low-temperature thermoelectric properties of bilayer grapheneth.if.uj.edu.pl › ~adamr ›...

Low-temperature thermoelectric propertiesof bilayer graphene

Adam Rycerz, Dominik Suszalski, and Grzegorz RutMarian Smoluchowski Institute of Physics, Jagiellonian University,

Kraków, Poland

StoCP 2018, Zakopane, Poland, September 16–21, 2018

Rycerz, Suszalski, Rut, Low-temperature thermoelectric properties . . .

IntroductionLandauer-Büttiker formalism

Thermoelectric properties

Emergent Dirac fermions in grapheneTunable band gap and trigonal warping in BLGQuantum size effects

Why graphene?

Atomically-thin2D

crystals

Dirac materialsgraphene, silicene(?), ... vdW heterostr.

h-BN, h-AlN(?)MoS2, HfSe2, ...Black-PTiSe2 (CWD+SC!)

Artificial graphenes & analogsTopolog. insulators: HgTe/CdTe,Bi1−xSbx , Bi2Se3, Bi2Te3, ...d-wave superconductorsWeyl semimetals

For a review of the topic, see: Wehling et al., Adv. Phys. 76, 1 (2014).Theory of SC in TiSe2: Chen et al., arXiv:1806.08064.





Emergent Dirac fermions in graphene

What differs graphene from a collection of carbon atoms?

The valley pseudospin (⇒ fermion doubling)

Time-reversal symmetry breaking at zero magnetic field

Pseudodiffusive charge transport, . . . , SC in twisted-BLG (!)





Tunable band gap and trigonal warping

E = "L

k`

"L =�1

4

✓�3

�0

◆2

⇡ 1 meV

k` =2p3

�1�3

a�20

⇡ 0.05 nm�1

[ Top: IEEE Spectrum, 2009; bottom: McCann & Koshino, 2013 ]





[55] Zhang et al., Phys. Rev. B 78, 235408 (2008).[56] Li et al., Phys. Rev. Lett. 102, 037403 (2009).[67] Dresselhaus & Dresselhaus, Adv. Phys. 51, 1 (2002).[76] Malard et al., Phys. Rev. B 76, 201401(R) (2007).[80] Kuzmenko et al., Phys. Rev. B 80, 165406 (2009).[82] Taychatanapat et al., Nature Phys. 7, 621 (2011).

Yankowitz et al., (2014): γ1 = 0.39 eV, γ4 = 0.22 eV, but γ3 not determined ...

⇒ Novel phenomena governed solely by γ3 desired!





Where to expect TW effects on transport?

Time-energy uncertainty relation limits the energy resolution:

∆E >~

∆t∼ ~vF

L,

where we set ∆t ∼ tflight ≈ L/vF .�E / 1/L

The condition ∆E . 2εL leads to

L &~vF

2εL=√

3γ3

0

γ1γ23

a ≈ 400 nm,

where γ0 = 3.16 eV, γ1 = 0.38 eV, γ3 = 0.3 eV, a = 0.246 nm.

⇒ Effects of γ3 6= 0 on transport prop-s may depend on thesystem size; the crossover range: 100 nm < L < 1µm.




Brief overviewUniversal conductivity of MLGEmergent conductivity of BLG

Electric and thermal currents

where gs = gv = 2 denote spin and valley degeneracies,T (E) ≡ Tr(tt†), and fL, fR are Fermi-Dirac distributions.





Monolayer at the charge-neutrality point

G =2e2

h

∞∑

n=−∞

1cosh2[π(n + α)L/W ]

W�L' 4e2

πhWL,

where α = 12 for the mass confinement (ΨA|y=0 = ΨB|y=0,

ΨA|y=W= −ΨB|y=W ).

The conductivity: σ = GL/W = 4e2/(πh) ≡ σ0 .

Theory: Katsnelson (2006); Tworzydło et al. (2006)Experiment: Miao et al., (2007); Danneau et al., (2008).





Emergent conductivity of BLG

Landauer conductivity [Snyman& Beenakker, 2007], γ3 = γ4 = 0:

σbilayer = GL/W = 2σ0

Kubo conductivity [Cserti et al.,2007], γ3 6= 0:

σbilayer = 6σ0 ( ! )

Experiment: [Mayorov et al.,2011] σbilayer ' 5σ0.

[ with σ0 ≡ (4/π)e2/h – universalconductivity of a monolayer. ]





EXP: Mayorov et al, 2011(�BLG ⇡ 5�0 )

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[ Notation: t⊥ ≡ γ1, t ′ ≡ γ3 ]

Non-cummuting limits:lim

L→∞lim

d→∞σBLG = 2σ0

limd→∞

limL→∞

σBLG = 6σ0 (! )

[ γ1∼γ3∼e−Bd/d0 , εL∼(

e−Bd/d0)3

,and thus εL/γ1 → 0 with d →∞(the interlayer distance) ]

Surprising analogy withsystems showingspontaneous symmetrybreaking !

Grzegorz Rut & AR:⇒ PRB 89, 045421 (2014)⇒ EPL 107, 47005 (2014)




Quasiparticles in doped BLGSeebeck coefficient and Lorentz numberGapless vs gapped BLG

Quasiparticles in doped BLG

E/EL

0 0.5 1.0 1.5 2.00

0.5

1.0

DO

S[e

V�

1nm

�2]

mC / |E|

Massless !Dirac!fermions

Massive !chiral!fermions

Hea

vy fe

r-s

In a generic 2D system!

the cyclotronic mass:

mC(E) =

⇡~2

2DOS(E)

me↵ =t?2v2

F

⇡ 0.033 me

Transitions between !different populations !of quasiparticles!should have !signatures !in thermoelectric properties...

!





Why thermoelectrics?

The Mott formula for metals [Cutler & Mott, Phys. Rev. (1969)]

S ≈ π2k2BT

3e

[∂ lnσ(E)

∂E

]

E=EF

,

The Drude conductivity: σ = ne2τ/meff ∼ DOS(EF )× kBT(provided that the mobility µ̃(E) ∼ τ/meff ≈ const.).⇒ Any abrupt change in DOS(E) should has some signature in S.

The Goldsmid-Sharp energy gap in semiconductors|S|max ≈ Eg/(2eTmax) allows the thermoelectric-basedband-gap (Eg) determination, instead of employing the familiaractivation formula σ ∼ exp(−Eg/2kBT ).[Goldsmid & Sharp, J. Electron. Mater. (1999)] .





Zero-temperature conductivity (µ 6= 0)

0 20 40 60 80

100 120 140

0 0.2 0.4 0.6 0.8 1 1.2

Cond

uctiv

ity [4

e 2/h

]

μ [meV]

0 10 20 30 40 50 0

2

4

6

8

Cond

uctiv

ity [4

e2/h

]

μ [µeV]

t0 =

0.3eV

t0 = 0.2 eV

t0 = 0

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Fabry-Perrot resonances: ∆E ≈ 2π(~vF/L)√|E |/t⊥ for t ′ = 0, or

∆E ∼ π~v3/L = 2π(~vF/L)√

EL/t⊥ for t ′ 6= 0.Above: L = 104l⊥ = 17.7µm. [ Suszalski, Rut, AR, PRB (2018) ]





In the linear-responce regime:

with Ln given by

The Lorentz number

[LWF ≡ 13π

2 (kB/e)2 ⇔ T (E) =const;a model T (E) ∝ |E | ⇔ Lmax ≈ 2.37LWF.]

[ Suszalski, Rut, AR, PRB (2018) ]

0.8

1.2

1.6

2

2.4

0.30.60.91.21.5

T [K

]

0.0

0.2

0.4

0.6

0.8

1.0

1.2

0 0.2 0.4 0.6 0.8 1 1.2

0 0.3 0.6 0.9 1.2EL [meV]

S [k

B/e]

[meV]μ

L/L W

F

T=

2K

T = 0.515K

T = 0.2K

T = 80mK





⇒ One does not necessarily need to include interactions tounderstand some experimental data!

Lmax ⇡ 2.37 LWF<latexit sha1_base64="YZTnSqhVJaxct3vzIcd4+w2X5EE=">AAACDHicdVDLSgMxAMzWV62vqkcvwSJ4kGV3W/q4FQTx4KGCtYVuWbJptg3NPkiy0rLsN3jxV7wonhTxF/wbs21FKjoQGGYmJDNuxKiQhvGp5VZW19Y38puFre2d3b3i/sGtCGOOSRuHLORdFwnCaEDakkpGuhEnyHcZ6bjj88zv3BEuaBjcyGlE+j4aBtSjGEklOUU9sTFi8Cp1Epv70EeT1EZRxMNJklp6uWaf/QQ6F6lTLBl6o161KlVo6IZRMy0zI1atUq5AUykZSmCBllP8sAchjn0SSMyQED3TiGQ/QVxSzEhasGNBIoTHaEiSWZsUnihpAL2QqxNIOFOXcsgXYuq7KukjORK/vUz8y+vF0qv3ExpEsSQBnj/kxQzKEGbTwAHlBEs2VQRhTtUPIR4hjrBUAxZU9e9+8H/StvSGbl5XSk1rsUEeHIFjcApMUANNcAlaoA0weABP4BW8affao/asvcyjOW1x5xAsQXv/ApQ9mzQ=</latexit><latexit sha1_base64="YZTnSqhVJaxct3vzIcd4+w2X5EE=">AAACDHicdVDLSgMxAMzWV62vqkcvwSJ4kGV3W/q4FQTx4KGCtYVuWbJptg3NPkiy0rLsN3jxV7wonhTxF/wbs21FKjoQGGYmJDNuxKiQhvGp5VZW19Y38puFre2d3b3i/sGtCGOOSRuHLORdFwnCaEDakkpGuhEnyHcZ6bjj88zv3BEuaBjcyGlE+j4aBtSjGEklOUU9sTFi8Cp1Epv70EeT1EZRxMNJklp6uWaf/QQ6F6lTLBl6o161KlVo6IZRMy0zI1atUq5AUykZSmCBllP8sAchjn0SSMyQED3TiGQ/QVxSzEhasGNBIoTHaEiSWZsUnihpAL2QqxNIOFOXcsgXYuq7KukjORK/vUz8y+vF0qv3ExpEsSQBnj/kxQzKEGbTwAHlBEs2VQRhTtUPIR4hjrBUAxZU9e9+8H/StvSGbl5XSk1rsUEeHIFjcApMUANNcAlaoA0weABP4BW8affao/asvcyjOW1x5xAsQXv/ApQ9mzQ=</latexit><latexit sha1_base64="YZTnSqhVJaxct3vzIcd4+w2X5EE=">AAACDHicdVDLSgMxAMzWV62vqkcvwSJ4kGV3W/q4FQTx4KGCtYVuWbJptg3NPkiy0rLsN3jxV7wonhTxF/wbs21FKjoQGGYmJDNuxKiQhvGp5VZW19Y38puFre2d3b3i/sGtCGOOSRuHLORdFwnCaEDakkpGuhEnyHcZ6bjj88zv3BEuaBjcyGlE+j4aBtSjGEklOUU9sTFi8Cp1Epv70EeT1EZRxMNJklp6uWaf/QQ6F6lTLBl6o161KlVo6IZRMy0zI1atUq5AUykZSmCBllP8sAchjn0SSMyQED3TiGQ/QVxSzEhasGNBIoTHaEiSWZsUnihpAL2QqxNIOFOXcsgXYuq7KukjORK/vUz8y+vF0qv3ExpEsSQBnj/kxQzKEGbTwAHlBEs2VQRhTtUPIR4hjrBUAxZU9e9+8H/StvSGbl5XSk1rsUEeHIFjcApMUANNcAlaoA0weABP4BW8affao/asvcyjOW1x5xAsQXv/ApQ9mzQ=</latexit><latexit sha1_base64="YZTnSqhVJaxct3vzIcd4+w2X5EE=">AAACDHicdVDLSgMxAMzWV62vqkcvwSJ4kGV3W/q4FQTx4KGCtYVuWbJptg3NPkiy0rLsN3jxV7wonhTxF/wbs21FKjoQGGYmJDNuxKiQhvGp5VZW19Y38puFre2d3b3i/sGtCGOOSRuHLORdFwnCaEDakkpGuhEnyHcZ6bjj88zv3BEuaBjcyGlE+j4aBtSjGEklOUU9sTFi8Cp1Epv70EeT1EZRxMNJklp6uWaf/QQ6F6lTLBl6o161KlVo6IZRMy0zI1atUq5AUykZSmCBllP8sAchjn0SSMyQED3TiGQ/QVxSzEhasGNBIoTHaEiSWZsUnihpAL2QqxNIOFOXcsgXYuq7KukjORK/vUz8y+vF0qv3ExpEsSQBnj/kxQzKEGbTwAHlBEs2VQRhTtUPIR4hjrBUAxZU9e9+8H/StvSGbl5XSk1rsUEeHIFjcApMUANNcAlaoA0weABP4BW8affao/asvcyjOW1x5xAsQXv/ApQ9mzQ=</latexit>

