Conclusions

manner to ωln. In the previous work [213, 214] it was reported that the softening of the phonon DOS leads to the increase in the Tc of that material. In Ni2NbAl, the Tc plot under compression in our previous work [215] is different from this work. In our previous work we used ultrasoft pseu- dopotentials and also we did not find any softening nature in the phonon dispersion curve as well as phonon density of states. In the present work we have used norm-conserving pseudopotentials.

These are more difficult to converge and require more computational resources but are more reliable.

Figure 3.13: Logarithmic frequency, electron-phonon coupling and superconducting transition tem- perature under compression for Ni2NbX (X = Al, Ga and Sn) and Ni2VAl compounds.

Table 3.1: Calculated theoretical equilibrium lattice parameter (athe) compared with the experi- mental lattice parameter (aexp) (in the units of ˚A), bulk modulus (B) (in the units of GPa), density of states at the Fermi level (N(EF)) (states/eV/f.u.) and Somerfield coefficient (γ) (mJ/mol K²) of Ni2XAl (X=Ti, Zr, Hf, V, Nb, Ta), Ni2NbGa and Ni2NbSn.

Parameters athe aexp B N(EF) γ

Ni2TiAl 5.90 5.87^a 164 3.50 8.25

Ni²ZrAl 6.13 6.12^a 151 3.25 7.65

Ni2HfAl 6.10 6.08^a 158 2.81 6.63

Ni2VAl 5.80 5.80^b 181 3.51 8.27

Ni²NbAl 5.99 5.97^a 181 2.27 5.36

Ni2TaAl 5.98 5.96^b 191 2.13 5.02

Ni2NbGa 5.991 5.956^c 182 2.20 5.19

Ni²NbSn 6.202 6.157^c, 6.160^d 170 2.34 5.52 a: Ref. [85];b: Ref. [90];c: Ref. [91];d: Ref. [172]

Table 3.2: Calculated elastic constants (in the units of GPa) and derived quantities for Ni2XAl (X=Ti, Zr, Hf, V, Nb, Ta), Ni2NbGa and Ni2NbSn at the theoretical lattice constant. Where E is Young’s modulus(in GPa),vl,vt, vmare the longitudinal, transverse and mean sound velocities in the units of km/s. CP = Cauchy’s pressure. PR = Pugh’s ratio. ΘD = Debye temperature in the units of Kelvin.

Parameters Ti Zr Hf V Nb Ta Ni²NbGa Ni²NbSn

C11 223 217 211 200 212 229 194 188

C¹² 135 119 132 171 167 172 176 162

C44 104 81 90 109 98 109 95 72

E 193 174 171 138 150 172 111 104

A 2.36 1.65 2.27 7.69 4.37 3.89 10.99 5.57

CP 30.18 37.41 41.64 62.33 68.26 63.01 81.17 89.14

PR 0.45 0.44 0.41 0.28 0.30 0.33 0.22 0.22

σ 0.30 0.31 0.32 0.37 0.36 0.35 0.40 0.40

GH 74.00 66.37 64.85 50.43 54.99 63.79 39.88 37.19

vl 13.00 11.88 10.17 12.22 11.81 10.48 10.43 9.80

vt 6.89 6.25 5.24 5.51 5.48 5.04 4.29 4.03

vm 7.71 6.99 5.86 6.21 6.17 5.66 4.86 4.56

ΘD 617.89 539.17 454.73 506.42 487.11 446.99 241 219

ΘD(exp) 358^b 280^a, 300^b 240^a 206^a

a: Ref. [91];b: Ref. [90]

Table 3.3: Calculated Tc (in the units of K) andλep values with experimental reports for Ni2NbX (X = Al, Ga, Sn) and Ni2VAl.

Parameters Ni2NbAl Ni2NbGa Ni2NbSn Ni2VAl

Tc (experimental) 2.15^a 1.54^a 2.90^a, 3.4^b, 3.4^c –

Tc(this work withµ^∗= 0.13) 1.92 1.18 3.21 3.84

Tc(this work withµ^∗= 0.15) 1.40 0.79 2.60 3.09

λ(experiment) 0.52^a, 0.514^d 0.50^a 0.61^a –

λ(this work) 0.56 0.50 0.68 0.68

a: Ref. [91];b: Ref. [172];c: Ref. [73];d: Ref. [90]

Chapter 4

Electronic topological transitions in Nb ³ Y (Y = Al, Ga, In, Ge, Sn, Os, Ir and Pt) compounds

In the present chapter, first principles electronic structure calculations of A-15 type Nb3Y (Y = Al, Ga, In, Ge, Sn, Os, Ir and Pt) compounds are performed at ambient conditions and high pressures. Mechanical stability is confirmed in all the compounds both at ambient conditios as well as under compression from the calculated elastic constants. We have observed four holes and two electron Fermi surfaces (FS) for the compounds Nb3Y (Y = Al, Ga, In, Ge, Sn), two hole and two electron FS for Nb3Y (Y = Os, Ir) and one hole and three electron FS for Nb3Pt together with FS nesting feature observed at M and along X - Γ in all the compounds. A continuous change in the FS topology is observed under pressure in all the compounds which is also reflected in the calculated elastic constants and density of states under pressure indicating the electronic topological transitions (ETT). The ETT observed at around 21.5 GPa, 17.5 GPa in Nb3Al and Nb3Ga are in good agreement with the anomalies observed by the experiments around the same compression.

