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

Motivation

• Quantum open systems out of equilibrium: Transport and

dissipation.

• Quantum energy flow: Heat conduction in bosonic/fermionic systems.
• Nonlinear transport: diode, NDC
• Control: Pumping of heat
• Nanodevices: Understand and manipulate heat transfer in molecular

systems and nanoscale objects.

J ΔT J

ΔT

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

Quantum energy flow Vibrational heat flow Photonic heat conduction Electronic energy transfer

Introduction/Motivation

Vibrational energy flow in molecules

S S

Molecular electronics

Heating in nanojunctions.

Fourier law in 1 D.

• C. Van den Broeck, PRL (2006).

Nanomachines IVR

carbon nanotubes

Phonon mediated energy transfer

• G. Schultze et al. PRL 100,

136801 (2008)

STM tip Adsorbed molecules Metal

• 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.

J TR TL

Single mode heat conduction by photons

• D. R. Schmidt et al., PRL 93, 045901

(2004). Experiment: M. Meschke et al., Nature 444, 187 (2006).

2

The electromagnetic power (blackbody radiation) flowing in the device is given by: ( ) ( ) coupling coefficient 4 ( )

e B B e e

P r n n d R R r R R

γ γ γ γ

ω ω ω ω

∞

⎡ ⎤ = − ⎣ ⎦ = +

∫

Exchange of information

• K. Schwab Nature 444, 161 (2006)

Radiation of thermal voltage noise

The quantum thermal conductance is universal, independent of the nature

• f the material and the

particles that carry the heat (electrons, phonons, photons) .

2 2

3

B Q

k T G h π =

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

• scillations caused by

electronic coherences.

• II. Models: Energy flow in hybrid systems

S L R L R

H H H H V V = + + + +

S n n

H E n n =∑

µR

collection of phonons; electron-hole excitations; spins. Hν

, , n m n m

V F S n m

ν ν

=

∑

• 1. Harmonic system

J TR TL

( )( )

† 0 0 † , , † † , , , S L R L R S k k k k k k k k

H H H H V V H b b H b b V b b b b

ν ν ν ν ν ν ν

ω ω λ = + + + + = = = + +

∑ ∑

.

• 2. Two Level System

J TR TL

( )

† , , † , , ,

2 ; =

S L R L R S z k k k k x k k k k

H H H H V V B H H b b V F F b b

ν ν ν ν ν ν ν ν ν

σ ω σ λ = + + + + = = = +

∑ ∑

J TR ; µR TL ; µL

• 3. Energy transfer between metals

S L R L R

H H H H V V = + + + +

† 0 0 † , , † , ; , ' , , ' , ' S k k k k k k k k k k

H b b H c c V c c S

ν ν ν ν ν ν ν ν

ω ε λ = = =

∑ ∑

No charge transfer

• III. Static Case: Nonlinear effects

( )

n n a n

J T T α = Δ

∑

1 2 3

lim / Conductance ( ) ( ) Thermal rectification ( ) / 0 Negative differential

T

J T J T J T J T T α α α

Δ →

= Δ → Δ ≠ −Δ ≠ < → ∂ Δ ∂Δ < thermal conductance

TL TR

;

a L R L R

T T T T T T = + Δ = −

( ) ( ) ( )

L R B B

J n n d ω ω ω ω ω ⎡ ⎤ = − ⎣ ⎦

∫T

• D. Segal, A. Nitzan, P.

Hanggi, JCP (2003).

Harmonic model

2

( ) ( ) ( ) Thermal rectification

n n a n

J T T J T J T α α = Δ → Δ ≠ −Δ ≠

∑

Thermal rectification

Reed 1997 Electrical rectifier

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)

• C. W. Chang, D. Okawa, A. Majumdar,
• A. Zettl, Science 314, 1121 (2006).

C9H16Pt Non uniform axial mass distribution

sensor heater

Experiment: thermal rectifier

S L R L R

H H H H V V = + + + +

( )

2 2

cos(2 ) 2 2

S

p V H x m π π = −

( )

2 , 2 , , 1 , 2

1 ( ) cos(2 ) 2 2 2

L i L L L L i L i L i i i

p V H k x x x m π π

+

= + − −

∑

2 int ,

( ) 2

L L L N

k V x x = −

Simulations B. W. Li, L. Wang, G. Casati, PRL (2004)

Formalism: Master Equation

[ ]

( )

Heat current: Tr , 2

S

i J H H V

ν ν ν ρ

= −

, ,

Model: ; ; collection of phonons; electron-hole excitations; spins.

