Femtosecond dynamics of correlated many-body states in C 60 fullerenes
Sergey Usenko, Michael Schüler, Armin Azima, Markus Jakob, Leslie L. Lazzarino, Yaroslav Pavlyukh, Andreas Przystawik, Markus Drescher, Tim Laarmann, Jamal Berakdar
FFemtosecond dynamics of correlated many-bodystates in C fullerenes Sergey Usenko , Michael Sch¨uler , Armin Azima , , , MarkusJakob , , Leslie L Lazzarino , Yaroslav Pavlyukh , AndreasPrzystawik , Markus Drescher , , , Tim Laarmann , and JamalBerakdar Deutsches Elektronen-Synchrotron DESY, Notkestr. 85, 22607 Hamburg, Germany Institut f¨ur Physik, Martin-Luther-Universit¨at Halle-Wittenberg, 06099 Halle,Germany Department of Physics, University of Hamburg, 22761 Hamburg, Germany Center for Free-Electron Laser Science CFEL, DESY, 22607 Hamburg, Germany The Hamburg Centre for Ultrafast Imaging CUI, Luruper Chaussee 149, 22761Hamburg, GermanyE-mail: [email protected]
E-mail: [email protected]
Abstract.
Fullerene complexes may play a key role in the design of future molecularelectronics and nanostructured devices with potential applications in light harvestingusing organic solar cells. Charge and energy flow in these systems is mediated bymany-body effects. We studied the structure and dynamics of laser-induced multi-electron excitations in isolated C by two-photon photoionization as a function ofexcitation wavelength using a tunable fs UV laser and developed a correspondingtheoretical framework on the basis of ab initio calculations. The measured resonanceline width gives direct information on the excited state lifetime. From the spectraldeconvolution we derive a lower limit for purely electronic relaxation on the orderof τ el = 10 +5 − fs. Energy dissipation towards nuclear degrees of freedom is studiedin time-resolved techniques. The evaluation of the non-linear autocorrelation tracegives a characteristic time constant of τ vib = 400 ±
100 fs for the exponential decay. Inline with the experiment, the observed transient dynamics is explained theoreticallyby nonadiabatic (vibronic) couplings involving the correlated electronic, the nucleardegrees of freedom (accounting for the Herzberg-Teller coupling), and their interplay.PACS numbers: 32.80.Rm, 33.20.Lg, 33.20.Wr, 81.05.ub a r X i v : . [ phy s i c s . a t m - c l u s ] O c t emtosecond dynamics of correlated many-body states in C fullerenes
1. Introduction
Molecular junctions, molecular transistors and organic solar cells rely on chargetransport channels with negligible energy dissipation during the carriers propagationtime. In nanostructured materials and molecular complexes the characteristic timescaleis determined by the long-range polarization interaction and by the formation andbreaking of chemical bonds mediated by the electronic and nuclear motion. Transientstructures and dynamics on the femto and sub-femtosecond timescale is the focus ofultrafast spectroscopy. Time-resolved experiments using femtosecond (fs) laser pulsesunravel the dynamic response of promising materials that could serve for instance asmolecular building blocks for organic photovoltaics. Polymer solar cells are commonlycomposed of a photoactive film of a conjugated polymer donor and a fullerene derivativeacceptor [1, 2, 3], which makes use of the fullerenes’ unique ability to form stable C − anions. Electron correlation plays an important role in the formation of four boundstates of the fullerene anion [4, 5, 6]. In fact, electronic correlations are responsible forthe binding of the 2A g state, whereas the bindings of the states 2T , 2T and 2T are less affected by electronic correlations (cf. Ref. [4] and further references therein).With its special structure consisting of 174 nuclear degrees of freedom, 60 essentiallyequivalent delocalized π electrons, and 180 structure-defining localized σ electrons,neutral C serves as a model for a large – but still finite – molecular system with manyelectronic and nuclear degrees of freedom. Because of the large charge conjugation, itsfinite ”energy gap”, and quantum confinement of electronic states, C may be viewedas an interesting intermediate case between a molecule and a condensed matter system.In fact, applying solid-state concepts to the valence ”Bloch electrons” on the C sphereresults in an ”angular band structure” [7] from which other relevant quantities (suchas plasma frequencies and group velocities) can be extracted. Photophysical studies offullerenes using fs laser fields cover the whole range from atomic through molecular tosolid state physics [8, 9, 10]. The molecular response is truly a multi-scale phenomenon.It ranges from attosecond dynamics in electronic excitation and ionization to statisticalphysics describing thermalization processes. So, light-induced processes in fullerenescover more than 15 orders of magnitude in time [11].Using low-temperature scanning tunneling microscopy (LT-STM) of C moleculesdeposited on copper surfaces Feng et al. observed tunneling through electronic statesthat possess nearly atom-like character [12]. These ”superatom” molecular orbitals(SAMOs), also discussed below, have a well-defined symmetry and can be characterizedby the nodal structure (principle quantum numbers) n and angular momentum quantumnumbers L [13]. In addition, these virtual states show a remarkable stability [14], i.e.,their initial stage of decay proceeds substantially slower than other states (even LUMOor HOMO) which qualifies them as robust channel for hot electron transport. From amolecular physics point of view, SAMOs can be regarded as low-lying, mixed valenceRydberg states [15] that exhibit substantial electron density inside the hollow sphere.They form chemical bonds affected by hybridization when the system is excited optically emtosecond dynamics of correlated many-body states in C fullerenes fullerenes [20, 21, 22],where they are energetically located below the known high-lying Rydberg states [23, 24].It is of fundamental importance for designing fullerene derivatives as building blocksfor solid state chemistry to go beyond the characterization of static many-body electronicstructure [25]. In particular for optimization and control of the charge flow and energydissipation, rigorous dynamic studies on the fs time scale using ultrafast lasers areindispensable but, still in their infancy [26, 27]. Additionally, time-dependent density-functional theory (TDDFT) calculations of the absorption and photoelectron spectra,accounting for full structural analysis, were performed [28, 29] and constitute the basisfor the further development presented below.Accessing energy dissipation upon the excitation of correlated many-body statesin gas phase C became feasible recently which allows to connect the coherentquantum [28, 30], classical and statistical mechanisms [31]. It is known from experimentswith optical lasers that electron thermalization mediated by inelastic electron–electroncollisions takes place on a time scale of 50 −
