Langevin approach to non-adiabatic molecular dynamics

Transcription

1 Langevin approach to non-adiabatic molecular dynamics Jing Tao Lü 1, Mads Brandbyge 1, Per Hedegård 2 1. Department of Micro- and Nanotechnology, Technical University of Denmark 2. Niels Bohr institute, University of Copenhagen Joint ICTP-IAEA Workshop on Non-adiabatic Dynamics and Radiation Damage in Nuclear Materials, Trieste, Italy Semi-classical Langevin dynamics Jing Tao Lü (DTU) 1 / 45

2 Outline 1 Motivation: Adiabatic and beyond 2 Generalised Langevin equation Isolated system System plus the environment/bath 3 Coupling to electron bath Influence functional Langevin equation Application to molecular conductors Semi-classical Langevin dynamics Jing Tao Lü (DTU) 2 / 45

3 Outline 1 Motivation: Adiabatic and beyond 2 Generalised Langevin equation Isolated system System plus the environment/bath 3 Coupling to electron bath Influence functional Langevin equation Application to molecular conductors Semi-classical Langevin dynamics Jing Tao Lü (DTU) 3 / 45

4 Molecular electronics Seminal paper in molecular electronics Molecular Rectifiers, A. Aviram and M. A. Ratner, Chem. Phys. Lett. 1974, 29, 277 Semi-classical Langevin dynamics Jing Tao Lü (DTU) 4 / 45

5 Motivation: Adiabatic and beyond Generalised Langevin equation Coupling to electron bath Joule heating, Current-induced forces Molecular dynamics in the presence of current! Semi-classical Langevin dynamics Jing Tao Lü (DTU) 5 / 45

6 Adiabatic (Born-Oppenheimer) dynamics Hamiltonian for a system of electrons and ions: Ĥ = ˆT e + ˆT ph + V e (ˆr) + V eph (ˆr, ˆR) + V ph ( ˆR) Born-Oppenheimer approximation: Ĥ BO = Ĥ ˆT ph Semi-classical Langevin dynamics Jing Tao Lü (DTU) 6 / 45

7 Adiabatic (Born-Oppenheimer) dynamics Electrons only see static ions: Ĥ BO (R, r)φ i (R, r) = ε i (R)Φ i (R, r) Forces to the ions: m R = Φ 0 R Ĥ BO (R, r) Φ 0 Semi-classical Langevin dynamics Jing Tao Lü (DTU) 7 / 45

8 Adiabatic (Born-Oppenheimer) dynamics Electrons only see static ions: Ĥ BO (R, r)φ i (R, r) = ε i (R)Φ i (R, r) Forces to the ions: m R = Φ 0 R Ĥ BO (R, r) Φ 0 No energy transfer between electron and phonon subsystems! Semi-classical Langevin dynamics Jing Tao Lü (DTU) 7 / 45

9 Ehrenfest dynamics Time-dependent Schrödinger equation: i t Ψ(t) = ĤBO(R(t), r)ψ(t) Force to the ions: m R = Ψ(t) R Ĥ BO (R(t), r) Ψ(t) Semi-classical Langevin dynamics Jing Tao Lü (DTU) 8 / 45

10 Ehrenfest dynamics Time-dependent Schrödinger equation: Force to the ions: i t Ψ(t) = ĤBO(R(t), r)ψ(t) m R = Ψ(t) R Ĥ BO (R(t), r) Ψ(t) Energy transfer possible, but not good enough for Joule heating problem Semi-classical Langevin dynamics Jing Tao Lü (DTU) 8 / 45

11 Joule heating: Simple perturbation analysis Suppose we have one harmonic oscillator (phonon), coupling with the electronic environment. Energy transfer between electrons and the phonon mode could be modelled by a rate equation, Ṅ = B(N + 1) AN, with N the phonon occupation number. At steady state, Ṅ = 0 N = B A B = 1 A B 1. For equilibrium electrons with temperature T e, we have [ ] A ω B = exp. k B T e Semi-classical Langevin dynamics Jing Tao Lü (DTU) 9 / 45

12 Joule heating: Simple perturbation analysis Suppose we have one harmonic oscillator (phonon), coupling with the electronic environment. Energy transfer between electrons and the phonon mode could be modelled by a rate equation, Ṅ = B(N + 1) AN, with N the phonon occupation number. At steady state, Ṅ = 0 N = B A B = 1 A B 1. For equilibrium electrons with temperature T e, we have [ ] A ω B = exp. k B T e Semi-classical Langevin dynamics Jing Tao Lü (DTU) 9 / 45

13 Joule heating: Simple perturbation analysis Suppose we have one harmonic oscillator (phonon), coupling with the electronic environment. Energy transfer between electrons and the phonon mode could be modelled by a rate equation, Ṅ = B(N + 1) AN, with N the phonon occupation number. At steady state, Ṅ = 0 N = B A B = 1 A B 1. For equilibrium electrons with temperature T e, we have [ ] A ω B = exp. k B T e Semi-classical Langevin dynamics Jing Tao Lü (DTU) 9 / 45

