Electronic and lattice properties of non-centrosymmetric superconductors ThTSi (T = Co, Ir, Ni, and Pt)
A. Ptok K. Domieracki K. J. Kapcia J. Łażewski P. T. Jochym M. Sternik P. Piekarz D. Kaczorowski







[e-mail: ]aptok@mmj.pl
Institute of Nuclear Physics, Polish Academy of Sciences, 

ul. W. E. Radzikowskiego 152, PL-31342 Kraków, Poland


Institute of Low Temperature and Structure Research, Polish Academy of Sciences, ul. Okólna 2, PL-50950 Wrocław, Poland


Institute of Nuclear Physics, Polish Academy of Sciences, 

ul. W. E. Radzikowskiego 152, PL-31342 Kraków, Poland


Institute of Nuclear Physics, Polish Academy of Sciences, 

ul. W. E. Radzikowskiego 152, PL-31342 Kraków, Poland


Institute of Nuclear Physics, Polish Academy of Sciences, 

ul. W. E. Radzikowskiego 152, PL-31342 Kraków, Poland


Institute of Nuclear Physics, Polish Academy of Sciences, 

ul. W. E. Radzikowskiego 152, PL-31342 Kraków, Poland


Institute of Nuclear Physics, Polish Academy of Sciences, 

ul. W. E. Radzikowskiego 152, PL-31342 Kraków, Poland


Institute of Low Temperature and Structure Research, Polish Academy of Sciences, ul. Okólna 2, PL-50950 Wrocław, Poland



The theoretical studies on the electronic and lattice properties of the series of non-centrosymmetric superconductors ThTSi, where T = Co, Ni, Ir, and Pt are presented. The electronic band structure and crystal parameters were optimized within the density functional theory. 
The spin-orbit coupling leads to the splitting of the electronic bands and Fermi surfaces, with the stronger effect observed for the compounds with the heavier atoms Ir and Pt. 
The possible mixing of the spin-singlet and spin-triplet pairing in the superconducting state is discussed. 
The phonon dispersion relations and phonon density of states were obtained using the direct method. 
The dispersion curves in ThCoSi and ThIrSi exhibit the low-energy modes along the S-N-S_0 line with the tendency for softening and dynamic instability. 
Additionally, we calculate and analyse the contributions of phonon modes to lattice heat capacity. 



Electronic and lattice properties of non-centrosymmetric superconductors ThTSi 
 (T = Co, Ir, Ni, and Pt)
    D. Kaczorowski
    December 30, 2023
=========================================================================================================






§ INTRODUCTION



Unconventional superconductors, which exhibit an anisotropic pairing
and the presence of nodes in the superconducting gap, have been extensively studied over the last decades <cit.>. 
A departure from the standard BCS theory may result from  
the absence of an inversion center in the crystal structure, which gives rise to an antisymmetric spin-orbit coupling (SOC) <cit.>.
If the SOC is sufficiently large, it leads to a mixture of spin-singlet and spin-triplet components in the superconducting state <cit.>.
Since the discovery of the first heavy fermion compound CePt_3Si <cit.>,
a great number of non-centrosymmetric superconductors were found and investigated <cit.>.


The compounds from the ThTSi family (where T = Co, Ir, Ni, and Pt) crystallize with a non-centrosymmetric tetragonal structure of the LaPtSi-type (space group I4_1md, No. 109) <cit.> (see Fig. <ref>).
Below T_c=2.95 K, ThCoSi exhibits type-II moderate-coupling superconductivity with substantial Pauli pair-breaking effect <cit.>.
Low-temperature measurements of the zero-field specific heat shows a proportionality C ∼ T^2, instead of an
exponential temperature dependence predicted by the BCS theory. 
This finding suggests the existence of line nodes in the superconducting gap.
The experimental characterization of ThNiSi indicates weak-coupling type-II superconductivity below T_c=0.84 K with somewhat
abnormal temperature dependence of the upper critical field <cit.>.
Contrary to ThCoSi, the superconductivity in ThNiSi
is governed by orbital pair-breaking mechanism and can be well
described within the Ginzburg-Landau theory.
These two examples clearly show a crucial role of the transition metal atom T in determining the electronic properties of the ThTSi compounds. 








In this paper, we present results of our density functional theory (DFT) studies of the structural, electronic, and phonon properties on the ThTSi superconductors. 
We analysed the effect of the SOC on electronic bands and Fermi surfaces.
Phonon dispersion relations and phonon density of states were obtained by means of the direct method. 
The tendency for unstable soft-mode behavior was found for ThCoSi and ThIrSi. 
The lattice heat capacity is calculated within the harmonic approximation and compared with the experimental data.
The contribution of atomic components and the wagging modes to the heat capacity is analysed.


