Collective effects in photoionization of sodium clusters: plasmon resonance spill, induced attractive force and correlation minimum
aa r X i v : . [ phy s i c s . a t m - c l u s ] O c t Collective effects in photoionization of sodium clusters: plasmon resonance spill,induced attractive force and correlation minimum
Rasheed Shaik, Hari R. Varma, ∗ and Himadri S. Chakraborty † School of Basic Sciences, Indian Institute of Technology Mandi, Kamand, H.P. 175075, India Department of Natural Sciences, D.L. Hubbard Center for Innovation and Entrepreneurship,Northwest Missouri State University, Maryville, Missouri 64468, USA (Dated: October 13, 2020)Photoionization studies of Na and Na clusters are carried out in a framework of linear re-sponse density functional theory. Cross sections show substantial spillover of plasmon resonancesto the near-threshold ionization energies which are in reasonable agreements with measurements.The analysis of the oscillator strength, consumed by the cross section, lends further insights. Themany-body interaction induced self-consistent field from density fluctuations suggests the existenceof an attractive force. This may cause time-delayed plasmonic photoemissions in ultrafast measure-ments. At the waning end of the plasmon structure, a strong minimum in the cross sections from acorrelation-driven coherence effect is predicted which can possibly be observed by the photoelectronspectroscopy. I. INTRODUCTION
The physics of atomic clusters has gained broad impor-tance as a domain of study of new physical objects, oftentermed as super-atoms, over the past few decades [1].Such clusters are aggregates of atoms, ranging from twoto a few thousand of atoms, and provide the platformto study a new phase of matter intermediate to atomsor molecules and solids [2]. These systems also offervaluable opportunities to understand how the bulk prop-erties emerge from their individual constituent atoms [3].Possibilities of designing new class of materials with tai-lor made characteristics using clusters as building blocks,instead of atoms, have rendered this field diverse and vi-brant [4]. The extensive research on clusters over theyears has revealed many unusual properties which are ofinterests to the field of physical and chemical sciences,material sciences, and biological sciences, making theseexplorations interdisciplinary in nature [5].Many intriguing phenomena are associated with thephotoresponse spectrum of atomic clusters [6]. Theseinclude phenomena, such as, plasmon resonances [7–9], Fano type resonances [10], and diffractive modula-tions [11] in the photoelectron signal due to the largelywell-defined cluster edges [12]. One primary focus of thepresent work is the photoionizing response of the plasmonresonances - resonances that form due to the collectiveoscillations of the valence electron cloud. Excitation ofa lower energy giant surface plasmon resonance is knownto be a prominent feature in the photoresponse of alkalimetal clusters below the ionization threshold. This fea-ture in metal clusters has applications in nano-optical de-vices, chemical and biological sensing, and bio-medicineetc [13–15], besides its eminent role as “spectral labora-tories” to assess many-electron effects. Therefore, devel- ∗ [email protected] † [email protected] oping detailed and accurate understanding of the origin,the underlying dynamics, and related observable effectsof this resonance are a matter of significant priority.In addition, the presence of a higher-energy volumelike plasmon makes the photo-spectrum of atomic clus-ters [7] more robust compared to the spectrum of thecorresponding bulk metal. It is well known that due tothe translational invariance, the coupling of the volumeoscillation in bulk to light is not feasible [16]. However,the situation is different for the case of finite systems likemetal clusters due to the broken translational invariancethat may cause boundary reflections of the plasma wave.As a result, optically active volume plasmon resonancecan also become possible. The presence of such a volumeplasmon excitation is observed for the first time in the ex-perimental work of Xia et al. [7]. Their photon depletionmeasurements of Na and Na clusters showed a broadvolume plasmon resonance mounted on the decay ridgeof giant surface plasmon and peaking slightly above 4 eV.The tail of this combined structure was found extendedto the ionization region. The present calculation is moti-vated to address this spillover part of the plasmon struc-ture closely above the ionization threshold. In studyingthis, we also predict a universal minimum at the decayend of the structure from an interchannel coupled phase-coherence effect arising from many-electron correlations.Furthermore, an analysis of the many-body induced self-consistent field potential at the plasmon spillover energiespoint