100 fs (see [11] and references therein). This iswhere the present experimental and theoretical work comes into play. Here, the objectiveis to reduce the complexity of laser-induced multi-photon processes by populating highly-excited many-body states in a resonant one-photon transition in the ultraviolet (UV)spectral range at low laser peak intensity on the order of 3 . × W/cm . Thisallows for a rather detailed probing of the correlated electron dynamics in highly excitedstates. The study is based on a resonance-enhanced multi-photon ionization (REMPI)scheme, i.e., two-photon photoemission (2PPE) spectroscopy as depicted in Fig. 1(a).The photoionization yield recorded for resonant excitation is enhanced as comparedto an experiment performed in the off-resonance regime. Thereby, we trace the time-dependent electronic structure of intermediate states free from any perturbation causedby metallic substrates affecting the energetics in LT-STM experiments. Furthermore,REMPI on gas phase fullerenes provides information on the neutral molecule whereas2PPE and LT-STM essentially probe the binding energy and density of states of an aniondeposited on a metal surface. Our experiments are compared to ab initio calculations.This paper is organized as follows. In Sec. 2 many-body states below the C ionization threshold are calculated as guideline for the two-photon photoemissionexperiments. Sec. 3 describes some experimental details with a focus on the tunable fslaser system in the UV spectral range used for the time-resolved studies. Excitationenergy dependent mass spectra are evaluated in Sec. 4.1 and discussed in terms ofresonance-enhanced ionization and excited state lifetimes. Sec. 4.2 concentrates on thetime-resolved experiments. A detailed theoretical analysis of the experimental data isgiven in Sec. 5 followed by a short summary and outlook. In Appendix A we provide emtosecond dynamics of correlated many-body states in C fullerenes
2. Optical excitations
The first step of a REMPI or 2PPE experiment entails the calculation of excited states,which are typically more difficult to describe than ground-state properties. Often exactdiagonalization (full configuration interaction) is not feasible for large systems in whichcase one may resort to only a few methods: TDDFT [32], equation-of-motion (EOM)quantum chemistry methods [33], and many-body perturbation theory (MBPT) basedon a Green’s function formulation [34]. For its reduced computational cost as comparedto the other methods, we have employed the linear-response TDDFT approach (Casida’smethod) [35].As a first step we calculated the Kohn-Sham (KS) orbitals using the octopus package [36]. A modified version of the asymptotically corrected functional byLeeuwen and Baerends [37], which was shown to considerably improve excited-stateproperties [38]. In order to account for a multitude of highly-excited states, we havechosen a relatively large box to which all KS states φ i ( r ) are confined (a sphere withradius 12 ˚A with uniform grid spacing of 0.15 ˚A). This ensures that higher virtual orbitals(including the SAMOs) are well represented. After converging the ground-state andcomputing a sufficient number of virtual orbitals, we computed the singly-excited (i.e.,single particle-hole excitations) many-body excitations by Casida’s method. Formallythis procedure amounts to approximating the excited many-body states | Φ α (cid:105) by | Φ α (cid:105) = (cid:88) i ∈ occ (cid:88) j ∈ virt A αij ˆ c † j ˆ c i | Φ (cid:105) , (1)where | Φ (cid:105) denotes the determinant built by the ground-state KS orbitals and ˆ c i (ˆ c † i )is the annihilation (creation) operator with respect to the KS basis. The particle-hole amplitudes A αi,j are determined by Casida’s equation based on linear response.The major approximation hereby is related to the exchange-correlation (xc) kernel f xc ( r , r (cid:48) ; ω ), defined as the functional derivative of the KS potential with respect tothe density. We use the local-density approximation (LDA) for the xc kernel, as itis local and (within adiabatic TDDFT) frequency-independent. Casida’s equation isthus transformed into an eigenvalue problem. We computed the Casida vectors A αij with octopus , taking 75 occupied and 60 virtual orbitals into account, yielding well-converged results for excitation energies up to 10 eV. For testing purposes we alsocomputed the binding energies of the virtual orbitals analogously to Ref. [21]. Weobtained very similar results for the low-lying states relevant for the present experiments.The energies of the obtained many-body excitations are shown in Fig. 1(a), wherewe distinguish the states with vanishing dipole transition moment (these we refer to asdark states, DS) from the ground state GS, and optically accessible states (bright states,BS). The onset of the visible to UV (UV-vis) optical absorption is well documented (e.g.,cf. Ref. [39] for a review and a comparison with previous experiments). Three distinct emtosecond dynamics of correlated many-body states in C fullerenes continuumGSDSBS CEG excitation G (6T )-IP HOMO HOMO-1HOMO-2 L U M O L U M O + L U M O + L U M O + s - SA M O - SA M O (a ) (b ) ene r g y [ e V ] ene r g y [ e V ] -2-4-6-8-10-12 Figure 1. (a) Excitation spectrum of the C molecule starting from the groundstate (GS). Optically accessible excited states are denoted as bright states (BS), whileexcitations with vanishing dipole transition moment are referred to as dark states(DS). The two-photon REMPI experiment is sensitive to BSs between 5 and 6 eV.The first three excitations are labeled according to ref. [39]. (b) Excitation G resolvedwith respect to the constituting occupied (blue), virtual (orange) and SAMO (purple)electronic states. The thickness of the red arrows is proportional to the correspondingweight | A α =G ij | . absorption peaks (labeled according to literature as C, E and G bands, respectively)are found in the UV-vis region. An overall agreement between our calculations ofthe optical absorption with the experimental results is found, though the excitationenergies are slightly