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
EElectronic and lattice properties of non-centrosymmetric superconductors ThTSi(T = Co, Ir, Ni, and Pt)
A. Ptok, ∗ K. Domieracki, K. J. Kapcia, J. (cid:32)La˙zewski, P. T. Jochym, M. Sternik, P. Piekarz, and D. Kaczorowski Institute of Nuclear Physics, Polish Academy of Sciences,ul. W. E. Radzikowskiego 152, PL-31342 Krak´ow, Poland Institute of Low Temperature and Structure Research,Polish Academy of Sciences, ul. Ok´olna 2, PL-50950 Wroc(cid:32)law, Poland
The theoretical studies on the electronic and lattice properties of the series of non-centrosymmetricsuperconductors ThTSi, where T = Co, Ni, Ir, and Pt are presented. The electronic band structureand crystal parameters were optimized within the density functional theory. The spin-orbit couplingleads to the splitting of the electronic bands and Fermi surfaces, with the stronger effect observedfor the compounds with the heavier atoms Ir and Pt. The possible mixing of the spin-singlet andspin-triplet pairing in the superconducting state is discussed. The phonon dispersion relations andphonon density of states were obtained using the direct method. The dispersion curves in ThCoSiand ThIrSi exhibit the low-energy modes along the S-N-S line with the tendency for softening anddynamic instability. Additionally, we calculate and analyse the contributions of phonon modes tolattice heat capacity. I. INTRODUCTION
Unconventional superconductors, which exhibit ananisotropic pairing and the presence of nodes in the su-perconducting gap, have been extensively studied overthe last decades [1]. A departure from the standardBCS theory may result from the absence of an inver-sion center in the crystal structure, which gives rise toan antisymmetric spin-orbit coupling (SOC) [2, 3]. If theSOC is sufficiently large, it leads to a mixture of spin-singlet and spin-triplet components in the superconduct-ing state [4, 5]. Since the discovery of the first heavyfermion compound CePt Si [6], a great number of non-centrosymmetric superconductors were found and inves-tigated [7–18].The compounds from the ThTSi family (where T = Co,Ir, Ni, and Pt) crystallize with a non-centrosymmetrictetragonal structure of the LaPtSi-type (space group I4 md , No. 109) [19–22] (see Fig. 1). Below T c = 2 .
95 K,ThCoSi exhibits type-II moderate-coupling superconduc-tivity with substantial Pauli pair-breaking effect [23].Low-temperature measurements of the zero-field specificheat shows a proportionality C ∼ T , instead of an ex-ponential temperature dependence predicted by the BCStheory. This finding suggests the existence of line nodesin the superconducting gap. The experimental character-ization of ThNiSi indicates weak-coupling type-II super-conductivity below T c = 0 .
84 K with somewhat abnormaltemperature dependence of the upper critical field [24].Contrary to ThCoSi, the superconductivity in ThNiSiis governed by orbital pair-breaking mechanism and canbe well described within the Ginzburg-Landau theory.These two examples clearly show a crucial role of the ∗ e-mail: [email protected] transition metal atom T in determining the electronicproperties of the ThTSi compounds.In this paper, we present results of our density func-tional theory (DFT) studies of the structural, electronic,and phonon properties on the ThTSi superconductors.We analysed the effect of the SOC on electronic bandsand Fermi surfaces. Phonon dispersion relations andphonon density of states were obtained by means of thedirect method. The tendency for unstable soft-mode be-havior was found for ThCoSi and ThIrSi. The lattice heatcapacity is calculated within the harmonic approximationand compared with the experimental data. The contri-bution of atomic components and the wagging modes to FIG. 1. Relation between the conventional standard celland the primitive unit of the ThTSi compounds (marked bythe bold dashed line and the solid line with the transparentfilling, respectively). The image was rendered using
