Corrigendum: Electron pair escape from fullerene cage via collective modes

Scientific Reports 6: Article number: 24396; published online: 18 April 2016; updated: 25 August 2016 The original version of this Article contained a typographical error in the spelling of the author Paola Bolognesi, which was incorrectly given as Paolo Bolognesi. This has now been corrected in thePDF and HTML versions of the Article.

Scientific RepoRts | 6:24396 | DOI: 10.1038/srep24396 A standard single photoemission (SPE) theory usually relies on the hole spectral density, which accommodates so-called intrinsic energy losses, and the optical matrix elements. Plasmon-mediated processes are typical for extrinsic losses. These refer to all scattering events which the photoelectron undergoes before detection 24 . Formulating a theory for SPE valid for all types of electronic systems, proved to be an involved task. The perturbation theory for the transition dipole, as employed for atoms or molecules 25 is in principle able to incorporate both electron-electron scattering processes and also collective effects 26 . One may also attempt at a direct diagrammatic expansion of the observable photocurrent, as was put forward in ref. 27. A formal theory of DPE entails the use of many-body perturbation theory (MBPT) for two-particle propagators 15 and is thus even more involved. Based on the direct diagrammatic approach for the observable coincidence yield 28 we present here the first fully ab initio implementation for DPE accompanied by charge density fluctuations and compare with the first experiments of this kind on C 60 . Our approach is applicable to complex atoms such as Xe possessing strong collective resonances 29 , as well.
The emerging physical picture is illustrated in Fig. 2(a): (i) Photoabsorption promotes a valence electron to a high-energy state. (ii) This electron scatters inelastically from charge-density fluctuations (plasmon creation) that (iii) decay emitting a second valence electron (whose energy and angular correlations with the first one is measured in a coincidence set up, revealing so how charge-density fluctuations mediate e-e interaction). This three-step mechanism (3SM) emerges from a diagrammatic nonequilibrium Green's function (NEGF) approach as detailed in the supplementary information. It is already clear at this stage that DPE is qualitatively different from SPE in that, a) it delivers information on e-e interaction mediated by charge-density fluctuations, and b) as these plasmonic excitations are triggered by an electron a multitude of modes, e.g. volume plasmons, are involved.

