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Carbon nanotube quantum dots:SU(4) Kondo and the Mixed Valence regimes

Gleb Finkelstein

Duke University

Alex Makarovski Thanks:F. Anders, H. Baranger, M. Galpin, L. Glazman, K. LeHur, J. Liu, D. Logan, G. Martins, K. Matveev, E. Novais, M. Pustilnik, D. Ullmo

Support: NSF DMR-0239748

Electronic band structure, sample

Coulomb blockade

Kondo effect / Mixed Valence

-apparent Kondo behavior in the Mixed Valence regime

Samples

doped Si

SiO2

nanotube

Vsource-drain Vgate

A

Single-wall CNT ~2 nm in diameter

Measurement: differential conductance = dI/dV (Vgate)

Graphene (2D graphite): semi-metal

kx

ky

E

k(2D vector)

Energy dispersion E(k)

k

E“Physical Properties of Carbon Nanotubes” “Physical Properties of Carbon Nanotubes” Saito, Dresselhaus and Dresselhaus 1998.1998.

k⊥

Nanotubes: quantization of quasi-momentum

k||

k⊥

k||

C= 2πR

r + C coincides with r =>k⊥= 2πN/C = N/Rk|| - arbitrary

Nanotubes: metallic and semiconducting

E

k

k- quasi-momentum along the length of the nanotube

metal

• Metallic or semiconducting • Two degenerate bands

k

semiconductor

k

E

k

Degenerate orbitals: ‘shells’

kE

k

Two degenerate subbands => degenerate orbitals

E

shell

W.J. Liang, M. Bockrath, and H. Park, PRL (2002)M. R. Buitelaar et al., PRL (2002)

Quantization along length

Electronic band structure, sample

Coulomb blockade

Kondo effect / Mixed Valence

-apparent Kondo behavior in the Mixed Valence regime

Ambipolar semiconducting nanotube

-8 -6 -4 -2 0 2 4 6 8 10 12 14 16 180.00.51.01.52.02.5

C

ondu

ctan

ce (e

2 /h)

Vgate (Volts)

T=1.5K

band gap

max: 4e2/h

P-type N-type

holes electrons

Quantum Dot

Leads and island formed within the same nanotube

M.S. Choi, R. Lopez and R. Aguado, PRL (2005)

4 degenerate levels in the dotare coupled one-to-one to 4 modes (↑, ↓, ↑, ↓) in the leads

The modes are not mixed by tunnelingTunneling Hamiltonian has

SU(4) symmetry

Nanotube quantum dot

Tunneling grows with Vgate

2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 10.00.0

0.5

1.0

Con

duct

ance

(e2 /h

)

V gate

Energy gap

Quantum Dot

Γ grows

2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 10.00.0

0.5

1.0

Con

duct

ance

(e2 /h

)

V gate

Groups of 4 peaks: orbital degeneracy

4 degenerate states:2 degenerate orbitals x 2 spin projections

Energy gap

E

-k0 k0

k||“shell”

Groups of 4 peaks: orbital degeneracy

E

-k0 k0

k||

7.0 8.0 9.0 10.00.0

0.5

1.0

Con

duct

ance

(e2 /h

)

V gate

EC+∆E

EC=e2/C – Charging energy

∆E – QM level spacing

EC

Conductance map: G(Vgate, Vsd)8 12 16 20 24

2 3 4 5 6 7 8 9 10-20-10

01020

Vsd

(mV

)

Vgate (V)

2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 10.00.0

0.5

1.0

Con

duct

ance

(e2 /h

)

V gate

E

B||

Nanotubes in Magnetic field

B||

B⊥

E

B⊥

Orbital + Zeeman ZeemanAndo, SST (2000)

0 1 2 3 4 5 6

9

10

11

12

Vga

te (V

)Magnetic field (Tesla)

Nanotube in B||

B||

B||

E

cond

ucta

nce

Orbital + Zeeman

B⊥: Zeeman splitting

B⊥

E

Sz = 1S=0,1

0 2 4 6 8

-3.4

-3.3

G

ate

volta

ge (V

)

Magnetic field (Tesla)

Addition energy = peak spacing in Vgate converted to energy

0 1 2 3 4 5 6 7 8 94.5

5.0

5.5

6.0

6.5

7.0

7.5

Ead

d (m

eV)

B (T)

