Introduction

Majorana fermions are half integer spin particles that are their own antiparticles1. In a two-dimensional superconductor, a vortex with an odd number of Majorana quasiparticle excitations obeys non-Abelian rather than the Bose or Fermi statistics2,3,4. The states are robust against local noise and, hence, are considered as strong candidates for a fault tolerant quantum information processing4. Initially, Majorana quasiparticle excitations have been predicted to occur in topological superconducting phases in which the Cooper pairs have a spin-triplet px + ipy pairing symmetry5. This case is very rare in nature. The most promising system is the exotic p-wave superconductor Sr2RuO46,7 for which the experimental situation, however, is not clear8,9.

Later, they also have been proposed to appear in topological superconducting interface states. This would enable induction by the conventional s-wave superconducting proximity effect on the Dirac-like boundary states of topological insulators10,11, of half-metallic materials12,13 and of semiconductors with substantial Rashba spin-orbit coupling14,15. Recently, superconductivity has been reported for the prototypical three-dimensional topological insulators Bi2Se3 and Bi2Te316,17,18,19 under doping20,21,22,23,24 and under pressure25. Accordingly, topological superconductivity is predicted to be induced by the superconducting proximity effect on topologically protected Dirac-like surface states of the superconductor itself20,22. Such a self-induced topological surface superconductor is very promissing for the realization of Majorana quasiparticle exitations due to the absence of lattice and chemical potential mismatches21.

1T-TiSe2 is one of the early discovered materials with a charge density wave (CDW)26 and has seen a revival of interest due to recently discovered superconductivity27,28,29,30,31,32,33,34,35. It is a layered compound, consisting of slabs in which Ti is octahedrally coordinated by six Se atoms. Se-Ti-Se slabs with ionic bonding are separated by a van der Waals gap, leading to a quasi two-dimensional nature. A commensurate CDW with a 2 × 2 × 2 superstructure developes below 200 K26. No sign of superconductivity is found by lowering the temperature only. However, under controlled Cu-intercalation into the van der Waals gap the CDW is progressively suppressed and a superconducting state emerges with a maximum transition temperature of Tc = 4.15 K27. Suppression of the CDW and emergence of superconductivity have been discovered in pressurized 1T-TiSe2 with a maximum Tc of 1.8 K34.

In the following, we apply fully-relativistic all-electron ab-initio calculations to reveal a pressure induced trivial-nontrivial-trivial topological phase transition in the normal state of 1T-TiSe2. We argue that the pressure range of the topologically nontrivial phase overlaps with that of the superconductivity. Our findings suggest pressurized 1T-TiSe2 as another candidate for self-induced topological surface superconductivity for appropriate doping.

Results

Experimental values of the structure parameters have not been determined for pressurized 1T-TiSe2 and theoretical calculations encounter serious deficiencies in predicting the parameters for 1T-layered compounds under pressure36. Our calculations confirm this observation. The structure parameters of 1T-TiSe2 at ambient pressure show errors of −1% (2%), −3% (9%) and 4% (−7%) for the values of a, c and z, respectively, employing the local density (generalized gradient) approximation. These errors are problematic because of a high sensitivity of the electronic states to the structure, see also the text below. The generalized gradient approximation with explicitly treated van der Waals interaction leads to even worse errors of 5%, 8% and 5% for the three parameters. On the other hand, the experimental structure is available for 1T-TiS2 under pressure37. The similarity of the crystal structure and chemical composition38 justifies to adopt the values of a/a0, c/c0 and z of 1T-TiS2 for the normal state of pressurized 1T-TiSe2, which we make use of in the following. Note that the crystal symmetry of the high-temperature normal phase is adopted for the entire pressure range, although a CDW transition with lattice distortion appears at low temperature for a pressure below 2 GPa. We will argue that this assumption does not affect the conclusions of our work.

The fully-relativistic electronic band structure of the normal state of 1T-TiSe2 at ambient pressure is depicted in Fig. 1(b). The observed energy overlap of the valence band maximum (VBM) and conduction band minimum (CBM) indicates a semimetallic nature. The almost full valence and almost empty conduction bands show mainly Se 4p and Ti 3d characters, respectively, indicating an overall ionic bonding with charge transfer from Ti to Se. Near EF the electronic structure of 1T-TiSe2 is governed by two bands, giving rise to the hole-like Se 4p derived VBM at the Γ-point and the electron-like Ti 3d derived CBM at the L-point. The space inversion symmetry of 1T-TiSe2 allows us to calculate the band topological invariants by the parity check method proposed by Fu and Kane39. Due to the semimetallic nature, however, one needs to consider the valence states instead of the occupied states below EF. The topological invariant can be calculated directly from knowledge about the parity of each pair of Kramers degenerate bands up to the local VBMs at the eight time-reversal momenta (1Γ, 3M, 1A and 3L). A value of zero is obtained for all four topological invariants, reflecting a topologically trivial nature of the normal state of 1T-TiSe2 at ambient pressure.