S L R S n n L R n m n m

H H H H V H E n n V V V V F S n m F B H

ν ν ν ν ν ν

λ = + + + = = + = =

∑ ∑

[ ]

, , ,

Dynamics: Liouville equation in the interacation picture [ ( ), (0)] ( ), ( ), ( )

t m n m n m n

d i V t d V t V dt ρ ρ τ τ ρ τ ⎡ ⎤ = − − ⎣ ⎦

∫

Formalism: Master Equation

Liouville Equation Pauli Master equation

2 2 , , , ,

( ) ( ) ( ) ( ) ( )

n n m m m n n n m n m m m

P t S P t k T P t S k T

ν ν ν ν ν ν → →

= −

∑ ∑

&

,

2

( ) ( ); ( ) ( ) (0)

n m

iE n m T

k T f T f T d e B B

ν

τ ν ν ν ν ν ν ν ν ν

λ τ τ

∞ → −∞

= = ∫

2 , , ,

1 ( ) ( ) ( ) 2

L R m n n m n n m L n m R n m

J E S P t k T k T

→ →

⎡ ⎤ = − ⎣ ⎦

∑

Weak system-bath coupling limit; <Bρ(0)>=0; Factorization of the density matrix of the whole system; Markovian limit.

Sufficient conditions for thermal rectification

(1) ( ) ( ) The reservoirs have different mean energy

L L R R

T H T H ρ ρ ≠

2 2 2 2

( ) 1 1 ( ) 1 1 (2) ( ) ( ) The relaxation rates' temperature dependence should differ from the central unit occupation function, combined with some spatial asymmetry.

H C H L R C L R

n n f T f T ω ω λ λ λ λ ⎛ ⎞ ⎛ ⎞ − − − ≠ − ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ ⎝ ⎠

L.A. Wu and D. Segal, PRL (2009). L.A. Wu, C.X. Yu, and D. Segal arXiv: 0905.4015 Harmonic force field Anharmonic force field

,

2

( ) ( ); ( ) ( ) (0)

n m

iE n m T

k T f T f T d e B B

ν

τ ν ν ν ν ν ν ν ν ν

λ τ τ

∞ → −∞

= = ∫

g(TH) g(TC)

TH TC

Spin-boson thermal rectifier

• D. Segal, A Nitzan PRL (2005).

( )

L R L R B B B B L R B B

J n n ω Γ Γ = − Γ + Γ ( ) ( ) ( )

1 2 1 2

L R L R B B B B L L R R B B B B

n n J n n ω Γ Γ − = Γ + + Γ +

• 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 W J

Simple model: Stochastic heat pump

( )

† , , † , , ,

( ) 2 ; =

S L R L R S z k k k k x k k k k

H H H H V V B t H H b b V F F b b

ν ν ν ν ν ν ν ν ν

ε σ ω σ λ = + + + + + = = = +

∑ ∑

JR>0 JR<0 JL<0 JL>0

ωL ωR

TL TR

( )

2 ,

Spectral function of the reservoirs ( ) 2

k k k

gν

ν

ω π λ δ ω ω = −

∑

Mechanism: Random fluctuations catalyze heat flow

W

L

ω

R

ω

R

ω W

L

ω

• D. Segal, A. Nitzan, PRE (2006).

D Segal PRL (2008); JCP (2009).

The subsystem is coupled to both ends The subsystem is coupled to the left side only. TLS temperature is effectively high TTLS>TL>TR

Formalism: Population

Liouville equation Pauli Master equation

( ) ( )

. 1 1 1 1 1 1

( ) ( ) ( )

L R L R

P t k k P t k k P t

ε ε ε → → → →

= − + + +

( )

