Probing and manipulating valley coherence of dark excitons in monolayer WSe$_2$

TL;DR

This study demonstrates how magnetic fields and exchange interactions can be used to probe and control the valley coherence of dark excitons in monolayer WSe2, advancing understanding of their quantum properties.

Contribution

It introduces a method to manipulate valley coherence of dark excitons in WSe2 using polarization-resolved photoluminescence and magnetic fields.

Findings

Magnetic fields enable probing valley coherence.

Exchange interaction influences dark exciton properties.

Valley degree of freedom can be manipulated externally.

Abstract

Monolayers of semiconducting transition metal dichalcogenides are two-dimensional direct-gap systems which host tightly-bound excitons with an internal degree of freedom corresponding to the valley of the constituting carriers. Strong spin-orbit interaction and the resulting ordering of the spin-split subbands in the valence and conduction bands makes the lowest-lying excitons in WX$_2$ (X~being S or Se) spin-forbidden and optically dark. With polarization-resolved photoluminescence experiments performed on a WSe$_2$ monolayer encapsulated in a hexagonal boron nitride, we show how the intrinsic exchange interaction in combination with the applied in-plane and/or out-of-plane magnetic fields enables one to probe and manipulate the valley degree of freedom of the dark excitons.

Equations

$$H=\left[\begin{array}[]{cccc}\Delta+g_{\mathrm{b}}\mathcal{B}_{z}&0&g_{\|}\mathcal{B}_{-}&0\\ 0&\Delta-g_{\mathrm{b}}\mathcal{B}_{z}&0&g_{\|}\mathcal{B}_{+}\\ g_{\|}\mathcal{B}_{+}&0&\delta/2+g_{\mathrm{d}}\mathcal{B}_{z}&\delta/2\\ 0&g_{\|}\mathcal{B}_{-}&\delta/2&\delta/2-g_{\mathrm{d}}\mathcal{B}_{z}\end{array}\right].$$

$$I_{\perp }({\varphi }_a)\propto {\left(d_{\Vert }\frac{g_{\Vert }{\mu }_{\mathrm{B}}{B}_{\Vert }}{\Delta }\right)}^2\times$$

$$\times \left[({\chi }_{+}-{\chi }_{-}{)}^2+4{\chi }_{+}{\chi }_{-}{\sin }^2({\varphi }_a+{\varphi }_{\mathrm{d}}-{\varphi }_B)\right],$$

$$E_{\mathrm{G},\mathrm{D}}=E_{\mathrm{d}}\pm \frac{1}{2}\sqrt{{\delta }^2+(g_{\mathrm{d}}{\mu }_{\mathrm{B}}{B}_{\perp }{)}^2}.$$

$$\displaystyle|D_{h}\rangle=\mathcal{N}\Big{[}-\frac{g_{\parallel}\mathcal{B}_{\parallel}}{\Delta}e^{-i\phi_{B}}\sin\Big{(}\frac{\vartheta}{2}+\frac{\pi}{4}\Big{)},\,-\frac{g_{\parallel}\mathcal{B}_{\parallel}}{\Delta}e^{i\phi_{B}}\cos\Big{(}\frac{\vartheta}{2}+\frac{\pi}{4}\Big{)},\,\sin\Big{(}\frac{\vartheta}{2}+\frac{\pi}{4}\Big{)},\,\cos\Big{(}\frac{\vartheta}{2}+\frac{\pi}{4}\Big{)}\Big{]},$$

$$\displaystyle|D_{l}\rangle=\mathcal{N}\Big{[}-\frac{g_{\parallel}\mathcal{B}_{\parallel}}{\Delta}e^{-i\phi_{B}}\cos\Big{(}\frac{\vartheta}{2}+\frac{\pi}{4}\Big{)},\,\frac{g_{\parallel}\mathcal{B}_{\parallel}}{\Delta}e^{i\phi_{B}}\sin\Big{(}\frac{\vartheta}{2}+\frac{\pi}{4}\Big{)},\,\cos\Big{(}\frac{\vartheta}{2}+\frac{\pi}{4}\Big{)},\,-\sin\Big{(}\frac{\vartheta}{2}+\frac{\pi}{4}\Big{)}\Big{]},$$

$$\left(\begin{array}[]{c}E_{h}\\ E_{l}\end{array}\right)=\frac{\delta}{2}\pm\frac{1}{2}\sqrt{\delta^{2}+\big{(}g_{\mathrm{d}}\mu_{B}B_{\perp}\big{)}^{2}}-\frac{\big{(}g_{\parallel}\mu_{B}\mathbf{B}_{\parallel}\big{)}^{2}}{4\Delta}.$$

$$\displaystyle\langle 0|\mathbf{d}|D_{h}\rangle=i\mathcal{N}d_{\parallel}\frac{g_{\parallel}\mathcal{B}_{\parallel}}{\Delta}\Big{\{}e^{-i\phi_{B}}\sin\Big{(}\frac{\vartheta}{2}+\frac{\pi}{4}\Big{)}[\mathbf{e}_{x}+i\mathbf{e}_{y}]+e^{i\phi_{B}}\cos\Big{(}\frac{\vartheta}{2}+\frac{\pi}{4}\Big{)}[-\mathbf{e}_{x}+i\mathbf{e}_{y}]\Big{\}}+i\mathcal{N}\sqrt{2}d_{\perp}\cos\Big{(}\frac{\vartheta}{2}\Big{)}\mathbf{e}_{z},$$

