Coherence of Auger and inter-Coulombic decay processes in the photoionization of Ar@C60 versus Kr@C60
Maia Magrakvelidze, Ruma De, Mohammad H. Javani, Mohamed E. Madjet, Steven T. Manson, Himadri S. Chakraborty
aa r X i v : . [ phy s i c s . a t m - c l u s ] D ec EPJ manuscript No. (will be inserted by the editor)
Coherence of Auger and inter-Coulombic decay processes in thephotoionization of Ar@C versus Kr@C Maia Magrakvelidze , Ruma De , Mohammad H. Javani , Mohamed E. Madjet , Steven T. Manson , and HimadriS. Chakraborty D.L. Hubbard Center for Innovation and Entrepreneurship, Department of Chemistry and Physics, Northwest Missouri StateUniversity, Maryville, Missouri 64468, USA, e-mail: [email protected] Department Physics and Astronomy, Georgia State University, Atlanta, Georgia 30303, USA Qatar Environment and Energy Research Institute, Hamad Bin Khalifa University, Qatar Foundation, P.O. Box 5825, Doha,Qatar Received: date / Revised version: date
Abstract.
For the asymmetric spherical dimer of an endohedrally confined atom and a host fullerene,an innershell vacancy of either system can decay through the continuum of an outer electron hybridizedbetween the systems. Such decays, viewed as coherent superpositions of the single-center Auger and two-center inter-Coulombic (ICD) amplitudes, are found to govern leading decay mechanisms in noble-gasendofullerenes, and are likely omnipresent in this class of nanomolecules. A comparison between resultingautoionizing resonances calculated in the photoionization of Ar@C and Kr@C exhibits details of theunderlying processes. PACS.
For a single-center system (generally an atom), the de-cay of an innershell electronic vacancy through an outer-shell continuum is the standard Auger process where the intra -Coulombic correlation enables local energy transferfrom the de-excitation to the ionization process. For multi-centered systems, like molecules, dimers or polymers, anon-local energy transfer can dominate, namely, the de-cay of a hole at one center, inducing the emission of anelectron from another - the inter -Coulombic decay (ICD)process [1,2]. This process is stronger and cleaner if thebonding between monomers are weak. Over last severalyears, considerable theoretical [3] and experimental [4,5]efforts have gone into ICD studies using rare gas dimers[6], rare gas clusters [7], surfaces [8], and water droplets[9,10]. Ultrafast ICDs of a dicationic monomer in a clusterto produce a cluster tricataion [11] or multiply excited ho-moatomic cluster [12] were predicted. Relatively recently,ICD following the resonant Auger decay is identified inAr dimers using momentum resolved electron-ion-ion co-incidence spectroscopy [13,14]. Furthermore, experimentsare also possible nowadays to probe the temporal aspectsof ICD mechanism in matters [15]. In fact, time domain a Current address: Department of Physics, Kansas State Uni-versity, Manhattan, Kansas 66502, USA measurements of ICD in He [16] and Ne [17] dimers haverecently been achieved. In the context of medical applica-tions, specifically radio-oncology, the low-energy ICD pro-cess was discussed [18,19].In this paper, we are interested in the resonant ICD(RICD) process where the initial vacancy is induced by theabsorption of a photon causing an innershell photoexcita-tion. Contemporary research has addressed various smallclusters and dimers to unravel effects of photon-stimulatedRICD. A prediction of strong RICD activities following Ne2 s → np excitations in MgNe clusters was made about adecade ago [20]. Experimentally, evidence for RICD wasseen in the photoelectron spectroscopy of Ne clusters for2 s → np excitations [21], and also in the double photoion-ization of Ne dimers by monitoring the creation of ener-getic Ne + [22]. Strong enhancement of the HeNe + yield,as He resonantly couples with the radiation, was recentlydetected [23], confirming an earlier prediction [24].Atoms endohedrally confined in fullerene molecules,endofullerenes, being near-spherical, atom-cluster dimersof loose Van-der-Walls type bonding have attracted