Introduction to mesoscopic physics Markus Büttiker University of Geneva NiPS Summer School 2010: Energy Harvesting at the micro and nanoscale Avigliano, Umbro, August 1 August 6, (2010).
Mesoscopic Physics 2 Wave nature of electrons becomes important Webb et al., 1985 Yacoby et al. 1995
Mach-Zehnder Interferometers Neder, Heiblum, Levinson, Mahalu, Umansky, PRL 96, 016804 (2006) 3 Roulleau, Portier, Glattli, Roche, Faini, Gennser, and D. Mailly, PRL 100, 126802 (2008) Bieri, Schoenenberger, Oberholzer, et al. PRB 79. 245324 (2009). Litvin, Tranitz, Wegscheider and Strunk, PRB 75, 033315 (2007)
Probing mesoscopics on the nanoscale 4 M. J. Brukman and D. A. Bonnell, Physics Today, June 2008, p. 36
Grapehne: single and bilayer 5 @Jian Li unige
Length scales 6 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
7 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.
Lecture contents Conductance from transmission 1. Single channel conductors 2. Multichannel conductors 3. Multiprobe conductors 8 Thermoelectric transport Nonlinear transport Noise 1. Basic 2. Equilibrium noise 3. Shot-noise two-probe conductors Fluctuation relations
9 Conductance from Transmission 1. Single channel conductors
Conductance from scattering theory Heuristic discussion Fermi energy left contact Fermi energy right contact applied voltage transmission probability reflection probability 10 incident current density density of states independent of material!! «Landauer formula»
Drift and diffusion 11 at constant Einstein relation for space dependent
Scattering matrix 12 scattering state scattering matrix current conservation S is a unitray matrix In the absence of a magnetic field S is an orthogonal matrix
Conductance from transmission 13 conductance quantum resistance quantum dissipation and irreversibility boundary conditions
Persistent current (periodic boundary conditions) 14 Buttiker, Imry and Landauer, Phys. Lett. 96A, 365 (1983). Measured in 1990 by L. Levy et al, in 1991 by Webb et al..
Persistent current 15 A. C. Bleszynski-Jayich, W. E. Shanks, B. Peaudecerf, E. Ginossar, F. von Oppen, L.Glazman, and J. G. E. Harris,, Science 326, 272 (2009).
Tunable wave splitter 16 Buttiker, Imry, Azbel, Phys. Rev. A30, 1982 (1984)
Aharonov-Bohm conductance oscillations 17 Gefen, Imry, Azbel, PRL 2004 Buttiker, Imry, Azbel, Phys. Rev. A30, 1982 (1984)
Aharonov-Bohm oscillations 18
Conductance from Transmission 19 2. Two-probe multi-channel conductors
Multi-channel conductance: leads 20 asymptotic perfect translation invariant potential seprable wave function energy of transverse motion channel threshold energy for transverse and longitudnial motion scattering channel
Mulit-channel conductance 21 incident current in channel i density in channel i density of states in channel i independent of channel «Landauer formula»
Eigen channels 22 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
Conductance of a perfect wire 23 equilbrium electrochemical potential number of channels with threshold spin degeneracy Example: Single wall carbon nanotube:
Quantum Point Contact 24 van Wees et al., PRL 60, 848 (1988) Wharam et al, J. Phys. C 21, L209 (1988) gate 2D-electron gas gate
Saddle-point potential Quantized conductance: saddle Buttiker, Phys. Rev. B41, 7906 (1990) 25 Transmission probability
Quantized conductance-magnetic field Buttiker, Phys. Rev. B41, 7906 (1990) 26 magnetic field B
Chaotic cavity 27 for symmetric cavity with asmmetric cavity including weak localization: Baranger and Mello, 1994
Diffusive wire 28 Dorokhov-Mello-Pereyra-Kumar Universal conductance fluctuations Stone and Lee, Altschuler
Conductance from Transmission 29 3. Multi-probe conductors
Multi-probe conductors 30 Buttiker, PRL 57, 1761 (1986); IBM J. Res. Developm. 32, 317 (1988)
Four-probe resistances Buttiker, PRL 57, 1761 (1986); IBM J. Res. Developm. 32, 317 (1988) 31 G has eigenvalue zero! Current contacts Voltgae probes
Sub-determinants of conductance matrix 32 D is a sub-determinant of rank three of the conductance matrix. All sub-determinants are (up to a sign) equal. Proof: Expand total determinant into sub-determinants: The only solution without current at any terminal requires that all applied voltages are equal.