The paper is organized as follows. 
The details of ab initio methods are presented in Sec. <ref>.
The structural properties are studied in Sec. <ref>.
The band structure, Fermi surfaces, and density of states are presented and analysed in Sec. <ref>. 
In Sec. <ref>, the phonon dispersion curves and phonon density of states are discussed. 
Lattice heat capacity is presented in Sec. <ref>. 
The results are summarized in Sec. <ref>.










§ AB INITIO CALCULATIONS






 §.§ Calculation details




The first-principles calculations were performed using the projector augmented-wave (PAW) potentials <cit.> and generalized gradient approximation (GGA) in the Perdew, Burke and Ernzerhof (PBE) parametrization  <cit.>, as implemented in the VASP code <cit.>.
First, the crystal structure was optimized in the conventional cell, containing two primitive unit cells (Fig. <ref>). 
The calculations were performed with and without the spin-orbit coupling.
For the summation over the reciprocal space we used 15×15×6 k-point grid.
The energy cut-off for plane waves expansion was set to 350 eV.

Phonon calculations were performed in the supercell with 48 atoms containing  2×2×1 conventional cells.
The phonon dispersion curves and phonon density of states (DOS) were calculated within the direct method <cit.> implemented in the Phonon software <cit.>.
In this approach, atoms are displaced from their equilibrium positions and Hellmann-Feynman forces acting on all atoms in the supercell are calculated.
Then, the force-constant matrix elements and the dynamical matrix are constructed.
We used the displacements generated according to the thermal distribution or, 
in the case of ThIrSi, using atomic positions obtained from the molecular dynamics simulations at T=100 K <cit.>. 
The phonon contribution to the heat capacity was calculated within the harmonic approximation.




 §.§ Crystal structures




All the studied ThTSi compounds were modeled by imposing the symmetry restrictions of the I4_ 1md space group (No. 109) on the crystal structure.
The conventional crystallographic cell contains two primitive unit cells as it is shown in Fig. <ref>. 
In the body-centered tetragonal structure three non-equivalent atoms: Si, T and Th are placed at the crystallographic sites: 2a(0, 0, 0), 2a(0, 0, z_T) and 2a(0, 0, z_Th), respectively.
All atoms are located in the lattice positions with the same site symmetry (C_2v,  2mm), however, due to different mutual arrangement, thorium atoms have special coordination, other than Si and T atoms.


The lattice parameters and atomic positions obtained with and without inclusion of SOC are compared with the experimental values in Table <ref>.
In all cases, the SOC inclusion results in decreasing of the lattice parameters. 
Generally, in comparison to the experimental data, our calculations overestimate a and underestimate c lattice constants. 
For example, for ThCoSi, a is slightly longer than the experimental value a_exp=4.081 Å, while the value of c is smaller compared to c_exp=14.003 Å <cit.>. 
Similar trend was found for ThNiSi, where the experimental values read a_exp=4.070 Å and c_exp=14.090 Å, respectively <cit.>.
Due to the heavier atoms, the lattice constants in ThIrSi and ThPtSi are significantly larger but they show similar relations between the experimental and theoretical values.
It is also worth noting that in spite of substantial differences in masses of the T atoms in the considered systems, fractional coordinates z_T, unconstrained by any symmetry elements, are in all these phases nearly the same.




 §.§ Electronic properties




The Brillouin zone of the I4_1md structure together with the high symmetry points and the path along which the electronic band structure was calculated are presented in Fig. <ref>. 
All calculations were performed both in the absence and in the presence of the spin-orbit interaction. 
A substantial spin-orbit splitting of the band structure is clearly visible in Fig. <ref>.
The degeneracy along the high-symmetry direction Γ–Z results from the tetragonal symmetry. 
However, SOC removes the degeneracy of some energy levels at the high-symmetry points and between them. 
For example, one clearly observes this effect along the Z–Y_1 direction, where the absence of the spin-orbit interaction results in a quadruple degeneracy of electronic states (dashed red lines). 
Turning on the spin-orbit coupling reduces the degeneracy, which is twofold in this case (solid green lines).
From the observed splitting, the value of the spin-orbit coupling for a chosen momentum can be evaluated.  
These compounds are characterized by the relatively large value of the spin-orbit coupling (around 0.5 eV) in relation to the other compounds of this class <cit.>.
The existence of such a strong spin-orbit coupling can induce effectively a mixing of the singlet and triplet superconducting pairing <cit.>.
However, a spin-singlet component of pairing still dominates in these compounds <cit.>.
Moreover, for each compound at the point Y_1 (which connects the Z–Y_1 and Y_1–P directions), one can notice some splitting of each band.
Due to the crystal symmetry, the degeneracy along the Y_1–P line is twice lower than along the Z–Y_1 line.











