Topologically Nontrivial Valley States in Bilayer Graphene Quantum Point Contacts

We present measurements of quantized conductance in electrostatically induced quantum point contacts in bilayer graphene. The application of a perpendicular magnetic field leads to an intricate pattern of lifted and restored degeneracies with increasing field: at zero magnetic field the degeneracy of quantized one-dimensional subbands is four, because of a twofold spin and a twofold valley degeneracy. By switching on the magnetic field, the valley degeneracy is lifted. Because of the Berry curvature, states from different valleys split linearly in magnetic field. In the quantum Hall regime fourfold degenerate conductance plateaus reemerge. During the adiabatic transition to the quantum Hall regime, levels from one valley shift by two in quantum number with respect to the other valley, forming an interweaving pattern that can be reproduced by numerical calculations.

Figure 1

(a) Schematic of the device consisting of bilayer graphene encapsulated in hBN on top of a graphite back gate. On top of the device split gates were evaporated. A layer of ${\mathrm{Al}}_2{\mathrm{O}}_3$ serves to separate the split gates from the channel gate. (b) Atomic force microscopy image of the device. Green dashed lines denote the edges of the bilayer graphene flake. Contacts are labeled $S$ and $D$. Six pairs of split gates are situated between the contacts. (c) Conductance as a function of channel gate voltage for various combinations of the split gate and back gate voltage, showing conductance plateaus with a step size of $\mathrm{\Delta }G=4e^2/h$ for large quantum numbers.

Figure 2

(a) Conductance of QPC $M$ as a function of $V_{\mathrm{CH}}$ for various magnetic field strengths. Several quantization sequences are observed. (b) Transconductance of QPC $M$ as a function $V_{\mathrm{CH}}$ and magnetic field. A pattern of mode splittings (see green and blue dotted modes) is observed. Numbers in purple indicate the conductance values in the quantum Hall regime. (c) Transconductance of QPC $L$ as a function $V_{\mathrm{CH}}$ and magnetic field. A similar pattern of mode crossings is observed in a smaller magnetic field range than for QPC $M$.

Figure 3

(a) Band structure (conduction band) at $B=0\mathrm{T}$ of BLG in the presence of a confinement potential $U(x)$ and a modulated gap $\mathrm{\Delta }(x)$ as described in the text showing a discrete, valley degenerate mode spectrum. Inset: lowest conduction band of homogeneous gapped BLG ($K_{-}$ valley) for ${\mathrm{\Delta }}_0=150\mathrm{meV}$ with three minivalleys forming around the $K$ point. (b) Band structure (conduction band) of the channel at $B=2.2\mathrm{T}$, where symmetry between valleys is broken. The valley splitting at small magnetic fields is proportional to the magnetic field. Inset: Berry curvature $\mathrm{\Omega }$ of the corresponding states with nonzero peaks in the three minivalleys. (c) Magnetic field dependence of the subband edges of the conduction bands in the electron channel. The nontrivial Berry curvature of the zero-field states implies a nonzero orbital magnetic moment $M$ of the states, $M\propto \mathrm{\Omega }$, which induces the linear in magnetic field splitting at small magnetic fields. At high magnetic fields the levels evolve into the LLs of gapped BLG.

Figure 4

(a) Differential conductance $dG/dE$ of a 180 nm wide BLG nanoribbon, including a thermal smoothening of 1.7 K. (b) and (c) show separately the contributions from the two valleys $K_{+}$ and $K_{-}$ at low energies. The channel voltage is determined through the relation $V_{\mathrm{CH}}\propto \sqrt{E_F^2-(\mathrm{\Delta }/2{)}^2}$, with $\mathrm{\Delta }/2=25\mathrm{meV}$.