Spin-resolved electron-phonon coupling in FeSe and KFe${}_2$Se${}_2$

The effect of the static magnetic moments of iron on electron-phonon interactions in layered FeSe and KFe${}_2$Se${}_2$ is studied. First-principles techniques based on the pseudopotential density functional approach and the local spin density approximation are utilized to calculate the band structures, phonon dispersions, and electron-phonon coupling properties. Our results indicate that the introduction of iron magnetic moments leads to significant changes in electronic structure induced by Fe 3$d$ states near the Fermi level, to phonon frequency softening for several vibrational modes, and to a dramatic increase in electron-phonon coupling for specific modes. The increase in Brillouin-zone-averaged coupling is about twofold. Our estimates of superconducting transition temperatures based on the McMillan equation yield values closer to experimental results for the spin-resolved case. However, these values are not large enough to explain the observed transition temperature.

Figure 1

Top: Electronic band structure of FeSe for the non-spin-polarized case. Brighter colored bands are those crossing the Fermi level. To the right the corresponding projected densities of states (in states/eV, from 0 to 12) are plotted for all atoms (black solid), Fe 3$d$ states (red solid), and Se (blue dashed). Bottom: The corresponding Fermi surface. The high-symmetry points are given and correspond to the points along the band structure plot on top.

Figure 2

Top: Electronic band structure of FeSe for the checkerboard spin-polarized case. Brighter colored bands are those crossing the Fermi level. To the right the corresponding projected densities of states (in states/eV, from 0 to 11) are plotted for all atoms (black solid), Fe 3$d$ up states (red solid), Fe 3$d$ down states (green dashed), and Se (blue dot-dashed). Bottom: The corresponding Fermi surface.

Figure 3

Top: Electronic band structure of FeSe for the striped spin-polarized case. Brighter colored bands are those crossing the Fermi level. To the right the corresponding projected densities of states (in states/eV, from 0 to 15) are plotted for all atoms (black solid), Fe 3$d$ up states (red solid), Fe 3$d$ down states (green dashed), and Se (blue dot-dashed). Bottom: The corresponding Fermi surface. Note that, even though the striped spin configuration has a different BZ (rotated by $\pi /2$ and scaled by $\sqrt{2}$ in the $k_x$ and $k_y$ directions), we use the conventional cell, so the symmetry points on the wave vector path along the $x$ axis have the same coordinates when expressed in terms of reciprocal lattice vectors as for the nonmagnetic and checkerboard BZs. The stripe direction is toward the given $X$ point. The $Y$ point corresponding to a direction perpendicular to the stripe is not shown.

Figure 4

Top: Electronic band structure of KFe${}_2$Se${}_2$ for the non-spin-polarized case. To the right the corresponding projected densities of states (in states/eV, from 0 to 30) are plotted for all atoms (black solid), Fe 3$d$ states (red solid), and Se (blue dashed). Bottom: The corresponding Fermi surface.

Figure 5

Top: Electronic band structure of KFe${}_2$Se${}_2$ for the checkerboard spin-polarized case. To the right the corresponding projected densities of states (in states/eV, from 0 to 15) are plotted for all atoms (black solid), Fe 3$d$ up states (red solid), Fe 3$d$ down states (green dashed), and Se (blue dot-dashed). Bottom: The corresponding Fermi surface.

Figure 6

Top: Electronic band structure of KFe${}_2$Se${}_2$ for the striped spin-polarized case. Brighter colored bands are those crossing the Fermi level. To the right the corresponding projected densities of states (in states/eV, from 0 to 12) are plotted for all atoms (black solid), Fe 3$d$ up states (red solid), Fe 3$d$ down states (green dashed), Se (blue dot-dashed), and K (purple dot-dashed). Bottom: The corresponding Fermi surface. Note that the striped spin configuration has a monoclinic BZ due to stripe-induced lattice distortion. We use corresponding symmetry points on the wave vector path along the $x$ axis that have the same coordinates when expressed in terms of reciprocal lattice vectors as for the nonmagnetic and checkerboard BZs. The stripe direction is toward the given $X$ point.

Figure 7

Phonon dispersions of FeSe for the non-spin-polarized (top) and the checkerboard spin-polarized (bottom) configurations. The phonon density of states (solid) and Eliashberg spectral function (dashed) are given to the right (in states/cm${}^{-1}$, from 0 to 0.2, for the nonmagnetic and in states/cm${}^{-1}$ multiplied by 2, from 0 to 0.4, for the spin-resolved configuration). Bands showing the largest degree of softening are shown in red (brighter) color to guide the eye.

Figure 8

Phonon dispersions of FeSe for the striped spin-polarized configuration. Phonon density of states (solid) and Eliashberg spectral function (dashed) are given to the right (in states/cm${}^{-1}$ multiplied by 1.5 for the PDOS, from 0 to 0.3). Note that, even though the striped spin configuration has a different BZ, the symmetry points on the wave vector path along the $x$ axis have the same coordinates when expressed in terms of reciprocal lattice vectors as for the nonmagnetic and checkerboard BZs, so the symbols remain the same.

Figure 9

Phonon dispersions of KFe${}_2$Se${}_2$ for the non-spin-polarized (top) and the checkerboard spin-polarized (bottom) configurations. The phonon density of states (solid) and Eliashberg spectral function (dashed) are given to the right (in states/cm${}^{-1}$ multiplied by 1.5 for the nonmagnetic and in states/cm for the spin-resolved configuration, from 0 to 0.5). Bands showing the largest degree of softening are shown in red (brighter) color to guide the eye.

Figure 10

Color map of the differences in matrix elements $\delta M^2$ between the checkerboard spin-resolved and non-spin-resolved cases [according to (3)] given for FeSe (top) and KFe${}_2$Se${}_2$ (bottom). The values on the $x$ and $y$ axes are in units of $2\pi /a$, and on the $z$ axis in arbitrary units. The points (0.0 0.5) and (0.5 0.5) are shown using dashed circles.