Review of Kondo Experimentswith thanks to David Goldhaber-Gordon

Kondo modifies density of states, transport

Kondo ground state can “slosh”

Single-electron Transistor (SET)

• Quantum dot coupled to twoelectron reservoirs

• Fully tunable

• Current turns on and off each timeone electron is added

Current flows at fixed occupancy!

Goldhaber-Gordon, Kastner et al.,Nature 391, 156 (1998).

alsoCronenwett, Kouwenhoven et al.,Science 281, 540 (1998).

Even-odd enhanced conductance

G-G

Unitary conductance in SET

• Symmetric unitary limit of 2e2/hconductance has been achievedin a SET

• Conductance saturates at low T

van der Wiel, Kouwenhoven et al.Science 289, 2105 (2000).

Kondo resonance: Zero-bias peak in dI/dVds

• Kondo resonances pinned to EF oneither electrode.

• Theory: peaks broaden for larger Vdue to inelastic effects

• The bias suppresses conductance.

• Width of peak ~ 2 kBTK/e

Vds (mV)G-G, 1998

Conductance enhanced for odd occupancy

Sharp resonance indicatesmany-body state: lifetimelonger than single electron onthe quantum dot.

G-G, 1998

Applied Magnetic Fields

Resonances in the density of states undergoZeeman splitting.

Peak in dI/dV restored when ±eV = geffmBH.(geff can be less than bare g due to Knight shiftfrom interaction with electrons in the leads)

Can think of the peaks as due to Kondorenormalization of spin-flip scattering.

Theory: Meir Wingreen, Lee, PRL 70, 2601 (1993).

Vds (mV)-0.4

dI/d

Vds

(e2

/h)

G-G, 1998

Quantitative Testsof Kondo Theory

†

TK =GU2

exp pe0(e0 +U )GU

Ê Ë Á

ˆ ¯ ˜

G-G, Kastner et al., PRL 81, 5225 (1998)

More Complicated Line Shapes

In many systems, the Kondo effect has been observed as as a “Fano resonance”

Interference from localized and continuum(resonant and non-resonant) channels

Increased

con

tribu

tion

from

reson

ant ch

ann

el

Odom, Lieber

†

G = Ginc +G0˜ e + q( )2

˜ e 2 +1

†

˜ e =e -e0W /2

†

q µtransmission through resonant channel

transmission through nonresonant channel

Fano Lineshapes for Single Electron Transistors

weak coupling to leadsno Kondo (TK << T)

intermediate coupling to leadsKondo enhancement in valleys

strong coupling to leadsFano lineshapes

Gores, G-G, Kastner et al., PRB 62, 2188 (2000)

Scanning tunneling spectroscopy ofthe Kondo resonance

Tip

CeAg(111)

Berndt et al., PRL 80, 2893 (1998).-20 -10 0 10 20

1.0

1.1

1.2

1.3

1.4r =

10 Å

6 Å

5 Å

4 Å

3 Å

2 Å

1 Å

0 Å

[ dI /

dV

] /

[ dI /

dV

]V=0

Sample Bias V (mV)

CoCu(111)

r

Manoharan/Eigler et al.

afterMadhavan/Crommie)Science 280,567 (1998).

(q close to zero)

Zoom-in on the Kondo ResonanceZoom-in on the Kondo Resonance

-500 0 500 -50 0 50

dI /

dV

(a.u

.)

-150 0 150

Sample Bias V (mV)

Manoharan/Eigler

Imaging the Kondo ResonanceImaging the Kondo Resonance

Topograph

(V = 5 mV)

dI/dV map

(V = ±5 mV)

• Single Cobalt atom• Simultaneously acquired 35 Å square images

Manoharan

Spatial extend of screeningcloud is small, asmeasured by density ofstates near Fermi level

Kondo Effect: Multiple Co AtomsKondo Effect: Multiple Co Atoms

Topograph

dI/dV map

Monomer Dimer

Trimer

Manoharan

Jamneala, Madhavan, and CrommiePRL 87, 256804 (2001).