The Fermi surfaces (FSs) of the studied compounds obtained in the absence of the spin-orbit coupling are shown in Fig. <ref>. 
Remarkably, FSs of ThCoSi  are very similar to those of ThIrSi, while FSs of ThNiSi are almost the same as those of ThPtSi. 
The pairs of atoms, Co and Ir as well as Ni and Pt, belong to the same groups of the periodic table of elements.
The main difference in FSs caused by an exchange of Co into Ir or Ni into Pt is associated with an emergence of new small Fermi pockets around the Γ or X point, respectively, i.e., the occurrence of Lifshitz transition <cit.>.


An impact of the spin-orbit coupling on the Fermi surface is shown in Fig. <ref>.
Red and green lines correspond to the cross-sections of the Fermi surfaces by chosen planes in the case of the absence and the presence of the spin-orbit coupling, respectively.
The relatively small splitting of the profile is observed for compounds containing Co and Ni. 
However, this effect seems to be slightly larger for ThNiSi than for ThCoSi, especially, within the plane marked in Fig. <ref> by an orange rectangle. 
This is obviously in accordance with the results presented in Fig. <ref>, which demonstrate that the magnitude of the spin-orbit splitting at the Fermi level depends strongly on a direction in the Brillouin zone. 
For systems that contain a heavy T atom, (i.e. Ir and Pt), the differences between the Fermi surfaces obtained in the absence and in the presence of the spin-orbit interaction are much more pronounced.


The total and orbital-projected electron densities of states (DOS) are presented in Fig. <ref>. 
A common feature of the investigated compounds is a principal role of the d-type orbitals.
In the case of ThCoSi and ThNiSi, the d-states dominate at energies from -2 to -1 eV and from -2.5 to -1.5 eV, respectively, while in the case of ThIrSi and ThPtSi, they are mainly situated below -2 eV.
In all the compounds, the 5f-orbital electron states are concentrated far above the Fermi level (approximately above 2 eV)
The spin-orbit coupling included in the calculations only slightly changes the density of electron states
and both values of DOS at the Fermi level are almost equal.
The effect can be inferred by comparing gray dashed lines and gray areas, which correspond to total DOS in the absence and in the presence of the spin-orbit coupling, respectively.




 §.§ Lattice dynamics




In Fig. <ref>, we compare the phonon dispersion relations and phonon DOS calculated without the spin-orbit coupling for all four compounds.
The obtained phonon frequencies are real showing dynamical stability of the crystals.
However, the lowest energy phonon branch reaches a very low energy level along the S-N-S_0 line in ThCoSi and ThIrSi.
In ThIrSi, these modes have a particularly strong tendency for softening. 


The total and partial atom-projected phonon DOS are presented in the side panels in Fig. <ref>.
The phonon DOS of all compounds contains at least one energy gap separating upper Si-dominated band. Additionally, in Co, Ir and Pt compounds the lower Si-dominated band is also separated from the rest of the spectrum by a smaller, 2-5 meV gap. The top band in all compounds, except for ThPtSi, is further splited by another small gap of approximately 1 meV.











In ThCoSi, vibrations of the Th atoms dominate at energies below 13 meV. 
The Co atoms vibrate with energies mainly between 12–25 meV and have small contribution to the highest optical Si modes above 39 meV. 
In ThNiSi, the high energy phonon branches 
are slightly shifted to lower energies in comparison with ThCoSi. 
The higher gap is increased to ∼10 meV, while the two lower energy gaps are reduced to about 1 meV.
The Ni atoms vibrate mainly with energies between 13–25 meV, while the Si states dominate above 16 meV.
Interestingly, in the lowest energy range (up to 12 meV), the ThNiSi spectrum consists solely of Th vibrations, which completely separate themselves from other states. This is in  contrast to ThCoSi, where the contribution of Co is evident in this part of DOS although the masses of Co and Ni atoms are comparable. The significantly weaker binding of Th atoms in ThNiSi compound (or weaker Ni-Th interatomic forces) can explain this effect. 


The situation is quite different for ThIrSi and ThPtSi where due to much larger masses of Ir and Pt, vibrations of all atoms except Si have similar frequencies and they mix together in the lower part of the phonon spectra (bottom row of Fig. <ref>). 
The Th and Pt atoms contribute mainly to the lowest energy range below 15 meV.
There is a gap (∼3 meV) which separates these states from the Si-derived bands around 20 meV.
The higher energy gap increases to about 15 meV in ThIrSi and 18 meV in ThPtSi. The highest modes above 40 meV correspond mainly to the Si atom vibrations. 


