Strain Engineering of the Berry Curvature Dipole and Valley Magnetization in Monolayer ${\mathrm{MoS}}_2$

The Berry curvature dipole is a physical quantity that is expected to allow various quantum geometrical phenomena in a range of solid-state systems. Monolayer transition metal dichalcogenides provide an exceptional platform to modulate and investigate the Berry curvature dipole through strain. Here, we theoretically demonstrate and experimentally verify for monolayer ${\mathrm{MoS}}_2$ the generation of valley orbital magnetization as a response to an in-plane electric field due to the Berry curvature dipole. The measured valley orbital magnetization shows excellent agreement with the calculated Berry curvature dipole, which can be controlled by the magnitude and direction of strain. Our results show that the Berry curvature dipole acts as an effective magnetic field in current-carrying systems, providing a novel route to generate magnetization.

Figure 1

Strain-tunable monolayer ${\mathrm{MoS}}_2$. (a) $K$ and $K^{\prime }$ valleys of monolayer ${\mathrm{MoS}}_2$ in momentum space. (b) Device schematics of van der Waals heterostructure fabricated on flexible polymethyl methacrylate (PMMA) substrate with prepatterned $\mathrm{Cr}/\mathrm{Au}$ metal pads. (c) Microscope image of a device consisting of ${\mathrm{MoS}}_2$ (orange dashed line), graphene (black dashed line), and capping $h$-BN. (d) SHG measurement for identifying the crystal orientation of ${\mathrm{MoS}}_2$ shown in Fig. 1. The angles of maximum SH intensity repeating every 60° correspond to the armchair direction of the monolayer crystal.

Figure 2

Valley magnetization measurement on a monolayer ${\mathrm{MoS}}_2$. (a) PL with increasing strain. Inset: Strain dependence of the PL center wavelength (${\lambda }_c$). (b) I-V curve of the ${\mathrm{MoS}}_2$ channel at zero strain. Inset: Channel conductance as a function of strain. (c) ${\theta }_{\mathrm{KR}}$ measured at the center of the ${\mathrm{MoS}}_2$ channel as a function of bias voltage ($V_{ds}$) with increasing strain. (d) Strain dependence of ${\theta }_{\mathrm{KR}}$ normalized by the channel current density ($J$).

Figure 3

Strain direction dependence of the valley magnetization. (a) KR map measured under zero strain at a finite bias ($V_{ds}=3.5\mathrm{V}$). The boundaries of ${\mathrm{MoS}}_2$ and graphene are shown by yellow and gray dashed lines, respectively. (b) KR map when 0.55% strain is applied along ${\widehat{x}}_{\text{zigzag}}$. (c) KR map when 0.55% strain is applied along ${\widehat{y}}_{\text{armchair}}$. (d)–(f) Upper: Top view of the ${\mathrm{MoS}}_2$ crystal structure under variable strain, where purple denotes molybdenum and yellow denotes sulfur. Lower: Schematics of modified band structure and Berry curvature distribution under variable strain. Red and blue colors represent positive and negative signs of the Berry curvature, respectively. Polarization of the Berry curvature distribution produces the Berry curvature dipole ($\mathbf{D}$) (orange arrows). Application of an in-plane electric field ($\mathbf{E}$) (green arrows) generates Fermi level tilting in the conduction band.

Figure 4

Experimental and theoretical Berry curvature dipoles ($\mathbf{D}$) and normalized valley magnetizations ($\mathbf{M}/\mathbf{J}$). (a) $|\mathbf{D}|$ and $|\mathbf{M}/\mathbf{J}|$ measured from five different devices. For devices A and B, both strain and $\mathbf{E}$ are applied along ${\widehat{x}}_{\text{zigzag}}$. For devices C and D, strain is applied along ${\widehat{y}}_{\text{armchair}}$ and $\mathbf{E}$ is applied along ${\widehat{x}}_{\text{zigzag}}$. For device $\mathbf{E}$, strain is applied along ${\widehat{x}}_{\text{zigzag}}$ and $\mathbf{E}$ is applied along ${\widehat{y}}_{\text{armchair}}$. (b) Calculated $\mathbf{D}$ and $\mathbf{M}/\mathbf{J}$ when strain is applied along ${\widehat{x}}_{\text{zigzag}}$ (red circles) and ${\widehat{y}}_{\text{armchair}}$ (blue triangles).

Figure 5

Valley magnetization generated in a flexible ${\mathrm{MoS}}_2$ transistor. (a) ${\theta }_{\mathrm{KR}}$ (black circles) and channel current (red line) measured as functions of the gate voltage at $V_{ds}=1\mathrm{V}$ under 0.28% strain. Inset: Strained device schematics with a graphene top gate (dark blue), with ${\mathrm{MoS}}_2$ (orange), $h$-BN (light blue), graphene source/drain (black), $\mathrm{Cr}/\mathrm{Au}$ (yellow), and PMMA (gray). (b) ${\theta }_{\mathrm{KR}}/J$ as a function of the gate voltage at $V_{ds}=1\mathrm{V}$ under 0.28% strain. Inset: Microscope image of the fabricated transistor, with ${\mathrm{MoS}}_2$ (orange dashed line), graphene source and drain electrodes (black dashed line), and graphene top gate (blue dashed line). Scale bar is $5\mu \mathrm{m}$.