Inherently high valley polarizations of momentum-forbidden dark excitons in transition-metal dichalcogenide monolayers

However, it is widely known that the valley-polarization of a bright exciton in a TMD-ML is very likely depolarized by the intrinsic electron-hole (e-h) exchange interaction that couples the inter-band excitations in the distinct valleys. Yu et al. (2014); Yu and Wu (2014); Hao et al. (2016); Wang et al. (2014); Glazov et al. (2015); Selig et al. (2020) Despite the weak meV-scale coupling strength, the momentum-conserving e-h exchange interaction (EHEI) can efficiently intermix two exchange-free excitations in the $K$ and $K^{\prime}$ valleys that hold the same momentum and similar energy (quasi-degenerate). Yu et al. (2014); Qiu et al. (2015) Note that, at the most general level, the fundamental time-reversal symmetry (TRS) ensures the degeneracy only for the excitations in the opposite valleys that carry the opposite momentum. Hence, even without any spatial symmetries, the distinct degenerate valley-exciton states that hold the same nearly vanishing momentum, i.e. the bright exciton (BX) states, meet the both criteria and natively suffer from the exchange-induced valley depolarization. By contrast, the valley exciton states with the same finite momenta, i.e. momentum-forbidden dark exciton (MFDX) states, are not enforced by the TRS to be degenerate and, unlike the BXs, the valley polarizations of the MFDX states in TMD-MLs are purely dictated by the crystal symmetries and should be momentum-dependent.

In spite of violating the momentum selection rules, those MFDXs in TMD-MLs have drawn massive attention recently because of their essential involvement in various optical and dynamics phenomena. Li et al. (2019); Liu et al. (2020); He et al. (2020); Brem et al. (2020); Liu et al. (2019); Bao et al. (2020); Wu et al. (2019); Koitzsch et al. (2019); Hong et al. (2020); Feierabend et al. (2021); Jiang et al. (2021); Wallauer et al. (2020); Simbulan et al. (2021) Very recently, direct generation and probe of finite-momentum excitons in WSe₂-MLs have been realized by using the integrated technology of optical pump-probe and angle-resolved photo-emission spectroscopies Madéo et al. (2020); Middleton (2021). Moreover, recent cryogenic photo-luminescence (PL) measurements on high-quality tungsten-based TMD-ML samples have revealed the pronounced optical signatures of the inter-valley MFDXs that are significantly bright,Liu et al. (2020) highly polarizedLi et al. (2019); He et al. (2020) and long-lived Chen et al. (2020). Those long-lived and optically accessible MFDXs are attractive for the quantum applications and realizing excitonic Bose-Einstein condensation (BEC) Wang et al. (2019); Mazuz-Harpaz et al. (2019); Sigl et al. (2020); Combescot et al. (2017). With the recently achieved experimental advances in the investigation of MFDXs, it is timely demanded to establish a comprehensive theoretical understanding of those finite-momentum dark excitons in TMD-MLs over the extended momentum space, which yet remains rarely explored so far Deilmann and Thygesen (2019).

In this Letter, we present a theoretical investigation of the finite-momentum exciton states of WSe₂-MLs by numerically solving the DFT-based BSE under the guidance of symmetry analysis.Peng et al. (2019) The studies reveal the symmetry-dictated landscape of the momentum-dependent valley and optical properties of the finite-momentum exciton states over the full Brillouin zone and carry out the analysis of phonon-assisted PL, which explains the recently observed spectral brightness Li et al. (2019); Liu et al. (2020); He et al. (2020), high degree of optical polarization Li et al. (2019); He et al. (2020) and long-lived dynamics Chen et al. (2020) of the inter-valley MFDXs in tungsten-based TMD-MLs.

