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[Chi Fang](https://orcid.org/0000-0001-6827-1913), Caihua Wan, Xiaoyue Zhang, [Satoshi Okamoto](https://orcid.org/0000-0002-0493-7568), [Tianyi Ma](https://orcid.org/0000-0002-5987-6459), Jianying Qin, Xiao Wang, Chenyang Guo, Jing Dong, [Guoqiang Yu](https://orcid.org/0000-0002-7439-6920), [Zhenchao Wen](https://orcid.org/0000-0001-7496-1339), [Ning Tang](https://orcid.org/0000-0003-2576-523X), [Stuart S. P. Parkin](https://orcid.org/0000-0003-4702-6139), Naoto Nagaosa, [Yuan Lu](https://orcid.org/0000-0003-3337-8205), [Xiufeng Han](https://orcid.org/0000-0001-8053-793X)

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This document is the Accepted Manuscript version of a Published Work that appeared in final form in NANO LETTERS, copyright © 2023 American Chemical Society after peer review and technical editing by the publisher. To access the final edited and published work see https://doi.org/10.1021/acs.nanolett.3c03085[In Copyright](http://rightsstatements.org/vocab/InC/1.0/)

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[Observation of the fluctuation spin Hall effect in a low-resistivity antiferromagnet](https://mdr.nims.go.jp/datasets/f0e848be-21bc-4cce-bbde-08501ebd35b5)

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Template for Electronic Submission to ACS JournalsSupporting InformationObservation of Fluctuation Spin Hall Effect in Low-resistive AntiferromagnetChi Fang1,2, Caihua Wan1,7, Xiaoyue Zhang3,5, Satoshi Okamoto4, Tianyi Ma1,3, Jianying Qin1,3, Xiao Wang1, Chenyang Guo1, Jing Dong1, Guoqiang Yu1,7, Zhenchao Wen6, Ning Tang5, Stuart S. P. Parkin2, Naoto Nagaosa9, Yuan Lu3, Xiufeng Han1,7,81Beijing National Laboratory for Condensed Matter Physics, Institute of Physics, University of Chinese Academy of Sciences, Chinese Academy of Sciences, Beijing 100190, China2Max Planck Institute of Microstructure Physics, Halle (Saale) 06120, Germany3Université de Lorraine, CNRS, Institut Jean Lamour, UMR 7198, campus ARTEM, 2 Allée André Guinier, 54011 Nancy, France4Materials Science and Technology Division, Oak Ridge National Laboratory, Oak Ridge, TN 37831, USA5State Key Laboratory of Artificial Microstructure and Mesoscopic Physics, School of Physics, Peking University, Beijing 100871, China6National Institute for Materials Science NIMS, Tsukuba, Ibaraki 3050047, Japan7Songshan Lake Materials Laboratory, Dongguan, Guangdong 523808, China8Center of Materials Science and Optoelectronics Engineering, University of Chinese Academy of Sciences, Beijing 100049, China9RIKEN Center for Emergent Matter Science (CEMS), Wako, 351-0198, Japan.ContentI. Sample preparation and characterization A. Sample Preparation B. Magnetization CharacterizationC. Resistivity determination of CrII. Transport data of spin Hall effectsA. ISHE and DSHE with in-plane fieldB. ISHE and DSHE with out-of-plane magnetic fieldC. ISHE with different thickness of CrD. ISHE with different current amplitudeE. Exaction of RISHE or RDSHEF. RISHE data from other devicesG. Shunting factorH. Temperature dependence of spin Hall conductivityIII. Normalized Power Consumption IV. Theoretical consideration on the SHE by AFM spin fluctuationV. Fitting details of resistivity and spin Hall resistivityReferenceI. Sample preparation and characterization A. Samples Preparation The growth details are following: Before deposition, the MgO(001) substrate was annealed at 700 °C for one hour to degas the surface, followed by depositing a 10 nm MgO seed layer at 450 °C to block the C impurity diffusion. The Cr layer was then deposited at 31 °C and followed by an in-situ post-annealing at 600 °C to obtain a flat Cr (001) surface. The temperature was maintained at 78 °C and 57 °C to deposit the MgO and Fe layers. A second post-annealing at 480 °C was performed to improve the crystalline quality of Fe. In the end, a 5 nm Au capping layer was deposited at 88 °C to prevent the surface oxidation. Surface structures of each stack were monitored by reflected high-energy electron diffraction (RHEED) throughout the growth process as shown in Fig. S1.Figure S1.  