Electronic topological transition in LaSn 3 under pressure
Swetarekha Ram, V. Kanchana, G. Vaitheeswaran, A. Svane, S. B. Dugdale, N. E. Christensen
aa r X i v : . [ c ond - m a t . m t r l - s c i ] N ov Electronic topological transition in LaSn under pressure Swetarekha Ram , V. Kanchana , ∗ , G. Vaitheeswaran ,A. Svane , S. B. Dugdale and N. E. Christensen Department of Physics,Indian Institute of Technology Hyderabad,Ordnance Factory Estate,Yeddumailaram, Hyderabad-502 205,Andhra Pradesh, India. Advanced Centre of Research in High Energy Materials (ACRHEM),University of Hyderabad,Prof. C. R. Rao Road,Gachibowli, Hyderabad- 500 046,Andhra Pradesh, India. Department of Physics and Astronomy,Aarhus University,DK-8000 Aarhus C, Denmark. H. H. Wills Physics Laboratory,University of Bristol, Tyndall Avenue,Bristol BS8 1TL, United Kingdom. (Dated: January 30, 2018) bstract The electronic structure, Fermi surface and elastic properties of the iso-structural and iso-electronic LaSn and YSn intermetallic compounds are studied under pressure within the frame-work of density functional theory including spin-orbit coupling. The LaSn Fermi surface consistsof two sheets, of which the second is very complex. Under pressure a third sheet appears aroundcompression
V /V = 0 .
94, while a small topology change in the second sheet is seen at compression
V /V = 0 .
90. This may be in accordance with the anomalous behaviour in the superconductingtransition temperature observed in LaSn , which has been suggested to reflect a Fermi surfacetopological transition, along with a non-monotonic pressure dependence of the density of states atthe Fermi level. The same behavior is not observed in YSn , the Fermi surface of which alreadyincludes three sheets at ambient conditions, and the topology remains unchanged under pressure.The reason for the difference in behaviour between LaSn and YSn is the role of spin-orbit cou-pling and the hybridization of La - 4 f states with the Sn - p states in the vicinity of the Fermilevel, which is well explained using the band structure calculation. The elastic constants and re-lated mechanical properties are calculated at ambient as well as at elevated pressures. The elasticconstants increase with pressure for both compounds and satisfy the conditions for mechanicalstability under pressure. PACS numbers: 63.20.dk, 71.20.-b, 74.25.Jb, 74.25.Ld, 71.18.+y . INTRODUCTION The RX -type ( R=rare earth elements, X= In, Sn, Tl, Pb) intermetallic compounds,which crystallize in the simple cubic Cu Au structure, have been the subject of many ex-perimental and theoretical investigations because of their diverse properties. Many of thesecompounds are superconductors. LaSn and YSn are particularly significant as they arefound to have relatively high superconducting transition temperatures, T c . For LaSn ,T c = 6 . and for YSn T c = 7 . whereas LaPb , LaTl and LaIn have lower T c ’sof 4.05 K, 1.51 K and 0.71 K, respectively. Some of the RX compounds, such as PrSn andNdSn , are found to order antiferromagnetically at T N = 8.6 K and 4.5 K, respectively, andCeSn has been categorized as a dense Kondo compound exhibiting valence fluctuations. It is interesting to compare the properties of LaSn and YSn to reveal to which extent thesimilar valence electron configurations of Y and La influence the details of the electronicstructure. The LaX ( X= Sn, In, Tl, Pb) series and their alloys show an oscillatorydependence in their bulk properties (superconducting transition temperature, magnetic sus-ceptibility, thermoelectric power factor) as a function of average valence-electron number. At first, this behavior was interpreted in a nearly-free electron model as a reflection of theFermi surface crossing the Brillouin zone close to the X-point, a viewpoint later contestedby Grobman. The pressure dependence of the critical temperature of LaSn is anomalous,as shown by Huang et al., which these authors expect to be driven by a Fermi surfacetopology change.In the present study, we calculate the Fermi surface of LaSn and indeed observe achange in topology under pressure, where a third set of electron pockets appear and a minorpart of the complex second sheet transfers from a closed orbit to an open orbit region. Asimilar transition is not predicted for YSn , despite the overall similarity of their electronicstructures. A number of studies are available on the band structures of LaSn and YSn , while less efforts have been devoted to the pressure dependence of the electronic structure,Fermi surface and elastic properties for these compounds. Hence we focus our attentionin this paper on analysing the pressure induced Fermi surface topology change in LaSn ,which might be associated with the anomalous behaviour of T c under pressure, and presenta comparative study of LaSn and YSn under pressure. The remainder of the paper isorganized as follows: Section II describes the method of calculation used in this study. The3esults and discussions are presented in Section III, while section IV concludes the paper. II. METHOD OF CALCULATION
