Anisotropic surface-state-mediated RKKY interaction between adatoms

Motivated by recent numerical studies of Ag on Pt(111), we derive an expression for the RKKY interaction mediated by surface states, considering the effect of anisotropy in the Fermi edge. Our analysis is based on a stationary phase approximation. The main contribution to the interaction comes from electrons whose Fermi velocity v${}_F$ is parallel to the vector R connecting the interacting adatoms; we show that, in general, the corresponding Fermi wave vector k${}_F$ is not parallel to R. The interaction is oscillatory; the amplitude and wavelength of oscillations have angular dependence arising from the anisotropy of the surface-state band structure. The wavelength, in particular, is determined by the projection of this k${}_F$ (corresponding to v${}_F$) onto the direction of R. Our analysis is easily generalized to other systems. For Ag on Pt(111), our results indicate that the RKKY interaction between pairs of adatoms should be nearly isotropic and so cannot account for the anisotropy found in the studies motivating our work. However, for metals with surface-state dispersions similar to Be(10$\overline{1}$0), we show that the RKKY interaction should have considerable anisotropy.

Figure 1

Constant-energy curves [via Eq. (2)] for $E/\epsilon$ when $\theta =\pi /15$. The vertical dotted line points in the $k_y$ direction and is parallel to $\mathbf{R}$. The slanted dotted line shows the angle through which the constant-energy curves have been rotated. The black dashed contour is the boundary of the first Brillouin zone. The solid black curve connecting the origin to the $E/\epsilon =4$ contour intersects the point on each constant-energy contour for which $k_y$ is maximized, so that $d{k}_y/d{k}_x=0$ (which is the stationary phase condition for the $d{k}_x$ integration). The intersection of this curve and the Fermi edge marks the location of the surface states that dominate the RKKY interaction. The solid black curve was found by numerically solving Eqs. (2) and (11) as a function of $E/\epsilon$.

Figure 2

RKKY interaction $-\Delta /\left({\pi }^2\right|J_{{\epsilon }_F,{\epsilon }_F}{|}^2)$ [Eq. (9a)] on a (111) fcc surface as a function of $R$ (in units of $a=$ 0.289 nm for Ag and $a=$ 0.256 nm for Cu) at zero temperature when ${\epsilon }_F/\epsilon =3$. Values of $\theta$ are approximately the angles an adatom makes with its second, ninth, seventh, fourteenth, and first nearest neighbors, from bottom to top. All curves oscillate about zero, indicated for each curve by a dashed line; these lines are separated by intervals of $0.1$ along the vertical axis. Values of $k_s$ are in units of $a^{-1}$.

Figure 3

The function $\mathfrak{n}\left({\epsilon }_F\right)$ of the constant energy contour maxima for different angles [cf. Eqs. (5), (6), and (12)]. Note that for $\theta =0$ the amplitude diverges when ${\epsilon }_F/\epsilon \to 4$, so that Eq. (5) overestimates the number of surface states contributing to the interaction. For all other values of $\theta$, $\mathfrak{n}\left({\epsilon }_F\right)\to 0$ when ${\epsilon }_F/\epsilon \to 4$ since the maximum of the constant energy curves becomes infinitely sharp in this limit. The far-field regime in which Eq. (9a) is valid is given by $R/\mathfrak{n}\left({\epsilon }_F\right)\gg 1$.

Figure 4

Equations (2) and (13) as functions of $k$ for $\theta =0$ and $\theta ={30}^{\circ }$. For Eq. (13), we also magnify the leading-order anisotropy, $[1+\cos \left(6\theta \right)]{\left(a\right|\mathbf{k}\left|\right)}^6/3240$, by factors of $10,100,$ and 1000. For ${\epsilon }_F/\epsilon <0.5$ [corresponding to Ag(111)], the inset shows that even with a thousandfold enhancement, the leading-order anisotropy does not contribute significantly to the band structure in the vicinity of the Fermi energy. Consequently, our model predicts an isotropic RKKY interaction between adatoms on Ag(111).

Figure 5

The normalized interaction energies ${\mathcal{V}}_F\Delta /({\pi }^2|J_{{\epsilon }_F,{\epsilon }_F}{|}^2)$ for the elliptic Fermi edge of Be(10$\overline{1}$0) (shown, to scale, in the inset) drawn by choosing ${\breve{\mathbf{k}}}_F={\breve{\mathbf{k}}}_F^{{}_{\overline{A}}}$. Stars are Bravais lattice positions. The vector ${\mathbf{k}}_{\overline{A}}\equiv \frac{1}{2}{\mathbf{G}}_{\overline{\Gamma }\overline{A}\overline{\Gamma }}$ points from $\overline{\Gamma }$ to $\overline{A}$. In the inset, we show two choices of the spanning vector ${\breve{\mathbf{k}}}_F$, viz., ${\breve{\mathbf{k}}}_F^{{}_{\overline{A}}}$ and ${\breve{\mathbf{k}}}_F^{{}_{\overline{\Gamma }}}$, and the corresponding separation vector $\mathbf{R}$. Note that ${\breve{\mathbf{k}}}_F^{{}_{\overline{A}}}={\breve{\mathbf{k}}}_F^{{}_{\overline{\Gamma }}}-{\mathbf{k}}_{\overline{A}}$. In the main figure, we omit a circular region whose radius is roughly 1 nm since (i) the far-field approximation is not valid near the origin and (ii) the interaction energy diverges as $R\to 0$.