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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 . . .
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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.
Rycerz, Suszalski, Rut, Low-temperature thermoelectric properties . . .
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IntroductionLandauer-Büttiker formalism
Thermoelectric properties
Emergent Dirac fermions in grapheneTunable band gap and trigonal warping in BLGQuantum size effects
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 (!)
Rycerz, Suszalski, Rut, Low-temperature thermoelectric properties . . .
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IntroductionLandauer-Büttiker formalism
Thermoelectric properties
Emergent Dirac fermions in grapheneTunable band gap and trigonal warping in BLGQuantum size effects
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 ]
Rycerz, Suszalski, Rut, Low-temperature thermoelectric properties . . .
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IntroductionLandauer-Büttiker formalism
Thermoelectric properties
Emergent Dirac fermions in grapheneTunable band gap and trigonal warping in BLGQuantum size effects
[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!
Rycerz, Suszalski, Rut, Low-temperature thermoelectric properties . . .
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IntroductionLandauer-Büttiker formalism
Thermoelectric properties
Emergent Dirac fermions in grapheneTunable band gap and trigonal warping in BLGQuantum size effects
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.
Rycerz, Suszalski, Rut, Low-temperature thermoelectric properties . . .
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IntroductionLandauer-Büttiker formalism
Thermoelectric properties
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.
Rycerz, Suszalski, Rut, Low-temperature thermoelectric properties . . .
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IntroductionLandauer-Büttiker formalism
Thermoelectric properties
Brief overviewUniversal conductivity of MLGEmergent conductivity of BLG
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).
Rycerz, Suszalski, Rut, Low-temperature thermoelectric properties . . .
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IntroductionLandauer-Büttiker formalism
Thermoelectric properties
Brief overviewUniversal conductivity of MLGEmergent conductivity of BLG
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. ]
Rycerz, Suszalski, Rut, Low-temperature thermoelectric properties . . .
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IntroductionLandauer-Büttiker formalism
Thermoelectric properties
Brief overviewUniversal conductivity of MLGEmergent conductivity of BLG
EXP: Mayorov et al, 2011(�BLG ⇡ 5�0 )
