20-09-2010Dias 1

Jens PaaskeThe Niels Bohr Institute & Nano-Science Center

Cotunneling and Kondo effect in quantum dots

Part I/II

J. Paaske, NBI & NSC

Bad Honnef, September, 2010

20-09-2010Dias 2

J. Paaske, NBI

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

Lecture plan

20-09-2010Dias 3

J. Paaske, NBI

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.

Suggested literature

20-09-2010Dias 4

J. Paaske, NBI

20-09-2010Dias 5

Transistor Realizations …

In

J. Paaske, NBI

Molecular Transistor Realizations …

100 nm

1000 nm

10 nm

Present day Intel workhorse

Carbon nanotube (Delft)

B

E

C

Single molecule (NBI/NSC)

Bardeen, Brattain og Shockley, Bell Labs 1947

EC

B

20-09-2010Dias 6

J. Paaske, NBI

Basic (field effect) transistor setup

Current through the device (from source to drain)

- turns on (Logical 1)- turns off (Logical 0)

by adjusting electrical potential on the gate electrodes.

Vso

urce

drai

n

Vg

gate

I

? Field Effect Transistor

20-09-2010Dias 7

J. Paaske, NBI

Bias-spectroscopy of nanostructures

”Coulomb diamonds”

V

sour

ce

drai

n

Vg

gate

I

s

d

20-09-2010Dias 8

J. Paaske, NBI

Typical nanostructures of interestand many more ...

Heterostructure quantum dot(GaAs/AlGaAs)

Carbonnanotube

SemiconductorNano-wire

Organic molecule Metal complex

C60 Peapod

Single cell …

20-09-2010Dias 9

J. Paaske, NBI

Contacts of current interest

Normal metal (Au)

Ferromagnetic metal (Ni)

Superconducting metal (Ti/Al, Pd/Nb)

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

MAT

ERIA

LS

Mechanical break junctionsElectromigrationNAN

OG

APS

Electron Beam Lithography Au Nano-rods

New design!

2 nm gap

NSC®

20-09-2010Dias 10

J. Paaske, NBI & NSC

Charge conduction: Quantum tunneling + Classical charging

Source Drain

Filledstates

Potential-landscape:

V

sour

ce

drai

n

Vg

gate

I

Elctrostatic-landscape:

RS

CS CD

RD

CG

20-09-2010Dias 11

J. Paaske, NBI & NSC

The Harlequin diamond plot: Coulomb Blockade

Plotting conductance as a function of and

Chemical potential of dot or molecule:

0

N-1, N N, N+1

N-1 N N+1

C.B. C.B.C.B.C.B. C.B.

gives the slopes:

for .

Current thresholds:

0

D

S

I≠0

I=0

Addition energy:

20-09-2010Dias 12

J. Paaske, NBI

Steady state current (sequential tunneling)

Consider a single quantum level of energy

Occupations:

Source:

Drain:

Dot:

Tunneling rates:

Source and drain currents:

Steady state (nonequilibrium) occupation number of the level:

Steady state current:

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

D

S

20-09-2010Dias 13

J. Paaske, NBI

Bias dependence & level broadening

Steady state current:

But where is the voltage in Ohm’s law, ?

Tunneling broadens the quantum level and smears energy conservation:

,

This changes the current to:

,

Conductance through a single level cannot exceed the conductance quantum:

DS

Heisenberg!

20-09-2010Dias 14

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”.

J. Paaske, NBI

Summarizing:

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

20-09-2010Dias 15

J. Paaske, NBI

Taking a closer look inside the diamonds …

20-09-2010Dias 16

J. Paaske, NBI

Cotunneling: Lifting Coulomb blockade by quantum fluctuations

0

N-1 N N+1

Charging cost:

Finite current:

Cotunneling rate (2.-order PT):

Spinful dot (odd occ.) (∞-order PT):

”Kondo-effect”:

20-09-2010Dias 17

J. Paaske, NBI

Inelastic Cotunneling: Bias spectroscopy

0

Excited state spectroscopy !!

Specific signatures:

• spin-flip transitions(Kondo-sharpened!)

• vibrationally assistedtransitions (sidebands!)

Extra contribution to the current:

20-09-2010Dias 18

J. Paaske, NBI

Kondo effect ( )

Hamiltonian:

(conduction (lead) electron s)(localized (dot) spin S)

(exchange amplitude J)

Transition probability in 3rd order perturbation-theory:

Perturbative Renormalization Group (Poor man’s scaling [PWAnderson, ’64]):

Universal scaling curve:

( Van der Wiel, Science 2000)

Interaction induced energy-scale !

Integrate down to relevant energy-scale:

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

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)

20-09-2010Dias 19

Observing a Kondo peak ...

(Liang et al., Nature 2002)

Spin is screened when lowering temperature!

