.22°). As a comparison, the exciton density in MoSe 2 /WSe 2 heterobilayers where the experimental evidence for exciton condensation was observed (16) is in the order of 10 12 cm −2 .
The degeneracy temperature (T d ) at which the classical gases cross over to the degenerate Bose or Fermi gases is given by k B T d = 2 ℏ 2 _ M n, as shown by the red line. Here, M is the effective mass of the exciton and can be estimated as the sum of the electron and hole effective masses. In fig. S4, we display the relative contributions of various electron-hole transitions for the lowest intralayer exciton in WSe 2 / MoTeSe ( = 0.22°). We find that the lowest exciton comprises primarily the transition from the valence band maximum (VBM) to the conduction band minimum (CBM); thus, M is the sum of the effective masses of the electron in CBM and the hole in VBM. Last, the BEC transition temperature (T c ) is the degeneracy temperature at the critical Mott density n c . For the lowest intralayer exciton in WSe 2 /MoTeSe ( = 0.22°), T c is estimated as 128 K. Therefore, we predict that the intralayer or direct excitons in twisted WSe 2 /MoTeSe heterobilayers could condense at high temperatures. Our prediction is consistent with a recent experiment that has shown evidence of high-temperature (~150 K) condensation of intralayer excitons in monolayer MoS 2 (26). The intralayer excitons in monolayer MoS 2 are delocalized with much shorter lifetimes, as compared to the localized moiré excitons in WSe 2 /MoTeSe heterobilayers. One would thus expect that the intralayer moiré excitons could condense at a higher temperature. The high-temperature condensation in the experiment was attributed to strong repulsions among the excitons in MoS 2 (26). Thus, the built-in dipoles in the Janus bilayers could also play an important role in the condensation of moiré excitons.
To illustrate the dependence of BEC on the twist angle, we calculate the Bohr radius (R), exciton effective mass (M), exciton binding energy, and degeneracy temperature (T d ) for the intralayer excitons in WSe 2 /MoTeSe as a function of the twist angle . As shown in Fig. 4C, the exciton Bohr radius is strongly correlated to its binding energy. As the twist angle approaches 0° or 60°, the moiré potential becomes deeper, and the exciton becomes more localized with a larger binding energy and a smaller Bohr radius. Similarly, a deeper moiré potential results in flatter (or less dispersive) bands and, thus, greater effective masses for the exciton, as shown in table S1. In Fig. 4D, we display the dependence of n c and T c on the twist angle, and it is found that n c increases as  approaches 0° or 60°, and T c is tunable from 123 to 193 K.
We have also carried out similar calculations for the two other heterobilayers (WSe 2 /MoTe 2 and WSe 2 /MoSeTe with  = 0.22°), and the results are summarized in Fig. 4E. The lowest intralayer exciton in WSe 2 /MoTe 2 and the lowest interlayer exciton in WSe 2 /MoSeTe are considered. The BEC temperature for the intralayer exciton in WSe 2 /MoTe 2 is estimated as 130 K, and the interlayer exciton in WSe 2 / MoSeTe is estimated as 115 K. Last, the 2D excitons can also transform into a superfluid state via the Berezinskii-Kosterlitz-Thouless (BKT) transition below a critical temperature (15) given by k B T BKT =  ℏ 2 _ 2M n . In Fig. 4 (E and F), we summarize T BKT dependence on the exciton density for the three twisted heterobilayers ( = 0.22°). The predicted superfluid temperature is ~30 K for the three heterostructures.
In this work, we focused on the radiative recombination of excitons, but phonon-assisted nonradiative recombination should also be considered. According to previous theoretical calculations and experiments, the nonradiative lifetimes of excitons in TMDs are in the order of nanoseconds (31)(32)(33), which are sufficiently long for BEC (18). Furthermore, the nonradiative recombination rates depend on nonadiabatic coupling (NAC) between relevant conduction and valence bands (34), and NAC is determined primarily by the wave function overlap between these bands. As shown in Fig. 3C, the electron and the hole of the lowest exciton in twisted moiré superlattice ( = 0.22°) are well separated with little spatial overlap; thus, NAC is expected to be exceedingly small, leading to long nonradiative lifetimes in the moiré bilayers. Similarly, Auger-like processes are not believed to be important because only the lowest energy excitons are likely to be involved in BEC. On the other hand, the electric dipole in the Janus heterolayers is expected to modify the Coulomb interaction between the electron and the hole in the excitons, which, in turn, could affect their condensation. The present time-dependent density functional theory with optimally tuned, screened, and range-separated hybrid exchange-correlation functionals is capable of capturing the Coulomb interaction (see Methods). Thus, the exciton lifetime, binding energy, Bohr radius, and effective mass in the presence of the electric dipole can be correctly described in our first-principles calculations. However, commonly used theoretical models for exciton condensation, such as that of Fogler et al. (15, 17, 29) do not consider exciton trapping by either moiré or other potentials nor is exciton-exciton repulsion included in these models. To remedy these deficiencies, we further adopt a two-fluid mean-field model (35,36) to refine our predictions. In this model, the condensate wave function is described by Gross-Pitaevskii equation in which both a trapping potential and a repulsive interaction between excitons are considered. The kinetic energy of the moiré excitons can be ignored (Thomas-Fermi approximation) thanks to the flat exciton energy bands (see Methods for details). On the basis of this model, we can determine the BEC transition temperatures T c as a function of the moiré exciton density n. For the lowest intralayer exciton in MoTeSe/WSe 2 bilayer ( = 0.22°), T c is estimated as 144 K for n c = 1.23 × 10 13 cm −2 ; this is slightly higher than the temperature (128 K) obtained from the model of Fogler et al. Additional results from the two-fluid model can be found in the Supplementary Materials (figs. S6 and S7).

