Transient dynamics without time propagation

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1 Transient dynamics without time propagation Department of Physics University of Jyväskylä Aug 30th, 2012

2 1 Introduction

3 We study quantum transport of non-interacting electrons. Why? Ideal conditions (low temperature, nanoscale): We apply wide-band limit approximation (WBLA). Why? Typically, e.g. gold and benzene, energy continua is much wider than the discrete levels Large systems with Landauer Büttiker formalism

4 We study quantum transport of non-interacting electrons. Why? Ideal conditions (low temperature, nanoscale): Mean free path>system size We apply wide-band limit approximation (WBLA). Why? Typically, e.g. gold and benzene, energy continua is much wider than the discrete levels Large systems with Landauer Büttiker formalism

5 We study quantum transport of non-interacting electrons. Why? Ideal conditions (low temperature, nanoscale): Mean free path>system size Not too many scatterings We apply wide-band limit approximation (WBLA). Why? Typically, e.g. gold and benzene, energy continua is much wider than the discrete levels Large systems with Landauer Büttiker formalism

6 We study quantum transport of non-interacting electrons. Why? Ideal conditions (low temperature, nanoscale): Mean free path>system size Not too many scatterings Overall structure of the studied system does not change too much due to interactions We apply wide-band limit approximation (WBLA). Why? Typically, e.g. gold and benzene, energy continua is much wider than the discrete levels Large systems with Landauer Büttiker formalism

7 We study quantum transport of non-interacting electrons. Why? Ideal conditions (low temperature, nanoscale): Mean free path>system size Not too many scatterings Overall structure of the studied system does not change too much due to interactions We apply wide-band limit approximation (WBLA). Why? Typically, e.g. gold and benzene, energy continua is much wider than the discrete levels Large systems with Landauer Büttiker formalism

8 We study quantum transport of non-interacting electrons. Why? Ideal conditions (low temperature, nanoscale): Mean free path>system size Not too many scatterings Overall structure of the studied system does not change too much due to interactions We apply wide-band limit approximation (WBLA). Why? Typically, e.g. gold and benzene, energy continua is much wider than the discrete levels Large systems with Landauer Büttiker formalism

9 Investigate the following quantum transport setup Equilibrium and out-of-equilibrium Dynamics Metallic leads (α) are connected to a molecular device (C)

10 Investigate the following quantum transport setup Equilibrium and out-of-equilibrium Dynamics Metallic leads (α) are connected to a molecular device (C) No electron electron interactions either in the leads or in the central region

11 Investigate the following quantum transport setup Equilibrium and out-of-equilibrium Dynamics Metallic leads (α) are connected to a molecular device (C) No electron electron interactions either in the leads or in the central region Ballistic transport

12 Investigate the following quantum transport setup Equilibrium and out-of-equilibrium Dynamics Metallic leads (α) are connected to a molecular device (C) No electron electron interactions either in the leads or in the central region Ballistic transport Energy eigenvalues of the central region are well inside the energy spectrum of the leads

13 Investigate the following quantum transport setup Equilibrium and out-of-equilibrium Dynamics Metallic leads (α) are connected to a molecular device (C) No electron electron interactions either in the leads or in the central region Ballistic transport Energy eigenvalues of the central region are well inside the energy spectrum of the leads Wide-band limit approximation (WBLA)

14 Investigate the following quantum transport setup Equilibrium and out-of-equilibrium Dynamics Metallic leads (α) are connected to a molecular device (C) No electron electron interactions either in the leads or in the central region Ballistic transport Energy eigenvalues of the central region are well inside the energy spectrum of the leads Wide-band limit approximation (WBLA) The discrete energy spectrum of the central region is connected to the continua of the leads

15 Investigate the following quantum transport setup Equilibrium and out-of-equilibrium Dynamics The Hamiltonian is Ĥ = ǫ kαˆn kα,σ + + kα,σ T mnˆd m,σˆd n,σ mn,σ mkα,σ [ ] T mkαˆd m,σˆd kα,σ +T kαmˆd kα,σˆd m,σ Leads Molecule Coupling

16 Investigate the following quantum transport setup Equilibrium and out-of-equilibrium Dynamics Initially the system is in thermal equilibrium at inverse temperature β and in chemical potential µ ˆρ= 1 Z e β(ĥ µˆn)