Lmax ⇡ 2.37 LWF<latexit sha1_base64="YZTnSqhVJaxct3vzIcd4+w2X5EE=">AAACDHicdVDLSgMxAMzWV62vqkcvwSJ4kGV3W/q4FQTx4KGCtYVuWbJptg3NPkiy0rLsN3jxV7wonhTxF/wbs21FKjoQGGYmJDNuxKiQhvGp5VZW19Y38puFre2d3b3i/sGtCGOOSRuHLORdFwnCaEDakkpGuhEnyHcZ6bjj88zv3BEuaBjcyGlE+j4aBtSjGEklOUU9sTFi8Cp1Epv70EeT1EZRxMNJklp6uWaf/QQ6F6lTLBl6o161KlVo6IZRMy0zI1atUq5AUykZSmCBllP8sAchjn0SSMyQED3TiGQ/QVxSzEhasGNBIoTHaEiSWZsUnihpAL2QqxNIOFOXcsgXYuq7KukjORK/vUz8y+vF0qv3ExpEsSQBnj/kxQzKEGbTwAHlBEs2VQRhTtUPIR4hjrBUAxZU9e9+8H/StvSGbl5XSk1rsUEeHIFjcApMUANNcAlaoA0weABP4BW8affao/asvcyjOW1x5xAsQXv/ApQ9mzQ=</latexit><latexit sha1_base64="YZTnSqhVJaxct3vzIcd4+w2X5EE=">AAACDHicdVDLSgMxAMzWV62vqkcvwSJ4kGV3W/q4FQTx4KGCtYVuWbJptg3NPkiy0rLsN3jxV7wonhTxF/wbs21FKjoQGGYmJDNuxKiQhvGp5VZW19Y38puFre2d3b3i/sGtCGOOSRuHLORdFwnCaEDakkpGuhEnyHcZ6bjj88zv3BEuaBjcyGlE+j4aBtSjGEklOUU9sTFi8Cp1Epv70EeT1EZRxMNJklp6uWaf/QQ6F6lTLBl6o161KlVo6IZRMy0zI1atUq5AUykZSmCBllP8sAchjn0SSMyQED3TiGQ/QVxSzEhasGNBIoTHaEiSWZsUnihpAL2QqxNIOFOXcsgXYuq7KukjORK/vUz8y+vF0qv3ExpEsSQBnj/kxQzKEGbTwAHlBEs2VQRhTtUPIR4hjrBUAxZU9e9+8H/StvSGbl5XSk1rsUEeHIFjcApMUANNcAlaoA0weABP4BW8affao/asvcyjOW1x5xAsQXv/ApQ9mzQ=</latexit><latexit sha1_base64="YZTnSqhVJaxct3vzIcd4+w2X5EE=">AAACDHicdVDLSgMxAMzWV62vqkcvwSJ4kGV3W/q4FQTx4KGCtYVuWbJptg3NPkiy0rLsN3jxV7wonhTxF/wbs21FKjoQGGYmJDNuxKiQhvGp5VZW19Y38puFre2d3b3i/sGtCGOOSRuHLORdFwnCaEDakkpGuhEnyHcZ6bjj88zv3BEuaBjcyGlE+j4aBtSjGEklOUU9sTFi8Cp1Epv70EeT1EZRxMNJklp6uWaf/QQ6F6lTLBl6o161KlVo6IZRMy0zI1atUq5AUykZSmCBllP8sAchjn0SSMyQED3TiGQ/QVxSzEhasGNBIoTHaEiSWZsUnihpAL2QqxNIOFOXcsgXYuq7KukjORK/vUz8y+vF0qv3ExpEsSQBnj/kxQzKEGbTwAHlBEs2VQRhTtUPIR4hjrBUAxZU9e9+8H/StvSGbl5XSk1rsUEeHIFjcApMUANNcAlaoA0weABP4BW8affao/asvcyjOW1x5xAsQXv/ApQ9mzQ=</latexit><latexit sha1_base64="YZTnSqhVJaxct3vzIcd4+w2X5EE=">AAACDHicdVDLSgMxAMzWV62vqkcvwSJ4kGV3W/q4FQTx4KGCtYVuWbJptg3NPkiy0rLsN3jxV7wonhTxF/wbs21FKjoQGGYmJDNuxKiQhvGp5VZW19Y38puFre2d3b3i/sGtCGOOSRuHLORdFwnCaEDakkpGuhEnyHcZ6bjj88zv3BEuaBjcyGlE+j4aBtSjGEklOUU9sTFi8Cp1Epv70EeT1EZRxMNJklp6uWaf/QQ6F6lTLBl6o161KlVo6IZRMy0zI1atUq5AUykZSmCBllP8sAchjn0SSMyQED3TiGQ/QVxSzEhasGNBIoTHaEiSWZsUnihpAL2QqxNIOFOXcsgXYuq7KukjORK/vUz8y+vF0qv3ExpEsSQBnj/kxQzKEGbTwAHlBEs2VQRhTtUPIR4hjrBUAxZU9e9+8H/StvSGbl5XSk1rsUEeHIFjcApMUANNcAlaoA0weABP4BW8affao/asvcyjOW1x5xAsQXv/ApQ9mzQ=</latexit>

[ EXP: Crossno et al, Science (2016) ]





Chemical potential µ is not a directly-measurable quantity, but itis monotonically-dependent on the carrier density n.

Characteristic temperatures, at which the secondarymaximum of S (minimum of L) vanishes are:

T S? = 0.050× EL/kB,

TL? = 0.118× EL/kB.

(For t ′ = 0.3 eV, EL/kB ≈ 10 K.)

Also, a global maximum ofS corresponds to

T Smax = 0.20× EL/kB.

[ Suszalski, Rut, AR, PRB (2018) ]

0.0

0.1

0.2

0.3

0 0.5 1 1.5

μ max

[m

eV]

T [K]

0

20

40

60

80

(n-p

) max

[μm

-2]

0

20

40

60

80

(n-p

) max

[μm

-2]

0

20

40

60

80

(n-p

) max

[μm

-2]

0.7

0.8

0.9

1.0

1.1

S max

[k B

/e]

0 0.5 1 1.5 2

0.00

0.03

0.06

0.09

0.12

0.15

ZTm

ax

T [K]t0 =0

t0 =0.2 e

V

t0 =

0.3eV

(a)

(b)

(c)

(d)

Z T ( μ m

ax )





Remarks on intrinsic gap in BLG (1)

Free-standing (suspended) BLG:

[Credit: Lau lab, UC Riverside]

Width of resistivity peak matches the self-consistent(mean-field) gap:

∆i (T ) = ∆i (0) tanh

(1.74

√Tc

T− 1

)for T ≤ Tc ,

with ∆i (0) ≈ 1.5 meV, Tc ≈ 12 K (∆i (0)/kBTc ≈ 1.5).[Nam et al., 2D Mater 2016; Morpurgo, Graph. Week 2018]

BLG in v.d. Walls heterostructures (vdWH):No signatures of∆i >0 found as yet (!)

[Kraft et al., Nat. Comm 2018]





Remarks on intrinsic gap in BLG (2)

Possible explanations of the presentce/absence of ∆iin free-standing BLG / vdWH containing BLG:

Self-consistent gap strongly depends on effectiveHubbard-U: ∆i ∼ t0 exp(−const/Ueff) [Hirsch, 1985].

Supression of Ueff due to dielectric constant maydramatically reduce ∆i and Tc .

Lattice mismatch between BLG and h-BN induces aneffective potential (Moiré pattern) destabilizing quasi-longrange order (?)

. . . ?





