Towards scalable nano-engineering of graphene

By merging bottom-up and top-down strategies we tailor graphene's electronic properties within nanometer accuracy, which opens up the possibility to design optical and plasmonic circuitries at will. In a first step, graphene electronic properties are macroscopically modified exploiting the periodic potential generated by the self assembly of metal cluster superlattices on a graphene/Ir(111) surface. We then demonstrate that individual metal clusters can be selectively removed by a STM tip with perfect reproducibility and that the structures so created are stable even at room temperature. This enables one to nanopattern circuits down to the 2.5 nm only limited by the periodicity of the Moiré-pattern, i.e., by the distance between neighbouring clusters, and different electronic and optical properties should prevail in the covered and uncovered regions. The method can be carried out on micro-meter-sized regions with clusters of different materials permitting to tune the strength of the periodic potential.

The experiments were carried out in an ultra-high-vacuum (UHV) system whose base pressure is below 5 × 10 −11 Torr. The system is equipped with a variable temperature scanning tunneling microscope (VT-STM), low energy electron diffraction (LEED), Auger electron spectroscopy (AES), sample and STM tip transfer and heating capabilities, an STM tip cleaning system by field emission, several interchangeable evaporation cells, a quartz crystal microbalance, and an ion gun for sample cleaning purposes.
The STM experiments were performed with a home-built variable temperature instrument. Tips were made of W and prepared by electrochemically etching and subsequently annealing in UHV conditions. STM data were acquired with a fully automated workstation that incorporates digital feedback control based on DSP (digital signal processor) technology 1 .
All the surface manipulation experiments, data acquisition, and image processing were per- Under such conditions, small areas of the Ir(111) substrate remained intentionally uncovered by graphene, which allowed us to estimate the coverage of W or Ir used for the cluster formation (see below). W and Ir were evaporated from high purity filaments composed of each corresponding material. An accurate calibration of the deposition rate as a function of the filament temperature, measured by an infrared pyrometer, was performed by means of STM images acquired on areas of bare -uncovered by graphene-Ir(111). Figure S1 shows a representative STM topograph acquired after the adsorption of 0.50 ± 0.05 ML of W at a deposition rate of 8 × 10 −3 ML/s on a graphene/Ir(111) surface at room temperature. Here, 1 ML corresponds to the atomic density of the Ir(111) surface. As can be appreciated in Fig. S1, a W cluster superlattice with high structural perfection extending over very large areas of the graphene/Ir(111) surface is formed.

Extraction probability
We acquired more than 1000 curves in order to obtain the extraction probability shown in Fig. 2l of the main manuscript. Each single point in this graph corresponds to the probability of removing a W cluster by approaching the STM tip towards the sample a certain distance with a given applied sample voltage. For each point, this probability was obtained by performing 20 single attempts to extract 20 different clusters. In each attempt, we set the stabilization current to 70 pA at the selected sample voltage and then, with the selected voltage fixed, we approached the tip towards the surface the chosen distance at a constant rate. After each single attempt, we measured a STM image to check whether the selected cluster was removed or not. The extraction probability for each point of the graph corresponds then to the total number of clusters removed in these 20 attempts divided by 20. Error bars correspond to the standard deviation of a binomial distribution.

Conductance histogram
We constructed the conductance histogram shown in Fig. 2m with σ 0 = e 2 /(4 ) the universal conductivity and the Fermi velocity v F . With the dielectric and the (generally complex) static dielectric constant = + i , the energy loss function S = −Im −1 shows a maximum at x = 1 with we obtain a good fit to the experimentally observed sound velocity of v s ≈ 1.4v F for Iridium. 3 EELS experiments were also performed for graphene on SiC 5 and on Platinum. 6 Again, a characteristic peak in the loss function with linear dispersion was observed with sound velocities v s ≈ 1.4v F for a SiC-substrate and v s ≈ 1.15v F for a Platinum substrate. Applying the same theory as above, i.e., only including interband transitions, we obtain good fits to the experimental data for | | ≈ 3.5 (SiC) and | | ≈ 6.1 (Platinum), respectively. For a dielectric substrate, the dielectric response is real, | | ≈ , leading to the usual dielectric constant for SiC SiC ≈ 6, where we used = ( SiC +1)/2 valid for a substrate/graphene/air interface. This confirms that assuming interband plasmonic resonances leads to consistent results.
We can refine the above approach and also include the band gap ∆ that appears in the spectrum of graphene on Iridium(111). For that, we need to consider the conductivity of gapped graphene which also acquires an imaginary part. The longitudinal part that couples to charge fluctuations reads 7 On the left hand side of Fig. S3, the electron loss function S = −Im −1 with the conductivity of gapped graphene, Eqs. (4) and (5), is shown. We also plot the experi- for Platinum are considerably lower than expected for a typical metal which suggests that optics for frequencies ranging from THz to mid-infrared associated with finite wave numbers is possible even in the presence of a metallic substrate.

