Kondo resonance in the conductance of CoPc/Au(111) and TBrPP-Co/Cu(111)

Recent scanning tunnel microscopy experiments on transport through CoPc and TBrPP-Co molecules adsorbed on metallic surfaces have produced several interesting results: (i) a Kondo temperature much higher than that typically observed on undressed magnetic atoms adsorbed on metal surfaces, (ii) a Kondo resonance that shows up either as a Kondo peak (CoPc) or as a Fano dip (TBrPP-Co), (iii) in the CoPc/Au(111) system, the Kondo resonance shows up once the molecule has been distorted by cutting out eight peripheral hydrogens, and, (iv) in TBrPP-Co/Cu(111) the Kondo temperature depends strongly on the molecule conformation (either planar or saddle) and on the number of molecules around and close to the one through which the current flows. The aim of this work is to discuss these experiments within the framework of a simple model recently proposed by the authors which, capturing the main features of the inner structure of the molecule (four lobe orbitals plus a strongly correlated orbital), offers a physical interpretation for most of these observations.

Figure 1

(Color online) Cluster of atoms utilized to describe TBrPP-Co or CoPc adsorbed onto a metallic surface. The four sites on the square account for the four molecule lobes, while the atom at the center represents Co. The upper (lower) atom accounts for the apex of the tip of the STM microscope (metallic surface). To calculate the conductance a constant self-energy was attached to the top atom, the metallic atom, and the lobes (see text).

Figure 2

(Color online) Transmission $T\left(E\right)$ (in units of the conductance quantum) versus the energy $E$ (in eV) referred to the Fermi energy, calculated by means of the SB method (see text). The parameters used in the calculations are the following: $U=1.6\text{eV}$ (upper panels) and $U=2.3\text{eV}$ (lower panels), $t_{l,t}=0.06\text{eV}$ (left panels) and $t_{l,t}=0$ (right panels), $t_{l,l}=0.0\text{eV}$ (black broken line), $t_{l,l}=0.05\text{eV}$ (blue chain line), and $t_{l,l}=0.2\text{eV}$ (continuous red line). The rest of the model parameters are $t_{\text{Co},m}=0.0\text{eV}$, $t_{\text{Co},l}=-0.14\text{eV}$, $t_{\text{Co},t}=0.08\text{eV}$, $t_{l,m}=0.14\text{eV}$, and ${\rho }_m=5{\text{eV}}^{-1}$.

Figure 3

(Color online) Transmission $T$ (in units of the conductance quantum) versus energy $E$ (in eV) for the Co uncoupled from the gold surface, i.e., $t_{\text{Co},m}=0$. The rest of hoppings are as follows: $t_{\text{Co},t}=0.5\text{eV}$, $t_{\text{Co},l}=1.25\text{eV}$, $t_{l,m}=1\text{eV}$, and, $t_{l,l}=2\text{eV}$ (red) $t_{l,l}=-2\text{eV}$ (black). The energy is referred to the Fermi energy.

Figure 4

(Color online) Transmission $T\left(E\right)$ (in units of the conductance quantum) versus the energy $E$ (in eV) referred to the Fermi energy, calculated by means of the SB method (see text). The parameters varied in the calculations are $t_{l,l}=0.2\text{eV}$ (continuous red lines), $t_{l,l}=-0.2\text{eV}$ (broken red lines), and $t_{\text{Co},m}=0.0\text{eV}$ (a) and $t_{\text{Co},m}=0.08\text{eV}$ (b). The rest of the model parameters are $U=2.3\text{eV}$, $t_{l,t}=0.06\text{eV}$, $t_{\text{Co},l}=-0.14\text{eV}$, $t_{\text{Co},t}=0.08\text{eV}$, $t_{l,m}=0.14\text{eV}$, and ${\rho }_m=5{\text{eV}}^{-1}$.

Figure 5

(Color online) Transmission $T$ (in units of the conductance quantum) versus the energy $E$ (in eV) referred to the Fermi energy (A,B), phase difference $\Phi$ between the direct path from the gold surface to the STM tip going through the Co atom, and the path that passes through the molecule lobes (c) (see text) and charge into the molecule and spin-spin correlations between Co-lobes and Co-tip versus the lobe-lobe hopping (d). A: $t_{l,m}=0$, $t_{\text{Co},t}=t_{\text{Co},m}=0.25\text{eV}$ and $t_{\text{Co},l}=1.25\text{eV}$; $t_{l,l}=0$ (green continuous line), and $t_{l,l}=2\text{eV}$ (red dashed line). B: $t_{\text{Co},m}=0$, $t_{\text{Co},t}=0.25\text{eV}$, $t_{l,m}=1\text{eV}$, and, $t_{\text{Co},l}=1.25\text{eV}$; $t_{l,l}=0$ (green), and $t_{l,l}=2\text{eV}$ (red). The standard Kondo resonance, obtained with $t_{\text{Co},l}=0$ and $t_{\text{Co},t}=t_{\text{Co},m}=0.25\text{eV}$ (black), is plotted in both (A and B). C: phase difference calculated for the parameters used in A. D: spin-spin correlation for Co/lobes (continuous line) and Co/STM tip (broken line) and total charge on the lobes plus Co (continuous blue line) versus $t_{l,l}$, for the parameters used in A. In all cases $U=8\text{eV}$.

