Optimization of spin Hall magnetoresistance in heavy-metal/ferromagnetic-metal bilayers

We present experimental data and their theoretical description on spin Hall magnetoresistance (SMR) in bilayers consisting of a heavy metal (H) coupled to in-plane magnetized ferromagnetic metal (F), and determine contributions to the magnetoresistance due to SMR and anisotropic magnetoresistance (AMR) in five different bilayer systems: W/Co20Fe60B20\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hbox {W}/\text {Co}_{20}\text {Fe}_{60}\text {B}_{20}$$\end{document}, Co20Fe60B20/Pt\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\text {Co}_{20}\text {Fe}_{60}\text {B}_{20}/\hbox {Pt}$$\end{document}, Au/Co20Fe60B20\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hbox {Au}/\text {Co}_{20}\text {Fe}_{60}\text {B}_{20}$$\end{document}, W/Co, and Co/Pt. The devices used for experiments have different interfacial properties due to either amorphous or crystalline structures of constitutent layers. To determine magnetoresistance contributions and to allow for optimization, the AMR is explicitly included in the diffusion transport equations in the ferromagnets. The results allow determination of different contributions to the magnetoresistance, which can play an important role in optimizing prospective magnetic stray field sensors. They also may be useful in the determination of spin transport properties of metallic magnetic heterostructures in other experiments based on magnetoresistance measurements.

Co 20 Fe 60 B 20 -based bilayers. Figure 1 shows relative magnetoresistance as a function of heavy metal Dependence of magnetoresistance on heavy-metal thickness, with fixed t F = 5 nm , shown in Fig. 1a-c indcates, as expected, that SMR is the largest contribution to magnetoresistance in heterostructure with W as a heavy metal layer due to larger spin Hall angle of W, |θ SH | ≈ 21% , compared to Pt, |θ SH | ≈ 6% , and to Au, |θ SH | ≈ 4% . Consequently, in Pt and Au bilayers AMR dominates over SMR.
The dependence of magnetoresistance on ferromagnetic layer thickness is shown in Fig. 1d-f. For 5 nm-thick W as heavy metal layer, shown in Fig. 1d SMR is still the dominating contribution to the total magnetoresistance in the studied thickness range. For device with 3 nm-thick Pt, the SMR for t F 2 nm is smaller than AMR. Note that for both W-and Pt-based bilayers the model fit and theoretical prediction do not describe the behavior of MR for t F 2 nm , which can be attributed to strong dependence of the interfacial parameters such as spin-mixing conductance on thickness. Due to small spin Hall angle, SMR in Au-based is rather small and MR is dominated by AMR.
Co-based bilayers. Magnetoresistance and relative magnetoresistance in H/Co bilayers (H: W, Pt) are shown in Fig. 2 as a function of heavy metal (Fig. 2a,b) and ferromagnetic metal (Fig. 2c,d) layer thicknesses.
For W and Pt bilayers with varying heavy metal thickness and t F = 5 nm , shown in Fig. 2, the total magnetoresistance is mostly due to AMR, in contrast to Co 20 Fe 60 B 20 -based bilayers described in the previous subsection.
For bilayers with varying ferromagnetic metal (Co) thickness, shown in Fig. 2c,d, the total magnetoresistance is also largely dominated by AMR.
Due to existence of a magnetic dead layer, disoriented crystalline structure of Co, and due to the fact that magnetization of Co does not lie completely in-plane of the sample for small thicknesses we introduced the magnetically effective thickness, t F,eff , of Co layer. More details on some of these aspects can be found in Supplementary Information. Since the thickness of heavy-metal is fixed and the thickness of the ferromagnetic metal www.nature.com/scientificreports/ increases, the large differences in resistivities result in larger portion of charge current flowing into Co leading to negligible SMR-thus preventing proper estimation of spin Hall angle of both W and Pt.

Discussion
Our systematic analysis of magnetoresistance in in-plane magnetized heavy-metal/ferromagnetic-metal bilayers with crystalline Co and amorphous Co 20 Fe 60 B 20 has shown that proper choice of ferromagnetic-metal is crucial to the optimization of spin Hall magnetoresistance. As shown in previous section, although W has larger spin Hall angle than Pt and Au, the magnetoresistance of W/Co 20 Fe 60 B 20 and even W/Co (for thin W) bilayers can be lower than that of Co/Pt bilayer due to high AMR contribution in the latter. This also leads to possible underestimation of SMR contribution which is lower in W/ Co than in W/Co 20 Fe 60 B 20 bilayers, one of the main reasons for which is quite large difference in resistivities of both layers (see Table 1), due to the fact that here β -W phase is disoriented (amorphous-like) resulting in more current flowing through Co than W and on average smaller spin Hall effect (see Supplementary Information). Moreover, due to the fact that here W is mostly amorphous and Co crystalline, a different interface properties between these materials than between crystalline-crystalline or amorphous-amorphous bilayers can influence spin transport as well.
