Interactions and Magnetotransport through Spin-Valley Coupled Landau Levels in Monolayer ${\mathrm{MoS}}_2$

The strong spin-orbit coupling and the broken inversion symmetry in monolayer transition metal dichalcogenides results in spin-valley coupled band structures. Such a band structure leads to novel applications in the fields of electronics and optoelectronics. Density functional theory calculations as well as optical experiments have focused on spin-valley coupling in the valence band. Here we present magnetotransport experiments on high-quality $n$-type monolayer molybdenum disulphide (${\mathrm{MoS}}_2$) samples, displaying highly resolved Shubnikov–de Haas oscillations at magnetic fields as low as 2 T. We find the effective mass $0.7m_e$, about twice as large as theoretically predicted and almost independent of magnetic field and carrier density. We further detect the occupation of the second spin-orbit split band at an energy of about 15 meV, i.e., about a factor of 5 larger than predicted. In addition, we demonstrate an intricate Landau level spectrum arising from a complex interplay between a density-dependent Zeeman splitting and spin- and valley-split Landau levels. These observations, enabled by the high electronic quality of our samples, testify to the importance of interaction effects in the conduction band of monolayer ${\mathrm{MoS}}_2$.

Figure 1

(a) Device schematic. A single-layer ${\mathrm{MoS}}_2$ is encapsulated between two layers of hBN. Graphite flakes are used as bottom and top gates. $\mathrm{Ti}/\mathrm{Au}$ electrodes are evaporated on top of the bottom hBN before the ${\mathrm{MoS}}_2$ layer is transferred. (b) Optical micrograph of the sample. The ${\mathrm{MoS}}_2$ flake is highlighted with black dashed lines (scale bar is $10\mu \mathrm{m}$). Inset: $\mathrm{Ti}/\mathrm{Au}$ contacts to the ${\mathrm{MoS}}_2$ flake are numbered 1–4 (scale bar is $2\mu \mathrm{m}$). (c) Four-terminal resistance $R_{24,13}$ as a function of $B$ at $V_{\mathrm{BG}}=-2.2\mathrm{V}$, $n_{\mathrm{SdH}}\approx 2.9\times {10}^{12}{\mathrm{cm}}^2$, and $T\approx 100\mathrm{mK}$. SdH oscillations appear at $B\approx 2\mathrm{T}$. We observe a predominantly odd filling factor sequence $\nu =17$, 19, 21, 23. Inset: Linear $I-V_{\text{bias}}$ traces as a function of $V_{\mathrm{TG}}$ at $T\approx 100\mathrm{mK}$. The Ohmic contact regime is achieved for $V_{\mathrm{TG}}>2\mathrm{V}$. (d) SdH oscillations as a function of the magnetic field for different temperatures at $V_{\mathrm{BG}}=-1.5\mathrm{V}$, $n_{\mathrm{SdH}}\approx 4.1\times {10}^{12}{\mathrm{cm}}^2$. An even filling factor sequence $\nu =22$, 24, 26, 28 is measured. Inset: Effective mass $m^{*}$ calculated for the four samples as a function of electron density. Blue, red, green, and black markers correspond to samples $A$, $B$, $C$, and $D$, respectively.

Figure 2

Four-terminal resistance $R_{24,13}$ as a function of $V_{\mathrm{BG}}$ and magnetic field at $T\approx 100\mathrm{mK}$. We observe a pronounced change in the slope (cyan dashed lines) of the Landau fan diagram at $V_{\mathrm{BG}}\approx -1.6\mathrm{V}$, $n_{\mathrm{SdH}}\approx 4\times {10}^{12}{\mathrm{cm}}^2$ (black arrow). The Landau fan diagram can be divided in three different regions: (I) $V_{\mathrm{BG}}<-1.6\mathrm{V}$, (II) $-1.6<V_{\mathrm{BG}}<1\mathrm{V}$, and (III) $V_{\mathrm{BG}}>1\mathrm{V}$. Left and right white dashed lines correspond to the line cuts presented in Figs. 1 and 1, respectively. Inset: Sketch of the conduction band minima at the $K$ and $K^{\prime }$ points in the first Brillouin zone of monolayer ${\mathrm{MoS}}_2$. Because of the strong spin-orbit interaction the spin degeneracy is lifted and spin and valley degrees of freedom are locked. Black dashed lines represent the Fermi energy corresponding to regions (I), (II), and (III), respectively.

Figure 3

(a) Region (I). $R_{24,13}$ as a function of electron density $n_{\mathrm{SdH}}$ and magnetic field at $T\approx 100\mathrm{mK}$. Two sets of LLs corresponding to the $K$ spin-down and $K^{\prime }$ spin-up valleys can be distinguished (blue and red dashed lines, respectively). Anticrossings between not fully spin polarized LLs appear (white circles). (b)–(d) Four-terminal resistance $\mathrm{\Delta }{R}_{24,13}$, after subtracting a smooth background, as a function of LL filling factors at different electron densities. By increasing the electron density, we observe an interplay between predominantly odd and predominantly even filling factor sequences [yellow, orange, and red arrows in (a), respectively].

Figure 4

(a) Region (II). $R_{24,13}$, as a function of $V_{\mathrm{BG}}$ and magnetic field at $T\approx 100\mathrm{mK}$. Inset: Electron density as a function of $V_{\mathrm{BG}}$. Black dashed line indicates the total electron density determined from the capacitor model. Blue and green markers represent the lower and upper split bands electron densities, respectively. Green dashed lines in Fig. 4 indicate the Landau fan diagram originating from the upper spin-orbit split bands. (b) Region (III). $R_{24,13}$ as a function of $V_{\mathrm{BG}}$ and magnetic field at $T\approx 100\mathrm{mK}$. Multiple anticrossings through spin-valley coupled LLs are observed.