Heat and electron transport in nano-structured conductors: Insights from atomistic simulations Mads Brandbyge

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1 Heat and electron transport in nano-structured conductors: Insights from atomistic simulations Mads Brandbyge Dept. of micro and nanotech. & Center for Nanostructured Graphene (CNG) Technical University of Denmark (DTU) Quantum Phononics, Heraklion, Crete, May 27-29, 2015

2 Thanks Theoretical Nanoelectronics Group Dr. Daniele Stradi Dr. Tue Gunst Ph.D. stud. Jesper T. Rasmussen Ph.D. stud. Rasmus B. Christensen Ph.D. stud. Nick P. Andersen Research assist. Cand. polyt. Mattias L. N. Palsgaard Dr. Jing-Tao Lü, HUST, Wuhan, China Dr. Haldun Sevincli Funding: Prof. Per Hedegård, Copenhagen Univ. The Danish Research Councils for Natural/Technical Sciences J-S. Wang, NUS, Singapore T. N. Todorov, D. Dundas Q.U. Belfast, UK

3 Group research topics Electron transport (DFT-NEGF)/tight-binding Thermoelectrics Phonon transport (DFT, Empirical potentials) Electron-phonon interaction in nano-electronics/molecular electronics What can we expect when going to the extreme limit : Squeezing current through small junctions

4 Topics Thermoelectric properties of nano-structured graphene Anti-dot lattices Substrate/kinks Electron-phonon interaction: Phonons perturbing electronic current Inelastic phonon spectroscopy Electron-phonon interaction: Electronic current perturbing phonons The generalized Langevin equation approach Amplification of vibrations by current ( Phonon-laser ) Current-induced forces influence the excess heat distribution/flow

5 Thermoelectrics: Basics Seebeck effect (1822) Power generation Peltier effect Active cooling/heating Both effects are related to a single parameter: Seebeck coefficient = S [mv/k] Thermo-electric figure of merit: Needs strongly Energy dependence of electronic structure (e.g. gap or resonance) Can be taylored by nanostructuring Should be minimized

6 Graphene nanostructures Graphene: High thermal conductivity High tunability with nanostructuring/ sculpting Controllable Atomic Scale Patterning of Freestanding Monolayer Graphene at Elevated Temperature Xu et al., ACS-Nano 7, 1566 (2013) Thermoelectric effects in graphene nanostructures, P. Dollfus, V. H. Nguyen, J. Saint-Martin, J. Phys. Cond. Mat. 27, (2015) Correlating Atomic Structure and Transport in suspended Graphene Nanoribbons Qi et al. Nano Lett. 2014, 14, J. Bai et al., Nat. Nanotechnol. 5, 190 (2010). Scale bars = 100nm M. Kim et al., Nano Lett. 10, 1125 (2010). Electrical Transport Measured in Atomic Carbon Chains Cretu et al., Nano Lett. 13, 3487 (2013)

7 Electron transport in finite Graphene Anti-dot Lattice GUNST, MARKUSSEN, JAUHO, AND BRANDBYGE PHYSICAL REVIEW B 84, (2011) π-model tight-binding Structure/heat-transport: Brenner potential Tranport calculated by recursive GF method {10,3}arm Fast convergence of transport gap with no. repetitions (M)

8 Transport gap scaling: edge effects Antidot edges: Zigzag Armchair Mixed

9 Thermal transport in finite GAL Symbols: Different edge-types Fast convergence in M Thermoelectric figure-of-merit Less sensitive to edge-effects Further reduction by C-isotope structuring, substrate etc.

