closed questions in heat transport and conversion at the...
TRANSCRIPT
![Page 1: CLOSED questions in heat transport and conversion at the ...crepieux/stock/whitney-talk_rob_1.pdf · Laboratoire de Physique et Mode´lisation des Milieux Condense´s Univ. Grenoble](https://reader036.vdocument.in/reader036/viewer/2022071216/6048100d14c36359763b0df9/html5/thumbnails/1.jpg)
??
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
![Page 2: CLOSED questions in heat transport and conversion at the ...crepieux/stock/whitney-talk_rob_1.pdf · Laboratoire de Physique et Mode´lisation des Milieux Condense´s Univ. Grenoble](https://reader036.vdocument.in/reader036/viewer/2022071216/6048100d14c36359763b0df9/html5/thumbnails/2.jpg)
??
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!
![Page 3: CLOSED questions in heat transport and conversion at the ...crepieux/stock/whitney-talk_rob_1.pdf · Laboratoire de Physique et Mode´lisation des Milieux Condense´s Univ. Grenoble](https://reader036.vdocument.in/reader036/viewer/2022071216/6048100d14c36359763b0df9/html5/thumbnails/3.jpg)
NANOSCALE MACHINES
Photosynthetic molecule (experiment)
Gerster et al (2012)
![Page 4: CLOSED questions in heat transport and conversion at the ...crepieux/stock/whitney-talk_rob_1.pdf · Laboratoire de Physique et Mode´lisation des Milieux Condense´s Univ. Grenoble](https://reader036.vdocument.in/reader036/viewer/2022071216/6048100d14c36359763b0df9/html5/thumbnails/4.jpg)
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
![Page 5: CLOSED questions in heat transport and conversion at the ...crepieux/stock/whitney-talk_rob_1.pdf · Laboratoire de Physique et Mode´lisation des Milieux Condense´s Univ. Grenoble](https://reader036.vdocument.in/reader036/viewer/2022071216/6048100d14c36359763b0df9/html5/thumbnails/5.jpg)
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)
![Page 6: CLOSED questions in heat transport and conversion at the ...crepieux/stock/whitney-talk_rob_1.pdf · Laboratoire de Physique et Mode´lisation des Milieux Condense´s Univ. Grenoble](https://reader036.vdocument.in/reader036/viewer/2022071216/6048100d14c36359763b0df9/html5/thumbnails/6.jpg)
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?
![Page 7: CLOSED questions in heat transport and conversion at the ...crepieux/stock/whitney-talk_rob_1.pdf · Laboratoire de Physique et Mode´lisation des Milieux Condense´s Univ. Grenoble](https://reader036.vdocument.in/reader036/viewer/2022071216/6048100d14c36359763b0df9/html5/thumbnails/7.jpg)
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?
![Page 8: CLOSED questions in heat transport and conversion at the ...crepieux/stock/whitney-talk_rob_1.pdf · Laboratoire de Physique et Mode´lisation des Milieux Condense´s Univ. Grenoble](https://reader036.vdocument.in/reader036/viewer/2022071216/6048100d14c36359763b0df9/html5/thumbnails/8.jpg)
SIMPLER – NANOSCALE THERMOELECTRIC
Reddy group(2015)
SIMPLE MODEL
TL
TR
ELECTRON
RESERVOIR L
ELECTRON
RESERVOIR RDot
state
HOT COLD
![Page 9: CLOSED questions in heat transport and conversion at the ...crepieux/stock/whitney-talk_rob_1.pdf · Laboratoire de Physique et Mode´lisation des Milieux Condense´s Univ. Grenoble](https://reader036.vdocument.in/reader036/viewer/2022071216/6048100d14c36359763b0df9/html5/thumbnails/9.jpg)
SIMPLER – NANOSCALE THERMOELECTRIC
Acts as Energy FILTER
different dynamics above and below Fermi surface
FILTER
![Page 10: CLOSED questions in heat transport and conversion at the ...crepieux/stock/whitney-talk_rob_1.pdf · Laboratoire de Physique et Mode´lisation des Milieux Condense´s Univ. Grenoble](https://reader036.vdocument.in/reader036/viewer/2022071216/6048100d14c36359763b0df9/html5/thumbnails/10.jpg)