signif-icant attention as natural laboratories for ICD research.These materials are stable in the room temperature withinexpensive sustenance cost and their synthesis techniquesare also rapidly improving. The earliest attempt to predictICD in endofullerenes was made by calculating ICD ratesfor Ne@C [25]. This was followed by some studies of Maia Magrakvelidze et al.: Auger-ICD coherence in Ar@C vs Kr@C Coulomb-mediated energy transfer from atom to fullerenethat broaden the Auger lines [26,27]. Systems supportingregular RICD can be visualized as antenna-receiver pairsat the molecular scale [23] where the antenna couples tothe incoming radiation and transfers energy to the receiverto perform an act of emission. Very recently, a differentclass of resonances decaying into atom-fullerene hybridfinal state vacancies for the photoionization of Ar@C have been predicted; this arises from the competition ofthe intra-Coulombic Auger channel with an intrinsicallyconnected ICD channel [28]. The calculated features werefound to be remarkably stronger than both regular ICDand Auger resonances. Obviously, these processes can ac-centuate the system efficiency by enabling the antennato also contribute to the emission resonantly with the re-ceiver through the coherence. Therefore, given that thiseffect may have utilization in nanoscale antenna technol-ogy [29] besides its established basic-science context, it isof great interest to investigate if such coherence phenom-ena are a common place energy-transfer mechanism in thespectroscopy of endofullerenes.To this purpose, we extend our calculations to a num-ber of noble gas endofullerenes and find strong abundanceof such coherence in the spectral landscape of these sys-tems. In this paper, we compare between the results ofAr@C and Kr@C to uncover details of the process;particularly, the dependence of the spectral features onthe choice of the encapsulated atom. Section 2 carries twosubsections, providing a short account of the theoreticalmethod, predicting atom-fullerene hybridization, and aninterchannel coupling based description of the Auger-ICDcoherence. Section 3 presents the final numerical results ofthe resonances with discussions. We conclude the paper inSection 4. Kohn-Sham density functional theory is used to describethe ground state electronic structure of the compoundsusing same methodology employed earlier [30]. The C molecule is modeled by smearing sixty C ions into aspherical jellium shell, fixed in space, with an experimen-tally known C mean radius 3.5 ˚Aand thickness ∆ , aug-mented by a constant potential V . The nucleus of theconfined atom is placed at the center of the sphere. TheKohn-Sham equations for the system of a total of 240 + N electrons ( N = 18 for Ar, N = 36 for Kr and 240 delo-calized electrons from C ) are then solved to obtain theelectronic ground state properties in the local density ap-proximation (LDA). The gradient-corrected Leeuwen andBaerends exchange-correlation functional [LB94] [31] isused for the accurate asymptotic behavior of the groundstate radial potential V LDA ( r ) = − zr + Z d r ′ ρ ( r ′ ) | r − r ′ | + V XC [ ρ ( r )] , (1) which is solved self-consistently in a mean-field frame-work. The parameters V and ∆ are determined by requir-ing both charge neutrality and obtaining the experimentalvalue, 7.54 eV, of the first C ionization potential. Thisprocedure yields a value of ∆ of 1.3˚A, in agreement withthe value inferred from experiment [32].Significant ground state hybridization of atomic va-lence orbitals np ( n = 3 , p is found, resulting in (X np ± C p ) levels from thesymmetric and antisymmetric mixing similar to the bond-ing and antibonding states in molecules or dimers:X np ± C p = | φ ± i = √ α | φ np X i±√ − α | φ p C i , (2)where X denotes Ar and Kr. The radial wavefunctionscorresponding to these levels and their binding energiesare shown in Figure 1(a). Note that in the Ar case theanti-symmetric combination induces one fewer node andis more strongly bound compared to the symmetric, whilethe opposite is true for Kr@C . Such atom-fullerene hy-bridization was predicted earlier [33] and detected in aphotoemission experiment on multilayers of Ar@C [34].In fact, the hybridization gap of 1.52 eV between (Ar+C )and (Ar − C ) in our calculation is in good agreement withthe measured value of 1.6 ± nℓ @ to denote the levels of theconfined atom and @ nℓ to represent the levels of the dopedC .A time-dependent LDA (TDLDA) approach [35] is usedto calculate the dynamical response of the compound tothe external dipole field z . In this method, the photoion-ization cross section corresponding to a bound-to-continuumdipole transition nℓ → kℓ ′ is σ nℓ → kℓ ′ ∼ |h kℓ ′ | z + δV | nℓ i| , (3)where the matrix element M = D + h δV i , with D beingthe independent-particle LDA matrix element. Here δV represents the complex induced potentials that accountfor electron correlations. In the TDLDA, z + δV are pro-portional to the induced frequency-dependent changes inthe electron density [35]. This change is δρ ( r ′ ; ω ) = Z χ ( r , r ′ ; ω ) zd r , (4)where the full susceptibility χ builds the dynamical cor-relation from the LDA susceptibilities, χ ( r , r ′ ; ω ) = occ X nl φ ∗ nl ( r ) φ nl ( r ′ ) G ( r , r ′ ; ǫ nl + ω )+ occ X nl φ nl ( r ) φ ∗ nl ( r ′ ) G ∗ ( r , r ′ ; ǫ nl − ω ) (5) via the matrix equation χ = χ [1 − ( ∂V /∂ρ ) χ ] − in-volving the variation of the ground-state potential V withrespect to the ground-state density ρ . The radial compo-nents of the full Green’s functions in Eq. 5 are constructed aia Magrakvelidze et al.: Auger-ICD coherence in Ar@C vs Kr@C -0.300.30.60.9 E x c it e d w a v e f un c ti on ( a . u . ) Ar 4pAr 4p@ (hybrid)
Radial coordinate (a.u.) -0.9-0.6-0.300.30.6 G r ound e n e r gy / w a v e f un c ti on ( a . u . ) Ar 3p + C Ar 3s@ (b)(a)
Ar@C potential X 0.5 Ar+Ar- Kr+Kr- (-1)(Ar-) ~ Kr+
Fig. 1. (Color online) (a) Ground state radial wavefunctionsand binding energies of Ar@C and Kr@C hybrid levels;X ± are used for short-hand notations for these levels where Xis Ar or Kr. That the inverted Ar- is quite similar to Kr+ isnoted. The radial potential of Ar@C is also shown. (b) Theexcited 4 p wavefunctions of free and confined Ar, and the inner3 s wavefunction of confined Ar are plotted. with the regular ( f L ) and irregular ( g L ) solutions of thehomogeneous radial equation (cid:18) r ∂∂r r ∂∂r − L ( L + 1) r − V LDA + E (cid:19) f L ( g L )( r ; E ) = 0(6)as G L ( r, r ′ ; E ) = 2 f L ( r < ; E ) h L ( r > ; E ) W [ f L , h L ] (7)where W represents the Wronskian and h L = g L + i f L .Obviously, TDLDA thus includes the dynamical correla-tion by improving upon the mean-field LDA basis. The TDLDA matrix elements M for the dipole photoion-ization of (X ± C ) levels, in the interchannel couplingframework introduced by Fano [36], can be written as [37], M ± ( E ) = D ± ( E ) + M c − c ± ( E ) + M d − c ± ( E ) , (8)where the single electron (LDA) matrix element D ± ( E ) = h ks ( d ) | z | φ ± i ; M c − c and M d − c are respectively correctionsfrom continuum-continuum and discrete-continuum chan-nel couplings, accounting for h δV i in Eq. 3. M c − c consti-tutes rather smooth many-body contribution to nonreso-nant cross section, while the resonance structures originate Ar + C Fig. 2. (Color online) Schematic of coherent mixing of one-center Auger decay amplitudes (green) of core vacancies withcorresponding cross-center ICD amplitudes (red) in the spectraof the Ar-C hybrid electron. See text for a fuller description. from M d − c . Following [36], M d − c ± = X nℓ X ηλ h ψ nℓ → ηλ | | r ± − r nℓ | | ψ ± ( E ) i E − E nℓ → ηλ D nℓ → ηλ , (9)in which the | ψ i refer to interacting discrete (inner) nℓ → ηλ and continuum (outer) X ± C p → ks ( d ) channel wavefunctions; E nℓ → ηλ and D nℓ → ηλ are LDA bound-to-bound excitation energies and matrix elements, respec-tively. The two-body interchannel coupling matrix ele-ments (ICME) of the Coulomb interaction in Eq. 9 arethe conduits of the energy transfer process between chan-nels. The excited states of the system are found hybridized[see Fig. 1(b)], implying that