A Strategy to Modulate the Bending Coupled Microwave Magnetism in Nanoscale Epitaxial Lithium Ferrite for Flexible Spintronic Devices

Abstract With the development of flexible electronics, the mechanical flexibility of functional materials is becoming one of the most important factors that needs to be considered in materials selection. Recently, flexible epitaxial nanoscale magnetic materials have attracted increasing attention for flexible spintronics. However, the knowledge of the bending coupled dynamic magnetic properties is poor when integrating the materials in flexible devices, which calls for further quantitative analysis. Herein, a series of epitaxial LiFe5O8 (LFO) nanostructures are produced as research models, whose dynamic magnetic properties are characterized by ferromagnetic resonance (FMR) measurements. LFO films with different crystalline orientations are discussed to determine the influence from magnetocrystalline anisotropy. Moreover, LFO nanopillar arrays are grown on flexible substrates to reveal the contribution from the nanoscale morphology. It reveals that the bending tunability of the FMR spectra highly depends on the demagnetization field energy of the sample, which is decided by the magnetism and the shape factor in the nanostructure. Following this result, LFO film with high bending tunability of microwave magnetic properties, and LFO nanopillar arrays with stable properties under bending are obtained. This work shows guiding significances for the design of future flexible tunable/stable microwave magnetic devices.

hours, the LSMO layer got etched out, and the LFO layer was peeled off from the STO substrate. Hence, we got the LFO/PDMS samples, which are shown in Figure S1. The microstructure and morphology of the LFO nanopillar (NP-LFO) were investigated by transmission electron microscope (TEM JEM-2100) and scanning electron microscope (SEM Quanta F250). The NP-LFO arrays were firstly filled by carbon and then coated by Pt for better resolution in TEM test. The magnetic hystersis loops were measured at room temperature (300K) by the vibrating sample magnetometer of Lake Shore. The electron spin resonance (ESR) measurements were performed using the JEOL, JES-FA 300 spectrometer (X-band at 9.0 GHz, power of 1 mW). The ferromagnetic resonance (FMR) absorption spectra were measured using a lock-in technique based on sweeping of a static external magnetic field superimposed over the ac magnetic field. Measurements for the bending samples were carried out by attaching the samples to acrylonitrile butadiene styrene (ABS) plastic rods with different radius from 2.5 mm to 7.5 mm. The rods were cut as molds with less than 1 mm thick to ensure that all samples were put in a uniform field during the FMR test.
Bending Cycling: One side of the PDMS substrate is fixed on the plastic rod (radius=4 mm), while the other side is attached and then peeled as a bending cycling. Hence, we repeat this process for 1000 times.

Derivation of the F MAE , the F ME and the Fitting Equations
The Magnetocrystalline Anisotropy Energies F MAE : According to the previous reports, [4] the magnetocrystalline anisotropy energies F MAE can be expressed by the anisotropy constant K 1 and K 2 : .  Figure 1g), α u , α v and α w can be expressed as: α 1 /β 1 /γ 1 are the included angles between x and u/v/w. α 2 /β 2 /γ 2 are the included angles between y and u/v/w. α 3 /β 3 /γ 3 are the included angles between z and u/v/w. Thus, F MAE represents the anisotropy energies for the epitaxial films with different in-plane (IP) and out-of-plane (OOP) crystallographic orientations. Table S2 shows the expressions of α u , α v and α w with the crystal orientation.

The Fitting Equations:
The expression for the FMR frequency can be derived by Finally, we obtained the expressions in the main text as the Equation (2)  The Magnetoelastic Energies F ME : To simplify the result, here we only discussed the magnetoelastic energy for the LFO with (001) OOP orientation. The magnetoelastic energy was deduced using the orthogonal normal stress model by M. Gueyue et al. [5] : where σ xx , σ yy and σ zz are the orthogonal normal stresses along x, y and z axis, respectively. λ is the isotropic magnetostriction coefficient (λ 100 = -27.8ppm).
In this paper, the 150-nm-thick transferred LFO film on the PDMS substrate should be relaxed according to the XRD result and the former reports. [3] Therefore, the magnetoelastic energy should not contribute a lot to the FMR result of the unbent film. Here, we mainly discuss the pure strain induced by bending, in order to figure out whether it can influence the MA in FMR measurements or not. Normally, the strain induced by bending can be calculated as: Where t is the total thicknesses of the LFO film and the PDMS tape (t=150nm+100 μm≈100 μm), η is the thickness ratio of the LFO film to the PI substrate (η=0.0015), χ is elastic moduli ratio between the LFO film and the PI substrate (χ ≈ 17400). Then we can obtain the relationship between the bending radius r and the strain along the the bending axis y (ε yy ). In the bending model, ε xx =ε zz =-vε yy , where v≈0. 3. ε xx and ε zz present the strain along x and z axis, respectively. The Young's modulus Y along the [100] direction of the LFO is 1.74 × 10 12 dyne cm −2 . Hence, the F ME can be characterized as a function of r, θ H , φ M : By plugging the parameters for LFO (001) that we discussed above in Equation (11) and (12) in the Supporting Information, the equation can be further derived as: Then, based on the formulas, the simulation of the angular θ M dependent H r for the T-LFO (001) film under different degrees of pure tensile strain along y axis was present in Figure S12.
It reveals that the pure tensile strain along y axis can theoretically decrease the H r of the LFO film, which is similar to the result by W. Liu et al. However, the pure tensile strain might not have a huge influence on the property of the bending film, due to the poor transfer efficiency of strain. [6] Even for the film under a bending radius of ~2.6 mm, the effective strain ε eff is calculated to be less than 0.03%, and the corresponding change is much smaller than that calculated from bending induced misorientation effects. Therefore, we argue that the F ME should not be the main cause for the bending modulated H r in our experiment.

Discussion about the contribution from H k1 and H k2 to the MA of LFO films
H K1 and H K2 are the anisotropy fields that mainly depend on the magnetocrystalline anisotropy. Firstly, according to the Equations (2) and (3) in the manuscript, it can influence the MA of LFO films. As shown in Figure S13a, where we provide the simulation result for T-LFO (001) with and without considering the contribution from H K1 and H K2.
Secondly, according to Table S1, the different crystalline orientation could influence the MA properties in FMR, even with the same value of H K1 and H K2 . Figure S13b presents the simulation results for T-LFO (001) and T-LFO (110) using the same parameters from Figure   1h, which are different. The differences become larger when H eff becomes smaller, as shown in Figure S13c. In this paper, although films with the same thicknesses and different orientations are provided, the differences in M s are very large due to the different growth mechanism. Therefore, we argue that it is not a feasible method to modulate the MA in FMR by just changing the crystalline orientation of the film.