Magnetism of small Cr clusters: Interplay between structure, magnetic order, and electron correlations

The magnetic properties of small ${\text{Cr}}_N$ clusters $\left(N\le 6\right)$ are investigated in the framework of density-functional theory. The interplay between electron correlations, cluster structure, and magnetic order is quantified by performing fully spin-unrestricted calculations allowing for noncollinear spin arrangements within both the local-spin-density approximation (LSDA) and the generalized-gradient approximation (GGA). The possible transition or saddle-point states are identified by determining the vibrational frequencies from diagonalizing the dynamical matrix. In agreement with previous studies, a dimer-based growth pattern is found in all considered low-lying isomers with very short equilibrium bond lengths (typically $d_{eq}^{\text{GGA}}=1.55–1.65\text{Å}$) alternating with relative long ones (typically $d_{eq}^{\text{GGA}}=2.75–2.85\text{Å}$) in the relaxed geometries. Strong local magnetic moments ${\vec{\mu }}_i$ are, in general, obtained for the relaxed geometries (e.g., $\left|{\mu }_i^{\text{GGA}}\right|\simeq 2{\mu }_{\text{B}}$ in ${\text{Cr}}_4$), which show a collinear magnetic order with antiparallel (parallel) alignment of the ${\vec{\mu }}_i$ along the short (long) bonds. In contrast to the GGA, the LSDA yields vanishing magnetization density in some cases ($\left|{\mu }_i^{\text{LSDA}}\right|=0\forall i$ for $N=2$ and 4). Despite quantitative differences, both LSDA and GGA functionals always yield collinear ground-state solutions for the fully relaxed structures. In fact, noncollinear spin arrangements are found only for particular symmetric (nondimerized) geometries. However, the structures are not local minima and involve considerably large excitation energies. The results clearly indicate that the magnetic frustration, which one would physically expect in compact antiferromagnetic spin systems, is solved by dimerization rather than by noncollinearity of the local moments.

Figure 1

(Color online) Ground-state magnetization density distribution $\vec{m}\left(\vec{r}\right)$ inside the PAW spheres in ${\text{Cr}}_2$ as obtained by using (a) GGA and (b) LSDA to exchange and correlation functionals at their respective equilibrium distance.

Figure 2

(Color online) Local density of states ${\rho }_i\left(E\right)$ at one of the atoms of ${\text{Cr}}_2$, as well as orbital-projected density of states ${\rho }_i\left(E\right)$ for $\alpha =s$, $p$, and $d$. The results are obtained using (a) GGA and (b) LSDA.

Figure 3

(Color online) Illustration of the different isomers of ${\text{Cr}}_3$ found using the GGA and the LSDA. Results are given for the equilibrium bond lengths $d_e^{ij}$ (in $\text{Å}$) and for the energy difference $\Delta E$ (in eV) with respect to the optimal structure for each XC functional. The numbers label the different atoms, which local moments are given in Table .

Figure 4

(Color online) Illustration of the different isomers of ${\text{Cr}}_4$ found using the GGA and the LSDA. Results are given for the equilibrium bond lengths $d_e^{ij}$ (in $\text{Å}$) and for the energy difference $\Delta E$ (in eV) with respect to the optimal structure for each XC functional. The numbers label the different atoms, which local moments are given in Table .

Figure 5

(Color online) Illustration of the different isomers of ${\text{Cr}}_5$ found using the GGA and the LSDA. Results are given for the equilibrium bond lengths $d_e^{ij}$ (in $\text{Å}$) and for the energy difference $\Delta E$ (in eV) with respect to the optimal structure for each XC functional. The numbers label the different atoms, which local moments are given in Table .

Figure 6

(Color online) Illustration of the different isomers of ${\text{Cr}}_6$ found using the GGA and the LSDA. Results are given for the equilibrium bond lengths $d_e^{ij}$ (in $\text{Å}$) and for the energy difference $\Delta E$ (in eV) with respect to the optimal structure for each XC functional. The numbers label the different atoms, which local moments are given in Table .