5.3 Results and discussions
5.3.4 Vibrational and superconducting properties
was shown by Johannes et al [34]. Through calculating both the real and imaginary parts of the generalised susceptibility,χ(q), whereqis the wavevector of the perturbation, an assessment can be made as to the geometrical nesting properties of the Fermi surface (in the imaginary part(χ′′(q))) as well as the response that the electrons will have (through the real part(χ′(q))). A true nesting effect will be seen through singularities in both the real and imaginary parts at the same wavevector, showing the electronic response at a particular ‘q’ is a consequence of the geometry of the Fermi surface. χ(q) has been calculated for both the compounds at ambient conditions NaCl structures as shown in Fig. 5.6(a) and 5.6(c). In the NaCl structure, a peak is seen in both the real and imaginary parts at a wavevector of [0.0,0.0,0.1]×2π/ain both the compounds along Γ-X. indicating a strong electronic response that is driven by the shape of the Fermi surface.
and at L points at different frequencies. We also observe degenerate LA2 mode along L-Γ-X and this degeneracy is lifted out in other directions. An anomaly (dip) in the degenerate LA2 mode is observed along Γ-X direction which may have significant effect in the physical properties of the present compounds. The same anomaly is observed in some of the Heusler compounds such as Ni2MnGa [200], Ni2MnIn [203], Ni2MnX (X= Sn, Sb) [204], Ni2VAl and Ni2NbX (X=Al, Ga, Sn) [254], which are having same FCC structure. The softening in the acoustic mode leads to Kohn anomaly [31], which is due to the interaction of electronic states with phonon at the EF together with parallel sheets in the FS topology. The mechanism behind a Kohn anomaly is one of electronic screening [31] due to the perturbation. Under the action of a perturbation (in this case, a phonon), the electrons at the Fermi surface (which have access to unoccupied states) will attempt to screen it [255]. The ions of the lattice will then interact via this screened potential which modifies the phonon frequencies. The extent of this softening can therefore be dictated by how responsive the electrons are to the initial perturbation. Kohn summarised that the shape of the Fermi surface would play a vital role in this mechanism [31]. In the present compounds also this interaction might be a reason for the anomaly. This is confirmed from the susceptibility calculations as discussed above at ‘q’- vector around [0.0 0.0 0.1]×2π/a along Γ-X direction, where we observe the FS nesting and phonon softening.
By first looking at the total PDOS in the NaCl structure, the major peak is found near a frequency of 150 cm−1 and from the atom-projected phonon DOS in Fig. 5.7(c), it can be seen that this is derived from As. It is also observed that the higher frequency optical modes above the frequency 125 cm−1 are due to As and the remaining modes below 125 cm−1 are due to Sn. In the case of SnSb, from Fig. 5.7(b), major peak in the total PDOS observed around 120cm−1 is due to both Sn and Sb atoms, which are having almost equal contribution (which is due to the nearly same atomic mass) around this frequency range (see Fig. 5.7(d)).
Having determined the phonon structure as discussed above, the electron-phonon coupling can be calculated to explore the superconducting properties. As discussed in section 5.1, present compounds are found to have superconducting nature in NaCl structure which is the ground state. The electron phonon coupling constant (λep) is extracted from the Eliashberg function (α2F(ω)) which can be used to determine the superconducting transition temperature (Tc) of a conventional phonon mediated superconductor. The Tc of the present compound is calculated by using Allen-Dynes [205] formula,
Tc= ωln
1.2exp(− 1.04(1 +λep)
λep−µ∗(1 + 0.62λep)) (5.4) where ωln is logarithmically averaged phonon frequency, λep is electron phonon coupling constant andµ∗ is Coulomb pseudopotential. The representation ofα2F(ω) is
α2F(ω) = 1 2πN(ǫf)
X
qj
νqj
¯ hωqj
δ(ω−ωqj) (5.5)
This function is often very similar to the phonon DOS (F(ω) = P
qjδ(ω−ωqj)) and differs from the phonon DOS by having a weight factor 1/2πN(ǫf) inside the summation. In the above formula N(ǫf) is the electronic density of states at the EF and νqj is the phonon line width, which can be
(a) (b)
(c) (d)
Figure 5.7: Phonon dispersion along with total phonon density of states and Eliashberg function for (a) NaCl phase of SnAs at ambient conditions and 37 GPa, (b) NaCl phase of SnSb at ambient conditions and 13 GPa. (c) Atom projected phonon density of states for SnAs in NaCl phase at ambient conditions (solid lines) and 37 GPa (dotted lines). (d) Atom projected phonon density of states for SnSb in NaCl phase at ambient conditions (solid lines) and 13 GPa (dotted lines).
represented as
νqj = 2πωqj
X
knm
|g(k+q)m,knqj |2δ(εkn−εF)δ(ε(k+q)m−εF) (5.6) where Dirac delta function express the energy conservation conditions and ‘g’ is the electron phonon matrix element. λep can be expressed in terms ofα2F(ω) as
λep= 2 Z dω
ω α2F(ω) = Z
λ(ω)dω (5.7)
where
λ(ω) = 2α2F(ω)
ω (5.8)
The calculated Eliashberg function is plotted in Fig. 5.7(a) and 5.7(b) for SnAs and SnSb respec- tively, where we find peaks at frequency around 140, 120, 100 cm−1 in SnAs and around 120, 70 cm−1 in SnSb. The height of the peak indicate the higher phonon line width and higher electron phonon coupling constant at that frequency region and is found to decrease gradually to lower fre- quencies. The calculated Tc values of the investigated compounds are 3.08 and 3.08 K with a value ofλeparound 0.62 and 0.68 for SnAs and SnSb respectively by considering theµ∗value to be 0.13.
These calculated values are in good agreement with the other reported values [112, 113, 250] and are given in Table 5.3. Calculatedλepand Tc of the present compounds are almost same.
From the above discussions superconducting nature is confirmed in both the compounds at ambient conditions NaCl structure. From the calculated total energy and enthalpy calculations (as discussed from Fig. 5.2) phase transition is observed in both the compounds from NaCl to CsCl-type structure. So, it is quite reasonable to study the pressure effect on the above mentioned properties of SnSb which are presented in next section.