Microscopic studies of nonlocal spin dynamics and spin transport (invited)

The emerging technology of spintronics utilizes the spin of the electron to carry information, and should enable greater functionality and performance over conventional charge-based electronics.¹ Generating spin currents and understanding the behavior of spin transport is a vital step toward viable spintronic devices. Spin currents were first generated in all-metal systems,² using a charge current to drive a spin polarized current across the interface between a ferromagnet (FM) and a nonmagnetic (NM) metal, and later extended to metallic FM/semiconducting NM systems.³ More recently, pure spin currents have been observed in FM/NM bilayers in spin pumping⁴ experiments. In these experiments, the ferromagnet is brought out of equilibrium by exciting its magnetization at the ferromagnetic resonance (FMR) frequency, resulting in a pure spin current injected across the interface into the neighboring metal layer. First demonstrated in all-metallic FM/NM systems,^5,6 the spin transfer across the interface has been attributed to the short-ranged exchange coupling of itinerant electrons in the metal exerting a spin torque on the magnetization of the ferromagnet at the interface.⁴ The same effect has more recently been demonstrated in insulating FM/metallic FM systems,⁷ and the effect is also attributed to itinerant electrons in the metal and exchange coupling to the ferromagnet. In all cases, the efficiency of spin transport is determined by the interface. A microscopic understanding of the coupling mechanism between magnetization and spin current across the interface is necessary to optimize the spin current generation.

In a series of recent experiments,^8–11 we have investigated a variety of material interfaces involving insulating FMs and insulating NMs using an array of sensitive resonance tools. Using magnetic resonance techniques together with strong magnetic field gradients, it is possible to selectively and non-invasively investigate spin dynamics within microscopic subvolumes of a homogeneous macroscopic sample with a technique known as magnetic resonance force microscopy (MRFM).^12,13 Using MRFM, we measure the spin lifetimes of microscopic volumes within an insulating paramagnetic (PM) diamond spin wire⁹ and an insulating homogeneous yttrium iron garnet (YIG) ferromagnetic film⁸ and observe spin transport in both systems despite the absence of itinerant electrons. In addition, we have systematically studied the spin transport in a YIG/metallic NM system by carefully engineering the interface with interlayers to enable spin conductance matching.¹⁰ Finally, we have investigated an insulating FM/insulating PM system using the optically detected magnetic resonance (ODMR) of nitrogen vacancy (NV) centers in diamond and observe nonlocal spin dynamics in a YIG/NV diamond system.¹¹ Different coupling mechanisms are at play within all of these systems, for example, the mechanism for spin transport in paramagnetic insulators is mediated by long-range dipole fields, whereas spin pumping in ferromagnet-metal systems is mediated by short-range exchange interactions. In this paper, we describe the use of a variety of sensitive resonance experiments to study the microscopic effect of interfaces in spin transport across a variety of material interfaces to highlight the robust nature of nonlocal spin dynamics and to enable a more generalized understanding of spin transport in these systems.

MRFM is a powerful technique^12,13 for the microscopic study of spin physics, as it combines the spectroscopic precision of magnetic resonance imaging (MRI) with the spatial resolution of atomic force microscopy (AFM). In MRFM, we use the strong magnetic field gradient from a micron-size probe magnet to selectively measure microscopic subvolumes of spins within a macroscopic sample. Using this technique, we have measured the size dependence of spin lifetimes within paramagnetic and ferromagnetic material systems, and find that spin transport plays a role in both cases, despite the absence of conduction electrons.

In a ferromagnetic film, we can use the dipole field from a probe magnet to confine nanoscale spin wave modes¹⁴ and use MRFM detection as a non-invasive probe to measure the size dependence of damping within an unpatterned ferromagnetic film.⁸ As spintronic devices reach the nanometer scale, it is important to consider that such effects can dominate the dynamics of these systems, and are of particular relevance to spin-torque oscillator nanocontacts.¹⁵ Our sample consists of an unpatterned YIG film of thickness 25 nm grown by off-axis sputtering on a (111)-oriented Gd₃Ga₅O₁₂ substrate. The probe field is provided by a high coercivity 1.75 μm Sm₁Co₅ particle mounted on a diamond atomic force microscope cantilever. When the applied field is anti-parallel to the tip moment, the tip creates a confining field well in the sample that localizes discrete magnetization precession modes immediately beneath it.¹⁴ As the tip is brought closer to the film, the confining field well gets deeper and the radius of the confined mode becomes smaller. We find that the frequency-dependent linewidth of the mode increases as the mode radius decreases, as shown in Fig. 1(a). Specifically, the frequency dependence of the linewidth is a measure of relaxation in a ferromagnet, and is characterized by the Gilbert damping parameter α. Enhanced damping is identified with spin pumping.¹⁶ 

