I couldn't find the value of resistivity of copper at 2.73K on google
[Physics] the resistivity of copper at 3 kelvin?
electrical-resistance
Related Solutions
Related question on EE: Does perfect insulation exist? (especially the part about vacuum)
Insulators and conductors
The property of a material to carry charges from one point to another is what electric current is.
The difference between insulators and conductors lies in the electron band structure they posses. In conductors the Fermi-level (thermodynamically probable energy of an electron) is in a conducting(valence) band. This means most electrons will be ready to move and electricity will flow easily.
In an insulator the Fermi level is far away from the next conduction band, this means that very few electrons will be ready to move(because of a low probability being at a specific state). Other electrons can only be excited by very strong fields or other circumstances.
Therefore the conduction in an insulator can be described as by chance. As air is an insulator this is the case.
How does pressure change this behavior?
Very low, and very high pressure will reduce the number of charged particles that get transported. The first case is because there are only few particles, the second is because it is too stuffed and they collide and loose energy all the time. Actually you shouldn't look at pressure, but the product of pressure and electrode distance, as their product is the thing that is important. That means low and high pressure reduce conductivity.
Except for probabilities, in real life you have cosmic and other rays that can ionize air molecules. This is actually a very important part of the conduction mechanism for gasses, and can't be neglected.
Breakdown laws (but for voltages lower than the breakdown voltage)
As you were only interested in conductivity I will just tell you a few keywords regarding the other aspects of conduction during breakdown: Pashen's law, Townsend Mechanism, Streamers.
These mechanisms can be fed parameters which won't result in breakdown. In these cases they will describe the conduction mechanism (to some degree).
In copper there are mobile (free) electrons which are not attached to any particular nucleus and these free electrons are responsible for the conduction process in copper and other metals.
So you can think of a lump of copper as having copper ions held in position in a structure called a lattice and the ions vibrating about fixed positions.
The free electrons move around these ions within the metal just like gas molecules which are in a box.
When a voltage is applied across copper the free electrons start moving from the negative terminal to the positive terminal and in doing so gain kinetic energy.
However the free electrons do not have a free passage through the copper and collide with the vibrating copper ions losing some of their energy and making the copper ions vibrate more. Thus the temperature of the copper increases.
After a collision with a copper ion a free electron again gains kinetic energy and the process repeats itself.
The more the ions vibrate the greater the impediment to the passage of the free electrons.
So as the temperature of the copper gets less the copper ions vibrate less and so there are fewer impediments to the passage of the free electrons – the resistance of the copper is lower.
This is a simple model which does illustrate what happens but perhaps now it is better to make the model slightly more sophisticated and say that if all the copper ions in the lattice were arranged in perfect order then free electrons would not interact with the copper ions (lattice) and the resistance of the copper would be zero.
So you can think of the thermal vibrations of the copper ions as introducing irregularity in the copper lattice and these irregularities are responsible for the resistance of copper.
As your graph shows, reducing the temperature reduces the resistance and that is because the irregularities due to the ions vibrating are reduced.
The variation of resistance of copper with temperature can be predicted fairly accurately and it is found that theory so far and experiment agree until the temperatures get closer to 0 kelvin.
The predicted resistance due to the lattice vibrations becomes much less than the actual resistance which is measured.
So there must be other processes which are responsible for the copper having resistance at very low temperatures.
Again this resistance is due to irregularities in the copper lattice but not because the copper ions are vibrating.
The irregularities are there because there are impurity atoms in the copper and so where there should be a copper ion there is an atom of another element.
It is these impurities which interact with the free electrons and cause resistance.
But it is not just impurity atoms which are responsible for the resistance. It could be the lack of a copper ion at a particular location, it could be that the copper ions are not lined up perfectly as in a perfect crystal (these are called dislocations) and the copper is made up not just of one but many crystals, it is polycrystalline. The boundaries between the crystals are also irregularities which the free electrons interact with and this causes resistance.
So at very low temperature the purity and the structure of the copper are mainly responsible for the resistance of copper rather than the thermal vibrations of the ions.
It does not stop there because even if the sample of copper is very, very pure and one single almost perfect crystal the outer surfaces of the crystal would have an effect on the resistance of copper at very low temperature. the surface of the crystal being an irregularity.
You will find in more advanced texts that the lattice vibrations are thought of as bundles of energy and momentum and are called phonons. This is similar to calling a bundle of energy and momentum related to an electromagnetic wave a photon. The interactions between the lattice and the free electrons are thought of as being interactions/collisions between the phonons and the free electrons.
One of the reasons that copper is a better conductor than lead is that the phonon – electron interaction in copper is not as strong (weaker) than the phonon – interaction in lead.
This means that the lattice vibrations impede free electrons in copper less than in lead.
So it is rather strange that metals which are relatively poor conductors are more likely to become superconductors at low temperature?
Superconductivity is due to pairs of electron (Cooper pairs) coupling together with the aid of phonons. If the phonon – electron interaction is weak as it is in copper this means that Copper pairs are less likely to occur and so copper will never become a superconductor no matter how low the temperature. There are other reasons as to why superconductivity does or does not happen eg how many free electrons are produced by each atom. Copper only produces one free electron per atom whereas for lead there are more free electrons per atom available than in copper and the photon – electron interaction also being stronger gives lead more of a chance of becoming a superconductor.
Best Answer
There is a problem with answering your question in that at such a low temperature the resistivity of copper is very much determined by the impurities and crystallographic defects eg dislocations, voids etc, which may be present.
At low temperatures it is the scattering of free electrons by impurities and crystallographic defects which determine the resistivity rather than the thermal excitation of the copper ions.
The parameter which is often measured is the residual resistance ratio $\dfrac{R_{\text{273 K}}}{R_{4.2\,\rm K}}$ which for fairly pure copper wire as used for telephone lines might be of the order of $100$.
Large single crystals of very pure copper can be produced with residual resistance ratios in the thousands.
I have annotated the graph to illustrate the non-linear logarithmic scale and how to find the resistivity of copper at $3\, \rm K$.