Design and discovery of a novel Half-Heusler transparent hole conductor made of all-metallic heavy elements

Metallic conductors that are optically transparent represent a rare breed of generally contraindicated physical properties that are nevertheless critically needed for application where both functionalities are crucial. Such rare materials have traditionally been searched in the general chemical neighborhood of compounds containing metal oxides, expected to be wide gap insulators that might be doped to induce conductivity.Focusing on the family of 18 valence electron ABX compounds we have searched theoretically for the ability of the compound's electronic structure to simultaneously lead to optical transparency, in parallel with the ability of its intrinsic defect structures to produce uncompensated free holes.This led to the prediction of a stable, never before synthesized TaIrGe compound made of all-metal heavy atom compound as the"best of class"from the V-IX-IV group. Laboratory synthesis then found it to be stable in the predicted crystal structure and p-type transparent conductor with measured strong direct absorption of 3.36 eV and remarkably high (albeit not predicted) hole mobility of 2730 cm2/Vs at room temperature. This methodology opens the way to future searches of transparent conductors in unexpected chemical groups.

, represent the usually contraindicated functionalities of optical transparency (generally associated with electrical insulators) coexisting with electrical conductivity (generally associated with optically opaque metals) 6 . The search of TC's has generally focused on chemical groups characterized as light element wide gap oxides, such as Al-doped ZnO, 7 (for ntype) or CuAlO2, 8 and K-doped SrCu2O2, 9 (for p-type) TC's. Rather than attempt discovery within such a preconceived chemical neighborhood (either by highthroughput computation 10 or via combinatorial synthesis), the discovery of TC's might be guided instead by a set of physics based metrics that p-type TC's need to satisfy a priory 9,11-14 then searching not only light element oxides or nitrides but allowing for a broader range of chemistries to be inspected. These "Design Principles" include a combination of (a) electronic structure features of the perfect crystal and, in parallel, (b) specialized properties of the non-ideal defected lattice. In category (a) we require a wide (> 2.5 eV) direct band gap for optical transparency (but the non-absorbing indirect band gap can be significantly smaller), whereas good mobility requires rather light (< 0.5 m0) hole effective mass and a bulk hole wavefunction that avoids as much as possible the close neighborhood of the ions. In category (b) we require that the intrinsic defects that are hole-producers (acceptors such as cation vacancies) have low formation enthalpy (i.e., be abundant) and have shallow acceptor levels (i.e., be readily ionizeable), whereas the intrinsic defects that are 'hole-killers' (donors such as anion vacancies) must have high formation enthalpy or have levels that are electrically inactive (not ionizeable). In considering such design principles it is not obvious a-priori that only the traditionally sought light element oxides are eligible for satisfying these simultaneous conditions.

