Silvano De Franceschi

Laboratorio Nazionale TASC INFM-CNR, Trieste, Italy

Orbital Kondo effect in carbon nanotube quantum dots

http://www.tasc.infm.it/~defranceschis/SilvanoHP.htm

‘Simple’ and controllable systems can be obtained in nanostructured materials.

=> quantum coherent electronics

=> fundamental quantum phenomena

(spintronics, quantum computation, superconducting electronics…)

(quantum coherent dynamics, entanglement, strongly correlated systems,…)

| |

Spin ½ Kondo

| |

Spin ½ Kondo

TK ~ 10 K

TK ~ 1 K

Nygard et al., Nature (2000)

Liang et al. & J. Park et al. Nature (2002)

TK ~ 0.1 - 1 K

Goldhaber-Gordon et al., Nature (1998)Cronenwett et al., Science (1998)Schmid et al., Physica B (1998)

Semiconductor dots

Carbon nanotube dots

Single-molecule dots

In a metal with magnetic impurties:

In a quantum dotwith spin 1/2

| |

Spin ½ Kondo

TK

| |

Spin ½ Kondo

Gate-voltage control:=> Kondo effect in the unitary limit (G → 2e2/h)

[Nature 405, 764 (2000)]

[Science 289, 2105 (2000)]

Magnetic-field control:=> integer-spin Kondo effect at singlet-triplet degeneracy

[ Phys. Rev. Lett. 88, 126803 (2002)]

Bias-voltage control:

[Phys. Rev. Lett. 89, 156801(2002)]

=> Kondo effect out of equilibrium

++

| |

|+ |

+

Spin ½ Kondo

Orbital Kondo

|+, |+,

|, |,

++

++

| |

|+ |

+

Spin ½ Kondo

Orbital Kondo

=

SU(4) Kondo

L. Borda et al., Phys. Rev. Lett. (2003).G. Zaránd et al., Solid State Comm. (2003).

Theory proposals in 2DEG QDs Experiments in 2DEG QDsS. Sasaki et al., Phys. Rev. Lett. (2004)  

k||

E (

k ||)

()()

v v

Periodic boundary conditions:

Quantized

momentum around circumference

One-dimensional

subbands

Orbital magnetic moment Orbital magnetic moment

Finite length L

discrete spectrum: 4-fold shell structure at

B=0 (orbital+spin degeneracy)

k||

E (

k ||)

E(1)

E(2)

E(3)

E(1)

E(2)

E(3)

()()

Nanotube quantum dotNanotube quantum dot

Gate

VVG

I

SWNT

v v

Orbital splitting

k||

E (

k ||)

E(1)

E(2)

E(3)

E(1)

E(2)

E(3)

()()

B 0

Nanotube quantum dot

Gate

VVG

I

SWNT

v v

B B

Prediction: Ajiki & Ando, J. Phys. Soc. Jpn (1993)

discrete spectrum: 4-fold is lifted at B0

(orbital splitting >> spin splitting)

[Phys. Rev. Lett. 94, 156802 (2005)]

Kondo effect in a NT QD with 4-fold shell Kondo effect in a NT QD with 4-fold shell structurestructure

Four-fold shell structure at B=0

Each shell has two orbitals with opposite orbital magnetic moment

Orbitals in different shells cross each other at high B

B

E

|+,

|,

|,

|+,gB0

|+, > |, >

| , >|+, >

B = 0

|+, > | , >

| , >|+, >

B = B0

gB

0

Intra-shell 4-fold degeneracy

Inter-shell 2-fold degeneracy

SU(4) Kondo

Orbital Kondo

[Nature 434, 484 (2005)]

2 3 40

1

2

G (e

2/h

)

VG (V)

4

3

2

1

0

B (

T)

n = 3 n = 2 n = 1

half of 1st SHELL

2nd SHELL

3rd SHELL

Linear conductance of a small-band-gap CNT QD

VG(V)

T=8K

UU+

4

3

2

1

0

B (

T)

v v

orb 0.8 meV/T (>> B = 0.06 meV/T)

Consistent with theoretical predictions (Ajiki&Ando J.Phys.Soc. Jpn (1993))

and with recent experiments:Minot et al., Nature (2004); Zaric et al., Science (2004); Coskun et al., ibid.

