Author: Fan Zhang

Introduction

Bulk \(FeSe\) is a conventional superconductor with PbO structure while \(FeTe_{1-x}Se_{x}\) is a candidate for topological superconductor.[1][2] It is useful to perform first principle density functional theory (DFT) calculation on the band structure and density of states of \(FeSe\) and \(FeTe_{1-x}Se_{x}\) with various x values since they provide information such as the role of Te on the whole structure, carrier density of the whole material, whether there would be critical feature near Fermi level. In order to ensure our DFT calculation could provide any useful insight on the real material, it is essential to choose the correct exchange correlation (XC) functional. This work aims at testing the band structure and density of states calculation by different functionals to fix the best XC functional among chosen ones. Also provide a simple step on the effect of Te substitution on the band structure and density of state.

Method

The code used in this work is from CASTEP implementing plane-wave density functional theory. The XC functional used in the following calculations are Generalized Gradient Approximation (Perdew-Burke-Ernzerholf), Generalized Gradient Approximation (Perdew-Wang 91), and Local Density Approximation (Ceperley–Alder–Perdew–Zunger). The pseudopotential used is OTFG ultrasoft which treats 3d6 4s2 for Fe, 3d10 4s2 4p4 for Se, 5s2 5p4 for Te as the valence electrons. Spin polarization is taken into account in all cases. The self-consistent-field (SCF) tolerance of the calculated energy is \(2^{-6}\) eV with 400 maximum cycles. The smearing of the orbital occupancy is fixed at 1eV with 20% empty bands. K points sampling used in the calculation is 5x5x4 (12 irreducible k points for the most symmetric structure). The cut off energy is 440eV. The convergence of both k points sampling and cut off energy were achieved with  \(\Delta\) E \(\leq\) 0.01eV.

Result

1.Convergence of cut off energy and k points sampling

The convergence of cut off energy and k points sampling is done by using the optimized FeSe crystal with PbO structure. The k points configuration used is 5x5x4 mesh as default when performing the convergence of cut off energy. Then the ideal 440eV of cut off energy is used in the convergence of k points sampling. It is found that a 440eV cut off energy with 5x5x4 mesh k points sampling (corresponding to 12 irreducible k points in this structure) could give \(\leq\) 0.01eV. The corresponding graph of these data is also shown in Figure 1 and Figure 2.

2. Structures used in this work

This work used three different structures. Each structure is  geometrically optimized with convergence energy tolerance to bw \(2^{-5}\) eV/atom, the maximum force to be 0.05eV/A and the maximum displacement to be 0.002A.

The top view, side view, and overall view of the structure of optimized FeSe with PbO are shown in Figure3. In this structure, a=b=3.947A, c=4.988A,  \(\alpha\) = \(\beta\) = \(\gamma\) = \(90.000^{\circ}\). Fe atoms are put on sites (0, 0, 1) and (0.5, 0.5, 1). Se atoms are put on sites (0, 0.5, 1.2385) and (0.5, 1, -1.2385).

By replacing one \(Se\) atom by \(Te\), \(FeTe_{0.5}Se_{0.5}\) is achieved. The top view, side view, and overall view of the structure of optimized FeTe0.5Se0.5 are shown in Figure 4. In this structure, a=b=3.947A, c=4.988A,  \(\alpha\) = \(\beta\) = \(\gamma\) = \(90.000^{\circ}\). Fe atoms are put on sites (0, 0, 0) and (0.5, 0.5, 0). Se atom is put on site (0.5, 0, 0.7615). Te atom is put on site (0, 0.5, 0.2385).

By combining 4 conventional cell of \(FeSe\) into a supercell and replacing one of the eight \(Se\) atoms by \(Te\), \(FeTe_{0.125}Se_{0.875}\) is achieved. The top view, side view, and overall view of the structure of optimized FeTe0.125Se0.875 are shown in Figure 5. In this structure, a=b=7.894A, c=4.988A,  \(\alpha\) = \(\beta\) = \(\gamma\) = \(90.000^{\circ}\). Fe atoms are put on sites (0, 0, 0), (0.25, 0.25, 0), (0.5, 0, 0), (0.75, 0.25, 0), (0, 0.5, 0), (0.25, 0.75, 0), (0.5, 0.5, 0) and (0.75, 0.75, 0). Se atoms are put on site (0, 0.25, 0.2385), (0.25, 0, -0.2385), (0.75, 0, -0.2385), (0, 0.75, 0.2385), (0.25, 0.5, -0.2385), (0.5, 0.75, 0.2385), and (0.75, 0.5, -0.2385). Te atom is put on site (0.5, 0.25, 0.2385).

3.Band structure of \(FeTe_{0.5}Se_{0.5}\) calculated by different XC functionals

The reciprocal lattice and various directions used in this work is shown in Figure 6. In order to compare with previous work, we focus on GZ direction (usually called \(\Gamma\)Z direction).[2]

The band structure calculated by various XC functionals are shown in Figure 7 and the calculation done by Peng Zhang et al. are shown in Figure 8. [2]

GGA PBE result is the only one that shows a band cross feature along \(\Gamma\)Z direction ,the only one shows spin polarized structure, and is the only one does not have a band gap. It is thus clear that GGA PBE is the most suitable XC functionals among the three tested. The difference between this calculation and the previous work may caused by spin-orbit coupling (SOC). In this work, we did not count SOC while SOC is claimed to be the reason for the opened gap in the paper by Peng Zhang et al.[2]

For the following analysis, we will choose GGA PBE as our XC functional.

4.Effect of Te substitution on band structure and density of states

The band structure of \(FeSe\), \(FeTe_{0.125}Se_{0.875}\), and \(FeTe_{0.5}Se_{0.5}\) are shown in Figure 9.

The density of states of \(Fe\), \(Se\) and \(Te\) in \(FeSe\), \(FeTe_{0.125}Se_{0.875}\), and \(FeTe_{0.5}Se_{0.5}\) are shown in Figure 10, Figure 11, and Figure 12.

It could be seen clearly from the calculation that \(Te\) substitution greatly changes the band structure of original \(FeSe\) instead of just moving chemical potential. By substituting a small amount of \(Te\), density of states around fermi surface could be doubled, which indicate an increase in the carrier density. However, increasing \(Te\) will decrease the density of states around fermi surface again. This change in density of states is contributed mostly by p and d orbitals of \(Fe\).

5.Comparison of calculation time of various XC functionals

The calculation time of band structures and density of states by different XC functionals are shown in Table 1. Results are normalized with respect to the shortest time.

GGA PBE is again the best choice for doing this calculation as it requires the least time per iteration.

Conclusion

By comparing the band structure of \(FeTe_{0.5}Se_{0.5}\) calculated using various XC functionals with the result in previous work, GGA PBE is the best choice among all three XC functionals chosen here. Comparison of computational time also suggests GGA PBE to be the best choice. By the calculation done with GGA PBE, adding \(Te\) to \(FeSe\) can greatly change the band structure. As the weight of \(Te\) increasing, the carrier density is expected to increase following by a decrease. The difference between the band structure of this work and previous work indicate the importance of SOC in this material.

For the next step of this work, I would add SOC to my material and increase the number of k points when calculating band structures to allow a more detailed comparison. Also, I would proceed to more possible x values in \(FeTe_{1-x}Se_{x}\) to see how \(Te\) would affect the material. Then, transport properties could be calculated to allow more possible comparisons with experiments.

References