Currents and pseudomagnetic fields in strained graphene rings

We study the effects of strain on the electronic properties and persistent current characteristics of a graphene ring using the Dirac representation. For a slightly deformed graphene ring flake, one obtains sizable pseudomagnetic (gauge) fields that may effectively reduce or enhance locally the applied magnetic flux through the ring. Flux-induced persistent currents in a flat ring have full rotational symmetry throughout the structure; in contrast, we show that currents in the presence of a circularly symmetric deformation are strongly inhomogeneous, due to the underlying symmetries of graphene. This result illustrates the inherent competition between the “real” magnetic field and the “pseudo” field arising from strains, and suggests an alternative way to probe the strength and symmetries of pseudomagnetic fields on graphene systems.

Figure 1

Unstrained graphene ring. (a) Energy spectra vs magnetic flux for ring with internal and external radii given by $r_1=50$ and $r_2=100$ nm, for different quantum states: $m$ $[m^{\prime }=-(m+1)]$ integer denotes results for $K$ $\left(K^{\prime }\right)$ valley given by dashed (continuous) lines. [Thicker (red) lines near $E\approx 17$ meV show lower levels for the same ring, but with a Gaussian deformation.] (b) Flat ring eigenstate with $m=0$ ($K$ valley); electronic probability distribution along the ring for $\Phi =0$, and (c) $\Phi /{\Phi }_0=6$. Main panels show the two spinor components separately; insets show total distribution, $|{\psi }_A{|}^2+{\left|{\psi }_B\right|}^2$. On right column, corresponding current densities for $K$ valley. (d) Notice counterpropagating edge currents for $\Phi =0$, indicated by red arrows, evolve to a current distribution mostly on the outer radius for large $\Phi$ in (e).

Figure 2

Deformed graphene ring. Contour plot of the local density $|{\psi }_A{|}^2+{\left|{\psi }_B\right|}^2$ for $m=0$ state at (a) $\Phi /{\Phi }_0=0$, and (b) $\Phi =-0.5$. A negative flux, as shown, pushes density towards the inner ring, while a positive flux would shift weights to the outer radius. (c) Current distribution for $\Phi =0$ and $m=0$, for the $K$ valley. (d) Current variation, $\Delta \vec{J}=\vec{J}-{\vec{J}}_{\mathrm{flat}}$ ($K$-valley), where ${\vec{J}}_{\mathrm{flat}}$ is the current for a flat ring—shown in Fig. 1d—can be seen as the net effect produced by the deformation. Notice six vortices with alternating circulation and cores near maxima/minima of the pseudomagnetic field [see Fig. 3a].

Figure 3

(a) Map of pseudomagnetic field in Eq. (12). Traces show polar plots of the current with $\Phi /{\Phi }_0=-0.5$ for ring without (circular) and with (star) strain. (b) Angular profiles of current density through the ring for the lowest $K$ valley state and different fluxes $(\Delta \Phi /{\Phi }_0=0.1)$, from $\Phi =0$ to $\Phi /{\Phi }_0=-0.5$, as indicated by the arrow. Dashed flat line near bottom shows current density for unstrained ring at $\Phi =0$. Notice mean value of current increases with strain and flux. (c) Flux dependence of the current through the ring for the lowest state: $m=0$, $K$ valley from $\Phi /{\Phi }_0=-0.5$ to 0, and $m=-1$ for $K^{\prime }$ valley from $\Phi /{\Phi }_0=0$ to $0.5$, at different angles along the ring. Curves for strained rings are shifted up for clarity; dashed lines indicate zero current in each case. Bottom traces show angle-integrated current for both strained (solid) and unstrained (dotted) rings; notice similar slopes, although much smaller discontinuity near $\Phi =0$ with strain.