<latexit sha1_base64="HyCKdNKPWLxmGgR/I3vh85vKi1g=">AAACBnicdVDLSgMxAMzWV62vqkcvwSJUKGV3Xfu4FT3owUMFawvdsmTTtA3NPkiyYln6AV78FS+KJ8Wjv+DfmG3XQ0UHAsPMhGTGDRkVUte/tMzS8srqWnY9t7G5tb2T3927FUHEMWnhgAW84yJBGPVJS1LJSCfkBHkuI213fJ747TvCBQ38GzkJSc9DQ58OKEZSSU6+WLRLtqBDDzmxzT14dnUxtVEY8uA+np6mjm6Xjp18QS/XaxXTqkC9rOtVwzQSYlatEwsaSklQACmaTv7T7gc48ogvMUNCdA09lL0YcUkxI9OcHQkSIjxGQxLPekzhkZL6cBBwdXwJZ+pCDnlCTDxXJT0kR+K3l4h/ed1IDmq9mPphJImP5w8NIgZlAJNRYJ9ygiWbKIIwp+qHEI8QR1iq6XKq+k8/+D9pmeV62bi2Cg0z3SALDsAhKAIDVEEDXIImaAEMHsEzeAPv2oP2pL1or/NoRkvv7IMFaB/fd7CYYA==</latexit><latexit sha1_base64="HyCKdNKPWLxmGgR/I3vh85vKi1g=">AAACBnicdVDLSgMxAMzWV62vqkcvwSJUKGV3Xfu4FT3owUMFawvdsmTTtA3NPkiyYln6AV78FS+KJ8Wjv+DfmG3XQ0UHAsPMhGTGDRkVUte/tMzS8srqWnY9t7G5tb2T3927FUHEMWnhgAW84yJBGPVJS1LJSCfkBHkuI213fJ747TvCBQ38GzkJSc9DQ58OKEZSSU6+WLRLtqBDDzmxzT14dnUxtVEY8uA+np6mjm6Xjp18QS/XaxXTqkC9rOtVwzQSYlatEwsaSklQACmaTv7T7gc48ogvMUNCdA09lL0YcUkxI9OcHQkSIjxGQxLPekzhkZL6cBBwdXwJZ+pCDnlCTDxXJT0kR+K3l4h/ed1IDmq9mPphJImP5w8NIgZlAJNRYJ9ygiWbKIIwp+qHEI8QR1iq6XKq+k8/+D9pmeV62bi2Cg0z3SALDsAhKAIDVEEDXIImaAEMHsEzeAPv2oP2pL1or/NoRkvv7IMFaB/fd7CYYA==</latexit><latexit sha1_base64="HyCKdNKPWLxmGgR/I3vh85vKi1g=">AAACBnicdVDLSgMxAMzWV62vqkcvwSJUKGV3Xfu4FT3owUMFawvdsmTTtA3NPkiyYln6AV78FS+KJ8Wjv+DfmG3XQ0UHAsPMhGTGDRkVUte/tMzS8srqWnY9t7G5tb2T3927FUHEMWnhgAW84yJBGPVJS1LJSCfkBHkuI213fJ747TvCBQ38GzkJSc9DQ58OKEZSSU6+WLRLtqBDDzmxzT14dnUxtVEY8uA+np6mjm6Xjp18QS/XaxXTqkC9rOtVwzQSYlatEwsaSklQACmaTv7T7gc48ogvMUNCdA09lL0YcUkxI9OcHQkSIjxGQxLPekzhkZL6cBBwdXwJZ+pCDnlCTDxXJT0kR+K3l4h/ed1IDmq9mPphJImP5w8NIgZlAJNRYJ9ygiWbKIIwp+qHEI8QR1iq6XKq+k8/+D9pmeV62bi2Cg0z3SALDsAhKAIDVEEDXIImaAEMHsEzeAPv2oP2pL1or/NoRkvv7IMFaB/fd7CYYA==</latexit><latexit sha1_base64="HyCKdNKPWLxmGgR/I3vh85vKi1g=">AAACBnicdVDLSgMxAMzWV62vqkcvwSJUKGV3Xfu4FT3owUMFawvdsmTTtA3NPkiyYln6AV78FS+KJ8Wjv+DfmG3XQ0UHAsPMhGTGDRkVUte/tMzS8srqWnY9t7G5tb2T3927FUHEMWnhgAW84yJBGPVJS1LJSCfkBHkuI213fJ747TvCBQ38GzkJSc9DQ58OKEZSSU6+WLRLtqBDDzmxzT14dnUxtVEY8uA+np6mjm6Xjp18QS/XaxXTqkC9rOtVwzQSYlatEwsaSklQACmaTv7T7gc48ogvMUNCdA09lL0YcUkxI9OcHQkSIjxGQxLPekzhkZL6cBBwdXwJZ+pCDnlCTDxXJT0kR+K3l4h/ed1IDmq9mPphJImP5w8NIgZlAJNRYJ9ygiWbKIIwp+qHEI8QR1iq6XKq+k8/+D9pmeV62bi2Cg0z3SALDsAhKAIDVEEDXIImaAEMHsEzeAPv2oP2pL1or/NoRkvv7IMFaB/fd7CYYA==</latexit>
[ 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)
Rycerz, Suszalski, Rut, Low-temperature thermoelectric properties . . .
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IntroductionLandauer-Büttiker formalism
Thermoelectric properties
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...
!
Rycerz, Suszalski, Rut, Low-temperature thermoelectric properties . . .
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IntroductionLandauer-Büttiker formalism
Thermoelectric properties
Quasiparticles in doped BLGSeebeck coefficient and Lorentz numberGapless vs gapped BLG
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)] .
Rycerz, Suszalski, Rut, Low-temperature thermoelectric properties . . .