Results
In Fig. 1(b) the electron pair coincidence yield versus the binding energy ω = − − B 1 2 ε ε ε of the doubly charged ion is reported and compared with the Auger spectrum. The binding energy of the latter is determined by the energy of the secondary electron and the carbon 1 s core level (see methods). The Auger process, which one might expect to be comparable to DPE when plotting as a function of the binding energy, can be interpreted in terms of the joint density of states (JDOS) as determined by the convolution of the density of occupied states of the neutral system,  D( ), and that of the ionized molecule, . Our ab initio calculations in Fig. 1(b) confirm this picture (note, these same D( )  and D are also part of DPE and are calculated with the same code). For plasmon-mediated DPE the situation is different. As inferred from Fig. 2(a), (ii), the spectral width of the plasmon modes is a determining factor for the width of the DPE spectrum. Which mode is active (and what is its multipolar nature) is set by the momentum balance that in turn points to the momentum region of the involved plasmons. The full ab initio calculations of multipolar plasmons in C 60 in ref. 30 enter as a part (i.e. steps (ii)-(iii) Fig. 2(a)) of our DPE calculations.
The electron pair coincidence yield is calculated following the derivation in the supplementary information. From Fig. 2 one infers that the non-local, frequency dependent screened electron-electron interaction W = v + vχv is a central quantity for DPE (v is the bare Coulomb interaction). As expected from the scheme in Fig. 2 the density-density response function χ(r, r′; ξ) is also the key factor for the electron energy loss experiments [31][32][33] and also for the screening of the optical field 34,35 by charge-density fluctuations in SPE (in our experiment this effect is negligible because the optical frequency is higher than the relevant plasmon resonances). We write the effective e-e interaction in the form eV, the normalized coincidence yield versus C 60 binding energy (red squares with error bars) is compared to the Auger spectrum with ω = 340 eV (black dots). The latter is compared to our calculations of the joint density of states (JDOS) (shaded blue line). Here, ξ denotes the frequency dependence, while λ represents a screening parameter discussed below. The collective modes are well characterized by the multipolarity L and a radial quantum number ν 30,36 . We account for symmetric surface (SS), the anti-symmetric surface (AS) and volume (V) modes (L = 0). The Lehmann representation of the response function is expressible as are known as fluctuation densities, which can be interpreted as the spatial distribution of the density oscillation associated to a particular plasmon ( Fig. 2(b)). The corresponding frequency spectrum is represented by B ν,L (ξ). For the radial profiles R ν,L (r) and plasmon spectra B ν,L (ξ) we utilize our recent approach from ref. 30 that yielded EELS spectra in very good agreement to experiments 33 . The static part in eq. (1) is written as . Previous calculations 37 provided an insight into the value of λ. The two-electron coincidence yield, averaged over the initial orientations of C 60 , reads Here, is the partial single-ionization cross section for a photoelectron with energy  k . The momenta of the two photoelectrons are denoted by k 1 and k 2 . The sum over n runs over all occupied states of the singly ionized molecule and is proportional to the angle-integrated and orientationaveraged (indicated by 〈 … 〉 c ) electron-impact ionization cross section 25,38 as calculated from the two-body matrix . Inspecting eq. (3) one identifies the steps (i)-(iii) sketched in Fig. 2(a). Note, due to rearrangement of the ionic core, the energy levels of the neutral molecule ( n ) are lowered by Δ when The computed coincidence photocurrent for the experimental setup of   = = .
10 7 1 2 eV is presented in Fig. 3(a) along with the data from the experiment. The equal energy-sharing case has been chosen by the experience on atoms, where this represents the case where the effects of the correlation and symmetry play a dominant role. Further tests (see supplementary information) show that, in contrast to the Auger process [ Fig. 1(b)], all ingredients of Eq. (3) (and hence all steps in Fig. 2) are essential: matrix elements effects encoded in  σ ω ( , ) k 0 , plasmon dispersions B ν,L (ξ), radial profiles of the fluctuations densities R ν,L (r), and density of states  A ( ) n . Hence, DPE in the present case adds new aspects to DPE from, e.g., atomic targets, and is a useful sensor for the e-e interaction mediated by charge-density fluctuations.

Discussion
The mechanism behind the narrowing of the DPE as compared to the Auger spectrum [ Fig. 1(b)] can be unraveled by analyzing the electronic structure and the individual plasmon modes as they contribute to DPE [ Fig. 3(b-e)]. Resolving the DPE yield with respect to either σ or π orbitals 39 [ Fig. 3(b)] one realizes that the emission from the σ band [ Fig. 3(c)], which is mainly responsible for the DPE signal at photon energies   ω 55 eV, is suppressed by the energy selectivity of the plasmon excitation. In particular, the plasmon giving rise to the emission of the second electron at stage (iii) needs to provide sufficient energy to promote a certain initial state of the + C 60 molecule [ Fig. 3(c)] to the continuum. Hence, the limited spectral width of the SS plasmon modes suppresses the emission from deeper σ states [ Fig. 3(b)]. A test calculation replacing the plasmon spectra B ν,L (ξ) by a constant produced a significantly broader DPE spectrum. This confirms the picture outlined above.
Our theory permits also to selectively include different plasmonic modes in the calculation. It is known that plasmon excitation upon photoabsorption obeys optical dipole selection rules and allows for exciting the SS plasmon with L = 1 only (the dipolar resonance mostly addressed in the literature and manifested e.g. in plasmon-assisted SPE 40 ). On the contrary, the electron scattering (as in EELS) transfers a finite momentum meaning that SS and AS plasmons with any multipolarity can be excited 30,33,36 . In agreement with this Fig. 3(d) underpins the significant contribution of the AS plasmons and thus substantiates the physical picture of the 3SM, according to which the step ii) can be regarded as an inelastic electron scattering event that is inherent to the plasmon-assisted DPE process. Similarly, SS dipolar plasmon transitions play only a minor role, while the multipolar plasmons are responsible to a large extent for the coincidence yield [ Fig. 3(e)]. All these facets endorse that DPE mediated by charge-density fluctuations as the predominant channel for e-e correlations represents a new facet to the information what is extractable from SPE and Auger spectra.
To summarize, an ab initio scheme for this process has been implemented with results in line with the first DPE experiment resolved with respect to the electron pair energies. We identified the dominant pathway as the following: a valence electron absorbs the photon and rescatters inelastically from multipolar collective modes that mediate the coherent emission of a second electron. The dwell time for this quasi-resonant scattering may be accessed by attosecond time-delay experiments 41 . For plasmon-assisted DPE the average electronic density plays a decisive role. For metals the plasmonic energies (which can be estimated using a classical expression ω = e m r 3 / e s pl 2 3 with r s being the Wigner-Seitz radius) are too low for plasmons to lead to a direct electron emission, although these modes may likely contribute to the loss channel for DPE. In contrast, for confined systems such as Carbon-based fullerenes the density is much higher (r s ≈ 1.0 a B ) resulting in plasmonic peaks in the XUV range. Thus,

