Abstract
Local energy extrema of the bands in momentum space, or valleys, can endow electrons in solids with pseudospin in addition to real spin^{1,2,3,4,5}. In transition metal dichalcogenides this valley pseudospin, like real spin, is associated with a magnetic moment^{1,6} that underlies the valleydependent circular dichroism^{6} that allows optical generation of valley polarization^{7,8,9}, intervalley quantum coherence^{10} and the valley Hall effect^{11}. However, magnetic manipulation of valley pseudospin via this magnetic moment^{12,13}, analogous to what is possible with real spin, has not been shown before. Here we report observation of the valley Zeeman splitting and magnetic tuning of polarization and coherence of the excitonic valley pseudospin, by performing polarizationresolved magnetophotoluminescence on monolayer WSe_{2}. Our measurements reveal both the atomic orbital and lattice contributions to the valley orbital magnetic moment; demonstrate the deviation of the band edges in the valleys from an exact massive Dirac fermion model; and reveal a striking difference between the magnetic responses of neutral and charged valley excitons that is explained by renormalization of the excitonic spectrum due to strong exchange interactions.
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Main
In monolayer transition metal dichalcogenides (TMDs), there is a valley pseudospin 1/2 which describes the two inequivalent but energydegenerate band edges (the ±K valleys) at the corners of the hexagonal Brillouin zone^{1}. With broken inversion symmetry, electrons in the two valleys can have finite orbital contributions to their magnetic moments which are equal in magnitude but opposite in sign by timereversal symmetry. This orbital magnetic moment is thus linked to the valley pseudospin in the same way that the bare magnetic moment (gμ_{B}S) is linked to the real spin S, where μ_{B} is the Bohr magneton and g is the Lande gfactor. The orbital magnetic moment in turn has two parts: a contribution from the parent atomic orbitals, and a ‘valley magnetic moment’ contribution from the lattice structure^{1} (Fig. 1a, top). The latter is related to the Berry curvature that produces the valley Hall effect^{11}.
The valley magnetic moment results in a valleydependent optical selection rule in monolayer TMDs, where light of σ^{+} (σ^{−}) circular polarization excites electron–hole pairs exclusively in the +K (−K) valley. This enables optical manipulation of the valley pseudospin through its excitonic states^{7,8,9,10,14,15,16,17}, or valley excitons. The neutral and charged valley excitons, with their exceptionally strong Coulomb interaction^{18,19,20,21,22,23,24}, are subject to a momentumdependent gauge field arising from electron–hole exchange, or valley–orbit coupling, which at zero magnetic field is predicted to result in massless and massive dispersion, respectively, within the light cone^{25}. This implies the possibility of controlling excitonic valley pseudospin via the Zeeman effect in an external magnetic field.
Our measurements of polarizationresolved photoluminescence (PL) in a magnetic field perpendicular to the 2D plane are performed on mechanically exfoliated WSe_{2} monolayers. We have obtained consistent results from many samples. The data presented here are all taken from one sample at a temperature of 30 K. To resolve the splitting between the +K and −K valley excitons, which is significantly smaller than the exciton linewidth (∼10 meV), we both excite and detect with a single helicity of light. In this way we address one valley at a time, and the splitting can be determined by comparing the peak positions for different polarizations.
Figure 1b shows the normalized PL spectra for the neutral exciton peak at selected values of the magnetic field B. At zero field (middle) the PL from the +K valley exciton (blue, σ^{+}) is identical to that from the −K valley (red, σ^{−}), as expected from timereversal symmetry^{10}. In contrast, at high field the σ^{+} and σ^{−} components are split, with σ^{−} at a slightly higher energy than σ^{+} for +7 T (top) and lower for −7 T (bottom). The small variations in lineshape seen here in the σ^{−} emission are artefacts related to sample inhomogeneity. To minimize their effect we extract the splitting in two ways (see Supplementary Methods 1), neither of which involves fitting the peaks. The results of both are shown in Fig. 1c. The black squares are obtained from the maximal points of the peaks, whereas the red circles are obtained from their ‘centre of mass’. These two very different methods both yield a similar splitting proportional to B. The error bar also shown corresponds to the spectral resolution of the detection system. For a conservative estimate of the slope, we fit the rougher ‘max point’ data to a straight line, yielding a slope of −0.11 ± 0.01 meV T^{−1} = −(1.9 ± 0.2) μ_{B}.