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IntroductionLandauer-Büttiker formalism
Thermoelectric properties
Quasiparticles in doped BLGSeebeck coefficient and Lorentz numberGapless vs gapped BLG
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
6�0<latexit sha1_base64="n/HZEG646XDAVmRDmvhtdR3BMvY=">AAAB5XicdVDLSgMxFL1TX7W+qi7dBIvgasiMQx+7ghuXFRxbaUvJpJk2NPMgyQil9BfcKK4Uf8df8G/MtHVR0QOBwzkn3HtukAquNMZfVmFjc2t7p7hb2ts/ODwqH5/cqySTlPk0EYnsBEQxwWPma64F66SSkSgQrB1MrnO//cik4kl8p6cp60dkFPOQU6KN9FDtKT6KyAAPyhVsN+pV16sibGNcc1wnJ27Nu/KQY5QcFVihNSh/9oYJzSIWayqIUl0Hp7o/I1JzKti81MsUSwmdkBGbLdacowsjDVGYSPNijRbqWo5ESk2jwCQjosfqt5eLf3ndTIf1/ozHaaZZTJeDwkwgnaC8MxpyyagWU0MIldxsiOiYSEK1uUzJVP/ph/4nvms3bOfWqzTrqxsU4QzO4RIcqEETbqAFPlCI4Bne4N0aW0/Wi/W6jBas1Z9TWIP18Q3gMIu9</latexit><latexit sha1_base64="n/HZEG646XDAVmRDmvhtdR3BMvY=">AAAB5XicdVDLSgMxFL1TX7W+qi7dBIvgasiMQx+7ghuXFRxbaUvJpJk2NPMgyQil9BfcKK4Uf8df8G/MtHVR0QOBwzkn3HtukAquNMZfVmFjc2t7p7hb2ts/ODwqH5/cqySTlPk0EYnsBEQxwWPma64F66SSkSgQrB1MrnO//cik4kl8p6cp60dkFPOQU6KN9FDtKT6KyAAPyhVsN+pV16sibGNcc1wnJ27Nu/KQY5QcFVihNSh/9oYJzSIWayqIUl0Hp7o/I1JzKti81MsUSwmdkBGbLdacowsjDVGYSPNijRbqWo5ESk2jwCQjosfqt5eLf3ndTIf1/ozHaaZZTJeDwkwgnaC8MxpyyagWU0MIldxsiOiYSEK1uUzJVP/ph/4nvms3bOfWqzTrqxsU4QzO4RIcqEETbqAFPlCI4Bne4N0aW0/Wi/W6jBas1Z9TWIP18Q3gMIu9</latexit><latexit sha1_base64="n/HZEG646XDAVmRDmvhtdR3BMvY=">AAAB5XicdVDLSgMxFL1TX7W+qi7dBIvgasiMQx+7ghuXFRxbaUvJpJk2NPMgyQil9BfcKK4Uf8df8G/MtHVR0QOBwzkn3HtukAquNMZfVmFjc2t7p7hb2ts/ODwqH5/cqySTlPk0EYnsBEQxwWPma64F66SSkSgQrB1MrnO//cik4kl8p6cp60dkFPOQU6KN9FDtKT6KyAAPyhVsN+pV16sibGNcc1wnJ27Nu/KQY5QcFVihNSh/9oYJzSIWayqIUl0Hp7o/I1JzKti81MsUSwmdkBGbLdacowsjDVGYSPNijRbqWo5ESk2jwCQjosfqt5eLf3ndTIf1/ozHaaZZTJeDwkwgnaC8MxpyyagWU0MIldxsiOiYSEK1uUzJVP/ph/4nvms3bOfWqzTrqxsU4QzO4RIcqEETbqAFPlCI4Bne4N0aW0/Wi/W6jBas1Z9TWIP18Q3gMIu9</latexit><latexit sha1_base64="n/HZEG646XDAVmRDmvhtdR3BMvY=">AAAB5XicdVDLSgMxFL1TX7W+qi7dBIvgasiMQx+7ghuXFRxbaUvJpJk2NPMgyQil9BfcKK4Uf8df8G/MtHVR0QOBwzkn3HtukAquNMZfVmFjc2t7p7hb2ts/ODwqH5/cqySTlPk0EYnsBEQxwWPma64F66SSkSgQrB1MrnO//cik4kl8p6cp60dkFPOQU6KN9FDtKT6KyAAPyhVsN+pV16sibGNcc1wnJ27Nu/KQY5QcFVihNSh/9oYJzSIWayqIUl0Hp7o/I1JzKti81MsUSwmdkBGbLdacowsjDVGYSPNijRbqWo5ESk2jwCQjosfqt5eLf3ndTIf1/ozHaaZZTJeDwkwgnaC8MxpyyagWU0MIldxsiOiYSEK1uUzJVP/ph/4nvms3bOfWqzTrqxsU4QzO4RIcqEETbqAFPlCI4Bne4N0aW0/Wi/W6jBas1Z9TWIP18Q3gMIu9</latexit>
