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Superconductivity in Nanosystems 1
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 at nanoscale
1. How is superconductivity affected by the size of the sample on nanometre scale? Does superconductivity exist in a sample
of any size? What is Anderson Criterion for superconductivity?
Superconductivity in Nanosystems 2
Temperature dependence of superconducting critical temperature for Bi, Pb and MgB2 superconducting nanocrystals.
Size dependence of superconductivity
C.C. Yang, Q. Jiang, Acta Materialia 53 (2005) 3305–3311.
Temperature dependence of superconducting critical temperature for Nb superconducting nanocrystals.
Superconductivity in Nanosystems 3
Pseudogap: a higher Tc on nanoscale?
There are evidences that electrons form pairs at a temperature T* that can be much larger than the critical temperature Tc
where superconductivity appears.
ZBCP
1) 18, 2) 35.5, 3) 43.6, 4) 51, 5) 80.5, 6) 109, 7) 186, 8) 208, 9) 227, 10) 281 and 11) 286 K
2. What is the relation between pairing of electrons and superconductivity? Can the first exist on nanoscale without the second?
Introduction to superconductivity 4
Characteristic parameters of superconductors
Critical temperature Tc, the penetration depth λ(0), the Cooper-pair size ξ(0) andthe upper critical magnetic field Hc2 for type-II superconductors (for layered compounds, thein-plane values are given)
Superconductivity in Nanosystems 5
A superconductor with one, two or three dimensions smaller than ξ is in thequasi-two-dimensional (2D), quasi-one-dimensional (1D) or quasi zero dimensional(0D) regime, respectively. For most superconductors, ξ is of the order ofnanometers. Therefore, systems falling in the 2D, 1D or 0D category arenanoscale systems. According to Hohenberg-Mermin-Wagner theorem(Hohenberg, 1967; Mermin & Wagner 1966), in these reduced dimensionalitysystems, fluctuations should destroy superconducting order even at lowtemperatures. In 2D samples, the Berezinski-Kosterlitz-Thousless transitionoccurs, enabling superconducting order to exist at reduced temperatures.
0,00 0,02 0,04 0,06 0,08 0,10
0,0
0,2
0,4
0,6
0,8
1,0
n = 105n = 35 n = 15
n = 7
n = 5
n = 3
n = 2
I
V
V = In
n = 1
I-V curves of a superconductor
Superconductivity in Nanosystems 6
Tunneling effect
Ivar Giaever
1973
The Nobel Prize in Physics 1973Leo Esaki, Ivar Giaever, Brian D. Josephson
Prize motivation: "for their experimental discoveries regarding tunneling phenomena in semiconductors and superconductors, respectively"Field: condensed matter physics, semiconductors
Superconductivity in Nanosystems 7
S SN N
I
V
No single-electron tunneling possible until
Δ
At the S-N interface:
Energy gap in superconductors
3. Describe principle of tunnelling between superconductor and normal metal. What are the implications of this nanoscale phenomenon?
What could be role of spin in this process?
Superconductivity in Nanosystems 8
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
N/S boundary reflections
Superconductivity in Nanosystems 9
In the presence of the tunneling barrier the Andreev reflection provides an extra tunneling amplitude.
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.
Andreev reflection
4. What is Andreev reflection? On what interfaces does it take place? What is zero bias conductance peak (ZBCP)? On what
interfaces can it be suppressed?
Superconductivity in Nanosystems 10
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
N/S boundary reflections: simulations
Superconductivity in Nanosystems 11
Andreev reflection
Superconductivity in Nanosystems 12
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.
6. What is parity effect in superconducting quantum dot and the Coulomb blockade of Andreev reflection? Can parity be
measured at the total number of electrons of about 109?
Introduction 13
Gate
DotElectron
Attraction to the gate
Repulsion at the dot
Cost
Single-electron transistor (SET)
Coulomb blockade in a tunnel barrier
E (Ne) = E ( (N+1)e)
At
the energy cost vanishes !
Q = Ne
Q0 = VgC
Superconductivity in Nanosystems 14
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!
Coulomb blockade
Superconductivity in Nanosystems 15
Parity effect:
What happens in a superconductor?
Energy depends on the parity of the electron number!
Coulomb blockade in superconductors
Superconductivity in Nanosystems 16
The ground state energy for odd n is Δ above the minimum
energy for even n
Electron parity effect
Superconductivity in Nanosystems 17
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.
Electron parity effect: experiment
Quantum dots 18
V=10 μV
What one would expect for a QD device?
