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Tunneling Spectroscopy of Quantum Statesin Nanoparticles and Single Molecules

• Quantized Electronic States in Metals -- How Interactions Affectthe Spectra

Spin-Orbit Effects

Superconducting Interactions

Non-Equilibrium Effects and Electron-Electron Interactions

Ferromagnetism and Magnetic Anisotropy Forces

• Tunneling via a Single Cobalt Atom in one Molecule

Jason Petta, Mandar Deshmukh, Sophie Guéron, Chuck BlackAbhay Pasupathy, Jiwoong Park, Jonas Goldsmith

Héctor Abruña, Paul McEuen, Dan Ralph

Thanks to Piet Brouwer, Jan von Delft, many others

Introduction to tunneling spectroscopyMeasuring “electrons-in-a-box” levels in a metal nanoparticle

Particles 3-10 nm.

Aluminum

Insulatingmembrane

Aluminumoxide

Aluminum

VG

I

V

metal nanoparticle

V

dI/dV

Can resolve discretestates if δE>>kT

δE

eVEF atV=0.

-10

-5

0

5

10

T = 4.2 K

200

100

0-100 -50 0 50 100

V (mV)

DataTheory

-400

-200

0

200

400

I (pA

)

-15 -10 -5 0 5 10 15V (mV)

T = 300 mKH = 0.1 Tesla

806040200

I (pA

)108642

V (mV)

T = 300 mKH = 0.1 Tesla

(a)

(b)

Coulomb blockade (~15 mV)

level spacing (~0.5 mV)

Relationship to Coulomb-Blockade Physics

806040200dI

/dV

(1/G

ž)

111098765

V (mV)

Lower Temperatures, Better Electrical FilteringΓ << kT

Part of Coulomb Diamondfor Aluminum Particle

Color scale denotesdI/dV.

Magnetic-Field Dependence of Aluminum Levels

g = 2.0 ± 0.1 for Al.

Even vs. Odd Numbers of Electrons:

e n e rg y

M a g n e t i c f i e l d

o c c u p ie d u no c c u p ie d

e n e rg y

M a g n e t i c f i e l d

o c c u p ie d u no c c u p ie dodd number of electrons even number of electrons

01234567

0.25 0.50 0.75 1.00 1.25V (mV)

B (T

)

In general, copper is a little more complicated than aluminum.

Effects of Spin-Orbit Scattering (perturbative picture)

1) g< 2perturbation theory:

nth level

Due to fluctuations in matrix elements and level spacing, gn variesfrom level to level.

2) Avoided crossings

gn = 2 1− 2ψm↓ Hso ψn↑

2

En − Em( )2m≠n

ℜ

�ℜ�ℜ�ℜ

ℜ

ℜ

�ℜ�ℜ�ℜ

H =

ε1(B) HsoHso

* ε2 (B)ℜ�ℜ

ℜ

ℜ�ℜ

Random Matrix Theories: In the presence of spin-orbit interactions, the random fluctuations in orbital electron-in-a-box wavefunctions will affect the spin part of the wavefunction.

Brouwer, Waintal, Halperin PRL 85, 369 (2000).Matveev, Glazman, Larkin PRL 85, 2789 (2000).

The g factor for each energy level should be a tensor -- should vary depending on the direction of magnetic field.

The theories provide quantitative predictions for the distributions of g-factors.

δε µµ2

2

12

12

22

22

32

32

4= + +B g B g B g B( ) , where B1, B2, and B3 are along the principal axes.

A

B

A700

0

B (m

T)

0.25 0.75V (mV)

C

D

B

C

D

Anisotropy of g-factors

Variations from Quantum State to Quantum State

Red: Cu#1

Blue: Cu#2

gmax gmiddle gmin

Orientations of principal axes for the g-factor ellipsoids:

The ellipsoids seem to be oriented randomly, as expected for coupling ofthe spin to random orbital wavefunctions.

Excellent quantitative agreement with theory for g-factor statistics.sample s-o strength gmax exp gmax th gmid exp gmid th gmin exp gmin th

Cu#1 1.8 1.3 ± 0.3 1.25 0.8 ± 0.2 0.76 0.4 ± 0.2 0.52Cu#2 1.1 1.6 ± 0.3 1.59 1.2 ± 0.2 1.12 0.9 ± 0.3 0.96

0.00

0.25

0.50

0.75

1.00In

tegr

ated

Pro

babi

lity

g-factor2.01.51.00.50

Cu #1 Cu #2 Ag #1 Au #1

1.0

0.5

0

Inte

grat

ed P

roba

bilit

y

Statistics of g-factors in different samples

One Mystery: g-factors for Gold are Surprisingly Low

g lLso

2 3= +π

τ δ α α ~ 1, geometry dependent factor

spin contribution orbital contribution

In the limit of strong spin-orbit scattering, theory (Matveev et al.) predicts:

5 Å Au on TEM grid

We measure as small as 0.02 in ballistic Au.

