Slide 1

Theory of Quantum Dots as
Zero-dimensional Metallic Systems
Igor Aleiner (Columbia)
Collaborators:
B.L.Altshuler (Princeton)
P.W.Brouwer (Cornell)
V.I.Falko (Lancaster, UK)
L.I. Glazman (Minnesota)
I.L. Kurland (Princeton)
Physics of the Microworld Conference, Oct. 16 (2004)

Outline:
•
•

Quantum dot (QD) as zero dimensional metal
Random Matrix theory for transport in quantum dots

a)
b)

Non-interacting “standard models”.
Peculiar spin-orbit effects in QD based on 2D electron gas.

•

Interaction effects:

a)
b)
c)

Universal interaction Hamiltonian;
Mesoscopic Stoner instability;
Coulomb blockade (strong, weak, mesoscopic);

d)

Kondo effect.

“Quantum dot” used in two different contents:

For the rest of Number
the talk:
of electrons:

1)

“Artificial atom”
Description requires exact diagonalization.
(Kouwnehoven group (Delft))

2)
“Artificial nucleus”

(Marcus group (Harvard))

Statistical description is allowed !!!

Random Matrix Theory for Transport in Quantum Dots

2DEG

QD

2DEG

Energy scales

Level spacing

L

Thouless Energy

Conductance
Assume:

Statistics of transport is determined
only 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:

RMT

Confinement, disorder, etc

No magnetic field, no SO

Magnetic field, no SO

No magnetic field, strong SO

Magnetic field + SO

Conductance of chaotic dot
classical

Mesoscopic
fluctuations

Weak localization
V

I

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

Conductance of chaotic dot
classical

Mesoscopic
fluctuations

Weak localization
V

I

Universal quantum corrections
[Altshuler, Shklovskii (1986)]

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 QD
Aleiner, Fal’ko (2001)

T - invariance

But

Spin dependent flux

Spin relaxation rate

Meir, Gefen, Entin-Wohlman (1989)
Mathur, Stone (1992)
Lyanda-Geller, Mirlin (1994)

Khaetskii, Nazarov (2000)

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 field
Observed in
Folk, Patel, Birnbaum, Marcus, Duruoz, Harris, Jr. (2001)

Interaction Hamiltonian
Energies smaller than Thouless energy:

Random matrix

????

In nuclear physics:

from shell model
random

Universal Interaction Hamiltonian
Energies smaller than Thouless energy:

Kurland, Aleiner, Altshuler (2000)

Only invariants compatible with the circular symmetry

are NOT random !!!

Universal Interaction Hamiltonian
Energies smaller than Thouless energy:

Random matrix
Not random
Valid if:
1)

2) Fundamental symmetries
are NOT broken at larger energies

Universal Interaction Hamiltonian
Energies smaller than Thouless energy:

One-particle levels
Interaction
with
Zero
dimensional
Fermi
liquid
determined by
additional conservations
Wigner – Dyson statistics
Valid if:
1)

2) Fundamental symmetries
are NOT broken at larger energies

Universal Interaction Hamiltonian

Singlet electron-hole
channel.
Triplet electron-hole
channel.

Particle-particle
(Cooper) channel.

Analogy with soft modes in metals

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 of
The ground state S > ½.

Mesoscopic Stoner Instability
Kurland, Aleiner, Altshuler (2000)

Energy of the ground state:

Spin is finite even for NO randomness
Typical
vs.S:

NO

interactions

Does not scale with the size of the system
FM instability
Stoner (1935)

Also Brouwer, Oreg, Halperin (2000)

random with known from RMT
correlation functions

Universal Interaction Hamiltonian

Singlet Channel:
gate voltage
is NOT renormalized
But

Q: What is charge degeneracy of the ground state

(isolated dot)

degeneracy

- half-integer

gap

Otherwise

Coulomb blockade of electron transport
For tunneling contacts:

Charge
degeneracy

Charge gap
Term introduced by Averin and Likharev (1986);
Effect first discussed by C.J. Gorter (1951).

Small quantum dots (~ 500 nm)

conductance (e2/h)

M. Kastner, Physics Today (1993)
E.B. Foxman et al., PRB (1993)

gate voltage (mV)

In metals first observed in Fulton, Dolan, PRL, 59, 109, (1987)

Coulomb blockade (CB) (II)
Strong CB
Weak CB

Mesoscopic CB
(reflectionless
contacts)

Random phase but
not period.

Statistical description of strong CB:
Theory:
Peaks: Jalabert, Stone, Alhassid (1992);
Valleys: Aleiner, Glazman (1996);
Reasonable agreement,
But problems with values of the correlation
fields

Courtesy of C.Marcus

Mesoscopic Coulomb Blockade

Aleiner, Glazman (1998)
Based on technique suggested by:
Matveev (1995); Furusaki, Matveev (1995);
Flensberg (1993).

Experiment:

Suppression
By a factor of 5.3
Th: Predicted 4.

Cronenwett et. al. (1998)

Even-Odd effect due to Kondo effect

Predicted:
Glazman, Raikh (1988)
Ng, Lee (1988)

Spin degeneracy in odd valleys:
Effective Hamiltonian:
local spin density
of conduction electrons

magnetic impurity

Observation:
1998

D. Goldhaber-Gordon et al. (MIT-Weizmann)
S.M. Cronenwett et al. (TU Delft)
J. Schmid et al. (MPI @ Stuttgart)

200 nm
15 mK

800 mK

van der Wiel et al. (2000)

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”)


Slide 2

Theory of Quantum Dots as
Zero-dimensional Metallic Systems
Igor Aleiner (Columbia)
Collaborators:
B.L.Altshuler (Princeton)
P.W.Brouwer (Cornell)
V.I.Falko (Lancaster, UK)
L.I. Glazman (Minnesota)
I.L. Kurland (Princeton)
Physics of the Microworld Conference, Oct. 16 (2004)

Outline:
•
•

Quantum dot (QD) as zero dimensional metal
Random Matrix theory for transport in quantum dots

a)
b)

Non-interacting “standard models”.
Peculiar spin-orbit effects in QD based on 2D electron gas.

•

Interaction effects:

a)
b)
c)

Universal interaction Hamiltonian;
Mesoscopic Stoner instability;
Coulomb blockade (strong, weak, mesoscopic);

d)

Kondo effect.

“Quantum dot” used in two different contents:

For the rest of Number
the talk:
of electrons:

1)

“Artificial atom”
Description requires exact diagonalization.
(Kouwnehoven group (Delft))

2)
“Artificial nucleus”

(Marcus group (Harvard))

Statistical description is allowed !!!

Random Matrix Theory for Transport in Quantum Dots

2DEG

QD

2DEG

Energy scales

Level spacing

L

Thouless Energy

Conductance
Assume:

Statistics of transport is determined
only 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:

RMT

Confinement, disorder, etc

No magnetic field, no SO

Magnetic field, no SO

No magnetic field, strong SO

Magnetic field + SO

Conductance of chaotic dot
classical

Mesoscopic
fluctuations

Weak localization
V

I

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

Conductance of chaotic dot
classical

Mesoscopic
fluctuations

Weak localization
V

I

Universal quantum corrections
[Altshuler, Shklovskii (1986)]

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 QD
Aleiner, Fal’ko (2001)

T - invariance

But

Spin dependent flux

Spin relaxation rate

Meir, Gefen, Entin-Wohlman (1989)
Mathur, Stone (1992)
Lyanda-Geller, Mirlin (1994)

Khaetskii, Nazarov (2000)

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 field
Observed in
Folk, Patel, Birnbaum, Marcus, Duruoz, Harris, Jr. (2001)

Interaction Hamiltonian
Energies smaller than Thouless energy:

Random matrix

????

In nuclear physics:

from shell model
random

Universal Interaction Hamiltonian
Energies smaller than Thouless energy:

Kurland, Aleiner, Altshuler (2000)

Only invariants compatible with the circular symmetry

are NOT random !!!

Universal Interaction Hamiltonian
Energies smaller than Thouless energy:

Random matrix
Not random
Valid if:
1)

2) Fundamental symmetries
are NOT broken at larger energies

Universal Interaction Hamiltonian
Energies smaller than Thouless energy:

One-particle levels
Interaction
with
Zero
dimensional
Fermi
liquid
determined by
additional conservations
Wigner – Dyson statistics
Valid if:
1)

2) Fundamental symmetries
are NOT broken at larger energies

Universal Interaction Hamiltonian

Singlet electron-hole
channel.
Triplet electron-hole
channel.

Particle-particle
(Cooper) channel.

Analogy with soft modes in metals

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 of
The ground state S > ½.

Mesoscopic Stoner Instability
Kurland, Aleiner, Altshuler (2000)

Energy of the ground state:

Spin is finite even for NO randomness
Typical
vs.S:

NO

interactions

Does not scale with the size of the system
FM instability
Stoner (1935)

Also Brouwer, Oreg, Halperin (2000)

random with known from RMT
correlation functions

Universal Interaction Hamiltonian

Singlet Channel:
gate voltage
is NOT renormalized
But

Q: What is charge degeneracy of the ground state

(isolated dot)

degeneracy

- half-integer

gap

Otherwise

Coulomb blockade of electron transport
For tunneling contacts:

Charge
degeneracy

Charge gap
Term introduced by Averin and Likharev (1986);
Effect first discussed by C.J. Gorter (1951).

Small quantum dots (~ 500 nm)

conductance (e2/h)

M. Kastner, Physics Today (1993)
E.B. Foxman et al., PRB (1993)

gate voltage (mV)

In metals first observed in Fulton, Dolan, PRL, 59, 109, (1987)

Coulomb blockade (CB) (II)
Strong CB
Weak CB

Mesoscopic CB
(reflectionless
contacts)

Random phase but
not period.

Statistical description of strong CB:
Theory:
Peaks: Jalabert, Stone, Alhassid (1992);
Valleys: Aleiner, Glazman (1996);
Reasonable agreement,
But problems with values of the correlation
fields

Courtesy of C.Marcus

Mesoscopic Coulomb Blockade

Aleiner, Glazman (1998)
Based on technique suggested by:
Matveev (1995); Furusaki, Matveev (1995);
Flensberg (1993).

Experiment:

Suppression
By a factor of 5.3
Th: Predicted 4.

Cronenwett et. al. (1998)

Even-Odd effect due to Kondo effect

Predicted:
Glazman, Raikh (1988)
Ng, Lee (1988)

Spin degeneracy in odd valleys:
Effective Hamiltonian:
local spin density
of conduction electrons

magnetic impurity

Observation:
1998

D. Goldhaber-Gordon et al. (MIT-Weizmann)
S.M. Cronenwett et al. (TU Delft)
J. Schmid et al. (MPI @ Stuttgart)

200 nm
15 mK

800 mK

van der Wiel et al. (2000)

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”)


Slide 3

Theory of Quantum Dots as
Zero-dimensional Metallic Systems
Igor Aleiner (Columbia)
Collaborators:
B.L.Altshuler (Princeton)
P.W.Brouwer (Cornell)
V.I.Falko (Lancaster, UK)
L.I. Glazman (Minnesota)
I.L. Kurland (Princeton)
Physics of the Microworld Conference, Oct. 16 (2004)

Outline:
•
•

Quantum dot (QD) as zero dimensional metal
Random Matrix theory for transport in quantum dots

a)
b)

Non-interacting “standard models”.
Peculiar spin-orbit effects in QD based on 2D electron gas.

•

Interaction effects:

a)
b)
c)

Universal interaction Hamiltonian;
Mesoscopic Stoner instability;
Coulomb blockade (strong, weak, mesoscopic);

d)

Kondo effect.

“Quantum dot” used in two different contents:

For the rest of Number
the talk:
of electrons:

1)

“Artificial atom”
Description requires exact diagonalization.
(Kouwnehoven group (Delft))

2)
“Artificial nucleus”

(Marcus group (Harvard))

Statistical description is allowed !!!

Random Matrix Theory for Transport in Quantum Dots

2DEG

QD

2DEG

Energy scales

Level spacing

L

Thouless Energy

Conductance
Assume:

Statistics of transport is determined
only 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:

RMT

Confinement, disorder, etc

No magnetic field, no SO

Magnetic field, no SO

No magnetic field, strong SO

Magnetic field + SO

Conductance of chaotic dot
classical

Mesoscopic
fluctuations

Weak localization
V

I

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

Conductance of chaotic dot
classical

Mesoscopic
fluctuations

Weak localization
V

I

Universal quantum corrections
[Altshuler, Shklovskii (1986)]

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 QD
Aleiner, Fal’ko (2001)

T - invariance

But

Spin dependent flux

Spin relaxation rate

Meir, Gefen, Entin-Wohlman (1989)
Mathur, Stone (1992)
Lyanda-Geller, Mirlin (1994)

Khaetskii, Nazarov (2000)

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 field
Observed in
Folk, Patel, Birnbaum, Marcus, Duruoz, Harris, Jr. (2001)

Interaction Hamiltonian
Energies smaller than Thouless energy:

Random matrix

????

In nuclear physics:

from shell model
random

Universal Interaction Hamiltonian
Energies smaller than Thouless energy:

Kurland, Aleiner, Altshuler (2000)

Only invariants compatible with the circular symmetry

are NOT random !!!

Universal Interaction Hamiltonian
Energies smaller than Thouless energy:

Random matrix
Not random
Valid if:
1)

2) Fundamental symmetries
are NOT broken at larger energies

Universal Interaction Hamiltonian
Energies smaller than Thouless energy:

One-particle levels
Interaction
with
Zero
dimensional
Fermi
liquid
determined by
additional conservations
Wigner – Dyson statistics
Valid if:
1)

2) Fundamental symmetries
are NOT broken at larger energies

Universal Interaction Hamiltonian

Singlet electron-hole
channel.
Triplet electron-hole
channel.

Particle-particle
(Cooper) channel.

Analogy with soft modes in metals

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 of
The ground state S > ½.

Mesoscopic Stoner Instability
Kurland, Aleiner, Altshuler (2000)

Energy of the ground state:

Spin is finite even for NO randomness
Typical
vs.S:

NO

interactions

Does not scale with the size of the system
FM instability
Stoner (1935)

Also Brouwer, Oreg, Halperin (2000)

random with known from RMT
correlation functions

Universal Interaction Hamiltonian

Singlet Channel:
gate voltage
is NOT renormalized
But

Q: What is charge degeneracy of the ground state

(isolated dot)

degeneracy

- half-integer

gap

Otherwise

Coulomb blockade of electron transport
For tunneling contacts:

Charge
degeneracy

Charge gap
Term introduced by Averin and Likharev (1986);
Effect first discussed by C.J. Gorter (1951).

Small quantum dots (~ 500 nm)

conductance (e2/h)

M. Kastner, Physics Today (1993)
E.B. Foxman et al., PRB (1993)

gate voltage (mV)

In metals first observed in Fulton, Dolan, PRL, 59, 109, (1987)

Coulomb blockade (CB) (II)
Strong CB
Weak CB

Mesoscopic CB
(reflectionless
contacts)

Random phase but
not period.

Statistical description of strong CB:
Theory:
Peaks: Jalabert, Stone, Alhassid (1992);
Valleys: Aleiner, Glazman (1996);
Reasonable agreement,
But problems with values of the correlation
fields

Courtesy of C.Marcus

Mesoscopic Coulomb Blockade

Aleiner, Glazman (1998)
Based on technique suggested by:
Matveev (1995); Furusaki, Matveev (1995);
Flensberg (1993).

Experiment:

Suppression
By a factor of 5.3
Th: Predicted 4.

Cronenwett et. al. (1998)

Even-Odd effect due to Kondo effect

Predicted:
Glazman, Raikh (1988)
Ng, Lee (1988)

Spin degeneracy in odd valleys:
Effective Hamiltonian:
local spin density
of conduction electrons

magnetic impurity

Observation:
1998

D. Goldhaber-Gordon et al. (MIT-Weizmann)
S.M. Cronenwett et al. (TU Delft)
J. Schmid et al. (MPI @ Stuttgart)

200 nm
15 mK

800 mK

van der Wiel et al. (2000)

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”)


Slide 4

Theory of Quantum Dots as
Zero-dimensional Metallic Systems
Igor Aleiner (Columbia)
Collaborators:
B.L.Altshuler (Princeton)
P.W.Brouwer (Cornell)
V.I.Falko (Lancaster, UK)
L.I. Glazman (Minnesota)
I.L. Kurland (Princeton)
Physics of the Microworld Conference, Oct. 16 (2004)

Outline:
•
•

Quantum dot (QD) as zero dimensional metal
Random Matrix theory for transport in quantum dots

a)
b)

Non-interacting “standard models”.
Peculiar spin-orbit effects in QD based on 2D electron gas.

•

Interaction effects:

a)
b)
c)

Universal interaction Hamiltonian;
Mesoscopic Stoner instability;
Coulomb blockade (strong, weak, mesoscopic);

d)

Kondo effect.

“Quantum dot” used in two different contents:

For the rest of Number
the talk:
of electrons:

1)

“Artificial atom”
Description requires exact diagonalization.
(Kouwnehoven group (Delft))

2)
“Artificial nucleus”

(Marcus group (Harvard))

Statistical description is allowed !!!

Random Matrix Theory for Transport in Quantum Dots

2DEG

QD

2DEG

Energy scales

Level spacing

L

Thouless Energy

Conductance
Assume:

Statistics of transport is determined
only 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:

RMT

Confinement, disorder, etc

No magnetic field, no SO

Magnetic field, no SO

No magnetic field, strong SO

Magnetic field + SO

Conductance of chaotic dot
classical

Mesoscopic
fluctuations

Weak localization
V

I

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

Conductance of chaotic dot
classical

Mesoscopic
fluctuations

Weak localization
V

I

Universal quantum corrections
[Altshuler, Shklovskii (1986)]

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 QD
Aleiner, Fal’ko (2001)

T - invariance

But

Spin dependent flux

Spin relaxation rate

Meir, Gefen, Entin-Wohlman (1989)
Mathur, Stone (1992)
Lyanda-Geller, Mirlin (1994)

Khaetskii, Nazarov (2000)

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 field
Observed in
Folk, Patel, Birnbaum, Marcus, Duruoz, Harris, Jr. (2001)

Interaction Hamiltonian
Energies smaller than Thouless energy:

Random matrix

????

In nuclear physics:

from shell model
random

Universal Interaction Hamiltonian
Energies smaller than Thouless energy:

Kurland, Aleiner, Altshuler (2000)

Only invariants compatible with the circular symmetry

are NOT random !!!

Universal Interaction Hamiltonian
Energies smaller than Thouless energy:

Random matrix
Not random
Valid if:
1)

2) Fundamental symmetries
are NOT broken at larger energies

Universal Interaction Hamiltonian
Energies smaller than Thouless energy:

One-particle levels
Interaction
with
Zero
dimensional
Fermi
liquid
determined by
additional conservations
Wigner – Dyson statistics
Valid if:
1)

2) Fundamental symmetries
are NOT broken at larger energies

Universal Interaction Hamiltonian

Singlet electron-hole
channel.
Triplet electron-hole
channel.

Particle-particle
(Cooper) channel.

Analogy with soft modes in metals

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 of
The ground state S > ½.

Mesoscopic Stoner Instability
Kurland, Aleiner, Altshuler (2000)

Energy of the ground state:

Spin is finite even for NO randomness
Typical
vs.S:

NO

interactions

Does not scale with the size of the system
FM instability
Stoner (1935)

Also Brouwer, Oreg, Halperin (2000)

random with known from RMT
correlation functions

Universal Interaction Hamiltonian

Singlet Channel:
gate voltage
is NOT renormalized
But

Q: What is charge degeneracy of the ground state

(isolated dot)

degeneracy

- half-integer

gap

Otherwise

Coulomb blockade of electron transport
For tunneling contacts:

Charge
degeneracy

Charge gap
Term introduced by Averin and Likharev (1986);
Effect first discussed by C.J. Gorter (1951).

Small quantum dots (~ 500 nm)

conductance (e2/h)

M. Kastner, Physics Today (1993)
E.B. Foxman et al., PRB (1993)

gate voltage (mV)

In metals first observed in Fulton, Dolan, PRL, 59, 109, (1987)

Coulomb blockade (CB) (II)
Strong CB
Weak CB

Mesoscopic CB
(reflectionless
contacts)

Random phase but
not period.

Statistical description of strong CB:
Theory:
Peaks: Jalabert, Stone, Alhassid (1992);
Valleys: Aleiner, Glazman (1996);
Reasonable agreement,
But problems with values of the correlation
fields

Courtesy of C.Marcus

Mesoscopic Coulomb Blockade

Aleiner, Glazman (1998)
Based on technique suggested by:
Matveev (1995); Furusaki, Matveev (1995);
Flensberg (1993).

Experiment:

Suppression
By a factor of 5.3
Th: Predicted 4.

Cronenwett et. al. (1998)

Even-Odd effect due to Kondo effect

Predicted:
Glazman, Raikh (1988)
Ng, Lee (1988)

Spin degeneracy in odd valleys:
Effective Hamiltonian:
local spin density
of conduction electrons

magnetic impurity

Observation:
1998

D. Goldhaber-Gordon et al. (MIT-Weizmann)
S.M. Cronenwett et al. (TU Delft)
J. Schmid et al. (MPI @ Stuttgart)

200 nm
15 mK

800 mK

van der Wiel et al. (2000)

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”)


Slide 5

Theory of Quantum Dots as
Zero-dimensional Metallic Systems
Igor Aleiner (Columbia)
Collaborators:
B.L.Altshuler (Princeton)
P.W.Brouwer (Cornell)
V.I.Falko (Lancaster, UK)
L.I. Glazman (Minnesota)
I.L. Kurland (Princeton)
Physics of the Microworld Conference, Oct. 16 (2004)

Outline:
•
•

Quantum dot (QD) as zero dimensional metal
Random Matrix theory for transport in quantum dots

a)
b)

Non-interacting “standard models”.
Peculiar spin-orbit effects in QD based on 2D electron gas.

•

Interaction effects:

a)
b)
c)

Universal interaction Hamiltonian;
Mesoscopic Stoner instability;
Coulomb blockade (strong, weak, mesoscopic);

d)

Kondo effect.

“Quantum dot” used in two different contents:

For the rest of Number
the talk:
of electrons:

1)

“Artificial atom”
Description requires exact diagonalization.
(Kouwnehoven group (Delft))

2)
“Artificial nucleus”

(Marcus group (Harvard))

Statistical description is allowed !!!

Random Matrix Theory for Transport in Quantum Dots

2DEG

QD

2DEG

Energy scales

Level spacing

L

Thouless Energy

Conductance
Assume:

Statistics of transport is determined
only 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:

RMT

Confinement, disorder, etc

No magnetic field, no SO

Magnetic field, no SO

No magnetic field, strong SO

Magnetic field + SO

Conductance of chaotic dot
classical

Mesoscopic
fluctuations

Weak localization
V

I

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

Conductance of chaotic dot
classical

Mesoscopic
fluctuations

Weak localization
V

I

Universal quantum corrections
[Altshuler, Shklovskii (1986)]

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 QD
Aleiner, Fal’ko (2001)

T - invariance

But

Spin dependent flux

Spin relaxation rate

Meir, Gefen, Entin-Wohlman (1989)
Mathur, Stone (1992)
Lyanda-Geller, Mirlin (1994)

Khaetskii, Nazarov (2000)

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 field
Observed in
Folk, Patel, Birnbaum, Marcus, Duruoz, Harris, Jr. (2001)

Interaction Hamiltonian
Energies smaller than Thouless energy:

Random matrix

????

In nuclear physics:

from shell model
random

Universal Interaction Hamiltonian
Energies smaller than Thouless energy:

Kurland, Aleiner, Altshuler (2000)

Only invariants compatible with the circular symmetry

are NOT random !!!

