Realizing Valley-Polarized Energy Spectra in Bilayer Graphene Quantum Dots via Continuously Tunable Berry Phases

The Berry phase plays an important role in determining many physical properties of quantum systems. However, tuning the energy spectrum of a quantum system via Berry phase is comparatively rare because the Berry phase is usually a fixed constant. Here, we report the realization of an unusual valley-polarized energy spectra via continuously tunable Berry phases in Bernal-stacked bilayer graphene quantum dots. In our experiment, the Berry phase of electron orbital states is continuously tuned from about $\pi$ to $2\pi$ by perpendicular magnetic fields. When the Berry phase equals $\pi$ or $2\pi$, the electron states in the two inequivalent valleys are energetically degenerate. By altering the Berry phase to noninteger multiples of $\pi$, large and continuously tunable valley-polarized energy spectra are realized. Our result reveals the Berry phase’s essential role in valleytronics and the observed valley splitting, on the order of 10 meV at a magnetic field of 1 T, is about 100 times larger than Zeeman splitting for spin, shedding light on graphene-based valleytronics.

Figure 1

(a)–(c) Jump of the Berry phase in monolayer graphene QDs. (d)–(f) Continuously tunable Berry phase in Bernal-stacked BLG QDs. (a),(d) Sketches of monolayer graphene QD and BLG QD, respectively. (b),(e) Schematic charge trajectories in momentum space of monolayer graphene QD and BLG QD, respectively. The orange solid and black dashed lines in (b) represent trajectories in the magnetic fields above and below the $B_C$, respectively. The orange solid and black dashed lines in (e) represent trajectories in the magnetic fields of $10T$ and $0T$, respectively. (c),(f) Berry phase as a function of the magnetic fields $B$ in monolayer graphene QD and BLG QD, respectively.

Figure 2

(a) A $5\times 5{\mathrm{nm}}^2$ atomic-resolved STM image ($V_{\text{sample}}=800\mathrm{mV}$, $I=200\mathrm{pA}$) of the Bernal-stacked BLG. The triangular graphene lattices of the Bernal-stacked BLG are overlaid onto the image. (b) Landau level spectra of the BLG for various magnetic fields. Curves are shifted vertically for clarity. The Landau level peak indices are marked, and the gap is labeled by shadows. (c) The Landau level energies for different magnetic fields obtained from (b) against $\pm {[n(n-1)]}^{1/2}B$. (d) Calculated low-energy dispersions of the Bernal-stacked BLG. (e) An enlarged image of the calculated flat band dispersion in (d). (f) Experimental and calculated DOS of the top layer as a function of energy.

Figure 3

(a) A color plot of the $dI/dV$ spectra measured at different tip-sample distances $Z_{\mathrm{tip}}$. The tip height decreases by increasing the tunneling current $I$ with a fixed voltage bias. When the tip is approaching the BLG (as the tunneling current increases), the signals of the flat bands and the bound states become obvious. (b) Differential of the tunneling conductance map in (a). The feature of the bound states is more obvious. The sudden jump at about $I\approx 600\mathrm{pA}$ may arise from the slight variation of the doping in the BLG induced by the STM tip.

Figure 4

(a) Experimental (left panel) and calculated (right panel) differential conductance maps versus magnetic fields $B$ in the center of the BLG QD. Here, calculated differential conductance is proportional to the differential LDOS $-{\partial }^{2}D(E)/\partial E^2$ of the top layer. The red (black) dashed lines guide the trend of bound states for the valley $K$ ($K^{\prime }$). (b),(c) Four representative line cuts in experiment and theory at different magnetic fields in (a). The red lines guide the trend of the levels for the valley $K$. The black lines guide the trend of the levels for the valley $K^{\prime }$.