Pressure induced antiferromagnetic-tetragonal to nonmagnetic-collapse-tetragonal insulator-metal transition in ThMnAsN

In the context of high temperature superconductivity in Fe-based superconductors, correlation of local structural parameters like h_As, As-Fe-As angles etc. with T_c (superconducting transition temperature) and its effect in electron correlation and magnetism is well established Chen2014 ; Mizuguchi2010 ; MingY2017 . In Fig.2, we display the variation of various structural parameters like normalized lattice parameters (a/a₀, c/c₀), c/a ratio, volume of the unit cell (V) and local magnetic moment of Mn atom as a function of hydrostatic pressure. It is quite evident that there is an abrupt change in the normalized lattice parameters, c/a ratio and volume of the unit cell at 9 GPa pressure. The change in the c-axis as compared to the a-axis is more prominent. This anomalous behaviour of structural parameters with the hydrostatic pressure is attributed to a iso-structural phase transition (tetragonal to collapse tetragonal) that occurs simultaneously with the magnetic transition (AFM to PM phase). It is worth mentioning here that in LaMnPO, a iso-structural compound to ThMnAsN, pressure induced AFM to PM transition is observed experimentally at 34 GPa Guo2013 . Furthermore, we also study the variation of various bond lengths and bond angles with pressure as shown in Fig.3. From Fig.3a, we can see a pronounced change in the Mn-As bond distance at 9 GPa pressure as compared to the Mn-Mn bond distance. Added to that, Fig.3b(c), showing the variation of in plane and out of plane As-As distances (’anion height’) with pressure also indicate a significant out of plane modification (along the c-axis) that occurs at 9 GPa. Modifications of Mn-As tetrahedral bond angles ($\alpha$ and $\beta$) are depicted in Fig.3d. It is quite evident from Fig.3d that, a large distortion in the Mn-As tetrahedron transpire at 9 GPa pressure.

Thus, at around 9 GPa pressure, all the structural parameters show strong anomalous variations e.g, c-lattice parameter, out of plane As-As bond length, anion height (h_As) undergo drastic modification. In particular, above P_c the out of plane As-As bond length contracts by   9 $\%$ whereas the h_As shortens by $~{}14\%$; the tetrahedral angles $\alpha/\beta$ increases/decreases by   6.4 $\%/$  3.6 $\%$; the in-plane Mn-As bond shortens by $~{}8\%$. The out of plane As-As bond length becomes shorter than its planer counter part. These modifications of ”local structural correlation” influences the electronic coordination of otherwise localized magnetic moments of Mn atoms which gets collapsed (quenched) at the same critical pressure (9 GPa) indicating a magnetic phase transition from AFM to para-magnetic (PM) phase. The critical value of the pressure P_c = 9 Gpa is robust and does not get modified with increasing Hubbard correlation U (Mn-d) up to 5 eV. However, the value of the magnetic moment of Mn in the C-AFM phase for GGA+U calculations exceeds the experimental value (see discussion in SM) and hence not presented in the figure (for higher values of U). This indicates to the fact at higher pressures the value of the Hubbard correlation for ThMnAsN might have been reduced from that of the ambient pressure. The two different phase transitions, magnetic and structural one occurs simultaneously driven by each other. Such simultaneous magneto-structural transition is very similar to those observed experimentally in NaFe₂As₂, CaFe₂As₂ and several other Fe-based superconductors Kreyssig2008 ; Stavrou2015 ; Long2013 ; Uhoya2010 ; Mittal2011 ; Nakajima2015 except the fact that the in-plane and out-of-plane As-As bond distances are comparable for ThMnAsN. The ratio c/a of the tetragonal phase at which cT phase transition takes place is about 2.1 for ThMnAsN whereas about 2.9 for KFe₂As₂ / CaFe₂As₂ / BaFe₂As₂ compounds respectively. The volume contraction is about 4.0 $\%$ for ThMnAsN whereas is about 3.0$\%$ in CaFe₂As₂ / BaFe₂As₂ compounds respectively. This opens up a prospects for ThMnAsN to possibly superconduct at higher pressures (superconductivity is observed in KFe₂As₂ Nakajima2015 above the cT phase with pressure).

