Strong-field-driven electron dynamics in solids Lecture 2 Vladislav - - PowerPoint PPT Presentation

Strong-field-driven electron dynamics in solids Lecture 2

Vladislav S. Yakovlev

Max Planck Institute of Quantum Optics Laboratory for Attosecond Physics

Winter course on “Advances in strong-field electrodynamics” 4 February 2014, Budapest

Outline

• An overview of recent experiments
• An overview of theoretical concepts and approaches

– analytical approaches

– numerical approaches – Metallisation of dielectric nanofilms (M. Stockman’s work)

• Getting insight into optical-field-induced currents

– Wannier-Stark interpretation – interference of multiphoton channels – semiclassical interpretation

• Outlook

Recent experiments

HHG in ZnO with mid-IR pulses

• 0.6 V/Å

3.25 µm

HHG in ZnO with mid-IR pulses

• 0.6 V/Å

3.25 µm

HHG efficiency is sensitive to crystal orientation and laser ellipticity

HHG with THz pulses in GaSe

• 0.72 V/Å

10 µm ( 30 THz)

HHG with THz pulses

Franz-Keldysh at extreme intensities

Franz-Keldysh at extreme intensities

Franz-Keldysh at extreme intensities

( )

3/2 g

4 2 ( ) exp 3 m E e F α ω ω     ∝ − −       ℏ ℏ

Conventional Franz-Keldysh effect (below the bandgap):

• S. Ghimire et al., PRL 107, 167407 (2011)

Attosecond transient XUV absorption

Experiment vs theory: SiO2

• M. Schultze et al.,

Nature 493, 75 (2013)

measurement – blue theory – red Interpretations:

• Wannier-Stark…
• EIT

Transient XUV absorption

Controlling dielectrics with the electric field of light

• M. Schultze et al., Nature 493, 75 (2013)

A “by-product” from the same measurement campaign:

Optical-field-induced current in dielectrics

Energy x

Au Au

4 eV 5 eV

• SiO2

Optical-field-induced current in dielectrics

Laser pulses:

760 nm, 400 µJ, < 4 fs (< 1.5 cycles) F0 ≤ 2 V/Å 3 kHz rep.rate, stabilised CEP Spacing between Au electrodes: ~50 nm No bias applied Active material: SiO2 Direct bandgap of ~9 eV Optical breakdown at 2.5 x 1015 W/cm

Optical-field-induced current in dielectrics

Optical-field-induced current in dielectrics

Single-pulse experiment Two-pulse experiment

Schiffrin A., et al. Nature 493, 70–74 (2013).

Optical-field-induced current in dielectrics

F0 ≈ 1.7 V/Å

∆x

Electric current is induced in a dielectric with a rise time of ~ 1 fs

Optical-field-induced current in dielectrics

CEP-controlled current for the drive pulse ⟹ subcycle creation

• f charge carriers

F0

(i) ≈ 2 V/Å, F0 (d) ≈ 0.2 V/Å

delay

∆x

GaN samples

F0 = 0.4 V/Å

GaN Al2O3

TEM grid 5 nm Ti + 50 nm Au

GaN ~3.5 eV bandgap → 2-photon absorption

• f ~760 nm light (NIR)

More advanced lithographic techniques @ LBNL (Berkeley) and WSI → controllable gaps (~ 50 nm – 300 µm)

GaN samples

F0

(i) ≈ 0.4 V/Å

F0

(d) ≈ 0.06 V/Å

Technique successfully adapted to flat lithographic GaN samples

~2.5 fs

Solid-state light-phase detector

• T. Paasch-Colberg et al., Nature Photonics (2014)

Solid-state light-phase detector

• T. Paasch-Colberg et al., Nature Photonics (2014)

Solid-state light-phase detector

CEP-detection using one junction → phase-ambiguity subsequent measurements with slightly changed CEP values second junction for phase-disambiguation

An overview of theoretical concepts and approaches

The gauge choice

2 LG

ˆ ˆ ( ) ( ) , 2 p H U e t e m = + + ⋅ > r F r

( )

2 VG

ˆ ( ) ˆ ( ), ( ) ( ) 2 p e t H U t t m + ′ = + = − A r F A ˆ ( ) ( ) ( ) i t H t t t ψ ψ ∂ = ∂ ℏ

