Cotunneling and Kondo effect in quantum dots Part I/II

J. Paaske, NBI & NSC

Cotunneling and Kondo effect in quantum dots Part I/II

Jens Paaske The Niels Bohr Institute & Nano-Science Center Bad Honnef, September, 2010

20-09-2010 Dias 1

J. Paaske, NBI

Lecture plan Part I 1.

Basics of Coulomb blockade and quantum conductance - Quantum tunneling and classical charging

2.

From Anderson model to cotunneling - Schrieffer-Wolff transformation

3.

Elastic vs. inelastic cotunneling - Bias spectroscopy

4.

Exchange cotunneling and basic Kondo effect - Signatures of Kondo effect

5.

Tunneling renormalization of cotunneling thresholds - Ferromagnetic leads, quasi-degenerate systems

Part II 1.

The nonequilibrium Kondo problem - What’s the problem?

2.

Poor man’s scaling for nonequilibrium systems - Lineshapes for inelastic cotunneling?

3.

The effect spin-orbit coupling - Source of bias-asymmetry and angular dependence of B-field

20-09-2010 Dias 2

J. Paaske, NBI

Suggested literature 1.

H. Bruus & K. Flensberg, “Many-Body Quantum Theory in Condensed Matter Physics” Oxford University Press (2004).

2.

R. Hanson et al.: “Spins in few electron quantum dots”, Reviews of Modern Physics 79, 1217 (2007).

3.

E. L. Wolf, “Principles of Electron Tunneling Spectroscopy”, Oxford University Press (1985).

4.

J. Von Delft: “Kondo effect in metals and quantum dots”, lecture notes from The 4th Windsor Summer School on Condensed Matter Theory, Available at http://www.lancs.ac.uk/users/esqn/windsor07/programme.html

5.

Articles cited along the way.

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J. Paaske, NBI

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J. Paaske, NBI

Molecular Transistor Realizations … Bardeen, Brattain og Shockley, Bell Labs 1947

Present day Intel workhorse

In

100 nm

Carbon nanotube (Delft)

Single molecule (NBI/NSC)

E E

C B

B

C

10 nm 1000 nm

20-09-2010 Dias 5

J. Paaske, NBI

Basic

(field effect)

transistor setup

Vg

V

Field Effect Transistor

drain

gate

source

?

I

Current through the device (from source to drain) - turns on (Logical 1) - turns off (Logical 0) by adjusting electrical potential on the gate electrodes.

20-09-2010 Dias 6

J. Paaske, NBI

Bias-spectroscopy of nanostructures

s

Vg

drain

d gate

source

V

I

”Coulomb diamonds” 20-09-2010 Dias 7

J. Paaske, NBI

Typical nanostructures of interest and many more ...

Heterostructure quantum dot (GaAs/AlGaAs)

Carbon nanotube

Semiconductor Nano-wire

C60 Peapod Single cell …

20-09-2010 Dias 8

Organic molecule

Metal complex

J. Paaske, NBI

Contacts of current interest

MATERIALS

Normal metal (Au) Ferromagnetic metal (Ni) Superconducting metal (Ti/Al, Pd/Nb)

NANOGAPS

Various combinations: NDS, SDS, SDF, FDF, etc…

New design! NSC® 2 nm gap Electron Beam Lithography

20-09-2010 Dias 9

Electromigration

Mechanical break junctions

Au Nano-rods


Charge conduction: Quantum tunneling + Classical charging

Elctrostatic-landscape: RS

RD

CG

CD

gate

source

CS

drain

Potential-landscape:

Drain

Source

Vg Filled states

20-09-2010 Dias 10

V

I


The Harlequin diamond plot: Coulomb Blockade Chemical potential of dot or molecule: S D

Plotting conductance as a function of

and Current thresholds:

I≠0

N, N+1

N-1, N

gives the slopes:

I=0 N-1

0

N

N+1 for

C.B.

C.B.

C.B.

C.B.

C.B. Addition energy:

20-09-2010 Dias 11

.

J. Paaske, NBI

Steady state current

(sequential tunneling)

Consider a single quantum level of energy Occupations:

S D

Source: Drain: Dot:

Tunneling rates:

Steady state (nonequilibrium) occupation number of the level:

Source and drain currents: Steady state current:

20-09-2010 Dias 12

Current is flowing only when the level lies within the ”bias-window”

J. Paaske, NBI

Bias dependence & level broadening Steady state current: S

But where is the voltage in Ohm’s law,

D

?

Tunneling broadens the quantum level and smears energy conservation: ,

Heisenberg!

This changes the current to:

, Conductance through a single level cannot exceed the conductance quantum:

20-09-2010 Dias 13

J. Paaske, NBI

Summarizing: I.

Conductance through a single quantum level is limited by e2/h

II.

Current is blocked by Coulomb-repulsion except for special resonant values of V and Vg .

