Nodeless superconductivity in the type-II Dirac semimetal ${\mathrm{PdTe}}_2$: London penetration depth and pairing-symmetry analysis

Superconducting gap structure was probed in type-II Dirac semimetal ${\mathrm{PdTe}}_2$ by measuring the London penetration depth using the tunnel diode resonator technique. At low temperatures, the data for two samples are well described by a weak-coupling exponential fit yielding $\lambda (T=0)=230$ nm as the only fit parameter at a fixed $\mathrm{\Delta }\left(0\right)/T_c\approx 1.76$, and the calculated superfluid density is consistent with a fully gapped superconducting state characterized by a single gap scale. Electrical resistivity measurements for in-plane and inter-plane current directions find very low and nearly temperature-independent normal-state anisotropy. The temperature dependence of resistivity is typical for conventional phonon scattering in metals. We compare these experimental results with expectations from a detailed theoretical symmetry analysis and reduce the number of possible superconducting pairing states in ${\mathrm{PdTe}}_2$ to only three nodeless candidates: a regular, topologically trivial $s$-wave pairing, and two distinct odd-parity triplet states that both can be topologically nontrivial depending on the microscopic interactions driving the superconducting instability.

Figure 1

Powder x-ray diffraction (XRD) pattern of crushed single crystal of ${\mathrm{PdTe}}_2$ (black line). The red lines are calculated XRD peaks for ${\mathrm{PdTe}}_2$ with hexagonal structure $[P\overline{3}\mathrm{m}1$, 164]. Blue stars mark peaks of solidified Te flux.

Figure 2

Temperature-dependent in-plane electrical resistivity of ${\mathrm{PdTe}}_2$, with bottom left panel showing zoom of the superconducting transition. Note a range below approximately 10 K, where resistivity becomes temperature-independent, and linear increase above 40 K. Left top inset compares normalized temperature-dependent part of resistivity in ${\mathrm{PdTe}}_2$ and Ag wire [38]. Right inset shows temperature-dependent resistivity anisotropy ratio, ${\rho }_c/{\rho }_{\parallel }$, with error bars determined by uncertainty of experimental geometry.

Figure 3

Temperature variation of London penetration depth $\mathrm{\Delta }\lambda \left(T\right)$ measured in ${\mathrm{He}}^3$ TDR setup for samples $\mathrm{A}$ (red) and $\mathrm{B}$ (blue). Main panel shows data with best fit using power-law function $\mathrm{\Delta }\lambda \left(T\right)=A+B{T}^n$, with $n=4.3\left(\mathrm{A}\right)$ and $n=4.2\left(\mathrm{B}\right)$. Right bottom panel shows same data plotted as a function of $T^4$ to verify quality of the fit. Top inset shows data over the whole temperature range up to $T_c\sim 1.8$ K.

Figure 4

Top panel: Fit of the temperature variation of London penetration depth with exponential BCS function, enabling determination of $\lambda \left(0\right)=240$ nm (sample $\mathrm{A}$, red) and $\lambda \left(0\right)=220$ nm (sample $\mathrm{B}$, blue). Inset: Logarithmic plot of the BCS function, showing that the function fits the data well in the range $1/t\equiv T_c/T=2–4$, or $T=0.25T_c–0.5T_c$. Bottom panel shows calculated superfluid density ${\rho }_s\left(T\right)\equiv {[\lambda \left(0\right)/\lambda \left(T\right)]}^2$ using $\lambda \left(0\right)$ values extracted from the fit.

Figure 5

The triplet vector (black arrows) on the two Fermi surfaces (red and blue solid lines) enclosing the $\mathrm{\Gamma }$ point is shown in (a), for the same sign of the order parameter on the two Fermi surfaces, and (b), for opposite signs, yielding a topologically nontrivial $\left({\nu }_1+{\nu }_2\ne 0\right)$ and trivial state $\left({\nu }_1+{\nu }_2=0\right)$, respectively. In (c), we indicate the high-symmetry points, where the triplet vectors of the two candidate states, $A_{1u}$ and $e_{u(1,0)}$ in Table 1, have to vanish as a consequence of inversion symmetry (red dots) and rotation symmetries (blue dots). The gray arrows illustrate the (simplest) texture of the triplet vector of the $A_{1u}$ state in the $k_z=0$ plane (with minimal number of defects). Panel (d) shows the directional dependence (i.e., anisotropy) of the gap of the $e_{u(1,0)}$ state on a Fermi surface enclosing the $\mathrm{\Gamma }$ point. The distance of the surface to the origin is proportional to the magnitude of the gap. The state breaks the threefold rotation symmetry and its gap is, thus, generically anisotropic.