scattering theory of thermoelectric transport - scattering... · 2018. 11. 9. · scattering theory...
TRANSCRIPT
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Scattering theory of
thermoelectric transport
Markus Büttiker University of Geneva
Summer School "Energy harvesting at micro and nanoscales“,
Workshop "Energy harvesting: models and applications“,
Erice, Italy, July 23-27, 2012
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Mesoscopic Physics Wave nature of electrons becomes important
Webb et al., 1985
Yacoby et al. 1995
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Graphene: single and bilayer
@Jian Li unige
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Length scales
Phase coherence length
Elastic scattering length
Inelastic scattering length
Geometrical dimension
Macroscopic conductor
Mesoscopic conductor
(size of conductor)
(distance an electron travels before suffering a phase change of
(mean free path between elastic scattering events)
(distance an electron travels before loosing an energy kT)
)
Beenakker and van Houten, 1991
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Physics versus geometry
Mesoscopic physics = « Between mircoscopic and macroscopic »
Nano physics = on the geometrical length of a nanometer
Definition of mesoscopic physics is based on physical length scales.
In contrast, nanophysiscs, is a definition based on a geometrical length
scale.
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Lecture contents
Conductance from transmission 1. Single channel conductors
2. Multichannel conductors
3. Multiprobe conductors (omitted)
Thermoelectric transport
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1. Two-terminal coductors
3. Multiprobe conductors (omitted)
4. Magnetic field symmetry (omitted)
2. Thermoelectrics of a quantum dot
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Conductance from Transmission
1. Single channel conductors
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Conductance from scattering theory
Fermi energy right contact
applied voltage
Heuristic discussion
transmission probability
reflection probability
Fermi energy left contact
incident current
density
density of states
independent of material !!
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« Landauer formula »
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Scattering matrix
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scattering state
scattering matrix
current conservation S is a unitray matrix
In the absence of a magnetic field S is an orthogonal matrix
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Aharonov-Bohm oscillations 10
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Conductance from Transmission
2. Two-probe multi-channel conductors
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Multi-channel conductance: leads
asymptotic perfect translation invariant potential
seprable wave function
energy of transverse motion
energy for transverse and longitudnial motion
scattering channel
channel threshold
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Mulit-channel conductance
incident current in channel i
density in channel i
density of states in channel i
independent of channel
« Landauer formula »
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Eigen channels
hermitian matrix; real eigenvalues
hermitian matrix; real eigenvalues
are the genetic code of
mesoscopic conductors !!
Many single channel conductors in parallel.
All the properties we discussed for single-channel two-probe conductors apply
equally to many-channel multi-probe conductors: in particular
Eigen channels 14
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Quantum point contact
gate
gate
van Wees et al., PRL 60, 848 (1988)
Wharam et al, J. Phys. C 21, L209 (1988)
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Quntum point contact
Saddle-point potential
Transmission probability
Buttiker, Phys. Rev. B41, 7906 (1990)
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Conductance of resonant level
Transmission probability oof single level
High temperature limit
Level width
Energy of resonant level
Conductance
Low temperature limit
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Thermoelectric Transport
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1. Two terminal conductors
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Energy current
H. L. Engquist and P. W. Anderson, Phys. Rev. B24, 1151 (1981)
Energy flux in a quantum channel: reservoirs at T1 and T2:
Small temperature difference
Thermal quantum (independent of electron or channel properties!!)
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Lorentz factor (Sommerfeld theory)
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Heat current Heat current in perfect quantum channel, (linear response )
Heat current (elastic backscattering , linear response)
Thermoelectric transport (linear response)
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Connection with energy and electrical current
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Thermoelectric transport Fluxes in response to potentials
Current and temperature differences as driving forces
R resistance
S thermopower
Peltier
thermal conductance
Multi-terminal expressions:
P. N. Butcher , J. Phys.: Condensed Matter 2, 4869 (1990).
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Thermopower
zero temperature limit
Cutler-Mott formula
Sommerfeld integral
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Thermopower of a QPC Proetto, PRB 44, 9096 (1991)
Channel dependence
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Thermoelectric transport 2. Thermoelectric transport of a quantum dot
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Thermopower for resonant transmission P. Mani, N. Nakpathomkun, H. Linke, Journal of Electronic Materials 38, 1163
(2009).
Resonant transmission probability
High temperature limit
Level width
Universal ! But only as long as thermal energy is small compared to
the level separation.
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Thermopower for resonant transmission
Cutler-Mott
Low temperature limit of CM-formula
Resonant transmission probability
Note that this is independent of symmetry
P. Mani, N. Nakpathomkun, H. Linke,
Journal of Electronic Materials 38, 1163 (2009).
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Thermopower of a multilevel dot C. W. J. Beenakker and A. A. M. Staring Phys. Rev. B 46, 9667 (1992)
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Thermopower of a chaotic cavity S. F. Godijn, S. Möller, H. Buhmann, L. W. Molenkamp,
S. A. van Langen PRL 82, 2927–2930 (1999)
Cutler-Mott-formula
zero temperature limit
Probability distribution of the
thermopower of a chaotic cavity
one channel leads
S. A. van Langen, P. G. Silvestrov,
C. W. J. Beenakker, Supperlattice and
Microstructures, 23, 691 (1999).
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Efficiency of a single level dot
Efficiency
Power
Heat current
Efficency
Current
High temperature limit
Carnot efficiency is reached
when
Stall voltage
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Efficiency at maximum power Nakpathomkun, Xu, Linke, PRB 82, 235428 (2012)
------- efficiency at maximum power
------- maximum efficiency
------- maximum power
Maximization is with
regards to the position of
the resonant level position
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Summary
Brief review of scattering approach to elctrical conductance
Magnetic field symmetry of conductance
Brief review of scattering approach to thermoelectric transport
Power, efficiency and efficiency at maximum power
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Thermoelectric transport through a single level dot