# Spin Dependent Transport through Driven Magnetic System with Aubry-Andre-Harper Modulation
## 3 Numerical Results and Discussion
Almost 100% spin polarization can be achieved for the entire allowed energy zones by suitably adjusting the irradiation parameter. It gives a clear signature of achieving externally controlled spin polarization through a magnetic nanojunction. The above analysis gives rise to a very fundamental question that how the spin polarization can vary if we tune \(A_{x}\) continuously, instead of fixing it at some typical values. To demonstrate it, in Figure 3 we show the dependence of \(P\) as a function of \(A_{x}\) by varying it in a wide range. The results are presented for two distinct energies, \(E=-1\) eV and \(1\) eV. Both for these two energies, \(P\) shows a large oscillation with increasing amplitude for lower \(A_{x}\), and eventually saturates exhibiting 100% polarization for higher \(A_{x}\). At \(E=-1\) eV or \(1\) eV, there is a finite overlap between the two transmission functions when \(A_{x}=0\) and thus \(P\) becomes very small. The spectral properties and thus the transmission spectra get modified with the alteration of \(A_{x}\). However, the fact is that the NNH strength \(t\) does not monotonically decrease with \(A_{x}\) as it follows the zeroth order Bessel function of the first kind (see Equation (3)). Due to this reason, in some cases we get finite overlap between \(T_{\uparrow}\) and \(T_{\downarrow}\) which yields lesser \(P\). On the other hand when the overlap is less, higher \(P\) is  
Figure 2: Up and down spin transmission probabilities (\(T_{\uparrow}\rightarrow\) light green color and \(T_{\downarrow}\rightarrow\) black color) and spin polarization coefficient \(P\) as a function of energy \(E\) at four typical values of \(A_{x}\). Here we set \(\theta_{i}=0\)\(\forall\)\(i\) and AAH phase \(\phi_{v}=0\). achieved. This is the underlying mechanism to have oscillatory behavior of \(P\) with \(A_{x}\). For large \(A_{x}\), when \(T_{\uparrow}\) and \(T_{\downarrow}\) are practically separated we get the maximum SP. Depending on the dominating factor among \(T_{\uparrow}\) and \(T_{\downarrow}\), we get either 100% up or down spin polarization. These results clearly justify that the degree of SP can be tuned in a wide range by regulating \(A_{x}\), without altering any other physical parameters describing the system. The results discussed so far are computed for the magnetic chain where all the magnetic moments are aligned along \(+Z\) direction i.e., \(\theta=0\) (we refer \(\theta_{i}=\theta\) as we assume that all the moments are aligned in a particular direction). For \(\theta=0\), there is no spin flipping i.e., \(T_{\uparrow\downarrow}=T_{\downarrow\uparrow}=0\), since in this case \(\sigma_{x}\) and \(\sigma_{y}\) do not involve into the TB Hamiltonian Equation (2) and it becomes exactly diagonal. However, finite spin flipping occurs as long as the moments are aligned in a particular direction with respect to \(+Z\) axis, and to reveal the \(\theta\) dependence on SP, in Figure 4, we present \(P\)-\(\theta\) characteristics by changing \(\theta\) in a wide range, for two different values of \(A_{x}\), considering the identical energy values as taken in Figure 3. Both for the orange and black curves, associated with \(A_{x}=0.5\) and \(1\) respectively, the spin polarization co-efficient shows a complete phase reversal under rotating the magnetic moments. For the two typical values of \(\theta\), SP drops exactly to zero, as expected. With the change of \(\theta\) spectral behavior gets changed and hence the SP. Thus, the alignment of the magnetic moments has an important role in SP. To explore the explicit dependance of SP on both \(\theta\) and \(A_{x}\), in Figure 5, we present a density plot of \(P\) by varying these factors in a broad range fixing the energies at some specific values. From these spectra we get a clear hint about the range of physical parameters for which large degree of spin polarization can be obtained for this magnetic junction. The phase reversal is also clearly noticed.