Cr trimers on Au

• Spectroscopic Features Near EF

From the theoretical fit, the Co/SWNT Kondo temperature ~ 90 K

• Spatial dependence of dI/dV peak

Kondo Effect in Co clusters on SWNTs

Odom, LieberScience 290,1549 (2000)

Three-terminal SET for non-equilibrium studies

Leturcq, Ensslin et al., cond-mat/050511 (2005)

experiment theory

De Franceschi, Kouwenhoven et al., PRL 89, 156801 (2002).

Three-terminal SET for non-equilibrium studies

• Peaks in conductance whenprobe is aligned with eitherreservoir

• Peaks get weak out ofequilibrium (Kondosuppressed)

De Franceschi,Kouwenhoven

Three-terminal SET for non-equilibrium studies

S B

DB

VP

P

R barrier

R

RR

bridge

AA

referencearm

• S ….. source

• D ….. drain

• B ….. base

• R ….. reflector

• P ….. plunger gate

Four Terminal Interferometer with Kondo Quantum Dot

Ji, Heiblum et al. PRL 88, 076601 (2002)Science 290, 770 (2000).(following Schuster et al. Nature 385, 417 (1997))

Expected Phase Evolution

0 50 100 150 2000.0

0.2

0.4

0.6

0.8

1.0

0.0

0.2

0.4

0.6

0.8

1.0

Coulomb Blockade Regime

measured by R.Schuster et al. (‘97)

• Phase evolves across resonance as expected

• Phase lapse in the valley not understood

• Similarity of phase in all peaks not understood

Plunger Gate Voltage

Mag

nitu

de (

2e2 /

h) Phase (p)

p

0 50 100 150 2000.0

0.2

0.4

0.6

0.8

1.0

0.0

0.2

0.4

0.6

0.8

1.0

0 50 100 150 2000.0

0.2

0.4

0.6

0.8

1.0

0.0

0.2

0.4

0.6

0.8

1.0

Kondo Correlated Regime

predicted by U. Gerland et al. (PRL 84, 3710 (2000))

T = 0 K T > 0 K

• Phase shift is a constant p/2 in the Kondo valley

• Total phase shift through two peaks is p

Plunger Gate Voltage

Mag

nitu

de (

2e2 /

h)

Phase ( p)

Plunger Gate Voltage

Mag

nitu

de (

2e2 /

h)

Phase ( p)

p/2

p

unitary limit

Measured Phase Evolution in Kondo Pair

phase slip

phase slip

Plunger Gate Voltage, VP (mV)

-300 -280 -260 -240 -220 -200 -180

0.00

0.05

0.10

0.15

0.20

0.25V

isib

ility

-1.0

-0.5

0.0

0.5

1.0Phase (p)

CB Kondo CB

p

Ji, Sprinzak, Heiblum

Integer-Spin Kondo (0 <-> 1)

6

N=3

S=0S=1

-1.4

-0.4

-1.2

-0.2

Magnetic field (T)

4

7

5

8

9

10

0

S=1/2

I

Vg

Vsd

a I d

b

1/2

1/2

↑ ↑

↑Ø

↑Ø

↑Ø

c N=6D

Magnetic field

B0

DRAIN

SOURCE

At singlet-triplet degeneracySasaki, Tarucha, Kouwenhovenet al., Nature 405, 764 (2000)

Two-Stage Kondo for S=1

van der Wiel, Kouwenhoven et al., PRL 88, 126803 (2002)theory by Pustilnik et al., cond-mat/0010336

Singlet-triplet degeneracy with single-channel leads

Other Kondo Variants

Spin 1/2, simultaneous orbital and spin degeneracySasaki, Tarucha et al., PRL 93, 017205 (2004)

Spin 1/2, orbital degeneracy, with and without spin degeneracyJarillo-Herrero, Kouwenhoven et al. Nature 434, 484 (2005)