10 Topics Thermoelectric properties of nano-structured graphene Anti-dot lattices Substrate/kinks Electron-phonon interaction: Phonons perturbing electronic current Inelastic electron transport spectroscopy Electron-phonon interaction: Electronic current perturbing phonons The generalized Langevin equation approach Amplification of vibrations by current ( Phonon-laser ) Current-induced forces influence the excess heat distribution/flow

11 Bend/ Kinky graphene Graphene Low et al., PRL 108, (2012) Nanostructured substrate Very small effect of deformation on electron transport But effect of doping (depending of substrate) Tailoring of local reactivity Bunch et al., Nano Lett. 8, 2458 (2008), Scharfenberg et al., Appl. Phys. Lett. 100, (2012), Bunch and Dunn, Solid. Stat. Comm. In press

12 Reaction-barrier reduction Hydrogen adsorption at sites with different local curvature DFT SIESTA, PBE GGA, 10x1x10 k-grid, SZ/SZP basis set Cut-off at 500 Ry, max force 0.01 ev/å ~ 25% reduction of adsorption barrier for R ~ 25 Å Radius of curvature of C 60 : 3.5 Å

13 Kink-line From sp 2 (planar) to sp 3 (tetrahedral) configuration of atoms in a line: Kinks Also considered by Chernozatonskii et al., Applied Physics Letters 91, (2007) Kinks pin the carbon

14 Formation of ribbons N atomic lines N=17 k in the ribbon direction create ribbons without cutting the graphene sheet. Follows N = 3j-1 metallic-rule like ribbons.

15 Nanostructured substrate: Thermal transport Sevincli, Brandbyge, APPLIED PHYSICS LETTERS 105, (2014) Density-functional tight-binding (DFTB) calculations + Lennard-Jones potential [DFTB: B. Aradi, B. Hourahine, and T. Frauenheim, J. Phys. Chem. A 111, 5678, (2007)]

16 Nanostructured substrate: Thermal transport Sevincli, Brandbyge, APPLIED PHYSICS LETTERS 105, (2014) Density-functional tight-binding (DFTB) calculations + Lennard-Jones potential [B. Aradi, B. Hourahine, and T. Frauenheim, J. Phys. Chem. A 111, 5678, (2007) SiC substrate: Typical insulating substrates: Metallic substrates:

17 Nanostructured substrate: Thermal transport Sevincli, Brandbyge, APPLIED PHYSICS LETTERS 105, (2014) Density-functional tight-binding (DFTB) calculations + Lennard-Jones potential [B. Aradi, B. Hourahine, and T. Frauenheim, J. Phys. Chem. A 111, 5678, (2007) SiC substrate: Typical insulating substrates: Metallic substrates:

18 Topics Thermoelectric properties of nano-structured graphene Anti-dot lattices Substrate/kinks Electron-phonon interaction: Phonons perturbing electronic current Inelastic electron transport spectroscopy Electron-phonon interaction: Electronic current perturbing phonons The generalized Langevin equation approach Amplification of vibrations by current ( Phonon-laser ) Current-induced forces influence the excess heat distribution/flow

19 Model and approximations Bottle-neck Nano-Structure System: phonons System Electron reservoir System-reservoir coupling Expansion in electron-phonon coupling M Electron-phonon coupling and transport with DFT: T. Frederiksen et al., PHYS. REV. B 75, (2007) Code: Siesta/Transiesta + the Inelastica Python package

20 Electron-phonon interaction: Two topics Unperturbed phonons Perturbed electronic current Inelastic Electron Transport(Tunnel) Spectroscopy (IETS)

21 Electron-phonon interaction: Two topics Unperturbed phonons Perturbed electronic current Inelastic Electron Transport(Tunnel) Spectroscopy (IETS) Unperturbed electrons (Steady-state current) Perturbed vibrations/phonons Current-induced forces in semi-classical Langevin equation for atomic dynamics

22 IETS Inelastic Electron Transport Spectroscopy P. K. Hansma, Phys. Rep. 30, 145 (1977) µ L µ R Au Low bias µ L High bias µ R Experiment: N. Agraït et al., PHYS. REV. LETT. 88, (2002) Theory: Frederiksen et al. PHYS. REV. LETT. 93, (2004), PRB 75, (2007)

