Plan for Dan Ralph’s Lectures (Tentative)

1. Basic introduction to Coulomb blockade in quantum dots Simple spin physics, Zeeman spin splitting Spin-orbit effects on Zeeman splitting, including random-matrix-theory ideas

2. Interacting Electrons within Quantum Dots -- the Universal Hamiltonian (a) Weak exchange interactions -- non-zero spin states prior to the Stoner instability (b) Superconducting pairing (c) Ferromagnetic quantum dots

3. Review of Kondo Experiments Single-Molecule Devices

4. Nano-magnetics (in metals) (a) basics of “giant magnetoresistance” (GMR) (b) spin-transfer torques and magnetic dynamics

Micron-scaleAluminum SET

(continuous DOS)Hergenrother et al.

GaAs SET(discrete DOS)

Patel et al.

-10-505

10

T = 4.2 K

200

100

0-100 -50 0 50 100V (mV)

DataTheory

Coulomb-Staircase Curves

-50 -25 0 25 50

1.5

1.0

0.5

0.0

-0.5

-1.0

-1.5

-2.0

source drain voltage [mV]

gate voltage [V]

-750-500-250 0 250 500 750

di/dv [pA/mV]

CoronaB R5C4 line3 in LHe (5/3/03)

-1

-0.5

0.5

0

-1

0

1

2 1

Vg

(V)

-20 100-100 0 50-50

V (mV)

color scale: dI/dV

Effects of Gate Voltage on Coulomb Blockade

)

-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 One Molecule

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

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

-400-200

0200400

I (pA

)

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

T = 300 mKH = 0.1 Tesla

806040200

I (pA

)

108642V (mV)

T = 300 mKH = 0.1 Tesla

(a)

(b)

Coulomb blockade (~15 mV)

level spacing (~0.5 mV)

806040200

I (pA

)

T = 320 mKH = 0.1 Tesla

806040200dI

/dV

(1/G

Ω)

111098765V (mV)

Tunneling through Individual Quantum StatesG << kT

Part of Coulomb Diamondfor Aluminum Particle

Color scale denotesdI/dV.

Red lines - excited states of n+1 electrons

Blue lines - excited states of n electrons

Black lines - ground states

nelectrons

n+1electrons

Magnetic-Field Dependence of Aluminum Levels

g = 2.0 ± 0.1 for Al.

Bias Voltage (mV)

Mag

netic

Fie

ld (T

)

0.0 0.2 0.4 0.6 0.8 1.0

2.01.81.61.41.21.00.80.60.40.2

200

100

0

I (pA

)

7654V (mV)

0.03 Tesla 3 Tesla

1.5

1.0

0.5

0.0dI/d

V (1

/MΩ

)

5.25.04.84.64.44.24.0V (mV)

3 Tesla 2 Tesla 1 Tesla 0.03 Tesla

(a)

(b)

odd number of electrons even number of electrons

g = 1.30 to 1.82<g> = 1.58

0

1

2

3

4

5

6

7

0.25 0.50 0.75 1.00 1.25

V (mV)

B (

T)

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

Random Matrix Theory Predictions: Distributions of the principal g-factorsfor different strengths of spin-orbit interaction.

• With one fitting parameter per sample, both the average g-factor and thestandard deviation are described well.• Significant differences in spin-orbit strength even for particles made ofthe same metal.

0.00

0.25

0.50

0.75

1.00

Integrated Probability

g-factor2.01.51.00.50

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

0

0.5

1.0

integratedprobabilitydistribution

l=12.7

l=2.4

l=1.2

l=0.7

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

Principal-axis directionsg1 axis g2 axis g3 axis

Principal-axis directions are randomly oriented in space.

Cu #4

Cu #5

x

z

<g1>

<g2>

<g3>

1.6 1.59

1.2 1.12

0.9 0.96

Exp. RMT<g1>

<g2>

<g3>

1.3 1.25

0.8 0.76

0.4 0.52

Exp. RMT

Check agreement between experiment and RMT predictions

23g2

2g21g ++

Single parameter fit: l (spin-orbit scattering strength) is determined by matching the experimental and theoretical values of .

Cu #4l=1.8

Cu #5l=1.1

Theory and experiment are in excellent agreement.

One Mystery: The g factors in gold nanoparticles are smaller than expected

Strong spin-orbit scattering limit (Matveev et al.)

glLso

2 3= +

pt d a

h, where a ~ 1

spin contributionorbital contribution

From Matveev et al.PRL 85, 2789 (2000)

For a ballistic nanoparticle, l~L2 1≥ ≥g

For a diffusive nanoparticle, l<Lg £ 2

If we assume the orbital part does not contribute, the spin contribution to the averageg factor gives an estimate for tso consistent with weak localization measurements.

Are the nanoparticles much more disordered than we expect so that they can quench theorbital contribution, or is there some shortcoming in the theory?

We see <g2> as small as 1/50.