The observed magnetic spectral splitting can be explained by the combination of the magnetic moment of the tungsten dorbitals^{26} and the ‘valley magnetic moment’ m_{τ}, the lattice contribution associated with Berry curvature^{6}. The bottom panel of Fig. 1a shows the Zeeman shift of the band edges from each of these two contributions as well as that from the bare spin. The dashed (solid) lines are the conduction and valence band edges at zero (positive) magnetic field, with blue and red denoting spin up and down, respectively. Because of the giant spin splitting (∼0.4 eV) in the valence band, the valence band edge in the +K (−K) valley has only spinup (down) states. For the conduction band edge, on the other hand, the spin splitting is small (∼0.03 eV), with opposite sign in the two valleys^{26,27}, and both spin states are relevant. The overall Zeeman shift of each band is determined by the sum of these three contributions to magnetic moment, which leads to an effective gfactor different from the vacuum value of 2.
The Zeeman shift due to the spin magnetic moment (Δ_{s} = 2s_{z}μ_{B}B, black arrows) does not affect the optical resonances because optical transitions conserve spin so that the effect on the initial and final states is the same. The atomic orbital contribution, however, does affect them because the conduction band edges are mainly composed of dorbitals with m = 0, whereas the valence band edges are mainly dorbitals with m = 2 in the +K valley and m = −2 in the −K valley. This contributes no shift to the conduction band and a shift of Δ_{α} = 2τμ_{B}B to the valence band edge (purple arrows), resulting in a net shift of the optical resonances by −2τμ_{B}B, where τ = ±1 is the index for the ±K valleys.
The Zeeman shift due to the valley magnetic moment is Δ_{v} = m_{τ}B (green arrows), with m_{τ} = α_{i}τμ_{B}, where α_{i} is the valley gfactor for band i (i = c, v). The leading order k ⋅ p approximation for the bandedge carriers yields a massive Dirac fermion model^{1,6} with α = (m_{0}/m^{∗}), where m^{∗} is the effective mass, which is the same for both conduction and valence bands^{1}. Within this approximation, the valley magnetic moment therefore does not affect the exciton resonances, just as for the bare spin. However, corrections beyond leading order give different effective masses and different valley magnetic moments for the electrons and holes^{26,27,28}. The result is a valleydependent shift of the optical resonances by τΔαμ_{B}B, where Δα = α_{c} − α_{v}.
The net effect is a valleydependent linear shift of the exciton resonance by −τΔ(B)/2, where Δ(B) = 2(2 − Δα)μ_{B}B is the excitonic valley Zeeman splitting. Thus the +K valley exciton (τ = 1) should be redshifted with respect to the −K valley exciton (τ = −1) for B > 0, and blueshifted for B < 0, consistent with the observations in Fig. 1b. The best fit to the data (grey line) in Fig. 1c yields Δα = 1.1 ± 0.1. The average value of Δα found over five similar samples was 1.2. We note that there are some variations between samples (Supplementary Methods 1), which can be separated into two groups with mean splittings of 1.57 μB and 2.86 μB. The origin of this bimodal distribution in the gfactors is unclear owing to the lack of understanding of what external factors can affect the gfactors in these new materials. Future studies will be necessary to quantitatively determine the effect of variables such as strain, doping and substrate on the magnetic properties. Nevertheless, this measurement of a definite nonzero Δα directly implies the existence of finite valley magnetic moments (and therefore finite Berry curvature), in addition to a deviation from the massive Dirac fermion model (Supplementary Methods 2).
The splitting in the applied magnetic field breaks the valley degeneracy, enabling control of the valley polarization. To investigate this we measure the degree of PL polarization for both helicities of incident circular polarization. Figure 2a shows PL for σ^{−} excitation with σ^{−} (red) and σ^{+} (orange) detection at a field of −7 T. The suppression of the σ^{+} signal relative to the copolarized σ^{−} peak is a signature of optically pumped valley polarization^{7,8,9,10}. The degree of exciton valley polarization is clearly larger for σ^{+} excitation than for σ^{−} excitation (Fig. 2b). On the other hand, when the magnetic field is reversed to +7 T (Fig. 2c, d) the polarization becomes larger for σ^{−}. This observation implies that, whereas the sign of the valley polarization is determined by the helicity of the excitation laser, its magnitude depends on the relationship between the helicity and the magnetic field direction.
