Büttiker s probe in molecular electronics: Applications to charge and heat transport

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1 Büttiker s probe in molecular electronics: Applications to charge and heat transport Dvira Segal Department of Chemistry University of Toronto Michael Kilgour (poster)

2 Büttiker s probe in molecular electronics: Applications to charge and heat transport B 1 B 2 S F 1 F 2 D. Cahill, Nature Mat (2012), L. Venkataraman Nano Lett (2013)

3 Methods: open, quantum, many-body effects, multiple reservoirs out of equilibrium B 1 B 2 S F 1 F 2 Numerically exact methods (MC, Influence functional, MCTDH, HEOM) Perturbative approaches (QME, NEGF) Asymptotic approaches (bounds) Phenomenological tools (Büttiker s probes) Semiclassical, mixed q-c, hybrid 3

4 Outline 1. Introduction 30 years with Büttiker s probes years of self-consistent reservoirs 2. Charge transfer o Technical details: dephasing probe and voltage probe o Linear response o High voltage results o Applications: diodes 3. Vibrational heat transfer o Simple examples o Large scale simulations 4. Outlook 4

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8 Strong bounds on Onsager coefficients and efficiency for three-terminal thermoelectric transport in a magnetic field K. Brandner and U. Seifert, PRL 110, (2013). Local temperature of out-ofequilibrium quantum electron systems J. Meair, J. P. Bergfield, C. A. Stafford, Ph Jacquod PRB 90, (2014) 8

9 PRA 1, 1086 (1970) 9

10 Büttiker s probes in molecular electronics Goal: Mimic environmental effects, electron-vibration interaction effects (IETS signals?!...). Understand dephasing/voltage probe, low/high voltage. Mechanisms: o tunneling to hopping o ballistic motion to hopping length, temperature, and energetic dependence. Results should be meaningful: o experiments o other methods (quantum master equations, GF) Advantages of the probe method: o weak-to-strong metal-molecule coupling - elastic limit is included exactly o low-to-high bias simulations, low-to-high temperature o large scale simulations o Fundamental understanding over the role of incoherent effects (Fourier s law, diode effect) 10

11 Transition from tunneling to hopping transport in long, conjugated Oligo-imine wires connected to metals S. H. Choi et al (Frisbie lab) JACS (2010) 11

12 Thermoelectric effect and its dependence on molecular length and sequence in single DNA Molecules Y. Li et al., (Tao lab) Nature Comm

13 Molecular junction with probes γ L ν γ R T μ L ε B T μ R f L (ε) f R (ε) 13

14 Molecular junction with probes γ L ν γ R T μ L ε B T μ R f L (ε) f 1 (ε) f 2 (ε) f 3 (ε) f 4 (ε) f R (ε) 14

15 Molecular junction with probes γ L ν γ R T μ L ε B T μ R f L (ε) f 1 (ε) f 2 (ε) f 3 (ε) f 4 (ε) f R (ε) 15

16 Dephasing probe: Incoherent elastic scattering f L (ε) f R (ε) f 1 (ε) f 2 (ε) f 3 (ε) Probe condition: Linear equation for f n (ε), probe distribution

17 Voltage probe: Incoherent inelastic scattering Probe condition: Linear Response: µ L µ R µ 1 µ 2 µ 3 Linear equation for µ n, probe chemical potentials! Low temperature Pastawski formula

18 Voltage probe: Incoherent inelastic scattering Probe condition: Far-from-equilibrium µ L µ R µ 1 µ 2 µ 3 Nonlinear equation for µ n, probe chemical potentials! Use Newton Raphson method to get the roots

19 Working Expressions But probes do more than level broadening! Tunneling, ballistic motion, hopping, and crossover between regions.