Figure 1
figure 1

Normal state of 1T-TiSe2.

(a) Crystal structure. (b) Electronic band structure at ambient pressure. (c–e) Detailed band structures near the VBM under ambient pressure, 3 GPa and 5 GPa, respectively. It should be noted that the Fermi energy is located around 0.5 eV below the VBM.

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Due to the smaller energy gap around the Γ-point, as compared to the other seven time-reversal momenta, the band topology is mainly defined by the band order at the Γ-point, on which we focus in the following. A zoom of the VBM is displayed in Fig. 1(c) for ambient pressure. Below EVBM, we see two spin-orbit split Se 4px,y states: the lower doublet and the upper doublet , where the latter forms the VBM. Above EVBM, the spin-orbit coupling lifts the degeneracy of the Ti 3dxz,yz states and yields the lower doublet and the upper doublet . Under sufficient pressure, the odd parity Se 4p state is pushed on top of the Ti 3d state with even parity. As an example, Fig. 1(d) illustrates the band structure of the normal state of 1T-TiSe2 under a pressure of 3 GPa. We find that the band order is not changed at the other seven time-reversal invariant momenta.

According to the parity check method for calculating topological invariants, inversion of bands with a different parity at the Γ-point transforms 1T-TiSe2 into a topologically nontrivial state of (1;000) type. The same topological phase transition due to pressure induced band inversion has been observed in layered GaS and GaSe40. For higher pressure, the odd parity Se 4p state is pushed above the Ti 3d state with even parity, see the band structure in Fig. 1(e) for a pressure of 5 GPa. As a result, the band topology switches back to trivial. Therefore, we are confronted with a pressure induced trivial-nontrivial-trivial topological phase transition in the normal state of 1T-TiSe2.

The phase transition may be viewed as a result of the small energy gap at the VBM and, hence, the sensitivity of the band hierarchy against modifications of the structure. In addition, spin-orbit coupling is crucial for its occurrence and the appearance of a topologically nontrivial phase. Without spin-orbit coupling, the Se 4px,y and states would form a degenerate pair. Under pressure these odd parity states would simultaneously be pushed above the even parity Ti 3d states and no phase transition would be observed. Therefore, an energy difference between the and states is necessary for the emergence of a pressure induced topological phase transition. In the present case, the energy difference is caused by the spin-orbit coupling. It determines the pressure range of the topologically nontrivial phase. Specifically, the critical pressures of the trivial-nontrivial () and nontrivial-trivial () transitions, are related to the energy differences between and and and , respectively. Accordingly, the topological phase diagram of the normal state of 1T-TiSe2 under pressure can be derived from the band order. The result is shown in Fig. 2(a) for pressures between 1.5 and 4.5 GPa.

Figure 2
figure 2

(a) Topological phase diagram of the normal state of 1T-TiSe2 under pressure. and represent the energy differences of the and states and of the and states, respectively. The experimental pressure phase diagram of the ground state is also sketched. TNS, SC and CDW denote the topologically nontrivial state, the superconducting ground state and the charge density wave ground state, respectively. (b) Same as (a) but for another set of a/a0, c/c0 and z, see the text for details.

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Figure 2(a) includes the experimental pressure phase diagram for the ground state. We obtain and . On the other hand, experiments for sub-Kelvin temperature point to pressure induced phase transitions from CDW to superconductor to the normal state34. With increasing pressure, the superconductivity appears around 2 GPa and disappears beyond 4 GPa. The pressure range to overlaps with the range in which the superconducting ground state develops. Since the band hierarchy is sensitive to the structure parameters, the pressure range to could depend on uncertainties in the structure determination. Using the data set reported in Ref. 41 for a/a0, c/c0 and z, we obtain another topological phase diagram, which is depicted in Fig. 2(b). The pressure range to changes substantially with respect to Fig. 2(a), but still overlaps with the range of the superconducting ground state. As mentioned above, a hypothetical crystal symmetry without CDW distortion is used for pressures lower than 2 GPa, which, however, is in the range where our assumption on the crystal symmetry does not influence the conclusions. The discrepancy between the two experimental sets of lattice constants concerns mainly the value of c/c0. For a pressure of 3 GPa, for example, the difference is 0.1% for a/a0 but 0.7% for c/c0. This leads to a 0.1 GPa shift for and a much larger shift of 1 GPa for . Therefore, as long as is in the pressure range from 2 GPa to 4 GPa, the range of the nontrivial phase will overlap with the superconducting ground state. This means that our conclusions stay valid even when the discrepancy in the experimental lattice constant is 10 to 20 times larger than that of Refs. 37 and 41. One of the most important consequences of a topologically nontrivial state is the existence of a gapless surface state39,42,43,44,45,46,47,48, which is protected by the combination of spin-orbit coupling and time-reversal symmetry and shows a spin-textured Dirac-like behavior16,17,18,19,49. Therefore, our results indicate that a superconducting ground state coexists with a topologically protected spin-textured Dirac-like surface state near the VBM in pressurized 1T-TiSe2.

