Valleytronic properties of monolayer WSe₂ in external magnetic field

Valley pseudospin is a novel quantum degree of freedom of electrons in solids analogous to spin, which corresponds to the states near the energy extrema in electronic bands of solids.^1,2 Since inequivalent valleys have opposite electrical, magnetic and optical properties, valleytronics is very promising in applications of information processing and storage.^3,4 Transition metal dichalcogenide (TMD) monolayer MX₂ such as e.g., MoS₂, WSe₂ are excellent valleytronic materials under intensive exploration.^5,6 The honeycomb TMD monolayer has a central transition metal atom layer between two chalcogen atom layers. There are two inequivalent Dirac valleys (Dirac cones) at ±K points. Mo and W atoms in MX₂ are nonmagnetic and the time reversal symmetry is preserved. Owing to spatial inversion asymmetry of TMD monolayer and time reversal symmetry, the Dirac valleys have valley-contrasting Berry curvatures, which underlies the fascinating Berry phase related quantum effects such as valley Hall effect, valley contrasting magnetic moments and valley dependent circular dichroism.^7–9 It has been shown that the circularly polarized σ⁺ (σ⁻) photons can selectively excite electron-hole pairs only at K (–K) valley. TMDs have very strong spin-orbit coupling (SOC), producing giant spin-splittings. Time reversal symmetry mandates opposite spin-splittings at ±K valleys, and hence there is valley-spin coupling in TMDs. The two inequivalent Dirac valleys have opposite orbital and spin magnetic moments,⁶ suggesting a route to magnetic manipulation of valley pseudospins.

Application of magnetic field B is the most tunable approach to controlling valley pseudospins. One can readily change the magnitude and direction of B to study the response of TMD monolayers. In analogy to Zeeman effect in an atom, the valley Zeeman effect of valley pseudospin is expected to occur in magnetic field. Recently, the valley Zeeman effect has been corroborated in the polarization-resolved photoluminescence spectrum of monolayer MoS₂, MoSe₂, WSe₂, WS₂ and MoTe₂ in external magnetic field.^10–17 The once degenerating σ⁺ and σ⁻ excitation peaks are split by B and the splitting varies linearly with B, indicating a valley energy splitting and the linear dependence on B. It is also observed that the valley energy splitting is inverted if the magnetic field switches direction.

Although there were some theoretical models to understand the valley Zeeman effect, the explanation was basically qualitative. It is still unknown how the band structure, spin state, orbital magnetic moment and Berry curvature could possibly respond to the external magnetic field. Also, it is still to explore the optical properties of TMDs in magnetic field, which corresponds to the experimental photoluminescent spectra. To the best of our knowledge, there have not been first-principles study of the valley Zeeman effects of TMDs to date. To understand the magnetic effects on the quantum valley pseudospin, we performed first-principles calculations on WSe₂ monolayer in magnetic fields. By calculating the Bloch wave functions, electronic band structure, Berry curvature, k-resolved degree of optical polarization and dielectric function, we are able to reveal the magnetic effects on the electronic, optical and Berry phase related properties of TMDs.

The first principles calculations are carried out with the density functional theory code OpenMX.^18–20 The generalized gradient approximation (GGA-PBE) is used.²¹ The localized atom-like orbitals (Mo-2s3p3d, Se-2s3p3d) are used to as basis function.^22,23 Atomic relaxation is done and the ionic force is smaller than 0.005 eV/Å. A 16×16×1 k-grid centered at point is employed for Brillouin zone sampling. We considered spin-orbit coupling (SOC) throughout the calculation. The hexagonal monolayer WSe₂ is modeled by a slab in a super-cell. The optimized lattice constants are a=b=3.32 Å. In the supercell, the slab is separated from its periodic images by 20 Å. The Berry curvature Ω_n(k) of band n at k point is calculated according to Kubo formula²⁴ 

where $ψnk$ is the calculated Bloch state and E_nk is the corresponding energy eigenvalue. The transition matrix element is $Pmnk=ψmkP^ψnk$⁠, where $P^$ is the momentum operator. We calculated the optical conductivity σ(ω) using Wannier90,²⁵ and obtained the dielectric function ε(ω) based on the relation: ε(ω)=ε₀+4πiσ(ω)/ω, where ε₀ is the dielectric constant of vacuum and ω the angular frequency of photons. The k-mesh for calculation of σ(ω) is 36×36×1. The total orbital magnetic moments of the valence bands are also calculated with Wannier90.

