Maxwell + TDDFT multiscale descriptions for interactions of...
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Maxwell + TDDFT multiscale descriptions for interactions of intense pulsed light with dielectrics
Kazuhiro YABANA
Center for Computational Sciences, University of Tsukuba
7 th Time-Dependent Density Functional Theory Benasque, Spain
September 20-23, 2016
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Collaborators
Univ. Tsukuba Mitsuharu Uemoto Xiao-Min Tong Yuta Hirokawa Taisuke Boku QST(KPSI) Tomohito Otobe Univ. Tokyo Yasushi Shinohara Max-Planck Institute for Structure and Dynamics of Matter Shunsuke Sato Kyung-Min Lee
Univ. Washington George F. Bertsch Tech. Univ. Wien Georg Wachter Isabella Floss Joachim Burgdoerfer Max Planck Institute for Quantum Optics (Experiment) Annkatrin Sommer Martin Schultze Ferenc Krausz ETH (Experiment) Matteo Lucchini Ursula Keller
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10 µm SiO2 thin film
!ω =1.55eVλ=800nmI=7×1013W/cm2
Laser electric field, red (strong), blue (weak)
Expensive calculation: 80,000 cores, 20 hours at K-Computer, Japan
Maxwell + TDDFT multiscale simulation for intense and ultrashort laser pulse propagation in dielectrics
- Why and when it is necessary? - How it is done? - How it works?
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CONTENTS
1. Intense and ultrashort laser pulse
2. Real-time TDDFT in a unit cell of crystalline solid
3. Maxwell + TDDFT multiscale formalism
4. Applications 4-1. Dynamical Franz-Keldysh effect 4-2. Ultrafast energy transfer from light to electrons in solids 4-3. Laser ablation
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CONTENTS
1. Intense and ultrashort laser pulse
2. Real-time TDDFT in a unit cell of crystalline solid
3. Maxwell + TDDFT multiscale formalism
4. Applications 4-1. Dynamical Franz-Keldysh effect 4-2. Ultrafast energy transfer from light to electrons in solids 4-3. Laser ablation
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Intense and ultrashort laser pulse prohibits separation but requires combination of two descriptions (EM and QM).
Ordinary theory and calculation for light-matter interactions
D !r, t( ) = dt 't∫ ε t − t '( )E
!r, t '( )
Macroscopic electromagnetism for light propagation in matter
Quantum mechanical calculation for dielectric function
Constitutive relation connects two descriptions
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Intense and Ultrashort Laser Pulse
Atomic unit: me = e = ! =1
Proton
Electron
1 a.u. of electric field
= field felt by electron in classical hydrogen model
Extremely nonlinear electron dynamics
Intense electric field Ultrafast motion
1(2π) a.u. of time
= Period of electron motion in classical hydrogen model
Attosecond science:
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Intense laser pulse on solids
21513 W/cm1010 −
Laser intensity
1023W/cm22 W/cm.13660Solar constant Strongest laser pulse
Vacuum breakdown
Laser Damage
( )ωε( )( ) ( )( )321
321
2 ,,,, ωωωχωωχ
Perturbative linear and nonlinear responses 1028W/cm2
Extreme Nonlinear response
1a.u. = 3.51×1016W/cm2
High Harmonics Generation
Nonthermal processing
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Ultrashort pulse : Snap shot of electron motion
s10 15− Time
Nucleon motion in nucleus
s10 18− s10 23−s10 12−
AttosecondFemtosecondPicosecond
Real-time observation of valence electron motion E. Goulielmakis et.al, Nature 466, 739 (2010).
