In-gap states in superconducting ${\mathrm{LaAlO}}_3-{\mathrm{SrTiO}}_3$ interfaces observed by tunneling spectroscopy

We identified quasiparticle states at well-defined energies inside the superconducting gap of the electron system at the ${\mathrm{LaAlO}}_3\text{−}{\mathrm{SrTiO}}_3$ interface using tunneling spectroscopy. The states are found only in a number of samples and depend upon the thermal-cycling history of the samples. The states consist of a peak at zero energy and other peaks at finite energies, symmetrically placed around zero energy. These peaks disappear, together with the superconducting gap, with increasing temperature and magnetic field. We discuss the likelihood of various physical mechanisms that are known to cause in-gap states in superconductors and conclude that none of these mechanisms can easily explain the results. The conceivable scenarios are the formation of Majorana bound states, Andreev bound states, or the presence of an odd-frequency spin triplet component in the superconducting order parameter.

Figure 1

Sample design of sample A. Schematic of a single device illustrating the flow of tunnel current $I_T$ and in-plane current $I_{2DEL}$ The inner and outer Ti electrodes are separated into a large current electrode and a small voltage probe section each. Cross-sectional thicknesses are not to scale.

Figure 2

A typical tunneling spectrum of sample A from the first measurement run without in-gap features measured at base temperature of the cryostat (50 mK). The Dynes fit [86] used to extract the gap width $\mathrm{\Delta }$ and quasiparticle lifetime parameter $\mathrm{\Gamma }$ is shown in addition to the data. The deviation between data and fit around zero bias is attributed to a minor Altshuler-Aronov offset [3].

Figure 3

Tunneling spectra of samples A and B illustrating the emergence of in-gap features: (a) Evolution of in-gap features in sample A with temperature. (b) Evolution of in-gap features in sample A with magnetic field. (c) Evolution of in-gap features in sample A with back-gate voltage, data measured in a different measurement run than (a) and (b). (d) Evolution of in-gap features in sample B with back-gate voltage. In-gap features in (a) and (b) consist of a strong peak at zero bias and smaller peaks on either side inside the superconducting gap. In-gap features in (c) and (d) appear less systematic but can be seen to be qualitatively similar to those of the other measurements when a different distribution of spectral weight between central and side peaks is assumed. Curves are vertically shifted for visibility by 1 $\mu \mathrm{S}$ for (a)–(c) and by 200 $\mu \mathrm{S}$ for (d), respectively. The conductivity is calculated by numerical differentiation with adaptive smoothing which does not change the in-gap feature characteristics. Negative back-gate voltage corresponds to depletion of the interface, whereas positive back-gate voltage corresponds to carrier accumulation.

Figure 4

Illustration of the quantitative analysis used to characterize the in-gap peaks: A Dynes fit to the gap part without peaks is used to calculate the difference between expected and measured conductivity values. Three Gaussians are fitted to the subtracted peak to determine position, height, and FWHM of the central peak and the separation of the side peaks. The dotted yellow lines indicate the three single Gaussians, and the continuous yellow line indicates the sum of all three Gaussians. Note that because spectral weight is shifted from the coherence peaks into the in-gap peaks, the coherence peaks are not described well by the lifetime-broadened BCS fit. The red data points correspond to the 60 mK curve in Fig. 3.

Figure 5

Evolution of height and FWHM of the zero bias peak and of the separation of the side peaks observed in sample A. In (a) and (b) we plot both the relative height of the peak, i.e., the difference between data and BCS fit (blue squares), and the absolute size of the peak, i.e., the conductivity value at zero bias in the measured data (red diamonds). Error bars indicate values obtained for BCS fits where the sum of weighted residuals is double that of the optimal fit. Since the absolute height of the peak depends only marginally on the fit, error bars for the absolute peak are smaller than the data points. (a) Evolution with temperature. (b) Evolution with magnetic field. Whereas the relative peak height decreases with increasing magnetic field or temperature, the absolute peak height remains almost constant. In (c) and (d) we plot the evolution of the FHWM of the central peak (blue squares) and the separation of the side peaks (red diamonds). Again, error bars are obtained from data calculated for BCS fits with twice the minimum residual. (c) Evolution with temperature. (d) Evolution with magnetic field. Both the FWHM of the central peak and the spacing of the side peaks remain approximately constant over the field and temperature range in which the superconducting gap can be observed. Values of the side peak separation are only shown for those curves in which side peaks can be clearly discerned. The data shown here was obtained from that of Figs. 3 and 3 using the procedure illustrated in Fig. 4.