First, the electronic bands of the monolayer WSe₂ without external magnetic field is calculated for comparison. As depicted in Fig. 1(b), there is a direct band gap of 1.278 eV at Dirac valleys between bands of V1 and C1, being consistent with the previous calculation.²⁶ Since the heavy atom W produces strong SOC and the system has no spatial inversion symmetry, there is a giant spin splitting of 0.485 eV at the valence band maximum (VBM). However, the spin splitting is only 0.024 eV at the conduction band minimum (CBM).

We calculated wavefunction of the Bloch state ψ_nk at ±K points, and derived the expansion coefficients of the wavefunction with respect to the localized atom-like s, p and d orbitals. It is found that the valley states V1 and V2 at ±K valleys are predominantly from $W-dx2−y2$ and d_xy orbitals with a hybridization of $dx2−y2±idxy$⁠, resulting in an atomic magnetic quantum number of m = ±2. Whereas the valley states C1 and C2 at ±K valleys are mainly from $W-dz2$ orbitals with m=0. Accordingly, SOC leads to giant spin splitting in VBM, but a much smaller one in CBM. The calculated spin (⁠$s^$⁠) expectation values $ψnks^ψnk$ of the Bloch states ψ_nk show that the spin of the band edges at Dirac valley is normal to the monolayer and is opposite at the ±K valleys.⁶ Accordingly, band V1 has a down spin at K and an up spin at −K, as can be seen in Fig. 1(b). The time reversal symmetry requires that the Kramer’s pair at K and −K points and satisfies $En↓K=En↑−K$⁠, where ↑ and ↓ denote states with up and down spin, respectively. Therefore, the Bloch states ψ_nK and ψ_n−K are energetically degenerate and have opposite spin. We also calculated the total orbital magnetic moments μ^orb of the valence bands by using Wannier90. As shown in Fig. 2(a), it is found that μ^orb is mainly aligned along the z axis and also opposite at ±K valleys, i.e. $μKorb=−μ−Korb$⁠.

Now we address the valley Zeeman effect of WSe₂ induced by external magnetic field. Since optical transition can occur only between the states with the same spin in photoluminescence (PL) experiment, we are more interested in the optical energy gap between the bands of the same spin at ±K valleys, which is denoted as $ΔKC−V$ and $Δ−KC−V$⁠, respectively. The superscripts V and C represent valence and conduction band, respectively. The gap disparity $δ=Δ−KC−V−ΔKC−V$ is called valley splitting. The optical transition peaks correspond to gaps $ΔKC−V$ and $Δ−KC−V$ in PL spectrum will overlap if δ=0 and will be split into two if δ≠0. We will focus on the gap between bands V1 and C2, since they have the same spin at K as well as at −K [Fig. 1(b)]. When no magnetic field is applied, there is time reversal symmetry and valley degeneracy is guaranteed. Accordingly, our calculation gives identical energy gap $ΔKC2−V1=Δ−KC2−V1=1.3012 eV$⁠, in agreement with the experimental observation that the PL spectrum in zero magnetic field has only one peak and that there is no valley splitting (δ=0).^10–13 When a nonzero magnetic field B is applied along z direction, we take into account the Zeeman term μB in the calculation, where μ is the magnetic moment of the Bloch states. Our calculations show that the valley degeneracy is lifted. For instance, when B=10 Tesla, $ΔKC2−V1$ is reduced by 1.06 meV while $Δ−KC2−V1$ is increased by 1.08 meV, resulting in a nonzero valley splitting $δ=Δ−KC2−V1−ΔKC2−V1=2.14 meV$⁠. This is because that time reversal symmetry is broken by external magnetic field and E_n↓(K)= E_n↑(−K) does not hold. The energy of a Bloch state will be shifted by μB in a magnetic field. As have been shown above that the two inequivalent valleys have anti-parallel magnetic moments μ, the energy of the states at ±K valleys will be shifted by μ(K)B and −μ(K)B, respectively. Such opposite energy shift of the valley states gives rise to nonzero valley splitting δ. When the direction of B is reversed (B=−10 Tesla), the valley splitting δ takes the opposite value.