Shortest laser pulse 67as
Period of electron orbital in Hydrogen, 150as
Period of Ti-sapphire laser pulse 1.55eV
Period of Optical phonon (Si, 64 fs)
1 a.u. = 0.0242 fs
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CONTENTS
1. Intense and ultrafast laser pulse
2. Real-time TDDFT in a unit cell of crystalline solid
3. Maxwell + TDDFT multiscale formalism
4. Applications 4-1. Dynamical Franz-Keldysh effect 4-2. Ultrafast energy transfer from light to electrons in solids 4-3. Laser ablation
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Real-time TDDFT for electron dynamics in a unit cell of crystalline solid
under spatially uniform, time-dependent electric field
Time-dependent Kohn-Sham equation for Bloch orbitals
( ) ( ) ( ) ( )[ ] ( )
( ) ( ) 2
22
,,
,,,''
'21,
∑
∫
=
⎥⎥⎦
⎤
⎢⎢⎣
⎡+
−+⎟
⎠
⎞⎜⎝
⎛ ++=∂
∂
nkkn
knxckn
trutrn
trutrntrnrr
erdtAcekp
mtru
ti
!!
!!!!!
!!!!!"
!
!! µ
( ) ( )ttA
ctE
∂
∂−=
!! 1
G.F. Bertsch, J-I. Iwata, A. Rubio, and K. Yabana, Phys. Rev. B 62, 7998 (2000) 43.
Atoms are fixed at equilibrium positions
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Elec
tric
Fiel
d (a
.u.)
403020100Time (fs)
Example: crystalline silicon under intense laser pulse
Electron density
i! ∂∂tun!k!r, t( ) = 1
2m!p+!k + e
c!A t( )
⎛
⎝⎜
⎞
⎠⎟2
+ d!r ' e2!r − !r '∫ n !r ', t( )+µxc n
!r, t( )⎡⎣ ⎤⎦⎡
⎣⎢⎢
⎤
⎦⎥⎥un!k!r, t( )
direct bandgap 2.4 eV in LDA( )E = 27.5 V/nm!ω =1.55 eVTFWHM =7 fs
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Physical quantities from real-time TDDFT calculation
direct bandgap 2.4 eV in LDA( )E =1.23 V/nm!ω =1.55 eVTFWHM =7 fs
Silicon
Two-photon absorption
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Linear response: Real-time TDDFT for dielectric function (Si, LDA)
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( ) ( )tktJ σ=
( ) ( ) ( ) ( )ttAtktE θδ ∝= ,Impulsive electric field at t=0 Induced current = conductivity in time domain
( ) ( )tJedtk
ti∫= ωωσ1 ( ) ( )
ωωσπ
ωεi41+=
fs2≈τ
Time-frequency Fourier transformation
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Quality of the exchange-correlation potential may be assessed by dielectric function
Meta-GGA potential of Tran and Blaha (TBmBJ) reproduces band gap of insulators. PRL102, 226401 (2009)
Silicon GermaniumS.A. Sato et.al, JCP143, 224116 (2015)
(TBmBJ)
SiO2 (α-quartz)S.A. Sato et.al, PRB92, 205413 (2015)
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CONTENTS
1. Intense and ultrafast laser pulse
2. Real-time TDDFT in a unit cell of crystalline solid
3. Maxwell + TDDFT multiscale formalism
4. Applications 4-1. Dynamical Franz-Keldysh effect 4-2. Ultrafast energy transfer from light to electrons in solids 4-3. Laser ablation
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( ) ( )dttdA
ctE 1
−=
( ) ( ) ( ) ( )[ ] ( )
( ) ( ) 2
22
,,
,,,''
'21,
∑
∫
=
⎥⎥⎦
⎤
⎢⎢⎣
⎡+
−+⎟
⎠
⎞⎜⎝
⎛ −+=∂
∂
nkkn
knxckn
trutrn
trutrntrnrr
erdtAcekp
mtru
ti
!!
!!!!!
!!!!!"
!
!! µ
Electron dynamics calculation as Numerical constitutive relation
For a given electric field
( ) ( ) ( ) ( )'',,1 tJdttPtrjrdtJt !!!!!!∫∫ −=
Ω=
Ω
( ) ∑ ⎟⎟⎠
⎞⎜⎜⎝
⎛−⎟
⎠
⎞⎜⎝
⎛ +=i
ii ccAcep
mtrj ..
