In the case $ e^2/C_{\Sigma} \ll k_{\text{B}}T$, when the tunnel junction can serve as an element of a primary thermometer—that is, a thermometer that does not need to be calibrated beforehand.

In the simplest case, such a thermometer is a single-electron transistor — two tunnel junctions and a gate (see Fig.1). By recording the conditions and probabilities of tunneling, at sufficiently high temperatures, we can obtain an expression for the current flowing through the tunnel junctions
$$I = \dfrac{V}{2R_{\text{t}}} + \dfrac{e}{R_{\text{t}}C_{\Sigma}} \cdot \left[ f \left(\dfrac{eV}{k_{\text{B}}T}\right) - f \left(-\dfrac{eV}{k_{\text{B}}T}\right) \right],$$where $f(x) = \dfrac{1+(x-1)e^{x}}{(1-e^x)^2}.$
At very high temperatures (e.g., room temperature), the second term can be neglected, and Ohm's law remains valid for two tunnel junctions with resistance $R_{\text{t}}$ connected in series.

A1 Express the differential conductivity $G = dI/dV$ as
$$G/G_{\text{t}} = 1 - \dfrac{e^2/C_{\Sigma}}{k_{\text{B}}T}\cdot g\left(\dfrac{eV}{2k_{\text{B}}T}\right),$$where $G_{\text{t}} = (2R_{\text{t}})^{-1}$ — conductivity of the system at high (room) temperature.
Obtain the explicit form of the function $g(x)$ and plot its graph.

A2 Determine the depth of the differential conductivity gap$1 - G_{\text{min}}/G_{\text{t}}$ at $V=0$. Remember that $e^{x} = \sum x^n/n!$.

A3 Numerically determine the full width $x_{1/2}$ of $g(x)$ on the half-height:
$$g\left(\frac{1}{2}x_{1/2}\right) = \frac{1}{2} g(0).$$

A4 Obtain the numerical value of the width of the conductivity gap $V_{1/2}$ at the boiling point of liquid helium $T = 4.21~\text{ K}$.

Thus, the width of the differential conduction gap depends only on temperature, which allows a single-electron transistor to be used as a primary thermometer.

A5 The graph shows the dependence of normalized conductivity on half the voltage across the transistor ($V_{\text{BIAS}} = V/2$). For each of the four dependencies, determine the temperature at which it was studied.

A6 Determine the total capacitance $C_{\Sigma}$ of a single-electron transistor for which the above dependencies were obtained.