Application of the Real-Time Time-Dependent Density Functional Theoryto Excited-State Dynamics of Molecules and 2D Materials

Yoshiyuki Miyamoto1,2+ and Angel Rubio3,4

1Research Center for Computational Design of Advanced Functional Materials, National Instituteof Advanced Industrial Science and Technology (AIST), Central 2, Tsukuba, Ibaraki 305-8568, Japan

2TASC Graphene Division, Tsukuba, Ibaraki 305-8565, Japan3Max Planck Institute for the Structure and Dynamics of Matter and Center for Free-Electron Laser Science,

Luruper Chaussee 149, 22761 Hamburg, Germany4Nano-Bio Spectroscopy Group and ETSF, Universidad del País Vasco, CSIC-UPV=EHU-MPC,

20018 San Sebastián, Spain

(Received April 8, 2017; accepted July 11, 2017; published online January 26, 2018)

We review our recent developments in the ab initio simulation of excited-state dynamics within the framework oftime-dependent density functional theory (TDDFT). Our targets range from molecules to 2D materials, although themethods are general and can be applied to any other finite and periodic systems. We discuss examples of excited-statedynamics obtained by real-time TDDFT coupled with molecular dynamics (MD) and the Ehrenfest approximation,including photoisomerization in molecules, photoenhancement of the weak interatomic attraction of noble gas atoms,photoenhancement of the weak interlayer interaction of 2D materials, pulse-laser-induced local bond breaking ofadsorbed atoms on 2D sheets, modulation of UV light intensity by graphene nanoribbons at terahertz frequencies, andcollision of high-speed ions with the 2D material to simulate the images taken by He ion microscopy. We illustrate howthe real-time TDDFT approach is useful for predicting and understanding non-equilibrium dynamics in condensedmatter. We also discuss recent developments that address the excited-state dynamics of systems out of equilibrium andfuture challenges in this fascinating field of research.

1. Introduction

Density functional theory (DFT)1,2) can be used forpredicting material properties by dealing with the complexexchange correlation of many-body electronic systems as afunctional of the electron charge density. The one-to-onerelation between charge density and external potential holdsfor the electronic ground state. This one-to-one relation wasextended to time-dependent systems, resulting in time-dependent density functional theory (TDDFT),3) which hasbecome a powerful tool for studying excited-state dynamicsin many-body interacting systems.

Instead of the direct solution of the time-dependent Kohn–Sham equation,3)

iħ@ nðr; tÞ

@t¼ HKSðr; tÞ nðr; tÞ; ð1Þ

the main technique for the excited-state calculation remainedperturbation theory using the imaginary time scheme tocompute the real-space dielectric response function4,5) untilthe benefit of real-time propagation of electrons in computingthe (non-linear) optical responses was demonstrated.6) Inequation (1), nðr; tÞ and HKSðr; tÞ represent the time-dependent Kohn–Sham orbitals of the n-th state and theHamiltonian, respectively. The direct solution of the Kohn–Sham equation was also applied to excited-state moleculardynamics (MD)7–9) within the Ehrenfest approximation.10)

Photoexcited ultrafast dynamics within 1 ps (1 ps =10�12 s) have become a hot topic since the development ofthe pump-probe measurement technique using femtosecondlasers,11–13) and attosecond resolution has now been achiev-ed.14) For example, photoexcitation was found to cause thevery fast cis–trans transition of retinal with a time constantof 500 fs15) (1 fs = 1 � 10�15 s), whereas this process waspreviously believed to take several picoseconds. The ability

to control the chemical reactions of molecules16–18) has alsobeen demonstrated, and the amount of products obtainedfrom a laser-induced chemical reaction was used as feedbackto tune the pulse shape to maximize the product yield.19)

Structural changes in solids caused by femtosecond lasershave also been reported.20–22) Furthermore, the developmentof a short-pulse high-intensity laser source enabled us tofabricate nanoscale patterns in materials in a controlledmanner,23) although the underlying physics was not wellunderstood.

The ultrafast dynamics cannot be directly explained bythermal equilibrium dynamics; electrons are excited and arenot steady but transient with a finite lifetime before decayinginto lower-energy states (non-radiative decay). The pump-probe experiments require a full dynamical non-equilibriumdescription of a combined electronic-ionic system. Thepresent review will address some aspects of this issue. Itmay be a good approximation to treat electronic excited statesby introducing an electron temperature of several thousandkelvin alongside the lattice temperature.24) However, thisapproximation does not hold after laser excitation, as hasbeen measured experimentally.25) If the real-time dynamics ofelectrons can be considered throughout the simulation, thenon-equilibrium dynamics triggered by a strong laser can betreated.

In contrast, the DFT first-principles MD approach withinthe excited-state Born–Oppenheimer approximation assumesthe electronic states are steady states [obtained by eitherperturbation theory or constraint DFT (ΔSCF) techniques asmentioned later]. However, this scheme is not generallyapplicable for the case where non-adiabatic phenomenainduced by the laser pulse play a pivotal role.26) Therefore,a perturbation approach within the TDDFT scheme may beapplicable27,28) unless the perturbation is strong. However,using real-time TDDFT for Ehrenfest dynamics appears to be

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a useful approach, because electron dynamics are directlytreated throughout the MD simulation. Therefore, transientdynamics, such as alternation of the potential energy surface(PES) and modulation of the applied optical field, can beobserved throughout the MD simulation.

However, performing light-induced classical MD is anapproximation, and the appropriate choice of one of twoclassical MD approaches (Ehrenfest dynamics10) and surfacehopping29)) was concluded to be system dependent,30) andthat surface hopping is necessary when the system containsan avoided crossing in the PES. Trajectory-based methods,such as ab initio multiple spawning31) and Tully surfacehopping,29) can efficiently simulate non-adiabatic processesonce the appropriate Born–Oppenheimer potential energysurfaces (BOPESs) and non-adiabatic coupling constants(NACs) are provided. These BOPESs and NACs are,however, extremely difficult to obtain for large systems,and there is a great demand for practical methods that avoidcalculating the BOPESs and NACs. A leading method fordescribing the dynamics of large electron-nuclear systems isTDDFT coupled to classical nuclear trajectories throughEhrenfest dynamics because of its favorable system-sizescaling. The Ehrenfest method with TDDFT has beensuccessful for many applications, from photodissociationdynamics in small molecules9) to radiation damage in metals;its efficiency allows calculations on large systems for eventhousands of femtoseconds, as shown in one of the resultspresented in this overview.

