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.