Systematic study of the physical origin of ferromagnetism in ${\mathrm{CeO}}_{2-\delta }$ nanoparticles

We have carried out a Schrieffer-Wolff transformation on a general tight-binding Hamiltonian and obtained a $4f$-one-band effective Hubbard Hamiltonian to study the physical origin of ferromagnetism in ${\mathrm{CeO}}_{2-\delta }$ nanoparticle systems. For a low temperature regime and low concentrations of oxygen vacancies, isolated vacancies have previously been showed to form on the $\left\{100\right\}$ and $\left\{110\right\}$ surfaces and our studies indicate these will be in singlet and triplet states, respectively. This is sustained by a superexchange interaction between the $4f$ electrons of the two cerium atoms, which are the nearest neighbors of the vacancy, and ferromagnetism and antiferromagnetism can coexist. Moreover, increasing the vacancy concentration we found that pairs of vacancies, which have been previously shown to form on the $\left\{111\right\}$ surfaces, produce Nagaoka ferromagnetism and isolated vacancies in the bulk produce an antiferromagnetic sign. Furthermore, further oxygen vacancy increases are previously known to favor the formation of oxygen vacancy clusters. In this case, our results showed a weakening of the magnetic correlations with respect to temperature. Thus, at a fixed temperature, the magnetic moment is reduced when the concentration of vacancies is increased, which is in agreement with experimental results reported in the literature. Interestingly, at a room-temperature regime, the antiferromagnetic order is destroyed and only the ferromagnetic couplings, produced mainly by isolated vacancies on the $\left\{110\right\}$ surfaces, survive. Finally, as temperature is increased further, the paramagnetic behavior of $4f$ electrons dominates.

Figure 1

(a) Unit cell of ceria (${\mathrm{CeO}}_2$), where the black balls represent cerium atoms and the white balls represent oxygen atoms. Each oxygen atom is bound to four cerium atoms forming a tetrahedron and each cerium atom is bound to eight oxygen atoms forming a cube. (b) The (100) facet, (c) the (110) facet, (d) the (111) facet, and (e) a tetrahedron present in bulk ceria. The second figure shows the same structures but with one or two oxygen vacancies, where the cerium atoms which are the nearest neighbors of the vacancy are marked with a star. The third figure show the system considered in this work. The spin configurations indicated are only to illustrate out the number of electrons. Note that the hopping integral between occupied sites is denoted by $t^{{}^{\prime }}$ and that there are still two oxygen atoms between the four and five sites of the second system in (c) and (d) (see text). The difference between the systems presented in (c) and (d) is that in (c), some sites of this system are in the subsurface layer.

Figure 2

Minimal energy ($E_{\min }$) for each possible total spin ($S_{\mathrm{tot}}$) of the system as a function of $U/t$, for $t^{{}^{\prime }}/t=1.0$ and 1.1. (a) dimer and triangle with two electrons, as represented in Fig. 1 and in upper 1 or upper 1, respectively. (b) the system is a tetrahedron with two electrons, as represented in Fig. 1.

Figure 3

(a) Minimal energy ($E_{\min }$) for each possible total spin ($S_{\mathrm{tot}}$) of the system with $t^{\prime }=1.1t$ and $S_{\mathrm{tot}}$ of the ground state (energy $E_0$) as a function of $U/t$ for both $t^{\prime }=t$ and $1.1t$. The structures are depicted in figures, where the balls represent cerium atoms and the squares represent oxygen vacancies [see Figs. 1 and 1]. In (a), (b), and (d), $N_e=N_s-1$, and in (c), $N_e=N_s$.