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 µ m

HHG in ZnO with mid-IR pulses � ��� � 0.6 V/Å � � � 3.25 µ m HHG efficiency is sensitive to crystal orientation and laser ellipticity

� ��� � 0.72 V/Å � � � 10 µ m ( � � 30 THz) HHG with THz pulses in GaSe

HHG with THz pulses

Franz-Keldysh at extreme intensities

Franz-Keldysh at extreme intensities

Franz-Keldysh at extreme intensities S. Ghimire et al. , PRL 107 , 167407 (2011) Conventional Franz-Keldysh effect (below the bandgap):    m E  4 2 ( ) 3/2 α ω ∝ − − ω ( ) exp ℏ   g e F 3 ℏ    

Attosecond transient XUV absorption

Experiment vs theory: SiO 2 Transient XUV absorption M. Schultze et al ., Nature 493 , 75 (2013) measurement – blue theory – red Interpretations: • Wannier-Stark… • EIT

Controlling dielectrics with the electric field of light A “by-product” from the same measurement campaign: M. Schultze et al ., Nature 493 , 75 (2013)

Optical-field-induced current in dielectrics

Optical-field-induced current in dielectrics Energy � � � 4 eV � � x Au Au � � � 5 eV SiO 2

Optical-field-induced current in dielectrics Laser pulses: 760 nm, 400 µJ, < 4 fs (< 1.5 cycles) F 0 ≤ 2 V/Å 3 kHz rep.rate, stabilised CEP Spacing between Au electrodes: ~50 nm No bias applied Active material: SiO 2 � Direct bandgap of ~9 eV � Optical breakdown at 2.5 x 10 15 W/cm

Optical-field-induced current in dielectrics Two-pulse experiment Single-pulse experiment Schiffrin A., et al . Nature 493 , 70–74 (2013).

Optical-field-induced current in dielectrics ∆ x Electric current is induced in a dielectric with a rise time of ~ 1 fs F 0 ≈ 1.7 V/Å

Optical-field-induced current in dielectrics delay ∆ x (i) ≈ 2 V/Å, F 0 (d) ≈ 0.2 V/Å F 0 � CEP-controlled current � � � ��� � � � � � � ��� for the drive pulse � ⟹ subcycle creation of charge carriers

GaN samples 5 nm Ti + 50 nm Au TEM grid GaN F 0 = 0.4 V/Å Al 2 O 3 GaN ~3.5 eV bandgap → 2-photon absorption of ~760 nm light (NIR) More advanced lithographic techniques @ LBNL (Berkeley) and WSI → controllable gaps (~ 50 nm – 300 µm)

GaN samples (i) ≈ 0.4 V/Å F 0 (d) ≈ 0.06 V/Å F 0 ~2.5 fs Technique successfully adapted to flat lithographic GaN samples

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 ∂ ˆ TDSE: t ψ = ψ i t H t t ℏ ( ) ( ) ( ) ∂ 2 p ˆ ˆ = + r + F ⋅ r > length gauge: H U e t e ( ) ( ) , 0 LG m 2 good : F ( t ) is unambiguous bad : saw-tooth potential, coupled crystal momenta ( ) 2 + A p ˆ e t ( ) ˆ ′ = + r F = − A velocity gauge: H U t t ( ), ( ) ( ) VG m 2 good : periodic potential (dipole approximation) bad : the stationary problem must be solved accurately; time-dependent Hamiltonian for F ( t )=const i A r t ( ) ψ = ψ t e t ( ) ( ) ℏ the gauge transformation: LG VG

Houston functions   ˆ 2 p r k + φ = ε φ U ( ) ( )   Let !,# $%& be Bloch states: k k n n n , , m 2   Let’s consider instantaneous eigenstates of the velocity- gauge Hamiltonian: ( )   2 + A p ˆ e t ( ) r  +  ϕ = ε ϕ U t ɶ t t ( ) ( ) ( ) ( )   m 2   i − A r ( ) t ′ r r k F ϕ = φ = − t e t e t ( , ) ( ), ( ) ( ) ℏ ℏ Solution: k k n n t , , ( ) 0 k ε = ε ɶ t ( ( )) k n n , 0 Houston functions = accelerated Bloch states analogous to Volkov states

