Towards molecular controlled magnonics

Magnonics is an emerging field broadly recognized as a paradigm shift for information technologies based on the use of spin waves. However, the low flexibility and variety of the existing systems still hamper their applications. Herein, we propose an unprecedented chemical approach to magnonics based on the creation of hybrid molecular/2D heterostructures. We analyse the modulation of the magnetic properties, magnon dispersion and spin dynamics of a single layer of CrSBr after the deposition of sublimable organic molecules via first-principles calculations. Our results predict a modulation of magnetic exchange, a shift in the magnon frequencies and an enhancement of their group velocities up to ∼7%. Interestingly, we find a linear correlation between these effects and the donor character of the molecules. This will pave the way for the design of a new class of magnonic materials that can be selectively tailored by a chemical approach.


1.Adsorption energy
First, we test the stability of the heterostructures formed by CrSBr monolayer and the organic species TTF, perylene, coronene, and TCNQ by analysing their respective optimized geometries.To do so, we calculate the adsorption energy by applying the following expression: where ECrSBr+molecule, ECrSBr and Emolecule are the total energies of the hybrid molecular/2D heterostructure, pristine CrSBr and the isolated molecule, respectively.More negative adsorption energies indicate more stable heterostructures.Then, we study in plane rotations of the molecules in the previously determined preferential site to explore if the interaction strength can be increased by acquiring a different orientation (Fig. S1).S4.Vacuum energy (Evac), Fermi energy (EFermi) and work function (Ф) for pristine CrSBr and adsorption systems.EF increases and Ф decreases with the donor behaviour of molecules, respectively.Work function is calculated by: Ф = Evac -EF.
Next, we show the projected density of states (PDOS) of the Cr atoms of the substrate for the pristine and different adsorption systems.In addition, we show the DOS for the isolated molecules and adsorbed molecules.In the latter case we observe the splitting of the molecular states for the different cases (all Figures are plotted with RAD-tools 1 ).

Model Hamiltonian
Magnetic properties of the CrSBr are modelled with the Hamiltonian of the form: where double counting is present and spins are normalized to 1.In this study we investigate the effect of the molecule deposition on the isotropic part of exchange interaction.The exchange parameters are computed via energy mapping with the four configurations displayed in Fig. S17.Exchange parameters are computed as: where  %& ,  $% !,  $% # ,  $% " -are the energies of the corresponding configurations per one Cr atom.The computed values of the total energies and computed exchange parameters are listed in Table S5.

Effects on the structure
In order to describe the structural distortion of the CrSBr substrate caused by the deposited molecule we define the characteristic angles and bond lengths.The angles are depicted in the Fig. S18.Three distances are defined as the distances between two nearest Cr atoms along the direction of the first lattice vector a, second lattice vector b and between two atoms in the unit cell (ab).The values are presented in Table S7.

Charge transfer
We study the charge transfer flowing along the interface between CrSBr monolayer and the molecules by performing a Bader analysis as stated in the main text.The charge donation to the substrate can be depicted in the population of the dz 2 orbital (Figs.S19, S20).When the molecules are deposited on top of the substrate, they change the magnetic moments of the Cr atoms.In the model Hamiltonian all changes are absorbed in the J values, however here we show the evolution of magnetic moments directly.Table S9.Relative total energies DE (meV/ Cr atom) for the four magnetic configurations scaled with respect the energy of ferromagnetic configuration.The three magnetic exchange parameters J1, J2 and J3 (meV) are listed.Data corresponds to pristine CrSBr, TTF/CrSBr in a 6x6 supercell (low coverage) and TTF/CrSBr in a 4x2 supercell (high coverage).

Magnon dispersion
Modelling of the magnon dispersion commences with the Hamiltonian, which differs from the model Hamiltonian (see section 3.1): spins are considered not to be normalized.In order to achieve that we map the supercell calculation to the simple model of the magnetic lattice of Cr atoms in the high spin state (3/2), effectively absorbing all effects in the J values themselves.
Magnons are defined as the excitation with respect to the ferromagnetic ground state, however in the model Hamiltonian there is no anisotropic terms, which are required to form ferromagnetic state in the 2D limit.As we stated before, more realistic Hamiltonian includes these terms, however they are not important to the analysis, which we are about to perform.Exchange and single-ion anisotropies contribute to the gap opening of the magnon dispersion effectively shifting the dispersion for the whole Brillouin zone, which does not affect the group velocities.Therefore, the evolution of the isotropic exchange is sufficient for the determination of the propagation velocities.The magnon dispersion relation is obtained via well-known Holstein-Primakoff (HP) transformation 2 : where i runs in (1, 2, 3) for the nearest, next-nearest and next-next-nearest neighbors, ni is the amount of the neighbors (2, 4, 2 correspondingly),   (!) are the structural factors: where a and b are the lattice parameters; plus sign in the second structural factor corresponds to the acoustic (bottom) branch, minus sign refers to the optical (upper) branch.
In Fig. S25 we show a direct comparison between our calculations and the measured magnon spectrum for CrSBr. 3As we can observe our theoretical predictions reproduce accurately the experimental data.
In Figs.S26 and S27 the magnon dispersion is presented (i) considering solely the distortion of the substrate caused by the molecules and (ii) distortion of the substrate and electron doping.