The nonzero gap case (∆ > 0)

-0.05 0.00 0.05 0.10 0.150.0

1.0

2.0

3.0

G/W

[g0/µm

]

μ-Δ/2 [meV]

(e)

(a)

(b)

{µL, TL}

VI

=)IQ

{µR, TR}

=)IQ

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0 3 6 9 120

6

12

18

μ [meV]

G/W

[g0/µ

m]

0

5

10

15

20

S[k

B/e

]

(c)

(d)

U=

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U=20

meV

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U=

10m

eV

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U =150meV

<latexit sha1_base64="Tr51jNqbo2fHDKe5uPiQvWPychs=">AAAB9HicbVDLSsNAFL3xWesrKq7cpBbBhZSkKLoRCm5cVjBtoSllMp20QycPZm6KJfRP3Ii4UfA7/AX/xmmbTVsPDBzOOcO99/iJ4Apt+9dYW9/Y3Nou7BR39/YPDs2j44aKU0mZS2MRy5ZPFBM8Yi5yFKyVSEZCX7CmP3yY+s0Rk4rH0TOOE9YJST/iAacEtdQ1T12vdO+VnBvbu/KQvWAWssaka5btij2DtUqcnJQhR71r/ni9mKYhi5AKolTbsRPsZEQip4JNil6qWELokPRZNlt6Yl1oqWcFsdQvQmumLuRIqNQ49HUyJDhQy95U/M9rpxjcdTIeJSmyiM4HBamwMLamDVg9LhlFMdaEUMn1hhYdEEko6p6K+nRn+dBV0qhWHLviPF2Xa9W8hAKcwTlcggO3UINHqIMLFDJ4g0/4MkbGq/FufMyja0b+5wQWYHz/AfKZkAM=</latexit><latexit sha1_base64="Tr51jNqbo2fHDKe5uPiQvWPychs=">AAAB9HicbVDLSsNAFL3xWesrKq7cpBbBhZSkKLoRCm5cVjBtoSllMp20QycPZm6KJfRP3Ii4UfA7/AX/xmmbTVsPDBzOOcO99/iJ4Apt+9dYW9/Y3Nou7BR39/YPDs2j44aKU0mZS2MRy5ZPFBM8Yi5yFKyVSEZCX7CmP3yY+s0Rk4rH0TOOE9YJST/iAacEtdQ1T12vdO+VnBvbu/KQvWAWssaka5btij2DtUqcnJQhR71r/ni9mKYhi5AKolTbsRPsZEQip4JNil6qWELokPRZNlt6Yl1oqWcFsdQvQmumLuRIqNQ49HUyJDhQy95U/M9rpxjcdTIeJSmyiM4HBamwMLamDVg9LhlFMdaEUMn1hhYdEEko6p6K+nRn+dBV0qhWHLviPF2Xa9W8hAKcwTlcggO3UINHqIMLFDJ4g0/4MkbGq/FufMyja0b+5wQWYHz/AfKZkAM=</latexit><latexit sha1_base64="Tr51jNqbo2fHDKe5uPiQvWPychs=">AAAB9HicbVDLSsNAFL3xWesrKq7cpBbBhZSkKLoRCm5cVjBtoSllMp20QycPZm6KJfRP3Ii4UfA7/AX/xmmbTVsPDBzOOcO99/iJ4Apt+9dYW9/Y3Nou7BR39/YPDs2j44aKU0mZS2MRy5ZPFBM8Yi5yFKyVSEZCX7CmP3yY+s0Rk4rH0TOOE9YJST/iAacEtdQ1T12vdO+VnBvbu/KQvWAWssaka5btij2DtUqcnJQhR71r/ni9mKYhi5AKolTbsRPsZEQip4JNil6qWELokPRZNlt6Yl1oqWcFsdQvQmumLuRIqNQ49HUyJDhQy95U/M9rpxjcdTIeJSmyiM4HBamwMLamDVg9LhlFMdaEUMn1hhYdEEko6p6K+nRn+dBV0qhWHLviPF2Xa9W8hAKcwTlcggO3UINHqIMLFDJ4g0/4MkbGq/FufMyja0b+5wQWYHz/AfKZkAM=</latexit><latexit sha1_base64="Tr51jNqbo2fHDKe5uPiQvWPychs=">AAAB9HicbVDLSsNAFL3xWesrKq7cpBbBhZSkKLoRCm5cVjBtoSllMp20QycPZm6KJfRP3Ii4UfA7/AX/xmmbTVsPDBzOOcO99/iJ4Apt+9dYW9/Y3Nou7BR39/YPDs2j44aKU0mZS2MRy5ZPFBM8Yi5yFKyVSEZCX7CmP3yY+s0Rk4rH0TOOE9YJST/iAacEtdQ1T12vdO+VnBvbu/KQvWAWssaka5btij2DtUqcnJQhR71r/ni9mKYhi5AKolTbsRPsZEQip4JNil6qWELokPRZNlt6Yl1oqWcFsdQvQmumLuRIqNQ49HUyJDhQy95U/M9rpxjcdTIeJSmyiM4HBamwMLamDVg9LhlFMdaEUMn1hhYdEEko6p6K+nRn+dBV0qhWHLviPF2Xa9W8hAKcwTlcggO3UINHqIMLFDJ4g0/4MkbGq/FufMyja0b+5wQWYHz/AfKZkAM=</latexit>

[ Rut, Suszalski, AR, in preparation ]





Red solid line:Approx/asymptoticexpression [,T (E) / ⇥(|E|� 1

2�)]

|S|max = |S|GSmax�(kB/e)⇥

⇥12 ln x + ln 2 � 1

2 + O(x�1)⇤, with x = �/2kBT � 1.

<latexit sha1_base64="YKcNO02Q3CzFBqYFFbYR9oTICxM=">AAADDnicbVLLbhMxFPUMrxIeTWHJxiGDlKjkMdlQkCqVRwULJApNaKU4jRyPk1jxjEe2h07k+CPY8CtsQKxA/QX+Bk+SRR9czcjH5z6Ofa9HKWdKt9t/Pf/a9Rs3b23cLt25e+/+ZnnrwWclMklojwgu5PEIK8pZQnuaaU6PU0lxPOL0aDR7XfiPvlCpmEi6ep7SQYwnCRszgrWjhlteCyWCJRFNNCyhzAFZlDKIafiJRlAJziJYUPYFRKj0Mk2lyFtYzeNUC81IQdLcaapCA/YD9J6OtWSTqcZSilNjUaVb268jl+cy3LY7p RrXFvsLVGmgChpLTExoTceiN5RrXA8GrqT7gsXhYmiQjGGMc7t7YXeyRG8PbaM2G75q0TrSLKYKcafdP1+SJya328XSsY1zjm2DCObwg63lJ6YR2jpaHnkQPIWlU6anMMh3V+dpdZxCF00mxoZBEw7L1XazvTR4FYRrUAVrOxiWz1AkSBa7DhOOleqH7VQPDJaueZxa13RFU0xmeELNcp4WPnFUBMdCut/NZcleiMOxcgMYucgY66m67CvI//n6mR7vDAxL0kzThKyExhmHWsDiccCISUo0nzuAiWTFeMkUu6Zp94RK7urh5YteBb1O83kz/Nip7u2se7ABHoHHoAZC8AzsgXfgAPQA8b55P7zf3h//q//d/+n/WoX63jrnIbhg/tk/d1z3Fg==</latexit><latexit sha1_base64="YKcNO02Q3CzFBqYFFbYR9oTICxM=">AAADDnicbVLLbhMxFPUMrxIeTWHJxiGDlKjkMdlQkCqVRwULJApNaKU4jRyPk1jxjEe2h07k+CPY8CtsQKxA/QX+Bk+SRR9czcjH5z6Ofa9HKWdKt9t/Pf/a9Rs3b23cLt25e+/+ZnnrwWclMklojwgu5PEIK8pZQnuaaU6PU0lxPOL0aDR7XfiPvlCpmEi6ep7SQYwnCRszgrWjhlteCyWCJRFNNCyhzAFZlDKIafiJRlAJziJYUPYFRKj0Mk2lyFtYzeNUC81IQdLcaapCA/YD9J6OtWSTqcZSilNjUaVb268jl+cy3LY7p RrXFvsLVGmgChpLTExoTceiN5RrXA8GrqT7gsXhYmiQjGGMc7t7YXeyRG8PbaM2G75q0TrSLKYKcafdP1+SJya328XSsY1zjm2DCObwg63lJ6YR2jpaHnkQPIWlU6anMMh3V+dpdZxCF00mxoZBEw7L1XazvTR4FYRrUAVrOxiWz1AkSBa7DhOOleqH7VQPDJaueZxa13RFU0xmeELNcp4WPnFUBMdCut/NZcleiMOxcgMYucgY66m67CvI//n6mR7vDAxL0kzThKyExhmHWsDiccCISUo0nzuAiWTFeMkUu6Zp94RK7urh5YteBb1O83kz/Nip7u2se7ABHoHHoAZC8AzsgXfgAPQA8b55P7zf3h//q//d/+n/WoX63jrnIbhg/tk/d1z3Fg==</latexit><latexit sha1_base64="YKcNO02Q3CzFBqYFFbYR9oTICxM=">AAADDnicbVLLbhMxFPUMrxIeTWHJxiGDlKjkMdlQkCqVRwULJApNaKU4jRyPk1jxjEe2h07k+CPY8CtsQKxA/QX+Bk+SRR9czcjH5z6Ofa9HKWdKt9t/Pf/a9Rs3b23cLt25e+/+ZnnrwWclMklojwgu5PEIK8pZQnuaaU6PU0lxPOL0aDR7XfiPvlCpmEi6ep7SQYwnCRszgrWjhlteCyWCJRFNNCyhzAFZlDKIafiJRlAJziJYUPYFRKj0Mk2lyFtYzeNUC81IQdLcaapCA/YD9J6OtWSTqcZSilNjUaVb268jl+cy3LY7p RrXFvsLVGmgChpLTExoTceiN5RrXA8GrqT7gsXhYmiQjGGMc7t7YXeyRG8PbaM2G75q0TrSLKYKcafdP1+SJya328XSsY1zjm2DCObwg63lJ6YR2jpaHnkQPIWlU6anMMh3V+dpdZxCF00mxoZBEw7L1XazvTR4FYRrUAVrOxiWz1AkSBa7DhOOleqH7VQPDJaueZxa13RFU0xmeELNcp4WPnFUBMdCut/NZcleiMOxcgMYucgY66m67CvI//n6mR7vDAxL0kzThKyExhmHWsDiccCISUo0nzuAiWTFeMkUu6Zp94RK7urh5YteBb1O83kz/Nip7u2se7ABHoHHoAZC8AzsgXfgAPQA8b55P7zf3h//q//d/+n/WoX63jrnIbhg/tk/d1z3Fg==</latexit><latexit sha1_base64="YKcNO02Q3CzFBqYFFbYR9oTICxM=">AAADDnicbVLLbhMxFPUMrxIeTWHJxiGDlKjkMdlQkCqVRwULJApNaKU4jRyPk1jxjEe2h07k+CPY8CtsQKxA/QX+Bk+SRR9czcjH5z6Ofa9HKWdKt9t/Pf/a9Rs3b23cLt25e+/+ZnnrwWclMklojwgu5PEIK8pZQnuaaU6PU0lxPOL0aDR7XfiPvlCpmEi6ep7SQYwnCRszgrWjhlteCyWCJRFNNCyhzAFZlDKIafiJRlAJziJYUPYFRKj0Mk2lyFtYzeNUC81IQdLcaapCA/YD9J6OtWSTqcZSilNjUaVb268jl+cy3LY7p RrXFvsLVGmgChpLTExoTceiN5RrXA8GrqT7gsXhYmiQjGGMc7t7YXeyRG8PbaM2G75q0TrSLKYKcafdP1+SJya328XSsY1zjm2DCObwg63lJ6YR2jpaHnkQPIWlU6anMMh3V+dpdZxCF00mxoZBEw7L1XazvTR4FYRrUAVrOxiWz1AkSBa7DhOOleqH7VQPDJaueZxa13RFU0xmeELNcp4WPnFUBMdCut/NZcleiMOxcgMYucgY66m67CvI//n6mR7vDAxL0kzThKyExhmHWsDiccCISUo0nzuAiWTFeMkUu6Zp94RK7urh5YteBb1O83kz/Nip7u2se7ABHoHHoAZC8AzsgXfgAPQA8b55P7zf3h//q//d/+n/WoX63jrnIbhg/tk/d1z3Fg==</latexit>

