time-dependent density functional theory for ultrafast electron … · 2013-03-14 ·...

Kazuhiro YABANA

Center for Computational Sciences, University of Tsukuba

Time-dependent density functional theory for ultrafast electron dynamics in solid

Computational approaches to nuclear many-body problems and related quantum systems

Feb. 12-16, 2013, RIKEN, Wako, Japan

First-principles computational approach for Frontiers of Laser Sciences

TDDFT: First-principles tool for electron dynamics simulation

Absorption cross section

Collision of atoms and ions

E S0 S1

S2

+Ze

Electronic excitation + + + + + +

- - - - - -

E(t)

hν

3hν

e-

Multiple ionization

Intense, ultra-short Laser pulses

Linear response regime (Perturbation theory)

Nonlinear, nonperturbative regime (Initial value problem)

e- e-

soft X ray (High harmonic generation)

Dielectric function

Coherent phonon Electron-hole plasma Optical breakdown

Conductance of nano-wire

( )ωσ

Laser-solid interaction

Coulomb explosion

( ) ( )[ ] ( ) ( )

( ) ( )2,,

,,,,

∑=

+=∂∂

ii

iextKSi

trtr

trtrVtrhtrt

i

ψρ

ψρψ

Theoretical description of light-matter interactions

[ ] ( )EEDD ωε==

Constitution relation Electromagnetism: Maxwell equation for macroscopic fields, E, D, B, H

Quantum Mechanics: Perturbation theory to calculate linear susceptibilities, ε(ω)

Ultra-short laser pulses require real-time description

Strong electric field induces nonlinear electron dynamics

( ) ( ) ( )','', trEttdttrDt

−= ∫ ε

( ) ( ) ( ) ( )( ) ( ) ( )

+−−+−= ∫∫∫ '',','',''''4','', 2 trEtrEttttdtdttrEttdttrDttt

χπε

A standard description

In current frontiers of optical sciences,

Nanosized material requires nonlocal description in space ( ) ( ) ( )','',',', trEttrrdttrD

t −= ∫ ε

4

CONTENTS

1. A frontier in light - matter interaction : intense and ultrafast - Theories and computations required in current optical sciences - 2. Real-time and real-space calculation for electron dynamics in solid - time-dependent band calculation, treatment of boundaries. - 3. Electron dynamics in solid for a given electric field - Dielectric function and optical breakdown, … - 4. Coupled Maxwell + TDDFT multiscale simulation - First-principles simulation for macroscopic electromagnetic field in intense regime, Dense electron-hole plasma induced by intense laser pulses -

Frontiers in Optical Sciences: Intense field

Nuclear reaction, Particle Acceleration

QED Vacuum breakdown

Relativistic Classical

Nonrelativistic Quantum

Nonlinear electron dynamics

G. Mourou

Phenomena in intense laser pulses irradiated on bulk materials

21513 W/cm1010 − Laser intensity

Vacuum breakdown

Laser acceleration

Electron-hole plasma Optical breakdown Laser machining

Coherent phonon

HHG

Nonrelativistic Quamtum mechanics

Relativistic Classical mechanics

Nonlinear optics

eE(t)z z

nucleiby field electric Internal pulselaser by field electric External

≈

7

Frontiers in Optical Sciences: Ultrafast dynamics

s10 15− Time

Nucleon motion in nucleus

s10 18− s10 23−s10 12−

Attosecond Femtosecond Picosecond

Real-time observation of valence electron motion E. Goulielmakis et.al, Nature 466, 739 (2010).

Shortest laser pulse 80 as (2008)

Period of electron orbital in Hydrogen, 150as

Period of Ti-sapphire laser pulse

Period of Optical phonon (Si, 64 fs)

8

𝑇ph : Period of optical phonon

M. Hase, et al. Nature (London) 426, 51 (2003)

pump probe

𝛥𝛥 𝑡𝛥

∝ 𝑒−Γ𝑡 sin Ωph𝑡 + 𝜙

Pump-Probe Measurement: Coherent Phonon

Intense 1st pulse excites electrons and atoms. Weak 2nd pulse measure change of reflectivity.