Typical RHEED pattern of MTJ film. RHEED pattern of the annealed Cr, MgO and Fe layers further confirmed the epitaxial growth mode. The sharp streaky patterns indicate a high-quality epitaxial growth character of our sample. The 10 nm Cr for resistivity calibration is fabricated by etching the Fe/Au away monitored by in-situ HIDEN Analytical system installed in etching & depositing system (scia coat) and stoped at MgO layer. The AlOx is in-situ deposited to prevent oxidization.The high-resolution cross-sectional transmission electron microscope (HRXTEM) was used to check the MTJ structure (Figure 1b). The low-mag image validated uniform thickness for each layer well corresponding to the nominal one. The black field image (inset of Figure 1b) allows us to measure the lattice constant of each layer. The lattice constants of the bcc Cr and Fe near interfaces were determined as aCr=2.91  and aFe=2.88 , close to their bulk values 2.88 and 2.87 S1. aMgO=4.24   reported in Ref. 32 was near aCr=4.11 and aFe=4.06, which implied the epitaxial relationship: (001)MgO//(001)Cr/Fe and [100]MgO//[110]Cr/Fe 32. The electron energy loss spectroscopy (EELS) was performed to map the elementary distributions (Fig. S2). There was negligible interdiffusion for the Cr, Fe, Mg and Au elements between neighboring layers.Figure S2. The electron energy loss spectroscopic (EELS) mapping of the Cr, Fe, Mg and Au elements.B. Magnetization CharacterizationMagnetization of stacks is determined by superconducting quantum interference device (SQUID, Quantum Design MPMS) at room temperature and low temperature as shown in Fig. S3. The Magnetic hysteresis loop in Fig. 1(d) is measured by vibrating sample magnetometer(VSM, Lakeshore) at room temperature.Figure S3. Magnetization in multilayer. (a) Magnetic hysteresis loop from 10 K to 350 K in Cr(25)/MgO(2)/Fe(10)/Au(5 nm) stack. (b) Temperature dependence of saturation magnetization Ms extracted from (a). The calculated spin polarization magnitude is shown by the right axis. (c) Cr thickness tCr dependence of Ms of stacks at 300 K. The Ms is irrelevant to the tCr.As current is polarized in Fe layer, the spin polarization is assumed proportional to the saturation magnetization Ms of FeS2,S3. The Ms follows a T 3/2 lawS1 as Ms(T)= Ms(0)(1-C T3/2). Fitting the measured Ms-T gives Ms(0)=1827 emu/cc and C=1.11E-5. As P in single crystal Fe/MgO electrode is reported39 as 0.74 at 0.25 K by superconducting tunnelling spectroscopy (STS) technique. The spin polarization P is determined as P(T)=0.74 Ms(T)/Ms(0).C. Resistivity determination of CrAs the TEM Fig. S4 shows, the film maintains a well-defined single-crystal quality in large scale. So the contribution to resistivity in Cr from grain border is ignorable. Figure S4. TEM picture of stacks for larger scale.The main contribution to the resistivity in Cr with different thickness is the surface or interface in the stacks. In this case, we could follow Fuchs-Sondheimer’s theory to calibrate the resistivity of Cr layerS4,S5.Below the mean free path  of bulk material, the conductivity in thin films follow:      (R1)where  is the bulk resistivity and p the a parameter describing the fraction of surface collisions. The  is derived as 15.2 nm in resistive Cr filmsS6. The  vould be longer in single crystal Cr. We prepared 4-probe bar with length l=50 µm and width w=20 µm to measure the conductance of the full stacks.  A tCr= 15 nm sample is added for the resistivity calibration. Based on the parallel circuit model, the conductance of full stack device is      (R2)where  is the conductance of Cr layer depending on the Cr thickness as  and  is the conductance of the capping  staying the constant. Combing equation (R1) and (R2), we have where  and .  