Using polarization-resolved photoluminescence spectroscopy, we investigate the breaking of valley degeneracy by an out-of-plane magnetic field in back-gated monolayer ${\mathrm{MoSe}}_2$ devices. We observe a linear splitting of $-0.22\mathrm{meV}/\mathrm{T}$ between luminescence peak energies in ${\sigma }_{+}$ and ${\sigma }_{-}$ emission for both neutral and charged excitons. The optical selection rules of monolayer ${\mathrm{MoSe}}_2$ couple the photon handedness to the exciton valley degree of freedom; so this splitting demonstrates valley degeneracy breaking. In addition, we find that the luminescence handedness can be controlled with a magnetic field to a degree that depends on the back-gate voltage. An applied magnetic field, therefore, provides effective strategies for control over the valley degree of freedom.

We use a combined ab initio calculations and $\mathbf{k}\ifmmode\cdot\else\textperiodcentered\fi{}\mathbf{p}$ theory based approach to derive a low-energy effective Hamiltonian for monolayer ${\mathrm{MoS}}_{2}$ at the $K$ point of the Brillouin zone. It captures the features which are present in first-principles calculations but not explained by the theory of Xiao et al. [Phys Rev Lett 108, 196802 (2012)], namely the trigonal warping of the valence and conduction bands, the electron-hole symmetry breaking, and the spin splitting of the conduction band. We also consider other points in the Brillouin zone which might be important for transport properties. Our findings lead to a more quantitative understanding of the properties of this material in the ballistic limit.

We derive an effective Hamiltonian that describes the dynamics of electrons in the conduction band of monolayer transition metal dichalcogenides (TMDC) in the presence of perpendicular electric and magnetic fields. We discuss in detail both the intrinsic and the Bychkov-Rashba spin-orbit coupling induced by an external electric field. We point out interesting differences in the spin-split conduction band between different TMDC compounds. An important consequence of the strong intrinsic spin-orbit coupling is an effective out-of-plane $g$ factor for the electrons that differs from the free-electron $g$ factor $g\simeq 2$. We identify a new term in the Hamiltonian of the Bychkov-Rashba spin-orbit coupling that does not exist in III-V semiconductors. Using first-principles calculations, we give estimates of the various parameters appearing in the theory. Finally, we consider quantum dots formed in TMDC materials and derive an effective Hamiltonian that allows us to calculate the magnetic field dependence of the bound states in the quantum dots. We find that all states are both valley and spin split, which suggests that these quantum dots could be used as valley-spin filters. We explore the possibility of using spin and valley states in TMDCs as quantum bits, and conclude that, due to the relatively strong intrinsic spin-orbit splitting in the conduction band, the most realistic option appears to be a combined spin-valley (Kramers) qubit at low magnetic fields.

We propose an interband tunneling picture to explain and predict the interlayer twist-angle dependence of the induced spin-orbit coupling in heterostructures of graphene and monolayer transition metal dichalcogenides (TMDCs). We obtain a compact analytic formula for the induced valley Zeeman and Rashba spin-orbit coupling in terms of the TMDC band structure parameters and interlayer tunneling matrix elements. We parametrize the tunneling matrix elements with few parameters, which in our formalism are independent of the twist angle between the layers. We estimate the value of the tunneling parameters from existing density functional theory calculations at zero twist angle and we use them to predict the induced spin-orbit coupling at nonzero angles. Provided that the energy of the Dirac point of graphene is close to the TMDC conduction band, we expect a sharp increase of the induced spin-orbit coupling around a twist angle of ${18}^{\circ }$.

The strong spin-orbit coupling and the broken inversion symmetry in monolayer transition metal dichalcogenides results in spin-valley coupled band structures. Such a band structure leads to novel applications in the fields of electronics and optoelectronics. Density functional theory calculations as well as optical experiments have focused on spin-valley coupling in the valence band. Here we present magnetotransport experiments on high-quality $n$-type monolayer molybdenum disulphide (${\mathrm{MoS}}_2$) samples, displaying highly resolved Shubnikov–de Haas oscillations at magnetic fields as low as 2 T. We find the effective mass $0.7m_e$, about twice as large as theoretically predicted and almost independent of magnetic field and carrier density. We further detect the occupation of the second spin-orbit split band at an energy of about 15 meV, i.e., about a factor of 5 larger than predicted. In addition, we demonstrate an intricate Landau level spectrum arising from a complex interplay between a density-dependent Zeeman splitting and spin- and valley-split Landau levels. These observations, enabled by the high electronic quality of our samples, testify to the importance of interaction effects in the conduction band of monolayer ${\mathrm{MoS}}_2$.