Diamond stability diagram
SET
Coulomb blockade oscillations
Superconductivity in Nanosystems 19
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
7. Can ‘diamond’ features be seen in stability diagram of a Cooper-pair box? What other features can be seen there? Is
it possible to observe crossover from 2e periodicity to e periodicity in the Cooper-pair box?
Superconductivity in Nanosystems 20
Observed in external magnetic field that suppresses superconductivity
Observed in S’-S-S’ systems, where the physics of Coulomb blockade is similar to Andreev current blockade.
Crossover from 2e periodicity to e periodicity
Superconductivity in Nanosystems 21
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
Is it possible to convey Cooper pairs between superconductors?
19738. Describe stationary Josephson Effect. What kind of tunnelling does it represent? What is its nature
and what does its amplitude depend on? How is Josephson Effect linked with the phase of the order
parameter of superconductor?
Superconductivity in Nanosystems 22
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 current through the junction – the Josephson effect.
Principle of stationary Josephson effect
Superconductivity in Nanosystems 23
To calculate the current let us introduce an auxiliary small magnetic field with vector potential δA which penetrates the junction. Then
Josephson effect: derivation
9. Can you derive formula for Josephson Effect by treating overlap of wave functions and introducing small auxiliary magnetic field with
vector potential δA, which penetrates the junction?
Superconductivity in Nanosystems 24
Josephson effect: derivation
2e
Superconductivity in Nanosystems 25
Josephson interferometer
Denote:
Most sensitive magnetometer - SQUID
(after intergration)
10. Can you derive expression for current in Josephson interferometer using Ginzburg-Landau equations? What quantum of flux does
play role there?
Electron phase coherence
Let us make a confined tube of magnetic field
Will the interference pattern feel this magnetic field?
For a plane wave, the wave function
The phase gain along some way is then
As we know, in a magnetic field
Additional phase difference
1
2
Phase shift
26
Electron phase coherence
Φ
Aharonov-Bohm Effect for Nanowires
Magnetic flux quantum
Aharonov-Bohm Effect for Nanowires
t is transmission amplitude
Electron phase coherence
Phase coherence in ballistic systems
Split gates Interference pattern
Electrostatic Aharonov-Bohm Effect
A
28
Superconductivity in Nanosystems 29
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.
Another important application – detection of weak electromagnetic signals
Non-stationary Josephson effect
11. Can you derive formula for non-stationary Josephson Effect using an expression for the phase change due to electrical field? What is
the difference in this formula for superconductor comparable with normal metal?
Superconductivity in Nanosystems 30
Suppose that one modulates the voltage as
Then
Shapiro steps: derivation
Superconductivity in Nanosystems 31
Then one can show that at a
time-independent step appears in the I-V-curve, its
amplitude being
Different curves are measured for different amplitudes of microwave radiationand shifted along the x-axis.
Shapiro steps
V
Detector of electromagnetic waves for magnetoencephalography
t = 0 s t = 0.6 s
Magneto-optical imaging is crucial forchecking quality of superconductingtransformer
Superconductivity in Nanosystems 32
Superconducting music
Superconductivity in Nanosystems 33
Superconductivity in Nanosystems 34
Superconductivity market
16. How world market for superconductors is expected to growth and what is the role of nanophysics and nanotechnology in this growth?
In what areas is the biggest growth expected?
Superconductivity in Nanosystems 35
Applications
12. Describe use of superconductors in electronics. What is superconducting electronics’ advantage comparable with conventional
electronics?
Superconductivity in Nanosystems 36
Devices and applications
13. What are main devices superconducting electronics? Can you give details of one of them?
Superconductivity in Nanosystems 37
•Metrology, Volt standard
•High frequency applications
•Magnetometers, SQUIDs
•Amplifiers
•Imaging, MRI
Main Applications
Superconductivity in Nanosystems 38
Medicine, biophysics and chemistry applications
•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
Superconductivity in Nanosystems 39
Magnetoencephalography
Superconductivity in Nanosystems 40
SQUID gradiometer
Superconductivity in Nanosystems 41
Power applications
14. What are main power applications of superconductors and what is the role of nanotechnology in these applications? Please describe
in detail one of the applications.
