First-principles study of magnetization relaxation enhancement and spin transfer in thin magnetic films

The interface-induced magnetization damping of thin ferromagnetic films in contact with normal-metal layers is calculated from first principles for clean and disordered Fe/Au and Co/Cu interfaces. Interference effects arising from coherent scattering turn out to be very small, consistent with a very small magnetic coherence length. Because the mixing conductances which govern the spin transfer are to a good approximation real-valued, the spin pumping can be described by an increased Gilbert damping factor but an unmodified gyromagnetic ratio. The results also confirm that the spin-current-induced magnetization torque is an interface effect.

Figure 1

Reflection spin-mixing conductance (per unit area) of a Au/Fe/Au(001) trilayer with perfect interfaces as a function of the thickness $d$ of the Fe layer. In this and subsequent plots, mixing conductances expressed in terms of number of conduction channels per unit area are converted to ${\mathrm{\Omega }}^{-1}{m}^{-2}$ using the conductance quantum $e^2∕h$, i.e., $G_{\uparrow \downarrow }=(e^2∕h)g_{\uparrow \downarrow }$.

Figure 2

Transmission spin-mixing conductance of a Au/Fe/Au (001) trilayer with perfect interfaces as a function of the thickness $d$ of the Fe layer.

Figure 3

Reflection spin-mixing conductance of a Cu/Co/Cu (111) trilayer with perfect interfaces as a function of the thickness $d$ of the Co layer.

Figure 4

Transmission spin-mixing conductance of a Cu/Co/Cu (111) trilayer with perfect interfaces as a function of the thickness $d$ of the Co layer.

Figure 5

(Color) Plotted within the first Brillouin zone for the Cu/Co(111) interface are transmission probability for (a) majority spins and (b) minority spins. (c) The real and (d) imaginary parts of $r_{N\to N}^{\uparrow }{r}_{N\to N}^{\downarrow \mathrm{\star }}$. (e) The real and (f) imaginary parts of $t_{\mathrm{int}}^{\uparrow }{t}_{\mathrm{int}}^{\downarrow \mathrm{\star }}$ where $t_{\mathrm{int}}^{\sigma }=t_{F\to N}^{\sigma }\bullet t_{N\to F}^{\sigma }$ as discussed in the text. Note the different scales for panels (a) and (b) and for (c)–(f).

Figure 6

Spin-mixing conductances of a Au/Fe/Au (001) trilayer with disordered interfaces as a function of the thickness $d$ of the Fe layer.

Figure 7

Spin-mixing conductances of a Cu/Co/Cu (111) trilayer with disordered interfaces as a function of the thickness $d$ of the Co layer.

Figure 8

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. The filled circles (•) are the RT values measured in Ref. 29 and the open one (엯) is a low temperature value from Ref. 31. The theoretical predictions based on Eq. (12) for 0 K (with $\sigma \to \infty$) are shown as solid and the RT-corrected (with phonon scattering) ones as dashed lines. The results of 0 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 $$for a single Fe film is marked with an arrow.

Figure 9

The real part of $G_{\uparrow \downarrow }^t$ calculated for a free electron model with ${\epsilon }_F=7\mathrm{eV}$ (energy measured from the bottom of the parabolic conduction band in the normal metal) and various choices of the exchange splitting $\mathrm{\Delta }$. The interlayer distance is taken to be the same as for the Cu/Co(111) system. The results of the first-principles calculations (●) from Fig. 4 are included for comparison.