Two-Dimensional Metals for Piezoelectriclike Devices Based on Berry-Curvature Dipole

Piezotronics is an emerging field, which exploits strain to control the transport properties in condensed matters. At present, piezotronics research majorly focuses on insulators with tunable electric dipole by strain. Metals are excluded in this type of application due to the absence of the electric dipole. The recently discovered Berry-curvature dipole can exist in metals, consequently introduces the possibility of the piezoelectric phenomena in them. In this paper, we predict that strain can switch the Berry-curvature dipole, and lead to the nonlinear Hall effect in the two-dimensional (2D) $1H\text{−}{\mathrm{MX}}_2$ ($\mathrm{M}=\mathrm{Nb},\mathrm{Ta}$; $\mathrm{X}=\mathrm{S},\mathrm{Se}$). Based on symmetry analysis and first-principles calculations, we show these 2D monolayer metals have the desired piezoelectriclike property: without strain, the Berry-curvature dipole is eliminated by symmetry, prohibiting the nonlinear Hall effect; while uniaxial strain can effectively reduce the symmetry to introduce sizable Berry-curvature dipole, and it can generate observable Hall voltage under a reasonable experimental condition. Due to the nonlinear and topological properties, the piezoelectriclike property here is quite different from the traditional one based on the electric dipole. Compared with the traditional piezoelectric materials, which can only exist in insulators, we manifest that the 2D metallic $1H\text{−}{\mathrm{MX}}_2$ ($\mathrm{M}=\mathrm{Nb},\mathrm{Ta}$; $\mathrm{X}=\mathrm{S},\mathrm{Se}$) can be applied in the platform for piezoelectriclike devices such as strain sensors, terahertz detections, energy harvesters etc.

Figure 1

Metallic piezoelectriclike device based on Berry-curvature dipole $\mathbf{D}$. Charge current $J^{(\omega )}$ leads to (a) zero nonlinear Hall voltage without strain due to the vanishing of $\mathbf{D}$, and (b) finite nonlinear Hall voltage $V^{(2\omega )}$ with uniaxial strain due to the emerging of $\mathbf{D}$.

Figure 2

(a) Crystal structure of monolayer $1H\text{−}{\mathrm{MX}}_2$ ($\mathrm{M}=\mathrm{Nb},\mathrm{Ta}$; $\mathrm{X}=\mathrm{S},\mathrm{Se}$). The arrows represent the in-plane ${\mathcal{C}}_2$ symmetries, line segments represent vertical mirror symmetries and hollowed hexagon denotes for the ${\mathcal{C}}_{3z}$ and ${\mathcal{M}}_{xy}$ symmetries. The symmetries in black (blue) are broken (unbroken) when uniaxial strain is applied. (b) Band structure of monolayer $1H\text{−}{\mathrm{Nb}\mathrm{S}}_2$. The inset illustrates the Brillouin zone.

Figure 3

(a)–(d) Upper panel: (a) Fermi surfaces, (b) Berry curvature ${\mathrm{\Omega }}_z(\mathbf{k})$, (c) Berry-curvature density $d_{xz}(\mathbf{k})$ and (d) $d_{yz}(\mathbf{k})$ of pristine $1H\text{−}{\mathrm{Nb}\mathrm{S}}_2$. Lower panel: (e)–(h) are the same as (a)–(d) with a tensile strain of $\xi =4\mathrm{\%}$ along the armchair direction.

Figure 4

$D_{xz}$ of monolayer $1H\text{−}{\mathrm{Nb}\mathrm{S}}_2$ under (a) armchair (b) zigzag strain, for which the positive (negative) strain value represents the tensile (compressive) strain.

Figure 5

Relationships between the strain, Berry-curvature dipole $\mathbf{D}$, and nonlinear Hall current. $\mathbf{D}$ is always perpendicular to the polar axis under (a) armchair and (b) zigzag strain, while inverse to its direction in two kinds of strain. The nonlinear Hall current is proportional to ${\left|E\right|}^2{D}_{xz}\cos \theta$, so the nonlinear Hall effect is maximum when the charge current is parallel to $\mathbf{D}$.

Figure 6

Velocity relationships between the symmetry points around the $K$ point under the ${\mathcal{C}}_{3z}$ symmetry.

Figure 7

$D_{xz}$ of (a) $1H\text{−}{\mathrm{Nb}\mathrm{Se}}_2$, (b) $1H\text{−}{\mathrm{Ta}\mathrm{S}}_2$, and (c) $1H\text{−}{\mathrm{Ta}\mathrm{Se}}_2$ under different armchair strains.