Spatial and magnetic confinement of massless Dirac fermions

The massless Dirac fermions and the ease to introduce spatial and magnetic confinement in graphene provide us unprecedented opportunity to explore confined relativistic matter in this condensed-matter system. Here we report the interplay between the confinement induced by external electric fields and magnetic fields of the massless Dirac fermions in graphene. When the magnetic length $l_B$ is larger than the characteristic length of the confined electric potential $l_V$, the spatial confinement dominates and a relatively small critical magnetic field splits the spatial-confinement induced atomiclike shell states by switching on a π Berry phase of the quasiparticles. When the $l_B$ becomes smaller than the $l_V$, the transition from spatial confinement to magnetic confinement occurs and the atomiclike shell states condense into Landau levels (LLs) of the Fock-Darwin states in graphene. Our experiment demonstrates that the spatial confinement dramatically changes the energy spacing between the LLs and generates large electron-hole asymmetry of the energy spacing between the LLs. These results shed light on puzzling observations in previous experiments, which hitherto remained unaddressed.

Figure 1

(a) Sketch of the STM setup. The electric field of the STM tip induces band bending to confine the massless Dirac fermions. (b) A $40\times 25-\mathrm{n}{\mathrm{m}}^2$ STM image ($V_{\mathrm{sample}}=300\mathrm{mV}, I=500\mathrm{pA}$) of the D1 region. The topmost graphene sheet decouples from the underlying graphene layer through a large twist angle of about 12°. The left inset shows an atomic-resolved STM image in black rectangle area and the right inset shows the fast Fourier transform (FFT) image. The white circles are reciprocal lattices of graphene and the green circles are reciprocal moiré lattices. (c) STS spectra of monolayer graphene. Dirac points are marked with orange arrows. (d) Schematic diagrams of charge orbits in the circular graphene resonators under different applied magnetic fields. (e) Schematic charge trajectories in momentum space on the Dirac cone for magnetic fields below (blue) and above (orange) the $B_C$. The orange solid and dashed lines represent $m=+1/2$ and $m=-1/2$ modes, respectively. (f), (g) Differential conductance maps vs magnetic fields $B$. When magnetic field is larger than the $B_C$, there is a large and sharp jump in energy of the $m=+1/2$ state in panel (f). No resonances can be observed and the Landau levels are clearly observed in magnetic fields on the order of 0.1 T in the D2 region. The black dotted lines in panels (f) and (g) guide the trend of the experimental energy levels.

Figure 2

Panels (a) (left) and (b) (left) correspond to the spectra recorded in the D1 and D2 regions, respectively. Landau levels of massless Dirac fermions are clearly observed in the spectra recorded in high magnetic fields. Panels (a) (right) and (b) (right) show the calculated Landau quantization in the presence of different spatial confinements. The red dotted line is $B^2=\frac{16\kappa }{{\left(v_{F}e\right)}^2}\left(e{V}_b-E_D\right)$.

Figure 3

(a), (c) The Landau level energies against $\mathrm{sgn}\left(n\right){\left(\right|n\left|B\right)}^{1/2}$ recorded in the D1 and D2 regions, respectively. The squares represent the experimental data. Black dots correspond to theoretical results extracted from Figs. 2 and 2 Effective Fermi velocities and $v_F^h/v_F^e$ as a function of magnetic fields. The theoretical result without considering the confining potential is also plotted (dashed lines) for comparison.

Figure 4

Theoretical Fermi velocity as a function of the confining potential. The insets schematically show the circular confining potentials. The n-p-n and p-n-p confining junctions have opposite effects on the Landau quantization for electrons (solid symbols) and holes (open symbols), therefore, generating opposite electron-hole asymmetry.