# Superconductivity

Experiments:
• "An introduction to superconductivity" by Alfred Leitner originally published in 1965, source: www.alfredleitner.com/
• Isotope effect on the critical temperature. hyperphysics.phy-astr.gsu.edu/hbase/Solids/coop.html mentions that:
If electrical conduction in mercury were purely electronic, there should be no dependence upon the nuclear masses. This dependence of the critical temperature for superconductivity upon isotopic mass was the first direct evidence for interaction between the electrons and the lattice. This supported the BCS Theory of lattice coupling of electron pairs.
Lectures:
• Actually goes into the equations.
Notably, youtu.be/O_zjGYvP4Ps?t=3278 describes extremely briefly an experimental setup that more directly observes pair condensation.
Media:
• Cool CNRS video showing the condensed wave function, and mentioning that "every pair moves at the same speed". To change the speed of one pair, you need to change the speed of all others. That's why there's not energy loss.
Transition into superconductivity can be seen as a phase transition, which happens to be a second-order phase transition.

## High-temperature superconductivity (HTS)

As of 2020, basically means "liquid nitrogen temperature", which is much cheaper than liquid helium.

## Yttrium barium copper oxide (YBCO)

Upside: superconducting above 92K, which is above the 77K of liquid nitrogen, and therefore much much cheaper to obtain and maintain than liquid helium.
Downside: it is brittle, so how do you make wires out of it? Still, can already be used in certain circuits, e.g. high temperature SQUID devices.

## Bismuth strontium calcium copper oxide (BSCCO)

Discovered in 1988, the first high-temperature superconductor which did not contain a rare-earth element.

## Applications of superconductivity

Superconductivity is one of the key advances of 21st century technology:

## Superconductor I-V curve

TODO, come on, Internet!
Bibliography.

## BCS Theory

Main theory to explain Type I superconductors very successfully.
TODO can someone please just give the final predictions of BCS, and how they compare to experiments, first of all? Then derive them.
High level concepts:
• the wave functions of pairs of electrons (fermions) get together to form bosons. This is a phase transition effect, thus the specific sudden transition temperature.
• the pairs form a Bose-Einstein condensate
• once this new state is reached, all pairs are somehow entangled into one big wave function, and you so individual lattice imperfections can't move just one single electron off trajectory and make it lose energy

## Josephson effect

Discrete quantum effect observed in superconductors with a small insulating layer, a device known as a Josephson junction.
To understand the behaviour effect, it is important to look at the Josephson equations consider two cases separately:
A good summary from Wikipedia by physicist Andrew Whitaker:
at a junction of two superconductors, a current will flow even if there is no drop in voltage; that when there is a voltage drop, the current should oscillate at a frequency related to the drop in voltage; and that there is a dependence on any magnetic field
Bibliography:

## AC Josephson effect

This is what happens when you apply a DC voltage across a Josephson junction.
It is called "AC effect" because when we apply a DC voltage, it produces an alternating current on the device.
By looking at the Josephson equations, we see that a positive constant, then just increases linearly without bound.
Therefore, from the first equation: $$I(t)=Ic​sin(φ(t)) (1)$$ we see that the current will just vary sinusoidally between .
This meas that we can use a Josephson junction as a perfect voltage to frequency converter.
Wikipedia mentions that this frequency is , so it is very very high, so we are not able to view individual points of the sine curve separately with our instruments.
Also it is likely not going to be very useful for many practical applications in this mode.

## Inverse AC Josephson effect

If you shine microwave radiation on a Josephson junction, it produces a fixed average voltage that depends only on the frequency of the microwave. TODO how is that done more preciesely? How to you produce and inject microwaves into the thing?
It acts therefore as a perfect frequency to voltage converter.
The Wiki page gives the formula: en.wikipedia.org/wiki/Josephson_effect#The_inverse_AC_Josephson_effect You get several sinusoidal harmonics, so the output is not a perfect sine. But the infinite sum of the harmonics has a fixed average voltage value.
And en.wikipedia.org/wiki/Josephson_voltage_standard#Josephson_effect mentions that the effect is independent of the junction material, physical dimension or temperature.
All of the above, compounded with the fact that we are able to generate microwaves with extremely precise frequency with an atomic clock, makes this phenomenon perfect as a Volt standard, the Josephson voltage standard.
TODO understand how/why it works better.

## Josephson equations

Two equations derived from first principles by Brian Josephson that characterize the device, somewhat like an I-V curve: $$I(t)=Ic​sin(φ(t))dtdφ(t)​=ℏ2eV(t)​ (1)$$ where:
• : Josephson current
• : the Josephson phase, a function defined by the second equation plus initial conditions
• : input voltage of the system
• : current across the junction, determined by the input voltage
Note how these equations are not a typical I-V curve, as they are not an instantaneous dependency between voltage and current: the history of the voltage matters! Or in other words, the system has an internal state, represented by the Josephson phase at a given point in time.
To understand them better, it is important to look at some important cases separately:

## Josephson current ()

Maximum current that can flow across a Josephson junction, as can be directly seen from the Josephson equations.
Is a fixed characteristic value of the physical construction of the junction.

## Josephson phase ()

A function defined by the second of the Josephson equations plus initial conditions.
It represents an internal state of the junction.

## Josephson junction

A device that exhibits the Josephson effect.

## Josephson constant

The inverse of the magnetic flux quantum.

## Josephson voltage standard

The most practical/precise volt standard.
The wiki page en.wikipedia.org/wiki/Josephson_voltage_standard contains amazing schematics of the device, apparently made by the US Government.

## SQUID device

Can be used as a very precise magnetometer.
There are high temperature yttrium barium copper oxide ones that work on liquid nitrogen.

## DC SQUID

In Ciro's ASCII art circuit diagram notation:
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