Next we present the variation of elastic constants of tetragonal ThMnAsN as a function of hydrostatic pressure as depicted in Fig.4a. Mechanical stability of a crystal under external pressure requires to satify the generalised Born-Huang stability criteria. For a tetragonal system the Born-Huang stability criteria is defined as Gomis2014 : C₁₁-P$>$0, C₄₄-P$>$0, C₆₆-P$>$0, C₁₁-C₁₂-2P$>$0 and (C₃₃-P)(C₁₁+C₁₂)-2(C₁₃+P)${}^{2}>$0. All the above criteria is satisfied for ThMnAsN up to 14 GPa pressure. Fig.4a shows that all the elastic constants are more or less increasing with the increase of external pressure except C₄₄. It is also to be noted here that the elastic constant C₁₁ is always greater than C₃₃ up to 15 GPa pressure which indicates that it is easier to compress the system along 001 direction than in 100 direction. In order to understand the observed behaviour of elastic constants around 9 GPa pressure, we also calculate the Cauchy pressure for tetragonal system as a function of pressure (see Fig.4b). It is well known that for metallic bonding, the Cauchy pressure is positive, while for directional bonding, it is negative and larger negative Cauchy pressure indicates a more directional character in the bonds Pettifor1992 . As we can see from Fig.4b, Cauchy pressure C₁₃-C₄₄ has small negative value at lower pressure but it increases rapidly as we go above 9 GPa pressure. On the other hand, there is almost no variation of Cauchy pressure C₁₂-C₆₆ with the hydrostatic pressure. This is so, because the variations of elastic constants C₁₂ and C₆₆ with the pressure are not very substantial as compared to the other elastic constants where the pressure variation is noticeable. High positive values of Cauchy pressure C₁₃-C₄₄ above 9 GPa pressure indicate the enhancement of metallic behaviour. In Fig.5, we present the pressure variation of C₄₄-P. There is a sudden drop of C₄₄-P at around 9 GPa pressure and above 14 GPa pressure C₄₄-P becomes negative. When the value of C₄₄-P is less than zero, it indicates that the system is mechanically unstable or is on the threshold of structural phase transition Gomis2014 .

Further, we have also depicted the charge densities of ThMnAsN at various pressures along the 100 plane in Fig.6. We can clearly see that (see the black arrow in the Fig.6) at 9 GPa pressure, the overlap of charge densities along Mn-As bond is significantly increased if we compare the same with that at the 8 GPa pressure. All these results collectively stipulate the enhancement of metallic character above 9 GPa pressure. In Fig.7, we depict the iso-surface of electronic charge densities at ambient as well as 9 GPa pressure. At 9 GPa the chrage densities of the out of plane As atoms are getting very close to each other, however, no direct covalent bonding is formed unlike those observed in 122 family of Fe-based superconductors Kreyssig2008 ; Stavrou2015 ; Long2013 ; Uhoya2010 ; Mittal2011 ; Nakajima2015 . As the out of plane As-As distances become less than the in-plane As-As bond length, all 4-As atoms strongly overlap with the Mn atom. This leads to indirect overlapping between the Mn atoms via As atoms (see Fig. 7). This destroys the otherwise ”localized” nature of Mn moments and moments get completely quenched above P_c.

To corroborate the charge density plots at 1, 8 and 9 GPa, the calculated integral crystal orbital Hamilton population (ICOHP) is indicated on Table 2. Their sign indicates net bonding for the reported distances. The absolute value of the ICOHP is proportional to the strength of the bond and is increasing for all bonds with pressure from 1 to 9 GPa. The Mn-As absolute value shows an increase of 6.7 % from 1 to 8 GPa and of 8.9 % from 8 to 9 GPa, considering the up spin channel (similar trend is observed for the dn spin channel). Therefore, the evolution in the smaller range 8 to 9 GPa is more significant than the whole 1 to 8 GPa range. It signifies the structural transition described from 8 to 9 GPa and the enhancement of metallic character at 9 GPa. On the other hand, projected COHP is shown in Fig.8 for the bonds Mn-As. Positive COHP represents bonding interactions while a negative COHP represents anti-bonding interactions. Anti-bonding states are present at the top of the valence band just below the Fermi level for both 1 and 8 GPa. In these two cases [Fig.8a, b], the band gap is opened in the AFM state while the metallic state at 9 GPa gives rise to anti-bonding states at the Fermi level. Such situation is destabilizing and should lead to a structural distortion and/or magnetic transition.