The energy bands at B=−15, −10, −5, 0, 5, 10 and 15 Tesla are calculated to study the valley splitting dependence on the magnetic field. In Fig. 1(d), one can see that the valley splitting δ varies linearly with respect to B with a slope of 0.21 meV/Tesla, agreeing with the experiments well.¹² Since V1 and C2 bands have the same spin, the spin-induced energy shift of the two bands at K (−K) is the same. Hence, spin will not change the gap $Δ±KV1−C1$ and will not induce valley splitting. Therefore, the valley splitting δ induced by magnetic field is determined by the orbital magnetic moment $μorb$⁠, which includes the atomic contribution $mμB$ and intercellular or valley contribution $αμB$⁠, where m is the atomic magnetic quantum number and μ_B is the Bohr magneton (0.05788 meV/Tesla).^11–13 αμ_B is valley magnetic moment related to the Berry curvature.¹³ The orbital magnetic moments $μKorb,C2$ and $μKorb,V1$ of C2 and V1 band at K point is $mKC2+αKC2μB$ and $mKV1+αKV1μB$⁠, respectively. As have been obtained above $mKV1=2$ and $mKC2=0$⁠, the change of the optical gap $ΔKC2−V1$ in B is $μKorb,C2−μKorb,V1B=mKC2+αKC2−mKV1−αKV1μBB=−2+δαμBB$⁠, where $δα=αKC2−αKV1$ is the intercellular orbital magnetic moment difference between the V1 and C2 band at the valley. Since the orbital magnetic moments of C2 and V1 bands at ±K valleys are opposite, the change of the optical gap is $Δ−KC2−V1=−ΔKC2−V1=2−δαμBB$⁠. Therefore, a linear dependence of valley Zeeman splitting δ on B is obtained as $δ=22−δαμBB$⁠. Comparing the slope 2(2−δα)μ_B with the result 0.21 meV/Tesla obtained above, we get δ=3.63μ_BB, which means a small but non-negligible intercellular contribution 2δα=0.37, in good agreement with the experiment.^10–12 For band gap difference of V2 and C1 at ±K valleys, similar conclusion is reached.

One fascinating property of WSe₂ is the valley dependent optical selection rule. Due to the C₃ rotational symmetry of the monolayer, selective excitation of valley carriers can be realized by right or left handed polarized light.^9,27,28 We calculated the interband transition matrix elements $Pcvk=ΨckP^Ψvk$⁠, where $P^$ is the momentum operator, v and c represent valence and conduction bands, respectively. The optical transition matrix elements of circularly polarized light are $P±cvk=Pxcvk±iPycvk/2$⁠, in which ± stands for left (σ⁺) and right (σ⁻) circularly polarized light. It is found that the k-resolved degree of circular polarization $ηk=P+cvk2−P−cvk2P+cvk2+P−cvk2$ is almost exactly 1 and −1 at K and −K and their neighboring zones have similar η(k, ω_cv) values, indicating a clear valley contrasting circular dichroism. After applying magnetic field, the C₃ symmetry is retained. Our calculations show that the k-resolved degree of circular polarization η(k) almost does not change [Fig. 2(b)], indicating that the valley dependent circular dichroism maintains and hence the valley carriers can be selectively excited in magnetic field. Further considering the giant spin splitting at the valleys and the spin selection rule, in addition to the selective pumping of K and −K valley by σ⁺ and σ⁻ light, selective excitation of spin state can be realized by choosing proper frequency of the light, as illustrated in Fig. 1(c).

To study the Berry phase related physics, we calculated the Berry curvature Ω_z(k) of the system. Time reversal symmetry dictates that Ω_z(−k) = −Ω_z(k) whereas spatial inversion symmetry demands that Ω_z(−k) = Ω_z(k).²⁹ Berry curvature can take nonzero values for the system without spatial inversion symmetry, which is the case of monolayer WSe₂. Ω_z(k) of the top valence band V1 of WSe₂ is calculated, as shown in Fig. 3(a). When B=0, Ω_z takes opposite values at the two Dirac valleys so that Ω_z(−K)+Ω_z(K) = 0, and it has valley contrasting peaks. After magnetic field is applied, the Berry curvature does not strictly satisfy Ω_z(−K)=−Ω_z(K) because the time reversal symmetry is broken. As depicted in Fig. 3(b), Ω_z(−K)+Ω_z(K) deviates from zero and the deviation also grows linearly with the increase of magnetic field. Since the Berry curvature still has different signs at inequivalent valleys, the valley Hall effect is preserved after application of magnetic field.

