---
author:
- Norman Haag
- Daniel Lüftner
- Florian Haag
- Johannes Seidel
- 'Leah L. Kelly'
- Giovanni Zamborlini
- Matteo Jugovac
- Vitaliy Feyer
- Martin Aeschlimann
- Peter Puschnig
- Mirko Cinchetti
- Benjamin Stadtmüller
title: 'Signatures of an Atomic Crystal in the Band Structure of a Molecular Thin Film - Supplemental Material'
---

Sample Preparation Procedure
============================

The C$_{60}$ films were grown in situ on an Ag(111) single crystal. Prior to the deposition of C$_{60}$, the Ag(111) crystal surface was prepared by repeated cycles of argon ion bombardment and subsequent annealing. The quality and cleanness of the Ag(111) surface was confirmed by the existence of well defined diffraction spots in low energy electron diffraction (LEED) with narrow line profiles as well as by the presence of the Shockley surface state in momentum resolved photoemission spectroscopy. The C$_{60}$ films were subsequently grown by molecular beam epitaxy using a commercial Knudsen cell evaporator (Kentax GmbH) at a sublimation temperature of $633\,$K. The film thickness was controlled by evaporation time and molecular flux and verified after the deposition procedure by core level spectroscopy of the C1s and Ag3d levels. In our study, the film thickness was determined to be $(5.0\pm 0.7)\,$ML.

Crystal Structure of the C$_{60}$ thin film
===========================================

The crystalline structure of the C$_{60}$ thin film was investigated by LEED. An exemplary LEED pattern of this film is shown in Fig. 1a. The best agreement between our LEED data and theoretical simulations was obtained for a superposition of three different structures. The major part of the LEED pattern can be described by a $(2\sqrt{3}\times 2\sqrt{3})R30^\circ$ superstructure in agreement with previous studies [@Shi.2012; @Tamai.2005]. The simulated LEED pattern is superimposed onto the experimental data in Fig. 1b as blue circles. In addition, we find diffraction spots of a $(2\sqrt{3}\times 2\sqrt{3})R30^\circ$ superstructure rotated by $\pm18^\circ$ (see LEED simulation in Fig. 1c) and rotated by $\pm 30^\circ$ (see LEED simulation in Fig. 1d). Note that the intensity of the diffraction spots of the $(2\sqrt{3}\times 2\sqrt{3})R30^\circ$$\pm30^\circ$ structure is very low pointing to a marginal relative contribution of this structure to the C$_{60}$ thin film. In our further analysis, the latter domain can hence be neglected.

![LEED image for a multilayer of C$_{60}$ grown on Ag(111) using a beam energy of $25\,$eV (a).The structure is a superposition of two contributions: a $(2\sqrt{3}\times 2\sqrt{3})R30^\circ$ superstructure (blue circles in (b)) and two domains of a $(2\sqrt{3}\times 2\sqrt{3})R30^\circ$ superstructure rotated by $\pm18^\circ$ (orange circles in (c)). A minor contribution stems from additional domains rotated by $\pm30^\circ$ (green circles in (d)). []{data-label="fig:Fig1"}](Fig_SI1.pdf){height="5.25cm"}

Experimental Methods
====================

All photoemission experiments were conducted at the NanoEsca end station at the Elettra Synchrotron Light Source, Trieste. The momentum-resolved photoemission yield was recorded with the photoemission electron microscopy system NanoEsca (Focus GmbH) [@Kromker.2008] which was operated in k-space mode. All experiments were performed in a fixed experimental geometry, i.e., with a fixed angle of incidence of the synchrotron beam of $65^\circ$ with respect to the surface normal as shown in Fig. 2. Photoemission data were recorded with p- and s-polarized light. For p-polarization the electric field vector (blue arrow) is parallel to the plane of incidence and for s-polarized it is perpendicular to the plane of incidence, i.e., it is located parallel to surface plane.

![Experimental geometry of the polarization dependent photoemission experiments.[]{data-label="fig:Fig2"}](Fig_SI2.pdf){width="5cm"}