TDSE: length gauge: velocity gauge: the gauge transformation:

( ) LG VG

( ) ( )

i t

t e t ψ ψ =

A r ℏ

good: F(t) is unambiguous bad: saw-tooth potential, coupled crystal momenta good: periodic potential (dipole approximation) bad: the stationary problem must be solved accurately; time-dependent Hamiltonian for F(t)=const

Houston functions

Let’s consider instantaneous eigenstates of the velocity- gauge Hamiltonian:

( )

2

ˆ ( ) ( ) ( ) ( ) ( ) 2 p e t U t t t m ϕ ε ϕ   +   + =     A r ɶ

Let !,#$%& be Bloch states:

2 , ,

ˆ ( ) ( ) 2

n n n

p U m φ ε φ   + =    

k k

r k

Solution:

( ) , , ( )

( , ) ( ), ( ) ( )

i t n n t

t e t e t ϕ φ

−

′ = = −

A r k k

r r k F

ℏ

ℏ

,

( ( ))

n n

t ε ε =

k

k ɶ

Houston functions = accelerated Bloch states analogous to Volkov states

TDSE in the basis of Houston states

Ansatz:

ˆ ( ) ( ) ( ) i t H t t t ψ ψ ∂ = ∂ ℏ

( ( )) ( ) , , ( )

( ) ( )

t n

i i t dt t n n t n

t t e e

ε

ψ α φ

′ ′ − −

∫ =∑

k A r k k k ℏ ℏ

( )

( )

( ) , ,

( ) ( ) ( ) ( )

t nq

i t dt n q nq q

i t eF t t B t e

ε

α α

′ ′ ∆

∫ ′ = −

∑

k k k

k

ℏ

ℏ

( )

( ) ( ( )) ( ( ))

nq n q

t t t ε ε ε ′ ′ ′ ∆ = − k k k

( ) ( ) e t t = − k k A ℏ

Solution for a linearly polarized field:

• J. B. Krieger, G. J. Iafrate,

PRB 33, 5494 (1986) , ,

( ) ( )

i n n

e u r φ

⋅

=

k r k k

r

Blount’s matrix element:

* 3 , , unit cell uc

( ) ( ) ( )

x

nq n k q

i B u u d r V = ∂

∫

k k

k r r

An approximate solution

In the limit of small excitation probabilities,

( )

( )

( ) , ,

( ) ( ) ( ) ( )

t nq

i t dt n q nq q

i t eF t t B t e

ε

α α

′ ′ ∆

∫ ′ = −

∑

k k k

k

ℏ

ℏ

( )

( )

( ) ,

( ) ( ) ( )

t nq

i t t dt n nq

e t i dt F t B t e

ε

α

′

′′ ′′ ∆

∫ ′ ′ ′ ≈ −

∫

k k

k

ℏ

ℏ

( ) ( ) e t t = − k k A ℏ

, (0) n nq

α δ =

k

initial conditions: This is a convenient starting point for analytical methods.

Adiabatic perturbation theory

( )

ˆ ( ) ( ) ( ) i t H t t t ψ ψ ∂ = ∂ c ℏ

• slowly varying Hamiltonian
• the system remain in a

nondegenerate eigenstate

( )

( )

( ) ( )

( ) ( )

t t

i E t dt i t

t e e t

γ

ψ ϕ

′ ′ − ∫

=

c

c

ℏ

( )

ˆ ( ) ( ) ( ) H E ϕ ϕ = c c c c

Ansatz: ( ) ( ) ( ) ( )

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

t t

d d i t t t t i dt t t dt dt γ ϕ ϕ γ ϕ ϕ ′ ′ ′ ′ = − ⇒ = ′

∫

c c c c

( ) ( ) ( ) ( )

f

( ) ( ) ( )

t t

d i dt t t d i t t d ϕ ϕ ϕ γ ϕ = = ∇ ⋅

∫ ∫

c

c c c c c

• periodic motion:

Berry phase Berry connection

Berry phase in Bloch bands

Berry connection

2

ˆ ( ) ˆ ( ) 2

i i

p e He U m

− ⋅ ⋅

+ = +

k r k r

k r ℏ

Zak’s phase:

BZ Zak n n n

d u i u γ = ⋅ ∇

∫

k k k

k

• D. Xiao et al., RMP 82, 1959 (2010)

How to make the Hamiltonian k-dependent?

