Cotunneling theory and multiplet excitations: emergence of asymmetric line shape in inelastic scanning tunneling spectroscopy of correlated molecules on surfaces

Abstract

Recent advances in on-surface chemistry, combined with scanning probe microscopy, have enabled the synthesis of correlated molecules on surfaces and the characterization of their chemical and electronic properties with unprecedented spatial resolution. Low-energy magnetic excitations of individual molecules are frequently investigated by scanning tunneling spectroscopy (STS) and often appear as symmetric step-like features in the differential conductance as a function of bias voltage. The interpretation of such steps is well established within cotunneling theory and effective model Hamiltonians (e.g., Hubbard- and spin-based models). Here, we extend the cotunneling formalism to general multireference systems. We show that multireference character, together with orbital-dependent and strongly asymmetric tip/substrate couplings, can produce pronounced asymmetric line shapes in inelastic STS. These results provide an alternative microscopic mechanism for the asymmetric peaks and dips near the Fermi level frequently observed in STS experiments.

Figures (7)

• Figure 1: Cotunneling scheme for a molecular quantum dot. a) The representation of the ground state as a combination of configurations with different orbital occupations (i,j,k). The Tip and Surface couple to the orbital sites ($\alpha=i,j,k$). b) Representation of an excitation mediated by virtual charge transitions. The blue dot accounts for the state in which the system is found in the different $N$ and $N \pm1$ situations.
• Figure 2: Representation of the singlet state a as function of the energy level $\epsilon_2$ (LUMO) with respect to $\epsilon_1$ (HOMO) and its coefficients $\alpha, \beta$ in the expansion of configurations for $U=230$ meWhen $\epsilon_2 = \epsilon_1$, $\alpha^2 = \beta^2 =0.5$, and we have a perfect diradical with a superposition of two Slater determinants.ts. In the limit of large $\epsilon_2$, $\alpha^2 \approx 1$, $\beta^2 \approx 0$ and the singlet is closed-shell.
• Figure 3: $dI/dV$ spectra for a two levels extended Hubbard described by the Hamiltonian \ref{['eq: homogeneous hubbard dimer']} with a singlet ground state and an excited triplet state at 4 meV for different LUNO occupations. In a-d) the features as function of the relative coupling strength $\text{v}=V_{T1}/V_{T2}$ for $n_{LUNO}=1.00, \, 0.66, \, 0.41, \, 0.13$ respectively.
• Figure 4: Singlet-triplet excitation through the intermediate charged states as function of the coefficients $\alpha ,\ \beta$ describing the singlet. The initial singlet and coupling to the molecular orbitals filtrates (or preponderates in intensity) the charged and discharged states that then go to the degenerate triplets. For simplicity only the addition and removal of $S_z=1/2$ is plotted, the other spin is completely analogous.
• Figure 5: dI/dV spectroscopy for the triplet ground states with singlet situated at 4 meV. In a,b) $\epsilon_2 = 200$ meV and $\epsilon_2 = 500$ meV, so that $n_{LUNO}=0.52$ and $n_{LUNO}=0.13$ in the singlet excited states, respectively. The transitions happen from the degenerated triplets to the singlet, developing an asymmetric character similar to the ones in Fig. \ref{['fig: asymmetries']}. In c,d) an anisotropy of $D=2$ meV is introduced, splitting the triplets. The transitions occur now between the $S_z=0$ and $S_z= \pm1$ states inside of the triplet. Under this situation no asymmetries are observed.