17 Investigate the following quantum transport setup Equilibrium and out-of-equilibrium Dynamics Initially the system is in thermal equilibrium at inverse temperature β and in chemical potential µ ˆρ= 1 Z e β(ĥ µˆn) The lead energy levels are suddenly shifted with some constant value (ǫ kα ǫ kα +V α )

18 Investigate the following quantum transport setup Equilibrium and out-of-equilibrium Dynamics Initially the system is in thermal equilibrium at inverse temperature β and in chemical potential µ ˆρ= 1 Z e β(ĥ µˆn) The lead energy levels are suddenly shifted with some constant value (ǫ kα ǫ kα +V α ) External bias voltage for theα:th lead

19 Investigate the following quantum transport setup Equilibrium and out-of-equilibrium Dynamics Initially the system is in thermal equilibrium at inverse temperature β and in chemical potential µ ˆρ= 1 Z e β(ĥ µˆn) The lead energy levels are suddenly shifted with some constant value (ǫ kα ǫ kα +V α ) External bias voltage for theα:th lead The system is driven out of equilibrium

20 Investigate the following quantum transport setup Equilibrium and out-of-equilibrium Dynamics Initially the system is in thermal equilibrium at inverse temperature β and in chemical potential µ ˆρ= 1 Z e β(ĥ µˆn) The lead energy levels are suddenly shifted with some constant value (ǫ kα ǫ kα +V α ) External bias voltage for theα:th lead The system is driven out of equilibrium Charge carriers start to flow through the central region

21 Investigate the following quantum transport setup Equilibrium and out-of-equilibrium Dynamics The dynamical equations of motion for the charge carriers out of equilibrium are the Kadanoff Baym equations (KBE)

22 Investigate the following quantum transport setup Equilibrium and out-of-equilibrium Dynamics The dynamical equations of motion for the charge carriers out of equilibrium are the Kadanoff Baym equations (KBE) Pack the physical quantities into one object: the one-particle Green s function G(z,z )= i T γ [ ˆψ H (z) ˆψ H (z )]

23 Investigate the following quantum transport setup Equilibrium and out-of-equilibrium Dynamics The Green s function satisfies the (integro-differential) KBE, [i d ] dz h G(z,z ) = δ(z,z )1+ d zσ(z, z)g( z,z ) γ d G(z,z ) i dz h = δ(z,z )1+ d zg(z, z)σ( z,z ), γ with Kubo Martin Schwinger (KMS) boundary conditions.

24 Investigate the following quantum transport setup Equilibrium and out-of-equilibrium Dynamics Project to the subspace of the central molecule Extract the lesser Keldysh component Equal-time limit (locality) [1 3] i d dt G< (t,t) [h,g < (t,t)]= [ G R Σ < +G < Σ A +G Σ ] (t,t) Self-energy Memory for in tunnelings WBLA Σ A α,mn (ω)= 1 Memory kt m,kα and T ω ǫ kα V α damping iη kα,n= iγ 2 α,mn Assumed to be a purely imaginary constant with respect to the frequencyω; Initial definition preparation ofγ

25 Investigate the following quantum transport setup Equilibrium and out-of-equilibrium Dynamics Project to the subspace of the central molecule Extract the lesser Keldysh component Equal-time limit (locality) [1 3] i d dt G< (t,t) [h,g < (t,t)]= [ G R Σ < +G < Σ A +G Σ ] (t,t) Self-energy in WBLA Σ A α,mn (ω)= kt m,kα 1 ω ǫ kα V α iη T kα,n= i 2 Γ α,mn Assumed to be a purely imaginary constant with respect to the frequencyω; definition ofγ blaa blaa

26 Procedure: Solving the equation for G < (t,t) Calculating the current Take-home message 1 Σ A Σ M Σ, Σ < ig < (t,t) = dω 2π f(ω µ) { Aα (ω+v α ) α + V α [ e i(ω+v α h+ i 2 Γ)t G R (ω)a α (ω+v α )+h.c. ] + V 2 α e i(h i 2 Γ)t G R (ω)a α (ω+v α )G A (ω)e i(h+ i 2 Γ)t} with the spectral function A(ω)= A α (ω)= = G R (ω)γ α G A (ω) α α