$$\displaystyle\langle 0|\mathbf{d}|D_{l}\rangle=i\mathcal{N}d_{\parallel}\frac{g_{\parallel}\mathcal{B}_{\parallel}}{\Delta}\Big{\{}e^{-i\phi_{B}}\cos\Big{(}\frac{\vartheta}{2}+\frac{\pi}{4}\Big{)}[\mathbf{e}_{x}+i\mathbf{e}_{y}]-e^{i\phi_{B}}\sin\Big{(}\frac{\vartheta}{2}+\frac{\pi}{4}\Big{)}[-\mathbf{e}_{x}+i\mathbf{e}_{y}]\Big{\}}-i\mathcal{N}\sqrt{2}d_{\perp}\sin\Big{(}\frac{\vartheta}{2}\Big{)}\mathbf{e}_{z}.$$

$$P_{\alpha }=\frac{I_{\alpha }({\sigma }_{+})-I_{\alpha }({\sigma }_{-})}{I_{\alpha }({\sigma }_{+})+I_{\alpha }({\sigma }_{-})}.$$

$$I_{\alpha }({\sigma }_{\pm })\propto {\int }_0^{2\pi }d\phi {\int }_0^{{\theta }_0}\sin \theta d\theta |{\mathbf{d}}_{\alpha }\cdot {\mathbf{e}}_{\pm }(\theta ,\phi ){|}^2.$$

$${\mathbf{e}}_r=\sin \theta \cos \phi {\mathbf{e}}_x+\sin \theta \sin \phi {\mathbf{e}}_y+\cos \theta {\mathbf{e}}_z,$$

$${\mathbf{e}}_{\theta }=\cos \theta \cos \phi {\mathbf{e}}_x+\cos \theta \sin \phi {\mathbf{e}}_y-\sin \theta {\mathbf{e}}_z,$$

$${\mathbf{e}}_{\phi }=-\sin \phi {\mathbf{e}}_x+\cos \phi {\mathbf{e}}_y.$$

$${\mathbf{d}}_{\alpha }\cdot {\mathbf{e}}_{\pm }(\theta ,\phi )=-\frac{a_{\alpha }}{\sqrt{2}}(1\mp \cos \theta )e^{i\phi }-\frac{b_{\alpha }}{\sqrt{2}}(1\pm \cos \theta )e^{-i\phi }\mp \frac{c_{\alpha }}{\sqrt{2}}\sin \theta ,$$

$$P_{\alpha}=\frac{6(|b_{\alpha}|^{2}-|a_{\alpha}|^{2})\cos^{2}\big{(}\frac{\theta_{0}}{2}\big{)}}{(|b_{\alpha}|^{2}+|a_{\alpha}|^{2})(4+\cos\theta_{0}+\cos^{2}\theta_{0})+2|c_{\alpha}|^{2}\sin^{2}\big{(}\frac{\theta_{0}}{2}\big{)}(2+\cos\theta_{0})}.$$

$$\displaystyle P_{h}=\frac{-6\cos^{2}\big{(}\frac{\theta_{0}}{2}\big{)}\,\sin\vartheta}{4+\cos\theta_{0}+\cos^{2}\theta_{0}+\Big{(}\frac{4\Delta}{g_{\parallel}\mu_{B}B_{\parallel}}\frac{d_{\perp}}{d_{\parallel}}\Big{)}^{2}\sin^{2}\big{(}\frac{\theta_{0}}{2}\big{)}(2+\cos\theta_{0})\,\cos^{2}\big{(}\frac{\vartheta}{2}\big{)}},$$

$$\displaystyle P_{l}=\frac{6\cos^{2}\big{(}\frac{\theta_{0}}{2}\big{)}\,\sin\vartheta}{4+\cos\theta_{0}+\cos^{2}\theta_{0}+\Big{(}\frac{4\Delta}{g_{\parallel}\mu_{B}B_{\parallel}}\frac{d_{\perp}}{d_{\parallel}}\Big{)}^{2}\sin^{2}\big{(}\frac{\theta_{0}}{2}\big{)}(2+\cos\theta_{0})\,\sin^{2}\big{(}\frac{\vartheta}{2}\big{)}}.$$

Full text

Monolayers of transition metal dichalcogenides (TMDs), such as MX2 with M=Mo or W, and X=S, Se or Te, are two-dimensional direct-gap semiconductors Mak et al. (2010) which attract a lot of interest due to their unique physical properties and potential applications in optoelectronics, photonics and the development of valleytronics Wang et al. (2012); Eda and Maier (2013); Xia et al. (2014); Xu et al. (2014); Koperski2017 ; Wang et al. (2018). The direct bandgap in semiconducting TMDs (S-TMDs) is located at the two inequivalent $\mathbf{K}_{\pm}$ points (valleys) of the first Brillouin zone, related by time reversal symmetry. In monolayers, the tightly bound and optically bright excitons Chernikov et al. (2014); Ye et al. (2014); Ugeda et al. (2014); He et al. (2014) from $\mathbf{K}_{\pm}$ valley can efficiently couple to light with right/left circular polarizationYao et al. (2008); Xiao et al. (2012), respectively.

