Direct and Inverse Spin Splitting Effects in Altermagnetic RuO2

Abstract Recently, the altermagnetic materials with spin splitting effect (SSE), have drawn significant attention due to their potential to the flexible control of the spin polarization by the Néel vector. Here, the direct and inverse altermagnetic SSE (ASSE) in the (101)‐oriented RuO2 film with the tilted Néel vector are reported. First, the spin torque along the x‐, y‐, and z‐axis is detected from the spin torque‐induced ferromagnetic resonance (ST‐FMR), and the z‐spin torque emerges when the electric current is along the [010] direction, showing the anisotropic spin splitting of RuO2. Further, the current‐induced modulation of damping is used to quantify the damping‐like torque efficiency (ξ DL) in RuO2/Py, and an anisotropic ξ DL is obtained and maximized for the current along the [010] direction, which increases with the reduction of the temperature, indicating the present of ASSE. Next, by way of spin pumping measurement, the inverse altermagnetic spin splitting effect (IASSE) is studied, which also shows a crystal direction‐dependent anisotropic behavior and temperature‐dependent behavior. This work gives a comprehensive study of the direct and inverse ASSE effects in the altermagnetic RuO2, inspiring future altermagnetic materials and devices with flexible control of spin polarization.


Supplementary 5
The VISHE resulting from the spin pumping is directly proportional to the cross product of the spin current Js (parallel to the z-axis) and the spin polarization vector  (parallel to the M).In spin pumping experiments, the generation of Js is derived from momentum transfer of magnetization procession.Meanwhile, the component of dynamic magnetization (mainly contain my and mz) also be affected.One can estimate V ISHE by dynamic magnetization.Therefore, V ISHE can be expressed as follow: Where Re(my), Im(my), Re(mz) and Im(mz) are the real and imaginary parts of my and mz.H is the angle of the applied magnetic field H relative to the y-axis [Fig.4(b)].In the measurement configuration, the rectification voltage aligns with the inductive microwave current in Py jPy(t), which can be expressed as The dynamic magnetization is linked to the dynamic magnetic susceptibility and driving fields [1,2] , which can be written as Where  I (a I ) represents the complex diagonal (off-diagonal) dynamic magnetic susceptibility attributed to in-plane excitation,  O (a O ) is the complex diagonal (offdiagonal) dynamic magnetic susceptibility associated with out-of-plane excitation, and is the phase shift between the dynamic magnetization and ℎ Oe RuO 2 ( ℎ o ).
Applicable to the out-of-plane excitation used in Fig. 4  For the out-of-plane excitation employed, it is possible to distinguish the symmetric components of VISHE and AMR due to the distinct angular dependencies revealed in supplementary Eq. ( 6) and (7).Notably, VISHE reaches maximum at H = 0 (M ⊥ the stripes), while all other signals are equal to zero.This geometric configuration provides us with the opportunity to accurately measure pure spin pumping signals in the RuO2/Py system.Therefore, the expressions for VA,SP and VS,SP [Eqs.( 5) and ( 6) in the main text] are derived to fit the experimentally measured DC voltage.Theses equations perfectly explain the experimental data, explicitly demonstrating the angular dependence of the DC voltage resulting from both in-plane and out-of-plane excitations.

Supplementary 6
The IASSE is studied by the spin pumping measurement in the stack of ( 101  the anisotropic spin splitting in the band structure of RuO2 [3,4] .In the scenario of = 90, as shown in Fig. S8, a similar trend of VS,SP and α with T variation is demonstrated.The pronounced temperature-dependent spin-pumping signal observed in RuO2 stands in contrast to that seen in heavy metals or oxides. 10 4 ( m) -1 , which is consistent with the previously reported [5] .For a single-domain RuO2, it is predicted that    is more than an order of magnitude larger than    .In the future, the nanoscale devices approaching single domain limit will significantly improve the    resulting from the spin splitting effect, as predicted by A. Bose et al. [6] Supplementary 9

