Electron-phonon interaction. Can dispersionless phonons provide relaxation?

Electron-phonon interaction. Can dispersionless phonons provide relaxation? P. Gartner, J. Seebeck, F. Jahnke Institute for Theoretical Physics University of Bremen Kiel, 21

Introduction Self-assembled quantum dots S. Anders et al., Phys. Rev. B 66, 12539 (22) Kiel, 21 2

Introduction Self-assembled quantum dots Energy spectrum electrons bound (QD) extended (WL) holes S. Anders et al., Phys. Rev. B 66, 12539 (22) Kiel, 21 2

Introduction Discrete spectrum important for optoelectronics Energy spectrum electrons controlled by quantum dot geometry Carrier kinetics bound (QD) extended (WL) holes Kiel, 21 2

Introduction Discrete spectrum important for optoelectronics controlled by quantum dot geometry Capture and relaxation electrons p shell photon s shell Carrier kinetics holes Kiel, 21 2

Introduction Low carrier densities carrier-carrier scattering negligible carrier-lo-phonon interaction dominant Perturbation theory: capture and relaxation prevented phonon bottleneck Kiel, 21 3

Introduction Low carrier densities carrier-carrier scattering negligible carrier-lo-phonon interaction dominant Perturbation theory: capture and relaxation prevented WL QD ω LO ω LO phonon bottleneck U.Bockelmann, G. Bastard, PRB 42, 8947 (199) H. Benisty et al., PRB 44, 1945 (1991) Kiel, 21 3

Introduction Experiment: fast (ps) capture and relaxation Kiel, 21 4

Introduction Experiment: fast (ps) capture and relaxation Example: S. Trumm et al. APL 87, 153113 (25) rapid relaxation inside wetting layer rapid transfer to bound states characteristic times (ps) independent of excitation power Kiel, 21 4

Outline Boltzmann kinetics relaxation properties, in general and with dispersionless LO-phonons Carrier-phonon system in equilibrium carrier renormalization = the polaron Quantum kinetics 2-time quantum kinetics 1-time approximation Results (QWs and QDs) Numerical difficulties Kiel, 21 5

Boltzmann kinetics Carrier-phonon Hamiltonian: H = X i ǫ i a i a i + X q ω qb qb q + X i,j,q M i,j (q)a i a j(b q + b q ) ǫ j ω q ω q = ω LO, M i,j (q) = g q ϕ i e iqr ϕ j gq 2 α 1 Fröhlich coupling q 2 ǫ i Kiel, 21 6

Boltzmann kinetics Carrier-phonon Hamiltonian: H = X i ǫ i a i a i + X q ω qb qb q + X i,j,q M i,j (q)a i a j(b q + b q ) ǫ j ω q ω q = ω LO, M i,j (q) = g q ϕ i e iqr ϕ j gq 2 α 1 Fröhlich coupling q 2 ǫ i Time evolution of population f i = a i a i : f i t = X j {W i,j (1 f i )f j W j,i (1 f j )f i } Kiel, 21 6

Boltzmann kinetics Carrier-phonon Hamiltonian: H = X i ǫ i a i a i + X q ω qb qb q + X i,j,q M i,j (q)a i a j(b q + b q ) ǫ j ω q ω q = ω LO, M i,j (q) = g q ϕ i e iqr ϕ j gq 2 α 1 Fröhlich coupling q 2 ǫ i Time evolution of population f i = a i a i : f i t = X j {W i,j (1 f i )f j W j,i (1 f j )f i } W i,j = Golden Rule transition rate j i : W i,j = 2π X M i,j (q) 2 {N qδ(ǫ i ǫ j ω q) + (N q + 1)δ(ǫ i ǫ j + ω q)} q Kiel, 21 6

Boltzmann kinetics Relaxation properties (analytic): conserves positivity: f i () = f i (t) stationary solution: f i (t) = 1/[e β(ǫ i µ) + 1] = F(ǫ i ) for any f i () f i (t) F(ǫ i ) as t provided all states can be connected using paths with nonzero transition rates. Kiel, 21 7