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
![Page 11: CLOSED questions in heat transport and conversion at the ...crepieux/stock/whitney-talk_rob_1.pdf · Laboratoire de Physique et Mode´lisation des Milieux Condense´s Univ. Grenoble](https://reader036.vdocument.in/reader036/viewer/2022071216/6048100d14c36359763b0df9/html5/thumbnails/11.jpg)
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
![Page 12: CLOSED questions in heat transport and conversion at the ...crepieux/stock/whitney-talk_rob_1.pdf · Laboratoire de Physique et Mode´lisation des Milieux Condense´s Univ. Grenoble](https://reader036.vdocument.in/reader036/viewer/2022071216/6048100d14c36359763b0df9/html5/thumbnails/12.jpg)
TRADITIONAL versus QUANTUM
e�e����� ���rent
heat
S��� � � ���i�� i�
�������� ���rent
heat
H��
C���
H��
C���
b����!"#$
o"�"%�#ic
q&'()&*)t+,*-+.+/),0/
12345 67895:;34<=38<6>
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
![Page 13: CLOSED questions in heat transport and conversion at the ...crepieux/stock/whitney-talk_rob_1.pdf · Laboratoire de Physique et Mode´lisation des Milieux Condense´s Univ. Grenoble](https://reader036.vdocument.in/reader036/viewer/2022071216/6048100d14c36359763b0df9/html5/thumbnails/13.jpg)
QUANTUM THERMOELECTRICS FOR REFRIGERATOR
PELTIER FRIDGE
Energy filter 1
COLD
Energy filter 2
AMBIENT
Heat ?ow in from environment
POWER
SUPPLY
AMBIENT
![Page 14: CLOSED questions in heat transport and conversion at the ...crepieux/stock/whitney-talk_rob_1.pdf · Laboratoire de Physique et Mode´lisation des Milieux Condense´s Univ. Grenoble](https://reader036.vdocument.in/reader036/viewer/2022071216/6048100d14c36359763b0df9/html5/thumbnails/14.jpg)
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
![Page 15: CLOSED questions in heat transport and conversion at the ...crepieux/stock/whitney-talk_rob_1.pdf · Laboratoire de Physique et Mode´lisation des Milieux Condense´s Univ. Grenoble](https://reader036.vdocument.in/reader036/viewer/2022071216/6048100d14c36359763b0df9/html5/thumbnails/15.jpg)
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
![Page 16: CLOSED questions in heat transport and conversion at the ...crepieux/stock/whitney-talk_rob_1.pdf · Laboratoire de Physique et Mode´lisation des Milieux Condense´s Univ. Grenoble](https://reader036.vdocument.in/reader036/viewer/2022071216/6048100d14c36359763b0df9/html5/thumbnails/16.jpg)
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
![Page 17: CLOSED questions in heat transport and conversion at the ...crepieux/stock/whitney-talk_rob_1.pdf · Laboratoire de Physique et Mode´lisation des Milieux Condense´s Univ. Grenoble](https://reader036.vdocument.in/reader036/viewer/2022071216/6048100d14c36359763b0df9/html5/thumbnails/17.jpg)
LANDAUER SCATTERING
THEORY
![Page 18: CLOSED questions in heat transport and conversion at the ...crepieux/stock/whitney-talk_rob_1.pdf · Laboratoire de Physique et Mode´lisation des Milieux Condense´s Univ. Grenoble](https://reader036.vdocument.in/reader036/viewer/2022071216/6048100d14c36359763b0df9/html5/thumbnails/18.jpg)
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
![Page 19: CLOSED questions in heat transport and conversion at the ...crepieux/stock/whitney-talk_rob_1.pdf · Laboratoire de Physique et Mode´lisation des Milieux Condense´s Univ. Grenoble](https://reader036.vdocument.in/reader036/viewer/2022071216/6048100d14c36359763b0df9/html5/thumbnails/19.jpg)
