Half-Heuslers: Crystal structure and electronic structure

Today, the class of Heusler materials comprises more than 1500 different compounds (see Reference 14 and the references therein). Semiconducting behavior is mainly found in the subclass of the half-Heusler XYZ compounds. A half-Heusler material is a crystalline compound consisting of a covalent and an ionic part: the X and Y atoms have a distinct cationic character, whereas Z can be seen as the anionic counterpart. The most electropositive element X (usually a main group element, a transition metal, or a rare-earth element) is placed at the beginning of the formula. Generally, the half-Heusler phases crystallize in a non-centrosymmetric structure corresponding to the space group $F\bar 43m$. Within the lattice, atoms Y and Z form a covalent-type ZnS structure, while X and Z form the ionic NaCl-type substructure, as shown in Figure 1. Usually the crystal structure can be viewed as stuffing interstitial sites in a zinc-blende lattice (e.g., GaAs or HgTe) with a third element. For example, the XYZ crystal can be understood as stuffing the (YZ)^n– zinc-blende sublattice with an X ⁿ⁺ ion.

Figure 1.

Crystal structure of (a) zinc-blende and (b) equiatomic (filled zinc-blende) half-Heusler compounds XYZ. Green spheres represent Z atoms, red spheres Y atoms, and blue spheres X atoms. (c) Illustration of the 8- or 18-electron rule. Half-Heuslers that fulfill this rule are usually nonmagnetic semiconductors, providing a diverse family of topological insulator candidates.

The electronic properties of half-Heusler compounds can be predicted by counting the number of valence electrons. When the total number of valence electrons per formula unit is 8 or 18 (for the closed shell), dubbed as the 8- or 18-electron-rule, half-Heusler XYZ compounds can exhibit semiconducting properties that are close to those of classical semiconductors, such as GaAs. For example, LiMgAs with 8 valence electrons is a semiconductor with an energy gap (E _g) of 2.3 eV,^{Reference Kuriyama and Kushida22} and ScPtSb with 18 valence electrons is found to be a narrow gap semiconductor with E _g = 0.7 eV.^{Reference Rowe23} The electron count is illustrated in Figure 1c.

One of the most attractive aspects of half-Heusler compounds is the tunability of the insulating gap over a wide energy range, from about 4 eV (e.g., LiMgN) down to zero (e.g., ScPtBi), by choosing XYZ combinations with different electronegativity values of the constituents and with different lattice constants. Because of the similarity to the TI HgTe in both structural and electronic properties, half-Heusler materials offer a platform for the design of new TIs by realizing band structures that look similar and are actually topologically equivalent to that of HgTe.

Topological bulk states, the prototype TI HgTe

For understanding the topology of the bulk band structure of a half-Heusler material, the electronic structure of HgTe may serve as the starting point. The topology of the bulk band structure is characterized by an inverted order between a valence band and a conduction band that have opposite parities.^{Reference Bernevig, Hughes and Zhang4} The band order is inverted with respect to that of a normal insulator, such as CdTe. The energy bands nearest to the Fermi level in HgTe and CdTe are located around the Γ-point in the Brillouin zone, as shown in Figure 2. Like the common semiconductor GaAs, CdTe has a normal band ordering, with the Γ₆ band as the conduction band minimum (CBM) and Γ₈ as the valence band maximum (VBM). Bands Γ₆ with “−” parity and total angular momentum j = 1/2 and Γ₈ with “+” parity and j = 3/2 are labeled by the symmetry of the wave functions and constituted by Cd-s and Te-p orbitals, respectively. Compared to Cd, Hg is a heavier element and undergoes stronger relativistic effects. As a result, the Hg-s state (Γ₆) is much lower in energy than the Te-p state (Γ₈). Therefore, HgTe exhibits an “inverted” band ordering: the p-type Γ₈ band is situated above the s-type Γ₆ band (Figure 2, lower left). This kind of band inversion is essential for the determination of the topologically non-trivial band structure. The Γ₆–Γ₈ inversion (s–p inversion) takes place only around the Γ point, while the normal band ordering (Γ₆ above Γ₈) is preserved in the other regions of the Brillouin zone. In addition, the Fermi energy crosses the quadruply degenerate Γ₈ bands due to the cubic symmetry. This leads to a topological semimetal. In order to realize a real TI, an energy gap can be opened either by quantum confinement in lower dimensions, which induces a 2D TI,^{Reference Bernevig, Hughes and Zhang4,Reference Koenig, Wiedmann, Bruene, Roth, Buhmann, Molenkamp, Qi and Zhang5} or by applying strain to break the cubic symmetry, which results in a 3D TI.^{Reference Dai, Hughes, Qi, Fang and Zhang24,Reference Bruene, Liu, Novik, Hankiewicz, Buhmann, Chen, Qi, Shen, Zhang and Molenkamp25}

