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Superconductivity at nanoscale
Superconductivity is the result of the formation of a quantum condensate of
paired electrons (Cooper pairs).
In small particles, the allowed energy levels are quantized and for sufficiently
small particle sizes the mean energy level spacing becomes bigger than the
superconducting energy gap.
It is generally believed that superconductivity is suppressed at this point (the
Anderson Criterion)
Q: Is superconductivity important for nano-devices?
In which way superconductivity manifests itself at
nanoscale?
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Tunneling in superconductors
Generally,
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At the S-N interface,
S
S
N
N
No single-electron tunneling possible until
I
Δ
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Then, how the charge is transferred between the
superconductor and normal metal?
In a a normal metal
ε
Hole branch
Fermi
level
p
Electron-hole
representation
Hole-like excitation
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In a superconductor,
An electron can be reflected as a hole with opposite
group velocity. In this way the charge 2e is
transferred – Andreev reflection
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Reflected
Transmitted
Incident
An electron (red) meeting the interface between a
normal conductor (N) and a superconductor (S) produces
a Cooper pair in the superconductor and a retroreflected
hole (green) in the normal conductor. Vertical arrows
indicate the spin band occupied by each particle.
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In the presence of the tunneling barrier the Andreev reflection
contains an extra tunneling amplitude.
However, at
exponentially.
the single-particle tunneling is suppressed
Andreev reflection is a way to bring Cooper pairs to a
superconductor from a normal conductor in a coherent way.
For a perfect (non-reflecting) interface the
probability of Andreev reflection is 1.
In general case both reflection channels –
normal and Andreev – have finite probabilities.
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e
Cooper
pair
h
8
Normal Reflection in an
N/S Phase Boundary
between semi-infinite N
and S Layers
Total Andreev Reflection
in an N/S Phase
Boundary between semiinfinite N and S Layers
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Parity effect
How much we pay to
transfer N electrons
to the box?
Coulomb energy:
We have taken into account that the electron charge is
discrete.
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We have arrived at the usual diagram for Coulomb
blockade – at some values of the gate voltage the electron
transfer is free of energy cost!
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What happens in a superconductor?
Energy depends on the parity of the electron number!
Parity
effect:
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The ground state energy for odd n is Δ above the minimum
energy for even n
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Experiment (Tuominen et al., 1992, Lafarge et al., 1993)
Coulomb blockade of
Andreev reflection
The total number
of electrons at
the grain is about
109. However, the
parity of such big
number can be
measured.
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Stability diagram of
Cooper pair box
By Hergenrother
et al., 1993
SET
Superconductivity in small systems manifests itself
through energy scales of current-voltage curves
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Crossover from 2e periodicity to e periodicity can be
observed in external magnetic field suppressing
superconductivity
Observed in S’-S-S’
systems, where the
physics of Coulomb
blockade is similar
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How one can convey Cooper pairs
between superconductors?
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Stationary Josephson effect
S
I
V
S
Weak link – two superconductors
divided by a thin layer of
insulator or normal conductor
What is the resistance of the junction?
For small currents, the junction is a superconductor!
Reason – order parameters overlap
in the weak link
B. Josephson
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Amplitude
S
S
Since superconductivity is
the equilibrium state, the
overlap leads to the change
in the Gibbs free energy.
This energy difference is
sensitive to the phase
difference of the order
parameter (the order
parameter is complex).
We will show that it leads to
the persistent current
through the junction – the
Josephson effect.
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To calculate the current let us introduce an auxiliary
small magnetic field with vector potential δA which
penetrates the junction. Then
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Josephson interferometer
(after intergration)
Denote:
Most sensitive magnetometer - SQUID
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Josephson junctions in magnetic field
y
1
Narrow junction –> H= const
2
x
Penetrated regions
Therefore
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In a wide junction the magnetic field created by the Josephson current becomes
important. Then H and A become dependent on z
From the Maxwell equation
Ferrel-Prange equation
Josephson penetration length
Josephson vortices
Distribution of current in
narrow and wide contacts
(fluxons)
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Non-stationary Josephson effect
Due to the gauge invariance the electric potential in
a superconductor can enter only in combination
Thus, the phase acquires the additional factor
Here θ is the phase difference while V is voltage
across the junction.
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Thus, is the voltage V is kept constant, then
where
is the Josephson frequency
This equation allows to relate voltage and frequency,
which is crucial for metrology.
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Dynamics of a Josephson junction: I-V curve
A particle with
In a washboard potential
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Macroscopic quantum tunneling
A macroscopic Josephson
junction can escape from
its ground state via
quantum tunneling – like
the α-decay in nuclear
physics.
Quantum effects were
observed through the
shape of an I-V curve
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Josephson junction in an a. c. field
Important application – detection of electromagnetic signals
Suppose that one modulates the voltage as
Then
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Then one can easily show that at
a
time-independent step appears in the I-V-curve, its
amplitude being
Shapira steps
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Applications
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Main Applications
•Metrology, Volt standard
•High frequency applications
•Magnetometers, SQUIDs
•Amplifiers, SQUIDs
•Imaging, MRI, SQUIDs
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Medicine, biophysics and chemistry
•Biomagnetism
•Biophysics:
- Diagnostics by magnetic tagging of antibodies
-Special frequency characteristics, no rinsing
•MRI (Magetic Resonance Imaging)
- Low frequency, low noise amplifiers, sc solenoids
•NMR (Nuclear Magnetic Resonance)
-Low frequency, small fields, sc solenoids
•NQR (Nuclear Quadropole Resonance)
- Low frequency, low noise amplifiers, sc solenoids
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Summary
• Andreev reflection allows coherent
transformation of normal quasiparticles to
Cooper pairs.
• Cooper pairs can be transferred through
tunneling barriers via Josephson effect.
• Coulomb blockade phenomena manifest
themselves as specific parity effect in
superconductor grains.
• Manipulation Cooper pairs allow devices of
a new type, e. g., serving as building blocks
for quantum computation
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