Chiral interface states in graphene $p\text{−}n$ junctions

We present a theoretical analysis of unidirectional interface states which form near $p\text{−}n$ junctions in a graphene monolayer subject to a homogeneous magnetic field. The semiclassical limit of these states corresponds to trajectories propagating along the $p\text{−}n$ interface by a combined skipping-snaking motion. Studying the two-dimensional Dirac equation with a magnetic field and an electrostatic potential step, we provide and discuss the exact and essentially analytical solution of the quantum-mechanical eigenproblem for both a straight and a circularly shaped junction. The spectrum consists of localized Landau-like and unidirectional snaking-skipping interface states, where we always find at least one chiral interface state. For a straight junction and at energies near the Dirac point, when increasing the potential step height, the group velocity of this state interpolates in an oscillatory manner between the classical drift velocity in a crossed electromagnetic field and the semiclassical value expected for a purely snaking motion. Away from the Dirac point, chiral interface states instead resemble the conventional skipping (edge-type) motion found also in the corresponding Schrödinger case. We also investigate the circular geometry, where chiral interface states are predicted to induce sizable equilibrium ring currents.

Figure 1

Sketch of two types of graphene $p\text{−}n$ junctions in a constant $B$ field, including examples for semiclassical cyclotron and/or skipping-snaking interface trajectories. Blue (red) stands for $n$-doped ($p$-doped) regions with constant potential $V=-V_0$ ($V=+V_0)$. Left panel: straight junction; see Eq. (3). Right panel: circular junction; see Eq. (4).

Figure 2

Spectrum $E_{n,k}$ vs $k$ obtained from Eq. (12) for a straight $p\text{−}n$ junction, with units in Eq. (2): (a) $V_0=0.6$, (b) $V_0=1.2$, and (c) $V_0=2.4$. Red dashed curves are guides to the eye only and illustrate the central chiral interface state passing through $k=E=0$. Density profiles for the four $E=2$ ($E=0$) states labeled by green (orange) arrows are shown in Fig. 3 (in Fig. 4). The blue vertical arrow shows the avoided crossing studied in Fig. 5.

Figure 3

Spatial profile of the probability density ${\rho }_{n,k}\left(x\right)$ at $V_0=2.4$ for four different eigenstates with energy $E_{n,k}=+2$ and wave numbers $k\simeq -5.27,-4.48,-3.82,-3.21$. These states are indicated by green arrows in Fig. 2.

Figure 4

Spatial profile of the probability density ${\rho }_{n,k}\left(x\right)$ at $V_0=2.4$ for the four zero-energy states with wave numbers $k\simeq -2.46,-2.04,-0.97,$ and $k=0$, corresponding to the orange arrows in Fig. 2.

Figure 5

Spatial profile of the probability density ${\rho }_{n,k}\left(x\right)$ at $V_0=2.4$ for $n=-1$ (red dotted) and $n=0$ (black solid curves) states with wave numbers $-4.364\le k\le -4.363$ near the avoided crossing indicated by the blue vertical arrow in Fig. 2.

Figure 6

Velocity $v_s=-v_{n=0,k=0}$ of the central chiral interface state at low energy scales vs $p\text{−}n$ potential strength $V_0$. The solid black curve shows the analytical result in Eq. (19). The dotted red line denotes the large-$V_0$ limit, $v_s/v_F=2/\pi \simeq 0.63$. The dashed blue line gives the drift velocity $v_s/v_F=2\sqrt{2/\pi }{V}_0$ expected for $V_0\ll 1$.

Figure 7

Current density profile $J_{n,k}\left(x\right)$, see Eq. (14), for $n=0$ eigenstates of a straight graphene $p\text{−}n$ junction with $V_0=0.6$ and several wave numbers $k$.

Figure 8

Energy spectrum $E_{n,j}$ vs angular momentum $j$ for a circular $p\text{−}n$ junction of radius $R=3.3$ with $V_0=0.6$ (main panel) and $V_0=1.2$ (inset), using the units in Eq. (2). Different colors and symbols correspond to different values of $n_r=\left|n\right|-(j+1/2)\mathrm{\Theta }\left(j\right)$: $n_r=0$ (black squares), $n_r=1$ (red upward-pointing triangles), $n_r=2$ (green downward-pointing triangles), and $n_r=3$ (blue diamonds).

Figure 9

Current density in azimuthal direction, $J_{n,j}^{\left(\phi \right)}\left(r\right)$, vs radial distance $r/l_B$ for a circular $p\text{−}n$ junction; see Eq. (29). The shown results are for $n=0$ and several $j<0$, with $V_0=0.6$ and $R=3.3$ as in the main panel of Fig. 8.

Figure 10

Dimensionless coefficients $-C_{n=0,j<0}$ vs $j$ determining the ring currents in Eq. (31) for a circular $p\text{−}n$ junction with $R=3.3$ (main panel) and $R=10$ (inset). The shown results follow from first-order perturbation theory in $V_0$; see Eq. (B5) in Appendix pp2. Dotted curves are guides to the eye only. In the inset, one clearly sees that the maximum of $|I_{0,j}|$ in Eq. (31) is reached for half-integer $j=j_0\approx -R^2/2=-50$.

Figure 11

Spectrum $E_{n,k}$ vs $k$ for a straight $p\text{−}n$ junction of Schrödinger fermions. Energies (wave numbers) are in units of ${\omega }_c$ ($l_B^{-1}$); cf. Eq. (2). The shown results are for $V_0/{\omega }_c=2.4$ and follow from Eq. (A4). Density profiles are illustrated in Fig. 12 for the states with $E_{n,k}=0,1,2,3$ indicated by arrows.

Figure 12

Density profiles for a straight Schrödinger $p\text{−}n$ junction with $V_0/{\omega }_c=2.4$. The shown states are indicated by arrows in Fig. 11, with $x$ in units of $l_B$. (a) Energy $E_{n,k}=0$ with $k=-1.00205$ (black solid) and $k=0.445883$ (red dashed). (b) Energy $E_{n,k}=1$ with $k=-1.54421$ (black solid), $k=-0.322044$ (red dashed), and $k=0.953544$ (green dot-dashed). (c) Energy $E_{n,k}=2$ for $k=-1.99022$ (black solid), $k=-0.899269$ (red dashed), and $k=1.37079$ (blue dot-dashed). (d) Energy $E_{n,k}=3$ with $k=-2.35387$ (black solid) and $k=-0.519357$ (red dashed).