Figure 2.

Comparison of the electronic band structure of CdTe and HgTe with calculations for the half-Heusler alloys, ScPtSb and ScPtBi. Red color indicates bands with Γ₈ symmetry, blue with Γ₆ symmetry. The comparison reveals close similarity between the binary systems and their ternary equivalents: both CdTe and ScPtSb are trivial insulators, with Γ₆ situated above Γ₈, which sits at the Fermi energy (set to zero). Both HgTe and ScPtBi are topological insulators with inverted band order; the band with Γ₆ symmetry is situated below Γ₈. Figure adapted from Reference 15.

For the electron structure calculations of HgTe-type compounds, we use the Bernevig-Hughes-Zhang (BHZ) type of Hamiltonian,^{Reference Bernevig, Hughes and Zhang4} which is based on the k·p perturbation expansion up to the quadratic term,

(1)$$H = \varepsilon \left( k \right){\rm{I}}_{4 \times 4} + \left[ {\matrix{ {\matrix{ {M(k)} & {Ak_ + } \cr {Ak_ - } & { - M(k)} \cr } } & 0 \cr 0 & {\matrix{ {M(k)} & { - Ak_ - } \cr { - Ak_ + } & { - M(k)} \cr } } \cr } } \right]$$

Here ε (k) = C − D k², M (k) = M ₀ + B k², k _± = k _x ± ik _y and I_4×4 is the 4 × 4 unitary matrix, and 2M ₀ corresponds to the energy gap. The states are expressed in the basis $|{\rm{\Gamma }}_6^ - ,{m_j} = {1 \over 2} >$, $|{\rm{\Gamma }}_8^ + ,{m_j} = {3 \over 2} >$, $|{\rm{\Gamma }}_6^ - ,{m_j} = - {1 \over 2} >$ and $|{\rm{\Gamma }}_8^ - ,{m_j} = - {3 \over 2} >$, where “+/−” represents the parity eigenvalues, and m _j is the angular momentum projection. Besides for HgTe, the BHZ model has also been generalized to describe other TIs, such as Bi₂Se₃-type TIs,^{Reference Zhang, Liu, Qi, Dai, Fang and Zhang9} the perovskite oxide TI BaBiO₃,^{Reference Yan, Jansen and Felser26} and the KHgSb-type weak TIs.^{Reference Yan, Müchler and Felser27} As we will see, the BHZ model also applies to half-Heusler TIs, since half-Heusler materials exhibit the same Γ₆−Γ₈ band inversion as HgTe.

For a given material, the parameters in the effective model can be determined by fitting the energy spectrum of the effective Hamiltonian to that of first-principles calculations. However, one can still obtain insight in the band structure without knowing concrete parameter values. Without loss of generality, we can omit the identity term ε(k) and focus on the second 4 × 4 matrix in Equation 1. The diagonal terms ±M(k) represent Γ₆ and Γ₈ bands with electron and hole-like parabolic dispersions, respectively, assuming B > 0. One can find that the energy gap at the Γ point is 2M ₀. The off-diagonal linear terms Ak _± correspond to the hybridization between Γ₆ and Γ₈ bands that have opposite parities. When M ₀ > 0, the Γ₆ bands are always higher in energy than the Γ₈ bands, which correspond to the case of a trivial insulator such as CdTe. In contrast, when M ₀ < 0, one finds the Γ₆ bands are lower in energy than Γ₈ at the Γ point, which corresponds to the case of a TI such as HgTe. For the purpose of studying the topological properties, it is sometimes convenient to project the continuum k·p model to a lattice model (i.e., a tight-binding representation), which gives the energy spectrum over the entire Brillouin zone. We can obtain a simplified lattice model by replacing k _x,y,z and $k_{x,y,z}^2$ in Equation 1 with ${1 \over a}{\rm{sin}}({k_{x,y,z}}a)$ and ${2 \over {{a^2}}}(1 - \cos \left( {{k_{x,y,z}}a} \right))$, respectively, where a is the lattice constant and usually taken as 1 for simplicity. The corresponding bulk band structures are shown in Figure 3.