Boltzmann kinetics Relaxation properties (analytic): conserves positivity: f i () = f i (t) stationary solution: LO-phonons + continuous spectrum (QW).5.4 f i (t) = 1/[e β(ǫ i µ) + 1] = F(ǫ i ) f e.3.2 for any f i ().1 f i (t) F(ǫ i ) as t provided all states can be connected using paths with nonzero transition rates. 4 t(fs) 2-2 5 2 15 1 E(meV) 25 phonon cascade Kiel, 21 7

Boltzmann kinetics Relaxation properties (analytic): conserves positivity: f i () = f i (t) stationary solution: LO-phonons in quantum dots WL ω LO f i (t) = 1/[e β(ǫ i µ) + 1] = F(ǫ i ) for any f i () f i (t) F(ǫ i ) as t provided all states can be connected using paths with nonzero transition rates. QD ω LO phonon bottleneck Kiel, 21 7

Boltzmann kinetics Relaxation properties (analytic): conserves positivity: f i () = f i (t) stationary solution: LO-phonons in quantum dots WL ω LO f i (t) = 1/[e β(ǫ i µ) + 1] = F(ǫ i ) for any f i () QD ω LO f i (t) F(ǫ i ) as t provided all states can be connected using paths with nonzero transition rates. Proposed solutions: LO+LA: T.Inoshita, H.Sakaki, PRB.46, 726 (1992) LO+LO: A. Knorr group (Berlin) in press phonon bottleneck Kiel, 21 7

Boltzmann vs. quantum kinetics Experiment: fast capture and relaxation (ps) Boltzmann kinetics: weak coupling slow evolution long time no memory (Markovian) Kiel, 21 8

Boltzmann vs. quantum kinetics Experiment: fast capture and relaxation (ps) Boltzmann kinetics: weak coupling slow evolution long time no memory (Markovian) Quantum kinetics: strong coupling fast evolution both early and later times memory effects included Kiel, 21 8

Boltzmann vs. quantum kinetics Experiment: fast capture and relaxation (ps) Boltzmann kinetics: weak coupling slow evolution long time no memory (Markovian) Quantum kinetics: strong coupling fast evolution both early and later times memory effects included Relaxation properties (numeric) is a steady state reached? steady state = thermal equilibrium? Kiel, 21 8

Polarons + + + + + + + + + + + + electron polarizes crystal Kiel, 21 9

Polarons + + + + + + + + + + free carrier retarded GF G R α(t) = θ(t) i e i ǫαt G R α(ω) = 1 ω ǫ α + iδ + + electron polarizes crystal spectral function bg α(ω) = 2Im G R α(ω) = 2π δ( ω ǫ α) Kiel, 21 9

Polarons + + + + + + + + + + electron polarizes crystal + + free carrier retarded GF G R α(t) = θ(t) i e i ǫαt G R α(ω) = spectral function 1 ω ǫ α + iδ bg α(ω) = 2Im G R α(ω) = 2π δ( ω ǫ α) polaron retarded GF G R α(ω) = 1 ω ǫ α + Σ R α(ω) Σ R α(ω) self-energy for carrier-phonon interaction bg α(ω) =? Kiel, 21 9

Quantum well polaron Self-consistent Random Phase Approximation j i ff τ ǫ k Gk R (τ) = = Z τ dτ Σ R k (τ )G R k (τ τ ) 1.8.6.4.2 G R Σ = 1 2 E k 2 (mev) 3 8 12 16 2 4 τ(fs) arbitrary order in α multi-phonon processes GaAs at T= 3K α =.6, ω LO = 36 mev Kiel, 21 1

Quantum well polaron 1 ^G Spectral function 1-2 1-4 1-6 1-8 ^ G k= (ω) InGaAs/GaAs T=3K -4-3 -2-1 1 2 3 Energy (h _ ω-e G ) / h _ ω LO Kiel, 21 11

Quantum well polaron 1 ^G Spectral function 1-2 1-4 1-6 1-8 ^ G k= (ω) InGaAs/GaAs T=3K polaron shift energy broadening phonon satellites -4-3 -2-1 1 2 3 Energy (h _ ω-e G ) / h _ ω LO J. Seebeck, T.R. Nielsen, P. Gartner, F. Jahnke, Phys. Rev. B 71, 125327 (25) Kiel, 21 11

