Valley isospin of interface states in a graphene $pn$ junction in the quantum Hall regime

In the presence of crossed electric and magnetic fields, a graphene ribbon has chiral states running along sample edges and along boundaries between $p$-doped and $n$-doped regions. We here consider the scattering of edge states into interface states, which takes place wherever the $pn$ interface crosses the sample boundary, as well as the reverse process. For a graphene ribbon with armchair boundaries, the evolution of edge states into interface states and vice versa is governed by the conservation of valley isospin. Although valley isospin is not conserved in simplified models of a ribbon with zigzag boundaries, we find that arguments based on isospin conservation can be applied to a more realistic modeling of the graphene ribbon, which takes account of the lifting of electron-hole degeneracy. The valley isospin of interface states is an important factor determining the conductance of a graphene $pn$ junction in a quantizing magnetic field.

Figure 1

Schematic picture of a graphene $pn$ junction in the first quantized Hall plateau. (a) For an armchair nanoribbon chiral states coming in from the left and right (going out to the left and right) have opposite valley isospin ${\mathit{\nu }}_{\mathrm{in}}$ and $-{\mathit{\nu }}_{\mathrm{in}}$ (${\mathit{\nu }}_{\mathrm{out}}$ and $-{\mathit{\nu }}_{\mathrm{out}}$), respectively. For sufficiently smooth potentials the valley isospin $\mathit{\nu }$ is conserved and the conductance is determined by the overlap of valley isospins ${\mathit{\nu }}_{\mathrm{in}}$ and ${\mathit{\nu }}_{\mathrm{out}}$ of incoming and outgoing scattering states, see Eq. (1). (b) For a lattice model of a zigzag nanoribbon with nearest-neighbor hopping only, the incoming (outgoing) edge states on the left and right have the same isospin ${\mathit{\nu }}_{\mathrm{edge},\mathrm{in}}$ (${\mathit{\nu }}_{\mathrm{edge},\mathrm{out}}^{\prime }$). Valley isospin is not conserved in the transition between edge states and interface states. The conductance of the $pn$ junction is determined by the overlap of the isospins ${\mathit{\nu }}_{\mathrm{in}}$ and ${\mathit{\nu }}_{\mathrm{out}}$ of the interface states connected to incoming and outgoing scattering states in the same ($p$) side of the junction.

Figure 2

Hexagonal lattice with its two-atom unit cell. The primitive lattice vectors are denoted ${\mathit{a}}_1$ and ${\mathit{a}}_2$. The right panel shows a choice of the unit cell that is rotated anticlockwise by an angle $2\pi /3$.

Figure 3

Choice of the origin $O$ and orientation of the two-atom unit cell corresponding to the boundary conditions (9) and (10) for zigzag and armchair boundary conditions.

Figure 4

Zigzag $pn$ junctions with two high-symmetry positions of the $pn$ interfaces (dashed) and with mirror axis parallel to edge (dot-dashed). The unit cell labeled $O$ is compatible with the boundary condition (9) at both boundaries. The unit cells are positioned symmetrically with respect to mirror inversion in the $pn$ interface. Valley isospin directions of edge states and interface states are indicated for a nearest-neighbor tight-binding model. The isospin directions of the interface states are fixed by symmetry arguments, see main text, up to an overall sign, which is determined by numerical simulations (see Sec. 3).

Figure 5

Armchair $pn$ junctions with high-symmetry positions of the $pn$ interfaces (dashed) and with mirror axis parallel to the edge (dot-dashed). Valley isospin labels are for a nearest-neighbor tight-binding model. They are given with respect to the origins $O$ and $\overline{O}$ for incoming and outgoing scattering states, respectively. (The origins $O$ and $\overline{O}$ are interchanged under mirror reflection in the mirror axis parallel to the edge.) Isospin directions are fixed by symmetry arguments, see main text, up to an overall sign, which is determined by numerical simulations (see Sec. 3).

Figure 6

Zigzag $pn$ junctions with an inversion center. The integer $n$ counts the number of half hexagons between the intersection of the $pn$ interface (dashed) and the left and right sample boundaries. Origins $O$ and $\overline{O}$ compatible with the boundary conditions at the left and right sample boundaries and symmetrically placed with respect to the $pn$ interface are shown for $\delta x=3/2$ (left) and $\mathrm{\Delta }x=1$ (right).