innershell electrons from purelevels are excited to the hybrid levels. But we do not ex-pect significant differences in D s → ηp between free andconfined Ar. This is because, even though hybrid excitedstate wavefunctions have induced structures in the vicinityof the C shell, the Ar 3 s @ wavefunction being un-mixedand localized on Ar [Fig. 1(b)] means that its overlap withthe excited state wavefunctions is largely unaffected by thehybridization. An identical reason also ensures that thedoped C ’s innershell excitation LDA matrix elementsare also essentially unchanged.Following Eq. 2, the hybridization of the continuumchannels in Eq. 9 assumes the form | ψ ± i = √ α | ψ np @X i ± √ − α | ψ @3 p C i , (10)where ψ np @X and ψ @3 p C are the wavefunctions of thechannels arising, respectively, from the valence np level ofthe atom and the 3 p level of C . In Eqs. (10) we used@ to indicate the continuum waves in confined Ar anddoped C . Using Eq. (10) in (9), and recognizing that theoverlap between a pure Ar and a pure C bound state is Maia Magrakvelidze et al.: Auger-ICD coherence in Ar@C vs Kr@C negligible, we separate the atomic and fullerene regions ofintegration as M d − c ± ( E ) = X nℓ X ηλ " √ α h ψ nℓ → ηλ | | r ± − r nℓ | | ψ np @X ( E ) i E − E nℓ → ηλ ± √ − α h ψ nℓ → ηλ | | r ± − r nℓ | | ψ @3 p C ( E ) i E − E nℓ → ηλ D nℓ → ηλ . (11)Eq. 11 can be schematically understood using Figure2 with the example of Ar@C . If nℓ → ηλ produces anAr innershell hole, then the de-excitation process (blackarrow) releases energy that can transfer to a hybrid levelas if into two branches (thick curved arrows): (i) The first(green) is a local transfer that liberates the atomic part ofthe hybrid electron denoted by the first term in Eq. (11).This represents the ordinary Auger decay in Ar. (ii) Thesecond (red) is a non-local Ar-to-C ICD energy transferthat knocks off the C part of the same electron repre-sented by the second term in Eq. (11). The partial elec-trons are denoted by the checkered spheres in Fig. 2. Forthe photoionization cross section, which involves the mod-ulus squared of the matrix element, these Auger and ICDcomponents of the amplitude coherently mix to induce theresonance, resulting in a shared (hybridized) outershell va-cancy. Likewise, for the de-excitation (blue arrow) of anoriginal C innershell hole, the first and second termsin Eq. (11) indicate the coherence between, respectively,a C -to-Ar ICD and a C Auger process. Hence, thesedecay pathways can be termed a resonant hybrid Auger-inter-Coulombic decay (RHA-ICD).
Figure 3(a) presents some selected Auger resonances forfree atoms and the empty C molecule. For each atom,the two lowest ns → np resonances in the valence crosssections are denoted by A and B. These are Ar 3 s → p, p and Kr 4 s → p, p . These resonances are characteris-tically near-symmetric, window-type. Note that for eachatom, the resonance width becomes significantly smallerwith higher final states. This behavior relates to the prop-erty of the excited orbitals, bulk of whose amplitudes pro-gressively move farther in the radial coordinate to weakenthe overlaps in ICME similar in Eq. 9, because it involvesa two-electron Coulomb operator. Furthermore, this prop-erty of decreasing width with increasing excitation followsdirectly from quantum defect theory [38] which shows thatthe widths drop off as 1/n ∗ with increasing n ∗ , the ef-fective principal quantum number. Note further that fiveC Auger resonances are also identified in Fig. 3(a) in thecross section of C p (that participates in the hybridiza-tion [Eq. 2] in endofullerenes) which are of diverse shapeswith 1-4 being strongly asymmetric and a near-symmetric5. As a rule, C Auger resonances are narrower thanatomic resonances [35] which follows directly from the de-localized behavior of C electrons since their being dif-fused in radial space produces smaller rates via the ICME. Figure 3(b) displays the photoionization cross sections,over the same energy range as Fig. 3(a), of the endofullerenehybrid levels (X ± C ). The labeled