Full quantum trajectory methods describe full nuclearquantum effects on the fly, but require an exhaustiveknowledge of the BOPES landscape.32–34) In addition, thering-polymer phase space was shown to provide a practicalMD framework to approximate the effects of quantumfluctuations, including zero-point oscillation and tunnelingeffects (see the review in Ref. 35 for details of this method).Recently, two alternative frameworks have been developedto describe non-adiabatic nuclear MD by either an exactfactorization of the many-body electron-nuclear wavefunction34,36,37) or through an efficient conditional wavefunction decomposition.38–40)

Instead of reviewing the excellent work reported by manygroups, here we present a selected overview of our work, tohighlight some important issues and concepts that we havestudied over recent years, and discuss some of the openquestions that still need to be answered. We first brieflyintroduce the theoretical framework and how this can beimplemented in massive high-performance computing archi-tectures to allow handling of complex molecular nanostruc-tures and materials. The main theme of this review is thedescription of the non-equilibrium dynamics for some finiteand extended systems. For molecular systems, we discussthe photoisomerization process of azobenzene in Sect. 3.1,in which the initial condition was set as an excited statecomputed by the ΔSCF scheme by promoting electronoccupation numbers according to the photoexcited states. Wethen consider the dynamics in molecules and 2D materialsunder an alternating electric field (E-field), which mimics theoptical E-field of lasers. The effect of the E-field varies fromincreasing the dipole moment (Sect. 3.2) to breaking localbonds, depending on the time dependence and intensity ofthe E-field (Sects. 3.3 and 3.4). Even when ion dynamics

are negligible, non-equilibrium electron dynamics play animportant role in generating a terahertz (THz) signal(Sect. 3.5). Finally, we consider the high-speed collision ofHe+ with graphene to illustrate the mechanisms of secondaryion emission (Sect. 3.6), which has been applied in a newmicroscopic technique.

2. Computational Scheme

Our main scheme is summarized in Ref. 41. The real-timeTDDFT calculations can be performed by using a plane-wavebasis set7,8) and for periodic=non-periodic systems with areal-space grid.6,9) In the discussion below, all calculationswere performed with a plane-wave basis set using norm-conserving pseudopotentials42) in separable forms43) toexpress electron-ion interactions through the dynamics ofvalence electrons. A numerically reliable scheme for solvingequation (1) is the split-operator method,44,45) in which thetime propagator [exponential of HKSðr; tÞ] is expressed asproducts of exponentials of the kinetic energy operator, localpotential, and non-local part of the pseudopotentials. Severalmathematical schemes for solving the time-dependent Kohn–Sham equation are reported in Ref. 46 for readers interestedin further details. A scheme to apply the split-operatormethod to the plane-wave basis set and pseudopotentialmethod as well as keeping a self-consistent field is describedin Refs. 7, 8, and 47.

Because the Kohn–Sham Hamiltonian [HKSðr; tÞ] is afunctional of charge density �ðr; tÞ, consisting of the sum ofthe norm of Kohn–Sham orbitals, KS

n ðr; tÞ, for all occupiedstates, a self-consistent relation between the time propagatorand the Kohn–Sham orbital must be maintained throughoutthe real-time propagation. To maintain this self-consistentrelation, a predictor-corrector algorithm must be used, inwhich the time variation of the Hartree-exchange–correlationpotential is extrapolated and interpolated by a railwaysmoothing scheme (see Refs. 7, 8, and 47 for more detail).

Ionic forces are computed by the Hellmann–Feynmanforce scheme,48) and the MD simulation is performed. Theinterval for the time step of the MD simulation was set ascommon to that for computing electron dynamics, which ison the sub-attosecond timescale (1 attosecond = 10�18 s).This short time interval is necessary for MD simulations, asthe force field shows high-frequency fluctuations arising fromthe electron dynamics due to equation (1).

In the examples discussed below, the real-time TDDFT-MD simulations were performed by the following procedure.First, we execute the DFT calculation in the electronicground state and optimize the structure of the system usingthe Hellmann–Feynman forces.48) For the initial conditionof the excited-state dynamics, we choose either a solutionobtained by a ΔSCF calculation to mimic the opticallyexcited state, or the electronic ground state, which is underan alternating optical field. Then, we start to move ionsaccording to the computed force field and begin timepropagation of all Kohn–Sham orbitals.

When the initial condition of the electronic state is set as theground state and the optical field is considered in the sim-ulation, equation (1) is modified with the Coulomb gauge as

iħ@ nðr; tÞ

@t¼ ðHKSðr; tÞ þ Vextðr; tÞÞ nðr; tÞ; ð2Þ

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in which the optical field is given by a negative value of thespatial gradient of the external potential, Vextðr; tÞ. To performMD coupled with equation (2), the introduction of fictitiouscharge �extðr; tÞ to express external potential as

Vextðr; tÞ ¼Z�extðr0; tÞjr0 � rj dr

0 ð3Þ

is convenient to compute the total energy to check the energyconservation rule.49)

To perform a real-time TDDFT-MD simulation, paralleli-zation is necessary to accelerate the computational speed.One of the merits of the split-operator method44,45) forparallelization is the automatic preservation of the orthonor-mal condition for all Kohn–Sham orbitals, KS

n ðr; tÞ, asZ KS�n ðr; tÞ KS

m ðr; tÞ dr ¼ �n,m; ð4Þ

where �n,m is the Kronecker delta. Therefore, parallelcomputing allocating the tasks of computing time propaga-tion for each Kohn–Sham orbital does not need to distributethe data for KS

n ðr; tÞ among the processors to build theorthonormal relation expressed by equation (4).

