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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
-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)
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)
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
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.