Universal Interaction Hamiltonian
Energies smaller than Thouless energy:

Random matrix
Not random
Valid if:
1)

2) Fundamental symmetries
are NOT broken at larger energies

Universal Interaction Hamiltonian
Energies smaller than Thouless energy:

One-particle levels
Interaction
with
Zero
dimensional
Fermi
liquid
determined by
additional conservations
Wigner – Dyson statistics
Valid if:
1)

2) Fundamental symmetries
are NOT broken at larger energies

Universal Interaction Hamiltonian

Singlet electron-hole
channel.
Triplet electron-hole
channel.

Particle-particle
(Cooper) channel.

Analogy with soft modes in metals

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 of
The ground state S > ½.

Mesoscopic Stoner Instability
Kurland, Aleiner, Altshuler (2000)

Energy of the ground state:

Spin is finite even for NO randomness
Typical
vs.S:

NO

interactions

Does not scale with the size of the system
FM instability
Stoner (1935)

Also Brouwer, Oreg, Halperin (2000)

random with known from RMT
correlation functions

Universal Interaction Hamiltonian

Singlet Channel:
gate voltage
is NOT renormalized
But

Q: What is charge degeneracy of the ground state

(isolated dot)

degeneracy

- half-integer

gap

Otherwise

Coulomb blockade of electron transport
For tunneling contacts:

Charge
degeneracy

Charge gap
Term introduced by Averin and Likharev (1986);
Effect first discussed by C.J. Gorter (1951).

Small quantum dots (~ 500 nm)

conductance (e2/h)

M. Kastner, Physics Today (1993)
E.B. Foxman et al., PRB (1993)

gate voltage (mV)

In metals first observed in Fulton, Dolan, PRL, 59, 109, (1987)

Coulomb blockade (CB) (II)
Strong CB
Weak CB

Mesoscopic CB
(reflectionless
contacts)

Random phase but
not period.

Statistical description of strong CB:
Theory:
Peaks: Jalabert, Stone, Alhassid (1992);
Valleys: Aleiner, Glazman (1996);
Reasonable agreement,
But problems with values of the correlation
fields

Courtesy of C.Marcus

Mesoscopic Coulomb Blockade

Aleiner, Glazman (1998)
Based on technique suggested by:
Matveev (1995); Furusaki, Matveev (1995);
Flensberg (1993).

Experiment:

Suppression
By a factor of 5.3
Th: Predicted 4.

Cronenwett et. al. (1998)

Even-Odd effect due to Kondo effect

Predicted:
Glazman, Raikh (1988)
Ng, Lee (1988)

Spin degeneracy in odd valleys:
Effective Hamiltonian:
local spin density
of conduction electrons

magnetic impurity

Observation:
1998

D. Goldhaber-Gordon et al. (MIT-Weizmann)
S.M. Cronenwett et al. (TU Delft)
J. Schmid et al. (MPI @ Stuttgart)

200 nm
15 mK

800 mK

van der Wiel et al. (2000)

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”)


Slide 6

Theory of Quantum Dots as
Zero-dimensional Metallic Systems
Igor Aleiner (Columbia)
Collaborators:
B.L.Altshuler (Princeton)
P.W.Brouwer (Cornell)
V.I.Falko (Lancaster, UK)
L.I. Glazman (Minnesota)
I.L. Kurland (Princeton)
Physics of the Microworld Conference, Oct. 16 (2004)

Outline:
•
•

Quantum dot (QD) as zero dimensional metal
Random Matrix theory for transport in quantum dots

a)
b)

Non-interacting “standard models”.
Peculiar spin-orbit effects in QD based on 2D electron gas.

•

Interaction effects:

a)
b)
c)

Universal interaction Hamiltonian;
Mesoscopic Stoner instability;
Coulomb blockade (strong, weak, mesoscopic);

d)

Kondo effect.

“Quantum dot” used in two different contents:

For the rest of Number
the talk:
of electrons:

1)

“Artificial atom”
Description requires exact diagonalization.
(Kouwnehoven group (Delft))

2)
“Artificial nucleus”

(Marcus group (Harvard))

Statistical description is allowed !!!

Random Matrix Theory for Transport in Quantum Dots

2DEG

QD

2DEG

Energy scales

Level spacing

L

Thouless Energy

Conductance
Assume:

Statistics of transport is determined
only 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:

RMT

Confinement, disorder, etc

No magnetic field, no SO

Magnetic field, no SO

No magnetic field, strong SO

Magnetic field + SO

Conductance of chaotic dot
classical

Mesoscopic
fluctuations

Weak localization
V

I

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

Conductance of chaotic dot
classical

Mesoscopic
fluctuations

Weak localization
V

I

Universal quantum corrections
[Altshuler, Shklovskii (1986)]

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 QD
Aleiner, Fal’ko (2001)

T - invariance

But

Spin dependent flux

Spin relaxation rate

Meir, Gefen, Entin-Wohlman (1989)
Mathur, Stone (1992)
Lyanda-Geller, Mirlin (1994)

Khaetskii, Nazarov (2000)

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 field
Observed in
Folk, Patel, Birnbaum, Marcus, Duruoz, Harris, Jr. (2001)

Interaction Hamiltonian
Energies smaller than Thouless energy:

Random matrix

????

In nuclear physics:

from shell model
random

Universal Interaction Hamiltonian
Energies smaller than Thouless energy:

Kurland, Aleiner, Altshuler (2000)

Only invariants compatible with the circular symmetry

are NOT random !!!

Universal Interaction Hamiltonian
Energies smaller than Thouless energy:

Random matrix
Not random
Valid if:
1)

2) Fundamental symmetries
are NOT broken at larger energies

Universal Interaction Hamiltonian
Energies smaller than Thouless energy:

One-particle levels
Interaction
with
Zero
dimensional
Fermi
liquid
determined by
additional conservations
Wigner – Dyson statistics
Valid if:
1)

2) Fundamental symmetries
are NOT broken at larger energies

Universal Interaction Hamiltonian

Singlet electron-hole
channel.
Triplet electron-hole
channel.

Particle-particle
(Cooper) channel.

Analogy with soft modes in metals

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 of
The ground state S > ½.

Mesoscopic Stoner Instability
Kurland, Aleiner, Altshuler (2000)

Energy of the ground state:

Spin is finite even for NO randomness
Typical
vs.S:

NO

interactions

Does not scale with the size of the system
FM instability
Stoner (1935)

Also Brouwer, Oreg, Halperin (2000)

random with known from RMT
correlation functions

Universal Interaction Hamiltonian

Singlet Channel:
gate voltage
is NOT renormalized
But

Q: What is charge degeneracy of the ground state

(isolated dot)

degeneracy

- half-integer

gap

Otherwise

Coulomb blockade of electron transport
For tunneling contacts:

Charge
degeneracy

Charge gap
Term introduced by Averin and Likharev (1986);
Effect first discussed by C.J. Gorter (1951).

Small quantum dots (~ 500 nm)

conductance (e2/h)

M. Kastner, Physics Today (1993)
E.B. Foxman et al., PRB (1993)

gate voltage (mV)

In metals first observed in Fulton, Dolan, PRL, 59, 109, (1987)

Coulomb blockade (CB) (II)
Strong CB
Weak CB

Mesoscopic CB
(reflectionless
contacts)

Random phase but
not period.

Statistical description of strong CB:
Theory:
Peaks: Jalabert, Stone, Alhassid (1992);
Valleys: Aleiner, Glazman (1996);
Reasonable agreement,
But problems with values of the correlation
fields

Courtesy of C.Marcus

Mesoscopic Coulomb Blockade

Aleiner, Glazman (1998)
Based on technique suggested by:
Matveev (1995); Furusaki, Matveev (1995);
Flensberg (1993).

Experiment:

Suppression
By a factor of 5.3
Th: Predicted 4.

Cronenwett et. al. (1998)

Even-Odd effect due to Kondo effect

Predicted:
Glazman, Raikh (1988)
Ng, Lee (1988)

Spin degeneracy in odd valleys:
Effective Hamiltonian:
local spin density
of conduction electrons

magnetic impurity

Observation:
1998

D. Goldhaber-Gordon et al. (MIT-Weizmann)
S.M. Cronenwett et al. (TU Delft)
J. Schmid et al. (MPI @ Stuttgart)

200 nm
15 mK

800 mK

van der Wiel et al. (2000)

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”)


Slide 7

Theory of Quantum Dots as
Zero-dimensional Metallic Systems
Igor Aleiner (Columbia)
Collaborators:
B.L.Altshuler (Princeton)
P.W.Brouwer (Cornell)
V.I.Falko (Lancaster, UK)
L.I. Glazman (Minnesota)
I.L. Kurland (Princeton)
Physics of the Microworld Conference, Oct. 16 (2004)

Outline:
•
•

Quantum dot (QD) as zero dimensional metal
Random Matrix theory for transport in quantum dots

a)
b)

Non-interacting “standard models”.
Peculiar spin-orbit effects in QD based on 2D electron gas.

•

Interaction effects:

a)
b)
c)

Universal interaction Hamiltonian;
Mesoscopic Stoner instability;
Coulomb blockade (strong, weak, mesoscopic);

d)

Kondo effect.

“Quantum dot” used in two different contents:

For the rest of Number
the talk:
of electrons:

1)

“Artificial atom”
Description requires exact diagonalization.
(Kouwnehoven group (Delft))

2)
“Artificial nucleus”

(Marcus group (Harvard))

Statistical description is allowed !!!

Random Matrix Theory for Transport in Quantum Dots

2DEG

QD

2DEG

Energy scales

Level spacing

L

Thouless Energy

Conductance
Assume:

Statistics of transport is determined
only 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:

RMT

Confinement, disorder, etc

No magnetic field, no SO

Magnetic field, no SO

No magnetic field, strong SO

Magnetic field + SO

Conductance of chaotic dot
classical

Mesoscopic
fluctuations

Weak localization
V

I

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

Conductance of chaotic dot
classical

Mesoscopic
fluctuations

Weak localization
V

I

Universal quantum corrections
[Altshuler, Shklovskii (1986)]

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 QD
Aleiner, Fal’ko (2001)

T - invariance

But

Spin dependent flux

Spin relaxation rate

Meir, Gefen, Entin-Wohlman (1989)
Mathur, Stone (1992)
Lyanda-Geller, Mirlin (1994)

Khaetskii, Nazarov (2000)

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 field
Observed in
Folk, Patel, Birnbaum, Marcus, Duruoz, Harris, Jr. (2001)

Interaction Hamiltonian
Energies smaller than Thouless energy:

Random matrix

????

In nuclear physics:

from shell model
random

Universal Interaction Hamiltonian
Energies smaller than Thouless energy:

Kurland, Aleiner, Altshuler (2000)

Only invariants compatible with the circular symmetry

are NOT random !!!

Universal Interaction Hamiltonian
Energies smaller than Thouless energy:

Random matrix
Not random
Valid if:
1)

2) Fundamental symmetries
are NOT broken at larger energies

Universal Interaction Hamiltonian
Energies smaller than Thouless energy:

One-particle levels
Interaction
with
Zero
dimensional
Fermi
liquid
determined by
additional conservations
Wigner – Dyson statistics
Valid if:
1)

2) Fundamental symmetries
are NOT broken at larger energies

Universal Interaction Hamiltonian

Singlet electron-hole
channel.
Triplet electron-hole
channel.

Particle-particle
(Cooper) channel.

Analogy with soft modes in metals

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 of
The ground state S > ½.

Mesoscopic Stoner Instability
Kurland, Aleiner, Altshuler (2000)

Energy of the ground state:

Spin is finite even for NO randomness
Typical
vs.S:

NO

interactions

Does not scale with the size of the system
FM instability
Stoner (1935)

Also Brouwer, Oreg, Halperin (2000)

random with known from RMT
correlation functions

Universal Interaction Hamiltonian

Singlet Channel:
gate voltage
is NOT renormalized
But

Q: What is charge degeneracy of the ground state

(isolated dot)

degeneracy

- half-integer

gap

Otherwise

Coulomb blockade of electron transport
For tunneling contacts:

Charge
degeneracy

Charge gap
Term introduced by Averin and Likharev (1986);
Effect first discussed by C.J. Gorter (1951).

Small quantum dots (~ 500 nm)

conductance (e2/h)

M. Kastner, Physics Today (1993)
E.B. Foxman et al., PRB (1993)

gate voltage (mV)

In metals first observed in Fulton, Dolan, PRL, 59, 109, (1987)

Coulomb blockade (CB) (II)
Strong CB
Weak CB

Mesoscopic CB
(reflectionless
contacts)

Random phase but
not period.

Statistical description of strong CB:
Theory:
Peaks: Jalabert, Stone, Alhassid (1992);
Valleys: Aleiner, Glazman (1996);
Reasonable agreement,
But problems with values of the correlation
fields

Courtesy of C.Marcus

Mesoscopic Coulomb Blockade

Aleiner, Glazman (1998)
Based on technique suggested by:
Matveev (1995); Furusaki, Matveev (1995);
Flensberg (1993).

Experiment:

Suppression
By a factor of 5.3
Th: Predicted 4.

Cronenwett et. al. (1998)

Even-Odd effect due to Kondo effect

Predicted:
Glazman, Raikh (1988)
Ng, Lee (1988)

Spin degeneracy in odd valleys:
Effective Hamiltonian:
local spin density
of conduction electrons

magnetic impurity

Observation:
1998

D. Goldhaber-Gordon et al. (MIT-Weizmann)
S.M. Cronenwett et al. (TU Delft)
J. Schmid et al. (MPI @ Stuttgart)

200 nm
15 mK

800 mK

van der Wiel et al. (2000)

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”)


Slide 8

Theory of Quantum Dots as
Zero-dimensional Metallic Systems
Igor Aleiner (Columbia)
Collaborators:
B.L.Altshuler (Princeton)
P.W.Brouwer (Cornell)
V.I.Falko (Lancaster, UK)
L.I. Glazman (Minnesota)
I.L. Kurland (Princeton)
Physics of the Microworld Conference, Oct. 16 (2004)

Outline:
•
•

Quantum dot (QD) as zero dimensional metal
Random Matrix theory for transport in quantum dots

a)
b)

Non-interacting “standard models”.
Peculiar spin-orbit effects in QD based on 2D electron gas.

•

Interaction effects:

a)
b)
c)

Universal interaction Hamiltonian;
Mesoscopic Stoner instability;
Coulomb blockade (strong, weak, mesoscopic);

d)

Kondo effect.

“Quantum dot” used in two different contents:

For the rest of Number
the talk:
of electrons:

1)

“Artificial atom”
Description requires exact diagonalization.
(Kouwnehoven group (Delft))

2)
“Artificial nucleus”

(Marcus group (Harvard))

Statistical description is allowed !!!

Random Matrix Theory for Transport in Quantum Dots

2DEG

QD

2DEG

Energy scales

Level spacing

L

Thouless Energy

Conductance
Assume:

Statistics of transport is determined
only 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:

RMT

Confinement, disorder, etc

No magnetic field, no SO

Magnetic field, no SO

No magnetic field, strong SO

Magnetic field + SO

Conductance of chaotic dot
classical

Mesoscopic
fluctuations

Weak localization
V

I

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

Conductance of chaotic dot
classical

Mesoscopic
fluctuations

Weak localization
V

I

Universal quantum corrections
[Altshuler, Shklovskii (1986)]

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 QD
Aleiner, Fal’ko (2001)

T - invariance

But

Spin dependent flux

Spin relaxation rate

Meir, Gefen, Entin-Wohlman (1989)
Mathur, Stone (1992)
Lyanda-Geller, Mirlin (1994)

Khaetskii, Nazarov (2000)

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 field
Observed in
Folk, Patel, Birnbaum, Marcus, Duruoz, Harris, Jr. (2001)

Interaction Hamiltonian
Energies smaller than Thouless energy:

Random matrix

????

In nuclear physics:

from shell model
random

Universal Interaction Hamiltonian
Energies smaller than Thouless energy:

Kurland, Aleiner, Altshuler (2000)

Only invariants compatible with the circular symmetry

are NOT random !!!

Universal Interaction Hamiltonian
Energies smaller than Thouless energy:

Random matrix
Not random
Valid if:
1)

2) Fundamental symmetries
are NOT broken at larger energies

Universal Interaction Hamiltonian
Energies smaller than Thouless energy:

One-particle levels
Interaction
with
Zero
dimensional
Fermi
liquid
determined by
additional conservations
Wigner – Dyson statistics
Valid if:
1)

2) Fundamental symmetries
are NOT broken at larger energies

Universal Interaction Hamiltonian

Singlet electron-hole
channel.
Triplet electron-hole
channel.

Particle-particle
(Cooper) channel.

Analogy with soft modes in metals

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 of
The ground state S > ½.

Mesoscopic Stoner Instability
Kurland, Aleiner, Altshuler (2000)

Energy of the ground state:

Spin is finite even for NO randomness
Typical
vs.S:

NO

interactions

Does not scale with the size of the system
FM instability
Stoner (1935)

Also Brouwer, Oreg, Halperin (2000)

random with known from RMT
correlation functions

Universal Interaction Hamiltonian

Singlet Channel:
gate voltage
is NOT renormalized
But

Q: What is charge degeneracy of the ground state

(isolated dot)

degeneracy

- half-integer

gap

Otherwise

Coulomb blockade of electron transport
For tunneling contacts:

Charge
degeneracy

Charge gap
Term introduced by Averin and Likharev (1986);
Effect first discussed by C.J. Gorter (1951).

Small quantum dots (~ 500 nm)

conductance (e2/h)

M. Kastner, Physics Today (1993)
E.B. Foxman et al., PRB (1993)

gate voltage (mV)

In metals first observed in Fulton, Dolan, PRL, 59, 109, (1987)

Coulomb blockade (CB) (II)
Strong CB
Weak CB

Mesoscopic CB
(reflectionless
contacts)

Random phase but
not period.

Statistical description of strong CB:
Theory:
Peaks: Jalabert, Stone, Alhassid (1992);
Valleys: Aleiner, Glazman (1996);
Reasonable agreement,
But problems with values of the correlation
fields

Courtesy of C.Marcus

Mesoscopic Coulomb Blockade

Aleiner, Glazman (1998)
Based on technique suggested by:
Matveev (1995); Furusaki, Matveev (1995);
Flensberg (1993).

Experiment:

Suppression
By a factor of 5.3
Th: Predicted 4.

Cronenwett et. al. (1998)

Even-Odd effect due to Kondo effect

Predicted:
Glazman, Raikh (1988)
Ng, Lee (1988)

Spin degeneracy in odd valleys:
Effective Hamiltonian:
local spin density
of conduction electrons

magnetic impurity

Observation:
1998

D. Goldhaber-Gordon et al. (MIT-Weizmann)
S.M. Cronenwett et al. (TU Delft)
J. Schmid et al. (MPI @ Stuttgart)

200 nm
15 mK

800 mK

van der Wiel et al. (2000)

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”)


Slide 9

Theory of Quantum Dots as
Zero-dimensional Metallic Systems
Igor Aleiner (Columbia)
Collaborators:
B.L.Altshuler (Princeton)
P.W.Brouwer (Cornell)
V.I.Falko (Lancaster, UK)
L.I. Glazman (Minnesota)
I.L. Kurland (Princeton)
Physics of the Microworld Conference, Oct. 16 (2004)

Outline:
•
•

Quantum dot (QD) as zero dimensional metal
Random Matrix theory for transport in quantum dots

a)
b)

Non-interacting “standard models”.
Peculiar spin-orbit effects in QD based on 2D electron gas.

•

Interaction effects:

a)
b)
c)

Universal interaction Hamiltonian;
Mesoscopic Stoner instability;
Coulomb blockade (strong, weak, mesoscopic);

d)

Kondo effect.

“Quantum dot” used in two different contents:

For the rest of Number
the talk:
of electrons:

1)

“Artificial atom”
Description requires exact diagonalization.
(Kouwnehoven group (Delft))

2)
“Artificial nucleus”

(Marcus group (Harvard))

Statistical description is allowed !!!

Random Matrix Theory for Transport in Quantum Dots

2DEG

QD

2DEG

Energy scales

Level spacing

L

Thouless Energy

Conductance
Assume:

Statistics of transport is determined
only 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:

RMT

Confinement, disorder, etc

No magnetic field, no SO

Magnetic field, no SO

No magnetic field, strong SO

Magnetic field + SO

Conductance of chaotic dot
classical

Mesoscopic
fluctuations

Weak localization
V

I

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

Conductance of chaotic dot
classical

Mesoscopic
fluctuations

Weak localization
V

I

Universal quantum corrections
[Altshuler, Shklovskii (1986)]

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 QD
Aleiner, Fal’ko (2001)

T - invariance

But

Spin dependent flux

Spin relaxation rate

Meir, Gefen, Entin-Wohlman (1989)
Mathur, Stone (1992)
Lyanda-Geller, Mirlin (1994)

Khaetskii, Nazarov (2000)

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 field
Observed in
Folk, Patel, Birnbaum, Marcus, Duruoz, Harris, Jr. (2001)

Interaction Hamiltonian
Energies smaller than Thouless energy:

Random matrix

????

In nuclear physics:

from shell model
random

Universal Interaction Hamiltonian
Energies smaller than Thouless energy:

Kurland, Aleiner, Altshuler (2000)

Only invariants compatible with the circular symmetry

are NOT random !!!

Universal Interaction Hamiltonian
Energies smaller than Thouless energy:

Random matrix
Not random
Valid if:
1)

2) Fundamental symmetries
are NOT broken at larger energies

Universal Interaction Hamiltonian
Energies smaller than Thouless energy:

One-particle levels
Interaction
with
Zero
dimensional
Fermi
liquid
determined by
additional conservations
Wigner – Dyson statistics
Valid if:
1)

2) Fundamental symmetries
are NOT broken at larger energies

Universal Interaction Hamiltonian

Singlet electron-hole
channel.
Triplet electron-hole
channel.

Particle-particle
(Cooper) channel.

Analogy with soft modes in metals

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 of
The ground state S > ½.

Mesoscopic Stoner Instability
Kurland, Aleiner, Altshuler (2000)

Energy of the ground state:

Spin is finite even for NO randomness
Typical
vs.S:

NO

interactions

Does not scale with the size of the system
FM instability
Stoner (1935)

Also Brouwer, Oreg, Halperin (2000)

random with known from RMT
correlation functions

Universal Interaction Hamiltonian

Singlet Channel:
gate voltage
is NOT renormalized
But

Q: What is charge degeneracy of the ground state

(isolated dot)

degeneracy

- half-integer

gap

Otherwise

Coulomb blockade of electron transport
For tunneling contacts:

Charge
degeneracy

Charge gap
Term introduced by Averin and Likharev (1986);
Effect first discussed by C.J. Gorter (1951).

Small quantum dots (~ 500 nm)

conductance (e2/h)

M. Kastner, Physics Today (1993)
E.B. Foxman et al., PRB (1993)

gate voltage (mV)

In metals first observed in Fulton, Dolan, PRL, 59, 109, (1987)

Coulomb blockade (CB) (II)
Strong CB
Weak CB

Mesoscopic CB
(reflectionless
contacts)

Random phase but
not period.

Statistical description of strong CB:
Theory:
Peaks: Jalabert, Stone, Alhassid (1992);
Valleys: Aleiner, Glazman (1996);
Reasonable agreement,
But problems with values of the correlation
fields

Courtesy of C.Marcus

Mesoscopic Coulomb Blockade

Aleiner, Glazman (1998)
Based on technique suggested by:
Matveev (1995); Furusaki, Matveev (1995);
Flensberg (1993).

Experiment:

Suppression
By a factor of 5.3
Th: Predicted 4.