to a correlation driven attractive force that suggestspossible time delays of the emerging photoelectron.The dynamical response of the Na clusters to the exter-nal electromagnetic radiation is calculated using a linearresponse density functional theory (DFT) approach. Afairly competent method belonging to this class is thejellium based time-dependent local density approxima-tion (TDLDA) [17]. In the past, application of TDLDAto atomic clusters is found to be successful in explainingthe collective phenomena occurring at low energies [8]and also the diffractive oscillations present at high ener-gies [12]. The ease and transparency of a jellium basedmodel enable physicists to interpret the key physics thatdetermines the dynamics. A recent study on fullerenesindicated that a better agreement with experiment can beobtained by using one such DFT approach with Leeuwen-Baerends (LB94) exchange-correlation functional thatproduces the correct ground state asymptotic proper-ties [18]. Another study has tested the efficacy of themethod for Na clusters [19]. Hence, this method is cho-sen in the present work.The paper is organized in the following way. A basicdescription of the theoretical methodology is provided inSection II which has two parts. The details of groundstate structure in the spherical jellium formalism alongwith a brief account of the LB94 parametrization schemeis provided in Sec. II A. A brief description of the methodthat incorporates electron correlations in response to theradiation is given in Sec. II B . Section III constitutes theresults and discussions of: photoionization cross sectionsand its comparison with experiment (Sec. III A), thephotoelectron oscillator strength (Sec. IIIB), the many-body correlation induced potential in the spillover regionof plasmon (Sec. III C), and the prediction of a “corre-lation minimum” at the high-energy end of the plasmonstructure (Sec. III D). Finally, Section IV concludes thestudy. II. THEORETICAL METHODOLOGYA. Ground states of Na and Na To investigate the ground state electronic structure ofNa n ( n =20 and 92), a DFT approach is adopted. For suchsystems of closed-shell configurations the jellium modelserves as a very good approximation. In this model, thejellium potential ( V jel ( r )) replaces the ionic core of 20and 92 Na ions, respectively for Na and Na , by po-tentials constructed after homogeneously smearing theirpositive charges into jellium spheres. The radius of eachcluster is determined by the number n of ions presentin the system. The radius of Na is calculated to be10.67 a.u. and that of Na to be 17.74 a.u. using theformula R c = r s n / , where r s is the Wigner-Seitz radius(3.93 a.u.) of a Na atom. The Kohn-Sham equations for n delocalized valence electrons, a 3s electron from eachNa atom, are solved to obtain the ground state structuresof Na and Na . It is to be noted that to match thevalence ionization thresholds with the experimental val-ues [20], suitable constant pseudo potentials are added.In terms of the single-particle density ρ ( r ), the groundstate self-consistent field LDA potential reads as, V LDA ( r ) = V jel ( r ) + Z d r ′ ρ ( r ′ ) | r − r ′ | + V xc [ ρ ( r )] , (1)where the second and third terms on the RHS arerespectively the direct and exchange-correlation (xc) -7-6-5-4-3-2-1 0 5 10 15 20 25 30 (b) R a d i a l po t e n ti a l ( e V ) Radial co-ordinate (a.u.) Na LB94Na LB94 -0.8-0.6-0.4-0.2 0 0.2 0.4 0.6 0.8 (a) R a d i a l o r b it a l ( a . u . ) HOMO(2s),Na LB94HOMO(1h),Na LB94 (a)
HOMO-1(1d),Na LB94HOMO-1(3s),Na LB94
FIG. 1. a) Ground-state radial wavefunctions for HOMO andHOMO-1 levels calculated for Na and Na . b) Ground-state radial potentials and energy levels calculated for Na and Na . potentials. Approximate form for V xc has to be usedsince its exact form is unknown. In the present work,we approximate the V xc with the LB94 functionalwhich provides an accurate asymptotic description ofthe ground-state properties. This functional belongsto a gradient corrected class of approximation by vanLeeuwen and Baerends, parameterized in terms of the re-duced density and its gradient ∇ ρ ( r ) [21], to be given by, V LB = − β [ ρ ( r )] / ( ξX ) βξX sinh − ( ξX ) , (2)where β =0.01 is a fitting parameter and X = [ ∇ ρ ] /ρ / .The parameter ξ arises due to the change of systemfrom spin-polarized to spin-unpolarized form [22]. Thisscheme of gradient correction of density is derived from agradient expansion series that naturally eliminates self-interactions and produces the correct asymptotic behav-ior.The calculated binding energies of the highest occupiedmolecular orbitals (HOMO) and (HOMO-1) of these sys-tems, of 2s and 1d character respectively for Na and 1h TABLE I. Binding energies (BE) of HOMO and HOMO-1levels of Na and Na in the harmonic oscillator notation.BE HOMO (eV) BE
HOMO-1 (eV)Na s − .
