42
2 Basic Elements of Spintronics
Equation (2.32) is quite significant in the sense it clearly exhibits that λ sd and
λ f are indeed related but not equal. For instance, by doping silver with increasing
levels of a gold, i.e., by a non-magnetic impurity, λ sd has been found to decrease.
The reason for this drop in λ sd originates from a decrease in the electronic mean free
path λ f with increasing impurity concentration. Furthermore, larger concentration of
gold impurities is supposed to increase spin–orbit scattering of electrons because of
heavy gold atoms. This in turn reduces spin-flip time τ s and hence causes decrease
in λ sd with increasing levels of a gold.
From Eq. 2.31, it follows that
τ s
τ m
= N
(2.33)
where τ m is the momentum relaxation time. From Eq. 2.33, it comes out that the
spin relaxation time τ s is directly proportional to the momentum relaxation time τ m .
Now, let us define the spin polarization α of the injected current as
α(x) = (J ↑ (x) − J ↓ (x))/J
(2.34)
where J is the total current density J↑(x) + J↓(x) injected across the interface, x being
the direction of current flowing perpendicular to the interface. Net spin accumulation
in the non-magnet/paramagnet for a current J flowing through the junction can be
found in the following manner.
As can be generally understood, J does not depend on x; hence, we may equate
the net spin injection across the interface given by
d(n ↑ −n ↓)
dt
x=0
=
Aα(0)J
e
(2.35)
(where A = cross-sectional area at the ferromagnet/non-magnet junction) to the rate
of decrease of the total spin concentration in the whole volume of the paramagnet.
In this attempt, we obtain
d(n ↑ −n ↓)
dt
x=0
=
A
τ S
∫
0
n ↑ − n ↓
dx
(2.36)
Since the decay of the spin accumulation has been assumed to be exponential
with characteristic decay constant or spin accumulation length λ sd , therefore, we
may write
n ↑ (x) − n ↓ (x) = n 0 e
−x/λ sd
(2.37)
where n 0 = n ↑ (0) − n ↓ (0).
Thus from Eqs. 2.35 to 2.37, we get
2 Basic Elements of Spintronics
Equation (2.32) is quite significant in the sense it clearly exhibits that λ sd and
λ f are indeed related but not equal. For instance, by doping silver with increasing
levels of a gold, i.e., by a non-magnetic impurity, λ sd has been found to decrease.
The reason for this drop in λ sd originates from a decrease in the electronic mean free
path λ f with increasing impurity concentration. Furthermore, larger concentration of
gold impurities is supposed to increase spin–orbit scattering of electrons because of
heavy gold atoms. This in turn reduces spin-flip time τ s and hence causes decrease
in λ sd with increasing levels of a gold.
From Eq. 2.31, it follows that
τ s
τ m
= N
(2.33)
where τ m is the momentum relaxation time. From Eq. 2.33, it comes out that the
spin relaxation time τ s is directly proportional to the momentum relaxation time τ m .
Now, let us define the spin polarization α of the injected current as
α(x) = (J ↑ (x) − J ↓ (x))/J
(2.34)
where J is the total current density J↑(x) + J↓(x) injected across the interface, x being
the direction of current flowing perpendicular to the interface. Net spin accumulation
in the non-magnet/paramagnet for a current J flowing through the junction can be
found in the following manner.
As can be generally understood, J does not depend on x; hence, we may equate
the net spin injection across the interface given by
d(n ↑ −n ↓)
dt
x=0
=
Aα(0)J
e
(2.35)
(where A = cross-sectional area at the ferromagnet/non-magnet junction) to the rate
of decrease of the total spin concentration in the whole volume of the paramagnet.
In this attempt, we obtain
d(n ↑ −n ↓)
dt
x=0
=
A
τ S
∫
0
n ↑ − n ↓
dx
(2.36)
Since the decay of the spin accumulation has been assumed to be exponential
with characteristic decay constant or spin accumulation length λ sd , therefore, we
may write
n ↑ (x) − n ↓ (x) = n 0 e
−x/λ sd
(2.37)
where n 0 = n ↑ (0) − n ↓ (0).
Thus from Eqs. 2.35 to 2.37, we get