Observation of a topologically non-trivial surface state in half-Heusler PtLuSb (001) thin films

The discovery of topological insulators, materials with bulk band gaps and protected cross-gap surface states in compounds such as Bi2Se3, has generated much interest in identifying topological surface states (TSSs) in other classes of materials. In particular, recent theoretical calculations suggest that TSSs may be found in half-Heusler ternary compounds. If experimentally realizable, this would provide a materials platform for entirely new heterostructure spintronic devices that make use of the structurally identical but electronically varied nature of Heusler compounds. Here we show the presence of a TSS in epitaxially grown thin films of the half-Heusler compound PtLuSb. Spin- and angle-resolved photoemission spectroscopy, complemented by theoretical calculations, reveals a surface state with linear dispersion and a helical tangential spin texture consistent with previous predictions. This experimental verification of topological behaviour is a significant step forward in establishing half-Heusler compounds as a viable material system for future spintronic devices.


Supplementary Note 1 | Theory Calculation Alignment
By examining ARPES snapshots for more negative binding energies, the bulk Γ7 band position can be tracked and aligned with the theory calculations. Supplementary Fig. 2b-2d show such an alignment for snapshots with an incident photon energy of 16 eV, 17 eV, and 18 eV, respectively. Good agreement can be found by shifting the theory calculation Fermi level -0.35 eV (with an approximate error on the order of 0.05 eV). Notably, this alignment deviates more for low photon energies, binding energies far from the Fermi level, and angles far from normal emission due to the divergence between the measured constant photon energy snapshots and the calculated constant kz bulk band structure projection (a limitation in the typical sudden-approximation free-electron-like final-state photoemission model 1 ).
The extracted surface state position and calculated bulk surface projections near the Γ point can be seen in Supplementary Fig. 2a. The surface state position lies such that it connects the envelope of the bulk Γ6 band projections to the envelope of the bulk Γ8 band projections. This agrees with the expectation of a TSS which connects these two inverted bands.

Supplementary Note 2 | Rashba-like Surface State Analysis
By examining ARPES snapshots for additional orientations and constant energy surfaces at lower binding energies, the Rashba-like surface state can be seen more clearly. Supplementary Fig. 3a and 3b highlight the split hole, double-arched, appearance expected for a Rashba or mixed Rashba/Dresselhaus surface state. Supplementary Fig. 3c and 3d show the constant energy surface at a binding energy of -0.74 eV. We note the presence of two generally concentric contours that align with the measured spin polarization locations. There is insufficient resolution to confirm that the observed state has the canonical pure-Rashba appearance but the measured data is consistent with the idea of a Rashba-like trivial surface state. Qualitatively, this state has strong parallels to that seen by Liu et al. 2 However, as would be expected, quantitatively, a different binding energy maxima is seen due to the difference in surface orientation and elemental components (i.e. PtLuBi and PtGdBi).

Supplementary Note 3 | Topological Surface State Peak Position Extraction and Extrapolation
Surface state peak position was extracted from several photon energies. Multi-peak fitting of individual photon energies was conducted with four or six peaks depending on the number of bulk bands expected for the kz value. Voigt peak functions with a linear background were used to identify the peak positions and individual momenta error in order to capture both the theoretical line shape and experimental broadening. Finally, a linear fit, based on peak positions for the TSS at a number of binding energies, was used to extrapolate toward the Dirac point (Fig. 3). The Voigt peak fit errors were incorporated through the linear curve fits. Utilizing these assumptions and neglecting systematic errors, the statistical error, which includes the intersection error, was computed as: (1) where 1 + 1 , 1 + 1 , 2 + 2 , and 2 + 2 are the two sets of linear fit coefficients and their corresponding errors. However, systematic errors in both the experiment, such as imperfect calibration of the energy scale or alignment of the Fermi level, and analysis, such as the exact functional form of the topological state, cannot be numerically quantified well. Consequently, the real error is most likely much larger than the computed value of ±0.02 eV.

Supplementary Note 4 | Additional Spin-ARPES Characterization
Spin ARPES measurements were performed at the I3 beamline of the MAX-IV laboratory, Sweden 3 . This beamline employs a Scienta R4000 hemispherical electron analyzer, configured to output either to a 2D MCP detector for spin-integrated measurements or to a mini-Mott spin detector. The latter permits simultaneous detection of two spin components by measuring the intensity difference between two channeltron detector pairs, one sensitive to in-plane polarization (along the analyzer slit direction) and one sensitive to out-of-plane polarization.
After scaling the channeltrons by known, constant sensitivity factors and subtracting a dark-count background, spin polarization was computed as 4 : where the value of the Sherman function S = 0.15, and IA and IB are the counts recorded by the channeltron pair. Error bars are based on counting statistics, and computed as: where N is the total intensity measured by a given detector pair.
In Supplementary Fig. 6, we show additional spin-resolved spectra acquired at a photon energy of 18 eV. Despite changing the photon energy, the spin texture observed in the main manuscript is unchanged, confirming that the observed spin polarizations originate from the surface bands, rather than the bulk bands.