Spin Hall torques generated by rare-earth thin films

We report an initial experimental survey of spin Hall torques generated by the rare-earth metals Gd, Dy, Ho, and Lu, along with comparisons to first-principles calculations of their spin Hall conductivities. Using spin torque ferromagnetic resonance (ST-FMR) measurements and dc-biased ST-FMR, we estimate lower bounds for the spin Hall torque ratio, ${\xi }_{\mathrm{SH}}$, of $\approx 0.04$ for Gd, $\approx 0.05$ for Dy, $\approx 0.14$ for Ho, and $\approx 0.014$ for Lu. The variations among these elements are qualitatively consistent with results from first principles [density-functional theory (DFT) in the local density approximation with a Hubbard-$U$ correction]. The DFT calculations indicate that the spin Hall conductivity is enhanced by the presence of the partially filled $f$ orbitals in Dy and Ho, which suggests a strategy to further strengthen the contribution of the $f$ orbitals to the spin Hall effect by shifting the electron chemical potential.

Figure 1

A naive estimate of the spin-orbit coupling based on Russel-Saunders or LS coupling for estimates of the orbital and spin angular momentum components of the total angular momentum for Hund's rules for a simple picture of $d$ and $f$ orbital fillings.

Figure 2

Representative ST-FMR resonance curves at 9 GHz for (a) a Gd(10)/Hf(1.5)/Py(5)/${\mathrm{AlO}}_x$ sample (numbers in parentheses are thicknesses in nm), (b) a Dy(10)/Hf(1.5)/Py(5)/${\mathrm{AlO}}_x$ sample, and (c) a Ho(10)/Hf(1.5)/Py(5)/${\mathrm{AlO}}_x$ sample. (d) Representative trace for a Fe(5)/Lu(10)/${\mathrm{AlO}}_x$ sample at 11 GHz. The higher frequency used for Lu is due to the larger effective magnetization of the Fe layer in the Fe/Lu as compared to the Py used for all other samples. The sign of the signals generated by the Lu sample can be compared directly to the others despite the inverted order of the Lu sample (ferromagnet on the bottom, which would ordinarily cause a sign change), because the sign of the anisotropic magnetoresistance in Fe is also reversed relative to the Py in the other samples. The consistent signs of the symmetric components indicate that all four rare-earth systems exhibit a positive spin Hall effect (the same sign as Pt). The sign of the antisymmetric component of the resonance is reversed in the Ho sample relative to the others.

Figure 3

Evolution of damping of a Ho(10)/Hf(1.5)/Py(5)/${\mathrm{AlO}}_x$ stack at 6 GHz with applied dc current. The linear component yields information about the spin Hall effect, while the quadratic background indicates heating.

Figure 4

Experimental estimates of the in-plane spin Hall (blue circles) and out-of-plane effective field (red squares) torque ratios in the Ho (10 nm)/Hf ($t_{\mathrm{Hf}}$ nm)/Py (5 nm)/${\mathrm{AlO}}_x$ multilayer structure using ST-FMR. For comparison, we also plot the in-plane torque ratios for measurements of a Hf ($t_{\mathrm{Hf}}$ nm)/Py (5 nm)/${\mathrm{AlO}}_x$ multilayer for $t_{\mathrm{Hf}}=1.5$ nm and 5 nm.

Figure 5

For the $f$-in-core picture: (a) Band structure for Lu (with Fermi energy ${\epsilon }_f$ at 0 eV) and spin Berry curvature ${\mathrm{\Omega }}_{\alpha ,\beta }^s\left(k\right)$ (b) calculated spin Hall conductivities.

Figure 6

For Lu in the $f$-in-band picture: (a) Projected density of states [(PDOS) (states/eV-atom)] (b) the band structure (top) and spin Berry curvature ${\mathrm{\Omega }}_{\alpha ,\beta }^s\left(k\right)$ (bottom) with $U_f=5.5$ eV and Fermi energy ${\epsilon }_f$ at 0 eV (c) calculated spin Hall conductivity $[{\sigma }_S^{\mathrm{DFT}}$ ($\hslash /2e\mathrm{m}{\mathrm{\Omega }}^{-1}{\mathrm{cm}}^{-1}$)] vs Fermi level.

Figure 7

For Ho in the $f$-in-band picture: (a) Projected density of states (‘PDOS’ (states/eV-atom)) (b) the band structure (top) and spin Berry curvature ${\mathrm{\Omega }}_{\alpha ,\beta }^s\left(k\right)$ (bottom) with $U_f=4.9$ eV and Fermi energy ${\epsilon }_f$ at 0 eV (c) calculated spin Hall conductivity (${\sigma }_S^{\mathrm{DFT}}$ ($\hslash /2e\mathrm{m}{\mathrm{\Omega }}^{-1}{\mathrm{cm}}^{-1}$)) vs. Fermi level.

Figure 8

The calculated spin Hall conductivity, ${\sigma }_s^{\mathrm{DFT}}$, for Dy, Ho and Lu using DFT-LDA with varying $U_f$ ($f$-in-band). As the $4f$ states become more localized with increasing $U_f$, we find that the calculated ${\sigma }_s^{\mathrm{DFT}}$ decreases for both Dy and Ho. For Gd and Lu we find only a modest change in the calculated ${\sigma }_s^{\mathrm{DFT}}$. The yellow region spans the values of ${\sigma }_s^{\mathrm{DFT}}$ calculated in the $f$-in-core picture [Fig. 5].

Figure 9

(a) Projected density of states of Gd, Dy, Ho, and Lu. (b)–(e) ${\sigma }_s^{\mathrm{DFT}}$ ($\hslash /2e\mathrm{m}{\mathrm{\Omega }}^{-1}{\mathrm{cm}}^{-1}$) vs Fermi level ${\epsilon }_f$ for each element.

Figure 10

Experimentally measured (a) spin Hall torque ratio and (b) spin Hall torque conductivity of Gd, Dy, Ho, and Lu. The Gd, Dy, and Ho devices have a 1.5 nm layer of Hf between the RE metal and ferromagnet. (c) Spin Hall conductivities for the same elements calculated using DFT+$U$ with values of $U_{\mathrm{f}}$ taken from the work of Topsakal and Wentzcovitch [48].

Figure 11

Damping of Ho (10 nm)/Hf ($t_{\mathrm{Hf}}$)/Py (5 nm)/${\mathrm{AlO}}_x$ stacks for $t_{\mathrm{Hf}}=0.5$, 1, 1.5, 2, and 4 nm (red circles). Fitting to $\alpha =\mathrm{offset}+\mathrm{A}{e}^{-t/{\lambda }_{\mathrm{sd}}}$ (blue line) yields offset = 0.0071, A = 0.0087, and ${\lambda }_{\mathrm{sd}}=0.81$ nm.