For materials with stronger spin-orbit coupling (W and Pt) and comparable resistivities to Co 20 Fe 60 B 20 , one obtains higher magnetoresistance response with thinner ferromagnet. In the case of Au-based bilayer, whose resistivity is smaller than that of Co 20 Fe 60 B 20 , one obtains higher magnetoresistance with thicker ferromagnet. The predicted SMR contribution for Au/Co 20 Fe 60 B 20 can be higher than the SMR contribution for Co 20 Fe 60 B 20 /Pt due to the fact that larger current density flows through Au than through Pt, thus increasing the spin Hall response.
The estimation of SMR in the case of metallic bilayers is hindered by large differences in resistivity of the constituent metallic layers. Since in this case AMR is strongly dominating, the total MR increases with increasing effective thickness of Co.
In conclusion, we have developed an extended model of magnetoresistance for magnetic metallic bilayers with in-plane magnetized ferromagnets, which explicitly includes SMR and AMR contributions. The model was then fitted to experimental data on magnetoresistance in: W/Co 20 Fe 60 B 20 , Co 20 Fe 60 B 20 /Pt , Au/Co 20 Fe 60 B 20 , W/Co,  Table 1 20 ) and crystalline ferromagnet (Co) on total magnetoresistance and analyzed the dependence of magnetoresistance on ferromagnet's thickness, which allows for better optimization of magnetic bilayers. These results allow for a more accurate estimation of different contributions to magnetoresistance in magnetic metallic systems, which is important for applications in, e. g., spintronic SOT-devices 42 or in other experimental schemes that rely on magnetoresistance measurements in evaluation of the spin transport properties.

Methods
Experiment. Table 1 shows the multilayer systems that were produced for SMR studies. The magnetron sputtering technique was used to deposit multilayers on the Si/SiO 2 thermally oxidized substrates. Thickness of wedged layers were precisely calibrated by X-ray reflectivity (XRR) measurements. The details of sputtering deposition parameters as well as structural phase analysis of highly resistive W and Pt layers can be found in our recent papers 38,41 . Au in Au/Co 20 Fe 60 B 20 bilayers is (111) fcc textured similarly as Pt in Co 20 Fe 60 B 20 /Pt bilayer 38 . In turn, structure analysis of the hcp-Co crystal phases grown on disoriented β -W can be found in the Supplementary Information.
After deposition, multilayered systems were nanostructured using either electron-beam lithography or optical lithography, ion etching and lift-off. The result was a matrix of Hall bars and strip nanodevices for further electrical measurements. The sizes of produced structures were: 100 µm x 10 µm or 100 µm x 20 µm . In order to ensure good electrical contact with the Hall bars and strips, Al(20)/Au(30) contact pads with dimensions of 100 µm x 100 µm were deposited. Appropriate placement of the pads allows rotation of the investigated sample and its examination at any angle with respect to the external magnetic field in a dedicated rotating probe station using a four-points probe. The constant magnetic field, controlled by a gaussmeter exceeded magnetization saturation in plane of the sample and the sample was rotated in an azimuthal plane from −120 • to +100 • .
The resistance of the system was measured with a two-and four-point technique using Keithley 2400 sourcemeters and Agilent 34401A multimeter. As shown in Supplementary Information, resistances of bilayers with amorphous ferromagnet Co 20 Fe 60 B 20 are about one order higher than these with polycrystalline Co. The same results were obtained using both methods. The thickness-dependent resistivity of individual layers was determined by method described in Ref. 7, and by a parallel resistors model. For more details on resistivity measurements we refer the reader to Supplementary Information.  Table 1 www.nature.com/scientificreports/ Theory. To properly assess all contributions to magnetoresistance one should find first the average current density flowing through the whole heterostructure. This approach, in contrast to the one described in Ref. 5 allows one to properly describe magnetoresistance in more complicated heterostructures, where ad hoc addition of consitutent terms might lead to oversimplification and improper determination of different components in the magnetoresistance. Moreover, calculating average current density allows for a phenomenological description of how various magnetoresistance effects depend on thicknesses of the constituent layers. The drawback, however, is the necessary simplification of fitting parameters, which we discuss in more detail in the next subsection devoted to fitting procedure. Only the component flowing along the normal to interfaces is relevant and will be taken into account in the following, i.e.
