Numerical renormalization group calculations of ground-state energy: Application to correlation effects in the adsorption of magnetic impurities on metal surfaces

The ground-state energy of a quantum impurity model can be calculated using the numerical renormalization group with a modified discretization scheme, with sufficient accuracy to reliably extract physical information about the system. The approach is applied to study binding of magnetic adsorbates modeled by the Anderson-Newns model for chemisorption on metal surfaces. The correlation energy is largest in the valence-fluctuation regime; in the strong-coupling (Kondo) regime the Kondo-singlet formation energy is found to be only a minor contribution. As an application of the method to more difficult surface-science problems, we study the binding energy of a magnetic atom adsorbed near a step edge on a surface with a strongly modulated surface-state electron density. The zero-temperature magnetic susceptibility is determined from the field dependence of the binding energy, thereby providing an independent result for the Kondo temperature $T_K$, which agrees very well with the $T_K$ extracted from a thermodynamic calculation.

Figure 1

(Color online) $\Lambda$ dependence of the calculated binding energy of a noninteracting adsorbate. Exact binding energy $\Delta E_{\text{exact}}$ is subtracted from the numerical results $\Delta E_{\text{NRG}}\left(\Lambda \right)$ (circles). Full line is a linear fit to results in the interval $\Lambda ∊\left[1.6:2\right]$. The error of the extrapolated $\Lambda \to 1$ value is $3.3\times {10}^{-9}$. The standard deviation ${\sigma }_{\Delta E}$ characterizes the spread of the results for different parameters $z$. An example of $\Delta E_{\text{NRG}}\left(z\right)$ for $\Lambda =12$ is shown in the inset. $N_z=32$ different values of $z$ were used, while the parameter $E_{\text{cutoff}}=10{\omega }_N$ defines the truncation cutoff in the NRG iteration (Ref. 14).

Figure 2

(Color online) Binding energy $\Delta E$ (right vertical axis) and numerical error $\Delta E_{\text{NRG}}-\Delta E_{\text{exact}}$ (left vertical axis) of a noninteracting adsorbate as a function of the hybridization $\Gamma$. The proportionality coefficient $\Delta E/\Gamma =-0.70$ is extracted in the interval $\Gamma ∊\left[0:0.01\right]$. For reference, the inset shows $\Delta E/\Gamma$ as a function of $ϵ$.

Figure 3

(Color online) (a) Binding energy, (b) correlation energy $E_c=\Delta E-\Delta E_{\text{HF}}$, Kondo temperature $T_K$ (right vertical axis), and (c) charge fluctuations of the single-impurity Anderson model as a function of the electron-electron interaction $U$. The Kondo temperature is extracted from the thermodynamic properties of the model according to Wilson’s prescription $k_B{T}_K{\chi }_{\text{imp}}\left(k_B{T}_K\right)/{\left(g{\mu }_B\right)}^2=0.07$, where ${\chi }_{\text{imp}}$ is the impurity magnetic susceptibility (Ref. 9).

Figure 4

(Color online) Binding energy and spin polarization of a magnetic adsorbate in an external magnetic field (expressed in reduced units of $x=g{\mu }_{B}B/k_B{T}_K$). The inset shows the spin-resolved impurity-spectral function in a strong field.

Figure 5

(Color online) $\Lambda$ dependence of the binding energy of a magnetic adsorbate hybridized to a band with energy-dependent density of states $\rho \left(\omega \right)$. The hybridization function is $\Gamma \left(\omega \right)={\Gamma }_0\rho \left(\omega \right)$ with (a) sharp-step function $\rho \left(\omega \right)=1+\theta \left(\omega -{\omega }_0\right)$ with ${\omega }_0=-0.1$, (b) rounded-step function $\rho \left(\omega \right)=1+\left\{1/2+\left(1/\pi \right){\text{tan}}^{-1}\left[\pi \left(\omega -{\omega }_0\right)/\Delta \right]\right\}$, where $\Delta =0.001$, and (c) oscillatory $\rho \left(\omega \right)=1+\left(1/2\right)\text{cos}\left[\left(9/2\right)\pi \left(1+\omega \right)\right]$. The $\Lambda =1.6$ results are used as reference values and subtracted from $\Delta E\left(\Lambda \right)$.

Figure 6

(Color online) Properties of a magnetic adsorbate on a surface at a distance of $x$ from a step edge modeled as a hard-wall potential scatterer. Fitting by oscillatory power-law functions $A+B\text{cos}\left(2k_{F}x+\delta \right)/x^{\alpha }$ provides the phase shifts $\delta$ (shown in the figure) and decay constants $\alpha$: the binding energy $\Delta E$ decays as $1/x^{3/2}$, the occupancy as $1/x^{1.19}$, and the Kondo temperature as $1/x^{1.16}$. The fitting procedure was performed with the data in the asymptotic region $x∊\left[40:80\right]\text{Å}$. The inset shows the hybridization function at $x=16\text{Å}$. ${\omega }_0=-0.039$ and $k_F=0.217{\text{Å}}^{-1}$.