VESTA software [25]. a r X i v : . [ c ond - m a t . m t r l - s c i ] O c t TABLE I. The calculated lattice constants and atomic positions (in fractional coordinates) obtained with (w/) or without(w/o) spin-orbit coupling (SOC) compared with the experimental data. Experimental results for compound with Co and Niare taken from Ref. [23] and [24], respectively, whereas these with Ir and Pt are obtained within the same method.system ThCoSi ThNiSi ThIrSi ThPtSiw/o SOC w/ SOC w/o SOC w/ SOC w/o SOC w/ SOC w/o SOC w/ SOC a (˚A) 4.1060 4.0947 4.1204 4.1153 4.2340 4.2272 4.2570 4.2515 c (˚A) 13.7348 13.6970 13.7830 13.7658 14.1631 14.1405 14.2399 14.2217z Th T a (˚A) 4.081 4.0697 4.141 4.154 c (˚A) 14.003 14.089 14.270 14.582z Th T the heat capacity is analysed.The paper is organized as follows. The details of abinitio methods are presented in Sec. II A. The structuralproperties are studied in Sec. II B. The band structure,Fermi surfaces, and density of states are presented andanalysed in Sec. II C. In Sec. II D, the phonon dispersioncurves and phonon density of states are discussed. Lat-tice heat capacity is presented in Sec. II E. The resultsare summarized in Sec. III. II. AB INITIO CALCULATIONSA. Calculation details
The first-principles calculations were performed us-ing the projector augmented-wave (PAW) potentials [27]and generalized gradient approximation (GGA) in thePerdew, Burke and Ernzerhof (PBE) parametrization[28], as implemented in the VASP code [29–31]. First,the crystal structure was optimized in the conventionalcell, containing two primitive unit cells (Fig. 1). The cal-culations were performed with and without the spin-orbit
FIG. 2. The Brillouin zone of the I md structure with thehigh symmetry points [26]. The red line demonstrates thepath used for the electronic band structure presentation. coupling. For the summation over the reciprocal spacewe used 15 × × k -point grid. The energy cut-off forplane waves expansion was set to 350 eV.Phonon calculations were performed in the supercellwith 48 atoms containing 2 × × T = 100 K [34]. The phonon contributionto the heat capacity was calculated within the harmonicapproximation. B. Crystal structures
All the studied ThTSi compounds were modeled byimposing the symmetry restrictions of the I4 md spacegroup (No. 109) on the crystal structure. The conven-tional crystallographic cell contains two primitive unitcells as it is shown in Fig. 1. In the body-centered tetrag-onal structure three non-equivalent atoms: Si, T andTh are placed at the crystallographic sites: 2 a (0 , , a (0 , , z T ) and 2 a (0 , , z Th ), respectively. All atomsare located in the lattice positions with the same sitesymmetry ( C v , mm ), however, due to different mutualarrangement, thorium atoms have special coordination,other than Si and T atoms.The lattice parameters and atomic positions obtainedwith and without inclusion of SOC are compared withthe experimental values in Table I. In all cases, the SOCinclusion results in decreasing of the lattice parameters.Generally, in comparison to the experimental data, our FIG. 3. The band structure of studied systems (as labeled) obtained in the absence and in the presence of the spin-orbitcoupling (dashed red and solid green lines, respectively). The Fermi level is located at zero energy. calculations overestimate a and underestimate c latticeconstants. For example, for ThCoSi, a is slightly longerthan the experimental value a exp = 4 .
081 ˚A, while thevalue of c is smaller compared to c exp = 14 .
003 ˚A [23].Similar trend was found for ThNiSi, where the experi-mental values read a exp = 4 .
070 ˚A and c exp = 14 .
