Tunable topological Nernst effect in two-dimensional transition-metal dichalcogenides

Two-dimensional semiconducting transition metal dichalcogenides (TMDs) exhibit an intrinsic Ising spin-orbit coupling (SOC) along with a valley contrasting Berry curvature, which can generate a purely anomalous spin and valley Nernst signal driven by a thermal gradient. We show that a small Bychkov-Rashba coupling, which is present in gated TMDs, can enhance the valley Nernst signal by at least 1–2 orders of magnitude. We find that the Nernst signal in these materials is dominated by the anomalous geometrical contribution, and the conventional contribution is much weaker. Importantly, the Nernst signal is also highly tunable by external gating. Although the total Nernst signal vanishes due to time reversal (TR) symmetry, a small magnetic coupling lifts the valley degeneracy and generates an amplified Nernst response. Additionally, we also discuss the Nernst response of bilayer TMDs, and show a similar enhancement and modulation of the Nernst signal due to Rashba SOC. Our predictions are highly pertinent to ongoing experimental studies in TMDs. The generated large anomalous Nernst signal can directly probe the presence of a large Berry curvature in these materials, and may serve as a promising tunable platform for caloritronics applications.

Figure 1

Tunable Berry curvature in TMDs. (Top left) The Berry curvature of the conduction bands can be tuned by varying ${\alpha }_R$. (Top right) The valence bands do not show significant variation with ${\alpha }_R$ because ${\alpha }_R\ll l_v$. (Bottom left) Berry curvature near the valley point for ${\alpha }_R=0$. (Bottom right) Berry curvature near the valley point for ${\alpha }_R=4$ meV displaying a higher magnitude and a much sharper peak at the valley point $\mathbf{K}$. The parameters here are chosen for ${\mathrm{MoS}}_2$.

Figure 2

(Top left) Anomalous Nernst coefficient (in the units $k_{B}e/\hslash )$ at the valley $\tau =1$ as a function of chemical potential $\mu$ for various TMDs at $T=50$ K. (Top right) Anomalous Nernst coefficient (in the units $k_{B}e/\hslash K)$ at valley $\tau =1$ for ${\mathrm{MoS}}_2$ at various temperatures. The Rashba SOC parameter was chosen to be ${\alpha }_R=0$. (Bottom left) Anomalous Nernst coefficient (units $\text{S}\text{nV}{\text{K}}^{-1})$ at $\tau =1,{\alpha }_R=0$, and $T=100$ K. (Bottom right) Conventional $B$-dependent Nernst coefficient (units $\text{S}\text{nV}{\text{K}}^{-1})$ at $\tau =1,T=100$ K, and $B=3$ T. The contrast in the magnitudes can be noted.

Figure 3

(Top) Anomalous Nernst coefficient (in the units $k_{B}e/\hslash )$ at valley $\tau =+1$ as a function of the chemical potential $\mu$ calculated for ${\mathrm{WS}}_2$. For ${\alpha }_R>0$, the Nernst coefficient shows a significantly enhanced signal compared (by at least one order of magnitude) to the Nernst coefficient contribution from only intrinsic SOC. (Bottom) Density plot for the anomalous Nernst coefficient for various values of ${\alpha }_R$ and $\mu$. The sharp peaks when $\mu$ is tuned to partially fill up one of the conduction bands is a striking feature which emerges from a finite Rashba SOC.

Figure 4

(Top left) The magnitude of the sum of Berry curvatures (in units ${\text{Å}}^2)$ in the two valleys for a particular conduction band in ${\mathrm{MoS}}_2$ when $B=15$ T and ${\alpha }_R=0$. For $B=0$, we do not expect a net Berry curvature due to TR symmetry. (Top right) Same as the left plot but for ${\alpha }_R=4$ meV and $B=1$ T. The Berry curvature now shows a much sharper peak around $\mathbf{K}$, which is more than one order of magnitude greater the former case even for a very small $B$ field. (Bottom) Density plot of the sum of peak Berry curvature for various $B$ and ${\alpha }_R$ values.

Figure 5

Total (summing contributions from both valleys) anomalous Nernst coefficient calculated for ${\mathrm{MoS}}_2$. A small $B$ field breaks TR symmetry and the valley symmetry resulting in a large net Nernst response when the chemical potential is tuned appropriately. We chose $T=50$ K and ${\alpha }_R=4$ meV. A large net Nernst signal as a result of a small $B$ field is a distinctive feature in these materials even though the $B$ field by itself is not expected to produce such a large Nernst signal.

Figure 6

(Top panel) Berry curvature (in units ${\text{Å}}^2)$ of four out of the eight bands of a 3R stacked bilayer ${\mathrm{MoS}}_2$. The Berry curvature is easily tunable with Rashba spin-orbit coupling and can be enhanced up to two orders of magnitude for typical ${\alpha }_R$ values. (Bottom) The total anomalous Nernst coefficient for a magnetic field of 5 T, tunable with the chemical potential and ${\alpha }_R$.

Figure 7

(Top) Conduction band spectrum of 2H stacked bilayer ${\mathrm{MoS}}_2$ for ${\alpha }_R=0$ (left) and ${\alpha }_R=8$ meV (right). The band degeneracy is lifted by a finite Rashba SOC. (Bottom) The conduction band Berry curvature (left) at the $\mathbf{K}$ point as a function of ${\alpha }_R$, which displays a giant peak for small finite ${\alpha }_R$. Anomalous valley Nernst coefficient (right) ${\alpha }_{xy}$ as a function of $\mu$ and ${\alpha }_R$.

Figure 8

(Top) Conduction energy bands (left) and the Berry curvature in units of ${\text{Å}}^2$ (right) for 2H stacked bilayer TMD at large values of Rashba SOC. At certain value of ${\alpha }_R$, two of the bands cross each other leading to a peak in the Berry curvature along with a sign change. (Bottom) Total anomalous Nernst coefficient $\left({\alpha }_{xy}\right)$ of a 2H stacked bilayer ${\mathrm{MoS}}_2$ for a magnetic field of $B=1$ T.