Fitting the  relation linearly gives  as show in Fig. S5a. To give the two terms of b respectively, we use the resistivity of single 10 nm Cr layer. The  gives a negative resistivity in 7 nm Cr. So the Cr could be oxidized partially at the surface. We further fabricate a 10 nm single Cr by etching the  capping layers away and depositing AlOx capping layer on it to prevent oxidation. The etching process, as the tunnel junction fabrication, is monitored by HIDEN Analytical system and stopped in the middle of MgO layer. The  is determined in 10 nm Cr at 300 K. So the [µ-1cm-1] or   [µ cm] where  is similar to bulk value of Cr 13 µ cm at room temperatureS7. . By substracting  from , the resistivity is determined as 17.59 µ cm in 25 nm Cr and 15.51 µ cm in 50 nm Cr as shown in Fig. S5b.Figure S5. Calibration of Cr resistivity. (a) Thickness dependence of total conductance of Cr/MgO/Fe/Au stacks. (b) Estimated resistivity in Cr with different thicknesses.II. Transport data of spin Hall effectsA. ISHE and DSHE with in-plane fieldOriginal data of ISHE and DSHE is plotted in Fig. S6a and b. The magnetic field is applied in-plane along [110]MgO direction as depicted in Fig. 2a. A random background resistance exists with each curve and is subtracted, which does not involve with the value of RISHE(DSHE).Figure S6. (a) ISHE and (b) DSHE in 10 nm Cr with in-plane magnetic field along MgO [110] at different temperatures. The RISHE(DSHE) values are extracted and plotted in Fig. 2d.B. ISHE and DSHE with out-of-plane magnetic fieldOne could control the output voltage  or resistance () by MFe with an external field H. When H is applied out-of-plane along the film normal,  vanishes because of the zero  term. Figure S7. Spin Hall signal with out-of-plane magnetic field H. (a) ISHE. (b) DSHEC. ISHE with different thickness of CrFigure S8. ISHE resistance in Cr at 300 K with different thicknesses. The RISHE(DSHE) values of these data and ones of more sample are extracted and plotted in Fig. S11, whose average value is plotted in Fig. 2e. D. ISHE with different current amplitudeFigure S9. Spin Hall signal with different magnitude of injection current. dVISHE/dJ is irrelevant to the applied current as desired, indicating the signal results from the intrinsic properties of the Cr/MgO/Fe junction instead of any J-related artefacts such as heating. E. Exaction of RISHE or RDSHEFollowing we show the procedure to extract the RISHE from data in Fig. 2b.  By fitting the two resistance platform linearly, two intercept are given with their uncertainty, (0.00419 ± 1.00297E-4)  and (-0.00351 ± 1.408E-4)  as shown in Fig. S10. RISHE equals to the average of the two values (0.00419+0.00351)/2=0.00385 , and its uncertainty is the sum of two uncertainty 1.00297E-4+1.408E-4 = 0.00024, so the RISHE = (3.85 ±  0.24) m. The uncertainty of spin Hall angle is given by equation (1) calculating with this RISHE value. RDSHE is extracted in the same way.Figure S10. Fitting and extraction of RISHE.F. RISHE data from other devicesFigure S11. Inverse Spin Hall signal in Cr with different thickness.Figure S12. Inverse Spin Hall signal in more samples. (a) Sample 2. (b) Sample 3.G. Shunting factorWe measured at least two sample of each thickness of Cr at 300 K shown in Fig. S11 and fitting it to give =20.03±1.84 nm. We use the same value as the room temperature one. It is not reasonable enough to still use the equation(2) to fit the spin diffusion length near TN or below it as the spin Hall angle in different thickness could have significant variation resulting from the domination of spin fluctuation. The spin fluctuation relates to the TN which increases as the thickness increases in thin films so the  could be different in Cr with different thickness even at the same temperature. The  is reported monotonously increasing as the temperature decreases47-49, which could bring less than 2% variation of shunting factor   as it saturates to 0.5 as  increases as shown in Fig. S13, which could not explain the enhancement of  over 100%. Keeping the same  will brings about at most 2% overestimation of .Figure S13.  