The explicit definitions of the matrix elements of $V_{\boldsymbol{k}_{ex}}^{d}$ and $V_{\boldsymbol{k}_{ex}}^{x}$ in terms of the Bloch wave functions are given in Ref.Sup Throughout this work, the screening in the direct e-h Coulomb interaction is modeled by using the Keldysh formalism. Keldysh (1979); Sup ; Cudazzo et al. (2011); Wu et al. (2015); Berkelbach et al. (2013); Stier et al. (2018); Ridolfi et al. (2018); Trolle et al. (2017) Following the comuptational methodology developed in Ref. Peng et al. (2019), we establish the BSE in the Wannier tight binding scheme Kośmider et al. (2013); Scharf et al. (2016); Lado and Fernández-Rossier (2016) based on the DFT-calculated electronic structures Mostofi et al. (2008, 2014); Sup and solve the exciton states and band structures by means of $\boldsymbol{k}$-grid discretization and direct matrix diagonalization.

It is known that the direct e-h Coulomb interaction makes the predominant contribution to the large binding energy of exciton in a TMD-ML, but no direct effect on the inter-valley couplings for exciton. Qiu et al. (2013); Chernikov et al. (2014) By contrast, the meV-scaled EHEIs make the inter-valley excitonic couplings that are especially efficient between the distinct valley excitons with the same momentum and similar energy (quasi-degeneracy). As previously stated, the degeneracies of exciton states over the momentum space are dictated by the enforcement the crystal symmetries, so are the degree of valley polarizations of the exciton states.

Below, we conudct the symmetry analysis to predict the degeneracies of the exchange-free exciton states of TMD-MLs under the $D_{3h}$-group symmetry.Robert et al. (2017) Consider two distinct free e-h pair states with the same $\boldsymbol{k}_{ex}$ excited from the different valence states at $\boldsymbol{k}$ and $\boldsymbol{k}^{\prime}$. They are degenerate as $\epsilon_{c,{\boldsymbol{k}+\boldsymbol{k}_{ex}}}-\epsilon_{v,\boldsymbol{k}}=\epsilon_{c,\boldsymbol{k}^{\prime}+\boldsymbol{k}_{ex}}-\epsilon_{v,\boldsymbol{k}^{\prime}}$, which can generally hold only if $\epsilon_{v,\boldsymbol{k}}=\epsilon_{v,\boldsymbol{k}^{\prime}}$ and $\epsilon_{c,\boldsymbol{k}+\boldsymbol{k}_{ex}}=\epsilon_{c,\boldsymbol{k}^{\prime}+\boldsymbol{k}_{ex}}$. From the theory of group representations, the above two equations hold when the space group symmetry of the TMD-ML satisfies the both equations, $\boldsymbol{k}^{\prime}=\hat{U}\boldsymbol{k}$ and $\boldsymbol{k}^{\prime}+\boldsymbol{k}_{ex}=\hat{U}\left(\boldsymbol{k}+\boldsymbol{k}_{ex}\right)$, for any symmetry operator $\hat{U}\in D_{3h}=\{E,C_{3},C_{3}^{-1},\sigma_{h},S_{3},S_{3}^{-1},C_{2,1}^{\prime},C_{2,2}^{\prime},C_{2,3}^{\prime},\sigma_{v,1},\sigma_{v,2},\sigma_{v,3}\}$. Accordingly, we find the criterion for the formation of valley-degeneracy of two distinct e-h pairs carrying the same $\boldsymbol{k}_{ex}$, i.e.

As an illustrative instance, Fig. 1c exemplifies the two distinctive e-h pair excitations with the same $\boldsymbol{k}_{ex}$ along the $k_{y}$-direction, i.e. $\overline{\Gamma_{ex}M_{ex,2}}$, which are excited from the valence $K$ towards the conduction $Q^{\prime}_{2}$ valley (red arrow line) and from the valence $K^{\prime}$ towards the conduction $Q_{2}$ valley (blue arrow line), respectively. In Fig. 1 b, one can identify the transition energies of the two free valley-excitations to be the same. For comparative illustration, we consider another set of two inter-band transitions excited from the distinctive valence valleys with the common $\boldsymbol{k}_{ex}$ along $\overline{\Gamma_{ex}K_{ex}^{\prime}}$, as depicted in Fig. 1d. With the misaligned $\boldsymbol{k}_{ex}$ from $\overline{\Gamma_{ex}M_{ex,i}}$ the transition energies are apparently different as predicted by analysis and seen in Figure 1b.