14 Why Ehrenfest fails? The energy transfer predicted from Ehrenfest dynamics Ṅ = BN AN < 0 The phonon keeps losing energy! E. J. McEniry, Y. Wang, D. Dundas, T. N. Todorov, L. Stella, R. P. Miranda, A. J. Fisher, A. P. Horsfield, C. P. Race, D. R. Mason, W. M.C. Foulkes and A. P. Sutton, EPJB, 77, 305 (2010) Semi-classical Langevin dynamics Jing Tao Lü (DTU) 10 / 45

15 Outline 1 Motivation: Adiabatic and beyond 2 Generalised Langevin equation Isolated system System plus the environment/bath 3 Coupling to electron bath Influence functional Langevin equation Application to molecular conductors Semi-classical Langevin dynamics Jing Tao Lü (DTU) 11 / 45

16 Semi-classical dynamics from influence functional Influence functional approach: Start from the fully quantum-mechanical problem Split into the system (ions), and the bath/environment (electrons) Given configuration of the system, trace out the bath (electrons), and get an effective action for the system Semi-classical Langevin equation References: Phonon bath: Feynman&Vernon1963,Caldeira&Leggett1983, Schmid1982, Electron bath: Chang&Chakravarty1985,Hedegård&Caldeira1987 Semi-classical Langevin dynamics Jing Tao Lü (DTU) 12 / 45

17 Semi-classical dynamics from influence functional Influence functional approach: Start from the fully quantum-mechanical problem Split into the system (ions), and the bath/environment (electrons) Given configuration of the system, trace out the bath (electrons), and get an effective action for the system Semi-classical Langevin equation References: Phonon bath: Feynman&Vernon1963,Caldeira&Leggett1983, Schmid1982, Electron bath: Chang&Chakravarty1985,Hedegård&Caldeira1987 Semi-classical Langevin dynamics Jing Tao Lü (DTU) 12 / 45

18 Semi-classical dynamics from influence functional Influence functional approach: Start from the fully quantum-mechanical problem Split into the system (ions), and the bath/environment (electrons) Given configuration of the system, trace out the bath (electrons), and get an effective action for the system Semi-classical Langevin equation References: Phonon bath: Feynman&Vernon1963,Caldeira&Leggett1983, Schmid1982, Electron bath: Chang&Chakravarty1985,Hedegård&Caldeira1987 Semi-classical Langevin dynamics Jing Tao Lü (DTU) 12 / 45

19 Semi-classical dynamics from influence functional Influence functional approach: Start from the fully quantum-mechanical problem Split into the system (ions), and the bath/environment (electrons) Given configuration of the system, trace out the bath (electrons), and get an effective action for the system Semi-classical Langevin equation References: Phonon bath: Feynman&Vernon1963,Caldeira&Leggett1983, Schmid1982, Electron bath: Chang&Chakravarty1985,Hedegård&Caldeira1987 Semi-classical Langevin dynamics Jing Tao Lü (DTU) 12 / 45

20 Semi-classical dynamics from influence functional Influence functional approach: Start from the fully quantum-mechanical problem Split into the system (ions), and the bath/environment (electrons) Given configuration of the system, trace out the bath (electrons), and get an effective action for the system Semi-classical Langevin equation References: Phonon bath: Feynman&Vernon1963,Caldeira&Leggett1983, Schmid1982, Electron bath: Chang&Chakravarty1985,Hedegård&Caldeira1987 Semi-classical Langevin dynamics Jing Tao Lü (DTU) 12 / 45

21 Outline 1 Motivation: Adiabatic and beyond 2 Generalised Langevin equation Isolated system System plus the environment/bath 3 Coupling to electron bath Influence functional Langevin equation Application to molecular conductors Semi-classical Langevin dynamics Jing Tao Lü (DTU) 13 / 45

22 Hamiltonian Hamiltonian for an ion moving in some potential: Ĥ = ˆp2 2m + V (ˆx) Semi-classical Langevin dynamics Jing Tao Lü (DTU) 14 / 45

23 Evolution operator with path integral Evolution operator in coordinate basis: b e iĥt/ a = ( lim b e iĥ t/ ) N a N + N 1 = lim dx j x j e iĥ t/ x j 1 N + j=1 with x N = b, x 0 = a. Semi-classical Langevin dynamics Jing Tao Lü (DTU) 15 / 45

24 Evolution operator with path integral x j e iĥ t/ x j 1 Semi-classical Langevin dynamics Jing Tao Lü (DTU) 16 / 45

25 Evolution operator with path integral x j e iĥ t/ x j 1 x j e iˆp2 t/2m e iv (ˆx) t/ x j 1 Trotter Semi-classical Langevin dynamics Jing Tao Lü (DTU) 16 / 45