A unique feature of S-TMD monolayers is the so-called spin-valley locking Xiao et al. (2012): strong spin-orbit interaction lifts the degeneracy between the two spin projections $s=\uparrow,\downarrow$ in each valley, leaving only the Kramers degeneracy between the opposite valleys, $\mathbf{K}_{+},s\leftrightarrow\mathbf{K}_{-},-s$. While the valence band spin-orbit splitting $\Delta_{v}$ is very large (several hundred meV Xiao et al. (2012)), its conduction band counterpart $\Delta_{c}$ is by an order of magnitude smaller Kośmider et al. (2013); Kormányos et al. (2015); Echeverry et al. (2016); Molas et al. (2017); Zhang et al. (2017), thus allowing some degree of manipulation by an in-plane magnetic field. In tungsten-based S-TMDs, $\Delta_{c}$ has the same sign as $\Delta_{v}$, leading to the spin subband ordering shown in Fig. 1(a). In each valley, the optically bright exciton has a higher energy than the dark exciton which is composed of the conduction and valence electronic states with opposite spin projections. This, among other things, results in a strong temperature dependence of the photoluminescence (PL) efficiency, as the bright states become more populated at higher temperatures Arora et al. (2015); Zhang2015 ; Wang et al. (2015).

Dark excitons can couple to light only via a residual spin-flip dipole matrix element $d_{\perp}$, whose direction is perpendicular to the monolayer plane Glazov et al. (2014); Slobodeniuk and Basko (2016); Wang et al. (2017). In consequence, the emission of dark excitons is directed predominantly along the monolayer plane, in contrast to the out-of-plane emission of bright excitons characterized by strong in-plane optical dipoles, $d_{\|}$, see Fig. 1(b). Notably, the valley degeneracy of dark excitons is lifted by the exchange interaction which mixes the valleys and produces two eigenstates with different energies. The higher energy component takes up the whole oscillator strength due to $d_{\perp}$ (so we call it “grey”), while the low-energy component is completely decoupled from light and truly “dark” Dery and Song (2015); Slobodeniuk and Basko (2016).

The possibility to control the valley degree of freedom for bright excitons by optical polarization Zeng et al. (2012); Mak et al. (2012); Jones et al. (2013) and external magnetic field Aivazian2015 ; Wang et al. (2016a); Smoleński et al. (2016) is impaired by the quick valley relaxation due to the exchange interaction Glazov et al. (2014); Hao et al. (2016), because the latter (i) depends on the exciton center-of-mass momentum, and (ii) is considerably strong, being governed by the same (large) in-plane dipole moment $d_{\|}$ as the optical transitions. For dark excitons, the exchange interaction is (i) a short-range local field effect, so it weakly depends on momentum, (ii) strong enough to produce a sizeable dark- grey splitting $\delta\approx{0}.6\>\mbox{meV}\approx{7}\>\mbox{K}$ in WSe2 Robert2017 which makes the dark component dominant at low temperatures, and (iii) at the same time weak enough to enable a control of the splitting and of the valley content by a perpendicular magnetic field $B_{\perp}$ of a few Tesla Robert2017 . Importantly, by applying an in-plane magnetic field $B_{\|}$, one can transfer the in-plane oscillator strength from the higher-energy bright states to the dark-grey doublet Molas et al. (2017); Zhang et al. (2017), thus boosting the out-of-plane emission of light from both components of this doublet.

In this work we present an experimental implementation of the described “dark exciton valley toolkit” in a WSe2 monolayer. Namely, we investigate the influence of the in-plane and/or out-of-plane magnetic fields on dark excitons emission. In particular, we experimentally demonstrate the coherent valley-superposition nature of the dark and grey excitons by transferring the oscillator strength from the bright excitons with an in-plane magnetic field and detecting a linearly polarized luminescence. We use a WSe2 sample encapsulated in hexagonal boron nitride (hBN), which is known to significantly enhance the sample quality Wang et al. (2016b); Cadiz et al. (2017); Ajayi et al. (2017); Manca et al. (2017); Vaclavkova et al. (2018). In our experiment, the encapsulation leads to a very strong brightening effect, much more pronounced than observed earlier Molas et al. (2017); Zhang et al. (2017), and enables us to spectrally resolve the fine structure of the brightened dark excitons. We observed that the combination of the in-plane and out-of-plane magnetic fields leads to circularly-polarized emissions due to the dark and grey excitons.

The valley structure of intravalley excitons in a magnetic field of arbitrary orientation can be described using the model of Refs 23; 17; 10. Let us work in the basis of the four intravalley A-exciton states with zero center-of-mass momentum at zero magnetic field. These states $|\mathbf{K}_{\pm},s_{c}\rangle$ can be identified by the valley $\mathbf{K}_{\pm}$ and the conduction band spin projection $s_{c}=\uparrow,\downarrow$. The valence band spin projection is fixed for the A exciton: $s_{v}=\uparrow$ in $\mathbf{K}_{+}$ valley and $\downarrow$ in $\mathbf{K}_{-}$. The states $|\mathbf{K}_{+},\uparrow\rangle$, $|\mathbf{K}_{-},\downarrow\rangle$ are bright and are denoted by $|\mathbf{K}_{\pm},\mbox{b}\rangle$, respectively. Their transition dipole matrix elements lie in the plane and correspond to the two circular polarizations, $\langle 0|\hat{\mathbf{d}}|\mathbf{K}_{\pm},\mathrm{b}\rangle=-id_{\|}(\pm\mathbf{e}_{x}+{i}\mathbf{e}_{y})$. The states $|\mathbf{K}_{+},\downarrow\rangle$, $|\mathbf{K}_{-},\uparrow\rangle$ are denoted as $|\mathbf{K}_{\pm},\mbox{d}\rangle$ and have transition dipoles $\langle 0|\hat{\mathbf{d}}|\mathbf{K}_{\pm},\mathrm{d}\rangle=id_{\perp}\mathbf{e}_{z}$, perpendicular to the plane Glazov et al. (2014); Slobodeniuk and Basko (2016); Wang et al. (2017); Robert2017 ; Slobodeniuk and Basko (2019), $d_{\perp}/d_{\|}\sim 0.01-0.1$ Slobodeniuk and Basko (2016).

