M.Kiselev Explicit and hidden symmetries in Quantum Dots ...

Institut für Theoretische Physik und Astrophysik
Universität Würzburg

M.Kiselev

Explicit and hidden symmetries in Quantum Dots and
Quantum molecules

Outline

• Quantum Dot Devices
• Kondo effect in Quantum Dots
• Exotic Kondo effect in complex dots
• From quantum dots to quantum chains
• Large quantum dots: level statistics
• Stoner instability in mesoscopic systems
• Perspectives

Quantum Dot devices

Artificial atoms

Kondo Effect in Quantum Dots

Universal Scaling

G / G0 ∝ ln−2 (max[T / TK ])

TK = 1 (ΓU )1/2 exp ⎛ πε 0 ε0 +U ⎞
⎜⎝ ⎟⎠
2 ΓU

Tunneling and co-tunneling

Elastic Electron-like
co-tunneling

Hole-like
Inelastic

Common sense:
• Kondo effect exists if the total number
of electrons in a dot is odd
• Kondo effect is destroyed by external
magnetic field
• Relaxation effects associated with the
non-equilibrium conditions eliminate
the Kondo peak

Parallel and Serial Dots:
from artificial atoms to molecules

R.Lopez, R.Aguado, and G.Platero, PRL 2002 Competition between Kondo and RKKY interactions

Kondo effect in an Artificial
Quantum Dot Molecule

Zero-bias maximum

Exotic symmetries

SU(4) symmetry

J. Von Delft et al (2003)

Quantum Phase Transition in
a two-channel quantum dot

Electrons in large dot provide an
additional channel for Kondo effect

Two-stage Kondo effect

L.Glazman et al (2003)

Singlet-triplet transition in
a magnetic field

L.Glazman et al (2001)

Single-wall carbon nanotubes

Transition is driven by Zeeman splitting

Parallel Double Quantum Dot

T-shape geometry

Symmetric and asymmetric double dots

Spin Rotator (SR) Model

∑Hint = [(JαTαT' S + J ST P) ⋅ sαα' + JαSα ' X SS nαα ' ]
αα '

∑+1
∑0

-1
c c+
kασ σσ ' k 'α 'σ '
triplet τsαα ' = nαα ' = ck+ασ 1 ck 'α 'σ

kk ' kk '

ET = ES + δ S+

singlet

SO(4) P+

Hidden symmetry in a Coulomb problem

Hydrogen atom: H = p2 − e2 En = − µe4 1
2n2
2µ r 2

( )A = 1 p × L − L × p − e2 r Runge-Lenz vector
2µ r

⎣⎡H , L⎤⎦ = 0 ⎡⎣H , A⎤⎦ = 0 ( AiL) = 0

SU (2) ⊗ SU (2) = SO(4)

Non-equilibrium Kondo effect in
Double Quantum Dot

Energy TK /τ d

∝| J ST |2 G / G0 ∝ ln−2 (T / TK )
LR
G(eV ,T )

G0 = 2e2 / h Dimensionless coupling

AC Voltage

DC Voltage ( )Gpeak ∝ G Vac cos (ωt )

( )G / G0 ∝ ln−2 max ⎣⎡(eV − δ ),T ⎦⎤ / TK G peak / G0 ∝ ln −2 ⎛ ⎞
⎜ ⎟
MNK, K.Kikoin and L.W.Molenkamp, PRB 68, 155323 (2003) ⎝ τ TK ⎠

Basic inequalities

{eVdc , eVdot , eVac} < {Ed ,U − Ed } Validity of effective Hamiltonian

TK << eV ≤ δ << D Absence of Kondo effect in equilibrium

| eV − δ |<< TK Condition of Kondo resonance in nonequilibrium

δ ⎛ δ ⎞2 << TK << δ DC decoherence rate effects are irrelevant
⎝⎜ ⎠⎟ AC decoherence rate effects are irrelevant
D

/τ d << TK

Messages

• Kondo effect exists in quantum dots in both
situations when the total number of electrons
is odd (usual) and even (exotic)
• Kondo effect may not be destroyed by an external
magnetic field
• Kondo effect may not be destroyed by the
relaxation effects associated with non-equilibrium
conditions

Reason: explicit and hidden symmetries in complex quantum dots

From complex dots to quantum chains

• Haldane Gap
• Charge Density Waves
• Exciton propagation

K.Kikoin, Y.Avishai and MNK cond-mat/0309606

Random Matrices: Wigner-Dyson statistics

({ }) ∑PEµ∝ ⎛ β ln ⎡| Eµ − Eν |⎤⎞
exp ⎜ ⎢ ⎥⎟
2 ν ≠µ ⎣ δ ⎦⎠
⎝
β = 1 Orthogonal (GOE)

β = 2 Unitary (GUE)

GOE GUE β =4 Simplectic (GSE)

Metallic QD: Universal Hamiltonian

Electron-electron interactions in isolated metallic grains

Mean-level spacing δ = Eα +1 − Eα (kinetic energy)

Thouless energy ET ∼ D ⋅ L−2 diffusive regime

Eα +1 g = ET / δ 1 ET ∼ vF L−1 ballistic regime
Eα GUE

e2 H int = Ec (n − N )2 − J (S)2 +
2C
− λBCS T T

Ec = charge spin Superconducting

Coulomb blockade Short-range interaction Ec ∼| J |∼ δ
Scaling:
Coulomb interaction Ec / δ = rs (kF L)d −1
|J|

Zero-bias anomaly in finite-size systems
Hint = Ec (n − N )2

ZBA ∼ e−Ec /(8T )

U (1) gauge

Mesoscopic Stoner Instability

Hint = Ec (n − N )2 − J (S )2DOS

S z Stoner Instability

Transverse spin fluctuations are important

Coupling constant J / δ

SU (2)

Temperature

MNK and Y.Gefen (in preparation)

Y.Alhassid and T.Rupp (2003)