??

Laboratoire de Physique et Modelisation des Milieux Condenses

Univ. Grenoble Alpes & CNRS, Grenoble, France

CLOSED questions

in heat transport and conversion

at the nanoscale

Robert S. Whitney

Review – Benenti, Casati, Saito, R.W.

Physics Reports 694, 1 (2017)

Marseille – Nov 2018

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OVERVIEW

(I) Laws of thermodynamics same in quantum

• where does entropy come from?Schrodinger Eq. = Reversible

• Carnot efficiency

(II) Extra constraints from quantum

• quantized heat flow

(III) Practical considerations

• can’t control heat!

NANOSCALE MACHINES

Photosynthetic molecule (experiment)

Gerster et al (2012)

NANOSCALE MACHINES

Photosynthetic molecule (experiment)

Gerster et al (2012)

SIMPLE MODEL

T0 T0

ELECTRON

RESERVOIR L

ELECTRON

RESERVOIR R

1

HOT PHOTON

RESERVOIR

Tph

2

NANOSCALE MACHINES

T0 T0

ELECTRON

RESERVOIR L

ELECTRON

RESERVOIR R

1

HOT PHOTON

RESERVOIR

Tph

2

FOR THIS TALK, I ASSUME:

♣ Nanoscale/quantum system between macroscopic reservoirs

• Long times (steady state) ⇒“macroscopic” work output

cf. fluctuation theorems

♣ Thermal reservoir states:

• No squeezed reservoir states cf. J. Roßnagel,et al PRL (2014)

• No non­equil. reservoir states

cf. Sanchez, Splettstoesser & Whitney "Demon preprint" (2018)

REVERSIBLE OR NOT

quantum

system

RESERVOIR

interactions

REVERSIBLEdynamics

REVERSIBLE

irreversible

RESERVOIR

irreversible

region modelled

interactionsREVERSIBLE

REVERSIBLE or NOT?

• Where is entropy produced?

• 2nd law of thermodynamics?

REVERSIBLE OR NOT

quantum

system

RESERVOIR

interactions

REVERSIBLEdynamics

REVERSIBLE

irreversible

RESERVOIR

irreversible

region modelled

interactionsREVERSIBLE

REVERSIBLE or NOT?

• Where is entropy produced?

• 2nd law of thermodynamics?

SIMPLER – NANOSCALE THERMOELECTRIC

Reddy group(2015)

