Electric-Field-Tunable Valley Zeeman Effect in Bilayer Graphene Heterostructures: Realization of the Spin-Orbit Valve Effect

We report the discovery of electric-field-induced transition from a topologically trivial to a topologically nontrivial band structure in an atomically sharp heterostructure of bilayer graphene (BLG) and single-layer ${\mathrm{WSe}}_2$ per the theoretical predictions of Gmitra and Fabian [Phys. Rev. Lett. 119, 146401 (2017)]. Through detailed studies of the quantum correction to the conductance in the BLG, we establish that the band-structure evolution arises from an interplay between proximity-induced strong spin-orbit interaction (SOI) and the layer polarizability in BLG. The low-energy carriers in the BLG experience an effective valley Zeeman SOI that is completely gate tunable to the extent that it can be switched on or off by applying a transverse displacement field or can be controllably transferred between the valence and the conduction band. We demonstrate that this results in the evolution from weak localization to weak antilocalization at a constant electronic density as the net displacement field is tuned from a positive to a negative value with a concomitant SOI-induced splitting of the low-energy bands of the BLG near the $K(K^{\prime })$ valley, which is a unique signature of the theoretically predicted spin-orbit valve effect. Our analysis shows that quantum correction to the Drude conductance in Dirac materials with strong induced SOI can only be explained satisfactorily by a theory that accounts for the SOI-induced spin splitting of the BLG low-energy bands. Our results demonstrate the potential for achieving highly tunable devices based on the valley Zeeman effect in dual-gated two-dimensional materials.

Figure 1

(a) Schematic of the device structure. The BLG and single-layer ${\mathrm{WSe}}_2$ stack is sandwiched between two $h$-BN flakes of thickness $\sim 20\mathrm{nm}$. (b) Optical image of the device without top-gate structure—the gray rectangle shows the position of the top gate. (c) Contour plot of four-probe resistance measured in the $V_{\mathrm{bg}}\text{−}{V}_{\mathrm{tg}}$ plane on logarithmic scale. The blue and yellow arrows indicate the directions of increasing $D$ and $n$, respectively.

Figure 2

(a) A two-dimensional contour plot of $R$ as a function of $n$ and $D$. The asymmetric feature around the primary Dirac point (marked by the dashed line) is a consequence of the fact that only one of the bands (VB for negative $D$ and CB for positive $D$) gets split due to induced SOI [11]. The data were acquired at $T=20\mathrm{mK}$ and $B=0\mathrm{T}$. The labels I–IV mark the four quadrants in the $n\text{−}D$ plane. The corresponding regimes are marked in the two schematic band structures in Fig. 3. The weak localization or weak antilocalization measurements reported in this Letter were performed along the green line (some of the points are labeled). (b) Plots of $R$ versus $n$ for $D=-0.2\mathrm{V}/\mathrm{nm}$ (blue line) and $D=0.4\mathrm{V}/\mathrm{nm}$ (red line). The insets show the data enlarged around the secondary peaks (marked in the main panel by dashed ovals) arising from SOI-induced splitting of the bands.

Figure 3

(a) Schematic showing the single-layer ${\mathrm{WSe}}_2$ and the two layers of the BLG. The low-energy bands in the BLG arise from hopping between the orbitals on the nondimer pair of sites formed of $B_1$ sublattice of the lower layer (one such atom is marked in red) and the $A_2$ sublattice of the top layer (one such atom is marked in blue) [33]. The system lies in the $x\text{−}y$ plane, while the positive direction of $D$ (indicated by the green arrow) is the positive $z$ axis. (b) Schematic of the bulk band structure near the $K$ valley for positive values of $D$. The electrons localized in the bottom layer ($B_1$ orbital) of the BLG form the spin-split CB, and those in the top layer ($A_2$ orbital) of the BLG form the VB [11]. (c) Schematic of the bulk band structure plotted near the $K$ valley for negative $D$. The electrons localized in the $B_1$ orbital of the BLG now form the spin-split VB, and those in the $A_2$ orbital of the BLG form the CB. The BLG thus undergoes a band inversion as $D$ changes sign [11].

Figure 4

(a) Plots showing the dependence of the ensemble averaged magnetoconductance data on $B$ for $n=-1.8\times {10}^{16}$ ${\mathrm{m}}^{-2}$ at different $D$. The data were collected along the green dotted line in Fig. 2. For $D<0$, WAL is observed which changes to WL as the $D$ is made sufficiently positive. The blue data points (measured at point B in Fig. 2) show the magnetoconductance near $D\sim 0.01\mathrm{V}/\mathrm{nm}$. (b),(c) The open circles are the ensemble averaged WL data taken for $D>0$ and $n<0$ (regime IV in Figs. 2 and 3). The solid red lines are the fits to the data using Eq. (1). (d),(e) The open circles are the ensemble averaged WAL data taken for $D<0$ and $n<0$ (regime II in Figs. 2 and 3). The solid red lines are the fits to the data using Eq. (2). The letters C, D, I, and J refer to the points in the $n\text{−}D$ plane in Fig. 2 at which the data were collected.

Figure 5

Shubnikov–de Haas oscillations (measured at $n\sim 3\times {10}^{16}{\mathrm{m}}^{-2}$) plotted versus $1/B$ for (a) $D\sim -0.15\mathrm{V}/\mathrm{nm}$ and (b) $D\sim 0.15\mathrm{V}/\mathrm{nm}$. The beating in the oscillations observed for positive values of $D$ indicates the presence of two close-by frequencies. (c) The red and blue curves show, respectively, the Fourier transform of the SdH oscillations for $D\sim 0.15\mathrm{V}/\mathrm{nm}$ and $D\sim -0.15\mathrm{V}/\mathrm{nm}$. One can see a clear change in the Fermi surface topology from single (regime I in Figs. 2 and 3) to doubly split (regime III in Figs. 2 and 3) on changing $D$ from negative to positive.