# Fileset

[Kulkarni_supplementary.pdf](https://mdr.nims.go.jp/filesets/8191e34c-c056-40d5-9b8b-632053537a74/download)

## Creator

[Prabhanjan D. Kulkarni](https://orcid.org/0000-0002-4605-5256), [Tomoya Nakatani](https://orcid.org/0000-0001-9590-216X)

## Rights

This article may be downloaded for personal use only. Any other use requires prior permission of the author and AIP Publishing. This article appeared in Prabhanjan D. Kulkarni, Tomoya Nakatani; Tunnel magnetoresistive sensors with non-hysteretic resistance–magnetic field curves using noncollinear interlayer exchange coupling through RuFe spacers. Appl. Phys. Lett. 14 October 2024; 125 (16): 162405, and may be found at https://doi.org/10.1063/5.0231451.[In Copyright](http://rightsstatements.org/vocab/InC/1.0/)

## Other metadata

[Tunnel magnetoresistive sensors with non-hysteretic resistance-magnetic field curves using noncollinear interlayer exchange coupling through RuFe spacers](https://mdr.nims.go.jp/datasets/79a14d49-3d8d-4d7e-bc80-bacf2749f756)

## Fulltext

Microsoft Word - Supplementary_Kulkarni.docx1  Supplementary material  Tunnel magnetoresistive sensors with non-hysteretic resistance-magnetic field curves using noncollinear interlayer exchange coupling through RuFe spacers  Prabhanjan D. Kulkarni, and Tomoya Nakatani†  Research Center for Magnetic and Spintronic Materials, National Institute for Materials Science, 1-2-1 Senen, Tsukuba, Ibaraki 305-0047, Japan  †Corresponding author: nakatani.tomoya@nims.go.jp    1. Calculation of R-H curve The R-H curves of the spin-valve structure shown in Fig. S1(a) were calculated by minimizing the total magnetic energy in the single magnetic domain regime, the Stoner-Wohlfarth model. Figure 1(b) shows the magnetization configuration under an external magnetic field, H, in the +x direction. mxx denotes the magnetization of each ferromagnetic (FM) layer. mPL1 and mPL3 are pinned in the +x direction, and mRL is pinned in the –x direction through the antiparallel IEC through the Ru spacer between RL and PL3. mPL2 is pinned in either ±θud titled direction from mPL1 by the noncollinear IEC through the RuFe spacer. Then, FL receives a unidirectional magnetic anisotropy in the same direction as mPL2 by the orange-peel FM coupling through the AgSn spacer. In addition, the FL has a uniaxial anisotropy in the ±θua titled direction from mPL1 which is induced by the annealing under a magnetic field. An external magnetic field H is applied in the ±x direction.   2    FIG. S1. (a) Layer structure and the energy terms. (b) Configuration of the magnetizations (m) of the ferromagnetic layers. θua and θud are the angles between the easy axes of the uniaxial and unidirectional anisotropies, respectively, and mPL1.   We define M as the saturation magnetization, t as the layer thickness, and θ as the angle between the magnetization and H for all the FM layers: PL1, PL2, FL, RL, PL3. We considered the following energy terms. (1) Zeeman energy: 𝐸 =  −𝐻(𝑀 𝑡 cos 𝜃 + 𝑀 𝑡 cos 𝜃 + 𝑀 𝑡 cos 𝜃 +𝑀 𝑡 cos 𝜃 + 𝑀 𝑡 cos 𝜃 ).   (S1) (2) Exchange bias to PL1 and PL3: 𝐸 =  − 𝐽 cos(𝜃 ) − 𝐽 cos(𝜃 ),   (S2) where 𝐽   and 𝐽   represent the exchange bias energies at the IrMn/PL1 and PL3/IrMn interfaces. (3) Noncollinear IEC between PL1 and PL2 through the RuFe spacer: 𝐸 =  − 𝐽 cos(𝜃 − 𝜃 ) − 𝐽 cos (𝜃 − 𝜃 ), (S3) where, J1 and J2 are the bilinear and biquadratic IEC energies, respectively. (4) Orange-peel FM coupling between PL2 and FL through the AgSn spacer: 𝐸 =  − 𝐽 cos(𝜃 − 𝜃 ),     (S4) FL barrierIrMnPL1RuFe spacerRLPL3RuIrMnPL2AgSn spacer(a)exchange bias: Jk1noncollinear IEC: J1, J2FM coupling: JfAFM coupling: J1Ruexchange bias: Jk2uniaxial anisotropy; KumRL(b)ymPL1, PL3mFLθHθuaθududxθududmPL23  where Jf is the bilinear IEC energy. (5) Uniaxial anisotropy of FL: 𝐸 =  𝐾 𝑡 sin (𝜃 − 𝜃 ),    (S5) where, 𝐾  is the anisotropy energy. (6) Antiferromagnetic IEC between RL and PL3 through the Ru spacer: 𝐸 =  − 𝐽 cos(𝜃 − 𝜃 ),    (S6) where, J1Ru is the bilinear IEC energy.   For simplicity, we assume that the magnetizations of PL1, RL, and PL3 are pinned in the original directions under the applied H; therefore, 𝜃 = 𝜃 = 0, and 𝜃 = 𝜋. Then, the total magnetic energy equation is simplified to  𝐸 = −𝐻(𝑀 𝑡 − 𝑀 𝑡 + 𝑀 𝑡 + 𝑀 𝑡 cos 𝜃 +𝑀 𝑡 cos 𝜃 ) + 𝐾 𝑡 sin (𝜃 − 𝜃 ) −  𝐽 cos 𝜃 − 𝐽 cos 𝜃 − 𝐽 cos(𝜃 − 𝜃 ) .                    (S7)  Based on the experimental data, we used the following values: 𝑀 𝑡  = 49.5 T nm, 𝑀 𝑡 = 𝑀 𝑡 = 𝑀 𝑡 = 𝑀 𝑡 = 6 T nm (1 T nm = 7.95×10-5 emu/cm2), 𝐽  = 0.1 mJ/m2, and 𝐾 𝑡  = 0.03 mJ/m2, corresponding to an anisotropy field (Hk) of 1.5 mT from the relationship of 𝐾 𝑡 = 𝐻 𝑀 𝑡 /2. For the noncollinear IEC through the RuFe spacer, J1 was varied, while J2 was fixed to –0.35 mJ/m2. This J2 value is somewhat larger than the experimentally obtained values of about –0.3 mJ/m2 for the Ru100-xFex (x = 54–70 at. %) spacers in the CIP-GMR films (Fig. 1). The choice of J2 = –0.35 mJ/m2 for the simulations was because mRL is slightly rotated for –0.3 mJ/m2, which complicates the interpretation of the results. Since the anisotropy field of the FL is determined only by Jf, the use of a larger value of J2 than that by the experiment does not significantly affect the simulation results.   Recalling that θud is the angle between the mPL1 and mPL2, and the mPL1 is pinned in the +x direction, we obtain 𝜃 = (𝜃 − 𝜃 ) = (𝜃 ) . In addition, 𝜃  follows [1, 2] 𝜃 = 0   when 𝐽 /𝐽 < −2, 𝜃 = 𝜋   when 𝐽 /𝐽 > 2, 4  𝜃 = cos (−𝐽 /2𝐽 ) when 2 ≥ 𝐽 /𝐽 ≥ −2. (S8) Therefore, for the calculations of the R-H curve, 𝜃  was varied from 0 to 𝜋 by changing J1/J2 from –2 to 2.   By minimizing Etotal in Eq. (S7), we obtained θFL at different H. The R-H curves were calculated by  𝑅 =  1/ +  cos(𝜃 − 𝜃 ) ,  (S9) where, 𝐺  and 𝐺  are the maximum and minimum values of the tunnel conductance in the parallel and antiparallel magnetization state between mFL and mRL, respectively, and 𝜃 = 𝜋. In this paper, we used Gmin = 1 and Gmax = 2.8 (i.e., TMR ratio = 180%). The calculated R-H curves for variations of 𝜃  and 𝜃  were numerically differentiated, and the sensitivity, , was calculated.  2. Supplementary data: Hysteresis of the TMR device shown in Fig. 4(b)  FIG. S2. (a) Full TMR curve of the device shown in Fig. 4(b) (θud ~ ±120° and θua = 60°), showing negligible magnetic hysteresis. (b) The same curve zoomed in small Hx range.   References [1] E.E. Fullerton and S.D. Bader, Phys. Rev. B 53, 5112 (1996). [2] P.D. Kulkarni, T. Nakatani, T. Sasaki, and Y. Sakuraba, J. Appl. Phys. 129, 213901 (2021). -10 -5 0 5 10050100150200TMR ratio (%)m0Hx (mT)-0.2 -0.1 0.0 0.1 0.23540455055TMR ratio (%)m0Hx (mT)(a) (b)H sweepdirection