Noncollinear magnetic configurations and substrate-mediated interactions in Mn trimers on the $\mathrm{GaN}\left(000\overline{1}\right)$ surface

Collinear and noncollinear calculations based on density functional theory are carried out to elucidate the magnetic ordering of Mn trimers on a $\mathrm{GaN}\left(000\overline{1}\right)$ substrate. These trimers had previously been observed in $3a\times 3a$ surface reconstructions through Mn deposition onto the N-polar face of wurtzite $\mathrm{GaN}\left(000\overline{1}\right)$. We start off by studying the effect of spin orbit coupling for the case of monomers and dimers of Mn atoms on top of a GaN surface. Based on an effective spin Hamiltonian, we estimate the magnetic anisotropy energy (MAE) for those cases and found that it is four orders of magnitude weaker than the exchange magnetic coupling between Mn adatoms. In the Mn trimer case, the magnetic ground state has the Mn spins in-plane with the GaN surface in which the relative spin orientation within each trimer is noncollinear due to the competition between the two antiferromagnetic interactions that affect each Mn spin in the trimer, which leads to the found energy minimum with 120 degree angles between the spins. By exploring the nature and fundamental mechanisms for the magnetic interaction among the Mn trimers, we find that the surface states of the substrate play a key role, involving a Ruderman-Kittel-Kasuya-Yosida (RKKY)-type interaction. We report on an electron-mediated long-distance exchange coupling between localized magnetic moments on a $\mathrm{GaN}\left(000\overline{1}\right)$ surface.

Figure 1

Structural representations of the $\mathrm{GaN}\left(000\overline{1}\right) 1\times 1$ surface. The color coding for the different atoms is stated in the legend where ${\mathrm{Ga}}_s$ stands for the Ga atoms at the surface. Left panel: Top view with the two high-symmetry sites marked H3 and T4. Right panel: Side view of the structure stating the distance between Ga and N monolayers.

Figure 2

Band structure (left) and projected density of states (PDOS) (right) for the bare $\mathrm{GaN}\left(000\overline{1}\right) 1\times 1$ surface along $\mathrm{\Gamma }\to K\to M\to \mathrm{\Gamma }$ points. The Fermi level is marked by a red dashed line. The PDOS is projected onto the different atoms of the structure. The major contribution of the electronic states close to the Fermi level corresponds to the ${\mathrm{Ga}}_s$ atomic levels.

Figure 3

Total density of states and projected density of states (PDOS) for one Mn adatom on a $\mathrm{GaN}\left(000\overline{1}\right)$ substrate. The positive (negative) DOS and PDOS represent majority (minority) states. The major contribution of the electronic states close to the Fermi level corresponds to the ${\mathrm{Ga}}_s$ atomic levels. The double degeneracy of the Mn $d$ orbitals is shown in the inset.

Figure 4

Structural model of the relaxed structures for the monomer case, a Mn adatom at an H3 site on top of the $\mathrm{GaN}\left(000\overline{1}\right)$ surface. Top panel displays the top view of the structure while the bottom panels indicate the magnetic spin orientations of the Mn adatom with respect to the GaN substrate (top view). (a) out-of-plane; (b) in-plane spin configurations.

Figure 5

Relaxed model structure for two Mn adatoms on top of the $\mathrm{GaN}\left(000\overline{1}\right)$ surface. Top panel displays the top view of the structure and bottom panel presents the magnetic configurations employed to estimate the main magnetic interaction parameters, (a) $J_i$, (b) $D_i$.

Figure 6

Total density of states and projected density of states (PDOS) for two Mn adatoms on the $\mathrm{GaN}\left(000\overline{1}\right)$ substrate. The major contribution of the electronic states close to the Fermi level corresponds to the ${\mathrm{Ga}}_s$ atomic levels. The lift of the double degeneracy of the Mn $d$ orbitals is shown in the inset.

Figure 7

Top panel: (a) Top and (b) side views corresponding to the lowest energy structural model for Mn trimers on top of the $\mathrm{GaN}\left(000\overline{1}\right)$ substrate. The color coding for each atom is stated in the legend. ${\mathrm{Ga}}_s$ stands for the Ga atom at the surface. Bottom panel: (c) High resolution STM image (in derivative mode) revealing a round ball-shaped protrusion at each trimer $3a\times 3a$ lattice site ($V_s=-1.03\mathrm{V}, I_t=47\mathrm{pA}$); several envisioned trimer units are overlaid on top of the image; (d) higher resolution STM image (in derivative mode) of the $3a\times 3a$ trimer lattice acquired at $V_s=-1.21\mathrm{V}, I_t=63\mathrm{pA}$; (e) $dI/dV$ image (in derivative mode) acquired simultaneously with (c) at bias voltage $V_s=-1.03\mathrm{V}$.

Figure 8

Total density of states and projected density of states (PDOS) for the trimer of Mn adatoms on the $\mathrm{GaN}\left(000\overline{1}\right)$ substrate. The major contribution of the electronic states close to the Fermi level corresponds to the ${\mathrm{Ga}}_s$ atomic levels. The Mn $d$ orbitals are shown in the inset.

Figure 9

Different magnetic configurations for a $3a\times 3a$-trimer Mn on the $\mathrm{GaN}\left(000\overline{1}\right)$ surface. (a) and (b) correspond to collinear magnetic states. (c) and (d) are the two lowest energy noncollinear magnetic configurations. The ground state corresponds to ${\mathrm{AF}}_{-1}$.

Figure 10

Magnetic configurations considered to estimate the $J$ values for the Mn trimers. The GaN substrate is omitted in the image to simplify the visualization of the magnetic configurations.

Figure 11

Noncollinear magnetic configurations for two Mn trimers on top of the $\mathrm{GaN}\left(000\overline{1}\right)$ surface. The GaN substrate is omitted in the image to simplify the visualization of the magnetic configurations. The distance between trimers is 9.57 Å, as in the $3a\times 3a$ reconstruction reported in Ref. [18]. The ground state corresponds to configuration A.

Figure 12

Top panel: Top view of the supercell considered in the calculations. The two Mn adatoms are depicted with AF alignment and are located at H3 sites on top of the $\mathrm{GaN}\left(000\overline{1}\right)$ surface. Different calculations are carried out by progressively increasing the distance $d_{ij}$ between the Mn adatoms, following a zigzag trajectory (green line). For each distance, both the FM and the AF spin configurations are calculated. Bottom panel: Energy difference between the AF and FM configurations, obtained with ab initio calculations, for different separations $d_{ij}$ for the Mn adatom pair. The magnetic interaction presents a damped oscillatory behavior with $d_{ij}$. The smooth curving (blue) line is a fit to the calculated data points using Eq. (6).

Figure 13

Top panel: Fermi surface of the $1\times 1\text{−}\mathrm{GaN}\left(000\overline{1}\right)$ substrate. Bottom panel: Fragment of B2 Fermi edge (black line) and free electronlike band state behavior, centered at the M point (dashed red line). The anisotropic character of B2 can be clearly evidenced comparing these two surface states.