underestimated (which is typical for DFT calculations). Note, thespecific experimental conditions may affect the spectral positions of the absorption peaks(as discussed in Ref. [39]). In particular, the present experiment probes the opticalproperties of isolated molecules by ultrafast pulses and thus potentially eliminatesenergy-loss channels (e.g. collisions and inter- or intra molecular decay) that mightshift the absorption peaks to higher energy. For these reasons, we base the subsequentcalculations on our DFT results without any adjustments. As detailed below, very goodagreement to the present experiment is achieved, justifying this approach a posteriori .According to our calculations, excitation G represents one triply degenerate many-body state, which is identified as 6T u excitation [40]. The projection onto the KSbasis is depicted in Fig. 1(b), where we illustrate the relative weight of the excitationfrom occupied ( i ) to virtual ( j ) orbitals by the thickness of the corresponding arrows.The relevant orbitals are also presented in Fig. 2. As becomes clear from Fig. 1(b)and Fig. 2, excitation G is predominantly composed of the transitions from (i) HOMOto virtual states above the first SAMOs with dominant angular momentum of L = 8and L = 6, and (ii) HOMO-1 to LUMO+2. The angular-momentum analysis of theindividual orbitals (Fig. 2) clarifies why the transition to the many-body state associatedto excitation G is optically allowed. Analogously one can conclude that neither the 3s– emtosecond dynamics of correlated many-body states in C fullerenes
0 2 4 6 8 10 12
HOMO-2HOMO-1
0 2 4 6 8 10 12
LUMOHOMO LUMO+1LUMO+2LUMO+3 3s-SAMO3p-SAMO
0 2 4 6 8 10 12 angular momentum L angular momentum L angular momentum L r e l a t i v e w e i gh t r e l a t i v e w e i gh t r e l a t i v e w e i gh t (a) (b) (c) Figure 2.
Occupied and virtual KS states close the negative ionization potential (IP)ordered (from bottom to top) according to Fig. 1(b). The respective orbital characteris illustrated by the relative weight of each angular-momentum component L . Onlyone representative of degenerate states is shown. (a) Occupied states, (b) virtual statesbelow the SAMOs, and (c) orbitals including the SAMOs and higher states. SAMO nor the 3p–SAMO can be populated in a direct single-photon dipole transitionfrom the HOMO. This is consistent with a full symmetry analysis of the initial orbitalsand the ab initio calculations.
3. Experimental setup
In order to study electronic transitions into highly excited many-body states close tothe ionization potential, fs pulses at ultraviolet (UV) frequencies are required [41].The present time-resolved mass-spectrometric study is based on second-order UVautocorrelation making use of a time-of-flight (TOF) mass spectrometer and state-of-the-art nonlinear optics. The laser setup for generating fs pulses in the range of216–222 nm comprises a Ti:Sa laser system, an optical parametric amplifier (OPA) andseveral frequency mixing stages. The outline of the system is sketched in Fig. 3. TheTi:Sa laser (Amplitude Technologies) is the backbone of the overall generation schemeand provides 35 fs (FWHM) pulses with a pulse energy of 12 mJ behind the compressorat a repetition rate of 25 Hz and 800 nm central wavelength. This output is split intoseveral arms. First, the beam is split in a 90:10 ratio. The more intense fraction issent to a commercial OPA (Light Conversion, TOPAS-C + HE-TOPAS). The OPA iscontinuously tunable in the infrared spectral range (1140–3500 nm) and pumped by theTi:Sa laser. For the subsequent frequency conversion in the UV the output of the OPAis tuned to 1200 nm. The 11 mJ of the 800 nm pump pulse are converted to ≈ . emtosecond dynamics of correlated many-body states in C fullerenes Figure 3.
A schematic layout of the femtosecond laser pulse generation scheme in theultraviolet spectral range. into two branches. One half is recombined with the 1200 nm output of the TOPASin a β -barium borate (BBO, 0.2 mm thick) crystal to generate 480 nm light by sumfrequency generation (SFG). The second half is frequency doubled in another BBO, andthen together with the 480 nm beam directed to the third SFG stage (40 µ m thick) tofinally generate the 5.7 eV (218 nm) photons. The UV pulse energy was measured usinga calibrated XUV photodiode and a pyro detector to be ≈ µ J.The output UV beam is directed into a vacuum chamber where it is split into twopulses by a reflective split-and-delay unit (SDU) in order to generate two synchronizedpulse replicas. The complete nonlinear optical setup was simulated using the softwarepackage LAB2 [42] including dispersion induced by UV pulse propagation in air andthrough the 2 mm thick entrance window of the vacuum chamber. According to thecalculation the 218 nm pulse duration in the interaction region is of the order of 100 fsFWHM with a spectral bandwidth of 2.8 nm. A coarse cross-correlation measurementperformed between the 400 nm and 480 nm pulses of 150 fs FWHM supports the derivedUV laser beam parameters.The SDU consists of a Si split-mirror with one half mounted on a delay stage whichcan displace the mirror along its normal to set the time delay between the pump andprobe pulses. The SDU is followed by a focussing mirror which spatially overlaps thetwo pulse replicas in the laser–sample interaction area. The laser beams are focusedonto the C molecular beam with a spherical mirror ( f = 300 mm). Its reflectivityis above 80% in the 200–245 nm range. The beam waist in the interaction area ison the order of 150 µ m. The maximum peak intensity reached in the experiments isapproximately 3 . × W/cm , which is derived from first principles based on Gaussianbeam propagation, pulse energy, pulse duration and the far-field laser profile.The molecular beam is produced by evaporation of gold grade C powder ina resistively heated oven at 775 K. The UV laser beams are focused perpendicularto both, the effusive molecular beam and the TOF spectrometer axis. The ionscreated in the intersection volume are extracted by a static electric field (Wiley–McLaren configuration [43]), directed onto multichannel plates, and finally counted afteramplification and discrimination by a digital oscilloscope. The mass resolution of theTOF spectrometer is 0.2% at M/q=720. emtosecond dynamics of correlated many-body states in C fullerenes wavelength [nm] g = 0.5g = 0.75g = 1.0g = 1.128 theoryexperiment data pointsfit
212 214 216 218 220 222 224 no r m a li z ed y i e l d C + g = 1.5g = 1.75g = 2.0 Figure 4.
