32
2 Basic Elements of Spintronics
conductivity mismatch problem. Such problem is quite well-known and has been
attempted to overcome by incorporation of some novel spin-injecting materials, such
as diluted magnetic semiconductors, half-metallic ferromagnets etc. These proposed
spin injector materials offer a scenario, where r F ≈ r N , thus yielding η ≈ P σ F ,
implying reasonably good spin injection.
Case (iii): Tunnel/Schottky barrier at metallic ferromagnetic and paramagnetic
semiconductor interface
In this case, r i r F , r N , thus yielding η ≈ P σ i . For a tunnel barrier (discussed in
Chap. 4), the wavefunctions and hence the transmission probabilities are different
for spin-up and spin-down electrons at E F of the injector ferromagnetic materials.
Thus, at the tunnel barrier the conductivities corresponding to the two spin species
are different, which means P σ i = 0. This in turn again signifies efficient spin injection through the barrier (Rashba 2000). However, if spin-independent thermionic
emission process results in carrier injection over the barrier, then P σ i ≈ 0.
Physical picture of spin injection from ferromagnetic injector to paramagnetic
material
Whenever electronic current flows from a ferromagnetic injector into a paramagnetic
material, there is a change in distribution of up-spin and down-spin carriers. We found
similar phenomenon occurring across a superconductor and normal metal interface,
where normal current converts into supercurrent. In that case, quasiparticles and
Cooper pairs have a difference in electrochemical potential near the interface (Jedema
et al. 2001). In order to analyse spin injection phenomenon considering difference in
electrochemical potential between up-spin and down-spin electrons, let us consider a
ferromagnetic and normal metal interface in the plane x = 0 assuming that the system
is homogeneous in the y and z directions. As shown in Fig. 2.5, the ferromagnet
stretches in the region x < 0 and the normal metal in the region x > 0. Thicknesses
of both materials are such that those are greater than their respective spin diffusion
lengths.
Let us suppose that the positive terminal of the voltage source (battery) is
connected to the normal metal and the negative terminal to the ferromagnetic contact
so that the current of electrons flows from the left to right in the x-direction, i.e.,
from the ferromagnet to normal metal only. Now, considering the situation that the
rate of electron scattering events that do not flip spin is far larger than the spin-flip
scattering rate at any arbitrary coordinate point x, we can define individual spatially
varying electrochemical potential for the up-spin (μ ↑ ) and down-spin (μ ↓ ) electron
channels. Specifically, at the ferromagnetic and normal metal interface these electrochemical potentials are expected to be quite different from each other and here lies
the origin of spin injection at ferromagnet and normal metal interface. To quantify
such difference, the conductivity and current density associated with the up-spin and
down-spin electrons are expressed as follows:
σ
↑
(x) = α(x)σ,
(2.9)
2 Basic Elements of Spintronics
conductivity mismatch problem. Such problem is quite well-known and has been
attempted to overcome by incorporation of some novel spin-injecting materials, such
as diluted magnetic semiconductors, half-metallic ferromagnets etc. These proposed
spin injector materials offer a scenario, where r F ≈ r N , thus yielding η ≈ P σ F ,
implying reasonably good spin injection.
Case (iii): Tunnel/Schottky barrier at metallic ferromagnetic and paramagnetic
semiconductor interface
In this case, r i r F , r N , thus yielding η ≈ P σ i . For a tunnel barrier (discussed in
Chap. 4), the wavefunctions and hence the transmission probabilities are different
for spin-up and spin-down electrons at E F of the injector ferromagnetic materials.
Thus, at the tunnel barrier the conductivities corresponding to the two spin species
are different, which means P σ i = 0. This in turn again signifies efficient spin injection through the barrier (Rashba 2000). However, if spin-independent thermionic
emission process results in carrier injection over the barrier, then P σ i ≈ 0.
Physical picture of spin injection from ferromagnetic injector to paramagnetic
material
Whenever electronic current flows from a ferromagnetic injector into a paramagnetic
material, there is a change in distribution of up-spin and down-spin carriers. We found
similar phenomenon occurring across a superconductor and normal metal interface,
where normal current converts into supercurrent. In that case, quasiparticles and
Cooper pairs have a difference in electrochemical potential near the interface (Jedema
et al. 2001). In order to analyse spin injection phenomenon considering difference in
electrochemical potential between up-spin and down-spin electrons, let us consider a
ferromagnetic and normal metal interface in the plane x = 0 assuming that the system
is homogeneous in the y and z directions. As shown in Fig. 2.5, the ferromagnet
stretches in the region x < 0 and the normal metal in the region x > 0. Thicknesses
of both materials are such that those are greater than their respective spin diffusion
lengths.
Let us suppose that the positive terminal of the voltage source (battery) is
connected to the normal metal and the negative terminal to the ferromagnetic contact
so that the current of electrons flows from the left to right in the x-direction, i.e.,
from the ferromagnet to normal metal only. Now, considering the situation that the
rate of electron scattering events that do not flip spin is far larger than the spin-flip
scattering rate at any arbitrary coordinate point x, we can define individual spatially
varying electrochemical potential for the up-spin (μ ↑ ) and down-spin (μ ↓ ) electron
channels. Specifically, at the ferromagnetic and normal metal interface these electrochemical potentials are expected to be quite different from each other and here lies
the origin of spin injection at ferromagnet and normal metal interface. To quantify
such difference, the conductivity and current density associated with the up-spin and
down-spin electrons are expressed as follows:
σ
↑
(x) = α(x)σ,
(2.9)