Optical microscope image of waveguides written inside bulk glass by a 25-MHz train of 5-nJ sub-100-fs pulses, C.B. Schaffer et.al, OPTICS LETTERS 26, 93 (2001)

R.R. Gattass, E. Mazur, Nature Photonics 2, 220 (2008).

Micromachining – waveguide- Nanosurgery

Ablation of a single mitochondrion in a living cell. N. Shen et.al, Mech. Chem. Biosystems, 2, 17 (2005).

Femto-technology: nonthermal machining by femtosecond laser pulses

10

CONTENTS

1. A frontier in light - matter interaction : intense and ultrashort - Theories and computations required in current optical sciences - 2. Real-time and real-space calculation for electron dynamics in solid - time-dependent band calculation, treatment of boundaries. - 3. Electron dynamics in solid for a given electric field - Dielectric function and optical breakdown, … - 4. Coupled Maxwell + TDDFT multiscale simulation - First-principles simulation for macroscopic electromagnetic field in intense regime, Dense electron-hole plasma induced by intense laser pulses -

3D real-space grid representation, high order finite difference

Time evolution: 4-th order Taylor expansion

( ) ( ) ( ) ( ) ( ) 4,!

1exp0

=

∆

≈

∆

=∆+ ∑=

Nti

tthk

ti

tthtt iKS

N

ki

KSi ψψψ

Calculation in real-time and real-space

Time-dependent Kohn-Sham equation

( ) ( )[ ] ( ) ( ) ( )2,,,,, ∑==∂∂

iiiKSi trtrntrtrnhtr

ti ψψψ

( )[ ] ( ) ( )[ ] ( )trVtrnrr

erdrVm

trnh extxcionKS ,,'

'2

,2

22

++−

++∇−= ∫ µ

Norm-conserving pseudopotential

12

Two key issues

Use (spatially uniform) vector potential instead of (spatially linear) scalar potential to express macroscopic electric field.

Boundary effect (macroscopic shape effect) needs to be considered from outside of the theory.

We assume a long wavelength limit, lattice const. << light wavelength

Electron dynamics in a unit cell under time-dependent, spatially uniform field

Treatment of optical electric field

We can manage them by the appropriate choice of gauge.

Employing a vector potential replacing the scalar potential, one may recover the periodicity of the Hamiltonian

( ) ( ) ( )ztEdtctAztE

tA

cE

t

ˆ''

1

∫=⇔=

∂∂

−∇−=

φ

φ

( ) ( ) ( ) ( )ttrVtAcep

mt

ti ψψ

+

−=

∂∂ ,

21 2

Electron dynamics in crystalline solid under spatially uniform field

( )zteE

z

( )zVScalar potential spoils spatial periodicity of the Hamiltonian

( )zteE

( ) )(rharh ≠+

( ) ( ) ( )tretartrhtarh nkaki

nk ,,),(,

ψψ =+=+

Time-dependent, spatially uniform electric field

Time-dependent Bloch function

a

14

( ) ( ) ( ) ( )

( ) ( )2

2

,,

,,21,

∑=

+

−+=

∂∂

nkkn

knkn

trutrn

trutrVtAcekip

mtru

ti

( ) ( )( ) ( )trutaru

truetr

nknk

nkrki

nk

,,,,

=+=ψ

Equation for time-dependent Bloch function

( ) ( ) ( ) ( )ttrVtAcep

mt

ti nknk ψψ

+

−=

∂∂ ,

21 2

( ) ( )ztEdtctAtA

cE

t

ˆ''1∫=

∂∂

−=

Transverse

+ + + + + + + + + + + + + + +

－－－－－－－－－－－－－－ Longitudinal

)()( tAtA ext= ( )tAtAtA onpolarizatiext += )()(

Treatment of Polarization (Macroscopic Shape of Sample) - We solve electron dynamics in a unit cell inside a crystalline solid. - The electric field in the unit cell depends on macroscopic geometry of the sample. - We need to specify the ‘macroscopic geometry’ in the calculation.

