Lecture 1 Vladislav S. Yakovlev Max Planck Institute of Quantum - - PowerPoint PPT Presentation

Strong-field-driven electron dynamics in solids Lecture 1

Vladislav S. Yakovlev

Max Planck Institute of Quantum Optics Laboratory for Attosecond Physics

Winter course on “Advances in strong-field electrodynamics” 3 February 2014, Budapest

Outline

• Motivation – new opportunities and frontiers
• An overview of strong-field phenomena in solids

How long does it take to switch electric current?

1 s t 

How long does it take to switch electric current?

9

10 s 1 ns t



 

How long does it take to switch electric current?

radiated THz wave DC bias photoconductive switch femtosecond

• ptical beam

12

10 s 1 ps t



 

How long does it take to switch electric current?

• Is it possible to switch on electric current

within a femtosecond = 10-15 s?

• Is it possible to manipulate electric

currents with sub-femtosecond accuracy?

Towards petahertz electronics

Utilize reversible nonlinear phenomena Investigate nonpertubative nonlinear effects Learn how to control electron motion with light Process signals at optical frequencies

Grand questions

• How to manipulate signals at petahertz

frequencies?

• How to reversibly change the electrical and/or
• ptical properties of a solid within a fraction of a

femtosecond?

• How do electrons in a solid respond to a laser pulse

at field strengths where the conventional nonlinear

• ptics is inapplicable?
• Which insights into nonperturbative effects may

come from attosecond measurements?

A brief history of strong-field phenomena in solids

• 1928 Bloch theory (Bloch oscillations?)

Felix Bloch

A brief history of strong-field phenomena in solids

• 1928 Bloch theory (Bloch oscillations?)
• 1933 Zener tunnelling

Clarence Zener

A brief history of strong-field phenomena in solids

• 1928 Bloch theory (Bloch oscillations?)
• 1933 Zener tunnelling
• 1958 Franz-Keldysh effect

Leonid Keldysh Walter Franz

A brief history of strong-field phenomena in solids

• 1928 Bloch theory (Bloch oscillations?)
• 1933 Zener tunnelling
• 1958 Franz-Keldysh effect
• 1961 second harmonic generation

A brief history of strong-field phenomena in solids

• 1928 Bloch theory (Bloch oscillations?)
• 1933 Zener tunnelling
• 1958 Franz-Keldysh effect
• 1961 second harmonic generation
• 1964 Keldysh theory

Leonid Keldysh

A brief history of strong-field phenomena in solids

• 1928 Bloch theory (Bloch oscillations?)
• 1933 Zener tunnelling
• 1958 Franz-Keldysh effect
• 1961 second harmonic generation
• 1964 Keldysh theory
• 1988 observation of Wannier-Stark

localisation in superlattices

A brief history of strong-field phenomena in solids

• 1928 Bloch theory (Bloch oscillations?)
• 1933 Zener tunnelling
• 1958 Franz-Keldysh effect
• 1961 second harmonic generation
• 1964 Keldysh theory
• 1988 observation of Stark

localisation (Bloch oscillations)

• 2011 high-harmonic generation in

solids

• 2013 optical-field-controlled current
• 2013 time-resolved XUV absorption

new!

Extreme nonlinear optics

Why did it take so long?

 Intense few-cycle pulses  Control over electric field (carrier-envelope phase)  Few-cycle pulses in the mid-infrared spectral range  A clear idea of what to look for

As of 1998:

Attosecond technology

• Intense few-cycle pulses with controlled carrier-

envelope phase – from terahertz to soft X-ray.

• Pump-probe measurements with attosecond

accuracy.

• Powerful measurement and reconstruction

techniques.

