Multiple magnetic impurities on surfaces: Scattering and quasiparticle interference

We study systems of multiple interacting quantum impurities deposited on a metallic surface in a three-dimensional host. For the real-space two-impurity problem, using numerical renormalization group calculations, a rich range of behavior is shown to arise due to the interplay between Kondo physics and effective Ruderman-Kittel-Kasuya-Yosida interactions—provided the impurity separation is small. Such calculations allow identification of the minimum impurity separation required for a description in terms of independent impurities, and thereby the onset of the “dilute-impurity limit” in many-impurity systems. A “dilute-cluster” limit is also identified in systems with higher impurity density, where interimpurity interactions are important only within independent clusters. We calculate the quasiparticle interference due to two and many impurities, and explore the consequences of the independent impurity and cluster paradigms. Our results provide a framework to investigate the effects of disorder due to interacting impurities at experimentally relevant surface coverages.

Figure 1

Two equivalent magnetic impurities on the 3D cubic (100) surface with $U/{\mathrm{\Gamma }}_0=28$ (see Table 1). NRG results for the local impurity spectrum $-\text{Im}\left[t{G}_d^{11}\left(\omega \right)\right]$ vs $\omega /t$ at $T=0$ (upper panels) and the temperature dependence of the impurity contribution to the entropy $S_{\text{imp}}\left(T\right)$ vs $T/t$ for the same systems (lower panels). Interimpurity vector $\mathbf{R}$, with $R_y=0$ (left panels) and $R_y=R_x$ (right), where $R_x/a_0=1,2,3,4$ for black, red, blue, and green lines. Circles show the single-impurity result.

Figure 2

Evolution of impurity spectra $-{\mathrm{\Gamma }}_0\text{Im}{G}_d^{11}\left(\omega \right)$ with decreasing $U/{\mathrm{\Gamma }}_0$, obtained by NRG. Plotted at $T=0$ vs $\omega /t$ for $R_x=R_y=1a_0,2a_0,3a_0$, and $4a_0$ as the black, red, blue, and green lines. Circles are the corresponding single-impurity results.

Figure 3

QPI for two equivalent magnetic impurities on the 3D cubic (100) surface, separated by vectors $\mathbf{R}/a_0=(0,1),(0,2)$, and $(1,1)$, as shown in the left, center, and right column panels, respectively. The full QPI $\left|\mathrm{\Delta }\rho \right(\mathbf{q},\omega \left)\right|$ is shown in the top row panels, while the local and nonlocal contributions, obtained from Eqs. (22) and (23), are shown in the center and bottom row panels. Obtained for $U/{\mathrm{\Gamma }}_0=28$ as per Fig. 1, with scanning energy $\omega ={10}^{-6}t\equiv 0.1T_K^{1\text{imp}}$.

Figure 4

Average fraction of impurity clusters of size $N_c=1,2,3$, and $\ge 4$, vs impurity coverage on a cubic lattice surface.

Figure 5

Independent magnetic impurities deposited randomly on a 3D cubic lattice surface. Upper panels: real-space LDOS maps. Lower panels: corresponding QPI. Left panels are for $N=100$ impurities within a $500\times 500$ sample region, while right panels show a $100\times 100$ sub-block containing $N=5$ impurities. Calculated for $U/{\mathrm{\Gamma }}_0=28$ and scanning energy $\omega ={10}^{-6}t\equiv 0.1T_K^{1\text{imp}}$.

Figure 6

Left panel: Block-averaged and symmetrized QPI, obtained by combining $M=25$ sub-blocks of size $100\times 100$ from the original $500\times 500$ sample shown in the left panels of Fig. 5. The right panel shows the pristine single-impurity result for comparison.

Figure 7

$N=100$ impurities within a $500\times 500$ surface region of a 3D cubic lattice. Impurity distribution and parameters as in Fig. 5. Left panel: Spatially resolved LDOS difference $|\mathrm{\Delta }\rho \left(\mathbf{r}\right)-\mathrm{\Delta }{\rho }_{\text{dil}}\left(\mathbf{r}\right)|$, where $\mathrm{\Delta }\rho \left(\mathbf{r}\right)$ is calculated within the independent cluster approximation (treating interimpurity interactions in the two $N_c=2$ clusters exactly), while $\mathrm{\Delta }{\rho }_{\text{dil}}\left(\mathbf{r}\right)$ is calculated assuming the dilute limit within an independent impurity picture. Right panel: Corresponding QPI difference $|\mathrm{\Delta }\rho \left(\mathbf{q}\right)-\mathrm{\Delta }{\rho }_{\text{dil}}\left(\mathbf{q}\right)|$.