III. Varying V and Vg leads to characteristic ’Coulomb-diamonds’ for the conductance: ”Single-electron transistor”.

InAs-wire based Quantum Dot, T. Sand Jespersen, NBI

20-09-2010 Dias 14

J. Paaske, NBI

Taking a closer look inside the diamonds …

20-09-2010 Dias 15

J. Paaske, NBI

Cotunneling: Lifting Coulomb blockade by quantum fluctuations Cotunneling rate

(2.-order PT):

Finite current:

Spinful dot (odd occ.) ”Kondo-effect”:

Charging cost:

N-1 0

20-09-2010 Dias 16

(∞-order PT):

N

N+1

J. Paaske, NBI

Inelastic Cotunneling: Bias spectroscopy

Extra contribution to the current:

Excited state spectroscopy !! Specific signatures:

0

• spin-flip transitions (Kondo-sharpened!) • vibrationally assisted

transitions

(sidebands!)

20-09-2010 Dias 17

J. Paaske, NBI

Kondo effect

(

J. Kondo, Prog. Theor. Phys. 32, 37 (1964) L. Glazman, M. Raikh, JETP Lett. 47, 452 (1988) T. K. Ng, P. A. Lee, Phys. Rev. Lett. 61, 1768 (1988)

)

(exchange amplitude J)

Hamiltonian:

(conduction (lead) electron s)

(localized (dot) spin S)

Transition probability in 3rd order perturbation-theory:

Perturbative Renormalization Group

(Poor man’s scaling

[PWAnderson, ’64])

: Universal scaling curve:

Integrate down to relevant energy-scale:

( Van der Wiel, Science 2000)

Interaction induced 20-09-2010 energy-scale ! Dias 18

Strong coupling regime: Landau Fermi Liquid Fixed Point [K.G. Wilson, ’71; P. Nozières, ’74]

J. Paaske, NBI

Observing a Kondo peak ...

(Liang et al., Nature 2002)

”Nucleon”

”Quark” Weak coupling:

Strong coupling: - Singlet (S=0)

Binding energy TK ~4K

- Doublet (S=1/2)

Spin is screened when lowering temperature! 20-09-2010 Dias 19

J. Paaske, NBI

Dot/lead-Hamiltonian

(2nd quantized many-body Hamiltonian)

Single-orbital Anderson model S

Kondo-regime:

Charge fluctuations are strongly suppressed! (Considered as a weak perturbation to Coulomb blockade) 20-09-2010 Dias 20

D

J. Paaske, NBI

Projecting out charge-fluctuations

(

odd)

The Schrieffer-Wolff transformation

Perform unitary transformation perturbatively:

Construct

so as to cancel the tunneling term

Satisfied with

, where:

J. R. Schrieffer, P. A. Wolff, Phys. Rev. 149, 491 (1966). 20-09-2010 Dias 21

P.-O. Löwdin, J. Chem. Phys. 19, 1396 (1951).

:

J. Paaske, NBI

Effective exchange-cotunneling

(Kondo)

model

Finishing the Schrieffer-Wolff transformation:

With (exchange-)cotunneling amplitudes:

(AFM exchange coupling)

(Potential scattering)

J. Appelbaum, Phys. Rev. Lett. 17, 91 (1966). P. W. Anderson 20-09-2010 , Phys. Rev. Lett. 17, 95 (1966). Dias 22

J. Paaske, NBI

Cotunneling current

(2nd order PT, finite B-field)

for Cotunneling-rates:

Nonequilibrium spin-occupation numbers:

20-09-2010 Dias 23

J. Paaske, NBI

Cotunneling conductance

(2nd, and 3rd order order PT, finite B-field)

M. R. Wegewijs, Y. Nazarov, arXiv: cond-mat/0103579 J. Paaske, A.20-09-2010 Rosch, P. Wölfle, Phys. Rev. B 69, 155330 (2004). V. N. Golovach, D. Loss, Phys. Rev. B 69, 245327 (2004). Dias 24


Inelastic cotunneling (typical experiments) Goldhaber-Gordon [GaAs/AlGaAs]

Zumbühl [GaAs/AlGaAs]

Ralph [Charge-trap]

Kogan [GaAs/AlGaAs]

Babic [CNT]

Nygård [CNT]

Cronenwet [GaAs/AlGaAs]

Schmid [GaAs/AlGaAs]

Osorio [OPV5]

…Zzz

Osorio [Mn2+]

20-09-2010 Dias 25

zz

zz

z ...

…z

zzz

zz

zZ…

J. Paaske, NBI

Contacting a single molecule

(Electromigration: gold wire)

and a bit of chemistry...

(Herre van der Zant et al., TU-Delft) 20-09-2010 Dias 26

J. Paaske, NBI

The completed single-molecule transistor ...