1 1

Transition rates: ( ) 1 ( ) ( ); ( ) ( ) ( ) k d g n I B k d g n I B

ν ν ν ν ν ν

ω ω ω ω ω ω ω ω

∞ ∞ → → −∞ −∞

= + − = −

∫ ∫

Spectral lineshape of the Kubo oscillator: 1 ( ) exp ( ') ' 2

t i t

I e i t dt d

ω ε

ω ε ω π

∞ −∞

=

∫ ∫

Spectral lineshape of the Kubo oscillator: 1 ( ) exp ( ') ' 2

t i t

I e i t dt d

ω ε

ω ε ω π

∞ −∞

=

∫ ∫

2 1 1 1 2 1 2 1 2

exp ( ) ( ) ( ) ( ) ( ) ( ) ( )

t t t

K t K t i dt t i dt dt t t t t

ε ε ε ε

ε ε ε ε ε = ⎡ ⎤ = + − + ⎣ ⎦

∫ ∫ ∫

K

Formalism: Random Frequency modulations

(Kubo Oscillator)

2 2

For a Gaussian process in the fast modulation limit / Define ' ( ) ( ') Obtain: ( ) dt t t t I γ π γ ε ε ω ω γ

∞

≡ + = +

∫

[ ] [ ]

1 1 1 1

( ) 1 ( ) ( ) ( ) 1 ( ) ( ) ( ) ( ) ( ) ( ) k d g n I B k g B n B k d g n I B k g B n B

γ ν ν ν ν ν ν γ ν ν ν ν ν ν

ω ω ω ω ω ω ω ω

∞ → → → −∞ ∞ → → → −∞

= + − → = + = − → =

∫ ∫

Formalism: Transition rates

2 2

/ For a Gaussian process in the fast modulation limit ( ) I γ π ω ω γ = +

Field-free vibrational relaxation rates Kubo oscillator transition rates

Formalism: Current

[ ]

ˆ Current operator: ( ) , 2

R S R R

i J H t H V = −

1 1 1

Master equation description:

R R R

J P f P f

ε ε ε → →

= −

[ ]

1 1

( ) ( ) ( ) ( ) ( ) ( ) 1 f d g I B n f d g I B n

ν ν ν ν ν ν

ωω ω ω ω ωω ω ω ω

∞ → −∞ ∞ → −∞

= − = − +

∫ ∫

1 0 1

B k B k

γ ν γ ν → → → →

→ →

Kubo oscillator transition rates Field-free vibrational relaxation rates

( ) ( )

. 1 1 1 1 1 1

Population: ( ) ( ) ( )

L R L R

P t k k P t k k P t

ε ε ε → → → →

= − + + + ( )

1 1

Transition rates: ( ) 1 ( ) ( ); ( ) ( ) ( ) k d g n I B k d g n I B

ν ν ν ν ν ν

ω ω ω ω ω ω ω ω

∞ ∞ → → −∞ −∞

= + − = −

∫ ∫

1 1 1

Heat current:

R R R

J P f P f

ε ε ε → →

= −

[ ]

1 1

( ) ( ) ( ) ( ) ( ) ( ) 1 f d g I B n f d g I B n

ν ν ν ν ν ν

ωω ω ω ω ωω ω ω ω

∞ → −∞ ∞ → −∞

= − = − +

∫ ∫

Formalism: Summary

Numerical Results

γ B0 2 4 6 12 14 16 18 20 22

0.1 0.2 Colormap of the heat flux at the Right contact JR>0 JR<0 JL<0 JL>0

ωL ωR

TL TR RL ωL =200, ωR=3, TL=TR=25.

( )

2 2

/ ( ) ( ) exp / I g A

ν ν ν

γ π ω ω γ ω ω ω ω = + = −

γ TL-TR 2 4 6 8 2 4 6 8 10 12

0.1 0.2

Colormap of the heat flux at the Right contact ωL =200, ωR=3, B0=15 TR=25.

Numerical Results

Pumping JR>0 JR<0 JL<0 JL>0

ωL ωR

TL TR

Proof of principle for a dichotomous noise

L

ω

R

ω

R

ω

L

ω

E1-E0=B0-Ω

( ) ( )

Dichotomous noise 1 ( ) ~ 2 I ω δ ω δ ω − Ω + + Ω ⎡ ⎤ ⎣ ⎦

( )

2 ,

( ) 2 Assumption: The R Reservoir spectral denisty strongly varies within the noise spectral window ( ) ( )

k k k R R

g g B g B

ν ν

ω π λ δ ω ω = − + Ω << − Ω

∑

E1-E0=B0+Ω

If TL=TR, it can be shown that current is catalyzed from the R side into the L side when the following condition is satisfied

( ) ( )[

] [ ]

( ) ( ) ( ) ( ) 1 ( ) 1 ( ) 1 Or ( ) ( )