Normalized C +60 ion yield (black scatter) as a function of excitationwavelength showing the resonance-enhanced two-photon photoionization. The fit tothe data points (black line) is a convolution of a Gaussian profile representing thelaser pulse spectrum and a Lorentzian profile, representing the natural linewidth ofthe resonance, respectively. The experimental data is compared with calculations (seeSubsec. 5.3) for different values of the electron-vibron coupling strength g .
4. Two-photon photoemission
The photoionization signal recorded for resonant excitation is enhanced compared toan experiment performed off-resonance. The spectral width of the resonance yieldsinformation on the excited state lifetime. A UV wavelength scan was performed bytuning the IR wavelength of the OPA. The populated many-body state in the neutralmolecule is subsequently ionized during the pulse duration of 100 fs. A full massspectrum is accumulated over ≈
250 laser shots for each excitation wavelength. The C +60 yield was normalized to the relative pulse energy monitored by a photodiode. The C +60 ion yield as a function of laser wavelength in the range of 216–222 nm is shown in Fig. 4.The cut-off at 216 nm corresponds to the OPA’s lower wavelength limit of 1140 nm.Relatively large error bars at short wavelengths result from the corresponding low pulseenergy and thus poor statistics. The C +60 signal disappears for excitation wavelengthslonger than 222 nm, thus representing the low-energy threshold of the resonance. Thewavelength scan clearly indicates resonance-enhanced two-photon photoionization at λ exc = 218 ± . +5 − fs. The lifetime estimate is derived from the deconvolution of the observedresonance with a Gaussian laser pulse spectrum of 2.8 nm (FWHM) and a Lorentziandescribing the homogeneous broadening. emtosecond dynamics of correlated many-body states in C fullerenes Time-resolved mass spectrometry traces the excited state dynamics in neutral C molecules directly in the time domain. The transient electronic structure is initiated by aUV 100 fs pump pulse and followed (probed) by a delayed pulse replica that photoionizesthe molecule (see Fig. 1(a)). The single-shot mass spectra are taken at varying delaytimes between the UV pulses ranging from -60 fs to 900 fs with ≈ +60 counts for each delay and normalizing it to therelative pulse energy monitored as the UV stray light peak in the TOF spectrum. Thepump–probe scan is repeated two times.In the most general case the ion signal from a three-level system with a transientintermediate electronic state exposed to resonant excitation will result from twoexcitation pathways: direct two-photon photoionization from the ground state to thecontinuum and the ionization via the transient state (REMPI). Therefore, the totalion signal S tot can be described as a sum of three components [44]: the coherent term S ac (coherent artifact), the incoherent term S inc and a constant background a bg . Thecoherent artifact reflects the direct nonlinear ionization process and is proportional tothe autocorrelation (AC) function of the laser pulse: S ac ( τ ) = + ∞ (cid:90) −∞ I ( t − τ ) I ( t ) dt (2)where I ( t ) is the laser intensity and τ is the delay between the pump and probe pulses.The incoherent term S inc carries information about the population dynamics of thetransient state. It is a convolution of the laser pulse AC with a symmetric decay function.In case of an exponential decay with a characteristic time constant τ vib the incoherentterm is given by: S inc ( τ ) = + ∞ (cid:90) −∞ S ac ( t − τ ) exp( −| t | /τ vib ) dt (3)The constant background a bg consists of contributions from each excitation pulseindividually and is independent of the pump–probe delay. The total signal reads as: S tot ( τ ) = a ac S ac ( τ ) + a inc S inc ( τ ) + a bg (4)with a i being the relative amplitudes of the different components. In general theamplitudes have a ratio depending on the spatial overlap of the two pulses andthe ionization pathway of the system. To extract τ vib from the measurement, theexperimental data is fit by the least squares method using expression (4). Thebackground is set to a bg = 1 and the laser pulses are assumed to be Gaussianwith τ FWHM = 100 fs. Other variables, i.e., a ac , a inc and τ vib , are free fitparameters. The best fit curve (black line in the bottom graph of Fig. 5) yieldsa time constant τ vib = 400 ±
100 fs (95% confidence band) and an amplitude ratio a ac : a inc : a bg = 0 .
24 : 1 .
13 : 1. The laser intensities in the interaction region in the emtosecond dynamics of correlated many-body states in C fullerenes r e l a t i v e i n t en s i t y pump-probe delay [fs] no r m a li z ed i on c oun t r a t e data points experiment fit theory g = 0.5g = 0.75g = 1.0 g = 1.128 g = 1.5g = 1.75 Figure 5.
Top graph: normalized stray light signal on the TOF detector inducedby the excitation pulse as a function of the pump-probe delay, which monitors theUV laser stability throughout the experiment. The straight dotted line designates 1.Bottom graph: normalized C +60 counts as a function of pump-probe delay for resonantexcitation at λ exc = 218 nm. The black scatter are the data points and the curve is a fitobtained from eq. (4) (for details see the text). The normalized ion counts derived fromtheory are compared to the experimental results for different scaling factors g of theelectron-vibron interaction strength (see Subsec. 5.3) in the limit of long pump-probedelays ( ≥
300 fs). The bold value indicates the best agreement between theory andexperiment. present experiment are as low as 3 . × W/cm which makes the contribution fromthe direct (nonlinear) two-photon process small. The ratio a inc : a bg is close to 1 asexpected for single photon ionization from an occupied transient state.The observed exponential time constant τ vib = 400 ±
100 fs is significantly longerthan the characteristic electron–electron interaction time derived from pump–probespectroscopy [45] and pulse duration dependent studies [46, 47] in the optical spectralrange. It seems that electron thermalization mediated by inelastic electron–electroncollisions on a time scale of 50 −
100 fs does not play a key role when high lying correlatedmany-body states are excited directly at rather low peak intensity. In the followinga detailed theoretical description of the observed resonance and its time-dependentstructure is discussed.