Computational Methods
=====================

The electronic structure calculations and the simulations of the momentum maps are based on ab-initio computations within the framework of density functional theory (DFT) employing the VASP code [@Kresse1993; @Kresse1999]. The C$_{60}$ film is modeled by a free-standing layer of C$_{60}$-molecules in a hexagonal unit cell with an in-plane lattice parameter of [19.85Å]{} containing four C$_{60}$ molecules with an additional vacuum layer of about [15Å]{} in the out-of-plane direction. This structure corresponds to a (111)-cut through the low-temperature bulk crystal structure of C$_{60}$. For the geometry relaxations of the internal ionic degrees of freedom, the generalized gradient approximation (GGA) is used in conjunction with the Perdew-Burke-Ernzerhof (PBE) functional [@Perdew1996] and the van-der-Waals corrections according to the Tkatchenko-Scheffler method are added.[@Tkatchenko2009] Using the projector augmented wave (PAW) method,[@Bloechl1994] a plane-wave cutoff of $500\,$eV is employed. For $k$-point sampling, a $\Gamma$-centered grid of $8 \times 8 \times 1$ points is used and a first-order Methfessel- Paxton smearing of $0.1\,$eV is utilized. Based on the relaxed adsorption geometries, we have computed the (projected) density of states. The Kohn-Sham eigenvalues and eigenstates are also the basis for the simulations of the photoemission intensity within the framework of photoemission tomography. Here, we have approximated the final state of the photoemission process by a plane-wave[@Luftner.2017] and assumed an inner potential $V_0$ of $13\,$eV[@Hasegawa.1998]. For the simulations of the constant binding energy momentum maps and the band maps, an $8 \times 8 \times 4$ sampling of the Brillouin zone and Gaussian broadenings of $0.05\,$Å$^{-1}$ and $0.1\,$eV in the momentum and energy axes have been chosen, respectively.

Rotational Domains in Photoemission Tomography Simulations
==========================================================

![(a) Structural model of the $(2\sqrt{3}\times 2\sqrt{3})R30^\circ$ structure with four C$_{60}$ molecules per unit cell. (b) Constant energy map of the $(2\sqrt{3}\times 2\sqrt{3})R30^\circ$ structure at E$_{\mathrm{B}}=3.5\,$eV simulated by PT. The same CE map is superimposed with the lattice of the surface Brillouin zones of the C$_{60}$ structure in (c). The CE maps of the $(2\sqrt{3}\times 2\sqrt{3})R30^\circ$$\pm18^\circ$ are shown in (d) and (e). (f) Total momentum-resolved photoemission yield calculated by adding up the CE maps of the different structural domains.[]{data-label="fig:Fig2"}](Fig_SI3.pdf){height="9cm"}

The high accuracy of our photoemission tomography (PT) simulations for the valence bands of C$_{60}$ relies on a proper treatment of the additional structural domains observed in our LEED data. The band structure calculation and the PT simulations were performed for a freestanding $(2\sqrt{3}\times 2\sqrt{3})R30^\circ$ structure with four C$_{60}$ molecules per unit cell, see section *Computational Methods* above. A structural model of the unit cell used in the simulations is shown in Fig. 3a. An exemplary constant energy (CE) map of the simulated momentum resolved photoemission yield is shown in Fig. 3b for one energy within the HOMO-1 band (E$_{\mathrm{B}}=3.5\,$eV). This binding energy corresponds to one of the binding energy of the HOMO-1 CE maps discussed in Fig. 2 of the main manuscript. The CE map consists of a regular arrangement of hexagonal emission features which follow the periodicity of the $(2\sqrt{3}\times 2\sqrt{3})R30^\circ$ superstructure in momentum space. The different hexagonal emission pattern represent the C$_{60}$ valence band structure in higher surface Brillouin zones as clearly visible in Fig. 3c where the surface Brillouin zones are superimposed onto the same CE map as white hexagons. The directions and high symmetry points of the surface Brillouin zones are indicated in the inset.\

![(a) The simulated momentum resolved photoemission yield of the $(2\sqrt{3}\times 2\sqrt{3})R30^\circ$ at E$_{\mathrm{B}}=3.9\,$eV, $3.5\,$eV and $2.9\,$eV. (b) Total momentum-resolved photoemission yield of the C$_{60}$ thin film at the same binding energies.[]{data-label="fig:Fig4"}](Fig_SI4.pdf){height="9cm"}

The momentum resolved photoemission yield of the $(2\sqrt{3}\times 2\sqrt{3})R30^\circ$$\pm18^\circ$ superstructure can be obtained by rotating the CE maps of the $(2\sqrt{3}\times 2\sqrt{3})R30^\circ$ (Fig. 3b) superstructure by $\pm18^\circ$, see Fig. 3d and e. The total momentum resolved photoemission yield is finally simulated by adding up the contributions of the three structural domains: $$I(k_x,k_y,E_B)=\alpha \times I_{ 0^\circ} (k_x,k_y,E_B) + \beta\times \left (I_{ +18^\circ} (k_x,k_y,E_B)+ I_{-18^\circ} (k_x,k_y,E_B) \right )$$ Here, $\alpha$, and $\beta$ denote the relative contributions of the two different structural domains. The best agreement between our PT simulations and the experiment was obtained for an almost equal ratio of the $\pm0^\circ$ and the $\pm18^\circ$ domains with $\alpha=1$ and $\beta=0.9$. The corresponding CE map is shown in Fig. 3f and in Fig. 2 of the main manuscript. Note that no spectroscopic signature of the $(2\sqrt{3}\times 2\sqrt{3})R30^\circ$$\pm30^\circ$ structure was observed in our momentum-resolved photoemission data. This is in line with the extremely weak intensity of the diffractions spots of this particular rotational domain in our LEED data discussed above. We therefore neglect any contribution of the $(2\sqrt{3}\times 2\sqrt{3})R30^\circ$$\pm30^\circ$ superstructure in our PT simulations.