Spectrum of mass-confined graphene quantum dots
In this section, we discuss the spectrum of spherical graphene quantum dots. Electronic localization is obtained due to a different mass-profile presumably provoked by the presence or absence of the metallic nanoclusters.
The removal of Ir-clusters on top of graphene on Iridium(111) presumably creates Dirac quantum dots with nanoscale dimensions due to mass confinement. Dirac carriers localized by a variable mass profile lead to well-defined discrete states with high intrinsic lifetime due to the absence of typical edge disorder. We model the confined region by a circular quantum dot of radius R where a step-like change in the mass term 2mv 2 F = ∆ 1 θ(R − r) + ∆ 2 θ(r − R) with ∆ 1 = 0.05eV for the dot and ∆ 2 = 0.4eV for the bulk region. The Hamiltonian of a gapped graphene sheet in polar coordinates readŝ With l ∈ Z, the eigenfunctions can be written in the general form For the given mass profile and ∆ 1 < 2|E| < ∆ 2 , we explicitly have with j l the Bessel and h (1) l = j l + iy l the Hankel function where y l denotes the Neumann function. We further have with and B is a normalization constant. The matching conditions on the frontier R yield the following equation: Its solutions are given by the discrete bound state energies. These are shown in Fig. S4, where the black curves correspond to l ≥ 0 and the red curves to l < 0. On the left hand side of Fig. S5, the eigenfunctions of Eq. (8) and (9) are plotted for the first three electronic energy levels of a dot with radius R = 15nm and on the right hand side the probability density |ψ E,l ( r)| 2 .
There is a bound state at E ≈ 0.2eV for a radial quantum dot even with radius R = 2.5nm. This energy is lowered by excitonic effects and also due to a smooth effective mass evolution which tends to decrease the electron confinement. Changes might also occur due to strain effects leading to effective magnetic fields. 8 We thus predict a bound exciton even in small quantum dots with radius R = 2.5nm to be verified by photoluminescence or nearfield spectroscopy. For larger, not necessarily circular quantum dots, we also expect at least one bound state in the conduction as well as in the valence band, see Fig. S4. For quantum dots with more than one bound state, the relaxation mechanisms to the ground state due to various mechanisms such as optical/acoustic phonons or Auger electrons can be discussed in detail as the dot size can be increased till the two-dimensional limit has been reached.

Excitonic bands in arbitrary quantum dot arrays
In this section, we will discuss possible extensions and future research directions that our approach offers. For explicit calculations, we assume a non-dissipative substrate which needs to be improved in subsequent calculations after the screening influence of the metal has been experimentally investigated.
Graphene optics on a metallic substrate is challenging since the induced electric dipoles in the graphene layer are usually strongly quenched by the metallic substrate. Nevertheless, the previous analysis on the acoustic plasmonic excitations showed that the influence for finite q-numbers is considerably less than expected, especially for Iridium, leading to a static dielectric constant | | ≈ 3.5. This means that screening due to induced particle- In the following, we will thus neglect intrinsic damping and assume a real dielectric constant . Then, only the spontaneous decay of the excitons must be considered when evaluating their lifetimes. This decay rate is given by 9 where ω and µ are the energy and the dipole moment of the exciton, respectively, ε 0 the vacuum permittivity and the real (average) dielectric constant, i.e., = ( substrate + 1)/2 for a substrate/graphene/air interface. For graphene, we can write this as Let us now assume a quantum dot in the first excited (excitonic) state next to a second quantum dot at distance D of the same dimension that is in its ground state. Since they have the same excitation spectrum there is the possibility for efficient energy (Förster) transfer from one dot to the other, see process 2 in Fig. S6. If this non-radiative process is considerably faster than the spontaneous decay rate of the exciton, a hopping mechanism for the exciton is induced and arbitrary two-dimensional lattices could be designed with the life-time limited only by intrinsic dissipative effects like phonon and bulk-electron scattering.
The transition probability t n of an exciton hopping from one quantum dot to another mediated by the inter-dot Coulomb interaction V C shall be estimated using Fermi's Golden rule, where |m and |n are excitonic states. The delta function is usually replaced by the absorption and emission spectra of the donor and acceptor. We will thus assume homogeneously broadened excitonic states ρ n ( ) = π −1 Γ/(( − n ) 2 + Γ 2 ) and replace the δ-function by δ n,m = d ρ n ( )ρ m ( ). 10 Conventional overlap of the wave functions of quantum dots leads to an additional hopping mechanism.
We model the excitonic state as product state of the electron and hole wave function.
The matrix element is thus given by El r 2 ; E l r 1 |V C |El r 1 ; E l r 2 , where E < 0, E > 0 and V C the Coulomb interaction with The two (direct and exchange) processes are depicted in Fig. S6.
Let us consider the special case of only one excitonic bound state, i.e., quantum dots with radius R 10nm and we drop the subindex t n → t. For typical broading Γ = 10meV and dot radius R = 5nm, the hopping probabilities for the two non-radiative processes, depicted in Fig. S6, as well as for conventional wave function overlap are plotted in Fig. S7 as a function of the inter-dot distance D.
The Förster energy transfer is dominant for D > 6R displaying the typical algebraic decay t ∼ D −6 obtained from a multipolar expansion. In this regime, the excitonic hopping induced by this mechanism is two orders of magnitude greater than the spontaneous decay rate γ. As a consequence, a well-defined band of Frenkel excitons is expected with Wannier functions well approximated by the individual quantum dot wave functions. On the other hand, for D < 6R, Wannier functions describing the exciton will be more spread and the resultant excitons would be more delocalized. A different approach including the array of quantum dots from the beginning is then necessary.
To conclude, our results demonstrate that depending on the dimensions of the quantum dots and the distance between them, we can engineer lattices hosting different kinds of excitons (Frenkel vs. Wannier) and with tunable excitonic hoppings between the dots. This would provide an alternative platform to study the dynamics in quantum mechanical tightbinding models and also the possibility of discussing exciton-polariton condensation 11 in a controllable system.