5. Simulations
In order to the describe the response of the molecule upon pulsed laser irradiation, theinterplay between the electronic excitations and the vibrationally hot molecule has tobe taken into account (for an overview on this topic we refer to the book [48] and furtherreferences therein). In full generality, an ab initio description for both the electronicand nuclear degrees of freedom and their coupling is not feasible currently. Hence, one emtosecond dynamics of correlated many-body states in C fullerenes molecule [54]; merging such schemes with a treatment of the electronic excitationsbeyond the KS level is computationally too demanding for our system. Alternatively, onecan treat the electrons in a single-particle atomic basis within a tight-binding model [55],removing the adiabaticity constraint with respect to the many-body states. Besides theinevitable empirical ingredient, such theory is also not directly compatible with the abinitio description of the many-body states in Sec. 2, i. e. electronic correlations can onlybe taken into account by great effort. In order to elucidate the laser- and vibration-induced dynamics we take a different angle.Since a considerable amount of energy is stored in thermally activated vibrations, whichcan only be transferred to the electronic subsystem in small portions, the vibrons can betreated as an effective heat bath for the electrons. A similar model has successfully beenemployed for incorporating the influence of the vibrations on charge-transfer processesin organic photovoltaic systems based on C [56]. To construct an appropriate modelfor our case, several ingredients are required. For the vibrations we restrict ourselvesto the harmonic approximation of the bottom of the BO surfaces. The vibroniceigenmodes along with their eigenfrequencies and reduced masses were computed usingthe Octopus code, as well. The resulting density of states (DOS) of the vibronic modesis shown in Fig. 6. Our results compare very well with those tabulated in the literature,for instance table 6.2 in the book [48].As inferred from Fig. 6 the high-energy modes (which affect the electrons the most)are only weakly populated for T = 775 K. Therefore, the oscillations of the nuclei aroundtheir equilibrium positions can be considered small. Hence, the Herzberg-Teller (HT)expansion [57] of the full Hamiltonian, including electrons and nuclei, yields a reasonabledescription for both subsystems and their interaction. The first-order HT Hamiltonianamounts to approximating the electron–vibron coupling as linear in the mode amplitudesˆ Q ν . On the KS level, the electron–vibron matrix elements are thus given by k νij = (cid:104) φ i | ∂v KS ∂Q ν | φ j (cid:105) (cid:12)(cid:12)(cid:12) Q ν =0 . (5)Details on the evaluation of eq. (5) are provided in Appendix A. Since we are optingfor a model in the many-body basis of the excitations discussed in Sec. 2, the matrixelements (5) are transformed according to eq. (1) (see Appendix A). We thus obtain the emtosecond dynamics of correlated many-body states in C fullerenes DOSBose distribution energy [meV] s pe c t r a l den s i t y [ / m e V ] o cc upa t i on Figure 6.
Vibrational density of states (DOS) along with the occupation accordingto the Bose distribution for T = 775 K. model Hamiltonian (atomic units are used throughout)ˆ H ( t ) = ˆ H el ( t ) + ˆ H el − vib + ˆ H vib , (6)where ˆ H el ( t ) = (cid:88) α E α | Φ α (cid:105)(cid:104) Φ α | + f ( t ) (cid:88) αβ M αβ | Φ α (cid:105)(cid:104) Φ β | , (7)ˆ H el − vib = g (cid:88) αβ (cid:88) ν K ναβ | Φ α (cid:105)(cid:104) Φ β | ˆ Q ν , (8)ˆ H vib = 12 (cid:88) ν (cid:32) ˆ P ν M ν + k ν ˆ Q ν (cid:33) . (9)Here, M αβ are the dipole transition matrix elements from our TDDFT calculations,while f ( t ) comprises the time-dependent fields. The prefactor g in eq. (8) is introducedas an overall scaling factor for the strength of the vibronic coupling. Ideally, g = 1should be fixed; however, due to the perturbative description resulting from the HTexpansion, the electron-vibron interaction might be underestimated. Hence, g is keptas a parameter.Instead of describing the dynamics of the full density matrix according toHamiltonian (6) (which is a formidable task), we treat the vibrations, as explainedabove, as a heat bath. This allows to obtain a master equation for the densitymatrix in the electronic subspace only. Here we assume Markovian dynamics and thusemploy the Lindblad master equation, following the standard derivation and formulationfrom Ref. [58]. More details are presented in Appendix B. This procedure requires anadditional parameter: the vibronic broadening η , which corresponds to the lifetime ofthe vibrational modes. Note that the laser-induced dynamics is, besides the electron-vibron interaction, basically treated on ab initio level endorsing the predictive power of emtosecond dynamics of correlated many-body states in C fullerenes time [fs] popu l a t i on ene r g y [ e V ] GS l a s e r e xc i t a t i on v i b r a t i ona l r e l a x a t i on thermalization GSBSDS DSDS/SAMOs BSDSDSDS DS/SAMOs (a) (b)3s 3p
Figure 7. (a) Population dynamics induced by the pump pulse (sketched in thebackground). The color coding of ground state (GS), bright states (BS) and darkstates (DS) is identical to Fig. 1(a). The insets show the weight of the 3s– and 3p–SAMOs in the dominantly populated states. The DSs involving the strongest SAMOexcitations is highlighted by the purple color. (b) Dominant population mechanismsfor the dynamics in (a). the approach. Note that the electron-vibron coupling (8) includes, in principle, Jahn-Teller and Herzberg-Teller couplings, which where identified as the main mechanismsfor vibrational coupling in fullerenes [59, 60, 61].The time evolution of the occupation of the states depicted in the level schemeFig. 1(a) is presented in Fig. 7(a). The driving pulse f ( t ) is chosen as a Gaussianpulse with a FWHM of 100 fs as in the experiment, while the central frequency isadjusted to the vertical excitation energy ∆ E = 5 .
