Electrical valley filtering in transition metal dichalcogenides

This work investigates the feasibility of electrical valley filtering for holes in transition metal dichalcogenides. We look specifically into the scheme that utilizes a potential barrier to produce valley-dependent tunneling rates, and perform the study with both a $k·p$-based analytic method and a recursive Green's function–based numerical method. The study yields the transmission coefficient as a function of incident energy and transverse wave vector, for holes going through lateral quantum barriers oriented in either armchair or zigzag directions, in both homogeneous and heterogeneous systems. The main findings are the following: (1) The tunneling current valley polarization increases with increasing barrier width or height; (2) both the valley-orbit interaction and band structure warping contribute to valley-dependent tunneling, with the former contribution being manifest in structures with asymmetric potential barriers, and the latter being orientation dependent and reaching maximum for transmission in the armchair direction; and (3) for transmission $\sim 0.1$, a tunneling current valley polarization of the order of $10\%$ can be achieved.

Figure 1

(a) Top view of $2H$-monolayer TMDC in real space, where large pink disks represent metal atoms $M$ and small brown disks represent the chalcogen atoms $X$. (b) Side view of $2H$-monolayer TMDC in real space. (c) The first Brillouin zone.

Figure 2

(a) A one-barrier structure with the gray area indicating a barrier lying along the armchair direction. (b) A one-barrier structure with the gray area showing a barrier lying along the zigzag direction. (c) The inverted valence band edge diagram of a one-barrier structure with the barrier height $V_0$ and the source-drain bias $V_1=0\left(V_1\ne 0\right)$ for a symmetric (asymmetric) structure.

Figure 3

The schematic diagram showing incident and reflected waves in the two regions, labeled as I and II, of the one-barrier structure.

Figure 4

(a) A lateral quantum structure in a two-dimensional square lattice. Unit cells are represented by circles. The gray area indicates a barrier. The region bounded by vertical yellow lines denotes a column of unit cells. $H_j\left(j=\mathrm{integer}\right)$ is the on-site part of the tight-binding Hamiltonian for a cell with coordinates $(i,j)$. $t_{\pm },u_{\pm },v_{\pm }$, and $w_{\pm }$ are hopping matrices. (b) Reduction of the two-dimensional lattice to a one-dimensional chain in the mixed r-k space, with ${\tilde{H}}_j$ and ${\tilde{t}}_{\pm }$ being, respectively, the effective on-site cell Hamiltonian and nearest-neighbor hoppings.

Figure 5

The chalcogen atoms are denoted by the dashed circle, and their atomic orbitals are ignored in the three-orbital model. (a) The hopping terms considered in the calculation, with $t_{\mathrm{nn}}$ the nearest-neighbor hopping, $t_{\mathrm{nnn}}$ the next-nearest-neighbor hopping, and $t_{\mathrm{tnn}}$ the third-nearest-neighbor hopping. (b) Supercell for barriers along the armchair direction. (c) Supercell for barriers along the zigzag direction.

Figure 6

Hole transmissions and corresponding tunneling current valley polarizations, i.e., $P_v$, as functions of total carrier energy, in one-barrier structures with barriers lying along the armchair direction. Results in (a)–(c) are calculated using the parameters of $\mathrm{WS}{\mathrm{e}}_2$. (a) Valley-averaged transmissions for various $k_x$’s with barrier height $V_0=0.01\mathrm{eV}$, source-drain bias $V_1=0.01\mathrm{eV}$, and barrier width $W$ nearly $100\AA$. (b) Tunneling current valley polarizations for various $k_x$’s in the same structure considered in (a). (c) Tunneling current valley polarization for various barrier heights $\left(V_0\right)$ and biases $\left(V_1\right)$ at $k_x=0.05a^{-1}$. (d) Tunneling current valley polarization for two TMDCs with $V_0=0.01\mathrm{eV},V_1=0.01\mathrm{eV},W\approx 100\AA$, and $k_x=0.05a^{-1}$.

Figure 7

Hole transmissions and corresponding tunneling current valley polarizations, i.e., $P_v$, as functions of total carrier energy, in symmetric, one-barrier structures with barriers lying along the zigzag direction and the source-drain bias $V_1=0\mathrm{eV}$. For (a)–(c), the results are obtained with the tight-binding parameters of $\mathrm{WS}{\mathrm{e}}_2$. (a) Valley-averaged transmissions for various transverse wave vectors $k_y$’s with barrier height $V_0=0.01\mathrm{eV}$ and width $W$ nearly $100\AA$. $a=\mathrm{lattice}$ constant. (b) Tunneling current valley polarizations for various $k_y$’s in the same structure considered in (a). (c) Tunneling current valley polarizations for various barrier heights $\left(V_0\right)$ and widths $\left(W\right)$ at $k_y=0.05a^{-1}$. (d) Tunneling current valley polarizations for two TMDCs with $V_0=0.01\mathrm{eV},V_1=0\mathrm{eV},W\approx 100\AA$, and $k_y=0.05a^{-1}$.

Figure 8

The comparison of tunneling current valley polarization between a $\mathrm{W}{\mathrm{S}}_2/\mathrm{Mo}{\mathrm{S}}_2/\mathrm{W}{\mathrm{S}}_2$ heterostructure and a homogeneous $\mathrm{Mo}{\mathrm{S}}_2$ quantum structure with the same barrier width and height. Curves labeled “hetero” refer to the tunneling current valley polarization in the heterostructure and those without the label refer to that in the homogeneous structure. (a) Tunneling current valley polarization for barriers along the armchair direction with $V_0=0.45\mathrm{eV},W\approx 10\AA$, and $k_x=0.1a^{-1}$, for $V_1=0\mathrm{eV}$ or 0.1 eV. (b) Tunneling current valley polarization for barriers along the zigzag direction with $V_0=0.45\mathrm{eV},W\approx 10\AA$, and $k_y=0.1a^{-1}$, for $V_1=0\mathrm{eV}$ or 0.1 eV. In (a), since the VOI effect is dominant, the polarization vanishes at $V_1=0\mathrm{eV}$.