Dephasing in ferromagnetic nanowires: Role of spin waves

We present a calculation of the dephasing time of electrons in a ferromagnet relevant for the conductance fluctuations. We focus on the contribution from the interaction with spin waves. Explicit results are presented for quasi-one-dimensional systems. Going beyond previous calculations, we do not restrict ourselves to the limit of a small exchange splitting compared to the electronic elastic scattering time, nor does our calculation rely on the diffusion approximation to describe electronic transport.

Figure 1

Illustration of our semiclassical calculation. (a) A spin-up electron propagates ballistically through the ferromagnet, its direction of propagation regularly being randomized by scattering off impurities (black circles). At time $t$, it interacts with the fluctuating field of the $d$-band electrons and its spin is flipped down. After a time $\tau$, the electron has traveled over a distance $\mathit{\rho }$ and it interacts again with the $d$-band electrons, its spin being flipped up again. Since the interactions occur randomly in time and space, the phase of the electron becomes randomized while propagating. (b) The part of the trajectory important for dephasing is when the electron carries spin down. When averaging over the displacement $\mathit{\rho }\left(\tau \right)$ during this part, we sum over trajectories with no scattering events ($n=0$), with one scattering event ($n=1$), and so forth.

Figure 2

(a) Diagrammatic representation of the bare structure factor ${\Pi }_{\sigma ,\sigma }^{\left(0\right)}$. The solid blue (dashed red) line represents a retarded (advanced) Green function, and the dotted line with the cross represents a correlated impurity scattering. (b) Diagrams representing the correction to the structure factor from spin-wave-induced dephasing. Correlated excitation and absorption of a spin wave changes temporarily the energy, momentum, and spin of one of the propagators. The shaded block represents a ladder of impurity scatterings.

Figure 3

Diagrams representing the spin-wave correction to the diffuson propagator. (Impurity lines are not shown in the diagrams in this figure.) (a) Diagrams in which the spin-wave propagator does not connect the retarded and advanced propagators in the diffusion. (b) Diagrams in which retarded and advanced propagators are connected by a spin-wave propagator. Only diagrams shown in part (a) are relevant for the calculation of the dephasing rate.

Figure 4

(a) Diagrams representing the spin-wave correction to the disorder-averaged structure factor. (b) Diagrammatic definition of the Hikami box appearing in some of the diagrams of part (a). (c) Definition of the dressed vertices appearing in some of the diagrams of part (a).