75 1 d − . h − .
47 3 s − . and 3s respectively for Na , are given in Table I and theirradial wavefunctions are shown in Fig. 1(a). It can beseen that HOMO level of Na has a single node whereasNa is nodeless. The situation reverses for HOMO-1.As a result, the radial structures of HOMO and HOMO-1 wavefunctions significantly differ in these two systems.Fig. 1(b) shows the ground-state radial potentials of Na and Na clusters and the level energies. B. Dynamical Response
The dynamical response of the cluster subjected to theexternal dipole field ‘ z ’ can be calculated by employing alinear-response DFT scheme, known as time-dependentLDA (TDLDA) modified by LB94 [23]. The dipole fieldcan induce a frequency-dependent complex change in theelectron density δρ [24], which, within a linear-responseperturbative framework, can be written as: δρ ( r ′ ; ω ) = Z χ ( r , r ′ ; ω ) zd r , (3)where χ is the full susceptibility of the system whichincorporates all the dynamical electron correlations. Theabove equation can be re-written in terms of independentparticle (IP) susceptibility, χ , and the complex field δV as, δρ ( r ′ ; ω ) = Z χ ( r , r ′ ; ω ) δV ( r ; ω ) d r , (4)where χ is constructed by the ground-state single-electron orbitals φ nl and energies ǫ nl [25]. The IP sus-ceptibility χ is related to χ by the matrix equation: χ = χ [1 − ( ∂V /∂ρ ) χ ] − . (5)The total field, δV , is given by the following relation: δV ( r ; ω ) = z + V ind ( r ; ω ) , (6)where V ind ( r ; ω ) = Z δρ ( r ′ ; ω ) | r − r ′ | d r ′ + h ∂V XC ∂ρ i ρ = ρ δρ ( r ; ω ) . (7)Using the matrix inversion method [26], Eq. 5 is finallysolved for χ which in turn is used for obtaining δρ andthereby δV by making use of Eq. 3 and Eq. 6 in a self-consistent way. In this formalism, the photoionization cross sectioncorresponding to a bound-to-continuum dipole transition nℓ → kℓ ′ is then calculated using σ nℓ → kℓ ′ ∼ |h kℓ ′ | δV | nℓ i| . (8)It is clear from the above equation that in addition tothe external perturbation z , the calculation involves thecomplex induced field V ind driven by the many-electroncorrelations. Obviously, setting δV = z yields the inde-pendent particle LDA cross section that ignores corre-lations. A comparison of LDA and TDLDA, therefore,easily possible in this method to study the role of many-electron effects in the photoionization process. −8 −6 −4 −2
10 100 Na (b) Photon Energy (eV) T o t a l c r o ss − s ec ti on ( a . u . ) −8 −6 −4 −2 Na (a) TDLDA−LB94LDA−LB94
FIG. 2. TDLDA and LDA cross sections of Na (a) and Na (b). III. RESULTS AND DISCUSSIONA. Photoionization cross-sections and comparisonwith measurements
Total photoionization cross-section of Na and Na clusters, calculated in TDLDA+LB94 and the corre-sponding LDA results, are shown in Fig. 2(a) andFig. 2(b) respectively. As seen, the TDLDA cross sectionprofiles, as a function of photon energies, strongly differfrom the LDA profiles by (i) significant enhancementsnear the ionization threshold region, (ii) the presence ofautoionization resonances, and (iii) the appearance of aminimum at about 7.20 eV for Na and 6.22 eV for Na .The near-threshold enhancement in TDLDA cross sec-tions combines the remnant of the giant surface plasmonresonance, that flourishes below the ionization thresh-old, and a major portion of the broader volume plasmonresonance, both emerging from the collective electronicmotions when subjected to an external electromagneticfield. This resonance spillover from the discrete to contin-uum spectrum is embedded by a host of narrow spikes,which are the autoionization resonances resulting fromthe degeneracy of ionization channel with the inner-levelsingle-electron discrete excitations [27]. As seen, thesefeatures are completely missing in the LDA predictionswhich neglect the electron correlations. Structures in theLDA profiles in this near-threshold region are due to thegradual openings of inner-level ionization channels. −6 −5 −4 −3 −2 −1
2 3 4 5 6 7 8 Na (b) Photon Energy (eV) T o t a l c r o ss − s ec ti on p e r a t o m ( Å ) −6 −5 −4 −3 −2 −1 Na (a) Expt. PRL102,156802(2009)TDLDA−LB94Lorentzian−fit