DISCUSSION
Because of their light masses and high binding energies, excitons in 2D semiconductors are expected to condense at higher temperatures. In this work, by means of first-principles calculations, we predict that both direct and indirect moiré excitons in twisted Janus heterobilayers could realize tunable high-temperature BEC and superfluid transitions. Two unique features that render the Janus heterobilayers particularly attractive are the formation of deep moiré potentials and the presence of out-of-plane electric dipoles. The former can provide potential traps necessary for exciton condensation. The availability of deep tunable trap potentials is the most distinctive advantage of the Janus heterobilayers for exciton condensation. The electric dipoles in the Janus bilayers, on the other hand, can enhance excitonexciton repulsions, which are critical for BEC when the exciton density is high. The out-of-plane electric dipoles are particularly important to intralayer excitons whose in-plane interactions are normally weak. As suggested in the recent experiment (26), strong exciton-exciton repulsions are the driving force for high-temperature condensation of intralayer excitons in monolayer MoS 2 . The electric dipoles also lead to charge-transfer moiré excitons with exceptionally long lifetimes (up to milliseconds) that are tunable by the twist angle. The electric dipoles are found to be responsible for the deep moiré potentials in the Janus heterobilayers, which, in turn, yield isolated flat bands. Both interlayer and intralayer moiré excitons are predicted to have large binding energies (~0.40 eV), which is necessary for high-temperature BEC (15,16). The key parameters for exciton condensation, including exciton Bohr radius, binding energy, effective mass, and critical Mott density are examined as a function of the twist angle in WSe 2 /MoTeSe heterobilayer for the intralayer exciton. Last, the exciton phase diagrams are constructed from which one can estimate the BEC (>100 K) and superfluid (~30 K) transition temperatures. We find that the direct intralayer excitons in WSe 2 /MoTeSe heterobilayers can condense at high temperatures (123 to 193 K), in support of the recent experiment (26). Our work lays the foundation for future experimental exploration of moiré excitons for tunable and high-temperature BEC.

First-principles ground state calculations
The ground state properties, including single-particle band structures and optimized atomic structures of various heterobilayers examined in this work were determined using Vienna Ab initio Simulation Package (37,38). The Perdew-Burke-Ernzerhof exchange-correlation functional (39) along with projector-augmented wave potentials (40) was used in these calculations. The vdW interaction was considered via vdW-D2 functional (41). For the band alignment diagrams shown in Fig. 2B, spin-orbit coupling was included in the calculations. To determine the band structures of the moiré superlattices with  = 0.22°, five special k-points were sampled along each of the high symmetry directions in the Brillouin zone. The atomic geometry in each moiré superlattice was fully optimized until the residual force on each atom was less than 0.01 eV Å −1 . A 20 Å vacuum layer was included in the supercells to separate the periodic images.