2

ˆ ˆ ( ) 2 p H U m = + r

The eigenstates of the transformed Hamiltonian are the envelope functions '(! ) '(!$) *&

Length-gauge analysis

Periodic potential +$,&, homogeneous constant electric field

Let $, -& be an eigenstate with energy .- Wannier-Bloch states:

2

ˆ ( , ( ) ( ) ) 2 p U z eF m U z E U z a z ϕ ϕ   + +     = + =

( ; ) ( )

ikla l

b z k e z la ϕ

∞ =−∞

= −

∑

2 WB

ˆ ( ) ( ; ) ( ) ( ; ) 2

n n n

p U z eF z i b z k E k b z k m k   ∂   + + + =     ∂    

For simplicity, 1D Wannier-Stark Hamiltonian:

Wannier-Bloch states and TDSE

Neglecting interband transitions, approximate solutions of the TDSE can be constructed as

( , ) ; exp

t n n n

eF i eF z t b z k t E k t dt ψ       ′ ′ = − − −            

∫

ℏ ℏ ℏ

Fourier analysis:

WS

WS

( , ) ( )

n

it E n nl l

z t z e ψ ψ

−

= ∑

ℏ 2

ˆ ( ) ( ; ) ( ) ( ; ) 2

n n n

p U z eF z i b z k E k b z k m k   ∂   + + + =     ∂    

Wannier-Stark states

eigenstates of the Wannier-Stark Hamiltonian in the single-band approximation localised on site /; localisation length: Δ1/$.||& form a basis Zener tunnelling adds an imaginary part to WS energies Wannier-Stark ladder:

/ WS /

( ) 2

a n n a

a E dk E k leaF

π π

π

−

= +

∫

Wannier-Stark states:

/ WS /

( ) ( ; ) 2

a ilak n nl a

a z dk b z k e

π π

ψ π

− −

=

∫

Kane functions

Wannier-Stark Hamiltonian:

2 2

ˆ ( ) 2 d H U z eFz m dz = − + + ℏ

, , , , ,

( ) ( ) / 2 , ( )

ikz n k n k n k n k n n

z e u z k k φ π φ φ δ δ

′ ′ ′

′ = = −

[S. Glutsch, PRB 69, 235317 (2004)]

Bloch functions: Ansatz:

/ , /

( ) ( ) ( ) 2

a n k a n n k

a z dk z

π π

ϕ ϕ φ π

−

=∑

∫

ɶ

, , ,

( )

n k n k n n

d eFz ieF k k dk φ φ δ δ

′ ′ ′

′ = − +

/2 * , , /2

( ) ( ) ( )

a n k k n k a

i dzu z u k a e k z Fδ

′ −

′ + − ∂

∫

Blount’s matrix element (Berry connection)

2 , , 2

( ) ( ) 2

n k n n k

d U z E k m dz φ φ   − + =     ℏ

Kane functions

/ * , , /

( ) ( ) ( )

a nq n k k q k a

i B k dk u z u z a

π π −

= ∂

∫

2 2

( ) 2 d U z eFz E m dz ϕ ϕ   − + + =     ℏ

/ , /

( ) ( ) ( ) 2

a n k a n n k

a z dk z

π π

ϕ ϕ φ π

−

=∑

∫

ɶ

,

( ) ( ) ) ( ) ( ) (

n n q q n q n

d E k ieF eF B k E d k k k k ϕ ϕ ϕ   + + =    

∑

ɶ ɶ ɶ boundary conditions:

( ) ( 2 / )

n n

k k a ϕ ϕ π = + ɶ ɶ

Approximation: neglect coupling between different bands

/ , / , ,

( ) , ; 2 ( ) exp ( ) ;

a n l n a k n l n l n

a E dk E k leaF l i k dk E E k eF

π π

π ϕ

−

′ ′ = + ∈   ′ ′   = − −      

∫ ∫

ℤ ɶ

/ , , , /

( ) ( ) ( ) ( ) ( ) 2 ( )

n n n a n l n n l n k a

a E k E k eFB z dk k k z

π π

ϕ ϕ φ π

−

= + =

∫

ɶ (Kane functions)