Dotted line:The reference(Goldsmid-Sharp)value:

|S|GSmax = �

2eT<latexit sha1_base64="me3uk8OOYXRW3msCasY892ZU01E=">AAACbXicbVHRTtswFHXCYKwbrGPSXpgmay0CHqiSvoxNmoQ0JPbI1HYg4VI5zg21sOPIvkGrQr5hv7Vf2F/sZe9zQl+AXcnS8Tn32j7HSaGkwyj6HYQrT1bXnq4/6zx/sbH5svtq67szpRUwEUYZe55wB0rmMEGJCs4LC1wnCs6S6y+NfnYD1kmTj3FRwFTzq1xmUnD01Kz7k+VG5inkSDus9MA2J1VMIj02iJDSZl9/oox1xnOgFjKwkAtoiL2KJRk9MSp1WqYHozm3Rb3vFXrDVQntULP6t6PbWcWsppr/qC9bdDKqP7PMc lGxY1DI62oI47o/6/aiQdQWfQziJeiRZZ3Our9YakSpvQOhuHMXcVTgtOIWpVBQe1MOCi6u+RVUbVw13fFUSjNj/fK+W/ZeH9fOLXTiOzXHuXuoNeT/tIsSs8NpJfOiRB/R3UVZqSga2mRPU2lBoFp4wIWV/oVU+My4QP9DHW89fmj0MZgMBx8H8bdh7+hwmcE62SbvyR6JyQdyRL6SUzIhgvwNtoN+sBP8Cd+Eb8N3d61hsJx5Te5VuPsPmd+5Vw==</latexit><latexit sha1_base64="me3uk8OOYXRW3msCasY892ZU01E=">AAACbXicbVHRTtswFHXCYKwbrGPSXpgmay0CHqiSvoxNmoQ0JPbI1HYg4VI5zg21sOPIvkGrQr5hv7Vf2F/sZe9zQl+AXcnS8Tn32j7HSaGkwyj6HYQrT1bXnq4/6zx/sbH5svtq67szpRUwEUYZe55wB0rmMEGJCs4LC1wnCs6S6y+NfnYD1kmTj3FRwFTzq1xmUnD01Kz7k+VG5inkSDus9MA2J1VMIj02iJDSZl9/oox1xnOgFjKwkAtoiL2KJRk9MSp1WqYHozm3Rb3vFXrDVQntULP6t6PbWcWsppr/qC9bdDKqP7PMc lGxY1DI62oI47o/6/aiQdQWfQziJeiRZZ3Our9YakSpvQOhuHMXcVTgtOIWpVBQe1MOCi6u+RVUbVw13fFUSjNj/fK+W/ZeH9fOLXTiOzXHuXuoNeT/tIsSs8NpJfOiRB/R3UVZqSga2mRPU2lBoFp4wIWV/oVU+My4QP9DHW89fmj0MZgMBx8H8bdh7+hwmcE62SbvyR6JyQdyRL6SUzIhgvwNtoN+sBP8Cd+Eb8N3d61hsJx5Te5VuPsPmd+5Vw==</latexit><latexit sha1_base64="me3uk8OOYXRW3msCasY892ZU01E=">AAACbXicbVHRTtswFHXCYKwbrGPSXpgmay0CHqiSvoxNmoQ0JPbI1HYg4VI5zg21sOPIvkGrQr5hv7Vf2F/sZe9zQl+AXcnS8Tn32j7HSaGkwyj6HYQrT1bXnq4/6zx/sbH5svtq67szpRUwEUYZe55wB0rmMEGJCs4LC1wnCs6S6y+NfnYD1kmTj3FRwFTzq1xmUnD01Kz7k+VG5inkSDus9MA2J1VMIj02iJDSZl9/oox1xnOgFjKwkAtoiL2KJRk9MSp1WqYHozm3Rb3vFXrDVQntULP6t6PbWcWsppr/qC9bdDKqP7PMc lGxY1DI62oI47o/6/aiQdQWfQziJeiRZZ3Our9YakSpvQOhuHMXcVTgtOIWpVBQe1MOCi6u+RVUbVw13fFUSjNj/fK+W/ZeH9fOLXTiOzXHuXuoNeT/tIsSs8NpJfOiRB/R3UVZqSga2mRPU2lBoFp4wIWV/oVU+My4QP9DHW89fmj0MZgMBx8H8bdh7+hwmcE62SbvyR6JyQdyRL6SUzIhgvwNtoN+sBP8Cd+Eb8N3d61hsJx5Te5VuPsPmd+5Vw==</latexit><latexit sha1_base64="me3uk8OOYXRW3msCasY892ZU01E=">AAACbXicbVHRTtswFHXCYKwbrGPSXpgmay0CHqiSvoxNmoQ0JPbI1HYg4VI5zg21sOPIvkGrQr5hv7Vf2F/sZe9zQl+AXcnS8Tn32j7HSaGkwyj6HYQrT1bXnq4/6zx/sbH5svtq67szpRUwEUYZe55wB0rmMEGJCs4LC1wnCs6S6y+NfnYD1kmTj3FRwFTzq1xmUnD01Kz7k+VG5inkSDus9MA2J1VMIj02iJDSZl9/oox1xnOgFjKwkAtoiL2KJRk9MSp1WqYHozm3Rb3vFXrDVQntULP6t6PbWcWsppr/qC9bdDKqP7PMc lGxY1DI62oI47o/6/aiQdQWfQziJeiRZZ3Our9YakSpvQOhuHMXcVTgtOIWpVBQe1MOCi6u+RVUbVw13fFUSjNj/fK+W/ZeH9fOLXTiOzXHuXuoNeT/tIsSs8NpJfOiRB/R3UVZqSga2mRPU2lBoFp4wIWV/oVU+My4QP9DHW89fmj0MZgMBx8H8bdh7+hwmcE62SbvyR6JyQdyRL6SUzIhgvwNtoN+sBP8Cd+Eb8N3d61hsJx5Te5VuPsPmd+5Vw==</latexit>

|S|G

Sm

ax�

|S| m

ax

[kB

/e]

<latexit sha1_base64="KSGBZTJvfc/4ccuvBiKsL1DuvLc=">AAACFHicbVDLSgMxAMz6rPW16tFLahE8aN0tih6LHvRYqX1Ad12yabYNzT5IsmLZ7m948Ve8iAii4N2/MW330taBwDAzIZlxI0aFNIxfbWFxaXllNbeWX9/Y3NrWd3YbIow5JnUcspC3XCQIowGpSyoZaUWcIN9lpOn2r0d+85FwQcPgXg4iYvuoG1CPYiSV5Ojnw9rQSSzuQx89pQ/JmFrHN7U0tQonVmHKtmC771ydEtvRi0bJGAPOEzMjRZCh6uifVifEsU8CiRkSom0akbQTxCXFjKR5KxYkQriPuiQZl0rhoZI60Au5OoGEY3Uqh3whBr6rkj6SPTHrjcT/vHYsvUs7oUEUSxLgyUNezKAM4Wgh2KGcYMkGiiDMqfohxD3EEZZqx7yqbs4WnSeNcsk0SubdWbFSzkbIgX1wAI6ACS5ABdyCKqgDDF7AG/gC39qz9qq9ax+T6IKW3dkDU9B+/gCufp2k</latexit><latexit sha1_base64="KSGBZTJvfc/4ccuvBiKsL1DuvLc=">AAACFHicbVDLSgMxAMz6rPW16tFLahE8aN0tih6LHvRYqX1Ad12yabYNzT5IsmLZ7m948Ve8iAii4N2/MW330taBwDAzIZlxI0aFNIxfbWFxaXllNbeWX9/Y3NrWd3YbIow5JnUcspC3XCQIowGpSyoZaUWcIN9lpOn2r0d+85FwQcPgXg4iYvuoG1CPYiSV5Ojnw9rQSSzuQx89pQ/JmFrHN7U0tQonVmHKtmC771ydEtvRi0bJGAPOEzMjRZCh6uifVifEsU8CiRkSom0akbQTxCXFjKR5KxYkQriPuiQZl0rhoZI60Au5OoGEY3Uqh3whBr6rkj6SPTHrjcT/vHYsvUs7oUEUSxLgyUNezKAM4Wgh2KGcYMkGiiDMqfohxD3EEZZqx7yqbs4WnSeNcsk0SubdWbFSzkbIgX1wAI6ACS5ABdyCKqgDDF7AG/gC39qz9qq9ax+T6IKW3dkDU9B+/gCufp2k</latexit><latexit sha1_base64="KSGBZTJvfc/4ccuvBiKsL1DuvLc=">AAACFHicbVDLSgMxAMz6rPW16tFLahE8aN0tih6LHvRYqX1Ad12yabYNzT5IsmLZ7m948Ve8iAii4N2/MW330taBwDAzIZlxI0aFNIxfbWFxaXllNbeWX9/Y3NrWd3YbIow5JnUcspC3XCQIowGpSyoZaUWcIN9lpOn2r0d+85FwQcPgXg4iYvuoG1CPYiSV5Ojnw9rQSSzuQx89pQ/JmFrHN7U0tQonVmHKtmC771ydEtvRi0bJGAPOEzMjRZCh6uifVifEsU8CiRkSom0akbQTxCXFjKR5KxYkQriPuiQZl0rhoZI60Au5OoGEY3Uqh3whBr6rkj6SPTHrjcT/vHYsvUs7oUEUSxLgyUNezKAM4Wgh2KGcYMkGiiDMqfohxD3EEZZqx7yqbs4WnSeNcsk0SubdWbFSzkbIgX1wAI6ACS5ABdyCKqgDDF7AG/gC39qz9qq9ax+T6IKW3dkDU9B+/gCufp2k</latexit><latexit sha1_base64="KSGBZTJvfc/4ccuvBiKsL1DuvLc=">AAACFHicbVDLSgMxAMz6rPW16tFLahE8aN0tih6LHvRYqX1Ad12yabYNzT5IsmLZ7m948Ve8iAii4N2/MW330taBwDAzIZlxI0aFNIxfbWFxaXllNbeWX9/Y3NrWd3YbIow5JnUcspC3XCQIowGpSyoZaUWcIN9lpOn2r0d+85FwQcPgXg4iYvuoG1CPYiSV5Ojnw9rQSSzuQx89pQ/JmFrHN7U0tQonVmHKtmC771ydEtvRi0bJGAPOEzMjRZCh6uifVifEsU8CiRkSom0akbQTxCXFjKR5KxYkQriPuiQZl0rhoZI60Au5OoGEY3Uqh3whBr6rkj6SPTHrjcT/vHYsvUs7oUEUSxLgyUNezKAM4Wgh2KGcYMkGiiDMqfohxD3EEZZqx7yqbs4WnSeNcsk0SubdWbFSzkbIgX1wAI6ACS5ABdyCKqgDDF7AG/gC39qz9qq9ax+T6IKW3dkDU9B+/gCufp2k</latexit>