First-principles excited state calculations
To determine the energies and the many-body wave functions of the excitons in the heterobilayers, we used a recently developed first-principles approach based on linear-response time-dependent density functional theory (LR-TDDFT) (42,43) with optimally tuned, screened, and range-separated hybrid (OT-SRSH) exchange-correlation functionals (22,24,(44)(45)(46)(47). Unlike the traditional LR-TDDFT method with local and semilocal functionals, the TDDFT method using the OT-SRSH functionals can capture the correct long-range electronelectron and electron-hole interactions in solids by choosing appropriate parameters (48)(49)(50)(51)(52). The LR-TDDFT methods with hybrid exchange-correlation functionals have been used extensively to study optical and excitonic properties in solids; see, for example, (53)(54)(55) and references therein, in addition to (48)(49)(50)(51)(52). In this work, we solve the following non-Hermitian eigenvalue equations of Casida (56) to determine the excitation energies  I where the pseudo-eigenvalue  I is the Ith exciton energy level and X I,ij and Y I,ij are eigenvectors, which are used to determine the exciton wave functions. The matrix elements of A and B based on Kohn-Sham (KS) orbitals (ij) are given by Here, K ij,lk is the coupling matrix whose indices i and k indicate the occupied KS orbitals, and j and l represent the virtual KS orbitals (57). Following the assignment ansatz of Casida (56), the many-body wave function of the excited state I can be written as where z I,ij = ( X I,ij + Y I,ij ) / √ _  I . a i (a † i ) is the annihilation (creation) operator acting on the ith KS orbital with spin  and  0 is the ground state many-body wave function taken as the single Slater determinant of the occupied KS orbitals.
In the TDDFT-OT-SRSH method, there are three parameters-, , and -needed to be specified.  controls the short-range exact exchange, and  is chosen to satisfy the requirement  +  = 1/, where  is the scalar dielectric constant. The optimal set of the parameters ( = 0.025,  = 0.975, and  = 0.03) was determined by fitting the quasiparticle gap of nontwist WSe 2 /MoTe 2 bilayer from the DFT-OT-SRSH calculations to that from the GW calculations. This is a typical procedure used in TDDFT-OT-SRSH calculations (48)(49)(50)(51)(52). With this set of parameters, we obtained an optical gap of 1.13 and 1.09 eV for WSe 2 /MoTe 2 heterobilayer and MoTe 2 monolayer, respectively, which agrees well to experimental results (49,(58)(59)(60). The exciton binding energy E b defined as the difference between the quasiparticle gap and the optical gap is calculated as 0.49 and 0.51 eV, respectively, which is also in reasonable agreement with previous results (0.58± 0.08 eV) (60) for the intralayer exciton in MoTe 2 monolayer. To further validate our method, we compute the exciton charge density and the electronic part of the excitonic wave function for the lowest intralayer moiré exciton in the angle-aligned WS 2 /WSe 2 heterobilayer (the moiré superlattice is formed owing to the lattice mismatch of the layers). The results shown in fig. S5 agree very well to those obtained from GW-BSE calculations (61). Additional validations of the method can be found in previous publications on various 2D materials and their heterostructures (22,24,44,46). Last, as very large supercells are used in the TDDFT calculations for the moiré excitons, only the  point of the Brillouin zone is sampled. Thus, the exciton momentum is zero.

The optical dipole moment calculations
To determine the optical dipole moment for each moiré exciton, we first examine the electron-hole transitions involved in the exciton. As mentioned above, the many-body wave function of the exciton  I is expressed as a linear combination of electron-hole transitions, and Z I,ij represents the corresponding electron-hole transition amplitude. The optical dipole moment ( I ) of the exciton I is calculated as where p ij is the transition dipole moment between the KS occupied state φ i and unoccupied state φ j . e is the electron charge, and r is the position operator of the electron.

The exciton radiative lifetime calculations
As only the  point is used in the calculations of excitonic properties with large moiré supercells, the radiative recombination rate of an exciton is given by (62) Here, A uc is the area of the in-plane unit cell, e is the electron charge,  I is the exciton energy, and  I is the optical dipole moment. ϵ 0 is the vacuum permittivity, and c is the speed of light. The radiative lifetime of the exciton is given as The two-fluid mean-field BEC model In this model, the BEC condensate wave function (r) is described by the Gross-Pitaevskii equation below − ℏ 2 ─ 2m ∇ 2 (r ) + V(r ) (r ) + 2g n 1 (r ) (r ) + g  3 (r ) = (r) (5) where g represents the interaction strength of exciton-exciton repulsion and V(r) represents the moiré potential. n 1 (r) is the distribution function of noncondensed excitons, and  is the chemical potential determined by the equilibrium of the condensed and noncondensed excitons. In the model, the periodic moiré potential takes the form V(r ) = 2V ∑ j=1,3,5 cos( b j • r − φ) (10), where b j are the moiré reciprocal lattice vectors and φ is a phase that specifies the potential extrema. Near the potential center, the potential is assumed to be harmonic, i.e., V(r ) = 1 _ 2 m  2 r 2 , where ℏ denotes the average spacing of the quantized energy spectrum of the moiré excitons (10). For the intralayer moiré exciton in MoTeSe/WSe 2 Janus bilayer ( = 0.22°), we obtain ℏ = 10 meV, which is similar to the value reported in (10). Here, g represents strength of the exciton-exciton repulson, which can be approximated as E repul = ne 2 d _  = gn (12); thus, g = e 2 d _  , where n is the exciton density, d is the size of the electric-hole dipole, and  is the dielectric constant (~5.2). d can be approximated by the distance between the electron and the hole (~3 Å) in the Janus monolayer. Hence, we arrive at g = 1.09 a. u. = 3.1 meV • a 2 (a = 5.14 nm).
In Eq. 5, the kinetic energy of the moiré excitons can be ignored (Thomas-Fermi approximation) owing to the flat exciton bands. The Thomas-Fermi approximation is also used by others in the two-fluid model (35,36). Here, we define N 0 = ∫  2 (r)d 2 r as the number of condensed excitons, and N is the total number of excitons, including both condensed (N 0 ) and noncondensed excitons. Following the approach in (35), we can obtain the following equations These equations can be solved self-consistently to determine T c and other relevant quantities. More specifically, T c is defined as the critical temperature below which the exciton condensation begins (i.e., N 0 = 0). The solution of the above equations depends on the exciton density n =N/A (A is the area of the 2D system). The results of T c are shown in fig. S6, and the temperature dependence of condensation fraction is shown in fig. S7.