Figure 7

Scattering geometry used to numerically calculate the valley isospin $\mathit{\nu }$ of the interface states. The lead region is shown in red. The gray background schematically indicates the height of the potential $U\left(\mathit{r}\right)$, which determines the position of the $pn$ interface. In the ideal lead, the $pn$ interface is parallel to the lead's zigzag boundary; in the scattering region, the $pn$ interface is slightly curved, such that it intersects the zigzag boundary of the scattering region at an angle $\alpha$ to the boundary normal.

Figure 8

Dispersion relation of the quantum Hall edge states (solid curves) along a zigzag edge in the lowest quantized Hall plateau for an edge without sublattice antisymmetry (a), for an edge with sublattice antisymmetry-breaking perturbation $0<\delta U<E_1$ (b), and for an edge with $\delta U>E_1$ (c), where $E_1$ is the energy of the first Landau level. The solid horizontal lines give the energies of the Landau levels. The chiral edge states in $p$- and $n$-type regions have equal valley isospin in case (a) but opposite isospin in cases (b) and (c).

Figure 9

The $z$-component ${\nu }_z$ of the isospin of interface states originating from the $p$-type region, as a function of $l_{\mathrm{pn}}\delta U/U_0$. We have chosen $U_0=0.2t$ and the values of $l_{\mathrm{pn}}$ (measured in units of the hexagon width) are given in the inset. The value of the magnetic field is such that the first Landau level is at $E_1=0.2t$ and the size of the scattering region $L_x=360$ and $L_y=10l_{\mathrm{pn}}$. The sublattice antisymmetry is broken by onsite potential of magnitude $\delta U$ that is added locally on the outermost sites of the zigzag edge.

Figure 10

Main panel: Valley isospin component ${\nu }_z$ of the interface state originating from the $p$-type region as a function of the angle $\alpha$ between the $pn$ interface and the boundary normal. The dashed line shows the linear approximation (20). Right inset: The azimuthal angle $\phi$ of the valley isospin $\mathit{\nu }$ at $\alpha =0$, as a function of the $x$ coordinate of the intersection of the $pn$ junction and the zigzag sample boundary. The coordinate $x$ is measured in units of the hexagon width; the origin $x=0$ and the reference unit cell are chosen at one of the outermost lattice sites, as shown in the left inset.

Figure 11

(a) Scattering region with two different zigzag terminations at the upper edge. The valley polarization of the chiral edge states is ${\mathit{e}}_z$ ($K$ valley) for the “horizontal” edges and $-{\mathit{e}}_z$ ($K^{\prime }$ valley) for the “tilted” edge, as indicated in the figure. (b) Valley isospin ${\nu }_z$ of the interface state originating from the (left) $p$-type region. The numbers “1,” “2,” “3,” and “4” indicate positions of the $pn$ interface shown in panel (a).

Figure 12

A nanoribbon with sections having zigzag/zigzag (1), zigzag/armchair (2), and armchair/armchair termination (3). With broken sublattice symmetry, the conductance $G$ is different for the three combinations of boundary termination but does not depend on the precise location or orientation of the $pn$ interface within these three sections. In contrast, in the presence of sublattice symmetry, $G$ depends strongly on the precise orientation (in region 1) or position (in region 2) of the $pn$ interface. The left panel shows a lattice in which the numbers $n_{\mathrm{hex}}$ of hexagons in section 1 is even and the number $n_{\mathrm{hex}}^{\prime }$ of half hexagons in section 3 is a multiple of 3 plus one. The right panel shows a lattice in $n_{\mathrm{hex}}$ and $n_{\mathrm{hex}}^{\prime }$ are odd and not a multiple of 3 plus one, respectively.

Figure 13

Conductance $G$ if a nanoribbon with mixed boundary conditions, as shown schematically in the left panel (solid) and right panel (dashed) of Fig. 12. The top and bottom panels are for broken and unbroken sublattice antisymmetry, respectively. The labels “1,” “2,” and “3” refer to the three regions shown in Fig. 12: A ribbon with two zigzag edges, a ribbon with one zigzag and one armchair edge, and a ribbon with two armchair edges, respectively. The inset shows how $G$ depends on the orientation of the $pn$ interface for a ribbon with two zigzag edges (region “1” in Fig. 12) in the case of unbroken sublattice antisymmetry.