Is the orbital contribution to the g-factor somehow suppressed by interactions?

g2

Superconducting Interactions Favor the Formation of Electron Pairs

Even-to-odd tunneling: Extra gap for tunneling compared to normal state

EF

∆

unoccup ie d s t a t e st unne ling e -

Odd-to-even tunneling: First two tunneling states are separated in energy by ~ 2∆∆∆∆

EF

∆

EF

∆

Coope r Pa ir

t u n n e lin g

e -t u n n e lin g

e -

1.5

1.0

0.5

0.076543210

H (Tesla)

transitions from even to odd # of electrons

∆∆∆∆

1.5

1.0

0.5

0.076543210

H (Tesla)

transitions from odd to even # of electrons

∆∆∆∆

Spectra from a Superconducting Aluminum Nanoparticle

δE

eVEF atV=0.

Non-Equilibrium + Electron-Electron Interactions

Non-equilibrium excitations can begenerated during current flow if eV is greater than δE.

They can shift the allowed energiesfor the next tunneling electron.

Results: Broadened and shifted resonances at large V, or even new resonances

(Oded Agam et al., PRL)

Gat

e vo

ltage

V (mV)

I

II

Line I, expect no non-equilibrum Line II, non-equilibrium expected

sharpresonances

broaderresonances

~10 nm diameterAl particle.

|V| (mV)0 1 2

Effe

ctiv

e Te

mp.

(K)

0.0

0.2

0.4

II

start of non-eq for II

I

Line I, expect no non-equilibrum Line II, non-equilibrium expected

For very small particles (< 4 nm diameter for Al), individual non-equilibriumtransitions can be resolved, producing clusters of resonances.

Ferromagnetic nanoparticlesLow field behavior

MH

H

schematicsketch:

• No more simple Zeeman splitting -- internal magnetic field.

• Strong coupling between energy levels and magnetic moment.

• Every state shifts differently -- state-to-state variations in magnetic anisotropy energy.

• More levels than expected for particle in a box, due to the effects of spin waves.

Why is the level spacing (~ 0.2 meV) much smaller than the independent electron picture estimate (~ 2 - 40 meV)?

n0+1electrons

n0 electrons

Non-equilibrium spin excitations produce strong shifts in the energies of the tunneling resonances.

The rate of spin relaxation << the rate of e flow.

Superconducting leads, no magnetic field

Most of the transitions can be turned off by tuning the gate voltage.

“Single-atom” transistors

oCo Co

SH

SH

HS

HS

N NN

NNN

N NN

NNN

13 Å

24 Å

Related measurements: Hongkun Park (Harvard)

Cyclic voltammetry indicates that the charge on the Co ioncan be changed at low voltage

Co2+Co3+

I

V

electrolyte

• After breaking, the gap width can be estimated from the tunneling resistance.

• Typically 1-3 nm.

Flexible way to make gated nanojunctions.Can stick many things in the gap, after breaking in situ.

100 nm

AFM image

Electromigration Break Junctions (Park, McEuen, 1999)

Au Au

The Device

• On breaking the wires, a fraction of them have the two thiol groups of the molecule bridging the gap between the electrodes - the signature of this is Coulomb blockade.

•Control experiments on gold wires, and gold wires with only the tpy-SH molecules attached to them do not exhibit Coulomb blockade.

• Unbroken wires are immersed in Co-(tpySH) solution for 1-3 days

• Wires are broken at low temperatures (4.2 Kelvin and below) - this is essential to create small gaps when the electrodes are broken, and to reduce diffusion.

Source Drain

GateV

Vg

I

I-V traces at different gate voltages - Coulomb blockade

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

-50 0 50 100-1.0

-0.5

0.5

I(nA

)

V (mV)

0

-100

Excited quantum levels in Co(tpy(CH2)5SH)2 V

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)

Effect of vibrational states on tunneling

• In resonant tunneling, we get a step in current each time the voltage on the source sweeps past an energy level on the molecule

• Tunneling can be assisted by phonons. When the difference between the source energy and the energy level on the molecule matches a vibrational mode’s energy, there is a step in current.

• In support of vibrational modes is the fact that the structure of the levels is similar for both charged states.

ResonantTunneling

Inelastic tunneling

7 meV normal mode

Visualizing low-energy molecular vibrations

Zeeman Splitting in a Magnetic Field

0 2 4 60.0

0.5

Peak

spl

ittin

g (m

eV)

Magnetic field (T)-0.50

6

3

0

-3

-6-0.40

V(m

V)

Vg (V)

Co3+ Co2+

-0.45

g = 2.1±0.21.0

magnetic field = 6 Tesla

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

Higher-energy excitations.

E ~ 25 meV

Co

SH

HS

N NN

N NN

Shorter Linker Molecules -- Increased Coupling to Electrodes

Kondo-Assisted Tunneling via the Cobalt Atom

0T4T5T6T7T8T9T10T

0

1.0

1.2

dI/d

V(e

2 /h)

V (mV)-5 5 0-5 5

1.5K

18K0.8

1.4

1 10

1.4

1.2

1.0

0.8

T (K)

dI/d

V(e

2 /h)

0 3 6 9

-2.0

-1.0

0.0

1.0

2.0

Magnetic field (T)

V (m

V)

12 21 30Differential conductance (µS)

Zeeman Splitting of Kondo Resonance

ConclusionsMeasurements of electron-in-a-box energy levels provide a way to study in detail the forces acting on electrons

• Spin-orbit coupling

• Superconducting interactions

• Fluctuations in e-e interactions

• Ferromagnetism

Can make single-molecule transistors using designer molecules

• These molecules exhibit quantized electronic and vibrational states.

• Changing the length of the molecule changes the transistor characteristics.