SIMPLE MODEL

TL

TR

ELECTRON

RESERVOIR L

ELECTRON

RESERVOIR RDot

state

HOT COLD

SIMPLER – NANOSCALE THERMOELECTRIC

Acts as Energy FILTER

different dynamics above and below Fermi surface

FILTER

SIMPLER – NANOSCALE THERMOELECTRIC

Acts as Energy FILTER

different dynamics above and below Fermi surface

HOT

Fermi sea

COLD

Fermi sea

thermoelectricsystem'stransmissionCOLD

Fermi sea

Other thermoelectricsystem's transmission

Heatsource

FILTERFILTER ABSORBER

NANOSCALE THERMOELECTRIC

Acts as Energy FILTER

different dynamics above and below Fermi surface

HOT

Fermi sea

COLD

Fermi sea

thermoelectricsystem'stransmissionCOLD

Fermi sea

Other thermoelectricsystem's transmission

Heatsource

FILTERFILTER ABSORBER

TRADITIONAL versus QUANTUM

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heat

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heat

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Traditional Quantum

(1) Filters ≪ scale of thermalisation ⇒ Quantum thermoelectric

(2) Central region (absorber) also ≪ scale of thermalisation

⇒ Quantum thermocouple

• Often filters­absorber combined in in quantum thermocouples

cf. simple model of photosynthetic molecule

QUANTUM THERMOELECTRICS FOR REFRIGERATOR

PELTIER FRIDGE

Energy filter 1

COLD

Energy filter 2

AMBIENT

Heat ?ow in from environment

POWER

SUPPLY

AMBIENT

EFFICIENCIES AND ZT

HEAT­ENGINE efficiency

ηengine =work doneheat used

Carnot limit

ηCarnotengine = 1− Tcold

/

Thot

LINEAR RESPONSE: ηengine = ηCarnotengine ×

√ZT+1 −1√ZT+1 +1

ZT =(Electric Conductance) (Seebeck)2 T

Thermal Conductance

ZT= ∞⇒ Carnot

ZT= 3 ⇒ Carnot/3

ZT= 1 ⇒ Carnot/6

EFFICIENCIES AND ZT

HEAT­ENGINE efficiency

ηengine =work doneheat used

Carnot limit

ηCarnotengine = 1− Tcold

/

Thot

LINEAR RESPONSE: ηengine = ηCarnotengine ×

√ZT+1 −1√ZT+1 +1

ZT =(Electric Conductance) (Seebeck)2 T

Thermal Conductance

ZT= ∞⇒ Carnot

ZT= 3 ⇒ Carnot/3

ZT= 1 ⇒ Carnot/6

FRIDGE efficiency

ηfridge =heat removed

work used

Carnot limit

ηCarnotfridge = Tcold

/

(Tambient−Tcold)

“LINEAR” RESPONSE: Max T difference :∆T

T≤ 1

2ZT

MODELLING METHODS

• Landauer scattering theory ⇐= No e-e Interactions

see chapters 4­6 of Benenti et al, Physics Reports 694, 1 (2017)

• Rate Equations ⇐= Weak Coupling

see chapters 8­9 of Benenti et al, Physics Reports 694, 1 (2017)

• Keldysh field theory ⇐= “everything” often need approx similar to above

OPEN questions ask Rosa & David, or Fabienne & Adeline ...

• Schrodinger eq. for system+environment

LANDAUER SCATTERING

THEORY

LANDAUER SCATTERING

THEORY

(TWO TERMINALS)

quantum

system

RESERVOIR R

RESERVOIR L

Currents into R

Particle currentinto R

≡ ddtNR =

∫

dE T (E)[

f(

E−µL

TL

)

− f(

E−µR

TR

)]

Energy currentinto R

≡ ddtER =

∫

dE E T (E)[

f(

E−µL

TL

)

− f(

E−µR

TR

)]

UNITS: h = kB = 1

LANDAUER SCATTERING

THEORY

(TWO TERMINALS)

quantum

system

RESERVOIR R

RESERVOIR L

Currents into R

Particle currentinto R

≡ ddtNR =

∫

dE T (E)[

f(

E−µL

TL

)

− f(

E−µR

TR

)]

Energy currentinto R

≡ ddtER =

∫

dE E T (E)[

f(

E−µL

TL

)

− f(

E−µR

TR

)]

UNITS: h = kB = 1

• Electric current into R = e ddtNR

• Work into R= µRddtNR =⇒ power generated = sum L & R

= (µR − µL)ddtNR

• Heat current into R = ddtER − µR

ddtNR

• Entropy current into R = Heat current/

TR ⇐ CLAUSIUS LAW (1855)

LANDAUER SCATTERING

THEORY

(TWO TERMINALS)

quantum

system

RESERVOIR R

RESERVOIR L

Currents into R

see section 6.4 of Phys. Rep. 694, 1 (2017)

• Rate of change of entropy ddtS

= (Entropy current into L) + (Entropy current into R) ≥ 0

⇒ 2nd law proven for all T (E), TL, TR, ...

