Time-dependent density functional theory for a unified description of ultrafast dynamics: pulsed light, electrons, and atoms in crystalline solids
aa r X i v : . [ phy s i c s . c o m p - ph ] O c t Time-dependent density functional theory for a unified description of ultrafastdynamics: pulsed light, electrons, and atoms in crystalline solids
Atsushi Yamada ∗ and Kazuhiro Yabana Center for Computational Sciences, University of Tsukuba,1-1-1 Tennodai, Tsukuba, Ibaraki 305-8577, Japan (Dated: October 16, 2018)We have developed a novel multiscale computational scheme to describe coupled dynamics of lightelectromagnetic field with electrons and atoms in crystalline solids, where first-principles moleculardynamics based on time-dependent density functional theory is used to describe the microscopicdynamics. The method is applicable to wide phenomena in nonlinear and ultrafast optics. To showusefulness of the method, we apply it to a pump-probe measurement of coherent phonon in diamondwhere a Raman amplification takes place during the propagation of the probe pulse.
Nonlinear optics in solids is the study of the interac-tion of intense laser light with bulk materials [1–3]. Itis intrinsically a complex phenomena arising from a cou-pled nonlinear dynamics of light electromagnetic fields,electrons, and phonons. They are characterized by twodifferent spatial scales, micro-meter for the wavelengthof the light and less than nano-meter for the dynamics ofelectrons and atoms.In early development, nonlinear optics has devel-oped mainly in perturbative regime and in frequencydomain[4, 5]. However, it has changed rapidly and dras-tically. Nowadays, measurements are carried out quiteoften in time domain using pump-probe technique as atypical method and the time resolution reaches a fewtens of attosecond[6, 7]. Extremely nonlinear phenom-ena has attracted interests such as high harmonic gen-eration in solids[8, 9], ultrafast control of electrons mo-tion in dielectrics that aims for future signal processingusing pulsed light[10–12], and ultrafast coherent opti-cal phonon control[13–22] and photoinduced structuralphase transition of materials[23–26].In this paper, we report a progress to develop first-principles computational method to describe nonlinearoptical processes in solids that arise from coupled dy-namics of light electromagnetic fields, electrons, andatoms in crystalline solids. In condensed matter physicsand materials sciences, first-principles computational ap-proaches represented by density functional theory havebeen widely used and recognized as an indispensabletool[27]. Development of first-principles approaches inoptical sciences is, however, still in premature stage dueto the complexity of the phenomena and the requirementof describing time-dependent dynamics.Our method utilize time-dependent density func-tional theory (TDDFT) for microscopic dynamics ofelectrons[28, 29]. The TDDFT is an extension of the den-sity functional theory so as to be applicable to electrondynamics in real time[30]. In microscopic scale, ultra-fast dynamics of electrons have been successfully exploredsolving the time-dependent Kohn-Sham (TDKS) equa-tion, the basic equation of TDDFT, in real time[31–33].We have further developed a multiscale scheme coupling
FIG. 1. Schematic illustration of the multiscale scheme fora light propagation through a material. the light electromagnetic fields and electrons in solids[34],and applied it to investigate extremely nonlinear opti-cal processes in dielectrics using few-cycle femtosecondpulses[11, 12]. Here we extend the approach to incor-porate atomic motion, combining first-principles molec-ular dynamics approach[35]. It will then be capable ofdescribing vast nonlinear optical phenomena involvingatomic motion such as stimulated Raman scattering[1–3] that will be discussed later in this paper.We consider a problem of an ultrashort pulsed light ir-radiation normally on a thin dielectric film, as illustratedin Fig.1. In our multiscale description [34], we introducetwo coordinate systems, the macroscopic coordinate X that is used to describe the propagation of the light elec-tromagnetic fields, and the microscopic coordinates r forthe dynamics of electrons and atoms.The light electromagnetic field is expressed by using avector potential A X ( t ) that satisfies the Maxwell equa-tion, (cid:20) c ∂ ∂t − ∂ ∂X (cid:21) A X ( t ) = 4 πc J X ( t ) , (1)where J X ( t ) is the electric current density at the point X .We will use a uniform grid for the coordinate X tosolve Eq.