Superconductivity in Nanosystems 42
Superconducting motors
43
http://www.aftenposten.no/amagasinet/Hvorfor-Tilbake-til-Fremtiden-II-tok-feil-7885865.html
Superconducting train
Superconductivity in Nanosystems 43
Superconductivity in Nanosystems 44
Transmission lines
R. L. Garwin and J. Matisoo, “Superconducting Lines for the Transmission of Large Amounts
of Electrical Power over Great Distances,” Proc. IEEE 55, 538, 1967. Nb3Sn (Tc = 18.3 K),
liquid He
J. R. Bartlit, F. J. Edeskuty and E. F. Hammel, "Multiple Use of Cryogenic Fluid Transmission
Lines," Proc. ICEC4, Eindhoven, 24/26 May, 1972. Cu, liquid H2 (20 K), liquid Natural Gas
(111 K)
D. E. Haney and R. Hammond, "Refrigeration and Heat Transfer in Superconducting Power
Lines," Stanford Report 275.05-75-2, April, 1975. Superconductors (14 K), liquid H2
S. M. Schoenung, W. V. Hassenzahl and P. M. Grant, "System Study of Long Distance Low
Voltage Transmission Using High Temperature Superconducting Cable," EPRI Report
WO8065-12, March, 1997 (Work performed as an EPRI Exploratory Research "Public
Benefit" project). HTS, liquid N2
P. M. Grant, "Will MgB2 Work?" The Industrial Physicist, October - November, 2001, p. 22.
HTS/MgB2, liquid H2
Short history of supergrid
Superconductivity in Nanosystems 45
15. What is supergrid and what are its prospects for solving current ecological and energy problems? What could be main
superconducting material for the supergrid?
V. S. Vysotsky, A. A. Nosov, S. S. Fetisov, G. G. Svalov, V. V. Kostyuk, E. V. Blagov, I. V. Antyukhov, V. P.
Firsov, B. I. Katorgin, and A. L. Rakhmanov, Hybrid energy transfer line with liquid hydrogen and
superconducting MgB2 cable — first experimental proof of concept, IEEE transactions on applied
superconductivity, 23 (2013) 5400906.
Cable version of supergrid: proof of concept
Inner diameter
12 mm
Total length12 m,
height 2.5 m
Superconductivity in Nanosystems 46
Cylinders for shielding
by Mg infiltration
P. Mikheenko, Superconductivity for hydrogen economy, Journal of Physics: Conference Series 286
012014 (2011). MgB2 pipes, liquid H2
Towards pipeline version of supergrid
Superconductivity in Nanosystems 47
P. Mikheenko, Superconductivity for hydrogen economy, Journal of Physics: Conference Series 286
012014 (2011). MgB2 pipes, liquid H2
Hydrostatic extrusion of MgB2 pipes
MgB2
W
MgB2
FORCE
800 C
Superconductivity in Nanosystems 48
Magneto-optical imaging of bulk MgB2
Magneto-optical image of a MgB2 bar at
magnetic field of 85 mT and temperatures
of 20 K (a), 37.4 K (b) and 38.4 K (c),
respectively. There is no visible
penetration of magnetic flux in the joints
(shown by arrows) at liquid hydrogen
temperature (a).
MOI images of the cross-section of an MgB2
joint at different temperatures.
P. Mikheenko, V. V. Yurchenko and T. H. Johansen, Magneto-
optical imaging of superconducting MgB2 joints, Supercond. Sci.
Technol. 25 045009 (2012)
Superconducting MgB2 joints
Superconductivity in Nanosystems 49
20 22 24 26 28 30 32 34 36
-3
-2
-1
0
1
2
StainlessSteel_1layer
Ta_1layer
StainlessSteel_2layers
m',
5m
'' (1
0-4 e
mu
)
T (K)
h = 0.5 Oe
20 22 24 26 28 30 320
20
40
60
80
100
StainlessSteel_2layers
I c-w
(A
/cm
)T (K)
Ta_1layer
H = 1000 Oe
MgB2 paint technology
Temperature dependence of real (lower part of the
plot) and imaginary (upper part) AC magnetic
moment of thick paint cover on Ta (black squares)
and stainless steel (blue and red squares).
Total critical current per centimetre of width Ic-w for
the layers shown in the left figure with the matching
colours, in 0.1 T magnetic field. The Ic-w of first MgB2
layer on stainless steel is not shown.
MgB2 paint-coating is simple and cheap method for liquid-hydrogen superconducting applications
Superconductivity in Nanosystems 50
MOI images of the screened magnetic flux in superconducting paint cover at temperatures 3.7 K (a) and 20
K (b). The images were recorded after zero field cooling and application of the magnetic field of 8.5 mT.
Magneto-optical imaging of MgB2 paint coatings
MgB2 paint-coating is simple and cheap method for liquid-hydrogen superconducting applications
a) b)
Superconductivity in Nanosystems 51
Superconductivity in Nanosystems 52
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 by Cooper pairs allows devices of a new type, e. g., serving as building blocks for quantum computation.
• There are multiple nanophysics-based applications of superconductivity. Market for superconductors is growing fast. Superconductivity promises solution of current ecological and energy problems.