Cronenwett et. al. (1998)

Even-Odd effect due to Kondo effect

Predicted:
Glazman, Raikh (1988)
Ng, Lee (1988)

Spin degeneracy in odd valleys:
Effective Hamiltonian:
local spin density
of conduction electrons

magnetic impurity

Observation:
1998

D. Goldhaber-Gordon et al. (MIT-Weizmann)
S.M. Cronenwett et al. (TU Delft)
J. Schmid et al. (MPI @ Stuttgart)

200 nm
15 mK

800 mK

van der Wiel et al. (2000)

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”)


Slide 10

Theory of Quantum Dots as
Zero-dimensional Metallic Systems
Igor Aleiner (Columbia)
Collaborators:
B.L.Altshuler (Princeton)
P.W.Brouwer (Cornell)
V.I.Falko (Lancaster, UK)
L.I. Glazman (Minnesota)
I.L. Kurland (Princeton)
Physics of the Microworld Conference, Oct. 16 (2004)

Outline:
•
•

Quantum dot (QD) as zero dimensional metal
Random Matrix theory for transport in quantum dots

a)
b)

Non-interacting “standard models”.
Peculiar spin-orbit effects in QD based on 2D electron gas.

•

Interaction effects:

a)
b)
c)

Universal interaction Hamiltonian;
Mesoscopic Stoner instability;
Coulomb blockade (strong, weak, mesoscopic);

d)

Kondo effect.

“Quantum dot” used in two different contents:

For the rest of Number
the talk:
of electrons:

1)

“Artificial atom”
Description requires exact diagonalization.
(Kouwnehoven group (Delft))

2)
“Artificial nucleus”

(Marcus group (Harvard))

Statistical description is allowed !!!

Random Matrix Theory for Transport in Quantum Dots

2DEG

QD

2DEG

Energy scales

Level spacing

L

Thouless Energy

Conductance
Assume:

Statistics of transport is determined
only 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:

RMT

Confinement, disorder, etc

No magnetic field, no SO

Magnetic field, no SO

No magnetic field, strong SO

Magnetic field + SO

Conductance of chaotic dot
classical

Mesoscopic
fluctuations

Weak localization
V

I

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

Conductance of chaotic dot
classical

Mesoscopic
fluctuations

Weak localization
V

I

Universal quantum corrections
[Altshuler, Shklovskii (1986)]

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 QD
Aleiner, Fal’ko (2001)

T - invariance

But

Spin dependent flux

Spin relaxation rate

Meir, Gefen, Entin-Wohlman (1989)
Mathur, Stone (1992)
Lyanda-Geller, Mirlin (1994)

Khaetskii, Nazarov (2000)

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 field
Observed in
Folk, Patel, Birnbaum, Marcus, Duruoz, Harris, Jr. (2001)

Interaction Hamiltonian
Energies smaller than Thouless energy:

Random matrix

????

In nuclear physics:

from shell model
random

Universal Interaction Hamiltonian
Energies smaller than Thouless energy:

Kurland, Aleiner, Altshuler (2000)

Only invariants compatible with the circular symmetry

are NOT random !!!

Universal Interaction Hamiltonian
Energies smaller than Thouless energy:

Random matrix
Not random
Valid if:
1)

2) Fundamental symmetries
are NOT broken at larger energies

Universal Interaction Hamiltonian
Energies smaller than Thouless energy:

One-particle levels
Interaction
with
Zero
dimensional
Fermi
liquid
determined by
additional conservations
Wigner – Dyson statistics
Valid if:
1)

2) Fundamental symmetries
are NOT broken at larger energies

Universal Interaction Hamiltonian

Singlet electron-hole
channel.
Triplet electron-hole
channel.

Particle-particle
(Cooper) channel.

Analogy with soft modes in metals

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 of
The ground state S > ½.

Mesoscopic Stoner Instability
Kurland, Aleiner, Altshuler (2000)

Energy of the ground state:

Spin is finite even for NO randomness
Typical
vs.S:

NO

interactions

Does not scale with the size of the system
FM instability
Stoner (1935)

Also Brouwer, Oreg, Halperin (2000)

random with known from RMT
correlation functions

Universal Interaction Hamiltonian

Singlet Channel:
gate voltage
is NOT renormalized
But

Q: What is charge degeneracy of the ground state

(isolated dot)

degeneracy

- half-integer

gap

Otherwise

Coulomb blockade of electron transport
For tunneling contacts:

Charge
degeneracy

Charge gap
Term introduced by Averin and Likharev (1986);
Effect first discussed by C.J. Gorter (1951).

Small quantum dots (~ 500 nm)

conductance (e2/h)

M. Kastner, Physics Today (1993)
E.B. Foxman et al., PRB (1993)

gate voltage (mV)

In metals first observed in Fulton, Dolan, PRL, 59, 109, (1987)

Coulomb blockade (CB) (II)
Strong CB
Weak CB

Mesoscopic CB
(reflectionless
contacts)

Random phase but
not period.

Statistical description of strong CB:
Theory:
Peaks: Jalabert, Stone, Alhassid (1992);
Valleys: Aleiner, Glazman (1996);
Reasonable agreement,
But problems with values of the correlation
fields

Courtesy of C.Marcus

Mesoscopic Coulomb Blockade

Aleiner, Glazman (1998)
Based on technique suggested by:
Matveev (1995); Furusaki, Matveev (1995);
Flensberg (1993).

Experiment:

Suppression
By a factor of 5.3
Th: Predicted 4.

Cronenwett et. al. (1998)

Even-Odd effect due to Kondo effect

Predicted:
Glazman, Raikh (1988)
Ng, Lee (1988)

Spin degeneracy in odd valleys:
Effective Hamiltonian:
local spin density
of conduction electrons

magnetic impurity

Observation:
1998

D. Goldhaber-Gordon et al. (MIT-Weizmann)
S.M. Cronenwett et al. (TU Delft)
J. Schmid et al. (MPI @ Stuttgart)

200 nm
15 mK

800 mK

van der Wiel et al. (2000)

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”)


Slide 11

Theory of Quantum Dots as
Zero-dimensional Metallic Systems
Igor Aleiner (Columbia)
Collaborators:
B.L.Altshuler (Princeton)
P.W.Brouwer (Cornell)
V.I.Falko (Lancaster, UK)
L.I. Glazman (Minnesota)
I.L. Kurland (Princeton)
Physics of the Microworld Conference, Oct. 16 (2004)

Outline:
•
•

Quantum dot (QD) as zero dimensional metal
Random Matrix theory for transport in quantum dots

a)
b)

Non-interacting “standard models”.
Peculiar spin-orbit effects in QD based on 2D electron gas.

•

Interaction effects:

a)
b)
c)

Universal interaction Hamiltonian;
Mesoscopic Stoner instability;
Coulomb blockade (strong, weak, mesoscopic);

d)

Kondo effect.

“Quantum dot” used in two different contents:

For the rest of Number
the talk:
of electrons:

1)

“Artificial atom”
Description requires exact diagonalization.
(Kouwnehoven group (Delft))

2)
“Artificial nucleus”

(Marcus group (Harvard))

Statistical description is allowed !!!

Random Matrix Theory for Transport in Quantum Dots

2DEG

QD

2DEG

Energy scales

Level spacing

L

Thouless Energy

Conductance
Assume:

Statistics of transport is determined
only 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:

RMT

Confinement, disorder, etc

No magnetic field, no SO

Magnetic field, no SO

No magnetic field, strong SO

Magnetic field + SO

Conductance of chaotic dot
classical

Mesoscopic
fluctuations

Weak localization
V

I

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

Conductance of chaotic dot
classical

Mesoscopic
fluctuations

Weak localization
V

I

Universal quantum corrections
[Altshuler, Shklovskii (1986)]

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 QD
Aleiner, Fal’ko (2001)

T - invariance

But

Spin dependent flux

Spin relaxation rate

Meir, Gefen, Entin-Wohlman (1989)
Mathur, Stone (1992)
Lyanda-Geller, Mirlin (1994)

Khaetskii, Nazarov (2000)

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 field
Observed in
Folk, Patel, Birnbaum, Marcus, Duruoz, Harris, Jr. (2001)

Interaction Hamiltonian
Energies smaller than Thouless energy:

Random matrix

????

In nuclear physics:

from shell model
random

Universal Interaction Hamiltonian
Energies smaller than Thouless energy:

Kurland, Aleiner, Altshuler (2000)

Only invariants compatible with the circular symmetry

are NOT random !!!

Universal Interaction Hamiltonian
Energies smaller than Thouless energy:

Random matrix
Not random
Valid if:
1)

2) Fundamental symmetries
are NOT broken at larger energies

Universal Interaction Hamiltonian
Energies smaller than Thouless energy:

One-particle levels
Interaction
with
Zero
dimensional
Fermi
liquid
determined by
additional conservations
Wigner – Dyson statistics
Valid if:
1)

2) Fundamental symmetries
are NOT broken at larger energies

Universal Interaction Hamiltonian

Singlet electron-hole
channel.
Triplet electron-hole
channel.

Particle-particle
(Cooper) channel.

Analogy with soft modes in metals

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 of
The ground state S > ½.

Mesoscopic Stoner Instability
Kurland, Aleiner, Altshuler (2000)

Energy of the ground state:

Spin is finite even for NO randomness
Typical
vs.S:

NO

interactions

Does not scale with the size of the system
FM instability
Stoner (1935)

Also Brouwer, Oreg, Halperin (2000)

random with known from RMT
correlation functions

Universal Interaction Hamiltonian

Singlet Channel:
gate voltage
is NOT renormalized
But

Q: What is charge degeneracy of the ground state

(isolated dot)

degeneracy

- half-integer

gap

Otherwise

Coulomb blockade of electron transport
For tunneling contacts:

Charge
degeneracy

Charge gap
Term introduced by Averin and Likharev (1986);
Effect first discussed by C.J. Gorter (1951).

Small quantum dots (~ 500 nm)

conductance (e2/h)

M. Kastner, Physics Today (1993)
E.B. Foxman et al., PRB (1993)

gate voltage (mV)

In metals first observed in Fulton, Dolan, PRL, 59, 109, (1987)

Coulomb blockade (CB) (II)
Strong CB
Weak CB

Mesoscopic CB
(reflectionless
contacts)

Random phase but
not period.

Statistical description of strong CB:
Theory:
Peaks: Jalabert, Stone, Alhassid (1992);
Valleys: Aleiner, Glazman (1996);
Reasonable agreement,
But problems with values of the correlation
fields

Courtesy of C.Marcus

Mesoscopic Coulomb Blockade

Aleiner, Glazman (1998)
Based on technique suggested by:
Matveev (1995); Furusaki, Matveev (1995);
Flensberg (1993).

Experiment:

Suppression
By a factor of 5.3
Th: Predicted 4.

Cronenwett et. al. (1998)

Even-Odd effect due to Kondo effect

Predicted:
Glazman, Raikh (1988)
Ng, Lee (1988)

Spin degeneracy in odd valleys:
Effective Hamiltonian:
local spin density
of conduction electrons

magnetic impurity

Observation:
1998

D. Goldhaber-Gordon et al. (MIT-Weizmann)
S.M. Cronenwett et al. (TU Delft)
J. Schmid et al. (MPI @ Stuttgart)

200 nm
15 mK

800 mK

van der Wiel et al. (2000)

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”)


Slide 12

Theory of Quantum Dots as
Zero-dimensional Metallic Systems
Igor Aleiner (Columbia)
Collaborators:
B.L.Altshuler (Princeton)
P.W.Brouwer (Cornell)
V.I.Falko (Lancaster, UK)
L.I. Glazman (Minnesota)
I.L. Kurland (Princeton)
Physics of the Microworld Conference, Oct. 16 (2004)

Outline:
•
•

Quantum dot (QD) as zero dimensional metal
Random Matrix theory for transport in quantum dots

a)
b)

Non-interacting “standard models”.
Peculiar spin-orbit effects in QD based on 2D electron gas.

•

Interaction effects:

a)
b)
c)

Universal interaction Hamiltonian;
Mesoscopic Stoner instability;
Coulomb blockade (strong, weak, mesoscopic);

d)

Kondo effect.

“Quantum dot” used in two different contents:

For the rest of Number
the talk:
of electrons:

1)

“Artificial atom”
Description requires exact diagonalization.
(Kouwnehoven group (Delft))

2)
“Artificial nucleus”

(Marcus group (Harvard))

Statistical description is allowed !!!

Random Matrix Theory for Transport in Quantum Dots

2DEG

QD

2DEG

Energy scales

Level spacing

L

Thouless Energy

Conductance
Assume:

Statistics of transport is determined
only 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:

RMT

Confinement, disorder, etc

No magnetic field, no SO

Magnetic field, no SO

No magnetic field, strong SO

Magnetic field + SO

Conductance of chaotic dot
classical

Mesoscopic
fluctuations

Weak localization
V

I

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

Conductance of chaotic dot
classical

Mesoscopic
fluctuations

Weak localization
V

I

Universal quantum corrections
[Altshuler, Shklovskii (1986)]

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 QD
Aleiner, Fal’ko (2001)

T - invariance

But

Spin dependent flux

Spin relaxation rate

Meir, Gefen, Entin-Wohlman (1989)
Mathur, Stone (1992)
Lyanda-Geller, Mirlin (1994)

Khaetskii, Nazarov (2000)

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 field
Observed in
Folk, Patel, Birnbaum, Marcus, Duruoz, Harris, Jr. (2001)

Interaction Hamiltonian
Energies smaller than Thouless energy:

Random matrix

????

In nuclear physics:

from shell model
random

Universal Interaction Hamiltonian
Energies smaller than Thouless energy:

Kurland, Aleiner, Altshuler (2000)

Only invariants compatible with the circular symmetry

are NOT random !!!

Universal Interaction Hamiltonian
Energies smaller than Thouless energy:

Random matrix
Not random
Valid if:
1)

2) Fundamental symmetries
are NOT broken at larger energies

Universal Interaction Hamiltonian
Energies smaller than Thouless energy:

One-particle levels
Interaction
with
Zero
dimensional
Fermi
liquid
determined by
additional conservations
Wigner – Dyson statistics
Valid if:
1)

2) Fundamental symmetries
are NOT broken at larger energies

Universal Interaction Hamiltonian

Singlet electron-hole
channel.
Triplet electron-hole
channel.

Particle-particle
(Cooper) channel.

Analogy with soft modes in metals

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 of
The ground state S > ½.

Mesoscopic Stoner Instability
Kurland, Aleiner, Altshuler (2000)

Energy of the ground state:

Spin is finite even for NO randomness
Typical
vs.S:

NO

interactions

Does not scale with the size of the system
FM instability
Stoner (1935)

Also Brouwer, Oreg, Halperin (2000)

random with known from RMT
correlation functions

Universal Interaction Hamiltonian

Singlet Channel:
gate voltage
is NOT renormalized
But

Q: What is charge degeneracy of the ground state

(isolated dot)

degeneracy

- half-integer

gap

Otherwise

Coulomb blockade of electron transport
For tunneling contacts:

Charge
degeneracy

Charge gap
Term introduced by Averin and Likharev (1986);
Effect first discussed by C.J. Gorter (1951).

Small quantum dots (~ 500 nm)

conductance (e2/h)

M. Kastner, Physics Today (1993)
E.B. Foxman et al., PRB (1993)

gate voltage (mV)

In metals first observed in Fulton, Dolan, PRL, 59, 109, (1987)

Coulomb blockade (CB) (II)
Strong CB
Weak CB

Mesoscopic CB
(reflectionless
contacts)

Random phase but
not period.

Statistical description of strong CB:
Theory:
Peaks: Jalabert, Stone, Alhassid (1992);
Valleys: Aleiner, Glazman (1996);
Reasonable agreement,
But problems with values of the correlation
fields

Courtesy of C.Marcus

Mesoscopic Coulomb Blockade

Aleiner, Glazman (1998)
Based on technique suggested by:
Matveev (1995); Furusaki, Matveev (1995);
Flensberg (1993).

Experiment:

Suppression
By a factor of 5.3
Th: Predicted 4.

Cronenwett et. al. (1998)

Even-Odd effect due to Kondo effect

Predicted:
Glazman, Raikh (1988)
Ng, Lee (1988)

Spin degeneracy in odd valleys:
Effective Hamiltonian:
local spin density
of conduction electrons

magnetic impurity

Observation:
1998

D. Goldhaber-Gordon et al. (MIT-Weizmann)
S.M. Cronenwett et al. (TU Delft)
J. Schmid et al. (MPI @ Stuttgart)

200 nm
15 mK

800 mK

van der Wiel et al. (2000)

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”)


Slide 13

Theory of Quantum Dots as
Zero-dimensional Metallic Systems
Igor Aleiner (Columbia)
Collaborators:
B.L.Altshuler (Princeton)
P.W.Brouwer (Cornell)
V.I.Falko (Lancaster, UK)
L.I. Glazman (Minnesota)
I.L. Kurland (Princeton)
Physics of the Microworld Conference, Oct. 16 (2004)

Outline:
•
•

Quantum dot (QD) as zero dimensional metal
Random Matrix theory for transport in quantum dots

a)
b)

Non-interacting “standard models”.
Peculiar spin-orbit effects in QD based on 2D electron gas.

•

Interaction effects:

a)
b)
c)

Universal interaction Hamiltonian;
Mesoscopic Stoner instability;
Coulomb blockade (strong, weak, mesoscopic);

d)

Kondo effect.

“Quantum dot” used in two different contents:

For the rest of Number
the talk:
of electrons:

1)

“Artificial atom”
Description requires exact diagonalization.
(Kouwnehoven group (Delft))

2)
“Artificial nucleus”

(Marcus group (Harvard))

Statistical description is allowed !!!

Random Matrix Theory for Transport in Quantum Dots

2DEG

QD

2DEG

Energy scales

Level spacing

L

Thouless Energy

Conductance
Assume:

Statistics of transport is determined
only 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:

RMT

Confinement, disorder, etc

No magnetic field, no SO

Magnetic field, no SO

No magnetic field, strong SO

Magnetic field + SO

Conductance of chaotic dot
classical

Mesoscopic
fluctuations

Weak localization
V

I

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

Conductance of chaotic dot
classical

Mesoscopic
fluctuations

Weak localization
V

I

Universal quantum corrections
[Altshuler, Shklovskii (1986)]

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 QD
Aleiner, Fal’ko (2001)

T - invariance

But

Spin dependent flux

Spin relaxation rate

Meir, Gefen, Entin-Wohlman (1989)
Mathur, Stone (1992)
Lyanda-Geller, Mirlin (1994)

Khaetskii, Nazarov (2000)

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 field
Observed in
Folk, Patel, Birnbaum, Marcus, Duruoz, Harris, Jr. (2001)

Interaction Hamiltonian
Energies smaller than Thouless energy:

Random matrix

????

In nuclear physics:

from shell model
random

Universal Interaction Hamiltonian
Energies smaller than Thouless energy:

Kurland, Aleiner, Altshuler (2000)

Only invariants compatible with the circular symmetry

are NOT random !!!

Universal Interaction Hamiltonian
Energies smaller than Thouless energy:

Random matrix
Not random
Valid if:
1)

2) Fundamental symmetries
are NOT broken at larger energies

Universal Interaction Hamiltonian
Energies smaller than Thouless energy:

One-particle levels
Interaction
with
Zero
dimensional
Fermi
liquid
determined by
additional conservations
Wigner – Dyson statistics
Valid if:
1)

2) Fundamental symmetries
are NOT broken at larger energies

Universal Interaction Hamiltonian

Singlet electron-hole
channel.
Triplet electron-hole
channel.

Particle-particle
(Cooper) channel.

Analogy with soft modes in metals

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 of
The ground state S > ½.

Mesoscopic Stoner Instability
Kurland, Aleiner, Altshuler (2000)

Energy of the ground state:

Spin is finite even for NO randomness
Typical
vs.S:

NO

interactions

Does not scale with the size of the system
FM instability
Stoner (1935)

Also Brouwer, Oreg, Halperin (2000)

random with known from RMT
correlation functions

Universal Interaction Hamiltonian

Singlet Channel:
gate voltage
is NOT renormalized
But

Q: What is charge degeneracy of the ground state

(isolated dot)

degeneracy

- half-integer

gap

Otherwise

Coulomb blockade of electron transport
For tunneling contacts:

Charge
degeneracy

Charge gap
Term introduced by Averin and Likharev (1986);
Effect first discussed by C.J. Gorter (1951).

Small quantum dots (~ 500 nm)

conductance (e2/h)

M. Kastner, Physics Today (1993)
E.B. Foxman et al., PRB (1993)

gate voltage (mV)

In metals first observed in Fulton, Dolan, PRL, 59, 109, (1987)

Coulomb blockade (CB) (II)
Strong CB
Weak CB

Mesoscopic CB
(reflectionless
contacts)

Random phase but
not period.

Statistical description of strong CB:
Theory:
Peaks: Jalabert, Stone, Alhassid (1992);
Valleys: Aleiner, Glazman (1996);
Reasonable agreement,
But problems with values of the correlation
fields

Courtesy of C.Marcus

Mesoscopic Coulomb Blockade

Aleiner, Glazman (1998)
Based on technique suggested by:
Matveev (1995); Furusaki, Matveev (1995);
Flensberg (1993).

Experiment:

Suppression
By a factor of 5.3
Th: Predicted 4.

Cronenwett et. al. (1998)

Even-Odd effect due to Kondo effect

Predicted:
Glazman, Raikh (1988)
Ng, Lee (1988)

Spin degeneracy in odd valleys:
Effective Hamiltonian:
local spin density
of conduction electrons

magnetic impurity

Observation:
1998

D. Goldhaber-Gordon et al. (MIT-Weizmann)
S.M. Cronenwett et al. (TU Delft)
J. Schmid et al. (MPI @ Stuttgart)

200 nm
15 mK

800 mK

van der Wiel et al. (2000)

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”)


Slide 14

Theory of Quantum Dots as
Zero-dimensional Metallic Systems
Igor Aleiner (Columbia)
Collaborators:
B.L.Altshuler (Princeton)
P.W.Brouwer (Cornell)
V.I.Falko (Lancaster, UK)
L.I. Glazman (Minnesota)
I.L. Kurland (Princeton)
Physics of the Microworld Conference, Oct. 16 (2004)

Outline:
•
•

Quantum dot (QD) as zero dimensional metal
Random Matrix theory for transport in quantum dots

a)
b)

Non-interacting “standard models”.
Peculiar spin-orbit effects in QD based on 2D electron gas.

•

Interaction effects:

a)
b)
c)

Universal interaction Hamiltonian;
Mesoscopic Stoner instability;
Coulomb blockade (strong, weak, mesoscopic);

d)

Kondo effect.

“Quantum dot” used in two different contents:

For the rest of Number
the talk:
of electrons:

1)

“Artificial atom”
Description requires exact diagonalization.
(Kouwnehoven group (Delft))

2)
“Artificial nucleus”

(Marcus group (Harvard))

Statistical description is allowed !!!

Random Matrix Theory for Transport in Quantum Dots

2DEG

QD

2DEG

Energy scales

Level spacing

L

Thouless Energy

Conductance
Assume:

Statistics of transport is determined
only 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:

RMT

Confinement, disorder, etc

No magnetic field, no SO

Magnetic field, no SO

No magnetic field, strong SO

Magnetic field + SO

Conductance of chaotic dot
classical

Mesoscopic
fluctuations

Weak localization
V

I

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

Conductance of chaotic dot
classical

Mesoscopic
fluctuations

Weak localization
V

I

Universal quantum corrections
[Altshuler, Shklovskii (1986)]

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 QD
Aleiner, Fal’ko (2001)

T - invariance

But

Spin dependent flux

Spin relaxation rate

Meir, Gefen, Entin-Wohlman (1989)
Mathur, Stone (1992)
Lyanda-Geller, Mirlin (1994)

Khaetskii, Nazarov (2000)

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 field
Observed in
Folk, Patel, Birnbaum, Marcus, Duruoz, Harris, Jr. (2001)

Interaction Hamiltonian
Energies smaller than Thouless energy:

Random matrix

????

In nuclear physics:

from shell model
random

Universal Interaction Hamiltonian
Energies smaller than Thouless energy:

Kurland, Aleiner, Altshuler (2000)

Only invariants compatible with the circular symmetry

are NOT random !!!

