2.4 Spin Accumulation
37
2.4 Spin Accumulation
2.4.1 What Is Spin Accumulation?
As a consequence of spin injection from ferromagnet to paramagnet in a ferromagnet/paramagnet heterostructure, a particular species of spin will accumulate in
the paramagnet near the interface (Jedema et al. 2001). Describing the ferromagnet
based on the Stoner–Wohlfarth model, let us assume that the up-spin electrons are
the majority in the ferromagnet. Thus, the density of up-spin electrons in the ferromagnet is much larger than that of down-spin electrons and consequently, up-spin
electrons become the major contributors of current injected by the ferromagnet.
This results in an excess of up-spin electrons and a net magnetic moment of such
spin-polarized ensemble within the paramagnet near the paramagnet/ferromagnet
interface. This phenomenon is referred to as spin accumulation. Since this is a nonequilibrium phenomenon, such accumulated spins are not expected to spread over
the entire paramagnet forever because of spin-flip scattering events, which might
convert some of the up-spin electrons of the ensemble into down-spin electrons. This
in turn brings the non-equilibrium phenomenon into equilibrium one, where at the
steady state the population of up-spin and down-spin electrons becomes the same
far from the interface into the bulk of the paramagnet. Thus, spin accumulation is
quite likely to gradually decay with distance away from the interface. In this context,
we may define and extract a ‘spin accumulation length’ which is the characteristic
distance over which the accumulated spin decays to 1/e times its magnitude at the
interface.
In order to discuss spin accumulation and its decay, let us consider an interface
between a ferromagnet (FM) and a non-magnetic material (NM) (Fig. 2.8a). In either
a bulk metallic or semiconducting non-magnetic material, the energy dispersion
relation could be simply considered as parabolic. As already discussed, if the material
is non-magnetic, the two spin channels have the same mobility.
Thus, under the application of a small electric field, Fermi energy surface for the
up-spin and down-spin channels on the (k x , k y ) plane will be shifted by an equal
amount, as shown in Fig. 2.9a. Denoting Δk as the shift in momentum space for both
spin Fermi surfaces, it must satisfy the following equation:
F = −eE =
dk
dt
=
k
τ m
,
(2.23)
where F is the force acting on the electron, E is the applied electric field, e is the
charge on the electron and τ m is the scattering time of the electron. τ m is related to
the mobility of the carrier (μ) by
μ =
eτ m
m ∗
(2.24)
37
2.4 Spin Accumulation
2.4.1 What Is Spin Accumulation?
As a consequence of spin injection from ferromagnet to paramagnet in a ferromagnet/paramagnet heterostructure, a particular species of spin will accumulate in
the paramagnet near the interface (Jedema et al. 2001). Describing the ferromagnet
based on the Stoner–Wohlfarth model, let us assume that the up-spin electrons are
the majority in the ferromagnet. Thus, the density of up-spin electrons in the ferromagnet is much larger than that of down-spin electrons and consequently, up-spin
electrons become the major contributors of current injected by the ferromagnet.
This results in an excess of up-spin electrons and a net magnetic moment of such
spin-polarized ensemble within the paramagnet near the paramagnet/ferromagnet
interface. This phenomenon is referred to as spin accumulation. Since this is a nonequilibrium phenomenon, such accumulated spins are not expected to spread over
the entire paramagnet forever because of spin-flip scattering events, which might
convert some of the up-spin electrons of the ensemble into down-spin electrons. This
in turn brings the non-equilibrium phenomenon into equilibrium one, where at the
steady state the population of up-spin and down-spin electrons becomes the same
far from the interface into the bulk of the paramagnet. Thus, spin accumulation is
quite likely to gradually decay with distance away from the interface. In this context,
we may define and extract a ‘spin accumulation length’ which is the characteristic
distance over which the accumulated spin decays to 1/e times its magnitude at the
interface.
In order to discuss spin accumulation and its decay, let us consider an interface
between a ferromagnet (FM) and a non-magnetic material (NM) (Fig. 2.8a). In either
a bulk metallic or semiconducting non-magnetic material, the energy dispersion
relation could be simply considered as parabolic. As already discussed, if the material
is non-magnetic, the two spin channels have the same mobility.
Thus, under the application of a small electric field, Fermi energy surface for the
up-spin and down-spin channels on the (k x , k y ) plane will be shifted by an equal
amount, as shown in Fig. 2.9a. Denoting Δk as the shift in momentum space for both
spin Fermi surfaces, it must satisfy the following equation:
F = −eE =
dk
dt
=
k
τ m
,
(2.23)
where F is the force acting on the electron, E is the applied electric field, e is the
charge on the electron and τ m is the scattering time of the electron. τ m is related to
the mobility of the carrier (μ) by
μ =
eτ m
m ∗
(2.24)