The calculations were performed using the full-potential linear augmented plane wave(FP-LAPW) method as implemented in the WIEN2K computer code, based on densityfunctional theory (DFT), which has been shown to yield reliable results for the electronicand structural properties of crystalline solids. Spin-orbit coupling (SOC) was included. Forthe exchange-correlation functional, both the local density approximation (LDA) as proposedby Ceperley and Alder and generalized gradient approximation (GGA) according to thePerdew-Burke-Ernzerhof parametrization was used. In order to achieve energy eigenvalueconvergence, the wave functions in the interstitial region were expanded using plane waveswith a cutoff of R MT K max = 9, where K max is the plane wave cut-off, and R MT is the smallestof all atomic sphere radii. The charge density was Fourier expanded up to G max =18 (a.u.) − .The maximum ℓ value for the wave function expansion inside the atomic spheres was confinedto ℓ max = 10. Convergence tests were carried out using higher G max and R MT K max values,giving no significant changes in the calculated properties. The muffin-tin radii were chosenas 2.75 a.u. for both La and Y and 2.83 a.u. for Sn. A ( 32 × ×
32 ) Monkhorst-pack k -point mesh was used resulting in 396 k -points in the irreducible part ofthe Brillouin zone. The self-consistent calculations were considered to be converged whenthe total energy of the system was stable within 10 − Ry. The Birch-Murnaghan equationof states was used to fit the total energy as a function of unit cell volume to obtain theequilibrium lattice constants and bulk moduli for the investigated systems. For the Fermisurfaces of RSn a ( 64 × ×
64 ) mesh was used to ensure accurate determination ofthe Fermi level and smooth interpolation of the bands crossing the Fermi level. The threedimensional (3D) Fermi surface plots were generated with the help of the Xcrysden molecularstructure visualization program. The elastic constants have been calculated within the total-energy method, where the unitcell is subjected to a number of small amplitude strains along several directions. The elasticconstants of solids provide links between the mechanical and dynamical properties of thecrystals. In particular, they provide information on the stability and stiffness of materials.It is well known that a cubic crystal has only three independent elastic constants C ,4 and C . From these one may obtain the Hill’s shear modulus G H , (which is thearithmetic mean of the Reuss and Voigt approximations), Young’s modulus E , and thePoisson’s ratio σ by using stardard relations. Furthermore, the Debye temperature maybe obtained in terms of the mean sound velocity v m :Θ D = hk B (cid:18) nρN A πM (cid:19) / v m , (1)where h , k B and N A are Planck’s, Boltzmann’s constants, and Avogadro’s number, respec-tively. ρ is the mass density, M the molecular weight, and n the number of atoms in theunit cell. The mean sound velocity is defined as: v m = (cid:20) (cid:18) v t + 1 v l (cid:19)(cid:21) − / , (2)where v l and v t are the longitudinal and transverse sound velocities, which may be obtainedfrom the shear modulus G H and bulk modulus B as: v l = s ( B + G H ) ρ (3)and v t = s G H ρ . (4) III. RESULT AND DISCUSSIONA. Ground state properties
The RSn compounds crystallize in the Cu Au type structure with space group
