Influence of topological edge states on the properties of $\mathrm{Al}/{Bi}_2{\mathrm{Se}}_3/\mathrm{Al}$ hybrid Josephson devices

In superconductor-topological insulator-superconductor hybrid junctions, the barrier edge states are expected to be protected against backscattering, to generate unconventional proximity effects, and, possibly, to signal the presence of Majorana fermions. The standards of proximity modes for these types of structures have to be settled for a neat identification of possible new entities. Through a systematic and complete set of measurements of the Josephson properties we find evidence of ballistic transport in coplanar $\mathrm{Al}-{Bi}_2{\mathrm{Se}}_3-\mathrm{Al}$ junctions that we attribute to a coherent transport through the topological edge state. The shunting effect of the bulk only influences the normal transport. This behavior, which can be considered to some extent universal, is fairly independent of the specific features of superconducting electrodes. A comparative study of Shubnikov–de Haas oscillations and scanning tunneling spectroscopy gave an experimental signature compatible with a two-dimensional electron transport channel with a Dirac dispersion relation. A reduction of the size of the ${\mathrm{Bi}}_2{\mathrm{Se}}_3$ flakes to the nanoscale is an unavoidable step to drive Josephson junctions in the proper regime to detect possible distinctive features of Majorana fermions.

Figure 1

(a) $G_{xx}$ after subtraction of the background as a function of the inverse of the magnetic field. The line is a spline interpolation of experimental data, and it is a guide for the eye. The inset shows $R_{xx}$ before the background subtraction. (b) Landau levels indexes inferred from conductance minima, and plotted as a function of $1/B$. The red line shows the fit obtained by fixing $F$ to the value $F_{\text{max}}$ extracted from the Fourier transform of $\Delta G_{xx}$ (see text for details), as shown in the top left inset. $F_{\text{max}}=179.3\mathrm{T}$ leads to an intercept $\beta =0.46$. The black line is obtained while keeping both $F$ and the intercept free for the linear fit $F=175.1\mathrm{T}$ and the intercept $\beta$ is 0.89. The bottom right inset shows an optical picture of a flake contacted in a Hall bar configuration.

Figure 2

(a) Shubnikov–de Haas oscillations as a function of the magnetic field ($B$) at different angles $\theta$ between the direction of $B$ and the plane of the current transport. The magnetoresistance background has been subtracted. Curves for various $\theta$ have been shifted for clarity. Experimental data are in blue, while the red lines are the fits using Eq. (4). The inset shows the values of the oscillation frequency inferred from the fit at different angles $\theta$ (blue points). Here the red line is the cosinusoidal behavior while the dashed green line is the expected dependence in the case of an ellipsoid shaped Fermi surface (two axes of the ellipsoid are equal and the third axis is 2 times bigger). This is the expected shape of the Fermi surface for the bulk band structure of Bi${}_2$Se${}_3$ [50, 51]. (b) 2D and 3D Hall coefficients extracted from the linear fit of the $R_{xy}\left(B\right)$ curve between $-5$ and $5\mathrm{T}$. The two coefficients show a clear trend with the flake thickness. A dependence of both 2D and 3D coefficient indicates a parallel transport through the bulk of the crystal and the 2D edge state.

Figure 3

Tunneling conductance spectrum acquired at the surface of a Bi${}_2$Se${}_3$ crystal. The spectrum has been measured with a bias modulation amplitude $V_{\text{mod}}$ = $2.5\mathrm{m}{\mathrm{V}}_{\mathrm{rms}}$. The V-shaped spectrum is consistent with the Dirac dispersion with the Dirac point at $V$ = $-350\mathrm{m}\mathrm{V}$ (minimum of the spectrum indicated by the arrow and labeled as $E_D$). The inset is an atomic resolved topographic image showing triangular shaped defects. The imaging conditions are $V=+0.1$ V, $I$ = 10 pA, scan area is 14.5 nm $\times 14.5$ nm (scale bar is 1 nm).

Figure 4

(a) Tunneling conductance spectra acquired in the presence of a magnetic field applied perpendicular to the sample surface. The spectra at different fields have been shifted for clarity. The tunneling spectra above 5 T show a series of peaks associated with the formation of Landau levels. (b) $dI/dV$ spectra at 7 T (in blue) and 8 T (in red) in (a). After the subtraction the Landau levels are clearly visible and they can be labeled. (c) Landau levels energy $E_n$ as a function of $\sqrt{nB}$. The dashed line is the linear fit of the data.

Figure 5

(a) $I\text{−}V$ curves of a Josephson junction at the temperature of $300\mathrm{m}\mathrm{K}$ for different powers of the applied microwaves at the output of the microwave source, with a frequency of $2\mathrm{GHz}$. The curves have been shifted for clarity. (b) Experimental and simulated (inset) data showing the modulation of the critical current and of the amplitude of the two first Shapiro steps as a function of the applied microwave power. Here we have assumed a conventional $I=I_{c}sin\phi$ CPR.

Figure 6

(a) Critical current as a function of the external magnetic field in a Al/Bi${}_2$Se${}_3$/Al Josephson junction at $300\mathrm{m}\mathrm{K}$. The red line is the reference curve appropriate for a small junction with a uniform critical current density. The inset shows a sketch of the device and an optical image. (b) Critical current modulations as a function of the magnetic field of a Al/Bi${}_2$Se${}_3$/Al SQUID (blue curve), compared to that of a reference SQUID (red curve). Both measurements have been performed at $300\mathrm{m}\mathrm{K}$. The two devices have modulations with the same periodicity. The different modulation depth is related to the different value of the kinetic inductance of the loops.

Figure 7

(a) $I\text{−}V$ curves of a Josephson junction as a function of the temperature. (b) Critical current of a Josephson junction as a function of the temperature. The analytical fit of the data to Eq. (6) leads to a Thouless energy of about $E_{\text{th}}=100\mu \mathrm{e}\mathrm{V}$ (shown in green). The numerical Eilenberger fit is shown in red. In the inset the critical current as a function of temperature is reported, in normalized units, for a junction with a Ti buffer layer and a junction of Pt buffer layer. Both devices show the same trend.