A discussion of the importance of the Fermi surface with some examples is given in the next section. In the case of Sr3Ru2O7, a metamagnetic behavior is observed [21] and also a connection between field-induced changes in the Fermi surface via a metamagnetic transition (most likely due to a van Hove singularity near the Fermi energy) and the presence of a spin density wave (SDW) is observed [22].
Application of pressure and it’s importance on the physical properties of solids
In the next section we discussed how applying pressure will affect the physical properties of the systems. In this thesis we discussed the importance of the Fermi surface to understand the physical properties of the studied systems and how these properties will change when pressure is applied.
Overview of the compounds of present study
- Heusler compounds
- A-15 compounds
- Binary Sn-based compounds
- Mn-based Cu 2 Sb type magnetic compounds
In the compounds III-V (AlSb, GaP, GaAs, GaSb, InP, InAs, InSb) and II-VI (ZnS, ZnSe, ZnTe, CdTe) a phase transition is observed from the diamond or zinc blende type to the β- Sn or NaCl type structure under pressure [109]. In the present work we have used MnAlGe, MnGaGe and MnZnSb compounds which have the same tetragonal Cu2Sb type.
Overview of the thesis
Born-Oppenheimer approximation
This approximation states that nuclei with heavier mass can be considered fixed in space from the point of view of electrons [136]. This essentially decouples the kinetic energy of the nuclei as zero and the potential energy term (VbN+VbN−e≈Vext) as a constant from the many-body equation.
Hartree approximation
With this approach, the complexity of the many body problems is reduced, but solving this also requires more effort. Although this makes the many-body problem simple, it failed to include the antisymmetric nature of the electron (Fermionic) and also the electron-electron interaction.
Hartree-Fock approximation
In this regard, density functional theory has given an optimal solution to the many-body problem which is explained in detail in the next section.
Introduction to density functional theory
Thomas-Fermi theory
Thomas-Fermi took the first step in substituting the density of electrons for the wave function, which alone is a rough way of approximating the term kinetic energy. For the practical implementation of this, further developments are carried out using the Hohenberg and Kohn theorems.
Hohenberg-Kohn theorems
Kohn-Sham method
The last term Excis the exchange-correlation energy which goes beyond the Hartree approximation for a better description of the system. The final minimum value gives the ground state energy and the density of the system.
Exchange-correlation functionals
The local-density approximation (LDA)
The first approximation describing the correlative exchange function is the local density approximation given by Kohn and Sham [138]. Hereǫhomxc is the per-particle exchange-correlation energy of the homogeneous interacting electron gas of densityn(r).
The generalised gradient approximation (GGA)
Methods
Linearized Augmented Plane Wave (LAPW) Method
From this comparison it is clear that the wave function with in the atomic spheres (MT) was described by radial functions using spherical harmonics, while in the interstitial it was formulated using the plane waves. In the case of LAPW, we also use the matching wave functions in the MT and the interstitial regions to determine coefficients of Alm and Blm.
Pseudopotential method
In the current dissertation, we used LAPW (FP-LAPW) and APW+lo full potential methods as implemented in WIEN2k code [153] to evaluate the electronic structure properties. In the superconducting Ni2VAL and Ni2NbY compounds (Y = Al, Ga and Sn), these Fermi surfaces exhibit nesting and lead to Kohn anomaly in the phonon dispersion relation for the transverse acoustic mode TA2 along the (1,1,0) direction.
Method of calculations
All electronic structure calculations are performed with 44×44×44 k-mesh in the Monkhorst-Pack [187] scheme, which yields 2168k points in the irreducible part of the Brillouin Zone (BZ). The functional exchange correlation GGA-PBE is used for all compounds in the present calculations.
Results and discussion
- Ground state, electronic structure properties
- Elastic constants
- Vibrational properties
- Superconductivity of Ni 2 NbAl, Ni 2 NbGa, Ni 2 NbSn and Ni 2 VAl
The calculated Sommerfield coefficient γ is also given in Table 3.1, which is proportional to the density of states at the Fermi level. These calculated values are in the range of current Ni2NbX (X= Al, Ga, Sn) and other Ni-based superconducting Heusler compounds ZrNi2Ga [185].
Pressure effect on electronic structure, elastic constants, vibrational and supercon-
From these plots, all compounds show a non-monotonic variation in Tc and the electron-phonon coupling constant under pressure is behaving in the opposite direction. In previous work it was reported that attenuation of the phonon DOS leads to the increase of Tc of that material.
Conclusions
In all compounds, a continuous change in FS topology under pressure is observed, which is also reflected in the calculated elastic constants and density of states under pressure, indicating electronic topological transitions (ETTs). These compounds crystallize in the A15-type crystal structure (see Figure 4.1), where the X atoms form three mutually orthogonal chain structures parallel to the edges of the unit cell.
Method of calculations
The charge density wave, Fermi surface slot, and Peierls instability may be correlated in these compounds. The 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. We have predicted Fermi surface nesting for all compounds under ambient conditions as well as under pressure.