Beyond the non-interacting e-h pair states, the above symmetry analysis remains valid for the exchange-free exciton states. Figure 2a shows the calculated energy band dispersions of exchange-free exciton, $E_{S,\boldsymbol{k}_{ex}}^{X(0)}$, of a WSe₂-ML with the $\boldsymbol{k}_{ex}$ along $\overline{\Gamma_{ex}M_{ex,2}}$ and $\overline{\Gamma_{ex}K_{ex}^{\prime}}$ directions, solved from the exchange-free BSE including the direct part of Coulomb interaction only. The lowest exchange-free exciton band over the full BZ is presented in Figure 1e and 1f, where $\Gamma_{ex}$, $Q_{ex}/Q_{ex}^{\prime}$ and $K_{ex}/K_{ex}^{\prime}$ are identified as the major low-lying excitonic valleys in a WSe₂-ML. In the absence of EHEI, the energy bands of the lowest exciton doublet along $\overline{\Gamma_{ex}M_{ex,2}}$ does remain degenerate while the ones along $\overline{\Gamma_{ex}K_{ex}^{\prime}}$ are shown valley-split. Figure 2b presents the energy splitting of the lowest exchange-free spin-like exciton doublet, $\Delta_{+-}^{(0)}(\boldsymbol{k}_{ex})\equiv E_{+,\boldsymbol{k}_{ex}}^{X(0)}-E_{-,\boldsymbol{k}_{ex}}^{X(0)}$ (where the subscript $+/-$ indicates the upper/lower band, and the superscript (0) indicates the absence of EHEI), as a function of $\boldsymbol{k}_{ex}$ over the BZ, indeed showing $\Delta_{+-}^{(0)}(\boldsymbol{k}_{ex})=0$ (magenta lines) along the three $\overline{\Gamma_{ex}M_{ex,i}}$ paths.

Figure 2d shows the calculated exciton bands of the WSe₂-ML with the full consideration of the both e-h direct and exchange interactions. Under the effect of EHEI, the exciton bands along the $\overline{\Gamma_{ex}M_{ex,i}}$ paths no longer remain degenerate, as seen in Fig. 2e. For an valley-mixed exciton state written as $|S,\boldsymbol{k}_{ex}\rangle=\sum_{\tau=K,K^{\prime}}\alpha_{S,\boldsymbol{k}_{ex}}^{\tau}|\tau,\boldsymbol{k}_{ex}\rangle$, in terms of the well-specified valley components, $\{|\tau,\boldsymbol{k}_{ex}\rangle\}$, the degree of valley polarization (DoV) of the exciton is quantified by $P_{S,\boldsymbol{k}_{ex}}^{v}\equiv\frac{|\alpha_{S,\boldsymbol{k}_{ex}}^{K}|^{2}-|\alpha_{S,\boldsymbol{k}_{ex}}^{K^{\prime}}|^{2}}{|\alpha_{S,\boldsymbol{k}_{ex}}^{K}|^{2}+|\alpha_{S,\boldsymbol{k}_{ex}}^{K^{\prime}}|^{2}}$. As seen in Fig. 2f, the exciton states lying on the $\overline{\Gamma_{ex}M_{ex,i}}$ paths under the impact of EHEI are featured with $P_{-,\boldsymbol{k}_{ex}}^{v}\sim 0$, while the MFDX ones apart from the $\overline{\Gamma_{ex}M_{ex,i}}$ paths, like those around $Q_{ex,i}/Q_{ex,i}^{\prime}$ and $K_{ex}/K_{ex}^{\prime}$ valleys, retain the high DoV of $\mid P_{-,\boldsymbol{k}_{ex}}^{v}\mid\lesssim 100\%$. Hereafter, we shall pay the main attention on the lowest spin-allowed MFDX states at $K_{ex}/K_{ex}^{\prime}$ (or named by inter-valley $KK^{\prime}/K^{\prime}K$ exciton) whose optical signatures featured with the atractive high degree of polarization were observed in recent cryognic PL measurements. Li et al. (2019); He et al. (2020)

