2.4 Spin Accumulation
39
∆k
E ≠ 0
k x
k y
E = 0
Semiconductor/Metal
k F
∆k ↑
E ≠ 0
k x
k y
E = 0
Ferromagnet
k F
∆k ↓
(a)
(b)
Fig. 2.9 a Shifting of Fermi surface in a paramagnetic metal or semiconductor in the presence of a
constant external electric field. This in turn leads to current conduction. b As per the generalization
of Mott model of the ferromagnet where up-spin and down-spin electrons have different mobilities
(Figure adapted and redrawn from Bandyopadhyay and Cahay 2008)
ˆ
J
S,Di f f usion
= −D ↑ ∇∇ s ↑ − D ↓
∇∇ s ↓
J
S,Di f f usion
= −D ↑
∇ρ ↑ − D ↓
∇ρ ↓
(2.25)
where D ↑ /D ↓ is the diffusion constant of electrons in up (↑)/down (↓) spin subchannels. Relation between diffusion constants and conductivities, as proposed by
Einstein, for metals and semiconductors takes the form like:
σ ↑↓ = N ↑↓ e
2 D ↑↓ : Metals
σ ↑↓ =
N ↑↓ e
2 D ↑↓
k B T
: N on−degenerate semiconductor
(2.26)