disordered electron systems i
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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. - PowerPoint PPT PresentationTRANSCRIPT
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 correctionsG. 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 exponentCuMn 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