FIG. 3. TDLDA cross sections along with experimental datafrom [7] for Na (a) and Na (b). Lorentzian fits to theexperimental cross sections are also shown. The existence of the volume plasmon resonance and
TABLE II. Resonance positions ( E o ), FWHM (Γ) of thehigher energy plasmon (HEP) [7]. E o (eV) Γ (eV)Na HEP 4.04 1.19Na HEP 4.20 1.16 the extended tail of the surface resonance to the ioniza-tion part is experimentally seen in the work of [7]. Com-parisons of our calculations with these measured datais shown in Fig. 3(a) for Na and Fig. 3(b) for Na .The data covers the range of 2 eV to 5.64 eV. FollowingRef. [7], fits to the experimental data as a sum of fourLorentzian profiles for Na and a sum of three for Na are also shown in Fig. 3. The lower energy Lorentziansin both cases represent the giant portion of the plasmonbelow the first ionization threshold that our photoion-ization results can not access. The present TDLDA cal-culations span the ionization part of the spectrum onlyand, as noted, overlap very well with the higher energyLorentizian fits used for both the clusters. The peak en-ergy position (E ) and width (Γ) of the fourth Lorentiziancurve for the higher energy plasmon resonance (HEP) forNa and the corresponding position and width for thethird Lorentzian curve for Na are shown in Table II. Itis clear from the figure that the steady background partsof TDLDA cross sections agree well with the experimentdescribing the spillover of the plasmon resonances. Asthe size of the cluster increases the peak positions of thevolume plasmon shifts to higher energies (blue shifted).Curiously, this size effect is found to be opposite in thecase of fullerenes (C and C ) where peak positionsshifted to lower energies as the size increased [18]. It isto be noted that the narrow autoionization resonances aremissing in the experimental data. This is due to the finitetemperature effects of the metal clusters in experimentalconditions which lead to the coupling of electronic motionwith the temperature induced vibrational and rotationalmodes [28], as was also found and discussed earlier forC [23]. B. Oscillator strength
From the experimental data shown in Fig. 3, it is quiteclear that the bulk of the absorption oscillator strength(OS) is consumed in the excitation part of the spectrumbelow the ionization threshold. In order to get a quan-titative measure of the fraction of the OS exhausted inthe ionization process, we introduce an “accumulative”OS as a function of photon energy given by the followingrelation: OS t ( E f ) = OS b + Z E f E th σ ( E ) dE, (9) Na OS b =0.4352 (b)Photon Energy (eV) O s c ill a t o r s t r e ng t h p e r a t o m Na OS b =0.5230 (a) TheoryExpt.fit
FIG. 4. Accumulative oscillator strengths for Na (a) andNa (b) where the theoretical results are calculated using thepresent TDLDA+LB94 cross sections and the experimentalOS are calculated using fitted values to the data in [7]. where OS b is defined as the baseline OS that is calcu-lated by integrating the experimental cross section [7]from the starting photon energy of the measurement tothe theoretical ionization thresholds (E th ). OS b there-fore corresponds to the oscillator strength exhausted bythe plasmon resonances below the threshold. The secondterm in Eq. 9 corresponds to the incremental addition tothe OS b , cumulatively with energy, due to the plasmonspill to the ionization region from E th to E f . Here E f will vary from E th onwards allowing us to calculate OS t as a function of E f . Note that until 5.64 eV, the pho-ton energy corresponding to the last experimental datum,E f ’s are the various energies for which experimental mea-surements are available. In Eq. 9, σ ( E ) corresponds toTDLDA total cross section for the calculations of theo-retical OS t . For calculating the experimental OS t , on theother hand, fitted cross section data between two succes-sive measurements are used.Figure 4 shows OS on a zero-to-one scale, because itrepresents OS per atom, that is the total OS divided by n . For Na , the OS b =0.5230 and for Na , OS