Transition-metal adatoms on 2D-GaAs: a route to chiral magnetic 2D materials by design

We first calculated the adsorption energy of a single TM adatom (Mn, Co, Mo and Os) on a 551-GaAs supercell and examined the stability of our systems using the following expression

$$\begin{equation}{E}_{\mathrm{a}\mathrm{d}}={E}^{\left(\mathrm{G}\mathrm{a}\mathrm{A}\mathrm{s}\right)}+{E}^{\left(\mathrm{T}\mathrm{M}\right)}-{E}^{\left(\mathrm{T}\mathrm{M}\text{--}\mathrm{G}\mathrm{a}\mathrm{A}\mathrm{s}\right)},\end{equation} \tag{ 4 }$$

where E^(TM–GaAs) and E^(GaAs) are the total energies of the system with adsorbed TM atom and 551-GaAs supercell respectively, and E^(TM) is the energy of a single isolated TM atom. More stable structures have a larger adsorption energy. We found that the preferential location of all the TM adatoms studied in our work is the hollow site (within the hexagonal Ga–As tile), followed by the site on top of As as the second most favourable. Os atom is metastable on both sites, with an energy difference of ∼284 meV. We have chosen the hollow site as the location for adsorption of the TM atoms, because it combines the best energetic and structural stability. Table 1 displays the adsorption energies on hollow site for comparison. The energy stability order of our studied configurations is as follows: Os (4.967 eV/adtom), Co (3.539 eV/adatom), Mo (2.799 eV/adatom), and Mn (2.166 eV/adatom). We note that all configurations exhibit comfortably large energetic stability.

Subsequently, we investigated the local structural distortions introduced by the adatom in the supercell. The optimised structural parameters for pristine 2D-GaAs crystal structure and TM on 551-GaAs are given in table 1. In addition, figure 1 depicts a top view for TM centered above one hollow site (middle of Ga–As hexagonal tile) of the 551-GaAs supercell (figure 1(a)), as well as the side view of the relaxed buckling for Ga and As atoms (Δ_As–Ga) located at the hollow site for 551-GaAs supercell in absence of any adatom (figure 1(d)). Furthermore, the optimised placements for Mn, Co, Mo and Os atoms (Δ_Ga–TM) are shown in figures 1(b), (c), (e), and (f), respectively. One sees from table 1 that the optimised structural parameters for pristine 2D-GaAs (a₀) are in good agreement with previous theoretical reports [5, 42]. When one TM atom is adsorbed on the 551-GaAs supercell, the relaxed unit-cell lattice parameter of TM on 551-GaAs system (a₀' = a_sup/5) is reduced slightly when compared with the one of pristine unit-cell, i.e. by 1.21, 1.26, 1.19, and 1.41%, for Mn, Co, Mo and Os adatom, respectively. This can be explained by the larger buckling parameter (Δ_As–Ga) for TM adsorbed on 551-GaAs when compared with the one of pristine 2D-GaAs. The nearest As–Ga bond (d_As–Ga) is larger for TM on 551-GaAs than the one in pristine 2D-GaAs by 4.78, 3.94, 6.19, and 12.42%, for Mn, Co, Mo and Os atom, respectively.

From the analysis above, one can see that Os (Z = 76) and Mo (Z = 42) atoms, which have higher atomic numbers and are larger, result in a larger buckling (Δ_As–Ga) and extend As–Ga bonding length (d_As–Ga) compared to the cases of Mn (Z = 25) and Co (Z = 27). The optimised placements for Mn, Co, Mo and Os atoms (Δ_Ga–TM) are shown in figuress 1(b), (c), (e), and (f), respectively. The Mn and Mo atoms remain above the hollow site at a vertical distance of, respectively, 0.362 Å and 0.566 Å from the Ga-plane, while Co is almost in Ga-plane (0.087 Å from Ga-plane) and Os is located between As-plane and Ga-plane, nearer to the Ga-plane (Δ_Os–Ga = 0.421 Å and Δ_As–Os = 0.916 Å). These differences in TM distance from Ga-plane (As-plane) can be intuitively explained by the electronegativity difference between TM and Ga (As) atoms. Mo (Os) and Mn (Co) are the most (least) electropositive atoms, thus Ga–Ga (As–As) in-plane distance increases (decreases) due to charge repulsion (attraction) when a more electropositive atom is adsorbed. This expansion (contraction) of Ga–Ga (As–As) in-plane distance can be evidenced by the increase (decrease) of the angle between neighbouring bonds (θ_GaAsGa) for Mo (2.797%) and Mn (0.175%) (Os: −11.888% and Co: −2.797%) when compared to the one of the pristine system, as shown in table 1. As a result, Mo and Mn atoms experience a lower net electrostatic force towards the Ga-plane (As-plane) when compared to Co and Os atoms. In table 1 we also give the charge located on the TM adatom (Δρ), which is consistent with the difference of electronegativity between the constituent elements.

