Kondo effect in transport through molecules adsorbed on metal surfaces: From Fano dips to Kondo peaks

The Kondo effect observed in recent scanning tunneling microscopy (STM) experiments on transport through CoPc and TBrPP-Co molecules adsorbed on Au(111) and Cu(111) surfaces, respectively, is discussed within the framework of a simple model [G. Chiappe and E. Louis, Phys. Rev. Lett. 97, 076806 (2006)]. Both molecules show a four lobed structure with Co placed in its center, the active $\mathrm{Co}d$ orbital lying at lower energy in the TBrPP-Co molecule. In the Kondo regime, and by varying the ratio $r$ of STM tip/Co to STM tip/lobes hoppings, it is possible to produce a crossover from a conductance Kondo peak (CoPc, high $r$) to a conductance Fano dip (TBrPP-Co, $r$ around 1). The essential role of the internal structure of the molecule in controlling even the entrance into the Kondo regime is highlighted. In particular, it is shown that the crossover from Fano dip to Kondo peak can also be produced, changing the sign of the lobe/lobe hopping while keeping that ratio constant. In the case of $\mathrm{TBrPP}\text{−}\mathrm{Co}∕\mathrm{Cu}\left(111\right)$, we show that varying the density of states at the metal surface, or distorting the molecule, changes the shape of the Fano dip and the Kondo temperature, and shifts the dip minimum. These features are in line with experimental observations, indicating that our simple model contains the essential physics underlying the transport properties of such complex molecules.

Figure 1

(Color online) Cluster of atoms utilized to describe the CoPc molecule adsorbed on a Au(111) surface and the TBrPP-Co adsorbed on a Cu(111) 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 scanning tunneling microscope (the gold surface). To calculate the conductance, semi-infinite chains were attached to the top atom and to either the gold atom or the lobes (see text).

Figure 2

(Color online) Upper panels: transmission $T\left(E\right)$ (in units of the conductance quantum) versus the energy $E$ (in eV) referred to as the Fermi energy, calculated by means of the (a) SB and (b) ECA methods (see text). Results for $t_{l,t}=0$ (violet/dark gray chain line) and $t_{l,t}=0.08\mathrm{eV}$ (continuous red/gray 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 $t_{\mathrm{Co},t}=0.08\mathrm{eV}$, $t_{\mathrm{Co},l}=-0.14\mathrm{eV}$, $t_{\mathrm{Co},m}=0$, $t_{l,l}=-0.2\mathrm{eV}$, $t_{l,m}=0.14\mathrm{eV}$, $U=1.6\mathrm{eV}$, and ${\rho }_m=5{\mathrm{eV}}^{-1}$.

Figure 3

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

Figure 4

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

Figure 5

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