Finite temperature magnetic properties of small Fe chains and clusters on Pt(111)

The magnetic properties of Fe chains and clusters on Pt(111) are investigated in the framework of a functional-integral theory of itinerant magnetism. The considered nanostructures show a ferromagnetic (FM) ground state with nearly saturated Fe local magnetic moments ${\mu }_{\text{Fe}}^0\simeq 3.15{\mu }_{\text{B}}$. In addition, small moments ${\mu }_{\text{Pt}}^0\simeq 0.1–0.3{\mu }_{\text{B}}$ are induced at the Pt substrate, which depend sensitively on the number of Fe atoms in their nearest-neighbor (NN) shell. The spin-fluctuation (SF) energies $\mathrm{\Delta }{F}_l\left(\xi \right)$ at the different atoms $l$ are calculated as a function of the local exchange fields ${\xi }_l$, by using a real-space recursive expansion of the local Green's functions. Results for the temperature dependence of the average magnetization per atom ${\overline{\mu }}_N$, local magnetic moments ${\mu }_l$, and spin correlation functions ${\gamma }_{lk}$ are derived. At the Fe atoms the dominant magnetic excitations are fluctuations of the local-moment orientations. The spin-flip energies $\mathrm{\Delta }{F}_l\left(\xi \right)$ in the deposited Fe clusters are found to be about $50\%$ smaller than in free-standing clusters of comparable size. This results in flatter SF-energy landscapes and in a weaker stability of the FM order at $T>0$. The effective exchange interactions between the Fe local moments, which are derived from the electronic calculations, reveal competing FM and antiferromagnetic couplings at different distances. In contrast to Fe, the main spin excitations at the Pt atoms are fluctuations of the size of the induced local magnetic moments. The interplay between the different types of spin excitations and their effect on the temperature-dependent magnetic properties is discussed.

Figure 1

Low-temperature limit of the local spin-fluctuation energy $\mathrm{\Delta }{F}_l^{\prime }\left(\xi \right)$ as a function of the exchange field (XF) $\xi$ at the central atom of ${\mathrm{Fe}}_N$ linear chains deposited on Pt(111). All other XFs remain fixed to their corresponding ground-state values [see Eq. (24)]. The dashed curve shows $\mathrm{\Delta }{F}_l^{\prime }\left(\xi \right)$ at the central atom of a free-standing ${\mathrm{Fe}}_7$ linear chain.

Figure 2

Effective exchange coupling constants $J_{0m}$ in ${\mathrm{Fe}}_N$ chains on Pt(111) between the central atom $l=0$ and its $m\mathrm{th}$ NN. For comparison, results are also given for the free-standing ${\mathrm{Fe}}_7$ chain (dashed line). The lines connecting the points are a guide to the eye.

Figure 3

Illustration of the considered ${\mathrm{Fe}}_N$ clusters on Pt(111). Dark (light) spheres indicate Fe (Pt) atoms. The numbers label the different sites $l$.

Figure 4

Local SF energy $\mathrm{\Delta }{F}_l^{\prime }\left(\xi \right)=F_l^{\prime }\left(\xi \right)-F_l^{\prime }\left({\xi }^0\right)$ at the different Fe and Pt atoms $l$ of ${\mathrm{Fe}}_N$ clusters deposited on Pt(111): (a) triangle, (b) rhombus, and (c) tetrahedron. The cluster structures together with the atom labels are illustrated in Fig. 3. The dashed curves correspond to free-standing ${\mathrm{Fe}}_N$ clusters having the same structure and bond lengths as the deposited ones.

Figure 5

Local SF energy $\mathrm{\Delta }{F}_l^{\prime }\left(\xi \right)$ at the different Fe and Pt atoms $l$ of (a) a hexagonal ${\mathrm{Fe}}_7$ cluster and (b) the Fe (ML) deposited on Pt(111) (see Fig. 3). Results for the corresponding free-standing configurations are shown by the dashed curves.

Figure 6

Spin-fluctuation energy landscape of linear ${\mathrm{Fe}}_3$ deposited on Pt(111). In (a) $\mathrm{\Delta }{F}^{\prime }({\xi }_1,{\xi }_2)=F^{\prime }({\xi }_1,{\xi }_2,{\xi }_3^0)-F^{\prime }({\xi }_1^0,{\xi }_2^0,{\xi }_3^0)$ is given as a function of the XFs ${\xi }_1$ and ${\xi }_2$ at the central atom and at an edge atom [see Fig. 3]. In (b) the corresponding contour plot is shown. The XFs at the other edge Fe atom ($l=3$) and at the Pt substrate are kept equal to their ground-state values.

Figure 7

Temperature dependence of the average magnetic moment per atom ${\overline{\mu }}_N$, given by Eq. (27), for the triangular ${\mathrm{Fe}}_3$ (triangles), linear ${\mathrm{Fe}}_3$ (circles), and rhombohedral ${\mathrm{Fe}}_4$ (squares) on Pt(111) (see Fig. 3). The XFs $\xi$ at the Pt atoms are either fixed to the ground-state values (open symbols) or computed self-consistently (full symbols) for each XF configuration of the Fe cluster.

Figure 8

Temperature dependence of the average magnetization per atom ${\overline{m}}_N$, given by Eq. (23), for triangular ${\mathrm{Fe}}_3$ (triangles), linear ${\mathrm{Fe}}_3$ (circles), and rhombohedral ${\mathrm{Fe}}_4$ (squares) on Pt(111). The XFs at the Pt atoms are either fixed to the ground-state values (open symbols) or calculated self-consistently for each XF configuration of the Fe cluster (full symbols).

Figure 9

Temperature dependence of the local magnetic moments ${\mu }_l=\sqrt{{\gamma }_{ll}}$ at different atoms $l$ of ${\mathrm{Fe}}_N$ clusters deposited on Pt(111): (a) triangle and chain, and (b) rhombus. The XFs at the Pt atoms are calculated self-consistently (SC) for each XF configuration of the Fe cluster. Assuming fixed ground-state exchange splittings ${\xi }_l^0$ at the substrate atoms yields almost the same results.

Figure 10

Temperature dependence of the pair correlation functions ${\gamma }_{lk}$ between different atoms $l$ and $k$ of ${\mathrm{Fe}}_N$ clusters deposited on Pt(111): (a) triangle and chain, and (b) rhombus. The cluster structures and atom labeling are shown in the inset of Fig. 9. As in Fig. 9 the results are obtained by treating the XFs at the Pt substrate atoms self-consistently. The NSC approach yields very similar results.