Abstract
Magnetic insulators (MIs) attract tremendous interest for spintronic applications due to low Gilbert damping and the absence of Ohmic loss. Spinorbit torques (SOTs) on MIs are more intriguing than magnetic metals since SOTs cannot be transferred to MIs through direct injection of electron spins. Understanding of SOTs on MIs remains elusive, especially how SOTs scale with the MI film thickness. Here, we observe the critical role of dimensionality on the SOT efficiency by studying the MI layer thicknessdependent SOT efficiency in tungsten/thulium iron garnet (W/TmIG) bilayers. We show that the TmIG thin film evolves from twodimensional to threedimensional magnetic phase transitions as the thickness increases. We report the significant enhancement of the measured SOT efficiency as the TmIG thickness increases, which is attributed to the increase of the magnetic moment density. We demonstrate the currentinduced SOT switching in the W/TmIG bilayers with a TmIG thickness up to 15 nm.
Introduction
The interplay between heavy metals (HMs) and magnetic insulators (MIs) in heavy metal/magnetic insulator (HM/MI) bilayer systems has attracted tremendous attention from both fundamental research and practical applications^{1,2,3,4}. First, the HM/MI bilayer benefits from the low Gilbert damping in the MI. In contrast to magnetic metal, MIs only allow spin information to propagate through magnons, instead of itinerant electrons, due to their large electronic bandgaps. The absence of Ohmic loss from the magnetic layer makes HM/MI bilayers more energy efficient than HM/magnetic metal bilayers.
The second advantage of the HM/MI bilayer is that the spinorbit coupling in the HM or at the HM/MI interface allows the efficient generation of spinorbit torques (SOTs) on the MI layer through the spin Hall effect (SHE) or Rashba–Edelstein effect^{5,6,7,8,9}. These SOTs enable efficient manipulation of magnetization dynamics in the MI layer. Although the MI layer is electrically insulating, SOTdriven magnetization dynamics of MIs can be detected through anomalous Hall resistance (AHR) and spin Hall magnetoresistance (SMR) in the HM layer^{10,11,12,13}. By probing the AHR, currentinduced magnetization switching (CIMS) was observed in both Pt/BaFe_{12}O_{19}^{14} and Pt/Tm_{3}Fe_{5}O_{12} (TmIG) bilayers^{15,16}. However, whether SOTs in Pt/MI bilayers are from SHE remains ambiguous due to the potential existence of the Rashba–Edelstein effect^{16}. It remains unclear whether the switching direction will be opposite when we utilize HMs with opposite spin Hall angles. Moreover, the observed dampinglike SOT efficiency (ξ_{DL}) in the Pt/TmIG that is responsible for switching is still much lower than those in the Pt/ferromagnetic metals (FMs)^{15,17,18}. To understand the origin of SOTs and to increase the value of ξ_{DL} in HM/MI bilayers, we utilize a HM with a large spin Hall angle opposite to that of Pt in a HM/MI bilayer, demonstrate magnetization switching, and analyze the contributions to the SOT.
In this article, we study the ξ_{DL} and CIMS in tungsten (W)/TmIG heterostructures with different TmIG layer thicknesses (t_{TmIG}). The thickness dependence of the dampinglike SOT allows us to understand the interplay between spin current and magnetism in TmIG. Here, W is chosen since it is reported to give the largest spin Hall angle among elemental HMs and its sign is opposite to that of Pt^{19}. When the TmIG film thickness is reduced from 15 to 3.2 nm, the effective exchange coupling is strongly reduced due to longwavelength thermal fluctuations, resulting in a dimensional crossover from threedimensionlike to twodimensionlike magnetic phase transitions. We quantify ξ_{DL} by using secondharmonic Hall measurements^{20,21}. The ξ_{DL} increases with the t_{TmIG} in W/TmIG bilayers; this is attributed to the enhanced magnetic moment density due to suppression of thermal fluctuations. We then demonstrate the CIMS in W/TmIG bilayers up to t_{TmIG} = 15 nm; for t_{TmIG} = 15 nm, the switching current density is as low as 8 × 10^{10} A/m^{2}. The estimated current switching efficiency enhances as t_{TmIG} increases, which is consistent with the increase of ξ_{DL} with t_{TmIG}. Importantly, the switching direction of our W/TmIG devices is indeed opposite to that of the Pt/TmIG device^{15}; this contrast confirms the important role of SHE in CIMS of MIs.