Having presented in detail the effect of hydrostatic pressures on the structural parameters and associated modifications in elastic constants, it is imperative to present the effects of such changes of structural parameters on electronic band structure, density of states and Fermi surface at ambient as well as at higher pressure in this subsection. As mentioned earlier, in order to create C-AFM spin configuration a supercell $\sqrt{2}\times\sqrt{2}\times 2$ structure is used. With a larger real-space cell, the reciprocal-space Brillouin Zone (BZ) will become smaller. As a result, the band structure will fold into the small supercell BZ. Electronic eigen states calculated for the supercell can be unfolded into the reciprocal space of the primitive cell directly in VASP, these technicalities are described in Dirnberger2021 . The electronic (as well as orbital projected) band structures at ambient and higher pressures are presented in Figs. 9, 10, 11 and 12 respectively. We depict the unfolded band structure in Fig.9 for ThMnAsN with C-AFM spin configuration at ambient pressure. A band gap of $\sim$ 0.4 eV is observed in ThMnAsN at ambient pressure and it can be compared with the experimentally measured band gap of 1.2 eV for LaMnAsO, a compound iso-structural to ThMnAsN McGuire2016 . Contribution of various Mn-d orbitals in the band dispersions are presented in the bottom Fig.9b. Mn d_xz+yz, d_xy, d${}_{z^{2}}$ and d${}_{x^{2}-y^{2}}$ orbitals are denoted by blue, red, magenta and green colours respectively. The Mn d_xz+yz and d${}_{x^{2}-y^{2}}$ derived bands approach the Fermi Level both from the conduction and valance band, reducing the band gap as the pressure is increased (cf Figs. 9, 10, 11). At 9 GPa pressure, there is no band gap and the system is metallic. However a closer look at different orbital selective bands in Figs. 9, 10, suggest orbital selective doping; the green band i.e, d${}_{x^{2}-y^{2}}$ band gets influenced similar to as being hole doped whereas the magenta coloured band i.e, d${}_{z^{2}}$ derived band near $\Gamma$ point suffers the effect of electron doping. This is a precursor to orbital selective Lifshitz transition usually observed as well as predicted in Fe-based superconductors MingY2017 ; Sen2020PRM ; Ghosh2020 ; Ghosh2021 ; Ghosh2022 ; Craco2021 .

In Fig.11, we depict our calculated band structure of non magnetic ThMnAsN at 9 GPa. Positions of Fermi level at different chemical potentials are also indicated in the band structure with a dotted line which is used to mimic the electron doping in the non magnetic ThMnAsN system within the rigid band model. Visibly different nature of electronic band structure above the magneto-structural transition is worth noticing. With the value of chemical potential $\mu$=0.25, we find a electronic band dispersion very similar to that of the generic band dispersion of Fe based superconductors. We also find that orbital nature of the low energy bands of electron doped ThMnAsN (within rigid band approximation with $\mu=0.25$) at 9 GPa pressure are also quite similar to the low energy band dispersion of most of the Fe based superconductors (see Fig.12). Moreover, domination of Mn d_xz+yz and d${}_{x^{2}-y^{2}}$ orbitals at the Fermi level at $\Gamma$ and M points which is quite similar to the band structure of stochiometric ThFeAsN 2Wang2016 ; Singh2016 ; Sen2020PRB , a Fe based superconductor iso-structural to ThMnAsN.

Our calculated density of states at different hydrostatic pressures are presented in Fig.13. We can clearly see, with the increasing pressure metallic behaviour is increasing as the energy gap is decreasing and at 9 GPa pressure, no energy gap exists at the Fermi level (cf. Fig. 13). This suggests that ThMnAsN is at the threshold of a metal insulator transition which is triggered by the collapse of local Mn moment and this transition may occur with the variation of pressure, temperature or may be even with the doping. We found a contrasting behaviour in LaMnPO, a iso-structural compound to ThMnAsN, where AFM to PM and insulator to metallic phase transitions occur at different pressures (not simultaneous). In LaMnPO, insulating AFM to metallic AFM transition occurs at 20 GPa pressure and metallic AFM to PM transition occurs at 34 GPa pressure Guo2013 . However, in this work, we have not studied the effect of doping and temperature in ThMnAsN. Nevertheless, we find three possible phase transitions (AFM to PM, tetragonal to collapse tetragonal and insulator to metal) that take place at around 9 GPa pressure. We also depict our calculated Mn-d orbital projected density of states for ThMnAsN at different hydrostatic pressure in Fig.14. The orbital selective band gaps (cf. Fig. 14) at finite hydrostatic pressures are one of the important special features of ThMnAsN (this suggests possibility of orbital selective optics up to 8 Gpa). In this context, it is worth mentioning that the orbital selective modulation of electronic structure is also observed in ThFeAsN Sen2020PRM ; Craco2021 .