2�0<latexit sha1_base64="0AMYOxZyrfYdJ6GrGWC1XFlvqOA=">AAAB5XicdVDLSsNAFL3xWeur6tLNYBFchSTGPnYFNy4rGFtpS5lMJ+3QmSTMTIRS+gtuFFeKv+Mv+DdO2rqo6IGBwzlnuPfcMOVMacf5stbWNza3tgs7xd29/YPD0tHxvUoySWhAEp7IdogV5SymgWaa03YqKRYhp61wfJ37rUcqFUviOz1JaU/gYcwiRrA20oPXVWwocN/pl8qOXa9VPL+CHNtxqq7n5sSr+pc+co2SowxLNPulz+4gIZmgsSYcK9VxnVT3plhqRjidFbuZoikmYzyk0/maM3RupAGKEmlerNFcXclhodREhCYpsB6p314u/uV1Mh3VelMWp5mmMVkMijKOdILyzmjAJCWaTwzBRDKzISIjLDHR5jJFU/2nH/qfBJ5dt91bv9y4Wt6gAKdwBhfgQhUacANNCICAgGd4g3drZD1ZL9brIrpmLf+cwAqsj2/ZRYu2</latexit><latexit sha1_base64="0AMYOxZyrfYdJ6GrGWC1XFlvqOA=">AAAB5XicdVDLSsNAFL3xWeur6tLNYBFchSTGPnYFNy4rGFtpS5lMJ+3QmSTMTIRS+gtuFFeKv+Mv+DdO2rqo6IGBwzlnuPfcMOVMacf5stbWNza3tgs7xd29/YPD0tHxvUoySWhAEp7IdogV5SymgWaa03YqKRYhp61wfJ37rUcqFUviOz1JaU/gYcwiRrA20oPXVWwocN/pl8qOXa9VPL+CHNtxqq7n5sSr+pc+co2SowxLNPulz+4gIZmgsSYcK9VxnVT3plhqRjidFbuZoikmYzyk0/maM3RupAGKEmlerNFcXclhodREhCYpsB6p314u/uV1Mh3VelMWp5mmMVkMijKOdILyzmjAJCWaTwzBRDKzISIjLDHR5jJFU/2nH/qfBJ5dt91bv9y4Wt6gAKdwBhfgQhUacANNCICAgGd4g3drZD1ZL9brIrpmLf+cwAqsj2/ZRYu2</latexit><latexit sha1_base64="0AMYOxZyrfYdJ6GrGWC1XFlvqOA=">AAAB5XicdVDLSsNAFL3xWeur6tLNYBFchSTGPnYFNy4rGFtpS5lMJ+3QmSTMTIRS+gtuFFeKv+Mv+DdO2rqo6IGBwzlnuPfcMOVMacf5stbWNza3tgs7xd29/YPD0tHxvUoySWhAEp7IdogV5SymgWaa03YqKRYhp61wfJ37rUcqFUviOz1JaU/gYcwiRrA20oPXVWwocN/pl8qOXa9VPL+CHNtxqq7n5sSr+pc+co2SowxLNPulz+4gIZmgsSYcK9VxnVT3plhqRjidFbuZoikmYzyk0/maM3RupAGKEmlerNFcXclhodREhCYpsB6p314u/uV1Mh3VelMWp5mmMVkMijKOdILyzmjAJCWaTwzBRDKzISIjLDHR5jJFU/2nH/qfBJ5dt91bv9y4Wt6gAKdwBhfgQhUacANNCICAgGd4g3drZD1ZL9brIrpmLf+cwAqsj2/ZRYu2</latexit><latexit sha1_base64="0AMYOxZyrfYdJ6GrGWC1XFlvqOA=">AAAB5XicdVDLSsNAFL3xWeur6tLNYBFchSTGPnYFNy4rGFtpS5lMJ+3QmSTMTIRS+gtuFFeKv+Mv+DdO2rqo6IGBwzlnuPfcMOVMacf5stbWNza3tgs7xd29/YPD0tHxvUoySWhAEp7IdogV5SymgWaa03YqKRYhp61wfJ37rUcqFUviOz1JaU/gYcwiRrA20oPXVWwocN/pl8qOXa9VPL+CHNtxqq7n5sSr+pc+co2SowxLNPulz+4gIZmgsSYcK9VxnVT3plhqRjidFbuZoikmYzyk0/maM3RupAGKEmlerNFcXclhodREhCYpsB6p314u/uV1Mh3VelMWp5mmMVkMijKOdILyzmjAJCWaTwzBRDKzISIjLDHR5jJFU/2nH/qfBJ5dt91bv9y4Wt6gAKdwBhfgQhUacANNCICAgGd4g3drZD1ZL9brIrpmLf+cwAqsj2/ZRYu2</latexit>
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) ]
Rycerz, Suszalski, Rut, Low-temperature thermoelectric properties . . .
