Disordered Electron Systems I.Roberto Raimondi

•Introduction•Scaling theory•Microscopic theory•Non-interacting case

Thanks to C. Di CastroC. Castellani

Workshop on Disorder and Interactions Savoyan Castle, Rackeve, Hungary

4-6 april 2006

•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 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 PhysicsProceedings 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 modesii. Inverse conductivity as expansion parameter

Scaling theory

Thouless’s argument

Control parameter: dimensionless conductance

Edwards and Thouless 1972

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

Real space

Fourier space

Basic tool: linear response theory

Observables

Charge conservation Gauge invariance

Castellani, Di Castro, Forgacs, Tabet 1983

Response functions and Ward identities

Dressed vertex

Ward identity

Bare vertex

Check: free case

Consequences of W.i.

Phenomenological theory obeys all !

Dynamic part

DOS

Microscopic theory: Green function

Task: recover semiclassical approach as the zeroth order in

Disorder expected effect

Disorder model: Gaussian random variable

Finite lifetimeQuasi-particle pole

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.

Recover the semiclassical result!

Langer, Neal 1976

How to go beyond and keep interference processes

Role of crossed diagrams

Expansion parameter

Enhanced backscattering due to time-reversed paths

Maximally crossed diagrams

Ladder self-energy

Weak localization correction

Correction to response function

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

WL seen in films and wires

Experiments?

•Dolan Osheroff PRL ‘79•Giordano et al PRL’79

Agreement

AuPd

•Dynes, Geballe, Hull, Garno PRB 83

InSb

Compensated Smc and alloys•Thomas et al PRB ‘82 GeSb•Hertel et al PRL ‘83 Nb Si

•Rhode Micklitz al PRB ‘87 BiKr

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 BUncompensated SiP

Problems

Si As n-doped, Si B p-doped

Anomalous B-dependence of critical exponent

CuMn Magnetic impurities ?

Okuma et al ‘87AlGaAs Si

Katsumoto et al JPSJ ‘87

Si Au Strong Spin Orbit

Nishida et al SSP ‘84

•Dai et al et al PRB ‘93 Si P

Singularity in DOS

Unexpected anomalies

•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

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?

Last but no least: 2D MIT in Si-MOSFETs and heterostructures

Kravchenko and Sarachik Rep. Progr. Phys. 67, 1 (2004)

•Unexpected with non-interacting theory•Strong magnetoresistance in parallel field•Open issue whether there is a MIT

Key parameter:

Quantum effects

MOSFET:

End of part I.

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