3.6 Quantitative Explanation of GMR
93
To summarize, GMR characterizes the gradual transit of magnetization of adjacent
ferromagnetic layers from the antiparallel alignment in zero magnetic field to the
parallel alignment under the application of a strong magnetic field. In order to obtain
a large GMR, one must seek a good alignment of energy bands, as much as possible,
between the ferromagnetic layers and the spacer layer in one spin channel; on the
other hand, a large mismatch between them to the best of the possibility in another
spin channel. This means that R ↑ should be as small as possible and R ↓ should be
as large as possible, which in turn means large spin asymmetry. Also, it needs to be
remembered that spin-flip scattering processes are detrimental to obtain the GMR
effect as it causes mixing of the ↑ and ↓ spin channels.
3.7 Magnetoresistance Theory
3.7.1 Resistor Network Theory of GMR
Although conceptually correct, simple resistor model of GMR in trilayer structure
should be transformed into a quantitative theory so that one could possibly explain
the differences between the observed GMR effect on CIP and CPP geometries, the
observed dependence of GMR on the layer thicknesses of the device and also on the
material.
In order to determine the effect of spin-dependent scattering on electrical conduction in the multilayer films, spin-dependent scattering process needs to be incorporated into the Boltzmann Equation. In general, electrons in a metal experience a
constant force F = −e E, where e is the electron charge, under the application of an
electric field E. The scattering, suffered by electrons in a metal from its imperfections, is supposed to modify the simple accelerated motion. Now, during mean free
time, i.e., the time interval when each electron with mass m moves without scattering,
electrons continue to accelerate and finally acquire a velocity v in the direction of E.
It is well known that the distance covered during this time is called as mean free path.
After experiencing each scattering events, electrons undergo acceleration process to
again attend its velocity v. This in turn leads to a steady-state condition where all the
electrons flow in the direction of E with a velocity v given by
v = −
eEτ
m
(3.8)
Considering n electrons per unit volume, the current density becomes
j = −nev = −
ne
2 Eτ
m
(3.9)
This equation is like Ohm’s law. Again, electrical conductivity (σ ) is defined as j
= σ E. Hence, resistivity ρ can be clearly given by
93
To summarize, GMR characterizes the gradual transit of magnetization of adjacent
ferromagnetic layers from the antiparallel alignment in zero magnetic field to the
parallel alignment under the application of a strong magnetic field. In order to obtain
a large GMR, one must seek a good alignment of energy bands, as much as possible,
between the ferromagnetic layers and the spacer layer in one spin channel; on the
other hand, a large mismatch between them to the best of the possibility in another
spin channel. This means that R ↑ should be as small as possible and R ↓ should be
as large as possible, which in turn means large spin asymmetry. Also, it needs to be
remembered that spin-flip scattering processes are detrimental to obtain the GMR
effect as it causes mixing of the ↑ and ↓ spin channels.
3.7 Magnetoresistance Theory
3.7.1 Resistor Network Theory of GMR
Although conceptually correct, simple resistor model of GMR in trilayer structure
should be transformed into a quantitative theory so that one could possibly explain
the differences between the observed GMR effect on CIP and CPP geometries, the
observed dependence of GMR on the layer thicknesses of the device and also on the
material.
In order to determine the effect of spin-dependent scattering on electrical conduction in the multilayer films, spin-dependent scattering process needs to be incorporated into the Boltzmann Equation. In general, electrons in a metal experience a
constant force F = −e E, where e is the electron charge, under the application of an
electric field E. The scattering, suffered by electrons in a metal from its imperfections, is supposed to modify the simple accelerated motion. Now, during mean free
time, i.e., the time interval when each electron with mass m moves without scattering,
electrons continue to accelerate and finally acquire a velocity v in the direction of E.
It is well known that the distance covered during this time is called as mean free path.
After experiencing each scattering events, electrons undergo acceleration process to
again attend its velocity v. This in turn leads to a steady-state condition where all the
electrons flow in the direction of E with a velocity v given by
v = −
eEτ
m
(3.8)
Considering n electrons per unit volume, the current density becomes
j = −nev = −
ne
2 Eτ
m
(3.9)
This equation is like Ohm’s law. Again, electrical conductivity (σ ) is defined as j
= σ E. Hence, resistivity ρ can be clearly given by