Kane functions

• Eigenfunctions of the Wannier-Stark Hamiltonian in the

single-band approximation

• Orthonormal and complete
• !,5$,& is localised at lattice site /
• Adiabatic evolution in the basis of Kane states:

( )

Bloch , , ( )

( ) exp ( )

t n n n t

i t E t dt ψ φ

+

  ′ ′ = −    

∫

k k A

k ℏ

( )

( )

Houston ,

exp ( ) ( ) ( )

t n n

i t E t dt t ϕ   ′ ′ = ⋅ −    

∫

k

A r k ℏ

Metallization of nanofilms

• M. Durach et al., PRL 150, 086803 (2010)

Kronig-Penney model:

• M. Durach et al., PRL 150, 086803 (2010)

Metallization of nanofilms

• M. Durach et al., PRL 150, 086803 (2010)

Metallization of nanofilms

• M. Durach et al., PRL 107, 086602 (2011)

Metallization of nanofilms

Numerical approaches

Quantum kinetics

• Many-body density

matrix

• Non-equilibrium

Green’s functions

Schrödinger equation

• Single-electron TDSE

+ effective potential

• Time-dependent

density-functional theory (TDDFT)

Semi-classical models

• Boltzmann equation

Time-dependent density functional theory

Kazuhiro Yabana (Tsukuba University)

Time-dependent density-functional theory (TDDFT) combined with solving Maxwell equations

• Many-electron dynamics
• One multiscale simulation for SiO2: 100000 core-hours

Getting insight into optical-field-induced electric currents in dielectrics

Optical-field-induced current in dielectrics

Single-pulse experiment Two-pulse experiment

• A. Schiffrin. et al. Nature 493, 70 (2013)

Numerical model #1

Tight-binding approximation (Stockman, Apalkov)

• A. Schiffrin. et al. Nature 493, 70 (2013)

Numerical model #2

Multiband optical Bloch equations

( ) ( )

( )

( ) ( ) ( )

* * * *

d 1 , d i d 1 , d i d 1 , d i

j jj j j j j j j j j j j j j j jj j j jj j j j j j j j j j j j

c t t d t ρ ρ ρ ρ ρ ρ ρ ρ ρ ρ ρ ρ ρ ρ ρ ρ

′ ′ ′ ′ ′ ′ ′ ′ ′ ′ ′ ′ ′′ ′′ ′′ ′′ ′ ′ ′ ′′ ′ ′ ′′ ′′ ′′ ′′ ′ ′ ′ ′′

  − + − +       = − + −       = = = − + −    

∑ ∑ ∑ ∑ ∑ ∑

ℓ ℓ ℓ ℓℓ ℓ ℓ ℓ ℓ ℓ ℓ ℓℓ ℓ ℓ ℓℓ ℓ ℓ ℓ ℓ ℓ ℓ ℓ ℓ ℓ ℓ ℓ ℓ ℓ ℓ ℓ ℓ

ℏ ℏ ℏ E M E M M E E M M E E M M

† †

, , , ,

jj j j

c c d d δ ρ ρ

′ ′ ′ ′ ′ ′ ′

= + = =

ℓℓ ℓℓ ℓ ℓ ℓ ℓ ℓ ℓ ℓ

E ε M

Field screening: instantaneous linear versus self-consistent

L 3 VB , BZ

d 4 , d d 2 ( , )Re ( ) ( ) d

fi fi f i

t e d k t N e t t m π ρ ≡ − = − +   ≡ = −    

∑ ∫

A E E P P J k p k A ∓

L

ε = E E

VS

courtesy of S. Kruchinin

Experiment vs theory: SiO2

• S. Yu. Kruchinin et al.,
• Phys. Rev. B 87, 115201 (2013)
• A. Schiffrin et al.,

Nature 493, 70 (2013)

Transferred charge vs pulse amplitude

Experiment vs theory: SiO2

• S. Yu. Kruchinin et al.,
• Phys. Rev. B 87, 115201 (2013)
• A. Schiffrin et al.,a

Nature 493, 70 (2013)