P m ¯3 m (No. 221). The calculated equilibrium lattice parameters a and zero pressure bulk modulus B are listed in Table I. The results are in good agreement with available experimental dataand other calculations. It often occurs that LDA underestimates and GGA overestimatesthe equillibrium lattice constants, which is also found in the present case, where the LDAvalue is ∼ ∼ . Elastic constants and mechanical properties The calculated elastic constants (C , C and C ) at ambient pressure for LaSn andYSn are presented in Table II, together with quantities related to the elastic constants. Thecalculations were performed at the equilibrium lattice constant as calculated with the LDA.The elastic constants extracted from the experimental phonon dispersion curves are listedfor comparison. The theoretical values lie systematically 20-30 % above the experimentalvalues, which partly is due to the too low equilibrium volume obtained in the LDA. Thefact that the experiments are done at room temperature, while the calculations pertainto zero temperature also contribute to this discrepancy. To the best of our knowledge noexperimental determinations of the elastic constants of YSn have been reported. Neitherhave any theoretical determinations been reported. From the calculated values of the elasticconstants, it can be seen that they satisfy the mechanical stability criteria for a cubic crystali.e. C > C , C >
0, and C + 2 C >
0, consistent with the elastic stability of thesematerials. Pugh has proposed a simple relationship in which the ductile/brittle propertiesof materials could be related empirically to their elastic constants by the ratio G H /B . If G H /B < .
57, the materials behave in a ductile manner, and brittle otherwise. From thevalues of G H /B reported in Table II it emerges that both compounds are of ductile character,and that YSn is more ductile than LaSn . Cauchy’s pressure ( C p = C − C ) is anotherindex to determine the ductile/brittle nature of metallic compounds, where a positive valueof Cauchy’s pressure indicates ductile nature, while a negative value indicates a brittle natureof the compounds. The calculated positive values of the Cauchy’s pressure reported in TableII corroborate the ductile nature of LaSn and YSn . The Young’s modulus E also reflectsthe ductility. The larger the value of E , the stiffer is the material, and as the covalent natureof the compounds increases E also increases. Another important parameter describing theductile nature of solids is the Poisson’s ratio σ (see Table II), which is calculated using theformula given in Ref. 34. For ductile metallic materials σ is typically around 0.33. So theductility of these compounds is confirmed by the calculated values of σ reported in TableII. The anisotropy A is defined as the ratio between C and ( C − C ) /
2, which becomesunity for an isotropic system. According to this definition, LaSn and YSn are elasticallyanisotropic.The elastic constants and bulk modulus increase monotonously under compression, ful-6lling the mechanical stability criteria also at higher pressures. Having calculated Young’smodulus E, the bulk modulus B, and the shear modulus G, one may derive the Debye tem-perature using Eq. (1). The calculated sound velocities ( v l , v t and v m ) and the Debyetemperature ( Θ D ) are included in Table II. The experimental Debye temperatures are almostthe same for LaSn and YSn , while the present calculations find the Debye temperature ofYSn substantially lower than that of LaSn , a consequence of the smaller calculated valuesof the elastic constants of YSn . The sound velocities increase with pressure for LaSn andYSn , reflecting the increase in the relevant acoustic phonon frequencies with pressure. C. Band structure and Density of States under pressure
The band structures of LaSn and YSn are illustrated in Fig. 1. The electronic levelsare calculated along the high-symmetry directions with and without spin-orbit coupling andat the LDA equilibrium volumes. The band structure of LaSn with SOC compares wellwith earlier work. Overall, the band structures of LaSn and YSn are very similar, asnoted by Ref. 19, which also discussed the effect of SOC. The major difference betweenthe two compounds in the vicinity of the Fermi level, E F , occurs at the X-point, where aband crosses the E F for LaSn , but stays below E F for YSn . This gives rise to a small holepocket around the X -point in LaSn , which does not appear for YSn (and which would notbe if SOC is not included). A second, less significant feature, is a very dispersiveband along Γ − R , which dips below E F for YSn , but stays above E F for LaSn .This is illustrated in the inset of Fig. 1. This band contributes to the third Fermisurface of YSn , which in LaSn only appears under pressure.The main interesting pressure effect on the electronic structure of these compounds is theopposite movement of the valence band and the conduction band. Along all high symmetrydirections the valence bands move down, while an upward shift of the conduction bandsunder compression is seen. This is illustrated for LaSn in Fig. 2, which shows the bandstructure for V /V = 1 . V /V = 0 .