Addition energies in B⊥

∆E

EC

Zeeman

5.0 6.00.0

0.5

Con

duct

ance

(e2 /h

)

V gate

1 2 3

Exchange is negligible

0 1 2 3 4 5 6 7 8 94.5

5.0

5.5

6.0

6.5

7.0

7.5

Ead

d (m

eV)

B (T)

Perpendicular magnetic field: Zeeman splitting

∆E

EC

Zeeman

Addition energy

B⊥

2

1, 3Exchange

Y. Oreg, K. Byczuk, and B.I. Halperin, PRL (2000)S-H. Ke, H.U. Baranger, and W. Yang, PRL (2003).

Electronic band structure, sample

Coulomb blockade

Kondo effect / Mixed Valence

-apparent Kondo behavior in the Mixed Valence regime

Kondo effect in Quantum Dots

G

Vgate

Conductance grows at low temperatures in valleys with a non-degenerate ground state

1e 2e 3e

Theory: Glazman and Raikh (1988)Ng and Lee (1988)

Experiment: Goldhaber-Gordon et al., (1998) Cronenwett et al., (1998)Schmid et al., (1998)

Kondo effect in Nanotubes

SU(4) theories for 1 electron:Double dots, dots with symmetries:D. Boese et al., PRB (2002)L. Borda et al., PRL (2003)K. Le Hur and P. Simon, PRB (2003)G. Zarand et al., SSC (2003)W. Izumida et al., J.P.Soc.Jpn. (1998)A. Levy Yeyati et al., PRL (1999)Nanotubes:M.S. Choi et al., PRL (2005)

1 electron SU(4) Kondo experiment:Quantum dots: S. Sasaki et al., PRL (2004)Nanotubes: P. Jarillo-Herrero et al., Nature (2005)

G

1e 2e 3eVgate

Kondo effect in Nanotubes

G

Vgate1e 2e 3e

2 electron SU(4) theoryM.R. Galpin, D.E. Logan and H.R. Krishnamurthy, PRL (2005)C.A. Busser and G.B. Martins, PRB (2007)

2-e Kondo in nanotubes – triplet or SU(4) ?W.J. Liang, M. Bockrath and H. Park, PRL (2002)B. Babic, T. Kontos and C. Schonenberger, PRB (2004)

Interactions ↑↓ = ↑↓ = ↑↓Exchange ↑↑ is small

Kondo effect in Nanotubes

G

Vgate1e 2e 3e

Friedel sum rule: Σ δi = πNKondo singlet: δ↑=δ↓=δ↑=δ↓1e δi = π/42e δi = π/2

G(2e) = 2G(1e)

This type of arguments may be found for the SU(2) case in Glazman and Pustilnikcond-mat/0501007

G=G0 Σ sin2(δi )

Temperature dependence of conductance

4 5 6 7 80.0

0.5

1.0

1.5

2.0

2.5

Vgate (Volts)

Temperature 15.0 K

Con

duct

ance

(e2 /h

)

1 2 3 1 2 3

barrier transparency grows

0/4 Ec, ∆ ~ 100K

Shell spacing

Temperature dependence of conductance

4 5 6 7 80.0

0.5

1.0

1.5

2.0

2.5

Vgate (Volts)

Temperature 1.3 K 3.3 K 6.4 K 10.4 K 15.0 K

Con

duct

ance

(e2 /h

)

Ec, ∆ ~ 100K

12

3 1

2 3

Growth of the signal in the valleys due to the Kondo effect

4 5 6 7 80.0

0.5

1.0

1.5

2.0

2.5

Vgate (Volts)

Temperature 1.3 K

Con

duct

ance

(e2 /h

)

Shape of the 4-electron clusters at low T and high transparency

Single-electron features are washed away

4e

4e

1

2 3

4 5 6 7 80.0

0.5

1.0

1.5

2.0

2.5

Vgate (Volts)

Temperature 1.3 K

Con

duct

ance

(e2 /h

)

Fabry- Perot (single-particle interference) does not work!