090 ˚A,respectively [24]. Due to the heavier atoms, the latticeconstants in ThIrSi and ThPtSi are significantly largerbut they show similar relations between the experimen-tal and theoretical values. It is also worth noting that inspite of substantial differences in masses of the T atomsin the considered systems, fractional coordinates z T , un-constrained by any symmetry elements, are in all thesephases nearly the same. C. Electronic properties
The Brillouin zone of the I md structure togetherwith the high symmetry points and the path along whichthe electronic band structure was calculated are pre-sented in Fig. 2. All calculations were performed bothin the absence and in the presence of the spin-orbit in-teraction. A substantial spin-orbit splitting of the bandstructure is clearly visible in Fig. 3. The degeneracyalong the high-symmetry direction Γ–Z results from thetetragonal symmetry. However, SOC removes the degen-eracy of some energy levels at the high-symmetry pointsand between them. For example, one clearly observesthis effect along the Z–Y direction, where the absenceof the spin-orbit interaction results in a quadruple de- generacy of electronic states (dashed red lines). Turn-ing on the spin-orbit coupling reduces the degeneracy,which is twofold in this case (solid green lines). Fromthe observed splitting, the value of the spin-orbit cou-pling for a chosen momentum can be evaluated. Thesecompounds are characterized by the relatively large valueof the spin-orbit coupling (around 0 . (which connects theZ–Y and Y –P directions), one can notice some splittingof each band. Due to the crystal symmetry, the degen-eracy along the Y –P line is twice lower than along theZ–Y line.The Fermi surfaces (FSs) of the studied compoundsobtained in the absence of the spin-orbit coupling areshown in Fig. 4. Remarkably, FSs of ThCoSi are verysimilar to those of ThIrSi, while FSs of ThNiSi are almostthe same as those of ThPtSi. The pairs of atoms, Co andIr as well as Ni and Pt, belong to the same groups ofthe periodic table of elements. The main difference inFSs caused by an exchange of Co into Ir or Ni into Pt isassociated with an emergence of new small Fermi pocketsaround the Γ or X point, respectively, i.e., the occurrenceof Lifshitz transition [37].An impact of the spin-orbit coupling on the Fermi sur-face is shown in Fig. 5. Red and green lines correspond tothe cross-sections of the Fermi surfaces by chosen planes FIG. 4. The Fermi surfaces of studied systems (as labeled)obtained in the absence of the spin-orbit coupling. in the case of the absence and the presence of the spin-orbit coupling, respectively. The relatively small splittingof the profile is observed for compounds containing Coand Ni. However, this effect seems to be slightly largerfor ThNiSi than for ThCoSi, especially, within the planemarked in Fig. 5 by an orange rectangle. This is obvi-ously in accordance with the results presented in Fig. 3,which demonstrate that the magnitude of the spin-orbitsplitting at the Fermi level depends strongly on a di-rection in the Brillouin zone. For systems that containa heavy T atom, (i.e. Ir and Pt), the differences be-tween the Fermi surfaces obtained in the absence and inthe presence of the spin-orbit interaction are much morepronounced.The total and orbital-projected electron densities ofstates (DOS) are presented in Fig. 6. A common featureof 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 − − − . − . − f -orbital electron states are con-centrated far above the Fermi level (approximately above2 eV) The spin-orbit coupling included in the calculationsonly slightly changes the density of electron states andboth values of DOS at the Fermi level are almost equal.The effect can be inferred by comparing gray dashed linesand gray areas, which correspond to total DOS in theabsence and in the presence of the spin-orbit coupling,respectively. FIG. 5. Cross-sections of the Fermi surfaces of studied sys-tems (as labeled) in two chosen planes shown in the top panel(cf. with Fig. 2). Results obtained in the absence and inthe presence of the spin-orbit coupling (red and green lines,respectively).
FIG. 6. The density of states (DOS) with projections on the orbitals (as indicated in labels) for the studied systems (aslabeled). The Fermi level is located at zero energy.