As shown in table 1, the adsorbed single adatom induces a magnetic ground state in the nonmagnetic 2D-GaAs system, with a net magnetic moment of 3.0 μ_B, 1.0 μ_B, 4.0 μ_B, and 2.0 μ_B per TM atom for the case of Mn, Co, Mo and Os adatom, respectively. Table 1 also shows the calculated MAE per TM atom of our systems. The positive (negative) value of the MAE implies that the easy magnetisation axis is out-of-plane (in-plane) to the surface of the 2D structures. One sees that Os atom induces the largest MAE (−6.930 meV), followed by Mo (3.460 meV), Co (−0.950 meV), and Mn (0.530 meV). However, as would be expected, we find that a higher magnetic moment does not necessarily lead to a higher MAE, e.g., Os with a moment of 2.00 μ_B has a higher MAE than Mo (4.00 μ_B) and Mn (3.00 μ_B). The physical reasons that lead to MA in 2D TM-adsorbed GaAs will be analyzed in the next paragraphs. For Os adsorbed on the metastable As-top site, we found a strikingly large MAE of −28 meV.

In order to provide fundamental insights into the interaction of a TM adatom with the 2D-GaAs host semiconductor (TM = Mn, Co, Mo and Os), and how those interactions can induce magnetocrystalline anisotropy, both the electronic band structure of pristine 2D-GaAs, and the ones of Mn, Co, Mo and Os adsorbed on the hollow site of 2D-GaAs were calculated for comparison. The parametric projected electronic band structures for both the pristine 2D-GaAs sheet and the adsorbed TM on 2D-GaAs are shown in figures 2 and 3, respectively. Notice in figure 2 that 2D-GaAs has a K–Γ indirect band gap of 1.06 eV. In the projected plots, the colour intensity corresponds to the degree of overlapping of sp orbitals between Ga and As atoms. The yellow lines, near and below the Fermi level, represent the contributions of As-4p_z orbitals. The unfilled band, near and above the Fermi level (green lines), represents the contribution of Ga-4p_z and Ga-4s orbitals. The buckling reduces the sp² (p_z orbitals) hybridization, and increases the sp³ one (p_z and planar s). Figure 3 displays the parametric orbital projected electronic structure for TM on 2D-GaAs [TM = Mn (a), Co (b), Mo (c) and Os (d)]. Electronic structure analysis shows that the semiconducting nature of pristine 2D-GaAs is retained by Mn (Γ–Γ), Co (M–Γ) and Os (M–M) systems, with bandgap energy reduced by 45%, 37% and 85%, respectively. On the other hand, adsorption of Mo on 2D-GaAs changes its nature to a half-metallic one. One sees in figure 3 that the TM adatom creates localised states near the Fermi level. In the projected plots, the colour intensity corresponds to the degree of contribution of the d-electrons of the TM adatom. The Ga and As atoms surrounding the adatom lead to a crystal field splitting of the d orbitals of the TM adatom. Due to the buckling of 2D-GaAs, and the different electronegativities and atomic numbers between TM, Ga and As atoms, the crystal-field splitting of the d orbitals of the TM on 2D-GaAs deviates from the known planar trigonal-level splitting. In the case of crystal-field splitting of trigonal planar geometry, if the ligand atoms lie in the x–y plane, the d_xy and ${d}_{{x}^{2}-{y}^{2}}$ orbitals have the highest energy because their electron density is concentrated in the x–y plane. The ${d}_{{z}^{2}}$ orbital has higher energy than the d_xz and d_yz orbitals. The physical reason of this feature is the ring of electron density that ${d}_{{z}^{2}}$ orbital has in the x–y plane. The d_xz and d_yz orbitals of the central TM adatom have the lowest energies because they have all their electron density out-of-plane.

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For the cases of Co and Os, which exhibit magnetisation in the in-plane direction, the orbitals of higher energy levels are d_xy and ${d}_{{x}^{2}-{y}^{2}}$, as shown in figures 3(b) and (d), respectively. Due to different vertical position of Co and Os adatom with respect to the Ga-plane and As-plane, see figures 1(c) and (f), respectively, the energies for ${d}_{{z}^{2}}$ and d_xz (d_yz ) orbitals differ. While the ${d}_{{z}^{2}}$ orbital for Co has a somewhat higher energy than that of the d_xz and d_yz orbitals, for Os we find the opposite, the energies for d_xz and d_yz orbitals are higher than that of ${d}_{{z}^{2}}$. Another difference between these cases is the large overlap among the d_xz and d_yz orbitals of Co and 4-p_x p_y p_z -GaAs orbitals as compared with those for Os, as shown in figure 3(b) by green lines. This can be explained by the closer distance between Co and Ga-planes as compared to distance of Os to the Ga-plane. Regarding the electronic structure of Mn and Mo on 2D-GaAs, one notices in figures 3(a) and (c), respectively, that d_xz and d_yz are the occupied states of higher energy, followed by d_xy and ${d}_{{x}^{2}-{y}^{2}}$. This can explain the out-of-plane magnetisation axis for both those TM adatoms on 2D-GaAs.