Quantum well polaron 1.E 1 Spectral function 1 1.E 2 1.E 4 1.E 6 1.E 8 1.E 1 1.E 12 8 1.E 2 1.E 3 1.E 4 1.E 5 1.E 6 1.E 7 1.E 8 6 1.E 9 4 1.E 1 1 k 2 4 2 2 ω E G / ω LO 4 1.E 11 ^G Spectral function 1-2 1-4 1-6 1-8 ^ G k= (ω) InGaAs/GaAs T=3K -4-3 -2-1 1 2 3 Energy (h _ ω-e G) / h _ ω LO Kiel, 21 12

Quantum well polaron 8 1.E 1 7 1.E 2 1.E 3 6 1.E 4 5 1.E 5 k 4 1.E 6 3 1.E 7 1.E 8 2 1.E 9 1 1 3 2 1 1 2 3 hw/hwlo 1.E 1 1.E 11 ^G Spectral function 1-2 1-4 1-6 1-8 ^ G k= (ω) InGaAs/GaAs T=3K -4-3 -2-1 1 2 3 Energy (h _ ω-e G) / h _ ω LO Kiel, 21 12

Quantum dot polaron LO-phonons + quantum dot states only Self-consistent Random Phase Approximation T.Inoshita, H.Sakaki, PRB 56, 261 (1998) K.Král, Z.Khás, PRB 57, 4355 (1998) finite number of phonons = diagonalization O.Verzelen, R.Fereira, G.Bastard, PRB 62, 489 (2) T.Stauber, R.Zimmermann, H. Castella, PRB 62, 7336 (2) BUT wetting layer states important for both kinetics and spectrum Kiel, 21 13

Quantum dot polaron LO-phonons + quantum dot + wetting layer states Kiel, 21 14

Quantum dot polaron LO-phonons + quantum dot + wetting layer states 1-2 QD p-shell 1-4 Spectral function 1-6 1-8 1-2 QD s-shell 1-4 1-6 1-8 InGaAs/GaAs T = 3K -5-4 -3-2 -1 1 2 Energy (h _ ω - E G ) / h _ ω LO E = 1.1 ω LO Kiel, 21 14

Quantum dot polaron LO-phonons + quantum dot + wetting layer states p 1-2 QD p-shell ω LO 1-4 s Spectral function 1-6 1-8 1-2 QD s-shell 1-4 Hybridization 1-6 1-8 InGaAs/GaAs T = 3K -5-4 -3-2 -1 1 2 Energy (h _ ω - E G ) / h _ ω LO E = 1.1 ω LO Kiel, 21 14

Quantum kinetics Quantity of interest: ρ α(t) = @ f e α(t) ψ α(t) ψ α(t) 1 f h α(t) 1 A = i G < α (t, t) electrons p shell 2x2 matrix in the band index, α = k, s, p photon s shell Methods: Equation of Motion hierarchy problem Nonequilibrium Green s Functions polaronic effects 2-time quantities holes Kiel, 21 15

Quantum kinetics Quantity of interest: 1 fα(t) e ψ α(t) ρ α(t) = @ A = i G < ψα(t) 1 fα(t) h α (t, t) 2x2 matrix in the band index, α = k, s, p Methods: Equation of Motion hierarchy problem Nonequilibrium Green s Functions polaronic effects 2-time quantities Nonequilibrium Green s Functions Closed Equations for G(t 1, t 2 ) = 1 D h ie T ψ(t 1 )ψ (t 2 ) i Procedure: solve for G α(t 1, t 2 ) recover physical data from the time-diagonal t 1 = t 2 Kiel, 21 15

Quantum kinetics 2-time Kadanoff-Baym equations: j i ff H (1) G α(t 1, t 2 ) = t 1 t 2 = {Σ G} α (t 1, t 2 ) t 1 Kiel, 21 16

Quantum kinetics 2-time Kadanoff-Baym equations: j i ff H (1) G α(t 1, t 2 ) = t 1 t 2 = {Σ G} α (t 1, t 2 ) j i ff + H (2) G α(t 1, t 2 ) = t 2 t 1 = {G Σ} α (t 1, t 2 ) Kiel, 21 16

