Pressure driven Weyl-topological insulator phase transition in Weyl semimetal SrSi$_{2}$

Using DFT-based first-principles calculations, we demonstrate the tuning of the electronic structure of Weyl semimetal SrSi$_{2}$ via external uniaxial strain. The uniaxial strain facilitates the opening of bandgap along $\Gamma$-X direction and subsequent band inversion between Si $p$ and Sr $d$ orbitals. Z$_{2}$ invariants and surface states reveal conclusively that SrSi$_{2}$ under uniaxial strain is a strong topological insulator. Hence, uniaxial strain drives the semimetallic SrSi$_{2}$ into fully gapped topological insulating state depicting a semimetal to topological insulator phase transition. Our results highlight the suitability of uniaxial strain to gain control over the topological phase transitions and topological states in SrSi$_{2}$.

Topological order may be tuned via changing the chemical composition and thereby modifying both the strength of spin orbit coupling (SOC) and the lattice parameters. Another strategy is to apply external strain [30,31] which has been theoretically shown for a number of narrow band gap cubic semiconductors such as grey tin ( -Sn) [32] and HgTe [32,33], InSb [34], KNa 2 Bi [35], TaAs [36], Co 3 Sn 2 S 2 [37]. (Colour online) Weyl-TI TPT in a non-magnetic system with intraband annihilation of chiral Weyl node [37] Hence, compressive/tensile strain is utilised to close/reopen the bandgap in some topological materials, which may be accompanied by strain-induced topological phase transitions [30,[38][39][40]. Recent studies indicate that SrSi 2 is a robust double-Weyl semimetal due to the absence of inversion symmetry. SrSi 2 is WSM in its ground state even without the requirement of spin-orbit coupling. SrSi 2 is also predicted to undergoes topological phase transition from Weyl semimetal to trivial insulator by increasing its lattice constant uniformly in all three directions [23]. Hence, triaxial strain is unsuitable to retain the topological behaviour in SrSi 2 . In comparison to triaxial strain, the application of uniaxial strain may offer an additional advantage in terms of introducing asymmetry into the lattice. It was earlier shown that uniaxial strain [30,35,41] is an effective tool to realize a topological insulating state by introducing a bulk band gap. Hence, using the uniaxial strain, it is possible to gain better control of topological properties and reveal topological phase transitions of SrSi 2 .
In this work, we employ first principle calculations to study the effect of applying uniaxial compressive strain on SrSi 2 . We discover a topological semimetal to topological insulator quantum phase transition and obtain a fully gapped quantum spin Hall state (QSH) in SrSi 2 at 5% strain. Here, we consider the new Weyl-TI phase transition which includes two separate topological non-trivial phases as well as the metal-insulator transition. The chiral Weyl nodes annihilate in pairs during phase transition if time-reversal symmetry is preserved [37] (figure 1). We predict a new d-p band inversion in SrSi 2 using band structure calculations. We confirm the stability of a strained structure using phonon band structure and also compute the Z 2 invariants and surface band structure to confirm its topological insulating behaviour. The organization of this paper is as follows. In section 2 we provide the crystal structure of bulk SrSi 2 and details of computational methodology. Section 3 is split into four parts. In section 3.1, electronic band structures of bulk SrSi 2 are studied. In section 3.2, strain-influenced structures of SrSi 2 and their electronic band structures are studied. In section 3.3, we discuss the band inversion in the straininfluenced structures and confirm their topological insulating behaviour by computing Z 2 invariants. Later in section 3.4, we discuss the surface states of the strain-influenced structures. In section 4, we discuss the primary conclusions drawn from the present work.  For bulk and strained SrSi 2 , the density functional theory (DFT) based calculations were performed using Quantum Espresso code [42], with the standard frozen-core projector augmented-wave (PAW) method. The exchange correlation of generalized gradient approximation in the Perdew-Burke-Ernzerhof (GGA-PBE) format was employed. We used kinetic energy cut-off for wave function as 45 Ry and 12 × 12 × 12 k-point grid to sample the Brillouin zone for optimizing the crystal structure and selfconsistent calculations. For the calculation of topological invariants and the surface states, we used Wannier wave functions [43][44][45] as implemented in the Wannier90 package [46]. Phonon calculation is done using the phonopy code and 2 × 2 × 2 supercell to obtain the phonon dispersion by means of density functional perturbation theory method [47].