The same procedure is repeated for the second binding energy of the valence band structure shown in Fig. 2 of the main manuscript. The momentum-resolved photoemission yield of the $(2\sqrt{3}\times 2\sqrt{3})R30^\circ$ superstructure as well as the total photoemission yield including both rotated domains is shown in Fig. 4 for three characteristic binding energies of the HOMO-1 band.

Challenges in Photoemission Tomography Simulations of Localized Molecular Orbitals
==================================================================================

![Experimental (a) and simulated (b) energy vs. momentum cut through the 3D ARPES data cube extracted along the $\overline{\Gamma}$ $\overline{\mathrm{M}}$ $\overline{\Gamma}'$-direction of the surface Brillouin zone of a crystalline C$_{60}$ thin film in the binding energy range of the $\sigma$-states. The yellow line indicates the intramolecular band dispersion as guide-to-the-eye.[]{data-label="fig:Fig5"}](Fig_SI5.pdf){height="6cm"}

Despite the overall excellent qualitative agreement between our momentum-resolved photoemission data and the PT simulations of the localized $\sigma$-state of C$_{60}$ at large binding energies E$_{\mathrm{B}}>5\,$eV, there are also minor but distinct deviations between experiment and simulation in Fig. 4 of the main manuscript. For instance, we observe a slightly different energy and momentum position of the molecular $\sigma$- states in the energy vs. momentum cuts in Fig. 5 leading to a different slope of the almost linear intramolecular dispersion curve. This discrepancy also coincides with a different radius and relative photoemission intensity of the concentric molecular emission features for different E$_\mathrm{B}$ in experiment and theory. We attribute these deviations to the strong k$_\mathrm{\bot}$ dependency of the 3D Fourier transform of the localized molecular states of non-planar molecules. To support this conclusion, we depict the k$_\mathrm{\bot}$ dependency of the 3D Fourier transform of the highest occupied molecular orbital (HOMO) of a free C$_{60}$ molecule in Fig. 6. Fig. 6a shows a 2D cut through the 3D Fourier transform of the C$_{60}$ HOMO in the k$_\mathrm{||}$- k$_\mathrm{\bot}$ plane. This 2D cut already illustrates the complex intensity pattern of the 3D Fourier transform with which varies as a function of both k$_\mathrm{||}$ and k$_\mathrm{\bot}$. Consequently, small variations of the total momentum k$_\mathrm{final}$ of the electrons in the photoemission final state result in a different spherical cut through the 3D Fourier transform of the localized molecular orbitals. This in turn severely alters the theoretically predicted CE emission characteristics of the HOMO as shown for three exemplary CE maps of the HOMO simulated for three different momentum k$_\mathrm{final}$ in Fig. 6b. In contrast, the k$_\mathrm{\bot}$ dependency of the 3D Fourier transform is almost neglectable for planar molecules. Fig. 6c shows a k$_\mathrm{||}$- k$_\mathrm{\bot}$ cut through the 3D Fourier transform of the HOMO of the planar molecule PTCDA. The 3D Fourier transformed reveals only a weak intensity modulation along the k$_\mathrm{\bot}$-direction. Consequently, the simulated momentum-dependent photoemission yield of the PTCDA HOMO is almost independent of the total electron momentum k$_\mathrm{final}$ in the photoemission final state k$_\mathrm{final}$. The latter was recently demonstrated experimentally by Weiss et al. for PTCDA/Ag(110)[@Weiss.2015]. This comparison clearly underlines the crucial role of the final state momentum k$_\mathrm{final}$ for the PT of 3D molecules. The latter can be influenced either by the experimental uncertainty of the inner potential V$_0$ of the material or by small deviations of the initial state energy of molecular states in the band structure calculations and the experiment. In the case of C$_{60}$, the self-interaction errors in the band structure calculations result in a significant shift of the $\sigma$-states with respect to the experiment. We hence propose that this effect is responsible for the qualitative difference observed for the PT simulations and the experimentally obtained CE maps of the C$_{60}$ $\sigma$-states.

![(a) 2D k$_\mathrm{||}$ - k$_\mathrm{\bot}$ cut through the 3D Fourier transform of the C$_{60}$ HOMO. (b) Momentum-dependent photoemission yield of the C$_{60}$ HOMO for different total momentum k$_\mathrm{final}$ of the electrons in the photoemission final state. (c) and (d) show similar plots for the HOMO of the planar model molecule PTCDA.[]{data-label="fig:Fig6"}](Fig_SI6.pdf){height="9cm"}

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