Resonant impurity states in chemically disordered half-Heusler Dirac semimetals

We address the electron transport characteristics in bulk half-Heusler alloys with their compositions tuned to the borderline between topologically nontrivial semimetallic and trivial semiconducting phases. Accurate first-principles calculations based on the coherent potential approximation (CPA) reveal that all the studied systems exhibit sets of dispersionless impurity-like resonant levels, with one of them being located at the Dirac point. By means of the Kubo-Bastin formalism we reveal that the residual conductivity of these alloys is strongly suppressed by impurity scattering, whereas the spin Hall conductivity exhibits a rather complex behavior induced by the resonant states. In particular for ${\mathrm{LaPt}}_{0.5}{\mathrm{Pd}}_{0.5}\mathrm{Bi}$ we find that the total spin Hall conductivity is strongly suppressed by two large and opposite contributions: the negative Fermi-surface contribution produced by the resonant impurity and the positive Fermi-sea term stemming from the occupied states. At the same time, we identify no conductivity contributions from the conical states.

Figure 1

(a) Compositional tuning. By taking LaPtBi as a starting point, we generate the series of $\mathrm{La}({\mathrm{Pt}}_{1-x}{\mathrm{Pd}}_x)\text{Bi}$ alloys, in which the topological phase transition occurs at $x\approx 0.5$. Analogously, within the $\mathrm{LaPt}({\mathrm{Bi}}_{1-y}{\mathrm{Sb}}_y)$ series, the transition occurs at $y\approx 0.8$. These two compositions generate a continuous set of borderline materials $\mathrm{La}({\mathrm{Pt}}_{1-x}{\mathrm{Pd}}_x\left)\right({\mathrm{Bi}}_{1-y}{\mathrm{Sb}}_y)$: $y\approx 0.8–1.6x$, whose compositions are placed along the red line separating the nontrivial and trivial systems (light-red and light-blue colored areas, respectively). Panels (b), (c), and (d) represent the BSF's, calculated for three borderline compositions: $\mathrm{La}({\mathrm{Pt}}_{0.5}{\mathrm{Pd}}_{0.5})\text{Bi}$, $\mathrm{LaPt}({\mathrm{Bi}}_{0.2}{\mathrm{Sb}}_{0.8})$, and $\mathrm{La}({\mathrm{Pt}}_{0.75}{\mathrm{Pd}}_{0.25}\left)\right({\mathrm{Bi}}_{0.6}{\mathrm{Sb}}_{0.4})$, respectively. The spectral intensity distribution corresponds to the logarithmic color scale, shown at the right side of panel (d). All electronic structures are shown within the same energy windows $\left(-1<E-E_{\mathrm{F}}<1\text{eV}\right)$.

Figure 2

Schematic effect of randomness on electronic structure: Bloch eigenstates of an ordered reference system acquire a finite Lorentzian broadening (colored with blue). The summed intensity is shown by the black thick line. Those eigenstates which are situated sparsely (with small DOS, $n)$ produce the impurity states which are still well distinguished; those which are dense enough (large $n)$ result into a broadened Gaussian continuum, i.e., disordered conventional metal regime.

Figure 3

Comparison of the electronic structures of the same $\mathrm{La}({\mathrm{Pt}}_{0.5}{\mathrm{Pd}}_{0.5})\text{Bi}$ composition, but related to the different chemical orders. Corresponding crystal structure units are shown on the left. Red and yellow spheres mark La and Bi atoms, and gray and dark-gray mark Pd and Pt, respectively. (a) The smallest possible ordered structure (8 nonequivalent atoms within unit cell) of the given stoichiometry, in which half of the tetrahedral sites are occupied by Pt and half by Pd. (b) More complicated ordered variant of the same composition with 24 nonequivalent atoms which includes more environmental combinations. (c) The system with fully random Pd/Pt disorder (random occupation of the tetrahedral sites by Pt/Pd is indicated by the half-dark-gray/half-light-gray spheres, respectively).

Figure 4

(a) Calculated BSF for $\mathrm{La}({\mathrm{Pt}}_{0.5}{\mathrm{Pd}}_{0.5})\text{Bi}$ alloy (the spectral intensity distribution is the same as in Fig. 1) and (b) the corresponding $s$-, $p$-, and $d$-projected DOS shown using a logarithmic scale $(n_{s,p,d}$ in gray, red, and blue, respectively). (c) Diagonal conductivity $\sigma$, Fermi-surface part of the spin Hall conductivity ${\sigma }_{\mathrm{SH}}^{\mathrm{I}}$, and their ratio ${\sigma }_{\mathrm{SH}}^{\mathrm{I}}/\sigma$ (spin Hall angle), calculated as function of the energy $E$. The black dashed line corresponds to the small imaginary energy offset $\delta =1$ meV, and the thick blue line correponds to $\delta =0.1$ meV.

Figure 5

(a) Longitudinal conductivity $\sigma$, (b) spin Hall conductivities: ${\sigma }_{\mathrm{SH}}^{\mathrm{I}}$ (Fermi-surface contribution, blue) and ${\sigma }_{\mathrm{SH}}={\sigma }_{\mathrm{SH}}^{\mathrm{I}}+{\sigma }_{\mathrm{SH}}^{\mathrm{II}}$ (total, red), together with the corresponding spin Hall angles (c), calculated as functions of energy $E$. The imaginary energy offset is $\delta =0.1$ meV.

Figure 6

The electronic structure in $\mathrm{La}({\mathrm{Pt}}_{1-x}{\mathrm{Pd}}_x)\text{Bi}$ series shown for the nontrivial (gapless $x=0$ and $x=0.4)$ and the trivial (gapped $x=0.6$ and $x=1.0)$ compositions. Topological phase transition, corresponding to the Dirac dispersion at $E_{\mathrm{F}}$ in the $\mathrm{\Gamma }$ point, occurs at $x\approx 0.5$.