Laboratoire de Physique et Modelisation ´ des Milieux Condenses ´ Univ. Grenoble & CNRS, Grenoble, France
Nonlinear Thermoelectricity : Cooling, Catastrophes and Carnot Robert S. Whitney
Pl´ eni` ere — GDR Physique Quantique M´ esoscopique [Aussois Oct 2012]
SUMMARY PART 0:
What is thermoelectricty — and why care?
♠ good refrigerator rarely in linear-response regime
PART I:
“Cooling & Catastrophes”
R.W. arXiv:1208.6130
♠ NON-LINEAR scattering theory: fridge with point-contacts ♠ RESULT = catatstrophe aids cooling “thermoelectric quality” 6= figure-of-merit
PART II:
“Carnot and other Constraints”
R.W. work in progress
♠ Thermodynamic & quantum constraints for refrigeration in “arbitrary” quantum systems
WHAT is THERMOELECTRICITY? Power-generation: heat flow → electrical current
Refrigeration: electrical current → heat flow
revisionworld.co.uk
ENERGY-DEPENDENCE of TRANSMISSION L lead HOT
n−type semi− conductor
HOT Fermi sea
R lead COLD
sub-Kelvin expt: semicond→supercond
T hot
semiconductor density of states 0000 1111
1111 0000 0000 1111 0000 1111 0000 1111 1111 0000 0000 1111 0000 1111 0000 1111
COLD Fermi sea
S-N-S THERMOELECTRIC COOLING n−type
heat
Rajauria, Luo, Fournier,Hekking, Courtois, and Pannetier (2007) p−type sub-Kelvin : S-N-S Courtois and co−workers (2007−2009) Point being cooled heat
metal
charge current
superconductor
superconductor
Refrigeration : 300mK → 100mK real semiconductor fridge element
nature.com
TEXT-BOOK THEORY for REFRIGERATION
I
minimal energy-conserving theory ⇒ nearly-linear response e.g. Goldsmid (2009) “Intro. to Thermoelectricity”
ambient temp,
T0
JC
thermo− electric device
heat flow at cold: JC = Θ (T0 − TC ) − Π I |{z}
+
Peltier effect (1834)
1.2
TC /T0
COLD side,
TC
1 2 2 RI
| {z }
Joule heating (1840s)
1.0 COOLING 0.8 0.6
• non-linear • non-conserving
0.4
minimum: TC /T0 = 1 − 12 ZT0
0.2 HEATING 0.0 0.0
0.5
1.0
1.4
I
with ZT0 = Π2 /(RΘT0 )
REFRIGERATION and ZT ZT =
GΠ2 (Θelectron + Θother )T
Π is Peltier coefficient G & Θ are electric & thermal conductance
♣ Currently: best bulk semiconductors ZT ∼ 1.5 − 2 ♣ Theory proposals: ZT ∼ 10
Casati, Mej´ıa-Monasterio, Prosen (2008) Nozaki, Sevin¸ cli, Li, Guti´ errez, Cuniberti (2010) Saha, Markussen, Thygesen, Nikoli´ c (2011) Wierzbicki, Swirkowicz (2011) Karlstr¨ om, Linke, Karlstr¨ om, Wacker (2011) Gunst, Markussen, Jauho, Brandbyge (2011) Rajput, Sharma (2011). Trocha, Barna´ s (2012)
... but good fridges rarely in linear-response regime ♣ Non-linear theory for refrigeration with S-N-S Rajauria, Gandit, Hekking, Pannetier, Courtois (2007). Vasenko, Bezuglyi, Courtois, Hekking, (2009).