Figure 3.

Illustration of band structures of (a) a trivial insulator (M ₀ > 0) and (b) a topological insulator (M ₀ < 0). The band dispersion is obtained from a lattice model based on the effective Hamiltonian in Equation 1. Blue and red lines represent Γ₆ (“−” parity) and Γ₈ (“+” parity) bands, respectively. A band inversion exists clearly for M ₀ < 0. (c) The parity eigenvalues at the Γ (0 0 0) point, the 3 X (0.5 0 0) points, the 3 M (0.5 0.5 0) points, and the T (0.5 0.5 0.5) point. The last line lists the topological Z ₂ invariant ν ₀ that can be estimated from Equation 2. The non-trivial Z ₂ invariant is attributed to the inverted band structure in (b).

The topological invariant Z ₂

A commonly used criterion to determine whether a crystalline material is a 3D topological insulator is by its Z ₂-valued topological invariant.^{Reference Fu, Kane and Mele7} A TI is characterized by 4 Z ₂ invariants: ν ₀ and ν ₁, ν ₂, ν ₃. The most important index is ν ₀ : ν ₀ = 1 representing a strong TI, while ν ₀ = 0 is a trivial insulator (or a weak TI when at least one of the other ν _i’s is 1). The calculation of ν ₀ is simplified when the crystal structure possesses inversion symmetry:^{Reference Fu and Kane28}

(2)$${\left( { - 1} \right)^{{\nu _0}}} &#x003D; \mathop \prod \limits_{i &#x003D; 1}^8 {\delta ^i}$$

where ${\rm{\delta }}^i &#x003D; \prod\nolimits_{i &#x003D; 1}^N {{\rm{\xi }}_{2n} (K^i )} $ is the product of the parity eigenvalues ξ_2n(K ⁱ) of all valence bands (with total number N) at the ith time-reversal invariant k-point K ⁱ, and the final product runs over all eight K ⁱ points in the Brillouin zone. Each time-reversal conjugate pair (i.e., the 2nth and (2n–1)th bands) only counts once in the parity product. These eight k-points are K ⁱ (0/0.5, 0/0.5, 0/0.5) in units of the reciprocal lattice vectors and satisfy K ⁱ ≡ −K ⁱ and TH (K ⁱ)T ⁻¹ = H (−K ⁱ) = H (K ⁱ) (i.e., the Hamiltonian remains invariant under the time-reversal symmetry operation T at K ⁱ). Even for systems without inversion symmetry, one can apply this criterion by adiabatically changing the material to a centrosymmetric system. For example, we can assign the parity eigenvalues to the HgTe wave functions by adiabatically connecting HgTe to the diamond lattice. For the model in Equation 1, ν ₀ = 0 for M ₀ > 0 because all valence bands are “+” in parity for all eight K ⁱ. However, ν ₀ = 1 for M ₀ < 0 because the valence band contributes a “–” sign at the Γ point, while the other seven K ⁱ points are “+.” Consequently, HgTe is found to be a TI due to the band inversion, as illustrated in Figure 3.