Quantum kinetics 2-time Reparametrize: (t 1, t 2 ) = (t, t τ) t 2 t t - main time τ - relative time j i ff t H (1) + H (2) G α(t, t τ) = t 1 = {Σ G} α (t, t τ) {G Σ} α (t, t τ) τ Kiel, 21 16

Quantum kinetics 2-time Polaronic Green s Function: G R α(t, t τ) = G R α(τ) t 2 t initial condition natural memory cutoff equil. (t, t τ) t 1 τ Kiel, 21 16

Quantum kinetics 2-time i t GR,< k (t, t τ) = Σ δ k(t)g R,< k (t, t τ) G R,< k (t, t τ)σ δ k(t τ) + + i t GR,< k (t, t τ) ; τ coll Instantaneous self-energy: Σ δ k(t) = ǫ c k d vc E(t) d cv E(t) ǫ v k «The optical excitation connects the two bands. Kiel, 21 17

Quantum kinetics 2-time i t GR,< k (t, t τ) = Σ δ k(t)g R,< k (t, t τ) G R,< k (t, t τ)σ δ k(t τ) + + i t GR,< k (t, t τ) ; τ coll RPA self-energies: Σ R k (t, t ) = i X q g 2 q h i D > (t t )Gk q(t, R t ) + D R (t t )G k q(t, < t ) Σ < k (t, t ) = i X q g 2 q D < (t t )G < k q(t, t ) i D > (τ) = (N LO + 1) e i ω LO τ + N LO e i ω LO τ = i D < ( τ) Kiel, 21 17

Quantum kinetics 2-time Collision terms for G R : i t GR k (t, t τ) Z t = coll t τ h i dt Σ R k (t, t )Gk R (t, t τ) Gk R (t, t )Σ R k (t, t τ) t 2 t t Σ t τ G t 1 Kiel, 21 18

Quantum kinetics 2-time Collision terms for G R : i t GR k (t, t τ) Z t = coll t τ h i dt Σ R k (t, t )Gk R (t, t τ) Gk R (t, t )Σ R k (t, t τ) t 2 t t Σ t τ G t 1 Kiel, 21 18

Quantum kinetics 2-time Collision terms for G < : t 2 G t τ t Z t i t G< k (t, t τ) coll = dt h Σ R k (t, t )G < k (t, t τ) +Σ < G A G R Σ < G < (t, t )Σ Ai t memory depth G t 1 Σ Kiel, 21 19

Quantum kinetics 2-time Collision terms for G < : t 2 G t τ t Z t i t G< k (t, t τ) coll = dt h Σ R k (t, t )G < k (t, t τ) +Σ < G A G R Σ < G < (t, t )Σ Ai t memory depth G Σ G < (t 1, t 2 ) = ˆG < (t 2, t 1 ) h i G A (t 2, t 1 ) = G R (t 1, t 2 ) t 1 Kiel, 21 19

Quantum kinetics 2-time Initial conditions - the polaron Before the pulse = equilibrium «G R g R (t, t τ) = c (τ) gv R (τ) G < (t, t τ) = g R v (τ) «m e m o r y d e p t h t g < c (τ) = g > v (τ) = g R v (τ) = g > v (τ) g < v (τ) τ Kiel, 21 2

Quantum kinetics 2-time No t-evolution before excitation: i t GR,< k (t, t τ) = Σ δ k(t)g R,< k (t, t τ) G R,< k (t, t τ)σ δ k(t τ) + + i t GR,< k (t, t τ) ; τ coll Σ δ ǫ c k(t) = k ǫ v k «Σ δ k G R,< k (t, t τ) G R,< k (t, t τ)σ δ k = Kiel, 21 21

Quantum kinetics 2-time No t-evolution before excitation: t 2 G t τ t Z t i t G< k (t, t τ) coll = dt h Σ R k (t, t )G < k (t, t τ) +Σ < G A G R Σ < G < (t, t )Σ Ai t memory depth G t 1 Σ Kiel, 21 22

Quantum kinetics 2-time No t-evolution before excitation: t 2 G t τ t Z t i t G< k (t, t τ) coll = dt h Σ R k (t, t )G < k (t, t τ) +Σ < G A G R Σ < G < (t, t )Σ Ai t memory depth G Σ G R (t, t τ) = G < (t, t τ) = g R c (τ) g R v (τ) g R v (τ) ««t 1 Kiel, 21 22