Universal Interaction Hamiltonian
Energies smaller than Thouless energy:

Random matrix
Not random
Valid if:
1)

2) Fundamental symmetries
are NOT broken at larger energies

Universal Interaction Hamiltonian
Energies smaller than Thouless energy:

One-particle levels
Interaction
with
Zero
dimensional
Fermi
liquid
determined by
additional conservations
Wigner – Dyson statistics
Valid if:
1)

2) Fundamental symmetries
are NOT broken at larger energies

Universal Interaction Hamiltonian

Singlet electron-hole
channel.
Triplet electron-hole
channel.

Particle-particle
(Cooper) channel.

Analogy with soft modes in metals

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 of
The ground state S > ½.

Mesoscopic Stoner Instability
Kurland, Aleiner, Altshuler (2000)

Energy of the ground state:

Spin is finite even for NO randomness
Typical
vs.S:

NO

interactions

Does not scale with the size of the system
FM instability
Stoner (1935)

Also Brouwer, Oreg, Halperin (2000)

random with known from RMT
correlation functions

Universal Interaction Hamiltonian

Singlet Channel:
gate voltage
is NOT renormalized
But

Q: What is charge degeneracy of the ground state

(isolated dot)

degeneracy

- half-integer

gap

Otherwise

Coulomb blockade of electron transport
For tunneling contacts:

Charge
degeneracy

Charge gap
Term introduced by Averin and Likharev (1986);
Effect first discussed by C.J. Gorter (1951).

Small quantum dots (~ 500 nm)

conductance (e2/h)

M. Kastner, Physics Today (1993)
E.B. Foxman et al., PRB (1993)

gate voltage (mV)

In metals first observed in Fulton, Dolan, PRL, 59, 109, (1987)

Coulomb blockade (CB) (II)
Strong CB
Weak CB

Mesoscopic CB
(reflectionless
contacts)

Random phase but
not period.

Statistical description of strong CB:
Theory:
Peaks: Jalabert, Stone, Alhassid (1992);
Valleys: Aleiner, Glazman (1996);
Reasonable agreement,
But problems with values of the correlation
fields

Courtesy of C.Marcus

Mesoscopic Coulomb Blockade

Aleiner, Glazman (1998)
Based on technique suggested by:
Matveev (1995); Furusaki, Matveev (1995);
Flensberg (1993).

Experiment:

Suppression
By a factor of 5.3
Th: Predicted 4.

Cronenwett et. al. (1998)

Even-Odd effect due to Kondo effect

Predicted:
Glazman, Raikh (1988)
Ng, Lee (1988)

Spin degeneracy in odd valleys:
Effective Hamiltonian:
local spin density
of conduction electrons

magnetic impurity

Observation:
1998

D. Goldhaber-Gordon et al. (MIT-Weizmann)
S.M. Cronenwett et al. (TU Delft)
J. Schmid et al. (MPI @ Stuttgart)

200 nm
15 mK

800 mK

van der Wiel et al. (2000)

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”)


Slide 15

Theory of Quantum Dots as
Zero-dimensional Metallic Systems
Igor Aleiner (Columbia)
Collaborators:
B.L.Altshuler (Princeton)
P.W.Brouwer (Cornell)
V.I.Falko (Lancaster, UK)
L.I. Glazman (Minnesota)
I.L. Kurland (Princeton)
Physics of the Microworld Conference, Oct. 16 (2004)

Outline:
•
•

Quantum dot (QD) as zero dimensional metal
Random Matrix theory for transport in quantum dots

a)
b)

Non-interacting “standard models”.
Peculiar spin-orbit effects in QD based on 2D electron gas.

•

Interaction effects:

a)
b)
c)

Universal interaction Hamiltonian;
Mesoscopic Stoner instability;
Coulomb blockade (strong, weak, mesoscopic);

d)

Kondo effect.

“Quantum dot” used in two different contents:

For the rest of Number
the talk:
of electrons:

1)

“Artificial atom”
Description requires exact diagonalization.
(Kouwnehoven group (Delft))

2)
“Artificial nucleus”

(Marcus group (Harvard))

Statistical description is allowed !!!

Random Matrix Theory for Transport in Quantum Dots

2DEG

QD

2DEG

Energy scales

Level spacing

L

Thouless Energy

Conductance
Assume:

Statistics of transport is determined
only 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:

RMT

Confinement, disorder, etc

No magnetic field, no SO

Magnetic field, no SO

No magnetic field, strong SO

Magnetic field + SO

Conductance of chaotic dot
classical

Mesoscopic
fluctuations

Weak localization
V

I

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

Conductance of chaotic dot
classical

Mesoscopic
fluctuations

Weak localization
V

I

Universal quantum corrections
[Altshuler, Shklovskii (1986)]

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 QD
Aleiner, Fal’ko (2001)

T - invariance

But

Spin dependent flux

Spin relaxation rate

Meir, Gefen, Entin-Wohlman (1989)
Mathur, Stone (1992)
Lyanda-Geller, Mirlin (1994)

Khaetskii, Nazarov (2000)

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 field
Observed in
Folk, Patel, Birnbaum, Marcus, Duruoz, Harris, Jr. (2001)

Interaction Hamiltonian
Energies smaller than Thouless energy:

Random matrix

????

In nuclear physics:

from shell model
random

Universal Interaction Hamiltonian
Energies smaller than Thouless energy:

Kurland, Aleiner, Altshuler (2000)

Only invariants compatible with the circular symmetry

are NOT random !!!

Universal Interaction Hamiltonian
Energies smaller than Thouless energy:

Random matrix
Not random
Valid if:
1)

2) Fundamental symmetries
are NOT broken at larger energies

Universal Interaction Hamiltonian
Energies smaller than Thouless energy:

One-particle levels
Interaction
with
Zero
dimensional
Fermi
liquid
determined by
additional conservations
Wigner – Dyson statistics
Valid if:
1)

2) Fundamental symmetries
are NOT broken at larger energies

Universal Interaction Hamiltonian

Singlet electron-hole
channel.
Triplet electron-hole
channel.

Particle-particle
(Cooper) channel.

Analogy with soft modes in metals

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 of
The ground state S > ½.

Mesoscopic Stoner Instability
Kurland, Aleiner, Altshuler (2000)

Energy of the ground state:

Spin is finite even for NO randomness
Typical
vs.S:

NO

interactions

Does not scale with the size of the system
FM instability
Stoner (1935)

Also Brouwer, Oreg, Halperin (2000)

random with known from RMT
correlation functions

Universal Interaction Hamiltonian

Singlet Channel:
gate voltage
is NOT renormalized
But

Q: What is charge degeneracy of the ground state

(isolated dot)

degeneracy

- half-integer

gap

Otherwise

Coulomb blockade of electron transport
For tunneling contacts:

Charge
degeneracy

Charge gap
Term introduced by Averin and Likharev (1986);
Effect first discussed by C.J. Gorter (1951).

Small quantum dots (~ 500 nm)

conductance (e2/h)

M. Kastner, Physics Today (1993)
E.B. Foxman et al., PRB (1993)

gate voltage (mV)

In metals first observed in Fulton, Dolan, PRL, 59, 109, (1987)

Coulomb blockade (CB) (II)
Strong CB
Weak CB

Mesoscopic CB
(reflectionless
contacts)

Random phase but
not period.

Statistical description of strong CB:
Theory:
Peaks: Jalabert, Stone, Alhassid (1992);
Valleys: Aleiner, Glazman (1996);
Reasonable agreement,
But problems with values of the correlation
fields

Courtesy of C.Marcus

Mesoscopic Coulomb Blockade

Aleiner, Glazman (1998)
Based on technique suggested by:
Matveev (1995); Furusaki, Matveev (1995);
Flensberg (1993).

Experiment:

Suppression
By a factor of 5.3
Th: Predicted 4.

Cronenwett et. al. (1998)

Even-Odd effect due to Kondo effect

Predicted:
Glazman, Raikh (1988)
Ng, Lee (1988)

Spin degeneracy in odd valleys:
Effective Hamiltonian:
local spin density
of conduction electrons

magnetic impurity

Observation:
1998

D. Goldhaber-Gordon et al. (MIT-Weizmann)
S.M. Cronenwett et al. (TU Delft)
J. Schmid et al. (MPI @ Stuttgart)

200 nm
15 mK

800 mK

van der Wiel et al. (2000)

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”)


Slide 16

Theory of Quantum Dots as
Zero-dimensional Metallic Systems
Igor Aleiner (Columbia)
Collaborators:
B.L.Altshuler (Princeton)
P.W.Brouwer (Cornell)
V.I.Falko (Lancaster, UK)
L.I. Glazman (Minnesota)
I.L. Kurland (Princeton)
Physics of the Microworld Conference, Oct. 16 (2004)

Outline:
•
•

Quantum dot (QD) as zero dimensional metal
Random Matrix theory for transport in quantum dots

a)
b)

Non-interacting “standard models”.
Peculiar spin-orbit effects in QD based on 2D electron gas.

•

Interaction effects:

a)
b)
c)

Universal interaction Hamiltonian;
Mesoscopic Stoner instability;
Coulomb blockade (strong, weak, mesoscopic);

d)

Kondo effect.

“Quantum dot” used in two different contents:

For the rest of Number
the talk:
of electrons:

1)

“Artificial atom”
Description requires exact diagonalization.
(Kouwnehoven group (Delft))

2)
“Artificial nucleus”

(Marcus group (Harvard))

Statistical description is allowed !!!

Random Matrix Theory for Transport in Quantum Dots

2DEG

QD

2DEG

Energy scales

Level spacing

L

Thouless Energy

Conductance
Assume:

Statistics of transport is determined
only 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:

RMT

Confinement, disorder, etc

No magnetic field, no SO

Magnetic field, no SO

No magnetic field, strong SO

Magnetic field + SO

Conductance of chaotic dot
classical

Mesoscopic
fluctuations

Weak localization
V

I

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

Conductance of chaotic dot
classical

Mesoscopic
fluctuations

Weak localization
V

I

Universal quantum corrections
[Altshuler, Shklovskii (1986)]

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 QD
Aleiner, Fal’ko (2001)

T - invariance

But

Spin dependent flux

Spin relaxation rate

Meir, Gefen, Entin-Wohlman (1989)
Mathur, Stone (1992)
Lyanda-Geller, Mirlin (1994)

Khaetskii, Nazarov (2000)

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 field
Observed in
Folk, Patel, Birnbaum, Marcus, Duruoz, Harris, Jr. (2001)

Interaction Hamiltonian
Energies smaller than Thouless energy:

Random matrix

????

In nuclear physics:

from shell model
random

Universal Interaction Hamiltonian
Energies smaller than Thouless energy:

Kurland, Aleiner, Altshuler (2000)

Only invariants compatible with the circular symmetry

are NOT random !!!

Universal Interaction Hamiltonian
Energies smaller than Thouless energy:

Random matrix
Not random
Valid if:
1)

2) Fundamental symmetries
are NOT broken at larger energies

Universal Interaction Hamiltonian
Energies smaller than Thouless energy:

One-particle levels
Interaction
with
Zero
dimensional
Fermi
liquid
determined by
additional conservations
Wigner – Dyson statistics
Valid if:
1)

2) Fundamental symmetries
are NOT broken at larger energies

Universal Interaction Hamiltonian

Singlet electron-hole
channel.
Triplet electron-hole
channel.

Particle-particle
(Cooper) channel.

Analogy with soft modes in metals

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 of
The ground state S > ½.

Mesoscopic Stoner Instability
Kurland, Aleiner, Altshuler (2000)

Energy of the ground state:

Spin is finite even for NO randomness
Typical
vs.S:

NO

interactions

Does not scale with the size of the system
FM instability
Stoner (1935)

Also Brouwer, Oreg, Halperin (2000)

random with known from RMT
correlation functions

Universal Interaction Hamiltonian

Singlet Channel:
gate voltage
is NOT renormalized
But

Q: What is charge degeneracy of the ground state

(isolated dot)

degeneracy

- half-integer

gap

Otherwise

Coulomb blockade of electron transport
For tunneling contacts:

Charge
degeneracy

Charge gap
Term introduced by Averin and Likharev (1986);
Effect first discussed by C.J. Gorter (1951).

Small quantum dots (~ 500 nm)

conductance (e2/h)

M. Kastner, Physics Today (1993)
E.B. Foxman et al., PRB (1993)

gate voltage (mV)

In metals first observed in Fulton, Dolan, PRL, 59, 109, (1987)

Coulomb blockade (CB) (II)
Strong CB
Weak CB

Mesoscopic CB
(reflectionless
contacts)

Random phase but
not period.

Statistical description of strong CB:
Theory:
Peaks: Jalabert, Stone, Alhassid (1992);
Valleys: Aleiner, Glazman (1996);
Reasonable agreement,
But problems with values of the correlation
fields

Courtesy of C.Marcus

Mesoscopic Coulomb Blockade

Aleiner, Glazman (1998)
Based on technique suggested by:
Matveev (1995); Furusaki, Matveev (1995);
Flensberg (1993).

Experiment:

Suppression
By a factor of 5.3
Th: Predicted 4.

Cronenwett et. al. (1998)

Even-Odd effect due to Kondo effect

Predicted:
Glazman, Raikh (1988)
Ng, Lee (1988)

Spin degeneracy in odd valleys:
Effective Hamiltonian:
local spin density
of conduction electrons

magnetic impurity

Observation:
1998

D. Goldhaber-Gordon et al. (MIT-Weizmann)
S.M. Cronenwett et al. (TU Delft)
J. Schmid et al. (MPI @ Stuttgart)

200 nm
15 mK

800 mK

van der Wiel et al. (2000)

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”)


Slide 17

Theory of Quantum Dots as
Zero-dimensional Metallic Systems
Igor Aleiner (Columbia)
Collaborators:
B.L.Altshuler (Princeton)
P.W.Brouwer (Cornell)
V.I.Falko (Lancaster, UK)
L.I. Glazman (Minnesota)
I.L. Kurland (Princeton)
Physics of the Microworld Conference, Oct. 16 (2004)

Outline:
•
•

Quantum dot (QD) as zero dimensional metal
Random Matrix theory for transport in quantum dots

a)
b)

Non-interacting “standard models”.
Peculiar spin-orbit effects in QD based on 2D electron gas.

•

Interaction effects:

a)
b)
c)

Universal interaction Hamiltonian;
Mesoscopic Stoner instability;
Coulomb blockade (strong, weak, mesoscopic);

d)

Kondo effect.

“Quantum dot” used in two different contents:

For the rest of Number
the talk:
of electrons:

1)

“Artificial atom”
Description requires exact diagonalization.
(Kouwnehoven group (Delft))

2)
“Artificial nucleus”

(Marcus group (Harvard))

Statistical description is allowed !!!

Random Matrix Theory for Transport in Quantum Dots

2DEG

QD

2DEG

Energy scales

Level spacing

L

Thouless Energy

Conductance
Assume:

Statistics of transport is determined
only 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:

RMT

Confinement, disorder, etc

No magnetic field, no SO

Magnetic field, no SO

No magnetic field, strong SO

Magnetic field + SO

Conductance of chaotic dot
classical

Mesoscopic
fluctuations

Weak localization
V

I

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

Conductance of chaotic dot
classical

Mesoscopic
fluctuations

Weak localization
V

I

Universal quantum corrections
[Altshuler, Shklovskii (1986)]

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 QD
Aleiner, Fal’ko (2001)

T - invariance

But

Spin dependent flux

Spin relaxation rate

Meir, Gefen, Entin-Wohlman (1989)
Mathur, Stone (1992)
Lyanda-Geller, Mirlin (1994)

Khaetskii, Nazarov (2000)

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 field
Observed in
Folk, Patel, Birnbaum, Marcus, Duruoz, Harris, Jr. (2001)

Interaction Hamiltonian
Energies smaller than Thouless energy:

Random matrix

????

In nuclear physics:

from shell model
random

Universal Interaction Hamiltonian
Energies smaller than Thouless energy:

Kurland, Aleiner, Altshuler (2000)

Only invariants compatible with the circular symmetry

are NOT random !!!

Universal Interaction Hamiltonian
Energies smaller than Thouless energy:

Random matrix
Not random
Valid if:
1)

2) Fundamental symmetries
are NOT broken at larger energies

Universal Interaction Hamiltonian
Energies smaller than Thouless energy:

One-particle levels
Interaction
with
Zero
dimensional
Fermi
liquid
determined by
additional conservations
Wigner – Dyson statistics
Valid if:
1)

2) Fundamental symmetries
are NOT broken at larger energies

Universal Interaction Hamiltonian

Singlet electron-hole
channel.
Triplet electron-hole
channel.

Particle-particle
(Cooper) channel.

Analogy with soft modes in metals

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 of
The ground state S > ½.

Mesoscopic Stoner Instability
Kurland, Aleiner, Altshuler (2000)

Energy of the ground state:

Spin is finite even for NO randomness
Typical
vs.S:

NO

interactions

Does not scale with the size of the system
FM instability
Stoner (1935)

Also Brouwer, Oreg, Halperin (2000)

random with known from RMT
correlation functions

Universal Interaction Hamiltonian

Singlet Channel:
gate voltage
is NOT renormalized
But

Q: What is charge degeneracy of the ground state

(isolated dot)

degeneracy

- half-integer

gap

Otherwise

Coulomb blockade of electron transport
For tunneling contacts:

Charge
degeneracy

Charge gap
Term introduced by Averin and Likharev (1986);
Effect first discussed by C.J. Gorter (1951).

Small quantum dots (~ 500 nm)

conductance (e2/h)

M. Kastner, Physics Today (1993)
E.B. Foxman et al., PRB (1993)

gate voltage (mV)

In metals first observed in Fulton, Dolan, PRL, 59, 109, (1987)

Coulomb blockade (CB) (II)
Strong CB
Weak CB

Mesoscopic CB
(reflectionless
contacts)

Random phase but
not period.

Statistical description of strong CB:
Theory:
Peaks: Jalabert, Stone, Alhassid (1992);
Valleys: Aleiner, Glazman (1996);
Reasonable agreement,
But problems with values of the correlation
fields

Courtesy of C.Marcus

Mesoscopic Coulomb Blockade

Aleiner, Glazman (1998)
Based on technique suggested by:
Matveev (1995); Furusaki, Matveev (1995);
Flensberg (1993).

Experiment:

Suppression
By a factor of 5.3
Th: Predicted 4.

Cronenwett et. al. (1998)

Even-Odd effect due to Kondo effect

Predicted:
Glazman, Raikh (1988)
Ng, Lee (1988)

Spin degeneracy in odd valleys:
Effective Hamiltonian:
local spin density
of conduction electrons

magnetic impurity

Observation:
1998

D. Goldhaber-Gordon et al. (MIT-Weizmann)
S.M. Cronenwett et al. (TU Delft)
J. Schmid et al. (MPI @ Stuttgart)

200 nm
15 mK

800 mK

van der Wiel et al. (2000)

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”)


Slide 18

Theory of Quantum Dots as
Zero-dimensional Metallic Systems
Igor Aleiner (Columbia)
Collaborators:
B.L.Altshuler (Princeton)
P.W.Brouwer (Cornell)
V.I.Falko (Lancaster, UK)
L.I. Glazman (Minnesota)
I.L. Kurland (Princeton)
Physics of the Microworld Conference, Oct. 16 (2004)

Outline:
•
•

Quantum dot (QD) as zero dimensional metal
Random Matrix theory for transport in quantum dots

a)
b)

Non-interacting “standard models”.
Peculiar spin-orbit effects in QD based on 2D electron gas.

•

Interaction effects:

a)
b)
c)

Universal interaction Hamiltonian;
Mesoscopic Stoner instability;
Coulomb blockade (strong, weak, mesoscopic);

d)

Kondo effect.

“Quantum dot” used in two different contents:

For the rest of Number
the talk:
of electrons:

1)

“Artificial atom”
Description requires exact diagonalization.
(Kouwnehoven group (Delft))

2)
“Artificial nucleus”

(Marcus group (Harvard))

Statistical description is allowed !!!

Random Matrix Theory for Transport in Quantum Dots

2DEG

QD

2DEG

Energy scales

Level spacing

L

Thouless Energy

Conductance
Assume:

Statistics of transport is determined
only 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:

RMT

Confinement, disorder, etc

No magnetic field, no SO

Magnetic field, no SO

No magnetic field, strong SO

Magnetic field + SO

Conductance of chaotic dot
classical

Mesoscopic
fluctuations

Weak localization
V

I

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

Conductance of chaotic dot
classical

Mesoscopic
fluctuations

Weak localization
V

I

Universal quantum corrections
[Altshuler, Shklovskii (1986)]

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 QD
Aleiner, Fal’ko (2001)

T - invariance

But

Spin dependent flux

Spin relaxation rate

Meir, Gefen, Entin-Wohlman (1989)
Mathur, Stone (1992)
Lyanda-Geller, Mirlin (1994)

Khaetskii, Nazarov (2000)

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 field
Observed in
Folk, Patel, Birnbaum, Marcus, Duruoz, Harris, Jr. (2001)

Interaction Hamiltonian
Energies smaller than Thouless energy:

Random matrix

????

In nuclear physics:

from shell model
random

Universal Interaction Hamiltonian
Energies smaller than Thouless energy:

Kurland, Aleiner, Altshuler (2000)

Only invariants compatible with the circular symmetry

are NOT random !!!

Universal Interaction Hamiltonian
Energies smaller than Thouless energy:

Random matrix
Not random
Valid if:
1)

2) Fundamental symmetries
are NOT broken at larger energies

Universal Interaction Hamiltonian
Energies smaller than Thouless energy:

One-particle levels
Interaction
with
Zero
dimensional
Fermi
liquid
determined by
additional conservations
Wigner – Dyson statistics
Valid if:
1)

2) Fundamental symmetries
are NOT broken at larger energies

Universal Interaction Hamiltonian

Singlet electron-hole
channel.
Triplet electron-hole
channel.

Particle-particle
(Cooper) channel.

Analogy with soft modes in metals

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 of
The ground state S > ½.

Mesoscopic Stoner Instability
Kurland, Aleiner, Altshuler (2000)

Energy of the ground state:

Spin is finite even for NO randomness
Typical
vs.S:

NO

interactions

Does not scale with the size of the system
FM instability
Stoner (1935)

Also Brouwer, Oreg, Halperin (2000)

random with known from RMT
correlation functions

Universal Interaction Hamiltonian

Singlet Channel:
gate voltage
is NOT renormalized
But

Q: What is charge degeneracy of the ground state

(isolated dot)

degeneracy

- half-integer

gap

Otherwise

Coulomb blockade of electron transport
For tunneling contacts:

Charge
degeneracy

Charge gap
Term introduced by Averin and Likharev (1986);
Effect first discussed by C.J. Gorter (1951).

Small quantum dots (~ 500 nm)

conductance (e2/h)

M. Kastner, Physics Today (1993)
E.B. Foxman et al., PRB (1993)

gate voltage (mV)

In metals first observed in Fulton, Dolan, PRL, 59, 109, (1987)

Coulomb blockade (CB) (II)
Strong CB
Weak CB

Mesoscopic CB
(reflectionless
contacts)

Random phase but
not period.

Statistical description of strong CB:
Theory:
Peaks: Jalabert, Stone, Alhassid (1992);
Valleys: Aleiner, Glazman (1996);
Reasonable agreement,
But problems with values of the correlation
fields

Courtesy of C.Marcus

Mesoscopic Coulomb Blockade

Aleiner, Glazman (1998)
Based on technique suggested by:
Matveev (1995); Furusaki, Matveev (1995);
Flensberg (1993).

Experiment:

Suppression
By a factor of 5.3
Th: Predicted 4.