3. Results: Application of Real-time TDDFT

In this section, we describe some approaches using real-time TDDFT coupled with MD simulation. In Sect. 3.1, theexcited-state MD simulation of molecules (photoisomeriza-tion) is presented. This simulation requires the initialcondition of the excited state prepared by a static DFTcalculation within the ΔSCF scheme by promoting electronoccupation numbers from the occupied molecular orbital tothe unoccupied molecular orbital. In Sect. 3.2, a method forincreasing the attractive van der Waals interaction in the Heatom dimer and sheets of hexagonal boron nitride (hBN) ispresented. The physics of the increased attraction is attributedto dipole–dipole attraction caused by laser-induced electronmotion and lattice vibrations. Then, in Sects. 3.3 and 3.4, aproposal for using a laser with an anisotropic field profile as afunction of time is presented to remove hydrogen from onlyone side of graphene terminated with H on both sides (i.e.,graphane) and to reduce graphene oxide. In Sect. 3.5, aproposal for applying graphene nanoribbons to the THzradiation source is presented. The optical property ofgraphene nanoribbons that causes amplitude modulation ofUV light is discussed. Finally, in Sect. 3.6, the application toelectronic excitation in 2D sheets by high-speed ions ispresented and the principles for imaging using a He ionmicroscope are discussed.

3.1 Photoisomerization of azobenzene using Ehrenfestdynamics

We have investigated the photoisomerization dynamics ofthe azobenzene molecule,50) which are shown schematicallyin Fig. 1(a). By using the ΔSCF scheme, the S1 excited statewas approximated as the highest molecular orbital (HOMO)and the lowest unoccupied molecular orbital (LUMO) beinghalf occupied. The S2 state was approximated as the HOMO-1 state and LUMO state being half occupied. The azobenzenemolecule showed optical S0 → S2 excitation and subsequentnon-adiabatic S2 → S1 decay.51) After reaching the S1 state,the azobenzene molecule exhibits trans to cis isomeriza-

tion52) which was reproduced by local-density approxima-tion (LDA) calculations within the ΔSCF scheme.53) Weperformed a real-time TDDFT-MD simulation of S2 excitedazobenzene and examined whether the S2 → S1 transition andthe trans–cis isomerization occurred. For the calculation, theLDA54) was adopted, as the PES of azobenzene because theLDA agrees with the results of other many-body approx-imations according to a previous theoretical study.52)

In the case of the typical Ehrenfest dynamics, in whichelectron dynamics and ion dynamics are treated individually,the MD trajectories are considered to follow a PES locatedbetween either the S2 and S1 states or the S1 and S0 states, asargued by Tully.26) However, by performing the real-timeTDDFT-MD simulation starting with the S2 excited state, thepotential energy obtained by TDDFT initially followed thepotential energy of the S2 state and smoothly transferred tothat of the S1 state,50) as depicted in Fig. 1(b).

Figure 1(b) was obtained by the following procedure.First, we plotted the time evolution of the potential energyalong with the real-time TDDFT-MD trajectory. Then,several snapshots of the trajectory were chosen, for whichthe static DFT calculations were performed to compute thepotential under the ground state and S1 and S2 excited stateswithin the ΔSCF scheme. Several MD trajectories startingfrom different atomic coordinates were examined50) and theS2 → S1 transition always occurred, rather than the mixing ofthese states.

TransCis

S0

S1

S2

Reaction coordinates

ygrenE laitneto P

(a)

(b)

Fig. 1. (Color online) (a) Schematic PESs of the S0 (ground) state and theexcited S1 and S2 states of azobenzene. The structures of trans- and cis-azobenzene are also displayed. The S0 → S2 photoexcitation is shown and thesubsequent dynamics showing the S2 → S1 transition during the trans–cisphotoisomerization are indicated by the broken arrow. (b) Numericalcomparison of the potential energy obtained by TDDFT and thoseobtained by static DFT. The potential with static DFT was computed usingatomic coordinates on snapshots of the Ehrenfest dynamics by TDDFT, seeRef. 50.

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When this simulation was continued to 500 fs, trans-azobenzene showed photoisomerization to cis-azobenzene[the corresponding structures are shown in Fig. 1(a)]. Incontrast, the potential obtained by real-time TDDFT-MDsimulation merged with that of S1 without undergoing mixingof the S0 and S1 states. The isomerization was completed bythe rotation of the two benzene rings of azobenzene, whichagreed with previous theoretical works performed under thecomplete active space self-consistent field scheme.55) Thissmooth transfer was the case with several classical MDtrajectories.50) The PESs also did not mix for ethylene (C2H4)molecules under � ! �� excitation, which causes C–Cstretching and twisting of the two CH2 units of the ethylenemolecule. The twisting makes the π–π bonding and π–�� anti-bonding states equivalent at 45° rotation around the C–C axisof the ethylene molecule.50) Such photoinduced dynamicscan also be reproduced by real-time TDDFT in organicmolecules,56,57) and thus we expect that real-time TDDFT isapplicable to many subfields of photochemistry.

So far, the absence of PES-mixing was just found byperforming TDDFT Ehrenfest dynamics, and still needsrigorous theoretical proof as the subject of future works.Meanwhile, photoisomerization is not followed by thedisintegration of molecules, in which the charge is redis-tributed across the fragments. After molecular disintegration,the fragments should possess integer numbers of electrons,whereas TDDFT can give fragments with non-integernumbers of electrons. We expect that the mixing of severalPESs could occur in these situations. This was argued for thecase of scattering of ions on metal surfaces.26) For morediscussion of the validation of Ehrenfest dynamics withinTDDFT, interested readers are directed to Ref. 41.

