34
2 Basic Elements of Spintronics
(ii) At the interface
At the interface, due to the difference in conductivities of the two materials, α(x) is
expected to change abruptly. However, in the absence of spin-flip scattering mechanisms at the interface, β(x) must be continuous. Consequently, at and at the close
vicinity of the interface, dβ/dx = 0 and μ ↑ (x) = μ ↓ (x). Now, introducing the weighted
electrochemical potential
μ 0 = α(x)μ ↑ (x) + (1 − α)μ ↓ (x),
(2.15)
it can be shown that
d
dx
μ 0 = −
e
σ (x)J
(2.16)
where σ(x) = σ ↑ (x) + σ ↓ (x) and J denotes the total current density flowing through
the interface. We understand that μ 0 (x) is the electrochemical potential value when
there is equilibrium distribution between the up-spin and down-spin electrons in the
current. As discussed earlier, at and in the vicinity of the interface, the discontinuity
of μ 0 leads to a boundary resistance defined as follows:
R b =
μ
e J
=
μ F, 0 (x = 0−) − μ N, 0 (x = 0 + )
e J
(2.17)
where
μ F, 0 (x = 0−) = α F μ F, ↑ (x = 0−) + (1 − α F ) μ F, ↓ (x = 0−),
μ N, 0 (x = 0+) = α N μ N, ↑ (x = 0+) + (1 − α N ) μ N, ↓ (x = 0+)
(2.18)
This electrochemical potential difference is the driving force for the spin current
conversion across the interface
In this way, we use the concept of electrochemical potentials for the up-spin and
down-spin channels to calculate the amount of spin polarization of the current in
a semiconductor in close contact with a ferromagnetic injector or in a spin valve
consisting of a semiconductor sandwiched between two ferromagnetic contacts.
(ii) Spin injection from one ferromagnetic (FMI) to another ferromagnetic
(FM2) material
Although less discussed, we present here description of spin injection process from
a ferromagnetic material to another. Let us do suitable biasing so that net electrons
flow takes place from one ferromagnetic to another ferromagnetic material as shown
in Fig. 2.6, then subsequently the spins of the first ferromagnet (FM1) are injected
into the second one (FM2) through the interface. This is followed by an exchange
interaction between injected electron spins with the spins of the electrons of the host
2 Basic Elements of Spintronics
(ii) At the interface
At the interface, due to the difference in conductivities of the two materials, α(x) is
expected to change abruptly. However, in the absence of spin-flip scattering mechanisms at the interface, β(x) must be continuous. Consequently, at and at the close
vicinity of the interface, dβ/dx = 0 and μ ↑ (x) = μ ↓ (x). Now, introducing the weighted
electrochemical potential
μ 0 = α(x)μ ↑ (x) + (1 − α)μ ↓ (x),
(2.15)
it can be shown that
d
dx
μ 0 = −
e
σ (x)J
(2.16)
where σ(x) = σ ↑ (x) + σ ↓ (x) and J denotes the total current density flowing through
the interface. We understand that μ 0 (x) is the electrochemical potential value when
there is equilibrium distribution between the up-spin and down-spin electrons in the
current. As discussed earlier, at and in the vicinity of the interface, the discontinuity
of μ 0 leads to a boundary resistance defined as follows:
R b =
μ
e J
=
μ F, 0 (x = 0−) − μ N, 0 (x = 0 + )
e J
(2.17)
where
μ F, 0 (x = 0−) = α F μ F, ↑ (x = 0−) + (1 − α F ) μ F, ↓ (x = 0−),
μ N, 0 (x = 0+) = α N μ N, ↑ (x = 0+) + (1 − α N ) μ N, ↓ (x = 0+)
(2.18)
This electrochemical potential difference is the driving force for the spin current
conversion across the interface
In this way, we use the concept of electrochemical potentials for the up-spin and
down-spin channels to calculate the amount of spin polarization of the current in
a semiconductor in close contact with a ferromagnetic injector or in a spin valve
consisting of a semiconductor sandwiched between two ferromagnetic contacts.
(ii) Spin injection from one ferromagnetic (FMI) to another ferromagnetic
(FM2) material
Although less discussed, we present here description of spin injection process from
a ferromagnetic material to another. Let us do suitable biasing so that net electrons
flow takes place from one ferromagnetic to another ferromagnetic material as shown
in Fig. 2.6, then subsequently the spins of the first ferromagnet (FM1) are injected
into the second one (FM2) through the interface. This is followed by an exchange
interaction between injected electron spins with the spins of the electrons of the host