resonances in this panelare the RHA-ICD “avatars” of the free-system Auger struc-tures in Fig. 3(a), and they occur almost at the same en-ergies for each hybrid. This means that features A, B,and C (C only for Ar) in Fig. 3(b) are resonances thatemerge from the decay of Ar 3 s @ → p @ , p @ , p @ and Kr4 s @ → p @ , p @ excitations through the continua of thesehybrid levels. Remarkably, they are significantly stronger,particularly for anti-symmetric (X-C ), than their Augercounterparts seen in Fig. 3(a). In addition, another dra-matic effect is evident: The resonances 1-5 in Fig. 3(b),decaying through the hybrid continua, grow to an orderof magnitude larger than the Auger resonances in emptyC [Fig. 3(a)]. In essence, Ar and C innershell vacanciesdecay significantly more powerfully through the photoion-ization continua of the X ± C hybrid levels than they dothrough the continua of pure levels. To understand whythis happens, note that both the terms in Eq. (11) arelarge, owing to the substantial overlaps between inner-shell bound states and (X ± C )3 p channel wavefunctions.But there is more. The resonances in the matrix element M d − c ± also coherently interfere with the nonresonant part D ± + M c − c ± , [Eq. (8)] which is generally stronger for hy-brid levels [39]. This interference, consequently, enhancesRHA-ICD resonances compared to their Auger counter-parts in free A or empty C channels, as seen in Fig. 3(b).Resonances A and B [Fig. 3(b)] of the confined Krmove lower in energy than the corresponding resonances[Fig. 3(a)] for the free Kr, and this shift is greater for res-onance B. For Ar too, such an energy red-shift is notedin moving from free to confined, but, in contrast to Kr,the shift in A in Ar is substantially larger. In order tounderstand this behavior, note first in Fig. (3) that theeffect of confinement blue-shifts the inner Ar 3 s @ whichis opposite to Kr 4 s @ that red-shifts under confinement.This can be explained as follows: In the compound sys-tem, the atom and C exert mutual perturbations oneach other. As a result, the general shift of energy lev-els from their “unperturbed” values is a function of twoeffects: (i) the addition of two attractive potentials shouldtend to make the levels more bound and (ii) the repul-sion between the atomic and C electron-groups in theself-consistent mean field induces just the opposite effect.It turns out that in our LDA ground state calculations,the former wins for Ar@C but the later for Kr@C ,since Kr adds a significantly larger number of electrons tothe combined system. This enables Ar 3 s @ (-30.1 eV) andKr 4 s @ (-26.5 eV) to become, respectively, more and lessbound than their free results (-29.6 eV and -27.6 eV, re-spectively). In fact, this effect also shifts the entire excita-tion spectra in a similar fashion that can be seen in Table1 where we determine the excited state energies from theresonance positions in Fig. (3) both for free and confinedatoms. Note that free Ar and Kr excited state energiesare essentially equal since their p-wave quantum defectsdiffer by almost exactly 1 [40,41,42], so it is only the con-finement that somewhat complicates the results in Table aia Magrakvelidze et al.: Auger-ICD coherence in Ar@C vs Kr@C Ar 3p + C
25 26 27 28 29 30Photon energy (eV)10 Ar 3pKr 4pC A - K r B - K r K r s A r s K r s @ A r s @ A - A r B - A r C r o ss s ec ti on ( M b ) A - K r B - K r A - A r B - A r C - A r (a)(b) Fig. 3. (Color online) (a) Photoionization cross sections of free Ar 3 p and Kr 4 p featuring two lowest Auger resonances for Ar3 s → p, p (A-Ar, B-Ar) and Kr 4 s → p, p (A-Kr, B-Kr) innershell vacancies. Five Auger resonances, labeled 1-5, in C p cross section are also shown. (b) The RHA-ICD counterparts of the resonances of panel (a) in the cross sections of hybridelectrons. Color-coded label tags (black for Ar and red for Kr) are used to guide the eye in identifying these resonances. Freeand confined Ar 3 s and Kr 4 s ionization thresholds, to which the excitation series converge, are indicated in the respectivepanels. Table 1.
