Valley pseudospin, a novel quantum degree of freedom, is expected to show valley Zeeman effect in analogy to real spin in magnetic field B. By performing first-principles calculations, we studied the magnetic effect on valley pseudospin in monolayer WSe_{2}. With the application of B, the time reversal symmetry is broken. Our calculation shows that the valley energy degeneracy is broken and the valley Zeeman splitting varies linearly with B, agreeing well with the experiments. It is found that the valley Zeeman splitting is contributed mainly from the atomic orbital magnetic moment of W atom, but the valley contribution is still appreciable. The Berry curvatures of the two inequivalent valleys of monolayer WSe_{2} are opposite and their change induced by B also depends linearly on B. The calculated circular dichroism and dielectric function reveal that the optical valley-dependent selection rule is preserved and the original single peak in polarization-resolved photoluminescence spectrum will be split into two peaks after the application of B. Our studies demonstrate the possibility of magnetic manipulation of the valley pseudospin.

## I. INTRODUCTION

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 MX_{2} such as e.g., MoS_{2}, WSe_{2} 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 MX_{2} 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 MoS_{2}, MoSe_{2}, WSe_{2}, WS_{2} and MoTe_{2} 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 WSe_{2} 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.

## II. METHODS

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 WSe_{2} 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 formula^{24}

where $\psi nk$ is the calculated Bloch state and *E*_{nk} is the corresponding energy eigenvalue. The transition matrix element is $Pmnk=\psi mkP^\psi nk$, 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*)=*ε*_{0}+4π*iσ*(*ω*)/*ω*, where *ε*_{0} 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.

## III. RESULTS AND DISCUSSION

First, the electronic bands of the monolayer WSe_{2} 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 *V*1 and *V*2 at ±K valleys are predominantly from $W-dx2\u2212y2$ and *d*_{xy} orbitals with a hybridization of $dx2\u2212y2\xb1idxy$, resulting in an atomic magnetic quantum number of *m* = ±2. Whereas the valley states *C*1 and *C*2 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 $\psi nks^\psi nk$ 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 *V*1 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 $En\u2193K=En\u2191\u2212K$, 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. $\mu Korb=\u2212\mu \u2212Korb$.

Now we address the valley Zeeman effect of WSe_{2} 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 $\Delta KC\u2212V$ and $\Delta \u2212KC\u2212V$, respectively. The superscripts *V* and *C* represent valence and conduction band, respectively. The gap disparity $\delta =\Delta \u2212KC\u2212V\u2212\Delta KC\u2212V$ is called valley splitting. The optical transition peaks correspond to gaps $\Delta KC\u2212V$ and $\Delta \u2212KC\u2212V$ 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 *V*1 and *C*2, 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 $\Delta KC2\u2212V1=\Delta \u2212KC2\u2212V1=1.3012\u2009eV$, 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, $\Delta KC2\u2212V1$ is reduced by 1.06 meV while $\Delta \u2212KC2\u2212V1$ is increased by 1.08 meV, resulting in a nonzero valley splitting $\delta =\Delta \u2212KC2\u2212V1\u2212\Delta KC2\u2212V1=2.14\u2009meV$. 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 *V*1 and *C*2 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 $\Delta \xb1KV1\u2212C1$ and will not induce valley splitting. Therefore, the valley splitting *δ* induced by magnetic field is determined by the orbital magnetic moment $\mu orb$, which includes the atomic contribution $m\mu B$ and intercellular or valley contribution $\alpha \mu B$, 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 $\mu Korb,C2$ and $\mu Korb,V1$ of *C*2 and *V*1 band at K point is $mKC2+\alpha KC2\mu B$ and $mKV1+\alpha KV1\mu B$, respectively. As have been obtained above $mKV1=2$ and $mKC2=0$, the change of the optical gap $\Delta KC2\u2212V1$ in B is $\mu Korb,\u2009C2\u2212\mu Korb,\u2009V1B=mKC2+\alpha KC2\u2212mKV1\u2212\alpha KV1\mu BB=\u22122+\delta \alpha \mu BB$, where $\delta \alpha =\alpha KC2\u2212\alpha KV1$ is the intercellular orbital magnetic moment difference between the *V*1 and *C*2 band at the valley. Since the orbital magnetic moments of *C*2 and *V*1 bands at ±*K* valleys are opposite, the change of the optical gap is $\Delta \u2212KC2\u2212V1=\u2212\Delta KC2\u2212V1=2\u2212\delta \alpha \mu BB$. Therefore, a linear dependence of valley Zeeman splitting *δ* on B is obtained as $\delta =22\u2212\delta \alpha \mu BB$. Comparing the slope 2(2−*δα*)*μ*_{B} with the result 0.21 meV/Tesla obtained above, we get *δ*=3.63*μ*_{B}*B*, which means a small but non-negligible intercellular contribution 2*δα*=0.37, in good agreement with the experiment.^{10–12} For band gap difference of *V*2 and *C*1 at ±*K* valleys, similar conclusion is reached.

One fascinating property of WSe_{2} is the valley dependent optical selection rule. Due to the C_{3} 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 $Pcvk=\Psi ckP^\Psi vk$, 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 $P\xb1cvk=Pxcvk\xb1iPycvk/2$, in which ± stands for left (σ^{+}) and right (σ^{−}) circularly polarized light. It is found that the k-resolved degree of circular polarization $\eta k=P+cvk2\u2212P\u2212cvk2P+cvk2+P\u2212cvk2$ 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 C_{3} 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 WSe_{2}. Ω_{z}(*k*) of the top valence band *V*1 of WSe_{2} 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 WSe_{2} 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*)=*ε*_{0}+4*πi*σ(*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.

## IV. CONCLUSIONS

In conclusion, the valleytronics of WSe_{2} 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.

## ACKNOWLEDGMENTS

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).

## REFERENCES

*ab initio*programs, see http://www.openmx-square.org.