3.3 Results and discussion
3.3.1 Ground state, electronic structure properties
The basic ground state properties are calculated using the Birch-Murnaghan equation of state and the results are reported in Table 3.1, along with the available experimental results. The ground state properties of Ni2XAl (X=Ti, Zr, Hf, V, Nb, and Ta), Ni2NbGa and Ni2NbSn are evaluated using the experimental lattice parameter and atomic positions [85, 90, 91, 172, 73, 190]. The calculated equilibrium lattice constant and bulk modulus values are presented in Table 3.1 along with the available experimental and other theoretical work and a good agreement is seen between the present values and earlier reports. The calculated bulk moduli is higher for Ni2TaAl and lower for Ni2ZrAl.
We have calculated the band structure along the high symmetry directions in the irreducible Brillouin zone with and without inclusion of SOC. From Fig. 3.1 (given for only Ni2NbSn) it is seen that the SOC effect is very small around the Fermi level and we have proceeded with the further calculations excluding SOC. We have also checked the effect of Hubbard ‘U’ and found no appreciable changes in the band structure as these are metallic systems and are not correlated. This is consistent with the recent studies on Heusler based compounds where the authors also concluded the same [191].
The band structures for all the compounds are given in Fig. 3.2. The overall profile of all these Ni2XAl compounds are the same, whereas the number of bands crossing the EF is not the same for all the compounds. For Ni2TiAl, Ni2ZrAl and Ni2HfAl three bands cross the EF, two of them crossing at the L-point from valence band to conduction band (Here the conduction bands refer to
Figure 3.1: Band structure of Ni2NbSn with and without inclusion of spin-orbit coupling(SOC) at the theoretical equilibrium volume.
the bands above the Fermi level and they are primarily X-derived states) and the third band crosses the EF from conduction band to valence band at X-point (band structure of Ni2TiAl is shown in Fig. 3.2). For Ni2VAl, we observe two bands to cross the Fermi level at X-point from conduction band to valence band. For Ni2NbAl, Ni2TaAl and Ni2NbGa compounds, we find only one band to cross the EF from conduction band to valence band at X-point at ambient conditions. In addition to that, we have an extra band at the X point in Ni2NbSn.
The electronic density of states (DOS) is shown in Fig. 3.3 along with the atom projected DOS.
Even though these compounds are composed of different elements from different rows in the periodic table, the total DOS for all the compounds looks similar reflecting the similar band profiles. For all the compounds we observe valleys at energies around -6 eV, -1 eV, 0.5 eV. In the case of Ni2NbSn there is another valley at around -3 eV. This feature indicates that the interaction between the constituent atoms is strong [85]. From the Fig. 3.3, it is evident that the contribution at EF is mainly dominated by Ni-‘deg’ states with an admixture of X-‘dt2g’ and Al/Ga/Sn-‘p’ states. The states at nearly -6 eV is mainly derived from the Al/Ga/Sn-‘s’ states. For all the compounds, the bonding and the anti-bonding regions are well separated from the non-bonding region and our calculations agree well with the earlier studies [85]. As we move to compounds containing X from V-B elements, we could observe the states to shift below EF due to band filling and is clearly evident from Fig. 3.2. Apart from this, our calculated density of states at the Fermi level show a decreasing trend as we move from top to bottom of the periodic table. Among all the compounds Ni2VAl has the highest value of DOS at EF with 3.51 states/eV/f.u. and Ni2TaAl has lowest value with 2.13 states/eV/f.u. From the atom projected DOS we have observed that the primary contribution to the total DOS at EF is due to Ni atom (‘deg’ states), the secondary contribution is due to X atom (‘dt2g’ states) and the least contribution arises from Al/Ga/Sn atom (‘p’ states). The calculated Sommerfield coefficient γ is also given in the Table 3.1, which is proportional to the density of states at the Fermi level. In the case of conventional superconductors, Tc value is propotional toγ.
Increase in theγleads to increase in the Tcof that material. In this chapter, the order of Tcvalues
(a) (b)
(c) (d)
(e) (f)
(g) (h)
Figure 3.2: Band structure at ambient conditions for (a) Ni2TiAl (b) Ni2VAl (c) Ni2ZrAl, (d) Ni2NbAl, (e) Ni2HfAl, (f) Ni2TaAl, (g) Ni2NbGa and (h) Ni2NbSn respectively.
follows as Ni2VAl>Ni2NbSn>Ni2NbAl>Ni2NbGa (3.84>3.21>1.92>1.18). The same order also followed byγ values as 8.27>5.52>5.36>5.19.
Experimental specific heat can be calculated using the expression, (1 +λ)×γth. The λin the enhancement factor is related to but not identical to the superconducting ‘λ’ and includes in addition contributions from spin fluctuations and other interactions if present. The inferred values are in the range∼0.5-1 for Ni2NbAl and Ni2VAl, in reasonable accord with the calculated superconducting ‘λ’
which is explained in the subsequent section. The values for Ni2NbGa and especially Ni2NbSn are anomalously low. The origin of this is not clear, and warrants further investigation. Site disorder in samples is one possibility. In any case, we also note that the N(EF) are not high enough to place any of the compounds near Stoner criterion for ferromagnetism.
The van Hove singularity is observed in both valence and conduction bands at the L-point close to EF around 1 eV and -1 eV energy range. From the earlier available reports [74, 184, 185], one saddle point is observed in Pd based compounds, at L point. But in the case of present compounds, we have two saddle points in both valence and conduction regions near the EF. The flat bands associated with the van Hove singularity at the L-point result in a maximum density of states.
In addition, the Fermi surfaces (FS) of all the investigated compounds at ambient conditions are shown in Fig. 3.4, for the corresponding band which crosses the EF as shown in Fig. 3.2. We observe the Fermi surface topology to be quite similar for Ni2TiAl, Ni2ZrAl and Ni2HfAl indicating the dominating nature of the Ni-‘d’ states with small contribution from X-‘d’ states at EF. For Ni2TiAl, Ni2ZrAl and Ni2HfAl compounds which contain IV-B elements, we find the Fermi surface to be of electron character at X-point and hole character at L-point respectively and is also evident from the band structure plot from Fig. 3.2, whereas in Ni2VAl, Ni2NbAl and Ni2TaAl compounds which contain V-B elements, we find the band to cross only at X-point resulting in the electron pocket at the same point. From Fig. 3.4(j,k) it is evident that Ni2VAl has two FS as a result of two
(a) (b)
(c) (d)
(e) (f)
(g) (h)
Figure 3.3: Density of states at ambient conditions for (a) Ni2TiAl (b) Ni2ZrAl (c) Ni2HfAl (d) Ni2VAl (e) Ni2NbAl (f) Ni2TaAl (g) Ni2NbGa, (h) Ni2NbSn compounds respectively.
bands crossing the EF (see Fig. 3.2(b)) and the remaining two compounds Ni2NbAl, Ni2TaAl have only one FS, due to a single band crossing the EF (see Fig. 3.2(d,f)). Ni2NbGa and Ni2NbSn have one and two FS correspondingly equivalent to the bands that cross the EF. In Ni2TiAl, first two FS have hole nature and the last one has electronic nature. Overall, in Ni2TiAl, Ni2ZrAl and Ni2HfAl compounds first two FS have hole nature and last FS has electronic nature and in the remaining Ni2VAl, Ni2NbAl, Ni2TaAl, Ni2NbGa and Ni2NbSn compounds, we have only electron FS. In all the compounds we have observed parallel sheets in FS which indicate a nesting feature.