3.2 Laser-induced enhancement of weak dispersioninteraction

Van der Waals forces are often represented as dynamicaldipole–dipole interactions (London forces).58) According tothe dynamical dipole–dipole picture, photoenhancement ofthe dipole moment on each molecule (atom) can increase theattractive force between them. We tested this idea byintroducing a simple model [Fig. 2(a)]. In this model, thereare two molecules each consisting of a cation (þe) and ananion (−e) connected by springs. We assumed that theinteractions between molecules consist of Coulomb inter-actions between ions. In contrast, we restricted the inter-actions inside each molecule to occurring only via the spring.We assumed that the mass of the cation (anion) ion is theatomic mass of B (N), and the spring constant is tuned to theresonance optical phonon frequency with a correspondingwavenumber of 780 cm−1. We then applied an alternatingE-field by a sinusoidal function with a frequency either 10%lower than the resonant frequency or equal to the resonantfrequency. Figure 2(b) shows the case with the frequency10% below the resonant frequency. The dynamics of thecation and anion showed the modulation of the amplitudeof vibration and there was no major reduction of theintermolecular distance. Figure 2(c) shows that when thefrequency of the E-field was exactly equal to the resonance,the amplitude of vibration increased and the dipole of eachmolecule also increased. As a result of the increased dipole,the intermolecular distance was greatly reduced.

This idea was verified by performing the real-timeTDDFT-MD simulation of the He dimer under irradiationwith UV light, which can excite electrons in the He atoms.59)

Suppose two He atoms are under an alternating E-field withthe polarization direction parallel to the He–He axis. Theelectrons on each of the He atoms attempt to follow themotion of the alternating E-field, causing a dynamical dipolemoment. If the frequency of the alternating field matches theresonant frequency of the electron motion (electron-cloudmotion) the amplitude of the dipole moment grows on eachHe atom in the parallel direction, increasing the dipole–dipoleattraction.

We computed the change of He–He distance within thegeneralized gradient approximation with the Perdew–Burke–Ernzerhof functional,60) including the correction for thevan der Waals force.61) A cutoff energy of 115Ry was used

(a)

(b)

(c)

E

―

―

Fig. 2. (Color online) (a) A model of two molecules consisting of an anionand a cation connected by springs. The optical field is denoted by the arrowlabeled “E”. Time variation of the positions of the anion with light-gray (red)and cation with dark-gray (blue) in the two molecules under an external fieldwith (b) a frequency 10% lower than the resonance and (c) resonantfrequency.

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for the plane-wave basis set. Figure 3(a) shows a snapshot ofthe time variation of the charge density of electrons aroundthe two He atoms, which is the charge density at t ¼ 28:15 fsminus the initial charge density (t ¼ 0). The frequency ofthe alternating E-field of 0.2068 fs corresponds to a photonenergy of 20 eV, which is close to the photoexcitation energyof the He atom. The charge density shown in Fig. 3(a) showsthe parallel orientation of the dipole that should increase theattractive force on the He atoms.

The increasing force was identified by the time variation ofthe He–He distance compared with the He–He distance in theabsence of an optical field. Figure 3(b) shows the timevariation of the He–He distance under alternating E-fieldswith several photon energies around 20 eV. The He–Hedistance decreased substantially with a photon energy of20 eV, whereas the speed of the He–He contraction becameslower when the photon energy was either above or below20 eV. According to the estimation by Newtonian dynamics,the increase of the force was 7 pN,59) which is comparable tothe order of force measured by atomic force microscopy.

Next, we discuss the physics behind the increasing He–Heattraction. The He dimer possesses orbital hybridization thatgenerates bonding and anti-bonding molecular levels con-sisting of He 1s orbitals. If photoexcitation from the bondingto the anti-bonding orbitals occurs, the He–He repulsive forceshould take effect, which is not consistent with the results. Incontrast, the photon energy of 20 eV, under which the He–Heattraction increased, does not coincide with the excitation intothe anti-bonding orbital. TDDFT can cause the unphysical

dynamical detuning effect of the optical response in the Heatom,62) even under exact matching between the photonenergy and excitation energy. This detuning effect comesfrom the adiabatic approximation of exchange–correlationfunctional currently used in the TDDFT simulations. Insteadof the sharp excitation into the He–He anti-bonding state, thisexcitation dynamically modulates the 1s orbital on each Heatom into a synchronized phase, as depicted in the timevariation of the charge density shown in Fig. 3(a). Thus, theincrease in the attractive He–He interaction was rationalizedas dipole–dipole attraction.

The optical enhancement of weak attraction is alsoobserved in extended systems such as layered materials(2D materials). Here we focus on hBN, which is an analogueof graphite that possesses B and N atoms instead of the Csub-lattice of the graphitic layer [Fig. 4(a)]. Because of thegreater electron affinity of N atoms compared with B atoms,the out-of-plane displacement of B and N atoms shown inFig. 4(b) causes a dipole moment perpendicular to the 2Dsheet, which increases the attraction between the hBN sheets.The corresponding optically active phonon has the A2u modewith a wavenumber of ∼780 cm−1.

(a)

, = 28.16fs − ( , = 0fs)ℏ = 20 eV

(b)

Fig. 3. (Color online) (a) Time variation of the charge density of the Hedimer at t ¼ 28:15 fs under an alternating E-field with a correspondingphoton energy of 20 eV. The thick (red) and thin (blue) contour lines showthe electron-rich region and electron-depleted region, respectively, comparedwith the charge density at t ¼ 0 fs. Reprinted from [Appl. Phys. Lett. 104,201107 (2014); doi: 10.1063=1.4875108], with the permission of AIPPublishing. (b) Time variation of the He–He distance in the presence andabsence of an optical field. The dashed (black) line is the time evolution inthe absence of light. The solid (green), dash-dotted (violet), and dotted (red)lines show the time evolution in the presence of optical fields correspondingto photon energies of 20, 20.32, and 19.89 eV, respectively.

(a)

(b)

B

N

(c)

E

Fig. 4. (Color online) (a) Planar structure of hBN layers. B and N atomsare depicted. (b) Atomic displacement of hBN under optical phonon mode.B and N atoms show out-of-plane displacement in opposite directions togenerate a dipole moment perpendicular to the hBN sheets. (c) Timevariation of the heights of the B and N atoms of four hBN layers underirradiation with an IR laser with a FWHM of 400 fs, a wavenumber of789.44 cm−1, and an optical E-field intensity of 1V=Å. The light-gray (red)and dark-gray (blue) lines represent the heights of the B and N atoms,respectively.