26 Evolution operator with path integral x j e iĥ t/ x j 1 x j e iˆp2 t/2m e iv (ˆx) t/ x j 1 Trotter dp j dp j x j p j p j e iˆp2 t/2m p j p j e iv (ˆx) t/ x j 1 Semi-classical Langevin dynamics Jing Tao Lü (DTU) 16 / 45

27 Evolution operator with path integral x j e iĥ t/ x j 1 x j e iˆp2 t/2m e iv (ˆx) t/ x j 1 Trotter dp j dp j x j p j p j e iˆp2 t/2m p j p j e iv (ˆx) t/ x j 1 dp j x j p j e ip j 2 t/2m p j x j 1 e iv (xj 1) t/ x j p j = e ip j x j / Semi-classical Langevin dynamics Jing Tao Lü (DTU) 16 / 45

28 Evolution operator with path integral x j e iĥ t/ x j 1 x j e iˆp2 t/2m e iv (ˆx) t/ x j 1 Trotter dp j dp j x j p j p j e iˆp2 t/2m p j p j e iv (ˆx) t/ x j 1 dp j x j p j e ip j 2 t/2m p j x j 1 e iv (xj 1) t/ x j p j = e ip j x j / dp j e ip j 2 t/2m e iv (xj 1) t/ e i(x j x j 1 )p j / Semi-classical Langevin dynamics Jing Tao Lü (DTU) 16 / 45

29 Evolution operator with path integral x j e iĥ t/ x j 1 x j e iˆp2 t/2m e iv (ˆx) t/ x j 1 Trotter dp j dp j x j p j p j e iˆp2 t/2m p j p j e iv (ˆx) t/ x j 1 dp j x j p j e ip j 2 t/2m p j x j 1 e iv (xj 1) t/ x j p j = e ip j x j / dp j e ip j 2 t/2m e iv (xj 1) t/ e i(x j x j 1 )p j / { [ ( ) } 2 i m xj x j 1 N exp V (x j 1)] t 2 t Semi-classical Langevin dynamics Jing Tao Lü (DTU) 16 / 45

30 Evolution operator with path integral Evolution operator in coordinate basis: N 1 [ b e iĥt/ i N ( ) 2 m xj x j 1 a = lim dx j exp V (x N + j 1)] 2 ɛ j=1 j=1 { ( )} i 1 Dx exp dt 2 mẋ 2 V (x) { } i Dx exp S(ẋ, x) Semi-classical Langevin dynamics Jing Tao Lü (DTU) 17 / 45

31 Semi-classical approximation Evolution operator in coordinate basis: { ( )} i 1 b e iĥt/ a = Dxexp dt 2 mẋ 2 V (x) Pick out the classical path: x = x cl + ξ, we get { } i b e iĥt a exp S cl { i Dξexp dt ( 1 2 m ξ 2 V (x cl )ξ 2 )} + O(ξ 3 ) In the limit of ξ 2, classical path contributes the most. Semi-classical Langevin dynamics Jing Tao Lü (DTU) 18 / 45

32 Semi-classical approximation Evolution operator in coordinate basis: { ( )} i 1 b e iĥt/ a = Dxexp dt 2 mẋ 2 V (x) Pick out the classical path: x = x cl + ξ, we get { } i b e iĥt a exp S cl { i Dξexp dt ( 1 2 m ξ 2 V (x cl )ξ 2 )} + O(ξ 3 ) In the limit of ξ 2, classical path contributes the most. Semi-classical Langevin dynamics Jing Tao Lü (DTU) 18 / 45

33 Evolution of Density matrix Density matrix: ˆρ(t) = e iĥt/ ˆρ(0)e iĥt/ Evolution in coordinate basis: x 1 ˆρ(t) y 1 = x 1 e iĥt/ x 0 x 0 ˆρ(0) y 0 y 0 e iĥt/ y 1 = D[x, y]e i(s(x) S(y))/ x 0 ˆρ(0) y 0 K (x 1, y 1 ; x 0, y 0 ) x 0 ˆρ(0) y 0 x y Semi-classical Langevin dynamics Jing Tao Lü (DTU) 19 / 45

34 Evolution of Density matrix Density matrix: ˆρ(t) = e iĥt/ ˆρ(0)e iĥt/ Evolution in coordinate basis: x 1 ˆρ(t) y 1 = x 1 e iĥt/ x 0 x 0 ˆρ(0) y 0 y 0 e iĥt/ y 1 = D[x, y]e i(s(x) S(y))/ x 0 ˆρ(0) y 0 K (x 1, y 1 ; x 0, y 0 ) x 0 ˆρ(0) y 0 x y Semi-classical Langevin dynamics Jing Tao Lü (DTU) 19 / 45

35 Semi-classical approximation Define R 1 2 (x + y)(classical),ξ x y(quantum), we get { } i K (x 1, y 1 ; x 0, y 0 ) = D[x, y]exp (S(x) S(y)) { D[R, ξ]exp i } ξ(t)(m R(t) + V (R(t)))dt + O(ξ 3 ) Only odd in ξ terms contribute! Think of ikx dk e 2π = δ(x), integrate out ξ, we get L 0 (t) = m R(t) + V (R(t)) = 0. Average path follows classical equation of motion Exact for quadratic potential Semi-classical Langevin dynamics Jing Tao Lü (DTU) 20 / 45