SOC plays a fundamental role in the magnetic behavior of the TM-adsorbed on 2D-GaAs semiconductor. The SOC mechanism works as follows: figure 2 displays s and p mixed orbitals of pristine 2D-GaAs near the Fermi level, both in the valence and in the conduction bands. Structural buckling reduces the overlapping between p_z orbitals. As a result, the sp³ hybridization near Fermi level becomes stronger than sp². On the other hand, the outermost d-orbital of TM ions are half-filled and, by Hund's rule, they are aligned parallel. When TM ion is adsorbed on 2D-GaAs semiconductor, the local magnetic moment of d-TM states hybridizes with the sp bands near the Fermi level of 2D-GaAs (spin–orbit interaction). Therefore, the sp–d exchange interaction between the sp band electrons of 2D-GaAs and the d electrons associated with the TM atoms, as well as the lack of inversion symmetry of the crystal environment, are the responsible of the magnetic behavior of TM-adsorbed on 2D-GaAs semiconductor.

The calculated electronic spin–charge density for systems with the easy magnetisation axis parallel [(a) Os and (b) Co] and perpendicular [(c) Mn and (d) Mo] to the surface of the considered 2D structures is shown in figure 4. The colour intensity represents the spin–charge density, where blue and red indicate high and low density, respectively. The magnetic moment in each of these systems mainly originates from the adatom, i.e. 63, 95, 68 and 53% of total magnetisation for Mn, Co, Mo and Os atom, respectively. Conversely, the magnetic moment stemming from the nearest neighbours at the hexagonal Ga–As tile where adatom resides is smaller than that found in adatom case, i.e. 4.9, 4.5, 10 and 37% of total magnetisation for Mn, Co, Mo and Os system, respectively. Thus, according to our results, the MA of TM on 2D-GaAs system depends robustly on the broken crystal symmetry, the orbital character of the states in the vicinity of the Fermi level, and the magnitude of the interaction of the spins with the field generated by the electron orbital motion in the crystal.

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During experimentally performed adsorption, one expects multiple adatoms to reside in close proximity to each other. The study of small clusters is of special scientific interest due to their unique magnetic properties can be tailored for desired applications in magneto-electronics by choosing a special size or composition of the cluster. That in turn opens questions about the nature of magnetic interaction arising between the adsorbed atoms in a close range from each other.

Modern DFT-based calculations are able to describe the electronic and magnetic properties of the transition metal clusters in excellent agreement with the value obtained from the experimental ones. For instance, the experimental study of magnetic properties of Co atoms adsorbed on graphene [60] agrees very well with the ones predicted by DFT [61]. In addition, Stroppa et al reported that computational images of Mn defects on GaAs predicted by DFT are consistent with the experimental ones [62].

We therefore set out to determine the magnetic exchange interaction between two neighbouring transition-metals adatoms on 2D-GaAs, by calculating the generalised 3 × 3 matrix according to equation (3). We will primarily focus on two components of the magnetic exchange interaction: the isotropic exchange and the DMI.

In order to examine the magnetic exchange interaction between two adsorbed transition metal atoms in close proximity to each other, we first relaxed the structures for the first (4.05 Å) and second nearest distance (7.01 Å) between TM atoms (MnMn, CoCo, and MoMo) adsorbed on 2D-GaAs. The second nearest distance was chosen as subject of this study, as shown in figure 5, instead of the nearest one, because its geometry was not distorted at the hexagon site after relaxation as for the nearest case. Then, we calculate the magnetic exchange parameters for the most stable geometry (${J}_{12}^{ij}$) using DFT methodology described in section 2. All nine elements of the matrix (${J}_{12}^{ij}$), as well as the DMI parameters ${D}_{12}^{x}$, ${D}_{12}^{y}$, ${D}_{12}^{z}$, the total DMI parameter |D|, and the ratio between |D| and the average of the diagonal exchange parameters ${J}_{12}^{xx}$, ${J}_{12}^{yy}$ and ${J}_{12}^{zz}$, are listed in table 2. Ratio |D|/|J_diag| gives one insight in how strong D is as compared to the isotropic exchange J [yielding standard (A)FM behavior].

