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Disordered Electron Systems I.

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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Disordered Electron Systems I.

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  1. Savoyan Castle, Rackeve, Hungary Workshop on Disorder and Interactions DisorderedElectron Systems I. • Introduction • Scaling theory • Microscopic theory • Non-interacting case Roberto Raimondi Thanks to C. Di Castro C. Castellani 4-6 april 2006

  2. 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

  3. Semiclassical theory: Drude-Boltzmann-Sommerfeld Random walk of step Diffusive motion Response function and Einstein’s relation Fermi gas case:

  4. Quantum corrections: self-intersecting trajectories Return probability Self-intersection probability Summing all times Task for microscopic theory: Diffusion modes as critical modes Inverse conductivity as expansion parameter

  5. Scaling theory Thouless’s argument Edwards and Thouless 1972 Control parameter: dimensionless conductance

  6. Scaling hypothesis: Depends on g only Fixed point: Critical exponent: Abrahams, Anderson, Licciardello, Ramakrishnana 1979

  7. Power behavior of physical quantities Correlation length Scaling law Metallic side expansion Time reversal invariance B-field or magnetic impurities

  8. Basic tool: linear response theory Castellani, Di Castro, Forgacs, Tabet 1983 Real space Fourier space Charge conservation Gauge invariance Observables

  9. Response functions and Ward identities Bare vertex Dressed vertex Ward identity

  10. Check: free case Consequences of W.i. Dynamic part DOS Phenomenological theory obeys all !

  11. 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

  12. Self-consistent Born approximation Key approximation: Self-consistent solution, only position of the pole matters Abrikosov, Gorkov, Dzyaloshinski

  13. Microscopic theory: response functions “Rainbow” for “Ladder” for W. I. Langer, Neal 1976 Recover the semiclassical result!

  14. How to go beyond and keep interference processes Role of crossed diagrams Expansion parameter Maximally crossed diagrams Enhanced backscattering due to time-reversed paths

  15. Correction to response function Ladder self-energy Weak localization correction Gorkov, Larkin, Khmelnitskii 1979

  16. What about B? Crossed diagrams in real space B enters via a “mass” in the diffusion propagator

  17. Magnetoresistance and dephasing time Crossover when Measure of

  18. Spin effects: magnetic impurities and spin-orbit coupling “Mass” Singlet and Triplet channels Antilocalizing

  19. Experiments? Agreement • Dolan Osheroff PRL ‘79 • Giordano et al PRL’79 WL seen in films and wires InSb AuPd • Dynes, Geballe, Hull, Garno PRB 83

  20. Thomas et al PRB ‘82 GeSb • Hertel et al PRL ‘83 Nb Si • Rhode Micklitz al PRB ‘87 BiKr Compensated Smc and alloys

  21. Problems 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 Uncompensated SiP Si As n-doped, Si B p-doped

  22. 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

  23. Unexpected anomalies Singularity in DOS • McMillan Mochel PRL ‘81 Ge Au • Hertel et al PRL ‘83 Nb Si

  24. 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

  25. Low-T enhancement of spin susceptibility • Ikeata et al SSC ‘85 • Paalanen et al PRL ‘86 • Alloul Dellouve PRL ‘87 • Hirsch et al PRL ‘92 • Schlager et al EPL ‘97 Key issue: how e-e interaction changes the game?

  26. Last but no least: 2D MIT in Si-MOSFETs and heterostructures Kravchenko and SarachikRep. Progr. Phys.67, 1 (2004) Quantum effects Key parameter: • Unexpected with non-interacting theory • Strong magnetoresistance in parallel field • Open issue whether there is a MIT MOSFET:

  27. End of part I. • Program for next lecture • Explore perturbative effects of interaction • Landau Fermi-liquid formulation • Renormalizability of response function • RG equations

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