We find that the Gilbert damping α of the confined modes scales with the surface-volume ratio of the mode, as shown in the main panel of Fig. 1, indicating a relaxation process due to intralayer spin transport at the edge of the mode, as outlined in the schematic of Fig. 1(b). Intralayer spin transport has been previously predicted¹⁷ and observed¹⁸ in metallic ferromagnets due to the interaction of itinerant electrons with non-uniform magnetization precession resulting in a damping-like torque. This spin pumping theory for itinerant electrons predicts an enhanced damping that scales as k², where k is the wavevector of the spin wave mode. It is revealing that we see a similar intralayer spin transport effect in an insulating ferromagnet. In our experiment, the damping does not scale as k² but rather manifests as an interfacial effect, where the interface is defined by the field and is the boundary between the confined precessing magnetization and the surrounding material that is off-resonance. Interfacial spin transport in ferromagnet/normal-metal bilayers is known as spin pumping and is characterized by a spin-mixing conductance parameter $g↑↓$⁠. We characterize our observed effect with a spin-mixing conductance parameter by measuring the surface-volume ratio dependence of the enhanced Gilbert damping parameter, and find $g↑↓=5.3×1019m−2$ for this YIG-YIG system. This is only slightly larger than the spin-mixing conductance $g↑↓=3.4×1019m−2$ previously measured for YIG-Pt.¹⁹ The larger spin-mixing conductance between the field-defined interface of YIG-YIG and the growth-defined interface of YIG-Pt might be explained by two effects: First, the field-defined interface is free of defects and second that the YIG-YIG exchange coupling is stronger than the exchange coupling across a ferromagnet/normal-metal bilayer.

We have previously demonstrated a similar experiment using MRFM to measure the lifetime of a microscopic volume of implanted nitrogen (P1) centers in a paramagnetic diamond spin wire⁹ and here we also find that spin transport plays a clear role in the measured lifetime. The dipole field from a micron-size magnet is used to create a well-defined volume in which the magnetic resonance condition is met, allowing for a microscopic study of spin lifetimes from a confined volume.²⁰ This particular experiment differs from many other resonance experiments as the spin system is not brought out of equilibrium by a microwave field. Rather, for this case of a few spins, it is the statistical fluctuations of the spin system about equilibrium—the spin noise—which is measured. The autocorrelation time of this spin noise signal is a direct measure of the lifetime of the system obtained without perturbing the spins away from thermal equilibrium. We find that the lifetime of spins in a microscopic volume is dominated by spin transport out of the volume.⁹ The spin signal and correlation time are plotted in Fig. 2.

As the MRFM detection volume enters the wire the spin signal grows, and this can be easily understood as an increase in the overlap of the detection volume with the nanowire volume. The correlation time, however, shows a more complex behavior: when the detection volume first enters the nanowire (position ∼50 nm in Fig. 2) the spins inside the volume easily interact with nearby outside neighbors within the spin wire, resulting in a relatively short correlation time. As the detection volume moves further into the spin wire, spins must diffuse further to exit the detection volume and change the overall magnetization of the ensemble, thus increasing the measured lifetime. The lifetime then starts to decrease when the second edge of the detection volume reaches the spin wire volume (position ∼100 nm Fig. 2), as spins can now diffuse out of either side of the detection volume, reducing the correlation time by a factor of ∼2. Scanning deeper into the wire results in no further change because the measurement geometry becomes translationally invariant.

