# Strontium Ferromolybdate-Based Magnetic Tunnel Junctions
## 2 Theoretical Considerations
### Tunnel Magnetoresistance  
The TMR of FM/insulator/FM tunnel junctions can be qualitatively described by means of the Julliere model [40]. The model considers tunneling between magnetically aligned and anti-aligned electrodes. Thereby, the spin is supposed to be conserved. Assuming the same matrix element for the tunneling of the majority and minority spin electrons, the TMR between these configurations arises from the degree of spin polarization \(P\) of the carriers [8]:  
\[P=\frac{n_{\uparrow(E_{F})-n_{\downarrow}(E_{F})}}{n_{\uparrow(E_{F})}+n_{ \downarrow}(E_{F})} \tag{12}\]  
where \(n_{\uparrow(\downarrow)}(E_{F})\) is the density of states of spin-up (spin-down) electrons at the Fermi level \(E_{F}\). Defining the TMR as  
\[TMR=\frac{R_{AP}-R_{P}}{R_{P}}, \tag{13}\]where \(R_{AP}\) is the electrical resistance in the anti-parallel and \(R_{P}\) is that in the parallel state, the TMR is given in terms of the degrees of the conduction electron spin polarization \(P_{1}\) and \(P_{2}\) in the two magnetic layers as:  
\[TMR=\frac{2P_{1}P_{2}}{1-P_{1}P_{2}}. \tag{14}\]  
The spin polarization resembles the spontaneous magnetization behavior at low temperatures known as [PERSON]'s \(T^{3/2}\) law [41]:  
\[P(T)=P_{0}(1-\alpha T^{3/2}). \tag{15}\]  
Here, \(\alpha\) is a fitting parameter that is generally larger for the surface than for the bulk magnetization [42], being very sensitive to surface contaminants [43]. The TMR is then determined by [44]:  
\[TMR=\frac{2P_{0}^{2}\left(1-\alpha T^{3/2}\right)^{2}}{1-P_{0}^{2}\left(1- \alpha T^{3/2}\right)^{2}}. \tag{16}\]  
Assuming tunneling via a magnetically disordered layer at the interface possessing angles between the spin orientation at the interface and the magnetization in the range of 0 to \(\pi\), Equation (16) becomes:  
\[TMR=\frac{2P_{0}^{2}\left(1-\alpha T^{3/2}\right)^{2}}{3-P_{0}^{2}\left(1- \alpha T^{3/2}\right)^{2}}. \tag{17}\]  
In this case, the TMR estimated for SFMO lowers to about 6%, slightly smaller than the experimental value [6]. The combined action of inelastic spin-independent tunneling and tunneling via a magnetically disordered layer at the interface may be expected in the presence of iron-deficient interface layers [3; 4]. If one takes into account the transmission probability by tunneling, one has to replace the FM polarization with the effective spin polarization of the FM-barrier couple [45]:  
\[P_{fb}=\frac{\left(k_{\uparrow}-k_{\downarrow}\right)}{\left(k_{\uparrow}+k_ {\downarrow}\right)}\cdot\frac{\left(\kappa^{2}-k_{\uparrow}k_{\downarrow} \right)}{\left(\kappa^{2}+k_{\uparrow}k_{\downarrow}\right)}, \tag{18}\]  
where \(\kappa\) is the reciprocal localization length of the wave function inside the barrier:  
\[\kappa=\sqrt{\frac{2m_{e}^{*}V_{0}}{\hbar^{2}}}. \tag{19}\]  
Here \(m_{e}^{*}\) is the effective electron mass, \(V_{0}\) is the barrier height, \(\hbar\) is the Planck constant expressed in J-s/radian, and \(k_{\uparrow(\downarrow)}\) is the wave vector of the wave function of the tunneling spin-up (spin-down) electrons. Thus, theory [45] predicts a decrease in spin polarization with barrier height. With regard to Equation (16), the Julliere model [40] then represents the limit of a very small wavefunction overlap, i.e., it is valid for sufficiently thick barriers. SFMO is half-metallic where only spin-down states should be active at the Fermi surface (100% spin polarization) [1]. In this case, \(k_{\uparrow}\) becomes imaginary, and the MT) represents a perfect magnetic valve with zero conductivity for antiparallel spin orientation in the two ferromagnetic layers. On the other hand, the calculation of \(k_{\downarrow}\) for SFMO is cumbersome. For FM Fe, the Fermi surface of the itinerant \(d_{\text{i}}\)-electrons can be approximated by a sphere of radius \(k_{F}=(3\pi n_{e})^{1/3}\), where \(n_{e}\) is the total electron density [46].