36 Semi-classical approximation Define R 1 2 (x + y)(classical),ξ x y(quantum), we get { } i K (x 1, y 1 ; x 0, y 0 ) = D[x, y]exp (S(x) S(y)) { D[R, ξ]exp i } ξ(t)(m R(t) + V (R(t)))dt + O(ξ 3 ) Only odd in ξ terms contribute! Think of ikx dk e 2π = δ(x), integrate out ξ, we get L 0 (t) = m R(t) + V (R(t)) = 0. Average path follows classical equation of motion Exact for quadratic potential Semi-classical Langevin dynamics Jing Tao Lü (DTU) 20 / 45

37 Outline 1 Motivation: Adiabatic and beyond 2 Generalised Langevin equation Isolated system System plus the environment/bath 3 Coupling to electron bath Influence functional Langevin equation Application to molecular conductors Semi-classical Langevin dynamics Jing Tao Lü (DTU) 21 / 45

38 Density matrix for system+bath model System variables: x, y, Bath variables: X, Y. The density matrix is x 1, X 1 ˆρ(t) y 1, Y 1 = x 1, X 1 e i Ĥt ˆρ(0)e i Ĥt y 1, Y 1 Since we are only interested in the system, we want to look at the evolution of the reduced density matrix for the system only ˆρ s Semi-classical Langevin dynamics Jing Tao Lü (DTU) 22 / 45

39 Density matrix for system+bath model System variables: x, y, Bath variables: X, Y. The density matrix is x 1, X 1 ˆρ(t) y 1, Y 1 = x 1, X 1 e i Ĥt ˆρ(0)e i Ĥt y 1, Y 1 Since we are only interested in the system, we want to look at the evolution of the reduced density matrix for the system only ˆρ s x 1 ˆρ s (t) y 1 = Tr X1 [ x 1, X 1 ˆρ(t) y 1, Y 1 ] Semi-classical Langevin dynamics Jing Tao Lü (DTU) 22 / 45

40 Density matrix for system+bath model System variables: x, y, Bath variables: X, Y. The density matrix is x 1, X 1 ˆρ(t) y 1, Y 1 = x 1, X 1 e i Ĥt ˆρ(0)e i Ĥt y 1, Y 1 Since we are only interested in the system, we want to look at the evolution of the reduced density matrix for the system only ˆρ s x 1 ˆρ s (t) y 1 = Tr X1 [ x 1, X 1 ˆρ(t) y 1, X 1 ] Semi-classical Langevin dynamics Jing Tao Lü (DTU) 22 / 45

41 Decoupling scheme The total Hamiltonian Ĥ = Ĥs + Ĥb + Ĥint Assume at t = 0, then ρ(0) = ρ s (0) ρ b (0) x 1 ρ s (t) y 1 = F(x, y)u s (x 1, x 0 ) x 0 ρ s (0) y 0 U s (y 0, y 1 ) with the influence functional (F), phase (Φ) Semi-classical Langevin dynamics Jing Tao Lü (DTU) 23 / 45

42 Decoupling scheme The total Hamiltonian Ĥ = Ĥs + Ĥb + Ĥint Assume at t = 0, then ρ(0) = ρ s (0) ρ b (0) x 1 ρ s (t) y 1 = F(x, y)u s (x 1, x 0 ) x 0 ρ s (0) y 0 U s (y 0, y 1 ) with the influence functional (F), phase (Φ) Semi-classical Langevin dynamics Jing Tao Lü (DTU) 23 / 45

43 Decoupling scheme The total Hamiltonian Ĥ = Ĥs + Ĥb + Ĥint Assume at t = 0, then ρ(0) = ρ s (0) ρ b (0) x 1 ρ s (t) y 1 = F(x, y)u s (x 1, x 0 ) x 0 ρ s (0) y 0 U s (y 0, y 1 ) with the influence functional (F), phase (Φ) F (x, y) e iφ(x,y)/ = Tr b [ U b (x)ρ b (0)U b (y) ] Semi-classical Langevin dynamics Jing Tao Lü (DTU) 23 / 45

44 Properties of F /Φ From the definition of F we have F(x, y) e iφ(x,y)/ = Tr b [ U b (x)ρ b (0)U b (y) ], F(x, y) = F (y, x), F(x, x) = 1, or in terms of classical (R) and quantum (ξ) paths F(R, ξ) = F (R, ξ), F(R, 0) = 1, Semi-classical Langevin dynamics Jing Tao Lü (DTU) 24 / 45

45 Properties of F /Φ From the definition of F we have F(x, y) e iφ(x,y)/ = Tr b [ U b (x)ρ b (0)U b (y) ], F(x, y) = F (y, x), F(x, x) = 1, or in terms of classical (R) and quantum (ξ) paths F(R, ξ) = F (R, ξ), F(R, 0) = 1, Semi-classical Langevin dynamics Jing Tao Lü (DTU) 24 / 45