For the analysis of the phonon-assisted indirect PL, we consider the extended exciton-photon-phonon system with the Hamiltonian $\hat{H}=\hat{H}_{X}+\hat{H}_{\nu}+\hat{H}_{ph}+\hat{H}_{X-\nu}+\hat{H}_{X-ph}$, where $\hat{H}_{X}=\sum_{S\boldsymbol{k}_{ex}}E_{S,\boldsymbol{k}_{ex}}^{X}\hat{X}_{S,\boldsymbol{k}_{ex}}^{\dagger}\hat{X}_{S,\boldsymbol{k}_{ex}}$ stands for the single-exciton Hamiltonian, $\hat{X}$ ($\hat{X}^{\dagger}$) is the operator annihilating (creating) an exciton, $\hat{H}_{\nu}=\sum_{\boldsymbol{\epsilon}}\sum_{\boldsymbol{k}_{\nu}}\hbar\omega_{\boldsymbol{k}_{\nu}}^{\boldsymbol{\epsilon}}\hat{a}_{\boldsymbol{\epsilon},\boldsymbol{k}_{\nu}}^{\dagger}\hat{a}_{\boldsymbol{\epsilon},\boldsymbol{k}_{\nu}}$ ($\hat{H}_{ph}=\sum_{\lambda}\sum_{\boldsymbol{q}}\hbar\Omega_{\boldsymbol{q}}^{\lambda}\hat{b}_{\lambda,\boldsymbol{q}}^{\dagger}\hat{b}_{\lambda,\boldsymbol{q}}$) is the Hamiltonian of photon (phonon) reservoir, $\omega_{\boldsymbol{k}_{\nu}}^{\boldsymbol{\epsilon}}$ is the frequency of the $\boldsymbol{\epsilon}$-polarized photon with the wave vector $\boldsymbol{k}_{\nu}$, $\Omega_{\boldsymbol{q}}^{\lambda}$ is the frequency of the $\lambda$-kind phonon with the wave vector $\boldsymbol{q}$,Jin et al. (2014) and $\hat{a}/\hat{a}^{\dagger}$ ($\hat{b}/\hat{b}^{\dagger}$) are the particle operators that annihilate/create a photon (phonon). $\hat{H}_{X-\nu}$ ($\hat{H}_{X-ph}$) represents the Hamiltonian of exciton-photon (exciton-phonon) interaction in terms of the $\boldsymbol{\epsilon}$($\lambda$)-dependent coupling constants $\eta_{S,\boldsymbol{k}_{ex}}^{\boldsymbol{\epsilon},\boldsymbol{k}_{\nu}}$ ($g_{S^{\prime},\boldsymbol{k}_{ex}^{\prime};S,\boldsymbol{k}_{ex}}^{\lambda,\boldsymbol{q}}$) (See Ref.Sup for the explicit expressions).

