Tight-binding calculations of optical matrix elements for conductivity using nonorthogonal atomic orbitals: Anomalous Hall conductivity in bcc Fe

We present a general formula for the tight-binding representation of momentum matrix elements needed for calculating the conductivity based on the Kubo-Greenwood formula using atomic orbitals, which are in general not orthogonal to other orbitals at different sites. In particular, the position matrix element is demonstrated to be important for delivering the exact momentum matrix element. This general formula, applicable to both orthonormal and nonorthonormal bases, solely needs the information of the position matrix elements and the ingredients that have already contained in the tight-binding representation. We then study the anomalous Hall conductivity in the standard example, ferromagnetic bcc Fe, by a first-principles tight-binding Hamiltonian. By assuming the commutation relation $\widehat{\vec{p}}=(i{m}_e/\hslash )[\widehat{H},\widehat{\vec{r}}]$, the obtained frequency-dependent Hall conductivity is found to be in good agreement with existing theoretical and experimental results. Better agreement with experiments can be reached by introducing a reasonable bandwidth renormalization, evidencing the strong correlation among $3d$ orbitals in bcc Fe. Since a tight-binding Hamiltonian can be straightforwardly obtained after finishing a first-principles calculation using atomic basis functions that are generated before the self-consistent calculation, the derived formula is particularly useful for those first-principles calculations.

Figure 1

First-principles band structure of ferromagnetic bcc Fe obtained by adopting 13 pseudoatomic orbitals per Fe atom with spin-orbit coupling. Only the bands near the Fermi energy, which is shifted to zero as denoted by $E_F$, are shown. Blue and red circles indicate the contribution of Fe $3d{e}_g$ and $t_{2g}$ orbitals, respectively. The other contribution ($s+p$) is presented by green circles. The modified band structure (Mod) with rescaled hopping integrals between $d$ orbitals ($80\%$) and adjusted onsite energies is presented by the black curves.

Figure 2

Frequency-dependent Hall conductivity in bcc Fe. (a) The imaginary part of $\omega {\sigma }_{xy}$, the magnetic circular dichroism spectrum, is compared with the experimental data presented by open circles from Ref. [28] (Exp) as reproduced in Ref. [15] and Ref. [16]. The black curve shows the result calculated using Eqs. (1) and (7) (Pos), and the green curve shows the result by allowing only $70\%$ strength of the spin-orbit coupling in the $d$ orbitals (Soc). The result obtained from the modified band structure with rescaled hopping integrals between $d$ orbitals ($80\%$) and adjusted onsite energies is presented by the red dashed curve (Mod). (b) The real part of ${\sigma }_{xy}$ is shown and compared with the dc experiment value [27] (solid circle) and the theoretical results for 0 K (open square) and 300 K (open circle), as discussed in Ref. [15].