Pressure induced Lifshitz transition in ThFeAsN

We start with the calculated pressure dependent structural parameters. In Fig.2, we present our calculated lattice parameters, volume of unit cell, anion height and bond angle ($\alpha$) as a function of hydrostatic pressure. For tetragonal ThFeAsN compound, lattice parameters a and c both gradually decrease with the increasing pressure. As a result of this, unit cell volume also decreases with the increasing pressure (see Fig.2c). Anion height is also reduced as we move in the higher values of hydrostatic pressure. Not only volume, but c/a ratio also monotonically decreases with the increasing external pressure. On the other hand, As-Fe-As bond angle increases with the pressure. Structural parameters like anion height and As-Fe-As bond angle play an important role in superconductivity of Fe-based superconductors Konzen2017 ; Mizuguchi2010 . Our calculated pressure dependencies of these structural parameters are consistent with the experiments Wang2018 .

Variations of anion height and As-Fe-As bond angle with external pressure follow the experimental trends only qualitatively. Optimized value of anion height within density functional theory (using GGA-PBE exchange-correlation functional) at ambient pressure is much lower than the experimentally measured value of anion height. This well addressed discrepancy of anion height calculation of Fe-based superconductors using ab initio density functional theory is due to the magnetic fluctuation associated with the Fe atoms, present in the Fe-based superconductors Mazin2008 ; Sharma2015 ; Singh2008A . Our current system has no long range magnetic order but tends to show strong magnetic fluctuation above 35 K Shiroka2017 . Therefore, the underestimation of anion height is quite justified. However, at higher pressure our calculated anion heights as well as As-Fe-As bond angles resemble with that of the experimental values. This also may give an indication of reduction of magnetic fluctuation in the system at higher pressure. Incidentally, superconducting transition temperature also decreases abruptly at higher pressure indicating that magnetic fluctuation may have influence the superconducting properties in ThFeAsN. Therefore, it demands further investigation to find if there is any connection between magnetic fluctuation and superconductivity in this material. We also show the behaviour of various bond lengths as a function of pressure. In Fig.3, we depict the variation of Fe-Fe, Fe-As, As-As (in plane and out of plane) and Th-As bond lengths with pressure. All the bond lengths shrink as we move towards higher pressure. Smooth behaviour of the structural parameters as a function of hydrostatic pressure suggests that there is no structural transition in ThFeAsN at higher pressure up to 35 GPa. Experimentally also, no structural phase transition is observed in ThFeAsN compound at higher pressure (up to 29.4 GPa of hydrostatic pressure). Variation of out of plane As-As distance is more prominent than that of the in plane one. This indicate that there is a possibility of significant modifications in the electronic structure along the z axis as compared to that in the xy plane. We also calculate the elastic constants of the tetragonal ThFeAsN system at various hydrostatic pressures.

In Fig.4, we depict our calculated elastic constants of ThFeAsN as a function of pressure. There are six elastic constants in tetragonal ThFeAsN system. All the six elastic constants (C₁₁, C₁₂, C₁₃, C₃₃, C₄₄, C₆₆) increase monotonically with pressure. All the elastic constants are positive and obey the well known Born criterion of mechanical stability Born throughout the pressure range that we considered. Absence of anomalies in the pressure variation of elastic constants indicate that there is no structural disorder (like collapse tetragonal phase as observed in some of the other Fe-based superconductors Quader2015 ).

In the next section, we present our calculated electronic structure at various external pressures.

Our electronic structure calculation at different hydrostatic pressures consists of density of states (DOS), electronic band dispersions and Fermi surfaces (FSs). First, we display our calculated total density of states at different hydrostatic pressures. In Fig.5, we present our calculated total density of states for 4, 10, 15, 20 and 25 GPa of hydrostatic pressure along with the ambient one.

In the naked eye, there are no observable changes in the DOS with pressure. But a closer look, reveals that the chemical potential shift towards the unoccupied states as we go towards higher pressure. This scenario is very similar to the case of electron doping in the system within rigid band model. Since Fe-d orbitals and As-p orbitals mainly constitute the low energy electronic structure of most of the Fe-based superconductors, we study the Fe-d and As-p orbital-projected DOS at different pressures. In Fig.6, we display our calculated Fe-d an As-p orbital-projected DOS at ambient and 4, 10, 15, 20, 25 GPa of hydrostatic pressure. We observe that, As-p orbital has very little contribution in the low energy electronic density of states at all pressure ranging from 0 to 35 GPa as compared to the contributions from the Fe-d orbitals. We also calculate the DOS at Fermi level for Fe-d states N${}_{E_{F}}(Fe)$ at each pressure and in Fig.8, we show the pressure dependencies of N${}_{E_{F}}(Fe)$.