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<latexit sha1_base64="KSGBZTJvfc/4ccuvBiKsL1DuvLc=">AAACFHicbVDLSgMxAMz6rPW16tFLahE8aN0tih6LHvRYqX1Ad12yabYNzT5IsmLZ7m948Ve8iAii4N2/MW330taBwDAzIZlxI0aFNIxfbWFxaXllNbeWX9/Y3NrWd3YbIow5JnUcspC3XCQIowGpSyoZaUWcIN9lpOn2r0d+85FwQcPgXg4iYvuoG1CPYiSV5Ojnw9rQSSzuQx89pQ/JmFrHN7U0tQonVmHKtmC771ydEtvRi0bJGAPOEzMjRZCh6uifVifEsU8CiRkSom0akbQTxCXFjKR5KxYkQriPuiQZl0rhoZI60Au5OoGEY3Uqh3whBr6rkj6SPTHrjcT/vHYsvUs7oUEUSxLgyUNezKAM4Wgh2KGcYMkGiiDMqfohxD3EEZZqx7yqbs4WnSeNcsk0SubdWbFSzkbIgX1wAI6ACS5ABdyCKqgDDF7AG/gC39qz9qq9ax+T6IKW3dkDU9B+/gCufp2k</latexit><latexit sha1_base64="KSGBZTJvfc/4ccuvBiKsL1DuvLc=">AAACFHicbVDLSgMxAMz6rPW16tFLahE8aN0tih6LHvRYqX1Ad12yabYNzT5IsmLZ7m948Ve8iAii4N2/MW330taBwDAzIZlxI0aFNIxfbWFxaXllNbeWX9/Y3NrWd3YbIow5JnUcspC3XCQIowGpSyoZaUWcIN9lpOn2r0d+85FwQcPgXg4iYvuoG1CPYiSV5Ojnw9rQSSzuQx89pQ/JmFrHN7U0tQonVmHKtmC771ydEtvRi0bJGAPOEzMjRZCh6uifVifEsU8CiRkSom0akbQTxCXFjKR5KxYkQriPuiQZl0rhoZI60Au5OoGEY3Uqh3whBr6rkj6SPTHrjcT/vHYsvUs7oUEUSxLgyUNezKAM4Wgh2KGcYMkGiiDMqfohxD3EEZZqx7yqbs4WnSeNcsk0SubdWbFSzkbIgX1wAI6ACS5ABdyCKqgDDF7AG/gC39qz9qq9ax+T6IKW3dkDU9B+/gCufp2k</latexit><latexit sha1_base64="KSGBZTJvfc/4ccuvBiKsL1DuvLc=">AAACFHicbVDLSgMxAMz6rPW16tFLahE8aN0tih6LHvRYqX1Ad12yabYNzT5IsmLZ7m948Ve8iAii4N2/MW330taBwDAzIZlxI0aFNIxfbWFxaXllNbeWX9/Y3NrWd3YbIow5JnUcspC3XCQIowGpSyoZaUWcIN9lpOn2r0d+85FwQcPgXg4iYvuoG1CPYiSV5Ojnw9rQSSzuQx89pQ/JmFrHN7U0tQonVmHKtmC771ydEtvRi0bJGAPOEzMjRZCh6uifVifEsU8CiRkSom0akbQTxCXFjKR5KxYkQriPuiQZl0rhoZI60Au5OoGEY3Uqh3whBr6rkj6SPTHrjcT/vHYsvUs7oUEUSxLgyUNezKAM4Wgh2KGcYMkGiiDMqfohxD3EEZZqx7yqbs4WnSeNcsk0SubdWbFSzkbIgX1wAI6ACS5ABdyCKqgDDF7AG/gC39qz9qq9ax+T6IKW3dkDU9B+/gCufp2k</latexit><latexit sha1_base64="KSGBZTJvfc/4ccuvBiKsL1DuvLc=">AAACFHicbVDLSgMxAMz6rPW16tFLahE8aN0tih6LHvRYqX1Ad12yabYNzT5IsmLZ7m948Ve8iAii4N2/MW330taBwDAzIZlxI0aFNIxfbWFxaXllNbeWX9/Y3NrWd3YbIow5JnUcspC3XCQIowGpSyoZaUWcIN9lpOn2r0d+85FwQcPgXg4iYvuoG1CPYiSV5Ojnw9rQSSzuQx89pQ/JmFrHN7U0tQonVmHKtmC771ydEtvRi0bJGAPOEzMjRZCh6uifVifEsU8CiRkSom0akbQTxCXFjKR5KxYkQriPuiQZl0rhoZI60Au5OoGEY3Uqh3whBr6rkj6SPTHrjcT/vHYsvUs7oUEUSxLgyUNezKAM4Wgh2KGcYMkGiiDMqfohxD3EEZZqx7yqbs4WnSeNcsk0SubdWbFSzkbIgX1wAI6ACS5ABdyCKqgDDF7AG/gC39qz9qq9ax+T6IKW3dkDU9B+/gCufp2k</latexit>

0

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�3 ⌘ t0 = 0.3 eV<latexit sha1_base64="v8M4KGTmQhq+41cyA3u1iRrJAxA=">AAAB/3icbVBNS8MwAE3n15xfVY9eokP0IKN1gvMgDLx4nGC3wTpKmqVbWNLUJB2O0osX/4oXxZPiP/Av+G/stl42fRB4vPdC8p4fMaq0Zf0YhaXlldW14nppY3Nre8fc3WsqEUtMHCyYkG0fKcJoSBxNNSPtSBLEfUZa/vBm4rdGRCoqwns9jkiXo35IA4qRziTPPHT7iHPkVV3yENNRkuqTa6tSdc9c7ovHhDRTzyxbFWsK+JfYOSmDHA3P/HZ7AsechBozpFTHtiLdTZDUFDOSltxYkQjhIeqTZFoghceZ1IOBkNkJNZyqcznElRpzP0typAdq0ZuI/3mdWAe1bkLDKNYkxLOHgphBLeBkDdijkmDNxhlBWNLshxAPkERYZ5uVsur2YtG/xDmvXFXsu4tyvZZvUAQH4AicAhtcgjq4BQ3gAAyewSv4AJ/Gk/FivBnvs2jByO/sgzkYX79YaZWP</latexit><latexit sha1_base64="v8M4KGTmQhq+41cyA3u1iRrJAxA=">AAAB/3icbVBNS8MwAE3n15xfVY9eokP0IKN1gvMgDLx4nGC3wTpKmqVbWNLUJB2O0osX/4oXxZPiP/Av+G/stl42fRB4vPdC8p4fMaq0Zf0YhaXlldW14nppY3Nre8fc3WsqEUtMHCyYkG0fKcJoSBxNNSPtSBLEfUZa/vBm4rdGRCoqwns9jkiXo35IA4qRziTPPHT7iHPkVV3yENNRkuqTa6tSdc9c7ovHhDRTzyxbFWsK+JfYOSmDHA3P/HZ7AsechBozpFTHtiLdTZDUFDOSltxYkQjhIeqTZFoghceZ1IOBkNkJNZyqcznElRpzP0typAdq0ZuI/3mdWAe1bkLDKNYkxLOHgphBLeBkDdijkmDNxhlBWNLshxAPkERYZ5uVsur2YtG/xDmvXFXsu4tyvZZvUAQH4AicAhtcgjq4BQ3gAAyewSv4AJ/Gk/FivBnvs2jByO/sgzkYX79YaZWP</latexit><latexit sha1_base64="v8M4KGTmQhq+41cyA3u1iRrJAxA=">AAAB/3icbVBNS8MwAE3n15xfVY9eokP0IKN1gvMgDLx4nGC3wTpKmqVbWNLUJB2O0osX/4oXxZPiP/Av+G/stl42fRB4vPdC8p4fMaq0Zf0YhaXlldW14nppY3Nre8fc3WsqEUtMHCyYkG0fKcJoSBxNNSPtSBLEfUZa/vBm4rdGRCoqwns9jkiXo35IA4qRziTPPHT7iHPkVV3yENNRkuqTa6tSdc9c7ovHhDRTzyxbFWsK+JfYOSmDHA3P/HZ7AsechBozpFTHtiLdTZDUFDOSltxYkQjhIeqTZFoghceZ1IOBkNkJNZyqcznElRpzP0typAdq0ZuI/3mdWAe1bkLDKNYkxLOHgphBLeBkDdijkmDNxhlBWNLshxAPkERYZ5uVsur2YtG/xDmvXFXsu4tyvZZvUAQH4AicAhtcgjq4BQ3gAAyewSv4AJ/Gk/FivBnvs2jByO/sgzkYX79YaZWP</latexit><latexit sha1_base64="v8M4KGTmQhq+41cyA3u1iRrJAxA=">AAAB/3icbVBNS8MwAE3n15xfVY9eokP0IKN1gvMgDLx4nGC3wTpKmqVbWNLUJB2O0osX/4oXxZPiP/Av+G/stl42fRB4vPdC8p4fMaq0Zf0YhaXlldW14nppY3Nre8fc3WsqEUtMHCyYkG0fKcJoSBxNNSPtSBLEfUZa/vBm4rdGRCoqwns9jkiXo35IA4qRziTPPHT7iHPkVV3yENNRkuqTa6tSdc9c7ovHhDRTzyxbFWsK+JfYOSmDHA3P/HZ7AsechBozpFTHtiLdTZDUFDOSltxYkQjhIeqTZFoghceZ1IOBkNkJNZyqcznElRpzP0typAdq0ZuI/3mdWAe1bkLDKNYkxLOHgphBLeBkDdijkmDNxhlBWNLshxAPkERYZ5uVsur2YtG/xDmvXFXsu4tyvZZvUAQH4AicAhtcgjq4BQ3gAAyewSv4AJ/Gk/FivBnvs2jByO/sgzkYX79YaZWP</latexit>

