Graphene-multiferroic interfaces for spintronics applications

Graphene and magnetoelectric multiferroics are promising materials for spintronic devices with high performance and low energy consumption. A very long spin diffusion length and high carrier mobility make graphene attractive for spintronics. The coupling between ferroelectricity and magnetism, which characterises magnetoelectrics, opens the way towards unique device architectures. In this work, we combine the features of both materials by investigating the interface between graphene and BaMnO3, a magnetoelectric multiferroic. We show that electron charge is transferred across the interface and magnetization is induced in the graphene sheet due to the strong interaction between C and Mn. Depending on the relative orientation of graphene and BaMnO3, a quasi-half-metal or a magnetic semiconductor can be obtained. A remarkably large proximity induced spin splitting of the Dirac cones (~300 meV) is achieved. We also show how doping with acceptors can make the high-mobility region of the electronic bands experimentally accessible. This suggests a series of possible applications in spintronics (e.g. spin filters, spin injectors) for hybrid organic-multiferroic materials and reveals hybrid organic-multiferroics as a new class of materials that may exhibit exotic phenomena such as the quantum anomalous Hall effect and a Rashba spin-orbit induced topological gap.

FIG. S1: The egg box effect: the total energy changes by displacing the coordinates by a fraction of the unit cell length. The effect can be visualised by plotting the difference between the total energy E and its average ∆(E) = E − (Emax + Emin)/2 for every coordinate shift, at each value of the energy cutoff (a). The accuracy on total energy calculations is given by the width of the egg box oscillation Emax − Emin for a given cutoff (panel b).

Bulk BaMnO3
The LDA functional and the converged computational parameters reproduce the correct energetic ordering between the paraelectric (P 6 3 /mmc) and ferroelectric (P 6 3 cm) phases of bulk hexagonal BaMnO 3 (2H-BaMnO 3 ). The calculations have been performed in the P 6 3 cm primitive cell, which contains 30 atoms and is three times larger than the primitive P 6 3 /mmc (10 atoms). The lattice parameters and atomic positions were relaxed until the maximal force on atoms was smaller than 0.001 eV/Å and the stress was smaller than 0.01 GPa. The converged lattice parameters are reported in Table S2, together with the computed macroscopic polarization and magnetic moment on Mn atoms.
Atomic relaxation of 2H-BaMnO 3 in the P 6 3 cm phase was also performed imposing a ∼2% in-plane strain to match the 4 × 4 relaxed graphene cell (in-plane lattice constant 9.848Å) in view of building the graphene-BaMnO 3 interface. The relaxed out-of plane lattice constant is 4.685Å. Within the present computational framework, the strained 2H-BaMnO 3 is a semiconductor with indirect electronic band gap of 1.55 eV between the L and A points of the Brillouin Zone ( Figure S2).
The macroscopic polarization of bulk BaMnO 3 has been computed with the Berry phase formalism 3 , correctly predicting no polarization for the paraelectric phase (P 6 3 /mmc) and P = 0.267 µC/cm 2 for the ferroelectric phase (P 6 3 cm). The polarization of strained BaMnO 3 is 19.037 µC/cm 2 . These values have also been checked with the Wannier-function method. The discrepancy with the experimental value 4 (1.47 µC/cm 2 ) is due to strain and to the underestimation of the lattice constant in LDA.   In order to model the graphene-BaMnO 3 interface, one can cut a stoichiometric slab along the (0001) plane out of bulk BaMnO 3 which is terminated on one side by Mn and, on the other, by BaO 3 atoms. However, the resulting slab is polar and will artificially become metallic (polar catastrophe) due to excess electronic charge on either side of the slab: Each Mn atom has a +4 and each BaO 3 has a −4 electron charge. To avoid this computational artefact, symmetric slabs have been considered, which our calculations correctly predict to be semiconducting (see main text). However, after atomic relaxation, these slabs return to a centrosymmetric structure, losing any initial ferroelectric polarization. For this reason, the present work focuses on the electronic and magnetic interaction between graphene and BaMnO 3 .
To avoid spurious interaction between the two opposite surfaces it is necessary to model a slab consisting of 13 atomic layers, i.e. 3 times the BaMnO 3 unit cell plus a Mn layer.
Graphene on the Mn-terminated BaMnO3 slab: LDA+U In Mn-based compounds the strong correlation of Mn 3d electrons limits the validity of the LDA and GGA approximations to the exchange-correlation functional. These approximations induce a spurious electrostatic self-interaction of the electron charge that overestimates the delocalization of Mn d states 7 . To test how this affects the results presented in this work, we computed magnetization and charge transfer for the graphene-BaMnO 3 ground state using the LDA+U approach. We have used the U value obtained by Hong et al 8 by fitting BaMnO 3 total energies to Heyd-Scuseria-Ernzerhof (HSE) calculations, i.e. U = 2.7 eV. These calculations show that the inclusion of U does not bring qualitative changes with respect to the LDA results, justifying the use of the LDA approach. The magnetization on Mn atoms is slightly increased with respect to the LDA case (Table S3). The largest change concerns the Mn at the interface with graphene and amounts at about 0.8 µ B . Most important, charge and magnetization transfer to graphene follow the same trend as in LDA. The difference in the computed magnetization and charge transfer on C atoms is at most 0.015 µ B and 0.008 e − , respectively (Table S4).   More specifically, the [-1.25 eV, 0.25 eV] region is almost exclusively contributed by the interface atoms, that is graphene and the atoms of the surface BaMnO 3 layer. The fat bands analysis including only the interface atoms of the g-BMO slab is illustrated in Fig. S3.
FIG. S3: Electronic band structure projected on the atomic orbitals (fat bands) for the graphene-BaMnO3 ground state structure: contribution of surface atoms only. The orbital projection highlights that the energetic levels in the vicinity of EF are exclusively due to the atoms at the interface between graphene and BaMnO3. The Fermi energy of the hybrid system is taken as reference energy.
FIG. S4: Ball-and-stick model (a) and electronic band structure (b) of a metastable phase of the symmetric graphene-BaMnO3 slab. The graphene/BaMnO3 distance is ∼1.834Å which allows for a strong C-Mn interaction. The energy difference between this configuration and the ground state is 0.237 eV per unit cell. This metastable structure is a magnetic semiconductor, with an energy gap of 324 meV and 205 meV for majority (red lines) and minority (black lines) carriers. The Dirac cone splitting amounts to 200 meV. The Fermi energy is taken as reference.
Graphene on the BaO3-terminated BaMnO3 slab A symmetric BaMnO 3 slab terminated with the BaO 3 surface has also been studied and found to be semiconducting. However, in this case, there is no spin polarization at the surface since all the Mn bonds are saturated by O atoms. Next, a 4 × 4 graphene supercell was placed on both sides of such a slab, as illustrated in Fig. S5.a. After atomic relaxation, the average distance between graphene and the BaO 3 layer is 2.82Å. The binding energy is −106 meV per C, indicating a weaker bonding with respect to the Mn-terminated surface (binding energy −274 meV per C). The magnetic moment at the surface of the BaO 3 -terminated slab are quite small, and the magnetization induced on C atoms is negligible. The electronic band structure of the relaxed slab is reported in Fig. S5.b.