If bulk superconductivity develops in pressurized 1T-TiSe2 at low temperature, Cooper pairs can tunnel from the bulk states into the surface states. An s-wave pairing symmetry is to be expected, because the bulk superconducting phase is governed by electron-phonon coupling35. This leads to the induction of a s-type superconducting energy gap in the Dirac-like surface states10. Note that the superconducting ground state persists for slight Cu-intercalation in pressurized 1T-TiSe231,34. By Cu-intercalation, the tunneling process can be enhanced because EF is shifted close to the VBM, hence to the Dirac-like surface states. The resulting two-dimensional superconducting surface state resembles the spinless px + ipy topological superconductor, except that the time-reversal symmetry is not violated10. Like the px + ipy superconductor, the induced surface superconductor will have a Majorana quasiparticle excitation bound to a vortex10. We thus propose pressurized 1T-TiSe2 as a material in which topological superconductivity is induced by the superconducting proximity effect on topologically protected spin-textured Dirac-like surface states of the superconductor itself. Such a self-induced surface topological superconductivity is promising as it is free of lattice and chemical potential mismatches. Self-induced topological surface superconductivity before has been predicted for Bi2Se3 and Bi2Te3 under both doping and pressure20,22. The semimetallic nature and formation of a hole pocket at the VBM found in our calculations indicate that the surface Dirac-like states are not located at EF, making electron doping necessary. Assuming a rigid band model, a doping of 0.7 electrons per formular unit is required to shift EF by 0.5 eV. A shift of more than 0.1 eV has been demonstrated for a formal doping of 0.065 electrons without losing the superconductivity30. Preservation for higher doping calls for experimental confirmation.

Discussion

In conclusion, we have applied density functional theory to demonstrate a pressure induced trivial-nontrivial-trivial topological phase transition for the normal state of 1T-TiSe2. The transition is ascribed to inversion of Se 4p and Ti 3d bands with different parity at the Γ-point. It is very likely that the pressure range in which the topologically nontrivial phase emerges overlaps with the one of the superconductivity, leading to self-induced topological surface superconductivity in pressurized 1T-TiSe2. As a consequence, the formation of topological superconducting surface states becomes accessible to an experimental investigation in the present material for appropriate electron doping. The surface state is promising for the realization of Majorana quasiparticle exitations because of the absence of lattice and chemical potential mismatches. We emphasize that the existence of the topological phase transition is guaranteed by spin-orbit coupling. For obtaining the pressure range of the self-induced topological surface superconductivity more exactly further experiments are required to determine the structure parameters under pressure with high accuracy. As s-type pairing is needed for topological superconducting surface states to occur, a more careful study is required to verify the conclusions of Ref. 35.

Methods

The high temperature normal structure of pristine 1T-TiSe2 shows the space group (No. 164) where the Ti and Se atoms occupy 1a [(0, 0, 0)] and 2d [(1/3, 2/3, z)] sites, respectively, see Fig. 1(a). Experiments under ambient pressure yield structure parameters of a = 3.54 Å, c = 6.008 Å and z = 0.2550450. Our full-potential linearized augmented plane wave calculations have been performed using the WIEN2k package51. We employ a threshold energy of −6.0 Ry for separating the valence and the core states, a muffintin radius of Rmt = 2.2 aB, high values of RmtKmax = 10 and ℓmax = 10, as well as a 20 × 20 × 10 k-mesh. Moreover, a combination of the modified Becke-Johnson exchange potential and the local density approximation of the correlation potential is used in order to predict band gaps and band orders with an accuracy similar to hybrid functionals and Gutzwiller calculations52. Spin-orbit coupling is treated applying a second-variational method with a scalar relativistic basis. States up to 10 Ry above the Fermi energy (EF) are included in the basis expansion.