Half-Heusler topological insulators

The diversity of half-Heusler materials allows one to find topological non-trivial band structures similar to HgTe. Based on first-principles calculations, tens of half-Heusler compounds show inverted band structures,^{Reference Chadov, Qi, Kübler, Fecher, Felser and Zhang15,Reference Lin, Wray, Xia, Xu, Jia, Cava, Bansil and Hasan16} as illustrated in Figure 4. For example, ScPtBi exhibits band inversion similar to HgTe, while ScPtSb displays a normal band ordering similar to CdTe, as shown in Figure 2. Hence we conclude that ScPtBi is a TI, while ScPtSb is a normal insulator. Here the conclusion based on the Γ₆–Γ₈ band inversion is consistent with that of directly calculating the Z ₂ invariant.^{Reference Xiao, Yao, Feng, Wen, Zhu, Chen, Stocks and Zhang29} First-principles calculations using the optimized exchange-correlation potential^{Reference Al-Sawai, Lin, Markiewicz, Wray, Xia, Xu, Hasan and Bansil30,Reference Feng, Xiao, Zhang and Yao31} and GW approach,^{Reference Vidal, Zhang, Yu, Luo and Zunger32} a method to calculate the self-energy of many-body systems—where G stands for the Green’s function and W for the dynamically screened Coulomb potential—have confirmed that a large number of half-Heuslers are TIs with inverted band structures, and at the same time have refined some of the topological features at the topological phase transition boundary.

Figure 4.

The band inversion strength calculated by the difference Γ₆–Γ₈ for various half-Heuslers as a function of the average spin–orbit coupling (SOC) strength represented by the average nuclear charge $ &#x0026;lt; Z &#x0026;gt; &#x003D; {1 \over N}\sum\nolimits_{i &#x003D; 1}^N {Z(x)} $, where Z(x) is the nuclear charge, and N is the number of elements in the compound. HgTe and CdTe binaries are shown for comparison. Open squares mark the systems that have not been experimentally realized. As elements become heavier with increasing SOC, the band inversion strength increases. Figure adapted from Reference 15.

Since the s-p band inversion determines the topological nature of HgTe and half-Heuslers, the interesting question arises as to what kind of mechanism produces the inversion. For the (Bi,Sb)₂(Se,Te)₃ family of TIs, the answer is clear. ^{Reference Zhang, Liu, Qi, Dai, Fang and Zhang9,Reference Xia, Qian, Hsieh, Wray, Pal, Lin, Bansil, Grauer, Hor, Cava and Hasan10} The bulk energy gap usually opens without SOC. By theoretically switching on SOC in calculations, the original gap closes, and a new energy gap is re-opened by inverting the conduction and valence bands. For half-Heusler compounds and HgTe, however, band structure calculations commonly show an s-p inverted band structure without SOC.^{Reference Wang, Dai, Jia, Wang, Liu, Cui and Liu33} This inversion can be attributed to other relativistic effects: the mass-velocity and Darwin corrections that are commonly considered in state-of-the-art density-functional theory calculations. One can understand these effects in the following simple picture.^{Reference Pyykkö34}

Inside an atom of the heavy element, the inner electrons move very fast, with speeds close to the speed of light. Einstein’s theory of relativity tells us that this leads to a mass (m) increase. The increased mass induces a smaller Bohr radius a ₀ (e.g., a ₀ = ħ / me ² for a hydrogen-like atom). This leads to a relativistic orbital contraction and energy decrease of all s and most of the p orbitals, the so-called direct relativistic effect. Moreover, because the contraction of the inner s and p shells results in strong screening of the nuclear charge, the outer d and f shells expand and destabilize, which reduces the nuclear screening. As an indirect consequence, the modified nuclear shielding stabilizes the outermost valence s shell (e.g., Hg-5s) by the enhanced attractive potential. Thus, both direct and indirect relativistic effects, which are included as mass-velocity and Darwin corrections in density-functional theory calculations, reduce the energy of the valence s electrons in heavy elements. Thus the s-p band inversion in HgTe and half-Heuslers is due to relativistic effects, just like SOC in the (Bi,Sb)₂(Se,Te)₃-type systems.

Topological surface states

One of the hallmarks of topological surface states is the linear energy dispersion (Dirac cone), similar to that in graphene. Moreover, the spin is locked to the momentum, which leads to a helical spin texture inside the Dirac cone. The ideal experimental tool to measure the Dirac cone and spin structure is angle-resolved photoemission spectroscopy (ARPES).^{Reference Hasan and Kane1} In addition, other surface sensitive techniques can be used, such as scanning tunneling microscopy and spectroscopy (STM and STS).^{Reference Roushan, Seo, Parker, Hor, Hsieh, Qian, Richardella, Hasan, Cava and Yazdani35} Another powerful experimental tool is magnetotransport (i.e., resistance measurements in a magnetic field) (see Reference 12). Quantum oscillations in the resistance of the material as a function of the magnetic field (the Shubnikov–de Haas effect) may identify the surface states and their topological nature by inspecting the phase of the oscillations via the geometrical phase or Berry’s phase. However, since magnetotransport probes the interior of the sample (the bulk) and the surface, prerequisites are a low bulk conductivity and a high transport mobility of the surface carriers. Another way to investigate the presence of Dirac dispersion of the surface states is by resistance measurements in very high magnetic fields, which is predicted to have a linear field variation.^{Reference Abrikosov36}

