Understanding the Unique Electronic Properties of Nano Structures Using Photoemission Theory

Newly emerging experimental techniques such as nano-ARPES are expected to provide an opportunity to measure the electronic properties of nano-materials directly. However, the interpretation of the spectra is not simple because it must consider quantum mechanical effects related to the measurement process itself. Here, we demonstrate a novel approach that can overcome this problem by using an adequate simulation to corroborate the experimental results. Ab initio calculation on arbitrarily-shaped or chemically ornamented nano-structures is elaborately correlated to photoemission theory. This correlation can be directly exploited to interpret the experimental results. To test this method, a direct comparison was made between the calculation results and experimental results on highly-oriented pyrolytic graphite (HOPG). As a general extension, the unique electronic structures of nano-sized graphene oxide and features from the experimental result of black phosphorous (BP) are disclosed for the first time as supportive evidence of the usefulness of this method. This work pioneers an approach to intuitive and practical understanding of the electronic properties of nano-materials.


A. Fourier transform of the initial-state wave function
An initial eigen-function of a molecule can be expressed by using linear combination of atomic orbitals (LCAO) centered at each atomic position, Rα. (S1) If we take Fourier transform over the both sides of equation (S1) Podolsky, Pauling, and Gadzuk developed the mathematical tool for Fourier transformation of atomic orbitals. S1, S2 For simplicity, if we consider only 2pz atomic orbital, which is the case of π states of poly aromatic hydrocarbons (PAH), equation (S2) reduces to equation ( where f(k) is a function of absolute value of k, and θk, φk are polar and azimuthal angles of vector k.

B. Plane wave approximation to the final state wave function of photo-emitted electrons
Equation (S4) denotes the photoelectron emission intensity at measured angle and energy of kf, and Ekf, respectively, from an initial state of ψi, where Ekf = (h/2π) 2 k 2 f/2me. If the final state wave function ψf is approximated to a plane wave, From equation (S4) and (S5) we obtain the following.
Substituting equation (S3), we get the following for any kf satisfying Ψ = Ψ + ℎ , However, this expression does not contain any information related to the quantum mechanical transition probability from initial to final state, which should include dipole selection rule. This equation just reflects the special distribution of molecular orbital of the initial state modulated with the incident light polarization effects on the final state electrons. S3 Even though this simple approach can give useful visual guidance to the experimental results, the experimental result can be significantly different from these simple results, in most of the situations.
Therefore, for more realistic correlation between experimental and theoretical results, the exact interaction between photon and electron in the atomic or molecular situation should be considered. S4, S5 4

C. Photoemission intensity from a molecule obtained by quantum mechanical calculation using IAC approximation
The PES cross section of the nth Kohn-Sham (K.-S.) energy level, which corresponds to ψi in equation (S7), can be defined by (S9) where ⃗⃗⃗ , and ⃗⃗⃗ , represent position vectors of detector and the ath atom in a molecule with respect to origin, respectively. Ekin, and ⃗ denote kinetic energy and the momentum vector of a photoemitted electron, respectively. S6 ∑ , , (̂) is an atomic factor that represents the PES probability from the ath atom, and can be written as, where, Rs is the radial dipole matrix element of s orbital, ± is the overall phase shift of the atom indexed by "a", and ± , are the functions defined in reference S7.
Cna represent the coefficients of ath atom to the nth molecular orbital wave function, which is obtained using pbeh1pbe/sto-3g level of theory, in which sto-3g uses minimum atomic orbitals.

D. Angle resolved photoemission simulation
Based on equation (S7) or (S8), the photoemission yields can be obtained as a function of momentum vectors of emitted electrons. The momentum vectors include the information on the kinetic energy and the direction of emitted electrons, which are polar and azimuthal angles.
Where hv is the incident photon energy and Φ is the work function of the system. Using equation S11 and S12, binding energy of an electron with momentum Kx and Ky can be easily calculated if the signal intensity is measured for each detector geometry (θ and φ) and kinetic energy of the electron. After conducting simulated experiments using equation (S7) or (S8) The same PES intensity can also be obtained using PES simulation equipped with the formalism exploiting the IAC approximation as shown in equation (S9). To show the validity and the usefulness of this method, the IAC simulated PES intensity on the well-known graphene nano structures is compared to that using Fourier transform and plane wave final states. Figure S1 shows two different types of nanoscale graphenes, which are armchair and zigzag edged graphenes respectively. The two different approaches using FT and IAC produce almost the same results, which had been predicted by the reports of Puchnig et al. S3 The most prominent feature is the different contribution of the edge induced effects on the original electronic structure. The zigzag edge induced features are obviously seen in Figure S1c and d, which correspond to the unique features interconnecting K points near the Fermi surface. On the contrary, armchair edged graphene exhibits significantly low surface state intensity at the highest occupied energy states, as shown in Figure S1a and b.

E. ARPES experiments on HOPG and comparison to the photoemission simulation
ARPES experiments were conducted under UHV of 2.0 × 10 −10 Torr at the 8A2 undulator beam line of the Pohang Accelerator Laboratory (PAL) in Korea using a high-resolution electron analyzer, VG Scienta SES 2002 with a 2D-CCD detector. Spectra of the valence bands were obtained using linearly polarized photons with energies ranging from 100 eV to 400 eV. All the experiments were performed at room temperature. The HOPG crystal was cleaved outside the vacuum chamber using scotch tape and introduced into the chamber immediately. The sample was degassed at 400 o C for more than 5 hours to remove any adsorbed molecules. The cleanness of the sample was confirmed by the measurements of the core levels of nitrogen, oxygen, and carbon.
To simulate HOPG as a molecule, an armchair edged hexagonally PAH with carbon numbers of 762 is used as shown in Figure 2. All the calculations were performed on the platform of Gaussian 09 package. S8 The geometry was fully optimized using PBEh1PBE/sto-3g level of theory. The K.-S. energy scale is compressed to compensate for the approximation of electronic relaxation and 8 correlation effects. In our calculation, we chose a compensation factor of 0.972. This value was obtained by scaling the binding energy of the calculated band structure to coincide with the experiment. And the overall energies are shifted so as for the midpoint between HOMO and LUMO to be zero. Figure S2 shows ARPES spectra using various photon energies with the experimental geometry shown in Figure 1c. The angle resolved electron measurements are made along the x-axis and indicated as α's. To simulate the randomly oriented graphite domains of HOPG, the total PES intensities are averaged over those of PAHs with 360 equally spaced azimuthal angles, ϕ's. The same with (a), but the x-axis is the acceptance angle in the range of ± 10˚. Right panel: Experimental data which is the same with Figure 1b and c except the tilt angle θ = 0.