The change in lifetime due to spin transport out of the detection volume in the paramagnetic diamond experiment is similar in phenomenology to the enhanced damping at the edge of the confined spin mode in a ferromagnet, as discussed previously. In the paramagnetic system, the spin transport occurs in the absence of itinerant electrons via flip-flop transitions, in which a pair of anti-aligned neighboring spins exchange their spin orientations via dipolar interactions.²¹ Successive flip-flops result in pure angular momentum or spin transport out of the detection volume without any associated electron charge transport. In the ferromagnetic system, the details of the spin transport out of the confined volume are not yet fully understood, but transport likely occurs via spin waves or impurity centers. Both of these studies indicate the importance of understanding spin transport across field-defined interfaces. They demonstrate spin transport in the absence of conduction electrons and represent relaxation mechanisms that are important to consider when spintronic devices reach the nanoscale.

We now turn our attention to spin transport across heterogeneous material interfaces. We have studied spin transport from an insulating ferromagnet into metallic paramagnets using the inverse spin Hall effect (ISHE) and from an insulating ferromagnet into insulating diamond NV spins.

To study the effect of interface engineering, we measure spin currents at room temperature from Y₃Fe₅O₁₂ or YIG in trilayers of YIG(20 nm)/Cu(t_Cu)/Pt(5 nm) and YIG(20 nm)/Cu(t_Cu)/W(5 nm), where the Cu thickness t_Cu is varied from 0 to 20 nm.¹⁰ The YIG magnetization is made to precess at its FMR frequency by introducing the sample to microwaves from a microwave cavity, while the voltage across the Pt or W layer is measured electrically. An external magnetic field is applied in the plane of the film and on resonance the precessing YIG magnetization transfers a spin current into Cu. Since the Cu thickness t_Cu is much smaller than the spin diffusion length $λSD$⁠, spin accumulation in the Cu spacer drives spin current into the Pt or W layer. The spin current is converted to a charge current by the ISHE in Pt and W, resulting in a measurable voltage $VISHE$⁠. The spin current J_s can be calculated from²² 

where $tNM$ and $θSH$ are the thickness and spin Hall angle of Pt or W, R and w are the total resistance and width of the trilayers, respectively. We find, as expected, that the normalized spin current $Js/Js(0)$ plotted in Fig. 3(a) as a function of Cu spacer thickness t_Cu for YIG/Cu/Pt decreases as t_Cu increases. The spin current for YIG/Cu/W, shown in Fig. 3(c), however, increases up to a value of 4.5 times J_s(0) (where J_s(0) is the spin current without a Cu spacer) when the spacer thickness is increased. This result clearly indicates that the spin current that reaches the W layer is enhanced by insertion of a Cu spacer.

In addition, the efficiency of spin pumping in these trilayers can be quantified by measuring the effective spin-mixing conductances, which are obtained from the enhanced damping of the YIG resonance. This is obtained by measuring the linewidth of the FMR line as a function of microwave frequency. The results are shown for YIG/Cu/W in Fig. 3(d) and it is clear that introduction of the Cu spacer increases the damping enhancement of the YIG. This is in agreement with the result that the spin current measured by ISHE increases when the spacer is present. The effective spin-mixing conductance for a YIG/Cu/NM trilayer has contributions from the two interfaces and the spin resistance of Cu

where the spin resistance $RCu=2e2tCuhσ$ (σ is the electrical conductivity). Using a measured resistivity of $4.0×10−8 Ωm$⁠, we obtain $RCu=6.2×10−20m2$⁠. To determine the intrinsic value for $gYIG/Cu↑↓$⁠, we grow a 2 μm thick Cu layer on YIG to reduce the backflow of spin current and obtain from the enhancement of the FMR linewidth a spin-mixing conductance of $gYIG/Cu↑↓=1.8×1019m−2$⁠. The result is that we can now obtain a value for the spin conductance of the Cu/NM interface and compare this with the spin-mixing conductance of the YIG/NM interface. We find that Cu/Pt has a spin conductance of $gCu/Pt=2.8×1018m−2$⁠, while YIG/Pt has a spin-mixing conductance of $gYIG/Pt↑↓=3.9×1018m−2$⁠. Therefore, the insertion of the Cu spacer leads to an effective spin-mixing conductance of $geff,trilayer↑↓=2.1×1018m−2$⁠, reducing the spin current transmitted from YIG to Pt. On the other hand, Cu/W has a spin conductance of $gCu/Pt=7.6×1018m−2$⁠, while YIG/W has a spin-mixing conductance of $gYIG/Pt↑↓=3.9×1018m−2$⁠. This results in an effective spin-mixing conductance of $geff,trilayer↑↓=4.0×1018m−2$⁠, which enhances the spin current reaching the W from YIG. This effect can be understood as the first step towards spin current optimization by spin-mixing conductance matching, and explains why the electrically detected spin current is larger when a Cu spacer is inserted between YIG and W, but not between YIG and Pt.

