Theory of STM junctions for $\pi$-conjugated molecules on thin insulating films

A microscopic theory of the transport in a scanning tunneling microscope (STM) setup is introduced for $\pi$-conjugated molecules on insulating films, based on the density matrix formalism. A key role is played in the theory by the energy dependent tunneling rates which account for the coupling of the molecule to the tip and to the substrate. In particular, we analyze how the geometrical differences between the localized tip and extended substrate are encoded in the tunneling rate and influence the transport characteristics. Finally, using benzene as an example of a planar, rotationally symmetric molecule, we calculate the STM current-voltage characteristics and current maps and analyze them in terms of few relevant angular momentum channels.

Figure 1

(a) Sketch of the investigated STM setup. A $\pi$-conjugated molecule, here benzene, is separated by a metal substrate (yellow) only through an ultrathin insulating film (red). A bias voltage is applied between the substrate and the tip. (b) Schematic illustration for the sum of the potentials of the substrate, the molecule, and the tip ($v=v_{\mathrm{sub}}+v_{\mathrm{m}}+v_{\mathrm{tip}}$) along the $z$ direction. We choose the energy of the vacuum between the molecule and the tip, as well as the energy of the tunneling barrier between molecule and substrate to be zero. The energies at the bottom of the conduction band of tip and substrate are ${\varepsilon }_0^{S/T}=-{\Phi }_0^{S/T}-{\varepsilon }_F^{S/T}$, where ${\varepsilon }_F^{S/T}$ are the Fermi energies measured from the band bottom and ${\Phi }_0^{S/T}$ are the work functions for the tip and the substrate. The work functions are shifted by the applied bias voltage.

Figure 2

Energy levels of the Hückel Hamiltonian and the corresponding values of the angular momentum $l$.

Figure 3

Tunneling rate ${\Gamma }_l^S$ describing substrate-molecule tunneling processes for different angular momentum quantum numbers $l$. The thickness of the substrate barrier is $d=3$ Å, while work function and Fermi energy are, respectively, ${\phi }_0^S=4$ eV and ${\varepsilon }_F^S=7$ eV.

Figure 4

Diagonal elements ${\Gamma }_l^T$ of the tip tunneling rate matrix ${\Gamma }_{l{l}^{\prime }}^T$ for the different angular momentum states. The rates are calculated assuming $z_{\mathrm{tip}}-d=3.5$ Å, ${\phi }_0^T=4$ eV, ${\varepsilon }_F^T=7$ eV, and $\hslash \omega =4$ eV. The presence of the harmonic confinement explains also the different energy limits with respect to the ones of Fig. 3. The lower limit is at $-{\varepsilon }_F^T+\hslash \omega$, while the upper limit is at $-{\varepsilon }_F^T+2\hslash \omega$.

Figure 5

Phase ${\phi }_l$ of the tunneling rate matrix ${\Gamma }_{l{l}^{\prime }}^T$ [Eq. (34)]. The phase is almost constant if the tip is moved along the radii outgoing from the center of the molecule. The carbon atoms are labeled by $\alpha =0,...,5$.

Figure 6

Together with a change in the energy, the transition from the six-particle ground state to the seven-particle (five-particle) ground states is also associated with a change in the angular momentum of $\Delta l=\pm 2$ ($\Delta l=\pm 1$).

Figure 7

Current-voltage characteristics and current maps associated with the neutral-anion transition. The current maps are calculated with $z_{\mathrm{tip}}-d=5$ Å. Notice that the map in the Coulomb blockade region is just a rescaling of the one at resonance.

Figure 8

Current-voltage characteristics and current maps associated with the neutral-cation transition. The current maps are calculated with $z_{\mathrm{tip}}-d=5$ Å. Notice that the value of the current at resonance is much higher than the one relative to the neutral-anion case (see Fig. 7).

Figure 9

Current-voltage characteristics and current maps associated with the neutral-anion transition. Interestingly the current map in the interference blockade region shows novel topographic features if compared with other maps involving the same states (see also Fig. 7). In the inset a zoom on the interference current peak is presented.

Figure 10

Constant current topographic images. The left panel refers to the neutral-cation resonance (${\phi }_0=7$ eV, $V_b=2.156$ V, $I=300$ pA), the right panel, instead, to the neutral-anion resonance (${\phi }_0=5$ eV, $V_b=-1.688$ V, $I=100$ pA).

Figure 11

Scheme of a one-dimensional, finite potential well with borders $a$ and $b$ and depth $V_0$.

Figure 12

Tunneling rates obtained by using Slater-type orbitals (solid lines) and Gaussian orbitals (dashed lines). The rates are calculated for a substrate-molecule distance $d=3$ Å. In the Slater-type orbital $Z_{\mathrm{eff}}=2$.