Multi-probe conductors: scattering matrix Buttiker, PRL 57, 1761 (1986); IBM J. Res. Developm. 32, 317 (1988) 33 magnetic field symmetry
Reciprocity 34 From and
Reciprocity: Benoit et al. Benoit, Washburn, Umbach, Laibowitz, Webb, PRL 57, 1765 (1986) 35
Reciprocity: van Houten et al. 36 skipping orbit electron focusing van Houten et al., Phys. Rev. B39, 8556 (1989)
Historical remarks 37 Plane-parallel barriers J. Frenkel, Phys. Rev. 36, 1604 (1930) W. Ehrenberg and H. Hoenel, Z. f. Physik 68, 289 (1931) A. Sommerfeld and H. Bethe, Handbuch der Physik (1945) R. Landauer, IBM J. Res. Developm. 1, 223 (1957) Single-channel transport R. Landauer, Phil. Mag. 21, 863 (1970) H. L. Engquist and P. W. Anderson, Phys. Rev. B24, 1151 (1981) Multi-channel conductors Anderson, Economou and Soukoulis, Azbel, Fisher and Lee, Buttiker, Imry and Landauer, Buttiker
Success and limitations Success: Magntic field symmetry : Reciprocity relations Negative four probe resistances, «uphill voltages» Widely applied to ballistic, chaotic and metallic diffusive relatively open conductors Theory of the Quantum Hall effect (edge state transport): probably the most stringent test of the approach 39 Range of application probably the same as DFT (!!) Limitations: Kondo effects, conductance anomalies,.. however extensions to incorporate inelastic scattering, dephasing, time-dependent potentials, etc. exist
Thermoelectric Transport 40
Energy current 40 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)
Heat current 41 Heat current in perfect quantum channel (linear response) Heat current (elastic backscattering, linear response) Thermoelectric transport
Thermoelectric transport 42 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).
Thermopower S. F. Godijn, S. Möller, H. Buhmann, L. W. Molenkamp, S. A. van Langen PRL 82, 2927 2930 (1999) Cutler-Mott-formula 43 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).
Nonlinear transport 44
Rectification 45 Scattering matrix: Weakly nonlinear transport: where 18 elements M. Büttiker, J. Phys.: Condens. Matter 5, 9361 (1993); T. Christen and M. Büttiker, Europhys. Lett. 35, 523 (1996)
Characteristic potentials M. Buttiker, J. Phys. Condensed Matter 5, 9361-9378 (1993). 46 Voltage partial DOS: injectivity emissivity Magnetic-field symmetry: Poisson equation: injectivity is source of
Magnetic field asymmetry of rectification Elastic transport: 47 Naive expectation: since T is even in the two-probe case, nonlinear I-V is also even Correct only in linear regime: reciprocity of s-matrix hinges on symmetry of U EQULIBRIUM AWAY FROM EQUILIBRIUM: Interaction effect At equilibrium microreversibilty is sufficient to dictate symmetry of transport coefficients: Away from equilibrium boundary conditions become important
Second order conductance of a chaotic dot D. Sanchez and M. Buttiker, PRL 93, 10602 (2004) M. Polianski and M. Buttiker, PRL 96, 1056804 (2006) 48 Unitray limit Numerical RMT
Carbon nanotubes Rectification: experiments I 49 Cavities J. Wei et al., PRL 95, 256601 (2005) Rings D. M. Zumbuhl et al, PRL 96, 206802 (2006) Theory agrees with experiment for N > 4 R. Leturcq et al., PRL 96, 126801 (2006)
Rectification: experiments II 50 D. Hartmann, L. Worschech, A. Forchel, PRB 78, 113306 (2008).
