In this section we will briefly discuss the two most popular methods that are used to calculate the electronic band structure, mechanical properties and dynamical properties of the investigated compounds in the present thesis work. These are (i) the linearized augmented plane wave (LAPW) method and (ii) pseudo-potentials in combination with the plane wave basis set.
2.4.1 Linearized Augmented Plane Wave (LAPW) Method
Before discussing the LAPW method we will first explain the Augmented plane wave method.
Augmented plane wave method (APW)
According to the APW method the unit cell is divided into two regions: atomic sphere (non- overlapping) also known as Muffin-Tin (MT) spheres and interstitial (I) [153]. The potential in
these regions are expressed as follows:
V (r) =
(PlmVlm(r)Ylm(r) (r∈M T) P
GVGeiG.r (r∈I)
(2.28)
Based on the above mentioned two regions, different basis sets are used. In the atomic regions the wave functions will vary rapidly, whereas in the interstitial region the wave functions are smoothly varying which are shown as followingly:
φ
AP Wkn(r, ǫ
l) =
(PlmAlm,knul(r,ǫl)Ylm(r) (r∈M T)
√1
Veikn.r (r∈I)
(2.29)
wherekn=k+Gn,Gnare the reciprocal lattice vectors,kis the wave vector inside the first Brillouin zone andV is the volume of unit cell. From this equation, it is evident that the wave function with in the atomic spheres (MT) were described by radial functions using spherical harmonics, whereas in the interstitial it is formulated using the plane waves. The coefficients Alm are calculated by matching the wave functions of the atomic sphere and the interstitial regions. Further the augmented plane wave function will be used to expand the Kohn-Sham wave function (ψ) as followingly:
ψ
k(r) = X
n
c
nφ
kn(r)(2.30)
The disadvantage of the APW method is the unknown parameter ‘ǫl’ in 2.29, which will not give the eigen values from a single diagonalization. For this we use the function ul(r, ǫl), which results in an eigenvalue problem which is non-linear in energy and are needed to be solved self-consistently which makes the APW method computationally inefficient.
Linearized Augmented Plane Wave (LAPW) method
LAPW method was introduced to solve the problems that occured in APW method due to the non-linearity oful(r, ǫl). Taylor series is used to expandǫlin LAPW method as shown below:
u
l(r, ǫ
l) = u
l(r, ǫ
1l) + (ǫ
l− ǫ
1l) ˙ u
l(r, ǫ
1l) + O((ǫ
l− ǫ
1l)
2) (2.31)
where ˙ul =∂ul/∂ǫl. The ǫ1l is a fixed point energy around which the Taylor expansion is carried out. The basis set for the LAPW method defined as:
φ
LAP Wkn(r) =
(Plm[Alm,knul(r,ǫl)+Blm,knu˙l(r,ǫl)]Ylm(r) (r∈M T)
√1
Veikn.r (r∈I)
(2.32)
From the above equation it is understood that there is no difference in the interstitial basis set of APW and LAPW methods, and the main difference is only in Muffin-Tin spheres, in which the basis set will depend both on energyul and its energy derivative, ˙ul. The inclusion of energy and its derivative in LAPW method will increase the accuracy compared to the APW method. In the case of LAPW, we also employ the matching wave functions in the MT and the interstitial regions to determine coefficients ofAlm andBlm.
APW+lo method
Further improvement on the linearization of APW method is achieved with the inclusion of local orbitals (lo). The defined basis set for the APW+lo method is as follows:
φ
AP Wlm +lo(r) =
([Almul(r,ǫl)+Blmu˙l(r,ǫl)]Ylm(r) (r∈M T) 0 (r∈I)
(2.33)
The APW+lo method gives a small basis set like the APW method but with the same accuracy as that of the LAPW method. Normalisation is used to evaluate the coefficients ofAlm andBlm
using the condition that the local orbital is zero at the Muffin-Tin boundary. In the present thesis we have used full-potential LAPW (FP-LAPW) and APW+lomethods as implemented in WIEN2k code [153] to evaluate the electronic structure properties.
2.4.2 Pseudopotential method
In order to reduce the complexity and computational time involved with the all-electron methods, we have used the pseudopotential method to evaluate the dynamical properties. A system contains both core and valence electrons, which describe the materials properties in all-electron methods.