D. Lattice dynamics
In Fig. 7, we compare the phonon dispersion relationsand phonon DOS calculated without the spin-orbit cou-pling for all four compounds. The obtained phonon fre-quencies are real showing dynamical stability of the crys-tals. However, the lowest energy phonon branch reachesa very low energy level along the S-N-S line in ThCoSiand ThIrSi. In ThIrSi, these modes have a particularlystrong tendency for softening.The total and partial atom-projected phonon DOS arepresented in the side panels in Fig. 7. The phonon DOSof all compounds contains at least one energy gap sepa-rating upper Si-dominated band. Additionally, in Co, Irand Pt compounds the lower Si-dominated band is alsoseparated from the rest of the spectrum by a smaller,2-5 meV gap. The top band in all compounds, exceptfor ThPtSi, is further splited by another small gap ofapproximately 1 meV.In ThCoSi, vibrations of the Th atoms dominate at en-ergies below 13 meV. The Co atoms vibrate with energiesmainly between 12–25 meV and have small contributionto the highest optical Si modes above 39 meV. In ThNiSi,the high energy phonon branches are slightly shifted tolower energies in comparison with ThCoSi. The highergap is increased to ∼
10 meV, while the two lower energygaps are reduced to about 1 meV. The Ni atoms vibratemainly with energies between 13–25 meV, while the Sistates dominate above 16 meV. Interestingly, in the low-est energy range (up to 12 meV), the ThNiSi spectrum consists solely of Th vibrations, which completely sepa-rate themselves from other states. This is in contrast toThCoSi, where the contribution of Co is evident in thispart of DOS although the masses of Co and Ni atomsare comparable. The significantly weaker binding of Thatoms in ThNiSi compound (or weaker Ni-Th interatomicforces) can explain this effect.The situation is quite different for ThIrSi and ThPtSiwhere due to much larger masses of Ir and Pt, vibrationsof all atoms except Si have similar frequencies and theymix together in the lower part of the phonon spectra(bottom row of Fig. 7). The Th and Pt atoms contributemainly to the lowest energy range below 15 meV. Thereis a gap ( ∼ ◦ . Two sets of perpendicular T–Si–T zig-zag FIG. 7. Phonon dispersion curves and density of states of studied systems (as labeled). chains form the three-dimensional T–Si framework of theThTSi crystal structure. The interchain T-Si distancesare close enough to the distances between chains. Tho-rium atoms are placed in the existing T–Si cages. The de-scribed mode comprise the silicon atoms vibrating out of ac b
FIG. 8. An example of atomic displacements in the waggingmode with the lowest energy at the N point in ThIrSi. Thesolid lines represent schematically the three-dimensional Si-Tframework based on the perpendicular T-Si-T zigzag chains.The image was rendered using
VESTA software [25]. the T–Si–T planes while the T atoms remain almost sta-tionary. The atomic movements resemble the wagging-like modes of the Si atoms. Similar wagging modes occurwhen T atoms are moving and other atoms stay almostmotionless. The energies of T wagging modes, apparentlylower because of larger masses of T, are the lowest ener-gies in the partial T-derived DOS. In the case of ThCoSiand ThIrSi, they are the lowest energy modes along theS-N-S line. A scheme of atomic displacements in thewagging mode at the N point obtained for ThIrSi is pre-sented in Fig. 8. E. Heat capacity
In Fig. 9, the total heat capacity measured for allcompounds at constant pressure ( C p ) is compared withthe phonon contribution calculated at constant volume( C V ). Experimental data were obtained by using poly-crystalline samples, as it was described in Refs. [23, 24].Below 100 K, a reasonable agreement was found forThCoSi and ThNiSi. The largest discrepancy betweenthe experimental and computed data is observed forThIrSi, which is connected with the low energy modesaround the S, N, and S points. At higher temperatures,the difference between C p and C V increases according tothe formula C p − C V = T α V /K , where α and K are thethermal expansion coefficient and isothermal bulk mod-ulus, respectively. The bifurcation between the experi-mental and theoretical curves, clearly observed for each T (K) C V ( J / K m o l / d . o . f . ) ThCoSi