�3 ⌘ t0 = 0.3 eV<latexit sha1_base64="v8M4KGTmQhq+41cyA3u1iRrJAxA=">AAAB/3icbVBNS8MwAE3n15xfVY9eokP0IKN1gvMgDLx4nGC3wTpKmqVbWNLUJB2O0osX/4oXxZPiP/Av+G/stl42fRB4vPdC8p4fMaq0Zf0YhaXlldW14nppY3Nre8fc3WsqEUtMHCyYkG0fKcJoSBxNNSPtSBLEfUZa/vBm4rdGRCoqwns9jkiXo35IA4qRziTPPHT7iHPkVV3yENNRkuqTa6tSdc9c7ovHhDRTzyxbFWsK+JfYOSmDHA3P/HZ7AsechBozpFTHtiLdTZDUFDOSltxYkQjhIeqTZFoghceZ1IOBkNkJNZyqcznElRpzP0typAdq0ZuI/3mdWAe1bkLDKNYkxLOHgphBLeBkDdijkmDNxhlBWNLshxAPkERYZ5uVsur2YtG/xDmvXFXsu4tyvZZvUAQH4AicAhtcgjq4BQ3gAAyewSv4AJ/Gk/FivBnvs2jByO/sgzkYX79YaZWP</latexit><latexit sha1_base64="v8M4KGTmQhq+41cyA3u1iRrJAxA=">AAAB/3icbVBNS8MwAE3n15xfVY9eokP0IKN1gvMgDLx4nGC3wTpKmqVbWNLUJB2O0osX/4oXxZPiP/Av+G/stl42fRB4vPdC8p4fMaq0Zf0YhaXlldW14nppY3Nre8fc3WsqEUtMHCyYkG0fKcJoSBxNNSPtSBLEfUZa/vBm4rdGRCoqwns9jkiXo35IA4qRziTPPHT7iHPkVV3yENNRkuqTa6tSdc9c7ovHhDRTzyxbFWsK+JfYOSmDHA3P/HZ7AsechBozpFTHtiLdTZDUFDOSltxYkQjhIeqTZFoghceZ1IOBkNkJNZyqcznElRpzP0typAdq0ZuI/3mdWAe1bkLDKNYkxLOHgphBLeBkDdijkmDNxhlBWNLshxAPkERYZ5uVsur2YtG/xDmvXFXsu4tyvZZvUAQH4AicAhtcgjq4BQ3gAAyewSv4AJ/Gk/FivBnvs2jByO/sgzkYX79YaZWP</latexit><latexit sha1_base64="v8M4KGTmQhq+41cyA3u1iRrJAxA=">AAAB/3icbVBNS8MwAE3n15xfVY9eokP0IKN1gvMgDLx4nGC3wTpKmqVbWNLUJB2O0osX/4oXxZPiP/Av+G/stl42fRB4vPdC8p4fMaq0Zf0YhaXlldW14nppY3Nre8fc3WsqEUtMHCyYkG0fKcJoSBxNNSPtSBLEfUZa/vBm4rdGRCoqwns9jkiXo35IA4qRziTPPHT7iHPkVV3yENNRkuqTa6tSdc9c7ovHhDRTzyxbFWsK+JfYOSmDHA3P/HZ7AsechBozpFTHtiLdTZDUFDOSltxYkQjhIeqTZFoghceZ1IOBkNkJNZyqcznElRpzP0typAdq0ZuI/3mdWAe1bkLDKNYkxLOHgphBLeBkDdijkmDNxhlBWNLshxAPkERYZ5uVsur2YtG/xDmvXFXsu4tyvZZvUAQH4AicAhtcgjq4BQ3gAAyewSv4AJ/Gk/FivBnvs2jByO/sgzkYX79YaZWP</latexit><latexit sha1_base64="v8M4KGTmQhq+41cyA3u1iRrJAxA=">AAAB/3icbVBNS8MwAE3n15xfVY9eokP0IKN1gvMgDLx4nGC3wTpKmqVbWNLUJB2O0osX/4oXxZPiP/Av+G/stl42fRB4vPdC8p4fMaq0Zf0YhaXlldW14nppY3Nre8fc3WsqEUtMHCyYkG0fKcJoSBxNNSPtSBLEfUZa/vBm4rdGRCoqwns9jkiXo35IA4qRziTPPHT7iHPkVV3yENNRkuqTa6tSdc9c7ovHhDRTzyxbFWsK+JfYOSmDHA3P/HZ7AsechBozpFTHtiLdTZDUFDOSltxYkQjhIeqTZFoghceZ1IOBkNkJNZyqcznElRpzP0typAdq0ZuI/3mdWAe1bkLDKNYkxLOHgphBLeBkDdijkmDNxhlBWNLshxAPkERYZ5uVsur2YtG/xDmvXFXsu4tyvZZvUAQH4AicAhtcgjq4BQ3gAAyewSv4AJ/Gk/FivBnvs2jByO/sgzkYX79YaZWP</latexit>

T = 1K<latexit sha1_base64="9ruOToFeRGKbw8XDYfczG8P7rkU=">AAAB6XicbVDLSsNAFL2pr1pfVZduBovgQkoiBetCKLgR3FRobKEJZTKdtEMnmTgzEUvoT7hRXCn+jL/g3zhts2nrgYHDOWe499wg4Uxp2/61CmvrG5tbxe3Szu7e/kH58OhRiVQS6hLBhewEWFHOYupqpjntJJLiKOC0HYxup377mUrFRNzS44T6ER7ELGQEayP5rRvHu/CiQLxk95NeuWJX7RnQKnFyUoEczV75x+sLkkY01oRjpbqOnWg/w1Izwumk5KWKJpiM8IBms1Un6MxIfRQKaV6s0UxdyOFIqXEUmGSE9VAte1PxP6+b6rDuZyxOUk1jMh8Uphxpgaa9UZ9JSjQfG4KJZGZDRIZYYqLNdUqmurNcdJW4l9XrqvNQqzTq+Q2KcAKncA4OXEED7qAJLhB4gjf4hC+LW6/Wu/Uxjxas/M8xLMD6/gM72I0Q</latexit><latexit sha1_base64="9ruOToFeRGKbw8XDYfczG8P7rkU=">AAAB6XicbVDLSsNAFL2pr1pfVZduBovgQkoiBetCKLgR3FRobKEJZTKdtEMnmTgzEUvoT7hRXCn+jL/g3zhts2nrgYHDOWe499wg4Uxp2/61CmvrG5tbxe3Szu7e/kH58OhRiVQS6hLBhewEWFHOYupqpjntJJLiKOC0HYxup377mUrFRNzS44T6ER7ELGQEayP5rRvHu/CiQLxk95NeuWJX7RnQKnFyUoEczV75x+sLkkY01oRjpbqOnWg/w1Izwumk5KWKJpiM8IBms1Un6MxIfRQKaV6s0UxdyOFIqXEUmGSE9VAte1PxP6+b6rDuZyxOUk1jMh8Uphxpgaa9UZ9JSjQfG4KJZGZDRIZYYqLNdUqmurNcdJW4l9XrqvNQqzTq+Q2KcAKncA4OXEED7qAJLhB4gjf4hC+LW6/Wu/Uxjxas/M8xLMD6/gM72I0Q</latexit><latexit sha1_base64="9ruOToFeRGKbw8XDYfczG8P7rkU=">AAAB6XicbVDLSsNAFL2pr1pfVZduBovgQkoiBetCKLgR3FRobKEJZTKdtEMnmTgzEUvoT7hRXCn+jL/g3zhts2nrgYHDOWe499wg4Uxp2/61CmvrG5tbxe3Szu7e/kH58OhRiVQS6hLBhewEWFHOYupqpjntJJLiKOC0HYxup377mUrFRNzS44T6ER7ELGQEayP5rRvHu/CiQLxk95NeuWJX7RnQKnFyUoEczV75x+sLkkY01oRjpbqOnWg/w1Izwumk5KWKJpiM8IBms1Un6MxIfRQKaV6s0UxdyOFIqXEUmGSE9VAte1PxP6+b6rDuZyxOUk1jMh8Uphxpgaa9UZ9JSjQfG4KJZGZDRIZYYqLNdUqmurNcdJW4l9XrqvNQqzTq+Q2KcAKncA4OXEED7qAJLhB4gjf4hC+LW6/Wu/Uxjxas/M8xLMD6/gM72I0Q</latexit><latexit sha1_base64="9ruOToFeRGKbw8XDYfczG8P7rkU=">AAAB6XicbVDLSsNAFL2pr1pfVZduBovgQkoiBetCKLgR3FRobKEJZTKdtEMnmTgzEUvoT7hRXCn+jL/g3zhts2nrgYHDOWe499wg4Uxp2/61CmvrG5tbxe3Szu7e/kH58OhRiVQS6hLBhewEWFHOYupqpjntJJLiKOC0HYxup377mUrFRNzS44T6ER7ELGQEayP5rRvHu/CiQLxk95NeuWJX7RnQKnFyUoEczV75x+sLkkY01oRjpbqOnWg/w1Izwumk5KWKJpiM8IBms1Un6MxIfRQKaV6s0UxdyOFIqXEUmGSE9VAte1PxP6+b6rDuZyxOUk1jMh8Uphxpgaa9UZ9JSjQfG4KJZGZDRIZYYqLNdUqmurNcdJW4l9XrqvNQqzTq+Q2KcAKncA4OXEED7qAJLhB4gjf4hC+LW6/Wu/Uxjxas/M8xLMD6/gM72I0Q</latexit>