20-09-2010 Dias 27

J. Paaske, NBI

OligoPhenyleneVenylene5 • Chemical synthesis (Bjørnholm et al. NSC-Copenhagen)

• Low temperature bias-spectroscopy in electromigrated gold-junction (van der Zant et al., TU-Delft)

17_megah_lockin.dat

dI/dV (nS) 12000

80 10000 60 8000 40 6000 Vb (mV)

20 4000

0 -20

2000

-40

0

-60

-2000

-80

-4000 -2.5

-2

-1.5

20-09-2010 Dias 28

-1

-0.5

0 0.5 Vg (V)

1

1.5

2

2.5

The perfect void for inelastic cotunneling involving low-energy excitations! Compare: Molecule 100 meV

CNT-dot 5 meV

J. Paaske, NBI

[Mn(terpy-O-(CH ) -SAc) )]2+ 2 6

2

• Chemical synthesis (Bjørnholm et al. NSC-Copenhagen)

• Low temperature bias-spectroscopy in electromigrated gold-junction (van der Zant et al., TU-Delft) Al2O3 gate SiO2

AuPd Au

AuPd

S=5/2 High-Spin

20-09-2010 Dias 29

S=1/2 2 m

Low-Spin

Electrical Spin Control !

S=1/2

S=1

S=5/2

S=0

N=5

N=6

J. Paaske, NBI

Spectroscopic fine-structure in carbon nanotubes: Tunneling renormalization

20-09-2010 Dias 30

Maria-Alm, Austria, January 2008

CNT Coulomb-blockade diamonds

(bias-spectroscopy)

Adding 285 electrons, one by one...

88 odd-occupied charge states with zero-bias Kondo peak.

20-09-2010 Dias 31


The standard diamond

Shell-filling

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H

He

Li

Be

B

C

N

O

F

Ne

Na

Mg


The standard diamond Elastic cotunneling (Kondo-peak)

Inelastic cotunneling

20-09-2010 Dias 33

J. Paaske, NBI

20-09-2010 Dias 34

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Tunneling induced level-shifts in nanotube QD

[Ni leads]

Spin-polarized leads:

N=1

(tunneling out)

Gate-dependent spin-splitting:

(tunneling in)

(

)

N=0

N=2

(Bethe logarithms …)

N=1

20-09-2010 Dias 35

J. Martinek et al., Phys. Rev. Lett. 91, 127203 (2003). J. Martinek et al., Phys. Rev. Lett. 72, 121302(R) (2005). M. Sindel et al., Phys. Rev. B 76, 045321 (2007).

J. Paaske, NBI

20-09-2010 Dias 36

J. Paaske, NBI

Gate-dependent exchange-field

0

1

(tunneling induced ”Lamb-shift”)

2

Findings and prospects: • Electrical spin-control (not via induction fields!) Allows for much faster switching (Spintronics)

• Extremely localized ’magnetic field’ of order 1T Single electron spin control (Qubit initialization)

20-09-2010 Dias 37

(even 70 Tesla !!!)

J. Paaske, NBI

Tunneling induced level-shifts in nanotube QD Different tunneling-amplitudes to different orbitals:

N=1

N=1

N=0

N=2

N=1

20-09-2010 Dias 38

N=0

N=2

N=1

[Au leads]


17

21

Gate-dependent excitation energy

7

20-09-2010 Dias 39


Tunneling-induced level shifts

(2nd order PT)

tunneling out

tunneling in

Energy of dot-state with i electrons in orbital 1 and j in orbital 2: Tunneling rate for orbital i=1,2 to lead =source, drain:

Γ1 ¿ Γ2

20-09-2010 Dias 40

(∝

Vg )

J. Paaske, NBI

Gate-dependent excitation energies

Strong coupling sub-gap structure … Unresolved ?!

20-09-2010 Dias 41

J. Paaske, NBI

Inelastic cotunneling in quantum dots and molecules with weakly broken degeneracies G. Begemann et al., Phys. Rev. B 82, 045316 (2010)

Gate-dependent 20-09-2010 line-shapes Dias 42

J. Paaske, NBI

Lecture plan Part I 1.

Basics of Coulomb blockade and quantum conductance - Quantum tunneling and classical charging

2.

From Anderson model to cotunneling - Schrieffer-Wolff transformation

3.

Elastic vs. inelastic cotunneling - Bias spectroscopy

4.

Exchange cotunneling and basic Kondo effect - Signatures of Kondo effect

5.

Tunneling renormalization of cotunneling thresholds - Ferromagnetic leads, quasi-degenerate systems

Part II 1.

The nonequilibrium Kondo problem - What’s the problem?

2.

Poor man’s scaling for nonequilibrium systems - Lineshapes for inelastic cotunneling?

3.

The effect spin-orbit coupling - Source of bias-asymmetry and angular dependence of B-field

20-09-2010 Dias 43