Weak coupling:

- Doublet (S=1/2)

Strong coupling:

- Singlet (S=0)Binding energy TK ~4K

”Quark”

”Nucleon”

J. Paaske, NBI

20-09-2010Dias 20

J. Paaske, NBI

Dot/lead-Hamiltonian(2nd quantized many-body Hamiltonian)

Single-orbital Anderson modelD

S

Charge fluctuations are strongly suppressed!(Considered as a weak perturbation to Coulomb blockade)

Kondo-regime:

20-09-2010Dias 21

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

20-09-2010Dias 22

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 , Phys. Rev. Lett. 17, 95 (1966).

20-09-2010Dias 23

J. Paaske, NBI

Cotunneling current (2nd order PT, finite B-field)

Cotunneling-rates:

Nonequilibrium spin-occupation numbers:

for

20-09-2010Dias 24

J. Paaske, NBI

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

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

20-09-2010Dias 25

J. Paaske, NBI & NSC

Inelastic cotunneling (typical experiments)

Osorio [OPV5]

Ralph [Charge-trap] Goldhaber-Gordon [GaAs/AlGaAs]

Cronenwet [GaAs/AlGaAs]

Schmid [GaAs/AlGaAs]

Zumbühl [GaAs/AlGaAs]

… Z z z z z z z z ... … z z z z z z z Z …

Kogan [GaAs/AlGaAs]

Osorio [Mn2+]

Nygård [CNT]

Babic [CNT]

20-09-2010Dias 26

J. Paaske, NBI

Contacting a single molecule (Electromigration: gold wire)

and a bit of chemistry...

(Herre van der Zant et al., TU-Delft)

20-09-2010Dias 27

J. Paaske, NBI

The completed single-molecule transistor ...

20-09-2010Dias 28

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

Vg (V)

Vb

(mV

)

-2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5

-80

-60

-40

-20

0

20

40

60

80

-4000

-2000

0

2000

4000

6000

8000

10000

12000dI/dV (nS)

The perfect void for inelastic cotunneling involving low-energy excitations!

Compare:

Molecule CNT-dot100 meV 5 meV

20-09-2010Dias 29

[Mn(terpy-O-(CH2)6-SAc)2)]2+

J. Paaske, NBI

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

SiO2

Al2O3 gate

AuPd

AuPd

Au

2 m

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

S=5/2 S=0

S=1S=1/2

N=5 N=6

S=1/2

Low-Spin

S=5/2

High-Spin

Electrical Spin Control !

20-09-2010Dias 30

J. Paaske, NBI

Spectroscopic fine-structure in carbon nanotubes:

Tunneling renormalization

20-09-2010Dias 31

Maria-Alm, Austria, January 2008

CNT Coulomb-blockade diamonds (bias-spectroscopy)

Adding 285 electrons,one by one...

88 odd-occupiedcharge states withzero-bias Kondo peak.

20-09-2010Dias 32

Maria-Alm, Austria, January 2008

The standard diamond

H He Li Be

B C N O

F Ne Na Mg

Shell-filling

20-09-2010Dias 33

Maria-Alm, Austria, January 2008

The standard diamond

Inelastic cotunneling

Elastic cotunneling (Kondo-peak)

20-09-2010Dias 34

J. Paaske, NBI

20-09-2010Dias 35

J. Paaske, NBI

N=1

Spin-polarized leads:

N=0N=2

N=1

Tunneling induced level-shifts in nanotube QD [Ni leads]

(tunneling out) (tunneling in)

Gate-dependent spin-splitting: ( )

(Bethe logarithms …)

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).

20-09-2010Dias 36

J. Paaske, NBI

20-09-2010Dias 37

J. Paaske, NBI

Gate-dependent exchange-field (tunneling induced ”Lamb-shift”)

0 21

Findings and prospects:

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

• Extremely localized ’magnetic field’ of order 1T (even 70 Tesla !!!)Single electron spin control (Qubit initialization)

20-09-2010Dias 38

J. Paaske, NBI

N=1

Different tunneling-amplitudes to different orbitals:

Tunneling induced level-shifts in nanotube QD [Au leads]

N=0N=2

N=1

N=1

N=0N=2

N=1

20-09-2010Dias 39

Maria-Alm, Austria, January 2008

Gate-dependentexcitation energy

17

21

7

20-09-2010Dias 40

Maria-Alm, Austria, January 2008

Tunneling-induced level shifts (2nd order PT)

tunneling out

tunneling in

Γ1 ¿ Γ2

Tunneling rate for orbital i=1,2 to lead =source, drain:

Energy of dot-state with i electrons in orbital 1 and j in orbital 2: ( )∝ Vg

20-09-2010Dias 41

J. Paaske, NBI

Strong coupling sub-gap structure … Unresolved ?!

Gate-dependent excitation energies

20-09-2010Dias 42

J. Paaske, NBI

Inelastic cotunneling in quantum dots and moleculeswith weakly broken degeneracies

Gate-dependent line-shapes

G. Begemann et al., Phys. Rev. B 82, 045316 (2010)

20-09-2010Dias 43

J. Paaske, NBI

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

Lecture plan