SiCo waggingCo not-waggingCo z Thaverageexperiment 0 50 100 150 200 250 300 T (K) C V ( J / K m o l / d . o . f . ) ThNiSi
SiNi waggingNi not-waggingNi z Thaverageexperiment0 50 100 150 200 250 300 T (K) C V ( J / K m o l / d . o . f . ) ThIrSi
SiIr waggingIr not-waggingIr z Thaverageexperiment 0 50 100 150 200 250 300 T (K) C V ( J / K m o l / d . o . f . ) ThPtSi
SiPt waggingPt not-waggingPt z Thaverageexperiment
FIG. 9. The theoretical phonon heat capacity C V (black dashed lines), averaged over all degrees of freedom (d.o.f.), comparedwith the measured total heat capacity C p (red open squares) of studied systems. Additionally, calculated partial C V per degreeof freedom are presented: averaged for Th (turquoise dotted) and Si (magenta dash-dotted) atoms as well as all componentsfor T atoms (green, violet, and orange solid lines correspond to xy wagging type, others xy , and z contributions, respectively). compound above about 100 K, can be attributed to thiseffect.The calculated individual atom contributions (deter-mined per degree of freedom) to the total phonon heatcapacity C V of the ThTSi phases are also presented inFig. 9. For T atoms, three separate components: the pro-jection 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 werefound very stable with well defined contribution to theheat capacity, generally independent of the T atom type.This finding can be directly associated with the quitesymmetric coordination polyhedrons of these atoms. Onthe contrary, the Si atoms yield very small contributionto C V . This is a consequence of their relatively low mass,which implies high-energetic phonon spectra, well sepa-rated from other modes. The situation is completely dif-ferent for the T atoms, where partial C V component con-nected with wagging modes is visibly detached from theother ones. The contributions of the Ir and Pt atoms areeven larger than those originating from the much heavierTh atoms. III. SUMMARY
To summarize, we have presented the results of theo-retical studies on the series of non-centrosymmetric su-perconductors ThTSi, where T = Co, Ni, Ir, and Pt. Us-ing the DFT, we optimized the crystal parameters andcompared them with the experimental values. For therelaxed systems, the electronic bands and Fermi surfaceswere calculated. Similar characteristic features of theelectronic structures in the compounds, which containthe T atoms from the same groups of the Periodic Ta-ble, were found. The SOC removes the degeneracy of theelectronic states and splits the spin-up and spin-downstates. The stronger impact of the SOC is found for thecompounds with the heavier atoms Ir and Pt.For each ThTSi compound, we found that the Thatoms are located in quite symmetric cages built of theT and Si atoms. Their surroundings can be modified notonly by changing the T atoms but also by proper dopingwhich can tune dynamical properties of the crystal. Thecalculated phonon dispersion curves and density of statesindicate the dynamical stability of all the studied mate-rials, however, the tendency for phonon softening alongthe S-N-S line in the Brillouin zone is found in ThCoSiand ThIrSi. In all investigated compounds strong sep-aration of low-dispersion Si vibration bands have beenfound, which can be attributed to large mass contrastand particular arrangement of Si atoms separated andsurrounded by much heavier atoms. The lattice heat ca-pacity was obtained and compared with the experimentaldata. The analysis of the partial C V of the Th, T, and Siatoms revealed a large contribution of the wagging modesinvolving the T atoms, especially the heavy atoms Ir andPt, to the heat capacity. ACKNOWLEDGMENTS