90, where V denotes the experimental equilibriumvolume. Owing to this opposite movement of bands under pressure the number of electronstates in the pockets around the M-points, as well as the number of hole states around theX-points increase.The narrow bands around 1.5-2 eV above the Fermi level (see Fig. 1(a)) in LaSn are7he La - 4 f bands. These unocupied La - 4 f bands overlap with the La - 5 d bands, withsome influence on the energy band structure in the vicinity of the Fermi energy. This isillustrated with the density of states, which is shown in Fig. 3, for both LaSn and YSn .In both compounds, the dominating character around the Fermi level is from Sn 5 p , withappreciable admixture of Y 4 d or La 5 d . But for LaSn the tail of the 4 f (blue line) inFig. 3(a) also crosses the Fermi level. The behaviour of the total density of states at theFermi level under compression is shown in Fig. 4. While the total density of states at theFermi energy, N ( E F ), decreases smoothly for YSn under compression, it is more irregularfor LaSn . In LaSn , N ( E F ) passes through a minimum at V /V = 0 .
94 (pressure of 1 GPaaccording to LDA). This behaviour of LaSn is caused by the occurrence of the third Fermisheet under pressure, which is discussed in the next subsection.In Table III the density of states at the Fermi level is compared to the experimental valueas derived from the Sommerfeld coefficient. From the difference, the average electron-phononcoupling constant λ ep may be estimated, assuming γ expt γ calc = 1 + λ ep . (5)For LaSn this reaches a value of λ ep = 0 .
86, while for YSn a value of λ ep = 0 .
34 isfound. Note that the latter is lower than the value deduced by Dugdale, Ref. 19, becausethe inclusion of spin-orbit coupling in the calculation increases N ( E F ) by ∼
20 %. Thevalue obtained here for LaSn is in good agreement with the value found by Ref. 13. Thesuperconducting transition temperature for LaSn may be estimated using the McMillanformula : T c = Θ D .
45 exp (cid:18) − . λ ep ) λ ep − µ ∗ (1 + 0 . λ ep ) (cid:19) . (6)Using a typical value of µ ∗ = 0 .
12 (Ref. 19) and the Debye temperature of LaSn of Θ D = 205K, this leads to a calculated value of T c = 8 . T c = 6 . D. Fermi surface under pressure
The Fermi surfaces of LaSn and YSn at the experimental equilibrium volumes are shownin Fig. 5. The similarity of the band structures lead to nearly identical topology of the Fermi8urfaces for the two compounds. The major difference is that for LaSn two bands crossthe Fermi level, whereas for YSn three bands cross the Fermi level. The first Fermi surfacesheet, Fig. 5(a) and 5(c), is a hole pocket centered on the Γ point. The second sheet, Fig.5(b) and 5(d), is a very complex surface, which we discuss in detail for LaSn below. Incomparison between the two compounds, a small hole pocket is seen around the X-point inFig. 5(b), which is absent in Fig. 5(d), which reflects the difference in band structure at X as discussed in the previous section. Finally, the third surface of YSn is a small electronpart close to the X-point. In Fig. 6 we illustrate for LaSn the complexity of the secondsheet by showing horizontal cuts through the three-dimensional Fermi surface. It is to benoted that without taking into account the SOC three bands would cross the Fermi leveleven at ambient volume, and the corresponding Fermi surfaces would be similar for LaSn and YSn , and furthermore, for YSn the Fermi surface would be topologically the samewith and without SOC.The most striking change in the Fermi surface of LaSn under pressure is the appearance ofthe third surface, already seen in YSn at ambient conditions. This appears at a compressionof V /V = 0 .