Although the topological nature of bulk half-Heusler materials has been put on a firm footing by theory, providing conclusive experimental evidence remains a challenging task. Hitherto, ARPES experiments were performed on LnPtBi, where Ln = Lu, Dy, and Gd.^{Reference Liu, Lee, Kondo, Mun, Caudle, Harmon, Bud’ko, Canfield and Kaminski37} According to the calculations, LuPtBi should be a strong TI (see Figure 4). For the measured [111] surface, metallic states were observed. However, the issue of whether these states are topological in nature could not be settled because of their complexity. The complexity is possibly due to the coexistence of many trivial dangling bond states near the Fermi energy. By counting how often the surface states cross the Fermi energy, one may distinguish topological (odd number of crossings) and trivial (even number of crossings) surface states. This simple rule (see Reference 1) has been applied to the surface states of, for example, Bi_1–xSb_x.^{Reference Hsieh, Qian, Wray, Xia, Hor, Cava and Hasan8} However, it requires a finite bulk energy gap, while the energy gap of unstrained half-Heusler materials is zero. Thus, one cannot naively employ the same rule to the half-Heusler surface.

High-field magnetoresistance measurements have been carried out on a number of compounds, including YPtSb, LuPdSb, LaPtBi, YPtBi,^{Reference Shekhar, Ouardi, Nayak, Fecher, Schnelle and Felser38,Reference Shekhar, Nicklas, Nayak, Ouardi, Schnelle, Fecher, Felser and Kobayashi39} and YPdBi.^{Reference Wang, Du, Xu, Zhang, Liu, Liu, Shi, Chen, Wu and Zhang40} These materials all exhibit a non-saturating magnetoresistance with a linear high-field term and a high charge-carrier mobility, which is in line with linear dispersion and topological surface states. In the case of LaPtBi,^{Reference Wosnitza, Goll, Bianchi, Bergk, Kozlova, Opahle, Elgazzar, Richter, Stockert, v Löhneysen, Yoshino and Takabatake41} YPtBi,^{Reference Butch, Syers, Kirshenbaum, Hope and Paglione18} and YPdBi,^{Reference Wang, Du, Xu, Zhang, Liu, Liu, Shi, Chen, Wu and Zhang40} Shubnikov-de Haas oscillations have been reported, but the angular variation of the oscillation frequency (by measuring them as a function of the magnetic field angle) shows these originate from small 3D Fermi surface volumes (pockets) and not from surface states.

We can obtain some useful insights into half-Heusler TIs from previous experimental and theoretical work on the HgTe surface, considering the close similarities between half-Heuslers and HgTe in both crystal structure and band structure. Under a relative large strain to break the cubic symmetry, the HgTe surface was found to exhibit Dirac cone-like topological surface states inside the bulk energy gap by calculations^{Reference Dai, Hughes, Qi, Fang and Zhang24,Reference Yan and Zhang42–Reference Wu, Yan and Felser44} that are also sensitive to the surface terminations.^{Reference Yan and Zhang42,Reference Wu, Yan and Felser44,Reference Virot, Hayn, Richter and van den Brink45} Under a small strain or no strain, however, it is revealed by both the k·p model Hamiltonian^{Reference Chu, Shan, Lu and Shen43} and first-principle^{Reference Wu, Yan and Felser44} calculations that the topological surface states are buried inside the bulk valence bands, with the Dirac point lying below the VBM at the Γ point. Although they strongly hybridize with bulk bands, it is still possible to distinguish these surface states in recent ARPES experiments.^{Reference Bruene, Liu, Novik, Hankiewicz, Buhmann, Chen, Qi, Shen, Zhang and Molenkamp25,Reference Yao, Zhou, Lu, Liu, Chen, Analytis, Brüne, Dang, Mo, Shen, Fisher, Molenkamp, Peng, Hussain and Chen46} Regarding the strain-free half-Heusler TIs measured by ARPES, we speculate that the surface Dirac cone also merges into the bulk valence bands in these compounds. Thus, one may find the topological surface states near the Γ point inside the valence bands (i.e., below the Fermi energy rather than at the Fermi surface). This calls for accurate surface band-structure calculations and high-resolution ARPES experiments to definitely settle the topological nature of half-Heuslers.