Optical response of thin film

～ onpolarizatiext EEE +=

( )tEext

( )tAtAtA onpolarizatiextr += )()(

Time-dependent Kohn-Sham equation

Case of longitudinal geometry: Treatment of induced polarization

( ) ( )trjrddt

tAd

cell

onpolarizati ,4

)(2

2

∫ΩΩ

=π

2

2

21

∑=

+−

+∇−=

∂∂

ii

ixc

iii

n

nEeA

cei

mti

ψ

ψδ

δφψψψ

∑

−

+=

iii ccA

cep

mj ..

21 * ψψ

Equation for polarization

dielectrics “parallel-plate-capacitor with dielectrics”.

( ) ')'(

)(1)(

∫−=

∂∂

−=

tdttEctA

ttA

ctE

( )trj ,

( )tA

Bertsch, Iwata, Rubio, Yabana, Phys. Rev. B62(2000)7998.

16

17

CONTENTS


Electron dynamics in bulk silicon under pulse light

Ground state density (110) Density change from the ground state (110)

I= 3.5×1014 W/cm2, T=50 fs, ħω=0.5 eV Laser photon energy is much lower than direct bandgap.

Elec

tric

Fiel

d (a

.u.)

403020100Time (fs)

-Blue0Green

Red +

18

19

Linear response: Two methods to calculate dielectric function from real-time electron dynamics

Transverse geometry Longitudinal geometry

( ) ( )

( ) ( )

( ) ( )ω

ωσπωε

σωσ

σ

ω

i

tedt

dtdA

cttdttJ

ti

t

41

1''

+=

=

−−=

∫

∫ ( ) ( ) ( )( ) ( )

( ) ( ) ( )

( )( )( )tAedt

tAedt

tAttdttA

tJdt

tAd

tAtAtA

extti

totti

ext

t

tot

pol

polexttot

ω

ω

ωε

ε

π

∫∫

∫

=

−=

=

+=

−

1

'''

4

1

2

2

For an impulsive external field,

( ) ( ) ( ) ( )tkctAtktE θδ −== ,

( ) ( )tktJ σ= ( ) ( )tkdt

dAc

tE tottot

11 −=−= ε

Comparison of numerical results for dielectric function of Si by two methods - Green : longitudinal - Red : transverse Re[ε]

Im[ε] J(t)/A0

Spatial grid: 16^3 k-points : 8^3 Distortion strength: A0 = 0.01 (longitudinal) 0.0005 (transverse)

Dielectric function of Si in TDDFT (ALDA) Transverse calculation

Dielectric function by TDDFT (ALDA) is not accurate enough. 21

Instantaneous pulse field is applied at t=0, the induced current as a function of time.

G. Onida, L. Reining, A. Rubio, Rev. Mod. Phys. 74(2002)601.

PRL107, 186401 (2011) S. Sharma, J.K. Dewhurst, A. Sanna, E.K.U. Gross, Bootstrap approx. for the exchange-correlation kernel of time-dependent density functional theory Limited to linear response regime.

Most successful but Beyond Kohn-Sham framework Dots: experiment Dash-dotted: RPA Solid: Bethe-Salpeter

TDDFT (ALDA)

GW approx. + Bethe-Salpeter

TDDFT with hybrid (nonlocal) functional

Good results for small gap materials, Less satisfactory for large gap materials. J. Paier, M. Marsman, G. Kresse, Phys. Rev. B78,121201 (2008)

Intense laser pulse on solid

21513 W/cm1010 − Laser intensity

Vacuum breakdown

Laser acceleration

Electron-hole plasma Optical breakdown Laser machining

Coherent phonon

HHG

Nonrelativistic Quamtum mechanics

Relativistic Classical mechanics

Nonlinear optics

eE(t)z z

nucleiby field electric Internal pulselaser by field electric External

≈

5 x 1014 W/cm2

1 x 1015 W/cm2

2 x 1015 W/cm2

Calculation for diamond laser frequency: 3.1eV, below bandgap (4.8 eV)

Eext(t) vs Etot(t) As the laser intensity increases,

-0.6

-0.4

-0.2

0.0

0.2

0.4

0.6

Electric field (au)

