(3) Spin-dependent scattering of electrons in multilayers:

We shall first discuss the different types of scatterings that the electrons may experience in magnetic multilayers. In the Boltzmann equation approach we are mainly concerned with elastic (energy conserving) scattering, i.e., in each scattering process only the direction of propagation of electrons changes. However, in real situation, the electrons may experience different types of scattering. Therefore, one needs to distinguish between spin-dependent scattering, which causes the GMR and spin-flip scattering, which is detrimental to the GMR. The two types of scattering are illustrated in Figure 13.4. In the case of spin-dependent scattering the orientation of the electron spin is conserved in each scattering event but the probabilities of scattering for electrons with ↑ and ↓ spin projections are different.

On the other hand, when an electron undergoes a spin-flip scattering, its spin orientation changes from or vice versa. There are several sources responsible for the spin-flip scattering:

1. During the fabrication process, some of the magnetic atoms may enter the NM spacer layer to form magnetic impurities. Hence, when an electron is scattered off by a magnetic impurity the spins of the electron and that of the impurity in the NM layer can interchange provided the impurity spin is free to rotate. This occurs when the impurity spin is not strongly coupled to the spins of the FM layers, i.e., the impurity is located considerably far away from the FM / NM interface.

2. Electrons can also be scattered by spin waves in the FM layers. Spin waves are quasi-particles with spin one and, therefore, creation (annihilation) of a spin wave in a collision with an electron leads to a flip of the electron spin. Since it involves the spin-wave energy, this is an inelastic process which is important only at elevated temperatures.

3. When impurities with a strong spin-orbit interaction, such as gold, are present in the multilayer, the spin of an electron scattered by such impurity may be reversed due to the spin-orbit interaction. Since all these processes mix the ↑ and ↓ spin channels, they are detrimental to GMR.

Now let us get back in to the spin-dependent scattering which conserves the electron spin. The key feature here is that electrons with different spin orientations (↑, ↓) are scattered at different rates when they enter into FM layers. Following that the electrons obey the Pauli exclusion principle, an electron can be scattered from an impurity atom only to quantum states that are not occupied by other electrons. At zero (low) temperatures, all the states with energies E below the Fermi energy E_F are occupied and those with E>E_F are empty. Since scattering from impurities is elastic, electrons at the Fermi level (which carry the current) can be scattered only to the states in the immediate vicinity of the Fermi level. For example, the Fermi level in copper (and other noble metals) intersects only the conduction band whose density of states D(E_F) is low (see Figure 13.5).