108
4 Tunnelling Magnetoresistance (TMR)
4.3.3 Tunnelling Process
The TMR effect can be understood using the same two-channel electron spin model
as that for GMR. While GMR occurs due to the spin-dependent scattering asymmetry
through the entire structure, “TMR is due to the difference between the tunnelling
conductances in the two spin channels, i.e., up-spin and down-spin channel.”
A Transfer Matrix Model
In general, interfaces between ferromagnetic contacts and the non-magnetic spacer
layers have a profound effect in generating spin valve MR of any spin valve device. In
this direction, we may adopt a simple one-dimensional model for clear understanding
of the effect of interfacial layer on spin valve or TMR effect (Bandyopadhyay and
Cahay 2008). In case of TMR, the intermediate non-magnetic semiconductor or
insulator spacer layer is kept very thin so that tunnelling of conduction electrons
must take place.
Therefore, such spacer layer is approximately referred to as a delta scatterer,
as shown in Fig. 4.5a. The scatterer and accordingly the scattering potential are
expected to have a spin-independent and a spin-dependent part of strength Γ and Γ
,
respectively. The spin valve MR is calculated employing transfer matrix formalism,
considering spin transport to take place in purely one-dimension. Let us suppose that
only one energy sub-band is occupied in case of one-dimensional semiconductor
quantum wire. Noteworthy, in order to evaluate TMR spin valve, we may assume
absolute zero temperature. Furthermore, for the calculation of TMR magnetoresistance, the conductance of the majority and minority spin sub-bands are estimated
using Landauer conductance formula.
Let us consider two separate cases:
Case 1: No spin-scattering at the interface
In this case, let us set both Γ and Γ
equal to zero. Thus, for parallel configuration
of magnetization of the two ferromagnetic electrodes, the conductance for both the
majority and minority spins is given by
G ↑ = G ↓ = G P ,
(4.4)
and
G P = 2 e
2
/h.
(4.5)
Now, let us consider the antiparallel configuration of magnetization of the two
ferromagnetic electrodes. In this case, the majority spin electrons originating from
the left ferromagnetic electrode, while transmitting across the interface, must experience an energy barrier of height , which is actually the exchange splitting energy.
However, for the electrons deriving from the minority spin band from the same ferromagnetic electrode, such energy barrier appears as a step down of size (Fig. 4.5b).
4 Tunnelling Magnetoresistance (TMR)
4.3.3 Tunnelling Process
The TMR effect can be understood using the same two-channel electron spin model
as that for GMR. While GMR occurs due to the spin-dependent scattering asymmetry
through the entire structure, “TMR is due to the difference between the tunnelling
conductances in the two spin channels, i.e., up-spin and down-spin channel.”
A Transfer Matrix Model
In general, interfaces between ferromagnetic contacts and the non-magnetic spacer
layers have a profound effect in generating spin valve MR of any spin valve device. In
this direction, we may adopt a simple one-dimensional model for clear understanding
of the effect of interfacial layer on spin valve or TMR effect (Bandyopadhyay and
Cahay 2008). In case of TMR, the intermediate non-magnetic semiconductor or
insulator spacer layer is kept very thin so that tunnelling of conduction electrons
must take place.
Therefore, such spacer layer is approximately referred to as a delta scatterer,
as shown in Fig. 4.5a. The scatterer and accordingly the scattering potential are
expected to have a spin-independent and a spin-dependent part of strength Γ and Γ
,
respectively. The spin valve MR is calculated employing transfer matrix formalism,
considering spin transport to take place in purely one-dimension. Let us suppose that
only one energy sub-band is occupied in case of one-dimensional semiconductor
quantum wire. Noteworthy, in order to evaluate TMR spin valve, we may assume
absolute zero temperature. Furthermore, for the calculation of TMR magnetoresistance, the conductance of the majority and minority spin sub-bands are estimated
using Landauer conductance formula.
Let us consider two separate cases:
Case 1: No spin-scattering at the interface
In this case, let us set both Γ and Γ
equal to zero. Thus, for parallel configuration
of magnetization of the two ferromagnetic electrodes, the conductance for both the
majority and minority spins is given by
G ↑ = G ↓ = G P ,
(4.4)
and
G P = 2 e
2
/h.
(4.5)
Now, let us consider the antiparallel configuration of magnetization of the two
ferromagnetic electrodes. In this case, the majority spin electrons originating from
the left ferromagnetic electrode, while transmitting across the interface, must experience an energy barrier of height , which is actually the exchange splitting energy.
However, for the electrons deriving from the minority spin band from the same ferromagnetic electrode, such energy barrier appears as a step down of size (Fig. 4.5b).