Results
Dimensional crossover of magnetism
To access SOT and realize CIMS, we prepare highquality TmIG thin films with different t_{TmIG} and characterize their magnetic properties. These TmIG(111) thin films were grown on substrate Nd_{3}Ga_{5}O_{12}(111) by pulsed laser deposition^{13}. All TmIG thin films show an atomically flat surface with mean roughness as low as 0.1 nm (Fig. 1a), providing a sharp interface for efficient spin momentum transfer. The Gilbert damping of TmIG thin films increases as the thickness decreases (see Supplementary Note 1). The large lattice mismatch between the TmIG and the Nd_{3}Ga_{5}O_{12} provides the tensile strain to generate perpendicular magnetic anisotropy in all TmIG thin films. The nature of perpendicular magnetic anisotropy is confirmed using magnetization hysteresis loops of TmIG thin films as a function of an outofplane magnetic field (Fig. 1b), from which we can determine saturation magnetization (M_{S}). We observe a strong t_{TmIG} dependence of the M_{S} at room temperature (Fig. 1c); the M_{S} reduces significantly from the bulk M_{S} (110 emu/cm^{3})^{22} with decreasing film thickness. Note that the estimated dead layer thickness is less than 1 nm (see Fig. 1c inset and Supplementary Note 2), which also suggests a sharp interface between TmIG and substrate^{23}. The reduction of the M_{S} at room temperature is attributed to finite size effect, strong thermal fluctuation and strong surface modification effect in ultrathin magnetic films^{24,25,26}. Following ref. ^{25}, we extract the critical exponents β for magnetic phase transitions in these TmIG thin films using temperature dependence of magnetic moment (M–T). The M–T curves follow the M = M_{0}(1 − T/T_{C})^{β} (Fig. 1d), where zerotemperature magnetic moment (M_{0}) and Curie temperature (T_{C}) are fitting parameters. The t_{TmIG}dependent β is better illustrated using log–log plots as shown in Fig. 1e and the results are summarized in Fig. 1f. We see a clear increase of β from 0.16 ± 0.06 to 0.42 ± 0.02 when the t_{TmIG} increases from 3.2 to 15 nm, where the uncertainty is coming from the fitting. This increase of β suggests a dimensional crossover from twodimensionlike to threedimensionlike magnetism since 2D Ising model and 3D Heisenberg model predict β to be 0.125 and 0.365, respectively^{26,27}. The dimensional crossover happens at around 6 nm, which is one order of magnitude larger than the typical transition thickness around 1 nm for magnetic metals^{25,26,27}. In the following sections, we point out that the reduction of M_{S} due to dimensional crossover has a major influence on the magnitude of the SOT and switching efficiency, which has been neglected in the previous experiments.
SOT measurement
To perform resistance, SOT, and CIMS measurements, we fabricate W(5 nm)/TmIG(t_{TmIG}) thin films into Hall bar devices (Fig. 2a). By using fourprobe resistance measurements in different Hall bar devices, we determine the W resistivity to be 155 ± 15 µΩ·cm, where the uncertainty is estimated from the multiple (>20) device measurements. According to ref. ^{19}, pure αW has resistivity around 20 µΩ·cm, and 6 nmthick W with mixed α and βphases has a resistivity as high as 170 µΩ·cm. So, most likely, our 5 nmthick W thin films have mixed α and βphases. The AHR in the W/TmIG is accurately determined by the sharp anomalous Hall hysteresis at low fields (Fig. 2b). The transverse planar Hall resistance (PHR) accompanying the longitudinal SMR is measured by rotating the magnetization in the xyplane (Fig. 2c). The observation of sizeable AHR and PHR (SMR) indicates that there is a significant spin current being transmitted across the W/TmIG interface or a sizable spin mixing conductance^{11} (see Supplementary Note 3).