23 IETS on molecules: Tunnel regime Alkanethiol: Okabayashi et al., PRL (2010), Nano Lett (2010) OPV/OPE/Alkane: Paulsson et al., Nano Lett (2006) Alkanedithiol: Paulsson et al., Nano Lett (2009) CO: Garcia-Lekue et al., PRB (2010) Alkanedithiol: Arroyo et al., PRB (2010)

24 Topics Thermoelectric properties of nano-structured graphene Anti-dot lattices Substrate/kinks Electron-phonon interaction: Phonons perturbing electronic current Inelastic phonon spectroscopy Electron-phonon interaction: Electronic current perturbing phonons The generalized Langevin equation approach Amplification of vibrations by current ( Phonon-laser ) Current-induced forces influence the excess heat distribution/flow

25 Electron-phonon interaction in nano/molecular electronics Joule heating Phonon dynamics with electronic current Current-induced forces Dell Laptop Fires Thermoelectrics: Drag-effects

26 Current-driven single-atom memory C. Schirm et al. Nature Nanotechnology 8, (2013) M. Matt et al., Nature Nanotechnology 8, 645 (2013) Theory G 0 = 2e2 h ¼ (13k ) 1

27 Semi-classical Generalized Langevin approach Joule heating Current-induced forces non-conservative Semi-classical Generalized Langevin equation (SGLE) Berry-phase force = Lorentz-like (no work) Equilibrium: MD with electronic frictions at equilibrium, M. Head-Gordon, J. C. Tully, J. Chem. Phys. 103 (1995) Negative friction Used for quantum thermal transport by J.-S. Wang, Phys. Rev. Lett. 99, (2007) Including electronic current: Brandbyge, Hedegård, Phys. Rev. Lett. 72, 2919 (1994) Lü, Brandbyge, Hedegård, Nano Lett. 10, (2010) Hussein, Metelman, Zedler, Brandes, Phys. Rev. B 92, (2010) Bode, Kusminskiy, Egger, v. Oppen, Phys. Rev. Lett. 107, (2011) J.-T. Lü, M. Brandbyge, P. Hedegård, T. N. Todorov, D. Dundas, Phys. Rev. B 85, (2012)

28 Method Electrons unperturbed by the phonons, steady-state current flow Landauer picture DFT-NEGF level Phonons perturbed by electrons (incl. Current) Lowest order in interaction (M) Effects of non-equilibrium electron occupation/current on the phonon dynamics Path-integral derivation following Schmid, A. J. Low Temp. Phys., 49, 609 (1982); Lü, Brandbyge, Hedegård, Nano Lett. 10, (2010), supplementary material The semi-classical Langevin equation: A physically intuitive (and practical) approach to the current-induced effects

29 Semi-classical Langevin equation Retarded self-energy Random noise force

30 Electronic wide-band : Modes Quasi-classical Langevin equation for the vibrational modes (Q): Non-conservative water-wheel forces: Anti-symmetric dynamic matrix Berry force Electronic friction Fluctuating force (Joule heating): Quantum fluctuations in electron reservoir J. T. Lü et al. Phys. Rev. B, 85, , (2012). J.-T. Lü, M. Brandbyge, P. Hedegård, Nano Lett. 10, 1657 (2010).

31 Simple 2D model: modes Non-conservative Berry y Water-Wheel Dundas et al., Nature Nano. 4, 99 (2009) x May obtain growing solutions: runaway Q-factor<0

32 Current-induced forces Energy non-conserving water-wheel forces Dundas, McEniry, Todorov, Nature Nano. 4, 99, (2009) Todorov, Dundas, Lü, Brandbyge, Hedegård, Eur. J. Phys (2014) How do these forces manifest themselves? Excess heating energy transfer Heat flow/heat distribution momentum transfer J.-T. Lü, R. B. Christensen, J-S. Wang, P. Hedegård, M. Brandbyge, Phys. Rev. Lett. 114, (2015)