At 9 GPa pressure, atom projected density of states suggest that among all other atoms in ThMnAsN, Mn atoms have mostly contributed to the total density of states around the Fermi level (see Fig.15a). Therefore, Mn d-orbital projected density of states for non magnetic ThMnAsN at 9 GPa pressure is presented in Fig.15b. All five Mn-d orbitals are present at the Fermi level. Among them, Mn d_xz+yz and d${}_{x^{2}-y^{2}}$ orbitals dominate the Fermi level.

To illustrate the metallic behaviour of ThMnAsN at 9 GPa pressure more closely, we calculate the FSs of ThMnAsN. Fermi surfaces of ThMnAsN at 9 GPa pressure in the non magnetic phase for different chemical potentials which are depicted in Fig.16. Fermi velocities are also indicated in all the Fermi surfaces. Fermi surface topology, specially with $\mu=0.25$, closely resembles with the generic Fermi surface of the Fe based superconductor Sen2014 ; Sen2015 ; Paglione2010 ; Sen2017s . It contains three hole like Fermi surfaces in the centre of the Brillouin zone ($\Gamma$ point) and two electron like Fermi surfaces at the corner of the Brillouin zone (M point). In Fig.17 and 18, we display the Fermi surface sections of ThMnAsN in 100 and 110 (k_z dispersion) plane respectively. From Fig.17 and 18, it is quite evident that with the higher values of chemical potential the Fermi surfaces become more 2 dimensional (2D) in nature. Appearance and disappearance of various Fermi surfaces in the BZ with the variation of chemical potential (introduced to mimic the effect of electron doping) bears the signature of orbital selective Lifshitz transition. 2D Nature of these Fermi surfaces are also favourable for nesting (with electron doping) which plays a very important role in pnictides superconductors to give rise to various exotic phases like spin density wave order (SDW), orbital density wave order (ODW) etc., including superconductivity Terashima2019 ; Sen2015 ; Ghosh2017 ; 1Sen2015 ; Sunagawa2015 . From Figs. 16, 17 and 18, it is clear that there is formation of electron type cylinders at around $\Gamma$ point whereas hole type cylindrical FSs at around X point. Strong reconstruction in electronic structure (and hence transport properties) similar to KFe₂As₂ Stavrou2015 ; with slight electron doping, electron like charge carriers are missing around M point whereas with further electron doping electron FSs again grow around the BZ corners (M points) and there is possibility of nesting between the inner hole to electron FSs. Since Fermi surface topology and low energy electronic structure are believed to be the main controlling factors of superconductivity in Fe based superconductors, ThMnAsN at high pressure has a great potential to become a high temperature superconductor with unconventional superconducting properties.

To probe the dynamical stability of ThMnAsN at various pressure we perform phonon calculations. In Fig.19, 20 and 21, we present our calculated phonon dispersions and phonon density of states at 0, 5 and 10 GPa pressures respectively for ThMnAsN. For phonon calculations at ambient and 5 GPa pressure, we use C-AFM spin configuration for ThMnAsN. On the other hand, for ThMnAsN at 10 GPa pressure, we performed non-magnetic phonon calculation. Absence of any imaginary frequencies clearly appraise the dynamical stability of the system at various pressures. The gap between the optical and the acoustic phonon branch in phonon density of states decreases with pressure that has important bearings on the phonon thermal conductivity and hot carrier relaxation processes. The phonon dispersion becomes flatter in $\Gamma-Z$ direction at higher pressures. The phonon density of states decreases in the low phonon frequencies at high pressures.