66 eV between ground state and theBSs corresponding to the G peak. The peak amplitude amounts to the intensity of3 . × W/cm . The values for g and η are chosen to match the time-dependentpump-probe signal observed in the experiment (see Subsec. 5.3).As one can infer from Fig. 7, the population transfer between the ground state andthe bright excited states is clearly not in a perturbative regime, as two Rabi cycles areapparent during the 100 fs UV pulse interaction. The relaxation dynamics, transferringpart of the excitation to the dark states, takes place on two time scales: for shortdelays one can observe a rapid energy transfer, while for longer times the distributionthermalizes. The depletion dynamics of the laser-excited BS primarily takes place dueto the coupling to two lower-energy states at ≈ . emtosecond dynamics of correlated many-body states in C fullerenes In order to compute the pump-probe signals from the dynamics of the density matrix ρ αβ ( t ), as discussed above, an extension to the scattering states is required. However,a straightforward implementation of the Lindblad equation including both bound andunbound many-body states is not feasible. This is due to the large dimension of theHamiltonian after the inevitable discretization of the continuum. Hence, we opt for aperturbation description which allows to compute the pump-probe dynamics from thetime-dependent density matrix without incorporating the ionization dynamics explicitly.This is achieved by a method known from time-resolved photoelectron spectroscopy [62].Adopting a straightforward derivation for the many-body case, one obtains N k ∝ Re (cid:88) αβ (cid:90) ∞−∞ d t (cid:90) t −∞ d t (cid:48) F probe ( t ) F probe ( t (cid:48) ) e − i( E α + ω − E + β − (cid:15) k )( t − t (cid:48) ) | M β k ,α | ρ αα ( t (cid:48) ) (10)for the number N k of released photoelectrons with momentum k and energy (cid:15) k . Theprobe laser pulse is assumed as f probe ( t ) = F probe ( t ) e − i ωt with the pulse envelop F probe .The matrix element M β k ,α = (cid:104) Φ β, k | ˆ D | Φ α (cid:105) ( ˆ D denotes the dipole operator) describesthe transition from the intermediate states | Φ α (cid:105) to the final states | Φ β, k (cid:105) with onephotoelectron and the ion state labeled by β (energy E + β + (cid:15) k ). Note that only thepopulation of excited intermediates states and not the full density matrix enters eq. (10).This is an approximation, which relies on the fact that pathway interferences play onlya minor role for the considered two-step ionization process.For the excited state with one electron in the continuum state | k (cid:105) , | Φ β, k (cid:105) , we writethe usual anti-symmetrized product ansatz | Φ β, k (cid:105) = ˆ c † k | Φ + β (cid:105) , (11)where | Φ + β (cid:105) is an eigenstate of the ionized system with energy E + β . In this case thephotoemission matrix element reduces to (cid:104) Φ β, k | ˆ D | Φ α (cid:105) = (cid:104) k | ˆ D | φ D αβ (cid:105) . (12)Here, φ D αβ ( r ) stands for the corresponding Dyson orbital. As the sum over all excitedstates β is implied, a (computationally expensive) precise calculation of the Dysonorbitals can be omitted by approximating them by simple hole states, i.e., by assuming | Φ + β =( m,α ) (cid:105) ≈ ˆ c m | Φ α (cid:105) , E + β ≈ E α − (cid:15) m . (13)Here, (cid:15) m stands for the KS eigenvalue of orbital φ m ( r ). The matrix element (12)can thus be evaluated in terms of a superposition of matrix elements with respectto the KS orbitals. The scattering states | k (cid:105) were computed with respect to thespherically averaged KS potential [28], which is known to be an adequate approximationfor angle-integrated quantities. Asymptotic corrections ensuring the r − behavior areincorporated by smoothly interpolating between the short-range and long-range regimes.Further orthogonalization with respect to the bound KS orbitals is performed.To reflect the experimental situation, the integration over all photoelectron stateshas to be performed. Furthermore, we note that eq. (10) balances spectral resolution emtosecond dynamics of correlated many-body states in C fullerenes e − i∆ Et ,∆ E = E α + ω − E + β − (cid:15) k , with the envelop function. Due to very long pulses ascompared to one oscillation period, this convolution practically yields a Dirac δ -functionwith respect to the energy balance. Taking this into account, the total ionization pump-probe signal S ( τ ) = (cid:82) d k N k simplifies to S ( τ ) ∝ (cid:88) αβ (cid:90) ∞−∞ d t (cid:90) t −∞ d t (cid:48) F probe ( t − τ ) F probe ( t (cid:48) − τ ) P αβ ( E α + ω ) ρ αα ( t (cid:48) ) , (14)where P αβ ( (cid:15) ) = k (cid:82) dΩ k | M β k ,α | with k = √ (cid:15) is proportional to the energy-resolvedionization probability with respect to the initial state | Φ α (cid:105) and final state | Φ + β (cid:105) . As inSec. 4.2, τ denotes the pump-probe delay. The pump-probe signal based on the laser-driven and vibron-coupled dynamics of thedensity matrix can now be compared to the experiment. We remark that Eq. (14)assumes that the two pulses can be separated and does not account for the scenario ofoverlapping pulses. We thus limit the comparison between the theoretical calculationsand the experiment, presented in Fig. 5, to the region of exponential decay for τ ≥
300 fs.Furthermore, the dynamics presented in Fig. 7 indicates that the laser intensity exceedsthe perturbative regime. However, even with stronger pulses, the bright states around5 .