In the process of indirect PL from a WSe₂-ML, an exciton initially in the lowest inter-valley MFDX state $|S=-,\boldsymbol{k}_{ex}=\boldsymbol{K}_{ex}/\boldsymbol{K}_{ex}^{\prime}\rangle\equiv|D\rangle$ is virtually transferred to the intermediate BX states, $|S^{\prime},\boldsymbol{k}_{ex}^{\prime}\in L.C.\rangle\equiv|S^{\prime},B\rangle$, assisted by the phonon with $\boldsymbol{k}_{ex}-\boldsymbol{k}_{ex}^{\prime}\equiv\boldsymbol{q}_{0}$ and then, from $|S^{\prime},B\rangle$, spontaneously emits a photon of wave vector $\boldsymbol{k}_{\nu}$ whose in-plane projection matches the 2D-exciton wave vector, $\boldsymbol{k}_{\nu,\parallel}=\boldsymbol{k}_{ex}^{\prime}$ (See Fig.3 for schematic illustration). For simplicity, throughout this work we focus on the specific kind of phonons that makes most the couplings between the inter-valley MFDX states and intra-valley BX ones, which was shown to be the $K_{3}$ mode in Ref.He et al. (2020); Liu et al. (2020), i.e. $\lambda=K_{3}$, and consider only the lowest BX doublet, $|S^{\prime}=\pm,B\rangle$, as the major intermediate states that are intrinsically valley-mixed by the EHEI $\tilde{\Delta}_{KK^{\prime}}$ and split by $\Delta_{+-}(\boldsymbol{k}_{ex}^{\prime})=2|\tilde{\Delta}_{KK^{\prime}}(\boldsymbol{k}_{ex}^{\prime})|$.

From the Fermi’s golden rule (as detailed in Ref.Sup ), we show that the averaged transition rates ($\overline{\gamma}_{D}^{(2)}$ and $\overline{\gamma}_{B}^{(1)}$) of the polarization-unresolved indirect PL from $|D\rangle$ and the direct PL from $|\pm,B\rangle$ are explicitly related by $\overline{\gamma}_{D}^{(2)}/\overline{\gamma}_{B}^{(1)}=\pi(\frac{a_{0}}{\lambda})^{2}\left|\frac{\hbar g_{BD}\sqrt{N}}{\Delta E_{BD}^{X}+\hbar\Omega_{\boldsymbol{q}_{0}}^{K_{3}}}\right|^{2}$, where $a_{0}=0.3316$nm is the lattice constant of WSe₂-ML Peng et al. (2019), $\lambda\approx 750$nm is the wavelength of the emitted light from $1s$ exciton, $N$ is the total number of primitive cells of the material, $\hbar g_{BD}=\frac{12.3}{\sqrt{N}}$meV is evaluated as the effective exciton-phonon coupling between $|D\rangle$ and $|B\rangle$,Sup ; Jin et al. (2014) $\Delta E_{BD}^{X}\equiv E_{B}^{X}-E_{D}^{X}\approx 29$meV, and $\hbar\Omega_{\boldsymbol{q}_{0}}^{K_{3}}\approx 26$meV He et al. (2020). With the use of those parameters, we count $\overline{\gamma}_{D}^{(2)}/\overline{\gamma}_{B}^{(1)}\gtrsim 10^{-8}$, indicating an extremely slow rate of the indirect PL transition. The transition rate of indirect PL rate is so low since only a very small portion of intermediate exciton states in the $\boldsymbol{k}_{ex}$ Brillouin zone that are bright to yield the PL, which is measured by the area ratio of the light-cone and the entire zone, i.e. $\left(\frac{a_{0}}{\lambda}\right)^{2}=(\frac{k_{c}}{|\boldsymbol{G}|})^{2}$, where $|\boldsymbol{G}|=\frac{2\pi}{a_{0}}$ and $k_{c}=\frac{2\pi}{\lambda}$.

Note that the intensity of a PL from an exciton state is determined by the transition rate as well as the exciton population therein, i.e. $I_{B/D}^{(1)/(2)}\propto{\bar{\gamma}_{B/D}}^{(1)/(2)}\overline{n}_{B/D}^{X}(T)$, where $\overline{n}_{B/D}^{X}(T)\propto e^{-E_{B/D}^{X}/k_{B}T}$ follows the Boltzmann statistics. For WSe₂-ML, the lowest $K_{ex}/K_{ex}^{\prime}$ MFDX states are so much lower than the BX ones by $\Delta E_{BD}^{X}\approx 29$meV and host the tremendously high exciton population at low temperature. Figure 3c presents the ratio of $\overline{n}^{X}_{D}/\overline{n}^{X}_{B}$ of a WSe₂-ML as a function of $T$, showing $\overline{n}^{X}_{D}/\overline{n}^{X}_{B}\gtrsim 10^{8}$ and $I_{D}^{(2)}/I_{B}^{(1)}>1$ at low temperatures $T<19.5$K. This accounts for the observed pronounced indirect PL peaks from the inter-valley MFDXs in WSe₂-MLs in cryogenic PL experiments. Li et al. (2019); Liu et al. (2020); He et al. (2020)