Energies (in eV) of free and confined excited statesarising from ground Ar 3 s and Kr 4 s levels calculated from thepositions of A and B in Fig. (3).Ar Ar@ Kr Kr@A -2.4 (4p) -4.8 -2.4 (5p) -1.5B -1.0 (5p) -1.7 -1.0 (6p) -0.5
1. We should also keep in mind that the excited stateenergies for the compound systems are affected also bythe orbital hybridization, resisting any simple systematicsin the shift and, thus, in the positions of resonances Aand B in Fig. 3(b). Furthermore, Figs. 3(a) versus (b) in-dicates that the positions of 1-5 RHA-ICD resonances forAr@C practically reproduce the positions of correspond-ing Auger lines, while those for Kr@C are systematicallyred-shifted. Obviously, this is also due to the significantlylarger electron-repulsion effects resulting in the lowering ofground-state level-energies in C when the central atomis Kr, as discussed above.It is also evident in Fig. 3 that the RHA-ICD reso-nances A-C roughly retain the symmetric window shapesof their free atom Auger counterparts. This is becausetheir atomic Auger components are playing the dominantrole in the coherence. But the effect of the ICD com-ponents is also evident, for instance, in the narrowing of the width of resonances A from their free atom re-sults; a forthcoming study of the Fano-shape fitting of allthe resonances will reveal the details [43]. On the otherhand, significant variations are noted for resonances 1-5 from the coherence. For resonances 1-4, (Ar-C ) and(Kr+C ) exhibit strong asymmetric shapes similar tothe resonances in empty C . But for other two hybridpartners, (Ar+C ) and (Kr-C ), shapes are more nearlysymmetric, while the former produces a minimum andthe later a maximum. For resonance 5, all four RHA-ICDresonances are asymmetric, although the shape similar-ity between the bonding hybrid of one system with theanti-bonding of another is retained. Since the excitationchannel nl → ηλ in Eq. 11 is unchanged, this behavior ofshape equivalence between (Ar ∓ C ) and (Kr ± C ) for 1-5 must depend on the properties of the continuum chan-nel to affect the ICME. Indeed, the primary reason forthis behavior lies in the approximate reflection symmetrybetween the corresponding hybrid wavefunctions. As ev-ident in Fig. 1(a), multiplying the (Ar-C ) wavefunctionby negative 1inverts it to a shape which is close to that of(Kr+C ). One can easily check that this reflection prop-erty also holds between the other two hybrids. Obviously,the choice of the caged atom alters the details of the hy-bridization which subsequently influences the RHA-ICDcoherence to determine the resonance shapes. Maia Magrakvelidze et al.: Auger-ICD coherence in Ar@C vs Kr@C In conclusion, we used the TDLDA methodology to calcu-late a class of innershell-excitation single-electron autoion-izing resonances in the photoionization of Ar@C andKr@C , decaying into atom-fullerene hybrid final statevacancies. It is demonstrated that these resonances, aris-ing from the interference of the intra-Coulomb autoion-izing channel with a coherently admixed inter-Coulombchannel. These resonances are found to be significantlystronger than both regular ICD and Auger resonances,which make them well amenable for experimental detec-tion. The detailed analysis of the results divulge variousspectral similarities and differences in the position andshape of the resonances as a function of the central atom.The results indicates that such coherent energy transferprocesses must exist across the periodic table when the el-ement supports endofullerene formations, since atom-C hybridization is likely to be the rule, not the exception,in the electronic structure of these materials [33,39,44].The current work addresses the participant decay pro-cesses where the excited electron itself drops on to thecore-hole. However, the decay of a hole annihilated bya different electron, the spectator process, can also con-tribute in a RHA-ICD pathway, suggesting its generality.Further, these hybrid decay processes are also likely topervade in the ionization continuum of molecules, nano-dimers and -polymers, and fullerene onion systems thatsupport hybridized electrons as well. In a related context,the attosecond time delay studies of the photoemission ofthese RHA-ICD resonances can lead to the understandingof the role of electron correlation from a temporal frame-work which attracted some recent interest [45]. This work is supported by NSF and DOE, Basic Energy Sci-ences.
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