46 Effect of the bath to the evolution Recall in the semi-classical approximation, for system only, we have { K (x 1, y 1 ; x 0, y 0 ) D[R, ξ]exp i } ξ(t)l 0 (t)dt. What is the effect of the bath? { K (x 1, y 1 ; x 0, y 0 ) D[R, ξ]exp i } (ξ(t)l 0 (t) Φ(t)) dt Semi-classical Langevin dynamics Jing Tao Lü (DTU) 25 / 45

47 Effect of the bath to the evolution Recall in the semi-classical approximation, for system only, we have { K (x 1, y 1 ; x 0, y 0 ) D[R, ξ]exp i } ξ(t)l 0 (t)dt. What is the effect of the bath? { K (x 1, y 1 ; x 0, y 0 ) D[R, ξ]exp i } (ξ(t)l 0 (t) Φ(t)) dt Semi-classical Langevin dynamics Jing Tao Lü (DTU) 25 / 45

48 Expansion of the influence phase Using F(R, 0) = 1, to 2nd order: F(R, ξ) = F (R, ξ), we expand Φ over ξ Φ(R, ξ) = Φ(R, 0) A(R; t, t )ξ(t) + i 2 ξ(t )B(R; t, t )ξ(t) + O(ξ 3 ) Put this back to K : K (x 1, y 1 ; x 0, y 0 ) exp { D[R, ξ]exp i { 1 2 ξ(t)b(t, t )ξ(t ) } ξ(t)(l 0 (t) + A(t, t ))dt } Modifies the equation of motion of R Introduces a O(ξ 2 ) term Semi-classical Langevin dynamics Jing Tao Lü (DTU) 26 / 45

49 Expansion of the influence phase Using F(R, 0) = 1, to 2nd order: F(R, ξ) = F (R, ξ), we expand Φ over ξ Φ(R, ξ) = Φ(R, 0) A(R; t, t )ξ(t) + i 2 ξ(t )B(R; t, t )ξ(t) + O(ξ 3 ) Put this back to K : K (x 1, y 1 ; x 0, y 0 ) exp { D[R, ξ]exp i { 1 2 ξ(t)b(t, t )ξ(t ) } ξ(t)(l 0 (t) + A(t, t ))dt } Modifies the equation of motion of R Introduces a O(ξ 2 ) term Semi-classical Langevin dynamics Jing Tao Lü (DTU) 26 / 45

50 Probability interpretation Using Gaussian integral: we have dξe ilξ e 1 2 aξ2 df dξe i(l+f )x e 2a 1 f 2 dkδ(l + k)e 1 2a k2, K (x 1, y 1 ; x 0, y 0 ) exp { D[R, ξ]exp i { 1 2 ξ(t)b(t, t )ξ(t ) } ξ(t)(l 0 (t) + A(t, t ))dt } Semi-classical Langevin dynamics Jing Tao Lü (DTU) 27 / 45

51 Probability interpretation Using Gaussian integral: we have dξe ilξ e 1 2 aξ2 df dξe i(l+f )x e 2a 1 f 2 dkδ(l + k)e 1 2a k2, K (x 1, y 1 ; x 0, y 0 ) exp { D[R, ξ, f ]exp i { 1 2 f (t)b 1 (t, t )f (t ) } ξ(t)(l 0 (t) + A(t, t ) + f (t))dt } Semi-classical Langevin dynamics Jing Tao Lü (DTU) 27 / 45

52 Probability interpretation Using Gaussian integral: we have dξe ilξ e 1 2 aξ2 df dξe i(l+f )x e 2a 1 f 2 dkδ(l + k)e 1 2a k2, K (x 1, y 1 ; x 0, y 0 ) D[R, f ] (L 0 (t) + A(t, t ) + f (t)) { exp 1 } 2 f (t)b 1 (t, t )f (t ) Semi-classical Langevin dynamics Jing Tao Lü (DTU) 27 / 45

53 Probability interpretation Using Gaussian integral: we have dξe ilξ e 1 2 aξ2 df dξe i(l+f )x e 2a 1 f 2 dkδ(l + k)e 1 2a k2, K (x 1, y 1 ; x 0, y 0 ) D[R, f ] (L 0 (t) + A(t, t ) + f (t)) { exp 1 } 2 f (t)b 1 (t, t )f (t ) f (t ) plays a role of a stochastic force acting on the equation of motion, with a Gaussian correlation. Semi-classical Langevin dynamics Jing Tao Lü (DTU) 27 / 45

54 Generalised Langevin equation Langevin equation: 2 t t 2 R(t) + V (R) = dt A(t, t )R(t ) f (t). with the noise correlation f (t)f (t ) = B(t, t ). Semi-classical Langevin dynamics Jing Tao Lü (DTU) 28 / 45

