Spin-orbit interaction induced in graphene by transition metal dichalcogenides

We report a systematic study on strong enhancement of spin-orbit interaction (SOI) in graphene induced by transition-metal dichalcogenides (TMDs). Low-temperature magnetotoransport measurements of graphene proximitized to different TMDs (monolayer and bulk ${\mathrm{WSe}}_2,{\mathrm{WS}}_2$, and monolayer ${\mathrm{MoS}}_2)$ all exhibit weak antilocalization peaks, a signature of strong SOI induced in graphene. The amplitudes of the induced SOI are different for different materials and thickness, and we find that monolayer ${\mathrm{WSe}}_2$ and ${\mathrm{WS}}_2$ can induce much stronger SOI than bulk ${\mathrm{WSe}}_2,{\mathrm{WS}}_2$, and monolayer ${\mathrm{MoS}}_2$. The estimated spin-orbit (SO) scattering strength for graphene/monolayer ${\mathrm{WSe}}_2$ and graphene/monolayer ${\mathrm{WS}}_2$ reaches $\sim 10$ meV, whereas for graphene/bulk ${\mathrm{WSe}}_2$, graphene/bulk ${\mathrm{WS}}_2$, and graphene/monolayer ${\mathrm{MoS}}_2$, it is around 1 meV or less. We also discuss the symmetry and type of the induced SOI in detail, especially focusing on the identification of intrinsic (Kane-Mele) and valley-Zeeman (VZ) SOI by determining the dominant spin relaxation mechanism. Our findings pave the way for realizing the quantum spin Hall (QSH) state in graphene.

Figure 1

(a) Magnetoconductivity correction $\left[\mathrm{\Delta }\sigma \right(B)\equiv \sigma (B)-\sigma (0\left)\right]$ for mono ${\mathrm{WSe}}_2$ averaged over 50 curves corresponding to 50 values of $V_g$ between 50 and 60 V at 250 mK. A clear WAL peak and flat tails for higher-$B$ regions are observed. The solid curve represents the fit based on the theoretical formula (1). The left inset shows the optical microscope image of the mono ${\mathrm{WSe}}_2$ device. The gate voltage dependence of resistance is displayed in the right inset. (b) $\mathrm{\Delta }\sigma \left(B\right)$ curves at different temperatures, averaged over 50 curves with $V_g$ between $-20$ and $-30$ V. Similar tendency as in (a) can be seen in the shape of each curve. The solid lines are theoretical fits.

Figure 2

$\mathrm{\Delta }\sigma \left(B\right)$ curves for mono ${\mathrm{WSe}}_2$ for different gate voltage ranges at 1 K. Clear WAL peaks are observed in all $V_g$ ranges. (b) Comparison of $\mathrm{\Delta }\sigma \left(B\right)$ curves from mono ${\mathrm{WSe}}_2$ and bulk ${\mathrm{WSe}}_2$ in the same $V_g$ range. While the curve for mono ${\mathrm{WSe}}_2$ is characterized by flat tails for high-$B$ region, that of bulk ${\mathrm{WSe}}_2$ exhibit instead a striking upturn with magnetic field following the small WAL peak around $B=0$. This demonstrates that stronger SOI is induced in graphene for mono ${\mathrm{WSe}}_2$ than for bulk ${\mathrm{WSe}}_2$. In the inset, we show $R$ vs $V_g$ curve for bulk ${\mathrm{WSe}}_2$.

Figure 3

(a) $\mathrm{\Delta }\sigma \left(B\right)$ for different gate voltage ranges obtained from mono ${\mathrm{WS}}_2$ A at 1 K. The clear WAL peaks demonstrate the strong induced SOI. Interestingly, the peak is sharper for $-50$ V $<V_g<-60$ V than for 50 V $<V_g<60$ V. All curves exhibit flat tails for high-$B$ range. The solid lines are theoretical fits. The inset shows the $V_g$ dependence of $R$. (b) Comparison of $\mathrm{\Delta }\sigma \left(B\right)$ between mono ${\mathrm{WS}}_2$ A and bulk ${\mathrm{WS}}_2$ B. As seen for the $\mathrm{graphene}/{\mathrm{WSe}}_2$ samples, bulk ${\mathrm{WS}}_2$ shows upturns following the WAL peak. The inset displays $V_g$ dependence of $R$. Theoretical fits are shown by the solid lines.

Figure 4

(a) An optical microscope image of mono ${\mathrm{MoS}}_2$ B sample. Graphene is deposited on a CVD grown monolayer ${\mathrm{MoS}}_2$. (b) Resistance as a function of gate voltage at 100 mK of mono ${\mathrm{MoS}}_2$ B. (c) $\mathrm{\Delta }\sigma \left(B\right)$ for $V_g$ between 10 and 20 V, and between $-20$ and $-30$ V at 70 mK. In contrast to mono ${\mathrm{WSe}}_2$ and mono ${\mathrm{WS}}_2$, both curves exhibit large upturns when $B$ is large. The solid lines show theoretical fits.

