Nonlocal magnetization dynamics in ferromagnetic heterostructures

Two complementary effects modify the GHz magnetization dynamics of nanoscale heterostructures of ferromagnetic and normal materials relative to those of the isolated magnetic constituents. On the one hand, a time-dependent ferromagnetic magnetization pumps a spin angular-momentum flow into adjacent materials and, on the other hand, spin angular momentum is transferred between ferromagnets by an applied bias, causing mutual torques on the magnetizations. These phenomena are manifestly nonlocal: they are governed by the entire spin-coherent region that is limited in size by spin-flip relaxation processes. This review presents recent progress in understanding the magnetization dynamics in ferromagnetic heterostructures from first principles, focusing on the role of spin pumping in layered structures. The main body of the theory is semiclassical and based on a mean-field Stoner or spin-density-functional picture, but quantum-size effects and the role of electron-electron correlations are also discussed. A growing number of experiments support the theoretical predictions. The formalism should be useful for understanding the physics and for engineering the characteristics of small devices such as magnetic random-access memory elements.

Figure 1

Spin-valve schematic: two monodomain ferromagnets $\left(F\right)$ separated by a normal $\left(N\right)$ spacer and attached to normal leads. In this current perpendicular to the interface plane (CPP) configuration, the current flowing between two normal reservoirs sequentially traverses two magnetic layers. Also shown are two effective circuits discussed in Sec. 2A for the semiclassical regime where the interlayer exchange coupling vanishes.

Figure 2

Magnetic scatterer (interconnector) connecting two chaotic normal-metal $\left(N\right)$ nodes via ballistic leads that support a quantized number of transverse channels at the Fermi energy. The scatterer includes a ferromagnetic $\left(F\right)$ region characterized by a uniform magnetic direction $\mathbf{m}$ (but not necessarily uniform magnitude of the exchange spin splitting along $\mathbf{m}$), which is depicted as the gray box in the center. The dark-gray areas of the ferromagnetic region near both $F\mid N$ interfaces mark the extent of the transverse spin coherence characterized by ${\lambda }_{\mathrm{sc}}$, Eq. (13). Each of the two normal-metal regions is divided into a reservoir characterized by the electrochemical potential ${\mu }_c$ and (vector) spin accumulation ${\mathbf{\mu }}_s$, a ballistic lead with a fixed number of transport channels, and possibly a disordered region incorporated in the interconnector (depicted by wavy lines), which accounts for relevant normal-metal scattering processes. The interconnector is described by spin-dependent reflection and transmission coefficients forming the scattering matrix, Eq. (21), for quantum channels in the normal leads. The purpose of the formalism is to calculate the nonequilibrium charge and spin flows in the leads, $I_c$ and ${\mathbf{I}}_s$, induced by spin accumulations and electrochemical-potential imbalance in the nodes.

Figure 3

Contact between a ferromagnetic and a normal node. The notation is analogous to Fig. 2. Here the charge and spin currents in the normal lead depend on the conductance matrix $g^{\sigma {\sigma }^{\prime }}$ defined in terms of the spin-dependent reflection coefficients from the normal-metal side, as before; see Eq. (23). The nonequilibrium spin accumulation in the ferromagnetic node is collinear with the magnetization $\mathbf{m}$.

Figure 4

The ferromagnetic magnetization $\mathbf{m}$ rotates around the effective field ${\mathbf{H}}_{\mathrm{eff}}$. In the steady state, the spin pumping ${\mathbf{I}}_s^{\text{pump}}$ cancels the spin backflow ${\mathbf{I}}_s^{\mathrm{back}}$ that accompanies the nonequilibrium spin accumulation ${\mathbf{\mu }}_s=\hslash \gamma {\mathbf{H}}_{\mathrm{eff}}$.

Figure 5

Symbols represent the data points derived from FMR experiments on $N\mid \mathrm{Py}\mid N$ sandwiches with $N=\mathrm{Pt}$, Pd, or Cu (105, 106). The lines show fits by $G+a∕d$. $G=0.97\times {10}^8$, $1.06\times {10}^8$, and $1.31\times {10}^8{\mathrm{s}}^{-1}$, where, following Eq. (63), $a$ corresponds to $g_r^{\uparrow \downarrow }∕S=0.2$, 15.3, and $25.8{\mathrm{nm}}^{-2}$ for Cu, Pd, and Pt, respectively, taking a $g$ factor of 2.1 for Py films (105, 106). The horizontal axis uses a reciprocal scale.