We thank Krzysztof Parlinski for valuable commentsand discussions. This work was supported by the Na-tional Science Centre (NCN, Poland) under grants UMO-2017/25/B/ST3/02586 (A.P., J.(cid:32)L., P.T.J., M.S. andP.P.), and UMO-2017/24/C/ST3/00276 (K.J.K.). [1] M. Sigrist and K. Ueda, “Phenomenological theory ofunconventional superconductivity,” Rev. Mod. Phys. ,239 (1991).[2] E. Bauer and M. Sigrist, Non-centrosymmetric supercon-ductors: introduction and overview , Vol. 847 (SpringerScience & Business Media, 2012).[3] S. Yip, “Noncentrosymmetric superconductors,” Annu.Rev. Condens. Matter Phys. , 15 (2014).[4] L. P. Gor’kov and E. I. Rashba, “Superconducting 2Dsystem with lifted spin degeneracy: Mixed singlet-tripletstate,” Phys. Rev. Lett. , 037004 (2001).[5] M. Smidman, M. B. Salamon, H. Q. Yuan, and D. F.Agterberg, “Superconductivity and spin–orbit couplingin non-centrosymmetric materials: a review,” Rep. Prog.Phys. , 036501 (2017).[6] E. Bauer, G. Hilscher, H. Michor, Ch. Paul, E. W.Scheidt, A. Gribanov, Yu. Seropegin, H. No¨el, M. Sigrist,and P. Rogl, “Heavy fermion superconductivity and mag-netic order in noncentrosymmetric CePt Si,” Phys. Rev.Lett. , 027003 (2004).[7] T. Akazawa, H. Hidaka, T. Fujiwara, T. C. Kobayashi,E. Yamamoto, Y. Haga, R. Settai, and Y. ¯Onuki,“Pressure-induced superconductivity in ferromagneticUIr without inversion symmetry,” J. Phys.: Condens.Matter , L29 (2004).[8] K. Togano, P. Badica, Y. Nakamori, S. Orimo, H. Takeya,and K. Hirata, “Superconductivity in the metal richLi-Pd-B ternary boride,” Phys. Rev. Lett. , 247004(2004).[9] T. Klimczuk, F. Ronning, V. Sidorov, R. J. Cava, andJ. D. Thompson, “Physical properties of the noncen-trosymmetric superconductor Mg Ir B ,” Phys. Rev.Lett. , 257004 (2007).[10] E. Bauer, R. T. Khan, H. Michor, E. Royanian, A. Gryt-siv, N. Melnychenko-Koblyuk, P. Rogl, D. Reith, R. Pod-loucky, E.-W. Scheidt, W. Wolf, and M. Marsman,“BaPtSi : A noncentrosymmetric BCS-like superconduc-tor,” Phys. Rev. B , 064504 (2009).[11] I. Bonalde, R. L. Ribeiro, K. J. Syu, H. H. Sung,and W. H. Lee, “Nodal gap structure in the noncen-trosymmetric superconductor LaNiC from magnetic-penetration-depth measurements,” New J. Phys. ,123022 (2011).[12] Y. Nishikubo, K. Kudo, and M. Nohara, “Supercon-ductivity in the honeycomb-lattice pnictide SrPtAs,” J.Phys. Soc. Jpn. , 055002 (2011).[13] R. P. Singh, A. D. Hillier, B. Mazidian, J. Quintanilla,J. F. Annett, D. McK. Paul Paul, G. Balakrishnan, andM. R. Lees, “Detection of time-reversal symmetry break-ing in the noncentrosymmetric superconductor Re Zrusing muon-spin spectroscopy,” Phys. Rev. Lett. , 107002 (2014).[14] Z. Sun, M. Enayat, A. Maldonado, C. Lithgow, E. Yel-land, D. C. Peets, A. Yaresko, A. P. Schnyder, andP. Wahl, “Dirac surface states and nature of supercon-ductivity in noncentrosymmetric BiPd,” Nat. Commun. , 6633 (2015).[15] J. A. T. Barker, D. Singh, A. Thamizhavel, A. D. Hillier,M. R. Lees, G. Balakrishnan, D. McK. Paul, and R. P.Singh, “Unconventional superconductivity in La Ir re-vealed by muon spin relaxation: Introducing a new familyof noncentrosymmetric superconductor that breaks time-reversal symmetry,” Phys. Rev. Lett. , 267001 (2015).[16] M. Sakano, K. Okawa, M. Kanou, H. Sanjo, T. Okuda,T. Sasagawa, and K. Ishizaka, “Topologically protectedsurface states in a centrosymmetric superconductor β -PdBi ,” Nat. Commun. , 8595 (2015).[17] B. Li, C. Q. Xu, W. Zhou, W. H. Jiao, R. Sankar, F. M.Zhang, H. H. Hou, X. F. Jiang, B. Qian, B. Chen, A. F.Bangura, and X. Xu, “Evidence of s-wave superconduc-tivity in the noncentrosymmetric La Ir ,” Sci. Rep. ,651 (2018).[18] E. M. Carnicom, W. Xie, T. Klimczuk, J. Lin,K. G´ornicka, Z. Sobczak, N. P. Ong, and R. J. Cava,“TaRh B and NbRh B : Superconductors with a chi-ral noncentrosymmetric crystal structure,” Sci. Adv. ,eaar7969 (2018).[19] K. Klepp and E. Parth´e, “ R PtSi phases ( R = La, Ce,Pr, Nd, Sm and Gd) with an ordered ThSi derivativestructure,” Acta Crystallogr. B , 1105 (1982).[20] W. X. Zhong, W. L. Ng, B. Chevalier, J. Etourneau, andP. Hagenmuller, “Structural and electrical properties ofnew silicides: ThCoxSi − x (0 ≤ x ≤
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