Orbital magnetic moment

VG(V)

Orbital magnetic momentOrbital magnetic moment

Colour scale x100

3

4

5

7

1

2

6

500

600

Vg (m

V)

700

800

B (T)0 3.6

E. Minot et al. Nature 428, 536 (2004)

0.3 0.4 0.5

-30

30

Vsd

(m

V)

0

Vg (V)

1 2 3 4

0.6

G (e2/h)

0

0.2

No 4-fold degeneracy No link between spectrum & B-evolution of QD states

They measured large orbital magnetic moments orb = DevF/4 ~ 0.7meV/T ~ 12 B

Problems:

Small band gap semic. nanotube

4

3

2

1

0

B (

T)

v v

orb 0.8 meV/T (>> B = 0.06 meV/T)

Consistent with theoretical predictions (Ajiki&Ando J.Phys.Soc. Jpn (1993))

and with recent experiments:Minot et al., Nature (2004); Zaric et al., Science (2004); Coskun et al., ibid.

Orbital magnetic moment

VG(V)

4

3

2

1

0

B(T

)

x20

IV I II III IV I II

A B C D E F

B1 C1

C2

D1

D2

E1

E2

F1

F2

G1

A’ B’ C’ D’ E’ F’

01/2 0

1

1/20

1

1/2

1/2 1/2

0

13.0 3.5 4.0VG(V)2.5

QD orbital & spin configurationQD orbital & spin configuration

k||

E (

k ||)

E(1)

E(2)

E(3)

E(1)

E(2)

E(3)

k||

E (

k ||)

E(1)

E(2)

E(3)

E(1)

E(2)

E(3)

k||

E (

k ||)

E(1)

E(2)

E(3)

E(1)

E(2)

E(3)

k||

E (

k ||)

E(1)

E(2)

E(3)

E(1)

E(2)

E(3)

k||

E (

k ||)

E(1)

E(2)

E(3)

E(1)

E(2)

E(3)

k||

E (

k ||)

E(1)

E(2)

E(3)

E(1)

E(2)

E(3)

4

3

2

1

0

B(T

)

x20

IV I II III IV I II

A B C D E F

B1 C1

C2

D1

D2

E1

E2

F1

F2

G1

A’ B’ C’ D’ E’ F’

01/2 0

1

1/20

1

1/2

1/2 1/2

0

13.0 3.5 4.0VG(V)2.5

QD orbital & spin configurationQD orbital & spin configuration

B

E

(AA’):

(CC’):

(DD’):

(EE’):

(FF’):

AA’

(BB’):BB1 B1B’

,

CC1 C2C’

,C1C2,

,DD1 D2D’,D1D2

,EE1E2E’

,E1E2

,FF1 F2F’

,F1F2

–orb(2) – – gB

12

orb(2) – – gB

12 –orb

(2) + – gB12

–orb(2) + – gB

12

orb(2) – – gB

12 –orb

(1) – – gB12

orb(2) + – gB

12

–orb(1) – – gB

12

orb(2) – – gB

12

–orb(1) – – gB

12

orb(2) + – gB

12

–orb(1) + – gB

12

orb(1) – – gB

12 –orb

(1) + – gB12

orb(2) + – gB

12

E(3)

E(2)

E(1)

E+(3)

E+(2)

E+(1)

E

B

+,1+,2

+,3

,3,2

,1

[Phys. Rev. Lett. 94, 156802 (2005)]

3 electrons 1 electron

4

3

2

1

0

B(T

)

x20

IV I II III IV I II

A B C D E F

B1 C1

C2

D1

D2

E1

E2

F1

F2

G1

A’ B’ C’ D’ E’ F’

