Energy transfer at the nanoscale: diodes and pumps Dvira Segal - PowerPoint PPT Presentation

Energy transfer at the nanoscale: diodes and pumps Dvira Segal Chemical Physics Theory Group University of Toronto

J Motivation Δ T • Quantum open systems out of equilibrium: Transport and dissipation. • Quantum energy flow: Heat conduction in bosonic/fermionic systems. • Nonlinear transport: diode, NDC J • Control: Pumping of heat Δ T • Nanodevices: Understand and manipulate heat transfer in molecular systems and nanoscale objects.

Outline I. Motivation II. Models for studying the fundamentals of quantum heat flow. III. Static case: Nonlinear effects Thermal rectification-diode 1. Experiment 2. Formalism 3. Sufficient conditions for thermal rectification IV. Dynamic case: Active control Stochastic heat pumps 1. Mechanism 2. Formalism 3. Examples: Control of the noise properties/ solid characteristics. 4. Efficiency: Approaching the Carnot limit V. Summary and Outlook

Introduction/Motivation Quantum energy flow Vibrational heat flow Photonic heat conduction Electronic energy transfer

Vibrational energy flow in molecules Fourier law in 1 D. IVR carbon nanotubes Nanomachines Molecular electronics Heating in nanojunctions. S S C. Van den Broeck, PRL (2006).

Phonon mediated energy transfer STM tip T L T R J Adsorbed molecules Metal G. Schultze et al. PRL 100, 136801 (2008) Z. Wang, et al., Science 317, 787 (2007) Strong laser pulse gives rise to strong increase of the electronic temperature at the bottom metal surface. Energy transfers from the hot electrons to adsorbed molecule.

Single mode heat conduction by photons The electromagnetic power (blackbody radiation) flowing in the ∞ ∫ ⎡ ⎤ γ e = ω ω − ω ω device is given by: P r n ( ) n ( ) d ⎣ ⎦ γ B B 0 R R D. R. Schmidt et al., PRL 93, 045901 γ e = coupling coefficient r 4 ( (2004). Experiment: M. Meschke et 2 + R R ) al., Nature 444, 187 (2006). γ e

Exchange of information Radiation of thermal voltage noise 2 2 π k T = B G Q 3 h The quantum thermal conductance is universal, independent of the nature of the material and the particles that carry the heat (electrons, phonons, photons) . K. Schwab Nature 444, 161 (2006)

Electronic energy transfer Coherence EET in poly- conjugated polymers (Collini and Scholes Science 323, 369 (2009) ) . The lines show the characteristic anticorrelation theoretically predicted for oscillations caused by electronic coherences.

II. Models: Energy flow in hybrid systems = + + + + H H H H V V S L R L R µ R = ∑ H E n n S n n H ν collection of phonons; electron-hole excitations; spins. ∑ = V F S n m ν ν n m , n m ,

. 1. Harmonic system = + + + + H H H H V V S L R L R T L T R J † = ω H b b S 0 0 0 ∑ † = ω H b b ν ν ν k , k , k k ∑ ( )( ) † † = λ + + V b b b b ν ν ν ν , k , k , k 0 0 k

2. Two Level System = + + + + H H H H V V S L R L R T L T R B J = σ H S z 2 ∑ † = ω H b b ν ν ν k , k , k k ∑ ( ) † = σ λ + V F ; F = b b ν ν ν ν ν ν x , k , k , k k

3. Energy transfer between metals T L ; µ L T R ; µ R J No charge † = ω H b b transfer S 0 0 0 ∑ † = ε H c c ν ν ν k , k , k = + + + + H H H H V V S L R L R k ∑ † = λ V c c S ν ν ν ν ν , ; , ' k k , k , ' k k k , '

III. Static Case: Nonlinear effects ∑ n = α Δ J ( T ) T n a n T L T R = + Δ = − T T T ; T T T a L R L R α = Δ lim / J T Conductance 1 Δ → T 0 α ≠ → Δ ≠ −Δ 0 J ( T ) J ( T ) Thermal rectification 2 α < → ∂ Δ ∂Δ < 0 J ( T ) / T 0 Negative differential 3 thermal conductance

Harmonic model ∫ T D. Segal, A. Nitzan, P. ⎡ ⎤ L R = ω ω − ω ω ω J ( ) n ( ) n ( ) d ⎣ ⎦ Hanggi, JCP (2003). B B

Thermal rectification ∑ Electrical rectifier n = α Δ J ( T ) T n a n → Δ ≠ −Δ α ≠ 0 J ( T ) J ( T ) Thermal rectification 2 Reed 1997 Asymmetry + Anharmonicity Thermal Rectification M. Terraneo, M. Peyrard, G. Casati, PRL (2002); B. W. Li, L. Wang, G. Casati, PRL (2004); D. Segal and A. Nitzan, PRL (2005),JCP (2005). B. B. Hu, L. Yang, Y. Zhang, PRL (2006) G. Casati, C. Mejia-Monasterio, and T. Prosen, PRL (2007) N. Yang, N. Li, L. Wang, and B. Li, PRB (2007) N. Zeng and J.-S. Wang, PRB (2008)

Experiment: thermal rectifier C 9 H 16 Pt Non uniform axial mass sensor distribution C. W. Chang, D. Okawa, A. Majumdar, A. Zettl, Science 314 , 1121 (2006). heater

