# Spin Dependent Transport through Driven Magnetic System with Aubry-Andre-Harper Modulation
## 3 Numerical Results and Discussion  
Based on the above theoretical framework now we present the results. Our primary goal is to achieve a high degree of spin polarization and tuning its phase by means of irradiation. All the results are worked out in the high frequency limit which is defined as \(\hbar\Omega>>t\), and in this limiting condition only the lowest order Floquet band, i.e., \(p=q=0\) contributes. For the chosen parameter values, the frequency becomes \(\Omega\geq 10^{15}\) Hz which denotes the far-infrared (FIR) region. The intensity of the irradiation is of the order of \(10^{7}\,\mathrm{W/m^{2}}\) which is within the experimental limit. The corresponding electric (\(\mathcal{E}\)) and magnetic (\(B=\mathcal{E}/c\)) fields are \(10^{5}\,\mathrm{V/m}\) and \(10^{-4}\,\mathrm{Tesla}\), respectively. The other common set of parameter values that we choose for our calculations are as follows. In the electrodes \(S\) and \(D\) we take \(\varepsilon_{0}=0\) and \(t_{0}=2\,\mathrm{eV}\), and they are coupled to the magnetic chain via the coupling strengths \(t_{S}=t_{D}=1\,\mathrm{eV}\). In the magnetic chain we choose \(t=1\,\mathrm{eV}\), AAH modulation strength \(W=1\,\mathrm{eV}\), spin flip parameter \(h_{i}=1\,\mathrm{eV}\)\(\forall\,i\), \(\varphi_{i}=0\)\(\forall\,i\). Unless specified, we fix the total number of sites in the magnetic chain \(N=30\). The parameter values those are not constant are given in the appropriate places, and all the other energies are also measured in unit of \(\mathrm{eV}\). Throughout the calculations we restrict ourselves in the zero temperature limit. This is a realistic approximation as long as the average energy level spacing is higher than the thermal energy, and for small scale systems (even for \(N<200\)) this condition can be easily achieved. It is also important to note that one cannot increase the chain length as much as it is possible, since we need to confine the system size within the spin coherence length. Otherwise no such phenomenon will be observed. Let us begin with Figure 2 where spin dependent transmission probabilities \(T_{\sigma}\) along with spin polarization co-efficient \(P\) are shown as a function of energy \(E\) for some specific values of \(A_{x}\). Several key features are emerged those are analyzed as follows. At a first glance we see that the transmission spectrum is highly fragmented and gapped in nature. This is solely due to the cosine modulation in site energies, as gapped energy spectrum is the generic feature of an AAH system. The transmission spectrum is a direct manifestation of the energy values. Unlike the perfect magnetic chain where energy spectrum is not gapped, for the AAH case we have a finite probability to get non-zero spin polarization at different energy zones. More importantly, even near the energy band centre we can get a reasonably large spin polarization. This feature is always desirable since one can easily place the Fermi level close to the central region, apart from placing it near the energy band edges. The role of irradiation is of course fascinating. For \(A_{x}=0\), a finite overlap between up and down spin transmission probabilities takes place, following the up and down spin energy channels, for a broad energy region. Therefore, for these energy zones spin polarization becomes too small. Whereas, the transmission spectra start to get shifted with \(A_{x}\) and they are almost separated for higher \(A_{x}\), which is clearly visible by comparing the spectra given in the left column of Figure 2.  
The shifting of transmission probabilities with \(A_{x}\) is entirely due to the modification of energy eigenvalues of the magnetic chain, since the NNH strength gets renormalized in presence of the irradiation (see Equation (3)). As effective hopping decreases compared to the irradiation free case, we get reduced allowed energy windows for up and down spin electrons, and thus the transmission spectra. All the characteristic features are directly reflected in the \(P\)-\(E\) spectra (see right column of Figure 2).