Supplementary 7
where, ΔH is FMR linewidth, ΔH0 is the inhomogeneous linewidth, which is independent of microwave frequency,  = 2f is microwave angle frequency, γ is the gyromagnetic ratio and  is the dimensionless Gilbert damping constant.The fitting coefficients show ΔH0 = 2.527 Oe and  = 0.00777.For the transition of DL and V SP sym at low temperature, we have performed Raman spectroscopy measurements at different temperatures with a micro-Raman spectrometer (Horiba LabRAM HR Evolution).A solid-state laser at 532 nm and below 100 μW is focused onto the samples along the z direction by a × 100 objective with a spot size less than 1 μm.The spectrum is collected at different temperatures for TiO2//RuO2(15 nm), as shown in Fig. S13.
Each cell of TiO2 with rutile structure contains two TiO2 molecules, which belong to the P42/mnm space group and Raman vibration is A1g + B1g + Eg.The vibration of the 143 cm -1 peak B1g is weak, the vibration of the 447 cm -1 peak Eg and the vibration of the 612 cm -1 peak A 1g is strong, which are the characteristic peaks of rutile TiO 2 //RuO 2 [7, 8] .As for the wide spectral band at 239 cm, Hara et al [9] believe that it is induced by the large lattice disorder of rutile, and it may also be caused by multi-level scattering or distortion.The position of the 612 cm -1 peak A1g and the 143 cm -1 peak B1g show no change basically, while the 447 cm -1 peak Eg appears as a distinct blue shift.From the point of view of the peak intensity of the three peaks, the peak intensity gradually decreases with the temperature decreasing from 300 K to 140 K, and reaches the weakest point at the critical temperature with 140 K.Then, with the further decrease of  Further to clarify the detailed relationship between the crystal phase transition of the TiO₂ substrate and the altermagnetic spin splitting effect, first-principles calculations of the spin splitting energy in band of RuO2 is performed.First-principles calculations were performed based on density functional theory (DFT) [10] as implemented in the Vienna ab initio simulation package (VASP). [11,12] he pseudopotentials were described using the projector augmented wave (PAW) method, [13] and the exchange-correlation functional was treated within the generalized gradient approximation (GGA) developed by Perdew-Burke-Ernzerhof (PBE). [14]In the calculations for RuO2, the cutoff energy for the plane-wave expansion was set to 500 eV, and the k-point grid was set to 16 × 16 × 16 to sample the irreducible Brillouin zone.Besides, The GGA+U [15,16] method with Ueff = 2 eV on Ru 4d orbitals was employed.The lattice parameters and the atomic coordinates were relaxed until the force on each atom was less than 0.001 eV/Å for the calculation of the band structure.
We have conducted calculations with a 2% expansion/compression of the optimized lattice to investigate the influence of the unit cell volume on the spin splitting in band structure of RuO2.As depicted in Fig. S14 S14(c)], the E decreases to 0.89 eV with the SST effect decreases.This is consistent with the results obtained from our experiments.As the temperature decreases from 300 K to 100 K, the TiO2 substrate induces a compressive strain on RuO2, causing an increase in its spin-splitting energy band, and the damping-like torque efficiency and IASSE of RuO2 are increase.At temperatures below 100 K, the TiO2 substrate induces tensile strain on RuO2, leading to a decrease in its spin-splitting energy band and a reduction in the damping-like torque efficiency and IASSE.

Fig
Fig. S3 shows the dependence results on the crystal angle C.The resistivity xx is ~ 130  cm for 15 nm thick (101)-RuO2 film at room temperature.

Fig. S5 shows
Fig. S5 shows the ST-FMR measurement for (101)-RuO2/Py prepared using the PLD method.Fig. S5(b) shows the detected ST-FMR signal Vmix,ST as a function of the applied in-plane magnetic field H for the (101)-oriented RuO2(15)/Py(8)/MgO(2)/Ta(3) (thicknesses in nanometers) sample at H = 40and C = 0, with a microwave of 18 dBm and 7 GHz.Figs.S5(c) and S5(d) depict the results of V S and V A measurements at C = 90, respectively.The obvious y-spin polarization (τ y DL ) ∝ cos  H sin 2 H from the conventional spin Hall effect (SHE) is due to the strict orthogonal relationship among the applied charge current, generated spin current and the spin polarization.The value of |S x DL /S y DL | and |S z FL /S y DL | is 0.015 and 0.14, respectively, indicating that the major contribution from y.However, for the applied current along the [010] direction with C = 0, the corresponding data in Figs.S5(e) and S5(f) cannot be fitted by only considering the S y DL cos φ sin 2φ term.By fitting the Eq.(2), the enhanced amplitude )-RuO2(15)/Py(20)/MgO(3)/Ta(3), where a 20 nm Py film is used to ensure a sufficiently strong signal in FMR measurement, as shown in Fig.S6a.The VS,SP signal as a function of H is extracted from the Lorentzian line-shape to acquire the VS,O,SP by Eq. (6), as shown in FigsS6(c) and S6(d), corresponding to C = 90 and 0, respectively.For the pure SP contribution, the obtained value of V S,O,SP is 0.56 V in Fig.S6cwith  C = 90 owing to the conventional spin-to-charge (σy(x)) conversion with the x-axis spin polarization, without the spin polarization component from N in yz-plane (the schematic diagram of Fig.S6(a).Note that the Js is along z-axis and the detected DC voltage V90° is along the y-axis.In contrast, the VS,O,SP in the case with C = 0 is increased to 1.07V in Fig.S6dwhen V0° is detected along the x-axis.Here, the spin polarization in y-axis (σy) includes the conventional σy(y) and N-dependent σNy(y), indicating that the enhanced charge current arises from the IASSE-induced spin-to-charge conversation in RuO2 layer.