The core electrons can be treated as inactive and can be assumed that they do not participate in evaluating the materials properties, and only valence electrons are used to describe the material properties based on the pseudopotential method. The wave functions in the pseudopotential method must not have any radial nodes within the core region and at a certain distancerc(radius of cut-off) the pseudo wave function becomes equal to the real wave function. The condition that need to be satisfied in pseudopotential method is that pseudo wave function and pseudopotential should be identical to the all electron wave function and potential outside a radius of cut-off rc. The other condition is that pseudo wave functions and its first and second derivatives must be continuous at rc. The Schr¨odinger equation in pseudopotential method will be,
( 1
2 ∇
2+ V )ψ = εψ (2.34)
Here ψ is the wave function for the all electron (AE) atomic system with angular momentum componentl. The pseudo wave function is of the form
ψ
psl= X
ni=1
α
ij
l(2.35)
Hereαiis the fitting parameter andjlare the spherical Bessel functions. In the present thesis, we have used the pseudopotential method to evaluate the structural optimization and phonon dispersion calculations as implemented in PWSCF [154] code.
Chapter 3
Pressure effect on electronic structure and vibrational
properties of Ni 2 XAl (X=Ti, Zr, Hf, V, Nb, and Ta), Ni 2 NbGa and Ni 2 NbSn compounds
In this chapter we have studied the electronic structure, mechanical, vibrational properties of Ni- based Heusler compounds, Ni2XAl (X=Ti, Zr, Hf, V, Nb, and Ta), Ni2NbGa and Ni2NbSn, both at ambient conditions and under compression. Calculated ground state properties agree well with other available data. All the studied compounds are found to be mechanically and dynamically stable at ambient conditions. Among the mentioned compounds, Ni2NbAl, Ni2NbGa and Ni2NbSn are experimentally reported as superconductors at ambient conditions and our calculated supercon- ducting transition temperature (Tc) and electron-phonon coupling constant (λep) values are in good agreement with the experiments. We have also predicted superconducting nature in Ni2VAl with λep = 0.68, which leads to Tc =∼4 K (assuming a Coulomb pseudopotentialµ∗ = 0.13), which is a relatively high transition temperature for Ni based Heusler alloys and also higher when compared with other Ni2NbY (Y = Al, Ga and Sn) compounds. From the Fermi surface calculations, flat Fermi sheets are observed along X-Γ direction in all the compounds. In the superconducting Ni2VAl and Ni2NbY (Y = Al, Ga and Sn) compounds these Fermi surface exhibit nesting and leads to Kohn anomaly in the phonon dispersion relation for the transverse acoustic mode TA2 along the (1,1,0) direction. Under compression change in the number of Fermi surfaces is observed in Ni2NbAl and a corresponding non-linear decrease in the total density of states is observed under compression in this compound. Under compression hardening in the phonon modes is observed in all the compounds.
But in the superconducting Ni2VAl and Ni2NbY (Y = Al, Ga and Sn) compounds, Kohn anomaly observed in acoustic TA2 mode, is found to soften with pressure. As a consequence, Tcandλepvary non-monotonically under pressure in superconducting compounds.
3.1 Introduction
Heusler alloys are ternary intermetallic compounds of the form X2YZ, where X is generally a tran- sition metal, Y is yet another transition metal from group VIIIB-IB and Z is a ‘sp’ metal or a metalloid. These compounds display a wide range of physical properties including halfmetallicity [68, 155, 156, 157], magnetic ordering [158, 159], heavy fermion behaviour [160, 161, 162], shape memory effect [163, 164] and thermoelectricity [165, 166].
Ternary intermetallic Heusler alloys like Ni2XAl (X=Ti, Zr, Hf, V, Nb, and Ta), Ni2NbGa and Ni2NbSn are having chemical formula A2YZ and crystallize in cubicL21- structure, where A and Y are the transition metals and Z is a non-magnetic element [167]. Though the Heusler compounds are well known for their ferromagnetic properties, it is interesting to note the existence of paramagnetic compounds, one among them is the above mentioned series Ni2XAl [85]. Ni2TiAl and many other Ni-based compounds have been explored from various perspective, ever since it was found that the high temperature creep resistance of NiAl could be improved by Ti addition [86], which stabilized the compounds in the Heusler type structure. The structural properties of Ni2TiAl was further studied experimentally by Taylor and Floyd [87], using X-rays and electron spectroscopy and Umakoshi et al. [88] further confirmed Ni2TiAl to have small lattice mismatch with B2 phase and found the same to possess good stability. This series attracted further interest and a complete phase diagram of Ni2TAl (T = Zr, Nb and Ta) was obtained by Raman and Schubert [89]. Theoretical insight on the above mentioned series was also provided by Lin and Freeman [85] further substantiating the stability of L21 structure of Ni2XAl (X = Zr, Nb and Ta). Further, Da Rocha et al. [90] have reported the specific heat and electronic structure of the above mentioned series. Heusler alloys ever remain a perennial resource of compounds with plethora of studies available explaining the transport, electrical conductivity, magnetic and many other properties owned by them [168, 169, 170].