Nonlocal spin dynamics have also been recently measured in a ferromagnet/paramagnet system, where both materials are insulators. A novel material system of NV paramagnetic spins and YIG was chosen for this experiment.¹¹ The NV spins exhibit a polarization-dependent photoluminescence (PL) that enables very sensitive detection of their spin state using optical readout. Typically, the NV spin state is manipulated using microwave excitation at its resonance frequency.^23,24 Surprisingly in our experiment,¹¹ we find that the polarization of the NV centers is reduced when a neighboring YIG film is on resonance, as seen in Fig. 4. The effect is clearly a long-range effect as the NV diamond consists of nanodiamonds dispersed on a film, where the interface has not been carefully engineered. Nevertheless, the YIG FMR condition can be clearly seen through its impact on the photoluminescence of the NV, indicating a change in the polarization of the NV spin state when the neighboring YIG is on resonance even when the microwave frequency is far from the NV resonance. The signal at the YIG resonance condition persists when the laser spot is focused on NV diamonds that are not in direct contact with the YIG but separated by a ∼300-nm-thick microwire, as seen in Fig. 4(c), confirming the long-range nature of the coupling.

While the mechanism of this coupling is still under investigation, it is useful to estimate the efficiency of this spin transport by calculating a spin current for comparison with the spin pumping experiments discussed earlier. Steady state spin populations are reached when the polarization rate due to the laser matches the depolarizing spin current from the YIG. This leads to the following phenomenological equation for the spin current defined as the number of spins relaxing per unit time and cross-sectional area in the diamond in units of A/m²:

where, e is the charge of an electron, n is the density of NV spins in diamond, Δp is the reduction in polarization of the NV spins when the YIG is on resonance, R_L is the optical pumping rate of the laser, and t is the thickness of the diamond film.

For this experiment, we assume values within reasonable estimates that give us an upper bound for the spin current. Using $n=5×1024 m−3, Δp=10 %$ (obtained from the 1% change in PL intensity), $RL=107 spins/s$ (from Manson et al.²⁵), and t = 500 nm, we obtain a spin current of $Js=4×105A/m2$⁠. This spin current, whose magnitude is set by the strength of the coupling of the YIG magnetization to that of the diamond, is only two orders of magnitude smaller than in YIG/metal bilayers. This is not unexpected, given the long range coupling between YIG and diamond in these experiments in contrast to the interfacial exchange coupling between YIG and a neighboring metal in spin pumping experiments. While the mechanism is yet to be understood, this provides further evidence for efficient spin transfer in the absence of itinerant electrons, and demonstrates that NV centers in diamond are a sensitive tool for the study of spin dynamics and transport.

In conclusion, we observe that spin transport occurs between a broad variety of spatially separated spin systems. Nonlocal spin dynamics have been observed in diverse settings and material contexts demonstrating that non-equilibrium angular momentum transfer is a robust and general phenomenon. Both exchange and dipole interactions can provide substantial coupling and relaxation effects. Current theory does not yet provide a comprehensive explanation for the observed effects. It seems clear, however, that the engineering of interfaces by the insertion of interlayers can be used to optimize spin transport. Finally, sensitive local and selective probes, such as MRFM, NV center photoluminescence, and electrical detection, are needed to provide a complete picture of the physics underlying spin transport.

This work was supported by the U.S. Department of Energy (DOE), Office of Science, Basic Energy Sciences (BES), under Award No. DE-FG02-03ER46054 (FMR measurements) and Award No. DE-SC0001304 (sample synthesis), the Army Research Office under Award No. W911NF-09-1-0147 (paramagnetic MRFM), the Center for Emergent Materials, an NSF-funded MRSEC under Award No. DMR-1420451 (structural characterization), and Lake Shore Cryotronics, Inc. (magnetic characterization). We also acknowledge technical support and assistance provided by the NanoSystems Laboratory at the Ohio State University.