L R L L R L

g g n B g n B n B n B g g n B g n B n B n B

− − + − − +

+ − Ω + + Ω − Ω < − Ω + + − Ω + + + Ω + + Ω < − Ω

( ) g g B

ν ν ± =

± Ω

Efficiency: Approaching the Carnot limit

2

Assume Einstein solids with g ( ) 2 ( )

ν ν ν

ω πλ δ ω ω = −

max

[ ( ) ( )] Pumping condition: Work: ( ) [ ( ) ( )] Cooling efficiency =

L L R R L R L R R R R R L L L R R R R R L R L R

J n n T T J T W n n J T W T T

ν ν ε ε ε ε

ω ω ω ω ω ω ω ω ω ω ω η η ω ω = − − − < → < = − − − = < = − − T฀ T฀ ωL ωR

TL TR

• Noise processes in nanomechanical resonators: Adsorption-desorption

noise, temperature fluctuations.

(Clealand and Roukes, J. App. Phys., 92 2758 (2002), Y.T. Yang et al. Nano Lett (2006). ).

• The resonance frequency can be tuned by the gate- voltage fluctuations

change the characteristic modes.

(Sazanova et al. Nature 431, 285 (2004).

Physical Realizations

Exciton stochastic pump

(a) The Metals have different band structure. Delta-like DOS will lead to the Carnot efficiency. Related ideas, showing reversible particle transfer, were considered in Humphrey and Linke PRL 2005.

Pumping of heat by modulating the reservoirs temperatures

Computer simulation of a heat pump where the temperature of one reservoirs is modulated periodically.

N.Li, B. Li and P. Hanggi,EPL (2008)

We studied quantum heat transfer in minimal models, seeking to connect the transport characteristics with the microscopic description (H J ?)

• Static transport: We discussed sufficient conditions for

thermal rectification.

• Dynamical control: We studied pumping of heat due to

the shaping of the reservoirs properties, given that the system is suffering a stochastic noise.

Summary

Formal issues:

• Going beyond the perturbative treatment.
• Add coherent effects
• Consider more realistic models.

Basic challenges:

• Static case: Understand other nonlinear phenomena,

e.g. negative differential thermal conductance, from first principles. Understand the ballistic- diffusive heat- flow crossover (Fourier law).

• Dynamic case: Control heat flow by modulating the

reservoirs temperatures.

Outlook

Rectifiers: Abraham Nitzan Tel Aviv University Lianao Wu University of Toronto & The Basque Country University at Bilbao Claire Yu University of Toronto Pumps: Abraham Nitzan Tel Aviv University

Thermal rectification:

• D. Segal and A. Nitzan, PRL 94, 034301 (2005); JCP 122, 194704 (2005).
• D. Segal, PRL 100 105901 (2008).

L.A. Wu and D. Segal, PRL 102, 095503 (2009); L.A. Wu, C. X. Yu, and D. Segal arXiv 0905:4015. Molecular heat pumps:

• D. Segal and A. Nitzan, PRE 73, 02609 (2005).
• D. Segal PRL 101 260601 (2008); JCP 130, 134510 (2009).

Thanks-

,

2

( ) ( ); ( ) ( ) (0)

n m

iE n m T

k T f T f T d e B B

ν

τ ν ν ν ν ν ν ν ν ν

λ τ τ

∞ → −∞

= = ∫

2

Harmonic bath, bilinear coupling ( ) ( ) 2 ( )

B j j

k T n

ν ν ν ν

ω πλ δ ω ω ⎡ ⎤ = − − − ⎢ ⎥ ⎣ ⎦

∑

Noninteracting spin bath ( ) ( ) ( )

S S

k T n

ν ν ν ν

ω ω = − Γ

The rate constant is reflecting the bath properties

2 ,

Noninteracting spinless electrons ( ) ( ) 2 ( )[ ( ) ( )]

B i j F i F i i j

k T n n n

ν ν ν ν ν ν

ω πλ δ ε ε ω ε ε ω ⎡ ⎤ = − − − + − + ⎢ ⎥ ⎣ ⎦

∑

/ 1

( ) [ 1]

T B

n e

ν

ω ν ω −

= −

/ 1

( ) [ 1]

T S

n e

ν

ω ν ω −

= +

( )/ 1

( ) [ 1]

T F

n e

ν ν

ω µ ν ω − −

= +