Methods
Experiment. The experiments have been performed using the multi-coincidence end station 42 of the Gas Phase Photoemission beam line 43 at Elettra, where fully linearly polarized radiation in the photon energy range 13-1000 eV is available. The vacuum chamber hosts two independent turntables, holding respectively three and seven electrostatic hemispherical analyzers at 30° with respect to each other ( Fig. 1(a)). The three spectrometers of the smaller turntable, are mounted at 0°, 30°, and 60° with respect to the polarization vector of the light in the plane perpendicular to the propagation direction of the radiation. The larger turntable rotates in the same plane and its seven analyzers can be used to measure the angular distribution of the correlated electrons. The ten analyzers have been set to detect electrons of kinetic energy   = = .
10 7 1 2 eV. The energy resolution and the angular acceptance were ∆ = 300  meV and Δθ 1,2 = ± 3°, respectively. The photon energy resolution was about 150 meV. At variance with previous works [44][45][46] where the di-cation yield was measured versus photon energy, here the energy spectrum of the C 60 di-cation states is reconstructed by detection of photoelectron-photoelectron pairs in coincidence as the photon energy is scanned. In order to improve the statistical accuracy of the experimental results, the coincidence signals were added up, after a careful energy calibration of the non-coincidence spectra independently collected by the ten analyzers. The C 60 source is collinear with the photon beam 47 , which passes through the hollow core of the source before interacting with the molecular beam and ending up on the photodiode. Six apertures drilled into the closure piece of the crucible and pointing to the interaction region increase the molecule density therein.
In the Auger measurements the photon energy was fixed at ω = 340 eV and Auger electrons with kinetic energy    = − Auger C 1s B , where  C1s is the binding energy of the carbon 1 s core state and B  stands for the binding energy in Fig. 1(b) ranging from 15 to 45 eV, were measured. (3) is derived from the diagrammatic approach to photoemission 28 based on the nonequilibrium Green's function formalism. The full derivation is presented in the supplementary information. For an ab initio implementation of eq. (3) we rely on density functional theory (DFT) to compute the Kohn-Sham (KS) bound orbitals and their energies n  . We used the local density approximation (LDA) with self-interaction corrections. They improve the asymptotic behavior of the KS potential that is utilized to compute scattering states. The IPs and the core rearrangement shift Δ enter as experimentally determined 44,48

Theory. Equation
is computed by the driven-scattering approach 49 , yielding excellent agreement with literature data 50,51 in the relevant energy range [ Fig. 4(a)] of   ω 40 eV. Note that incorporating many-body effects is not required here (as they mainly influence the cross section around the plasmon resonances). The multipolar plasmon modes entering eq. (2) needed for computing the effective interaction (1) is parameterized according to previous calculations 30 and tested against EELS measurements in Fig. 4(b). Describing the Auger spectrum in Fig. 1(b) simply by the JDOS, thus neglecting plasmonic and other correlation effects, is justified by the large kinetic energy of the Auger electron, ruling out matrix-element effects in the considered energy window. Particularly, dynamical screening effects are strongly suppressed for a swift Auger electron due to the momentum-dependence of the density-density response function.
The accurate description of these central ingredients for describing DPE endorses the predictive power of the current theory. Full details on the calculations is provided by the supplementary information. (2) (symbols) compared to experimental data 33 (solid lines) for different scattering angles θ. The prefactor between theory and experiment was fixed for θ = 3° and kept constant for θ = 4° and θ = 5°.