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Because the interlayer interaction of hBN is well describedby the LDA because of a fortuitous error cancellationbetween the exchange and correlation functionals,63) weadopted the LDA for studying the effect of phonon excitationon the interlayer interaction in hBN. We used a 1 � 1 unit cellof the double-layer hBN sheets and applied an alternating E-field with wavenumbers of 710.50, 789.44, and 868.38 cm−1

to solve equation (2). These values are determined by thephonon frequency of the A2u mode by LDA and its variationby �10%.

When we set the laser power to 1:327 � 1013W=cm2 witha corresponding E-field of 1V=Å and a frequency of789.44 cm−1, an interlayer contraction of over 12% of theoriginal interlayer distance was observed.64) However, afterreaching the minimum interlayer distance, the two hBNsheets repelled one another, which was associated withelectronic excitation due to the high intensity of the opticalE-field.

We continued our work in Ref. 64 by using a pulsed laserwith an full width at half-maximum (FWHM) of 400 fsinstead of using a continuous-wave (CW) IR laser, becausepulsed sources are more accessible for high-intensity IRlasers than CW sources. The same wavenumber of 789.44cm−1 and the same E-field intensity of 1V=Å were assumed.For the hBN layers, we extended the model size from twolayers to four layers to determine whether the increase ininterlayer attraction also occurs for multiple layers. Weobserved a substantial interlayer contraction, as shown bythe time variation of the heights of the B and N atoms ineach of the four hBN layers [Fig. 4(c)]. The transientcontraction of the interlayer distances of the four hBN layersoccurs up to 1 ps. After 1 ps, the outer (top and bottom) andinner layer motions show different periods for the change inthe positions of their centers of mass. The A2u optical phononremained after the optical E-field completely faded out,although it should decay as a result of thermal dissipationinto the environment, which was not considered in thesimulation.

3.3 Laser-induced dehydrogenation of graphane(hydrogenated graphene)

Lasers with particularly short pulses have been consideredas a tool for the selective cleavage of chemical bonds.Graphane, which is graphene terminated with H atoms onboth sides of the sheet, was theoretically predicted,65) andexperimental work into cyclic hydrogenation and dehydro-genation has been reported.66) Because of the chemicalequivalence of the C–H bonds, the selective dehydrogenationfrom one side of graphane is impossible. However, we haveproposed a theoretical way to remove H atoms from only oneside of graphane by using ultrashort laser pulses.67)

Figure 5(a) depicts the ultrashort pulse laser used in thesimulation, which had a strong E-field in one directionaround t ¼ 2 fs, whereas the intensity of the E-field in theopposite direction was much smaller. Because of thisparticular pulse shape, the graphene electron cloud, whichis shown by the contour maps in Figs. 5(b)–5(e), showed adownward shift. Because of this shift, the electron chargedensity in the upper region was lower than that in the lowerregion [Fig. 5(e)], even after the decay of the optical E-field.The timescale in Figs. 5(b)–5(e) was too short to see

substantial ion motion. However, this unbalanced chargedensity caused different strengths of the C–H bonds anddynamics of the H atoms over a longer timescale, as shown inFig. 5(f ), in which only the H atoms in the upper region wereremoved from the graphane, whereas those in the lowerregion remained. The dehydrogenated side would becomechemically active, and we proposed the introduction ofhalogen gas immediately after irradiation with the ultrashortpulse laser to induce bonding of the halogen atoms to thedehydrogenated side, thereby fabricating a heterogeneouslyterminated dipole layer with monoatomic thickness.67) Suchheterogeneously terminated sheets have many intriguingpossible applications, and one example is proposed inRef. 67.

Up to this point, we have focused our attention on thedynamics induced by ultrashort (several femtosecond) laserpulses. Such short pulses are, however, challenging toachieve using current laser technology. At present, thetypical FWHMs are a few hundreds of femtoseconds forpractical applications in material fabrication68,69) with thermalwelding localized at the laser spot. However, when theFWHM is less than 100 fs, the welding is expected to occurnon-thermally.70) In the next section, we set the FWHM at10 fs, which is a compromise between current practical valuesand the theoretical value in this section, and discuss TDDFT-MD simulations for such systems.

0 fs H

HC

2 fs

H

H

C

4 fs H

HC

H

6 fs H

C

(a)

(b) (c) (d) (e)

30fs

Upper hydrogen

Lower hydrogen

(f)

Fig. 5. (Color online) (a) Pulse shape of the ultrashort pulse laser used inthe graphane simulation. (b)–(e) Change in the valence charge density ofgraphane under the ultrashort laser field. The superimposed downward arrow(in yellow) in (c) denotes the direction of the optical field that forceselectrons downward. (f ) Dynamics of upper and lower H atoms under theultrashort laser field. These figures are adapted with permission from Ref. 67.Copyrighted by the American Physical Society.

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3.4 Simulation of the reduction of graphene oxide by afemtosecond pulsed laser

Graphene oxide is a widely used material that dissolves insolvents, allowing it to be used as an ink to print circuits.After printing, the key processes are reduction of thegraphene oxide to produce conductivity and repairing thedamage caused by the reduction.71) In our previous work,72)

we focused on the epoxy structure, in which each O atom wasconnected to two C atoms on a graphene sheet, and thehydroxyl structure, in which each OH group was connectedto one C atom on a graphene sheet, assuming the very shortpulse shown in Fig. 5(a). For both the epoxy and hydroxylstructures, the simulation suggested that irradiation with anultrashort pulse laser can reduce graphene oxide withoutdamaging the graphene sheet. Chemical methods usinghydrazine also completely reduce graphene oxide (N2H4).73)

However, the use of N2H4 is not practical because of itstoxicity, and thus alternatives to chemical methods are inhigh demand. We have proposed laser-assisted reduction,which can be used without damaging the graphene.