Variation of Fe-d DOS at Fermi level with pressure follows the same trend as that of the superconducting T_c measured experimentally Wang2018 at different pressures. DOS at Fermi level drops significantly with pressure and a sharp fall around 20 GPa of pressure is observed. This in turn reduce the possibility of electron pairing at higher pressure. This may be one of the reason that superconducting T_c decreases with the increasing pressure.

We also calculate the variation of Fe-d orbital resolved DOS with pressure. In Figs.7(a-d), 7(e-h), 7(i-l), we depict our calculated Fe d_xy, d_yz+xz, d${}_{z^{2}}$, d${}_{x^{2}-y^{2}}$ orbital-projected DOS for ambient, 15 GPa and 25 GPa pressure respectively. It is quite evident from Figs.7c, 7g, and 7k that Fe-d${}_{z^{2}}$ orbital derived partial DOS are modified remarkably with the pressure as the Fermi level shifts away from a van Hove singularity as we go towards higher pressure. On the other hand, we observe exactly opposite scenario for the degenerate d_yz+xz orbital projected DOS. This certainly indicates that orbital characters around Fermi level at higher pressure and ambient pressure are different from each other. Next, we see the modifications in the electronic band structure with external pressure. In Fig.9, we present our calculated low energy (-1 eV to 1 eV) electronic band structures for tetragonal ThFeAsN at different pressures. A number of noticeable modifications are found in the band structures at various hydrostatic pressures.

We can clearly see from Fig.9, electronic bands around $\Gamma$ point are significantly modified as we gradually increase the pressure. On the other hand, hydrostatic pressure hardly modifies the electronic bands near (M/X) point. In Fig.10 (Fig.11), we present the orbital-projected band structures of ThFeAsN around $\Gamma$ point (M point) for various hydrostatic pressures. Various Fe-d orbital characters are depicted by different colours (d_yz+xz, d${}_{z^{2}}$, d${}_{x^{2}-y^{2}}$ orbitals are indicated by blue, magenta and brown colours respectively). In Fig.12, we depict the variation of different band energies around $\Gamma$ point with pressure. The band with d${}_{z^{2}}$ orbital character goes above the Fermi level as we move from ambient to higher pressure. This is recognized as Lifshitz transition occurring at around 25 GPa pressure. Degenerate bands with d_yz+xz orbital characters around $\Gamma$ point also move downwards and touch the Fermi level at 30 GPa of hydrostatic pressure. But the band with d${}_{x^{2}-y^{2}}$ orbital character is hardly modified by pressure. Therefore, this also can be considered as orbital selective Lifshitz transition. This in turn may leads to an orbital selective pairing in ThFeAsN as observed in some other Fe-based superconductors Nica2017 ; Kreisel2017 .

On the other hand, no such modifications (LT/ETT) in the low energy electronic band dispersion are observed in the vicinity of M/X point. Therefore, LTs precisely occur in the hole like bands at the centre of the Brillouin zone ($\Gamma$ point). Since the effect of pressure in the electronic structure of ThFeAsN resembles with the case of electron doping in the system, our results are in full agreement with the general trends of Fe-based SCs Khan2014 ; Ghosh2017 ; Liu2011 . However, there are some dissimilarities in the nature of LT in ThFeAsN with the other Fe-based SCs that we will discuss in the later part of this section. We see in the structural parameters as a function of pressure, in plane As-As distance varies very little in comparison with the out of plane As-As distance (see Fig.3c). Therefore, orbital that extend in the xy plane (d${}_{x^{2}-y^{2}}$) remains almost the same as we increase the pressure. On the other side, orbitals that extend along the z axis (d_yz+xz, d${}_{z^{2}}$,) are modified largely by the external pressure. A closer look at d${}_{z^{2}}$ band around $\Gamma$ point, reveals that the nature of band dispersion also changes with the pressure. More precisely, electron like d${}_{z^{2}}$ band becomes hole like band at higher pressure.

