Transcript Disordered Electron Systems
Workshop on Disorder and Interactions Savoyan Castle, Rackeve, Hungary
Disordered
Electron Systems I.
Roberto Raimondi
• Introduction • Scaling theory • Microscopic theory • Non-interacting case Thanks to C. Di Castro C. Castellani 4-6 april 2006
Key problem: metal-insulator transition (MIT) • MIT from interplay of disorder and interaction • Metallic side in terms of Fermi liquid Aim: describe MIT as continuous phase transition Tasks:identify couplings and critical modes 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 Diffusive motion Response function and Einstein’s relation Fermi gas case:
Quantum corrections: self-intersecting trajectories Return probability Self-intersection probability Summing all times Task for microscopic theory: i.
Diffusion modes as critical modes ii. Inverse conductivity as expansion parameter
Scaling theory Thouless’s argument Edwards and Thouless 1972 Control parameter: dimensionless conductance
Scaling hypothesis: Depends on g only Fixed point: Critical exponent: Abrahams, Anderson, Licciardello, Ramakrishnana 1979
Power behavior of physical quantities Correlation length Scaling law Metallic side expansion Time reversal invariance B-field or magnetic impurities
Basic tool: linear response theory Castellani, Di Castro, Forgacs, Tabet 1983 Real space Fourier space Charge conservation Gauge invariance Observables
Response functions and Ward identities Bare vertex Dressed vertex Ward identity
Check: free case Consequences of W.i.
Dynamic part Phenomenological theory obeys all !
DOS
Microscopic theory: Green function Task: recover semiclassical approach as the zeroth order in Disorder expected effect Finite lifetime Quasi-particle pole Disorder model: Gaussian random variable
Self-consistent Born approximation Key approximation: Self-consistent solution, only position of the pole matters Abrikosov, Gorkov, Dzyaloshinski
Microscopic theory: response functions “Rainbow” for “Ladder” for W. I.
Langer, Neal 1976 Recover the semiclassical result!
How to go beyond and keep interference processes Role of crossed diagrams Expansion parameter Maximally crossed diagrams Enhanced backscattering due to time-reversed paths
Correction to response function Ladder self-energy 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 Crossover when Measure of
Spin effects: magnetic impurities and spin-orbit coupling Singlet and Triplet channels “Mass” Antilocalizing
Experiments? WL seen in films and wires Agreement • Dolan Osheroff PRL ‘79 • Giordano et al PRL’79 AuPd InSb • Dynes, Geballe, Hull, Garno PRB 83
Compensated Smc and alloys • Thomas et al PRB ‘82 GeSb • Hertel et al PRL ‘83 Nb Si • Rhode Micklitz al PRB ‘87 BiKr
Problems Uncompensated SiP Si-P critical exponent puzzle • Rosenbaum et al PRL ‘80, PRB ‘83 • Stupp et al PRL ‘93 • Shafarman et al PRB ‘89 Si As • Dai et al PRB ‘93 Si B Si As n-doped, Si B p-doped
Anomalous B-dependence of critical exponent CuMn Magnetic impurities ?
AlGaAs Si Okuma et al ‘87 Katsumoto et al JPSJ ‘87 • Dai et al et al PRB ‘93 Si P Si Au Strong Spin Orbit Nishida et al SSP ‘84
Unexpected anomalies Singularity in DOS • McMillan Mochel PRL ‘81 Ge Au • Hertel et al PRL ‘83 Nb Si
Low-T enhancement of specific heat • Kobayashi et al SSC ‘79 Si P • Thomas et al PRB ‘81 Si P • Paalanen et al PRL ‘88 Si P • Lakner et al PRL ‘89 Si P
• Ikeata et al SSC ‘85 • Paalanen et al PRL ‘86 • Alloul Dellouve PRL ‘87 • Hirsch et al PRL ‘92 • Schlager et al EPL ‘97 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 Sarachik Rep. Progr. Phys. 67 , 1 (2004) Quantum effects Key parameter: MOSFET: • Unexpected with non-interacting theory • Strong magnetoresistance in parallel field • Open issue whether there is a MIT
End of part I.
Program for next lecture • Explore perturbative effects of interaction • Landau Fermi-liquid formulation • Renormalizability of response function • RG equations