The electronic band structure of ZrOsSi was computed by first-principles calculations based on density functional theory using the full-potential linearized augmented plane wave (FP-LAPW) method, as implemented in the WIEN2k program suite [37]. Here, we use the exchange-correlation potential from the generalized gradient approximation (GGA) [38], with the plane-wave cutoff being defined by the condition rk _max = 7.0, with r being the minimum LAPW sphere radius and k _max being the plane-wave-vector cutoff. For ZrOsSi, the lattice parameters, optimized using DFT, are: a = 6.43 (9) Å, b = 4.08 (0) Å, and c = 7.50 (7) Å, consistent with previous reports [15]. The Wyckoff positions corresponding to the structure of ZrOsSi are: Zr 4c (0.01253, 0.25, 0.81533), Si 4c (0.21227, 0.75, 0.62165), Os 4c (0.16157, 0.25, 0.44187). The lattice parameters for TaOsSi [14] are: a = 6.26 (5) Å, b = 3.89 (3) Å, and c = 7.25 (9) Å; and for NbOsSi [14] are: a = 6.28 (1) Å, b = 3.89 (7) Å, and c = 7.27 (3) Å. Clearly, replacing Ta or Nb in (Ta, Nb)OsSi by Zr leads to an expansion of the unit cell.

The electronic band structure of ZrOsSi, with and without considering the effects of SOC, are shown in Figures 2A, B, respectively. We note that several bands cross the Fermi level, leading to the electron- and hole Fermi surfaces shown in Figure 3 (without SOC). Inside the Brillouin zone, one can identify extended regions where several Fermi-surface sheets are parallel and close to each other. This feature is also present in the Fermi surfaces of (Ta, Nb)OsSi, where it was argued to influence the nature of their superconducting ground states [14]. The reason for the closeness of the different Fermi surface sheets and the relatively small sizes of the Fermi surfaces of (Ta, Nb, Zr)OsSi can be traced back again to the nonsymmorphic symmetries of these materials, similar to that of others with the same space-group symmetry [13, 39].

From Figure 2, we note that SOC causes a significant band splitting. In particular, it results in a maximum band-splitting of ∼ 100  meV along the TY direction, where an extra Fermi-surface sheet emerges due to SOC. Most importantly, from Figure 2B, we note that due to SOC there is a continuous direct energy gap throughout the Brillouin zone between the highest occupied “valence” band (shown in red) and the lowest unoccupied “conduction” band (shown in blue). As a result, ZrOsSi can be thought of as a semimetal, for which the classification of band insulators in three dimensions (3D), based on the Z ₂ topological invariants (ν ₀; ν ₁ ν ₂ ν ₃), is also applicable [40]. From the topological point of view, such identification is possible since its bands can be continuously deformed into those of an insulating system, characterized by a full gap throughout the Brillouin zone. In general, the Z ₂ topological invariants can be computed by calculating the evolution of the Wannier charge centers (WCCs) for the six time-reversal invariant planes: k _x = 0, π; k _y = 0, π; and k _z = 0, π [41, 42] as shown in Figure 4. In the ZrOsSi case, the Z ₂ invariants are 1 for the k _x = 0, k _y = 0, and k _z = 0 planes, and 0 for the k _x = π, k _y = π, and k _z = π planes. This implies that, in ZrOsSi, the Z ₂ topological invariants are (1; 000), thus indicating that ZrOsSi is a strong topological material.

Since ZrOsSi has a centrosymmetric structure, we can confirm the above classification scheme by computing the Z ₂ topological invariants from the product of the parity eigenvalues of the filled valence bands (up to the red band in Figure 2B) at the eight time-reversal invariant momentum (TRIM) points [40] with SOC. These parity products up to the red band in Figure 2B are −1 at the Γ point and +1 at all the other TRIM points. This leads to the Z ₂ topological invariants (1; 000) for ZrOsSi, the same as those obtained by the WCC method described above. As a result, we conclude that ZrOsSi is a Z ₂ topological metal, which can be continuously deformed into a strong Z ₂ topological insulator. In this sense, the physics of ZrOsSi is expected to be similar to that of a doped topological insulator [35].