b =0.4352. These suggest that about 52% of OS is exhausted be-low the ionization threshold of Na , while it is about44% for Na . The implication is that more fraction ofelectrons are available to participate in the ionizationprocess for the larger cluster. Hence, the size of plas-mon spill increases for Na . The rise of the theoreticalOS curve above the experimental curve is due primarilyto the narrow autoionizing resonances in the theoreticalspectrum which add strengths. This accounts roughlyabout 20% of OS for Na , while it is 30% for Na . Theexperimental measurements cannot access such narrowsingle particle resonances due to the finite temperatureeffects [28]; similar distinction was also noted for the plas-mon resonances in C [23]. However, the missing partof the oscillator strengths in the measurements may bedistributed over higher photon energies, in which caseit is expected that the two curves may eventually inter-sect at higher energies. One comment should be madeabout the measurements though. The energy depositedinto the cluster will not only dissipate through ionizationthat we calculate, but also be shared with thermalizationand evaporation channels as well. Even though the cor-responding branching ratio is unknown, the ionization isknown to be a much faster process (in attoseconds) com-pared to thermalization/evaporation (in femtoseconds).Therefore, one may reasonably assume that the maincontribution to the measured data above the thresholdis likely from the ionization process. C. Collectivity induced field
TDLDA dipole matrix element h kℓ ′ | z + V ind | nℓ i inEq. (8) requires knowledge of V ind , which is the com-plex induced field driven by electron correlations and issingularly responsible for the plasmonic enhancement inthe cross section. The behavior of the real and imagi-nary parts of V ind across the collective resonance regionis well known [29, 30]. Im( V ind ) characteristically shows awell-type shape across the energy range of the resonancewhere the minimum of the well occurs near the energyof the resonance peak. On the other hand, Re( V ind ) exe-cutes an oscillation by switching the sign over this rangewhere it sluices through the zero at the resonance peak.These two distinct behaviors can be combined in a uni-fied picture. Im( V ind ) has a predominant collective char-acter, while Re( V ind ) represents effects of the externalfield. Therefore, as the resonance builds with increas-ing energy and approaches its peak, the effect of externalfield reduces while the collective motion grows. At thepeak, the effect of external field is negligible, where thecollective response dominates. At the waning part of theresonance this trend reverses.In the current study, however, we access the remnantof the plasmon structure spilling over to the photoioniza-tion channel which is fragmented by many single electronnarrower resonances mixing coherently with the plasmoneffect. This coherence indicates that there are interfer- Na P ho t on E n e r gy ( e V ) R a d i a l c o - o r d i n a t e ( a . u . ) R e ( V i nd ) ( a . u . ) -60-40-20 0 20 40 3.4 3.6 3.8 4 4.2 4.4 4.6 4.8 0 5 10 15 20-100 0 100 200 300 400 500 600 700 800 Na P ho t on E n e r gy ( e V ) R a d i a l c o - o r d i n a t e ( a . u . ) R e ( V i nd ) ( a . u . ) -100 0 100 200 300 400 500 600 700 800 3.6 3.8 4 4.2 4.4 4.6 4.8 0 5 10 15-500-400-300-200-100 0 100 P ho t on E n e r gy ( e V ) R a d i a l c o - o r d i n a t e ( a . u . ) I m ( V i nd ) ( a . u . ) -500-400-300-200-100 0 100 3.4 3.6 3.8 4 4.2 4.4 4.6 4.8 0 5 10 15 20-700-600-500-400-300-200-100 0 100 P ho t on E n e r gy ( e V ) R a d i a l c o - o r d i n a t e ( a . u . ) I m ( V i nd ) ( a . u . ) -700-600-500-400-300-200-100 0 100 FIG. 5. The real and imaginary part of radial self-consistent field potential, V ind ( r ), for Na (left panels) and Na (rightpanels). ences between the single electron Auger and collectiveplasmon. So the simple mechanism expressed above be-comes complex, the induced field becomes structured,but an interesting general trend should remains that wenow explain.Figure 5 