55 Example: Harmonic oscillators For a harmonic oscillator coupling linearly (in displacement) with harmonic oscillator bath, the semi-classical approximation is exact: A(t t ) = Ck 2 sin ω k (t t ), 2mω k k B(t t ) = k C 2 k 2mω k coth ω k 2k B T cos ω k(t t ). Feynman&Vernon, Caldeira&Leggett, Semi-classical Langevin dynamics Jing Tao Lü (DTU) 29 / 45

56 Outline 1 Motivation: Adiabatic and beyond 2 Generalised Langevin equation Isolated system System plus the environment/bath 3 Coupling to electron bath Influence functional Langevin equation Application to molecular conductors Semi-classical Langevin dynamics Jing Tao Lü (DTU) 30 / 45

57 Outline 1 Motivation: Adiabatic and beyond 2 Generalised Langevin equation Isolated system System plus the environment/bath 3 Coupling to electron bath Influence functional Langevin equation Application to molecular conductors Semi-classical Langevin dynamics Jing Tao Lü (DTU) 31 / 45

58 Adiabatic expansion We have the time-dependent Schrödinger equation i t Ψ(r, R(t)) = ĤBO(r, R(t))Ψ(r, R(t)) Within the adiabatic (Born-Oppenheimer) basis: Expand Ψ(r, R(t)): Ĥ BO (r, R)φ n (r, R) = ε n (R)φ n (r, R), Ψ(r, R(t)) = n a n (R(t))φ n (r, R(t)). Including correction O(V 2 ) V = i φ m φ n O(ξ 2 ). Semi-classical Langevin dynamics Jing Tao Lü (DTU) 32 / 45

59 Adiabatic expansion We have the time-dependent Schrödinger equation i t Ψ(r, R(t)) = ĤBO(r, R(t))Ψ(r, R(t)) Within the adiabatic (Born-Oppenheimer) basis: Expand Ψ(r, R(t)): Ĥ BO (r, R)φ n (r, R) = ε n (R)φ n (r, R), Ψ(r, R(t)) = n a n (R(t))φ n (r, R(t)). Including correction O(V 2 ) V = i φ m φ n O(ξ 2 ). Semi-classical Langevin dynamics Jing Tao Lü (DTU) 32 / 45

60 Adiabatic expansion We have the time-dependent Schrödinger equation i t Ψ(r, R(t)) = ĤBO(r, R(t))Ψ(r, R(t)) Within the adiabatic (Born-Oppenheimer) basis: Expand Ψ(r, R(t)): Ĥ BO (r, R)φ n (r, R) = ε n (R)φ n (r, R), Ψ(r, R(t)) = n a n (R(t))φ n (r, R(t)). Including correction O(V 2 ) V = i φ m φ n O(ξ 2 ). Semi-classical Langevin dynamics Jing Tao Lü (DTU) 32 / 45

61 Influence functional Influence functional from electron bath: [ ] F e = Det [G 0 ] 1 + V { [ ]} exp Tr lng 1 0 { exp {Tr [G 0 V ]}exp 1 } 2 Tr [G 0VG 0 V ] Semi-classical JTL, M. Langevin Brandbyge, dynamicsp. Jing Hedegård, Tao Lü (DTU) Nano. Letters, 10, 1657 (2010) 33 / 45

62 Influence functional Influence functional from electron bath: [ ] F e = Det [G 0 ] 1 + V { [ ]} exp Tr lng 1 0 { exp {Tr [G 0 V ]}exp 1 } 2 Tr [G 0VG 0 V ] The adiabatic Green s function: ( i ) t ε n(r) Gnn(t, 0 t ) = δ(t t ) Semi-classical JTL, M. Langevin Brandbyge, dynamicsp. Jing Hedegård, Tao Lü (DTU) Nano. Letters, 10, 1657 (2010) 33 / 45

63 Influence functional Influence functional from electron bath: [ ] F e = Det [G 0 ] 1 + V { [ ]} exp Tr lng 1 0 { exp {Tr [G 0 V ]}exp 1 } 2 Tr [G 0VG 0 V ] The adiabatic Green s function: ( i ) t ε n(r) Gnn(t, 0 t ) = δ(t t ) Adiabatic correction: V mn = i φ m φ n Semi-classical JTL, M. Langevin Brandbyge, dynamicsp. Jing Hedegård, Tao Lü (DTU) Nano. Letters, 10, 1657 (2010) 33 / 45

64 Outline 1 Motivation: Adiabatic and beyond 2 Generalised Langevin equation Isolated system System plus the environment/bath 3 Coupling to electron bath Influence functional Langevin equation Application to molecular conductors Semi-classical Langevin dynamics Jing Tao Lü (DTU) 34 / 45

65 Langevin equation (Equilibrium electrons) 2 t 2 R(t) = V γ eh (t)ṙ(t) + f (t) Semi-classical Langevin dynamics Jing Tao Lü (DTU) 35 / 45