Coupling of Kondo state with microwave photonsKogan, Amasha, and Kastner, Science 304, 1293 (2004)

Coupling of Kondo state with vibrational modes in molecular devicestheory: Paaske and Flensberg, PRL 94, 176801 (2005)

2-Channel Kondo effect in quantum dotstheory: Oreg and Goldhaber-Gordon, PRL 90, 136602 (2003)

Kondo effect in single-molecule transistorsKondo effect with magnetic electrodes

(more on these later)

Two-impurity Kondo

Dot 1 Dot 2

Craig, Marcus et al.Science 304, 565 (2004)

M is odd.When Dot 2 contains an unpairedelectron, the Kondo effect in Dot 1

is suppressed.

Two-impurity Kondo

Craig, Marcus et al.Science 304, 565 (2004)

The coupling between the twodots is tunable.

Odd number of electrons in both dots

Making Molecular-Scale Junctions

Techniques for Contacting Molecular-Scale Structuresß Scanning-Probe Techniquesß Adjusting Gap Sizes Controllablyß Nanoholes, and Devices made from Self-Assembled Molecular Layers (SAMs)ß Mechanical Break Junctionsß Electromigration Break Junctions

Some Physics Results with Electromigration Junctionsß C60 and C140 devices:

Single-electron transistors and molecular vibrationsß Transition-metal complexes:

Using chemistry to control coupling to electrodesCoulomb blockade and the Kondo EffectKondo Effect with Magnetic Electrodes

Limits of Electron-Beam Lithography

5 nm

Disadvantages:Reproducibility from device to device is about 5 nm

-- not quite to the molecular scaleHard to maintain clean surfaces.

1. Scanning-Probe Techniques

AdvantagesGreat flexibilityCan manipulate the sample and do

spectroscopy.Spatial InformationForce Information

DisadvantagesHard to make gates to tune energy levelsHard to imagine making circuitsVibration and driftHard to cool to low T for best spectroscopyElectrical lines must be filtered for best spectroscopy

B. Xu and N. J. Tao, Science 301, 1221 (2003)X. Xiao, B. Xu, and N. J. Tao, Nano Lett. 4, 267 (2004)B. Q. Xu, X. Y. Xiao, X. M. Yang, and N. J. Tao, J. Am. Chem. Soc. 127, 2386 (2005)

recent demonstration ofelectrochemical gating

2. Adjusting the Size of The GapElectrodeposition (Morpurgo et al., Appl. Phys. Lett. 74, 2084 (1999))Electrochemical Etching (Y. V. Kervennic et al., Appl. Phys. Lett. 83, 3782 (2003))

0 50 100 150 200 250 300 350

100

1000

10000

100000

1000000

1E7

~12kW

Sample 108Initial separation~70nm

Res

ista

nce

(ohm

s)Time (seconds)

Advantages: Easy to do.With electrodeposition, can make electrodes from different materials.

Disadvantages: Difficult to maintain surface cleanliness

Resis

tanc

e (O

hms)

Y. V. Kervennic et al., Appl. Phys. Lett. 83, 3782 (2003).

Electrochemical etching of a thin gold layer.

Exponential dependence of resistance as a function of time suggests a constant etch rate, even on near-atomic scales.

Controlled gap spacing?

3. Nanoholes

Si3N4

5-10 nm

Ralls, Buhrman (1989)

Wide variety of uses:10 nm

metal interfaces,spin filtering,Andreev reflection

quantizedstates innanoparticles

DNA transport(Brandon,Golovchenko)

self-assembledmonolayers(Reed)

Disadvantages of Studying Molecular Layers with Nanoholes

Hard to measure single moleculesNo gate to tune molecular energy levelsWhat happens to the monolayer when metal is deposited on top?Must be very careful to avoid leakage currents through nitride above 77 K.