(1). At each grid point X , we consider aninfinitely periodic medium composed of electrons andatoms whose motion are described using the microscopiccoordinate r . Since the wavelength of the pulsed lightis much longer than the typical spatial scale of the mi-croscopic dynamics of electrons and atoms, we assumea dipole approximation: At each point X , the elec-trons and atoms evolve under a spatially-uniform electricfield, E X ( t ) = − (1 /c )( ∂A X ( t ) /∂t ). Then we may applythe Bloch theorem at each time t in the unit cell, andmay describe the electron motion using Bloch orbitals u n k ,X ( r , t ) with the band index n and the crystallinemomentum k [36]. Atomic motion can also be describedby atomic coordinates in the unit cell, R α,X ( t ), wherethe index α distinguishes different atoms in the unit cell.The Bloch orbitals satisfy the TDKS equation, i ~ ∂∂t u n k ,X ( r , t ) = (cid:20) m n − i ~ ∇ r + ~ k + ec A X ( t ) o − eφ X ( r , t )+ δE XC [ n e,X ] δn e,X + ˆ v ion,X ( r ; { R α,X ( t ) } ) (cid:21) u n k ,X ( r , t )(2)where n e,X is the electron density given by n e,X ( r , t ) = P n, k | u n k ,X ( r , t ) | . φ X ( r , t ) and E XC [ n e,X ] are theHartree potential and the exchange-correlation energy,respectively. ˆ v ion,X ( r ; { R α,X ( t ) } ) is the electron-ion po-tential for which we use norm-conserving pseudopotential[37].To describe the dynamics of atoms, we use a so-calledEhrenfest dynamics [28] where the atomic motion is de-scribed with mean-field from the electrons by the Newtonequation, M α d R α,X dt = − eZ α c d A X dt + ∂∂ R α,X Z d r [ en ion,X φ X ](3)where M α is the mass of the α -th ion, n ion,X is thecharge density of ions given by n ion,X ( r , { R α,X ( t ) } ) = P α Z α δ ( r − R α,X ( t )), with Z α the charge number of the α -th ion.The electric current density at point X , J X ( t ), consistsof electronic and ionic contributions, J X ( t ) = J e,X ( t ) + J ion,X ( t ) . (4)The electronic component J e,X ( t ) is given by the Blochorbitals u n k ,X ( r , t ) [34]. The ionic component is givenby J ion,X ( t ) = e P α Z α ( d R α,X ( t ) /dt ) / Ω, where Ω is thevolume of the unit cell.We solve Eqs. (1) - (4) simultaneously to obtain thewhole dynamics at once with the initial condition that theelectronic state at each point X is set to the ground statesolution of the static density functional theory, atomicpositions are set to their equilibrium position in the elec-tronic ground state, and the vector potential of the inci-dent pulsed light is set in the vacuum region in front ofthe thin film. We note that the present scheme naturally includes the ordinary macroscopic electromagnetism in aweak field limit, since solving the TDKS equation (2) isequivalent to utilizing the linear constitutive relation fora weak field.Now, we apply the method to describe the pump-probemeasurement that aims to investigate the generation ofcoherent optical phonons [19]. We consider a diamondthin film of 6 µ m thickness, and two linearly-polarizedpulses are irradiated successively and normally on thesurface. We set the crystalline abc axes of cubic diamondto coincide with the xyz axes of the Cartesian coordi-nates, respectively.We implemented our scheme in the open-source soft-ware SALMON [38, 39] which utilizes real-space uniformgrid representation to express orbitals. Since a numberof microscopic dynamics are calculated simultaneously inthe multiscale scheme, we use a massively parallel super-computer with an efficient implementation of paralleliza-tion [40]. Adiabatic local density approximation is usedfor the exchange-correlation potential [41]. The X co-ordinate from