Quantum interference in topological insulator Josephson junctions

Using nonequilibrium Green's functions, we studied numerically the transport properties of a Josephson junction, superconductor-topological insulator-superconductor hybrid system. Our numerical calculation shows first that proximity-induced superconductivity is indeed observed in the edge states of a topological insulator adjoining two superconducting leads and second that the special characteristics of topological insulators endow the edge states with an enhanced proximity effect with a superconductor but do not forbid the bulk states to do the same. In a size-dependent analysis of the local current, it was found that a few residual bulk states can lead to measurable resistance, whereas because these bulk states spread over the whole sample, their contribution to the interference pattern is insignificant when the sample size is in the micrometer range. Based on these numerical results, it is concluded that the apparent disappearance of residual bulk states in the superconducting interference process as described by Hart et al. [Nat. Phys. 10, 638 (2014)] is just due to the effects of size: the contribution of the topological edge states outweighs that of the residual bulk states.

Figure 1

(a) Schematic diagram for the superconductor-TI-superconductor device. (b) and (c) The energy spectra of the ribbon of HgTe/CdTe quantum wells in the absence of a magnetic field and in the presence of a magnetic field with the hopping phase $\mathrm{\Phi }=0.001$. The ribbon width is $W=400$ nm and the effective mass is $M=-10$ meV.

Figure 2

(a) and (b) The superconductor order parameters in the central TI region in the absence of a magnetic field and in the presence of a magnetic field ($\mathrm{\Phi }=0.001$), respectively. Note that ${\mathbf{i}}_y$ represents the $y$ index of the site located at the middle cross section of HgTe/CdTe QWs. The ribbon width is $W=300$ nm and the length of the central region is $L=400$ nm.

Figure 3

Interference pattern shown in critical current for the HgTe/CdTe hybrid system. (a), (b), and (c) The Fermi energies of $E_F=1$ meV, $E_F=12$ meV, and $E_F=60$ meV, respectively. The ribbon width is $W=500$ nm, the length of the central region is $L=100$ nm, and the effective mass is $M=-10$ meV. Note that $\mathrm{\Phi }$ represents the additional hopping phase due to the external magnetic field, which is defined in Sec. 2a.

Figure 4

(a) Distribution of Cooper pair and (b) magnitude of local current along the cross section of the central TI sample, where $J_{\mathbf{i}}=\left|J_{\mathbf{i}\to \mathbf{i}+\widehat{\mathbf{x}}}\right|$, and ${\mathbf{i}}_y$ represents the $y$ index of the site locating at the middle cross section of the HgTe/CdTe QWs. Noted that a definite resistance (about $h/8e^2$) are preserved by choosing a suitable Fermi energy no matter what the width of sample is. The length of the central region is taken as $L=100$ nm, and the effective mass is $M=-10$ meV.