Figure 5

(a) $\mathrm{\Delta }\sigma \left(B\right)$ for $\left|B\right|<35$ G. The fits with no symmetric SOI $(B_{\mathrm{sym}}=0$) and the symmetric SOI equal to the asymmetric SOI $(B_{\mathrm{sym}}=B_{\mathrm{asy}})$ deviate from the best fit $(B_{\mathrm{sym}}=70B_{\mathrm{asy}})$ especially close to the peak. Note that the two fits $(B_{\mathrm{sym}}=0$ and $B_{\mathrm{sym}}=B_{\mathrm{asy}})$ almost overlap in the figure. (b) In high-field region, there is no striking difference between the fits with different ratio of $B_{\mathrm{sym}}$ to $B_{\mathrm{asy}}$.

Figure 6

Removal of temperature independent components. (a) The original data at 500 mK and that at 4 K from mono ${\mathrm{WS}}_2$ A. In the inset, we show $\mathrm{\Delta }\sigma \left(B\right)$ obtained from mono ${\mathrm{WS}}_2$ A after the subtraction of the background signal. The subtraction procedure is written in the main text, and the subtracted regions are marked by the rectangles. (b) Comparison of the $\mathrm{\Delta }\sigma \left(B\right)$ curves at 70 mK and 4 K for mono ${\mathrm{MoS}}_2$ B. It is clear that even for higher-$B$ region, the shape of the two curves is drastically different at the two temperatures, indicating that there does not exist any temperature independent components. Theoretical fits are shown by the solid lines. (Inset) The same curve of $\mathrm{\Delta }\sigma \left(B\right)$ at 4 K as in the main figure with the fit based only on the weak localization term in (1) (namely, ${\tau }_{\mathrm{asy}}\to \infty$ and ${\tau }_{\mathrm{so}}\to \infty )$. The upturn is well reproduced therefore it can be attributed to the weak localization contribution.

Figure 7

Comparison of the spin-orbit energy $\left(E_{\mathrm{so}}\right)$ estimated from the theoretical fits for each graphene/TMD heterostructure. The eight samples can be categorized into the two groups, the group with $E_{\mathrm{so}}\gtrsim 10$ mV (mono ${\mathrm{WSe}}_2$ and ${\mathrm{WS}}_2)$ and the one with $E_{\mathrm{so}}\lesssim 1$ meV (mono ${\mathrm{MoS}}_2$, bulk ${\mathrm{WSe}}_2$, and ${\mathrm{WS}}_2)$. In the inset, we show the temperature dependence of ${\tau }_{\varphi }$ for the sample mono ${\mathrm{MoS}}_2$ B as an example. The experimental data are consistent with the relation ${\tau }_{\varphi }^{-1}\propto T$.

Figure 8

(a) Simulated curves of $\mathrm{\Delta }\sigma \left(B\right)$ based on equation (2) with $B_{\varphi }=\hslash /4eD{\tau }_{\varphi }=0.01$ with different ratio of ${\tau }_{\varphi }/{\tau }_{iv}$. The shape of $\mathrm{\Delta }\sigma \left(B\right)$ dramatically changes for ${\tau }_{iv}\gtrsim {\tau }_{\varphi }$. (b) Experimental data of $\mathrm{\Delta }\sigma \left(B\right)$ from mono ${\mathrm{WS}}_2$ A. The flat tails in high-field region are observed over a broad range of temperature, indicating that the system is in the limit ${\tau }_{iv}\ll {\tau }_{\varphi }$. The solid lines are the fits based on (1).

Figure 9

Sensitivity of the fits to the ratio ${\tau }_{\mathrm{sym}}/{\tau }_{\mathrm{asy}}$. Different fits are shown for different values of ${\tau }_{\mathrm{sym}}/{\tau }_{\mathrm{asy}}$: When there is no symmetric contribution (light blue curve), the fit strongly deviates from the experimental points. In contrast, when ${\tau }_{\mathrm{sym}}\ll {\tau }_{\mathrm{asy}}$ and $E_{\mathrm{so}}\gtrsim 12$ meV, we can well reproduce the experimental data. (Inset) Temperature dependence of 1/${\tau }_{\varphi }$ for mono ${\mathrm{WSe}}_2$ for $V_g$ between 50 and 60 V.

Figure 10

Extraction of ${\mathrm{\Delta }}_{\mathrm{EY}}$ and ${\mathrm{\Delta }}_{\mathrm{DP}}$ to identify the dominant SOI for each heterostructure. (a) Logarithmic-scale plot of ${\varepsilon }_F^2{\tau }_p/{\tau }_{\mathrm{so}}$ as a function of ${\varepsilon }_F^2{\tau }_p^2$. The nonzero $y$-axis intercept and the deviation from the linearity are due to the EY-type spin relaxation. The error bars are obtained by considering the standard deviation of the Fermi energy $\left(\delta E_F\right)$ due to electron-hole puddles. (b) Using the same experimental data as (a) we plot them in a different way based on (4) to clarify the EY contribution. The positive slope is clearly seen as a result of the EY contribution for the three samples. The inset shows the same plot for mono ${\mathrm{MoS}}_2$ B.