Dirac metal to topological metal transition at a structural phase change in ${\mathrm{Au}}_2\mathrm{Pb}$ and prediction of ${\mathbb{Z}}_2$ topology for the superconductor

Three-dimensional Dirac semimetals (DSMs) are materials that have massless Dirac electrons and exhibit exotic physical properties. It has been suggested that structurally distorting a DSM can create a topological insulator but this has not yet been experimentally verified. Furthermore, Majorana fermions have been theoretically proposed to exist in materials that exhibit both superconductivity and topological surface states. Here we show that the cubic Laves phase ${\mathrm{Au}}_2\mathrm{Pb}$ has a bulk Dirac cone that is predicted to gap on cooling through a structural phase transition at 100 K. The low temperature phase can be assigned a ${\mathrm{Z}}_2=-1$ topological index, and this phase becomes superconducting below 1.2 K. These characteristics make ${\mathrm{Au}}_2\mathrm{Pb}$ a unique platform for studying the transition between bulk Dirac electrons and topological surface states as well as studying the interaction of superconductivity with topological surface states, combining many different properties of emergent materials—superconductivity, bulk Dirac electrons, and a topologically nontrivial ${\mathrm{Z}}_2$ invariant.

Figure 1

Characterization of the superconductivity and normal state resistivity in ${\mathrm{Au}}_2\mathrm{Pb}$. (a) Critical field determination of ${\mathrm{Au}}_2\mathrm{Pb}$, the blue line shows the linear fit to determine the slope $d{H}_{c2}/dT=-0.0928$ T/K. The inset shows the resistivity close to $T_c$ for different fields. (b) $\rho \left(T\right)$ between 300 and 25 K. Three discontinuities can be seen, at 97, 51, and 40 K. The lower inset shows the derivative which shows the discontinuities more clearly. Blue lines represent data measured on cooling while red lines represent data measured on heating.

Figure 2

Specific heat of ${\mathrm{Au}}_2\mathrm{Pb}$, confirming the bulk superconductivity and a phase transition at 50.4 K. The upper inset shows the electronic specific heat vs temperature of ${\mathrm{Au}}_2\mathrm{Pb}$. The sharp jump indicates bulk superconductivity. The value of $\mathrm{\Delta }C/\left(\gamma T_c\right)=1.95$ is higher than the expected BCS value. The lower inset shows the linear fit above $T_c$ that was used to determine the Sommerfeld coefficient $\gamma$ and the coefficient for the phononic contribution to the heat capacity $\beta$.

Figure 3

Structural information on ${\mathrm{Au}}_2\mathrm{Pb}$. (a) Crystal structure of the high temperature cubic Laves phase ${\mathrm{Au}}_2\mathrm{Pb}$. (b) Crystal structure of the low temperature orthorhombic, distorted phase of ${\mathrm{Au}}_2\mathrm{Pb}$. (c) Synchrotron diffraction patterns at different temperatures showing the presence of three different phases in ${\mathrm{Au}}_2\mathrm{Pb}$. Newly appearing peaks are circled, peaks associated with the orthorhombic phase are circled in red. Blue circled peaks belong to the intermediate phase. Impurity phases are marked with asterisks; those peaks remain present across the whole temperature regime measured. Patterns are displaced for clarity. (d) Rietveld refinement of orthorhombic ${\mathrm{Au}}_2\mathrm{Pb}$ at 7 K, the insert shows the region shown in (c); all the newly appearing peaks are captured by the structural model.

Figure 4

(a) Electronic structure of high temperature cubic ${\mathrm{Au}}_2\mathrm{Pb}$. (b) A view of the electronic structure of the high temperature phase along the (100) direction. The cubic electronic structure has a 3D Dirac cone along $\mathrm{\Gamma }-X$ (circled), which gets gapped out when the compound becomes distorted at low temperatures (Fig. 5). Despite the metallic pockets the Dirac point should be visible in APRES measured on the (100) surface. (c) 2D projection of the 3D band structure of cubic ${\mathrm{Au}}_2\mathrm{Pb}$, along a cubic axis such as (0,0,1). Color brightness shows the intensity of the signal that would be observed (e.g., in ARPES) by probing the bulk along this cubic axis, at energies near the energy of the Dirac cone, below the Fermi energy. The features associates with the Dirac cone are well separated, even in this 2D projection, from the Fermi surfaces of other band crossings.

Figure 5

(a) Electronic structures of low temperature orthorhombic ${\mathrm{Au}}_2\mathrm{Pb}$. (b) Emphasizes the continuous gap in the electronic structure of orthorhombic ${\mathrm{Au}}_2\mathrm{Pb}$. An energy dependent density of states (DOS) plot is shown for orthorhombic ${\mathrm{Au}}_2\mathrm{Pb}$ in (c). Orthorhombic ${\mathrm{Au}}_2\mathrm{Pb}$ has a continuous gap and can be viewed as a semiconductor with bent bands. The calculated DOS at $E_F$ matches well with the measured value.