21, * ψψ
!!""
provides electric current and polarization
Electron dynamics calculation in a unit cell of solid
!P!R, t( ) =
!P!E!R, t( )⎡
⎣⎤⎦
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z
At each macroscopic grid point, we solve real-time electron dynamics in parallel
Maxwell + TDDFT multiscale approach
Macroscopic grid for Maxwell equation
Microscopic grid for TDDFT electron dynamics
( ) ( ) ( )tJc
tAZ
tAtc ZZZ
π412
2
2
2
2 =∂
∂−
∂
∂ i! ∂∂tuZ ,n!k
!r, t( ) = h!AZ t( ),nZ
!r, t( )⎡⎣ ⎤⎦uZ ,n!k!r, t( )
!JZ t( ) = 1
V!jZ
V∫ !r, t( )
K. Yabana et.al, Phys. Rev. B85, 045134 (2012).
µm scale
nm scale
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I=1010W/cm2
Z=0 [µm]
[µm]
Z=0.8 [µm] Z=1.6 [µm]
Propagation of weak pulse (transparent) (Linear response regime, ordinary linear optics applies)
Vacuum SiLaser frequency:1.55eV: lower than direct bandgap 2.4eV(LDA)
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A/c
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I=1010W/cm2
[µm]
Vacuum SiLaser frequency:1.55eV: lower than direct bandgap 2.4eV(LDA)
Electron excitation energy per unit cell volume
A/c
Propagation of weak pulse (transparent) (Linear response regime, ordinary linear optics applies)
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Z=0 [µm]
[µm]
Z=0.8 [µm] Z=1.6 [µm]
I =5 x 1012W/cm2
Vacuum Si
More intense laser pulse (absorptive) Maxwell and TDKS equations no more separate.
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I = 5 x 1012W/cm2
More intense pulse (absorptive) (2-photon absorption dominates)
[µm]
Vacuum SiLaser frequency:1.55eV: lower than direct bandgap 2.4eV(LDA)
Electron excitation energy per unit cell volume
A/c
[µm]
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CONTENTS
1. Intense and ultrafast laser pulse
2. Real-time TDDFT in a unit cell of crystalline solid
3. Maxwell + TDDFT multiscale formalism
4. Applications 4-1. Dynamical Franz-Keldysh effect 4-2. Ultrafast energy transfer from light to electrons in solids 4-3. Laser ablation
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Application 1: Change of dielectric property of solid under intense field
Elec
tric
Fiel
d (a
.u.)
403020100Time (fs)
At each time T, how dielectric function changes?
ε ω,T( )
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Under a static electric field: Franz-Keldysh effect
w E-field w/o E-field
Energy [eV] Energy [eV]
Imε ω( ) Δ Imε ω( ) = ImεE ω( )− ImεE=0 ω( )
bandgap
( ) ( ) ( ){ }
energy icalElectroopt2
AiAi',Im
3/1222
222
2/1
⎟⎟⎠
⎞⎜⎜⎝
⎛=Θ
Θ
−=
−Θ
∝
µ
ωξ
ξξξω
ωε
!
!
Fe
E
F
gap Airy function →Quantum tunneling
Analytic formula for a two-band model
W. Franz, Z. Naturforsch. 13, 484 (1958) L.V. Keldysh, Sov. Phys. JETP 34, 788 (1958)
Conduction band
Valence band
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Pump 1.55eV 6.5x1012W/cm2
Probe 〜42eV 250as
50nm diamond thin film
Spectral decomposition of probe pulse to see changes of absorption profile.
Probe strength (w/o film)
Probe strength (w film)
Experiment (ETH group) and Calculation (Maxwell + TDDFT) Pump-probe method for diamond thin film
- Irradiate IR strong pump-pulse (1.55 eV) and XUV weak probe pulse (42eV) simultaneously. - Examine change of probe abosorption as a function of time-delay.
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Pump electric field
Δabs(ω,T) Exp.