Quantum kinetics 1-time Generalized Kadanoff-Baym ansatz (GKBA) G α (t, t ) i G R α(t t ) G α (t, t ) t > t Z d t fα(t) = 2Re dt X dt M α,β (q) 2 Gβ(t R t ) ˆG α(t R t ) β q n f α(t ) ˆ1 f β (t ) D > q (t t ) +ˆ1 f α(t ) f β (t )D < q (t t ) o D q (t) = N LO e iω LO t + (1 + N LO ) e ±iω LO t renormalized quasiparticles (polarons) beyond Markov approximation (memory effects) Kiel, 21 23

Quantum kinetics 1-time Generalized Kadanoff-Baym ansatz (GKBA) G α (t, t ) i G R α(t t ) G α (t, t ) t > t Z d t fα(t) = 2Re dt X dt M α,β (q) 2 e iǫ β(t t ) ˆe iǫα(t t ) β q n f ˆ1 α(t) f β (t) D q > (t t ) +ˆ1 f α(t) f β (t)d < q (t t ) o D q (t) = N LO e iω LO t + (1 + N LO ) e ±iω LO t bare particles Markov approximation (no memory effects) Kiel, 21 23

Quantum well results: polarization CdTe α =.31 T = 3K p p.16.12.8.4.16.12.8.4 5 1 15 2 E(meV) 25 8 4 t(fs) 4 5 1 15 E(meV) 2 25 8 4 4 t(fs) 2-time 1-time Strong dephasing Kiel, 21 24

Quantum well results: electron population CdTe α =.31 T = 3K f e f e.4.3.2.1.12.8.4 5 1 15 E(meV) 2 25 8 12 4 t(fs) 4 5 1 15 E(meV) 2 25 8 12 4 t(fs) 4 2-time 1-time Steady state reached Not the same! Kiel, 21 25

Kubo-Martin-Schwinger condition Exact relation in thermal equilibrium KMS condition Z f α = d( ω) 2π F(ω) b G α(ω) Kiel, 21 26

Kubo-Martin-Schwinger condition Exact relation in thermal equilibrium KMS condition Z f α = d( ω) 2π F(ω) b G α(ω) Non-interacting system b G α(ω) = 2π δ( ω ǫ α) f α = F(ǫ α) Kiel, 21 26

Kubo-Martin-Schwinger condition Exact relation in thermal equilibrium KMS condition Z f α = d( ω) 2π F(ω) b G α(ω) Non-interacting system G b α(ω) = 2π δ( ω ǫ α) f α = F(ǫ α) Interacting system Generalization of Fermi-Dirac distribution Kiel, 21 26

Kubo-Martin-Schwinger test in quantum wells CdTe, α =.31 GaAs, α =.6 f e.4.3.2.1 2 time 1 time KMS.6.4 f e.2.1.1 1e 6 1e 8 1 2 2 time 1 time KMS 3 4 5 1 15 E ~k 2 (mev) 2 25 5 1 15 2 25 E ~k 2 (mev) 3 35 4 P. Gartner, J. Seebeck, F. Jahnke, Phys. Rev. B 73, 11537 (26) Kiel, 21 27

Kubo-Martin-Schwinger test in quantum dots CdTe α =.31 E = 2.4 ω LO p-level excitation electrons 2-time calculation photon p shell s shell f e.2.1 QD s state QD p state WL k= relaxation s shell p shell holes 4 8 12 t (fs) Kiel, 21 28

Kubo-Martin-Schwinger test in quantum dots CdTe α =.31 E = 2.4 ω LO p-level excitation 1-time calculation 2-time calculation f e.2.1 QD s state QD p state WL k= f e.2.1 QD s state QD p state WL k= 2 4 6 t (fs) 4 8 12 t (fs) Kiel, 21 28

Numerics: instabilities.5.4 fefh+ct8 every :::79::79 u 2:3 fefh+ct8 every :::89::89 u 2:3.3.2.1.1 5 1 15 2 25 3 detuning = 12 mev t = 55fs t = 65fs abrupt cut of the collision integrals at memory depth = 5 fs Kiel, 21 29