33 DFT calculation: Au-wire WBA, no electrode phonons Au (100) Normal modes Runaway mode Parameters: DFT-NEGF: SIESTA / TranSIESTA Without n.c.f. With n.c.f. Soler et al., J. Phys.: Cond.. Mat. 14 (2002) Brandbyge et al., Phys Rev. B 65, (2002) Phonons/H ep : Inelastica Frederiksen et al., Phys Rev. B 75, (2007) J-T. Lü, M. Brandbyge, P. Hedegård, Nano Lett. 10, (2010)

34 Gold Breaking of Au Chains Early exp.: Yasuda, Sakai, PRB 56, 1069 (1997) Smit et al, Nanotechnology, 15, S472 (2004) Short chains ~ 3-5 atoms Long chains ~ 5-7 atoms Will the non-conservative force make wires break? Coupling to electrodes is important M. Engelund et al., Phys. Rev. B, 80, (2009)

35 Topics Thermoelectric properties of nano-structured graphene Anti-dot lattices Substrate/kinks Electron-phonon interaction: Phonons perturbing electronic current Inelastic phonon spectroscopy Electron-phonon interaction: Electronic current perturbing phonons The generalized Langevin equation approach Amplification of vibrations by current ( Phonon-laser ) Current-induced forces influence the excess heat distribution/flow

36 Current-driven amplification - phonon laser Phonon absorption Phonon emission Steady State: Damping/electronic friction el = A B

37 Donor-Acceptor system D pinned to L Filled A pinned to R Empty Equilibrium Cooling Amplification

38 DFT calculation Experimental setup: Functionalized STM-tip and adsorbed molecule D - A S NH 2 DFT-NEGF calculation using TranSIESTA A D D A J-T. Lü, P. Hedegård, M. Brandbyge, Phys. Rev. Lett. 107, (2011)

39 Negative friction: Graphene constriction T. Gunst, J-T. Lü, P. Hedegård, M. Brandbyge, PHYS. REV. B 88, R (2013) Beyond wide-band electronic structure: Full time kernel Include coupling to non-eq. electrons + phonons in the electrodes Density of phonon states (DOS) :

40 Topics Thermoelectric properties of nano-structured graphene Anti-dot lattices Substrate/kinks Electron-phonon interaction: Phonons perturbing electronic current Inelastic phonon spectroscopy Electron-phonon interaction: Electronic current perturbing phonons The generalized Langevin equation approach Amplification of vibrations by current ( Phonon-laser ) Current-induced forces influence the excess heat distribution/flow

41 Heating in atomic junctions Unsymmetrical hot electron heating in quasi-ballistic nanocontacts Makusu Tsutsui, Tomoji Kawai & Masateru Taniguchi, SCIENTIFIC REPORTS SCIENTIFIC REPORTS 2 : 217 (2012)

42 Asymmetry in heating: hot spots Only Joule Heat with non-cons. force Electron transport Hole transport J.-T. Lü, R. B. Christensen, J-S. Wang, P. Hedegaard, M. Brandbyge, Phys. Rev. Lett. 114, (2015)

43 Simple chain model Electron-like e-ph interaction region Two site e-ph interaction J.-T. Lü, R. B. Christensen, J-S. Wang, P. Hedegaard, M. Brandbyge, Phys. Rev. Lett. 114, (2015)

44 Simple chain model Hole-like e-ph interaction region Two site e-ph interaction J.-T. Lü, R. B. Christensen, J-S. Wang, P. Hedegaard, M. Brandbyge, Phys. Rev. Lett. 114, (2015)

45 Summary: Atomistic calculations Thermoelectric properties graphene A lot to gain by nano-structuring Electron-phonon interaction Spectroscopy in electronic current: molecular fingerprints Current can destabilize contacts ( water wheel force, laser ) Current-induced forces influence the excess heat distribution/flow