Hyperfine interactions at lanthanide impurities in Fe

The magnetic hyperfine field and electric-field gradient at isolated lanthanide impurities in an Fe host lattice are calculated from first principles, allowing a qualitative and quantitative understanding of an experimental data set collected over the past $40\text{years}$. It is demonstrated that the common local density approximation leads to quantitatively and qualitatively wrong results, while the $\mathrm{LDA}+\mathrm{U}$ method performs much better. In order to avoid pitfalls inherent to the $\mathrm{LDA}+\mathrm{U}$ method, a combination of free ion calculations and “constrained density matrix” calculations is proposed and tested. Quantitative results for the exchange field and crystal field parameters are obtained ($B_{\mathit{exc}}=+420\mathrm{T}$, $B_0^4=-1000{\mathrm{cm}}^{-1}$, $B_0^6=-800{\mathrm{cm}}^{-1}$), showing in particular how crystal field effects influence the hyperfine fields for the lightest and heaviest lanthanides. The hyperfine fields are shown to be dominated by the $4f$ orbital contribution, with small corrections due to the spin dipolar and Fermi contact fields. The latter is found to be constant for all lanthanides, a feature that is understood by a modified version of the well-known core polarization mechanism for $3d$ hyperfine fields. Spin dipolar fields and electric-field gradients have apart from a $4f$ contribution a surprisingly strong contribution due to the completely filled lanthanide $5p$ orbitals—the mechanism behind this is explained. The lanthanide $4f$ spin moment is found to couple antiparallel to the magnetization of the Fe lattice, in agreement with the model of Campbell and Brooks. There is strong evidence for a delocalization-localization transition that is shifted from Ce to at least Pr and maybe further up to Sm. This shift is interpreted in terms of the effective pressure felt by lanthanides in Fe. Implications for resolving ambiguities in the determination of delocalization in pure lanthanide metals under pressure are discussed. For the localized lanthanides, Yb is shown to be divalent in this host lattice, while all others are trivalent (including Eu). The temperature dependence of the hyperfine fields is discussed as well.

Figure 1

(Color online) Comparison between experiment and several types of calculations for the magnetic hyperfine field of lanthanides in Fe. Diamonds: most probable experimental data points [this is the solid line from (a)]. Gray (green) dotted line: LDA results for the ferrimagnetic orientation (Campbell-Brooks orientation) and for the ferromagnetic orientation when this one has the lowest energy. Solid lines and dashed lines: $\mathrm{LDA}+\mathrm{U}$ calculations for trivalent and divalent lanthanides (see text). Inset picture: experimental data set for the magnetic hyperfine fields of lanthanides in Fe. If the sign of the HFF is not measured, the data point is plotted at both positive and negative values. A distinction is made (see text) between highly reliable data for which the sign is measured (diamond), highly reliable data without sign measurement (square), less reliable data without sign (triangle), and data that are rather unreliable for the magnitude of the HFF but reliable for the sign (circle). For references and values, see text. The line connects the most likely values for all lanthanides. If multiple measurements with the same reliability were available, only one of them is given. More data can be found in the compilation of Rao (Ref. 1).

Figure 2

Schematic summary of the model of Campbell and Brooks, illustrating the ferromagnetic intraatomic $4f\text{−}5d$ coupling and the ferrimagnetic interatomic coupling between Fe $3d$ and Ln $5d$ moments (see text).

Figure 3

(Color online) Thin gray (green) line: the LDA ferromagnetic solution (FoM). Thick gray (green) line: the LDA ferrimagnetic solution (FiM). Black (red) lines: the contributions to the HFF obtained with the CDM method without considering the crystal field interaction (dashed line) and with crystal field effects taken into account (solid line). Mind the scale, which is 10 times larger in (a) compared to (b) and (c). (a) Orbital contribution to the HFF at the lanthanide site due to $4f$ electrons. (b) Dipolar contribution to the HFF. (c) Fermi contribution. The CF has no effect on the Fermi contribution; therefore, the two black (red) lines coincide. The dotted line represents the Fermi HFF for the free lanthanide ions. The Fermi HFF obtained with CDM is indistinguishable from the one obtained with LDA. The total HFF is the sum of (a), (b), and (c), and is almost undistinguishable from (a).

Figure 4

(Color online) $V_{zz}$ for lanthanides in Fe. Symbols: experimental data (Refs. 12, 43, 50, 60) (see text). The gray symbol for Ce indicates that this value is a—rather safe—guess (see text). Dotted line: LDA results. Solid and dashed lines: $\mathrm{LDA}+\mathrm{U}$ values for trivalent and divalent lanthanides (see text).