Figure 6

(Color online) Upper panels: transmission $T\left(E\right)$ (in units of the conductance quantum) versus the energy $E$ (in eV) referred to the Fermi energy, calculated by means of the SB (a) and the ECA (b) methods (see text). Results for $t_{l,t}=0$ (violet chain line) $t_{l,t}=0.08\text{eV}$ (continuous red line) are shown. (c) Nondiagonal elements of the density matrix versus $t_{l,t}$ as calculated by means of the SB approach. (d) Spin-spin correlation versus $t_{l,t}$ calculated by means of the ECA method. The rest of the parameters used in the calculations are $t_{\text{Co},t}=0.08\text{eV}$, $t_{\text{Co},l}=-0.14\text{eV}$, $t_{\text{Co},m}=0$, $t_{l,l}=-0.2\text{eV}$, $t_{l,m}=0.14\text{eV}$, $U=1.6\text{eV}$, and ${\rho }_m=5{\text{eV}}^{-1}$.

Figure 7

(Color online) Transmission $T\left(E\right)$ (a) and differential conductance $dI/dV$ (b) in units of the conductance quantum, versus either the energy $E$ referred to the Fermi energy or the bias potential $V_{\text{bias}}$ (in eV) as calculated by means of the SB method. Results for three values of the metal density of states ${\rho }_m$ are shown. The rest of the model parameters are $t_{\text{Co},m}=0.04\text{eV}$, $t_{\text{Co},l}=-0.14\text{eV}$, $t_{\text{Co},t}=0.04\text{eV}$, $t_{l,l}=-0.2\text{eV}$, $t_{l,m}=0.14\text{eV}$, $t_{l,t}=0.03\text{eV}$, and $U=1.6\text{eV}$. The dotted line in (a) depicts transmission results obtained with ${\rho }_m=6.25{\text{eV}}^{-1}$, $t_{l,t}=0.006\text{eV}$, and $t_{\text{Co},t}=0.008\text{eV}$ (the rest of the model parameters as above), amplified by a factor of 675.

Figure 8

(Color online) Differential conductance versus bias voltage $V$ (in eV) at small (left panels) and large (right panels) voltage scales. The potential was assumed to drop entirely at the tip/molecule contact. Results for $t_{\text{Co},m}=0$ (red broken lines) and $t_{\text{Co},m}=0.2\text{eV}$ (black continuous lines) are shown. Upper panels: planar configuration (lobe-lobe hopping $t_{l,l}=0.2\text{eV}$). Lower panels: saddle configuration (lobe-lobe hopping $t_{l,l}=0.56\text{eV}$ and 0.13 eV, see text). The rest of the model parameters are $t_{\text{Co},t}=t_{l,t}=-0.008\text{eV}$, $t_{\text{Co},l}=0.16\text{eV}$, and $U=1.6\text{eV}$.

Figure 9

(Color online) LDOS versus energy at Co (left panels) and lobes (right panels) orbitals. Upper panels: planar configuration (lobe-lobe hopping $t_{l,l}=-0.2\text{eV}$). Lower panels: saddle configuration (lobe-lobe hoppings $-0.56\text{eV}$ and $-0.13\text{eV}$, see text). The rest of the parameters are $t_{\text{Co},m}=0.04$, $t_{\text{Co},t}=t_{l,t}=-0.008\text{eV}$, $t_{\text{Co},l}=-0.16\text{eV}$, and $U=1.6\text{eV}$.

Figure 10

(Color online) Differential conductance versus bias voltage $V$ (in eV) for the planar (black continuous lines) and saddle (red broken lines) configurations. Left panel: the bias voltage drops 75% at the STM tip/molecule junction and 25% at the metal/molecule junction. Right panel: the bias voltage drops equally at the STM tip/molecule and metal/molecule contacts. In the planar configuration all lobe-lobe hoppings are $t_{l,l}=0.2\text{eV}$, while in the saddle the two hoppings are 0.56 eV and 0.13 eV (see text) The rest of the parameters are $t_{\text{Co},m}=0.04$, $t_{\text{Co},t}=t_{l,t}=-0.008\text{eV}$, $t_{\text{Co},l}=0.16\text{eV}$, and $U=1.6\text{eV}$.