Simulations B. W. Li, L. Wang, G. Casati, PRL (2004) = + + + + H H H H V V S L R L R 2 p V = 0 − 0 π H cos(2 x ) S 0 2 ( ) 2 m π 2 0 2 p 1 V ∑ L i , 2 = + − − π L H k ( x x ) cos(2 x ) + L L L i , L i , 1 L i , 2 ( ) 2 m 2 π 2 i i k 2 = int L − V ( x x ) L L N , 0 2

Formalism: Master Equation = + + + Model: H H H H V S L R ∑ = H E n n S n n ∑ = + = = λ V V V V ; F S n m ; F B ν ν ν ν ν L R n m , n m , H collection of phonons; electron-hole excitations; spins. ν i ( ) [ ] = − ν ρ Heat current: J Tr H H , V ν ν S 2 Dynamics: Liouville equation in the interacation picture ρ t d ∫ ⎡ [ ] ⎤ m n , = − ρ − τ τ ρ τ i V t [ ( ), (0)] d V t ( ), V ( ), ( ) ⎣ ⎦ m n , dt m n , 0

Formalism: Master Equation Liouville Equation � Pauli Master equation ∑ ∑ 2 2 & ν ν = − P t ( ) S P t k ( ) ( T ) P t ( ) S k ( T ) → ν → ν n n m , m m n n n m , n m ν ν , m , m ∞ = ∫ τ iE ν 2 = λ τ τ k ( T ) f T ( ); ( f T ) d e n m , B ( ) B (0) → ν ν ν ν ν ν ν ν n m T ν −∞ 1 ∑ 2 ⎡ ⎤ L R = − J E S P t ( ) k ( T ) k ( T ) ⎣ ⎦ → → m n , n m , n n m L n m R 2 n m , Weak system-bath coupling limit; <B ρ (0)>=0; Factorization of the density matrix of the whole system; Markovian limit.

Sufficient conditions for thermal rectification Harmonic force field Anharmonic force field ρ ≠ ρ (1) ( ) T H ( ) T H L L R R The reservoirs have different mean energy ⎛ ⎞ ⎛ ⎞ H C − ω − ω n ( ) 1 1 n ( ) 1 1 ⎜ − ⎟ ≠ ⎜ − ⎟ (2) 2 2 2 2 ⎝ λ λ ⎠ ⎝ λ λ ⎠ f T ( ) f T ( ) H L R C L R T H T C g(T H ) g(T C ) The relaxation rates' temperature dependence should differ from the central unit occupation function, combined with some spatial asymmetry. ∞ = ∫ τ iE ν 2 = λ τ τ k ( T ) f T ( ); ( f T ) d e n m , B ( ) B (0) → ν ν ν ν ν ν ν ν n m T ν −∞ L.A. Wu and D. Segal, PRL (2009). L.A. Wu, C.X. Yu, and D. Segal arXiv: 0905.4015

Spin-boson thermal rectifier ( ) L R L R Γ Γ − n n B B B B = ω J ( ) ( ) 0 L L R R Γ + + Γ + 1 2 n 1 2 n B B B B L R Γ Γ ( ) L R = ω − B B J n n 0 B B L R Γ + Γ B B D. Segal, A Nitzan PRL (2005).

III. Dynamic Case: Active control Until now: Heat was flowing from hot objects to cold objects. Question 1: Can we direct heat against a temperature gradient? Answer 1: Add (i) external forces (ii) asymmetry Heat pump moves heat from a cold bath to a high temperature bath. Question 2: Do we need to shape the external force in order to achieve the pumping operation? Answer 2: Random noise can lead to pumping. J J W

Simple model: Stochastic heat pump = + + + + H H H H V V S L R L R J L <0 J R >0 + ε B ( ) t ω L ω R = 0 σ H S z 2 T R T L ∑ † = ω H b b J L >0 J R <0 ν ν ν k , k , k k ∑ ( ) † = σ λ + V F ; F = b b ν ν ν ν ν ν x , k , k , k k Spectral function of the reservoirs ∑ ( ) 2 ω = π λ δ ω − ω g ν ( ) 2 ν , k k k

Mechanism: Random fluctuations catalyze heat flow W ω The subsystem is coupled to both L ends ω R W ω L The subsystem is coupled to the left ω side only. TLS temperature is R effectively high T TLS >T L >T R D. Segal, A. Nitzan, PRE (2006). D Segal PRL (2008); JCP (2009).

Formalism: Population Liouville equation � Pauli Master equation . ( ) ( ) L R L R = − + + + P t ( ) k k P t ( ) k k P t ( ) 1 → → → → 1 0 1 0 1 0 1 0 1 0 ε ε ε Transition rates: ∞ ∞ ∫ ∫ ( ) ν ν = ω ω + ω − ω = ω ω ω − ω k d g ( ) 1 n ( ) I B ( ); k d g ( ) n ( ) ( I B ) → ν ν → ν ν 1 0 0 0 1 0 −∞ −∞ Spectral lineshape of the Kubo oscillator: ∞ t 1 ∫ ∫ ω i t ω = ε ω I ( ) e exp i ( ') t dt ' d π 2 −∞ 0 ε