92
3 Giant Magnetoresistance (GMR)
3.6 Quantitative Explanation of GMR
Such discussion on the origin of GMR in a trilayer structure, consisting of two
ferromagnetic metallic layers separated by a non-magnetic metallic layer, can be
efficiently put in the form of a simple resistor model as shown in Fig. 3.11c, d.
Let us first consider the ferromagnetic (FM) configuration (Fig. 3.11a) of the
structure, i.e., when the magnetic moment of the ferromagnetic layers are parallel to
each other. In this case, the electrons with ↑ spin suffer weak scattering and the electrons with ↓ spin undergo strong scattering both in the first and second ferromagnetic
layers. This feature of scattering can be simulated in an equivalent resistor network,
where two small resistors (R ↑ ) represent ↑ spin channel and two large resistors (R ↓ )
represent that of ↓ spin channel, as shown in Fig. 3.11c. Here, R ↑ /R ↓ corresponds
to the resistance when spin of conduction electron is parallel/antiparallel to the
magnetization of the ferromagnetic layers. Obviously, R ↓ should be much larger
than R ↑ , since minority spin experiences large scattering compared to that of
majority spin electrons. Thus, in parallel configuration, resistance of ↑ spin channel
is much smaller compared to that of ↓ spin channel. For simplicity, consider the case
where R ↑ < < R ↓ , therefore the total resistance in the parallel configuration is
R ↑↑ =
2 R ↑ R ↓
R ↑ + R ↓
≈ 2 R ↑
(3.5)
Therefore, the total resistance in ferromagnetic configuration is determined by the
small-resistance in the ↑ spin channel, which ‘short-circuit’ the high-resistance ↓ spin
channel.
Let us now consider the antiferromagnetic (AFM) configuration of the structure
(Fig. 3.11b), i.e., when the magnetic moment of the ferromagnetic layers is antiparallel to each other. In this case, ↓ spin electrons suffer strong scattering in the first
ferromagnetic layer and weak scattering in the second ferromagnetic layer. Similarly,
↑ spin electrons suffer weak scattering in the first ferromagnetic layer and strong scattering in the second. In this case, the equivalent resistor network can be modelled by
associating one large (R ↓ ) and one small (R ↑ ) resistor corresponding to strong and
weak scattering, respectively, as shown in Fig. 3.11d. No shorting feasibility can be
sensed in this case and therefore, total resistance in this case becomes,
R ↓↑ =
R ↑ + R ↓
2
≈
R ↓
2
(3.6)
Thus, R ↑↓ > R ↑↑ . We can express GMR by defining a new term, spin asymmetry,
i.e., ∝ =
R ↑
R ↓
. Thus we get,
G M R (%) =
R
R
=
R ↓↑ − R ↑↑
R ↑↑
=
R ↑ − R ↓
2
4R ↑ R ↓
=
(∝ −1)
2
4 ∝
(3.7)
3 Giant Magnetoresistance (GMR)
3.6 Quantitative Explanation of GMR
Such discussion on the origin of GMR in a trilayer structure, consisting of two
ferromagnetic metallic layers separated by a non-magnetic metallic layer, can be
efficiently put in the form of a simple resistor model as shown in Fig. 3.11c, d.
Let us first consider the ferromagnetic (FM) configuration (Fig. 3.11a) of the
structure, i.e., when the magnetic moment of the ferromagnetic layers are parallel to
each other. In this case, the electrons with ↑ spin suffer weak scattering and the electrons with ↓ spin undergo strong scattering both in the first and second ferromagnetic
layers. This feature of scattering can be simulated in an equivalent resistor network,
where two small resistors (R ↑ ) represent ↑ spin channel and two large resistors (R ↓ )
represent that of ↓ spin channel, as shown in Fig. 3.11c. Here, R ↑ /R ↓ corresponds
to the resistance when spin of conduction electron is parallel/antiparallel to the
magnetization of the ferromagnetic layers. Obviously, R ↓ should be much larger
than R ↑ , since minority spin experiences large scattering compared to that of
majority spin electrons. Thus, in parallel configuration, resistance of ↑ spin channel
is much smaller compared to that of ↓ spin channel. For simplicity, consider the case
where R ↑ < < R ↓ , therefore the total resistance in the parallel configuration is
R ↑↑ =
2 R ↑ R ↓
R ↑ + R ↓
≈ 2 R ↑
(3.5)
Therefore, the total resistance in ferromagnetic configuration is determined by the
small-resistance in the ↑ spin channel, which ‘short-circuit’ the high-resistance ↓ spin
channel.
Let us now consider the antiferromagnetic (AFM) configuration of the structure
(Fig. 3.11b), i.e., when the magnetic moment of the ferromagnetic layers is antiparallel to each other. In this case, ↓ spin electrons suffer strong scattering in the first
ferromagnetic layer and weak scattering in the second ferromagnetic layer. Similarly,
↑ spin electrons suffer weak scattering in the first ferromagnetic layer and strong scattering in the second. In this case, the equivalent resistor network can be modelled by
associating one large (R ↓ ) and one small (R ↑ ) resistor corresponding to strong and
weak scattering, respectively, as shown in Fig. 3.11d. No shorting feasibility can be
sensed in this case and therefore, total resistance in this case becomes,
R ↓↑ =
R ↑ + R ↓
2
≈
R ↓
2
(3.6)
Thus, R ↑↓ > R ↑↑ . We can express GMR by defining a new term, spin asymmetry,
i.e., ∝ =
R ↑
R ↓
. Thus we get,
G M R (%) =
R
R
=
R ↓↑ − R ↑↑
R ↑↑
=
R ↑ − R ↓
2
4R ↑ R ↓
=
(∝ −1)
2
4 ∝
(3.7)