Noncollinear magnetism in transition metal nanostructures: Exchange interaction and local environment effects in free and deposited clusters

A self-consistent tight-binding theory of noncollinear magnetism in transition metal nanostructures is presented and applied to free and deposited clusters. The electronic structure is calculated by using a realistic rotational invariant tight-binding Hamiltonian and a real-space recursive expansion of the local Green’s functions. For free clusters with sizes $N⩽13$, results are given for the ground-state local magnetic moments, magnetic order, and average magnetic moments as a function of the Coulomb exchange integral $J$ and $d$-band filling. A variety of qualitatively different noncollinear spin arrangement solutions are obtained, which reveals the complex magnetic landscape in transition metal nanostructures. The onset of noncollinear magnetism is discussed by analyzing the stability of the various magnetic solutions. Structural effects are studied by considering representative compact and open geometries. The calculations are compared with available density-functional results. Substrate effects on the noncollinearity of the moments are discussed by considering small Fe clusters deposited on Pt(111). Extensions and limitations of our work are also pointed out.

Figure 1

Illustration of representative collinear and noncollinear magnetic configurations of a three-atom cluster having a triangular structure. The numbers label the atoms within the cluster. Results are given for $J/J_b=1$ and for several values of the $d$-band filling $n_d$.

Figure 2

Cluster average magnetic moment ${\mathrm{\mu ̄}}_N$ and difference of electronic energies between the noncollinear (NC) and collinear (C) magnetic configurations $\Delta E=E_{\mathrm{NC}}-E_{\mathrm{C}}$ as a function of the interaction parameter $J$ for a three-atom cluster (see Fig. 1 for the cluster structure). $J_b$ refers to the Fe bulk value. Results are given for several values of the $d$-band filling $n_d$: (a) and (d) for $n_d=7.0$, (b) and (e) for $n_d=6.0$, and (c) and (f) for $n_d=5.0$.

Figure 3

Illustration of representative collinear and noncollinear magnetic configurations of a five-atom cluster having a triangular bipyramid structure. The numbers label the atoms within the cluster. Results are given for $J/J_b=1$ and for several values of the $d$-band filling $n_d$.

Figure 4

Cluster average magnetic moment ${\mathrm{\mu ̄}}_N$ and difference of electronic energies between the noncollinear (NC) and collinear (C) magnetic configurations $\Delta E=E_{\mathrm{NC}}-E_{\mathrm{C}}$ as a function of the interaction parameter $J$ for a five-atom cluster (see Fig. 3 for the cluster structure). $J_b$ refers to the Fe bulk value. Results are given for several values of the $d$-band filling $n_d$: (a) and (d) for $n_d=7.0$, (b) and (e) for $n_d=6.0$, and (c) and (f) for $n_d=5.0$.

Figure 5

Band-filling dependence of the magnetic order in Fe${}_5$. The ground-state magnetic configurations are illustrated. The size of the arrows is proportional to the modulus of the local magnetic moments. The curve corresponds to the difference of electronic energies between the noncollinear (NC) and collinear (C) magnetic configurations $\Delta E=E_{\mathrm{NC}}-E_{\mathrm{C}}$. The average magnetic moment ${\mathrm{\mu ̄}}_N$ as a function of $n_d$ is shown in the inset. Here, we take the value of $J/J_b$ that corresponds to Fe.

Figure 6

Illustration of representative collinear and noncollinear magnetic configurations of a 13-atom cluster having an icosahedral structure. Results are given for $J/J_b=1$ and several values of the $d$-band filling $n_d$.

Figure 7

Cluster average magnetic moment $\overline{{\mu }_N}$ and difference of electronic energies between the noncollinear (NC) and collinear (C) magnetic configurations $\Delta E=E_{\mathrm{NC}}-E_{\mathrm{C}}$ as a function of the interaction parameter $J$ for a icosahedral 13-atom cluster (see Fig. 6 for the cluster structure). $J_b$ refers to the Fe bulk value. Results are given for several values of the $d$-band filling $n_d$: (a) and (d) for $n_d=7.0$, (b) and (e) for $n_d=6.0$, and (c) and (f) for $n_d=5.0$.

Figure 8

Illustration of representative collinear and noncollinear magnetic configurations of a 13-atom cluster having a fcc-like structure. Results are given for $J/J_b=1$ and for several values of the $d$-band filling $n_d$.

Figure 9

Cluster average magnetic moment $\overline{{\mu }_N}$ and difference of electronic energies between the noncollinear (NC) and collinear (C) magnetic configurations $\Delta E=E_{\mathrm{NC}}-E_{\mathrm{C}}$ as a function of the interaction parameter $J$ for a fcc-like 13-atom cluster (see Fig. 8 for the cluster structure). $J_b$ refers to the Fe bulk value. Results are given for several values of the $d$-band filling $n_d$: (a) and (d) for $n_d=7.0$, (b) and (e) for $n_d=6.0$, and (c) and (f) for $n_d=5.0$.

Figure 10

Illustration of representative collinear and noncollinear magnetic configurations of a 13-atom cluster having a bcc-like structure. Results are given for several values of the $d$-band filling $n_d$.

Figure 11

Cluster average magnetic moment ${\mathrm{\mu ̄}}_N$ and difference of electronic energies between the noncollinear (NC) and collinear (C) magnetic configurations $\Delta E=E_{\mathrm{NC}}-E_{\mathrm{C}}$ as a function of the interaction parameter $J$ for a bcc 13-atom cluster (see Fig. 10 for the cluster structure). $J_b$ refers to the Fe bulk value. Results are given for several values of the $d$-band filling $n_d$: (a) for $n_d=7.0$, (b) and (d) for $n_d=6.0$, and (c) and (e) for $n_d=5.0$.

Figure 12

Illustration of the magnetic solutions for Cr${}_{13}$ obtained within the self-consistent tight-binding approach by using the optimal structure given by DFT calculations (see text). The numbers in (a) refer to the equilibrium interatomic distances. The average magnetic moment ${\mathrm{\mu ̄}}_{13}$ and excitation energy $\Delta E$ are indicated for all the magnetic isomers.

Figure 13

(a) Illustration of the structure considered in the calculations on Fe${}_3$ clusters deposited on Pt(111). Open (filled) circles refer to Pt (Fe) atoms. The numbers label the different atomic sites $i$. (b) Magnetic-moments configuration (including spin and orbital contributions). Notice that the magnitude of the Pt magnetic moments is multiplied by a factor 10 for the sake of clarity.