Single Photon Emission from a Plasmonic Light Source Driven by a Local Field-Induced Coulomb Blockade

A hallmark of quantum control is the ability to manipulate quantum emission at the nanoscale. Through scanning tunneling microscopy-induced luminescence (STML), we are able to generate plasmonic light originating from inelastic tunneling processes that occur in the vacuum between a tip and a few-nanometer-thick molecular film of C60 deposited on Ag(111). Single photon emission, not of molecular excitonic origin, occurs with a 1/e recovery time of a tenth of a nanosecond or less, as shown through Hanbury Brown and Twiss photon intensity interferometry. Tight-binding calculations of the electronic structure for the combined tip and Ag–C60 system results in good agreement with experiment. The tunneling happens through electric-field-induced split-off states below the C60 LUMO band, which leads to a Coulomb blockade effect and single photon emission. The use of split-off states is shown to be a general technique that has special relevance for narrowband materials with a large bandgap.


Model
The C60 film is described as a few (n) layers of C60 molecules forming a (111) surface whose molecular positions and orientations are obtained from bulk crystalline C60 22 . The film's electronic structure is based on a tight-binding parameterization of 4n molecules per unit cell volume that is periodically continued to form an infinite film parallel to the surface. The factor 4 accounts for the four different molecular orientations of C60 that are the observed positions found to be frozen in at 4 K, the experimental temperature. One 2p orbital is placed on each C atom, pointing radially out from the molecule. The five HOMO (hu) and three LUMO (t1u) frontier orbitals of each molecule are then constructed from these 2p orbitals. The hopping integrals between the 2p orbitals are parameterized by fitting them to DFT calculations 9 from which the hopping integrals between the HOMO and LUMO orbitals on different molecules as well as the potential energies of these orbitals are then calculated.
This results in a very accurate representation of the LUMO DFT band structure. 9 The electronic structure S2 calculation of the full film then involves (8 × 4n) by (8 x 4n) matrices. The factor 8 accounts for the hu and t1u orbitals of C60.
Electrostatic effects are gradually added to the model, starting with the STM tip being absent. In this context the C60 layer is treated as a homogeneous dielectric medium with a dielectric constant of ε=4.4 35 that rests on a perfect metal surface. The C60 layer thickness is na/√3, where a is the C60 lattice parameter and the factor 1/√3 is due to the crystal face being (111). The image charges due to an electron in the LUMO or a hole in the HOMO are now considered. An electron in the LUMO on molecule feels an image potential − image , while a hole in the HOMO feels the potential + image . The effect of the vacuum-C60 and C60-metal interfaces are taken into account using image charges and the appropriate boundary conditions for these interfaces. Image charges are chosen such that the boundary conditions are satisfied for one surface at a time. This approach is iterated until the process converges and the boundary conditions are finally satisfied at both surfaces.
Next, we take the tip into account. The tip is described as a sphere with radius R = 1 nm whose surface is a distance d away from the vacuum-C60 interface. A charge = is added at the sphere's center to obtain the potential at the sphere's surface. Image charges are then introduced once again to restore the correct boundary conditions at the tip-vacuum, vacuum-C60, and C60-metal interfaces. The resulting potential at molecule is tip . Finally, we also include the image potential of an electron (in the LUMO) or a hole (in the HOMO) in the tip, obtaining slightly different tip potentials for electrons and holes.
image only depends on the layer index , while tip also has a strong lateral dependence and is assumed to be localized to an area that would encompass about 100 C60 molecules per layer.
The work function for a multilayer C60 film on polycrystalline Ag is 4.68 eV 33 and for Au it is 5.26 eV. 34 These values are used to construct and approximate the potential between the C60-Ag(111) sample and the Au tip. The difference between these work functions is 0.58 eV. While this difference is sensitive to Ag atoms coating the Au tip during the experimental tip preparation, this effect is not taken into account because the qualitative features of the resulting calculations are relatively insensitive to this difference.
Thus, when the bias is applied, we simply require that = + 0.58 eV.
The tight-binding band gap of C60 is 2.24 eV, reasonably close to the experimental result 2.3 eV. 38 We have then applied a small correction to the C60 HOMO band, ΔEg = -0.06 eV. This approach puts the bottom of the LUMO band for the four layer system at about 0.27 eV, slightly above = 0. This S3 alignment to is arbitrary, but leads to a LUMO bound state going through for approximately correct biases for one to four layer systems.
For a bias of < −3.32 V, a split-off LUMO state moves below and can be occupied. If this split-off state has the weight on molecule , we add these charges to the center of the molecule and calculate their images in the surfaces. This leads to an additional potential, filled , resulting from all the charges and their images, except that the potential at molecule from charge is eff . Here, eff = 1.3 V is an effective Coulomb interaction. 39 This calculated value has been shown to be in good agreement with experiment. 38 If the split-off state is assumed to be localized on one molecule in the outermost layers, is the hopping integral between orbital on molecule and orbital on molecule . If one split-off LUMO state is filled, we also add the potential filled from the electron in this state.

Numerical Methods
The electronic structure is calculated using a Green's function technique.

S6
Supporting Figure 1 shows the electronic structure of the HOMO and LUMO bands for two and three C60 layers. These structures were calculated for values of for which tunneling from a new split-off state or narrow resonance becomes possible. As a specific example, in order to tunnel through the first HOMO split-off state for 3 layers of C60, an applied bias of -2.60 V is required. The curve that is labeled with "#1: -2.60" corresponds to the calculated density of states for an applied bias of -2.60 V. These values of then correspond to structures in the / curves in Figure 3d. The positions of the structures agree well with experiment, except that the splitting between the peaks in the LUMO (and perhaps the HOMO) is too small compared with experiment (see Figure 3d in main text). This could be due to the neglect of crystal-field splittings in the model, specifically, the splitting of the t1u and hu states due to the presence of a surface, and the applied potential having a different effect on each orbital because they are oriented differently with respect to the tip and surface. In addition, the tunneling matrix elements may coincidentally cause some features to be obscured during measurements.

Qualitative behavior of HOMO and LUMO States in Figure 5 at small bias
As stated in the main text: "When a small bias is applied with a tip (-0.68 V in Figure 5b), the band edges stay fixed while the local DOS on the C60 directly below the tip is depleted at the upper band edge and accumulates at the lower edge. This causes the first moment to move to a lower potential, in this case, towards (Figure 5a-b). This is due to the combined effects of the electrostatic and image potentials from the tip." For the HOMO these contributions have opposite signs which happen to shift the HOMO slightly upwards.
The absolute energy shift of the density of states depends on both the electrostatic potential and the image potential, and their relative magnitudes. For Ag(111)-C60 it is the case that the HOMO DOS shifts positively in energy, in an opposite direction to the small negatively applied bias of -0.68 V.

Control experiment
Supporting Figure 2. Photon intensity correlation dip (a) present on the C60 layers, and (b) absent on Ag (111). Same tip used in both measurements, -3.5 V, 250 pA. Absolute event counts is the total number of events per bin, each bin being 24.4 ps.

S8
Tip-sample distance dependence of g (2) (t) Supporting Figure 3. Photon intensity correlation function g (2) (t) as a function of tip-sample distance, measured with the same tip for fixed bias -3.5 V. Correlation functions were measured at the same position as in Figure 2c of the main text, vertically offset by multiples of 2 for clarity. The top trace is identical to Figure 2c. From top to bottom, the tip is repositioned by the relative distance Δz indicated in the plot. Resulting dip depths d = 1 -g (2) (0) and recovery time constants τ as indicated. The values of τ decrease for increasing current and increasing electric field in the C60 layer.