Accurate bare susceptibilities from full-potential ab initio calculations

Electronic susceptibilities are a very popular tool to study electronic and magnetic properties of materials, both in experiment and theory. Unfortunately, the numerical evaluation of even the bare susceptibility, which depends on the computation of matrix elements and sums over energy bands, is very work intensive and therefore various approximations have been introduced to speed up such calculations. We present a reliable and efficient implementation of the tetrahedron method which allows us to accurately calculate both static and dynamic bare susceptibilities, based on full-potential density functional theory (DFT) calculations. In the light of the exact results we assess the effects of replacing the matrix elements by a constant and the impact of truncating the sum over the energy bands. Results will be given for representative and topical materials such as Cr, a classical transition metal, as well as the iron-based superconductor FeSe.

Figure 1

(a) Tetrahedral mesh of the irreducible wedge of an fcc lattice. The other panels show the mesh after the first (b) and after the second (c) cut with the Fermi surface (see text). For the sake of simplicity we use a parabolic dispersion $\epsilon \left(\mathbf{k}\right)={\left|\mathbf{k}\right|}^2$; $\mathbf{q}=(0.15,0,0)$ and $k_F=0.77$.

Figure 2

LDA band structure of Cr, decorated with partial characters: $s$ (green), $d-{\mathrm{t}}_2$ (blue), and $d-\mathrm{e}$ (red). The coordinates of the high-symmetry points are ${\Gamma }_0=(0,0,0),\mathrm{H}=(0,1,0),\mathrm{N}=(\frac{1}{2},\frac{1}{2},0)$, and $\mathrm{P}=(\frac{1}{2},\frac{1}{2},\frac{1}{2})$, all in units of $\frac{2\pi }{a}$ with $a$ being the lattice constant.

Figure 3

FS of Cr in the $k_z=0$ plane. The different colors indicate different orbital character as described in Fig. 2 and the boundary of the 1BZ is indicated by dotted lines.

Figure 4

Static bare susceptibility ${\chi }^0$ for Cr. CMA results (red, right axis) are compared with the exact results (black, left axis). ${\Gamma }_1$ stands for the $\Gamma$ point in the second BZ. The upper panel (a) shows the converged results, while in the lower panel (b) only the contributions of the bands at the Fermi level have been included in the susceptibility.

Figure 5

LDA band structure for FeSe at ambient pressure with the bands labeled according to their dominant orbital character. The five Fe $d$ bands that form the Fermi surface are shown in red, while all other Fe $d$ bands are depicted in blue. The Se $p$ are shown in black and the bands above Fe $d$ in green.

Figure 6

Static bare susceptibility in CMA ${\chi }_{\text{CMA}}^0$ for FeSe calculated by including different sets of bands, as shown in Fig. 5. From bottom to top the included bands are Fe $d$ that cross the Fermi energy (red), all Fe $d$ bands (blue), Se $p$ plus all Fe $d$ bands (black), and Se $p$ plus Fe $d$ plus all higher bands which are needed to ensure convergence of the full susceptibility ${\chi }^0$.

Figure 7

Static bare susceptibility ${\chi }^0$ for FeSe at ambient pressure calculated by including different sets of bands, as explained in Fig. 6. The left panel (a) shows the exact result while the right panel (b) contains the CMA result.

Figure 8

Contributions of the band transitions to ${\chi }^0$ for FeSe at ambient pressure at the $\mathrm{M}$ point. The red square contains transitions only between Fermi surface bands and the blue square contains all band transitions between the Fe $3d$ bands. Inside the black square are all Fe $d$ and Se $p$ bands. We employed a logarithmic color scale to visually enhance small values.

Figure 9

Imaginary part of the dynamic bare susceptibility ${\chi }^0$ for FeSe halfway between $\Gamma$ and $\mathrm{M}$. The black line represents the result obtained with exact matrix elements and the red line depicts the CMA result.

Figure 10

Imaginary part of the dynamic bare susceptibility for FeSe at ambient pressure. The left panel (a) shows $\text{Im}{\chi }^0$ whereas the right panel (b) contains the results for $\text{Im}{\chi }_{\text{CMA}}^0$. The $\mathbf{q}\mathrm{point}$ shown in Fig. 9 is indicated by a vertical white line.