Disordered Electron Systems I.

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Disordered Electron Systems I.
Savoyan Castle, Rackeve, Hungary
Workshop on Disorder and Interactions

• Roberto Raimondi

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

Thanks to C. Di Castro C. Castellani
4-6 april 2006
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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
• Tasksidentify couplings and critical modes

Key physicsquantum 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
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Semiclassical theory Drude-Boltzmann-Sommerfeld
Random walk of step
Diffusive motion
Response function and Einsteins relation
Fermi gas case
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Quantum corrections self-intersecting
trajectories
Return probability
Self-intersection probability
Summing all times
Task for microscopic theory

1. Diffusion modes as critical modes
2. Inverse conductivity as expansion parameter

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Scaling theory
Thoulesss argument
Edwards and Thouless 1972

Control parameter dimensionless conductance
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Scaling hypothesis
Depends on g only
Fixed point
Critical exponent
Abrahams, Anderson, Licciardello, Ramakrishnana
1979
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Power behavior of physical quantities
Correlation length
Scaling law
Metallic side expansion
Time reversal invariance
B-field or magnetic impurities
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Basic tool linear response theory
Castellani, Di Castro, Forgacs, Tabet 1983
Real space
Fourier space
Charge conservation
Gauge invariance
Observables
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Response functions and Ward identities
Bare vertex
Dressed vertex
Ward identity
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Check free case
Consequences of W.i.
Dynamic part
DOS
Phenomenological theory obeys all !
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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
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Self-consistent Born approximation
Key approximation
Self-consistent solution, only position of the
pole matters
Abrikosov, Gorkov, Dzyaloshinski
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Microscopic theory response functions
Rainbow for
Ladder for
W. I.
Langer, Neal 1976
Recover the semiclassical result!
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How to go beyond and keep interference processes
Role of crossed diagrams
Expansion parameter
Maximally crossed diagrams
Enhanced backscattering due to time-reversed paths
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Correction to response function
Ladder self-energy
Weak localization correction
Gorkov, Larkin, Khmelnitskii 1979
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What about B?
Crossed diagrams in real space
B enters via
a mass in the diffusion propagator
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Magnetoresistance and dephasing time
Crossover when
Measure of
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Spin effects magnetic impurities and spin-orbit
coupling
Mass
Singlet and Triplet channels
Antilocalizing
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Experiments?
Agreement

• Dolan Osheroff PRL 79
• Giordano et al PRL79

WL seen in films and wires
InSb
AuPd

• Dynes, Geballe, Hull, Garno PRB 83

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• Thomas et al PRB 82 GeSb
• Hertel et al PRL 83 Nb Si
• Rhode Micklitz al PRB 87 BiKr

Compensated Smc and alloys
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Si-P critical exponent puzzle
Problems

• 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
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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
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Unexpected anomalies
Singularity in DOS

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

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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

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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?
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Last but no least 2D MIT in Si-MOSFETs and
heterostructures
Kravchenko and Sarachik Rep. 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
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End of part I.

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