We quantify ξ_{DL} by using the secondharmonic analysis of both AHR and PHR (R_{AHE} and R_{PHE})^{20,21}. The secondharmonic Hall resistance (\(R_{\mathrm{H}}^{2\omega }\)) in a single domain subjected to an inplane magnetic field can be written as^{21,28}
where H_{K} and H_{ext} are perpendicular magnetic anisotropy effective field and inplane external field, respectively. In Eq. (1), \(R_{{\mathrm{FL}}}^{2\omega }\) and \(R_{{\mathrm{DL}}}^{2\omega }\) are the peak values of cos 2φsin φ and sin φ components in \(R_{\mathrm{H}}^{2\omega }\), which are fieldlike SOT and dampinglike SOT contributions, respectively. H_{FL} and H_{DL} are the currentinduced fieldlike and dampinglike effective fields, respectively. For example, when the H_{ext} = 2500 Oe, we observe significant contributions from both dampinglike and fieldlike SOTs, as reflected by the cos 2φsin φ and sin φ angle dependencies (see Fig. 2d and Supplementary Note 4). According to Eq. (1), slopes of linear fits to the \(R_{{\mathrm{DL}}}^{2\omega }\) as a function of 1/(H_{ext} − H_{K}) (Fig. 2e) give the information about H_{DL}, and the intercepts are the spin Seebeck resistances (or voltages), which is fieldindependent in the single domain case (see Eq. (1))^{21,29}.
We calculate ξ_{DL} using \(\xi _{{\mathrm{DL}}} = \frac{{2eM_{\mathrm{S}}t_{{\mathrm{TmIG}}}H_{{\mathrm{DL}}}}}{{\hbar J_{{\mathrm{ac}}}}}\)^{6}, where e is the electron charge, ħ is the reduced Planck constant, and J_{ac} is the applied current density. We observe a characteristic increase of ξ_{DL} as t_{TmIG} increases with a saturation length of 10 nm (see Fig. 2f). Similarly, previous experiments have revealed a saturation length around 1 nm in FM heterostructures^{18,30,31}. This saturation length is very close to the measured penetration depth of transverse spin current for FMs using spin pumping technique^{32,33,34}. Thus, the saturation length has been interpreted as an indicator of penetration depth^{33,34}. However, for our MI TmIG thin films, the scenario becomes complex since the electron spin cannot directly tunnel into the MI and the magnetism of MI thin films is strongly dependent on the MI thickness (Fig. 1). Note that the SOT efficiency (ξ_{DL}~0.02) in our W/TmIG (≥9 nm) devices is smaller than that in βW/CoFeB (ξ_{DL}~0.3)^{19}. There are two possible reasons. First, our W thin films are in mixed phases, which have a smaller spin Hall angle. Second, the material interfaces in W/magnetic metal and W/MI bilayers could be very different^{17}, which requires further investigations.
SOT switching
After quantifying the SOT efficiency, we perform the CIMS experiments for W/TmIGs with different t_{TmIG}. The switching is achieved in all devices with t_{TmIG} up to 15 nm and the switching phase diagrams are summarized in Fig. 3a. In the presence of an external field along the +y direction, a sufficiently large charge current along the +y direction will cause magnetization (AHR) switching from the +z direction to the −z direction (negative to positive). The required amount of charge current to flip the magnetization decreases as the external field increases. When we apply a sufficiently large charge current along the −y direction while keeping the external field along the +y direction, the magnetization (AHR) is switched from the −z direction to the +z direction (positive to negative) (upper panels in Fig. 3b, c). For the same current direction, the switching direction is opposite when we reverse the external field direction (lower panels in Fig. 3b, c). All of the above facts agree with the picture of SOTdriven magnetization switching. Note that the switching current density is as low as 6 × 10^{10} A/m^{2} for the W (5 nm)/TmIG (9.6 nm) (Fig. 3b), which is three times smaller than the Pt (5 nm)/TmIG (8 nm) case^{15}. This suggests that W enables more energy efficient magnetization switching.
The switching direction driven by currentinduced SOTs is consistent with the sign of the spin Hall angle of W, and it is opposite to that in the Pt/TmIG bilayer^{15}. Therefore, our work strongly suggests the dominant role of the SHE in the generation of SOTs and CIMS in HM/MI bilayers. However, we do notice that there could be an interfacial Rashba–Edelstein effect at the W/TmIG interface contributing to the SOTs by comparative analyses of SOTs and SMR (AHR) (see Supplementary Note 5).
