Johnson noise at low temperature

It has long been accepted that Johnson noise in metals is proportional to absolute temperature, and is materials-independent. This is a semi-classical result, based on the equipartition of energy. Johnson noise in copper below 10 mK suggests a departure from this relation. The data have been fitted empirically to T_J = T_o coth $\text{T}{}_{\mathrm{<mtext>o</mtext>}}\text{T}$, where T_J is the “Johnson noise” temperature, T is measured by CMN, and T_o is chosen for best fit and is about 2 mK. T_o is now identifiable with a zero-point frequency of electronic charge imbalance, which moves with average Fermi velocity ν from end to end of the resistor of length l. $\text{T}{}_{\mathrm{<mtext>o</mtext><mtext>\; =\; </mtext><mtext>h\nu </mtext><mtext>2kl\; \approx \; 2\; </mtext><mtext>mK</mtext>}}$, and so is materials- and geometry-dependent.