Signature of Hanle precession in trilayer ${\mathrm{MoS}}_2$: Theory and experiment

Valley-spin coupling in transition-metal dichalcogenides (TMDs) can result in unusual spin transport behaviors under an external magnetic field. Nonlocal resistance measured from two-dimensional materials such as TMDs via electrical Hanle experiments is predicted to exhibit nontrivial features compared with results from conventional materials due to the presence of intervalley scattering as well as a strong internal spin-orbit field. Here we report the all-electrical injection and nonlocal detection of spin-polarized carriers in trilayer ${\mathrm{MoS}}_2$ films. We calculate the Hanle curves theoretically when the separation between spin injector and detector is much larger than spin diffusion length ${\lambda }_s$. The experimentally observed curve matches the theoretically predicted Hanle shape under the regime of slow intervalley scattering. The estimated spin lifetime was found to be around 110 ps at 30 K.

Figure 1

(a) Schematic of the energy spectrum of trilayer ${\mathrm{MoS}}_2$ at $K$ and $K^{\prime }$ valleys. ${\mathrm{\Omega }}_{SO}$ is the spin splitting at the Fermi level; $\lambda$ is the spin splitting at valence band top, and $\mathrm{\Delta }$ is the band gap. The intervalley scattering rate ${\gamma }_v$ is determined by short-range impurities. (b) The bottom scheme indicates the internal spin orbital field $\pm B_{SO}$ induced by spin orbital coupling along the $z$ direction in the case of trilayer ${\mathrm{MoS}}_2$. Here we find it appropriate to mention that there is still no consensus about whether $K$ $\left(K^{\prime }\right)$ or the $Q$ point of the Brillouin zone should be the minima of the conduction band for trilayer ${\mathrm{MoS}}_2$. A recent report by Brumme et al. [22] showed that for relatively high electron doping ($1.69\times {10}^{14}$ ${\mathrm{cm}}^{-2})$, all eight minima in the conduction band may be populated. However, they showed that for a relatively lower doping level ($2.25\times {10}^{13}$ ${\mathrm{cm}}^{-2}$), only the minima at the $K$ $\left(K^{\prime }\right)$ point are occupied. Based on our Hall effect results which showed that the carrier concentration in our films was $1.3\times {10}^{13}$ ${\mathrm{cm}}^{-2}$ at 30 K, we assumed that the latter (i.e., electrons populating only the minima at $K$ and $K^{\prime }$ points) is the case.

Figure 2

(a) Raman spectra of trilayer ${\mathrm{MoS}}_2$. The separation of vibration frequency $E_{2g}^1$ and $A_{1g}$ indicates three monolayers of ${\mathrm{MoS}}_2$. (b) Schematic illustration of Hanle experiments performed through electrical methods on trilayer ${\mathrm{MoS}}_2$, three-dimensional view and top view. The measure of a nonlocal resistance is a voltage between the right two contacts (3 and 4), while the current flows through the first two contacts (1 and 2), which are made of 15-nm NiFe. The distance between the middle two contacts (2 and 3) $d$ (distance between the spin injector and detector) is 1 $\mu m$. We studied the Hanle curves under the condition of $d>{\lambda }_s$. The arrows in the FM indicate the magnetization direction and also the spin-polarization direction. Hanle experiments were performed under a normally oriented external magnetic field.

Figure 3

(a) $I-V$ curves of the trilayer ${\mathrm{MoS}}_2$ sample with NiFe contacts from 75 to 300 K. (b) Arrhenius plots, In ($I_{12}/T^{3/2}$) vs $1000T^{-1}$, at different $V_{12}$. (c) Extraction of Schottky barrier height ${\mathrm{\Phi }}_B$.

Figure 4

Theoretical Hanle curve calculated from Eq. (7) in the case of slow intervalley scattering. In all six panels the value of parameter $\mathrm{\Gamma }$ is 0.2, the distance $d$ is chosen to be $d=1\mu \mathrm{m}$, and the diffusion coefficient is equal to $D=2.6{\mathrm{cm}}^2/\mathrm{s}$: (a) $B_{SO}$ = 1 T, ${\tau }_s^{*}$ = 10 ps, $\tilde{d}$ = 19.7; (b) $B_{SO}$ = 1 T, ${\tau }_s^{*}$ = 8 ps, $\tilde{d}$ = 22.0; (c) $B_{SO}$ = 1 T, ${\tau }_s^{*}$ = 5 ps, $\tilde{d}$ = 27.8; (d) $B_{SO}$ = 0.1 T, ${\tau }_s^{*}$ = 200 ps, $\tilde{d}$ = 4.4; (e) $B_{SO}$ = 0.1 T, ${\tau }_s^{*}$ = 100 ps, $\tilde{d}$ = 6.2; and (f) $B_{SO}$ = 0.1 T, ${\tau }_s^{*}$ = 20 ps, $\tilde{d}$ = 13.9. Insets in (a)–(c) exhibit the zoom-in curve in a smaller field range, $\left|B\right|<0.2$ T.

Figure 5

Theoretical Hanle curve calculated from Eq. (11) $R\left(B\right)$ vs external field $B$ under conditions of $\mathrm{\Gamma }=2$ for different $B_{SO}$ and ${{\tau }_s^{*}}_{\pm }$. (a) $B_{SO}$ = 0.1 T, ${\tau }_{s+}^{*}$ = 3 ps, ${\tau }_{s-}^{*}$ = 68 ps, ${\tilde{d}}_{+}$ = 17.1, ${\tilde{d}}_{-}$ = 7.5 and (b) $B_{SO}$ = 1 T, ${\tau }_{s+}^{*}$ = 1.5 ps, ${\tau }_{s-}^{*}$ = 18 ps, ${\tilde{d}}_{+}$ = 50.8, ${\tilde{d}}_{-}$ = 14.9.

Figure 6

(a) Experimental results $R\left(B\right)$ in normal field (maximum 0.06 T) performed at 30 K. The red curve is a guide to the eye. (b) Calculated $R\left(B\right)$ curve using Eq. (7) in which the zoom-in portion of the curve matches the experimental result shown in (a).

Figure 7

Experimental Hanle data at (a) 25, (b) 55, (c) 35, (e) 40, (g) 45, and (i) 50 K. The calculated Hanle curves are also shown at various temperatures of (d) 35, (f) 40, (h) 45, and (j) 50 K over the magnetic field range of $\pm 0.25$ T. The red part of the curve shows the fitting of the experimental data to Eq. (7).