Known as well as 'missing' compounds in the ABX family:
We chose to focus on the 18-electron ABX compounds that represent diverse chemical groups such as A n B 10 X (8-n) as well as A (n+1) B 9 X (8-n) (n = 1, 2, 3, 4) with atoms spanning columns 1~17 in the Periodic Table. This broad range of atoms (albeit within the group of ABX) encompasses a broad range of chemistries including light as well as heavy elements; anions as well as cations. The previously synthesized members from these groups manifest extraordinary functionalities such as thermoelectricity, superconductivity, piezoelectricity, and topological insulation, but as yet no TC's. But not all atom combinations that by the current understanding of solid-state chemistry can plausibly lead to such ABX structures were in fact realized. For example, from a total of 483 possible 18 valence electron ABX compounds from the above noted chemical groups, only 83 are known, 15 whereas 400 are, in fact, "missing compounds" that might constitute an attractive playing ground for new materials with new functionalities. We 16,17 and others 18,19 have developed theoretical, first-principles techniques for examining the thermodynamic stability of missing compounds sorting out the 'missing and predicted unstable' from the 'missing and predicted stable'. Our first principles thermodynamics search includes the examination of (a) The lowest-energy crystalline form of the ABX phase, (b) its stability with respect to decomposition into any combination of elemental or binary phases of the A+B+X constituents and (c) the dynamic (phonon) stability of the lowest energy ternary phase. Note that in search (a) we do not limit the allowed structure to just a single structure-type (as in Carrete et al 18  where only previously documented oxide materials were examined. A succinct summary of the method we use is given in Supplementary Section I. The application of first principles thermodynamics based on density-functional methodology 16,17,20 to a total of 483 ABX compounds (400 missing as well as 83 known 15 ), we have predicted that the 54 of the 400 missing compounds are predicted stable in specific crystal structures and 346 are predicted unstable. As a test to our thermodynamic equilibrium methodology we applied it to ~30 known (previously synthesized and structurally characterized) ABX compounds listed in ICSD 15 . We find in all cases that these compounds are predicted to be stable in the observed crystal structures, validating the methodology. In the present work we do not focus on the broad picture of the prediction of stable new compounds 20 . Instead we wish to focus on the functionality of transparent conductivity. Figure 1a provides the calculated results on one such group V-IX-IV of 18-electron ABX compounds, with check marks indicating previously known compounds, plus signs (+) indicating previously missing and now predicted stable, and minus signs (-) indicate previously missing and now predicted unstable compounds. We next illustrate how a specific functionality is searched.

Design-principle guided search: (a) Properties of ideal bulk crystal.
Having supplemented the currently known groups of 18-electron ABX compounds by the previously missing and now predicted stable compounds, we next examine their systematic electronic properties using self-consistent hybrid functional (HSE06) 21 wavefunctions considering the design metrics for p-type transparent conductors.
We find a number of new structure-property relationships in this family that focused our attention to specific subgroups for prediction p-type transparent conductors.
First, we find that all 18-electron cubic Half-Heusler ABX compounds containing two transition metal atoms (e.g., TaIrSn and ZrIrSb) are non-metals (i.e., have a band gap between occupied and unoccupied bands), whereas the cubic 18-electron Half-Heusler compounds with one transition metal atom (e.g., AlNiP) are metals (see Supplementary Figs. S3 and S4). Thus, to look for candidate transparent materials we focus on the former group containing two transition elements. The finite band gap in the former case is explained by the strong d-d hybridization between the two transition metals (e.g. Ta and Ir) that are nearest neighbors in the Half-Heusler (LiAlSi-type) structure, as illustrated for TaIrGe in Fig. 1c.
Second, we find that the ABX compounds with heavy X atom tend to be stable in cubic Half-Heusler structures, hence can potentially have wide band gaps. For example, the predicted ABX compounds in groups IV-X-IV, IV-IX-V, and V-IX-IV with heavy X elements Sb, Bi, Sn, or Pb are all stable in cubic structure. In contrast, we find that ABX with light X atoms (i.e. O, S, Se, N, P, As, C, Si, and Ge) tends to have non-cubic structure and thus are often metallic, since for light X (e.g. O), the A and B metals are ionized and strongly repel each other when they are nearest neighbors as in the LiAlSi-type structure. We thus look for TC's in heavy X atom compounds.
In accordance with the above two structure-property relationships, we identify nine wide gap ( eV) members in the groups of cubic, two-transition element Half-Heusler ABX compounds: from V-IX-IV group we find TiCoSb, TiRhSb, TiIrSb, ZrRhSb, ZrIrSb and HfCoSb; from the IV-X-IV group we find HfPtSn, whereas from the V-IX-IV group we identify TaIrGe and TaIrSn. These are heavy, all-metal atom Half-Heusler insulators, rather unusual in that many heavy atom compounds tend to be narrow gap semiconductors 22

