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Fundamental aspects of transport in nanostructures and atomic physics Massimiliano Di Ventra Department of Physics, University of California, San Diego Collaborators Neil Bushong (NC) Yuriy Pershin (USC) Chih-Chun (LANL) Mike Zwolak (OSU) Roberto D’Agosta (Spain) Giovanni Vignale (U. Missouri) Outline • Introduction to the transport problem • Many-body effects related to viscosity of the electron liquid (large for structures with smaller transmissions) • Properties of steady states and predictions Theory: Microcanonical picture of transport Experiments: Atomic gases in optical lattices Outline • Introduction to the transport problem • Many-body effects related to viscosity of the electron liquid (large for structures with smaller transmissions) • Properties of steady states and predictions Theory: Microcanonical picture of transport Experiments: Atomic gases in optical lattices Field: nanoscale electronics Nanotubes/wires Atomic point contacts Molecular junctions Tans et al. (1997) Organic electronics Scheer et al. (1998) from Nitzan et al. (2003) Fast DNA sequencing Poly(3-hexylthiophene) source polymer drain insulator Doped Si Lagerqvist et al. (2006) Z.Q. Li et al. (2006) What do we want to describe ? R R 1 G dI dV Solved ? Not quite, especially at the atomic level ! Major difference with macro/mesoscopic systems ph e Pw I th e I th forces Large current densities increased e-e, e-ph scattering Why is the problem difficult (and interesting) ? • The system is out of equilibrium (non-equilibrium statistical mechanics is still an open subject; do we need to go beyond Hamiltonian dynamics?) • Interactions among electrons (Coulomb blockade, correlations, non-Fermi liquid behavior) • Interactions among electrons and ions (e.g., el-phonon scattering, current-induced forces) • Interaction with the environment (dissipation and dephasing) • Physical properties are quite sensitive to atomic details From experiment to model system Approximation 1: open quantum systems Closed system H H S H Battery Hint S T rBattery Open system: dynamical interaction with reservoirs From experiment to model system Approximation 1: open quantum systems H H S H Battery Hint S T rBattery Battery dense spectrum No initial correlations Small interaction t S S In general, no closed equation of motion for s From experiment to model system Approximation 2: ideal steady state Assume existence of at least one steady state solution ss I Tr S (t ) I Tr I I constant t Still many-body open quantum system ! From experiment to model system Approximation 3: “openness” vs boundary conditions Loss of information closedsystem(with battery) opensystem closed and infinitedifferentsystem From experiment to model system Approximation 4: mean-field approximation With or w/o interaction Non-interacting electrons Non-interacting electrons If leads are interacting NO closed equation of motion for the current ! From experiment to model system Approximation 5: independent channels and energy filling Non-interacting electrons Non-interacting electrons With or w/o interaction The Landauer current Non-interacting electrons From scattering theory I e e dE f ( E ) f ( E ) T ( E ) dE f ( E ) f ( E ) Tr G G L R L R R L This formula has nothing to do with NEGF !!! Interacting sample Non-interacting electrons Non-interacting electrons With interactions From NEGF I ie dE Tr ( E ) ( E ) G f ( E ) ( E ) f ( E ) G ( E ) G (E) L R L L R R 2 Meir and Wingreen, 1992 Physical origin of many-body corrections: linear-response theory ji (r) ij (r,r') E j (r') dr' j xˆ 2 V xˆ 3 Current conservation C2 i ij (r,r') 0 C3 Gauge invariance C1 C4 xˆ 1 xˆ 4 'j ij (r,r') 0 Gmn = - dr dr' xˆ mi ij (r,r') xˆ nj Cm Cn mn Physical origin of many-body corrections: linear-response theory ij (r, r') e lim 2 0 ˜ ij (r, r'; ) Im “Proper” currentcurrent response function ˜ ij (r,r'; ) j (r) j j (r') i + + For a non-interacting system ij (r,r') e 2 f ( ) Wi* (r)Wj (r') e e I dE f ( E ) f ( E ) T ( E ) dE f ( E ) f ( E ) Tr G G L R L R R L +… Outline • Introduction to the transport problem • Many-body effects related to viscosity of the electron liquid (large for structures with smaller transmissions) • Properties of steady states and predictions Theory: Microcanonical picture of transport Experiments: Atomic gases in optical lattices Interactions in the whole system: the microcanonical picture of transport j n t I exact jexactds jexactdv jKS dv jKS ds I KS S V V M. Di Ventra, T.N. Todorov, (J. Phys. Cond. Matt. 2004) S Fast relaxation of momentum w momentum relaxation time 1/nc tc ħ/E m w2/2ħ2 1 fs Bushong, Sai and M. Di Ventra (Nano Letters 2005) Comparison with Landauer formula Chen, Zwolak and Di Ventra , in preparation Entanglement entropy R L Gaussian L TrR Z S L Tr[ L log( L )] Binomial Exact C(t) = correlation matrix PL = projection operator Chen, Zwolak and Di Ventra , in preparation Approximate Klich and Levitov, PRL 2009 Electron flow Quasi-2D electron liquid, TDDFT V= 0.2V 2 KS 1 e i Axc (r , t ) Vext (r , t )k (r , t ) 0 i c t 2m Sai, Bushong, Hatcher, and Di Ventra , PRB 2007 Hydrodynamics of the electron liquid d (t ) i H (t ) dt Exact! n = density v = j/n Dt n n v 0 continuity mnDt v j i Pij n jVext 0 F=ma Information on all e-e interactions (generally unknown) A hydrodynamic formulation is more natural in QM than in classical physics Martin and Schwinger, Phys. Rev. (1959) Anticipates TDDFT by many years ! Fast relaxation of momentum w momentum relaxation time 1/nc tc ħ/E m w2/2ħ2 1 fs Bushong, Sai and M. Di Ventra (Nano Letters 2005) Quantum Navier-Stokes equations Dt n n v 0 mnDt v j i Pij n jVext 0 Pij F[Pijk F[Pijkl ]] 1) 1 I [ f ] pi p j Dt Pij Pij v Pik k v j Pkj k vi k Pijk(3) m 2) v /( L max(,n c )) 1 n 3) R. D’Agosta and M. Di Ventra, JPCM (2006) Conductance quantization from hydrodynamics Dt n n v 0 mnDt v j j P ~ij n jVext 0 1D, stationary, non-viscous fluid 2 P n n 2m 2 v P Vext const 2 n Bernoulli 2 2 e I V h D’Agosta and M. Di Ventra, JPCM (2006) Quantized conductance is the one of a 1D ideal, non-viscous charged fluid Electron Dynamics Electron dynamics in nanostructures similar to a viscous liquid Predictions turbulence electron heating effect on ion heating Turbulence in nanoscale systems QPCs molecules Actual atomic structure Approx. potential adiabatic D’Agosta and M. Di Ventra, JPCM (2006) non-adiabatic Turbulence in nanoscale systems mI Re e Adiabatic constrictions (e.g., QPC), laminar flow laminar turbulent e 10 air 2 D’Agosta and M. Di Ventra, JPCM (2006) Nonadiabatic constrictions (e.g., molecules), turbulent flow Time-Dependent Current DFT 2 KS 1 e i Axc (r , t ) Vext (r , t )k (r , t ) 0 i c t 2m xc,ij (r, t ) e Axc,i (r, t ) 1 ALDA eExc,i (r, t ) iVxc (r , t ) r c t n(r, t ) j j no memory, Markov approx. 