Bulk SrSi 2
The entire crystal volume was optimized in order to attain ground state lattice parameters for bulk SrSi 2 . The energy-volume ( − ) data was then fitted in the Birch-Murnaghan equation of state. The optimized lattice parameter was found to be = 6.563 Å, in good agreement with previous experimental [48,49] and theoretical [23] results. For the ground state bulk SrSi 2 , the electronic band structure from GGA and GGA+SOC (spin-orbit coupling) along high symmetry path was calculated as shown in figure 3(a) and (b). In consonance with previous reports, we notice that the valance band (VB) and conduction band (CB) touch each other at two distinct k points along Γ-X direction, clearly revealing the semimetallic nature of SrSi 2 . Our computed energy band structures are in good agreement with the previous results [23,24]. Due to the absence of inversion symmetry in SrSi 2 , the SOC is expected to lift the spin degeneracy. Having compared the electronic band structure with and without SOC, we find that the overall nature of band structure is identical due to the involvement of light elements of SrSi 2 . The inclusion of spin-orbit coupling leads to a gapless state at the Fermi level identical to the case when the spin-orbit coupling is not taken into account.

Strain influenced electronic structure of SrSi 2
Initially, we investigated the effect of triaxial strain, i.e., the lattice parameter "a" was reduced uniformly in all three directions. We found that with the application of triaxial strain, SrSi 2 retained its WSM behaviour, albeit the node separation increased progressively as the compressive strain was increased. The above was reported earlier by Bahadur et al [23]. The authors also report on the observation of a semimetal to trivial insulator transition with tensile triaxial strain [23]. Hence, SrSi 2 remains semimetallic with the application of compressive triaxial strain and becomes a trivial insulator with the application of tensile strain. Next, we applied uniaxial strain on the crystalline -axis while keeping lattice parameters along and axis unchanged [ figure 2(c)]. This geometry enforces anisotropy in an otherwise cubic SrSi 2 , thereby lowering the symmetry of lattice from cubic to tetragonal. We progressively increased the strain (compressive) and, at 5% value, we notice a slight opening of the band gap of SrSi 2 along the high-symmetry direction Γ-X. We conclude that the introduction of anisotropy in the lattice triggers the opening of the band gap near Fermi level which pushes the system into an insulating state. To address the question whether this insulating phase holds at increased values of strain, we applied a compressive strain till 10% and found that the gapped state is persistent although the band gap increases slightly. The electronic band structures shown in figure 3(c-f) are corresponding to two selected values of the strain out of several values: 5% (where the gapped state first appears) and 10% uniaxial strain, with lattice parameter = 6.236 Å and = 5.907 Å, respectively.  and without SOC. The observed band gap was 51 meV and 62 meV for 5% and 10% strained structure, respectively. After the inclusion of SOC, the bandgap reduction was observed, from 51 meV to 36 meV and from 62 meV to 46 meV, for 5% and 10% strained structure, respectively. Further, we computed the p and d orbitals projected band structures for 5% [ figure 4(a-f)] and 10% strain [ figure 4(g-l)], respectively. The red, green and blue coloured circles in figure 4 represent the , p and orbitals of Si and cyan, magenta and yellow coloured circles represent , and orbitals of Sr. From the figure, we observe that the contribution of and orbitals is relatively large near the Fermi level as compared to orbital. The figure also shows that t 2 orbitals (including , and orbitals) are present closer to the Fermi level (marked as zero) while e orbitals ( 2 − 2 and 2 ) reside far away from Fermi level. A similar trend was observed in the case of 10% strain. The total and partial electronic density of states (DOS) of SrSi 2 for bulk and strained structures are calculated and illustrated in figure 5(a) and figure 5(b) and 5(c), respectively, where the Fermi level is set to 0 eV. The contributions of , , orbitals of Sr and , orbitals of Si are displayed. For bulk SrSi 2 , a narrow intense peak near  the Fermi level is observed but the same feature was not observed for strained structures. In the vicinity of the Fermi level, the calculated values of partial DOS show that the contributions of Sr -states are predominant whereas the contributions of Si states are comparatively smaller. The large contributions of -states near the Fermi level are indicative of large electronic correlations and higher localization than the other known topological materials, which makes SrSi 2 a good platform for illustrating the effects of correlations on topology. We also examined the dynamical stability of strain impacted SrSi 2 by computing the phonon dispersion as shown in figure 6a) and b). There are total 36 phonon modes in the vibrational spectra, out of which 3 are the acoustical and rest are the optical modes. The absence of any imaginary frequencies along the high-symmetry points indicates that the structure is dynamically stable at the values of strain considered in this work.