Shape of the 4-electron clusters at low T and high transparency

4 5 6 7 80.0

0.5

1.0

1.5

2.0

2.5

Vgate (Volts)

Temperature 1.3 K

Con

duct

ance

(e2 /h

)

G ~ sin2(πN/4) where N varies linearly with Vgate

N0e 4e

Shape of the 4-electron clusters at low T and high transparency

Electronic band structure, sample

Coulomb blockade

Kondo effect / Mixed Valence

-apparent Kondo behavior in the Mixed Valence regime

Phase shifts of scattered waves δ

Low temperature singlet:

δ↑ = δ↓ = δ↑ = δ↓

Friedel’s sum rule Σ δi = πN

N=CVg/e (open dot)

G=G0Σ sin2(δi)= 4G0sin2(πCVg/4e)

δi = πN/4

Characteristic energy scale T0~3 K << Ec, ∆, Γ (all ~100 K)

Evidently, the argument works even for non-integer N

Temperature dependence of conductance

4 5 6 7 80.0

0.5

1.0

1.5

2.0

2.5

Vgate (Volts)

Temperature 1.3 K 3.3 K 6.4 K 10.4 K 15.0 K

Con

duct

ance

(e2 /h

)

Ec, ∆ ~ 100K

12

3 1

2 3

4 5 6 7 8

-10-505

10

V

sd (m

V)

Vgate (Volts)

Spectroscopy at finite bias

E

Vgate

Tunneling from the weakly coupled lead probes the density of states in the Kondo / Mixed Valence system 1e 2e 3e

ΓL >> ΓRTunneling density of states

EF

2.25e2/h0Width of the resonance: 10 K ~ T0

T = 3.3 K

4 5 6 7 8

-10-505

10

V

sd (m

V)

Vgate (Volts)

Spectroscopy at finite bias

E

Vgate

Tunneling from the weakly coupled lead probes the density of states in the Kondo / Mixed Valence system 1e 2e 3e

ΓL >> ΓRTunneling density of states

EF

1.5e2/h0

T = 10 K

Electronic band structure, sample

Coulomb blockade

Kondo effect / Mixed Valence

Numerical Renormalization Group calculations

F. Anders, M. Galpin, D. Logan, and G.F., PRL (2008)

Leads and island formed within the same nanotube

4 degenerate levels (↑, ↓, ↑, ↓)in the dot are coupled one-to-one to 4 modes (↑, ↓, ↑, ↓) in the leads

The modes are not mixed by tunneling; t amplitude does not depend on α or σ Hamiltonian has SU(4) symmetry

two leads ν = L, R

spin σ, orbital α N number of electrons

SU(4) tunneling Hamiltonian

NRG calculations of the SU(4) tunneling Hamiltonian

F. Anders, M. Galpin, D. Logan, and G.F. PRL (2008)

Γ= 0.5, 1, 2 meVT= 15, 8.5, 5, 3, 1.7 K(color sequence reversed)

Γ= 1 meV

Γ= 0.5 meV Γ= 2 meV

Zero-bias conductance – numerical results

0 1 2 3 40.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0 4e2/ h sin2(πN/4) 15 K 8.5 K 5 K 3 K 1.7 K

N

σ (e

2 /h)

Ngate

Γ= 0.5 meV

Zero-bias conductance – numerical results

0 1 2 3 40.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0 4e2/ h sin2(πN/4) 15 K 8.5 K 5 K 3 K 1.7 K

N

σ (e

2 /h)

Ngate

Γ= 1 meV

Zero-bias conductance – numerical results

0 1 2 3 40.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0 4e2/ h sin2(πN/4) 15 K 8.5 K 5 K 3 K 1.7 K

N

σ (e

2 /h)

Ngate

Γ= 2 meV

Universal scaling curve G(T/T0) for 2 electrons

0.1 10.0

0.2

0.4

0.6

0.8

1.0

SU(2) SU(4)

shell II III IV

G(T

) / G

(0)

T/T0

4 5 6 7 80.0

0.5

1.0

1.5

2.0

2.5

Vgate (Volts)

Con

duct

ance

(e2 /h

)

2e SU(4) curve is more steep than inthe 1e SU(2) case

4 5 6 7 8

-10-505

10

Vsd

(mV

)

Vgate (Volts)

Electronic band structure, sample

Coulomb blockade

Kondo effect / Mixed Valence

-apparent Kondo behavior in the Mixed Valence regime

-Kondo in magnetic field

Kondo in B⊥: SU(4) Kondo suppressed

-4 -3 -2 -1 0 1 2 3 40.000.010.020.030.040.050.060.070.080.090.100.110.120.13

Con

duct

ance

(e2 /h

)

