Magnetic exchange interactions in yttrium iron garnet: A fully relativistic first-principles investigation

Magnetic isotropic and Dzyaloshinskii-Moriya interactions in yttrium iron garnet have been obtained by ab initio fully relativistic calculations. The calculated coupling constants are in agreement with available experimental data. Using linear spin-wave theory, we are able to reproduce the experimental magnon spectrum including the spin-wave gap and stiffness. The way to calculate the exchange coupling constants using the Korringa-Kohn-Rostoker formalism for large magnetic systems such as complex oxides is discussed in detail.

Figure 1

YIG unit cell. The dodecahedrally coordinated Y ions (green) occupy the $24c$ Wyckoff sites, the octahedrally coordinated ${\mathrm{Fe}}^{\mathrm{O}}$ ions (gold) occupy the $16a$ sites, and the tetrahedrally coordinated ${\mathrm{Fe}}^{\mathrm{T}}$ ions (blue) occupy the $24d$ sites. The oxygen $96h$ sites (red) are shown. This figure was created using vesta [19].

Figure 2

Band gap (eV) of YIG as a function of $U_{\text{eff}}=U-J$. The theoretical values were obtained using different exchange-correlation functionals as implemented in vasp and sprkkr. Experimental values of the band gap are represented by the gray bar (see Table 4 in Appendix pp1).

Figure 3

Total ($m_{\mathrm{tot}}=m_{\mathrm{S}}+m_{\mathrm{L}}$, top panel) and orbital ($m_{\mathrm{L}}$, bottom panel) magnetic moments of Fe atoms as a function of $U_{\text{eff}}=U-J$. Note that for PAW calculations, $m_{\mathrm{tot}}=m_{\mathrm{S}}$.

Figure 4

One-fourth of the ferrimagnetic YIG unit cell. The solid lines represent the first six next-nearest-neighbor exchange interactions. The subscripts $aa, dd$, and $ad$ are ${\mathrm{Fe}}^{\mathrm{O}}-{\mathrm{Fe}}^{\mathrm{O}}, {\mathrm{Fe}}^{\mathrm{T}}-{\mathrm{Fe}}^{\mathrm{T}}$, and ${\mathrm{Fe}}^{\mathrm{O}}-{\mathrm{Fe}}^{\mathrm{T}}$ interactions, respectively. In the upper left corner, magnetic octahedral and tetrahedral sites are shown, and they compose the majority and minority spin sublattice of the ferrimagnet, respectively.

Figure 5

Monte Carlo (MC) results for the critical temperature as a function of $U_{\text{eff}}=U-J$. The experimental critical temperature is shown as a black solid line [67].

Figure 6

Spin-wave stiffness constant $D$ as a function of $U_{\text{eff}}=U-J$. The experimental values, obtained by different methods, are shown by a gray bar [70]. Experimental data are from Princep et al. [7], Shamoto et al. [10], and Man et al. [71]. Theoretical data are from Cherepanov et al. [1] and Xie et al. [11].

Figure 7

Magnetic exchange interactions (meV) as a function of the interatomic distance $R$ scaled with YIG's lattice constant $a$. The results have been obtained within TB-KKR, with (filled squares) and without (empty circles) the effective Fe-O interaction (Eff). Experimental (exp) data are from Plant [6] (1977), Plant [8] (1983), Princep et al. [7], and Shamoto et al. [10]. Theoretical data are from Cherepanov et al. [1], Xie et al. [11], and Barker et al. [12]. Vertical dashed lines show the coordination shells for Fe atoms. More information can be found in Table 3. NS fit refers to the neutron scattering (NS) data fit.

Figure 8

Top: calculated spin-wave spectrum compared with experimental data from Plant [6] (295 K, cyan filled squares; 83 K, magenta filled circles), Shamoto et al. [10] (295 K, white filled circles), and Man et al. [71] (10 K, green filled circles). Bottom: adiabatic spin-wave spectrum, obtained using exchange couplings from this paper (black thick lines) and from recent experimental studies by Shamoto et al. [10] (red lines) and Princep et al. [7] (blue lines). Here, r.l.u., reciprocal lattice units.

Figure 9

Adiabatic magnon spectrum: low-energy ferromagnetic acoustic mode in the ferrimagnetic insulator YIG, as a function of $U$.