We show the orbital-decomposed band structure of ZrOsSi in Figure 5, where the p-d band inversions along the Γ − X, Γ − Y, and Γ − Z directions are clearly visible, thus supporting our identification of ZrOsSi with a strong topological material. Due to the bulk-boundary correspondence, there will be topologically protected metallic surface states in ZrOsSi. We investigate the surface states by computing the surface spectral function or the surface-state spectrum using surface Green’s function and maximum localized Wannier functions, as implemented in the WannierTools package [42]. In ZrOsSi, we compute the surface-state spectrum for a (001) slab, as well as the spectral function for the corresponding bulk. In both cases, we consider the SOC effects. The bulk spectral function is shown in Figure 6A and the corresponding surface-state spectral function for the slab is shown in Figure 6B on the (001) surface Brillouin zone. For clarity, here we focus on the high-symmetry Γ ̄ − X ̄ and Γ ̄ − Y ̄ directions (motivated also by the evidence of band inversion along these directions, as shown in Figure 5).

Some key regions where the surface states are clearly visible are marked in Figure 6B, where we show also their zoomed versions. Note that, while in these regions the spectral function is nonzero for the slab, it is zero for the bulk [see Figure 6A]. This fact, together with the existence of nontrivial topological invariants and of band inversions described above, clearly indicate that, in ZrOsSi, the surface states are indeed due to its nontrivial bulk topology. The apparent asymmetry of the surface-state spectrum along the two high symmetry directions in Figure 6B is due to the direct gap moving up and down in energy, depending on the direction away from the Γ point, as shown in Figure 2B. We note that, just below the Fermi level, there are several topologically-protected surface Dirac cones. In our case, an odd number of them reflects the odd number of bulk band inversions, thus confirming the Z ₂ topological-metal nature [3] of ZrOsSi, as highlighted by the zoomed images in the rightmost panels in Figure 6B. The surface states relevant to our discussion are most evident at ∼ 0.05 eV below the Fermi level, at the X ̄ point, and at ∼ 0.4  eV below the Fermi level, at the Y ̄ point. Hence, the normal state of ZrOsSi, which is like a strong topological insulator, is qualitatively different from that of its isostructural counterparts (Ta,Nb)OsSi, which are nonsymmorphic symmetry-protected semimetals [11–14]. The basic difference between insulating and metallic character of these materials can be probed, for instance, by NMR experiments. Therefore, in the following section, we report about and discuss our systematic investigation of these materials using NMR.

NMR investigations of AOsSi (A = Ta, Nb and Zr) are, in principle, very suitable, considering the presence of three NMR-active nuclei: A, Os and Si. However, the large spin and/or quadrupole moments of ⁹³Nb and ¹⁸¹Ta, and the low-frequency, low-abundance of ⁹¹Zr, make the A nuclei inapt for NMR experiments. Similarly, ¹⁸⁹Os has a low natural abundance and a low gyromagnetic ratio. These factors suggest the I = 1/2 ²⁹Si nucleus as the only viable microscopic probe for a systematic study of the AOsSi series. Since the magnetic field required for the NMR experiments suppresses an already low superconducting T _c (of only ∼ 1.7 K in the ZrOsSi case [15]), we had to limit our investigations to the normal-state electronic properties of ZrOsSi. Yet, this is not a major limitation, since we can still investigate the consequences of the nontrivial band topology of ZrOsSi, as compared with the conventional TaOsSi and NbOsSi metals. Note that, for ²⁹Si to be an effective probe, the Si 3p orbitals should contribute to the inverted bands responsible for the topological properties (here, the red band in Figure 2). As clearly evidenced from the orbital resolved DOS shown in Figure 5, this is indeed the case, further justifying the use of ²⁹Si NMR as a probe of the topological properties of ZrOsSi.