X =0 to 6 µ m is discretized by 400 gridpoints with the spacing of 15 nm. The microscopic unitcell of diamond consists of eight carbon atoms in the cu-bic cell of the side length of 3.567 ˚A. The Bloch orbitalsare expressed using 16 uniform spatial grids in the unitcell and 12 of k -points in the Brillouin zone. The timestep is chosen as 2 as.We first show the calculation for the generation of thecoherent phonon by the pump pulse. Figure 2(a) showsthe propagation of the pump light and the displacementof the atoms at different macroscopic point X . The fre-quencies and the pulse duration of the pump pulse is setto 1.55 eV / ~ and 6.5 fs in FWHM, respectively. Thepulse duration is chosen to be shorter than the period ofthe optical phonon of diamond, 25 fs. The polarizationdirection of the pump pulse is chosen as [011] direction,which causes the optical phonon in [100] direction. Themaximum intensity of the pump pulse is set to 2 × W/cm .When the pump field propagates through the material,the harmonic motion of atoms is generated in turn ateach X position. In Fig. 2(a), a wave-like behavior ofatomic displacement is seen along the X axis. However,this is not an ordinary phonon wave described by thelattice dynamics. The period of the oscillation at each X position is the period of the optical phonon which is 25fs, and the speed of the propagation is equal to the speedof the pump pulse in the medium, c/n , with the index ofrefraction n .In Fig.2(b), the generation process of the opticalphonon at X = 2 µ m is shown as a function of time. Atfirst, the force is proportional to the square of the electricfield. As the phonon starts to move, the restoring forcebegins to work. These behavior of the force and the gen-eration process of the coherent phonon is consistent withthe picture of the impulsively stimulated Raman scatter- FIG. 2. (a) Snapshots of the electric field of the pump pulse(red lines) and the atomic displacement (green filled circles)are shown along the macroscopic position X . (b) Electricfield of the pump pulse (red line), the force acting on theatom (black line), and the atomic displacement (green line)are shown at X = 2 µ m as a function of time.FIG. 3. Coherent phonon generation by pump pulses ofthree different frequencies, ~ ω = (a)1.55, (b)3.5 and (c)6.0 eV.The atomic displacements as a function of time are shown atselected macroscopic positions of X (the left panels) and thosealong the coordinate X at specific times (the right panels). ing (ISRS) mechanism[19]. After the pump field passesaway, there is no driving force and a simple harmonicmotion of atoms continues without decay.We next show in Fig.3 the generation of coherentphonons by pulses of three different frequencies. Panel(a) shows the generation with ~ ω =1.55 eV, the same asthat shown in Fig.2(a). Here the ISRS mechanism isresponsible for the coherent phonon generation as men-tioned above, since the frequency is below the optical FIG. 4. Snapshots of the electric field of the probe pulse in[010] (red-line) and in [001] (blue-line) polarization directions,and the atomic displacment (green filled circles) are shownalong the macroscopic coordinate X . gap energy. At ~ ω =3.5 eV, the pump frequency is stillbelow the optical gap energy. However, Fig.3(b) showsthat the phonon amplitude decays with X whereas theamplitude at each position X does not decay with time.This is caused by the two photon absorption of the pumppulse. Since the pump pulse is rather strong with the in-tensity, 2 × W/cm , the pump pulse excites electronsand looses energy as it propagate through the medium.At ~ ω =6 eV that is above the optical gap energy, thegenerated phonon amplitude at the solid surface is oneorder of magnitude larger than the non-resonant cases,as seen in Fig.3(c), The displacement of atoms showsa harmonic motion of cosine shape, namely the oscilla-tion takes place around the shifted equilibrium position.This is due to the change of the generation mechanismfrom ISRS to the displacive excitation of coherent phonon(DECP) [19]. The amplitude of the phonon decays with X as the