T = 5K<latexit sha1_base64="RJWpAWqPADdEfRm1qsDliJZHJRc=">AAAB6XicbVDLSsNAFL2pr1pfVZduBovgQkoiinUhFNwIbio0ttCEMplO2qGTTJyZiCX0J9worhR/xl/wb5y22bT1wMDhnDPce26QcKa0bf9ahZXVtfWN4mZpa3tnd6+8f/CoRCoJdYngQrYDrChnMXU105y2E0lxFHDaCoa3E7/1TKViIm7qUUL9CPdjFjKCtZH85s2ld+ZFgXjJ7sfdcsWu2lOgZeLkpAI5Gt3yj9cTJI1orAnHSnUcO9F+hqVmhNNxyUsVTTAZ4j7NpquO0YmReigU0rxYo6k6l8ORUqMoMMkI64Fa9Cbif14n1WHNz1icpJrGZDYoTDnSAk16ox6TlGg+MgQTycyGiAywxESb65RMdWex6DJxz6vXVefholKv5TcowhEcwyk4cAV1uIMGuEDgCd7gE74sbr1a79bHLFqw8j+HMAfr+w9B5I0U</latexit><latexit sha1_base64="RJWpAWqPADdEfRm1qsDliJZHJRc=">AAAB6XicbVDLSsNAFL2pr1pfVZduBovgQkoiinUhFNwIbio0ttCEMplO2qGTTJyZiCX0J9worhR/xl/wb5y22bT1wMDhnDPce26QcKa0bf9ahZXVtfWN4mZpa3tnd6+8f/CoRCoJdYngQrYDrChnMXU105y2E0lxFHDaCoa3E7/1TKViIm7qUUL9CPdjFjKCtZH85s2ld+ZFgXjJ7sfdcsWu2lOgZeLkpAI5Gt3yj9cTJI1orAnHSnUcO9F+hqVmhNNxyUsVTTAZ4j7NpquO0YmReigU0rxYo6k6l8ORUqMoMMkI64Fa9Cbif14n1WHNz1icpJrGZDYoTDnSAk16ox6TlGg+MgQTycyGiAywxESb65RMdWex6DJxz6vXVefholKv5TcowhEcwyk4cAV1uIMGuEDgCd7gE74sbr1a79bHLFqw8j+HMAfr+w9B5I0U</latexit><latexit sha1_base64="RJWpAWqPADdEfRm1qsDliJZHJRc=">AAAB6XicbVDLSsNAFL2pr1pfVZduBovgQkoiinUhFNwIbio0ttCEMplO2qGTTJyZiCX0J9worhR/xl/wb5y22bT1wMDhnDPce26QcKa0bf9ahZXVtfWN4mZpa3tnd6+8f/CoRCoJdYngQrYDrChnMXU105y2E0lxFHDaCoa3E7/1TKViIm7qUUL9CPdjFjKCtZH85s2ld+ZFgXjJ7sfdcsWu2lOgZeLkpAI5Gt3yj9cTJI1orAnHSnUcO9F+hqVmhNNxyUsVTTAZ4j7NpquO0YmReigU0rxYo6k6l8ORUqMoMMkI64Fa9Cbif14n1WHNz1icpJrGZDYoTDnSAk16ox6TlGg+MgQTycyGiAywxESb65RMdWex6DJxz6vXVefholKv5TcowhEcwyk4cAV1uIMGuEDgCd7gE74sbr1a79bHLFqw8j+HMAfr+w9B5I0U</latexit><latexit sha1_base64="RJWpAWqPADdEfRm1qsDliJZHJRc=">AAAB6XicbVDLSsNAFL2pr1pfVZduBovgQkoiinUhFNwIbio0ttCEMplO2qGTTJyZiCX0J9worhR/xl/wb5y22bT1wMDhnDPce26QcKa0bf9ahZXVtfWN4mZpa3tnd6+8f/CoRCoJdYngQrYDrChnMXU105y2E0lxFHDaCoa3E7/1TKViIm7qUUL9CPdjFjKCtZH85s2ld+ZFgXjJ7sfdcsWu2lOgZeLkpAI5Gt3yj9cTJI1orAnHSnUcO9F+hqVmhNNxyUsVTTAZ4j7NpquO0YmReigU0rxYo6k6l8ORUqMoMMkI64Fa9Cbif14n1WHNz1icpJrGZDYoTDnSAk16ox6TlGg+MgQTycyGiAywxESb65RMdWex6DJxz6vXVefholKv5TcowhEcwyk4cAV1uIMGuEDgCd7gE74sbr1a79bHLFqw8j+HMAfr+w9B5I0U</latexit>

T = 10K<latexit sha1_base64="DqCo4FZpD1QjVQIm0KVncI/gf0Q=">AAAB6nicbVDLSgMxFL3js9ZX1aWbYBFcSJkRwboQCm4ENxU6ttCpJZNm2tBkMiQZsQz9CjeKK8WP8Rf8G9N2Nm09EDicc8K954YJZ9q47q+zsrq2vrFZ2Cpu7+zu7ZcODh+1TBWhPpFcqlaINeUspr5hhtNWoigWIafNcHg78ZvPVGkm44YZJbQjcD9mESPYWOmpceO5wXkgQvmS3Y+7pbJbcadAy8TLSRly1Luln6AnSSpobAjHWrc9NzGdDCvDCKfjYpBqmmAyxH2aTXcdo1Mr9VAklX2xQVN1LoeF1iMR2qTAZqAXvYn4n9dOTVTtZCxOUkNjMhsUpRwZiSbFUY8pSgwfWYKJYnZDRAZYYWLseYq2urdYdJn4F5XrivdwWa5V8xsU4BhO4Aw8uIIa3EEdfCCg4A0+4csRzqvz7nzMoitO/ucI5uB8/wGqG41K</latexit><latexit sha1_base64="DqCo4FZpD1QjVQIm0KVncI/gf0Q=">AAAB6nicbVDLSgMxFL3js9ZX1aWbYBFcSJkRwboQCm4ENxU6ttCpJZNm2tBkMiQZsQz9CjeKK8WP8Rf8G9N2Nm09EDicc8K954YJZ9q47q+zsrq2vrFZ2Cpu7+zu7ZcODh+1TBWhPpFcqlaINeUspr5hhtNWoigWIafNcHg78ZvPVGkm44YZJbQjcD9mESPYWOmpceO5wXkgQvmS3Y+7pbJbcadAy8TLSRly1Luln6AnSSpobAjHWrc9NzGdDCvDCKfjYpBqmmAyxH2aTXcdo1Mr9VAklX2xQVN1LoeF1iMR2qTAZqAXvYn4n9dOTVTtZCxOUkNjMhsUpRwZiSbFUY8pSgwfWYKJYnZDRAZYYWLseYq2urdYdJn4F5XrivdwWa5V8xsU4BhO4Aw8uIIa3EEdfCCg4A0+4csRzqvz7nzMoitO/ucI5uB8/wGqG41K</latexit><latexit sha1_base64="DqCo4FZpD1QjVQIm0KVncI/gf0Q=">AAAB6nicbVDLSgMxFL3js9ZX1aWbYBFcSJkRwboQCm4ENxU6ttCpJZNm2tBkMiQZsQz9CjeKK8WP8Rf8G9N2Nm09EDicc8K954YJZ9q47q+zsrq2vrFZ2Cpu7+zu7ZcODh+1TBWhPpFcqlaINeUspr5hhtNWoigWIafNcHg78ZvPVGkm44YZJbQjcD9mESPYWOmpceO5wXkgQvmS3Y+7pbJbcadAy8TLSRly1Luln6AnSSpobAjHWrc9NzGdDCvDCKfjYpBqmmAyxH2aTXcdo1Mr9VAklX2xQVN1LoeF1iMR2qTAZqAXvYn4n9dOTVTtZCxOUkNjMhsUpRwZiSbFUY8pSgwfWYKJYnZDRAZYYWLseYq2urdYdJn4F5XrivdwWa5V8xsU4BhO4Aw8uIIa3EEdfCCg4A0+4csRzqvz7nzMoitO/ucI5uB8/wGqG41K</latexit><latexit sha1_base64="DqCo4FZpD1QjVQIm0KVncI/gf0Q=">AAAB6nicbVDLSgMxFL3js9ZX1aWbYBFcSJkRwboQCm4ENxU6ttCpJZNm2tBkMiQZsQz9CjeKK8WP8Rf8G9N2Nm09EDicc8K954YJZ9q47q+zsrq2vrFZ2Cpu7+zu7ZcODh+1TBWhPpFcqlaINeUspr5hhtNWoigWIafNcHg78ZvPVGkm44YZJbQjcD9mESPYWOmpceO5wXkgQvmS3Y+7pbJbcadAy8TLSRly1Luln6AnSSpobAjHWrc9NzGdDCvDCKfjYpBqmmAyxH2aTXcdo1Mr9VAklX2xQVN1LoeF1iMR2qTAZqAXvYn4n9dOTVTtZCxOUkNjMhsUpRwZiSbFUY8pSgwfWYKJYnZDRAZYYWLseYq2urdYdJn4F5XrivdwWa5V8xsU4BhO4Aw8uIIa3EEdfCCg4A0+4csRzqvz7nzMoitO/ucI5uB8/wGqG41K</latexit>