A femtosecond laser with an FWHM of 100 fs was used toreduce graphene oxide;74) however, the Raman spectrumshowed that the reduced region was badly damaged.Therefore, laser reduction technology should be improvedby either changing the FWHM or the shape of the laser pulse.Recently, the phase-controlled mixing of two wavelengths,400 and 800 nm, was used to introduce asymmetric rotationaldynamics in gas molecules.75) The asymmetric motion of gasmolecules arose from asymmetry in the optical E-fieldpolarization of the phase-controlled mixing laser. Weinvestigated the applicability of this laser technique to thereduction of graphene oxide by performing real-timeTDDFT-MD simulations.

Similar to the pulse shape shown in Fig. 5(a), mixingwavelengths of 400 and 800 nm also produces an asymmetricoptical field with wider pulse widths (Fig. 6). Although thepulse width used in Ref. 75 was 130 fs, we considered ashorter pulse width of 10 fs as a future technology to increasethe strength of the optical E-field. The pulse shape shown inFig. 6 is expressed by

EðtÞ ¼ E0 exp �A t � t0�=2

� �2( )

ðsin!t þ sin 2!tÞ; ð5Þ

where A ¼ 1=2 ln 2, � ¼ 10 fs, and t0 ¼ 20 fs. The grapheneoxide model comprised stacked layers of graphene oxidewith an epoxy structure [Fig. 7(a)], in which O atoms areadsorbed on both sides of the graphene sheet. This modelwas proposed by Yan and Chou,76) suggesting preferableaggregation of two O atoms in the epoxy structure on bothsides of the graphene sheets according to DFT calculations.

Because of the presence of the epoxy structures on thegraphene sheet, the interlayer distance was increased to 6Å,yet these two sheets were still bound. Because the interlayerdistance became wider than that of pristine graphene sheets,the intercalation of water molecules occurred easily[Fig. 7(b)], with a binding energy of 77meV per watermolecule using the LDA functional of DFT. Figures 7(c)–7(e) show the time evolution of the structure in Fig. 7(b) 0,50, and 100 fs after illumination with the pulse in Fig. 6. Thepresence of the water molecule affected the dynamics ofreduction under the laser pulse. The laser field intensity was2.5V=Å with a corresponding power of 8:3 � 1013W=cm2.The polarization vector of the laser field is shown as an arrowin Fig. 7(c). The water molecule provided its H atom to theejected O atom, and two OH groups were formed. We alsoperformed the simulation without water molecules and foundthat the O atoms left the graphene layers as O radicals, whichsubsequently attacked the graphene. Therefore, we concludedthat the presence of water molecules is important to preventthe reaction of laser-generated O radicals with the graphenesheets.

Although the E-field polarization is asymmetric (Fig. 6),this simulation with the polarization vector parallel to thegraphene sheets did not show the asymmetry of thealternating optical field. We expect that the asymmetryoccurs when the optical polarization is perpendicular to thegraphene sheet, as discussed in Sect. 3.3.

So far, we have described the structural changes inmolecules and 2D sheets induced by electronic excitation,meanwhile real-time TDDFT can also be applied to electronmotion induced by electronic excitation with no structuralchange. This electron motions is useful for device applica-tions and the measurement of nanostructures. We describethese examples in the following subsections.

Fig. 6. (Color online) Pulse shape of a laser with mixed wavelengths of400 and 800 nm and FWHM of 10 fs.

(a) (b)H

O

H2O

(c) (d) (e)

Fig. 7. (Color online) (a) Model of stacked graphene oxide using armchairgraphene nanoribbons with the epoxy structure. (b) Same as (a), but withwater molecules between the graphene oxide layers. (c)–(e) Time evolutionof graphene oxide stacked layers with water molecules, at (c) 0, (d) 50, and(e) 100 fs. The superimposed vertical arrow (in red) in (c) shows thepolarization vector of the optical field, which is decayed in (d) and (e).

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3.5 Designing THz radiation devices using graphenenanoribbons

Because of the wide use of THz light for imaging77)

in high-speed communications, developing sources of THzradiation has become an important topic in fundamental andengineering research. By irradiating YBCO solids with afemtosecond laser, THz radiation can be obtained from laser-excited carriers.78) Moreover, irradiation of semiconductorscoupled by a strong magnetic field with a femtosecond laseralso produces THz radiation.79) In both cases, understandingthe electronic dynamics at THz frequencies is essential forgenerating THz radiation. We investigated the properties ofgraphene nanoribbons relevant to THz radiation, and foundthat the graphene nanoribbons can modulate the intensity oflight at THz frequencies.80)

We investigated the optical properties of armchairgraphene nanoribbons,80) which are semiconductors.81) Weconsidered the polarization direction of the optical fieldparallel to the graphene sheet and perpendicular to the ribbonaxis. The Fourier spectrum of the response to a short pulsewithin the TDDFT scheme6) revealed that the graphenenanoribbons were found to possess strong optical absorptionin the visible and UV regions. We solved equation (2) andplotted the spatial gradient of the self-consistent Hartree-Exchange–Correlation potential within the TDDFT schemeto monitor the induced field. The geometrical conditions ofthis simulation are shown schematically in Fig. 8(a) and thesum of the induced field and applied optical field gave the

generated field. The generated field was monitored at thecenterline and at the ribbon axis at a height of 3.34Å abovethe sheet. The field was spatially averaged along the period ofthe ribbon.

The graphene nanoribbon in Fig. 8(a) has seven C–Cbonds parallel to the ribbon axis along the direction of theribbon width, denoted by N ¼ 7 in Ref. 81. The N ¼ 7

nanoribbon showed strong modulation of light with photonenergies of 6.20 and 6.53 eV, both of which are in the UVregion. Although this ribbon had a peak in the visible lightregion, the peak did not show strong modulation in theoptical field. The modulation is shown in Fig. 8(b), where thegenerated field had the same frequency as the applied opticalfield, but its intensity increased, decreased, and then increas-ed again over time. This trend in the modulation of UV lightwas also observed in wider graphene nanoribbons (N ¼ 9 and11) according to our simulation,80) and the period of themodulation was around 100 fs, corresponding to a frequencyof 10THz. The intensity modulation could be converted intophotocurrent modulation at THz frequencies if a UV lightwere shone on a photoconducting semiconductor througha graphene nanoribbon. Connecting the photocurrent to anantenna could generate THz radiation.

