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Coulomb blockade in metallic islands and quantum dots
•Charging energy and chemical potential of a metallic island
•Coulomb blockade and single-electron transistors
•Quantum dots and the constant interaction model
•Finite bias and Coulomb diamonds
•Excited states: transport spectroscopy
•Calculating the current
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Different types of quantum dots
Gated 2-DEG
Metallic island
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Different types of quantum dots
InA
s
InA
s
InA
s
InP
InP
Semiconducting nanowire
Carbon nanotubeSingle molecule quantum dot
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Metallic island•Continuous density of states:
•Energy required to add one electron: EC � kBT
�E ⌧ kBT, . . .
VG
ΣCe2Sµ Dµ
Gα
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Quantum dot: quantized levels
100 nm dot 30 nm dot 10 nm dot
Single electron transistor (SET) limit: no energy quantization effects
Quantum dot limit: Quantum effects important
Δε
εΔ>>= ΣCeEC /2 εΔ>CE εΔ<CE
Constant interaction model peak spacing: e↵g�Vg = �µ = EC + "N � "N�1
EC ⇠ 1
C⌃⇠ 1
L
�" ⇠ 1
L2
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Finite bias voltage
Small width of resonance: � ⌧ kBT,EC
Current only when at least one is between andµN µL µR
Metallic island In general µN / �eX
i
↵iVi, ↵i =Ci
C⌃
With CL = CR, VL = �VR = V/2
is independent of µN Vand the diamonds are straight
Coulomb diamonds (zero current areas)
in general�µN
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Finite bias voltage
Stability diagram: differential conductance on color scale as a function of gate and bias voltages
dI(V, Vg)
dV
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Finite bias voltage
Quantum dot: even diamonds are larger because of spin degeneracy
"1,2"2,3
"4,5
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Finite bias voltage
Carbon nanotube quantum dots: four-fold patter because of additional orbital degeneracy
Leturcq, et. al. (2009)
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Finite bias voltage
What are all these other lines?
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Finite bias voltage: excited states
Energy needed to add electron N to the ground state, provided that the dot is in
the N-1 electron ground state.
Energy needed to add electron N to the excited state i, provided that the dot is in
the N-1 electron excited state j state.
Can also “tunnel through excited states”
increased current
conductance peak
Tunnel spectroscopy of quantum dot levels
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Finite bias voltage: excited states
Energy needed to add electron N to the ground state, provided that the dot is in
the N-1 electron ground state.
Energy needed to add electron N to the excited state i, provided that the dot is in
the N-1 electron excited state j state.
Tunneling can lead to occupation of excited states,also with energies larger than kBT
Current-induced “heating”. To calculate the current, we have to calculate the quantum dot state in nonequilibrium!
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Calculating the current
Single electron tunneling � ⌧ kBT,EC
Rate of electrons tunneling into state i from lead L, R: �L,R
X
j
f(µijN � µL,R) P (j)
Tunnel rate
Available electron in the lead?
Probability of state j being occupied
Rate of tunneling out to lead L, R, leaving the dot in state i:
Current: I = �L
X
ijN
n
f(µijN � µL,R)�
h
1� f(µjiN � µL,R)
io
P (j)
I = �L,R
X
j
h1� f(µji
N � µL,R)iP (j)
Need occupations! From thermal equilibrium (small V), or from solving rate (or master) equation
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Freaky diamonds and their usefulness
Spectroscopy of vibrational modes in OPV5 molecule Measuring lifetimes of
vibrations in suspended CNTs
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Freaky diamonds and their usefulness
Single-molecule magnet in a transport junction?
Can we see the peas in thepeapod?
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Freaky diamonds and their usefulness
Engineered Majorana fermions in a nanowire?