4.3 Physical Explanation
115
Substituting the values of G P and G AP and using the definitions of the spin polarizations (P 1 , P 2 ) of the ferromagnetic electrodes from Eqs. 4.21 and 4.22, the junction
magnetoresistance ratio (JMR) is found to be
JMR =
2 P 1 P 2
1 + P 1 P 2
.
(4.23)
Equation 4.23 gives JMR ratio as derived by Jullière. The parameters P 1 and P 2
can be determined separately from the measurements of the tunnelling current in
ferromagnet (FM)–insulator (I)–superconductor (S) junctions. This compact result,
as shown in Eq. 4.23, is the famous Jullière formula.
Similarly, the TMR ratio obtained using Jullière formula is as follows:
TMR =
2 P 1 P 2
1 − P 1 P 2
.
(4.24)
4.3.5 Simple Description of Tunnelling Phenomenon
Let us first recall and summarize from the above-made discussions, the two important
assumptions, based on which tunnelling effect has been explained:
First, spins of electrons remain conserved in the tunnelling process. Given the fact
that in ferromagnetic electrodes, densities of states of majority D ↑ (E) and minority
D ↓ (E) spin electrons are different; tunnelling of up- and down-spin electrons should
be two independent processes. Therefore, conductance is supposed to take place
in two independent spin channels, constituted by up-spin and down-spin electrons
(Fig. 4.4). It is obvious that electrons deriving from one kind of spin orientation
(either up or down spin) of the first ferromagnetic electrode are received by unoccupied energy levels of the same spin orientation of the second ferromagnetic electrode.
This means that when the magnetizations of the two ferromagnetic electrodes are
parallel, the minority spins (spin antiparallel to the magnetization) should tunnel
to the minority spin states, whereas the majority spins (spin parallel to the magnetization) tunnel to the majority spin states. However, in the opposite case when
magnetizations of the two ferromagnetic electrodes are antiparallel, the identity of
the majority- and minority-spin electrons is reversed. This means that if up-spin
electron is the majority spin electrons (spin parallel to the magnetization direction)
for the first ferromagnetic electrode, this will be the minority spin electrons (spin
antiparallel to the magnetization) for the second ferromagnetic electrode. In the
same picture, down-spin electrons acting as the minority spin electrons for the first
ferromagnetic electrode would be the majority spin electrons for the second ferromagnetic electrode. Therefore, it comes out that for the antiparallel alignment of
magnetizations of the ferromagnetic electrodes, majority spin electrons of the first
ferromagnetic electrode tunnel to the available minority spin states in the second
115
Substituting the values of G P and G AP and using the definitions of the spin polarizations (P 1 , P 2 ) of the ferromagnetic electrodes from Eqs. 4.21 and 4.22, the junction
magnetoresistance ratio (JMR) is found to be
JMR =
2 P 1 P 2
1 + P 1 P 2
.
(4.23)
Equation 4.23 gives JMR ratio as derived by Jullière. The parameters P 1 and P 2
can be determined separately from the measurements of the tunnelling current in
ferromagnet (FM)–insulator (I)–superconductor (S) junctions. This compact result,
as shown in Eq. 4.23, is the famous Jullière formula.
Similarly, the TMR ratio obtained using Jullière formula is as follows:
TMR =
2 P 1 P 2
1 − P 1 P 2
.
(4.24)
4.3.5 Simple Description of Tunnelling Phenomenon
Let us first recall and summarize from the above-made discussions, the two important
assumptions, based on which tunnelling effect has been explained:
First, spins of electrons remain conserved in the tunnelling process. Given the fact
that in ferromagnetic electrodes, densities of states of majority D ↑ (E) and minority
D ↓ (E) spin electrons are different; tunnelling of up- and down-spin electrons should
be two independent processes. Therefore, conductance is supposed to take place
in two independent spin channels, constituted by up-spin and down-spin electrons
(Fig. 4.4). It is obvious that electrons deriving from one kind of spin orientation
(either up or down spin) of the first ferromagnetic electrode are received by unoccupied energy levels of the same spin orientation of the second ferromagnetic electrode.
This means that when the magnetizations of the two ferromagnetic electrodes are
parallel, the minority spins (spin antiparallel to the magnetization) should tunnel
to the minority spin states, whereas the majority spins (spin parallel to the magnetization) tunnel to the majority spin states. However, in the opposite case when
magnetizations of the two ferromagnetic electrodes are antiparallel, the identity of
the majority- and minority-spin electrons is reversed. This means that if up-spin
electron is the majority spin electrons (spin parallel to the magnetization direction)
for the first ferromagnetic electrode, this will be the minority spin electrons (spin
antiparallel to the magnetization) for the second ferromagnetic electrode. In the
same picture, down-spin electrons acting as the minority spin electrons for the first
ferromagnetic electrode would be the majority spin electrons for the second ferromagnetic electrode. Therefore, it comes out that for the antiparallel alignment of
magnetizations of the ferromagnetic electrodes, majority spin electrons of the first
ferromagnetic electrode tunnel to the available minority spin states in the second