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 t 10 s 1 ns

How long does it take to switch electric current? photoconductive switch femtosecond optical beam radiated THz wave DC bias    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 Process signals at optical frequencies Utilize reversible nonlinear phenomena Investigate Learn how to nonpertubative control electron nonlinear effects motion with light

Grand questions • How to manipulate signals at petahertz frequencies? • How to reversibly change the electrical and/or optical 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 optics 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 Walter Franz 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

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

Extreme nonlinear optics

Why did it take so long? As of 1998:  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

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.

Few-cycle laser pulses Cosine waveform φ = 0 E ( t ) = E a ( t )cos( ω L t - φ ) Sine waveform φ =  /2

New opportunities Intense optical waveforms UV/VIS/NIR wavepackets 1eV 10 eV spectral channels synthesized light electric field E L ( t ) coherent superposition of wavepackets with varying phase & time [fs] amplitude -10 -5 0 5 10 courtesy of Ferenc Krausz

New opportunities Attosecond pulses and tools shortest isolated pulses: 67 attoseconds • spectral range: 1 eV – 1000 eV • electron energy analyzer XUV spectrometer filter imaging mirror courtesy of Ferenc Krausz

New opportunities High-repetition-rate laser sources

An overview of strong-field phenomena in solids

Conventional nonlinear optics external polarisation electric response field        (1) (2) 2 (3) 3 ( ) ( ) ( ) ( ) P t F t F t F t linear nonlinear susceptibility susceptibilities

A model 1D system Independent electrons • A homogeneous external electric field: 𝐺(𝑢) • A periodic potential 𝑉 𝑨 + 𝑏 = 𝑉(𝑨) • V(x) x

A model 1D system Independent electrons • A homogeneous external electric field: 𝐺(𝑢) • A periodic potential 𝑉 𝑨 + 𝑏 = 𝑉(𝑨) •  ˆ TDSE: t    i ( ) t H t ( ) ( ) t  ˆ 2 p ˆ      length gauge: H U ( ) r e F ( ) t r , e 0 LG 2 m   2  ˆ A ( ) p e t ˆ  velocity gauge:     ( ), r F ( ) A ( ) H U t t VG 2 m i A ( ) t r    the gauge transformation: ( ) ( ) t e t LG VG

A model 1D system V(x) Even though the electrons are x independent, the Pauli exclusion principle is not violated (within numerical accuracy)

Strong-field polarisation response linear response quantum beats, nonperturbative residual polarisation response 1D model of SiO 2 : bandgap = 9 eV

Multiphoton ionisation Energy Γ ∝ 𝐽 𝑂

Multiphoton interband transitions SiO 2 Γ ∝ 𝐽 𝑂

Tunnelling ionisation Energy −const 𝐺 Γ ∝ 𝑓     2 m        exp dx V x ( ) E 2       semiclassical approximation: ( ) x 2 m V x    ( ) E 4 2

Interband tunnelling Energy conduction band valence band

Interband tunnelling Energy −const 𝐺 Γ ∝ 𝑓 Clarence Zener

Bloch oscillations semi-classical motion in a perfect crystal (ballistic transport): ∆ b Bloch frequency:   Localisation length: b L e F

Bloch oscillations semi-classical motion: ∆ b Bloch frequency:   Localisation length: b L e F

Wannier-Stark localisation Wannier-Stark Hamiltonian (1D for simplicity):   2 d ˆ           H U z ( ) eFz E 2   2 m dz   ( ) ( ) U z a U z 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)  2   13 A necessary condition: 10 seconds T T  B scattering B   e F a B h V 8 F 10 m eaT scattering 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…

Bloch oscillations & Wannier-Stark localisation The existence of Bloch oscillations used to be a very controversial topic… until they were observed in superlattices E. E. Mendez et al. , PRL 60 , 2426 (1988)

A numerical example Extended-zone scheme

A numerical example 0 = 1 V/Å 𝐺 𝜇 L = 800 nm 1D model of SiO 2 excitation probability

A numerical example 0 = 1.5 V/Å 𝐺 𝜇 L = 800 nm 1D model of SiO 2 excitation probability

A numerical example 0 = 2 V/Å 𝐺 𝜇 L = 800 nm 1D model of SiO 2 excitation probability

A numerical example Initial state: a wave packet in the lowest conduction band 𝐺 0 = 0.7 V/Å 𝜇 L = 1600 nm 1D model of SiO 2 excitation probability

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: (the gap region is assigned to complex K)   C  2 ( ) x dx    2  p ( x ) / ( x ) e Tunnelling probability: B C B        C     Tunnelling rate: B p eFa h / exp 2 ( ) x dx  2 B

Zener vs Kane vs Keldysh Tunnelling rate in a strong constant external field:     1/2 3/2  m E e F a • Zener:    g   exp    2 V 2 e F   uc    1/2 3/2   2 2 1/2 m E e F m    • Kane: g   ex p  2 1/2   18 E 2 e F   g 5/2     3/2     1/2 3/2  E mE m E e F 2 • Keldysh:      g g g     exp    2 2 1/2 3/2     9 m E 2 e F     g 1      𝑛 : reduced mass 3 𝐹 𝑕 : band gap s m 𝐺 : electric field