Wannier-function approach to spin excitations in solids

We present a computational scheme to study spin excitations in magnetic materials from first principles. The central quantity is the transverse spin susceptibility, from which the complete excitation spectrum, including single-particle spin-flip Stoner excitations and collective spin-wave modes, can be obtained. The susceptibility is derived from many-body perturbation theory and includes dynamic correlation through a summation over ladder diagrams that describe the coupling of electrons and holes with opposite spins. In contrast to earlier studies, we do not use a model potential with adjustable parameters for the electron-hole interaction but employ the random-phase approximation. To reduce the numerical cost for the calculation of the four-point scattering matrix we perform a projection onto maximally localized Wannier functions, which allows us to truncate the matrix efficiently by exploiting the short spatial range of electronic correlation in the partially filled $d$ or $f$ orbitals. Our implementation is based on the full-potential linearized augmented-plane-wave method. Starting from a ground-state calculation within the local-spin-density approximation (LSDA), we first analyze the matrix elements of the screened Coulomb potential in the Wannier basis for the $3d$ transition-metal series. In particular, we discuss the differences between a constrained nonmagnetic and a proper spin-polarized treatment for the ferromagnets Fe, Co, and Ni. The spectrum of single-particle and collective spin excitations in fcc Ni is then studied in detail. The calculated spin-wave dispersion is in good overall agreement with experimental data and contains both an acoustic and an optical branch for intermediate wave vectors along the [1 0 0] direction. In addition, we find evidence for a similar double-peak structure in the spectral function along the [1 1 1] direction. To investigate the influence of static correlation we finally consider $\text{LSDA}+U$ as an alternative starting point and show that, together with an improved description of the Fermi surface, it yields a more accurate quantitative value for the spin-wave stiffness constant, which is overestimated in the LSDA.

Figure 1

Diagrammatic representation of (a) the magnetic response function and (b) the $T$ matrix.

Figure 2

(Color online) $e_{\text{g}}$-like ($3d_{3z^2-r^2}$ and $3d_{x^2-y^2}$) maximally localized Wannier functions for bcc Fe. The different tints denote regions with opposite sign.

Figure 3

(Color online) Convergence of the average on-site matrix elements of the bare and screened Coulomb interactions between the $3d$ orbitals as a function of (a) the number of $\mathbf{k}$ points and (b) the number of bands used in the construction of the MLWF basis for the NM and FM states of bcc Fe. (c) The same as (b) for the average spread of the $3d$ orbitals.

Figure 4

(Color online) Average bare and screened on-site direct $\left(\tilde{U}\right)$ Coulomb matrix elements between the $3d$ orbitals for the series of $3d$ transition metals. For comparison results from Ref. 49 (filled spheres) are given. We include six bands in the construction of the MLWFs.

Figure 5

(Color online) Spin-resolved density of states for bcc Fe, fcc Co, and fcc Ni for the NM and FM states. The vertical dashed lines denote the Fermi level. Note that positive values of DOS refer to the majority-spin electrons and negative values to the minority-spin electrons.

Figure 6

(Color online) LSDA and $\text{LSDA}+U$ ($U=1.9\text{eV}$ and $J=1.2\text{eV}$) band structures of FM fcc Ni for (a) majority and (b) minority spins.

Figure 7

(Color online) The Kohn-Sham magnetic response function for fcc Ni for selected wave vectors along the $\Gamma \text{-X}$ direction in the Brillouin zone. Since the Kohn-Sham system is noninteracting, the spectrum exhibits only single-particle Stoner excitations but no collective magnon modes at lower energies. In each panel the peak amplitudes are scaled to the same height.

Figure 8

Spin-wave dispersion for fcc Ni along high-symmetry lines $\left(\text{L-}\Gamma \text{-X}\right)$ in the Brillouin zone within (a) LSDA, (b) $\text{LSDA}+U$, and (c) LSDA with a reduced exchange splitting. The experimental dispersion is taken from Refs. 70, 86.

Figure 9

(Color online) Spin-wave excitation spectra for fcc Ni within LSDA and $\text{LSDA}+U$ for selected wave vectors along (a) the [1 0 0] and (b) the [1 1 1] directions in the Brillouin zone. The double-peak structure for certain wave vectors is clearly visible. All peak amplitudes are scaled to the same height.

Figure 10

(Color online) Decomposition of the single feature for $\mathbf{q}=\left(0.25,0,0\right)$ into two Lorentzian peaks. In the inset we compare with the original curve.

Figure 11

(Color online) Same as Fig. 9 for $\mathbf{q}=\left(0.15,0,0\right)$ and $\mathbf{q}=\left(0.2,0,0\right)$ for different exchange splitting within LSDA. The spectra for the original LSDA exchange splitting are included for comparison.