shows the 3D plots of real and imaginary partof radial induced field, V ind ( r ; ω ), for Na and Na . Allthe plots show many smaller structures, which are be-cause of single-electron resonances that become relativelystronger as the plasmon spill gradually weakens withincreasing energy. However, there appears wider well-shapes for Im ( V ind ) of both systems. This broader rangeof negative fields suggests the emergence of an attractivefield as a result of the electrons’ collective dynamics inthese plasmonic energies. Consequently, the liberatingphotoelectrons will experience a resistance against theirexit and will likely slow down. This can lead to observ-able time-delayed emissions of electrons [31] in the plas-mon spillover range. Particularly, this may find relevancein the context of the electron’s intrinsic Wigner-type de-lay properties [32]. Time delay studies in photoemissionbelong to a contemporary field of interest based on meth-ods of attosecond photoemission measurements in RA-BITT or streaking spectrometry [33]. Similar attractiveforce, driven by electronic collective interactions, has ear-lier been predicted at plasmon photoionization energies of C [34]. D. Correlation minimum
As seen in Fig. 2, the total photoionization cross sec-tions undergo oscillations both in LDA and TDLDAstarting from energies where the plasmon structure be-gins to fizzle. The oscillations far above the plasmon re-gion are well understood and are attributed to the diffrac-tion of the photoelectron waves from the edges of thecluster [12]. Since the feature is associated with the clus-ter geometry, it is seen even in the independent particleLDA results. The merging of the oscillations at higherenergies in TDLDA and LDA reflects on the fact that theoscillations have nothing to do with the electron correla-tions. However, the TDLDA cross sections reveal a ratherstrong additional minimum occurring at 7.20 eV for Na and 6.22 eV for Na which are clearly missing in theirLDA counterparts. Incidentally, this feature was also re-vealed in earlier calculations [9], but was never properlyinterpreted. If it were a diffractive feature, it would bepresent in the LDA results as well. Obviously, that is notbeing the case suggests that this lower energy minimummust be induced by an electron correlation effect. −8 −6 −4 −2
5 10 20 30 40 50 Na (TDLDA)(d) Photon Energy (eV)
Total1h3s2d C r o ss − s ec ti on ( a . u . ) −8 −6 −4 −2 Na (TDLDA)(c) Total2s1d1p −8 −6 −4 −2 Na (LDA)(b) Photon Energy (eV)
Total1h3s2d −8 −6 −4 −2 Na (LDA)(a) Total2s1d1p FIG. 6. Subshell cross-sections for Na , (a) and (c), and Na , (b) and (d), compared with corresponding total cross-sectionobtained using LDA (left-panels) and TDLDA (right panels). Further insights into this minimum can be harnessedby looking at the individual subshell cross sections. InFig. 6 are shown the cross sections from the three out-ermost subshells of Na and Na along with the to-tal cross sections obtained from LDA (Fig. 6(a) andFig. 6(b)) and TDLDA(Fig. 6(c) and Fig. 6(d)). It isclear from the TDLDA results that in each of the sub-shell cross sections for a cluster this particular minimumin consideration occurs dramatically at the same low pho-ton energy to create a rather emphatic minimum in thetotal cross section. But this feature is totally absent inthe LDA curves.For a subshell, while the creation of a minimum de-rives directly from the ionization amplitude, its positionin energy traces in to the ionization phase. The posi-tions of the minima at high energies are seen to be verysensitive to the ionizing subshells. It was shown ear-lier that the energy-positions of high energy minima area function of the phase of the individual subshell LDAtransition amplitudes [35], since the contribution of δV [Eq. (6)] is virtually zero at these energies. In addition,at very high energies the contributions to the scattering- phase from even the short-range and Coulomb potentialsare negligible. As a result, the final state wavefunctionis approximately of the form ψ f ≈ cos( kr − l ′ π ), where l ′ = l ± l ’ is the initial state