In the basis $|\mathbf{K}_{+},\mbox{b}\rangle$, $|\mathbf{K}_{-},\mbox{b}\rangle$, $|\mathbf{K}_{+},\mbox{d}\rangle$, $|\mathbf{K}_{-},\mbox{d}\rangle$, the Hamiltonian matrix has the form

[TABLE]

Counting energies from the dark exciton state, we denoted by $\Delta\approx 40\>\mbox{meV}$ Molas et al. (2017); Zhang et al. (2017); Robert2017 ; Wang et al. (2017) and $\delta\approx 0.6\>\mbox{meV}$ Robert2017 , respectively, the dark-bright and dark-grey splitting at zero field. The parameter $\delta$ characterizes the short-range exchange interaction which lifts the valley degeneracy of the dark excitons at $B_{\perp}=0$ and forms two new (grey and dark) eigenstates: |\mathrm{G}\rangle,|\mathrm{D}\rangle=\big{(}|\mathbf{K}_{+},\mathrm{d}\rangle\pm|\mathbf{K}_{-},\mathrm{d}\rangle\big{)}/\sqrt{2}.

Other terms in equation (1) describe the effect of the magnetic field, of the in-plane component, $B_{\|}$, and of the perpendicular one, $B_{\perp}$; we use the shorthand notation $\mathcal{B}_{z}\equiv\mu_{\mathrm{B}}B_{\perp}/2$, $\mathcal{B}_{\pm}\equiv\mu_{\mathrm{B}}(B_{x}\pm{i}B_{y})/2$, where $\mu_{\mathrm{B}}=57.88\>\mu\mbox{eV/T}$ is the Bohr magneton. The perpendicular field $B_{\perp}$ lifts the single-electron Kramers degeneracy between opposite spins in opposite valleys. This results in a valley-dependent shift of the excitonic levels with different Landé factors in the bright and dark-grey sectors. The in-plane magnetic field $\mathbf{B}_{\parallel}=(B_{x},B_{y})$ produces the Zeeman coupling ($g_{\|}\mu_{\mathrm{B}}B_{\|}$) of the electronic states with opposite spins in the same valley, and thus mixes the dark-grey and bright excitonic sectors, remaining diagonal in the valley index.

The magnetic field terms in Hamiltonian (1) provide the experimental tools which enable one to control and probe the dark exciton states. At $B_{\|}=0$, one can tune continuously the valley content of the dark and grey excitons by changing $B_{\perp}$. Then, turning on the  $B_{\|}$ field, one admixes the bright states to the dark-grey sector, thereby transferring the in-plane oscillator strength; since $\Delta$ is the largest energy scale in Hamiltonian (1), this admixture can be treated perturbatively. Then, it turns out that the valley content of the eigenstates in the dark-grey sector can be probed by measuring linear polarization of the light emitted now in the direction perpendicular to the monolayer plane. Consider an eigenstate $|\psi_{\mathrm{d}}\rangle$ at $B_{\|}=0$ in the form of a general linear combination, $|\psi_{\mathrm{d}}\rangle=\chi_{+}e^{i\phi_{\mathrm{d}}}|\mathbf{K}_{+},\mathrm{d}\rangle+\chi_{-}e^{-i\phi_{\mathrm{d}}}|\mathbf{K}_{-},\mathrm{d}\rangle$ with real $\chi_{\pm}$, $\chi_{+}^{2}+\chi_{-}^{2}=1$ and some phase $\phi_{\mathrm{d}}$. Let us apply $B_{\|}$ and detect the luminescence, emitted normally to the plane, with an analyzer selecting the electric field direction with the polar angle $\phi_{a}$. Then, calculating perturbatively in $1/\Delta$ the corresponding in-plane dipole matrix element, we obtain the emitted light intensity:

[TABLE]

where $\phi_{B}=\arctan(B_{y}/B_{x})$ is the polar angle of the in-plane magnetic field. Below we present the experimental implementation of the described tools.

