# Strontium Ferromolybdate-Based Magnetic Tunnel Junctions
## 2 Theoretical Considerations
### Low-Field Magnetoresistance  
In LSMO ceramics, extrinsic LFMR was found in 1996 [49]. It was shown that the MR in polycrystalline LSMO exhibits two distinct regions: large MR at low fields (\(\sim\)0.1 T) dominated by spin-dependent tunneling between grains, and high-field MR also obtained in single crystals. However, the initial expectation of using LSMO in RT spintronic applications was not fulfilled. Two years later, the discovery by [PERSON] et al. in 1998 [1] of LFMR and the half-metallicity of SFMO possessing a Curie temperature of about 420 K renewed the interest in double perovskites in the context of their potential applications in the field of spin electronics.  
At present, it is well established that the main contribution to LFMR in SFMO ceramics arises from spin-dependent electron tunneling across insulating grain boundaries (GBs) [50; 51; 52] and not from the existence of Fe/Mo antisite defects (ASD) in the bulk [53]. The LFMR is almost absent in single crystals [54]. A study of an epitaxial SFMO thin film grown on an STO (100) bicrystal boundary has proven that LFMR is caused by spin-dependent electron transfer across GBs and not by an intragranular effect [55]. Depending on deposition conditions, LFMR might be missing in SFMO thin films deposited by PLD onto STO (001) substrates [56; 57]. However, the tunneling barriers necessary for LFMR could be additionally created, e.g., by a post-deposition heat treatment at 475-500 \({}^{\circ}\)C in an ultrapure Ar (99.9995 %) atmosphere for 5 h [56] or by annealing in a reducing 5%H\({}_{2}\)/N\({}_{2}\) atmosphere for 10 h at 950 \({}^{\circ}\)C [58]. In comparison, neither magnetic nor magnetotransport nor structural properties were improved by post-deposition annealing of SFMO thin films at temperatures between 500 and 1100 \({}^{\circ}\)C in a vacuum, Ar, 5%H\({}_{2}\)/Ar, or air atmospheres for 5 or 10 h [59]. Obviously, grain boundary modification should be performed under carefully selected conditions.  
SFMO thin films grown by PLD on STO (111) substrates exhibit a larger LFMR effect in (111)-oriented films compared to those with the (001) orientation [60]. The origin of this phenomenon is suggested to be the presence of antiphase domain boundaries (APBs). In fact, APBs are nucleated at steps on the substrate surfaces if nucleation on the two terraces separated by a step edge starts at different atomic planes [61]. Atomically flat STO (111) surfaces show a clear step-and-terrace structure with a step height of about 0.24 nm [62]. This approximately corresponds to the plane separation \(d_{111}\) = 0.228 nm of SFMO in the (111) direction [63]. Note that the larger separation of (111) planes at the substrate surface favors the formation of FeO\({}_{6}\)-FeO\({}_{6}\) octahedra in comparison to FeO\({}_{6}\)-MoO\({}_{6}\) octahedra. Experimentally, a step height of about 0.45 nm was derived from AFM images on the surface of epitaxial (111)-oriented SFMO films obtained by growth mediated by screw dislocations on STO (111) substrates [60]. This value corresponds to 2\(d_{111}\) and provides evidence that film growth takes place with a unit cell as a growth species.  
The LFMR is favored in thin films subjected to large biaxial compressive strain at the film-substrate interface [57]. This was related to the formation of low-angle GBs.