and ‘ k ’ is the photo-electron momentum. Consequently, the squared transi-tion amplitudes differ by a phase of l ′ π . This explainswhy the oscillations in the cross sections of two subshellsof angular momenta differing by an odd integer, such as,s and p, p and d, s and h etc. are roughly out-of-phase,while those differing by an even integer, such as, s and d,p and f etc. are roughly in-phase [36]. This can be seenin both LDA and TDLDA subshell results at very highenergies. At not-so-high energies, these patterns are notseen to be exactly followed due to non-negligible short-range and Coulomb phases.However, the energy-concurrent minima in theTDLDA profiles occur at a low enough energy. Eventhough the diffractive mechanism forming oscillations isemerging, this range is still in the waning part of the plas-mon structure. Hence, the contribution from the correla-tion phase, that is the phase of the amplitude with signif-icant strength of δV , still dominates over the l ′ π in thisrange. The correlation phase, being collective in nature,is subshell independent, leading to all subshells experi-encing effectively the same phase [32]. This ensures thatthe minimum for all the subshells to appear coherentlyat the same energy. Since this coherence originates froma direct influence of the collective dynamics, we call thisfeature a correlation minimum, which can, in principle,be accessible by the photoelectron spectroscopy (PES). IV. CONCLUSIONS
In summary, the present jellium based linear responseDFT calculation describes the remnant of the giant sur-face and the bulk of the volume plasmon in the pho-toionization cross section of Na n ( n =20 and 92) clusters.Results show that an appreciable amount of plasmonspillover into the ionization continuum occurs in thesesystems. The steady background part of the cross sec-tion exhibits reasonably good agreement with previousexperimental results of the absorption of these clusters.However, a detailed scrutiny of the oscillator strength cal-culated from the cross section and its comparison withthe strength extracted from the measurements enables us to quantify the contribution of the single electron Auger-type resonances that the theory predicts. Such, rathernarrow, resonances shown in the theory, which does notincorporate the temperature and vibro-rotational effects,are missing in the measurements that include these ef-fects. Furthermore, a deeper scrutiny of the many-bodyinduced self-consistent DFT field reveals the presence ofan attractive force in the plasmon spillover energy region.We speculate that this force can cause an observable de-lay in the emission of photoelectrons at these energies,enabling these clusters as interesting candidates for time-delay measurements. The current study further uncoversthe presence of a correlation minimum in the cross sec-tions which appears at the waning range of the plasmonresonance structure and which may also attract experi-mental effort. To this end, the collective-motion spectralproperties divulged in the current DFT study of Na clus-ters are likely quite general and may apply at variousdegrees of prominence in other metallic clusters. ACKNOWLEDGMENTS
The research is supported by the SERB, India, GrantNo. EMR/2016/002695 (HRV) and by the US NationalScience Foundation Grant No. PHY-1806206 (HSC). [1] P. Jena and Q. Sun, “Super atomic clusters: design rulesand potential for building blocks of materials”, Chem.Rev. ,5755 (2018).[2] W.P. Halperin, “Quantum size effects in metal particles”,Rev. Mod. Phys. , 533 (1986).[3] S. Tony, “How small is a solid?”, Nature, , 116 (1988).[4] A.W. Castleman Jr and S.N. Khanna, “Clusters, super-atoms, and building blocks of new materials”, The Jour-nal of Physical Chemistry C, , 2664 (2009).[5] P.M. Dinh, P.G. Reinhard, and E. Suraud, in “An Intro-duction to Cluster Science” , (John Wiley and Sons, 2013),pp. 127-157.[6] U. Kreibig and M. Vollmer, in “Optical properties ofmetal clusters” , Vol. , (Springer Science & BusinessMedia, 2013 ).[7] C. Xia, C. Yin, and V.V. Kresin, “Photoabsorption byvolume plasmons in metal nanoclusters”, Phys. Rev.Lett. , 156802 (2009).