# Computational Atomistic Modeling in Carbon Flatland and Other 2D Nanomaterials
## 5 Magnetoresistance in 2D Tunnel Junctions  
Since the isolation of graphene, the library of synthesizable two-dimensional materials has been enriched with several hundreds items featuring versatile electronic properties. This catalog offers unprecedented design opportunities for the rapidly growing field of spintronic where the structural quality of interfaces often plays an instrumental role in determining the functionality and performance of devices. A prototypical spintronic device is the magnetic tunnel junction (MTJ) consisting of two ferromagnetic (FM) layers separated by a thin insulating (I) barrier. The instrumental property of MTJs is the modulation of their resistance by individually switching the magnetization of the FM electrodes via the application of an external magnetic field. MTJs are thus characterized by two states of electrical resistance associated with the parallel and anti-parallel magnetizations of the electrodes. This is quantified by the tunnel magnetoresistance (TMR) defined as:  
\[\text{TMR}\equiv(R_{AP}-R_{P})/R_{P}=(G_{P}-G_{AP})/G_{AP}, \tag{9}\]  
where \(R_{AP,P}\) and \(G_{AP,P}\) correspond to the resistance and conductance of the junction in the anti-parallel and parallel configurations. This effect was discovered by [PERSON] in 1975 [75] who proposed a simple model to rationalize the measured TMR signal based on the assumption that (i) the carriers spin is conserved upon tunneling and (ii) the conductance of a given spin channel is proportional to the product of the density of conducting states in the FM electrodes,  
\[G_{P}\propto\rho_{L}^{\uparrow}\rho_{R}^{\uparrow}+\rho_{L}^{ \downarrow}\rho_{R}^{\downarrow}\,\] \[G_{AP}\propto\rho_{L}^{\uparrow}\rho_{R}^{\downarrow}+\rho_{L}^{ \downarrow}\rho_{R}^{\uparrow}. \tag{10}\]  
Here, \(\rho_{L/R}^{\uparrow}\) (\(\rho_{L/R}^{\downarrow}\)) denotes the density of conducting states of majority (minority) spin in the left and right electrodes. Within this framework, the TMR can be expressed as:  
\[\text{TMR}=2\mathcal{P}_{L}\mathcal{P}_{R}/(1-\mathcal{P}_{L}\mathcal{P}_{R}), \tag{11}\]  
where \(\mathcal{P}_{L,R}\) are the spin polarization of the conducting electrons in the left and right electrodes defined as \(\mathcal{P}_{L,R}=(\rho_{L,R}^{\uparrow}-\rho_{L,R}^{\downarrow})/(\rho_{L,R} ^{\uparrow}+\rho_{L,R}^{\downarrow})\). The spin polarization of the tunneling electrons ultimately depends on the electronic structure of the interfaces. Over the last decade, significant research efforts have been dedicated to introduce 2D materials to control and tailor the magnetoresistive response of the interfaces [76]. Boosted by the theoretical prediction of perfect spin filtering by few of its layers [77; 78; 79; 80], graphene has been the first 2D material to be investigated [81; 82; 83], rapidly followed by insulating \(h\)-BN [84; 85; 86] and semi-conducting transition metal dichalcogenides (TMDCs) [87; 88; 89; 90]. While very promising experimental results have yet been reported, first-principles simulations provide a complementary tool to unravel the full potential of 2 DMs in tunneling junctions. This is illustrated here with the theoretical investigation of a MTJ consisting of a monolayer \(h\)-BN sandwiched between two Co ferromagnet electrodes. The in-plane lattice constant of the \(\langle 0001\rangle\) hcp Co matches almost perfectly the lattice constant of \(h\)-BN. This suggests that epitaxial Co/\(h\)-BN interface can be realized experimentally by CVD technique [91; 92]. As a consequence, the interface may be perfectly commensurate or not depending on the integration pathway. The epitaxial interface is illustrated in Figure 10 along with a misaligned configuration obtained by twisting the \(h\)-BN monolayer by \(21.78^{\circ}\). First-principles calculations reveal strikingly different spin behaviours for these two geometries. Structural optimization of the aforementioned interfaces indicates a stronger bonding of _h_-BN with Co in the aligned configuration with respect to the misaligned one.