Optical field enhancement can be understood as theresonance of the response of the electronic state with theapplied optical field, whereas the intensity modulation can beunderstood as the presence of several excited states thathave different eigenfrequencies. This is analogous to THzgeneration using quantum wells in semiconductors,82) inwhich the THz frequency is proportional to the inverse of thedifference in excitation energies in several excited states.

Our computational study80) suggested that THz radiationwith a frequency of 10THz can be generated from graphenenanoribbons, whereas the industrially required frequenciesare 0.5–5THz. Therefore, it is necessary to explore othermaterials that may induce light modulation that matches therequired frequencies.

We have focused on the dynamics induced by opticalexcitation. However, electronic excitation can also beachieved by collisions with high-speed ions. In the finalexample, we describe the use of our TDDFT scheme toexplain electronic excitation by high-speed ion collisions,which is applied to a new type of microscopy.

3.6 Interaction between high-speed He+ and condensedmatter: Principles of He ion microscopy

In this section, we describe the application of our real-timeTDDFT scheme to the interaction between high-speed ionsand condensed matter. When the ion velocity is comparableto the electron Fermi velocity, the collision of ions withcondensed matter can excite an electronic state. This wasfirst demonstrated by a real-time TDDFT-MD simulationof protons colliding with graphene sheets,83) causing ionstopping when the incident kinetic energy of the projectileproton was on the order of kiloelectron volts or higher. Theinteraction of high-speed ions with graphene also causeselectronic excitation in the projectile ion, which results influorescence from the traversing ions after the collision.84)

The electronic excitation upon the collision of high-speedions with condensed matter is also useful for measurementtechniques, such as He ion microscopy (HIM) using He ion

Applied field

(a)

Generated field

(b)

Fig. 8. (Color online) (a) Graphene nanoribbon under an applied opticalfield, shown by an arrow with dark-gray (blue). The generated field, shownby an allow in light-gray (red), is plotted on a central line 3.34Å above thegraphene nanoribbon sheet, and is averaged over the period along the ribbonaxis. (b) Time evolution of the applied and generated fields with opticalenergies of 6.20 and 6.53 eV. The horizontal black arrows indicate theperiods of enhancement in the generated field. (See, Ref. 80.)

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beams.85) In this technique, He+ is accelerated to 30 keV in anarrow beam, allowing them to traverse material at a speedcomparable to the electron Fermi velocity. Secondaryelectron emission occurs and the intensity of the emissioncurrent is measured by scanning the He+ beam to generate aspatial map of the yield of the secondary electron emission.Because the He beam is narrower than the electron beam afterthe collision with the target,86) the spatial resolution of HIMis superior to that of scanning electron microscopy.

It is important to understand why secondary electrons areemitted during the collision of the He+ beam, which has theopposite polarity to the electron beam. Thus, a first-principlessimulation was performed to understand the physical basis ofthe electron emission. However, simulating the collisions ofcharged ions and solid surfaces requires technical skillbecause of the potential diagram (Fig. 9). The diagram showsthat ions usually have their unoccupied levels below theFermi level of the solid; thus, the charge transfer shouldoccur in the self-consistent calculation in ordinary DFT nomatter how far the ion is from the solid surface. This chargetransfer has been avoided by dividing the space artificially todistribute the charge density,87) although for ion-solidcollisions, the artificial division of the space should be timedependent and is inappropriate for short distances betweenthe ion and solid.

We used a real-time TDDFT-MD simulation to demon-strate that the emission of the secondary electrons upon the

collision of He+ is dependent on the impact points, as well asto clarify the emission mechanisms.88) We prepared the ionand the target electronic states separately in the ground state,and then merged their states in a large unit cell includinga vacuum region to avoid overlap of the wave functions ofthe ion and the target,89) prior to the real-time TDDFT-MDsimulation of the collision.

As a case study, we chose a suspended graphene sheet andexamined the spatial resolution of HIM by comparing theamount of secondary electrons emitted at different impactpoints of the He+ beam.88) Figure 10(a) shows the valencecharge density of a graphene sheet, and Fig. 10(b) showsimpact points A–F of the incident He+ beam. The kineticenergy of He+ was set to 30 keV.85,86) For impact positionsA–F, the amount of secondary electrons emitted wascalculated by integrating the charge density at 8Å abovethe graphene.88) The electron intensity contour map is shownin Fig. 10(c). A difference of approximately threefold wasobserved between the maximum and minimum intensities.The emission profile as a function of the He+ impact pointwas similar to the valence charge profile, and we concludedthat the mechanism for the emission of the secondaryelectrons is impact ionization by the incident He+, as opposedto electron emission followed by the Auger process (Penningionization), which occurs at a lower kinetic energy90) thanHIM. The dynamics of the secondary electron emission weresimilar for collisions of neutral He atoms and a graphenesheet, also supporting the impact ionization mechanism.88)

He+ did not damage the graphene sheet despite the highkinetic energy of 30 keV, because the cross section for theHe–C collision was extremely small at impact points A–F inFig. 10(b).

The He+ beam has a finite diameter similar in size to a Hetrimer,85,91) which does not provide atomic-level resolution,but can visualize the edge of the graphene.88) The develop-ment of a stable He+ source with a narrower beam size isnecessary to achieve a spatial resolution for HIM that iscomparable to transmission electron microscopy. Real-timeTDDFT-MD can be used to analyze multi-scattering of thesecondary emitted electrons and sample damage by deaccel-erated He+ with increasing cross sections for C atoms.

4. Concluding Remarks

We have presented an overview of our recent applications

EF

SolidIon

Velocity

Charge transfer

Fig. 9. (Color online) Schematic diagram of the potential profiles of the(left to right) solid, vacuum, and ion. The dotted horizontal lines representunoccupied orbitals in the ion, whereas the solid horizontal lines representthe occupied orbital. The ion velocity is indicated by the green arrow,whereas the direction of the charge density is indicated by the red arrow.

