Page 1
Igor Aleiner (Columbia)
Theory of Quantum Dots as Zero-dimensional Metallic Systems
Physics of the Microworld Conference, Oct. 16 (2004)
Collaborators:B.L.Altshuler (Princeton)P.W.Brouwer (Cornell)V.I.Falko (Lancaster, UK)L.I. Glazman (Minnesota)I.L. Kurland (Princeton)
Page 2
Outline:• Quantum dot (QD) as zero dimensional metal• Random Matrix theory for transport in quantum dotsa) Non-interacting “standard models”.b) Peculiar spin-orbit effects in QD based on 2D electron gas.
• Interaction effects:a) Universal interaction Hamiltonian;b) Mesoscopic Stoner instability; c) Coulomb blockade (strong, weak, mesoscopic);
d) Kondo effect.
Page 3
“Quantum dot” used in two different contents:
“Artificial atom”
Description requires exact diagonalization.
“Artificial nucleus”
Statistical description is allowed !!!
(Kouwnehoven group (Delft))
Number of electrons:1)
(Marcus group (Harvard))
2)
For the rest of the talk:
Page 4
Random Matrix Theory for Transport in Quantum Dots
2DEGQD
L
Energy scales
2DEG Level spacing
Thouless Energy
ConductanceAssume:
Page 5
Statistics of transport is determinedonly by fundamental symmetries !!!
Reviews: Beenakker, Rev. Mod. Phys. 69, 731 (1997) Alhassid, Rev. Mod. Phys. 72, 895 (2000) Aleiner, Brouwer, Glazman, Phys. Rep., 309 (2002)
Original Hamiltonian: Confinement, disorder, etc
RMT
Page 6
No magnetic field, no SO
Magnetic field, no SO
No magnetic field, strong SO
Magnetic field + SO
Page 7
IV
Conductance of chaotic dotclassical Mesoscopic
fluctuations
Weak localization
Jalabert, Pichard, Beenakker (1994)Baranger, Mello (1994)
Page 8
IV
Conductance of chaotic dotclassical Mesoscopic
fluctuations
Weak localization
[Altshuler, Shklovskii (1986)]
Universal quantum corrections
Page 9
Peculiar effect of the spin-orbit interaction
Naively:SO
But the spin-orbit interaction in 2D is not generic.
Page 10
Spin-orbit interaction in GaAs/AlGaAs (001) 2DEG
- spin-orbit lengths
[001]
Rashba term
Dresselhaus term
Dyakonov-Perel spin relaxation
Page 11
Approximate symmetries of SO in QDAleiner, Fal’ko (2001)
T - invariance
But
Spin dependent flux Spin relaxation rate
Mathur, Stone (1992)
Khaetskii, Nazarov (2000)
Meir, Gefen, Entin-Wohlman (1989)
Lyanda-Geller, Mirlin (1994)
Page 12
Energy scales:
Brouwer, Cremers,Halperin (2002) May be violated for
Page 13
Effect of Zeeman splitting
Orthogonal, !!!
But no spin degeneracy; spins mixed:
New energy scale:
Page 14
6 possible symmetry classes:
Page 15
6 possible symmetry classes:
Page 16
Orbital effect of the magnetic field
Page 17
Orbital effect of the magnetic fieldObserved inFolk, Patel, Birnbaum, Marcus, Duruoz, Harris, Jr. (2001)
Page 18
Interaction HamiltonianEnergies smaller than Thouless energy:
Random matrix ????
In nuclear physics:
from shell model
random
Page 19
Universal Interaction HamiltonianEnergies smaller than Thouless energy:
are NOT random !!!
Kurland, Aleiner, Altshuler (2000)
Only invariants compatible with the circular symmetry
Page 20
Universal Interaction HamiltonianEnergies smaller than Thouless energy:
Valid if:
1)2) Fundamental symmetries
are NOT broken at larger energies
Random matrixNot random
Page 21
Universal Interaction HamiltonianEnergies smaller than Thouless energy:
Valid if:
1)2) Fundamental symmetries
are NOT broken at larger energies
One-particle levels determined by Wigner – Dyson statistics
Interaction with additional conservations
Zero dimensional Fermi liquid
Page 22
Universal Interaction Hamiltonian
Analogy with soft modes in metals
Singlet electron-holechannel.
Triplet electron-holechannel. Particle-particle
(Cooper) channel.
Page 23
Universal Interaction Hamiltonian
Cooper Channel:Renormalization:
Normal
Superconducting(e.g. Al grains)
Page 24
Universal Interaction Hamiltonian
Triplet Channel:is NOT renormalized
But may lead to the spin ofThe ground state S > ½.
Page 25
Mesoscopic Stoner Instability Kurland, Aleiner, Altshuler (2000)
Also Brouwer, Oreg, Halperin (2000)
vs.
Energy of the ground state:
NO randomness
NO interactions
FM instabilityStoner (1935)
random with known from RMTcorrelation functions
Spin is finite even for
Typical S:
Does not scale with the size of the system
Page 26
Universal Interaction Hamiltonian
Singlet Channel:is NOT renormalized
gate voltage
But
Q: What is charge degeneracy of the ground state
Page 27
- half-integer Otherwise
degeneracygap
(isolated dot)
Page 28
Coulomb blockade of electron transport
Term introduced by Averin and Likharev (1986);Effect first discussed by C.J. Gorter (1951).
For tunneling contacts:
Chargedegeneracy
Charge gap
Page 29
Small quantum dots Small quantum dots (~ (~ 500 nm500 nm))
M. Kastner, Physics Today (1993)E.B. Foxman et al., PRB (1993)
cond
ucta
nce
(e2 /
h)
gate voltage (mV)
In metals first observed in Fulton, Dolan, PRL, 59, 109, (1987)
Page 30
Coulomb blockade (CB) (II)
Strong CB
Weak CB
Mesoscopic CB
(reflectionlesscontacts) Random phase but
not period.
Page 31
Courtesy of C.Marcus
Statistical description of strong CB:
Theory:Peaks: Jalabert, Stone, Alhassid (1992);Valleys: Aleiner, Glazman (1996);Reasonable agreement,But problems with values of the correlationfields
Page 32
Mesoscopic Coulomb Blockade
Based on technique suggested by: Matveev (1995); Furusaki, Matveev (1995);Flensberg (1993).
Aleiner, Glazman (1998)
Page 33
Experiment:Cronenwett et. al. (1998)
SuppressionBy a factor of 5.3
Th: Predicted 4.
Page 34
Even-Odd effect due to Kondo effect
Spin degeneracy in odd valleys:
Effective Hamiltonian:
magnetic impuritylocal spin density of conduction electrons
Predicted:Glazman, Raikh (1988)Ng, Lee (1988)
Page 35
Observation:D. Goldhaber-Gordon et al. (MIT-Weizmann) S.M. Cronenwett et al. (TU Delft)J. Schmid et al. (MPI @ Stuttgart)
1998
van der Wiel et al. (2000)
200 nm
15 mK 800 mK
Page 36
Conclusions
1)Random matrix is an adequate description for the transport in quantum dots if underlying additional symmetries are properly identified.
2) Interaction effects are described by the Universal Hamiltonian (“0D Fermi Liquid”)