More on evaporating metals onto molecular monolayers:

A. V. Walker et al., Appl. Phys. Lett. 84, 4008 (2004)From IR spectroscopy and SIMS, determine that evaporated Ti reacts with self-assembled monolayer

Au evaporated at room temp canpenetrate barrier. Molecules floaton top.

J. W. P. Hsu et al., J. Vac. Sci. Technol. B 21, 1928 (2003):From analysis of transport through GaAs/octanedithiol SAM/metal devices, ~ 35% of layer has penetrating gold for evaporation with room-temp substrate~ 1% of layer has penetrating gold for evaporation with -15 C substrate

These are problems for nanoholes, but even more so for larger contactson top of molecular layers.

Even when the initial evaporation does not produce metallic shorts through amolecular layer, subsequent application of voltages can produce metal filaments.

An applied voltage can be used to switch a Pt/C18/Ti device between high and low conductance states (compare Heath, etc.), but the change is due to a local metal filament, not the molecules.

C. N. Lau, D. R. Stewart, R. S. Williams, Marc Bockrath, Nano Lett. 4, 569 (2004).

A Gentler Way for Putting Metal Contacts on Top of Molecular Layers:nanoTransfer Printing

Y.-L. Loo, D. V. Lang, J. A. Rogers, J. W. P. HsuNano Lett. 3, 913, (2003)

4. Mechanical Break Junctions

Advantage: Very mechanically stableCan pull on molecule, look for change in resistance

Disadvantages: ß Some molecule studies have used large electric fields -- diffusion of gold

is a worry.

Used for studying molecules(Reed, Saclay, Leiden, Karlsruhe)

Total displacement = 5 Åwith stability better than 1 pm.Energy shift by the gate = 160 meV

Alex Champagne et al., Nano Letters 5, 305 (2005)

Gated Mechanical Break Junctions

dI/dV(mS)0 0.05 0.10 0.15

dI/dV(mS)0 0.005 0.01

0 2.5 5.0

Vg(V)0 2.5 5.0

Vg(V)

V(m

V)

0

-20

-40

20

40

60

V(m

V)

0

-20

-40

20

40

60

X0 X0+2.8Å

V(mV)-15 0 15 30

I(nA

)

0

-1

1

2X0X0+0.17ÅX0+0.33ÅX0+0.50Å

C60 devices

Gate areas and bondingcontacts are defined via

photolithography.

Nanowires are generated by

e-beam lithography.

Depositmolecules onelectrodes

Pass large currents:electromigration – Induced Gap formation.(Park, et al. APL 75, 301)

200 nmC60

Cobalt

5. Electromigration Break Junctions (Park, McEuen, et al., Appl. Phys. Lett. 75, 301 (1999))

Electromigration Junctions

• After breaking, the gap width can beestimated from the tunneling resistance.• Typically 1-3 nm wide. Relaxation occurswhen the devices are warmed to roomtemperature.

Flexible way to make gated nanojunctions.

150 nm

BEFORE AFTER

nanoscalegap

Pt

104 106 108 1010 10120

2

4

6

8

R (Ω)

num

ber o

f dev

ices

Single electron transistors based on individual C60 molecules“a quantum ball and spring”

J. Park, H. Park, et al., Nature 407, 57 (2000).

I

Source Drain

GateV

Vg

0.7 nm

- 40 - 20 0 20 40 60

- 0.2

- 0.1

0

0.1

I (n

A)

V (mV)

0.2

- 60

Vg = 6.9 V

Vg = 7.7 V

Vg = 5.9 V

• I-V curves at different gate voltages

After forming the nm-scale gap,deposit C60 in toluene solution.

C60n-

C60(n+1)-

Reading excitation energies

Bias voltage at the intersections

excitation energy

4.5 5.0 5.5 6.0 6.5 7.0 7.5 8.0

50

25

0

-25

-50

Vg (V)

V (

mV

)

C60n-

C60(n+1)-

Red lines - excited states of C60(n+1)-

Blue lines - excited states of C60n-

Black lines - ground states

5.5 6.0 6.5

15

0

30

-30

-15

2.4 2.6 2.8

1.0 1.1 1.3 1.4

10

0

20

-20

-10

-9.0 -7.0 -5.0-11.0

• 5 meV excited levels in both charge states.• Multiple lines with an equal spacing.