(a) (b) (c) Minimum emissionMaximum emission

Fig. 10. (Color online) (a) Calculated valence charge density of graphene. (b) Lattice structure of graphene. A–F indicate the impact points of He+.(c) Contour map of the intensity of secondary emitted electrons interpolated from impact points A–F in (b) These figures are adapted with permission fromRef. 88. Copyrighted by the American Physical Society.

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of real-time TDDFT simulations coupled with MD simu-lations. The photoisomerization of molecules was reproducedwithout dynamical mixing of several PESs (Sect. 3.1), andwe have also simulated the optical enhancement of the weakdispersive force in the cases of He dimers and hBN sheets(Sect. 3.2), selective bond breaking and reduction on graph-itic sheets (Sects. 3.3 and 3.4), and the application of electronmotions for devices (Sect. 3.5) and microscopy (Sect. 3.6).

The computational schemes can be applied to manysystems, some of which were not included in the currentoverview, such as the dynamics of adatoms on solidsurfaces,92) carrier decay dynamics in carbon nanotubes,93)

laser exfoliation of graphene sheets from graphite,94) highharmonic generation in atoms95) and solids,96) observation ofelectron–photon dressing in WSe2,97) and creation of stableFloquet–Weyl semimetals.98)

However, there are several approaches to go beyondEhrenfest dynamics based on sampling classical trajecto-ries.38–40) Instead of Hellmann–Feynman forces, the formu-lation of generalized forces for NACs was reported28) withina scheme of the linear response theory and with accuratetreatment of the forces from pseudopotentials using theKohn–Sham derivative matrix. Extending this approachbeyond the perturbation theory is required when electronicexcitation is accompanied by charge transfer. Charge transferin strongly correlated systems can be approximated byincluding the Hubbard U term, and the combination of thismodel with the real-time TDDFT-MD has been demon-strated.99)

Another challenging subject for real-time TDDFT istreatment of the full quantum nature of the electromagneticfield. In the present overview, we used a long-wavelengthapproximation, in which the optical E-field is uniformeverywhere and light propagation was not directly treated.The penetration=reflection of light at solid surfaces can betreated by combining the classical Maxwell equation andreal-time TDDFT.100) An approach to combine quantumoptics and TDDFT in the quantum electrodynamical (QED)Hamiltonian has also been reported, demonstrating many-body electron–electron interactions mediated by photon–electron interactions,101) opening up the fields of QEDchemistry and materials.102)

Many new challenges can be addressed with the methodsthat have been described in this overview. Of course, furtherdevelopment of the theory (exchange–correlation functionals)and algorithms are needed to bridge the different time scales(attosecond to millisecond) and size (nanoscale to bulksolids) in normal and strongly correlated molecular andextended systems.

Acknowledgements All of the computations described in this paper wereperformed using the parallel-computing system at AIST, the supercomputingsystem in the Cybermedia Center at Osaka University, the Earth Simulator at theYokohama Institute of Earth Science, JAMSTEC, and the supercomputing systemin the Cyberscience Center at Tohoku University. Part of the presented work wasperformed in collaboration with Professor Hong Zhang, Professor Xinlu Cheng,Dr. Takehide Miyazaki, and Dr. Yoshitaka Tateyama. YM acknowledgesfinancial support from JSPS KAKENHI (Grant Numbers JP16H00925 andJP16K05049) and from a project supported by the New Energy and IndustrialTechnology Development Organization (NEDO). AR acknowledges financialsupport from the European Research Council (ERC-2015-AdG-694097), GruposConsolidados (IT578-13) of the UPV=EHU and Basque Government, theEuropean Union’s H2020 program under GA No. 676580 (NOMAD),MOSTOPHOS (GA No. 646259), and the JSPS Fellowship program in 2016.

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Yoshiyuki Miyamoto was born in Iwakuni city in1962. He joined NEC Fundamental ResearchLaboratories in 1987, and he received his Dr. ofEngineering (Solid State Physics) in 1991 fromOsaka University, Japan. He did postdoctoral workat University of California at Berkeley from 1993 to1995. After he returned NEC, he accepted specially-appointed professor of Osaka University in 2008. InFebruary 2001, he moved to National Institute ofAdvanced Science and Technology (AIST), Japan.

His research interests are rooted to semiconductor physics, physics of severalphases of carbons, diamond, graphene, nanotube, and so on. He contributedto the development of numerical tool for electron-ion dynamics in condensedmatters based on the time-dependent density functional theory.

Angel Rubio was born in Oviedo in 1965. Hereceived his Ph.D. in Physics in 1991 from theUniversity of Valladolid (UVA), Spain. During hisPh.D. studies, he spent time at the Max Planck inBerlin and at Universities of Palma de Mallorca,Santader, Barcelona (Spain), Osnabruck (Germany),Nancy (France). He did postdoctoral work atUniversity of California at Berkeley. After anAssociate Professorship at (UVA) in 1994, he movedin 2001 to University of the Basque Country (UPV=

EHU) as Full Professor=Chair of Condensed Matter Physics and director ofthe Nano-Bio Spectroscopy Group. In August 2014 he accepted to beDirector of the Max Planck Institute for the Structure and Dynamics ofMatter in Hamburg. In September 2017 he has been appointed DistinguishedResearch Scientist at the Flatiron institute of the Simons Foundation in NewYork. He has an excellent publication record: more than 300 publicationswith more than 29000 citations (Hirsch index 86). His research activity isinternationally recognized and he has received numerous honors and awards.His research interests are rooted to the modeling and theory of electronic andstructural properties of condensed matter, to the development of newtheoretical tools to investigate the electronic response of materials,nanostructures, biomolecules, and hybrid materials to external electro-magnetic fields.

J. Phys. Soc. Jpn. 87, 041016 (2018) Special Topics Y. Miyamoto and A. Rubio

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