Figs 2(a)-(c) illustrate the results of micro-magneto-PL measurements performed in a wide spectral range ($1.64-1.71$ eV), covering the whole emission spectrum of our sample, but at the expense of rather low spectral resolution (0.8  nm). In contrast, the results shown in Figs 2(d),(e) and in the inset of Fig. 2(a) refer to experiments with higher spectral resolution (0.1 nm). All spectra were measured in the configuration of the normal (to the sample plane) coincidence of the excitation and the collected beams (see Fig. 1(b) and Supplemental Material (SM) for details). The PL peaks, labelled as X${}^{\textrm{D}}$ and X${}^{\textrm{G}}$, are ascribed, correspondingly, to dark and grey states of the neutral exciton. Only X${}^{\textrm{G}}$ is observed in the spectra measured at zero magnetic field. We expect that the X${}^{\textrm{G}}$-emission is directed predominantly along the monolayer plane, thought it is still visible in the spectra, due to a relatively high numerical aperture (NA) of the lenses used to collect the emitted light. Both components of the X${}^{\textrm{D}}$/X${}^{\textrm{G}}$ doublet contribute to the emission spectrum when the magnetic field is applied (see, $e.g.$, Fig. 2(d)). Note that this doublet structure is not resolved in the spectra shown in Fig. 2(b), measured with low spectral resolution. These results are in good agreement with previous reports Cadiz et al. (2017); Courtade2017 ; Robert2017 ; Barbone2018 ; Chen2018 ; Li2018 ; Zhou2017; Wang et al. (2017); Robert2017 : (i) the grey exciton X${}^{\textrm{G}}$-line is red-shifted by $38$ meV from the emission peak X${}^{\textrm{B}}$ associated with the neutral bright exciton (see Fig. 2(a)), (ii) the energy separation between the grey (X${}^{\textrm{G}}$) and dark (X${}^{\textrm{D}}$) excitons is $\delta=660\>\mu\mbox{eV}$ (see Fig. 2(d)), and (iii) the X${}^{\textrm{D}}$ and X${}^{\textrm{G}}$ lines are much narrower ($\simeq 0.6$ meV) than the X${}^{\textrm{B}}$ line ($\simeq 4$ meV) what indicates the significantly longer lifetimes of dark and grey excitons as compared to the lifetime of the bright exciton (see SM for details).

Application of the in-plane magnetic field $B_{\|}$ leads to a strong brightening effect, which is remarkably pronounced in our encapsulated WSe2 monolayer, much more than in previously probed WSe2 monolayers on Si/SiO2 substrates Molas et al. (2017); Zhang et al. (2017). This is demonstrated in Fig. 2(b) showing PL spectra measured in a wide spectral range, as a function of $B_{\|}$. With increasing $B_{\|}$, the intensity of the grey/dark exciton doublet (separate components not resolved) increases significantly, while the intensity of the bright exciton and of the lower energy features of the spectrum, stays practically at the same level. Note that the T${}^{\textrm{D}}$ line observed at about 1.65 eV in Fig. 2(b) also becomes significantly brighter at the largest magnetic fields (B$$>10 T). Recently, this peak was ascribed to the emission of the dark negatively charged exciton Liu2019 . In Fig. 2(c), we plot the total emission intensity (integrated over the whole PL spectrum) as a function of $B_{\|}$. At $B_{\|}$=14 T, the total emission intensity is enhanced by an order of magnitude and practically the whole luminescence of the sample is due to the dark/grey excitons doublet.

The PL response of our hBN/WSe2/hBN sample, measured with better spectral resolution, enabled us, for the first time, to resolve the magnetic brightening of the dark and grey components separately. As can be seen in Fig. 2(d), while only the grey exciton is observed when $B_{\|}$=0, at fields above 1.6 T the X${}^{\textrm{D}}$ line becomes more intense. The $B_{\|}$ dependence of the intensity of each component is expected to be $I=I_{0}+\alpha{B}_{\|}^{2}$ (Refs 17; 18), where the zero-field intensity $I_{0}$ vanishes for the dark component but is nonzero and depends on the NA of the excitation/detection lens for the grey component. The results of the quadratic fit are shown in Fig. 2(e). The $\alpha$ parameter turns out to be strongly different for the two components: $\alpha_{\mathrm{D}}=0.45\>\mbox{T}^{-2}$, while $\alpha_{\mathrm{G}}=0.09\>\mbox{T}^{-2}$. This large difference is explained by the difference in the populations of the two states: at temperature $T$=4.2 K which corresponds to our experimental conditions, the population ratio is $e^{-\delta/k_{B}T}=0.16$, in reasonable agreement with the measured ratio $\alpha_{\mathrm{G}}/\alpha_{\mathrm{D}}=0.2$.

Now, one can investigate the polarization properties of magnetically brightened dark and grey excitons, see Fig. 2(f). We present in Fig. 2(g) the integrated PL intensity for each peak, as collected via the linear polarizer and measured as a function of the angle $\phi_{a}-\phi_{B}$ between the polarizer axis and the direction of the $\mathbf{B}_{\|}$ field. For the dark exciton, the polarization dependence fully agrees with equation (2): indeed, at $B_{\perp}=0$, we have $\chi_{+}=-\chi_{-}=1/\sqrt{2}$ and $\phi_{\mathrm{d}}=0$, resulting in the transition dipole oriented along $\mathbf{B}_{\|}$. Our polarization measurement thus reveals experimentally the nature of the dark exciton state, which is a coherent superposition of the states in the two valleys.

For the grey exciton, the measured polarization dependence corresponds to equation (2) plus a constant background. The latter is due to large NA of the lens used to collect the emitted light. Indeed, equation (2) describes emission perpendicular to the monolayer and thus is valid in the limit of small NA. Our large-NA lens picks up the light emitted by the out-of-plane component $d_{\perp}$ of the transition dipole moment, which has no in-plane polarization. For the dark exciton at $B_{\perp}=0$, this component is absent.