Candidate topological superconductors

The possibility of realizing a topological superconductor has attracted much attention. Topological superconductors are closely related to TIs due to the direct analogy between topological band theory and superconductivity: the Bogoliubov-de Gennes Hamiltonian for the quasiparticles of a superconductor resembles the Hamiltonian of a band insulator, with the superconducting gap in the role of the gap of the band insulator.^{Reference Hasan and Kane1,Reference Qi and Zhang2} Topological superconductors are predicted to be unconventional superconductors with mixed even and odd parity Cooper-pair states, which have surface/edge states that are robust against perturbation.^{Reference Sato47} Unfortunately, only a few topological superconductors have been proposed to date (e.g., Cu_xBi₂Se₃^{Reference Hor, Williams, Checkelsky, Roushan, Seo, Xu, Zandbergen, Yazdani, Ong and Cava48} and In_xSn_1–xTe),^{Reference Sasaki, Ren, Taskin, Segawa, Fu and Ando49} but the experimental evidence is not conclusive, and the search remains ongoing. In this light, it is important to examine the topological aspects of superconductivity in the half-Heusler series.

Table I lists candidate topological superconductors, which all belong to the platinum and palladium bismuthide series. The electrical resistivity shows, in all cases, weak metallic- or semi-metallic-like behavior with a broad maximum in the temperature range from 50–100 K, as illustrated in Figure 5 for YPtBi^{Reference Butch, Syers, Kirshenbaum, Hope and Paglione18,Reference Bay, Jackson, Paulsen, Baines, Amato, Orvis, Aronson, Huang and de Visser50} and ErPdBi.^{Reference Pan, Nikitin, Bay, Huang, Paulsen, Yan and de Visser20} All of the compounds listed in Table I are low carrier systems, with a carrier concentration of n ∼ 10¹⁹ cm^–3 at low temperatures. The superconducting transition temperature, T _c, ranges from 0.8 to 1.7 K.

Table I.

Superconducting transition temperature T _c and upper critical field B _c2 of half-Heusler platinum and palladium bismuthides.

* Superconducting volume fraction of only 8%.

Figure 5.

Semi-metallic-like electrical resistivity as a function of temperature of YPtBi (upper frame)^{Reference Bay, Jackson, Paulsen, Baines, Amato, Orvis, Aronson, Huang and de Visser50} and ErPdBi (lower frame).^{Reference Pan, Nikitin, Bay, Huang, Paulsen, Yan and de Visser20} Superconductivity is observed near 1 K.

A potential way to identify unconventional superconductivity is via the suppression of T _c in a magnetic field (i.e., by upper critical field measurements), B _c2(T). In general, a magnetic field will suppress the superconducting state via the interaction with the spins of the Cooper pair (Pauli limit, B ^P) and its orbital momentum (orbital limit, $B_{{\rm{c}}\,2}^{orb}$).^{Reference Werthamer, Helfand and Hohenberg51} If both contributions are present, B _c2(T→0) for a standard spin-singlet weak-coupling BCS superconductor is given by $B_{{\rm{c}}2} \left( 0 \right) &#x003D; B_{{\rm{c}}2}^{orb} {{\left( 0 \right)} \mathord{\left/ {\vphantom {{\left( 0 \right)} {\sqrt {1 &#x002B; {\rm{\alpha }}^2 } }}} \right. \kern-\nulldelimiterspace} {\sqrt {1 &#x002B; {\rm{\alpha }}^2 } }}$, where ${\rm{\alpha }}\, &#x003D; \,\sqrt 2 B_{{\rm{c}}2}^{orb} {{\left( 0 \right)} \mathord{\left/ {\vphantom {{\left( 0 \right)} {B^P }}} \right. \kern-\nulldelimiterspace} {B^P }}$ is the Maki parameter. This tells us the measured critical field should be lower than $B_{{\rm{c}}\,2}^{orb}$. An extensive study of YPtBi^{Reference Bay, Naka, Huang and de Visser52} shows its upper critical field is clearly at odds with this simple scenario, as shown in Figure 6, where B _c2(T) has been traced in reduced coordinates. In fact, the data also exceed the values of model calculations for an unconventional odd-parity Cooper pair state.^{Reference Scharnberg and Klemm53} This calls for more theoretical work to delineate the intricate behavior in a magnetic field of mixed even and odd parity Cooper pair states in topological superconductors.