Group velocities
Group velocities of magnon propagation are defined as a derivative of the frequency with respect to the chosen direction of k.Due to the orthorhombic symmetry of the lattice, we focus our attention on the dependency of the group velocity (vx) in the direction of the first lattice vector (a) with the k vector along G-X path and of the group velocity (vy) in the direction of the second lattice vector (b) with the k vector along G-Y path.The corresponding formulas are:

Fig. S1 .
Fig. S1.Evolution of the adsorption energy for the selected rotations according to symmetry matching between molecule and substrate.(a) TTF, (b) perylene, (c) coronene, (d) TCNQ.Note that TTF was rotated 180° instead of 90° as for the rest of the C2v symmetry molecules because of its tilted configuration.
Fig. S3.HOMO orbital of TTF molecule adsorbed on CrSBr.Blue (red) surfaces represent positive (negative) values of the molecular orbital.Isovalue is set to 0.01 eV•Å -3 .The tilting of TTF can be attributed to the favourable overlap between the highest occupied molecular orbital (HOMO) and the π orbitals of the close Br atoms from the substrate, thus intensifying the binding strength.
Fig. S17.(a) CrSBr monolayer marked with the three nearest neighbours exchange parameters depicted as J1, J2 and J3.On the right, the four computed magnetic configurations: (b) FM (c) AF1 (d) AF2 (e) AF3.Blue and red arrows represent Cr atoms with spin up and spin down state, respectively.Thin black lines indicate the unit cells.

Fig. S18 .
Fig. S18.Characteristic angles of CrSBr structure.The Cr, S and Br atoms are represented as cyan, yellow and pink spheres, respectively.

Fig. S19 .
Fig. S19.Orbital splitting of Cr 3+ ion in a distorted octahedron with C2v symmetry.Orange arrows represent the electrons that are filling the Cr d orbitals in (a) pristine CrSBr and (b) donor molecule/CrSBr heterostructure.

Fig. S22 .
Fig. S22.Evolution of exchange parameters for the hybrid molecular/CrSBr heterostructures, electron doping and structural distortion.Values of the pristine CrSBr are shown with dashed lines.

Fig. S23 .
Fig. S23.Distribution of the magnetic moments for the 6x6 supercell subjected to the molecules, electron doping and structural distortion.Dots represent magnetic moments of each individual Cr atom.Solid lines show the mean value for each corresponding magnetic configuration.Dashed lines illustrate magnetic moments of each configuration for the pristine CrSBr.

Fig. S24 .
Fig. S24.Variation of the Curie Temperature (ΔTC) for the different heterostructures with respect to pristine CrSBr (dashed line).The calculated TC of monolayer CrSBr is 140 K.

Fig. S25 .
Fig. S25.Direct comparison between the magnon spectrum of CrSBr obtained from our calculations and the experimental one reported by Scheie et al.3

Fig. S26 .
Fig. S26.Magnon dispersion for the structural distortions, caused by each molecule.Insert shows the k-path.

Fig. S27 .
Fig. S27.Magnon dispersion for the structural distortions and electron doping corresponding to each molecule.Insert shows the k-path.
Fig. S29.Group velocities of the magnon propagation for the CrSBr substrate with the molecules deposited on top.(a) Group velocity  2 4-7 (b) Group velocity  3 4-8 .Solid lines for the acoustic branch, dashed lines for the optical branch.

Fig. S31 .
Fig. S31.Effect of the (a) molecules and (b) electron doping on the group velocities with respect to the corresponding distortion.

Table S1 .
Adsorption energy values for the different sites.Values in bold indicate the preferential site for adsorption.

Table S2 .
Preferential site and adsorption energy for TTF and derivatives.

Table S5 .
Relative total energies DE (meV/ Cr atom) for the four magnetic configurations scaled with respect the energy of ferromagnetic configuration.The three magnetic exchange parameters J1, J2 and J3 (meV) are listed.Data for the calculations with molecules on top of CrSBr.

Table S6 .
Relative total energies DE (meV/ Cr atom) for the four magnetic configurations scaled with respect the energy of ferromagnetic configuration.The three magnetic exchange parameters J1, J2 and J3 (meV) are listed.Data for the calculations with the distortion in the CrSBr substrate, caused by each molecule, but without the molecules.

Table S7 .
Structural parameters.Bond lengths and angles between the specified atoms.They represent the average values where the considered Cr atoms in the heterostructures are the ones located in the substrate area that is under the molecule.Notation is referred to Fig.S18.

Table S8 .
Relative total energies DE (meV/ Cr atom) for the four magnetic configurations scaled with respect the energy of ferromagnetic configuration.The three magnetic exchange parameters J1, J2 and J3 (meV) are listed.Data for the calculations with the distortion in the CrSBr substrate, caused by each molecule and corresponding electron doping.

Table S10 .
Data and fit parameters of the relative velocity vs induced charge transfer.