Nenciu (2007) = proof for N reservoirs

• Carnot efficient system : reversible ddtS = 0

requires transmission is only non­zero atE−µL

TL= E−µR

TR

Humphrey, Newbury, Taylor, Linke (2002)

Quantum bound on heat flow

Maximum HEAT FLOW per transverse mode⇒maximum ENTROPY CHANGE

Bekenstein, Phys. Rev. Lett. (1981)

Pendry, J. Phys. A (1983)Heat current out of hot reservoir

Jhot ≤ Nπ2k2B6h

(

T 2hot − T 2

cold

)

with “quantum” N ∼ cross­sectionwavelength

Quantum bound on heat flow

Maximum HEAT FLOW per transverse mode⇒maximum ENTROPY CHANGE

Bekenstein, Phys. Rev. Lett. (1981)

Pendry, J. Phys. A (1983)Heat current out of hot reservoir

Jhot ≤ Nπ2k2B6h

(

T 2hot − T 2

cold

)

with “quantum” N ∼ cross­sectionwavelength

Quantum bound on power output

Theory: RW (2013) – Experiment: Chen et al (2018)

ALLOWED

P@ABD @EFGEF

IJJfKfLMKN

OQRTUt efVWXWYTXZ

FORBIDDEN

Pqb

Quantum bound (qb)

Pqb ∝ N (Thot−Tcold)2

Carnot ×(

1− α1

√

P/

Pqb + · · ·)

...do better with B­field/interactions cf. Brandner et al 2015, Lao et al 2018

Is quantum bound relevant for REAL applications?

Cross­section for 100W of power ?

with wavelength λF∼10−8m

♦ Minimal cross­section ∼ 4mm2

♦ 90% of Carnot requires > 0.4cm2

RATE EQUATIONS

RATE EQUATIONS

T0 T0

ELECTRON

RESERVOIR L

ELECTRON

RESERVOIR R

[\]\^ _

[\]\^ `

HOT PHOTON

RESERVOIR

Tph

(0,0)"empty"

(1,0)"state 1"

(0,1)"state 2"

L R

ph

Set of coupled equations for system evolution:

d

dtPb(t) =

∑

a

(

Γb←a Pa(t) − Γa←b Pb(t))

where Pb = prob. system is in state b

& Γb←a = rate a→b

Current into R = Γ(0,0)←(0,1) P(0,1) − Γ(0,1)←(0,0) P(0,0)

RATE EQUATIONS

T0 T0

ELECTRON

RESERVOIR L

ELECTRON

RESERVOIR R

acdcg h

acdcg j

HOT PHOTON

RESERVOIR

Tph

(0,0)"empty"

(1,0)"state 1"

(0,1)"state 2"

L R

ph

USEFUL TRICKS – look at loops:

• only Carnot efficient if no loop generates entropy

• if single­loop system ⇒ ηengine =Work in one cycle

Heat in one cycle

CAN’T CONTROL HEAT !!

HOT

Heatsource

COLD COLD

insulation

phonons photons

108

10-16

COPPER

GLASS

Electrical Conductivity

GLASS

DIAMOND

10-1

103

vacu

um

Thermal Conductivity

at 300K

COPPER

10-12

DIAMOND SI units: left S/m right W/Km

(a) (b)

Heat engine efficiency ηengine =work done

heat used + heat lost

CAN’T CONTROL HEAT !!

HOT

Heatsource

COLD COLD

insulation

phonons photons

108

10-16

COPPER

GLASS

Electrical Conductivity

GLASS

DIAMOND

10-1

103

vacu

um

Thermal Conductivity

at 300K

COPPER

10-12

DIAMOND SI units: left S/m right W/Km

(a) (b)

Heat engine efficiency ηengine =work done

heat used + heat lost

Power output, P

Effic

iency

0

strong phonons

no phonons

weak phonons

CARNOT

??

CONCLUSION: CLOSED† QUESTIONS†closed for theories simple enough to treat

≡ (a) Landauer & (b) Rate eqs.

(I) Laws of thermodynamics same at nanoscale

if reservoirs are macroscopic & internal equlibrium

• Entropy produced

• Carnot efficiency (in principle) ⇐ vanishing power

(II) Extra constraints from quantum

• quantized heat flow

(III) Practical considerations

• can’t control heat!