To quantitatively compare the switching efficiency of W/TmIG devices with different t_{TmIG}, we define an effective switching efficiency as \(\eta = \frac{{2eM_{\mathrm{S}}t_{{\mathrm{TmIG}}}H_{\mathrm{P}}}}{{\hbar J_{{\mathrm{sw}}}(H_{\mathrm{y}} \to 0)}}\)^{35}, where H_{P} is the domain wall depinning field estimated from the coercive field (see Supplementary Note 6) and J_{sw}(H_{y} → 0) is the zerofield limit of current density in the switching phase diagram. This formula is chosen because the CIMS is achieved through domain nucleation and domain wall motion in the Hall bar devices due to the large scale of our Hall bar devices, of which the channel width is 20 µm^{36}. We observe a dramatic increase of η with t_{TmIG} (Fig. 3d), for which we consider two reasons. First, the ξ_{DL} increases with t_{TmIG}, which means that the same amount of charge current in the W layer generates stronger dampinglike SOT on the TmIG layer. Thus, the increase of ξ_{DL} contributes to a lower J_{sw} and thus a larger η. Second, the Joule heating effect becomes much more significant when a larger charge current is applied, which is the case for switching a thicker TmIG. Joule heating causes reduction of thermal stability through decreasing the M_{S} and H_{P}; these two values will be smaller than those measured at the low current limit. Therefore, the M_{S} and H_{P} used to calculate η are overestimated, leading to a larger η.
Discussion
Here, we discuss the mechanism for the MI thickness dependence of ξ_{DL}. We propose that ξ_{DL} depends on M_{S} when M_{S} of the thin films is well below the corresponding bulk value. The Landau–Lifshitz–Gilbert equation in the presence of dampinglike SOT can be written as
where \(\hat m\) is the unit vector of magnetization, \(\hat \sigma\) is the unit vector of currentinduced spin polarization, γ is the gyromagnetic ratio, α is the Gilbert damping, t_{M} is the thickness of the magnetic layer, J_{C} is the charge current density, and \(\vec H_{{\mathrm{eff}}}( = \vec H_{\mathrm{K}} + \vec H_{{\mathrm{ext}}})\) is the total effective magnetic field acting on the magnetization. The last term on the righthand side of Eq. (2) arises due to the absorption of transverse spin current by the magnet, which is referred to as the currentinduced dampinglike (dissipative) SOT. Its strength is parameterized by dimensionless efficiency parameters ξ_{DL}. The origin of the SOT can be understood in a simple microscopic picture as follows. A charge current at the HM and ferromagnet interface induces an accumulation of spin density, \(\rho \hat \sigma\), due to the finite spinorbit interaction (for example, by SHE or Rashba–Edelstein effect). Here, ρ is the magnitude of the spin density, which is proportional to the strength of the spinorbit interaction. This spin density interacts with the ferromagnet via exchange interaction, of the form \(U_{{\mathrm{ex}}} \sim \rho M_{\mathrm{S}}\hat m \cdot \hat \sigma\), enabling the absorption of the spin current by the ferromagnet. In the perturbative treatment, the spin current absorbed by the ferromagnet can be obtained up to second order in the exchange interaction to yield the dampinglike SOT with \(\xi _{{\mathrm{DL}}}\sim M_{\mathrm{S}}^2\)^{37}. The positive correlation between ξ_{DL} and M_{S} is referred as the M_{S}effect; it has also been theoretically studied in the frame of spin pumping effect (in Appendix B of ref. ^{38}), which is the Onsager reciprocal process of the spin torque effect. The increase of spin mixing conductance with M_{S} is consistent with the calculation from first principles^{39} when the surface modification effect presents in the ultrathin regime^{26}.
Our experiments are the demonstrations of the M_{S}effect; we show that as the thickness increases, the SOT efficiency significantly increases with M_{S} in the low M_{S}regime (see Fig. 4), which is in qualitative agreement with the M_{S}effect. Also, we show that as the temperature decreases, the SOT efficiency increases with M_{S}, due to suppression of thermal fluctuations (see Supplementary Note 7). Intuitively, as the magnetic moment density (M_{S}) increases, the interfacial exchange interaction is enhanced, which allows more spin current to pass through the interface. As the thickness increases, the SOT efficiency saturates earlier than M_{S}, around half of the bulk magnetization (60 emu/cm^{3}), which suggests that the SOT is determined by the local magnetization that is saturated at a smaller thickness than the global magnetization M_{S}. Our experiments show the need for further investigation of the interaction between ultrathin magnetic films and HMs, which would include the spin physics of dimensional crossover.
