Renormalization of spin excitations and Kondo effect in open-shell nanographenes

We study spin excitations and the Kondo effect in open-shell nanographenes, motivated by recent scanning tunneling inelastic spectroscopy experiments. Specifically, we consider three systems: the triangulene, the extended triangulene with rocket shape, both with an $S=1$ ground state, and a triangulene dimer with $S=0$ on account of intermolecular exchange. We focus on the consequences of hybridization of the nanographene zero modes with a conducting substrate on the $dI/dV$ line shapes associated with spin excitations. The partially filled nanographene zero modes coupled to the conduction electrons in the substrate constitute multiorbital Anderson impurity models that we solve in the one-crossing approximation, which treats the coupling to the substrate to infinite order. We find that the coupling to the substrate leads to (i) renormalization of the spin flip excitation energies of the bare molecule, (ii) broadening of the spectral features, and (iii) the emergence of zero bias Kondo peaks for the $S=1$ ground states. The calculated substrate induced shift of the spin excitation energies is found to be significantly larger than their broadening, which implies that this effect has to be considered when comparing experimental results and theory.

Figure 1

Atomic structures with graphical representations of the ZMs [(a),(c),(e)] and corresponding single-particle spectra [(b),(d),(f)] for $t=-2.7\mathrm{eV}$ of the three NGs discussed in the paper: triangulene dimer [(a),(b)], single triangulene [(c),(d)], and “rocket” structure [(e),(f)]. The size of the red circles in the structures (a), (c), and (e) yields the weight $|{\psi }_{\alpha }{\left(i\right)|}^2$ of the $E=0$ wave function ${\psi }_{\alpha }$ on carbon site $i$. Note that the other two ZMs in the case of the dimer and the other ZM for the single triangulene have the same weights (but different phases) as the ones already depicted in (a) and (c).

Figure 2

OCA results for triangulene dimer for $t=-2.7\mathrm{eV}, U=1.9\left|t\right|$, and ${\varepsilon }_{\mathrm{C}}^{*}=-0.485\mathrm{eV}$. (a) Spectral function of zero modes for different values of the on-site energy shift $\delta {\varepsilon }_{\mathrm{C}}$ and for broadening $\mathrm{\Gamma }/\pi =52\mathrm{meV}$. (b) Spectral function at low energies for different values of the broadening $\mathrm{\Gamma }$ and for $\delta {\varepsilon }_{\mathrm{C}}=0.2\mathrm{eV}$. (c) Derivatives of spectral functions shown in (b) close to the inelastic spin excitation step. The dashed gray line marks the position of the bare excitation energy. (d) Shift and width of the ISTS step feature as a function of the broadening $\mathrm{\Gamma }$ for $\delta {\varepsilon }_{\mathrm{C}}=0.2\mathrm{eV}$. The dashed lines show power-law fits to the data, which yield exponents of $\alpha \sim 0.7$ for both shift and width as a function of $\mathrm{\Gamma }$. (e) Local spectral functions ${\rho }_i\left(\omega \right)$ for the two carbon sites of the NG shown in the inset for $\mathrm{\Gamma }/\pi =39\mathrm{meV}$ and $\delta {\varepsilon }_{\mathrm{C}}=0.2\mathrm{eV}$. (f) Density map of spatially resolved spectral function $\rho (\mathbf{r};{\omega }_{-})$ evaluated at $z=5$ Å above the molecular plane for ${\omega }_{-}=-15.5\mathrm{meV}$. Other parameters as in (e).

Figure 3

Spectral functions calculated within OCA for the triangulene monomer for $U=\left|t\right|$ and $t=-2.7\mathrm{eV}$ at ph symmetry (${\varepsilon }_{\mathrm{C}}=-0.26\mathrm{eV}$). (a) Spectral function of ZMs for different values of the coupling to the substrate $\mathrm{\Gamma }$ at low temperature $kT=0.4\mathrm{meV}\sim 5\mathrm{K}$. (b) Same as (a) but for a smaller energy scale showing the evolution of the Kondo peak as $\mathrm{\Gamma }$ decreases. (c) Local spectral function for the carbon sites marked in the atomic structure (d) in the corresponding color. (e) Density map of spatially resolved spectral function $\rho (\mathbf{r};{\omega }_0)$ evaluated at $z=5$ Å above the molecular plane for ${\omega }_0=0$.

Figure 4

Spectral functions calculated within the OCA for rocket structure for $U=\left|t\right|$ and $\mathrm{\Gamma }/\pi =13\mathrm{meV}$ at low temperature, $kT=0.1\mathrm{meV}\sim 1.2\mathrm{K}$. (a) Orbital-resolved ($A_1,A_2$) and total ($A_{\mathrm{T}}=A_1+A_2$) spectral functions at half-filling ($N=N_1+N_2=2$) at ${\varepsilon }_{\mathrm{C}}=-0.186\mathrm{eV}$. (b) Orbital-resolved spectral functions around the Kondo peak for different values of detuning $\delta {\varepsilon }_{\mathrm{C}}$ from half-filling. The dashed (solid) lines show the spectral functions of orbital 1 and 2, respectively. (c) Local spectral function ${\rho }_i\left(\omega \right)$ for carbon sites $i$ marked in corresponding color in the atomic structure shown in the inset for $\delta {\varepsilon }_{\mathrm{C}}=-40\mathrm{meV}$. (d) Density map of spatially resolved spectral function $\rho (\mathbf{r};{\omega }_0)$ evaluated at $z=5$ Å above the molecular plane for ${\omega }_0=0$.