66 eV are the only accessible channels, as absorbing another photon leads to immediateionization. This will, however, only affect the background signal. Therefore, we adjustthe background S bg = S ( τ → ∞ ) to the experiment. After solving the Lindblad masterequation for various values of the parameters g and η , we found the best fit for g = 1 . η = 0 .
77 meV. The latter corresponds to a vibrational lifetime of ≈ . g from unity underpins the predictive power of our treatment. Note that also g = 1results in a decay dynamics which closely resembles, apart from the background, theexperimental data, whereas varying g to smaller or larger values clearly deviates fromthe measurements.We also calculated the ionization signal for varying photon energy ω and comparedthe resulting spectra in Fig. 4. To match the laser spectral bandwidth to the experiment,the obtained curves were convolved with a Gaussian having a FWHM of 2.8 nm. Thetheoretical spectra are centered around the vertical excitation energy, which is lowerthan what is observed in the experiment. This kind of underestimating of bandgapsand thus excitation energies is typical for most DFT calculations and for the C molecular, in particular [40]. However, the theoretical and the experimental peak differonly by ≈
10 meV. Generally, both the experimental as well as the theoretical spectraare considerably sharper as compared to optical absorption measurements. This is amajor advantage of the current experimental setup: the dipole excitation dominatesover possible loss channels, if the pulse strength is increased, which leads to a narrowresonance. emtosecond dynamics of correlated many-body states in C fullerenes The interpretation of the present data on the relaxation dynamics requires the fullpicture including both, correlated electronic and nuclear degrees of freedom andtheir interactions [63, 64]. It is known that highly excited electronic and vibrationalC states are strongly mixed [65, 55]. In turn, relaxation channels open up thatdepopulate the electronically excited states by internal conversion close to conicalintersections similar to electron–phonon coupling in solid state materials. We notethat characteristic fragmentation patterns observed in TOF mass spectra using opticallasers as a function of pump–probe delay revealed (nonadiabatic) vibronic coupling ona time scale of τ vib = 200 −
300 fs [45], which is in good agreement with the presentfindings. Furthermore, nonadiabatic coupling of mixed valence and Rydberg states isubiquitous in polyatomic molecules affecting potential energy surfaces, energy relaxationand dissociation dynamics [21]. Similar processes have been considered to evoke the lossof small neutral fragments from C on a picosecond time scale [66].The identification of vibronic coupling playing a key role in the electronic energyloss of correlated many-body states may open new vistas for optical control of charge-transport phenomena in smart materials containing these nanospheres. For instance,coherently induced radial symmetric ”breathing” motion of the cage atoms stronglyimpacts the structure and dynamics of the molecule [67]. The carbon cage and theelectron system start to exchange many eV in energy periodically on a sub-100 fs periodtimescale. The coherent oscillation prevails for several cycles [67], which might beinteresting for novel ultrafast switching applications in molecular electronics.
6. Summary and Outlook
Ultrashort pulses in the ultraviolet spectral range excite correlated many-body states ofisolated C molecules below the ionization continuum. The population is followed bysubsequent UV pulses of the same wavelength that ionize the molecules. By recordingthe C +60 ion yield as a function of time delay between pump and probe pulses we observean exponential decay with a time constant of τ vib = 400 ±
100 fs which is explained by thecoupling of electronic excitation to nuclear motion in the neutral molecule. The initialelectronic relaxation can be as fast as τ el = 10 +5 − fs according to the evaluation of theresonance line width in single pulse experiments as a function of excitation wavelength.The experimental results are in good agreement with ab initio calculation of structureand dynamics including electronic correlation and vibronic coupling. However, the UVpulse duration of 100 fs did not allow to observe pure electron dynamics in time-resolvedexperiments. Future experimental work making use of shorter UV pulses shall reveal thepredicted laser-driven Rabi oscillation and time-resolved transformation of the electronicorbitals, i.e., the coupling between different electronic states. emtosecond dynamics of correlated many-body states in C fullerenes Acknowledgment
This research was supported by the Deutsche Forschungsgemeinschaft through theexcellence cluster ”The Hamburg Centre for Ultrafast Imaging (CUI) – Structure,Dynamics and Control of Matter at the Atomic Scale” (DFG-EXC1074), thecollaborative research centres ”Light induced dynamics and control of correlatedquantum systems” (SFB 925), ”Functionality of Oxide Interfaces” (SFB 762), theGrK 1355 and Grant Number PA 1698/1-1.
Appendix A. Electron-vibron matrix elements
In this Appendix we provide the details on how the first-order (Herzberg-Teller)electron-vibron coupling matrix elements, Eq. (5), were computed. After the calculationof the vibrational eigenmodes ν , one readily obtains the associated deformations ofthe fullerene cage. Denoting the collection of nuclear coordinates in equilibrium by { R (0) } = ( R (0)1 , . . . , R (0)60 ), the vibrational distortion is characterized by { R ( ν ) ( q ) } = ( R ( ν )1 ( q ) , . . . , R ( ν )60 ( q )) with R ( ν ) m ( q ) = R (0) m + q V ( ν ) m , (A.1)where V ( ν ) m is the displacement eigenvector. Based on Eq. (A.1), we performed aDFT calculation for each vibrational mode, fixing the magnitude of the distortions at δq = 0 .