In summary, we have systematically studied the dimensional crossover of magnetism and its effect on SOTs in ultrathin MI films with perpendicular magnetic anisotropy. The characteristic increase of SOT efficiency with the MI thickness can be understood from the enhancement of magnetic moment density and the suppression of thermal fluctuations. In addition, we have realized CIMS in W/TmIG devices with t_{TmIG} up to 15 nm. The switching current density for W/TmIG devices is lower or comparable with these for HM/FM despite the fact that the saturated ξ_{DL} is estimated to be only around 0.02 at this stage, which is much less than the 0.3 that is estimated for W in W/CoFeB bilayers^{19}. Further improvement of the ξ_{DL} could be done by spin mixing conductance matching^{40} and surface treatment^{41}. Our results presented here show the great potential of ultrathin MIbased spintronics.
Methods
Materials growth and characterization
All TmIG(111) films were grown on Nd_{3}Ga_{5}O_{12}(111) by pulsed laser deposition^{13} before transferring to a magnetron sputtering chamber in the ambient condition. At room temperature, we deposited a 5 nmthick W layer on top of TmIG followed by subsequent deposition of MgO (2 nm)/TaO_{x} (3 nm) layers to protect W from oxidization. Magnetization hysteresis loops as a function of an outofplane magnetic field were measured by a vibrating sample magnetometer and a superconducting quantum interference device. The nominal thin film area is 5 × 5 mm^{2}.
Devices fabrication and characterization
The films were patterned into Hall bar devices (Fig. 2a) by using standard photolithography and dry etching for the resistance, SOT, and switching measurements. The channel width is 20 µm, and the distance between two neighboring Hall contacts is 26 µm. We measured the secondharmonic Hall resistance by applying I_{ac,r.m.s} = 1 mA (J_{ac,r.m.s} = 10^{10} A/m^{2}) with a frequency ω/2π = 195.85 Hz. The magnetic field and angle controls were done in a physical properties measurement system. The CIMS experiments were performed in the ambient environment by applying a pulse current with 5 ms pulse width and reading Hall voltage subsequently.
Data availability
The data that support the findings of this study are available from the corresponding authors upon reasonable request.
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Acknowledgements
We acknowledge helpful discussions with Sadamichi Maekawa, ChiFeng Pai, Wei Zhang, Yi Li, Ke Xia, Yongxi Ou, and Can Onur Avci. This work is supported as part of the Spins and Heat in Nanoscale Electronic Systems (SHINES), an Energy Frontier Research Center funded by the US Department of Energy (DOE), Office of Science, Basic Energy Sciences (BES), under Award # DESC0012670. Q.S., G.Y., A.N., S.A.R., P.U., Q.L.H., L.P., Y.T., and K.W. are also very grateful to the support from the Function Accelerated nanoMaterial Engineering (FAME) Center and Center for Spintronic Materials, Interfaces and Novel Architectures (CSPIN), two of six centers of Semiconductor Technology Advanced Research network (STARnet), a Semiconductor Research Corporation (SRC) program sponsored by Microelectronics Advanced Research Corporation (MARCO) and Defense Advanced Research Projects Agency (DARPA). G.Y. acknowledges the financial support from the National Natural Science Foundation of China (NSFC)–Science Foundation Ireland (SFI) Partnership Programme [Grant No. 51861135104] and the 1000 Youth Talents Program.
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Author notes
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Contributions
Q.S. and G.Y. conceived the project. C.T., Y.L. and J.X.L. grew the thin films, performed the film characterizations and magnetic measurements. Q.S. and P.Z. performed the SOT measurements. Q.S., G.Y., C.H. and P.Y. performed the resistance measurements. Q.S, C.T. and G.Y. analyzed the transport data. Q.S., P.Z., G.Y. and C.H. performed the switching measurements. A.N. and C.Z. fabricated the Hall bar devices. P.U., S.K. and Y.T. proposed the M_{S}effect. All authors contributed to the discussion. Q.S., G.Y., C.T, J.S. and K.L.W. wrote the manuscript with the input from all other authors.
Competing interests
The authors declare no competing interests.
Corresponding authors
Correspondence to Guoqiang Yu or Kang L. Wang.
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