Band inversion and Z 2 invariants:
Topologically non-trivial materials could be well identified through the band-inversion phenomenon. Due to the relativistic impact from heavy elements, the s-orbits may be pushed underneath the -orbits leading to an inverted band character known as band-inversion. Hence, in order to check whether the insulating phase obtained at 5% strain is trivial or non-trivial, we study the signatures of band inversion. In this work, we predict d-p band inversion in strain induced SrSi 2 . The d-p band inversion is unconventional as compared to band inversions found in previous topological insulators. In Bi 2 Se 3 , topological band inversion occurs only due to -orbitals [50]. We find that, starting from 5% strain, the band structure reveals a band inversion along Γ-X direction when SOC is included in the calculation. The inverted band structure is obtained between the Sr-orbital and Si-orbitals as shown in figure 7 (for 5% and 10% strain), which may indicate TI phase. Figure 7(a) and 7(c) shows the Sr -states for 5% and 10% strained SrSi 2 . Here, the radii of red circles correspond to the proportion of Sr-electrons.We observe that the weight of Sr-orbital is higher for the valence band compared to the conduction band, in contrast to the ground state (not shown here) where CB is dominated by Sr -states. In figure 7(b) and 7(d) we plotted Si -states (the radii of green circles correspond to the proportion of Si-electrons) for 5% and 10% strained SrSi 2 , respectively, and find that the weight of Si-orbital is higher in the conduction band as compared to the valence band, while for the ground state, the VB is composed of Si -states. In 5% strain the band inversion happens in Sr -states (specifically in and orbitals as seen in figure 4) and Si -states (specifically in and orbitals as seen in figure 4), whereas in 10% strain, the band inversion happens in Sr -states (specifically in and orbitals) and Si -states (specifically in and orbitals). Thus, the d-p band inversion is apparent in the strained structures. Hence, we conclude that there may be a topological semimetal to topological insulator transition at 5% uniaxial strain. In many well-studied topological materials like HgTe/CdTe quantum wells and Bi 2 Se 3 , significant interplay between only and orbitals can be observed. The − inversion observed in the present work for SrSi 2 , was earlier observed in a few topological materials such as bismuth-based skutterudites [51].

Surface states
The presence of surface states is one of the most crucial indications of the topological insulating phase. Surface states are the extraordinary states that appear inside the bulk energy gap and permit a metallic conduction on the surface of the topological insulator. To explore the non-trivial surface states, we performed calculations for the projected band structure of a strained SrSi 2 on the (001) surface. We utilized the Wannier Tool [54] package based on a tight-binding model with Wannier wave functions built from the plane wave solution. In order to study the surface states, we consider the slab structure which is made up of a slab of 25 layers of SrSi 2 along (001) surface. The slab band structure for strain-induced SrSi 2 is shown in figure 8(a) and (b) for 5% and 10% uniaxial strain. The gapped bulk states are clearly seen in the slab band structure. This again confirms the prediction about d-p band inversion yielding a topological insulating phase in strained SrSi 2 . Figure 8(c) and (d) illustrates the topologically protected metallic surface states along¯−¯−¯direction over the surface Brillouin zone, near Fermi level.

Conclusion
In conclusion, using density functional theory-based calculations, we have demonstrated the suitability of external uniaxial strain to control the band structure and topological properties of the semimetallic SrSi 2 . We examined the modifications in the band structure of SrSi 2 by progressively increasing the uniaxial compressive strain along the crystalline -axis ( -direction). At 5% and beyond, a slight opening of the bandgap is observed. To confirm whether this insulating state is trivial or non-trivial, we calculated topological invariants and the surface band structure. We observe topologically protected surface states on the (001) surface of strained SrSi 2 . Hence, we conclude that strain-induced SrSi 2 is a strong topological insulator and undergoes a TSM ↔ TI phase transition under the effect of uniaxial strain. Our findings reveal that the topological behaviour of SrSi 2 could be well controlled with uniaxial strain and may prospectively trigger future interests in developing SrSi 2 -based electronic devices.

Acknowledgements
Author PS would like to thank NPSF C-DAC Pune for providing HPC facility.