Figure 2e shows the degree of PL polarization for both σ^{+} (blue) and σ^{−} (red) excitation as a function of B between −7 T and +7 T for the neutral exciton peak. It is linear in B, with a negative (positive) slope. This ‘X’ pattern implies that the valley Zeeman splitting induces an asymmetry in the intervalley scattering. (Note that the overall tilt of the ‘X’ pattern seen here signifies an asymmetry of the response of the entire experimental system to magnetic field whose origin we do not know, but it does not affect any of our conclusions.) In contrast, the PL polarization of the negative trion peak increases for either sign of B and shows a ‘V’ pattern (Fig. 2f).
These findings can be understood as resulting from magnetic tuning of the different dispersions of valley excitons and trions, as illustrated in Fig. 3. Exchange interactions between electrons and holes strongly couple the valley pseudospin to the exciton centreofmass wavevector k, splitting the exciton dispersion into two branches (Supplementary Methods 3 and 4; ref. 25). As shown in Fig. 3a, the upper branch has a steeper dispersion: its states have much smaller momenta than states at the same energy on the lower branch (Fig. 3a). At B = 0, the two branches touch at k = 0, where the two degenerate eigenstates are the −K and +K valley excitons, which emit σ^{−} and σ^{+} light, respectively (Fig. 3a, middle). A finite magnetic field lifts this degeneracy and opens a gap Δ(B). For B > 0 the centres of the upper and lower branches are the −K and +K valley excitons respectively^{25} (Fig. 3a, right). For B < 0 these are interchanged (Fig. 3a, left).
The ‘X’ pattern for neutral excitons results from the fact that the formation of σ^{−} excitons for B > 0 and σ^{+} excitons for B < 0 is much easier because of the magnetic tuning of excitonic dispersion. At B > 0 with σ^{+} excitation (Fig. 3b), electrons and holes are created at k = 0 in the +K valley and relax to form +K excitons (blue, centre of lower exciton branch) at a valleyconserving rate γ_{1} or −K excitons (red, centre of upper branch) at a valleyflipping rate γ_{2}. For σ^{−} excitation (Fig. 3c), the valleyconserving and valleyflipping processes result in −K and +K valley excitons, respectively, instead. The degree of PL polarization is determined by valley depolarization both during exciton formation (that is, the ratio γ_{2}/γ_{1}) and in the exciton ground state before recombination (r_{1}). The measurements in Fig. 2a–d illustrate our finding that in all cases the higherenergy exciton retains more valley polarization than the lowerenergy exciton. This is the opposite of what would result from thermal relaxation to the lowestenergy valleyexciton state. It implies that the PL polarization is largely determined during the exciton formation process.
The steeper dispersion of the upper exciton branch should facilitate formation of excitons in this branch relative to those in the lower branch, in which larger momentum transfers by scattering are required to reach the light cone (see Supplementary Fig. 2). Therefore, for B > 0, the carriers created by σ^{−} excitation have a larger valleyconserving rate γ_{1} and a smaller valleyflipping rate γ_{2} than for σ^{+}, leading to γ_{1}(σ^{−})/γ_{2}(σ^{−}) > γ_{1}(σ^{+})/γ_{2}(σ^{+}) and a larger valley polarization, as observed. For B < 0 the converse holds by timereversal symmetry. By solving the rate equations, taking the fielddependent valleyexciton formation process and measured valley Zeeman splitting (Fig. 1c) into account, the ‘X’ pattern can be fully reproduced, as shown by the solid lines in Fig. 2e (Supplementary Methods 5).
Within the same framework, the ‘V’ pattern seen for negative trions (X^{−}) can also be explained as resulting from their qualitatively different spectrum. With a second electron occupying either the lowestenergy spinup band in the +K valley or spindown band in the −K valley (top panel, Fig. 3d), at B = 0, X^{−} has two degenerate sets of valley–orbit coupled bands where the large exchange interaction with the extra electron opens a gap δ (∼6 meV; ref. 25) at k = 0 (bottom panel, Fig. 3d). As δ is already much larger than the achievable valley Zeeman splitting, the asymmetry in valleyexciton formation rates in the presence of a field does not dominate the B dependence of the X^{−} valley polarization. Instead, the valley Zeeman splitting breaks the energy degeneracy of the X^{−} ground states, which suppresses the valley relaxation channels (grey arrows in Fig. 3e) relative to their zerofield rates. This mechanism protects the valley polarization and increases the PL polarization for either sign of B. Taking into account the valley depolarization in the exciton formation process, as in the neutral exciton case, we can reproduce the ‘V’ pattern of X^{−} valley polarization, the result being the solid lines in Fig. 2f (Supplementary Methods 5 and 6).