[8] M.E. Madjet and H.S. Chakraborty,“Collective reso-nances in the photoionization of metallic nanoclusters”,J. Phys.: Conf. Ser. , 022103 (2009).[9] W. Ekardt, “Size-dependent photoabsorption and pho-toemission of small metal particles”, Phys. Rev. B ,6360, 1985.[10] A.E. Miroshnichenko, and S. Flach and Y.S. Kivshar,“Fano resonances in nanoscale structures”, Rev. Mod.Phys. , 2257 (2010).[11] K. J¨ank¨al¨a, M. Tchaplyguine, M.-H. Mikkel¨a, O.Bj¨orneholm, and M. Huttula, “Photon energy dependentvalence band response of metallic nanoparticles”, Phys.Rev. Lett. , 183401 (2011). [12] M.E. Madjet, H.S. Chakraborty, and J.M. Rost, “Spu-rious oscillations from local self-interaction correction inhigh-energy photoionization calculations for metal clus-ters”, J. Phys. B. , L345 (2001).[13] N.A. Mirin, K. Bao, and P. Nordlander, “Fano resonancesin plasmonic nanoparticle aggregates”, J. Phys. Chem. , 4028 (2009).[14] C. Loo, A. Lowery, N. Halas, J. West, and R. Drezek,“Immunotargeted nanoshells for integrated cancer imag-ing and therapy”, Nano letters. , 709 (2005).[15] H. Liao, C.L. Nehl, and J.H. Hafner, “Biomedical ap-plications of plasmon resonant metal nanoparticles”,Nanomedicine , 201 (2006).[16] R.A. Ferrell, “Predicted radiation of plasma oscillationsin metal films”, Physical Review , 1214 (1958).[17] G. Onida, L. Reining, and A. Rubio,“Electronic exci-tations: density-functional versus many-body Green’s-function approaches”, Rev. Mod. Phys. , 601 (2002).[18] J. Choi, E. Chang, D.M. Anstine, M.E. Madjet, and H.S.Chakraborty, “Effects of exchange-correlation potentialson the density-functional description of C versus C photoionization”, Phys. Rev. A , 023404 (2017).[19] R. Shaik, H.R. Varma, and H.S. Chakraborty, “Effects ofexchange-correlation functionals on the structure and thephotoionization dynamics of Na versus Na cluster”,J. Phys.: Conf. Ser. , 102009 (2020).[20] F. Chandezon, S. Bjørnholm, J. Borggreen, and K.Hansen, “Electronic shell energies and deformations inlarge sodium clusters from evaporation spectra”, Phys.Rev. B , 5485 (1997). [21] R. Van Leeuwen and E.J. Baerends, “Exchange-correlation potential with correct asymptotic behavior”,Phys. Rev. A , 2421 (1994).[22] G.L. Oliver and J.P. Perdew, “Spin-density gradient ex-pansion for the kinetic energy”, Phys. Rev. A , 397(1979).[23] M.E. Madjet, H.S. Chakraborty, J.M. Rost, and S.T.Manson, “Photoionization of C : a model study”, J.Phys. B. , 105101 (2008).[24] M.G.U.J. Petersilka, U.J. Gossmann, and E.K.U. Gross,“Excitation energies from time-dependent density-functional theory”, Phys. Rev. Lett. ,1212 (1996).[25] P.J. Feibelman, “Microscopic calculation of electromag-netic fields in refraction at a jellium-vacuum interface”,Phys. Rev. B , 1319 (1975).[26] G. Bertsch, “An RPA program for jellium spheres”,Comput. Phys. Commun. , 247 (1990).[27] U. Fano, “Effects of configuration interaction on intensi-ties and phase shifts”, Physical Review , 1866 (1961).[28] G.F. Bertsch and D. Tom´anek, “Thermal line broadeningin small metal clusters”, Phys. Rev. B , 2749 (1989).[29] G. Wendin, “Collective effects in atomic photoabsorptionspectra. III. Collective resonance in the 4d shell in Xe,J. Phys. B , 42 (1973). [30] A. Zangwill and P. Soven, “Density-functional approachto local-field effects in finite systems: Photoabsorption inthe rare gases”, Phys. Rev. A , 1561 (1980).[31] M. Schultze et al ., “Delay in Photoemission”, Science , 1658 (2010).[32] M. Magrakvelidze, M.E. Madjet, G. Dixit, M. Ivanov,and H.S. Chakraborty, “Attosecond time delay in va-lence photoionization and photorecombination of argon:A time-dependent local-density-approximation study”,Phys. Rev. A
1, 063415 (2015).[33] R. Pazourek, S. Nagele, and J. Burgd¨orfer, “Attosecondchronoscopy of photoemission”, Rev. Mod. Phys. , 765(2015), references therein.[34] M. Magrakvelidze, M.E. Madjet, and H.S. Chakraborty,“Correlation drives a strong attractive force on plasmonicphotoelectrons”, J. Phys.: Conf. Ser. , 072040(2020).[35] O. Frank and J.M. Rost, Chem. Phys. Lett. ”, J. Phys. B.41