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)
![Page 20: CLOSED questions in heat transport and conversion at the ...crepieux/stock/whitney-talk_rob_1.pdf · Laboratoire de Physique et Mode´lisation des Milieux Condense´s Univ. Grenoble](https://reader036.vdocument.in/reader036/viewer/2022071216/6048100d14c36359763b0df9/html5/thumbnails/20.jpg)
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)
![Page 21: CLOSED questions in heat transport and conversion at the ...crepieux/stock/whitney-talk_rob_1.pdf · Laboratoire de Physique et Mode´lisation des Milieux Condense´s Univ. Grenoble](https://reader036.vdocument.in/reader036/viewer/2022071216/6048100d14c36359763b0df9/html5/thumbnails/21.jpg)
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
![Page 22: CLOSED questions in heat transport and conversion at the ...crepieux/stock/whitney-talk_rob_1.pdf · Laboratoire de Physique et Mode´lisation des Milieux Condense´s Univ. Grenoble](https://reader036.vdocument.in/reader036/viewer/2022071216/6048100d14c36359763b0df9/html5/thumbnails/22.jpg)
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
![Page 23: CLOSED questions in heat transport and conversion at the ...crepieux/stock/whitney-talk_rob_1.pdf · Laboratoire de Physique et Mode´lisation des Milieux Condense´s Univ. Grenoble](https://reader036.vdocument.in/reader036/viewer/2022071216/6048100d14c36359763b0df9/html5/thumbnails/23.jpg)
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
![Page 24: CLOSED questions in heat transport and conversion at the ...crepieux/stock/whitney-talk_rob_1.pdf · Laboratoire de Physique et Mode´lisation des Milieux Condense´s Univ. Grenoble](https://reader036.vdocument.in/reader036/viewer/2022071216/6048100d14c36359763b0df9/html5/thumbnails/24.jpg)
RATE EQUATIONS
![Page 25: CLOSED questions in heat transport and conversion at the ...crepieux/stock/whitney-talk_rob_1.pdf · Laboratoire de Physique et Mode´lisation des Milieux Condense´s Univ. Grenoble](https://reader036.vdocument.in/reader036/viewer/2022071216/6048100d14c36359763b0df9/html5/thumbnails/25.jpg)
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)
![Page 26: CLOSED questions in heat transport and conversion at the ...crepieux/stock/whitney-talk_rob_1.pdf · Laboratoire de Physique et Mode´lisation des Milieux Condense´s Univ. Grenoble](https://reader036.vdocument.in/reader036/viewer/2022071216/6048100d14c36359763b0df9/html5/thumbnails/26.jpg)
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
![Page 27: CLOSED questions in heat transport and conversion at the ...crepieux/stock/whitney-talk_rob_1.pdf · Laboratoire de Physique et Mode´lisation des Milieux Condense´s Univ. Grenoble](https://reader036.vdocument.in/reader036/viewer/2022071216/6048100d14c36359763b0df9/html5/thumbnails/27.jpg)
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
![Page 28: CLOSED questions in heat transport and conversion at the ...crepieux/stock/whitney-talk_rob_1.pdf · Laboratoire de Physique et Mode´lisation des Milieux Condense´s Univ. Grenoble](https://reader036.vdocument.in/reader036/viewer/2022071216/6048100d14c36359763b0df9/html5/thumbnails/28.jpg)
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
![Page 29: CLOSED questions in heat transport and conversion at the ...crepieux/stock/whitney-talk_rob_1.pdf · Laboratoire de Physique et Mode´lisation des Milieux Condense´s Univ. Grenoble](https://reader036.vdocument.in/reader036/viewer/2022071216/6048100d14c36359763b0df9/html5/thumbnails/29.jpg)
??
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!