94, as shown in Fig. 7(c). A second, less drastic change occurs in the secondFermi surface of LaSn , where a change in topology is observed. This is most easily seen inthe two-dimensional contours of Fig. 6(b), where the hole pocket around the X-point, (themiddle point of the k z = π/a face) increases and merges with the surrounding triangularhole regions. In contrast, for V = V , Fig. 6(a), this pocket is detached from the larger holeregion, facilitating a small closed electron orbit. At the same time, the electron concentrationaround the M-point (the midpoints of all edges of the BZ) increases under compression,which eventually leads to the connection of all electron pockets in the k z = π/a face. Thishappens around V /V = 0 .
90 (see Fig. 7(b)). Altogether, under compression the electronconcentration at M and the hole concentration at X increase simultaneously in LaSn . Inthe case of YSn only the electron concentration at the M-point increases, while there is nohole pocket at X even at ambient volume, and therefore the Fermi surface topology of YSn remains unchanged under (modest) compression.The occurrence of a third Fermi surface sheet for LaSn under pressure leads to an increaseof the density of states at the Fermi level, as illustrated in Fig. 4. For comparison the samequantity for YSn is also shown in the figure, and it is seen to decrease monotonously underpressure. These results for LaSn are in accordance with the zero-pressure measurements of9he superconducting transition temperature in the (La,Th)Sn alloy system as investigatedby Havinga et al. These authors also speculated that their observed oscillatory behaviourof T c versus alloy composition might be due to a singular behaviour of the electronic densityof states in the vicinity of the Fermi level of LaSn . Huang et al. observed an irregularbehaviour of T c for LaSn under pressure. These authors reported an initial slight increasein T c with a maximum at a pressure around 0.8 GPa, beyond which T c gradually decreases.Within the BCS framework of superconductivity, the change in T c observed could reflect achange in the density of states at the Fermi level. At first, this seems in accordance withthe non-monotonic variation under compression of the density of states at the Fermi level inLaSn found in the present calculations. However, we find the opposite trend of an initiallydecreasing density of states at the Fermi level, and an increase only starts for compressionsbelow 0.94 V . Several other factors of course also influence the transition temperature. Thepressure dependence of T c arising from pressure-induced abrupt changes in the Fermi surfacetopology was theoretically analysed by Makarov and Baryakhtar . IV. CONCLUSION An ab initio study of the intermetallic compounds LaSn and YSn was performed withinthe local density approximation. The structural, electronic, elastic, and mechanical proper-ties as well as the Fermi surfaces were studied, including pressure effects. These compoundsare ductile in nature and their Cu Au crystal structure is stable even at high pressure. Theelastic constants and the bulk modulus increase monotonically with pressure. The densityof states near to the Fermi level are mainly Sn p-like states in both compounds, but a Fermisurface topology change is observed in LaSn at a compression of around V /V ∼ .