Multiple devices - 5 meV excitation

Excitations of a C60 molecule

Electronic excitation?- measured energy is too low- charge state dependent- multiple lines not explained

(Green, et al. J. phys. Chem. 100, 14892)

Rotational excitation? ~ 0.6 meV ( « 5 meV )

- measured energy is too large

Vibrational excitation…charge state independent…explains multiple lines

- Vibration against the surface - Internal vibration ( ~ 5 meV ) (> 33 meV )

h2 / 2I

Vg (V)

V (

mV

)

30

20

10

0-10

-20

-30-0.4 -0.2 0 0.2

1.4 1.6 1.8 2.0 2.2

20

10

0

-10

-20

-1.0 -0.8 -0.6

20

10

0

-10

-20

0.11 0.13 0.15

20

10

0

-10

-20

20

10

0

-10

-20-0.10 -0.05 0.0

0.00 0.04

20

10

0

-10

-20

0.02

C140 Transistors11 meV internal stretching mode

Excitations commonlyobserved near 11 meVand sometimes 22 meV.

Other lower-energystates are observed, too.

Sometimes the 11 meVline is absent.

ISource Drain

Al GateV

Vg

A. N. Pasupathy et al. Nano Lett. 5, 203 (2005).

SH

SH

HS

HS

N NN

NNN

N NN

NNN

13 Å24 Å

Designer molecules for making transistors

Co2+(tpy(CH2)5SH)2 Co2+(tpySH)2

• Longer molecule: Coulomb-blockade effects.

• Shorter molecule: Kondo effect.

Co Co

Related measurements, different molecules : H. Park (Harvard)

J. Park, A. N. Pasupathy et al., Nature 438, 457 (2003)

)

-50 0 50 100-1.0

-0.5

0.5

I (nA

V (mV)

Vg = -1.00V Vg = -0.86V Vg = -0.74V Vg = -0.56V Vg = -0.41V

0

-100

I

Source Drain

Silicon GateV

Vg

Coulomb-Blockade Effects in the Longer Molecule

• High resistance ( > megaOhms) - single electron charging.

• Coulomb blockade > 150 meV (unstable beyond this).

Excited quantum levels in the longer moleculeV

sd (m

V)

Vg (V)-0.50 -0.45 -0.40 -0.35

8

4

0

-4

-8

Co3+ Co2+

0.3 0.4

4

0

-4

2

-2

Vsd

(mV

)

Vg (V)

-2.10 -2.08 -2.06 -2.04

10

5

0

-5

-10

Vsd

(mV

)

Vg (V)-0.15 -0.10 -0.05 -0.00

20

10

0

-10

-20

Vsd

(mV

)

Vg (V)

Zeeman Splitting in a Magnetic Field

0 2 4 60.0

0.5

Pea

k sp

littin

g (m

eV

)Magnetic field (T)

-0.50

6

3

0

-3

-6-0.40

V (

mV

)

Vg (V)

Co3+ Co2+

-0.45

g = 2.1±0.2

1.0magnetic field = 6 Tesla

S=1/2 for Co2+, S=0 for Co3+.

0

1.0

1.2

(mV)V-5 5

c 1.5 K

18 K0.8

1.4

• Often high conductance ~ 2 e2/h• Peak height decreases logarithmically around TK• Peak splits as a function of magnetic field.• Kondo temperatures vary from < 1 K to > 50 K.

0 3 6 9

-2.0

-1.0

0.0

1.0

2.0

Magnetic field H (T)

V (

mV

)12 21 30

dI/dV (mS)

dI/d

V (e

2/h)

The Kondo Effect in the Shorter Co(tpySH)2

Abhay Pasupathy, Jiwoong Park, Jonas Goldsmith et al., Nature 417, 722 (2002).