Page 14
IntroductionLandauer-Büttiker formalism
Thermoelectric properties
Quasiparticles in doped BLGSeebeck coefficient and Lorentz numberGapless vs gapped BLG
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
Rycerz, Suszalski, Rut, Low-temperature thermoelectric properties . . .
Page 15
IntroductionLandauer-Büttiker formalism
Thermoelectric properties
Quasiparticles in doped BLGSeebeck coefficient and Lorentz numberGapless vs gapped BLG
⇒ 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) ]
Rycerz, Suszalski, Rut, Low-temperature thermoelectric properties . . .
Page 16
IntroductionLandauer-Büttiker formalism
Thermoelectric properties
Quasiparticles in doped BLGSeebeck coefficient and Lorentz numberGapless vs gapped BLG
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 )
Rycerz, Suszalski, Rut, Low-temperature thermoelectric properties . . .
Page 17
IntroductionLandauer-Büttiker formalism
Thermoelectric properties
Quasiparticles in doped BLGSeebeck coefficient and Lorentz numberGapless vs gapped BLG
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]
Rycerz, Suszalski, Rut, Low-temperature thermoelectric properties . . .
Page 18
IntroductionLandauer-Büttiker formalism
Thermoelectric properties
Quasiparticles in doped BLGSeebeck coefficient and Lorentz numberGapless vs gapped BLG
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 (?)
. . . ?
Rycerz, Suszalski, Rut, Low-temperature thermoelectric properties . . .
Page 19
IntroductionLandauer-Büttiker formalism
Thermoelectric properties
Quasiparticles in doped BLGSeebeck coefficient and Lorentz numberGapless vs gapped BLG
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
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[ Rut, Suszalski, AR, in preparation ]
Rycerz, Suszalski, Rut, Low-temperature thermoelectric properties . . .
Page 20
IntroductionLandauer-Büttiker formalism
Thermoelectric properties
Quasiparticles in doped BLGSeebeck coefficient and Lorentz numberGapless vs gapped BLG
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.
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Dotted line:The reference(Goldsmid-Sharp)value:
|S|GSmax = �
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|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>
|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>
0
10
20
0 10 20
0
1
2
3
0
1
2
3
0 10 20�/(2kBT ) �/(2kBT )
Sm
ax
[kB
/e]
(a) (b)
(c)
�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 ]
Rycerz, Suszalski, Rut, Low-temperature thermoelectric properties . . .
Page 21
IntroductionLandauer-Büttiker formalism
Thermoelectric properties
Quasiparticles in doped BLGSeebeck coefficient and Lorentz numberGapless vs gapped BLG
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.
Rycerz, Suszalski, Rut, Low-temperature thermoelectric properties . . .