Spin diffusion length  dependence of shunting factor  in equation (1) for t=10 nm.H. Temperature dependence of spin Hall conductivityFigure S14. Temperature dependence of spin Hall conductivity in sample 1.III. Normalized Power Consumption If one ignores the possibility of current shunting through the magnet, P =  where w, t and l is the dimensions of the device,  is the resistivity of NM layer,  is the critical current for spin-orbit torque (SOT) switching and  is the critical current density. This is consistent with the understanding of reviewer. With normalizing the device dimensions, P. For a type Z switching (magnetization out-of-plane),  is given by Ref. S8 and S9 asin which e, ℏ,  , ,  and  are the elementary charge, the reduced Planck constant, the saturation magnetization of FM layer, the thickness of FM layer, the effective spin Hall angle in NM layer, the effective anisotropy field and external magnetic field, respectively.By normalizing, the FM layer paremeters , Ms ，device dimensions and external field Hx are kept the same. So and . It worth mentioning this inference is based on the assumption that the thickness of NM stays below the spin diffusion length. Otherwise the effective , i.e. , usually decreases as the thickness increases and results in larger power consumption.  The values are from the references list in the table and the reviews.Table S1 Summary of the normalized power consumption of SOT materialsThe PN is calculated for different material categories with the magnitude of spin Hall angle or spin-orbit torque efficiency and resistivity. For a given  or  with a scale, the maximum  and the minimum  is adopted to give a low-estimation of PN. Structure (thickness in nm) Method SHA or SOT efficiency Resistivity (⁠μΩ⋅ cm) Normalized Power Consumption (μΩ·cm) Reference Non-magnetic metals(NM)   Ti(1) MOKE  0.074 21.2 3871.44 S10 Pt Second harmonic  0.12 20 ∼ 100  1388.89 S11 Pt(6) Second harmonic  0.09 36 4444.44 S12 Pt(10) SHTS 0.04∼ 0.09 27 3333.33 35 Ta(6.2) ST-FMR  -0.12 190 13194.44 10 Ta(3) Second harmonic  -0.006 178.5 4958333 10 Ta(10) SHTS -0.05∼ 0.11 210 17355.37 35 W(5.2) Critical current  -0.33 260 2387.51 15 W(6.2) Critical current  -0.18 80 2469.14 15 W(5) Critical current  -0.83 193 803.83 S13 Hf(3.5) Second harmonic  -0.02 199 497500 S14 Pd(8) Second harmonic  0.033 30 27548.21 S15 Ir  ST-FMR 0.03 21.1 23444.44 S16 Insert- or multi-layers  [Pt(0.75)/Ti(0.2)]n/Pt(0.75) Second harmonic  0.35 90 734.69 S17 Alloys  Au0.93W0.07(30) ST-FMR  -0.1 57 5700 S18 Au25Pt75(8) Second harmonic  0.35 80 653.06 7 PtAl(6) Second harmonic  0.14 75 3826.53 S12 Pt85Hf15(6) Second harmonic  0.16 110 4296.88 S12 Cu99.5Bi0.5 Lateral spin valve  -0.24 5.1 88.54 S19 Cu1−xPtx(6) ST-FMR  0.07 20∼ 70 4081.63 S20 Cu40Au60(8)  Second harmonic  0.097 29 3082.16 S21 Ni80Cu20(>7nm) Thermal spin injection 52.7 0.11 4355.37 25 Ni80Cu20(<7nm) Thermal spin injection 52.7 0.46 249.05 25 Antiferromagnets  FeMn Spin pumping  0.008 167.7 2620313 S22 PdMn Spin pumping  0.015 223 991111.1 S22 IrMn Spin pumping  0.022 269.3 556405 S22 PtMn Spin pumping  0.06 164 45555.56 S22 PtMn ST-FMR 0.064∼ 0.081 164.5 25072.4 S23 IrMn ST-FMR 0.053∼ 0.057 272.3 83810.4 S23 PdMn ST-FMR 0.028∼ 0.049 220 91628.49 S23 FeMn ST-FMR 0.022∼ 0.028 161.5 205994.9 S23 Ir22Mn78 ST-FMR 0.057  278 85564.79 S24 Ir25Mn75 ST-FMR 0.02 167∼ 278 417500 S25 Ir25Mn75 ST-FMR 0.1 183 18300 S16 Ir25Mn75 [111] ST-FMR 0.12 198 13750 S16 Ir25Mn75 [100] ST-FMR 0.2 160 4000 S16 Ir25Mn75 [100] annealed ST-FMR 0.35 160 1306.12 S16 Mn2Au ST-FMR 0.22 150 3099.17 S26 Mn3Sn ST-FMR 1 360 360 S27 Mn3Sn Lateral spin valve  0.053 1133 403346.39 S28 RuO2 [100] ST-FMR 0.0719 149 28822.29 S29 RuO2 [110] ST-FMR 0.0231 89 166788.48  S29 Topological insulators  Bi2Se3(8) ST-FMR  3.5 1755 143.27 18 Bi2Se3(7.4) Loop shift  0.16 1060 41406.25 17 Bi2Se3 Second harmonic  0.08 667 104218.8 16 Bi2Se3 SHTS 0.8 700 1093.75 S30 BixSe1−x(4) Second harmonic  18.62 12820 36.98 S31 Bi0.9Sb0.1(10) Coercivity  52 400 0.15 S32 (Bi,Sb)2Te3(8) Loop shift  0.4 4020 25125 17 (Bi,Sb)2Te3 Second harmonic  