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Our results show that the total magnetisation obtained for Co (1 μ_B/adatom) is smaller than those for Mn (3 μ_B/adatom) and Mo (4 μ_B/adatom) atoms. The pairwise distances between Co, Mn and Mo adatoms after relaxation are, respectively, 6.68 Å, 6.77 Å, and 7.16 Å. One can see that for our three magnetic systems, the magnetic exchange interaction is stronger as the TM adatoms reside closer to each other. The optimised position of the two adatoms has the same physical tendency as for the single adatom cases. The Mn and Mo atoms remain above the hollow site to a vertical distance of 0.469 Å (0.362 Å) and 0.611 (0.566 Å) from Ga-plane, respectively, while Co resides in the Ga-plane (0.087 Å from Ga-plane). The values given in parenthesis represent the corresponding distances for the case of a single adatom, for comparison. When two TM atoms are adsorbed on the 551-GaAs supercell, the relaxed unit-cell lattice parameter of TM on 551-GaAs system (a₀' = a_sup/5) is reduced slightly when compared with the one of the pristine unit-cell, i.e. by 1.48 (1.21), 1.58 (1.26) and 1.24 (1.19)%, for Mn, Co and Mo atoms, respectively (the values in parenthesis are for single adatom cases). The optimised structural parameters for a pair of adatoms therefore exhibit the same physical tendencies as the previous single-adatom cases.

As shown in table 2, the magnetic interaction between adjacent TM adatoms significantly changes the emergent magnetic properties. One sees that Co prefers the FM ground state, with the magnetisation along the x-axis, and exhibits a strong FM interaction in both x and z directions, as ${J}_{12}^{xx}$ = −8.05 meV and ${J}_{12}^{zz}$ = −6.73 meV. Such difference of diagonal elements points to strong anisotropy. Regarding the Mn system, FM interaction is also found. The difference between all diagonal elements differs only by 0.01 meV, meaning there is almost no anisotropy. On the other hand, there is an AFM interaction found between two Mo atoms on 2D-GaAs, with a slight anisotropy favouring the z-(out-of-plane) direction. We found for all three systems that ${J}_{12}^{xy}\enspace \ne \enspace {J}_{12}^{yx}$, ${J}_{12}^{yz}\enspace \ne \enspace {J}_{12}^{zy}$, and ${J}_{12}^{xz}\enspace \ne \enspace {J}_{12}^{zx}$, which indicates that the matrix J₁₂ is antisymmetric. Furthermore, the off-diagonal elements of the exchange matrix are all different from zero, in agreement with the antisymmetry considerations by Moriya [34]. The consequent magnitudes of |D| for Mn, Co, and Mo pairs of adatoms are, respectively, ∼17%, ∼42%, and ∼9% of the average symmetric exchange interaction, as shown in table 2. Hence, DMI is both stronger and more influential in the case of Co adatoms, compared to Mn and Mo adatoms. While having a weaker magnetic moment, Co hosts larger SOC and therefore enhances the DMI interaction. Table 2 also displays the calculated MAE with respect to the TM pair of our systems. The positive (negative) value of the MAE implies that the easy magnetization axis is out-of-plane (in-plane) to the surface of the 2D structures. We can see that Mo atom induces the largest MAE (2.89 meV), followed by Mn (1.03 meV) and Co (−0.54 meV). As for the single adatom cases, the easy magnetization axis of Co and Mo (Mn) atoms in two adatom cases remains in-plane and out-of-plane, respectively. Figure 5 depicts orientation vectors for both the spins at ground state (gold colour) and DMI (blue colour) of the magnetic exchange interaction for each studied pair of TM atoms absorbed on 2D-GaAs. The ground state spin configurations are obtained by solving Heisenberg spin Hamiltonian iteratively. We note once more that D vector tends to align the neighbouring spins orthogonally to each other, instead of the parallel or antiparallel spin alignments obtained by the usual Heisenberg exchange interaction. The direction of the induced magnetisation on 2D-GaAs semiconductor can be selectively manipulated depending on chosen TM adatoms. The induced DMI in the TM-adsorbed GaAs monolayer causes spins of adjacent TM atoms to deviate from perfect (anti)ferromagnetic alignment. As a consequence, spins prefer non-collinear, chiral magnetic ordering, enabling spin textures unattainable otherwise. That is in turn attractive for development of spintronic devices, for example those based on special properties of spin-waves, with dispersion tailored by local DMI.

Finally we observe that Os adatoms in close proximity to each other the magnetisation vanishes on each Os atom, and the system goes to a non-magnetic state. Therefore, Os atoms coupling neither FM nor AFM in a close range from each other. This result strongly differs from our calculations for single Os adatom, where the system is comfortably magnetic with 2 μ_B magnetisation. It is therefore clear from these results that magnetic properties of adatoms strongly depend on the density of adatoms, and that more studies are needed to fully characterize the magnetic properties of specific patterns and densities of TM adatoms on 2D-GaAs. Furthermore, taking into consideration that the magnetic interactions may change drastically with distance [63], the study of the distance dependent magnetic exchange interaction of TM-adsorbed 2D-GaAs will be explored in further research.