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