�3 ⌘ t0 = 0<latexit sha1_base64="oI2GJniA5NkbcumT+2F6Za9h3Kk=">AAAB8nicbVDLSsNAFJ3UV62v+Ni5GSyiq5KoYF0IBTcuKxhbaEqYTKft0Jkkztwp1NAvcaO4UvwQf8G/MW2zaeuBgcM5Z7j3njARXIPj/FqFldW19Y3iZmlre2d3z94/eNKxUZR5NBaxaoZEM8Ej5gEHwZqJYkSGgjXCwd3EbwyZ0jyOHmGUsLYkvYh3OSWQSYF95PeIlCS49Nmz4cN0DGe3TmCXnYozBV4mbk7KKEc9sH/8TkyNZBFQQbRuuU4C7ZQo4FSwcck3miWEDkiPpdOdx/g0kzq4G6vsRYCn6lyOSK1HMsySkkBfL3oT8T+vZaBbbac8SgywiM4GdY3AEONJAbjDFaMgRhkhVPFsQ0z7RBEKWU2l7HR38dBl4l1Ubiruw1W5Vs07KKJjdILOkYuuUQ3dozryEEUv6A19oi/LWK/Wu/Uxixas/M8hmoP1/QcXN5Bf</latexit><latexit sha1_base64="oI2GJniA5NkbcumT+2F6Za9h3Kk=">AAAB8nicbVDLSsNAFJ3UV62v+Ni5GSyiq5KoYF0IBTcuKxhbaEqYTKft0Jkkztwp1NAvcaO4UvwQf8G/MW2zaeuBgcM5Z7j3njARXIPj/FqFldW19Y3iZmlre2d3z94/eNKxUZR5NBaxaoZEM8Ej5gEHwZqJYkSGgjXCwd3EbwyZ0jyOHmGUsLYkvYh3OSWQSYF95PeIlCS49Nmz4cN0DGe3TmCXnYozBV4mbk7KKEc9sH/8TkyNZBFQQbRuuU4C7ZQo4FSwcck3miWEDkiPpdOdx/g0kzq4G6vsRYCn6lyOSK1HMsySkkBfL3oT8T+vZaBbbac8SgywiM4GdY3AEONJAbjDFaMgRhkhVPFsQ0z7RBEKWU2l7HR38dBl4l1Ubiruw1W5Vs07KKJjdILOkYuuUQ3dozryEEUv6A19oi/LWK/Wu/Uxixas/M8hmoP1/QcXN5Bf</latexit><latexit sha1_base64="oI2GJniA5NkbcumT+2F6Za9h3Kk=">AAAB8nicbVDLSsNAFJ3UV62v+Ni5GSyiq5KoYF0IBTcuKxhbaEqYTKft0Jkkztwp1NAvcaO4UvwQf8G/MW2zaeuBgcM5Z7j3njARXIPj/FqFldW19Y3iZmlre2d3z94/eNKxUZR5NBaxaoZEM8Ej5gEHwZqJYkSGgjXCwd3EbwyZ0jyOHmGUsLYkvYh3OSWQSYF95PeIlCS49Nmz4cN0DGe3TmCXnYozBV4mbk7KKEc9sH/8TkyNZBFQQbRuuU4C7ZQo4FSwcck3miWEDkiPpdOdx/g0kzq4G6vsRYCn6lyOSK1HMsySkkBfL3oT8T+vZaBbbac8SgywiM4GdY3AEONJAbjDFaMgRhkhVPFsQ0z7RBEKWU2l7HR38dBl4l1Ubiruw1W5Vs07KKJjdILOkYuuUQ3dozryEEUv6A19oi/LWK/Wu/Uxixas/M8hmoP1/QcXN5Bf</latexit><latexit sha1_base64="oI2GJniA5NkbcumT+2F6Za9h3Kk=">AAAB8nicbVDLSsNAFJ3UV62v+Ni5GSyiq5KoYF0IBTcuKxhbaEqYTKft0Jkkztwp1NAvcaO4UvwQf8G/MW2zaeuBgcM5Z7j3njARXIPj/FqFldW19Y3iZmlre2d3z94/eNKxUZR5NBaxaoZEM8Ej5gEHwZqJYkSGgjXCwd3EbwyZ0jyOHmGUsLYkvYh3OSWQSYF95PeIlCS49Nmz4cN0DGe3TmCXnYozBV4mbk7KKEc9sH/8TkyNZBFQQbRuuU4C7ZQo4FSwcck3miWEDkiPpdOdx/g0kzq4G6vsRYCn6lyOSK1HMsySkkBfL3oT8T+vZaBbbac8SgywiM4GdY3AEONJAbjDFaMgRhkhVPFsQ0z7RBEKWU2l7HR38dBl4l1Ubiruw1W5Vs07KKJjdILOkYuuUQ3dozryEEUv6A19oi/LWK/Wu/Uxixas/M8hmoP1/QcXN5Bf</latexit>

[ Rut, Suszalski, AR, in preparation ]





Thermoelectric figure of merit

ZT =σS2Tκel + κph

,

where σ is the electrical conductivity, S-the Seebeck coefficient, κel(κph) - electronic (phononic) part of the thermal conductivity.

In the linear-responce regime, one can roughly approximate themaximal energy conversion efficiency

η =Pel

IQ≈ 1

4ZT

∆TT

(=

ZT4η?

)

(for ∆T � T ), with Pel – the maximal (usefull) electric power and IQ –the thermal current [see: Kim et al., PNAS 112, 8205 (2015)].

For instance, ZT =3, ∆T =100 K and T =300 K gives η ≈ 0.25,comparable with η? = 0.33 for a perfect Carnot engine.





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0 2 4 6 8 10

ZT

µ [meV]

ZTmax

ZTmax

0.04

0.08

0.12

0.

ZT

0.

0.4

0.8

1.2 (a)

(b)

(c)

(d)

LA+TA

LA+TA

LA+TA+ZA

U=0

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LA+TA+ZA

0.0

1.0

2.0

3.0

0 50 100 150 2000.0

0.2

0.4

0.6

� [meV]<latexit sha1_base64="Hqz/fQBkPLtKTONDGI38mEYWqJk=">AAAB83icbVDLSsNAFJ34rPUVHzs3g0VwVRIRdFnUhcsK9gFNKJPpTTt08mDmRqyhX+JGxI2C/+Ev+DdO22zaemDgcM4Z7r0nSKXQ6Di/1srq2vrGZmmrvL2zu7dvHxw2dZIpDg2eyES1A6ZBihgaKFBCO1XAokBCKxjeTvzWEygtkvgRRyn4EevHIhScoZG69rF3BxKZRzsewjPmETTHfteuOFVnCrpM3IJUSIF61/7xegnPIoiRS6Z1x3VS9HOmUHAJ47KXaUgZH7I+5NOdx/TMSD0aJsq8GOlUncuxSOtRFJhkxHCgF72J+J/XyTC89nMRpxlCzGeDwkxSTOikANoTCjjKkSGMK2E2pHzAFONoaiqb093FQ5dJ86LqOlX34bJSuylKKJETckrOiUuuSI3ckzppEE5eyBv5JF9WZr1a79bHLLpiFX+OyBys7z8QOJDZ</latexit><latexit sha1_base64="Hqz/fQBkPLtKTONDGI38mEYWqJk=">AAAB83icbVDLSsNAFJ34rPUVHzs3g0VwVRIRdFnUhcsK9gFNKJPpTTt08mDmRqyhX+JGxI2C/+Ev+DdO22zaemDgcM4Z7r0nSKXQ6Di/1srq2vrGZmmrvL2zu7dvHxw2dZIpDg2eyES1A6ZBihgaKFBCO1XAokBCKxjeTvzWEygtkvgRRyn4EevHIhScoZG69rF3BxKZRzsewjPmETTHfteuOFVnCrpM3IJUSIF61/7xegnPIoiRS6Z1x3VS9HOmUHAJ47KXaUgZH7I+5NOdx/TMSD0aJsq8GOlUncuxSOtRFJhkxHCgF72J+J/XyTC89nMRpxlCzGeDwkxSTOikANoTCjjKkSGMK2E2pHzAFONoaiqb093FQ5dJ86LqOlX34bJSuylKKJETckrOiUuuSI3ckzppEE5eyBv5JF9WZr1a79bHLLpiFX+OyBys7z8QOJDZ</latexit><latexit sha1_base64="Hqz/fQBkPLtKTONDGI38mEYWqJk=">AAAB83icbVDLSsNAFJ34rPUVHzs3g0VwVRIRdFnUhcsK9gFNKJPpTTt08mDmRqyhX+JGxI2C/+Ev+DdO22zaemDgcM4Z7r0nSKXQ6Di/1srq2vrGZmmrvL2zu7dvHxw2dZIpDg2eyES1A6ZBihgaKFBCO1XAokBCKxjeTvzWEygtkvgRRyn4EevHIhScoZG69rF3BxKZRzsewjPmETTHfteuOFVnCrpM3IJUSIF61/7xegnPIoiRS6Z1x3VS9HOmUHAJ47KXaUgZH7I+5NOdx/TMSD0aJsq8GOlUncuxSOtRFJhkxHCgF72J+J/XyTC89nMRpxlCzGeDwkxSTOikANoTCjjKkSGMK2E2pHzAFONoaiqb093FQ5dJ86LqOlX34bJSuylKKJETckrOiUuuSI3ckzppEE5eyBv5JF9WZr1a79bHLLpiFX+OyBys7z8QOJDZ</latexit><latexit sha1_base64="Hqz/fQBkPLtKTONDGI38mEYWqJk=">AAAB83icbVDLSsNAFJ34rPUVHzs3g0VwVRIRdFnUhcsK9gFNKJPpTTt08mDmRqyhX+JGxI2C/+Ev+DdO22zaemDgcM4Z7r0nSKXQ6Di/1srq2vrGZmmrvL2zu7dvHxw2dZIpDg2eyES1A6ZBihgaKFBCO1XAokBCKxjeTvzWEygtkvgRRyn4EevHIhScoZG69rF3BxKZRzsewjPmETTHfteuOFVnCrpM3IJUSIF61/7xegnPIoiRS6Z1x3VS9HOmUHAJ47KXaUgZH7I+5NOdx/TMSD0aJsq8GOlUncuxSOtRFJhkxHCgF72J+J/XyTC89nMRpxlCzGeDwkxSTOikANoTCjjKkSGMK2E2pHzAFONoaiqb093FQ5dJ86LqOlX34bJSuylKKJETckrOiUuuSI3ckzppEE5eyBv5JF9WZr1a79bHLLpiFX+OyBys7z8QOJDZ</latexit>

T=5 K

T=1 K T=1 K

T=5 K

ZT = S2σT/κ , with κ = κel + κph. We have µZTmax ≈ ∆/2; in contrast:

µSmax ≈ 1

2 kBT ln(2∆/kBT )� ∆/2. In effect, ZTmax is sensitive to T (E)

details near the band boundary and does not grow monotonicallywith ∆ (like |S|max). Instead, ZT reaches a global maximum for∆max∼102kBT . [ Rut, Suszalski, AR, in preparation ]





Acknowlegments

The group:

Grzegorz Rut Dominik Suszalski

Former collaboration: Anton Akhmerov,Jens Bardarson, Carlo Beenakker, PatrikRecher, Klaus Richter, Björn Trauzettel,Jakub Tworzydło, Michael Wimmer, . . .

Funding:

Project web site: http://th.if.uj.edu.pl/∼adamr/sonata.html

See also: Suszalski et al., POSTER NO. 15Rut et al., POSTER NO. 13

THANK YOU!


low-temperature thermoelectric properties of bilayer grapheneth.if.uj.edu.pl › ~adamr ›...

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