Cronenwett et. al. (1998)

Even-Odd effect due to Kondo effect

Predicted:
Glazman, Raikh (1988)
Ng, Lee (1988)

Spin degeneracy in odd valleys:
Effective Hamiltonian:
local spin density
of conduction electrons

magnetic impurity

Observation:
1998

D. Goldhaber-Gordon et al. (MIT-Weizmann)
S.M. Cronenwett et al. (TU Delft)
J. Schmid et al. (MPI @ Stuttgart)

200 nm
15 mK

800 mK

van der Wiel et al. (2000)

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”)


Slide 19

Theory of Quantum Dots as
Zero-dimensional Metallic Systems
Igor Aleiner (Columbia)
Collaborators:
B.L.Altshuler (Princeton)
P.W.Brouwer (Cornell)
V.I.Falko (Lancaster, UK)
L.I. Glazman (Minnesota)
I.L. Kurland (Princeton)
Physics of the Microworld Conference, Oct. 16 (2004)

Outline:
•
•

Quantum dot (QD) as zero dimensional metal
Random Matrix theory for transport in quantum dots

a)
b)

Non-interacting “standard models”.
Peculiar spin-orbit effects in QD based on 2D electron gas.

•

Interaction effects:

a)
b)
c)

Universal interaction Hamiltonian;
Mesoscopic Stoner instability;
Coulomb blockade (strong, weak, mesoscopic);

d)

Kondo effect.

“Quantum dot” used in two different contents:

For the rest of Number
the talk:
of electrons:

1)

“Artificial atom”
Description requires exact diagonalization.
(Kouwnehoven group (Delft))

2)
“Artificial nucleus”

(Marcus group (Harvard))

Statistical description is allowed !!!

Random Matrix Theory for Transport in Quantum Dots

2DEG

QD

2DEG

Energy scales

Level spacing

L

Thouless Energy

Conductance
Assume:

Statistics of transport is determined
only 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:

RMT

Confinement, disorder, etc

No magnetic field, no SO

Magnetic field, no SO

No magnetic field, strong SO

Magnetic field + SO

Conductance of chaotic dot
classical

Mesoscopic
fluctuations

Weak localization
V

I

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

Conductance of chaotic dot
classical

Mesoscopic
fluctuations

Weak localization
V

I

Universal quantum corrections
[Altshuler, Shklovskii (1986)]

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 QD
Aleiner, Fal’ko (2001)

T - invariance

But

Spin dependent flux

Spin relaxation rate

Meir, Gefen, Entin-Wohlman (1989)
Mathur, Stone (1992)
Lyanda-Geller, Mirlin (1994)

Khaetskii, Nazarov (2000)

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 field
Observed in
Folk, Patel, Birnbaum, Marcus, Duruoz, Harris, Jr. (2001)

Interaction Hamiltonian
Energies smaller than Thouless energy:

Random matrix

????

In nuclear physics:

from shell model
random

Universal Interaction Hamiltonian
Energies smaller than Thouless energy:

Kurland, Aleiner, Altshuler (2000)

Only invariants compatible with the circular symmetry

are NOT random !!!

Universal Interaction Hamiltonian
Energies smaller than Thouless energy:

Random matrix
Not random
Valid if:
1)

2) Fundamental symmetries
are NOT broken at larger energies

Universal Interaction Hamiltonian
Energies smaller than Thouless energy:

One-particle levels
Interaction
with
Zero
dimensional
Fermi
liquid
determined by
additional conservations
Wigner – Dyson statistics
Valid if:
1)

2) Fundamental symmetries
are NOT broken at larger energies

Universal Interaction Hamiltonian

Singlet electron-hole
channel.
Triplet electron-hole
channel.

Particle-particle
(Cooper) channel.

Analogy with soft modes in metals

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 of
The ground state S > ½.

Mesoscopic Stoner Instability
Kurland, Aleiner, Altshuler (2000)

Energy of the ground state:

Spin is finite even for NO randomness
Typical
vs.S:

NO

interactions

Does not scale with the size of the system
FM instability
Stoner (1935)

Also Brouwer, Oreg, Halperin (2000)

random with known from RMT
correlation functions

Universal Interaction Hamiltonian

Singlet Channel:
gate voltage
is NOT renormalized
But

Q: What is charge degeneracy of the ground state

(isolated dot)

degeneracy

- half-integer

gap

Otherwise

Coulomb blockade of electron transport
For tunneling contacts:

Charge
degeneracy

Charge gap
Term introduced by Averin and Likharev (1986);
Effect first discussed by C.J. Gorter (1951).

Small quantum dots (~ 500 nm)

conductance (e2/h)

M. Kastner, Physics Today (1993)
E.B. Foxman et al., PRB (1993)

gate voltage (mV)

In metals first observed in Fulton, Dolan, PRL, 59, 109, (1987)

Coulomb blockade (CB) (II)
Strong CB
Weak CB

Mesoscopic CB
(reflectionless
contacts)

Random phase but
not period.

Statistical description of strong CB:
Theory:
Peaks: Jalabert, Stone, Alhassid (1992);
Valleys: Aleiner, Glazman (1996);
Reasonable agreement,
But problems with values of the correlation
fields

Courtesy of C.Marcus

Mesoscopic Coulomb Blockade

Aleiner, Glazman (1998)
Based on technique suggested by:
Matveev (1995); Furusaki, Matveev (1995);
Flensberg (1993).

Experiment:

Suppression
By a factor of 5.3
Th: Predicted 4.

Cronenwett et. al. (1998)

Even-Odd effect due to Kondo effect

Predicted:
Glazman, Raikh (1988)
Ng, Lee (1988)

Spin degeneracy in odd valleys:
Effective Hamiltonian:
local spin density
of conduction electrons

magnetic impurity

Observation:
1998

D. Goldhaber-Gordon et al. (MIT-Weizmann)
S.M. Cronenwett et al. (TU Delft)
J. Schmid et al. (MPI @ Stuttgart)

200 nm
15 mK

800 mK

van der Wiel et al. (2000)

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”)


Slide 20

Theory of Quantum Dots as
Zero-dimensional Metallic Systems
Igor Aleiner (Columbia)
Collaborators:
B.L.Altshuler (Princeton)
P.W.Brouwer (Cornell)
V.I.Falko (Lancaster, UK)
L.I. Glazman (Minnesota)
I.L. Kurland (Princeton)
Physics of the Microworld Conference, Oct. 16 (2004)

Outline:
•
•

Quantum dot (QD) as zero dimensional metal
Random Matrix theory for transport in quantum dots

a)
b)

Non-interacting “standard models”.
Peculiar spin-orbit effects in QD based on 2D electron gas.

•

Interaction effects:

a)
b)
c)

Universal interaction Hamiltonian;
Mesoscopic Stoner instability;
Coulomb blockade (strong, weak, mesoscopic);

d)

Kondo effect.

“Quantum dot” used in two different contents:

For the rest of Number
the talk:
of electrons:

1)

“Artificial atom”
Description requires exact diagonalization.
(Kouwnehoven group (Delft))

2)
“Artificial nucleus”

(Marcus group (Harvard))

Statistical description is allowed !!!

Random Matrix Theory for Transport in Quantum Dots

2DEG

QD

2DEG

Energy scales

Level spacing

L

Thouless Energy

Conductance
Assume:

Statistics of transport is determined
only 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:

RMT

Confinement, disorder, etc

No magnetic field, no SO

Magnetic field, no SO

No magnetic field, strong SO

Magnetic field + SO

Conductance of chaotic dot
classical

Mesoscopic
fluctuations

Weak localization
V

I

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

Conductance of chaotic dot
classical

Mesoscopic
fluctuations

Weak localization
V

I

Universal quantum corrections
[Altshuler, Shklovskii (1986)]

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 QD
Aleiner, Fal’ko (2001)

T - invariance

But

Spin dependent flux

Spin relaxation rate

Meir, Gefen, Entin-Wohlman (1989)
Mathur, Stone (1992)
Lyanda-Geller, Mirlin (1994)

Khaetskii, Nazarov (2000)

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 field
Observed in
Folk, Patel, Birnbaum, Marcus, Duruoz, Harris, Jr. (2001)

Interaction Hamiltonian
Energies smaller than Thouless energy:

Random matrix

????

In nuclear physics:

from shell model
random

Universal Interaction Hamiltonian
Energies smaller than Thouless energy:

Kurland, Aleiner, Altshuler (2000)

Only invariants compatible with the circular symmetry

are NOT random !!!

Universal Interaction Hamiltonian
Energies smaller than Thouless energy:

Random matrix
Not random
Valid if:
1)

2) Fundamental symmetries
are NOT broken at larger energies

Universal Interaction Hamiltonian
Energies smaller than Thouless energy:

One-particle levels
Interaction
with
Zero
dimensional
Fermi
liquid
determined by
additional conservations
Wigner – Dyson statistics
Valid if:
1)

2) Fundamental symmetries
are NOT broken at larger energies

Universal Interaction Hamiltonian

Singlet electron-hole
channel.
Triplet electron-hole
channel.

Particle-particle
(Cooper) channel.

Analogy with soft modes in metals

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 of
The ground state S > ½.

Mesoscopic Stoner Instability
Kurland, Aleiner, Altshuler (2000)

Energy of the ground state:

Spin is finite even for NO randomness
Typical
vs.S:

NO

interactions

Does not scale with the size of the system
FM instability
Stoner (1935)

Also Brouwer, Oreg, Halperin (2000)

random with known from RMT
correlation functions

Universal Interaction Hamiltonian

Singlet Channel:
gate voltage
is NOT renormalized
But

Q: What is charge degeneracy of the ground state

(isolated dot)

degeneracy

- half-integer

gap

Otherwise

Coulomb blockade of electron transport
For tunneling contacts:

Charge
degeneracy

Charge gap
Term introduced by Averin and Likharev (1986);
Effect first discussed by C.J. Gorter (1951).

Small quantum dots (~ 500 nm)

conductance (e2/h)

M. Kastner, Physics Today (1993)
E.B. Foxman et al., PRB (1993)

gate voltage (mV)

In metals first observed in Fulton, Dolan, PRL, 59, 109, (1987)

Coulomb blockade (CB) (II)
Strong CB
Weak CB

Mesoscopic CB
(reflectionless
contacts)

Random phase but
not period.

Statistical description of strong CB:
Theory:
Peaks: Jalabert, Stone, Alhassid (1992);
Valleys: Aleiner, Glazman (1996);
Reasonable agreement,
But problems with values of the correlation
fields

Courtesy of C.Marcus

Mesoscopic Coulomb Blockade

Aleiner, Glazman (1998)
Based on technique suggested by:
Matveev (1995); Furusaki, Matveev (1995);
Flensberg (1993).

Experiment:

Suppression
By a factor of 5.3
Th: Predicted 4.

Cronenwett et. al. (1998)

Even-Odd effect due to Kondo effect

Predicted:
Glazman, Raikh (1988)
Ng, Lee (1988)

Spin degeneracy in odd valleys:
Effective Hamiltonian:
local spin density
of conduction electrons

magnetic impurity

Observation:
1998

D. Goldhaber-Gordon et al. (MIT-Weizmann)
S.M. Cronenwett et al. (TU Delft)
J. Schmid et al. (MPI @ Stuttgart)

200 nm
15 mK

800 mK

van der Wiel et al. (2000)

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”)


Slide 21

Theory of Quantum Dots as
Zero-dimensional Metallic Systems
Igor Aleiner (Columbia)
Collaborators:
B.L.Altshuler (Princeton)
P.W.Brouwer (Cornell)
V.I.Falko (Lancaster, UK)
L.I. Glazman (Minnesota)
I.L. Kurland (Princeton)
Physics of the Microworld Conference, Oct. 16 (2004)

Outline:
•
•

Quantum dot (QD) as zero dimensional metal
Random Matrix theory for transport in quantum dots

a)
b)

Non-interacting “standard models”.
Peculiar spin-orbit effects in QD based on 2D electron gas.

•

Interaction effects:

a)
b)
c)

Universal interaction Hamiltonian;
Mesoscopic Stoner instability;
Coulomb blockade (strong, weak, mesoscopic);

d)

Kondo effect.

“Quantum dot” used in two different contents:

For the rest of Number
the talk:
of electrons:

1)

“Artificial atom”
Description requires exact diagonalization.
(Kouwnehoven group (Delft))

2)
“Artificial nucleus”

(Marcus group (Harvard))

Statistical description is allowed !!!

Random Matrix Theory for Transport in Quantum Dots

2DEG

QD

2DEG

Energy scales

Level spacing

L

Thouless Energy

Conductance
Assume:

Statistics of transport is determined
only 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:

RMT

Confinement, disorder, etc

No magnetic field, no SO

Magnetic field, no SO

No magnetic field, strong SO

Magnetic field + SO

Conductance of chaotic dot
classical

Mesoscopic
fluctuations

Weak localization
V

I

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

Conductance of chaotic dot
classical

Mesoscopic
fluctuations

Weak localization
V

I

Universal quantum corrections
[Altshuler, Shklovskii (1986)]

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 QD
Aleiner, Fal’ko (2001)

T - invariance

But

Spin dependent flux

Spin relaxation rate

Meir, Gefen, Entin-Wohlman (1989)
Mathur, Stone (1992)
Lyanda-Geller, Mirlin (1994)

Khaetskii, Nazarov (2000)

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 field
Observed in
Folk, Patel, Birnbaum, Marcus, Duruoz, Harris, Jr. (2001)

Interaction Hamiltonian
Energies smaller than Thouless energy:

Random matrix

????

In nuclear physics:

from shell model
random

Universal Interaction Hamiltonian
Energies smaller than Thouless energy:

Kurland, Aleiner, Altshuler (2000)

Only invariants compatible with the circular symmetry

are NOT random !!!

Universal Interaction Hamiltonian
Energies smaller than Thouless energy:

Random matrix
Not random
Valid if:
1)

2) Fundamental symmetries
are NOT broken at larger energies

Universal Interaction Hamiltonian
Energies smaller than Thouless energy:

One-particle levels
Interaction
with
Zero
dimensional
Fermi
liquid
determined by
additional conservations
Wigner – Dyson statistics
Valid if:
1)

2) Fundamental symmetries
are NOT broken at larger energies

Universal Interaction Hamiltonian

Singlet electron-hole
channel.
Triplet electron-hole
channel.

Particle-particle
(Cooper) channel.

Analogy with soft modes in metals

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 of
The ground state S > ½.

Mesoscopic Stoner Instability
Kurland, Aleiner, Altshuler (2000)

Energy of the ground state:

Spin is finite even for NO randomness
Typical
vs.S:

NO

interactions

Does not scale with the size of the system
FM instability
Stoner (1935)

Also Brouwer, Oreg, Halperin (2000)

random with known from RMT
correlation functions

Universal Interaction Hamiltonian

Singlet Channel:
gate voltage
is NOT renormalized
But

Q: What is charge degeneracy of the ground state

(isolated dot)

degeneracy

- half-integer

gap

Otherwise

Coulomb blockade of electron transport
For tunneling contacts:

Charge
degeneracy

Charge gap
Term introduced by Averin and Likharev (1986);
Effect first discussed by C.J. Gorter (1951).

Small quantum dots (~ 500 nm)

conductance (e2/h)

M. Kastner, Physics Today (1993)
E.B. Foxman et al., PRB (1993)

gate voltage (mV)

In metals first observed in Fulton, Dolan, PRL, 59, 109, (1987)

Coulomb blockade (CB) (II)
Strong CB
Weak CB

Mesoscopic CB
(reflectionless
contacts)

Random phase but
not period.

Statistical description of strong CB:
Theory:
Peaks: Jalabert, Stone, Alhassid (1992);
Valleys: Aleiner, Glazman (1996);
Reasonable agreement,
But problems with values of the correlation
fields

Courtesy of C.Marcus

Mesoscopic Coulomb Blockade

Aleiner, Glazman (1998)
Based on technique suggested by:
Matveev (1995); Furusaki, Matveev (1995);
Flensberg (1993).

Experiment:

Suppression
By a factor of 5.3
Th: Predicted 4.

Cronenwett et. al. (1998)

Even-Odd effect due to Kondo effect

Predicted:
Glazman, Raikh (1988)
Ng, Lee (1988)

Spin degeneracy in odd valleys:
Effective Hamiltonian:
local spin density
of conduction electrons

magnetic impurity

Observation:
1998

D. Goldhaber-Gordon et al. (MIT-Weizmann)
S.M. Cronenwett et al. (TU Delft)
J. Schmid et al. (MPI @ Stuttgart)

200 nm
15 mK

800 mK

van der Wiel et al. (2000)

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”)


Slide 22

Theory of Quantum Dots as
Zero-dimensional Metallic Systems
Igor Aleiner (Columbia)
Collaborators:
B.L.Altshuler (Princeton)
P.W.Brouwer (Cornell)
V.I.Falko (Lancaster, UK)
L.I. Glazman (Minnesota)
I.L. Kurland (Princeton)
Physics of the Microworld Conference, Oct. 16 (2004)

Outline:
•
•

Quantum dot (QD) as zero dimensional metal
Random Matrix theory for transport in quantum dots

a)
b)

Non-interacting “standard models”.
Peculiar spin-orbit effects in QD based on 2D electron gas.

•

Interaction effects:

a)
b)
c)

Universal interaction Hamiltonian;
Mesoscopic Stoner instability;
Coulomb blockade (strong, weak, mesoscopic);

d)

Kondo effect.

“Quantum dot” used in two different contents:

For the rest of Number
the talk:
of electrons:

1)

“Artificial atom”
Description requires exact diagonalization.
(Kouwnehoven group (Delft))

2)
“Artificial nucleus”

(Marcus group (Harvard))

Statistical description is allowed !!!

Random Matrix Theory for Transport in Quantum Dots

2DEG

QD

2DEG

Energy scales

Level spacing

L

Thouless Energy

Conductance
Assume:

Statistics of transport is determined
only 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:

RMT

Confinement, disorder, etc

No magnetic field, no SO

Magnetic field, no SO

No magnetic field, strong SO

Magnetic field + SO

Conductance of chaotic dot
classical

Mesoscopic
fluctuations

Weak localization
V

I

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

Conductance of chaotic dot
classical

Mesoscopic
fluctuations

Weak localization
V

I

Universal quantum corrections
[Altshuler, Shklovskii (1986)]

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 QD
Aleiner, Fal’ko (2001)

T - invariance

But

Spin dependent flux

Spin relaxation rate

Meir, Gefen, Entin-Wohlman (1989)
Mathur, Stone (1992)
Lyanda-Geller, Mirlin (1994)

Khaetskii, Nazarov (2000)

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 field
Observed in
Folk, Patel, Birnbaum, Marcus, Duruoz, Harris, Jr. (2001)

Interaction Hamiltonian
Energies smaller than Thouless energy:

Random matrix

????

In nuclear physics:

from shell model
random

Universal Interaction Hamiltonian
Energies smaller than Thouless energy:

Kurland, Aleiner, Altshuler (2000)

Only invariants compatible with the circular symmetry

are NOT random !!!

Universal Interaction Hamiltonian
Energies smaller than Thouless energy:

Random matrix
Not random
Valid if:
1)

2) Fundamental symmetries
are NOT broken at larger energies

Universal Interaction Hamiltonian
Energies smaller than Thouless energy:

One-particle levels
Interaction
with
Zero
dimensional
Fermi
liquid
determined by
additional conservations
Wigner – Dyson statistics
Valid if:
1)

2) Fundamental symmetries
are NOT broken at larger energies

Universal Interaction Hamiltonian

Singlet electron-hole
channel.
Triplet electron-hole
channel.

Particle-particle
(Cooper) channel.

Analogy with soft modes in metals

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 of
The ground state S > ½.

Mesoscopic Stoner Instability
Kurland, Aleiner, Altshuler (2000)

Energy of the ground state:

Spin is finite even for NO randomness
Typical
vs.S:

NO

interactions

Does not scale with the size of the system
FM instability
Stoner (1935)

Also Brouwer, Oreg, Halperin (2000)

random with known from RMT
correlation functions

Universal Interaction Hamiltonian

Singlet Channel:
gate voltage
is NOT renormalized
But

Q: What is charge degeneracy of the ground state

(isolated dot)

degeneracy

- half-integer

gap

Otherwise

Coulomb blockade of electron transport
For tunneling contacts:

Charge
degeneracy

Charge gap
Term introduced by Averin and Likharev (1986);
Effect first discussed by C.J. Gorter (1951).

Small quantum dots (~ 500 nm)

conductance (e2/h)

M. Kastner, Physics Today (1993)
E.B. Foxman et al., PRB (1993)

gate voltage (mV)

In metals first observed in Fulton, Dolan, PRL, 59, 109, (1987)

Coulomb blockade (CB) (II)
Strong CB
Weak CB

Mesoscopic CB
(reflectionless
contacts)

Random phase but
not period.

Statistical description of strong CB:
Theory:
Peaks: Jalabert, Stone, Alhassid (1992);
Valleys: Aleiner, Glazman (1996);
Reasonable agreement,
But problems with values of the correlation
fields

Courtesy of C.Marcus

Mesoscopic Coulomb Blockade

Aleiner, Glazman (1998)
Based on technique suggested by:
Matveev (1995); Furusaki, Matveev (1995);
Flensberg (1993).

Experiment:

Suppression
By a factor of 5.3
Th: Predicted 4.

Cronenwett et. al. (1998)

Even-Odd effect due to Kondo effect

Predicted:
Glazman, Raikh (1988)
Ng, Lee (1988)

Spin degeneracy in odd valleys:
Effective Hamiltonian:
local spin density
of conduction electrons

magnetic impurity

Observation:
1998

D. Goldhaber-Gordon et al. (MIT-Weizmann)
S.M. Cronenwett et al. (TU Delft)
J. Schmid et al. (MPI @ Stuttgart)

200 nm
15 mK

800 mK

van der Wiel et al. (2000)

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”)


Slide 23

Theory of Quantum Dots as
Zero-dimensional Metallic Systems
Igor Aleiner (Columbia)
Collaborators:
B.L.Altshuler (Princeton)
P.W.Brouwer (Cornell)
V.I.Falko (Lancaster, UK)
L.I. Glazman (Minnesota)
I.L. Kurland (Princeton)
Physics of the Microworld Conference, Oct. 16 (2004)

Outline:
•
•

Quantum dot (QD) as zero dimensional metal
Random Matrix theory for transport in quantum dots

a)
b)

Non-interacting “standard models”.
Peculiar spin-orbit effects in QD based on 2D electron gas.

•

Interaction effects:

a)
b)
c)

Universal interaction Hamiltonian;
Mesoscopic Stoner instability;
Coulomb blockade (strong, weak, mesoscopic);

d)

Kondo effect.

“Quantum dot” used in two different contents:

For the rest of Number
the talk:
of electrons:

1)

“Artificial atom”
Description requires exact diagonalization.
(Kouwnehoven group (Delft))

2)
“Artificial nucleus”

(Marcus group (Harvard))

Statistical description is allowed !!!

Random Matrix Theory for Transport in Quantum Dots

2DEG

QD

2DEG

Energy scales

Level spacing

L

Thouless Energy

Conductance
Assume:

Statistics of transport is determined
only 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:

RMT

Confinement, disorder, etc

No magnetic field, no SO

Magnetic field, no SO

No magnetic field, strong SO

Magnetic field + SO

Conductance of chaotic dot
classical

Mesoscopic
fluctuations

Weak localization
V

I

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

Conductance of chaotic dot
classical

Mesoscopic
fluctuations

Weak localization
V

I

Universal quantum corrections
[Altshuler, Shklovskii (1986)]

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 QD
Aleiner, Fal’ko (2001)

T - invariance

But

Spin dependent flux

Spin relaxation rate

Meir, Gefen, Entin-Wohlman (1989)
Mathur, Stone (1992)
Lyanda-Geller, Mirlin (1994)

Khaetskii, Nazarov (2000)

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 field
Observed in
Folk, Patel, Birnbaum, Marcus, Duruoz, Harris, Jr. (2001)

Interaction Hamiltonian
Energies smaller than Thouless energy:

Random matrix

????

In nuclear physics:

from shell model
random

Universal Interaction Hamiltonian
Energies smaller than Thouless energy:

Kurland, Aleiner, Altshuler (2000)

Only invariants compatible with the circular symmetry

are NOT random !!!

