Download Coulomb Blockade

WS 2012/2013 Nanosystems I
Chapter 06: Coulomb Blockade
(Part 1)
Eric Hoffmann
WSI, S313
Tel.: 289 11553
[email protected]
Charles Augustin de Coulomb
(1736 - 1806)
Nanosystems I / WS 12-13
06. Coulomb Blockade - 1
Coulomb Blockade Part 1: Content
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Single tunnel junction
Current through a single tunnel junction
Charging energy of a capacitor
Double tunnel junction and the quantum dot
The constant interaction model
Artificial atoms
Coulomb Blockade
Single Electron Tunneling
Size and temperature dependence
Current CB research
Energy scales
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Dispersion Relations from 3D to “0D”
QD
z
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Single Tunnel Junction
S
D
The single tunnel
junction can be
considered a leaky
capacitor with
resistance R and
capacitance C.
RC
circuit symbol
EB
EF
Energy diagram
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Single Tunnel Junction
EB
EF
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Single Tunnel Junction
Fermi’s Golden Rule:
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Single Tunnel Junction
Fermi’s Golden Rule:
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Current through a Single Tunnel Junction
Simplification:
Total current:
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Single Tunnel Junction: R and C
Ohm’s law is obeyed:
S
D
The tunneling resistance, RT, should not be confused with an Ohmic
resistance, because of the very different nature of charge transport through a
tunnel junction and an Ohmic resistor. The latter arises from scattering events
while the former with (incoherent) transitions between quantum states.
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Double Tunnel Junction
S
Dot
D
The double tunnel
junction forms a charge
island, called a “quantum
dot” or simply a “dot”.
Dot
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Tunnel Junctions are 3D Objects
d
A
The dot couples capacitively
to its environment.
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Capacitive Energy
A
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Capacitive Energy
Via nanoscale engineering, we
can observe the capacitive
charging of a single electron!
A
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Making a quantum dot electrostatically
Quantum energy levels appear,
which are not related to the
classical capacitive energy.
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Quin + Holleitner LMU ‘00
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A nanowire quantum dot
InP
x
InAs
px and py quantization
via nanowire surface
z
Quantum dot
y
pz quantization via barriers
55 nm diameter
5 nm barriers
10 nm dot
740 meV
Strong quantum effects
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Combining Capacitive and Quantum Energy
Constant interaction model:
• C is assumed constant as N changes.
• The En spectrum is also constant in N.
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Electrochemical potential
Classical (Coulomb)
contribution
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Quantum
contribution
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Addition Energy
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The electrochemical potential of a NW
En are the solutions to the
“cylindrical box” problem.
Note that En = 0 when En
and En - 1 are degenerate.
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Quantum Dots are Artificial Atoms
• Quantum dots create core-shell
structure just like atoms.
• The quantum levels of the dot
have spin and orbital momentum
degeneracy.
• Quantum dots can have source
and drain leads, unlike atoms.
N:
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Coulomb Blockade and Single Electron Tunneling
a) No energy level, , exists
between S and SD, so current
does not flow. This is Coulomb
blockade.
b) Current can flow from source to
drain via (N). The source/drain
tunnel rates, S/D, determine the
magnitude of the current. This is
single electron tunneling.
c) The current measured through
the dot, IDOT, as a function of
gate voltage, VG.
VG tunes (N) continuously
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Coulomb Diamonds
VS
VG
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Single Electron Transistor
VG
I
V
SET
CB
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The Effect of Dot Size
66 nm diameter
L: 190 nm dot
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55 nm diameter
L: 10 nm dot
L
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The Effect of Temperature
2
1
0.4
0.3
0.15
0.075
U. Meirav et al.
Phys. Rev. Lett. 65, 771 (1990)
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Theoretical Coulomb Diamonds
V (mV)
Experimental Data
Theoretical prediction
Gate Voltage (V)
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White = High conductance
Black = Zero conductance
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Filling Orbitals
Björk et al. Nano Lett. 4, 9, 1621 (2004)
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A Room-temperature Quantum Dot
• Dot diameter: ~ 1 nm
• EAdd = 1.6 eV
• E = 0.8 eV
A. Barreiro, H.S.J. van der Zant, L.M.K. Vandersypen,
“Quantum Dots at Room Temperature carved out from
Few-Layer Graphene”, arXiv:1211.4551 (21 Nov. 2012)
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Energy Scales of Various Systems
Charging
Energy
QM
Excitation
Typical Size
Contacts
Nanowires
~ 10 meV
~ 1 meV/L[m]
~ 10 - 100 nm
optical and
e-beam lithography
nanotubes
~ 50 meV
1.7 meV/L[m]
~ 1 m
optical and
e-beam lithography
~ 22-90 nm
optical and
e-beam lithography
~ 20-500 nm
e-beam lithography
~ 1-6 nm
break junction,
self assembly
standard IC design
mesoscopic circuits
1 - 5 meV
0.1 - 0.5 meV
colloidal nanostructures
~ 30 meV
~ 100 meV
single molecules
> 200 meV
~ eV
~ 0.1 - 2 nm
nanopores, STM,
break junction,
self assembly
Etched graphene
> 1 eV
~ eV
~ 1 nm
e-beam lithography
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Summary
Coulomb Blockade
Coulomb Diamonds
Coulomb Diagram
Coulomb Energy
Coulomb Island
Coulomb Gap
Coulomb Oscillations
…
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Additional topics
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Converting gate voltage to energy
Predicting resonant current theoretically
Excited states
Cotunneling
Electron turnstile and current standard
Photon-assisted tunneling
Coulomb blockade with
superconducting leads
• …
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Coulomb Blockade Reading Material
• R. Hanson, L. P. Kouwenhoven, J. R. Petta, S. Tarucha, L. M. K.
Vandersypen, “Spins in few-electron quantum dots”. (2006)
Section II – Basics of quantum dots
http://arxiv.org/abs/cond-mat/0610433
• H. van Houten, C.W.J. Beenakker, A.A.M. Staring, “CoulombBlockade Oscillations in Semiconductor Nanostructures”. (1992)
http://arxiv.org/abs/cond-mat/0508454
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