# Spin Dependent Transport through Driven Magnetic System with Aubry-Andre-Harper Modulation
## 3 Numerical Results and Discussion
Now we focus our attention to examine the critical role played by the AAH phase \(\phi_{v}\) on spin selective electron transfer. In Figure 6, we present the variation of spin polarization co-efficient as a function of \(\phi_{v}\) for two distinct energies. The orange curve is for \(E=-0.75\) eV, while the other one is for \(E=0.75\) eV. A reasonably large change is reflected from both these two curves. Regulating \(\phi_{v}\), that can be done externally, we can change the available energy channels between the electrodes as the effective site energies of the magnetic chain are modified, and therefore, the degree of spin polarization can be tuned. Thus, along with  
Figure 3: \(P\)-\(A_{x}\) characteristics at two distinct energies. All the other physical parameters kept unchanged as taken in Figure 2.  
the irradiation parameter, the AAH phase can also be considered as a suitable parameter for regulating the spin transfer. To have a more clear picture and to understand precisely the interplay between the light parameter and the AAH phase, in Figure 7, we present a density plot of SP by varying simultaneously \(A_{x}\) and \(\phi_{v}\) in a broad range. Some typical energies are selected like what we consider in Figure 5. Quite interestingly we find that, the degree of SP and its sign can be monitored selectively by adjusting \(A_{x}\) and \(\phi_{v}\). The phenomenon persists for a wide range of these parameters which suggests that extremely fine tuning is no longer required. It certainly gives us a confidence that the present findings can be tested in a suitable laboratory setup. Figure 4: \(P\) as a function of \(\theta\) (\(\theta_{i}=\theta\;\forall\;i\)) at two different energies, where the orange and black lines correspond to \(A_{x}=0.5\) and \(1\), respectively. The AAH phase \(\phi_{v}=0\). Finally, keeping in mind the possible experimental realization of our prescription it is indeed required to investigate the effect of system size on spin dependent transmission probabilities and the spin polarization co-efficient. To explore it, in Figure 8, we show the dependence of these physical quantities on system size \(N\) by varying it in a wide range at two typical energies. For both of these energies, \(T_{\sigma}\) and \(P\) exhibit large amplitude oscillation with \(N\). This is solely due to the effect of quantum interference among electronic waves, and can be observed for other energies as well, which we confirm through our detailed numerics. The crucial thing is that the oscillating nature persists even for a reasonably large system size, and therefore, we can safely verify our proposal in a suitable laboratory setup. In this context it is relevant to note that similar kind of oscillation in transport quantities by varying system size has also been reported in different contemporary works, and for the ordered systems it can be tested even analytically [65; 88; 89]. Figure 6: Spin polarization co-efficient \(P\) as a function of \(\phi_{\
u}\) at two typical energies, where the orange and black curves are for \(E=-0.75\,\)eV and \(E=0.75\,\)eV, respectively. Here, we consider \(A_{x}=0.75\,\) and \(\theta_{i}=0\,\,\forall\,i\). Figure 7: Simultaneous variation (density plot) of \(P\) with AAH phase \(\phi_{\
u}\) and light parameter \(A_{x}\) at some specific energies. Here we set \(\theta_{i}=0\,\,\forall\,i\).