Vsd (mV)

0 T2 T4 T6 T8 T

2eE

B⊥

Non-degenerate ground stateKondo suppressed

Kondo in B⊥: SU(4) to orbital SU(2)

E

0 T2 T4 T6 T8 T

-4 -3 -2 -1 0 1 2 3 40.000.010.020.030.040.050.060.070.080.090.100.110.120.13

Con

duct

ance

(e2 /h

)

Vsd (mV)

1e

B⊥

E

B⊥

0 T2 T4 T6 T8 T

Degenerate ground state =>Orbital SU(2) Kondo

0 1 2 3 4 5 6

9

10

11

12

Vga

te (V

)Magnetic field (Tesla)

Nanotube in B||

B||

3e

2e

1eE

B||

0 1 2 3 4 5 6 7 8 9

0 1 2 3 4 5 6 7 8 9

0 1 2 3 4 5 6 7 8 9

-6

-4

-2

0

2

4

6

Vsd

(mV

)

Magnetic field (Tesla)

E E E

B||

0 1 2 3 4 5 6 7 8 9

0 1 2 3 4 5 6 7 8 9

0 1 2 3 4 5 6 7 8 9

-6

-4

-2

0

2

4

6

Vsd

(mV

)

Magnetic field (Tesla)

1e 2e 3e 2e singlet ground state => Kondo disappears

SU(2) spin Kondo still possible for 1e and 3e

Kondo in B||

-10 -5 0 5 100.0

0.5

1.0

1.5

2.0

2.5

3.0

Con

duct

ance

(e2 /h

)

Vsd (mV)-10 -5 0 5 10

0.0

0.5

1.0

1.5

2.0

2.5

3.0

Con

duct

ance

(e2 /h

)

Vsd (mV)

3e2e

9T

0T

TKSU(2) < spin splitting singlet, non-degenerate

-10 -5 0 5 100.0

0.5

1.0

1.5

2.0

2.5

3.0

Con

duct

ance

(e2 /h

)

Vsd (mV)

1e

SU(4) => SU(2)SU(4) => no Kondo

TKSU(2) > spin splitting

SU(4) => no Kondo

-9 -6 -3 0 3 6 90.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

2.0

2.2

shell 1 shell 2

Con

duct

ance

(e2 /h

)

Vsd (mV)

1e

3e

2e

B|| =3.5 T

Kondo features in 2 successive shells

TKSU(2) > spin splittingSU(4) => SU(2)

singlet, non-degenerateSU(4) => no Kondo

TKSU(2) < spin splittingSU(4) => no Kondo

10.0 10.5 11.0 11.5 12.0

-10

-5

0

5

10

Vsd

(mV)

Vgate (Volts)

B|| = 3 T

1e 2e 3e 1e 2e 3e

Kondo features in 2 successive shells

The difference between 1e and 3e cases is persistent in 2 successive shells

B||

E

Spin-orbit

B||

E

B||

Without SO With SO

Large spin gap=> no Kondo

Small spin gap=> SU (2) Kondo

Kuemmeth et al., Nature (2008)

Summary

• Orbital degeneracy in nanotube Quantum Dots

• Kondo effect / Mixed Valence – a major modification of conductance at low T.

• Why does G ~ sin2(πN/4) work so well for noninteger N?

• What is the low energy scale?

• Lifting SU(4) symmetry by magnetic field:

B⊥ orbital SU(2) Kondo effect

B|| spin-orbit in 1e and 3e valleys

Determination of Γ

4 5 6 7 80.0

0.5

1.0

1.5

conductance at 15K 16 peak Lorentzian fit

Vgate (Volts)

Con

duct

ance

(e2 /h

)

0 2 4 6 8 10 12 14 160123456789

IVIIIIII

Γ (m

eV)

Peak Number

(E-E0)2-Γ21

10K

3.3K

Is the coupling of the two orbitals to the leads different?

10.0 10.5 11.0 11.5 12.00.0

0.2

0.4

0.6

0.8

1.0

1.2

3.5 T

765321

C

ondu

ctan

ce (e

2 /h)

Vgate (V)

The widths of the single-electron peaks at non-zero field for two orbitals in consecutive shells are about the same

What is the cause of the e-h asymmetry?

Spin-orbit

Positions of the resonances

T = 1.3 K

1.2e2/h0