Spin pumping during the antiferromagnetic–ferromagnetic phase transition of iron–rhodium

FeRh attracts intensive interest in antiferromagnetic (AFM) spintronics due to its first-order phase transition between the AFM and ferromagnetic (FM) phase, which is unique for exploring spin dynamics in coexisting phases. Here, we report lateral spin pumping by which angular momentum is transferred from FM domains into the AFM matrix during the phase transition of ultrathin FeRh films. In addition, FeRh is verified to be both an efficient spin generator and an efficient spin sink, by electrically probing vertical spin pumping from FM-FeRh into Pt and from Py into FeRh, respectively. A dramatic enhancement of damping related to AFM-FeRh is observed during the phase transition, which we prove to be dominated by lateral spin pumping across the FM/AFM interface. The discovery of lateral spin pumping provides insight into the spin dynamics of magnetic thin films with mixed-phases, and the significantly modulated damping advances its potential applications, such as ultrafast spintronics.

is along the x' direction, which generates an inductive current j Pt (t) and j FeRh (t) in Pt and FeRh in the same direction. The direction of the detected dc-voltage V is parallel to j FeRh (t), thus, a dc-voltage V originating from the anisotropic magneto-resistance effect of FeRh as well as from spin pumping should be considered. M(t) is the dynamic magnetization, M(t) = (M, m y , m z ).  is magnetization angle.
In the present measurement configuration, two main excitations are considered. One is the out-ofplane excitation due to the Oersted field from the CPW as shown in (b). In the coordinate system of (x, y, z), the driving field can be written as h = (0, 0, h O ). The other one is the in-plane Oersted field ℎ Oe Pt due to the inductive current in Pt j Pt (t) as shown in (c). ℎ Oe Pt is along the y' direction, and in the coordinate system of (x, y, z), it can be written as h = ℎ Oe Pt (cos, sin, 0). V ISHE induced by spin pumping is proportional to the cross product of the spin current J S (// z) and the spin polarization vector  ( ), i.e., [3][4][5][6][7][8][9] where Re(m y ), Im(m y ), Re(m z ), and Im(m z ) are the real and imaginary parts of m y and m z . For the present measurement configuration with the direction of the dc-voltage V parallel to the inductive microwave current in FeRh j FeRh (t), dc-voltages induced by the anisotropic magnetoresistance (AMR) of FeRh can be derived as The dynamic magnetization is connected with the dynamic magnetic susceptibility and the driving fields (for details see SI of Ref. 4,5), and can be written as where  I ( a I ) is the complex diagonal (off-diagonal) dynamic magnetic susceptibility due to the inplane excitation,  O ( a O ) the complex diagonal (off-diagonal) dynamic magnetic susceptibility due to 6 the out-of-plane excitation, and  1 ( 2 ) the phase shift between the dynamic magnetization and Thus, one can obtain the following conclusions: Especially, V ISHE is maximized at  = 0° (M is perpendicular to the stripe) and all other signals are zero. Thus, this geometry gives us a chance to determine V ISHE accurately. We have tested the above theory experimentally and have succeeded in probing ISHE signals induced by spin pumping in a Py/Pt bi-layer [3] and at an Fe/GaAs interface [8].
Therefore, we obtain the expressions of V a-sym and V sym (Equation 1 and 2 in the main text) to fit the experimentally obtained dc-voltages. As shown in Fig. 2c  The arrows indicate the temperature below which is the phase transition region.

Supplementary
where ω = 2πf is the angular frequency, γ = gμ B /ħ is the gyromagnetic ratio, g is the spectroscopic splitting factor, μ B is the Bohr magneton, ħ is the Dirac constant. Here As it is known that the intrinsic Gilbert damping α Gilbert scales linearly with U ⊥ due to the fact that both α Gilbert  Note that both of FM-FeRh and AFM-FeRh can act as spin sinks, and in both cases the spin mixing conductance is similar. The magnitude of eff ↑↓ is almost temperature independent in the investigated temperature range.
Furthermore, the spin Hall angle  SHE of FeRh can be quantified by where R is the resistance of the FeRh/Py bi-layer, I C the charge current induced by the inverse spin Hall effect, e the electronic charge, ħ the Dirac constant, w the width of the stripe,  the spin diffusion length of FeRh, d the thickness of FeRh, and J S the magnitude of spin current at the interface. Instead of  SHE , we use the product of  SHE and ,  SHE  in unit of nm, to determine the spin-to-charge efficiency [13]. Based on the condition that  of FeRh is assumed to be much smaller than d, which holds also for other AFMs [14], Supplementary Equation 9 can be rewritten as: The magnitude of J S can be determined by where  is the angular frequency ( = 2f), Re(m y ) (Re(m z )) the real part of the dynamic magnetization m y (m z ), and Im(m y ) (Im(m z )) the imaginary part of the dynamic magnetization m y (m z ).
As shown in Fig. 5d where A sym and A asym are the magnitudes of symmetric part and antisymmetric part respectively. In order to clarify the contribution of two-magnon scattering, the sample is measured during out-of-plane rotation from θ H = 0° to θ H = 90° where φ H is fixed to 45°. As shown in Supplementary Figure 18, the linewidth for the out-of-plane direction (θ H = 90°) is smaller than for the in-plane direction (θ H = 0°), which clearly proves the contribution of two-magnon scattering in the in-plane configuration [10,15,16]. The magnitude of two-magnon scattering is predicted to be enhanced during the phase transition, through comparing the length of the two arrows for the 310 K case and the 360 K case shown close to 0° in Supplementary Figure 18. This result is consistent with the increased α 2M (T) related to two-magnon scattering when decreasing the temperature, as discussed in Fig. 6d of the main text.
Supplementary Figure 19 displays the frequency dependent resonance field μ 0 H R and linewidth where μ 0 H R has a linear relationship with f. By fitting the data using Supplementary Equation 13, μ 0 M eff can be directly read as the intercept at 0 GHz. The increase of μ 0 M eff at lower T is consistent with that measured in the in-plane configuration (Fig. 4c of the main text), regardless of the slightly larger value. g is estimated to be in the range between 2.14 and 2.19 for both samples, without clear temperature dependence [18,19]. In addition, through linear fitting of f-dependent μ 0 ΔH in Supplementary Figure 19b  FeRh and Pt. Note that no full film-FMR signal is detected when t is reduced to 5 nm. The high sensitivity of spin pumping measurements presents a great advantage beyond standard full film-FMR