Here θ SH is the spin Hall angle, ρ H 0 is the bare resistivity of the heavy metal, and µ H s (z) is the spin accumulation that is generally z-dependent.
The charge current density in the heavy-metal (H) layer, in turn, can be written in the form and contains the bare charge current density and the current due to inverse spin Hall effect. Note, that the spin current in general can induce charge current also flowing along the axes x and y. However, due to lateral dimensions of the samples much larger than the layer thicknesses and spin diffusion lengths, those additional components can be neglected. Thus, one can write 28,29 : in which θ AMR is the AMR angle, defined as Charge current density in the ferromagnetic layer (F) can be written as 28,29 , Note, in the above equations the current densities in both H and F layers we assumed as linear response to electric field, i.e. we neglected the so-called unidirectional spin Hall magnetoresistance effect [24][25][26][27] . The spin current j HF s flowing through the heavy-metal/ferromagnet interface is given by the following expression [43]: and G ↑ and G ↓ denoting the interface conductance for spin-↑ and spin-↓ . Furthermore, G r ≡ ReG mix and G i ≡ ImG mix , where G mix is the so-called spin-mixing conductance. Note, that we neglect explicitly a contribution from the interfacial Rashba-Edelstein spin polarization 38 . A strong interfacial spin-orbit contribution which induces spin-flip processes can also be combined with the interfacial spin conductance G F as a spin-conductance reducing parameter 1 − η , with η = 0 for no interfacial spin-orbit coupling, and η = 1 for maximal spin-orbit coupling. Note, that this reduction could also be attributed to the magnetic proximity effect, especially in the case of Pt-based heterostructures 13 , however recent studies suggest its irrelevance for spin-orbit-torque-related experiments 22 . In the following discussion we assume η = 0 and treat G F as an effective parameter.
To find charge and spin currents we need to find first the spin accumulation at the H/F interface and also at external surface/interfaces. This can be found from the following boundary conditions: Having found electrochemical potential and spin accumulation from general solution www.nature.com/scientificreports/ where A F,H and B F,H are coefficients to be determined and F,H is the spin diffusion length in ferromagnet or heavy metal, one can find the longitudinal in-plane components of the averaged charge current j(m) from the formula: The total longitudinal charge current can be written down in the Ohm's-law form, where the longitudinal resistivity is defined as follows: with In the above expressions the following dimensionless coefficients have been introduced to simplify the notation: With the resistivity defined in Eq. (10) we can now define magnetoresistance, Taking into account Eqs. (11)-(13), the above formula can be written as, In order to compare the models with and without AMR, we define SMR as: which simplifies our model to that introduced by Kim et al. 5. We also define AMR coefficient Fitting procedure. In order to analyze the experimental data in light of our extended model, we fit Eq. (19) to the data on relative magnetoresistance. We have assumed some constant values according to literature and  39 . Note, that we have assumed constant effective spin diffusion lengths for the constituent layers obtained from our previous analyses 38,41 . In general, however, these parameters can depend on temperature or thickness of the layers 44,45 . This fact can lead to underestimation of spin diffusion lengths and overestimation of the spin Hall angles. One of the remedies might be to use effective thickness-dependent parameters 45 . However, such approaches are still mostly empirical and not based on proper theoretical grounding and as such have their own limitations. Moreover we have assumed transparent contacts for spin transport, i.e. G F → ∞ and G r → ∞ , and also assumed G i to be negligible. These assumptions are mostly valid for metallic interfaces. However, these parameters can also strongly depend on type of interface, i.e. they can differ in amorphous/crystalline (f.i. Co 20 Fe 60 B 20 /Pt ), crystalline/crystalline (Co/Pt), and amorphous-like/amorphous (f.i. W/Co 20 Fe 60 B 20 ) heterostructures.
We have assumed spin Hall angle θ SH and AMR coefficient θ AMR as fitting parameters and the results of fitting the model to the experimental data on magnetoresistance are gathered in Table 1. Morevoer, we have assumed anomalous Hall effect to be negligible in the in-plane magnetized systems considered in the paper. This effect might play an important role for ferromagnets with stronger spin-orbit coupling or ferromagnets tilted out of plane [28][29][30] .