Fractional Josephson effect in nonuniformly strained graphene

Nonuniform strain distributions in a graphene lattice can give rise to uniform pseudomagnetic fields and associated pseudo-Landau levels without breaking time-reversal symmetry. We demonstrate that by inducing superconductivity in a nonuniformly strained graphene sheet, the lowest pseudo-Landau levels split by a pairing gap can be inverted by changing the sign of the pairing potential. As a consequence of this inversion, we predict that a Josephson $\pi$ junction deposited on top of a strained graphene sheet exhibits one-dimensional gapless modes propagating along the junction. These gapless modes mediate single electron tunneling across the junction, giving rise to the $4\pi$-periodic fractional Josephson effect.

Figure 1

Josephson $\pi$ junction in a graphene sheet with nonuniform uniaxial strain in the $y$ direction ($\theta$: superconducting phase; $t$: nearest-neighbor electron hopping amplitude between atoms on $A$ and $B$ sublattices). We model the strain pattern by changing the hopping amplitudes on the vertical bonds (brown) by an amount $\delta t\propto y$ linearly increasing along the $y$ direction and constant along the $x$ direction. For the numerical diagonalization results (Fig. 3) we assume the junction is parallel to the zigzag chains (gray) indexed by an integer $j$. The $\pi$ phase difference leads to an inverted pseudo-Landau-level structure near the Dirac points of the graphene band structure. As a result, a pair of counterpropagating gapless 1D modes appears near the center of the junction (red and blue arrows).

Figure 2

Bogoliubov–de Gennes spectrum for the continuum model of strained graphene with a Josephson $\pi$ junction built on top, with $n$ the pseudo-Landau-level index and $p_x$ the momentum along the junction. Assuming the junction is at $y=0$, the superconducting phase varies from $\theta =0$ when $y>0$ to $\theta =\pi$ when $y<0$. The guiding center of the pseudo-Landau level is located at $y_0=-l_B^2{p}_x$, which moves from the $\theta =0$ region to the $\theta =\pi$ region as $p_x$ is varied from negative to positive. The $n=0$ pseudo-Landau level becomes inverted if either the pairing potential or $p_x$ change sign.

Figure 3

(a) Bogoliubov–de Gennes spectrum for the lattice model of a strained graphene strip (600 sites wide) with zigzag edges and a Josephson $\pi$ junction built on top, as a function of the momentum $k_x$ along the junction centered at $y=0$. The chosen strain field is equivalent to a 20 T pseudomagnetic field, and the pairing gap is $1\%$ of the nearest-neighbor hopping strength. ${\tilde{E}}_n$ is the energy in units of $v_F{l}_B^{-1}/\sqrt{2}$. The zeroth pseudo-Landau level is inverted at both the $K_{+}$ and $K_{-}$ valleys, with gapless chiral 1D modes inside the pairing gap $\tilde{\mathrm{\Delta }}$ appearing along the $\pi$ Josephson junction in both valleys. We plot the probability density for the gapless modes at the $K_{-}$ valley along the $y$ direction when the energy is (b) below the Dirac point, (c) at the Dirac point, and (d) above the Dirac point.

Figure 4

Bogoliubov–de Gennes spectrum at the Dirac point of a Josephson junction on a strained graphene strip as a function of the phase difference $\delta \theta$ across the junction, computed from the lattice model. The zeroth pseudo-Landau level exhibits a $4\pi$-periodic energy-phase relation, leading to a $4\pi$-periodic Josephson supercurrent. The other pseudo-Landau levels do not disperse with $\delta \theta$ and thus do not contribute to the supercurrent.