Workshop on Disorder and Interactions
Savoyan Castle, Rackeve, Hungary
Disordered Electron Systems I.
Roberto Raimondi
•Introduction
•Scaling theory
•Microscopic theory
•Non-interacting case
4-6 april 2006
Thanks to C. Di Castro
C. Castellani
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
Observables
Gauge invariance
Response functions and Ward identities
Bare vertex
Dressed vertex
Ward identity
Check: free case
Consequences of W.i.
Dynamic part
DOS
Phenomenological theory obeys all !
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?
Agreement
WL seen in films and wires
AuPd
•Dynes, Geballe, Hull, Garno PRB 83
•Dolan Osheroff PRL ‘79
•Giordano et al PRL’79
InSb
Compensated Smc and alloys
•Thomas et al PRB ‘82 GeSb
•Hertel et al PRL ‘83 Nb Si
•Rhode Micklitz al PRB ‘87 BiKr
Problems
Si-P critical exponent puzzle
Uncompensated SiP
•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 ?
Okuma et al ‘87
AlGaAs Si
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
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)
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