Savoyan Castle, Rackeve, Hungary. Workshop on Disorder and Interactions . Disordered Electron Systems I. Introduction Scaling theory Microscopic theory Non-interacting case. Roberto Raimondi. Thanks to C. Di Castro C. Castellani. 4-6 april 2006.
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Savoyan Castle, Rackeve, Hungary
Workshop on Disorder and Interactions
DisorderedElectron Systems I.
Thanks to C. Di Castro
4-6 april 2006
Key problem: metal-insulator transition (MIT)
Key physics:quantum interference corrections
G. Bergman Phys. Rep. 107, 1 (1984)
P.A. Lee and T.V. Ramakrishnan Rev. Mod. Phys. 57, 287 (1985)
B.L. Altshuler and A.G. Aronov in Electron-electron Interactions in Disordered Systems,
Eds. M.Pollak and A.L. Efros North-Holland, Amsterdam (1984) p.1
A.M. Finkelstein Sov. Sci. Rev.14, 1 (1990)
D. Belitz and T.R. Kirkpatrick Rev. Mod. Phys. Rep. 66, 261 (1994)
C. Di Castro and R. Raimondi in The Electron Liquid Paradigm in Condensed Matter Physics
Proceedings of the Inter. School of Physics E. Fermi,
Eds. G.F. Giuliani and G. Vignale IOP Press 20041. Cond-mat/0402203
Semiclassical theory: Drude-Boltzmann-Sommerfeld
Random walk of step
Response function and Einstein’s relation
Fermi gas case:
Quantum corrections: self-intersecting trajectories
Summing all times
Task for microscopic theory:
Diffusion modes as critical modes
Inverse conductivity as expansion parameter
Edwards and Thouless 1972
Control parameter: dimensionless conductance
Depends on g only
Abrahams, Anderson, Licciardello, Ramakrishnana 1979
Power behavior of physical quantities
Metallic side expansion
Time reversal invariance
B-field or magnetic impurities
Basic tool: linear response theory
Castellani, Di Castro, Forgacs, Tabet 1983
Response functions and Ward identities
Check: free case
Consequences of W.i.
Phenomenological theory obeys all !
Microscopic theory: Green function
Task: recover semiclassical approach as the zeroth order in
Disorder expected effect
Disorder model: Gaussian random variable
Self-consistent Born approximation
Self-consistent solution, only position of the pole matters
Abrikosov, Gorkov, Dzyaloshinski
Microscopic theory: response functions
Langer, Neal 1976
Recover the semiclassical result!
How to go beyond and keep interference processes
Role of crossed diagrams
Maximally crossed diagrams
Enhanced backscattering due to time-reversed paths
Correction to response function
Weak localization correction
Gorkov, Larkin, Khmelnitskii 1979
What about B?
Crossed diagrams in real space
B enters via
a “mass” in the diffusion propagator
Magnetoresistance and dephasing time
Spin effects: magnetic impurities and spin-orbit coupling
Singlet and Triplet channels
WL seen in films and wires
Compensated Smc and alloys
Si-P critical exponent puzzle
Si As n-doped, Si B p-doped
Anomalous B-dependence of critical exponent
CuMn Magnetic impurities ?
Okuma et al ‘87
Katsumoto et al JPSJ ‘87
Si Au Strong Spin Orbit
Nishida et al SSP ‘84
Singularity in DOS
Low-T enhancement of specific heat
Low-T enhancement of spin susceptibility
Key issue: how e-e interaction changes the game?
Last but no least: 2D MIT in Si-MOSFETs and heterostructures
Kravchenko and SarachikRep. Progr. Phys.67, 1 (2004)
End of part I.