Simulating time dependent thermoelectric transport with the t-kwant software

Simulating time dependent thermoelectric transport with the t-kwant software Adel Kara Slimane, Phillipp Reck, and Geneviève Fleury Service de Physique de l Etat Condensé (SPEC) CEA Saclay CNRS UMR 3680 Paris Saclay University

OUTLINE Motivations Theoretical predictions of improved thermoelectric performance in the dynamic regime, far from equilibrium Development of high-frequency nanoelectronics experiments Simulating quantum transport with the Kwant and t-kwant softwares Kwant: a flexible, efficient, user-friendly tool for DC electronic quantum transport Time dependent extension t-kwant Some applications Extension of t-kwant to dynamic thermoelectric transport Implementation of energy and heat currents (only electrons, no phonons!) Validity check of the code on the Resonant Level Model Perspectives

Motivation: promising (yet quasi unexplored) dynamic regime V(t) V(t) Abrupt variation of the energy level in the dot T L μ L Γ Γ T R μ R ΔV t (time) Enhanced thermopower S Enhanced thermoelectric efficiency η ΔV ΔV Crépieux et al., PRB 83, 153417 (2011) Zhou et al., Scientific Report 5, 14870 (2015)

Theoretical approaches to time-dependent thermoelectrics Extreme computational cost for integrating the integro-differential equations of the Non-Equilibrium Green s Function formalism Analytical Keldysh resolution of the Resonant Level Model: Crépieux et al., PRB (2011) Zhou et al., Scientific Report (2015) Daré et al., PRB (2016) Auxiliary mode approach Lehmann et al., Chemical Physics (2018) Response to the off-on switch of a voltage/temperature bias Eich, Di Ventra & Vignale, PRB (2016) Covito et al., J. Chem. Th. Comput. (2018) Floquet scattering theory for AC transport Battista et al., PRB (2014) Moskalets, PRL (2014) Need for a generic tool that can deal with realistic quantum devices

Motivation: emergence of experiments in time-resolved quantum nanoelectronics Possible on-demand injection of single electrons: Mesoscopic capacitor Glattli et al., Science 2006, @ LPA, ENS-Paris Levitons (with lorentzian voltage pulses) Glattli et al., Nature 2013, @ SPEC, CEA Saclay Single electrons driven with a surface acoustic wave Bäuerle, Meunier et al., Nature 2011, @ Néel, Grenoble Ritchie et al., Nature 2011, @ Cambridge Manipulation + detection of single electrons: towards electron quantum optics Experiments probing dynamic thermoelectric transport in the non-adiabatic regime are within reach

KWANT package for quantum transport calculations at equilibrium Kwant developers: C. Groth, X. Waintal (CEA Grenoble, INAC) A. Akhmerov, J. Weston, M. Wimmer (TU Delft) Arbitrary non-interacting tight-binding model: H = H ij c i i,j i = site + internal degrees of freedom (spin, atomic orbital, electron-hole) c j Abritrary lattice (square, honeycomb, ) Abritrary shape with disorder, spin-orbit, magnetic field, with superconductivity Solver based on the wavefunction formulation of the scattering problem (as opposed to the Green s function formulation) (Publicly available) Python package with a user-friendly interface https://kwant-project.org/ - Groth et al., New J. Phys. 16, 063065 (2014)

What can we calculate with Kwant? Transmission T ji (E) from lead i to lead j Quantum Hall conductance plateaus with disorder Local density of states (and local currents) Armchair graphene nanoribbon in the QH regime with superconductivity Bandstructure of a translationally invariant system Plot of the wavefunction in a closed system A growing community (several hundreds of users) More than 100 published papers using Kwant Used for designing experiments in the lab

t-kwant: time-dependent extension of KWANT H(t) = H ij (t)c i i,j c j Time-dependent magnetic field B(t) Time-dependent voltage V(t) in the lead absorbed in H 10 with a gauge transformation H 10 e i t t0 duv(u) H 10 Time-dependent on-site potential Fig. from X. Waintal t-kwant developers: B. Gaury, C. Groth, B. Rossignol, X. Waintal (CEA Grenoble, INAC) J. Weston (TU Delft) Gaury et al., Physics Reports 534, 1 (2014) Weston et al., PRB 93, 134506 (2016) & J. Comput. Electron. 15, 1148 (2016)