Fig. S6 .
Fig. S6.Angular dependence of DC voltages for the spin pumping measurement.(a) The actual depiction (top) and schematic diagram (bottom) of the spin pumping measurement for device configuration with an out-of-plane microwave excitation field HRF in (101)-RuO2/Py.H is the angle of the applied magnetic field H relative to the y-axis and the layout includes two orthogonal directional components V0° and V90°.N is angle between the N and the z-axis.(b) The DC voltage signals (Vmix,SP) obtained for (101)-RuO2(15)/Py(20)/MgO(3)/Ta(3) sample at H = 40 the spin pumping device layout along the C = 0, where the fitting results include symmetric and antisymmetric parts.Symmetric voltage amplitudes (VS,SP) as a function of angle H for (c) C = 90 and (d) C = 0, respectively.Black plots and curves show the raw data and fitting parameters, contributed by the VS,I (blue), VS,O (yellow), and VS,O,SP (red), fitting by the Eqs.(5) and (6).(e) The extracted pure spin pumping voltage signals (VS,O,SP) as a function of H.(f) Microwave power dependence of the extracted symmetric voltage component (V S,SP ) for both  C = 0 and  C = 90 orientations.Different from the samples with 8 nm Py above, here, the 20 nm Py is used.And the SP signal decreases relative to the AMR-spin rectification signal, due to the increase of Py layer thickness aligning with the experimental expectations.Indeed, one can see that the crystal orientation-dependent IASSE of RuO2 confirms the feature of IASSE with the Né el vector-dependent spin-to-charge conversion.The result validates the reliability of our data.Furthermore, the voltage signal with  H = 0 is predominantly

Fig. S8 . 8 Fig. S9 .E
Fig. S8.(a) Voltage signals recorded from the RuO2(15)/Py(20)/MgO(3)/Ta(3) sample for C = 90 at 8 GHz and 300 K under different microwave power levels ranging from 17 to 23 dBm.(b) Voltage signals acquired from the RuO2(15)/Py(20)/MgO(3)/Ta(3) sample across a range of RF frequencies from 4 to 8 GHz, under microwave power of 23 dBm, at 300 K. (c) Temperature-dependent of V mix,SP under 23 dBm RF excitation for C = 90.(d) A summary of the Hr values corresponding to different temperature in (c).(e) A representative plot of H-f at various temperatures and the corresponding

Fig. S11 .
Fig. S11.(a) FMR spectra at different microwave frequencies.(b) Hr dependence of the microwave frequency.(c) FMR linewidth as a function of microwave frequency.The dots are measured data, and the lines are fitting results.

Fig. S12 . 5 . 11 Fig. S13 .
Fig. S12.(a) Schematic illustration of the ST-FMR measurement in (101)-RuO2/Py.The heterostructures are patterned into long stripes within dashed blue rectangles.(b) Schematic illustration of the spin pumping measurement for device configuration with an out-of-plane microwave excitation field HRF in (101)-RuO2/Py.(c) The schematic of the spin pumping measurement for film with an in-plane HRF in (101)-RuO2/Py.
temperature, the peak intensity increases.The above results show that there is structural distortion or energy band change at the range of 140 K to 120 K, which leads to the change of Raman scattering peak intensity, which can further explain the change of DL and V SP sym value with temperature.
(a), with a lattice compression of ξ = -2%, the energy of spin splitting in the band (ΔE) is 1.11 eV, indicating a larger band splitting compared to the case of ξ = 0 (0.99 eV) [Fig.S14(b)], suggesting an increase in SST relative to the ξ = 0 case.However, for a lattice expansion of  = 2% [Fig.
in the main text.