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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

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

slide2

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

slide3

Semiclassical theory: Drude-Boltzmann-Sommerfeld

Random walk of step

Diffusive motion

Response function and Einstein’s relation

Fermi gas case:

slide4

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

slide5

Scaling theory

Thouless’s argument

Edwards and Thouless 1972

Control parameter: dimensionless conductance

slide6

Scaling hypothesis:

Depends on g only

Fixed point:

Critical exponent:

Abrahams, Anderson, Licciardello, Ramakrishnana 1979

slide7

Power behavior of physical quantities

Correlation length

Scaling law

Metallic side expansion

Time reversal invariance

B-field or magnetic impurities

slide8

Basic tool: linear response theory

Castellani, Di Castro, Forgacs, Tabet 1983

Real space

Fourier space

Charge conservation

Gauge invariance

Observables

slide9

Response functions and Ward identities

Bare vertex

Dressed vertex

Ward identity

slide10

Check: free case

Consequences of W.i.

Dynamic part

DOS

Phenomenological theory obeys all !

slide11

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

slide12

Self-consistent Born approximation

Key approximation:

Self-consistent solution, only position of the pole matters

Abrikosov, Gorkov, Dzyaloshinski

slide13

Microscopic theory: response functions

“Rainbow” for

“Ladder” for

W. I.

Langer, Neal 1976

Recover the semiclassical result!

slide14

How to go beyond and keep interference processes

Role of crossed diagrams

Expansion parameter

Maximally crossed diagrams

Enhanced backscattering due to time-reversed paths

slide15

Correction to response function

Ladder self-energy

Weak localization correction

Gorkov, Larkin, Khmelnitskii 1979

slide16

What about B?

Crossed diagrams in real space

B enters via

a “mass” in the diffusion propagator

slide17

Magnetoresistance and dephasing time

Crossover when

Measure of

slide18

Spin effects: magnetic impurities and spin-orbit coupling

“Mass”

Singlet and Triplet channels

Antilocalizing

slide19

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
slide20

Thomas et al PRB ‘82 GeSb

  • Hertel et al PRL ‘83 Nb Si
  • Rhode Micklitz al PRB ‘87 BiKr

Compensated Smc and alloys

slide21

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

slide22

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

slide23

Unexpected anomalies

Singularity in DOS

  • McMillan Mochel PRL ‘81 Ge Au
  • Hertel et al PRL ‘83 Nb Si
slide24

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
slide25

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?

slide26

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:

slide27

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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