Visualization by scanning SQUID microscopy of the intermediate state in the superconducting Dirac semimetal ${\mathrm{PdTe}}_2$

The Dirac semimetal ${\mathrm{PdTe}}_2$ becomes superconducting at a temperature $T_c=1.6$ K. Thermodynamic and muon spin rotation experiments support type-I superconductivity, which is unusual for a binary compound. A key property of a type-I superconductor is the intermediate state, which presents a coexistence of superconducting and normal domains at magnetic fields lower than the thermodynamic critical field $H_c$. We present scanning SQUID microscopy studies of ${\mathrm{PdTe}}_2$ revealing coexisting superconducting and normal domains of tubular and laminar shape as the magnetic field is more and more increased, thus confirming type-I superconductivity in ${\mathrm{PdTe}}_2$. Values for the domain wall width in the intermediate state have been derived. The field amplitudes measured at the surface indicate bending of the domain walls separating the normal and superconducting domains.

Figure 1

(a) The SQUID response after ZFC at three temperatures as indicated. In the normal phase (black solid line), the SQUID response is smooth. In the superconducting phase, we distinguish three different behaviors: (i) flat response, i.e., screening for $H<(1-N)H_c$, (ii) high density of $I_c$ jumps, i.e., penetration of magnetic flux tubes for $(1-N)H_c<H<H_f$, and (iii) smoother jumps, i.e., when flux tubes fuse into laminar structures $H_f<H<H_c$. The fields $H_p=(1-N)H_c, H_f$, and $H_c$ are indicated by arrows. In (b) the phase diagram is constructed from the gathered characteristic field values. The solid magenta line represents a BCS-fit $H_c\left(T\right)=H_c\left(0\right)[1-{(T/T_c)}^2]$ with ${\mu }_0{H}_c\left(0\right)=13.62$ mT and $T_c=1.57$ K, and the solid cyan line represents the equivalent fit with ${\mu }_0{H}_p\left(0\right)=3.83$ mT and $T_c=1.58$ K. The green squares indicate the field values when the flux changes become smoother, above which laminar structures appear. The vertical dashed lines indicate the temperatures at which the SQUID response is shown in (a).

Figure 2

(a)–(d) ZFC scanning SQUID images taken at a temperature of 900 mK ($H_c=9.1$ mT) at an applied field of 1 mT (reduced field, $h=H_a/H_c$, of 0.11), 3 mT (0.33), 4.5 mT (0.5), and 7.5 mT (0.82), respectively (all images from the same cool down). These images show the magnetic flux structures in the different regions of the phase diagram [Meissner, intermediate tubular, and intermediate laminar in Fig. 1]: dark gray (blue) regions are superconducting, and light gray (orange) are normal. Panel (a) contains a zoom on the weakest flux tube we observed. In panel (e), the inset shows the flux profile along line A of the flux tube in the zoom in panel (a), while the main panel shows the increase in collected flux as the magnetic field is summed up over areas with increasing lateral length, $L$. (f) Field profile along line $B$ as shown in (b). The inset in (f) represents the schematics of flux tube branching at the surface of the sample, neglecting NS interface bending.

Figure 3

Scanning SQUID images taken after field cooling under 3.5 mT (reduced field, $h=H_a/H_c$, of 0.34) and under 8 mT (0.61) in (b), at a temperature of 300 mK ($H_c=13.1$ mT). Points 1 and 2 indicate the maximal measured fields of each scan referred to in Fig. 4. The extended domains (open topology) are typical for field-cooled type-I superconductors. Dark gray (blue) regions are superconducting, and light gray (orange) regions are normal.

Figure 4

The maximum of the magnetic field threading the SQUID loop above the normal regions in the intermediate state divided by the critical field, $H_N/H_c$ (triangles), as a function of the applied field divided by the critical field, $H_a/H_c$, for ZFC (900 mK) (red/full) and FC (300 mK) (blue/empty) measurements. The green (dark gray) line represents a linear fit. The red dashed line traces the reduction of the maximal field at the surface in the case of the Landau laminar model as reported by Fortini et al. [35]. Points 1 and 2 are references to the corresponding points in the images of Fig. 3.