1000 800 600 400 200 0

time (au)

ext2d15_400 tot2d15_400

-0.2

-0.1

0.0

0.1

0.2

Electric field (au)

1000 800 600 400 200 0

time (au)

ext1d15_400 tot1d15_400

-0.2

-0.1

0.0

0.1

0.2

Electric field (au)

1000 800 600 400 200 0

time (au)

ext5d14_400 tot5d14_400 Weak field:

Dielectric dynamics

Threshold for breakdown

T. Otobe, M. Yamagiwa, J.-I. Iwata, K.Y. T. Nakatsukasa, G.F. Bertsch, Phys. Rev. B77, 165104 (2008)

At a certain intensity, resonant energy transfer is observed.

exttot EE 1−= ε

exttot EE 1−≠ ε

Band gap

valence band

conduction band X X

X X

Resonant energy transfer (1x1015 W/cm2, 3.1eV, 40fs)

Eext(t)

Etot(t)

Number of excited electrons

Energy transfer from laser to diamond

Incident laser frequency = plasma frequency of excited electrons Plasma frequency for 0.4/atom

( ) eV40

4≈

=

επω

mnex

p

exceeds the frequency of laser pulse, 3.1eV

diamond

25

Resonant intensity is higher than measured optical breakdown threshold

Resonance starts 7x1014 W/cm2 (16fs)

Two photon

Calculated threshold: 6J/cm2 Measured threshold: 0.63±0.15 J/cm2

D.H. Reitze et.al, Phys. Rev. B45, 2677(1992)

26

27

When nonlinear effects are important, neither longitudinal or transverse calculation is sufficient.

Up to now , we considered electron dynamics under a given field

Next step is to consider dynamics of both electrons and electromagnetic fields simultaneously.

28

CONTENTS


Usually, light-matter interaction is described by

[ ] ( )EEDD ωε==

Constitution relation Electromagnetism: Maxwell equation for macroscopic fields, E, D, B, H

Quantum Mechanics: Perturbation theory to calculate linear susceptibilities, ε(ω)

As the light intensity increases, nonlinear effects appear

[ ] ( ) ( ) ( ) +++== 3322 44 EEEEDD πχπχωε

Extreme intensity of laser pulse

We need to solve couple Maxwell + Schroedinger dynamics.

Quantum mechanics is useful to calculate nonlinear susceptibilities as well.

( )tri ,ψ( ) ( )trBtrE ,,,

external field

polarization

[μm] [nm] Different spatial scales

We introduce two spatial grids and achieve “multiscale simulation”

Maxwell equation describing electromagnetic field

Time-dependent density-functional theory describing electron dynamics

Two spatial scales

We need to solve couple Maxwell + Schroedinger dynamics.

Maxwell eq. Propagation z-direction, polarization x-direction

ｘ

ｚ ( ) ( ) ( )

( ) ( )xtzAtrA

tzAz

tzAtc

z

ˆ,,

0,, 2

2

2

2

2

=

=∂∂

−∂∂

ε

We consider a problem: Laser pulse irradiation normal on bulk Si

Index of refraction ε=n

2

11

+−

=nnR

Reflectance

Velocity of wave

ncv =

Vacuum

ε=16

eV55.1,nm800 == ωλ

Si

Laser frequency below direct bandgap

31

A/c

[µm]

Macroscopic grid

Multiscale simulation: Formalism

( ) ( ) ( )tZJc

tZAZ

tZAtc

,4,,12

2

2

2

2π

=∂∂

−∂∂

Macroscopic grid points（μm） to describe macroscopic vector potential

ZeionZ

Zixc

ZiZZiZi

enenn

EeAcei

mti

,2

,,,

2

,

421

−−=∇

+−

+∇−=

∂∂

πφ

ψδ

δψφψψ

( ) Anc

emi

j

jrdtZJ

Zei

ZiZiZiZiZe

Ze

,*,,,

*,,

,

42

),(

πψψψψ −∇−∇=

=

∑

∫Ω

At each macroscopic points, Kohn-Sham orbitals are prepared, and described in microscopic grids.

