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Superconductivity at nanoscale
Superconductivity in Nanosystems 2
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?
Superconductivity in Nanosystems 3
Tunneling in superconductorsGenerally,
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At the S-N interface,
S SN N
I
V
No single-electron tunneling possible until
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Then, how the charge is transferred between the superconductor and normal metal?
p
Fermilevel
Hole-like excitation
Hole branch
Electron-hole representation
In a a normal metal
Superconductivity in Nanosystems 6
In a superconductor,
An electron can be reflected as a hole with oppositegroup velocity. In this way the charge 2e is transferred Andreev reflection
Superconductivity in Nanosystems 7
An electron (red) meeting the interface between a normal conductor (N) and a superconductor (S) produces a Cooper pair in the superconductor and a retroreflectedhole (green) in the normal conductor. Vertical arrows indicate the spin band occupied by each particle.
Reflected
Incident
Transmitted
Superconductivity in Nanosystems 8
In the presence of the tunneling barrier the Andreev reflection contains an extra tunneling amplitude.
However, at the single-particle tunneling is suppressed exponentially.
Andreev reflection is a way to bring Cooper pairs to a superconductor from a normal conductor in a coherent way.
e
e h
Cooper pair
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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Total Andreev Reflection in an N/S Phase
Boundary between semi-infinite N and S Layers
Normal Reflection in an N/S Phase Boundary
between semi-infinite 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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Parity effect:
What happens in a superconductor?
Energy depends on the parity of the electron number!
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The ground state energy for odd n is 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 about109. However, the parity of such big number can be measured.
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By Hergenrother et al., 1993
Stability diagram of Cooper pair box
Superconductivity in small systems manifests itself through energy scales of current-voltage curves
SET
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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?
Superconductivity in Nanosystems 19
Stationary Josephson effect
What is the resistance of the junction?
IS S
V
Weak link two superconductors divided by a thin layer of insulator or normal conductor
For small currents, the junction is a superconductor!
Reason order parameters overlap in the weak link
B. Josephson
Superconductivity in Nanosystems 20
S S
AmplitudeSince 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 currentthrough the junction the Josephson effect.
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To calculate the current let us introduce an auxiliary small magnetic field with vector potential A which penetrates the junction. Then
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Josephson interferometer
Denote:
Most sensitive magnetometer - SQUID
(after intergration)
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y
x
Josephson junctions in magnetic field
1 2
Penetrated regions
Narrow junction > H= const
Therefore
Superconductivity in Nanosystems 25
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
Distribution of current in narrow and wide contacts
Josephson vortices
(fluxons)
Superconductivity in Nanosystems 26
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 is the phase difference while V is voltage across the junction.
Superconductivity in Nanosystems 27
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 -decay in nuclear
physics.
Quantum effects were observed through the shape of an I-V curve
Superconductivity in Nanosystems 30
Suppose that one modulates the voltage as
Then
Important application detection of electromagnetic signals
Josephson junction in an a. c. field
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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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Metrology, Volt standard
High frequency applications
Magnetometers, SQUIDs
Amplifiers, SQUIDs
Imaging, MRI, SQUIDs
Main Applications
Superconductivity in Nanosystems 35
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
Superconductivity at nanoscaleSlide Number 2Tunneling in superconductorsSlide Number 4Slide Number 5Slide Number 6Slide Number 7Slide Number 8Slide Number 9Slide Number 10Slide Number 11Slide Number 12Slide Number 13Slide Number 14Slide Number 15Slide Number 16Slide Number 17How one can convey Cooper pairs between superconductors?Slide Number 19Slide Number 20Slide Number 21Slide Number 22Slide Number 23Slide Number 24Slide Number 25Non-stationary Josephson effectSlide Number 27Slide Number 28Slide Number 29Slide Number 30Slide Number 31Slide Number 32Slide Number 33Slide Number 34Slide Number 35Slide Number 36Slide Number 37Slide Number 38Slide Number 39Slide Number 40Slide Number 41Slide Number 42Summary
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