Among the Heusler compounds, some are having superconducting nature with maximum su- perconducting transition temperature (Tc) around 4.7 K (YPd2Sn). Superconductivity was first reported for the Heusler alloys by Ishikawa et al. [171], where they focused mainly on the systems RPd2Sn and RPd2Pb with R being a rare-earth metal. Among the known Heusler superconductors, Ni-based alloys Ni2NbX (X = Al, Ga and Sn) have attracted much attention due to their interme- diate electron phonon coupling constant [91, 172, 73]. Interestingly, Ni2NbX (X = Al, Ga and Sn) compounds are paramagnets and are found to be superconductors although Ni is a ferromagnet and Ni compounds are often magnetic. Here we predict Ni2VAl to be a superconductor. Importantly, this provides an experimentally testable prediction that if confirmed and taken in conjunction with the correct predictions for the related compounds, would strongly restrict the possible role of spin- fluctuations associated with Ni magnetism in the superconducting properties of these phases. In Ni2NbX (X = Al, Ga and Sn) compounds, the presence of Nb works against magnetism associated with Ni leading to the paramagnetic ground states of these compounds [91, 73]. The same para- magnetic nature is recently observed in ScT2Al (T= Ni, Pd, Pt, Cu, Ag, Au) [173]. The Ni2NbX superconductors have Tcof 2.15 K (Ni2NbAl) [91], 1.54 K (Ni2NbGa) [91] and 2.9 K (ref. [91]) (3.4 K (ref. [172, 73])) (Ni2NbSn) with calculated electron phonon coupling constants [91] (λep) of 0.52, 0.50 and 0.61 respectively. Electronic structures and cohesive properties of Ni2NbAl and Ni2VAl were studied by Lin et al [85] and a large value of cohesive energy is observed with a pronounced hybridization between Ni-Nb/V-Al atoms. The interaction between these atoms creates deep valleys in density of states which separate bonding and anti-bonding region. It is this type of covalency
that works against magnetism.
Superconductivity in conventional intermetallics has been traditionally discussed in terms of the density of states at the Fermi level and the number of valence electrons per atom. From the Matthias rule [174, 175], the number of valence electrons per atom should be close to 5 or 7. Even though the Heusler superconducting compounds follow the above prescription, the superconducting transition temperatures of these compounds are relatively low. Among the compositional elements of Ni2VAl, vanadium is reported to be a superconductor with superconducting transition temperature (Tc) of 5.3 K [176] and 3.6 K [177]. From previous literatures [177], the total density of states (DOS) value of bcc Vanadium is 1.46 (states/eV/f.u.). Pressure effect on the Tc has been studied experimentally and theoretically for bcc Vanadium [176, 177, 178, 179] upto 120 GPa in which the Tc is found to increase linearly with pressure. However, high density of states also favours spin-fluctuations, which work against electron-phonon superconductivity [180]. In this work the total DOS of Ni2VAl is observed to be more than the value of vanadium and the number of electrons/atom is observed to be 7 in the present compound obeying the Matthias rule.
We also studied the pressure dependence, as pressure provides a tuning parameter that is very useful in understanding trends and mechanisms. Previous studies on ‘Hf’ based Heusler alloys have shown an increase in the superconducting transition temperature with decreasing lattice parameter [181], while certain other cases show an increase in Tcwith increase in lattice constant [181]. We note that the behaviour can also be more complex under pressure [182]. Fundamentally, superconductivity is an instability of the Fermi surface [183], and so pressure induced changes in Fermi surface can lead to non-trivial changes in superconducting properties.
Another interesting feature that connects with superconductivity is the presence of van Hove singularities [6], in the electronic structure, leading to peaks in the density of states as found in some of the Pd based Huesler compounds [74, 184, 185]. Interesting behaviour might be anticipated if the Fermi level can be brought to this peak by means of alloying. Another noteworthy point present in these compounds is the softening of the TA2 acoustic phonon modes, which can be well correlated with the FS nesting and the corresponding nesting vector decides the position of the Kohn anomaly present in these compounds. Further to these, we also find a softening in the acoustic phonons under pressure leading to the variation in the Tc, which is discussed in detail in the present chapter.
The organization of the chapter is as follows. Computational details are presented in the section 3.2. Results and discussions of ground state, electronic structure, mechanical, vibrational, super- conducting properties and pressure effect on these properties are presented in the section 3.3. The conclusions are given in section 3.5.