Superconductivity at nanoscale

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?

Superconductivity in Nanosystems

2

Tunneling in superconductors Generally,


3

At the S-N interface, S

S

N

N

No single-electron tunneling possible until I

Δ Superconductivity in Nanosystems

V 4

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


5

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 Superconductivity in Nanosystems

6

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. Superconductivity in Nanosystems

7

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.


e

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


9


10

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.


11

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!


12

What happens in a superconductor? Energy depends on the parity of the electron number!

Parity effect:


13

The ground state energy for odd n is Δ above the minimum energy for even n Superconductivity in Nanosystems

14

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.


15

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 Superconductivity in Nanosystems

16

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


17

How one can convey Cooper pairs between superconductors?


18

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 Superconductivity in Nanosystems

19

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. Superconductivity in Nanosystems

20

To calculate the current let us introduce an auxiliary small magnetic field with vector potential δA which penetrates the junction. Then


21


22

Josephson interferometer

(after intergration)

Denote:

Most sensitive magnetometer - SQUID Superconductivity in Nanosystems

23

Josephson junctions in magnetic field y

1

Narrow junction –> H= const

2 x

Penetrated regions

Therefore


24

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)


25

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.


26

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.


27

Dynamics of a Josephson junction: I-V curve

A particle with In a washboard potential


28

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


29

Josephson junction in an a. c. field Important application – detection of electromagnetic signals Suppose that one modulates the voltage as

Then


30

Then one can easily show that at

a

time-independent step appears in the I-V-curve, its amplitude being

Shapira steps


31

Applications


32


33

Main Applications •Metrology, Volt standard

•High frequency applications

•Magnetometers, SQUIDs

•Amplifiers, SQUIDs

•Imaging, MRI, SQUIDs Superconductivity in Nanosystems

34

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 Superconductivity in Nanosystems

35


36


37


38


39


40


41


42

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 in Nanosystems

43