Results on Properties of ideal bulk crystals: Thermodynamic stability and crystal structure of the heavy, all-metal atom insulators:
The ranges of thermodynamic stability are evaluated in the chemical potential space of A, B, and X considering the elemental, binary and ternary competing phases (see the inset of Fig. 3a and Supplementary Section II). The predicted allowed chemical potential range reflects the thermodynamic growth conditions, which can be used as a guide for experimental synthesis. For the newly predicted and never before synthesized ABX compounds (TiIrSb, ZrIrSb, TaIrGe and TaIrSn), we find that their stability range is located mainly around the boundary of X richest and A/B-poor condition.
The predicted lowest-energy structure-type of TaIrGe is shown in Fig. 1b . 2a). The calculated optical absorption coefficient of TaIrGe based on the GW approximation 24 for electron's self-energy is presented in Fig. 2c showing an optical transition onset of ~2.64 eV. Thus TaIrGe is transparent for most visible light with frequency below 2.64 eV. The first strong peak of absorption coefficient appears at 3.1 eV corresponding to the transition between the two parallel bands highlighted by green in Fig. 2a. The absorption coefficient is < 10 5 cm -1 for photon energy in the range of (2.64 eV, 2.9 eV) which is acceptable for thin film (a few hundred nm) transparent conductors.

Results on Properties of ideal bulk crystals: The nature of the hole orbitals
promises ideal carrier mobility. Three factors are noteworthy: First, the hole wavefunction (see Fig. 2d) is delocalized mainly along the interstitial channels (in between Ta-Ir and Ta-Ge atoms) that do not pass through ionic sites and hence have but a low potential to scatter the mobile carriers. Third, the predicted low electron and hole effective masses: It is generally difficult to find oxides with not so high hole masses because the valence band tends to be narrow and leading to large effective masses 10

Design-principle guided search: (b) Defect property and p-typeness.
Having 3 & 4a) and the carrier concentration predicted at a given temperature and chemical potential (Fig. 4b).

Results on Properties of doped p-type crystals: Hole producers. A most exciting
prediction from our intrinsic defect simulation is that TaIrGe displays a strong ptype behavior (nhole> 10 16 cm -3 , EF (eq) < EVBM + 0.2 eV) in the majority of its stability area (bright blue color in Fig. 4a and bright red color in Fig. 4b). From the simulation for crystal growth condition (compounding at 1200 K followed by a quench to room temperature), we find the maximum hole concentration of ~2.5×10 17 cm -3 when approaching the Pmax corner of the stability area (the most Tapoor and Ge-rich condition in Fig. 4b). The hole concentration could be further increased by introducing extrinsic dopants into TaIrGe.
The microscopic origin for p-type is revealed by inspecting formation energies of leading defects under the optimal chemical potential conditions (inset in Fig. 3a that have much higher formation energies (Fig. 3a).

Laboratory Synthesis and crystallographic structure determination:
Experimental realization of the hitherto missing (unreported) TaIrGe was accomplished by bulk synthesis methods, followed by optical and electronic characterization. The TaIrGe specimens were synthesized by vacuum annealing the stoichiometric mixture of pure elements (see details in Supplementary Section III). Figure 5a shows the X-ray diffraction pattern of TaIrGe, confirming that phase-pure bulk TaIrGe was successfully fabricated. By comparing the experimental X-ray pattern with the calculated diffraction pattern, positions and intensities of the experimental TaIrGe are in agreement with the theoretical prediction, and the refined lattice parameter is 5.9664(5) Å with a calculated density of 13.94 g/cm 3 .
The lattice parameters were accurately established using high purity silicon as an internal standard. Figure 5b displays a representative bright field TEM image and corresponding selected area electron diffraction (SAED) pattern, suggesting a grain size of around 100 nm, and with no secondary phase.

