Valley Subband Splitting in Bilayer Graphene Quantum Point Contacts

We report a study of one-dimensional subband splitting in a bilayer graphene quantum point contact in which quantized conductance in steps of $4e^2/h$ is clearly defined down to the lowest subband. While our source-drain bias spectroscopy measurements reveal an unconventional confinement, we observe a full lifting of the valley degeneracy at high magnetic fields perpendicular to the bilayer graphene plane for the first two lowest subbands where confinement and Coulomb interactions are the strongest and a peculiar merging or mixing of $K$ and $K^{\prime }$ valleys from two nonadjacent subbands with indices $(N,N+2)$, which are well described by our semiphenomenological model.

Figure 1

(a) Schematic illustrating the device layout. (b) Cross section of the device along the dashed line in (a), together with a sketch of the electronic band structure across the constriction defined by the SG.

Figure 2

(a) Differential conductance $G$ as a function of $V_{\mathrm{TG}}$ for different values of $V_{\mathrm{SG}}$ from $-11.0$ (left) to $-10.5\mathrm{V}$ (right) with an increment of 0.1 V and at constant $V_{\mathrm{BG}}=9\mathrm{V}$. The curves are shifted for clarity by 2 V between consecutive traces (the leftmost curve is not shifted). Well-quantized plateaus are observed in steps of $4e^2/h$. (b) Gray scale map of $dG/d{V}_{\mathrm{TG}}$ as a function of $V_{\mathrm{TG}}$ and $V_{\mathrm{SG}}$ at $V_{\mathrm{BG}}=9\mathrm{V}$. Small markers denote the position of line cuts shown in (a).

Figure 3

(a) Transconductance versus source-drain bias voltage $V_{\mathrm{bias}}$ and $V_{\mathrm{TG}}$. Minima in $dG/d{V}_{\mathrm{TG}}$ correspond to plateaus in the $G(V_{\mathrm{TG}})$ curves. The resulting checkerboard pattern reveals an increasing energy level spacing with increasing subband index. Effect of the Fabry-Pérot interferences is clearly visible as lines parallel to the 1D subband dispersion lines. Blue crosses highlight the subband edge crossings representing the energy spacing $\mathrm{\Delta }{E}_{N,N+1}$ between two consecutive 1D subbands of the QPC. (b) $\mathrm{\Delta }{E}_{N,N+1}$ showing a linear dependence as a function of the 1D subband indices $N$.

Figure 4

(a) Differential conductance $G$ as a function of $V_{\mathrm{TG}}$ for different values of magnetic field $B$ in steps of 100 mT at fixed $V_{\mathrm{BG}}=9$ and $V_{\mathrm{SG}}=-10.6\mathrm{V}$. The curves are shifted for clarity to the right by an offset of $2\mathrm{V}/\mathrm{T}$ (200 mV between consecutive curves). The thicker black line (not shifted) corresponds to the data acquired at $B=20\mathrm{mT}$, which is shown in Fig. 2. (b) Corresponding gray scale map of $dG/d{V}_{\mathrm{TG}}$ as a function of $V_{\mathrm{TG}}$ and $B$. Colored dashed lines at $B=0.9$ (orange), 1.5 (blue), 2.2 (green), and 8.0 T (red) denote the line cuts associated with the highlighted conductance traces shown in (a). Transitions across magnetoelectric subbands appear as dark lines. (c) Energy level diagram of the QPC at zero and high magnetic field. (d) Valley subband dispersion as a function of $B$ calculated with our model. The numbers displayed in the plot correspond to the quantized conductance values of the plateaus in units of $e^2/h$.