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78 -Expt. 51.5 e -TABLE I: Calculated lattice constant a (in ˚A) and bulk modulus B (in GPa) for LaSn and YSn ,as obtained with the GGA and LDA approximations for exchange and correlation. The bulkmodulus is evaluated at the theoretical equilibrium volumes. Experimental values are quoted forcomparison a: Ref. 35; b: Ref. 1; c: Ref. 36 d: Ref. 3; e: From elastic constants obtained in Ref.37 arameters LaSn YSn C (GPa) 97.3 (70.5 a ) 82.3C (GPa) 53.6 (42.0 a ) 64.6C (GPa) 44.2 (33.5 a ) 34.2 A G H (GPa) 33.3 20.0 E (GPa) 85.9 (64 b ) 54.9 (98 b ) σ G H /B C p (GPa) 9.4 30.3 v l (km/s) 3.77 3.59 v t (km/s) 2.05 1.63Θ D (K) 230 (205 c ) 188.4 (210 d )TABLE II: Elastic constants and derived quantites for LaSn and YSn , as calculated with LDA atequilibrium volume. A is the anisotropy factor, A = 2 C / ( C − C ), and C p = C − C is theCauchy pressure. Experimental values are given in parentheses where available. a: From phononmeasurements, Ref. 37; b: Ref. 38 c: Ref. 2; d: Ref. 3. OS(states/eV) γ (mJ/mol K ) λ ep T c (K)LaSn Theory a b c , d d ,10.96 e ,11.0 f c g ,6.02 h YSn Theory a i j - 7.0 j TABLE III: Calculated density of states at the Fermi level (evaluated at the experimental equi-librium volumes), together with derived Sommerfeld constants, γ , and electron-phonon couplingconstants, λ ep , for LaSn and YSn . Last column gives the superconducting transition tempera-ture. Experimental values are quoted for comparison. a: This work, LDA; b: Ref. 35 c: Ref. 13;d: Ref. 43; e: Ref. 15; f: Ref. 37; g: Ref. 1; h: Ref. 9; i: Ref. 19; j: Ref. 3; a) (b) FIG. 1: (Color online) Electronic band structures of (a): LaSn and (b): YSn . The solid lines (redcolour) show the electronic levels calculated with spin-orbit coupling included, while the dottedlines (blue colour) show the electronic levels as calculated without spin-orbit coupling. The energiesare given in eV relative to the Fermi level, E F , which is marked with the horizontal dashed line.The major difference between the two compounds around the Fermi level occurs in the vicinityof the X-point (for SOC included). A second, less significant feature, is a very dispersiveband along Γ − R , which stays above E F for LaSn , but dips below E F for YSn . Theinset illustrates this. Γ M -0.4-0.200.20.40.6 E n e r g y ( e V ) LaSn (V/V =1.0) E F (a) X Γ M -0.4-0.200.20.40.6 E n e r g y ( e V ) LaSn (V/V =0.90) E F (b) FIG. 2: Band structure of LaSn under compression (zoom-in on the vicinity of the Fermi level).The electron pocket at M and the hole pocket at X increases under pressure. (a) (b) FIG. 3: (Color online) Density of states of (a): LaSn and (b): YSn , as calculated at the exper-imental lattice constants. The total DOS as well as La, Sn and La-4 f partial contributions areshown in (a), the total DOS as well as Y and Sn partial contributions are shown in (b). The unitis states per eV and per formula unit. A factor of 2 for spin is included. .9 0.92 0.94 0.96 0.98 1 1.02 1.04V/V N ( E F ) e V - YSn LaSn (a) FIG. 4: (Color online) Density of states at the Fermi level, N ( E F ), for LaSn and YSn undercompression. The jump in density of states for LaSn around V /V = 0 .
94 is due to the appearanceof the third Fermi sheet. V denotes the respective experimental equilibrium volumes of LaSn andYSn . a) (b)(c) (d)(e) FIG. 5: (Color online) Fermi surface of (a), (b): LaSn . and (c), (d), (e): YSn (includingspin-orbit coupling and evaluated at the experimental equilibrium volumes). (a) and (c) are holepockets around Γ, (e) are electron pockets around the X-points. The complex second sheet of (b)is illustrated through two-dimensional cuts in Fig. 6(a). In (a) the BZ critical points are marked. a)(b)FIG. 6: (Color Online) Second Fermi surface of LaSn , 2-dimensional contours corresponding to k z = 0 . . π/a . (a): at the experimental equilibrium volume, and (b): at a volumeof 90 % of the experimental equilibrium volume. The shaded (red) areas correspond to occupiedstates. a) (b)(c) FIG. 7: (Color online) Fermi surface of LaSn at compression V /V = 0 .
90. V denotes theexperimental equilibrium volume.denotes theexperimental equilibrium volume.