The ²⁹Si NMR spectra, collected at selected temperatures in a 7-T field, are shown in Figure 7. Since the NMR line shapes of TaOsSi and NbOsSi are quite similar, here we compare the representative line shapes of TaOsSi in Figure 7A with those of ZrOsSi in Figure 7B. In TaOsSi we observe a narrow main peak (at 59.785 MHz), followed by a secondary hump of negligible magnitude to the right. The latter is most likely due to impurities—probably arising from unreacted components—beyond the detection threshold of XRD. The unique crystallographic position of Si nuclei inside the Pnma structure of TaOsSi is reflected in the single main peak. Its position is almost independent of temperature and it shows only a smooth increase in width as the temperature is lowered. The shift from the ²⁹Si reference frequency decreases with temperature, to saturate below 100 K. Such trends are more clearly seen in Figure 8, where the evolution of both shifts and widths with temperature is quite smooth. In particular, the limited increase in width as the temperature is lowered is typical of nonmagnetic materials.

Although normally silicon exhibits single narrow peaks, in ZrOsSi we observe a double NMR peak (centered at 59.8 MHz), compatible with its atomic coordination. Here, Si atoms reside in the 4c sites of the orthorhombic structure shown in Figure 1, bonded in a 10-coordinate geometry to 4-Os and 6-Zr atoms, on average 245 pm and 285 pm apart, respectively. Further, considering that Os is twice as electronegative as Zr, we expect a measurable line splitting into two groups with ∼ 2 : 3 ratio, as experimentally observed. This assignment, however, should explain also the TaOsSi lines, since Ta and Zr have similar electronegativities and the two compounds have identical Pnma crystal structures (albeit with slightly different lattice parameters [15]). The different NMR lineshapes of TaOsSi and ZrOsSi (see Figure 7) might instead arise from the different topological character of ZrOsSi compared to that of the standard metal TaOsSi.

This hypothesis is indirectly confirmed by previous ⁷⁷Se NMR studies of the Bi₂Se₃ topological insulator, which also shows two distinct peaks in the NMR spectra, of which the shoulder was assigned to the topologically nontrivial surface effects [28]. These results suggest that NMR may be used as a sensitive and effective probe of the nontrivial surface states in topological insulators, although a detailed theoretical backing is still missing.

As shown in Figure 8, in the normal state, ZrOsSi exhibits a²⁹Si-NMR shift of ca. 500 ppm and a line width of ca. 8.5 kHz, both of which decrease smoothly with temperature, to saturate above 200 K. In contrast with metallic TaOsSi, where the electronic environment of Si nuclei is more homogeneous, the twofold larger line width of ZrOsSi reflects its insulating nature. In general, the total shift K, consists of an orbital- (chemical) K _orb and a spin part K _s . Considering the ∼ 200  ppm chemical shift of sixfold coordinated Si [43], the shift we observe in ZrOsSi is essentially of chemical nature ( ∼ 350  ppm), with the spin contribution (arising from the conduction electrons) being marginal. This, too, is indicative of a poor metallic behavior, as confirmed also by electronic band-structure calculations, which indicate a deeply suppressed density of states at the Fermi level, here dominated by the Os- and Zr d-orbitals. We expect this to be reflected in very long spin-lattice relaxation times.

Indeed, this is confirmed by a comparison of the 1/T ₁ relaxation rates of ZrOsSi with that of its isostructural counterparts NbOsSi and TaOsSi, as shown in Figure 9 (Ta, Nb)OsSi show fast relaxation rates (even at low temperature) and a practically linear dependence of 1/T ₁ vs. T (Figure 9A). This is evident also from their almost constant Korringa product [44] 1/(T ₁ T) (=6.5 × 10⁻³ s⁻¹K⁻¹), shown in Figure 9B. The relatively small value of the Korringa product indicates a low electronic density of states at the Fermi level E _F of NbOsSi and TaOsSi, a result compatible with the electronic band-structure calculations [36], which suggest that, even in the presence of SOC, both compounds retain their metallic character. These results are in stark contrast with the relaxation rates and the 1/(T ₁ T) product of ZrOsSi, almost an order of magnitude smaller, typical of semimetals or insulating materials.