Results and discussions
Ground state properties and elastic constants
The real and imaginary part of the LindhardRe[χ(q)] forω= 0 response function are calculated directly from the energy eigenvalues using,. The calculated positive values of Cauchy pressure (C12-C44) indicate the ductile nature of the present composites and it is also confirmed by the calculated value of Pugh's ratio (GBH) [198]. From the calculated values of Poisson's ratio, we observed that all the compounds have the value closer to the upper limit indicating the stiffness of the actual compounds.
Density of states
This Pugh's ratio is less than 0.57 which is known as critical number to separate brittle and ductile nature. The Poisson's [199] ratio indicates the stability of the crystal against shear and takes the values between -1 to 0.5, where -1 and 0.5 serve as lower and upper limits respectively.
Band structure and Fermi surface topology
In the case of Nb3Ge, the first four-hole bands cross the EF only at the M point. The last two FS have sheets in the center of the BZ faces with a pocket at Γ point. In the case of Nb3Al, the first four FS have holes in nature and among them the first two FS have pockets at the M point (Fig.
Pressure effect on the electronic structure and elastic constants
Due to these two bands (53 and 54), a continuous change in the FS topology under pressure is observed. In the case of Nb3Sn, band structure topology is found to continuously change along M-R and R-X due to the bands and 54 under pressure. In the case of Nb3Sn, at R point, the peak observed at ambient conditions is found to be absent under compression.
Conclusions
In this chapter, the first electronic structure calculations are performed using density functional theory for SnA and SnSb. In addition, calculations of the overall susceptibility of SnA are performed using DFT to determine the role of the electronic structure in the softening of specific phonon modes. The Fermi surface of SnAs and SnSb is also presented for the first time together with a numerical analysis of its role in phonon softening.
Computational details
In addition to inducing superconductivity, pressure can have the effect of increasing Tc, prompting investigation of the effect pressure has on the superconducting properties of SnAs and SnSb, a structurally similar system to SnO that is already superconducting at ambient pressure. A Gaussian broadening of 0.005 Ry and a uniform grid of 8×8×8 'q' points are used for phonon calculations. Energy convergence up to 10−5 Ry is used for proper convergence of the self-consistent calculation.
Results and discussions
- Ground state properties and structural phase transition
- Electronic structure
- Elastic constants
- Vibrational and superconducting properties
Now we proceed further with the calculation of vibrational properties, which can be used to check the dynamic stability of compounds present in ambient conditions, the NaCl structure. The absence of imaginary phonon frequencies indicates the dynamic stability of compounds present at ambient conditions. An anomaly (dip) in the degenerate mode LA2 is observed along the Γ-X direction which may have a significant effect on the physical properties of the compounds present.
Pressure effect on the electronic structure, phase transition, vibrational and super-
We have calculated the sensitivity at the transition pressure in CsCl type for both the compounds and the same is plotted in figure. The calculated phonon dispersion for CsCl-type SnAs and SnSb at the transition pressure is given in figure. a reason for the high Tc in the CsCl-type structure compared to the NaCl-type structure.
Conclusions
In the case of Mn-based compounds, MnAlGe, MnGaGe and MnZnSb, a quasi-two-dimensional nature is observed from Fermi surface calculations. Under compression, a decrease in the magnetic moment of the Mn atom and a change in band and FS topology are observed in all Mn-based compounds. The non-linear nature of the elastic constants is observed in compression in all compounds.
Methodology
Results and discussions at ambient conditions
Zr 2 TiAl
The calculated band structures are given in Fig.6.3 for the AFM1, AFM2 and FM states in the supercell as shown in Fig.6.1. In the minority spin case we have single FSs (Fig.6.4(d)) which have a pocket around the Γ point due to the band crossing at the same Γ point from the valence band to the conduction band. To check the mechanical stability of the present composite we have calculated the elastic constants for both magnetic and non-magnetic cases and are given in Table 6.3 at ambient conditions.
Mn-based compounds
The first two (c, d) FSs belong to the majority spin case, while the remaining two (e, f) belong to the minority spin case. The first two (c, d) FSs belong to the majority spin case, while the remaining three (e, f, g) belong to the minority spin case. The first four (c, d, e, f) FS belong to the case of majority rotation, and the remaining two (g, h) belong to the case of minority rotation.
Results and discussions for systems under pressure
Zr 2 TiAl
From this we found small pockets along Γ-X in the first FS which is evident from Fig. 6.17., the changes in the band structure and FS topology are observed at L-point where the new bands are added as shown from the zoomed band structure in Fig. Non-monotonic variations in the DOS under compression in both majority spin and minority spin are found.
Mn-based compounds
In the same way, we found the absence of pockets at the X point in the case of minority rotation compared to the ambient one. A drastic decrease in the magnetic moment of the Mn atom in MnZnSb is also confirmed at V/V0=0.92 as shown in Fig. Similar to MnAlGe, we also observed changes in the FS topology in MnGaGe and MnZnSb and the plots are given in Fig. .
Conclusion
Density functional study of elastic and vibrational properties of Heusler-type alloys Fe2VAl and Fe2VGa. Phys. First-principles investigation of phonon damping and lattice instability in the Ni2MnGa shape memory system.Phys. Predicted superconductivity of Ni2VAl and pressure dependence of superconductivity in Ni2NbX (X= Al, Ga and Sn) and Ni2VAl.J.Phys.:Condens.