27 Procedure: Solving the equation for G < (t,t) Calculating the current Take-home message 1 Σ A Σ M Σ, Σ < 2 G M G, G R ig < (t,t) = dω 2π f(ω µ) { Aα (ω+v α ) α + V α [ e i(ω+v α h+ i 2 Γ)t G R (ω)a α (ω+v α )+h.c. ] + V 2 α e i(h i 2 Γ)t G R (ω)a α (ω+v α )G A (ω)e i(h+ i 2 Γ)t} with the spectral function A(ω)= A α (ω)= = G R (ω)γ α G A (ω) α α

28 Procedure: Solving the equation for G < (t,t) Calculating the current Take-home message 1 Σ A Σ M Σ, Σ < 2 G M G, G R 3 Calculate convolutions inside the square brackets ig < (t,t) = dω 2π f(ω µ) { Aα (ω+v α ) α + V α [ e i(ω+v α h+ i 2 Γ)t G R (ω)a α (ω+v α )+h.c. ] + V 2 α e i(h i 2 Γ)t G R (ω)a α (ω+v α )G A (ω)e i(h+ i 2 Γ)t} with the spectral function A(ω)= A α (ω)= = G R (ω)γ α G A (ω) α α

29 Procedure: Solving the equation for G < (t,t) Calculating the current Take-home message 1 Σ A Σ M Σ, Σ < 2 G M G, G R 3 Calculate convolutions inside the square brackets 4 Solve for G < dω ig < (t,t) = 2π f(ω µ) { Aα (ω+v α ) α + V α [ e i(ω+v α h+ i 2 Γ)t G R (ω)a α (ω+v α )+h.c. ] + V 2 α e i(h i 2 Γ)t G R (ω)a α (ω+v α )G A (ω)e i(h+ i 2 Γ)t} with the spectral function A(ω)= A α (ω)= = G R (ω)γ α G A (ω) α α

30 Procedure: Solving the equation for G < (t,t) Calculating the current Take-home message 1 Σ A Σ M Σ, Σ < 2 G M G, G R 3 Calculate convolutions inside the square brackets 4 Solve for G < dω ig < (t,t) = 2π f(ω µ) { Aα (ω+v α ) α + V α [ e i(ω+v α h+ i 2 Γ)t G R (ω)a α (ω+v α )+h.c. ] + V 2 α e i(h i 2 Γ)t G R (ω)a α (ω+v α )G A (ω)e i(h+ i 2 Γ)t} with the spectral function A(ω)= αa α (ω)= αg R (ω)γ α G A (ω)

31 Solving the equation for G < (t,t) Calculating the current Take-home message Time-dependent current is calculated from I α (t)=4qre { Tr [ Σ < α G A +Σ R α G < +Σ α G ] (t,t) } dω I α (t)= 2 2π f(ω µ) β Tr { Γ α G R (ω+v β )Γ β G A (ω+v β ) Γ α G R (ω+v α )Γ β G A (ω+v α ) +V β [ Γα e i(ω+v β h+ i 2 Γ)t G R (ω) ( iδ αβ G R (ω+v β )+A β (ω+v β ) ) + h +V 2 β Γ αe i(h i 2 Γ)t G R (ω)a β (ω+v β )G A (ω)e i(h+ i 2 Γ)t}

32 Solving the equation for G < (t,t) Calculating the current Take-home message Time-dependent current is calculated from I α (t)=4qre { Tr [ Σ < α G A +Σ R α G < +Σ α G ] (t,t) } dω I α (t)= 2 2π f(ω µ) β Tr { Γ α G R (ω+v β )Γ β G A (ω+v β ) Γ α G R (ω+v α )Γ β G A (ω+v α ) +V β [ Γα e i(ω+v β h+ i 2 Γ)t G R (ω) ( iδ αβ G R (ω+v β )+A β (ω+v β ) )] +V 2 β Γ αe i(h i 2 Γ)t G R (ω)a β (ω+v β )G A (ω)e i(h+ i 2 Γ)t}

33 Solving the equation for G < (t,t) Calculating the current Take-home message Non-interacting charge carriers with WBLA Closed and analytic formulae for time-dependent density and current Neither time-propagation nor self-consistency algorithms needed