Figure 6

Schematic view of a $F\mid N$ bilayer in which the magnetization direction $\mathbf{m}\left(t\right)$ of the ferromagnet $F$ rotates in an easy plane, pumping a spin current ${\mathbf{I}}_s^{\text{pump}}$ into the adjacent normal-metal layer $N$. The $N$ layer here is not an ideal reservoir but rather a film of the same cross section as the magnetic layer $F$. The spin pumping builds up a position $\left(x\right)$ dependent spin accumulation in $N$ that either relaxes by spin-flip scattering or drives a spin current back into the ferromagnet as ${\mathbf{I}}_s^{\left(0\right)}$.

Figure 7

Gilbert damping constants (circles) in $\mathrm{Cu}\mid \mathrm{Py}\left(3\mathrm{nm}\right)\mid \mathrm{Cu}\left(L\right)\mid \mathrm{Pt}\left(5\mathrm{nm}\right)$ and $\mathrm{Cu}\mid \mathrm{Py}\left(3\mathrm{nm}\right)\mid \mathrm{Cu}\left(L\right)$ structures measured by 107, 108 as a function of Cu-layer thickness $L$. Solid lines are the theoretical results according to Eq. (80) with parameters discussed in the text. The horizontal axis uses a logarithmic scale.

Figure 8

Similar to Fig. 6, but the normal-metal system is a $N_1\mid N_2$ bilayer. Ferromagnetic magnetization precession pumps spins into the first normal-metal layer $N_1$. The spin accumulation in $N_1$ either flows back into the ferromagnet $F$ as spin current ${\mathbf{I}}_{s1}^{\left(0\right)}$, relaxes in $N_1$, or leaves via the second normal-metal layer $N_2$ as spin current ${\mathbf{I}}_{s2}^{\left(0\right)}$. The spin accumulation in the ideal spin sink $N_2$ vanishes.

Figure 9

Gilbert damping measured in $\mathrm{Cu}\mid \mathrm{Py}\left(3\mathrm{nm}\right)\mid \mathrm{Cu}\left(L\right)\mid \mathrm{Pt}\left(2\mathrm{nm}\right)$ and $\mathrm{Cu}\mid \mathrm{Py}\left(3\mathrm{nm}\right)\mid \mathrm{Cu}\left(L\right)$ structures for several Cu-layer thicknesses $L$ at various temperatures. Solid lines are fits based on spin-pumping theory. Adapted from 172.

Figure 10

Enhancement of the Gilbert damping coefficient for an Fe∣Au∣Fe trilayer as a function of $1∕d$ where $d$ is the thickness of the excited Fe layer in monolayers (ML’s). The filled circles (●) are the room-temperature values measured by 162 and the open one (엯) is a low-temperature $\left(77\mathrm{K}\right)$ value from 59. The theoretical prediction based on Eq. (83) for $0\mathrm{K}$ (with $\sigma \to \infty$) is shown as solid line and the room-temperature (RT) corrected (with phonon scattering) one as dashed lines. The results of $0\mathrm{K}$ calculations for a Au∣Fe∣vacuum system are given by crosses (×) and stars (∗) for specular and disordered interfaces, respectively. The value of the Gilbert damping $\alpha$ for a single Fe film is marked with an arrow. From 177.

Figure 11

(Color online) A cartoon of the dynamic-coupling phenomenon. In the left drawing, the layer $F_1$ is at resonance and its precessing magnetic moment pumps spin current into the spacer, while $F_2$ is detuned from its FMR. In the right drawing, both films resonate at the same external field, inducing spin currents in opposite directions. The short arrows in $N$ indicate the instantaneous direction of the spin angular momentum $\propto {\mathbf{m}}_i\times {\dot{\mathbf{m}}}_i$ carried away by the spin currents. Darker areas in $F_i$ around the interfaces represent the narrow regions in which the transverse spin momentum is absorbed. From 57.

Figure 12

The main panel displays the real part of the two frequencies $\omega$ which solve Eq. (86) for linearized dynamics of a symmetric magnetic bilayer close to the parallel magnetization configuration. Solid lines are obtained by setting the static coupling to zero, ${\omega }_x=0$, and dashed lines for ${\omega }_x=0.01{\omega }_1$. The lower inset shows the corresponding imaginary part of $\omega$. When ${\omega }_x=0$ and near the frequency crossing, ${\omega }_1\approx {\omega }_2$, the symmetric mode has a smaller imaginary part (less damping) and only slightly larger real part. For ${\omega }_x<0$, the symmetric mode has a larger (real part of the) frequency than the antisymmetric mode, whereas the antisymmetric mode acquires a higher frequency for sufficiently large and positive ${\omega }_x$. In all cases, the antisymmetric mode remains more strongly damped. The upper inset shows the results for ${\omega }_x=0.01{\omega }_1$ but setting ${\alpha }_1=0$ while keeping a finite ${\alpha }_2$. This inset illustrates the effect of a bilayer asymmetry.

