Electronic spin-fluctuation theory of finite-temperature cluster magnetism: Size and environment dependence in ${\text{Fe}}_N$

The temperature dependence of the magnetic properties of ${\text{Fe}}_N$ clusters $\left(N\le 24\right)$ are determined in the framework of a functional-integral itinerant-electron theory. For each exchange-field configuration $\vec{\xi }$, the electronic structure is calculated by using a realistic $d$-band Hamiltonian and a real-space recursive expansion of the local Green’s functions. The statistical averages over all $\vec{\xi }$ are performed by implementing a parallel tempering exchange Monte Carlo method. Results are given for the average magnetic moment per atom ${\overline{\mu }}_N$, local magnetic moments ${\mu }_l$ at different atoms $l$ within the cluster, and interatomic spin-correlation functions ${\gamma }_{lk}$ as a function of temperature $T$. A remarkable dependence of ${\overline{\mu }}_N\left(T\right)$ on size and structure is observed that reflects the importance of the electronic structure to the cluster spin excitations. The correlation between local atomic environment and finite $T$ magnetism is analyzed in some detail by means of the spin-correlation functions. The role of bond-length relaxations on the temperature dependent properties is quantified. An interpretation of our electronic results in terms of Ising or Heisenberg models of localized magnetism reveals a strong dependence of the effective interatomic exchange couplings $J_{lk}$ on size and local coordination number, which defies straightforward transferability and easy generalizations.

Figure 1

(Color online) Autocorrelation function $q\left(\tau ,T_m\right)$ as a function of the Monte Carlo step $\tau$ for different temperatures $T_m$. Results are given for (a) ${\text{Fe}}_6$ and (b) ${\text{Fe}}_{15}$ clusters and for representative simulations temperatures.

Figure 2

Illustration of the considered cluster structures of ${\text{Fe}}_N$ clusters $\left(N\le 24\right)$. The numbers label the different atomic sites $l$.

Figure 3

Temperature dependence of the average magnetic moment per atom ${\overline{\mu }}_N\left(T\right)$ of ${\text{Fe}}_N$ clusters [see Eqs. (19, 20)]. The labels (a)–(i) refer to the structures shown in Fig. 2.

Figure 4

Temperature dependence of the average magnetization per atom ${\overline{m}}_N$ of ${\text{Fe}}_N$ clusters [see Eq. (18)]. The labels (a)–(g) refer to the structures shown in Fig. 2.

Figure 5

(Color online) Local magnetic moments ${\mu }_l$ and pair-correlation functions ${\gamma }_{lk}$ in small ${\text{Fe}}_N$ clusters having $N\le 4$ atoms [see Eqs. (22, 23)]. The numbers refer to the different atoms $l$ or pairs of atoms $lk$ as labeled in the inset figure (or in Fig. 2).

Figure 6

(Color online) Local magnetic moments ${\mu }_l$ and pair-correlation functions ${\gamma }_{lk}$ in ${\text{Fe}}_N$ clusters having $5\le N\le 15$ atoms [see Eqs. (22, 23)]. The numbers refer to the different atoms $l$ or pairs of atoms $lk$ as labeled in the inset figure or Fig. 2.

Figure 7

Bond-length dependence of the magnetization curves ${\overline{\mu }}_N\left(T\right)$ of Fe clusters [see Eqs. (19, 20)]. Results are given for ${\text{Fe}}_5$ (triangular bipyramid), ${\text{Fe}}_6$ (square bipyramid) and ${\text{Fe}}_{15}$ (bcc structure). The numbers indicate the ratio $r/r_b$ between the NN distance $r$ in the cluster and the bulk value $r_b$.