01. Instead of taking derivatives of the KS potential, we employ the equivalentformulation k νij (cid:39) δq (cid:104) φ i | ˆ h ν − ˆ h | φ j (cid:105) , ˆ h ν = − ∇ + v KS ( r , { R ( ν ) ( δq ) } ) , (A.2)where we have approximated the derivative by finite differences. The KS Hamiltoniandescribing the molecule in equilibrium is denoted by ˆ h , while ˆ h ν describes the distortedmolecule. For the evaluation of Eq. (A.2) we insert a (approximate) completenessrelation and obtain k νij (cid:39) δq (cid:104) (cid:104) φ i | ˜ φ ( ν ) k (cid:105) ˜ (cid:15) ( ν ) k (cid:104) ˜ φ ( ν ) k | φ j (cid:105) − (cid:15) i δ ij (cid:105) , ˆ h ν | ˜ φ ( ν ) k (cid:105) = ˜ (cid:15) ( ν ) k | ˜ φ ( ν ) k (cid:105) , ˆ h | φ i (cid:105) = (cid:15) i | φ i (cid:105) . (A.3)Note that the overlaps (cid:104) φ i | ˜ φ ( ν ) k (cid:105) incorporate the symmetry properties of the vibronicallyinduced transitions.The transformation into the many-body basis is accomplished by expressing theone-body coupling operator in second quantization, ˆ k ν = (cid:80) ij k νij ˆ c † i ˆ c j , and evaluating (cid:104) Φ α | ˆ k ν | Φ β (cid:105) according to the algebra of the fermionic creation and annihilation operators. Appendix B. Lindblad master equation
For the derivation of the Lindblad master equation in the weak-coupling limit, we followRef. [58]. For the Hamiltonian (7)–(9) one obtains the following EOM for the electronic emtosecond dynamics of correlated many-body states in C fullerenes ρ ( t ) = (cid:80) αβ ρ αβ ( t ) | Φ α (cid:105)(cid:104) Φ β | : dd t ˆ ρ ( t ) = − i (cid:104) ˆ H el ( t ) , ˆ ρ ( t ) (cid:105) + (cid:88) αβα (cid:48) β (cid:48) Γ αβα (cid:48) β (cid:48) (cid:16) ρ ββ (cid:48) ( t ) | Φ α (cid:105)(cid:104) Φ α (cid:48) | − {| Φ β (cid:48) (cid:105)(cid:104) Φ β | ˆ ρ ( t ) } (cid:17) . (B.1)The square (curly) brackets denote the commutator (anti-commutator). The vibronicbath enters intoΓ αβα (cid:48) β (cid:48) = (cid:88) ν γ ν ( E β − E α ) δ E β − E α ,E β (cid:48) − E α (cid:48) K ναβ K να (cid:48) β (cid:48) , (B.2)where γ ν ( E ) = ( N B ( E ) + 1) A ν ( E ). N B ( E ) denotes the Bose distribution(displayed in Fig. 6) which accounts for the occupation of the vibronic modes forthe given temperature. The vibrational frequencies determine the spectral function A ν ( E ) = 2 πδ ( E − Ω ν ), which we replace by the smeared form A ν ( E ) = (cid:113) π/η exp[ − ( E − Ω ν ) / η ] (B.3)to account for the finite lifetime of the vibrations. References [1] Sariciftci N S, Smilowitz L, Heeger A J and Wudl F 1992
Science
Science
Nat. Mater. J. Phys. Chem. Lett. Phys. Chem. Chem. Phys. Nano Lett. Chem. Phys. Lett.
Adv. At. Mol. Opt. Phys. Solid State Commun.
Solid State Commun.
J. Phys. B Science
Acc. Chem. Res. J. Chem. Phys.
Int. Rev. Phys. Chem. Phys. Rev. B Carbon Phys. Chem. Chem. Phys. Phys. Rev. Lett.
Phys. Rev. Lett.
ChemPhysChem J. Phys. Chem. A
Phys. Rev.Lett. Eur. Phys. J. D Science emtosecond dynamics of correlated many-body states in C fullerenes [26] Shchatsinin I, Laarmann T, Zhavoronkov N, Schulz C P and Hertel I V 2008 J. Chem. Phys.
Phys. Rev. Lett.
Phys. Rev. A Phys. Rev. A Sci. Rep. Phys. Rev. B Fundamentals ofTime-Dependent Density Functional Theory (Springer)[33] Bartlett R and Musia˚A M 2007
Rev. Mod. Phys. Rep. Prog. Phys. Recent Advances in Density Functional Methods (World Scientific)[36] Andrade X, Strubbe D, Giovannini U D, Larsen A H, Oliveira M J T, Alberdi-Rodriguez J, VarasA, Theophilou I, Helbig N, Verstraete M J, Stella L, Nogueira F, Aspuru-Guzik A, Castro A,Marques M A L and Rubio A 2015
Phys. Chem. Chem. Phys. Phys. Rev. A J. Chem. Phys.
J. Phys. B J. Am. Chem. Soc.
EPJ Web of Conferences [43] Wiley W C and McLaren I H 1955 Rev. Sci. Instrum. Phys. Rev. B J. Chem. Phys.
Phys. Rev. Lett. J. Chem. Phys.
The Jahn-Teller Effect in C and Other IcosahedralComplexes (Princeton University Press)[49] Leszczynski J 2012 Handbook of Computational Chemistry (Springer Science & Business Media)[50] Pittalis S, Delgado A, Robin J, Freimuth L, Christoffers J, Lienau C and Rozzi C A 2015
Adv.Funct. Mater. J. Phys. Chem. A
Rep. Prog. Phys. J. Comput. Chem. Phys. Rev. A Phys. Rev. B J. Chem. Phys.
The Jahn-Teller Effect: Fundamentals andImplications for Physics and Chemistry (Springer Science & Business Media)[58] Breuer H P and Petruccione F 2002
The Theory of Open Quantum Systems (Oxford UniversityPress)[59] Sassara A, Zerza G, Chergui M, Negri F and Orlandi G 1997
J. Chem. Phys. emtosecond dynamics of correlated many-body states in C fullerenes [60] Men´endez J and Page J B 2000 Vibrational spectroscopy of C Light Scattering in Solids VIII (Springer) 27–95[61] Orlandi G and Negri F 2002
Photochem. Photobiol. Sci. Phys. Rev. B Phys. Rev.Lett. J. Chem. Phys.
Phys. Rev. B J. Chem. Phys.
Phys. Rev. Lett.98