Application of a magnetic field $B_{\perp}$ in the direction perpendicular to the monolayer plane shifts the energies in the two valleys in the opposite directions. Thus, the eigenstates in the dark-grey sector resulting from valley mixing by exchange coupling have no longer equal weights in the two valleys. As a result, both grey and dark exciton components have a finite out-of-plane optical dipole moment even at $B_{\|}$=0, and are visible in the luminescence spectra due to large NA of the lens (see SM for details). Note that the intensity of the dark exciton lines is independent of the detected helicity ($\sigma_{+}$ or $\sigma_{-}$) of light.

In order to induce the $\sigma_{\pm}$ selectivity for the dark excitons emission, we propose to combine the effects of parallel ($B_{\|}$) and perpendicular ($B_{\perp}$) magnetic fields. $B_{\|}$ gives rise to linear polarizations of the grey and dark excitons (see Fig. 2(f) and (g)), which are subsequently transformed by $B_{\perp}$ into elliptical polarizations. The experimental evidence corroborating this scenario is presented in Fig. 3(a)-(c). Upon increasing of the $B_{\perp}$ component of the magnetic field, both dark and grey excitons gain circular $\sigma_{+}$ and $\sigma_{-}$ polarizations, respectively. This allows us to determine that the $g$-factor of dark and grey excitons is negative as the one of the bright neutral exciton. The gradual increase in the circular degree of polarization, illustrated in Fig. 3(a)-(c), can be described in terms of the competition between the Zeeman shift $g_{\mathrm{d}}\mu_{B}B_{\perp}$ and the initial splitting $\delta$ and taking into account a finite NA of excitation/collection lens used in experiment. The result of such competition is presented in Fig. 3(d) for the case of dark exciton emission, as an example. The blue dots (red curve) depicts the experimentally (theoretically) derived data of the circular polarization degree as a function of out-of-plane magnetic field $B_{\perp}$. Note that the value of the polarization degree saturates already at 60% and not at 100% as one can expect. The explanation of such phenomenon involves a finite contribution of the out-of-plane dipole components of X${}^{\textrm{D}}$ and X${}^{\textrm{G}}$ exciton states (see SM for details). To prove the dominant contribution of the circular polarization ($\sigma_{\pm}$) for the dark and grey exciton emissions, they were measured as a function of orientation of the $\lambda$/4 waveplate in detection, see Fig. 3(f).

In this work, we investigated the PL response of a hBN-encapsulated WSe2 monolayer subject to a magnetic field, whose direction lies in the monolayer plane, or perpendicular to it. Thanks to the encapsulation, the in-plane field leads to a very pronounced brightening effect on the dark lowest-energy excitons in WSe2, and most of the sample’s luminescence comes from these dark excitons. Moreover, in our encapsulated sample one can resolve the fine structure of the magnetically brightened dark excitons, split by the exchange interaction which produces two coherent valley superpositions as eigenstates. We were able to probe the coefficients of the superpositions by a polarization-resolved PL measurement. These coefficients can be changed by applying a perpendicular and/or parallel magnetic field. These findings open a perspective for controlled manipulation of the valley degree of freedom of long-lived dark excitons in monolayer S-TMDs.

The work has been supported by the ATOMOPTO project (TEAM programme of the Foundation for Polish Science, co-financed by the EU within the ERDFund), the EU Graphene Flagship project (no. 785219), the National Science Centre, Poland (grant no. UMO-2018/31/B/ST3/02111), the Nanofab facility of the Institut Néel, CNRS/UGA and the LNCMI-CNRS, a member of the European Magnetic Field Laboratory (EMFL). K.W. and T.T. acknowledge support from the Elemental Strategy Initiative conducted by the MEXT, Japan and and the CREST (JPMJCR15F3), JST.

S1 Samples and experimental setups

The active part of our sample consists of a monolayer of WSe2, which has been encapsulated in hexagonal boron nitride (hBN) and deposited on a bare Si substrate. It was fabricated by two-stage polydimethylsiloxane (PDMS)-based Gomez2014 mechanical exfoliation of WSe2 and hBN bulk crystals. Micro-magneto-photoluminescence (PL) measurements (spatial resolution $\sim$5 $\mu$m) were carried out at liquid helium temperature using two different experimental setups. In each setup, the sample was placed on top of a $x-y-z$ piezo-stage kept at $T$=4.2 K and was excited using laser diodes with either 515 nm wavelength (2.408 eV photon energy) or 647 nm wavelength (1.917 eV photon energy). The emitted light was dispersed with a 0.5 m long monochromator and detected with a charge coupled device (CCD) camera. The data presented in Figs 2(a)-(e) and S3 refer to measurements performed with the fiber-optic based setup and were carried out in magnetic fields up to 14 T. Unpolarized emission was measured in Voigt configuration (see Fig. 2(a)-(e)), whereas the $\sigma_{-}$ and $\sigma_{+}$-polarized components of the emitted light were resolved in experiments (see Fig. S3) performed in the Faraday configuration (polarizers placed in a close vicinity of the sample, in between the sample and optical fibers). The data presented in Figs 2(f)-(g) refer to experiments carried out using a split-coil (10 T) superconducting magnet with a free-beam-optics arrangement. Fig. 3 illustrates results obtained with the aid of a system of two split-coil superconducting magnets with a free-beam-optics arrangement allowing to apply independently in-plane and out-of-plane magnetic fields. The linear and circular polarizations of the emissions were analyzed using a set of polarizers and waveplates ($\lambda$/2 and $\lambda$/4) placed directly in front of the spectrometers. The data presented in Figs 2,  3,  S1 and  S3 refer to spectra measured with different spectral resolutions: 0.8 nm — Figs 2(a)-(c),  S1 and  S3; 0.2 nm — Fig. 2(f)-(g); 0.1 nm — the inset of Fig. 2(a), Fig. 2(d)-(e) and Fig. 3.