Page 22
IntroductionLandauer-Büttiker formalism
Thermoelectric properties
Quasiparticles in doped BLGSeebeck coefficient and Lorentz numberGapless vs gapped BLG
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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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U =0<latexit sha1_base64="k5jzTIH21PC+M0mjeazBWOFSp4U=">AAAB5HicbVBNS0JBFL3Pvsy+rJZtxiRoJfMkqE0gtGlp0FPBJzJvnKeD8z6YuS8Q8R+0iWhT0O/pL/RvGvVt1A4MHM45w73nBqmSBin9dQpb2zu7e8X90sHh0fFJ+fSsZZJMc+HxRCW6EzAjlIyFhxKV6KRasChQoh2MH+Z++0VoI5P4GSep6EVsGMtQcoZWant+5d6v0H65Smt0AbJJ3JxUIUezX/7xBwnPIhEjV8yYrktT7E2ZRsmVmJX8zIiU8TEbiuliyRm5stKAhIm2L0ayUFdyLDJmEgU2GTEcmXVvLv7ndTMM73pTGacZipgvB4WZIpiQeWMykFpwVBNLGNfSbkj4iGnG0d6lZKu760U3Satec2nNfbqpNur5EYpwAZdwDS7cQgMeoQkecBjDG3zClxM6r86787GMFpz8zzmswPn+A5ghihc=</latexit><latexit sha1_base64="k5jzTIH21PC+M0mjeazBWOFSp4U=">AAAB5HicbVBNS0JBFL3Pvsy+rJZtxiRoJfMkqE0gtGlp0FPBJzJvnKeD8z6YuS8Q8R+0iWhT0O/pL/RvGvVt1A4MHM45w73nBqmSBin9dQpb2zu7e8X90sHh0fFJ+fSsZZJMc+HxRCW6EzAjlIyFhxKV6KRasChQoh2MH+Z++0VoI5P4GSep6EVsGMtQcoZWant+5d6v0H65Smt0AbJJ3JxUIUezX/7xBwnPIhEjV8yYrktT7E2ZRsmVmJX8zIiU8TEbiuliyRm5stKAhIm2L0ayUFdyLDJmEgU2GTEcmXVvLv7ndTMM73pTGacZipgvB4WZIpiQeWMykFpwVBNLGNfSbkj4iGnG0d6lZKu760U3Satec2nNfbqpNur5EYpwAZdwDS7cQgMeoQkecBjDG3zClxM6r86787GMFpz8zzmswPn+A5ghihc=</latexit><latexit sha1_base64="k5jzTIH21PC+M0mjeazBWOFSp4U=">AAAB5HicbVBNS0JBFL3Pvsy+rJZtxiRoJfMkqE0gtGlp0FPBJzJvnKeD8z6YuS8Q8R+0iWhT0O/pL/RvGvVt1A4MHM45w73nBqmSBin9dQpb2zu7e8X90sHh0fFJ+fSsZZJMc+HxRCW6EzAjlIyFhxKV6KRasChQoh2MH+Z++0VoI5P4GSep6EVsGMtQcoZWant+5d6v0H65Smt0AbJJ3JxUIUezX/7xBwnPIhEjV8yYrktT7E2ZRsmVmJX8zIiU8TEbiuliyRm5stKAhIm2L0ayUFdyLDJmEgU2GTEcmXVvLv7ndTMM73pTGacZipgvB4WZIpiQeWMykFpwVBNLGNfSbkj4iGnG0d6lZKu760U3Satec2nNfbqpNur5EYpwAZdwDS7cQgMeoQkecBjDG3zClxM6r86787GMFpz8zzmswPn+A5ghihc=</latexit><latexit sha1_base64="k5jzTIH21PC+M0mjeazBWOFSp4U=">AAAB5HicbVBNS0JBFL3Pvsy+rJZtxiRoJfMkqE0gtGlp0FPBJzJvnKeD8z6YuS8Q8R+0iWhT0O/pL/RvGvVt1A4MHM45w73nBqmSBin9dQpb2zu7e8X90sHh0fFJ+fSsZZJMc+HxRCW6EzAjlIyFhxKV6KRasChQoh2MH+Z++0VoI5P4GSep6EVsGMtQcoZWant+5d6v0H65Smt0AbJJ3JxUIUezX/7xBwnPIhEjV8yYrktT7E2ZRsmVmJX8zIiU8TEbiuliyRm5stKAhIm2L0ayUFdyLDJmEgU2GTEcmXVvLv7ndTMM73pTGacZipgvB4WZIpiQeWMykFpwVBNLGNfSbkj4iGnG0d6lZKu760U3Satec2nNfbqpNur5EYpwAZdwDS7cQgMeoQkecBjDG3zClxM6r86787GMFpz8zzmswPn+A5ghihc=</latexit>
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 ]
Rycerz, Suszalski, Rut, Low-temperature thermoelectric properties . . .
Page 23
IntroductionLandauer-Büttiker formalism
Thermoelectric properties
Quasiparticles in doped BLGSeebeck coefficient and Lorentz numberGapless vs gapped BLG
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!
Rycerz, Suszalski, Rut, Low-temperature thermoelectric properties . . .