Figure 5

(a) Energy difference between the situations with the lanthanide $4f$ spin moment ferromagnetically aligned with the Fe $3d$ moment and with the $4f$ moment ferrimagnetically aligned $(E_{\mathrm{FoM}}-E_{\mathrm{FiM}})$. If this energy difference is positive, the ferrimagnetic situation has the lowest energy. Energy differences for LDA are compared with energy differences for $\mathrm{LDA}+\mathrm{U}$ with progressively larger $U$. The LDA result is equivalent to $U=0.0\mathrm{Ry}$. (b) Exchange field obtained from the energy differences from (a) as a function of $U$ $[B_{\mathit{exc}}=(E_{\mathrm{FoM}}-E_{\mathrm{FiM}})∕(\mid {\mu }_{4f}^{\mathrm{FoM}}\mid +\mid {\mu }_{4f}^{\mathrm{FiM}}\mid )]$.

Figure 6

The exchange-field-split ground-state multiplets of ${\mathrm{Ce}}^{3+}$ and ${\mathrm{Yb}}^{3+}$ ions and the influence of the crystal field on the energy levels $(B_0^6=0)$. The sizes of the various energy splittings are also indicated (in kelvin).

Figure 7

(Color online) Contribution of each $m$ orbital to the (a) $4f$-up, (b) $4f$-down orbital HFF and the contribution of the (c) $5p$-up, (d) $5p$-down, (e) $5p$ $\text{up}+\text{down}$ electrons to the orbital HFF of a lanthanide in Fe. Data points: results from calculations for lanthanides in Fe. Solid lines: linear fit through these data points. Dotted lines: linear fit through a complete set of calculations for free lanthanide ions.

Figure 8

(Color online) Contribution of each $m$ orbital to the (a) $4f$-up, (b) $4f$-down dipolar HFF and the induced (c) $5p$-up, (d) $5p$-down, (e) $5p$ $\text{up}+\text{down}$ contributions to the dipolar HFF of a lanthanide in Fe (the $5p$-up and $5p$-dn contributions correspond to the FiM solutions; for the FoM orientation the pictures for $5p$ up and down have to be interchanged). Data points: results from calculations for lanthanides in Fe. Solid lines: linear fit through these data points. Dotted lines: linear fit through a complete set of calculations for free lanthanide ions.

Figure 9

(Color online) Contribution of each $m$ orbital to the (a) $4f$-up, (b) $4f$-down to the total $V_{zz}$ and the induced (c) $5p$-up, (d) $5p$-down, (e) $5p$ $\text{up}+\text{down}$ to the total $V_{zz}$ of a lanthanide in Fe (the $5p$-up and $5p$-dn contributions correspond to the FiM solutions; for the FoM orientation the pictures for $5p$ up and down have to be interchanged). Data points: results from calculations for lanthanides in Fe. Solid lines: linear fit through these data points. Dotted lines: linear fit through a complete set of calculations for free lanthanide ions.

Figure 10

(Color online) The deviation of the total HFF from experiment (in tesla) as a function of the CF parameters for Dy (left) and Er (right). The grey (red) circles indicate the values we have chosen for the CF parameters.

Figure 11

The core polarization mechanism for $3d$ and $4f$ free ions.

Figure 12

(Color online) Fermi contact contribution to the HFF as a function of $4f$ spin moment for a large set of different solutions for different lanthanides in Fe. White circles: core contribution $\left(1s\text{−}4s\right)$. Black circles: valence contribution $\left(5s\text{−}6s\right)$. Gray circles: total Fermi contact contribution. The lines through the core and valence contributions are linear fits; the line through the total Fermi field is the sum of those two fits.

Figure 13

(a) The anisotropic $p$ density ${\rho }_{20}^{p\text{−}p}\left(r\right)$ for Tb in Fe (sum of up and down electrons, arbitrary units for $y$ scale). (b) Left axis (arbitrary units): ${\rho }_{20}^{p\text{−}p}\left(r\right)∕r^3$ for Tb in Fe. Right axis (arbitrary units): integral of ${\rho }_{20}^{p\text{−}p}\left(r\right)∕r^3$, which is called $V_{zz}^{p\text{−}p}$ (apart from a constant factor with negative sign). (c) and (d) The same, but for the $f\text{−}f$ contribution.

Figure 14

Total hyperfine field (in tesla) as a function of temperature.

Figure 15

The experimental transition pressure from localized $4f$ to delocalized $4f$ electrons for pure lanthanides (squares) and two possible choices for the effective pressure felt by lanthanides in Fe (solid and dotted lines) (for details see text).

Figure 16

(a) Experimental value for the HFF in free lanthanide ions, compared with $\mathrm{LDA}+\mathrm{U}$ predictions based on Figs. 7, 8, 12 both for divalent and trivalent lanthanides. (b) Experimental value for $V_{zz}$ in free lanthanide ions, compared with $\mathrm{LDA}+\mathrm{U}$ predictions based on Fig. 9, for both divalent and trivalent lanthanides.