Sine waveform φ = /2 Cosine waveform φ = 0

E(t) = Ea(t)cos(ωLt - φ)

Few-cycle laser pulses

New opportunities

spectral channels 1eV 10 eV UV/VIS/NIR wavepackets time [fs] synthesized light electric field EL(t)

• 10
• 5

5 10

coherent superposition

• f wavepackets with

varying phase & amplitude

Intense optical waveforms

courtesy of Ferenc Krausz

New opportunities Attosecond pulses and tools

• shortest isolated pulses: 67 attoseconds
• spectral range: 1 eV – 1000 eV

filter imaging mirror XUV spectrometer electron energy analyzer

courtesy of Ferenc Krausz

New opportunities High-repetition-rate laser sources

An overview of strong-field phenomena in solids

Conventional nonlinear optics

(1) (2) 2 (3) 3

( ) ( ) ( ) ( ) P t F t F t F t       

external electric field polarisation response linear susceptibility nonlinear susceptibilities

A model 1D system

x V(x)

• Independent electrons
• A homogeneous external electric field: 𝐺(𝑢)
• A periodic potential 𝑉 𝑨 + 𝑏 = 𝑉(𝑨)

A model 1D system

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     

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

( ) LG VG

( ) ( )

i t

t e t   

A r

• Independent electrons
• A homogeneous external electric field: 𝐺(𝑢)
• A periodic potential 𝑉 𝑨 + 𝑏 = 𝑉(𝑨)

A model 1D system

V(x) x

Even though the electrons are independent, the Pauli exclusion principle is not violated (within numerical accuracy)

Strong-field polarisation response

linear response quantum beats, residual polarisation nonperturbative response

1D model of SiO2: bandgap = 9 eV

Multiphoton ionisation

Energy

Γ ∝ 𝐽𝑂

Multiphoton interband transitions

SiO2

Γ ∝ 𝐽𝑂

Tunnelling ionisation

Energy

Γ ∝ 𝑓

−const 𝐺

semiclassical approximation:

   

2 4 2

2 exp ( ) ( ) 2 ( ) m dx V x E x m V x E               



Interband tunnelling

conduction band valence band

Energy

Interband tunnelling

Energy

Γ ∝ 𝑓

−const 𝐺

Clarence Zener

Bloch oscillations

semi-classical motion in a perfect crystal (ballistic transport):

Bloch frequency:

∆b

Localisation length:

b

L e F  

Bloch oscillations

semi-classical motion: Bloch frequency: Localisation length:

b

L e F  

∆b

Wannier-Stark localisation

2 2

ˆ ( ) 2 ( ) ( ) d H U z eFz E m dz U z a U z                

Wannier-Stark Hamiltonian (1D for simplicity): If such eigenstates exist, what are their properties?

• If 𝜔(𝑨) is an eigenstate with energy 𝐹, then 𝜔(𝑨 − 𝑏) is an

eigenstate with energy 𝐹 + 𝑓𝐺𝑏.

• If the field is not too strong, there must be a way to limit

dynamics to a single band. Such eigenstates must be localised in space.

• Bloch oscillations: the motion of a wave packet can be

constructed from such eigenstates.

Bloch oscillations in bulk solids

Bloch oscillations are suppressed by

• electron scattering
• interband transitions (Zener tunnelling)

A necessary condition:

13 B scattering B

2 10 seconds T T  





B 8 scattering

V 10 m e F a h F eaT  

A constant field of this strength would destroy most solids.

Bloch oscillations & Wannier-Stark localisation

The existence of Bloch oscillations used to be a very controversial topic…

The existence of Bloch oscillations used to be a very controversial topic… until they were observed in superlattices

Bloch oscillations & Wannier-Stark localisation

• E. E. Mendez et al., PRL 60, 2426 (1988)

A numerical example

Extended-zone scheme

A numerical example

𝐺

0 = 1 V/Å

𝜇L = 800 nm 1D model of SiO2

excitation probability

A numerical example

𝐺

0 = 1.5 V/Å

𝜇L = 800 nm 1D model of SiO2

excitation probability

A numerical example

𝐺

0 = 2 V/Å

𝜇L = 800 nm 1D model of SiO2

excitation probability

A numerical example

𝐺

0 = 0.7 V/Å

𝜇L = 1600 nm

1D model of SiO2

excitation probability Initial state: a wave packet in the lowest conduction band