Measured optical properties:
The optical absorption spectrum of bulk TaIrGe, obtained from the ultraviolet-visible diffuse reflectance measurements (see Fig. 6a and Supplementary Section III) indicated that TaIrGe exhibits two optical absorption thresholds at about 365 and 485 nm in the visible range (which correspond to band gap energies of 3.39 and 2.55 eV, respectively). Subsequent analyses of the diffuse reflectance using the Kubelka-Munk function [25][26][27] indicated an indirect transition of 1.64 eV (Fig. 6b) and a direct transition of 3.36 eV (Fig. 6c).
The difference between measured optical band gaps and predicted values (~3.1 eV) is ~0.2 eV. Thus, the absorption window of TaIrGe covers most of the visible solar spectrum (~ 350 to 700 nm). To ascertain about this optical band gap, we performed high vacuum thin film deposition using TaIrGe target in a pulsed laser deposition system and the obtained 30-nm-thick film shows transparent in visible light. Figure 6d shows the optical transmittance spectrum of the TaIrGe film grown on transparent quartz substrate, which varies from 85% to 92% in the wavelength region of 350 to 700 nm. This demonstrates that the TaIrGe film is transparent in the visible region. and a Hall mobility of ~+2730 cm 2 /Vs. The achieved Hall mobility is much higher than that of known p-type transparent conducting oxides 8,9 (e.g. ~10 cm 2 /Vs for CuAlO2).

Discussion
We illustrate that a rather rare functionality of p-type transparent conductivity can be discovered by inverse design in a never before made compound. (ii) Defect physics in the TaIrGe 18-electron ABX Half-Heusler prototype: The X IVon A V antisite hole-producer has low formation energy since both A and X sites are in a similar four-fold coordination (while B in eight-fold) and the size of X IV (Ge) is smaller than A V (Ta). In contrast to the favorable conditions for producing holes, the hole-compensating (electron producing) metal interstitials (Iri, Tai, Gei) that form by occupying the interstitial site in Half-Heusler structure (see Fig. 1b), require high formation energy due to the large atomic sizes (for Iri and Tai) or repulsion from isoatom nearest neighbors (for Tai and Gei) and thus do not block the acceptors.
(iii) Electronic structure conducive to high hole mobility: The previously studied ptype transparent conductors are prone to have low hole mobilities 8,9 (e.g. ~10 cm 2 /Vs for CuAlO2 and 0.46 cm 2 /Vs for SrCu2O2), which limited their application. In contrast, the Half-Heusler compound TaIrGe can have a hole mobility as high as the standard covalent semiconductor Ge (~1900 cm 2 /Vs at room temperature 28 ). The electronic structure features that enable good hole mobility include the distribution of the hole VBM wavefunction along interstitial channels that do not pass through ionic sites; the high degeneracy of the hole states in the 3D momentum space near VBM, and the relatively low hole effective mass due to significant hybridization (Fig.   2d). Thus, whereas TaIrGe has hole property similar to those of narrow gap semiconductors such as Ge, unlike the latter cases it has a wide band gap opened by the strong d-d hybridization between the metal atoms.
(iv) Inverse design of functional materials: The proposed approach in inverse design framework starts from the physically formulated design principles of material functionalities, without the preconceived limitations referring to the previous discoveries, considers both existing compounds as well as previously undocumented but now predicted stable members followed by evaluation of design metrics, thereby (significantly) increasing the playing field of discovery, and leads to the efficient selection of candidate materials for experimental realization. This approach, searching directly for specific target functionality (rather that making all compounds at the outset, as in combinatorial synthesis or its computational analogue-high throughput calculations), as demonstrated in current study, can find interesting candidate materials in unexpected chemical groups, of technologically significant functionalities that are hard to find and optimize by rational experiments (e.g. p-type TC), potentially opening a new direction to design functional materials (e.g. all-heavy-element materials for high-mobility TC).
Following this approach, one can also start the design from the physical formulation of design principles, thus eventually leads to discovery of new material functionalities from inverse design. The transmittance spectrum was determined at room temperature for both the substrate and film.

Methods
Electrical transport measurement. Four-probe resistivity was determined using the Van der Pauw method on the pellet samples by a Physical Measurement System (Quantum Design PPMS 6000). The Seebeck coefficient was obtained from the slope of the thermovoltage versus temperature gradient. The Hall coefficient (RH) was measured at room temperature using a five-probe configuration with the magnetic sweep between ±2 Tesla using the PPMS system.