Exchange of information by macroscopic current and macroscopic vector potential. ( )tZA ,

( )tZJ ,

Zi ,ψ

At each macroscopic grid point, We consider a unit cell and prepare microscopic grid.

K. Yabana, T. Sugiyama, Y. Shinohara, T. Otobe, G.F. Bertsch, Phys. Rev. B85, 045134 (2012).

I=1010W/cm2

Z=0 [µm]

[µm]

Z=0.8 [µm] Z=1.6 [µm]

Propagation of weak pulse (Linear response regime, separate dynamics of electrons and E-M wave)

Vacuum Si Laser frequency：1.55eV： lower than direct bandgap 2.4eV(LDA）

33

A/c

34

I=1010W/cm2

Propagation of weak pulse (Linear response regime, separate dynamics of electrons and E-M wave)

[µm]


Electron excitation energy per unit cell volume

A/c

35

Maxwell+TDDFT vs Maxwell only (constant ε)

Red = Maxwell + TDDFT Green = Maxwell

Si Vacuum

( ) ( ) 160,, 2

2

2

2

2 ==∂∂

−∂∂ εε tzA

ztzA

tc

εcv =

In linear regime, dispersion of dielectric function Induces Phase velocity vs group velocity Chirp effect is also seen

+

=

ωε

εωε

dd

cvg

21

[µm]

A/c

Z=0 [µm]

[µm]

Z=0.8 [µm] Z=1.6 [µm]

I =5 x 1012W/cm2

Vacuum Si

More intense laser pulse Maxwell and TDKS equations no more separate.

36

37

I = 5 x 1012W/cm2

More intense pulse (2-photon absorption dominates)

[µm]


Electron excitation energy per unit cell volume

A/c

38

Vector potential and electronic excitation energy after the laser pulse splits into reflected and transmitted waves

39

Dense electron-hole plasma generation at the surface modifies dielectric properties at the surface.

K. Sokoowski-Tinten, D. von der Linde, Phys. Rev. B61, 2643 (2000)

Si

Pump-probe

λ=625nm, 100fs pulse Strong pump-pulse excites electrons at the surface, forming dense electron-hole plasma

Drude model fit

( )

+

−=

τωω

πεε

imne

n phgsph

14*

2

m*=0.18, τ=1fs

Probe-pulse measures change of Dielectric properties. Reflectivity

change

Pump-pulse intensity

40

I=1x1011W/cm2, hv=1.55eV

“Numerical Pump-Probe Experiment”

Si Vacuum

Excited electron density at the surface

2-photon absorption

We can calculate excited electron density at the surface by pump-pulse / reflectivity of probe pulse

Reflectivity of probe pulse

( )

+

−=

τωω

πεε

imne

n phgsph

14*

2

Fit by Drude model

41

Performance at K-Computer in Kobe (in early access)

Computational Aspects Large computational resources required for multi-scale calculation

1,000 cores, 10 hours (SGI Altix, U. Tokyo) 20,000 cores, 20 min (K-computer, Kobe) macroscopic grid: 256 microscopic grid: 16^3 k-points 8^3 -> 80 (too small) time step: 16,000

Future Prospects: Multi-dimensional wave propagation

At present, 1-dim propagation (macroscopic grid) Si, diamond: 1,000 cores, 10 hours 20,000 cores, 20 min (K-computer, Kobe) SiO2 (α-quartz) 30,000 cores, 2 hours

Oblique incidence, 2-dim

50×

3-dim - Self focusing - Circular polarization A million of macro-grid points

000,1×need to wait next generation supercomputers

Electron dynamics in bulk periodic solid by real-time TDDFT - First-principles description of electron dynamics in femto- and atto-second time scale - Applications to - dielectric function (linear response) - coherent phonon generation - optical breakdown Coupled Maxwell + TDDFT multi-scale simulation - Promising tool to investigate laser-matter interaction - Requires large computational resources, a computational challenge Further developments necessary - Surface dynamics, electron emission - Collision effects (electron-electron, electron-phonon)

Summary

43

time-dependent density functional theory for ultrafast electron … · 2013-03-14 ·...

Documents