34 Numerical integrals: C++ with armadillo[4] and gsl[5] 1 Construct the single-particle Hamiltonian matrix h for the central molecule the coupling matrices T between the leads and the molecule 2 Calculate Green s functions, spectral functions, etc. 3 Use the derived formulae 4 Time steps trivial to parallelize Problem: Integrals are difficult

35 Six-site ring Flat-band lattices Other model systems 6-site ring, two 1D TB leads Symmetric bias Γ=2T 2 /b Formula for the current reduces to the one obtained by Stefanucci in [6] This case acts as a benchmark reference

36 Six-site ring Flat-band lattices Other model systems Six levels (two degenerate) Calculate right current Γ=2T 2 /b Formula for the current reduces to the one obtained by Stefanucci in [6] This case acts as a benchmark reference

37 Six-site ring Flat-band lattices Other model systems Six levels (two degenerate) Calculate right current Γ=2T 2 /b Small coupling: Increasing bias increases the effect from the 3rd term

38 Six-site ring Flat-band lattices Other model systems Six levels (two degenerate) Calculate right current Γ=2T 2 /b Small coupling: Increasing bias increases the effect from the 3rd term

39 Six-site ring Flat-band lattices Other model systems Six levels (two degenerate) Calculate right current Γ=2T 2 /b Larger coupling: Increasing bias increases the effect from the 3rd term Also the decay is slower

40 Six-site ring Flat-band lattices Other model systems Six levels (two degenerate) Calculate right current Γ=2T 2 /b Larger coupling: Increasing bias increases the effect from the 3rd term Also the decay is slower

41 Six-site ring Flat-band lattices Other model systems Six levels (two degenerate) Calculate right current Γ=2T 2 /b Small coupling: Increasing the bias window open new conduction channels Oscillations between the molecular levels

42 Six-site ring Flat-band lattices Other model systems Six levels (two degenerate) Calculate right current Γ=2T 2 /b Larger coupling: Increasing the bias window open new conduction channels Oscillations between the molecular levels Also the decay is slower

43 Six-site ring Flat-band lattices Other model systems Six levels (two degenerate) Calculate right current Γ=2T 2 /b Antisymmetric coupling: Different levels couple with different strengths More oscillations

Six-site ring Flat-band lattices Other model systems Flat-band lattice, two 1D TB leads Bias on the left Calculate right current Cosine band and flat band Some levels

44 Six-site ring Flat-band lattices Other model systems Flat-band lattice, two 1D TB leads Bias on the left Calculate right current Cosine band and flat band Some levels are conducting, some are not Flat-band states, constant dispersion relation, infinite effective mass, vanishing group velocity Fixµ insulator (independent of the bias)

45 Six-site ring Flat-band lattices Other model systems Flat-band lattice, two 1D TB leads Bias on the left Calculate right current All the levels available for conduction (µ=0.0) current Only the flat-band levels available for conduction (µ set between the bands) no current

46 Six-site ring Flat-band lattices Other model systems Multiterminal transport Graphene nanoribbons Carbon nanotubes

47 Six-site ring Flat-band lattices Other model systems Thank you for your attention!

48 [1] Myöhänen P, Stan A, Stefanucci G and van Leeuwen R 2009 Phys. Rev. B (Preprint arxiv: v1) [2] Myöhänen P, Stan A, Stefanucci G and van Leeuwen R 2010 J. Phys.: Conf. Ser (Preprint arxiv: v1) [3] Uimonen A M, Khosravi E, Stefanucci G, Kurth S, van Leeuwen R and Gross E K U 2010 J. Phys.: Conf. Ser (Preprint arxiv: v1) [4] Sanderson C 2010 Armadillo: An Open Source C++ Linear Algebra Library for Fast Prototyping and Computationally Intensive Experiments [5] Galassi M et al. GNU Scientific Library Reference Manual [6] Stefanucci G and Almbladh C O 2004 Phys. Rev. B (Preprint arxiv:cond-mat/ v4)

arxiv: v1 [cond-mat.mes-hall] 11 Oct 2017 Received: date / Accepted: date

arxiv: v1 [cond-mat.mes-hall] 11 Oct 2017 Received: date / Accepted: date Noname manuscript No. (will be inserted by the editor) Formal equivalence between partioned and partition-free quenches in quantum transport Michael Ridley Riku Tuovinen arxiv:1710.04141v1 [cond-mat.mes-hall]