Cyclic structure of Landau levels in transition metal dichalcogenide semiconductors

Abstract

Transition metal dichalcogenides (TMDs) exhibit unconventional Landau level (LL) spectra that cannot be fully captured by an effective mass approximation or a minimal two-band Dirac model. Namely, TMDs show an anomalous, upward-sloping zeroth LL in the valence band and an asymmetric orbital magnetization between electron and hole bands. In this paper, we employ a continuum three-band model to derive analytic constraints on the LL spectrum of the $K$ and $K'$ valleys at weak magnetic fields. This model highlights the cyclic structure of the LL spectrum inherited from $C_3$ symmetry, providing both analytical tractability and an accurate description of the band geometry in the low energy approximation of the valleys. We compare our results against numerical calculations using the three-band tight-binding model of Ref.[1] and a distorted kagome lattice model. We find that the Landau levels of the $K$ and $K'$ valleys show a cyclic structure which explains their anomalous slope and magnetization asymmetry. This asymmetry can be traced to the topological obstruction of TMD semiconductors. We further analyze the impact of disorder, finding that the zeroth LL exhibits partial robustness against certain off-diagonal perturbations, in contrast to the exact index-theorem protection of massive Dirac particles. Our results establish a direct link between orbital structure, band topology, and magnetic response in TMDs.

Figures (5)

• Figure 1: (a) Band structure for the three-band tight-binding model of MoTe$_2$, and (b) its LL spectrum as a function of magnetic flux $\phi$ in units of the flux quantum $\phi_0 = h / e$. The energy unit is 1 eV. (c), (e) Band structures and (d), (f) LL spectra of the distorted kagome lattice for $t = 0.5, t^\prime = 1$ and $t = 1.5, t^\prime = 1$, respectively, as functions of magnetic flux $\phi$ in units of the flux quantum. The color denotes the expectation value of $S = (3/2\pi) \arg Z$, where $S=0, 1, -1$ correspond to the $\omega^0 (\gamma=0), \omega^1 (\gamma=1)$, and $\omega^2 (\gamma=2)$ orbital characters, respectively. $K$ and $K'$ in the LL plots indicate which valley the lowest (highest) LL at the conduction (valence) band comes from.
• Figure 2: (a) Triangular lattice for the TMD model. The lobes demonstrate the Wannier function of the valence band to the lowest order approximation (Eq. \ref{['eq:Wannier_TMD']}). $\hat{a}, b, c$ labels the three inequivalent Wyckoff positions. Note that the chalcogen atoms are located at Wyckoff positions $c$, which misaligns with the Wannier center. (b) Distorted kagome lattice with $\alpha$ controlling the size of the unit cell comparing to the lattice spacing. $\hat{a}, B, C$ labels the three sites in one unit cell. $t$ ($t^\prime$) denotes the nearest neighbor intra-(inter-)cell hopping strength.
• Figure 3: The evolution of density of states (in arbitrary units, scaled for clarity) as a function of energy $E$ as the correlation length $\eta$ increases for graphene with $m = 0, \phi = 0.05\phi_0$ ((a)-(c)), graphene with $m = 0.4 t, \phi = 0.05\phi_0$ ((d)-(e)) and MoTe$_2$ with $\phi = 0.1\phi_0$ ((g)-(i)). For graphene, the color denotes the expectation value of $\sigma_z$. For TMDs, the color denotes the expectation value of $S = \frac{3}{2\pi} \operatorname{arg} Z$. The first, second and third column plots the results with on-site disorder ($\epsilon_i$), hopping disorder ($\delta_{ij}$) and magnetic fluctuation ($\theta_{ij}$), respectively. The density of state for clean samples are also plotted for comparison. For graphene the energy unit $t$ is the nearest neighbor hopping strength. For MoTe$_2$, the disorder strength is $\sigma = 0.1$ eV. The strength of disorder $\sigma = 0.3t$ is used for graphene. $N = 60$ for graphene and $N = 40$ for MoTe$_2$. We averaged over $5$ disorder realizations for a clear qualitative demonstration. This protection is topological in nature, as the number of zero modes of a chiral Dirac Hamiltonian is a topological invariant fixed by the Atiyah-Singer index theorem.
• Figure 4: (a) The LL spectra of MoTe$_2$ around the valence band top as a function of magnetic flux in one unit cell in unit of the flux quantum $\phi_0 = h / e$. The blue lines indicate the results of perturbation calculation (Eq. \ref{['eq:two_contribution']}), and the red dashed line denotes LL obtained from minimal substitution (Eq. \ref{['eq:C3_in_B']}), which show perfect agreement. $t = 1$eV is the energy unit used. (b) The slope of the LLs in (a) as a function of LL index. The interception is given by $1/M^* - g$ and the slope of this line is $2 / M^*$.
• Figure 5: The Hofstadter butterfly spectrum of the three-band model for (a) MoTe$_2$, (b) WSe$_2$ and the distorted kagome lattice model with (c) $t = 0.5, t^\prime = 1$ and (d) $t = 1.5, t^\prime = 1$. For the TMDs, the energy unit is set to $1$eV. $\phi_0 = h / e$ is the flux quantum. The color denotes the expectation value of $S = \frac{3}{2\pi} \operatorname{arg} Z$.