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Carbon nanotube quantum dots:

SU(4) Kondo and the Mixed Valence regimes

Gleb Finkelstein

Duke University

Alex Makarovski

Support: NSF DMR-0239748

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


Electronic band structure, sample

Coulomb blockade

Kondo effect / Mixed Valence

-apparent Kondo behavior in the Mixed Valence regime


Samples

V source-drain

V gate

A

nanotube

SiO 2

doped Si

Measurement: differential conductance

= dI/dV (Vgate)

Single-wall CNT ~2 nm in diameter


Graphene (2D graphite): semi-metal

k y

Energy dispersion E(k)

E

k x

E

k

k

(2D vector)

“Physical Properties of Carbon Nanotubes”

Saito, Dresselhaus and Dresselhaus 1998.


k ⊥

Nanotubes: quantization of quasi-momentum

k ||

k ⊥

k ||

C= 2πR

r + C coincides with r =>

k ⊥ = 2πN/C = N/R

k || - arbitrary


Nanotubes: metallic and semiconducting

k

k

E

E

k

k

metal

semiconductor

• Metallic or semiconducting

• Two degenerate bands

k- quasi-momentum

along the length of

the nanotube


Degenerate orbitals: ‘shells’

E

Quantization along length

k

E

k

Two degenerate subbands

=> degenerate orbitals

W.J. Liang, M. Bockrath, and H. Park, PRL (2002)

M. R. Buitelaar et al., PRL (2002)

shell


Electronic band structure, sample

Coulomb blockade

Kondo effect / Mixed Valence

-apparent Kondo behavior in the Mixed Valence regime


Ambipolar semiconducting nanotube

Conductance (e 2 /h)

2.5

2.0

1.5

1.0

0.5

0.0

max: 4e 2 /h

-8 -6 -4 -2 0 2 4 6 8 10 12 14 16 18

P-type

band

gap

Vgate (Volts)

N-type

T=1.5K

holes

electrons


Nanotube quantum dot

4 degenerate levels in the dot

are coupled one-to-one to

4 modes (↑, ↓, ↑, ↓) in the leads

Quantum Dot

Leads and island formed

within the same nanotube

The modes are not mixed by tunneling

Tunneling Hamiltonian has

SU(4) symmetry

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


Tunneling grows with Vgate

Quantum Dot

Conductance (e 2 /h)

1.0

0.5

Energy gap

Γ grows

0.0

2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 10.0

V gate


Groups of 4 peaks: orbital degeneracy

E

4 degenerate states:

2 degenerate orbitals x 2 spin projections

-k 0 k 0

k ||

“shell”

Conductance (e 2 /h)

1.0

0.5

Energy gap

0.0

2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 10.0

V gate


Groups of 4 peaks: orbital degeneracy

E

-k 0 k 0

k ||

∆E – QM level spacing

E C =e 2 /C – Charging energy

Conductance (e 2 /h)

1.0

0.5

E C

E C +∆E

0.0

7.0 8.0 9.0 10.0

V gate


Vsd (mV)

Conductance map: G(Vgate, Vsd)

10

20

-10 0

-20

8 12 16 20 24

2 3 4 5 6 7 8 9 10

Vgate (V)

Conductance (e 2 /h)

1.0

0.5

0.0

2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 10.0

V gate


Nanotubes in Magnetic field

B||

B⊥

E

E

B||

B⊥

Orbital + Zeeman

Ando, SST (2000)

Zeeman


Nanotube in B||

12

B||

E

Vgate (V)

11

10

B||

Orbital + Zeeman

conductance

9

0123456

Magnetic field (Tesla)


B ⊥ : Zeeman splitting

-3.3

E

Gate voltage (V)

-3.4

S=0,1

Sz = 1

B⊥

0 2 4 6 8

Magnetic field (Tesla)


Addition energies in B ⊥

7.5

0.5

7.0

Conductance (e 2 /h)

1 2 3

Eadd (meV)

6.5

6.0

5.5

∆E

Zeeman

0.0

5.0 6.0

V gate

5.0

4.5

0 1 2 3 4 5 6 7 8 9

B (T)

E C

Addition energy = peak spacing in Vgate converted to energy


Perpendicular magnetic field: Zeeman splitting

Addition energy

2

Exchange

1, 3

B ⊥

Exchange is negligible

Eadd (meV)

7.5

7.0

6.5

6.0

5.5

5.0

4.5

∆E

Zeeman

0 1 2 3 4 5 6 7 8 9

B (T)

E C

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

1e 2e 3e

Vgate

Conductance grows at low

temperatures in valleys with a

non-degenerate ground state

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

G

1e 2e 3e

Vgate

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)


Kondo effect in Nanotubes

G

1e 2e 3e

Vgate

Interactions ↑↓ = ↑↓ = ↑↓

Exchange ↑↑ is small

2 electron SU(4) theory

M.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)