Ferromagnetic electrode fabrication

• Tunneling MagnetoResistance (TMR)of a bare nickel point contact

• Shape anisotropy affects the switching fields– electrodes switch independently

For P = 0.31 for Ni, Julliere estimate isJMR = 2P2/(1-P2) = 21%

B (mT)-100 0 100

0.32

0.38

R (

MΩ

)

C Ni-NiAP

P

JMR = 19%

Au contacts

50 nm

1 mm

Nickel electrodes

gate

Splitting of zero-bias anomaly due to exchangeinteraction

Cannot be due to magnetic field.5 meV splitting would require >50 T.

• Au – C60 – Au

• Ni – C60 – Ni ferromagnetic sample

No splitting for B=0

B=0

B=10T

B=0

Ni-C60-Ni

parallelelectrodes

1.5 K

Abhay Pasupathy, Jan Martinek et al.Science 306, 86 (2004)

(see also Natelson group,Nano Lett. 4, 79 (2004))

Kondo effect in C60 moleculeswith and without magnetic electrodes

Kondo splitting depends onelectrode orientation• Parallel

• Antiparallel

• Large splitting for parallel moments

• Reduction of splitting when moments are antiparallel

• Gradual change corresponds to noncollinear geometry

Good agreement with theory of Kondo effect with magnetic electrodes(J. Martinek et al., PRL 2003)

Density of States of Quantum Dotwith Magnetic Electrodes

0.10.2 0.0 0.1 0.2

mRmL

d%ede

0.10.2 0.0 0.1 0.2

mRmL

d%ede

0.2P=/0.005TG=d/2G=-e/100DG=

Splitting is due to unequal hybridization between thequantum well state and the spin-polarized states in theelectrodes -- resulting in a very large effective field.

• Antiparallel Moments inElectrodes (equal coupling)• Parallel Moments in Electrodes

There are several complementary techniques for molecular-scale device fabrication.• Need more experimental “knobs” -- ways to make systematic studies of moleculartransport properties (e.g., combine gating, mechanical motion, light, in situ chemicalmodification, etc.)• Understanding/control of the metal-molecule connection is a challenge.• More experiments need to be reproduced.• Best to understand full distribution of devices, not just “best” ones.

It is possible to make electrical contact to individual molecules and to manipulate theenergy of their electronic states with a gate to give transistor action.

The strength of electronic coupling between molecular states and the leads can bechanged by adjusting the molecular structure:

• Weak coupling: Coulomb blockade, vibration-assisted tunneling• Strong coupling: Kondo effect, vibration signals blurred or absent (f ~ G)

Measurements so far provide new insights into Kondo physics, the coupling betweenelectronic and mechanical degrees of freedom in molecules, and spin transport througha molecule between magnetic electrodes.

Tools for Studying Electron and Spin Transport

in Single Molecules

Will molecular electronics ever be useful? -- Hard to say.Challenges:

• No gain• Devices not stable at room temperature yet • Relatively slow speeds (long RC times) • How to wire up many devices correctly?

At this stage, our main goal is scientific exploration.

Additional References

Experiments on Single-Molecule Transistors1. J. Park, H. Park, et al., Nature 407, 57 (2000).2. J. Park, A. N. Pasupathy, et al., Nature 417, 722 (2002).3. W. Liang et al., Nature 417, 725 (2002).4. L. H. Yu and D. Natelson, Nano Lett. 4, 79 (2004).5. A. N. Pasupathy et al., Nano Lett. 5, 203 (2005). Selected Theory of Vibration-Assisted Tunneling in Single Molecules1. K. Flensberg, Phys. Rev. B 68, 205323 (2003).2. S. Braig and K. Flensberg, Phys. Rev. B 68, 205324 (2003).3. A. Mitra, I. Aleiner, and A. J. Millis, Phys. Rev. B 69, 245302 (2004).