Superconducting Quantum Interference in Edge State Josephson Junctions

We study superconducting quantum interference in a Josephson junction linked via edge states in two-dimensional (2D) insulators. We consider two scenarios in which the 2D insulator is either a topological or a trivial insulator supporting one-dimensional (1D) helical or nonhelical edge states, respectively. In equilibrium, we find that the qualitative dependence of critical supercurrent on the flux through the junction is insensitive to the helical nature of the mediating states and can, therefore, not be used to verify the topological features of the underlying insulator. However, upon applying a finite voltage bias smaller than the superconducting gap to a relatively long junction, the finite-frequency interference pattern in the nonequilibrium transport current is qualitatively different for helical edge states as compared to nonhelical ones.

Figure 1

(a) Schematic of the long narrow biased Josephson junction formed by edge states of a bulk insulator pierced by a magnetic flux to realize an SQI setup. In a trivial insulator, nonhelical (continuous and dashed lines), whereas in a topological insulator, helical (only continuous lines) edge states may contribute to subgap transport. (b) and (c) Energy spectrum of the upper edge ($\tau =u$) with bare bandwidth $M$ and its renormalization to low energies $2\mathrm{\Delta }$, in case of helical and nonhelical edge states, respectively.

Figure 2

Most relevant processes in the critical current. (a) Dominant CAR process through the channels that have favorable propagation direction with respect to the bias. Present both in helical- and nonhelical systems, $\propto f_C^2{B}_h$. (b) Flux-dependent spin-singlet process in the preferable direction, only possible in nonhelical edges, $\propto A_{nh}\approx B_h$ in a long junction. (c) Overlap-type singlet process $\propto A_h$, the dominant flux-dependent term in equilibrium for both kinds of systems masking differences between them. (d) Spin-triplet process involving the same edge state for both electrons, the only possible flux-dependent contribution for long $\alpha =h$ junctions that is length independent in case of finite bias [42].

Figure 3

Different typical scenarios of the flux-dependent interference patterns in $I_{{\omega }_J}$ for helical ($\alpha =h$) and nonhelical ($\alpha =nh$) edge states in the long JJ regime, $L=20\xi$. Larger figures are normalized as $I_{{\omega }_J}(\mathrm{\Phi },p)/{\max }_{\mathrm{\Phi }}\{I_{{\omega }_J}(\mathrm{\Phi },p)\}$ for $p=T,V,K,K_c$, respectively, to compare qualitative changes in the shape of interference curves, while smaller ones compare relative amplitudes and ranges ($I_{\text{min}}$ to $I_{\text{max}}$) of $I_{{\omega }_J}$ as $p$ is swept, $\max (\min {)}_{\mathrm{\Phi }}\{I_{{\omega }_J}(\mathrm{\Phi },p)\}/{\max }_{\mathrm{\Phi },p}\{I_{{\omega }_J}(\mathrm{\Phi },p)\}$. Panels (a)–(d) and (g),(h) correspond to systems without interaction at finite temperatures, while (e),(f) explore the effect of interaction at $T=0$. Spin flips are excluded $(f_T=0)$ in all cases [42] and all plots are calculated with the detuning of the Fermi-level $k_{F}L\approx \pi (4n+3)/4$, to which only (c), (e), and (g) are sensitive, but not crucially [42]. The CAR strength is kept at $f_C=0.3$ throughout and quantities with dimensions are expressed in units of $\mathrm{\Delta }$. (a),(b) In equilibrium $V=0$ for temperatures $T=0,\dots ,0.2$. (c),(d) Same temperature sweep for biased junctions $V=0.1$. (e),(f) Biased junctions, $V=0.1$ with interactions $K,K_c=0.5,\dots ,1$ and $K_s=1$ at $T=0$. (g),(h) Bias sweep $V=0,\dots ,0.2$ at $T=0.1$. All nonindicated parameters remain unchanged.