66 Langevin equation (Equilibrium electrons) 2 t 2 R(t) = V γ eh (t)ṙ(t) + f (t) The adiabatic(born-oppenheimer) force V = j n F (ε j ) R ε j(t, R(t)) Semi-classical Langevin dynamics Jing Tao Lü (DTU) 35 / 45

67 Langevin equation (Equilibrium electrons) The electronic friction 2 t 2 R(t) = V γ eh (t)ṙ(t) + f (t) γ eh (ω) k,j 1 ω φ j R V ep φ k φ k R V ep φ j (n F (ε k µ) n F (ε j µ)) δ(ω (ε k ε j )) Generally friction not local in time! But very short memory for electrons. Excitation of electron-hole pairs! Semi-classical Langevin dynamics Jing Tao Lü (DTU) 35 / 45

68 Langevin equation (Equilibrium electrons) 2 t 2 R(t) = V γ eh (t)ṙ(t) + f (t) The equilibrium noise correlation ( ff (ω) = 2ωγ eh n B (ω, T ) + 1 ) colored noise 2 Fluctuations around the classical path: zero T, zero point motion, ωγ eh high T, thermal fluctuation, 2γ eh k B T Semi-classical Langevin dynamics Jing Tao Lü (DTU) 35 / 45

69 Langevin equation (Equilibrium electrons) 2 t 2 R(t) = V γ eh (t)ṙ(t) + f (t) The equilibrium noise correlation ( ff (ω) = 2ωγ eh n B (ω, T ) + 1 ) colored noise 2 Fluctuations around the classical path: zero T, zero point motion, ωγ eh high T, thermal fluctuation, 2γ eh k B T Ignoring f (t), phonons keep losing energy to electrons! (Ehrenfest) Semi-classical Langevin dynamics Jing Tao Lü (DTU) 35 / 45

70 Energy of harmonic phonons For a harmonic oscillator coupling weakly with an equilibrium electron bath at T e, we can calculate the energy of the phonons from the Langevin equations ( E = ω 0 n B (ω, T e ) + 1 ) 2 Semi-classical Langevin dynamics Jing Tao Lü (DTU) 36 / 45

71 Nonequilibrium electrons Nonequilibrium Green s function method: B(t t ) = i ( Π > + Π <), 2 A(t t ) = Π > (t t ) Π < (t t ) Semi-classical Langevin dynamics Jing Tao Lü (DTU) 37 / 45

72 Outline 1 Motivation: Adiabatic and beyond 2 Generalised Langevin equation Isolated system System plus the environment/bath 3 Coupling to electron bath Influence functional Langevin equation Application to molecular conductors Semi-classical Langevin dynamics Jing Tao Lü (DTU) 38 / 45

73 Motivation: Adiabatic and beyond Generalised Langevin equation Coupling to electron bath Joule heating, Current-induced forces Molecular dynamics in the presence of current! Semi-classical Langevin dynamics Jing Tao Lü (DTU) 39 / 45

74 Langevin equation (Nonequilibrium electrons) 2 t 2 R(t) = V (γ eh (ev ) + γ(t))ṙ(t) + f (t) Nonequilibrium corrections: B(eV )Ṙ(t) + A(eV )R(t) Joule heating: ff (ω) = ω(γ eh + γ) coth ω 2k B T + Π(ω) Nonequilibrium correction to the friction coefficient Berry-phase induced effective magnetic field - B anti-symmetric Non-conservative current-induced forces - A anti-symmetric JTL, M. Brandbyge, P. Hedegård, Nano. Letters, 10, 1657 (2010) JTL, P. Hedegård, M. Brandbyge, PRL, 107, (2010) Semi-classical Langevin dynamics Jing Tao Lü (DTU) 40 / 45

75 Langevin equation (Nonequilibrium electrons) 2 t 2 R(t) = V (γ eh (ev ) + γ(t))ṙ(t) + f (t) Nonequilibrium corrections: B(eV )Ṙ(t) + A(eV )R(t) Joule heating: ff (ω) = ω(γ eh + γ) coth ω 2k B T + Π(ω) Nonequilibrium correction to the friction coefficient Berry-phase induced effective magnetic field - B anti-symmetric Non-conservative current-induced forces - A anti-symmetric JTL, M. Brandbyge, P. Hedegård, Nano. Letters, 10, 1657 (2010) JTL, P. Hedegård, M. Brandbyge, PRL, 107, (2010) Semi-classical Langevin dynamics Jing Tao Lü (DTU) 40 / 45

76 Langevin equation (Nonequilibrium electrons) 2 t 2 R(t) = V (γ eh (ev ) + γ(t))ṙ(t) + f (t) Nonequilibrium corrections: B(eV )Ṙ(t) + A(eV )R(t) Joule heating: ff (ω) = ω(γ eh + γ) coth ω 2k B T + Π(ω) Nonequilibrium correction to the friction coefficient Berry-phase induced effective magnetic field - B anti-symmetric Non-conservative current-induced forces - A anti-symmetric JTL, M. Brandbyge, P. Hedegård, Nano. Letters, 10, 1657 (2010) JTL, P. Hedegård, M. Brandbyge, PRL, 107, (2010) Semi-classical Langevin dynamics Jing Tao Lü (DTU) 40 / 45