Universal Interaction Hamiltonian
Energies smaller than Thouless energy:

Random matrix
Not random
Valid if:
1)

2) Fundamental symmetries
are NOT broken at larger energies

Universal Interaction Hamiltonian
Energies smaller than Thouless energy:

One-particle levels
Interaction
with
Zero
dimensional
Fermi
liquid
determined by
additional conservations
Wigner – Dyson statistics
Valid if:
1)

2) Fundamental symmetries
are NOT broken at larger energies

Universal Interaction Hamiltonian

Singlet electron-hole
channel.
Triplet electron-hole
channel.

Particle-particle
(Cooper) channel.

Analogy with soft modes in metals

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 of
The ground state S > ½.

Mesoscopic Stoner Instability
Kurland, Aleiner, Altshuler (2000)

Energy of the ground state:

Spin is finite even for NO randomness
Typical
vs.S:

NO

interactions

Does not scale with the size of the system
FM instability
Stoner (1935)

Also Brouwer, Oreg, Halperin (2000)

random with known from RMT
correlation functions

Universal Interaction Hamiltonian

Singlet Channel:
gate voltage
is NOT renormalized
But

Q: What is charge degeneracy of the ground state

(isolated dot)

degeneracy

- half-integer

gap

Otherwise

Coulomb blockade of electron transport
For tunneling contacts:

Charge
degeneracy

Charge gap
Term introduced by Averin and Likharev (1986);
Effect first discussed by C.J. Gorter (1951).

Small quantum dots (~ 500 nm)

conductance (e2/h)

M. Kastner, Physics Today (1993)
E.B. Foxman et al., PRB (1993)

gate voltage (mV)

In metals first observed in Fulton, Dolan, PRL, 59, 109, (1987)

Coulomb blockade (CB) (II)
Strong CB
Weak CB

Mesoscopic CB
(reflectionless
contacts)

Random phase but
not period.

Statistical description of strong CB:
Theory:
Peaks: Jalabert, Stone, Alhassid (1992);
Valleys: Aleiner, Glazman (1996);
Reasonable agreement,
But problems with values of the correlation
fields

Courtesy of C.Marcus

Mesoscopic Coulomb Blockade

Aleiner, Glazman (1998)
Based on technique suggested by:
Matveev (1995); Furusaki, Matveev (1995);
Flensberg (1993).

Experiment:

Suppression
By a factor of 5.3
Th: Predicted 4.

Cronenwett et. al. (1998)

Even-Odd effect due to Kondo effect

Predicted:
Glazman, Raikh (1988)
Ng, Lee (1988)

Spin degeneracy in odd valleys:
Effective Hamiltonian:
local spin density
of conduction electrons

magnetic impurity

Observation:
1998

D. Goldhaber-Gordon et al. (MIT-Weizmann)
S.M. Cronenwett et al. (TU Delft)
J. Schmid et al. (MPI @ Stuttgart)

200 nm
15 mK

800 mK

van der Wiel et al. (2000)

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”)


Slide 24

Theory of Quantum Dots as
Zero-dimensional Metallic Systems
Igor Aleiner (Columbia)
Collaborators:
B.L.Altshuler (Princeton)
P.W.Brouwer (Cornell)
V.I.Falko (Lancaster, UK)
L.I. Glazman (Minnesota)
I.L. Kurland (Princeton)
Physics of the Microworld Conference, Oct. 16 (2004)

Outline:
•
•

Quantum dot (QD) as zero dimensional metal
Random Matrix theory for transport in quantum dots

a)
b)

Non-interacting “standard models”.
Peculiar spin-orbit effects in QD based on 2D electron gas.

•

Interaction effects:

a)
b)
c)

Universal interaction Hamiltonian;
Mesoscopic Stoner instability;
Coulomb blockade (strong, weak, mesoscopic);

d)

Kondo effect.

“Quantum dot” used in two different contents:

For the rest of Number
the talk:
of electrons:

1)

“Artificial atom”
Description requires exact diagonalization.
(Kouwnehoven group (Delft))

2)
“Artificial nucleus”

(Marcus group (Harvard))

Statistical description is allowed !!!

Random Matrix Theory for Transport in Quantum Dots

2DEG

QD

2DEG

Energy scales

Level spacing

L

Thouless Energy

Conductance
Assume:

Statistics of transport is determined
only 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:

RMT

Confinement, disorder, etc

No magnetic field, no SO

Magnetic field, no SO

No magnetic field, strong SO

Magnetic field + SO

Conductance of chaotic dot
classical

Mesoscopic
fluctuations

Weak localization
V

I

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

Conductance of chaotic dot
classical

Mesoscopic
fluctuations

Weak localization
V

I

Universal quantum corrections
[Altshuler, Shklovskii (1986)]

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 QD
Aleiner, Fal’ko (2001)

T - invariance

But

Spin dependent flux

Spin relaxation rate

Meir, Gefen, Entin-Wohlman (1989)
Mathur, Stone (1992)
Lyanda-Geller, Mirlin (1994)

Khaetskii, Nazarov (2000)

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 field
Observed in
Folk, Patel, Birnbaum, Marcus, Duruoz, Harris, Jr. (2001)

Interaction Hamiltonian
Energies smaller than Thouless energy:

Random matrix

????

In nuclear physics:

from shell model
random

Universal Interaction Hamiltonian
Energies smaller than Thouless energy:

Kurland, Aleiner, Altshuler (2000)

Only invariants compatible with the circular symmetry

are NOT random !!!

Universal Interaction Hamiltonian
Energies smaller than Thouless energy:

Random matrix
Not random
Valid if:
1)

2) Fundamental symmetries
are NOT broken at larger energies

Universal Interaction Hamiltonian
Energies smaller than Thouless energy:

One-particle levels
Interaction
with
Zero
dimensional
Fermi
liquid
determined by
additional conservations
Wigner – Dyson statistics
Valid if:
1)

2) Fundamental symmetries
are NOT broken at larger energies

Universal Interaction Hamiltonian

Singlet electron-hole
channel.
Triplet electron-hole
channel.

Particle-particle
(Cooper) channel.

Analogy with soft modes in metals

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 of
The ground state S > ½.

Mesoscopic Stoner Instability
Kurland, Aleiner, Altshuler (2000)

Energy of the ground state:

Spin is finite even for NO randomness
Typical
vs.S:

NO

interactions

Does not scale with the size of the system
FM instability
Stoner (1935)

Also Brouwer, Oreg, Halperin (2000)

random with known from RMT
correlation functions

Universal Interaction Hamiltonian

Singlet Channel:
gate voltage
is NOT renormalized
But

Q: What is charge degeneracy of the ground state

(isolated dot)

degeneracy

- half-integer

gap

Otherwise

Coulomb blockade of electron transport
For tunneling contacts:

Charge
degeneracy

Charge gap
Term introduced by Averin and Likharev (1986);
Effect first discussed by C.J. Gorter (1951).

Small quantum dots (~ 500 nm)

conductance (e2/h)

M. Kastner, Physics Today (1993)
E.B. Foxman et al., PRB (1993)

gate voltage (mV)

In metals first observed in Fulton, Dolan, PRL, 59, 109, (1987)

Coulomb blockade (CB) (II)
Strong CB
Weak CB

Mesoscopic CB
(reflectionless
contacts)

Random phase but
not period.

Statistical description of strong CB:
Theory:
Peaks: Jalabert, Stone, Alhassid (1992);
Valleys: Aleiner, Glazman (1996);
Reasonable agreement,
But problems with values of the correlation
fields

Courtesy of C.Marcus

Mesoscopic Coulomb Blockade

Aleiner, Glazman (1998)
Based on technique suggested by:
Matveev (1995); Furusaki, Matveev (1995);
Flensberg (1993).

Experiment:

Suppression
By a factor of 5.3
Th: Predicted 4.

Cronenwett et. al. (1998)

Even-Odd effect due to Kondo effect

Predicted:
Glazman, Raikh (1988)
Ng, Lee (1988)

Spin degeneracy in odd valleys:
Effective Hamiltonian:
local spin density
of conduction electrons

magnetic impurity

Observation:
1998

D. Goldhaber-Gordon et al. (MIT-Weizmann)
S.M. Cronenwett et al. (TU Delft)
J. Schmid et al. (MPI @ Stuttgart)

200 nm
15 mK

800 mK

van der Wiel et al. (2000)

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”)


Slide 25

Theory of Quantum Dots as
Zero-dimensional Metallic Systems
Igor Aleiner (Columbia)
Collaborators:
B.L.Altshuler (Princeton)
P.W.Brouwer (Cornell)
V.I.Falko (Lancaster, UK)
L.I. Glazman (Minnesota)
I.L. Kurland (Princeton)
Physics of the Microworld Conference, Oct. 16 (2004)

Outline:
•
•

Quantum dot (QD) as zero dimensional metal
Random Matrix theory for transport in quantum dots

a)
b)

Non-interacting “standard models”.
Peculiar spin-orbit effects in QD based on 2D electron gas.

•

Interaction effects:

a)
b)
c)

Universal interaction Hamiltonian;
Mesoscopic Stoner instability;
Coulomb blockade (strong, weak, mesoscopic);

d)

Kondo effect.

“Quantum dot” used in two different contents:

For the rest of Number
the talk:
of electrons:

1)

“Artificial atom”
Description requires exact diagonalization.
(Kouwnehoven group (Delft))

2)
“Artificial nucleus”

(Marcus group (Harvard))

Statistical description is allowed !!!

Random Matrix Theory for Transport in Quantum Dots

2DEG

QD

2DEG

Energy scales

Level spacing

L

Thouless Energy

Conductance
Assume:

Statistics of transport is determined
only 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:

RMT

Confinement, disorder, etc

No magnetic field, no SO

Magnetic field, no SO

No magnetic field, strong SO

Magnetic field + SO

Conductance of chaotic dot
classical

Mesoscopic
fluctuations

Weak localization
V

I

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

Conductance of chaotic dot
classical

Mesoscopic
fluctuations

Weak localization
V

I

Universal quantum corrections
[Altshuler, Shklovskii (1986)]

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 QD
Aleiner, Fal’ko (2001)

T - invariance

But

Spin dependent flux

Spin relaxation rate

Meir, Gefen, Entin-Wohlman (1989)
Mathur, Stone (1992)
Lyanda-Geller, Mirlin (1994)

Khaetskii, Nazarov (2000)

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 field
Observed in
Folk, Patel, Birnbaum, Marcus, Duruoz, Harris, Jr. (2001)

Interaction Hamiltonian
Energies smaller than Thouless energy:

Random matrix

????

In nuclear physics:

from shell model
random

Universal Interaction Hamiltonian
Energies smaller than Thouless energy:

Kurland, Aleiner, Altshuler (2000)

Only invariants compatible with the circular symmetry

are NOT random !!!

Universal Interaction Hamiltonian
Energies smaller than Thouless energy:

Random matrix
Not random
Valid if:
1)

2) Fundamental symmetries
are NOT broken at larger energies

Universal Interaction Hamiltonian
Energies smaller than Thouless energy:

One-particle levels
Interaction
with
Zero
dimensional
Fermi
liquid
determined by
additional conservations
Wigner – Dyson statistics
Valid if:
1)

2) Fundamental symmetries
are NOT broken at larger energies

Universal Interaction Hamiltonian

Singlet electron-hole
channel.
Triplet electron-hole
channel.

Particle-particle
(Cooper) channel.

Analogy with soft modes in metals

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 of
The ground state S > ½.

Mesoscopic Stoner Instability
Kurland, Aleiner, Altshuler (2000)

Energy of the ground state:

Spin is finite even for NO randomness
Typical
vs.S:

NO

interactions

Does not scale with the size of the system
FM instability
Stoner (1935)

Also Brouwer, Oreg, Halperin (2000)

random with known from RMT
correlation functions

Universal Interaction Hamiltonian

Singlet Channel:
gate voltage
is NOT renormalized
But

Q: What is charge degeneracy of the ground state

(isolated dot)

degeneracy

- half-integer

gap

Otherwise

Coulomb blockade of electron transport
For tunneling contacts:

Charge
degeneracy

Charge gap
Term introduced by Averin and Likharev (1986);
Effect first discussed by C.J. Gorter (1951).

Small quantum dots (~ 500 nm)

conductance (e2/h)

M. Kastner, Physics Today (1993)
E.B. Foxman et al., PRB (1993)

gate voltage (mV)

In metals first observed in Fulton, Dolan, PRL, 59, 109, (1987)

Coulomb blockade (CB) (II)
Strong CB
Weak CB

Mesoscopic CB
(reflectionless
contacts)

Random phase but
not period.

Statistical description of strong CB:
Theory:
Peaks: Jalabert, Stone, Alhassid (1992);
Valleys: Aleiner, Glazman (1996);
Reasonable agreement,
But problems with values of the correlation
fields

Courtesy of C.Marcus

Mesoscopic Coulomb Blockade

Aleiner, Glazman (1998)
Based on technique suggested by:
Matveev (1995); Furusaki, Matveev (1995);
Flensberg (1993).

Experiment:

Suppression
By a factor of 5.3
Th: Predicted 4.

Cronenwett et. al. (1998)

Even-Odd effect due to Kondo effect

Predicted:
Glazman, Raikh (1988)
Ng, Lee (1988)

Spin degeneracy in odd valleys:
Effective Hamiltonian:
local spin density
of conduction electrons

magnetic impurity

Observation:
1998

D. Goldhaber-Gordon et al. (MIT-Weizmann)
S.M. Cronenwett et al. (TU Delft)
J. Schmid et al. (MPI @ Stuttgart)

200 nm
15 mK

800 mK

van der Wiel et al. (2000)

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”)


Slide 26

Theory of Quantum Dots as
Zero-dimensional Metallic Systems
Igor Aleiner (Columbia)
Collaborators:
B.L.Altshuler (Princeton)
P.W.Brouwer (Cornell)
V.I.Falko (Lancaster, UK)
L.I. Glazman (Minnesota)
I.L. Kurland (Princeton)
Physics of the Microworld Conference, Oct. 16 (2004)

Outline:
•
•

Quantum dot (QD) as zero dimensional metal
Random Matrix theory for transport in quantum dots

a)
b)

Non-interacting “standard models”.
Peculiar spin-orbit effects in QD based on 2D electron gas.

•

Interaction effects:

a)
b)
c)

Universal interaction Hamiltonian;
Mesoscopic Stoner instability;
Coulomb blockade (strong, weak, mesoscopic);

d)

Kondo effect.

“Quantum dot” used in two different contents:

For the rest of Number
the talk:
of electrons:

1)

“Artificial atom”
Description requires exact diagonalization.
(Kouwnehoven group (Delft))

2)
“Artificial nucleus”

(Marcus group (Harvard))

Statistical description is allowed !!!

Random Matrix Theory for Transport in Quantum Dots

2DEG

QD

2DEG

Energy scales

Level spacing

L

Thouless Energy

Conductance
Assume:

Statistics of transport is determined
only 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:

RMT

Confinement, disorder, etc

No magnetic field, no SO

Magnetic field, no SO

No magnetic field, strong SO

Magnetic field + SO

Conductance of chaotic dot
classical

Mesoscopic
fluctuations

Weak localization
V

I

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

Conductance of chaotic dot
classical

Mesoscopic
fluctuations

Weak localization
V

I

Universal quantum corrections
[Altshuler, Shklovskii (1986)]

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 QD
Aleiner, Fal’ko (2001)

T - invariance

But

Spin dependent flux

Spin relaxation rate

Meir, Gefen, Entin-Wohlman (1989)
Mathur, Stone (1992)
Lyanda-Geller, Mirlin (1994)

Khaetskii, Nazarov (2000)

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 field
Observed in
Folk, Patel, Birnbaum, Marcus, Duruoz, Harris, Jr. (2001)

Interaction Hamiltonian
Energies smaller than Thouless energy:

Random matrix

????

In nuclear physics:

from shell model
random

Universal Interaction Hamiltonian
Energies smaller than Thouless energy:

Kurland, Aleiner, Altshuler (2000)

Only invariants compatible with the circular symmetry

are NOT random !!!

Universal Interaction Hamiltonian
Energies smaller than Thouless energy:

Random matrix
Not random
Valid if:
1)

2) Fundamental symmetries
are NOT broken at larger energies

Universal Interaction Hamiltonian
Energies smaller than Thouless energy:

One-particle levels
Interaction
with
Zero
dimensional
Fermi
liquid
determined by
additional conservations
Wigner – Dyson statistics
Valid if:
1)

2) Fundamental symmetries
are NOT broken at larger energies

Universal Interaction Hamiltonian

Singlet electron-hole
channel.
Triplet electron-hole
channel.

Particle-particle
(Cooper) channel.

Analogy with soft modes in metals

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 of
The ground state S > ½.

Mesoscopic Stoner Instability
Kurland, Aleiner, Altshuler (2000)

Energy of the ground state:

Spin is finite even for NO randomness
Typical
vs.S:

NO

interactions

Does not scale with the size of the system
FM instability
Stoner (1935)

Also Brouwer, Oreg, Halperin (2000)

random with known from RMT
correlation functions

Universal Interaction Hamiltonian

Singlet Channel:
gate voltage
is NOT renormalized
But

Q: What is charge degeneracy of the ground state

(isolated dot)

degeneracy

- half-integer

gap

Otherwise

Coulomb blockade of electron transport
For tunneling contacts:

Charge
degeneracy

Charge gap
Term introduced by Averin and Likharev (1986);
Effect first discussed by C.J. Gorter (1951).

Small quantum dots (~ 500 nm)

conductance (e2/h)

M. Kastner, Physics Today (1993)
E.B. Foxman et al., PRB (1993)

gate voltage (mV)

In metals first observed in Fulton, Dolan, PRL, 59, 109, (1987)

Coulomb blockade (CB) (II)
Strong CB
Weak CB

Mesoscopic CB
(reflectionless
contacts)

Random phase but
not period.

Statistical description of strong CB:
Theory:
Peaks: Jalabert, Stone, Alhassid (1992);
Valleys: Aleiner, Glazman (1996);
Reasonable agreement,
But problems with values of the correlation
fields

Courtesy of C.Marcus

Mesoscopic Coulomb Blockade

Aleiner, Glazman (1998)
Based on technique suggested by:
Matveev (1995); Furusaki, Matveev (1995);
Flensberg (1993).

Experiment:

Suppression
By a factor of 5.3
Th: Predicted 4.

Cronenwett et. al. (1998)

Even-Odd effect due to Kondo effect

Predicted:
Glazman, Raikh (1988)
Ng, Lee (1988)

Spin degeneracy in odd valleys:
Effective Hamiltonian:
local spin density
of conduction electrons

magnetic impurity

Observation:
1998

D. Goldhaber-Gordon et al. (MIT-Weizmann)
S.M. Cronenwett et al. (TU Delft)
J. Schmid et al. (MPI @ Stuttgart)

200 nm
15 mK

800 mK

van der Wiel et al. (2000)

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”)


Slide 27

Theory of Quantum Dots as
Zero-dimensional Metallic Systems
Igor Aleiner (Columbia)
Collaborators:
B.L.Altshuler (Princeton)
P.W.Brouwer (Cornell)
V.I.Falko (Lancaster, UK)
L.I. Glazman (Minnesota)
I.L. Kurland (Princeton)
Physics of the Microworld Conference, Oct. 16 (2004)

Outline:
•
•

Quantum dot (QD) as zero dimensional metal
Random Matrix theory for transport in quantum dots

a)
b)

Non-interacting “standard models”.
Peculiar spin-orbit effects in QD based on 2D electron gas.

•

Interaction effects:

a)
b)
c)

Universal interaction Hamiltonian;
Mesoscopic Stoner instability;
Coulomb blockade (strong, weak, mesoscopic);

d)

Kondo effect.

“Quantum dot” used in two different contents:

For the rest of Number
the talk:
of electrons:

1)

“Artificial atom”
Description requires exact diagonalization.
(Kouwnehoven group (Delft))

2)
“Artificial nucleus”

(Marcus group (Harvard))

Statistical description is allowed !!!

Random Matrix Theory for Transport in Quantum Dots

2DEG

QD

2DEG

Energy scales

Level spacing

L

Thouless Energy

Conductance
Assume:

Statistics of transport is determined
only 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:

RMT

Confinement, disorder, etc

No magnetic field, no SO

Magnetic field, no SO

No magnetic field, strong SO

Magnetic field + SO

Conductance of chaotic dot
classical

Mesoscopic
fluctuations

Weak localization
V

I

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

Conductance of chaotic dot
classical

Mesoscopic
fluctuations

Weak localization
V

I

Universal quantum corrections
[Altshuler, Shklovskii (1986)]

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 QD
Aleiner, Fal’ko (2001)

T - invariance

But

Spin dependent flux

Spin relaxation rate

Meir, Gefen, Entin-Wohlman (1989)
Mathur, Stone (1992)
Lyanda-Geller, Mirlin (1994)

Khaetskii, Nazarov (2000)

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 field
Observed in
Folk, Patel, Birnbaum, Marcus, Duruoz, Harris, Jr. (2001)

Interaction Hamiltonian
Energies smaller than Thouless energy:

Random matrix

????

In nuclear physics:

from shell model
random

Universal Interaction Hamiltonian
Energies smaller than Thouless energy:

Kurland, Aleiner, Altshuler (2000)

Only invariants compatible with the circular symmetry

are NOT random !!!

Universal Interaction Hamiltonian
Energies smaller than Thouless energy:

Random matrix
Not random
Valid if:
1)

2) Fundamental symmetries
are NOT broken at larger energies

Universal Interaction Hamiltonian
Energies smaller than Thouless energy:

One-particle levels
Interaction
with
Zero
dimensional
Fermi
liquid
determined by
additional conservations
Wigner – Dyson statistics
Valid if:
1)

2) Fundamental symmetries
are NOT broken at larger energies

Universal Interaction Hamiltonian

Singlet electron-hole
channel.
Triplet electron-hole
channel.

Particle-particle
(Cooper) channel.

Analogy with soft modes in metals

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 of
The ground state S > ½.

Mesoscopic Stoner Instability
Kurland, Aleiner, Altshuler (2000)

Energy of the ground state:

Spin is finite even for NO randomness
Typical
vs.S:

NO

interactions

Does not scale with the size of the system
FM instability
Stoner (1935)

Also Brouwer, Oreg, Halperin (2000)

random with known from RMT
correlation functions

Universal Interaction Hamiltonian

Singlet Channel:
gate voltage
is NOT renormalized
But

Q: What is charge degeneracy of the ground state

(isolated dot)

degeneracy

- half-integer

gap

Otherwise

Coulomb blockade of electron transport
For tunneling contacts:

Charge
degeneracy

Charge gap
Term introduced by Averin and Likharev (1986);
Effect first discussed by C.J. Gorter (1951).

Small quantum dots (~ 500 nm)

conductance (e2/h)

M. Kastner, Physics Today (1993)
E.B. Foxman et al., PRB (1993)

gate voltage (mV)

In metals first observed in Fulton, Dolan, PRL, 59, 109, (1987)

Coulomb blockade (CB) (II)
Strong CB
Weak CB

Mesoscopic CB
(reflectionless
contacts)

Random phase but
not period.

Statistical description of strong CB:
Theory:
Peaks: Jalabert, Stone, Alhassid (1992);
Valleys: Aleiner, Glazman (1996);
Reasonable agreement,
But problems with values of the correlation
fields

Courtesy of C.Marcus

Mesoscopic Coulomb Blockade

Aleiner, Glazman (1998)
Based on technique suggested by:
Matveev (1995); Furusaki, Matveev (1995);
Flensberg (1993).

Experiment:

Suppression
By a factor of 5.3
Th: Predicted 4.