At last, let us examine the optical polarization, $P_{D}^{o(2)}\equiv\frac{I_{D}^{(2)}(\boldsymbol{\epsilon}_{+})-I_{D}^{(2)}(\boldsymbol{\epsilon}_{-})}{I_{D}^{(2)}(\boldsymbol{\epsilon}_{+})+I_{D}^{(2)}(\boldsymbol{\epsilon}_{-})}$, of the indirect PL from the inter-valley valley-mixed MFDX state, $|D\rangle=\alpha_{D}^{K}|K,\boldsymbol{k}_{ex}\rangle+\alpha_{D}^{K^{\prime}}|K^{\prime},\boldsymbol{k}_{ex}\rangle$. In the second-order perturbation theory (as detailed in Ref. Sup ), we show that the optical polarization and the valley polarization of the initial MFDX state of indirect PL are related by

The last term in Eq.3 containing $\widetilde{\beta}_{BD}^{KK^{\prime}}\equiv-\frac{\widetilde{\Delta}_{KK^{\prime}}\left(\boldsymbol{k}_{\nu,\parallel}\right)}{\Delta E_{BD}^{X}+\hbar\Omega_{\boldsymbol{q}_{0}}^{K_{3}}}$ arises from the EHEI in the intermediate BX states, through which the $K$($K^{\prime}$)-valley component in $|D\rangle$ is corss-converted and the optical polarization could be degraded. However, the small value of $|\widetilde{\beta}_{BD}^{K_{\pm}K_{\mp}}\left(\boldsymbol{k}_{\nu,\parallel}\right)|\sim 10^{-2}$ is estimated with $|\widetilde{\Delta}_{K_{\pm}K_{\mp}}|\sim 1$meV and $(\Delta E_{BD}^{X}+\hbar\Omega_{\boldsymbol{q}_{0}}^{K_{3}})\approx 55$meV. This reveals the native suppression of exchange-induced depolarization in the second-order PL process and accounts for the experimentally observed high degree of polarizations in the phonon-assisted indirect PLs from WSe₂-MLs He et al. (2020); Li et al. (2019). Figure 3d presents the calculated $P_{D}^{o(2)}/P_{D}^{v}$ for the indirect PL from the MFDX state with varying $\Delta E_{BD}^{X}$, showing the near-unity degree of the valley-to-optical polarization conversion.

In conclusion, we present a comprehensive theoretical investigation of the full-zone band structures and the momentum-dependent valley polarizations of neutral excitons in TMD-MLs by means of DFT-based numerical computations and symmetry analysis. Our studies reveal that inter-valley MFDXs, unlike the well recognized BXs, in TMD-MLs are inherently well immune from the exchange-induced valley depolarization under the enforcement of the $D_{3h}$ crystal symmetry. Thus, the valley polarization of the inter-valley $K_{ex}/K_{ex}^{\prime}$ ($KK^{\prime}/K^{\prime}K$) MFDX states are predicted superiorly high and, importantly, fully transferable to the optical polarization of the resulting phonon-assisted indirect PLs. Those findings shed light on the prospective of the valley-based photonics with the utilization of those long-lived, optically accessible and highly valley-polarized inter-valley finite-momentum dark excitons.

P.Y.L. and S.J.C. thank M. Bieniek and P. Hawrylak for fruitful discussion. This study is supported by the Ministry of Science and Technology, Taiwan, under contracts, MOST 109-2639-E-009-001 and 109-2112-M-009 -018 -MY3, and by National Center for High-Performance Computing (NCHC), Taiwan.