2.4 Spin Accumulation
41
Ferromagnet
Metal
Spin injection
Spin flipped after
scatterings
Fig. 2.10 Spin injection of an up-spin electron from a ferromagnetic contact into a paramagnet is
demonstrated. Injected up-spin electron suffers N momentum relaxing scattering events (indicated
by crosses) after it undergoes random walk in the paramagnet. Eventually, spin of the injected
electron is flipped. Spin flip occurs at an average distance (over an ensemble of many injected
carriers) of roughly the spin diffusion length from the interface (Figure adapted and redrawn from
Bandyopadhyay and Cahay 2008)
will undergo N momentum-relaxing collisions before being flipped, as demonstrated
in Fig. 2.10.
We denote the corresponding mean free path, i.e., the average distance between
momentum scattering collisions by λ f and the average spin flip time by τ ↑↓ . Now,
the target up-spin electron is allowed to move randomly in three dimensions in
equal amount. Then after suffering each collision, the average distance in a direction
perpendicular to the interface that the electron penetrates into the non-magnetic
material, i.e., spin accumulation length λ sd is given by
λ sd = λ f
N /3
(2.30)
On the other hand, the total distance travelled by the injected up-spin electron is
N * λ f , which in turn equals the velocity of the injected electron at the Fermi level,
i.e., Fermi velocity, v F times the spin-flip time τ s (=τ ↑↓ =τ ↓↑ )
i.e., N ∗ λ f = v F ∗ τ s
(2.31)
Assuming elastic collisions here, the magnitude of the electron’s velocity is
invariant. Furthermore, we suppose that the carriers are injected with the Fermi
velocity v F . From Eqs. (2.30) and (2.31), we obtain the expression
λ sd =
λ f v F τ S
3
(2.32)
41
Ferromagnet
Metal
Spin injection
Spin flipped after
scatterings
Fig. 2.10 Spin injection of an up-spin electron from a ferromagnetic contact into a paramagnet is
demonstrated. Injected up-spin electron suffers N momentum relaxing scattering events (indicated
by crosses) after it undergoes random walk in the paramagnet. Eventually, spin of the injected
electron is flipped. Spin flip occurs at an average distance (over an ensemble of many injected
carriers) of roughly the spin diffusion length from the interface (Figure adapted and redrawn from
Bandyopadhyay and Cahay 2008)
will undergo N momentum-relaxing collisions before being flipped, as demonstrated
in Fig. 2.10.
We denote the corresponding mean free path, i.e., the average distance between
momentum scattering collisions by λ f and the average spin flip time by τ ↑↓ . Now,
the target up-spin electron is allowed to move randomly in three dimensions in
equal amount. Then after suffering each collision, the average distance in a direction
perpendicular to the interface that the electron penetrates into the non-magnetic
material, i.e., spin accumulation length λ sd is given by
λ sd = λ f
N /3
(2.30)
On the other hand, the total distance travelled by the injected up-spin electron is
N * λ f , which in turn equals the velocity of the injected electron at the Fermi level,
i.e., Fermi velocity, v F times the spin-flip time τ s (=τ ↑↓ =τ ↓↑ )
i.e., N ∗ λ f = v F ∗ τ s
(2.31)
Assuming elastic collisions here, the magnitude of the electron’s velocity is
invariant. Furthermore, we suppose that the carriers are injected with the Fermi
velocity v F . From Eqs. (2.30) and (2.31), we obtain the expression
λ sd =
λ f v F τ S
3
(2.32)