Numerics: instabilities Collision integral t t 2 G t τ t memory depth G Σ Kiel, 21 3

Numerics: instabilities Collision integral t t 2 G t τ integral abruptly cut at memory depth t G Σ memory depth Kiel, 21 3

Numerics: instabilities Collision integral t t 2 G t τ integral abruptly cut at memory depth t G Σ beyond the memory depth extend with equilibrium Σ, G memory depth Kiel, 21 3

Numerics: instabilities Collision integral t t 2 G t τ integral abruptly cut at memory depth t G Σ beyond the memory depth extend with equilibrium Σ, G memory depth smooth interpolation to equilibrium Σ, G Kiel, 21 3

Numerics: instabilities Collision integrals extended with equilibrium GFs.35.3 fefh+c14 every :::79::79 u 2:3 fefh+c14 every :::89::89 u 2:3.8.6 fefh+c14 every :::79::79 u 2:3 fefh+c14 every :::99::99 u 2:3.25.4.2.2.15.1.2.5.4 5 1 15 2 25 3.6 5 1 15 2 25 3 t = 55fs t = 65fs t = 55fs t = 75fs Kiel, 21 31

Numerics: instabilities Collision integrals extended with equilibrium GFs smooth change using an interpolating (Fermi-like) function.4.35 fefh+c25 every :::79::79 u 2:3 fefh+c25 every :::99::99 u 2:3.3.25.2.15.1.5 5 1 15 2 25 3 t = 55fs t = 75fs Kiel, 21 32

Low temperatures The polaron G R α(t) WL p-shell s-shell.8 1.6.4.2-3 -2-1 1 2 energy ǫ e α in hω LO 1.5 1.5 time in ps Polaronic GF: GaAs at T=1K Kiel, 21 33

Low temperatures The polaron spectral function Ĝα(ω) 1 1.1.1.1.1 WL, k= s-shell p-shell 1e-5-6 -4-2 2 4 energy in hω LO Spectral function: GaAs at T=1K Kiel, 21 33

Low temperatures Quantum kinetics 1-time Z d t fα(t) = 2Re dt X dt M α,β (q) 2 Gβ(t R t ) ˆG α(t R t ) β q n f α(t ) ˆ1 f β (t ) D q > (t t ) +ˆ1 f α(t ) f β (t )D < q (t t ) o Kiel, 21 34

Low temperatures Quantum kinetics 1-time Z d t fα(t) = 2Re dt X dt M α,β (q) 2 Gβ(t R t ) ˆG α(t R t ) β q n f α(t ) ˆ1 f β (t ) D q > (t t ) +ˆ1 f α(t ) f β (t )D < q (t t ) o 2 Gα(t) R = 1 i Θ(t) 4 X j 3 e i (ǫα α,j)t + Gα(t) R 5 G R α(t) for t Kiel, 21 34

Low temperatures WL electron population.5.4.3.2.1 population 1 2 energy in hω LO 2-time 3 4 1.5 time in ps.5.4.3.2.1 population 1 2 energy in hω LO 3 1-time 4 1.5 time in ps Kiel, 21 35

Low temperatures Electron population (long time) population.6.4.2.5 1 1.5 2 2.5 3 energy in hω LO 5 1 15 2 25 time in ps population.1.9.8.7.6.5.4.3.2.1 s-shell p-shell -5 5 1 15 2 25 time in ps Kiel, 21 36

Low temperatures KMS test of the steady state population.14.12.1.8.6.4.2 1.1.1.1.1 1e-5 KMS kinetics 1e-6-3 -2-1 1 2 3 4-3 -2-1 1 2 3 4 energy in hω LO no thermalization (asymptotic bottleneck) Kiel, 21 37

Summary and conclusions Carrier-LO-phonon interaction role - twofold: equilibrium: polaron kinetics: polaron-phonon scattering Relaxation properties (room temp.): Steady state reached both in 2-time and in 1-time kinetics Thermalization: 2-time kinetics: Yes! 1-time kinetics, only low α Problems with low temperatures: 2-time - only the early kinetics 1-time - no thermalization (asymptotic bottleneck?) Kiel, 21 38