Valley Zeeman splitting is manifested as the splitting between polarization-resolved photoluminescence peak energies of σ⁺ and σ⁻ emission observed in experiments. To study the optical spectra of WSe₂ in magnetic field, we calculated the dielectric function. The optical conductivity σ(ω) is first calculated by using Wannier90. The d and p orbitals are chosen as the projection functions for W and Se atoms. The interpolated electronic band structure obtained by Wannier90 almost coincides with that by DFT calculation. The dielectric function can be derived by ε(ω)=ε₀+4πiσ(ω)/ω. The imaginary part of dielectric function for absorption of circularly polarized light in different magnetic fields is depicted in Fig. 3(c), which corresponds to the experimental PL spectra. When B=0, the σ⁺ and σ⁻ absorption peaks overlap because valleys at ±K are energetically degenerate. The absorption spectrum in the nonzero magnetic field has two splitting σ+ and σ− peaks. When B is reversed, the two peaks is switched in position and the splitting turns opposite. Our results are in good accord with the experimental observations.^10–13 

It can be seen in Fig. 3(c) that there is a broadening in the calculated spectra. One may expect that the transition between the VBM and CBM edges at ±K points will only yield line spectra. In reality, the excitation light used in photoluminescence experiment has a broadening around a central frequency.³⁰ Therefore, transitions occur not only between the extrema at the VBM and CBM, but also between the states around the them, giving rise to a broadening in the photoluminescence spectra. In our calculation, the light energy is chosen to be continuum near the energy corresponding to the band gap, and hence the transitions between the states near the VBM and CBM are included. Therefore, the theoretical spectra in Fig. 3(c) are broadened as well.

Another feature of the spectra in magnetic field shown in Fig. 3(c) is that the left peak is higher than the right peak. The intensity of the theoretical spectrum at certain frequency depends not only on the transition probability (proportional to the interband transition matrix element), but also on the number of the states (joint density of states, JDOS) that can contribute to the interband transition at this frequency. We calculated the JDOS in magnetic field and found that the JDOS has two peaks with the left one higher than the right one as a result of energy shift induced by external magnetic field, as shown in the inset of Fig. 3(c). Therefore, there are more states participating the optical transitions around the left peak than those around the right peak, which tends to produce larger intensity at the left peak in the theoretical spectra. In the valley Zeeman effect experiments, however, the results are more complicated. The left peak in some photoluminescent spectra is found to be higher than the right one,^11,14 whereas in other experiments, the two peaks have similar intensity.^12,13 In addition to the optical properties of the materials, the experimental photoluminescent spectra depend on the light source and detectors as well. If the intensity of light and sensitivity of the detectors are uniform at the interested frequency range, one may observe that the left peak is higher than the right one. If the light has higher intensity and the detectors are more sensitive at higher frequency, one may observe that the two peaks have similar intensity. This may account for the different observations of the intensity of the two peaks in experiments.

In conclusion, the valleytronics of WSe₂ monolayer in external magnetic field has been studied by first-principles calculations. It is found that the two inequivalent Dirac valleys, which were energetically degenerate without magnetic field, exhibit valley Zeeman splitting when B ≠ 0 because of the breaking of time reversal symmetry. The valley splitting has a linear dependence on B, with a slope of 0.21 meV/Tesla, in good agreement with the experiments.^10–12 Our analysis has shown that the valley Zeeman splitting is mainly ascribed to the atomic orbital magnetic moment of W atom, but the valley contribution is nonnegligible. The change of the Berry curvatures is small and also linearly dependent on the magnetic field. The valley Hall effect and valley-selective circular dichroism are preserved. The calculated imaginary part of dielectric function agrees with the experimental PL spectrum well.^10–13 Our calculations manifest that valley pseudospin can be manipulated by magnetic field through valley Zeeman effect.

The authors acknowledge the support of the National Natural Science Foundation of China (Grant No. 11874315 and No. 11874316), National Basic Research Program of China (Grant No. 2015CB921103), and the Oak Ridge Institute for Science and Education (ORISE) HERE Program (J Z).