Cronenwett et. al. (1998)

Even-Odd effect due to Kondo effect

Predicted:
Glazman, Raikh (1988)
Ng, Lee (1988)

Spin degeneracy in odd valleys:
Effective Hamiltonian:
local spin density
of conduction electrons

magnetic impurity

Observation:
1998

D. Goldhaber-Gordon et al. (MIT-Weizmann)
S.M. Cronenwett et al. (TU Delft)
J. Schmid et al. (MPI @ Stuttgart)

200 nm
15 mK

800 mK

van der Wiel et al. (2000)

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”)


Slide 28

Theory of Quantum Dots as
Zero-dimensional Metallic Systems
Igor Aleiner (Columbia)
Collaborators:
B.L.Altshuler (Princeton)
P.W.Brouwer (Cornell)
V.I.Falko (Lancaster, UK)
L.I. Glazman (Minnesota)
I.L. Kurland (Princeton)
Physics of the Microworld Conference, Oct. 16 (2004)

Outline:
•
•

Quantum dot (QD) as zero dimensional metal
Random Matrix theory for transport in quantum dots

a)
b)

Non-interacting “standard models”.
Peculiar spin-orbit effects in QD based on 2D electron gas.

•

Interaction effects:

a)
b)
c)

Universal interaction Hamiltonian;
Mesoscopic Stoner instability;
Coulomb blockade (strong, weak, mesoscopic);

d)

Kondo effect.

“Quantum dot” used in two different contents:

For the rest of Number
the talk:
of electrons:

1)

“Artificial atom”
Description requires exact diagonalization.
(Kouwnehoven group (Delft))

2)
“Artificial nucleus”

(Marcus group (Harvard))

Statistical description is allowed !!!

Random Matrix Theory for Transport in Quantum Dots

2DEG

QD

2DEG

Energy scales

Level spacing

L

Thouless Energy

Conductance
Assume:

Statistics of transport is determined
only 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:

RMT

Confinement, disorder, etc

No magnetic field, no SO

Magnetic field, no SO

No magnetic field, strong SO

Magnetic field + SO

Conductance of chaotic dot
classical

Mesoscopic
fluctuations

Weak localization
V

I

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

Conductance of chaotic dot
classical

Mesoscopic
fluctuations

Weak localization
V

I

Universal quantum corrections
[Altshuler, Shklovskii (1986)]

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 QD
Aleiner, Fal’ko (2001)

T - invariance

But

Spin dependent flux

Spin relaxation rate

Meir, Gefen, Entin-Wohlman (1989)
Mathur, Stone (1992)
Lyanda-Geller, Mirlin (1994)

Khaetskii, Nazarov (2000)

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 field
Observed in
Folk, Patel, Birnbaum, Marcus, Duruoz, Harris, Jr. (2001)

Interaction Hamiltonian
Energies smaller than Thouless energy:

Random matrix

????

In nuclear physics:

from shell model
random

Universal Interaction Hamiltonian
Energies smaller than Thouless energy:

Kurland, Aleiner, Altshuler (2000)

Only invariants compatible with the circular symmetry

are NOT random !!!

Universal Interaction Hamiltonian
Energies smaller than Thouless energy:

Random matrix
Not random
Valid if:
1)

2) Fundamental symmetries
are NOT broken at larger energies

Universal Interaction Hamiltonian
Energies smaller than Thouless energy:

One-particle levels
Interaction
with
Zero
dimensional
Fermi
liquid
determined by
additional conservations
Wigner – Dyson statistics
Valid if:
1)

2) Fundamental symmetries
are NOT broken at larger energies

Universal Interaction Hamiltonian

Singlet electron-hole
channel.
Triplet electron-hole
channel.

Particle-particle
(Cooper) channel.

Analogy with soft modes in metals

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 of
The ground state S > ½.

Mesoscopic Stoner Instability
Kurland, Aleiner, Altshuler (2000)

Energy of the ground state:

Spin is finite even for NO randomness
Typical
vs.S:

NO

interactions

Does not scale with the size of the system
FM instability
Stoner (1935)

Also Brouwer, Oreg, Halperin (2000)

random with known from RMT
correlation functions

Universal Interaction Hamiltonian

Singlet Channel:
gate voltage
is NOT renormalized
But

Q: What is charge degeneracy of the ground state

(isolated dot)

degeneracy

- half-integer

gap

Otherwise

Coulomb blockade of electron transport
For tunneling contacts:

Charge
degeneracy

Charge gap
Term introduced by Averin and Likharev (1986);
Effect first discussed by C.J. Gorter (1951).

Small quantum dots (~ 500 nm)

conductance (e2/h)

M. Kastner, Physics Today (1993)
E.B. Foxman et al., PRB (1993)

gate voltage (mV)

In metals first observed in Fulton, Dolan, PRL, 59, 109, (1987)

Coulomb blockade (CB) (II)
Strong CB
Weak CB

Mesoscopic CB
(reflectionless
contacts)

Random phase but
not period.

Statistical description of strong CB:
Theory:
Peaks: Jalabert, Stone, Alhassid (1992);
Valleys: Aleiner, Glazman (1996);
Reasonable agreement,
But problems with values of the correlation
fields

Courtesy of C.Marcus

Mesoscopic Coulomb Blockade

Aleiner, Glazman (1998)
Based on technique suggested by:
Matveev (1995); Furusaki, Matveev (1995);
Flensberg (1993).

Experiment:

Suppression
By a factor of 5.3
Th: Predicted 4.

Cronenwett et. al. (1998)

Even-Odd effect due to Kondo effect

Predicted:
Glazman, Raikh (1988)
Ng, Lee (1988)

Spin degeneracy in odd valleys:
Effective Hamiltonian:
local spin density
of conduction electrons

magnetic impurity

Observation:
1998

D. Goldhaber-Gordon et al. (MIT-Weizmann)
S.M. Cronenwett et al. (TU Delft)
J. Schmid et al. (MPI @ Stuttgart)

200 nm
15 mK

800 mK

van der Wiel et al. (2000)

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”)


Slide 29

Theory of Quantum Dots as
Zero-dimensional Metallic Systems
Igor Aleiner (Columbia)
Collaborators:
B.L.Altshuler (Princeton)
P.W.Brouwer (Cornell)
V.I.Falko (Lancaster, UK)
L.I. Glazman (Minnesota)
I.L. Kurland (Princeton)
Physics of the Microworld Conference, Oct. 16 (2004)

Outline:
•
•

Quantum dot (QD) as zero dimensional metal
Random Matrix theory for transport in quantum dots

a)
b)

Non-interacting “standard models”.
Peculiar spin-orbit effects in QD based on 2D electron gas.

•

Interaction effects:

a)
b)
c)

Universal interaction Hamiltonian;
Mesoscopic Stoner instability;
Coulomb blockade (strong, weak, mesoscopic);

d)

Kondo effect.

“Quantum dot” used in two different contents:

For the rest of Number
the talk:
of electrons:

1)

“Artificial atom”
Description requires exact diagonalization.
(Kouwnehoven group (Delft))

2)
“Artificial nucleus”

(Marcus group (Harvard))

Statistical description is allowed !!!

Random Matrix Theory for Transport in Quantum Dots

2DEG

QD

2DEG

Energy scales

Level spacing

L

Thouless Energy

Conductance
Assume:

Statistics of transport is determined
only 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:

RMT

Confinement, disorder, etc

No magnetic field, no SO

Magnetic field, no SO

No magnetic field, strong SO

Magnetic field + SO

Conductance of chaotic dot
classical

Mesoscopic
fluctuations

Weak localization
V

I

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

Conductance of chaotic dot
classical

Mesoscopic
fluctuations

Weak localization
V

I

Universal quantum corrections
[Altshuler, Shklovskii (1986)]

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 QD
Aleiner, Fal’ko (2001)

T - invariance

But

Spin dependent flux

Spin relaxation rate

Meir, Gefen, Entin-Wohlman (1989)
Mathur, Stone (1992)
Lyanda-Geller, Mirlin (1994)

Khaetskii, Nazarov (2000)

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 field
Observed in
Folk, Patel, Birnbaum, Marcus, Duruoz, Harris, Jr. (2001)

Interaction Hamiltonian
Energies smaller than Thouless energy:

Random matrix

????

In nuclear physics:

from shell model
random

Universal Interaction Hamiltonian
Energies smaller than Thouless energy:

Kurland, Aleiner, Altshuler (2000)

Only invariants compatible with the circular symmetry

are NOT random !!!

Universal Interaction Hamiltonian
Energies smaller than Thouless energy:

Random matrix
Not random
Valid if:
1)

2) Fundamental symmetries
are NOT broken at larger energies

Universal Interaction Hamiltonian
Energies smaller than Thouless energy:

One-particle levels
Interaction
with
Zero
dimensional
Fermi
liquid
determined by
additional conservations
Wigner – Dyson statistics
Valid if:
1)

2) Fundamental symmetries
are NOT broken at larger energies

Universal Interaction Hamiltonian

Singlet electron-hole
channel.
Triplet electron-hole
channel.

Particle-particle
(Cooper) channel.

Analogy with soft modes in metals

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 of
The ground state S > ½.

Mesoscopic Stoner Instability
Kurland, Aleiner, Altshuler (2000)

Energy of the ground state:

Spin is finite even for NO randomness
Typical
vs.S:

NO

interactions

Does not scale with the size of the system
FM instability
Stoner (1935)

Also Brouwer, Oreg, Halperin (2000)

random with known from RMT
correlation functions

Universal Interaction Hamiltonian

Singlet Channel:
gate voltage
is NOT renormalized
But

Q: What is charge degeneracy of the ground state

(isolated dot)

degeneracy

- half-integer

gap

Otherwise

Coulomb blockade of electron transport
For tunneling contacts:

Charge
degeneracy

Charge gap
Term introduced by Averin and Likharev (1986);
Effect first discussed by C.J. Gorter (1951).

Small quantum dots (~ 500 nm)

conductance (e2/h)

M. Kastner, Physics Today (1993)
E.B. Foxman et al., PRB (1993)

gate voltage (mV)

In metals first observed in Fulton, Dolan, PRL, 59, 109, (1987)

Coulomb blockade (CB) (II)
Strong CB
Weak CB

Mesoscopic CB
(reflectionless
contacts)

Random phase but
not period.

Statistical description of strong CB:
Theory:
Peaks: Jalabert, Stone, Alhassid (1992);
Valleys: Aleiner, Glazman (1996);
Reasonable agreement,
But problems with values of the correlation
fields

Courtesy of C.Marcus

Mesoscopic Coulomb Blockade

Aleiner, Glazman (1998)
Based on technique suggested by:
Matveev (1995); Furusaki, Matveev (1995);
Flensberg (1993).

Experiment:

Suppression
By a factor of 5.3
Th: Predicted 4.

Cronenwett et. al. (1998)

Even-Odd effect due to Kondo effect

Predicted:
Glazman, Raikh (1988)
Ng, Lee (1988)

Spin degeneracy in odd valleys:
Effective Hamiltonian:
local spin density
of conduction electrons

magnetic impurity

Observation:
1998

D. Goldhaber-Gordon et al. (MIT-Weizmann)
S.M. Cronenwett et al. (TU Delft)
J. Schmid et al. (MPI @ Stuttgart)

200 nm
15 mK

800 mK

van der Wiel et al. (2000)

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”)


Slide 30

Theory of Quantum Dots as
Zero-dimensional Metallic Systems
Igor Aleiner (Columbia)
Collaborators:
B.L.Altshuler (Princeton)
P.W.Brouwer (Cornell)
V.I.Falko (Lancaster, UK)
L.I. Glazman (Minnesota)
I.L. Kurland (Princeton)
Physics of the Microworld Conference, Oct. 16 (2004)

Outline:
•
•

Quantum dot (QD) as zero dimensional metal
Random Matrix theory for transport in quantum dots

a)
b)

Non-interacting “standard models”.
Peculiar spin-orbit effects in QD based on 2D electron gas.

•

Interaction effects:

a)
b)
c)

Universal interaction Hamiltonian;
Mesoscopic Stoner instability;
Coulomb blockade (strong, weak, mesoscopic);

d)

Kondo effect.

“Quantum dot” used in two different contents:

For the rest of Number
the talk:
of electrons:

1)

“Artificial atom”
Description requires exact diagonalization.
(Kouwnehoven group (Delft))

2)
“Artificial nucleus”

(Marcus group (Harvard))

Statistical description is allowed !!!

Random Matrix Theory for Transport in Quantum Dots

2DEG

QD

2DEG

Energy scales

Level spacing

L

Thouless Energy

Conductance
Assume:

Statistics of transport is determined
only 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:

RMT

Confinement, disorder, etc

No magnetic field, no SO

Magnetic field, no SO

No magnetic field, strong SO

Magnetic field + SO

Conductance of chaotic dot
classical

Mesoscopic
fluctuations

Weak localization
V

I

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

Conductance of chaotic dot
classical

Mesoscopic
fluctuations

Weak localization
V

I

Universal quantum corrections
[Altshuler, Shklovskii (1986)]

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 QD
Aleiner, Fal’ko (2001)

T - invariance

But

Spin dependent flux

Spin relaxation rate

Meir, Gefen, Entin-Wohlman (1989)
Mathur, Stone (1992)
Lyanda-Geller, Mirlin (1994)

Khaetskii, Nazarov (2000)

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 field
Observed in
Folk, Patel, Birnbaum, Marcus, Duruoz, Harris, Jr. (2001)

Interaction Hamiltonian
Energies smaller than Thouless energy:

Random matrix

????

In nuclear physics:

from shell model
random

Universal Interaction Hamiltonian
Energies smaller than Thouless energy:

Kurland, Aleiner, Altshuler (2000)

Only invariants compatible with the circular symmetry

are NOT random !!!

Universal Interaction Hamiltonian
Energies smaller than Thouless energy:

Random matrix
Not random
Valid if:
1)

2) Fundamental symmetries
are NOT broken at larger energies

Universal Interaction Hamiltonian
Energies smaller than Thouless energy:

One-particle levels
Interaction
with
Zero
dimensional
Fermi
liquid
determined by
additional conservations
Wigner – Dyson statistics
Valid if:
1)

2) Fundamental symmetries
are NOT broken at larger energies

Universal Interaction Hamiltonian

Singlet electron-hole
channel.
Triplet electron-hole
channel.

Particle-particle
(Cooper) channel.

Analogy with soft modes in metals

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 of
The ground state S > ½.

Mesoscopic Stoner Instability
Kurland, Aleiner, Altshuler (2000)

Energy of the ground state:

Spin is finite even for NO randomness
Typical
vs.S:

NO

interactions

Does not scale with the size of the system
FM instability
Stoner (1935)

Also Brouwer, Oreg, Halperin (2000)

random with known from RMT
correlation functions

Universal Interaction Hamiltonian

Singlet Channel:
gate voltage
is NOT renormalized
But

Q: What is charge degeneracy of the ground state

(isolated dot)

degeneracy

- half-integer

gap

Otherwise

Coulomb blockade of electron transport
For tunneling contacts:

Charge
degeneracy

Charge gap
Term introduced by Averin and Likharev (1986);
Effect first discussed by C.J. Gorter (1951).

Small quantum dots (~ 500 nm)

conductance (e2/h)

M. Kastner, Physics Today (1993)
E.B. Foxman et al., PRB (1993)

gate voltage (mV)

In metals first observed in Fulton, Dolan, PRL, 59, 109, (1987)

Coulomb blockade (CB) (II)
Strong CB
Weak CB

Mesoscopic CB
(reflectionless
contacts)

Random phase but
not period.

Statistical description of strong CB:
Theory:
Peaks: Jalabert, Stone, Alhassid (1992);
Valleys: Aleiner, Glazman (1996);
Reasonable agreement,
But problems with values of the correlation
fields

Courtesy of C.Marcus

Mesoscopic Coulomb Blockade

Aleiner, Glazman (1998)
Based on technique suggested by:
Matveev (1995); Furusaki, Matveev (1995);
Flensberg (1993).

Experiment:

Suppression
By a factor of 5.3
Th: Predicted 4.

Cronenwett et. al. (1998)

Even-Odd effect due to Kondo effect

Predicted:
Glazman, Raikh (1988)
Ng, Lee (1988)

Spin degeneracy in odd valleys:
Effective Hamiltonian:
local spin density
of conduction electrons

magnetic impurity

Observation:
1998

D. Goldhaber-Gordon et al. (MIT-Weizmann)
S.M. Cronenwett et al. (TU Delft)
J. Schmid et al. (MPI @ Stuttgart)

200 nm
15 mK

800 mK

van der Wiel et al. (2000)

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”)


Slide 31

Theory of Quantum Dots as
Zero-dimensional Metallic Systems
Igor Aleiner (Columbia)
Collaborators:
B.L.Altshuler (Princeton)
P.W.Brouwer (Cornell)
V.I.Falko (Lancaster, UK)
L.I. Glazman (Minnesota)
I.L. Kurland (Princeton)
Physics of the Microworld Conference, Oct. 16 (2004)

Outline:
•
•

Quantum dot (QD) as zero dimensional metal
Random Matrix theory for transport in quantum dots

a)
b)

Non-interacting “standard models”.
Peculiar spin-orbit effects in QD based on 2D electron gas.

•

Interaction effects:

a)
b)
c)

Universal interaction Hamiltonian;
Mesoscopic Stoner instability;
Coulomb blockade (strong, weak, mesoscopic);

d)

Kondo effect.

“Quantum dot” used in two different contents:

For the rest of Number
the talk:
of electrons:

1)

“Artificial atom”
Description requires exact diagonalization.
(Kouwnehoven group (Delft))

2)
“Artificial nucleus”

(Marcus group (Harvard))

Statistical description is allowed !!!

Random Matrix Theory for Transport in Quantum Dots

2DEG

QD

2DEG

Energy scales

Level spacing

L

Thouless Energy

Conductance
Assume:

Statistics of transport is determined
only 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:

RMT

Confinement, disorder, etc

No magnetic field, no SO

Magnetic field, no SO

No magnetic field, strong SO

Magnetic field + SO

Conductance of chaotic dot
classical

Mesoscopic
fluctuations

Weak localization
V

I

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

Conductance of chaotic dot
classical

Mesoscopic
fluctuations

Weak localization
V

I

Universal quantum corrections
[Altshuler, Shklovskii (1986)]

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 QD
Aleiner, Fal’ko (2001)

T - invariance

But

Spin dependent flux

Spin relaxation rate

Meir, Gefen, Entin-Wohlman (1989)
Mathur, Stone (1992)
Lyanda-Geller, Mirlin (1994)

Khaetskii, Nazarov (2000)

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 field
Observed in
Folk, Patel, Birnbaum, Marcus, Duruoz, Harris, Jr. (2001)

Interaction Hamiltonian
Energies smaller than Thouless energy:

Random matrix

????

In nuclear physics:

from shell model
random

Universal Interaction Hamiltonian
Energies smaller than Thouless energy:

Kurland, Aleiner, Altshuler (2000)

Only invariants compatible with the circular symmetry

are NOT random !!!

Universal Interaction Hamiltonian
Energies smaller than Thouless energy:

Random matrix
Not random
Valid if:
1)

2) Fundamental symmetries
are NOT broken at larger energies

Universal Interaction Hamiltonian
Energies smaller than Thouless energy:

One-particle levels
Interaction
with
Zero
dimensional
Fermi
liquid
determined by
additional conservations
Wigner – Dyson statistics
Valid if:
1)

2) Fundamental symmetries
are NOT broken at larger energies

Universal Interaction Hamiltonian

Singlet electron-hole
channel.
Triplet electron-hole
channel.

Particle-particle
(Cooper) channel.

Analogy with soft modes in metals

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 of
The ground state S > ½.

Mesoscopic Stoner Instability
Kurland, Aleiner, Altshuler (2000)

Energy of the ground state:

Spin is finite even for NO randomness
Typical
vs.S:

NO

interactions

Does not scale with the size of the system
FM instability
Stoner (1935)

Also Brouwer, Oreg, Halperin (2000)

random with known from RMT
correlation functions

Universal Interaction Hamiltonian

Singlet Channel:
gate voltage
is NOT renormalized
But

Q: What is charge degeneracy of the ground state

(isolated dot)

degeneracy

- half-integer

gap

Otherwise

Coulomb blockade of electron transport
For tunneling contacts:

Charge
degeneracy

Charge gap
Term introduced by Averin and Likharev (1986);
Effect first discussed by C.J. Gorter (1951).

Small quantum dots (~ 500 nm)

conductance (e2/h)

M. Kastner, Physics Today (1993)
E.B. Foxman et al., PRB (1993)

gate voltage (mV)

In metals first observed in Fulton, Dolan, PRL, 59, 109, (1987)

Coulomb blockade (CB) (II)
Strong CB
Weak CB

Mesoscopic CB
(reflectionless
contacts)

Random phase but
not period.

Statistical description of strong CB:
Theory:
Peaks: Jalabert, Stone, Alhassid (1992);
Valleys: Aleiner, Glazman (1996);
Reasonable agreement,
But problems with values of the correlation
fields

Courtesy of C.Marcus

Mesoscopic Coulomb Blockade

Aleiner, Glazman (1998)
Based on technique suggested by:
Matveev (1995); Furusaki, Matveev (1995);
Flensberg (1993).

Experiment:

Suppression
By a factor of 5.3
Th: Predicted 4.

Cronenwett et. al. (1998)

Even-Odd effect due to Kondo effect

Predicted:
Glazman, Raikh (1988)
Ng, Lee (1988)

Spin degeneracy in odd valleys:
Effective Hamiltonian:
local spin density
of conduction electrons

magnetic impurity

Observation:
1998

D. Goldhaber-Gordon et al. (MIT-Weizmann)
S.M. Cronenwett et al. (TU Delft)
J. Schmid et al. (MPI @ Stuttgart)

200 nm
15 mK

800 mK

van der Wiel et al. (2000)

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”)


Slide 32

Theory of Quantum Dots as
Zero-dimensional Metallic Systems
Igor Aleiner (Columbia)
Collaborators:
B.L.Altshuler (Princeton)
P.W.Brouwer (Cornell)
V.I.Falko (Lancaster, UK)
L.I. Glazman (Minnesota)
I.L. Kurland (Princeton)
Physics of the Microworld Conference, Oct. 16 (2004)

Outline:
•
•

Quantum dot (QD) as zero dimensional metal
Random Matrix theory for transport in quantum dots

a)
b)

Non-interacting “standard models”.
Peculiar spin-orbit effects in QD based on 2D electron gas.

•

Interaction effects:

a)
b)
c)

Universal interaction Hamiltonian;
Mesoscopic Stoner instability;
Coulomb blockade (strong, weak, mesoscopic);

d)

Kondo effect.

“Quantum dot” used in two different contents:

For the rest of Number
the talk:
of electrons:

1)

“Artificial atom”
Description requires exact diagonalization.
(Kouwnehoven group (Delft))

2)
“Artificial nucleus”

(Marcus group (Harvard))

Statistical description is allowed !!!

Random Matrix Theory for Transport in Quantum Dots

2DEG

QD

2DEG

Energy scales

Level spacing

L

Thouless Energy

Conductance
Assume:

Statistics of transport is determined
only 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:

RMT

Confinement, disorder, etc

No magnetic field, no SO

Magnetic field, no SO

No magnetic field, strong SO

Magnetic field + SO

Conductance of chaotic dot
classical

Mesoscopic
fluctuations

Weak localization
V

I

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

Conductance of chaotic dot
classical

Mesoscopic
fluctuations

Weak localization
V

I

Universal quantum corrections
[Altshuler, Shklovskii (1986)]

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 QD
Aleiner, Fal’ko (2001)

T - invariance

But

Spin dependent flux

Spin relaxation rate

Meir, Gefen, Entin-Wohlman (1989)
Mathur, Stone (1992)
Lyanda-Geller, Mirlin (1994)

Khaetskii, Nazarov (2000)

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 field
Observed in
Folk, Patel, Birnbaum, Marcus, Duruoz, Harris, Jr. (2001)

Interaction Hamiltonian
Energies smaller than Thouless energy:

Random matrix

????

In nuclear physics:

from shell model
random

Universal Interaction Hamiltonian
Energies smaller than Thouless energy:

Kurland, Aleiner, Altshuler (2000)

Only invariants compatible with the circular symmetry

are NOT random !!!