Kondo effect in Nanotubes

G

1e 2e 3e

Vgate

Friedel sum rule: Σδ i = πN

Kondo singlet: δ↑=δ↓=δ↑=δ↓

1e δ i = π/4

2e δ i = π/2

G=G 0 Σ sin 2 (δ i )

This type of arguments may

be found for the SU(2) case

in Glazman and Pustilnik

cond-mat/0501007

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


Temperature dependence of conductance

Conductance (e 2 /h)

2.5

2.0

1.5

1.0

0.5

1 2 3

barrier transparency grows

1 2 3

Temperature

15.0 K

0/4 Ec, ∆ ~ 100K

0.0

4 5 6 7 8

Vgate (Volts)

Shell

spacing


Temperature dependence of conductance

Conductance (e 2 /h)

2.5

2.0

1.5

1.0

0.5

1

2

3

1

2 3

Temperature

1.3 K

3.3 K

6.4 K

10.4 K

15.0 K

Ec, ∆ ~ 100K

0.0

4 5 6 7 8

Vgate (Volts)

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


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

2.5

Temperature

1.3 K

Conductance (e 2 /h)

2.0

1.5

1.0

0.5

1

2 3

4e

4e

0.0

4 5 6 7 8

Vgate (Volts)

Single-electron features are washed away


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

Conductance (e 2 /h)

2.5

2.0

1.5

1.0

0.5

Temperature

1.3 K

0.0

4 5 6 7 8

Vgate (Volts)

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


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

Conductance (e 2 /h)

2.5

2.0

1.5

1.0

0.5

Temperature

1.3 K

0.0

4 5 6 7 8

Vgate (Volts)

0e

4e

N

G ~ sin 2 (πN/4) where N varies linearly with Vgate


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 Σδ i = πN

δ i = πN/4

N=CVg/e (open dot)

G=G 0 Σ sin 2 (δ i )= 4G 0 sin 2 (πCVg/4e)

Evidently, the argument works even for non-integer N

Characteristic energy scale T 0 ~3 K


Temperature dependence of conductance

Conductance (e 2 /h)

2.5

2.0

1.5

1.0

0.5

1

2

3

1

2 3

Temperature

1.3 K

3.3 K

6.4 K

10.4 K

15.0 K

Ec, ∆ ~ 100K

0.0

4 5 6 7 8

Vgate (Volts)


Vsd (mV)

Spectroscopy at finite bias

10

5

0

-5

-10

4 5 6 7 8

Vgate (Volts)

E

T = 3.3 K

Vgate

0

2.25e 2 /h

Width of the resonance: 10 K ~ T 0

Tunneling from the weakly coupled lead

probes the density of states in

the Kondo / Mixed Valence system 1e 2e 3e

E F

Γ L >> Γ R

Tunneling density of states


Vsd (mV)

Spectroscopy at finite bias

10

5

0

-5

-10

4 5 6 7 8

Vgate (Volts)

E

T = 10 K

Vgate

0

1.5e 2 /h

Tunneling from the weakly coupled lead

probes the density of states in

the Kondo / Mixed Valence system 1e 2e 3e

E F

Γ L >> Γ R

Tunneling density of states


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)


SU(4) tunneling Hamiltonian

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

spin σ, orbital α

N number of electrons

two leads ν = L, R

The modes are not mixed by tunneling;

t amplitude does not depend on α or σ

Hamiltonian has SU(4) symmetry


NRG calculations of the SU(4) tunneling Hamiltonian

Γ= 0.5 meV Γ= 2 meV

Γ= 1 meV

Γ= 0.5, 1, 2 meV

T= 15, 8.5, 5, 3, 1.7 K

(color sequence reversed)

F. Anders, M. Galpin,

D. Logan, and G.F.

PRL (2008)


Zero-bias conductance – numerical results

4.0

3.5

3.0

2.5

15 K

8.5 K

5 K

3 K

1.7 K

4e 2 / h sin 2 (πN/4)

N

Γ= 0.5 meV

σ (e 2 /h)

2.0

1.5

1.0

0.5

0.0

0 1 2 3 4

Ngate


Zero-bias conductance – numerical results

4.0

3.5

3.0

2.5

15 K

8.5 K

5 K

3 K

1.7 K

4e 2 / h sin 2 (πN/4)

Γ= 1 meV

N

σ (e 2 /h)

2.0

1.5

1.0

0.5

0.0

0 1 2 3 4

Ngate


Zero-bias conductance – numerical results

4.0

3.5

3.0

2.5

15 K

8.5 K

5 K

3 K

1.7 K

4e 2 / h sin 2 (πN/4)

N

Γ= 2 meV

σ (e 2 /h)