2 xc,ij (r , t ) (n, r , t ) j vi i v j ij k vk (n, r , t ) v ij 3 bulk viscosity Vignale and Kohn PRL 1996; Vignale, Ullrich and Conti (1997) Electron turbulence TDCDFT Closed system, quasi-2D electron liquid, TDCDFT Eddies size Rcr Re 0 ~ l 3/ 4 nanoscale, for fully developed turbulence Bushong, Gamble, and M. Di Ventra (Nano Lett. 2007); D’Agosta and Di Ventra JPCM (2006) NS 0.02 V (Laminar) Bushong, Gamble and M. Di Ventra (Nano Lett. 2007) 3 V (Turbulent) Bushong, Gamble and M. Di Ventra (Nano Lett. 2007) Possible exp. verification Laminar Bushong, Pershin and M. Di Ventra (Phys. Rev. Lett. 2007) Possible exp. verification Turbulent Bushong, Pershin and M. Di Ventra (Phys. Rev. Lett. 2007) Possible exp. verification Bushong, Pershin and M. Di Ventra (Phys. Rev. Lett. 2007) Electron Dynamics Electron dynamics in nanostructures similar to a viscous liquid Predictions turbulence electron heating effect on ion heating Electron heating: elementary considerations Power in the junction: Heat dissipated in the electrodes: 2 Vbias Pe , 1 R I th nTe2 I th Pe Te eVbias D’Agosta, Sai and M. Di Ventra, Nano Lett. (2006) Pw I th I th Electron heating from hydrodynamics Heat equation (no turbulence) ~ij jivi [kTe ] 0 Pw I th I th Thermal conductivity Te eVbias e.g. Au QPC e viscosity D’Agosta, Sai and M. Di Ventra, Nano Lett. (2006) e 65K / V Electron Dynamics Electron dynamics in nanostructures similar to a viscous liquid Predictions turbulence electron heating effect on ion heating Ionic Heating: elementary considerations Power in the junction: Heat dissipated in the electrodes: 2 Vbias Pw , 1 R I th nTw4 I th Pw Tw Vbias Y-C Chen, M. Zwolak, M. Di Ventra Nano Letters (2003) T.N. Todorov Phil. Mag, (1998) Pw I th I th Effect of heating on shot noise and current ? Tw Vbias G Vbias 0 G F S Vbias 2eI I th I th Chen, Di Ventra, PRB (2003); Phys. Rev. Lett. (2005) Agrait et al., Phys. Rev. Lett. (2002) Effect of el. and ion heating on inelastic scattering Exp. Djukic et al PRB (2005) H2 e--e- e--ph Th. D’Agosta and M. Di Ventra, J. Phys. Cond. Matt. (2008) Effect on the ionic temperature: theory Pw I th I th Tw [ 4e pV 2 4eV 4 ]1/ 4 e--ph D’Agosta, Sai and M. Di Ventra, Nano Lett. (2006) e--e- Effect on the ionic temperature: exp. c Teff [ 4e pV 2 4eV 4 ]1/ 4 325 a Teff /K 315 F * f ln(toff / f ) f ln rF (1) toff E t D exp( b ) K BTeff Teff Eb KB 1 f F* ln rF t D f t D 1.5 10 10 (2) 305 C6 C8 C10 295 0.0 s, Eb 0.52eV , f 0.26nN 0.4 0.8 Vbias /V Huang et al. Nano Lett. (2006); Huang et al. (Nature Natotech., 2007) 1.2 1.6 Outline • Introduction to the transport problem • Many-body effects related to viscosity of the electron liquid (large for structures with smaller transmissions) • Properties of steady states and predictions Theory: Microcanonical picture of transport Experiments: Atomic gases in optical lattices Interactions in the whole system: the microcanonical picture of transport j n t I exact jexactds jexactdv jKS dv jKS ds I KS S V V M. Di Ventra, T.N. Todorov, (J. Phys. Cond. Matt. 2004) S Formation of steady states w ~ G0 The formation of a steady state has nothing to do with the infinite nature of momentum relaxation time 1/nc tc ħ/E m w2/2ħ2 1 fs the electrodes Bushong, Sai and M. Di Ventra (Nano Letters 2005) Cold atoms are ideal systems for studying transport phenomena • You can choose: 1.fermions or bosons 2.harmonic trap, optical lattice, or both (Box-potential is coming soon!) 3.single or multi components or species 4.dimensions (3D, 2D, 1D, or mixed) • You can tune: 1.interactions among atoms (via Feshbach resonance) 2.trap depth or lattice constant 3.temperature 4.density / filling factor Experimental tests: cold atoms Observation of Fermi surfaces I.Bloch, Nat. Phys. 1, 23 (2005) Superfluid-Mott insulator transition Hambury-Brown-Twiss experiment Quantum ratchet Science 326, 1241 (2009) Density-induced transport 1. Loading non-interacting single-species atoms into the ground state. 