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)

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.

“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:

Random Matrix Theory for Transport in Quantum Dots

2DEGQD

L

Energy scales

2DEG Level spacing

Thouless Energy

ConductanceAssume:

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

No magnetic field, no SO

Magnetic field, no SO

No magnetic field, strong SO

Magnetic field + SO

IV

Conductance of chaotic dotclassical Mesoscopic

fluctuations

Weak localization

Jalabert, Pichard, Beenakker (1994)Baranger, Mello (1994)

IV

Conductance of chaotic dotclassical Mesoscopic

fluctuations

Weak localization

[Altshuler, Shklovskii (1986)]

Universal quantum corrections

Peculiar effect of the spin-orbit interaction

Naively:SO

But the spin-orbit interaction in 2D is not generic.

Spin-orbit interaction in GaAs/AlGaAs (001) 2DEG

- spin-orbit lengths

[001]

Rashba term

Dresselhaus term

Dyakonov-Perel spin relaxation

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)

Energy scales:

Brouwer, Cremers,Halperin (2002) May be violated for

Effect of Zeeman splitting

Orthogonal, !!!

But no spin degeneracy; spins mixed:

New energy scale:

6 possible symmetry classes:

6 possible symmetry classes:

Orbital effect of the magnetic field

Orbital effect of the magnetic fieldObserved inFolk, Patel, Birnbaum, Marcus, Duruoz, Harris, Jr. (2001)

Interaction HamiltonianEnergies smaller than Thouless energy:

Random matrix ????

In nuclear physics:

from shell model

random

Universal Interaction HamiltonianEnergies smaller than Thouless energy:

are NOT random !!!

Kurland, Aleiner, Altshuler (2000)

Only invariants compatible with the circular symmetry

Universal Interaction HamiltonianEnergies smaller than Thouless energy:

Valid if:

1)2) Fundamental symmetries

are NOT broken at larger energies

Random matrixNot random

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

Universal Interaction Hamiltonian

Analogy with soft modes in metals

Singlet electron-holechannel.

Triplet electron-holechannel. Particle-particle

(Cooper) channel.

Universal Interaction Hamiltonian

Cooper Channel:Renormalization:

Normal

Superconducting(e.g. Al grains)

Universal Interaction Hamiltonian

Triplet Channel:is NOT renormalized

But may lead to the spin ofThe ground state S > ½.

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

Universal Interaction Hamiltonian

Singlet Channel:is NOT renormalized

gate voltage

But

Q: What is charge degeneracy of the ground state

- half-integer Otherwise

degeneracygap

(isolated dot)

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

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)

Coulomb blockade (CB) (II)

Strong CB

Weak CB

Mesoscopic CB

(reflectionlesscontacts) Random phase but

not period.

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

Mesoscopic Coulomb Blockade

Based on technique suggested by: Matveev (1995); Furusaki, Matveev (1995);Flensberg (1993).

Aleiner, Glazman (1998)

Experiment:Cronenwett et. al. (1998)

SuppressionBy a factor of 5.3

Th: Predicted 4.

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)

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

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”)