S2 Low-temperature PL spectrum of the WSe2 monolayer

Fig. S1 demonstrates the photoluminescence spectrum measured on a WSe2 monolayer encapsulated in hBN flakes. The PL spectrum displays several emission lines with a similar characteristic pattern already reported in a number of previous works on WSe2 monolayers embedded in between hBN flakes Courtade2017 ; Li2018 ; Chen2018 ; Barbone2018 ; Paur2019 ; Liu2019 . In accordance with these reports, the assignment of the observed emission lines is as follows: X${}^{\textrm{B}}$ — a bright neutral exciton formed in the vicinity of the A exciton; XX — a neutral biexciton; T${}^{\textrm{S}}$ and T${}^{\textrm{T}}$ — singlet (intravalley) and triplet (intervalley) negatively charged excitons, respectively; X${}^{\textrm{G}}$ — a grey exciton; XX${}^{\textrm{-}}$ — a negatvely charged biexciton; T${}^{\textrm{D}}$ - a negatively charged dark exciton; L${}_{\textrm{1}}$ — a so-called localized exciton. The attribution of the P line is not clear, as in the most reported PL spectra this peak was absent Li2018 ; Chen2018 ; Barbone2018 ; Paur2019 ; Liu2019 , but was observed in the neutral regime (between $n$ and $p$-type doping) in Ref. 2. As the energy separation between X${}^{\textrm{B}}$ and P lines of about 31 meV is close to the longitudinal optical (LO) phonon energy ($E_{\textrm{LO}}$=32 meV Zhang2015 ), we ascribed tentatively this peak to the LO phonon replica of the bright neutral exciton.

S3 The influence of in-plane magnetic field on lifetimes of grey and dark excitons

To get more information on the grey and dark excitons, we measured their lifetimes by time-resolved photoluminescence spectroscopy, when an in-plane magnetic field is applied. Fig. S2(a) shows the X${}^{\textrm{G}}$ and X${}^{\textrm{D}}$ emissions kinetics following a pulsed laser excitation, while Fig. S2(b) summarizes their obtained decay times for a few values of magnetic fields. We found that the grey and dark excitons lifetimes are on the order of hundreds ps and are significantly longer as compared to the reported $\sim$2 ps lifetime of the bright neutral exciton in WSe2 monolayer Robert2016 ; Robert2017 . As can be appreciated in Fig. S3(b), we do not observe an effect of an in-plane magnetic field on the X${}^{\textrm{G}}$ and X${}^{\textrm{D}}$ lifetimes. Consequently, the extrapolated values of the grey and dark excitons at zero magnetic field are of about 220 ps and 250 ps, respectively. We note also that the obtained X${}^{\textrm{G}}$ lifetime is almost two time larger than the reported previously in Ref. 10.

S4 The effect of perpendicular magnetic field on grey and dark excitons emissions

The magneto-luminescence spectra for two opposite circular polarizations, measured in the Faraday configuration are shown in Fig. S3 a. The bright exciton emission splits into two circularly polarized components centered at energies $E_{\mathrm{b},\pm}=E_{\mathrm{B}}\pm{g}_{\mathrm{b}}\mu_{\mathrm{B}}B_{\perp}/2$, due to the valley Zeeman effect, as can be appreciated at the top of Fig. S3a,b. We obtain $g_{\mathrm{b}}=-4.2$, in agreement with previously reported values Srivastava2015 ; Aivazian2015 ; WangGlazov2015 ; Mitioglu2015 ; Robert2017 ; Koperski2019 .

The magnetic field evolution of the dark and grey exciton peaks is different. First, the intensity of dark(grey) exciton line is the same in both circular polarizations. Secondly, the peak energies are separated by the the exchange gap $\delta$:

[TABLE]

From the fit we extract $\delta=650\>\mu\mbox{eV}$ and $|g_{\mathrm{d}}|=9.6$, in agreement with Ref. 10 ($\delta=600\>\mu\mbox{eV}$ and $|g_{\mathrm{d}}|=9.4$) and our measurements in the in-plane field, described above ($\delta=660\>\mu\mbox{eV}$). In the single particle picture, the Zeeman shift in a perpendicular magnetic field has two contributions: the spin and the valley-orbital Aivazian2015 ; Srivastava2015 ; Li2014 ; Macneill2015 ; Koperski2017 ; Koperski2019 . The latter is the same for bright and dark excitons from the same valley. Hence, from the difference in the $g$ factors of dark and bright excitons we can extract the spin $g$ factor in the conduction band, which amounts to 2.7. Note that due to the results performed in combination of the in-plane and out-of-plane magnetic fields, we were able to determine that the grey-dark $g$-factor is negative similarly as for the bright neutral exciton and equals $g_{\mathrm{d}}=-9.4$.