Universal Interaction Hamiltonian
Energies smaller than Thouless energy:

Random matrix
Not random
Valid if:
1)

2) Fundamental symmetries
are NOT broken at larger energies

Universal Interaction Hamiltonian
Energies smaller than Thouless energy:

One-particle levels
Interaction
with
Zero
dimensional
Fermi
liquid
determined by
additional conservations
Wigner – Dyson statistics
Valid if:
1)

2) Fundamental symmetries
are NOT broken at larger energies

Universal Interaction Hamiltonian

Singlet electron-hole
channel.
Triplet electron-hole
channel.

Particle-particle
(Cooper) channel.

Analogy with soft modes in metals

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 of
The ground state S > ½.

Mesoscopic Stoner Instability
Kurland, Aleiner, Altshuler (2000)

Energy of the ground state:

Spin is finite even for NO randomness
Typical
vs.S:

NO

interactions

Does not scale with the size of the system
FM instability
Stoner (1935)

Also Brouwer, Oreg, Halperin (2000)

random with known from RMT
correlation functions

Universal Interaction Hamiltonian

Singlet Channel:
gate voltage
is NOT renormalized
But

Q: What is charge degeneracy of the ground state

(isolated dot)

degeneracy

- half-integer

gap

Otherwise

Coulomb blockade of electron transport
For tunneling contacts:

Charge
degeneracy

Charge gap
Term introduced by Averin and Likharev (1986);
Effect first discussed by C.J. Gorter (1951).

Small quantum dots (~ 500 nm)

conductance (e2/h)

M. Kastner, Physics Today (1993)
E.B. Foxman et al., PRB (1993)

gate voltage (mV)

In metals first observed in Fulton, Dolan, PRL, 59, 109, (1987)

Coulomb blockade (CB) (II)
Strong CB
Weak CB

Mesoscopic CB
(reflectionless
contacts)

Random phase but
not period.

Statistical description of strong CB:
Theory:
Peaks: Jalabert, Stone, Alhassid (1992);
Valleys: Aleiner, Glazman (1996);
Reasonable agreement,
But problems with values of the correlation
fields

Courtesy of C.Marcus

Mesoscopic Coulomb Blockade

Aleiner, Glazman (1998)
Based on technique suggested by:
Matveev (1995); Furusaki, Matveev (1995);
Flensberg (1993).

Experiment:

Suppression
By a factor of 5.3
Th: Predicted 4.

Cronenwett et. al. (1998)

Even-Odd effect due to Kondo effect

Predicted:
Glazman, Raikh (1988)
Ng, Lee (1988)

Spin degeneracy in odd valleys:
Effective Hamiltonian:
local spin density
of conduction electrons

magnetic impurity

Observation:
1998

D. Goldhaber-Gordon et al. (MIT-Weizmann)
S.M. Cronenwett et al. (TU Delft)
J. Schmid et al. (MPI @ Stuttgart)

200 nm
15 mK

800 mK

van der Wiel et al. (2000)

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”)


Slide 33

Theory of Quantum Dots as
Zero-dimensional Metallic Systems
Igor Aleiner (Columbia)
Collaborators:
B.L.Altshuler (Princeton)
P.W.Brouwer (Cornell)
V.I.Falko (Lancaster, UK)
L.I. Glazman (Minnesota)
I.L. Kurland (Princeton)
Physics of the Microworld Conference, Oct. 16 (2004)

Outline:
•
•

Quantum dot (QD) as zero dimensional metal
Random Matrix theory for transport in quantum dots

a)
b)

Non-interacting “standard models”.
Peculiar spin-orbit effects in QD based on 2D electron gas.

•

Interaction effects:

a)
b)
c)

Universal interaction Hamiltonian;
Mesoscopic Stoner instability;
Coulomb blockade (strong, weak, mesoscopic);

d)

Kondo effect.

“Quantum dot” used in two different contents:

For the rest of Number
the talk:
of electrons:

1)

“Artificial atom”
Description requires exact diagonalization.
(Kouwnehoven group (Delft))

2)
“Artificial nucleus”

(Marcus group (Harvard))

Statistical description is allowed !!!

Random Matrix Theory for Transport in Quantum Dots

2DEG

QD

2DEG

Energy scales

Level spacing

L

Thouless Energy

Conductance
Assume:

Statistics of transport is determined
only 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:

RMT

Confinement, disorder, etc

No magnetic field, no SO

Magnetic field, no SO

No magnetic field, strong SO

Magnetic field + SO

Conductance of chaotic dot
classical

Mesoscopic
fluctuations

Weak localization
V

I

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

Conductance of chaotic dot
classical

Mesoscopic
fluctuations

Weak localization
V

I

Universal quantum corrections
[Altshuler, Shklovskii (1986)]

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 QD
Aleiner, Fal’ko (2001)

T - invariance

But

Spin dependent flux

Spin relaxation rate

Meir, Gefen, Entin-Wohlman (1989)
Mathur, Stone (1992)
Lyanda-Geller, Mirlin (1994)

Khaetskii, Nazarov (2000)

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 field
Observed in
Folk, Patel, Birnbaum, Marcus, Duruoz, Harris, Jr. (2001)

Interaction Hamiltonian
Energies smaller than Thouless energy:

Random matrix

????

In nuclear physics:

from shell model
random

Universal Interaction Hamiltonian
Energies smaller than Thouless energy:

Kurland, Aleiner, Altshuler (2000)

Only invariants compatible with the circular symmetry

are NOT random !!!

Universal Interaction Hamiltonian
Energies smaller than Thouless energy:

Random matrix
Not random
Valid if:
1)

2) Fundamental symmetries
are NOT broken at larger energies

Universal Interaction Hamiltonian
Energies smaller than Thouless energy:

One-particle levels
Interaction
with
Zero
dimensional
Fermi
liquid
determined by
additional conservations
Wigner – Dyson statistics
Valid if:
1)

2) Fundamental symmetries
are NOT broken at larger energies

Universal Interaction Hamiltonian

Singlet electron-hole
channel.
Triplet electron-hole
channel.

Particle-particle
(Cooper) channel.

Analogy with soft modes in metals

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 of
The ground state S > ½.

Mesoscopic Stoner Instability
Kurland, Aleiner, Altshuler (2000)

Energy of the ground state:

Spin is finite even for NO randomness
Typical
vs.S:

NO

interactions

Does not scale with the size of the system
FM instability
Stoner (1935)

Also Brouwer, Oreg, Halperin (2000)

random with known from RMT
correlation functions

Universal Interaction Hamiltonian

Singlet Channel:
gate voltage
is NOT renormalized
But

Q: What is charge degeneracy of the ground state

(isolated dot)

degeneracy

- half-integer

gap

Otherwise

Coulomb blockade of electron transport
For tunneling contacts:

Charge
degeneracy

Charge gap
Term introduced by Averin and Likharev (1986);
Effect first discussed by C.J. Gorter (1951).

Small quantum dots (~ 500 nm)

conductance (e2/h)

M. Kastner, Physics Today (1993)
E.B. Foxman et al., PRB (1993)

gate voltage (mV)

In metals first observed in Fulton, Dolan, PRL, 59, 109, (1987)

Coulomb blockade (CB) (II)
Strong CB
Weak CB

Mesoscopic CB
(reflectionless
contacts)

Random phase but
not period.

Statistical description of strong CB:
Theory:
Peaks: Jalabert, Stone, Alhassid (1992);
Valleys: Aleiner, Glazman (1996);
Reasonable agreement,
But problems with values of the correlation
fields

Courtesy of C.Marcus

Mesoscopic Coulomb Blockade

Aleiner, Glazman (1998)
Based on technique suggested by:
Matveev (1995); Furusaki, Matveev (1995);
Flensberg (1993).

Experiment:

Suppression
By a factor of 5.3
Th: Predicted 4.

Cronenwett et. al. (1998)

Even-Odd effect due to Kondo effect

Predicted:
Glazman, Raikh (1988)
Ng, Lee (1988)

Spin degeneracy in odd valleys:
Effective Hamiltonian:
local spin density
of conduction electrons

magnetic impurity

Observation:
1998

D. Goldhaber-Gordon et al. (MIT-Weizmann)
S.M. Cronenwett et al. (TU Delft)
J. Schmid et al. (MPI @ Stuttgart)

200 nm
15 mK

800 mK

van der Wiel et al. (2000)

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”)


Slide 34

Theory of Quantum Dots as
Zero-dimensional Metallic Systems
Igor Aleiner (Columbia)
Collaborators:
B.L.Altshuler (Princeton)
P.W.Brouwer (Cornell)
V.I.Falko (Lancaster, UK)
L.I. Glazman (Minnesota)
I.L. Kurland (Princeton)
Physics of the Microworld Conference, Oct. 16 (2004)

Outline:
•
•

Quantum dot (QD) as zero dimensional metal
Random Matrix theory for transport in quantum dots

a)
b)

Non-interacting “standard models”.
Peculiar spin-orbit effects in QD based on 2D electron gas.

•

Interaction effects:

a)
b)
c)

Universal interaction Hamiltonian;
Mesoscopic Stoner instability;
Coulomb blockade (strong, weak, mesoscopic);

d)

Kondo effect.

“Quantum dot” used in two different contents:

For the rest of Number
the talk:
of electrons:

1)

“Artificial atom”
Description requires exact diagonalization.
(Kouwnehoven group (Delft))

2)
“Artificial nucleus”

(Marcus group (Harvard))

Statistical description is allowed !!!

Random Matrix Theory for Transport in Quantum Dots

2DEG

QD

2DEG

Energy scales

Level spacing

L

Thouless Energy

Conductance
Assume:

Statistics of transport is determined
only 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:

RMT

Confinement, disorder, etc

No magnetic field, no SO

Magnetic field, no SO

No magnetic field, strong SO

Magnetic field + SO

Conductance of chaotic dot
classical

Mesoscopic
fluctuations

Weak localization
V

I

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

Conductance of chaotic dot
classical

Mesoscopic
fluctuations

Weak localization
V

I

Universal quantum corrections
[Altshuler, Shklovskii (1986)]

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 QD
Aleiner, Fal’ko (2001)

T - invariance

But

Spin dependent flux

Spin relaxation rate

Meir, Gefen, Entin-Wohlman (1989)
Mathur, Stone (1992)
Lyanda-Geller, Mirlin (1994)

Khaetskii, Nazarov (2000)

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 field
Observed in
Folk, Patel, Birnbaum, Marcus, Duruoz, Harris, Jr. (2001)

Interaction Hamiltonian
Energies smaller than Thouless energy:

Random matrix

????

In nuclear physics:

from shell model
random

Universal Interaction Hamiltonian
Energies smaller than Thouless energy:

Kurland, Aleiner, Altshuler (2000)

Only invariants compatible with the circular symmetry

are NOT random !!!

Universal Interaction Hamiltonian
Energies smaller than Thouless energy:

Random matrix
Not random
Valid if:
1)

2) Fundamental symmetries
are NOT broken at larger energies

Universal Interaction Hamiltonian
Energies smaller than Thouless energy:

One-particle levels
Interaction
with
Zero
dimensional
Fermi
liquid
determined by
additional conservations
Wigner – Dyson statistics
Valid if:
1)

2) Fundamental symmetries
are NOT broken at larger energies

Universal Interaction Hamiltonian

Singlet electron-hole
channel.
Triplet electron-hole
channel.

Particle-particle
(Cooper) channel.

Analogy with soft modes in metals

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 of
The ground state S > ½.

Mesoscopic Stoner Instability
Kurland, Aleiner, Altshuler (2000)

Energy of the ground state:

Spin is finite even for NO randomness
Typical
vs.S:

NO

interactions

Does not scale with the size of the system
FM instability
Stoner (1935)

Also Brouwer, Oreg, Halperin (2000)

random with known from RMT
correlation functions

Universal Interaction Hamiltonian

Singlet Channel:
gate voltage
is NOT renormalized
But

Q: What is charge degeneracy of the ground state

(isolated dot)

degeneracy

- half-integer

gap

Otherwise

Coulomb blockade of electron transport
For tunneling contacts:

Charge
degeneracy

Charge gap
Term introduced by Averin and Likharev (1986);
Effect first discussed by C.J. Gorter (1951).

Small quantum dots (~ 500 nm)

conductance (e2/h)

M. Kastner, Physics Today (1993)
E.B. Foxman et al., PRB (1993)

gate voltage (mV)

In metals first observed in Fulton, Dolan, PRL, 59, 109, (1987)

Coulomb blockade (CB) (II)
Strong CB
Weak CB

Mesoscopic CB
(reflectionless
contacts)

Random phase but
not period.

Statistical description of strong CB:
Theory:
Peaks: Jalabert, Stone, Alhassid (1992);
Valleys: Aleiner, Glazman (1996);
Reasonable agreement,
But problems with values of the correlation
fields

Courtesy of C.Marcus

Mesoscopic Coulomb Blockade

Aleiner, Glazman (1998)
Based on technique suggested by:
Matveev (1995); Furusaki, Matveev (1995);
Flensberg (1993).

Experiment:

Suppression
By a factor of 5.3
Th: Predicted 4.

Cronenwett et. al. (1998)

Even-Odd effect due to Kondo effect

Predicted:
Glazman, Raikh (1988)
Ng, Lee (1988)

Spin degeneracy in odd valleys:
Effective Hamiltonian:
local spin density
of conduction electrons

magnetic impurity

Observation:
1998

D. Goldhaber-Gordon et al. (MIT-Weizmann)
S.M. Cronenwett et al. (TU Delft)
J. Schmid et al. (MPI @ Stuttgart)

200 nm
15 mK

800 mK

van der Wiel et al. (2000)

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”)


Slide 35

Theory of Quantum Dots as
Zero-dimensional Metallic Systems
Igor Aleiner (Columbia)
Collaborators:
B.L.Altshuler (Princeton)
P.W.Brouwer (Cornell)
V.I.Falko (Lancaster, UK)
L.I. Glazman (Minnesota)
I.L. Kurland (Princeton)
Physics of the Microworld Conference, Oct. 16 (2004)

Outline:
•
•

Quantum dot (QD) as zero dimensional metal
Random Matrix theory for transport in quantum dots

a)
b)

Non-interacting “standard models”.
Peculiar spin-orbit effects in QD based on 2D electron gas.

•

Interaction effects:

a)
b)
c)

Universal interaction Hamiltonian;
Mesoscopic Stoner instability;
Coulomb blockade (strong, weak, mesoscopic);

d)

Kondo effect.

“Quantum dot” used in two different contents:

For the rest of Number
the talk:
of electrons:

1)

“Artificial atom”
Description requires exact diagonalization.
(Kouwnehoven group (Delft))

2)
“Artificial nucleus”

(Marcus group (Harvard))

Statistical description is allowed !!!

Random Matrix Theory for Transport in Quantum Dots

2DEG

QD

2DEG

Energy scales

Level spacing

L

Thouless Energy

Conductance
Assume:

Statistics of transport is determined
only 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:

RMT

Confinement, disorder, etc

No magnetic field, no SO

Magnetic field, no SO

No magnetic field, strong SO

Magnetic field + SO

Conductance of chaotic dot
classical

Mesoscopic
fluctuations

Weak localization
V

I

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

Conductance of chaotic dot
classical

Mesoscopic
fluctuations

Weak localization
V

I

Universal quantum corrections
[Altshuler, Shklovskii (1986)]

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 QD
Aleiner, Fal’ko (2001)

T - invariance

But

Spin dependent flux

Spin relaxation rate

Meir, Gefen, Entin-Wohlman (1989)
Mathur, Stone (1992)
Lyanda-Geller, Mirlin (1994)

Khaetskii, Nazarov (2000)

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 field
Observed in
Folk, Patel, Birnbaum, Marcus, Duruoz, Harris, Jr. (2001)

Interaction Hamiltonian
Energies smaller than Thouless energy:

Random matrix

????

In nuclear physics:

from shell model
random

Universal Interaction Hamiltonian
Energies smaller than Thouless energy:

Kurland, Aleiner, Altshuler (2000)

Only invariants compatible with the circular symmetry

are NOT random !!!

Universal Interaction Hamiltonian
Energies smaller than Thouless energy:

Random matrix
Not random
Valid if:
1)

2) Fundamental symmetries
are NOT broken at larger energies

Universal Interaction Hamiltonian
Energies smaller than Thouless energy:

One-particle levels
Interaction
with
Zero
dimensional
Fermi
liquid
determined by
additional conservations
Wigner – Dyson statistics
Valid if:
1)

2) Fundamental symmetries
are NOT broken at larger energies

Universal Interaction Hamiltonian

Singlet electron-hole
channel.
Triplet electron-hole
channel.

Particle-particle
(Cooper) channel.

Analogy with soft modes in metals

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 of
The ground state S > ½.

Mesoscopic Stoner Instability
Kurland, Aleiner, Altshuler (2000)

Energy of the ground state:

Spin is finite even for NO randomness
Typical
vs.S:

NO

interactions

Does not scale with the size of the system
FM instability
Stoner (1935)

Also Brouwer, Oreg, Halperin (2000)

random with known from RMT
correlation functions

Universal Interaction Hamiltonian

Singlet Channel:
gate voltage
is NOT renormalized
But

Q: What is charge degeneracy of the ground state

(isolated dot)

degeneracy

- half-integer

gap

Otherwise

Coulomb blockade of electron transport
For tunneling contacts:

Charge
degeneracy

Charge gap
Term introduced by Averin and Likharev (1986);
Effect first discussed by C.J. Gorter (1951).

Small quantum dots (~ 500 nm)

conductance (e2/h)

M. Kastner, Physics Today (1993)
E.B. Foxman et al., PRB (1993)

gate voltage (mV)

In metals first observed in Fulton, Dolan, PRL, 59, 109, (1987)

Coulomb blockade (CB) (II)
Strong CB
Weak CB

Mesoscopic CB
(reflectionless
contacts)

Random phase but
not period.

Statistical description of strong CB:
Theory:
Peaks: Jalabert, Stone, Alhassid (1992);
Valleys: Aleiner, Glazman (1996);
Reasonable agreement,
But problems with values of the correlation
fields

Courtesy of C.Marcus

Mesoscopic Coulomb Blockade

Aleiner, Glazman (1998)
Based on technique suggested by:
Matveev (1995); Furusaki, Matveev (1995);
Flensberg (1993).

Experiment:

Suppression
By a factor of 5.3
Th: Predicted 4.

Cronenwett et. al. (1998)

Even-Odd effect due to Kondo effect

Predicted:
Glazman, Raikh (1988)
Ng, Lee (1988)

Spin degeneracy in odd valleys:
Effective Hamiltonian:
local spin density
of conduction electrons

magnetic impurity

Observation:
1998

D. Goldhaber-Gordon et al. (MIT-Weizmann)
S.M. Cronenwett et al. (TU Delft)
J. Schmid et al. (MPI @ Stuttgart)

200 nm
15 mK

800 mK

van der Wiel et al. (2000)

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”)


Slide 36

Theory of Quantum Dots as
Zero-dimensional Metallic Systems
Igor Aleiner (Columbia)
Collaborators:
B.L.Altshuler (Princeton)
P.W.Brouwer (Cornell)
V.I.Falko (Lancaster, UK)
L.I. Glazman (Minnesota)
I.L. Kurland (Princeton)
Physics of the Microworld Conference, Oct. 16 (2004)

Outline:
•
•

Quantum dot (QD) as zero dimensional metal
Random Matrix theory for transport in quantum dots

a)
b)

Non-interacting “standard models”.
Peculiar spin-orbit effects in QD based on 2D electron gas.

•

Interaction effects:

a)
b)
c)

Universal interaction Hamiltonian;
Mesoscopic Stoner instability;
Coulomb blockade (strong, weak, mesoscopic);

d)

Kondo effect.

“Quantum dot” used in two different contents:

For the rest of Number
the talk:
of electrons:

1)

“Artificial atom”
Description requires exact diagonalization.
(Kouwnehoven group (Delft))

2)
“Artificial nucleus”

(Marcus group (Harvard))

Statistical description is allowed !!!

Random Matrix Theory for Transport in Quantum Dots

2DEG

QD

2DEG

Energy scales

Level spacing

L

Thouless Energy

Conductance
Assume:

Statistics of transport is determined
only 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:

RMT

Confinement, disorder, etc

No magnetic field, no SO

Magnetic field, no SO

No magnetic field, strong SO

Magnetic field + SO

Conductance of chaotic dot
classical

Mesoscopic
fluctuations

Weak localization
V

I

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

Conductance of chaotic dot
classical

Mesoscopic
fluctuations

Weak localization
V

I

Universal quantum corrections
[Altshuler, Shklovskii (1986)]

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 QD
Aleiner, Fal’ko (2001)

T - invariance

But

Spin dependent flux

Spin relaxation rate

Meir, Gefen, Entin-Wohlman (1989)
Mathur, Stone (1992)
Lyanda-Geller, Mirlin (1994)

Khaetskii, Nazarov (2000)

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 field
Observed in
Folk, Patel, Birnbaum, Marcus, Duruoz, Harris, Jr. (2001)

Interaction Hamiltonian
Energies smaller than Thouless energy:

Random matrix

????

In nuclear physics:

from shell model
random

Universal Interaction Hamiltonian
Energies smaller than Thouless energy:

Kurland, Aleiner, Altshuler (2000)

Only invariants compatible with the circular symmetry

are NOT random !!!

Universal Interaction Hamiltonian
Energies smaller than Thouless energy:

Random matrix
Not random
Valid if:
1)

2) Fundamental symmetries
are NOT broken at larger energies

Universal Interaction Hamiltonian
Energies smaller than Thouless energy:

One-particle levels
Interaction
with
Zero
dimensional
Fermi
liquid
determined by
additional conservations
Wigner – Dyson statistics
Valid if:
1)

2) Fundamental symmetries
are NOT broken at larger energies

Universal Interaction Hamiltonian

Singlet electron-hole
channel.
Triplet electron-hole
channel.

Particle-particle
(Cooper) channel.

Analogy with soft modes in metals

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 of
The ground state S > ½.

Mesoscopic Stoner Instability
Kurland, Aleiner, Altshuler (2000)

Energy of the ground state:

Spin is finite even for NO randomness
Typical
vs.S:

NO

interactions

Does not scale with the size of the system
FM instability
Stoner (1935)

Also Brouwer, Oreg, Halperin (2000)

random with known from RMT
correlation functions

Universal Interaction Hamiltonian

Singlet Channel:
gate voltage
is NOT renormalized
But

Q: What is charge degeneracy of the ground state

(isolated dot)

degeneracy

- half-integer

gap

Otherwise

Coulomb blockade of electron transport
For tunneling contacts:

Charge
degeneracy

Charge gap
Term introduced by Averin and Likharev (1986);
Effect first discussed by C.J. Gorter (1951).

Small quantum dots (~ 500 nm)

conductance (e2/h)

M. Kastner, Physics Today (1993)
E.B. Foxman et al., PRB (1993)

gate voltage (mV)

In metals first observed in Fulton, Dolan, PRL, 59, 109, (1987)

Coulomb blockade (CB) (II)
Strong CB
Weak CB

Mesoscopic CB
(reflectionless
contacts)

Random phase but
not period.

Statistical description of strong CB:
Theory:
Peaks: Jalabert, Stone, Alhassid (1992);
Valleys: Aleiner, Glazman (1996);
Reasonable agreement,
But problems with values of the correlation
fields

Courtesy of C.Marcus

Mesoscopic Coulomb Blockade

Aleiner, Glazman (1998)
Based on technique suggested by:
Matveev (1995); Furusaki, Matveev (1995);
Flensberg (1993).

Experiment:

Suppression
By a factor of 5.3
Th: Predicted 4.

Cronenwett et. al. (1998)

Even-Odd effect due to Kondo effect

Predicted:
Glazman, Raikh (1988)
Ng, Lee (1988)

Spin degeneracy in odd valleys:
Effective Hamiltonian:
local spin density
of conduction electrons

magnetic impurity

Observation:
1998

D. Goldhaber-Gordon et al. (MIT-Weizmann)
S.M. Cronenwett et al. (TU Delft)
J. Schmid et al. (MPI @ Stuttgart)

200 nm
15 mK

800 mK

van der Wiel et al. (2000)

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”)