01/2 0

1

1/20

1

1/2

1/2 1/2

0

13.0 3.5 4.0VG(V)2.5

QD orbital & spin configurationQD orbital & spin configuration

B

E

Orbital crossing at B=3T

Orbital Kondo Effect Orbital Kondo Effect

0.90 0.95

10

8

6

4

2

0VG (V)

B (

T)

0

1

0

1/2 1/2

1/21/2

IIIII IV I II III

B = B0 6T

Orbital flip

B

E

|+,

|,

|,

|+,gB0

B

B

|,

|+,

|+,

|,

E

B0

Orbital Kondo EffectOrbital Kondo Effect

B

E

|+,

|,

|,

|+,gB0

0.90 0.95

10

8

6

4

2

0VG (V)

B (

T)

0

1

0

1/21/2

1/21/2

IIIII IV I II III

B = B0 6T

Orbital flipOrbital flip at eV=B

B = B0 6T

-1 0 1 20.7

1.4

dI/d

V (e

2/h

)

V (mV)

2gB0

2B

1.1

1.2

0.1 1 T (K)

dI/d

V (

e2 /h)

B = B0 6T

-1 0 1 20.7

1.4

dI/d

V (e

2/h

)

V (mV)

2gB0

2B

1.1

1.2

0.1 1 T (K)

dI/d

V (

e2 /h)

B = B0 6T

Low-impedance bipolar spin Low-impedance bipolar spin filterfilter

B

VG

I II IIIIV

I II III

Switch VG switch filter polarityOrbital Kondo effect low impedance

4

3

2

1

0

B(T

)

IV I II III IV I II

G1

01/2 0

1

1/20

1

1/2

1/2 1/2

0

13.0 3.5 4.0VG(V)2.5

Orbital+Spin Degeneracy => Strong Kondo Orbital+Spin Degeneracy => Strong Kondo (multilevel)(multilevel)

4

2

0

-2

-4

V (

mV

)

B = 0T

I II III0 IV

2.50 2.75 3.00 3.25 3.50VG (V)

Strong Kondo effect for 1 and 3 electrons in the shell

Strong triplet-singlet inelastic cotunneling peaks for 2 electrons in the shell [S. Sasaki, S. DF et al. Nature (2000)]

4

2

0

-2

-4

V (

mV

)

2.50 2.75 3.00 3.25 3.50

4

2

0

-2

-4

V (

mV

)

VG (V)

B = 0T

B = 1.5T

I II III0 IV

Multiple splitting @ finite Multiple splitting @ finite B B !!

The Kondo resonance for 1 electron splits in 4 peaks

V (

mV

)

-2 -1 0 1 2

2

1

0

-1

-2

B (T)

Four-fold splitting Four-fold splitting SU(4)- SU(4)-KondoKondo

Zeemansplitting

Orbitalsplitting

[Theory: Choi, Lopez and Aguado, cond-mat/0411665]

I

B

dI/dV

V

-1.21 -1.19

2

0

-2

VG (V)

V (

mV

)

Inelastic cotunneling spectroscopyInelastic cotunneling spectroscopy[PRL 86, 878 (2001)]

Step in dI/dV at V=level spacing

E

eV= +E

eV= -E

0 1 2 3

0.8

0.4

0

-0.4

-0.8

B (T)

V (

mV

)

B=0.7T

0.04

0.08

dI/

dV

(e2

/h)

0.5 0 -0.5

0.04

0.08

V (mV)

dI/

dV

(e2

/h)

B=80mT

0 1 2 3B (T)

0 1 2 3

-0.8

-0.4

0.0

0.4

0.8

V (

mV

)

B (T)

2gBB

gBB

4orbB

Zeeman, orbital, orbital + ZeemanZeeman, orbital, orbital + Zeeman

References

Pablo Jarillo-HerreroJing Kong Herre van der ZantCees DekkerLeo Kouwenhoven

Orbital Kondo effect [Nature 434, 484 (2005)]Magneto-transport spectroscopy [Phys. Rev. Lett. 94, 156802 (2005)]

Collaborators