Many-body theory of current-induced fluorescence in molecular junctions

Nonequilibrium superoperator Green’s function theory is used to calculate the fluorescence signal of molecules induced by currents in scanning tunneling microscope junctions. The spectrum of benzene and its variation with tip position and bias are simulated at the density functional theory level. The formal analogy with laser-induced fluorescence is pointed out. Many-body effects can be accounted for through self-energies and the Keldysh Dyson equations. The sum-over-orbital expressions obtained within density functional theory may not be expressed as an amplitude square. This is due to dephasing effects induced by the many-electron excitations, which act as a bath.

Figure 1

Double-sided Feynman diagrams corresponding to the six time-ordered interactions that contributing to LIF, Eq. (11). The three processes in (a) for $\tau >0$ are complex conjugates to $\tau <0$, shown in (b). Feynman diagrams for CIF from a negatively (positively) charged molecule can be obtained by replacing the two $B$ operators acting at times ${\tau }_1$ and ${\tau }_2$ with ${\psi }_k^{†}\left({\psi }_k\right)$ and ${\psi }_l\left({\psi }_l^{†}\right)$, respectively. For CIF, $\mid a⟩$, $\mid b⟩$, $\mid c⟩$, and $\mid d⟩$ correspond to states with different number of electrons (see text). Time increases from bottom to top.

Figure 2

Double-sided Feynman diagrams for CIF. The molecule is initially in the ground state with $N$ electrons. (a) Emission from a negatively charged molecule $\left({\mathcal{S}}_n\right)$. An electron is transferred from the tip to the molecular orbital $(\mathrm{LUMO}+1)$, creating an excited state of the $(N+1)$-electron system (represented by a star). The system then relaxes to the ground $N+1$ electron state by emitting a phonon. (b) Emission from a positively charged molecule $\left({\mathcal{S}}_p\right)$. An electron is transferred from the molecular orbital $\mathrm{HOMO}-1$ to the tip. This creates an excited state of the $(N-1)$-electron system. The system relaxes to the ground $N-1$ electron state by transferring an electron from HOMO to $\mathrm{HOMO}-1$ orbital accompanied by a photon emission.

Figure 3

Energy level scheme for several unoccupied molecular orbitals of neutral benzene. The states are labeled relative to the LUMO (L). The Fermi energy $E_F=2.6\mathrm{eV}$ is taken to be the midpoint between the HOMO and LUMO energy levels. Arrows depict the transitions which contribute strongly to the current-induced fluorescence involving the negative anion at positive bias voltage. The left ordinate axis depicts the orbital energies ${ϵ}_i$ while the right axis indicates the applied bias energy required to induce a transition from a given orbital, $eV=-E_F+{ϵ}_i$. The spatial variation of the emission originating from these transitions is examined in Figs. 7, 8, 9.

Figure 4

(Color online) Contour plots of the overlap coupling $J_i{({ϵ}_F+eV)}^2$ between 15 LUMOs and the tip orbital positioned at a height of 1Å above the molecular plane. The molecular structure of benzene is superimposed over the images as a visual reference. The threshold bias energy at which these orbitals will accept tunneling electrons are also given. The coupling strength ranges from (red) 0 to (magenta) $1.5\times {10}^{-5}$. $\mathrm{LUMO}+9$ through $\mathrm{LUMO}+14$ have the strongest coupling orbitals and are scaled by ${10}^{-1}$.

Figure 5

(Color online) Panels (1)–(9) show the two-dimensional current-induced fluorescence spectrum of benzene as a function of photon energy $\hslash {\omega }_s$ and bias energy $eV$ at different lateral tip positions [cf. Fig. 7a], at constant height 1Å above the molecule. The intensities are normalized so that from red to magenta is on the order of unity. The weaker spectra for positions (3), (4), and (9) have been scaled by five times.

Figure 6

(Color online) (a) and (b) show two families of one-dimensional spectra calcualted at different bias energies between the dashed lines of panels (5) and (7) of Fig. 5, respectively. The spectra demonstrate a blueshift in the peak maximum as the bias voltage increases and are offset vertically for visual clarity.

Figure 7

(Color online) (a) depicts a slice of the local density of states, Eq. (61), of benzene at a bias energy of 12.5 eV. White numbers indicate the lateral position of the tip for the calculated spectra shown in (b) which have been offset for visual clarity. The dashed lines (c)–(f) indicate the energies of the major peaks. Panels (c)–(f) show two-dimensional contour plots of the spatially resolved fluorescence intensity $d{S}_n∕dV$ at a potential bias of $eV=12.5\mathrm{eV}$ and photon energies $\hslash {\omega }_s=$ (c) 0.0, (d) 1.1, (e) 3.0, and (f) 5.8 eV. The emission signal is sliced at 1Å above the molecular plane.

Figure 8

(Color online) The spatial varation of ${\rho }_{\mathrm{LDOS}}$, Eq. (61), for benzene at $eV=16.7\mathrm{eV}$ is shown in (a) and is plotted on the same scale as Fig. 7a. (b) shows emission spectra calculated with the tip centered at positions 1–9 indicated in (a). Panels (c)–(f) show the spatial maps (at $z=1\mathrm{\AA }$) of the fluorescence peaks at this potential bias and photon energies $\hslash {\omega }_s$ (c) 0.6, (d) 4.3, (e) 5.6, and (f) 7.5 eV. The signal intesity in (e) and (f) has been scaled by two times in order to enhance the contrast.

Figure 9

(Color online) The panels are similar to those shown in Figs. 7, 8 except that the potential bias energy is set to $eV=24.5\mathrm{eV}$. Only two peaks contribute to the emission spectrum.