S5 Polarization properties of dark excitons emission

We consider the lowest energy (dark) exciton states of S-TMD monolayer in the presence of non-zero out-of-plane $B_{\perp}$ and in-plane $\mathbf{B}_{\parallel}$ magnetic fields. $B_{\perp}$ couples $|\mathbf{K}_{+},\mathrm{d}\rangle$ and $|\mathbf{K}_{-},\mathrm{d}\rangle$ dark states of opposite valleys. $\mathbf{B}_{\parallel}$ mixes bright $|\mathbf{K}_{\pm},\mathrm{b}\rangle$ and dark $|\mathbf{K}_{\pm},\mathrm{d}\rangle$ states within the same valley. Taking into account both effects and considering the bright-dark exciton coupling perturbatively in the parameter $g_{\parallel}\mathcal{B}_{\parallel}/\Delta\equiv g_{\parallel}\mu_{B}|\mathbf{B}_{\parallel}|/2\Delta\ll 1$, one derives the lowest energy exciton states

[TABLE]

written in the basis $\{|\mathbf{K}_{+},\mathrm{b}\rangle,|\mathbf{K}_{-},\mathrm{b}\rangle,|\mathbf{K}_{+},\mathrm{d}\rangle,|\mathbf{K}_{-},\mathrm{d}\rangle\}$. Here \vartheta=\arcsin\Big{[}g_{\mathrm{d}}\mu_{B}B_{\perp}/\sqrt{\delta^{2}+\big{(}g_{\mathrm{d}}\mu_{B}B_{\perp}\big{)}^{2}}\Big{]}\in[-\pi/2,\pi/2] and \mathcal{N}=1/\sqrt{1+\big{(}g_{\parallel}\mathcal{B}_{\parallel}/\Delta\big{)}^{2}}. The indices $h$ and $l$ represents the higher and lower energy branches of these new exciton states, with corresponding energies up to quadratic in $g_{\parallel}\mathcal{B}_{\parallel}/\Delta$ terms

[TABLE]

Note that $|D_{h}\rangle$ and $|D_{l}\rangle$ states turn into gray $|G\rangle$ and dark $|D\rangle$ ones respectively in the limit of the zero magnetic field. Therefore, one can call $|D_{h}\rangle$ and $|D_{l}\rangle$ also dark exciton states for brevity.

The new dark excitons are optically active and their radiation patterns can be determined from the corresponding transition dipole matrix elements

[TABLE]

Both dipole vectors have the structure $\mathbf{d}_{\alpha}\equiv\langle 0|\mathbf{d}|D_{\alpha}\rangle=a_{\alpha}[\mathbf{e}_{x}+i\mathbf{e}_{y}]+b_{\alpha}[-\mathbf{e}_{x}+i\mathbf{e}_{y}]+c_{\alpha}\mathbf{e}_{z},\,\alpha=h,l$ and can be considered simultaneously.

The polarization degree of dark exciton emission is defined as

[TABLE]

Here $I_{\alpha}(\sigma_{\pm})$ is the total intensity of the $\sigma_{\pm}$ polarized light passing through the lens (see Fig. S4(a)). $I_{\alpha}(\sigma_{\pm})$ can be estimated by calculating the number of $\sigma_{\pm}$ polarized photons propagating inside the solid angle, defined by the colatitude $\theta_{0}$. We assume that the sample is placed in the focus of the lens, its size is much smaller than the size of the lens, and that the circular polarization of a photon is not affected as it passes through the lens.

The amount of $\sigma_{\pm}$ polarized photons, which are emitted from the sample, depends on the direction of their propagation. This direction is defined by spherical angles $(\theta,\varphi)$ or radial unit vector $\mathbf{e}_{r}$ (see Fig. S4(b)). The corresponding total intensities of light with fixed polarization can be evaluated as the integral

[TABLE]

Here $\mathbf{e}_{\pm}(\theta,\varphi)$ are the polarization vectors of the $\sigma_{\pm}$ polarized photons propagating in the direction defined by the angles $(\theta,\varphi)$. In order to write $\mathbf{e}_{\pm}$ explicitly, we introduce the local orthogonal unit vectors in the directions of increasing $r$, $\theta$ and $\varphi$, respectively (see Fig.S4 b)

[TABLE]

The spherical basis $\{\mathbf{e}_{\theta},\mathbf{e}_{\varphi},\mathbf{e}_{r}\}$ has the same relative orientation as $\{\mathbf{e}_{x},\mathbf{e}_{y},\mathbf{e}_{z}\}$. Hence, it is natural to define $\mathbf{e}_{\pm}(\theta,\varphi)\equiv(\pm\mathbf{e}_{\theta}+i\mathbf{e}_{\varphi})/\sqrt{2}$. Taking the corresponding scalar products

[TABLE]

and then using (10) we obtain

[TABLE]

Finally, the expressions for polarization degrees of higher-energy and lower-energy states read

[TABLE]

We fit the experimental data for lower-energy dark excitons polarization degree using the following parameters: $\Delta=38$ meV, $\mu_{B}=0.05788$ meV/T, $B_{\parallel}=|\mathbf{B}_{\parallel}|=1.5$ T, $|g_{\mathrm{d}}|=9.6$, $g_{\parallel}=2$, $\delta=0.65$ meV. The best fit corresponds to $\theta_{0}=1.139\approx 65^{\circ}$ and $d_{\perp}/d_{\parallel}=0.0035$, and is represented by a blue curve in Fig. S5. Using the same parameters we plot the curve for polarization degree for higher-energy dark excitons, see an orange curve in Fig. S5. For the considered range of parameters, the second curve approximates badly the corresponding experimental points. Probably, in the latter case there are additional unaccounted mechanisms, which reduce strongly the polarization degree of higher-energy exciton states.