Δabs(ω,T) Maxwell-TDDFT
- Absorption increase at 42eV when pump field exists. - “Fish-bone” structure (time delay in response change)
- Calculation reproduce measured features. - Supports interpretation by dynamical Franz-Keldysh effect
ResultsM. Lucchini et.al, Science 353, 916-919 (2016)
T
T
ω
ω
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CONTENTS
1. Intense and ultrafast laser pulse
2. Real-time TDDFT in a unit cell of crystalline solid
3. Maxwell + TDDFT multiscale formalism
4. Applications 4-1. Dynamical Franz-Keldysh effect 4-2. Ultrafast energy transfer from light to electrons in solids 4-3. Laser ablation
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Application 2: laser pulse propagation through SiO2 thin film
Thin film of SiO2 10µm
Compare pulses of different intensities
EXP: Attosecond streaking measurements by Max-Planck Inst. Quantum Optics
Energy deposition from laser pulse to dielectrics
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Maxwell + TDDFT multiscale simulation : 10 µm SiO2
!ω =1.55eVλ=800nmI=7×1013W/cm2
Laser electric field, red (strong), blue (weak)
Energy transfer from laser pulse to electrons [eV/atom]
80,000 cores, 20 hours at K-Computer, Kobe
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Comparison of pulse shape after passing through SiO2 thin film
Maxwell + TDDFT
I=1.28×1014W/cm2I=7×1013W/cm2
Experiment
- Although very close to damage threshold, pulse shape is dominated by 3rd order nonlinearity. - Pulse shape change, phase shift well described by Maxwell + TDDFT calculation.
weak
strong
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I=5×1013W/cm2
I=7×1013W/cm2
I=1×1014W/cm2
Energy deposited to electrons increases rapidly as the laser intensity increases.
0.1 eV/atom
0.1 eV/atom
0.1 eV/atom
0 10µm
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Energy deposition from laser pulse to SiO2 at mid point (5µm) can be evaluated experimentally from the transmitted pulse.
TheoryExperiment
A. Sommer et.al, Nature 534, 86 (2016). (EXP: Max Planck Institute for Quantum Optics)
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CONTENTS
1. Intense and ultrafast laser pulse
2. Real-time TDDFT in a unit cell of crystalline solid
3. Maxwell + TDDFT multiscale formalism
4. Applications 4-1. Dynamical Franz-Keldysh effect 4-2. Ultrafast energy transfer from light to electrons in solids 4-3. Laser ablation
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Application 3: Estimate laser-ablation threshold and depth
Single shot laser pulse create ‘crator’.
Femtosecond laser pulse expected for nonthermal laser processing.
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Ablation depth 100 – 150 nm Ablation threshold 2 x 1014W/cm2
eV6=cohesionE
!ω =1.55eV (λ = 800nm),TFWHM = 7fsEgap = 9.0eV
Energy transfer from pulsed electric field to electrons [eV/atom]
Maxwell + TDDFT multiscale calculation for α-quartz (SiO2)
S.A. Sato et.al, Phys Rev. B92, 205413 (2015)
Depth from surface
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Ablation: threshold and depth, comparison with measurements
Crater formation: measurement, B. Chimier et al, Phys. Rev. B84, 094104 (2011)
S.A. Sato et.al, Phys Rev. B92, 205413 (2015)
Threshold and depth reasonably estimated.
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Real-time TDDFT in crystalline solid is useful to describe nonlinear and nonperturbative electron dynamics induced by intense and ultrashort laser pulse. Maxwell + TDDFT multiscale simulation provides numerical experiment platform for optical science frontiers - can describe propagation of intense laser pulse in the medium. - can provide physical quantities which can be compared with cutting-edge optical measurements. - will open first-principle investigation of nonthermal laser processing - is computationally challenging requiring cutting-edge supercomputers Future problems - collision (e-e, e-phonon), dephasing effects - improved functional (excitons, …) - 2D, 3D light pulse propagation (self-focusing, filamentation, light vortex…)
Summary
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