PART I
— COOLING & CATASTROPHES —
NON-LINEAR scattering theory Fridge made out of point-contacts
SCATTERING THEORY BEYOND LINEAR RESPONSE Linear response :
charge conductance Landauer & B¨ uttiker (1957-1980s) heat conductance Enquist & Anderson (1981) thermoelectric Sivan & Imry (1986), Butcher (1990)
decoherence as “extra leads” B¨uttiker (1980s)
beyond linear response: Hartree-like interactions included self-consistently point-contact Moskalets (1995) general Christen-B¨ uttiker (1996)
Ji =
Z
∞
−∞
dǫ ǫ − eVj (ǫ − eVi ) Tij (ǫ) f h kB Tj
Self-consistent loop:
Sij (ǫ)
† Sij Tij (ǫ) = tr Sij
potential-distrib. in system
FRIDGE using POINT-CONTACTS at PINCH-OFF Expt: suspended structures e.g. Heron, Fournier, Mingo, Bourgeois (2009)
I
e-
-eVL
J1
e-
I temperature Tisl
temperature T0
Long point-contact only parameter = Epc (minimal tunnelling)
⇒ interactions only modifies Epc
-eVL
-aeVL
screening by gates
potential's peak energy=Epc
FRIDGE using POINT-CONTACTS at PINCH-OFF nearly-linear “text-book” theory 1.2
Tisl /T0 1.0
L G AL N RO W
COOLING 0.8 0.6 0.4 0.2
HEATING 0.0 0.0
0.5
1.0
1.4
Fully non-linear theory: Exact result. 2 hJ π 2 Tisl = − 2 2 (kB T0 ) 12T 0 h(Imax (Tisl ) − I) −Li2 1−exp ekB T0
I
with I ≤ I max (Tisl ) = ekB Tisl ln[2]/h
Mathematically: “fold catastrophe” at IC . In principle: cooling to absolute zero (beyond catastrophe)
EFFECTS SUPPRESSING REFRIGERATION Effect of phonons/photons
ga
1.0 1.0
inc r ea sin
0.8 0.8
4 Jph = a(T04 − Tisl )
0.6 0.6
Stefan-Boltzmann Law
0.4 0.4
2
similar curves for T -photons
0.2 0.2
Pascal, Courtois, Hekking (2011)
0.0 0.0 0.0 0.0
0.1 0.1
0.2 0.2
0.3 0.3
0.4 0.4
mimicked by point-contact in parallel with resistance
Noting point-contact has ZT = 1.4 i.e. not a strong thermoelectric Proposed nanodevices have ZT ∼ 10
1.0 1.0 0.8 0.8 0.6 0.6
inc rea sin
Less strong thermoelectric effects
gg
hI/(ek BT0 )
g=1/4 nearly-linear approx. for g=1/4
0.4 0.4 0.2 0.2 0.0 0.0 0.0 0.0
g=0
0.2
nearly-linear approx. for g=0
0.4 0.4 0.6 0.8 0.8 1.0 1.2 1.2 1.4
hI/(ek BT0 )
0.5 0.5
PART II
— CARNOT & OTHER FUNDAMENTAL CONSTRAINTS —
FUNDAMENTAL CONSTRAINTS on THERMOELECTRICITY ZEROTH LAW EQUILIBRIUM
FIRST LAW P d i (Ji + Vi Ii ) = 0 dt E = SECOND LAW :
d dt S
=
P
i
Ji /Ti ≥ 0
two-leads Bruneau, Jakˇsic, Pillet (2012)
many leads including superconductor
QUANTUM CONSTRAINT : Ji ≥ − Stefan-Boltzmann for fermions:
π2 Ni (kB Ti )2 6h
N◦ channels,
Ni ∼
lead cross−section (wavelength)2
QUANTUM VS THERMODYNAMIC CONSTRAINTS CONSTRAINTS on heat flow out of object being refrigerated Carnot efficiency: Quantum:
Ex. I:
−JC ≤ Psupplied −JC ≤
Carnot
Ex. II:
⇒ −Jc ≤ 0.1pW ⇒ −Jc ≤ 0.01pW
kitchen freezer.
Carnot Quantum
Carnot (1824)
π2 NC (kB TC )2 6h
few channel nanostructure.
Quantum
TC T0 − TC
⇒ −Jc ≤ 13W ⇒ −Jc ≤ 3.6W per square-cm
Psupplied ≃ 1pW TC ≃ 0.1K & T0 ≃ 1K
Psupplied ≃ 100W TC ≃ 260K & T0 ≃ 300K NC ≃ 1010 per square-cm
SUMMARY PART 0:
What is thermoelectricty — and why care?
♠ good refrigerator rarely in linear-response regime
PART I:
“Cooling & Catastrophes”
R.W. arXiv:1208.6130
♠ NON-LINEAR scattering theory: fridge with point-contacts ♠ RESULT = catatstrophe aids cooling “thermoelectric quality” 6= figure-of-merit
PART II:
“Carnot and other Constraints”
R.W. work in progress
♠ Thermodynamic & quantum constraints for refrigeration in “arbitrary” quantum systems
— — — EXTRAS — — —
Appendix : Calculation for FRIDGE using POINT-CONTACTS Method.
Scattering theory gives:
(a) non-linear heat current J in terms of temperature Tisl & voltage, V (b) non-linear charge current I in term of temperature Tisl & voltage, V Invert (b) & substitute for V in (a)
⇒ J in terms of temperature Tisl & current I. Exact result. 2 h π 2 Tisl h(Imax (Tisl ) − I) J =− − Li2 1 − exp (kB T0 )2 12T02 ekB T0 with I ≤ I max (Tisl ) = ekB Tisl ln[2]/h