4.1 Introduction

Eversince the discovery of superconductivity in V3Si with Tc∼17 K in the year 1953 by Hardy and Hulm [93], the family of compounds with composition X3Y (X= V, Nb, Cr, Ti, Mo, Zr, Ta, W to Hf and Y= Al, Ga, Ge, In, Sn, Os, Ir, Pt etc) had attracted considerable attention of researchers as some of them possess quite high superconducting transition temperature (Tc). The interest in these compounds are not only due to the rather high Tc but also their high critical current density and critical magnetic field, along with acceptable mechanical properties make them viable for applications. These compounds crystallize in the A15 type crystal structure (see Fig. 4.1), where X atoms form three mutually orthogonal chain like structure parallel to the edges of the unit cell.

Some of these compounds undergo cubic to tetragonal martensitic transformation near to their superconducting transition temperatures Tc [95]. For example, the martensitic transition temper- atures of V3Si (21 K) and Nb3Sn (45 K) are close to their respective superconducting transition temperatures 17 and 18 K. Acoustic phonon instabilities were found to be responsible for martensitic transition in previous studies [96]. A similar behaviour was also seen in the Nb3AlxGe1−x[97], V3Ga [98, 99], V3Ge [99] and Nb3Al [100] compounds. Experiments [101] also indicated a dimerization of the transition-metal chains accompanied by a tetragonal distortion of the lattice during the transfor- mation. It has been proposed that the tetragonal transformation is driven by band Jahn-Teller like mechanism. These A15 compounds exhibit different behaviour in electronically derived properties at low temperatures such as knight shifts, electrical resistivity etc [102]. This unusual behaviour of various properties of A15 family compounds has been related to the sharp peak in electronic density of states near to the Fermi level arising from the ‘d’ states of the transition metal atoms [102]. Hence it is clear that many properties of these compounds are related to their electronic structures.

In these X3Y compounds, X atoms have low site symmetry and are responsible for the high Tc

values in this class of materials. X-‘d’ electrons are found to play dominant role in the electronic structure properties around the Fermi level (EF). In these compounds if Y is a transition metal, the Tc of these materials will be low because of the Y-‘d’ electrons which will be competing with the X-‘d’ bands. In some compounds, for a given X element Tc is found to be increase as decrease in the mass and size of Y element. In the compounds Nb3X(X= Os, Ir, Pt), the studies of X-ray photoemission spectra informed that the ‘4d’ and the ‘5d’ energy bands of these Nb3X compounds seems to be more and more separated with increasing atomic number of the X element [216].

Recent experiments [217] have explored the possible relationship between superconductivity and martensitic transition in V3Si and Nb3Sn compounds by measuring electrical resistance and specific heat under high pressure. They have observed that initially Tc increases with pressure and merges with martensitic transition temperature at high pressure, where further Tc increase is ceased and concluded that the martensitic transition play an important role in superconductivity of these com- pounds. So to understand this, a thorough understanding of electronic structures is necessary. So far only a few electronic structure studies are available [218, 219, 220, 221] but a systematic study especially under high pressure is still lacking. Some elements like Cd [222] and Co [223] show a continuous change in FS topology under pressure which highlights the ETT in these elements. In this work, we have carried out a systematic study of electronic properties under pressure for Nb based A15 compounds namely Nb3Y (Y = Al, Ga, In, Ge, Sn, Os, Ir and Pt).

Since any property that involves the conduction electrons in a metal must depend on the shape of the Fermi surface and on the wave functions of the electrons at or near the Fermi surface of that

(a) (b)

Figure 4.1: (a) Crystal structure of Nb3X (X= Al, Ga, In, Ge and Sn) and (b) Brillouin zone high symmetry points.

metal [224] it will be interesting to study the Fermi surface topology and their pressure variations for these compounds. Further, Bilbro and McMillan [225] have studied the interaction of charge density wave (CDW) and superconductivity (SC) in A15 materials within mean field approximation and predicted that both states compete with each other for developing their respective gaps in the same Fermi surface. In Nb3Sn, opening of a charge-density wave gap is observed and expected that it would be due to the dimerization of Nb atoms and nesting at the Fermi surface [226], and this further indicate the possibility of nesting feature in these type of compounds. Charge density wave, Fermi surface nesting and Peierls instability may be inter-related in these compounds. Calculation of the susceptibility [227]χ(~q, ω) for a given system is one of the way to understand the possibility of Fermi surface nesting and formation of CDW. Zero energy value of the Lindhard response function χ0(~q)≡χ0(~q, ω= 0) can be used to determine the presence of Fermi surface nesting. There should be a peak in the imaginary part of the response function Im

χ0(~q)

at the Fermi surface nesting vector.

The formation of charge density wave requires a large real part of the susceptibility, Re χ0(~q)

. Hence in this work we have also calculated the Lindhard response functionsχ0(~q) ≡χ0(~q, ω = 0) for these compounds to study the possible Fermi surface nesting in these systems.

We have predicted Fermi surface nesting for all the compounds at ambient conditions as well as under pressure. The Fermi surface nesting is found to be enhanced under pressure for all the compounds except for Nb3Sn, Nb3Os, Nb3Ir and Nb3Pt. We have also predicted an electronic topological transition (ETT) at different compressions for all the compounds.