Scattering theory of thermoelectric transport. Markus Büttiker University of Geneva
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1 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
2 Mesoscopic Physics 2 Wave nature of electrons becomes important Webb et al., 1985 Yacoby et al. 1995
3 Graphene: single and bilayer Li unige
4 Length scales 4 Geometrical dimension (size of conductor) Phase coherence length (distance an electron travels before suffering a phase change of ) Elastic scattering length (mean free path between elastic scattering events) Inelastic scattering length (distance an electron travels before loosing an energy kt) Beenakker and van Houten, 1991 Macroscopic conductor Mesoscopic conductor
5 5 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.
6 Lecture contents Conductance from transmission 1. Single channel conductors 2. Multichannel conductors 3. Multiprobe conductors (omitted) Thermoelectric transport 1. Two-terminal coductors 2. Thermoelectrics of a quantum dot 3. Multiprobe conductors (omitted) 4. Magnetic field symmetry (omitted) 6
7 7 Conductance from Transmission 1. Single channel conductors
8 Conductance from scattering theory Heuristic discussion Fermi energy left contact Fermi energy right contact applied voltage transmission probability reflection probability 8 incident current density density of states independent of material!! «Landauer formula»
9 Scattering matrix 9 scattering state scattering matrix current conservation S is a unitray matrix In the absence of a magnetic field S is an orthogonal matrix
10 Aharonov-Bohm oscillations 10
11 Conductance from Transmission Two-probe multi-channel conductors
12 Multi-channel conductance: leads 12 asymptotic perfect translation invariant potential seprable wave function energy of transverse motion channel threshold energy for transverse and longitudnial motion scattering channel
13 Mulit-channel conductance 13 incident current in channel i density in channel i density of states in channel i independent of channel «Landauer formula»
14 Eigen channels 14 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
15 Quantum point contact gate van Wees et al., PRL 60, 848 (1988) Wharam et al, J. Phys. C 21, L209 (1988) gate
16 Saddle-point potential Quntum point contact Buttiker, Phys. Rev. B41, 7906 (1990) Transmission probability
17 Conductance of resonant level Transmission probability oof single level Conductance Energy of resonant level Level width Low temperature limit High temperature limit
18 24 Thermoelectric Transport 1. Two terminal conductors
19 Energy current 25 Energy flux in a quantum channel: reservoirs at T1 and T2: Small temperature difference Thermal quantum (independent of electron or channel properties!!) H. L. Engquist and P. W. Anderson, Phys. Rev. B24, 1151 (1981) Lorentz factor (Sommerfeld theory)
20 Heat current Heat current in perfect quantum channel, (linear response ) 26 Heat current (elastic backscattering, linear response) Connection with energy and electrical current Thermoelectric transport (linear response)
21 Thermoelectric transport Fluxes in response to potentials 27 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).
22 Thermopower 28 Cutler-Mott formula Sommerfeld integral zero temperature limit
23 Thermopower of a QPC Proetto, PRB 44, 9096 (1991) Channel dependence
24 29 Thermoelectric transport 2. Thermoelectric transport of a quantum dot
25 Thermopower for resonant transmission P. Mani, N. Nakpathomkun, H. Linke, Journal of Electronic Materials 38, 1163 (2009). 30 Resonant transmission probability Level width High temperature limit Universal! But only as long as thermal energy is small compared to the level separation.
26 Thermopower for resonant transmission P. Mani, N. Nakpathomkun, H. Linke, Journal of Electronic Materials 38, 1163 (2009). Cutler-Mott 31 Low temperature limit of CM-formula Resonant transmission probability Note that this is independent of symmetry
27 Thermopower of a multilevel dot C. W. J. Beenakker and A. A. M. Staring Phys. Rev. B 46, 9667 (1992) 32
28 Thermopower of a chaotic cavity S. F. Godijn, S. Möller, H. Buhmann, L. W. Molenkamp, S. A. van Langen PRL 82, (1999) 33 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).
29 Efficiency of a single level dot 34 Efficiency Power Current Heat current High temperature limit Efficency Carnot efficiency is reached when Stall voltage
30 Efficiency at maximum power Nakpathomkun, Xu, Linke, PRB 82, (2012) efficiency at maximum power maximum efficiency maximum power 35 Maximization is with regards to the position of the resonant level position
31 36 Summary Brief review of scattering approach to elctrical conductance Magnetic field symmetry of conductance Brief review of scattering approach to thermoelectric transport Thermoelectric transport through a single level dot Power, efficiency and efficiency at maximum power
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