Transferred charge vs pulse length

Picture #1: Wannier-Stark localisation

Quantum-mechanical basis: Wannier-Stark states

Mark Stockman

reversibly enhanced polarisability

Picture #1: Wannier-Stark localisation

Picture #2: multiphoton interference

Quantum-mechanical basis: Bloch states

Stanislav Kruchinin

asymmetric population of conduction bands

Symmetry breaking by quantum interference

GaAs

ω + 2ω

• 1-photon amplitude is

an even function of k

• 2-photon amplitude is

an odd function of k

The amount of transferred charge correlates with the spectral overlap of multiphoton channels

Picture #2: multiphoton interference

• S. Yu. Kruchinin, M. Korbman, V. S. Y., Phys. Rev. B 87, 115201 (2013)

The amount of transferred charge increases with the applied field according to the interference of multiphoton channels (for 6 ≲ 1.5 V/Å)

89 :! ; 28:9<! ⋯

• S. Yu. Kruchinin, M. Korbman, V. S. Y., Phys. Rev. B 87, 115201 (2013)

Picture #2: multiphoton interference

Amplitude scan for 2-photon pulse Amplitude scan for 3-photon pulse

Picture #2: multiphoton interference

Picture #3: semiclassical motion

semiclassical electron displacement >$?6&

Peter Földi

Quantum-mechanical basis: Houston states

Verifying the semiclassical picture

take a very short cosine-pulse take the excitation probability from a quantum simulation multiply the probability with the semiclassical displacement

Verifying the semiclassical picture: GaN

Verifying the semiclassical picture: GaN

carrier injection: two-photon absorption

Verifying the semiclassical picture: GaN

carrier injection: two-photon absorption

Verifying the semiclassical picture: SiO2

carrier injection: interband tunnelling

Role of self-consistent screening: time dependence of polarization

E0 = 1.5 V/Å E0 = 2 V/Å E0 = 2.5 V/Å

Role of self-consistent screening: CEP dependence

Instantaneous linear screening Self-consistent screening

Bragg-like reflections in angle-resolved measurements

Recent ab initio simulations

• G. Wachter et al., arXiv:1401.4357 [cond-mat.mtrl-sci ] (2014)

• What is the simplest and yet appropriate physical

interpretation?

• Does momentum relaxation (electron-phonon

scattering) matter?

• What determines the

carrier-envelope phase that maximizes the transferred charge?

Induced current

• M. V. Fischetti et al., Phys. Rev. B 31, 8124 (1985)

A phenomenological model

• P. Földi et al.

New J. Phys. 15, 063019

CEP-dependent charge transfer

Scattering and dephasing do not do much harm

Carrier-envelope phase (radians) Transferred charge (arb. units)

@ dephasing rate K scattering rate

Semiclassical wave packet motion

E6 0.3 V/Å

Semiclassical wave packet motion

The anharmonicity of the electron motion has a major effect on the transferred charge

E6 1 V/Å

Transferred charge and Bloch oscillations

Bloch oscillations intensity-dependent phase shift > ?6 0 (Å)

Semiclassical electron displacement:

Pulse amplitude (V/Å)

Nick Karpowicz

CEP (radians) Pulse amplitude (V/Å) , E6 / E6 ;

Transferred charge and Bloch oscillations

Bloch oscillations intensity-dependent phase shift > ?6 0 (Å)

Semiclassical electron displacement:

Pulse amplitude (V/Å) CEP (radians) Pulse amplitude (V/Å) , E6 / E6 N

Quantum simulation

Semi-classical vs quantum

semi-classical quantum (Houston basis)

Outlook

To do

• Advanced 3D models
• 2D propagation simulations
• Quantum kinetic simulations

To find out

• What is the most straightforward experimental way to

reconstruct electron dynamics in strong fields?

• Can the Franz-Keldysh effect be used to modulate light with

light at petahertz frequencies?

• What is the role of disorder in strong-field phenomena?
• How to use the tools of attosecond physics to gain new insights

into electron relaxation and dephasing phenomena?

• How to control the electron motion in solids by controlling light?

Acknowledgements

Michael Korbman Stanislav Kruchinin Mark Stockman Nick Karpowicz Agustin Schiffrin Peter Földi Ferenc Krausz Tim Paasch- Colberg Kazuhiro Yabana Elisabeth Bothschafter