# Strontium Ferromolybdate-Based Magnetic Tunnel Junctions
## 2 Theoretical Considerations
### Neel Coupling at the Interface  
The tunnelling probability exponentially varies with the barrier thickness. Consequently, conformal coating of the bottom SFMO layer by a thin barrier layer is required. In this case, the Neel coupling--also called the \"orange-peel\" effect--appears due to nonzero surface roughness. Let us consider a laminar structure consisting of two FM layers, a fixed (pinned) layer and a free one, separated by a nonmagnetic barrier (NB) of thickness \(d_{b}\), with conformal waviness described by a two-dimensional (2D) sinusoidal wave with a wavelength \(\lambda\) and amplitude \(h\). Then, a magnetic offset field occurs in the free layer due to the magnetostatic coupling between the two in-plane polarized FM layers. In this case, a bulk, pinned hard-magnetic (HM) layer couples via the barrier to another bulk, free soft-magnetic (SM) layer. Accounting only for the first-order terms of \(h/d_{g}\), with \(d_{g}\) being the grain size, the coupling field is given in the limit of rigid isotropic in-plane magnetization by [14; 15]:  
\[H_{off}=\frac{\pi^{2}\hbar^{2}M_{h}}{\sqrt{2}\lambda d_{s}}M_{h}\exp\left(- \frac{2\pi\sqrt{2}d_{b}}{\lambda}\right) \tag{2}\]  
where \(d_{s}\) is the thickness of the SM layer and \(M_{h}\) is the magnetization of the HM layer. In thin-film structures, magnetic poles created at the outer surfaces of the magnetic layers result in additional FM and anti-FM interactions which must be taken into account. This yields an SM/NB/HM trilayer structure [16]:  
\[ \begin{split}& H_{off}=\frac{\pi^{2}\hbar^{2}M_{h}}{\sqrt{2} \lambda d_{s}}\exp\left(-\frac{2\pi\sqrt{2}d_{s}}{\lambda}\right)\times\\ &\times\left[1-\exp\left(-\frac{2\pi\sqrt{2}d_{s}}{\lambda} \right)\right]\cdot\left[1-\exp\left(-\frac{2\pi\sqrt{2}d_{s}}{\lambda}\right) \right].\end{split}  \tag{3}\]  
Experimentally, the inverse variation of \(H_{off}\) with the SM layer thickness and its exponential variation with the NB thickness were observed in [17; 18]. A coercivity enhancement in the free layer resulting from the orange-peel coupling was reported in [19]. Note that small offset fields require \(h<<\lambda\). In this case, both pulsed laser deposition (PLD) and magnetron sputtering (MSP) ensure conformal surface coating. PLD generates short time (\(\upmu\)s) fluxes of condensable species with large kinetic energies (a few eV up to several tens eV) [20]. This favors layer-by-layer growth and, therefore, film crystallinity and conformal edge coverage. Moreover, film growth is not disturbed by new incident species between the laser pulses. As a result, additional surface diffusion occurs. MSP is a deposition method possessing a more directed flux of film-forming species. Here, DC pulsing in the kHz range is convenient for suppressing arc formation. Ions with energies up to several hundred eV are impinging on the growing film surface [21], providing a layer-by-layer growth with \(h<<\lambda\) and a widely conformal substrate coverage at low incident fluxes. A review of SFMO film deposition and post-processing is provided in [22]. The case of uniform out-of-plane magnetization of the two FM layers was considered in [23]. Here, the magnetostatic anisotropy constant with an absolute value up to \(|\,K\,|=1.32\times 10^{6}\) J/m\({}^{3}\) will be negative, corresponding to a parallel alignment of the magnetizations in the two FM layers. The out-of-plane magnetization induces an anti-FM interaction between the layers due to a stray field resulting from the uniformly charged curved surface [24]. For arbitrary orientations of magnetization, the offset field is given by [24]:  
\[ \begin{split}& H_{off}=\frac{\pi^{2}h^{2}}{\sqrt{2}\lambda d_{i}}M_{ h}\exp\Bigl{(}-\frac{2\pi\sqrt{2}d_{h}}{\lambda}\Bigr{)}\times\\ &\times\Bigl{[}1-\exp\Bigl{(}-\frac{2\pi\sqrt{2}d_{h}}{\lambda} \Bigr{)}\Bigr{]}\cdot\Bigl{[}1-\exp\Bigl{(}-\frac{2\pi\sqrt{2}d_{h}}{\lambda} \Bigr{)}\Bigr{]}\times\\ &\times(2\sin\alpha\sin\beta-\cos\alpha\cos\beta),\end{split}