78
3 Giant Magnetoresistance (GMR)
Expression of the conductivity σ is given by
σ =
n s e
2
τ s
m ∗
s
+
n d e
2
τ d
m
∗
d
(3.2)
In Eq. 3.2, n s /n d = ratio of electrons’ numbers in the s and d bands, τ s /τ d =
scattering times of electrons in the s/d bands, where τ S ∼ τ d . Thus, considering
our discussion on the mass of electrons it comes out that the first term in Eq. 3.2
dominates σ , and s electrons mainly contribute to the conduction process. It is also
understandable that it is the s → d transition that mainly contributes to the transition
probability. Now at temperatures T T C , where T C = para-ferromagnetic transition
temperature, all the unoccupied d orbitals being antiparallel, resulting in only half
of the s electrons that could possibly undergo the transitions. However, for T > T C ,
since all the s electrons can undergo transitions, therefore, there is more scattering.
Thus from Eq. 3.2, below T C one may expect a decrease in the resistivity. Now, let
us consider a magnetic field is applied to Ni. Such applied field might possibly rise
the spin polarization and thereby allow less s → d transitions. This in turn leads to
decrease in the resistivity; therefore, a negative MR is observed.
3.2.3 Anisotropic Magnetoresistance (AMR)
of Ferromagnetic Transition Metals
The origin of AMR of a ferromagnetic transition metal lies on the fact that the
electrons would suffer more scattering while travelling along the applied magnetic
field than those travelling perpendicular to the applied field. This implies that in
ferromagnetic materials electrical conductivity is decided by the relative alignment
of the direction of the electrical current and the magnetization of the ferromagnetic
materials. As is depicted in Fig. 3.1, resistivity depends on the direction of flow of
current. This phenomenon, discovered by Lord Kelvin in the nineteenth century, is
also known as spontaneous magnetoresistance anisotropy (SMA).
Physical origin of this MR phenomenon in a ferromagnetic material lies on
the decrease of symmetry of the magnetized state compared to its non-magnetic
state. This is due to the simultaneous presence of the magnetization and spin–orbit
coupling. Resistivity of such ferromagnetic materials varies following cosine square
function if the angle between the direction of magnetization and that of applied electrical bias is varied. The consequence of such MR effect is that maximum resistivity
(ρ ) is achieved when the electric field and magnetic field are parallel to each other,
whereas the resistivity (ρ ) is minimum when they are perpendicular. As is commonly
known, the ratio (ρ –ρ )/ρ is called the MR ratio (%). The most common AMR
sensor material is permalloy (Ni 81 Fe 19 ), which has very high permeability (>2000).
Moreover, the AMR effect has been found to depend on the thickness of the film and
its deposition method. For instance, in case of permalloy film deposited by ion beam
3 Giant Magnetoresistance (GMR)
Expression of the conductivity σ is given by
σ =
n s e
2
τ s
m ∗
s
+
n d e
2
τ d
m
∗
d
(3.2)
In Eq. 3.2, n s /n d = ratio of electrons’ numbers in the s and d bands, τ s /τ d =
scattering times of electrons in the s/d bands, where τ S ∼ τ d . Thus, considering
our discussion on the mass of electrons it comes out that the first term in Eq. 3.2
dominates σ , and s electrons mainly contribute to the conduction process. It is also
understandable that it is the s → d transition that mainly contributes to the transition
probability. Now at temperatures T T C , where T C = para-ferromagnetic transition
temperature, all the unoccupied d orbitals being antiparallel, resulting in only half
of the s electrons that could possibly undergo the transitions. However, for T > T C ,
since all the s electrons can undergo transitions, therefore, there is more scattering.
Thus from Eq. 3.2, below T C one may expect a decrease in the resistivity. Now, let
us consider a magnetic field is applied to Ni. Such applied field might possibly rise
the spin polarization and thereby allow less s → d transitions. This in turn leads to
decrease in the resistivity; therefore, a negative MR is observed.
3.2.3 Anisotropic Magnetoresistance (AMR)
of Ferromagnetic Transition Metals
The origin of AMR of a ferromagnetic transition metal lies on the fact that the
electrons would suffer more scattering while travelling along the applied magnetic
field than those travelling perpendicular to the applied field. This implies that in
ferromagnetic materials electrical conductivity is decided by the relative alignment
of the direction of the electrical current and the magnetization of the ferromagnetic
materials. As is depicted in Fig. 3.1, resistivity depends on the direction of flow of
current. This phenomenon, discovered by Lord Kelvin in the nineteenth century, is
also known as spontaneous magnetoresistance anisotropy (SMA).
Physical origin of this MR phenomenon in a ferromagnetic material lies on
the decrease of symmetry of the magnetized state compared to its non-magnetic
state. This is due to the simultaneous presence of the magnetization and spin–orbit
coupling. Resistivity of such ferromagnetic materials varies following cosine square
function if the angle between the direction of magnetization and that of applied electrical bias is varied. The consequence of such MR effect is that maximum resistivity
(ρ ) is achieved when the electric field and magnetic field are parallel to each other,
whereas the resistivity (ρ ) is minimum when they are perpendicular. As is commonly
known, the ratio (ρ –ρ )/ρ is called the MR ratio (%). The most common AMR
sensor material is permalloy (Ni 81 Fe 19 ), which has very high permeability (>2000).
Moreover, the AMR effect has been found to depend on the thickness of the film and
its deposition method. For instance, in case of permalloy film deposited by ion beam