The NMR results in the normal state of (Ta, Nb)OsSi discussed above are in agreement with the resistivity measurements [14, 36], which show that the low-temperature resistivity follows a T ²-law, i.e., these materials exhibit a good Fermi-liquid behavior. Recently, (Ta, Nb)OsSi were reported to be nonsymmorphic symmetry-protected semimetals [11–14], with the SOC effects playing a crucial role in determining their topological properties. Although NMR is not a direct probe of the topological properties, the lack of any anomalous behavior in the electronic properties of (Ta, Nb)OsSi as seen by NMR is a bit surprising, since one expects to see a change in the NMR shift if SOC changes the electronic band structure [20–23]. In view of the strikingly different electronic behavior of (Ta, Nb)OsSi vs. ZrOsSi (see Figure 9), one might wonder whether SOC has some role to play, which prompts for a closer comparison of the different SOC strengths of these materials.

The calculation of SOC is not straightforward. However, thanks to the early efforts of Herman [45, 46], based on the Hartree-Fock method, as well as to more recent results (see, e.g., Figure 1 in Ref. [47]), we have reliable estimates of SOC for all the elements. Considering the almost ∼ Z ² dependence of SOC with the atomic number Z, we expect it to increase slightly from the 4d-element Zr (Z = 40) to the 4d-element Nb (Z = 41), and even more so to the 5d-element Ta (Z = 73). A similar trend is indeed found from band-structure calculations: while the SOC-induced maximum band splitting near the Fermi level is ∼ 100  meV in both ZrOsSi (see Figure 2) and NbOsSi [14], it is ∼ 140  meV in TaOsSi [36]. We note that since both Nb and Ta belong to group-5 of the periodic table, their comparison excludes the role of valence electrons and highlights that of SOC.

However, since the experimental results in Figure 9 show the NbOsSi and TaOsSi datasets to be basically clustered together, possibly the SOC effects are not the driving force for the observed phenomena. To justify these results we note that Zr exhibits a lower formal valency (+4), resulting in larger atomic and lattice parameters. Therefore, it does not come as a surprise that Zr acts as a lattice expander and moves the material towards the insulating side. Indeed, although the NbOsSi, TaOsSi, and ZrOsSi silicides share the same TiNiSi-type structure (space group Pnma, Z = 4), ZrOsSi is very different from the other two. (i) While NbOsSi and TaOsSi have practically the same cell parameters and unit-cell volume (differing by less than 0.5%), ZrOsSi shows an 8.6% increase in unit-cell volume, with a remarkable increase in all the cell parameters (a ∼ + 1.7%, b ∼ + 3.95%, c ∼ + 2.2%). (ii) The coordinates of Si (both relative and absolute) inside the unit cell change remarkably in ZrOsSi, implying an increase of ca. 3.5% in the interatomic distances [15].

The clear conjecture, therefore, is that the replacement of Nb (or Ta) with Zr corresponds to an expansion of the crystal lattice (i.e., to a negative chemical pressure) and consequently to a significant weakening of its metallic nature. These conclusions, are in good agreement with the atomic and covalent radii [48] of Nb and Ta (both being very similar and with the same formal valence of +5), to be contrasted with the 8% larger radius of Zr (with a formal valence +4 and here acting as lattice expander).

Incidentally, the negative chemical pressure occurring in ZrOsSi complements well our previous high-pressure study of TaOsSi [36], i.e., it allows us to understand how TaOsSi would have behaved under lattice expansion. As such, it would be interesting to investigate ZrOsSi further by means of other experimental techniques.