abhay pasupathy, jiwoong park, jonas goldsmith in … › talks › workshops › int_02_2 ›...
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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.