2.5 5464 874.24 16 SnTe Second harmonic  1.41 1835 922.99 16 Transition metal dichal-cogenides  WTe2(5.5) ST-FMR  0.029 385 457788.3 S33 WTe2(19.6) ST-FMR  0.51 580 2229.91 S34 PtTe2 ST-FMR  0.05 33 13200 S35 PtTe2 ST-FMR  0.15 333 14800 S35 MoTe2 ST-FMR  0.032 550 537109.4 S36 This work Cr(10) at 200 K  SHTS 0.35 20 163.3 This work Cr(10) at 300 K  SHTS 0.14 27 1377.6 This workIV. Theoretical consideration on the SHE by AFM spin fluctuationHere, we briefly discuss how the formalisms derived in Ref. 44 for the FM spin fluctuation should be modified when the AFM spin fluctuation is considered. The main difference from the FM fluctuation is that electrons at the Fermi surface are scattered by the AFM fluctuation with its spectral function given by Here,  measures the distance from the magnetic ordering as  at  and   with  being the ordered moment, which behaves as  near TN. A constant  is introduced so that  has the unit of energy, and  is the Landau damping. In contrast to the similar expression in Ref. 44, (i) the momentum dependence is given by  with  the magnetic wave vector characterizing the magnetic ordering, and (ii) the damping term  is independent of momentum. Because of (i), electrons that contribute to the SHE have to satisfy the nesting condition, i.e., momenta  and  have to be on the Fermi surface. This difference may lead to the following modified forms:for the skew scattering contribution, and for the side jump contribution. As in Ref. 44,  is given bywhere the momentum integral variable is changed from  to  by measuring it from the magnetic wave vector , and  is the Fermi velocity.  is the carrier lifetime, which is assumed to be independent of momentum that satisfies the nesting condition.  appearing in the expression of  is given by A similar expression is derived for the electron self-energy by the AFM spin fluctuation in Ref. S30. The detailed and comprehensive theoretical derivation can be referred to Ref. 50. Except for the vicinity of the magnetic transition temperature TN,  and  behave as  and , respectively. Thus,  and  behave as and respectively. As shown in Fig. 1(b), the resistivity shows a smooth crossover behavior from , where  is a constant, at low temperatures (Fig. S8a) to  at high temperatures. Thus, the carrier lifetime is dominated by the impurity or disorder and the electron-electron interaction at low temperatures and by the phonon scattering at high temperatures. Assuming that  remains constant away from TN and , one findsandMoreover, there is no limitation for the above microscopic processes to take place above or below the ordering temperature, in contrast to ferromagnetically ordered systems where the pure SHE is expected to exist only above the transition temperature because below the transition temperature the anomalous Hall effect appears. Therefore, the AFM order is necessary to observe a pure spin-fluctuation-originated contribution to the SHE, i.e., FSHE, without involvement with the anomalous Hall effect.V. Fitting details of resistivity and spin Hall resistivityFigure S14. Fitting of T-dependence of resistivity and spin resistivity. (a), Resistivity fitting. Blue line is duplicated from Fig. 1b. Fitting  gives the . (b), Spin resistivity fitting. 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B 2019, 100 (18), 184402121image3.jpegimage4.pngimage5.jpegimage6.emf-500 -250 0 250 500020406080100-500 -250 0 250 500-505101520253035Offset dVISHE /dJ (mW)H (Oe)300 K275 K250 K225 K200 K175 K150 K125 K100 K50 K75 KOffset dVDSHE /dJ (mW)H (Oe)250 K200 K150 K100 K(a)(b)image7.emf-600-400-2000200400600-6-3036dVISHE /dJ (mW)H (Oe) Out-of-plane10 nm  CrSample 1T=300 KISHE(a) (b)image8.emf-500 0 500-3-2-10123dVISHE/dJ (mW)H (Oe)T = 300 KtCr = 7 nm 10 nm 25 nm50 nm-500 0 500H (Oe)-500 0 500H (Oe)-500 0 500H (Oe)image9.pngimage10.pngimage11.jpegimage12.emf0501001502002503000123RISHE (mW)T (K)10 nmSample 2(a) (b)image13.pngimage14.emf50 100 150 200 250 30051015|sSH| (ħ/e) (103 S/cm)T (K)image15.jpegimage1.pngimage2.png