Bloch oscillations in optical lattices

• S. Longhi, J. Phys. B 45, 225504 (2012)

Zener theory of interband tunnelling

• Proc. Royal Soc. London A, 145, 523 (1934)

( ) K K E eFx  

Integrate on the complex plane:

C B

2 ( ) 2 C B

( ) / ( )

x dx

p x x e



 

 

 

Tunnelling probability: Tunnelling rate:

 

 

C B B

/ exp 2 ( ) 2 p eFa h x dx        

(the gap region is assigned to complex K)

Zener vs Kane vs Keldysh

Tunnelling rate in a strong constant external field:

• Zener:
• Kane:
• Keldysh:

1/2 3/2 5/2 3/2 g g 2 2 1/2 3/2 g g

2 9 exp 2 m E E mE e e F F m E                             

1/2 3/2 2 2 1/2 2 1/2 g g

ex 18 p 2 e F m E m E e F               

 

3

1 s m   

1/2 uc 3/2 g

exp 2 2 m E e V F e F a               

𝑛: reduced mass 𝐹𝑕: band gap 𝐺: electric field

Keldysh theory of interband tunnelling

Keldysh parameter:

g

mE e F   

𝛿 ≪ 1: tunnelling 𝛿 ≫ 1: multiphoton excitation

Keldysh excitation rate

electric field (V/Å) tunnelling rate (arb. units)

Keldysh theory & a few-cycle pulse

4-fs 800-nm pulse

Kane theory & a few-cycle pulse

4-fs 800-nm pulse

Measuring excitation probabilities

V.V. Temnov et al. PRL 97, 237403 (2006)

Method Ultrafast time-resolved imaging interferometry Pump pulse 50 fs, 800 nm Probe pulse 50 fs, 400 nm

Interband excitation in SiO2

Interband excitation in SiO2

Nonperturbative regime

• 𝛿Keldysh ≲ 1 (tunnelling regime)
• 𝑓 𝐺 𝑏 = ℏ𝜕B ≳ 𝐹g
• ℏΩRabi ≳ 𝐹g (Rabi flopping)
• ΩRabi ≳ 𝜕L (carrier wave Rabi flopping)

An incomplete list of cases where the medium response is highly nonperturbative—conventional perturbation theory (expansion in powers of 𝐺) breaks down:

Rabi Keldysh L

  

L g Keldysh

mE e F   

Solids meet vacuum

Interband tunnelling Schwinger effect

Energy

g

mE e F    mc e F   

Keldysh parameter:

1  1 

2 3

exp m c e F              

1/2 3/2 g

exp 2 m E e F              

2

4

2

mc

e F m c



        

g

2E

F m



        

PRL 101, 130404 (2008)

Franz-Keldysh effect

 

3/2 g

4 2 ( ) exp 3 m E e F                

Absorption below the bandgap:

• M. Wegener, “Extreme Nonlinear Optics”

Franz-Keldysh effect

• M. Wegener, “Extreme Nonlinear Optics”

 

3/2 g

4 2 ( ) exp 3 m E e F                

A general formula for the absorption in the external field: Absorption below the bandgap:

• H. Haug, S. W. Koch,

“Quantum Theory of the Optical and Electronic Properties of Semiconductors”

High-harmonic generation in solids

• Low intensities:
• High intensities: Rabi flopping

(1) (2) 2 (3) 3

( ) ( ) ( ) ( ) P t F t F t F t       

• T. Tritschler et al.,

PRA 68, 033404 (2003)

High-harmonic generation in solids

• Low intensities:
• High intensities: Rabi flopping
• High intensities + long wavelengths: Bloch oscillations
• S. Ghimire et al.,

Nature Physics 7, 138 (2011)

(1) (2) 2 (3) 3

( ) ( ) ( ) ( ) P t F t F t F t       