36
2 Basic Elements of Spintronics
ρ = (ρ + + ρ − ) = (−e)(n + + n − )
(2.21)
Here,
s + ( s − ) and ρ + (ρ − ) are the density of spin and charge related to the majority
(minority) spin electrons, respectively. It is known that the direction of angular
momentum and magnetization for the electrons is opposite with respect to each
other. As generally can be considered, the system is large enough compared to that
of the electronic mean free path. Therefore, flow of electrons takes place diffusively
by undergoing repeated scattering and then acceleration successively.
Figure 2.7 shows that a negatively charged electron (−e) is under the application
of an electric field
E, which in turn accelerates it towards right direction. After
undertaking a short duration of travel, the electron suffers scattering by a scattering
center and consequently, its velocity changes. Followed by this scattering, the electron
again accelerates until it experiences the next collision. It is well known that the
average distance, traversed by electrons, between two consecutive collisions is termed
as the mean free path, which is typically 10 nm for metal at 300 K. Hence, a physical
picture can be framed where in each spin sub-channel, diffusive transport of the
majority and minority spin electrons takes place independently. It should be noted
that in those two spin sub-channels, both the charge densities and drift velocities
( ν + ,
ν − ) of electrons are different. Therefore, electric conductivities (σ + , σ − ) (=
(charge density) × (mobility)) of electrons are also different in those two spin subchannels. Thus, under the application of an electric field, along with drift charge
current density,
J
Q,Dri f t , we also obtain a drift spin current density ˆ
J
S,Dri f t .
ˆ
J
S,Dri f t
∗ = =
ν +
s + + +
ν −
s − =
2
e spin
(σ + − σ − )
−e
E
J
Q,Dri f t
= =
ν + ρ + + +
ν − ρ − = (σ + + σ − )
E
(2.22)
Here, we have neglected the exchange mediated term (i.e., second term) for
simplicity.
E
Fig. 2.7 Drift motion of free electron in conductive material
2 Basic Elements of Spintronics
ρ = (ρ + + ρ − ) = (−e)(n + + n − )
(2.21)
Here,
s + ( s − ) and ρ + (ρ − ) are the density of spin and charge related to the majority
(minority) spin electrons, respectively. It is known that the direction of angular
momentum and magnetization for the electrons is opposite with respect to each
other. As generally can be considered, the system is large enough compared to that
of the electronic mean free path. Therefore, flow of electrons takes place diffusively
by undergoing repeated scattering and then acceleration successively.
Figure 2.7 shows that a negatively charged electron (−e) is under the application
of an electric field
E, which in turn accelerates it towards right direction. After
undertaking a short duration of travel, the electron suffers scattering by a scattering
center and consequently, its velocity changes. Followed by this scattering, the electron
again accelerates until it experiences the next collision. It is well known that the
average distance, traversed by electrons, between two consecutive collisions is termed
as the mean free path, which is typically 10 nm for metal at 300 K. Hence, a physical
picture can be framed where in each spin sub-channel, diffusive transport of the
majority and minority spin electrons takes place independently. It should be noted
that in those two spin sub-channels, both the charge densities and drift velocities
( ν + ,
ν − ) of electrons are different. Therefore, electric conductivities (σ + , σ − ) (=
(charge density) × (mobility)) of electrons are also different in those two spin subchannels. Thus, under the application of an electric field, along with drift charge
current density,
J
Q,Dri f t , we also obtain a drift spin current density ˆ
J
S,Dri f t .
ˆ
J
S,Dri f t
∗ = =
ν +
s + + +
ν −
s − =
2
e spin
(σ + − σ − )
−e
E
J
Q,Dri f t
= =
ν + ρ + + +
ν − ρ − = (σ + + σ − )
E
(2.22)
Here, we have neglected the exchange mediated term (i.e., second term) for
simplicity.
E
Fig. 2.7 Drift motion of free electron in conductive material