This recent paper shows a nice example of applying three different primary thermometry techniques to a single system, a puddle of electrons confined in 2d at a semiconductor interface, at about 6 mK. This is all the more impressive because of how easy it is to inadvertently heat up electrons in such 2d layers. All three techniques rely on our understanding of how electrons behave at low temperatures. According to our theory of electrons in metals (which these 2d electrons are, as far as physicists are concerned), as a function of energy, electrons are spread out in a characteristic way, the FermiDirac distribution. From the theory side, we know this functional form exactly (figure from that wikipedia link). At low temperatures, all of the electronic states below a highestfilledstate are full, and all above are empty. As \(T\) is increased, the electrons smear out into higher energy states, as shown. The three effects measured in the experiment all depend on \(T\) through this electronic distribution:
Fig. 2 from the paper, showing excellent, consistent agreement
between experiment and theory, showing electron temperatures
of ~ 6 mK.

 Current noise in a quantum point contact, the fluctuations in the average current. For this particular device, where conduction takes place through a small, controllable number of quantum channels, we think we understand the situation completely. There is a closedform expression for what the noise should do as a function of average current, with temperature as the only adjustable parameter (once the conduction has been measured).
 "Coulomb blockade" in a quantum dot. Conduction through a puddle of electrons connected to input and output electrodes by tunneling barriers ("pinched off" versions of the point contacts) shows a very particular form of currentvoltage characteristic that is tunable by a nearby gate electrode. The physics here is that, because of the mutual repulsion of electrons, it takes energy (supplied by either a voltage source or temperature) to get charge to flow through the puddle. Again, once the conduction has been measured, there is a closedform expression for what the conductance should do as a function of that gate voltage.
 "Environmental" Coulomb blockade in a quantum dot. This is like the situation above, but with one of the tunnel barriers replaced by a controlled resistor. Again, there is an expression for the particular shape of the \(IV\) curve where the adjustable parameter is \(T\).