Simply put, the difference between the ‘X’ and ‘V’ patterns reflects the different dominant valley depolarization process. If depolarization occurs during exciton formation, one sees an ‘X’ pattern because exciton formation favours the upper exciton branch. If, on the other hand, depolarization occurs predominantly after reaching the ground state, one sees a ‘V’ shape because either sign of magnetic field suppresses the groundstate intervalley relaxation.
Finally, we investigate the magnetic field dependence of valley coherence. Figure 4a shows the polarizationresolved PL spectrum at selected fields with vertically polarized excitation and vertically (purple) and horizontally (black) polarized detection. As shown previously^{10}, the observed linear polarization of the exciton PL is due to the creation of valley quantum coherence. As the magnetic field increases, the degree of linear polarization decreases and shows a ‘Λ’ pattern (Fig. 4b). This demonstrates that valley coherence is suppressed by magnetic field. The data for horizontally polarized excitation is also plotted, showing that the effect is isotropic and not due to any crystal anisotropy.
Like depolarization, valley decoherence can occur during both the exciton formation and its groundstate recombination processes. The magnetic field dependence in the latter case is the Hanle effect, in which spin precession quenches the linear polarization, and the halfwidth of the Hanle peak corresponds to the decoherence rate. Here, the linearly polarized exciton has an inplane valley pseudospin. The Hanle effect is qualitatively consistent with the ‘Λ’ pattern observed; however, the extracted valley decoherence and the exciton recombination times are both of the order of 1 ps (Supplementary Methods 7), which is much smaller than those deduced from timeresolved measurements^{17}. It is therefore likely that valley pseudospin precession and decoherence in the exciton formation process dominate. After linearly polarized excitation generates electron–hole pairs in a superposition of the two valleys, the +K and −K valleyexciton formation pathways (the red and blue wavy lines in Fig. 3b) become different as the magnetic field opens the gap (as inferred from the polarization data in Fig. 2e). This difference destroys the optically generated coherence during the formation of groundstate excitons, leading to reduced linear polarization of the PL.
During preparation of manuscript, we became aware of similar work on WSe_{2} by the ETH group^{28} and on MoSe_{2} by the Cornell group^{29}.
Methods
A monolayer sample of WSe_{2} is mechanically exfoliated onto 300 nm SiO_{2} on heavily doped Si. The samples are cooled typically to 30 K in a closedcycle cryostat with a 7 T superconducting magnet in the Faraday geometry. The samples are excited with a 1.88 eV laser focused to a 2 μm spot size with an aspheric lens. The photoluminescence was collected with same lens and freespace coupled to a spectrometer with a liquid nitrogen cooled CCD detector.
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Acknowledgements
We thank X. Li for helpful discussions. This work is mainly supported by the DoE, BES, Materials Sciences and Engineering Division (DESC0008145). Z.G. and W.Y. were supported by the Croucher Foundation (Croucher Innovation Award) and the RGC of Hong Kong (HKU705513P, HKU9/CRF/13G). D.C. is supported by US DoE, BES, Materials Sciences and Engineering Division (DESC0002197). J.Y. and D.G.M. were supported by US DoE, BES, Materials Sciences and Engineering Division. RL.C. and C.Z. are supported by ARO (W911NF1210334) and AFOSR (FA95501310045). X.X. acknowledges a Cottrell Scholar Award. Device fabrication was performed at the University of Washington Microfabrication Facility and NSFfunded Nanotech User Facility.
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X.X. and W.Y. conceived the project; G.A. performed the experiment, assisted by A.M.J., under the supervision of X.X.; G.A. and X.X. analysed the data; Z.G. and W.Y. provided the theoretical explanation, with input from RL.C. and C.Z.; J.Y. and D.G.M. synthesized and characterized the bulk WSe_{2} crystals; G.A., X.X., W.Y., D.C. and Z.G. wrote the paper. All authors discussed the results.
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Aivazian, G., Gong, Z., Jones, A. et al. Magnetic control of valley pseudospin in monolayer WSe_{2}. Nature Phys 11, 148–152 (2015). https://doi.org/10.1038/nphys3201
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DOI: https://doi.org/10.1038/nphys3201
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