No such electronic topological transitions occur in the bands near M/X point. Significant modification in the bands near $\Gamma$ point with pressure also indicate that FS topology of ThFeAsN also changes with the external pressure. We show our calculated FSs of ThFeAsN system at different hydrostatic pressures in Fig.13. At ambient pressure (0 GPa), there are three hole like FSs at the $\Gamma$ point and two electron like FSs at the M point. All the FSs at different pressures are also shown separately in Fig.14 for better insights. It is quite evident from Fig.14, that electron like FSs at M point are hardly affected by the external pressure. On the contrary, external pressure influences the hole like FSs around $\Gamma$ point remarkably. Hole like FSs labelled as 1, 2 and 3 completely changed topologically at higher pressure. This evolution of FSs with hydrostatic pressure directly affects the nesting of FS, which is believed to play a key role in the superconductivity of Fe-based SCs.

It is also well documented that two dimensional (2D) FS favours superconductivity in Fe-based SCs (nesting is stronger in 2D FSs as compared to the 3D FSs) Sen2015 ; Sunagawa2014 . But at higher pressure, we find more 3D like FSs (specially hole like FSs around $\Gamma$ point) in contrast to that at the lower pressure. See appendix for FSs calculated using experimental structural parameters where the transformation from 2D like FSs (ambient pressure) to more 3D like FSs (25 GPa pressure) is very clear. Therefore, we can conclude that pressure induced LT or ETT affects the low energy electronic structures significantly which control the superconducting properties in ThFeAsN system.

Experimentally it has been shown that superconducting transition temperature (T_c) decreases with the increasing pressure. T_c drops sharply to 0 K at around 25 GPa of pressure on the verge of LT. Therefore a close connection between superconductivity and LT is established in ThFeAsN system. In general, LTs in Fe-based superconductors are largely found to help in growing superconductivity Khan2014 ; Ghosh2017 ; Lei2016 . For example, in electron doped BaFe₂As₂, LTs occur in the hole like bands as the band with d_xy orbital character goes below the Fermi level (one of the FS disappears from the $\Gamma$ point) at the same doping concentration where T_c reaches its maximum value Ghosh2017 . However, in ThFeAsN superconducting T_c vanishes approximately at the same pressure where LT occurs. Therefore, it is worthy to mention here that appearance of the band at the Fermi level with d${}_{z^{2}}$ orbital character around $\Gamma$ point at higher pressure is a quite unique feature of ThFeAsN superconductor. This make ThFeAsN system an unique one, breaking the universal trend of superconductivity and LT/ETT in Fe-based SC. Therefore, orbital seductive LT can be a way to predict the trend of superconducting transition temperature (T_c) in Fe-based SCs in general.

In this section, we show the influence of spin-orbit coupling (SOC) in the low energy electronic band structures of ThFeAsN at higher pressure. In Fig.15, we depict our calculated band structures in the presence of SOC for ambient as well as 25 GPa of hydrostatic pressure. Orbital projected (Fe-d orbitals) band structures with SOC at ambient and 25 GPa pressures are presented in Fig.16a and Fig.16b respectively. It is very clear from Fig.15 and Fig.16 that with the introduction of SOC, low energy band structures around $\Gamma$ point are modified remarkably as degenerate d_yz+xz band near Fermi level splits into two bands with the same d_yz+xz orbital character for both the cases (ambient and 25 GPa pressure).

LT also occurs in presence of SOC with pressure as the band with d${}_{z^{2}}$ orbital character around $\Gamma$ point moves upward and crosses the Fermi level. The splitting of d_yz+xz band around $\Gamma$ point is very large at higher pressure (25 GPa) as compared to that at the ambient pressure.

From Fig.16, we can clearly see that, SOC at higher pressure modifies the energy ordering of the orbitals near $\Gamma$ point. SOC at higher pressure splits the d_yz+xz band around $\Gamma$ point such that one of the two bands goes below the Fermi level whereas in absence of SOC, degenerate d_yz+xz band (no splitting) goes below the Fermi level (see Fig.10). Since SOC modifies the energy ordering (or in other words orbital ordering) near the Fermi level at higher pressure, it will affect the orbital selective pairing of electron and orbital selective properties Nica2017 ; Kreisel2017 . This indicates that SOC may play a crucial role in superconductivity of ThFeAsN system.