Weak localization in boron nitride encapsulated bilayer ${\mathrm{MoS}}_2$

We present measurements of weak localization on hexagonal boron nitride encapsulated bilayer ${\mathrm{MoS}}_2$. From the analysis we obtain information regarding the phase coherence and the spin diffusion of the electrons. We find that the encapsulation with boron nitride provides higher mobilities in the samples, and the phase coherence shows improvement, while the spin relaxation does not exhibit any significant enhancement compared to nonencapsulated ${\mathrm{MoS}}_2$. The spin relaxation time is in the order of a few picoseconds, indicating a fast intravalley spin-flip rate. Lastly, the spin-flip rate is found to be independent from electron density in the current range, which can be explained through counteracting spin-flip scattering processes based on electron-electron Coulomb scattering and extrinsic Bychkov-Rashba spin-orbit coupling.

Figure 1

A high-mobility encapsulated ${\mathrm{MoS}}_2$ bilayer Hall bar and device characteristics. (a) Optical image of an h-BN/2L-${\mathrm{MoS}}_2$/h-BN stack, with prepatterned holes in the top h-BN. (b) Optical image of a completed Hall bar device. (c) Cross-sectional schematic of the device. (d) Electrical conductivity $\sigma$ and electron mobility $\mu$ as a function of the gate voltage $V_{\text{g}}$, obtained from Hall measurements at $T=2$ K.

Figure 2

Weak localization in bilayer ${\mathrm{MoS}}_2$. (a) Magnetoresistivity $[\mathrm{\Delta }\rho =\rho (B)-\rho (B=0\text{T})$, $\rho (B=0\text{T})=4.56\mathrm{k}\mathrm{\Omega }$ at $V_{\text{g}}=45$ V] as a function of the back-gate voltage ($V_{\text{g}}$) and the magnetic field ($B$) at $T=2$ K. The overlaid linecut shows the magnetoresistivity at a gate voltage of $V_{\text{g}}=45$ V. (b) Symmetrized magnetoconductivity $\mathrm{\Delta }{\sigma }^{*}\left(B\right)$ as a function of magnetic field for different electron densities $n$ measured at $T=2$ K. (c) Symmetrized magnetoconductivity as a function of magnetic field for different temperatures with $n=7.2\times {10}^{12}{\mathrm{cm}}^{-2}$. The drawn solid black lines correspond to fits using the HLN model. Data and fitted curves have been shifted vertically by $0.1e^2/\pi h$ for clarity.

Figure 3

Phase-coherence length (a) and phase-coherence time (b) as a function of the electron density for $T=2$ K. (c) Logarithmic plot of the phase-coherence length as a function of temperature, for two different back-gate voltages. The power law dependence with $a\sim 0.5$ suggests electron-electron scattering as the dephasing mechanism.

Figure 4

Spin relaxation properties of electrons in bilayer ${\mathrm{MoS}}_2$. (a) Energy-dispersion schematic illustrating spin relaxation due to intravalley spin-flip process. Different colors represent electron populations of different spin orientations. Spin relaxation length $L_{\text{so}}$ (b) and time ${\tau }_{\text{so}}$ (c) as a function of electron density ($T=2$ K), obtained from fitting the magnetoconductivity data to Eq. (1). The spin relaxation length increases with electron density, while the spin relaxation time is independent of the electron density.