Figure 6.

The upper critical field of YPtBi measured at ambient pressure and under pressure up to 2.51 GPa as indicated by the different symbols. In reduced coordinates (B _c2(T)/T _c)/(dB _c2/dT)|_Tc versus T/T _c, all datasets collapse onto a universal curve, which exceeds the values for a standard weak-coupling orbital-limited s-wave superconductor (red line). The blue solid line is a model calculation for a p-wave superconductor (see text). See Reference 52 for details.

We note that the half-Heusler compounds lack inversion symmetry and therefore may also be classified as non-centrosymmetric superconductors.^{Reference Bauer and Sigrist54} Here the lack of an inversion center in the crystal structure causes an electric field gradient, leading to the Rashba-type SOC effect (i.e., momentum-dependent spin-splitting of bands). This results in a splitting of the Fermi surface, which impedes spin-singlet or spin-triplet Cooper pairing of the conventional type. Instead, new pairing states, notably mixed even and odd parity Cooper pair states, are predicted to make up the superconducting condensate, just as in topological superconductors.

Yet another surprising observation was made for ErPdBi.^{Reference Pan, Nikitin, Bay, Huang, Paulsen, Yan and de Visser20} Here the rare-earth atom carries a magnetic moment. The magnetic susceptibility follows a local-moment Curie-Weiss law with an effective moment close to the Er³⁺ free-ion value of 9.58 µ_B and a small paramagnetic Curie temperature θ_P = –4.6 K. Transport and magnetic measurements show that the system orders antiferromagnetically at T _N = 1.1 K.^{Reference Pan, Nikitin, Bay, Huang, Paulsen, Yan and de Visser20} Nevertheless, ErPdBi becomes superconducting below T _c = 1.2 K, as shown in Figure 5. The coexistence of superconductivity and antiferromagnetic order is unusual. The magnetic and superconducting phase diagram for this compound is shown in Figure 7. Since T _N ≈ T _c, the interaction of superconductivity and magnetic order is expected to give rise to a complex ground state. Electronic structure calculations reveal a topologically non-trivial band inversion.^{Reference Pan, Nikitin, Bay, Huang, Paulsen, Yan and de Visser20} Accordingly, ErPdBi is advocated as a novel, unique platform to study the interplay of topological states, superconductivity, and magnetic order.

Figure 7.

Superconducting (T _c-SC) and antiferromagnetic (T _N-AFM) phase diagram of ErPdBi. The upper critical field (blue squares) shows an unusual linear behavior with values that exceed model calculations for a standard weak-coupling orbital-limited s-wave superconductor (blue line). The inset shows the half-Heusler crystal structure with Er, Pd, and Bi atoms. Figure adapted from Reference 20.

Concluding remarks

We have addressed a new functionality in the class of half-Heusler compounds: the topological insulating state. Electronic structure calculations provide solid grounds for a strong band inversion in selected materials that give rise to non-trivial, topological electron states. Detection of the topological state is not straightforward since the surface Dirac cone has probably merged with the bulk valence band. This calls for challenging experimental solutions, such as strain-engineering, to induce a TI gap and/or high-resolution ARPES. The great advantage of ternary half-Heusler compounds is the wide tunability of the electronic structure that can easily be accomplished by varying the constituents. For instance many of the ternary zero-gap semiconductors may also be formed with a magnetic rare-earth element (for instance ErPdBi), which offers new routes to explore the interplay of magnetism and topological phases. Likewise, some of the band-inverted half-Heuslers turn superconducting at low temperatures, which presents a new playground to investigate topological superconductivity.