Ground State of Magnetic Dimers on Metal Surfaces

We present model studies of the ground state for magnetic dimers on metal surfaces. We find it can be neither ferromagnetic nor antiferromagnetic, but is often canted for nearest neighbors. Thus, the system cannot be described using bilinear exchange. We give a criterion which can be used quite generally to interrogate the local stability of ferromagnetically or antiferromagnetically aligned dimers, and which also may be used to infer the canting angle when canted states are stable.

Figure 1

For our simulation of a pair of Cr atoms on the Mo surface, we show the dependence of the interaction energy with the angle $\theta$ between the magnetic moments. In (a) we have the ions on nearest neighbor sites, and the ground state is canted. In (b)–(d) we show results for the case where the ions are on second, third, and fourth neighbor sites. Here the effective exchange coupling is well approximated by the simple bilinear exchange whose sign alternates with distance. The solid curves are our full calculations, and the dashed curve in (a) is a fit to the form $-J_1(\cos \theta -1)-J_2({cos}^2\theta -1)$. In the calculations, the occupation numbers are as described in the text; we have taken $t^{\prime }=1$, and we have direct hopping between the adsorbates of strength $t_d=1$, but only for nearest neighbors.

Figure 2

The energy as a function of the canting angle of a dimer pair on a substrate with 1.88 electrons per orbital. In (a) the ions in the dimer have $n_d=0.9$ electrons in each $d$ shell, and in (b) the ions have $n_d=1.3$ electrons. The solid curves are calculated for the case $t^{\prime }=t_d=1$, the dashed curves for $t^{\prime }=1.2$ and $t_d=1$, and the dot-dashed curves for $t^{\prime }=1$ and $t_d=1.2$.

Figure 3

The frequency spectrum of small amplitude spin fluctuations out of the ferromagnetic state for one of the ions in the dimer pair considered in Fig. 1. We illustrate the frequency spectrum through the calculation of $\mathrm{Im}\{{\chi }_{+,-}^{11}(\Omega )\}$, as discussed in the text. In (a) we have the spectrum of the nearest neighbor pair, in (b) that of next nearest neighbors, and in (c) and (d) that of third and fourth neighbors, respectively.