TDSE in the basis of Houston states i i t ∫ ′ ′ − ε k − A r t dt t = ∑ ( ( )) ( ) ψ α n φ t t e e ( ) ( ) Ansatz: ℏ ℏ 0 k k k n n t , , ( ) 0 0 n e ∂ k k A = − t t ˆ ( ) ( ) t ψ = ψ i t H t t ( ) ( ) ( ) ℏ 0 ℏ ∂ Solution for a linearly polarized field: i t i ∫ ( ) ′ ′ ∆ ε k t dt ∑ ( ) ( ) ′ k α = − α nq t eF t t B t e ( ) ( ) ( ) ( ) ℏ 0 k k n q nq , , ℏ 0 0 q Blount’s matrix element: ( ) ′ ′ ′ k k k ∆ ε = ε − ε t t t ( ) ( ( )) ( ( )) nq n q r k r ⋅ φ = i e u r ( ) ( ) k k n , n , i ∫ J. B. Krieger, G. J. Iafrate, k r r = ∂ * 3 B u u d r ( ) ( ) ( ) k k nq n k q unit , , PRB 33 , 5494 (1986) V x cell uc

An approximate solution i t i ∫ ( ) k ′ ′ ∆ ε ∑ ( ) t dt ( ) ′ k α = − α nq t eF t t B t e ( ) ( ) ( ) ( ) ℏ 0 k k n q nq , , ℏ 0 0 q e k k A = − t t ( ) ( ) 0 ℏ In the limit of small excitation probabilities, i ′ t e ∫ ( ) ′′ ′′ k ∆ ε t t dt ( ) ( ) ∫ ′ ′ ′ k α ≈ − nq t i dt F t B t e ( ) ( ) ( ) ℏ 0 k n nq , ℏ 0 0 α = δ , (0) initial conditions: k n nq 0 This is a convenient starting point for analytical methods.

Adiabatic perturbation theory ∂ • slowly varying Hamiltonian ( ) ˆ c t ψ = ψ i t H t t ( ) ( ) ( ) ℏ ∂ • the system remain in a nondegenerate eigenstate ( ) ˆ c c c c ϕ = ϕ H E ( ) ( ) ( ) − ∫ i t ( ) c ′ ′ E t dt ( ) ( ) γ c ψ = i t ϕ ( ) t e e t ( ) ( ) ℏ t Ansatz: 0 d d t ( ) ( ) ( ) ( ) ∫ ′ c c ′ c ′ c ′ γ ϕ = − ϕ ⇒ γ = ϕ ϕ i t t t t i dt t t ( ) ( ) ( ) ( ) ( ) ( ) ′ dt dt t 0 periodic motion: d t ( ) ( ) ( ) ( ) ∫ ∫ c c c c c γ = ϕ ϕ = ⋅ ϕ ∇ ϕ f � t i dt t t d i ( ) ( ) ( ) c d t t 0 Berry phase Berry connection

Berry phase in Bloch bands 2 p ˆ ˆ r = + H U ( ) m 2 How to make the Hamiltonian k -dependent? k + 2 p ˆ ( ) ℏ ˆ − k r ⋅ k r ⋅ r = + i i e He U ( ) m 2 The eigenstates of the transformed Hamiltonian are the envelope functions ' (! ) � ' (! $) � *& ∫ k γ = ⋅ ∇ Zak d u i u Zak’s phase: k k k n n n BZ Berry connection D. Xiao et al. , RMP 82 , 1959 (2010)

Length-gauge analysis For simplicity, 1D Periodic potential +$,& , homogeneous constant electric field �   2 p ˆ Wannier-Stark + + ϕ = ϕ + = U z eF z E U z a U z ( ) , ( ) ( )   Hamiltonian: m 2   Let �$, � -& be an eigenstate with energy � � .�- ∞ ∑ = ikla ϕ − b z k e z la Wannier-Bloch states: ( ; ) ( ) =−∞ l   ∂ 2 p ˆ   + + + = WB U z eF z i b z k E k b z k ( ) ( ; ) ( ) ( ; )     n n n ∂ m k 2    

Wannier-Bloch states and TDSE   2 ∂ p ˆ   + + + = U z eF z i b z k E k b z k ( ) ( ; ) ( ) ( ; )     n n n ∂ m k 2     Neglecting interband transitions, approximate solutions of the TDSE can be constructed as   eF i eF     t ∫ ′ ′ ψ = − − − z t b z k t E k t dt ( , ) ; exp       n n n 0 0  ℏ  ℏ  ℏ    0 it E = ∑ WS − ψ ψ WS n z t z e ( , ) ( ) ℏ Fourier analysis: n nl l

Wannier-Stark states Wannier-Stark ladder: a π a / ∫ WS = + E dk E k leaF ( ) π n 2 n − π a / Wannier-Stark states: a π a / − ∫ WS ilak ψ = z dk b z k e ( ) ( ; ) π n 2 nl − π a / � eigenstates of the Wannier-Stark Hamiltonian in the single-band approximation � localised on site / ; localisation length: Δ 1 /$.|�|& � form a basis � Zener tunnelling adds an imaginary part to WS energies