Figure 13

Dissipation power $P$ of a symmetric bilayer in the presence of a uniform circularly polarized rf driving field at frequency $\omega$. The parameters are the same as those used to generate the dashed lines in Fig. 12.

Figure 14

(Color online) Dependence of the static FMR fields $H_1$ (circles) and $H_2$ (triangles) of the thin Fe film $\left(F_1\right)$ and the thick Fe film $\left(F_2\right)$, respectively, on the angle $\phi$ of the static magnetic field with respect to the Fe [100] crystallographic axis. The sketch of the in-plane measurement in the left inset shows how the rf magnetic field (double-pointed arrow) drives the magnetization (on a scale grossly exaggerated for easy viewing). The right inset shows the measured absorption peaks for layers $F_1$ and $F_2$ at $\phi =60°$. The absorption power is given by the imaginary part of the susceptibility of the rf magnetization component along the rf driving field, which is denoted by ${\chi }^{″}$. From 57.

Figure 15

(Color online) Comparison of theory (solid lines) with room-temperature measurements (symbols) close to and at the crossover of the FMR fields, marked by the shaded area in Fig. 14. The left and right frames show FMR signals for the field difference, $H_2-H_1$, of $-78$ and $+161\mathrm{Oe}$, respectively. The theoretical results are parametrized with the full set of magnetic parameters which were measured independently (162). The magnitude of the spin-pump current was determined by the linewidth at large separation of the FMR peaks and agrees well with that predicted theoretically; see Sec. 4C. The middle frame displays the effective FMR linewidth of magnetic layers for the signals fitted by two Lorentzians as a function of the external field. At $H_1=H_2$, the FMR linewidths reached their minimum values at the level of intrinsic Gilbert damping of isolated films. The calculations in the middle frame did not take small variations of the intrinsic damping with angle $\phi$ into account, which resulted in deviations between theory and experiment for larger $\mid H_1-H_2\mid$. Notice that $\Delta H_1$ first increases before attaining its minimum, which is due to a contribution of the antisymmetric collective mode. As a side comment, it should be noted that although fitting the absorption signal by two Lorentzians is a legitimate approach to comparing theoretical calculation with experimental curves, the analysis does not imply that the signal is always well approximated by the sum of two Lorentzians, which may not be the case very close to the FMR-field crossover. From 57.

Figure 16

A schematic view of the superlattice and its geometric parameters as considered in the text. A $F\mid N$ bilayer is repeated along the $x$ axis, with either ferromagnetic or antiferromagnetic alignment of the consecutive magnetic layers. The system is translationally invariant along the two remaining axes.

Figure 17

Schematic of a current-biased spin valve (a “dynamic” version of Fig. 1). The symbols are explained in the text.

Figure 18

Solid lines display the precession-cone angle dependence of the Gilbert damping parameter, Eq. (99), and the dotted lines show the extrapolated small-angle approximation (100). The lowest lines are representative for Co and the uppermost ones for Py, assuming magnetic-film thickness of $5\mathrm{nm}$. Fe and Ni are expected to be intermediate between these two cases.

Figure 19

Gilbert damping, Eq. (116), in units of ${\alpha }_{\max }=\gamma s_0∕2M_s$ as a function of the normalized spin-relaxation rate. Inset: Geometry of the model.

Figure 20

The real part of $G_{\uparrow \downarrow }^t=(e^2∕h)t^{\uparrow \downarrow }∕S$ as a function of the magnetic-layer thickness $d$ calculated from first principles (●) for realistic multiband electronic structures and that calculated for an isotropic free-electron gas with Fermi energy ${\epsilon }_F=7\mathrm{eV}$ (chosen to obtain the correct value of the Sharvin conductance in Cu) and various choices of the exchange splitting $\Delta$. From 177.

Figure 21

Spin-mixing conductances of a Cu∣Co∣Cu(111) structure with disordered interfaces as a function of the Co-layer thickness. From 177.