Main steps of the t-kwant algorithm Switch-on time-dependent perturbations at t > 0 1 - Calculate the stationnary scattering states with KWANT H t 0 ψ st αe st = Eψ αe 2 - Calculate the time-dependent scattering states iħ t ψ st αe t = H(t)ψ αe t with ψ αe t = 0 = ψ αe time 3 Calculate the observables by integrating over. the energy E and summing over all modes α e.g. the particle current between sites i and j : I ij t = α de 2π f α E 2Im [ψ αe (t)] i H ij (t)[ψ αe t ] j

t-kwant versus other methods for time-dependent transport t-kwant = wave function approach Equivalent to the Non Equilibrium Green s Functions approach (Wingreen, Jauho & Meir, PRB, 1993) G R t, t = iθ(t t de ) 2π ψ αe t ψ αe t α G < t, t de = i 2π f α(e)ψ αe t ψ αe t α but much more efficient from a computational point of view: In 3D, CPU tl 3 (t-kwant) versus CPU t 2 L 7 (NEGF) Smaller memory footprints More stable In practice, systems with Parallel algorithm > 10 5 sites are solved Equivalent to the Floquet scattering theory for time-periodic perturbations Equivalent to the Partition-Free approach (Cini, PRB, 1980 Kurth et al., PRB, 2005)

Some applications of t-kwant in time-resolved quantum nanoelectronics Dynamical control of interference patterns (Gaury et al., Nature Comm., 2014 & 2015) Dynamical control of electron trajectories (Gaury et al., PRB, 2014) Practical scheme to calculate quantum noise in t-kwant (Gaury et al., Physica E, 2016)

Calculation of energy currents in t-kwant Particle current in lead l: H l (t) H ls (t) + H sl (t) H s (t) I l N t = dn l H dt with N l H (t) = iεl c i (t)c i (t) I l N t = iεl jεs α de 2π f α E 2Im [ψ αe (t)] i H ij (t) [ψ αe t ] j Energy current in lead l : I l E t = dh l H dt = H l t i ħ H l H t, H H (t) I l E t = qεs i,jεl α de 2π f α E 2Im [ψ αe (t)] q H qi (t)h ij (t)[ψ αe t ] j

Calculation of heat currents in t-kwant «Ambiguous» definition of time-resolved heat currents in out-of-equilibrium systems In the static case I l h = I l E μ l I l N With time dependent central system H s t Ludovico, Moskalets et al., PRB 94, 035436 (2016) H l (t) H ls (t) + H sl (t) H s (t) I l h t = I l E t μ l I l N t + 0.5 I l Ec t with I l Ec t = dh ls+h sl dt To recover the 2 nd law S irr (t) 0 With time dependent central system H s t + time-dependent coupling H ls (t) + H sl (t) Haughian, Esposito et al., PRB 97, 085435 (2018) I l h t = I l E t μ l I l N t + 0.5 I l Ec t + I l Ec (t) Derived for the Resonant Level Model in the wide band limit (not yet in t-kwant)

L I h Γ2 2πħ 1 Validity check on the Resonant Level Model t-kwant discretized model γ γ γ c γ c γ γ V(t) V t = V 0 + ΔVθ(t) Recovering the Wide Band Limit with t-kwant Covito et al., J. Chem. Th. Comput. (2018) γ c = λ γ c γ = λ γ to keep fixed Γ = 4 γ c 2 γ and take the limit λ t ħ Γ 1 Crépieux et al., PRB 83, 153417 (2011)

What s next? Towards the simulation of realistic nanodevices Peltier cooling with Levitons in a Quantum point contact Thermoelectric transport in a dynamic Fabry-Perot interferometer Optimizing band edge thermoelectric transport in a dynamically gated (doped) semiconductor nanowire The long term Kwant project (X. Waintal s team, INAC, CEA Grenoble) + environment Coupling to the electromag. environment, to magnons, Lindblad solver to simulate decoherence & quantum measurements + thermoelectricity Time-dependent extension t-kwant Kwant + e-e interactions Self consistent Poisson solver (electrostatics) Diagrammatic Quantum Monte Carlo (ee-kwant)

Conclusion t-kwant extension to time-dependent thermoelectric transport Arbitrary (non-interacting) tight-binding models Arbitrary perturbations in time Fast: CPU time scales as t max N sites User-friendly interface Proper definition of the time resolved heat currents? looking for DC signatures of dynamic thermoelectric effects Validity check of the code on the Resonant Level Model Simulating realistic nanodevices for stimulating experiments Numerical tool to be shared with the community. We welcome new collaborations!