2. Remove particles on the right half using photons. T=0, N particles: fermions: bosons: Bosonic vs fermionic currents bosons (quasi-condensate) fermions (Fermi sea) More on the current • Fermionic QSSC: Robust against trap and T • Bosonic current: Never reaches a finite QSSC even in the thermodynamic limit at T=0 IN 1/ 2 Interaction-induced transport 1. Loading non-interacting twocomponent fermions into the ground state. 2. Turning on interactions on half of the lattice. Optically controlled collisions: Optical Feshbach resonance (coupling to auxiliary channels by photons. Realized in Yb and Sr, proposed for Li.): PRL 105, 050405; PRA 79, 021601; PRL 107,073202; PRL 108, 010401. Equations of motion Using Wick’s decompositions at 2-particle level or 3-particle level show similar results. QSSC and conducting-nonconducting transition Similar to the negative differential resistance in semiconductor devices. (Phys. Today 23, 35 (1970), IBM J. Res. Develop. 14, 61 (1970).) Mismatch of energy spectra 1. The switch-on of U changes the energy dispersion of the left-half lattice. 2. Moving a high-energy particle to a low-energy state or vice versa is forbidden by the underlying quantum dynamics. 3. The blockade is dynamical. (unlike Mott insulator in equilibrium which is due to energy minimization.) Conclusions • Microcanonical picture of transport • Many-body effects related to viscosity of the electron liquid (large for structures with smaller transmissions) • Predictions (some verifiable in atomic lattices): • Turbulence of the electron liquid • Quasi-steady state formation also in finite systems • Fermi distributions not necessary for the existence of a QSS • Conducting-nonconducting transition due to interactions Future direction: transport in a liquid environment Idea: Transverse Transport Voltage Biased translocaton DNA -A-C- T-G - ACTG FIB Milled Channels (10-50-nm width) AFM Formed Channel (1-5-nm width and length) (b) (a) (c) 750 nm Microchannels Microchannels STM Formed Nanoelectrodes 2-nm width Idea: Transverse Transport e- E E|| M. Zwolak and M. Di Ventra, Nano Lett. 5, 421 (2005) MD + Quantum Transport Hamiltonian Transverse Transport (dynamics) J. Lagerqvist, M. Zwolak, and M. Di Ventra, Nano Letters 2006 Current Distributions Accuracy 99.9 % 106 measurements/ s Genome seq. time < 3 days No parallelization 3 10 bases (70counts) s 3days 9 106 m easurem ents N Error 1 P I n X A,T ,C ,G J. Lagerqvist, M. Zwolak, and M. Di Ventra, Nano Letters 2006 X 4 P n X n 1 N N N N P P P P n A n 1 n T n 1 n C n 1 n G n 1 Idea: Transverse Transport Voltage Biased translocaton DNA -A-C- T-G - ACTG FIB Milled Channels (10-50-nm width) http://www.mcb.harvard.edu/branton/ Microchannels Microchannels STM Formed Nanoelectrodes 2-nm width AFM Formed Channel (1-5-nm width and length) (b) (a) (c) 750 nm M. Ramsey, UNC Branton et al., Harvard S. Lindsay T. Kawai S.Y. Chou U. Penn, IBM, Samsung…..