Current Noise in Mesoscopic Conductors 51 1. Basics
Fundamental sources of noise Buttiker, PRB 46, 12485 (1992) Thermal fluctuations of occupation numbers in the contacts 52 Nyquist-Johnson noise Quantum partition noise: kt = 0 occupation numbers: incident beam transmitted beam reflected beam averages: Each particle can only be either transmitted or reflected: Blanter and Buttiker, Phys. Rep. 336, 1 (2000)
Occupation number and current amplitudes Incident current at kt = 0 Buttiker, PRB 46, 12485 (1992) 53 Incident current at kt > 0 Occupation number Creation and annihilation operators < > = statistical average «Incident current» «Current amplitude»
Noise spectral density Spectral density S (noise power) 54 quantum statistical average of four creation and annihilation op. zero-frequency spectrum (white noise limit) equilibrium non-equilibrium fluctuation-dissipation theorem shot-noise Buttiker, PRL 65, 2901 (1990); PRB 46, 12485 (1992)
Current Noise in Mesoscopic 55 Conductors 2. Equilibrium Noise
Use Thermal current fluctuations 56 with for all auto-correlation cross-correlation QHE-plateau N:
Current Noise in Mesoscopic 57 Conductors 3. Shot Noise: Two-probe conductors
Shot-noise: two-terminal 60 Consider kt = 0, V>0, and a two-terminal conductor: Quantum partition noise If all Shottky (Poisson) Fano factor Khlus (1987) Lesovik (1989) Buttiker (1990)
Shot-noise: Qunatum point contact 61 Kumar, L. Saminadayar, D. C. Glattli, Y. Jin, B. Etienne, PRL 76, 2778 (1996) M. I. Reznikov, M. Heiblum, H. Shtrikman, D. Mahalu, PRL 75, 3340 (1996) Ideally only one channel contributes
Crossover from thermal to shot noise 62 tunnel junction H. Birk et al., PRL 75, 1610 (1995)
Current Noise in Mesoscopic 63 Conductors 4. Shot Noise: Correlations
Shot-noise correlations 64 Consider multi-terminal conductor at kt = 0, M source contacts with distribution voltage All other contacts grounded at voltage Correlation measured bewteen two grounded contacts: M =1, partition noise M =2, exchange effects, two paricle Aharonov-Bohm effect, orbital entanglement, violation of Bell inequality Samuelsson, Sukhorukov, Buttiker, PRL 92, 026805 (2004) Buttiker, Samuelsson, Sukhorukov, Physica E20, 33 (2003)
Beam splitter with noisy input state 65 Oberholzer et al. Physica E6, 314 (2000) Here Bias configuration:
Experiment of Oberholzer et al. 66 Oberholzer et al, Physica E6, 314 (2000) See also: Henny, et al., Science 284, 296 (1999); Oliver et al. Science 284, 299 (1999)
Review on Shot Noise 67 «Shot Noise in Mesoscopic Conductors» Ya. M. Blanter and M. Buttiker, Phys. Rep. 336, 1 (2000)
Fluctuation relations 68
Nonlinear transport and noise H. Forster and M. Buttiker, PISA, arxiv: 0903.1431 Fluctuation dissipation theorem 69 Fluctuation relation of Forster and Buttiker (microreversible only at eq.) Fluctation relation of Saito and Utsumi [General case: H. Forster and M. Buttiker, PRL 101, 136805 (2008) ]
Nonlinearity and noise H. Forster and M. Buttiker, arxiv: 0903.1431 70 Negative excess noise
Kobayashi s experiment Nakamura, Yamauchi, Hashisaka, Chida, Kobayashi, Ono, Leturcq, Ensslin, Saito, Utsumi, and Gossard, Phys. Rev. Lett. 104, 080602 (2010) 71
Nongaussian noise on macroscopic scales Nagaev, Ayvazyan, Sergeeva, and Buttiker, arxiv: 1004.5310 72 macroscopic!! potential dependence of conductance cyclotron-frequency times scattering time [Saito and Utsumi, 2008]
Summary Transport theory for coherent electron transport Conductance Thermal transport Non linear transport Noise Correlations Fluctuation relations