2.0

1.5

1.0

0.5

0.0

0 1 2 3 4

Ngate


Universal scaling curve G(T/T 0 ) for 2 electrons

1.0

0.8

Conductance (e 2 /h)

2.5

2.0

1.5

1.0

0.5

G(T) / G(0)

0.6

0.4

0.2

0.0

SU(2)

SU(4)

shell

II

III

IV

4 5 6 7 8

Vgate (Volts)

2e SU(4) curve is

more steep than in

the 1e SU(2) case

0.1 1

0.0


Vsd (mV)

10

5

0

-5

-10

4 5 6 7 8

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

Conductance (e 2 /h)

0.13

0.12

0.11

0.10

0.09

0.08

0.07

0.06

0.05

0.04

0.03

0.02

0.01

0.00

2e

0 T

2 T

4 T

6 T

8 T

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

E

B ⊥

Non-degenerate ground state

Kondo suppressed

Vsd (mV)


Kondo in B⊥: SU(4) to orbital SU(2)

Conductance (e 2 /h)

0.13

0.12

0.11

0.10

0.09

0.08

0.07

0.06

0.05

0.04

0.03

0.02

0.01

0.00

1e 0 T

2 T

0 T

4 T

2 T

6 T

4 T

8 T

6 T

8 T

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

E

B ⊥

E

Degenerate ground state =>

Orbital SU(2) Kondo

B ⊥

Vsd (mV)


Nanotube in B||

12

B||

11

3e

E

Vgate (V)

10

2e

1e

9

B||

0123456

Magnetic field (Tesla)


Vsd (mV)

6

4

2

0

-2

-4

1e 2e 3e

2e singlet ground

state => Kondo

disappears

SU(2) spin Kondo

still possible for

1e and 3e

-6

0123456789

0123456789

0123456789

Magnetic field (Tesla)

E E E

B||


Kondo in B||

3.0

2.5

1e

3.0

2.5

2e

0T

3.0

2.5

3e

Conductance (e 2 /h)

2.0

1.5

1.0

Conductance (e 2 /h)

2.0

1.5

1.0

Conductance (e 2 /h)

2.0

1.5

1.0

0.5

0.5

9T

0.5

0.0

-10 -5 0 5 10

0.0

-10 -5 0 5 10

0.0

-10 -5 0 5 10

Vsd (mV)

Vsd (mV)

Vsd (mV)

T K

SU(2)

< spin splitting singlet, non-degenerate

T K

SU(2)

> spin splitting

SU(4) => no Kondo

SU(4) => no Kondo

SU(4) => SU(2)


Kondo features in 2 successive shells

2.2

2.0

1.8

B|| =3.5 T

3e

T K

SU(2)

> spin splitting

SU(4) => SU(2)

1.6

Conductance (e 2 /h)

1.4

1.2

1.0

0.8

0.6

2e

1e

singlet, non-degenerate

SU(4) => no Kondo

T K

SU(2)

< spin splitting

0.4

SU(4) => no Kondo

0.2

0.0

shell 1

shell 2

-9 -6 -3 0 3 6 9

Vsd (mV)


Kondo features in 2 successive shells

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

10

5

B|| = 3 T

Vsd (mV)

0

-5

1e 2e 3e

1e 2e 3e

-10

10.0 10.5 11.0 11.5 12.0

Vgate (Volts)


Spin-orbit

B||

Kuemmeth et al., Nature (2008)

E

Without SO

E

With SO

Small spin gap

=> SU (2) Kondo

B||

B||

Large spin gap

=> no Kondo


Summary

• Orbital degeneracy in nanotube Quantum Dots

• Kondo effect / Mixed Valence – a major modification of

conductance at low T.

• Why does G ~ sin 2 (πN/4) work so well for noninteger N?

• What is the low energy scale?

• Lifting SU(4) symmetry by magnetic field:

B⊥

B||

orbital SU(2) Kondo effect

spin-orbit in 1e and 3e valleys


Determination of Γ

1.5

Conductance (e 2 /h)

1.0

0.5

conductance at 15K

16 peak Lorentzian fit

1

(E-E 0 ) 2 -Γ 2

Γ (meV)

0.0

9

8

7

6

5

4

3

2

1

0

4 5 6 7 8

Vgate (Volts)

I

II III IV

0 2 4 6 8 10 12 14 16

Peak Number


10K

3.3K


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

1.2

Conductance (e 2 /h)

1.0

0.8

0.6

0.4

0.2

1

2

3

3.5 T

5

6

7

The widths of the singleelectron

peaks at non-zero field

for two orbitals in consecutive

shells are about the same

0.0

10.0 10.5 11.0 11.5 12.0

Vgate (V)

What is the cause of the e-h asymmetry?

Spin-orbit


Positions of the resonances

T = 1.3 K

0

1.2e 2 /h

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