77 Langevin equation (Nonequilibrium electrons) 2 t 2 R(t) = V (γ eh (ev ) + γ(t))ṙ(t) + f (t) Nonequilibrium corrections: B(eV )Ṙ(t) + A(eV )R(t) Joule heating: ff (ω) = ω(γ eh + γ) coth ω 2k B T + Π(ω) Nonequilibrium correction to the friction coefficient Berry-phase induced effective magnetic field - B anti-symmetric Non-conservative current-induced forces - A anti-symmetric JTL, M. Brandbyge, P. Hedegård, Nano. Letters, 10, 1657 (2010) JTL, P. Hedegård, M. Brandbyge, PRL, 107, (2010) Semi-classical Langevin dynamics Jing Tao Lü (DTU) 40 / 45

78 Computational tools SIESTA for electronic structure calculation J. M. Soler, E. Artacho, J. D. Gale, A. García, J. Junquera, P. Ordejón, and D. Sánchez-Portal, J. Phys. Condens. Matter 14, 2745 (2002) TranSIESTA for electronic transport M. Brandbyge, J.-L. Mozos, P. Ordejón, J. Taylor, and K. Stokbro, Phys. Rev. B 65, (2002) Inelastica for phonons and e-ph coupling T. Frederiksen, M. Paulsson, M. Brandbyge, A.-P. Jauho, Phys. Rev. B 75, (2007) Semi-classical Langevin dynamics Jing Tao Lü (DTU) 41 / 45

79 Harmonic vs. Anharmonic Σ Σ MD: Follow how energy is redistributed by anharmonic couplings 28 DTU Nanotech, Technical University of Denmark Semi-classical Langevin dynamics Jing Tao Lü (DTU) 42 / 45

80 Carbon chain MD simulations Tight-binding + Brenner potential Importance of anharmonic couplings in energy redistribution Anharmonic Harmonic Tue Gunst, M.Sc. Thesis, DTU 29 DTU Nanotech, Technical University of Denmark Semi-classical Langevin dynamics Jing Tao Lü (DTU) 43 / 45

81 Take home message General first-principles way of deriving Langevin equation for system+bath model Applied to ions coupling with nonequilibrium, time-dependent (in the adiabatic sense) electron baths Non-adiabatic effect enters as friction and noise. The noise is crucial for the energy transfer. Application to radiation damage??? JTL, M. Brandbyge, P. Hedegård, Nano. Letters, 10, 1657 (2010), M. Brandbyge, P. Hedegård, PRL, 72, 2919 (1994), M. Brandbyge, P. Hedegård, T. F. Heinz, J. A. Misewich, and D. M. Newns, PRB, 52, 6042 (1995). Semi-classical Langevin dynamics Jing Tao Lü (DTU) 44 / 45

82 Take home message General first-principles way of deriving Langevin equation for system+bath model Applied to ions coupling with nonequilibrium, time-dependent (in the adiabatic sense) electron baths Non-adiabatic effect enters as friction and noise. The noise is crucial for the energy transfer. Application to radiation damage??? JTL, M. Brandbyge, P. Hedegård, Nano. Letters, 10, 1657 (2010), M. Brandbyge, P. Hedegård, PRL, 72, 2919 (1994), M. Brandbyge, P. Hedegård, T. F. Heinz, J. A. Misewich, and D. M. Newns, PRB, 52, 6042 (1995). Semi-classical Langevin dynamics Jing Tao Lü (DTU) 44 / 45

83 Take home message General first-principles way of deriving Langevin equation for system+bath model Applied to ions coupling with nonequilibrium, time-dependent (in the adiabatic sense) electron baths Non-adiabatic effect enters as friction and noise. The noise is crucial for the energy transfer. Application to radiation damage??? JTL, M. Brandbyge, P. Hedegård, Nano. Letters, 10, 1657 (2010), M. Brandbyge, P. Hedegård, PRL, 72, 2919 (1994), M. Brandbyge, P. Hedegård, T. F. Heinz, J. A. Misewich, and D. M. Newns, PRB, 52, 6042 (1995). Semi-classical Langevin dynamics Jing Tao Lü (DTU) 44 / 45

84 Take home message General first-principles way of deriving Langevin equation for system+bath model Applied to ions coupling with nonequilibrium, time-dependent (in the adiabatic sense) electron baths Non-adiabatic effect enters as friction and noise. The noise is crucial for the energy transfer. Application to radiation damage??? JTL, M. Brandbyge, P. Hedegård, Nano. Letters, 10, 1657 (2010), M. Brandbyge, P. Hedegård, PRL, 72, 2919 (1994), M. Brandbyge, P. Hedegård, T. F. Heinz, J. A. Misewich, and D. M. Newns, PRB, 52, 6042 (1995). Semi-classical Langevin dynamics Jing Tao Lü (DTU) 44 / 45