The common feature of all these phonon spectra is separation of flat, dispersionless, mostly Si-derived bands located above 40 meV. Their frequencies weakly depend on the masses of other constituent atoms. 
They are also very similar to those previously observed in the phonon spectrum of LaPtSi <cit.>. 


The lower vibration bands observed close to 30 meV for T = Co or Ni and 20 meV for T = Ir and Pt are related to the Si atoms that built the characteristic T–Si–T chains connecting Si atom with two nearest neighboring T atoms (Fig. <ref>).  The T–Si–T bond angles slightly deviate from 120. Two sets of perpendicular T–Si–T zig-zag chains form the three-dimensional T–Si framework of the ThTSi crystal structure.  The interchain T-Si distances are close enough to the distances between chains. Thorium atoms are placed in the existing T–Si cages. The described mode comprise the silicon atoms vibrating out of the T–Si–T planes while the T atoms remain almost stationary. The atomic movements resemble the wagging-like modes of the Si atoms.
Similar wagging modes occur when T atoms are moving and other atoms stay almost motionless. The energies of T wagging modes, apparently lower because of larger masses of T, are the lowest energies in the partial T-derived DOS. In the case of ThCoSi and ThIrSi, they are the lowest energy modes along the S-N-S_0 line. A scheme of atomic displacements in the wagging mode at the N point obtained for ThIrSi is presented in Fig. <ref>.





 §.§ Heat capacity



In Fig. <ref>, the total heat capacity measured for all compounds at constant pressure (C_p) is compared with the phonon contribution calculated at constant volume (C_V).   
Experimental data were obtained by using polycrystalline samples, as it was described in Refs. <cit.>.
Below 100 K, a reasonable agreement was found for ThCoSi and ThNiSi. The largest discrepancy between the experimental and computed data is observed for ThIrSi, which is connected with the low energy modes around the S, N, and S_0 points.
At higher temperatures, the difference between C_p and C_V increases according to the formula C_p-C_V = T α^2 V/K, where α and K are the thermal expansion coefficient and isothermal bulk modulus, respectively.
The bifurcation between the experimental and theoretical curves, clearly observed for each compound above about 100 K, can be attributed to this effect.


The calculated individual atom contributions (determined per degree of freedom) to the total phonon heat capacity C_V of the ThTSi phases are also presented in Fig. <ref>.
For T atoms, three separate components: the projection on a or b  direction due to wagging (T wagging) and non-wagging modes (T not-wagging) and c direction (T z) were considered.
Vibrations of the Th atoms were found very stable with well defined contribution to the heat capacity, generally independent of the T atom type. 
This finding can be directly associated with the quite symmetric coordination polyhedrons of these atoms.
On the contrary, the Si atoms yield very small contribution to C_V. This is a consequence
of their relatively low mass, which implies high-energetic phonon spectra, well separated from other modes. 
The situation is completely different for the T atoms, where partial C_V component connected with wagging modes is visibly detached from the other ones. The contributions of the Ir and Pt atoms are even larger than those originating from the much heavier Th atoms. 




§ SUMMARY



To summarize, we have presented the results of theoretical studies on the series of non-centrosymmetric superconductors ThTSi, where T = Co, Ni, Ir, and Pt. 
Using the DFT, we optimized the crystal parameters and compared them with the experimental values. For the relaxed systems, the electronic bands and Fermi surfaces
were calculated.  Similar characteristic features of the electronic structures in the compounds, which contain the T atoms from the same groups of the Periodic Table, were found.
The SOC removes the degeneracy of the electronic states and splits the spin-up and spin-down states. The stronger impact of the SOC is found for the compounds with the heavier atoms Ir and Pt.


For each ThTSi compound, we found that the Th atoms are located in quite symmetric cages built of the T and Si atoms. Their surroundings can be modified not only by changing the T atoms but also by proper doping which can tune dynamical properties of the crystal.
The calculated phonon dispersion curves and density of states indicate the dynamical stability
of all the studied materials, however, the tendency for phonon softening along the S-N-S_0 line in the Brillouin zone is found in ThCoSi and ThIrSi. In all investigated compounds strong separation of low-dispersion Si vibration bands have been found, which can be attributed to large mass contrast and particular arrangement of Si atoms separated and surrounded by much heavier atoms.
The lattice heat capacity was obtained and compared with the experimental data.
The analysis of the partial C_V of the Th, T, and Si atoms revealed a large contribution of the wagging modes involving the T atoms, especially the heavy atoms Ir and Pt, to the heat capacity.



We thank Krzysztof Parlinski for valuable comments and discussions.
This work was supported by the National Science Centre (NCN, Poland) under grants UMO-2017/25/B/ST3/02586 (A.P., J.Ł., P.T.J., M.S. and P.P.), and UMO-2017/24/C/ST3/00276 (K.J.K.).