field gets weaker by the absorption of the pulse.These results are consistent with Ref. [33] where gen-eration mechanisms of coherent phonons are discussedwithout describing the light propagation.We next proceed to the calculation of the propagationof probe pulses. Although it is possible to carry out thepump and the probe pulse propagations in a single calcu-lation, we separate them since the reflection of the pumppulse at the back surface of the thin film complicates theanalyses. In the probe pulse calculation, we prepare theinitial medium in which the coherent phonon already ex- FIG. 5. (a) Transmission wave in the right vacuum regionand (b) the Fourier transformed power spectrum with τ = 83.0fs. Those with τ = 89.5 fs are shown in (c) and (d). ists and irradiate the probe pulse from the vacuum. Thephonon motion is in [100] direction as discussed above,and we choose the polarization direction of the probepulse as [010]. Then we expect the emergence and theamplification of the stimulated Raman wave polarizedalong [001] direction due to the structure of the Ramantensor of the diamond [42]. The intensity of the probepulse is set sufficiently small, 10 W/cm , so that thereoccurs no significant nonlinearity related to the probepulse.We show the propagation of the probe pulse and theamplification process of the stimulated Raman wave inFig.4. In the figure, the probe pulse is irradiated with thepump-probe delay time of τ = 83 . π/ τ = 83 . τ = 89 . τ = 83 . π/
2. On the otherhand, in the case of τ = 89 . J Raman ( t ) ∝ Q ( t ) E probe ( t ) [19, 43], where E probe ( t ) is the electric field of the probe pulse and Q ( t ) isthe phonon amplitude given by the linear combination of { R α ( t ) } . When the probe pulse enters the medium at themaximum of the phonon amplitude, Q ( t ) may be roughlyregarded as a constant and J Raman ( t ) is mostly propor-tional to E probe ( t ) since the half period of the phonon islonger than the duration of the probe pulse. However,when the probe pulse moves with the nodal position ofthe phonon, the phonon amplitude is approximated by alinear function of time changing the sign. Then, we have J Raman ( t ) ∝ tE probe ( t ), and the phonon motion producesone extra node to the electric current density. This ex-plains the shape change of the stimulated Raman waveshown in Fig.5(b).We show the power spectra of the stimulated Ramanwaves in panels (b) and (d). Reflecting the differencein the time profile, the power spectra also show a dis-tinct structures: the double-peak structure appears inthe power spectrum of the stimulated Raman wave trav-eling with the nodal point of phonon. We note that suchdouble-peak structure is indeed related to recent pump-probe measurement of the coherent phonon in diamondand other insulators [14, 17]. We will report our analysisfor this problem in a separate publication.In summary, we have developed a computational ap-proach for nonlinear light-matter interaction in solidsbased on first-principles time-dependent density func-tional theory. A multiscale scheme is developed simul-taneously solving Maxwell equations for light propaga-tion, time-dependent Kohn-Sham equation for electrons,and Newton equation for atoms. As a test example, apump-probe measurement of coherent phonon generationin diamond is simulated where an amplification by thestimulated Raman scattering is observed for the probepulse. We expect the method will be useful for a widephenomena of nonlinear and ultrafast optics.We acknowledge the supports by JST-CREST un-der grant number JP-MJCR16N5, and by MEXT asa priority issue theme 7 to be tackled by using Post-K Computer, and by JSPS KAKENHI Grant Number15H03674. Calculations are carried out at Oakforest-PACS at JCAHPC through the Multidisciplinary Co-operative Research Program in CCS, University ofTsukuba, and through the HPCI System ResearchProject (Project ID: hp180088). ∗ [email protected][1] Robert W. Boyd. Nonlinear Optics, Third Edition . Aca- demic Press, Inc., Orlando, FL, USA, 3rd edition, 2008.[2] Y. R. Shen.
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