21 Molecular junction with probes γ L ν γ R T μ L γ d ε B T μ R f L (ε) f 1 (ε) f 2 (ε) f 3 (ε) f 4 (ε) f R (ε) ε B =0.5, ν= 0.05, γ L,R =0.2 [ev] Linear tilting of levels 21

22 Distance dependence M. Kilgour and DS JCP 143, (2015)

23 Distance dependence M. Kilgour and DS JCP 143, (2015)

24 Distance dependence M. Kilgour and DS JCP 143, (2015)

25 Distance dependence M. Kilgour and DS JCP 143, (2015)

26 Kramer s like turnover G γ d G 1/γ d G γ d 2 M. Kilgour and DS, JCP 143, (2015)

28 M. Kilgour and DS JCP 144, (2016)

29 outgoing incoming Voltage probe dephasing probe

30 Diode: Absence of rectification with dephasing probes Rectification with voltage probe beyond linear response (Michael s poster) from the dephasing probe condition

31 Tunneling Diodes Tunneling diodes under environmental effects M. Kilgour and DS JPC C 119, (2016)

32 Summary: Buttiker s probes in molecular electronics charge transport 1. Examine susceptibility of coherent phenomena to incoherent (elastic/inelastic) environmental scattering effects - Cheap at low bias and at low T. - Expensive high bias simulations: nonlinear function 2. Capture turnover between different transport mechanisms M. Kilgour and DS, JCP 143, (2015) M. Kilgour and DS JPC C 119, (2016) M. Kilgour and DS JCP 144, (2016)

33 Nano Lett., 2015, 15 (5), pp

34 Methods for quantum heat transfer: I. Landauer Formalism - harmonic force field II. Phenomenology of anharmonic effects: self consistent reservoirs III. Genuine anharmonic effects: Green s function, quantum master equation, Boltzmann s equation

35 PRA 1, 1086 (1970) 35

36 Anharmonicity: Phenomenological Approach Effective anharmonicity: overall energy is conserved, but not the number of quanta Harmonic model: simple, and not so interesting

37 Solve for T l : T 2, T 3 T N 1 Heat current:

38 Solve for T l : T 2, T 3 T N 1 Heat current:

39 Solve for T l : T 2, T 3 T N 1 Heat current:

40 N + Classical O Exact quantum Quantum linear response (dotted line: Exact quantum, no internal reservoirs) M = 1, γ = 0.2 M. Bandyopadhyay and D. Segal Phys. Rev. E 84, (2011)

Quantum heat diode J + β H = 0.1, β C = 0.2 γ = 0.2, M n = 0.2 + n 1 0.2 β H = 1, β C = 1.

41 Quantum heat diode J + β H = 0.1, β C = 0.2 γ = 0.2, M n = n β H = 1, β C = 1.2 β H = 1, β C = 5 J J + Quantum heat transfer in harmonic chains with self consistent reservoirs: Exact numerical simulations M. Bandyopadhyay and DS Phys. Rev. E 84, (2011)

42 Application: Self-consistent reservoirs Thermal transport through two dimensional constrictions of graphene K. Saaskilahti, J. Oksanen, J. Tulkki PRE 88, (2013).

capture the nature of the scatterer 3. carefully compare to other methods 4.

43 Conclusions-outlook Large-scale + microscopic mechanisms + analytical results Future directions: 1. simulation of large scale systems: monolayers, devices 2. capture the nature of the scatterer 3. carefully compare to other methods 4. dynamical effects The noneq Green s functions method and descendants: Ways to avoid and to go P. Greck et al IEEE 2010 Walter Schottky Institute

44 Canada Research Chair Program Hyehwang Kim (undergrad) Michael Kilgour (grad) CQIQC Malay Bandyopadhyay Pdoc IIT Bhubaneswar Hava Friedman (grad) Bijay Kumar Agarwalla (pdoc) 44

From exhaustive simulations to key principles in DNA nanoelectronics

From exhaustive simulations to key principles in DNA nanoelectronics Dvira Segal Department of Chemistry University of Toronto Roman Korol (undergrad) Hyehwang Kim (undergrad) Michael Kilgour (grad) Challenge: