In this problem, for simplicity, we will consider an electron to be a positively charged particle with a charge of $e=1.6\cdot 10^ {-19}$ C.

Part A. Tunnel junction and Coulomb's blockade

A tunnel junction can be imagined as a parallel connection of a capacitor and a nonlinear resistance. The current through the resistance describes the tunneling of elementary charges from one plate to another if this flow is energetically favourable. In this part of the problem, we will derive the condition for tunneling, that is, we will determine under what conditions the resistance is finite and equal to $R_{\text{t}}$, and when it becomes infinite.

Consider a tunnel junction disconnected from the voltage source.

A1 Write down the expression for the energy $W$ of a capacitor with capacitance $C$ (in the equivalent circuit of a tunnel junction) if the plates have a charge of $n$ electron charges. Now let the value $n$ increase or decrease by one. How will the energy of the capacitor change?

A2 Assuming that the value of $n$ is not necessarily an integer, determine the values of voltage across the capacitor at which tunneling of single electron is impossible in either the forward or reverse direction.

The existence of a voltage range at which electron tunneling is impossible is called Coulomb blockade.

Part B. Single-electron pump

Using tunnel junctions, it is possible to assemble a so-called single-electron pump, which is capable of generating a direct current.

The figure shows a diagram of such a pump, consisting of three tunnel junctions and two capacitors (« gates »). Two ideal ammeters are included in the circuit to monitor the current flow. There are two points in this circuit where the charge can only change due to tunneling through the junctions. We will call these points « islands » and assume that they can only contain an integer number of electrons: $n_1$ and $n_2$, respectively. Depending on the voltages $U_1$ and $U_2$ applied to the gates, different numbers of electrons on the islands will be in equilibrium (energy-efficient). Fig. 3b shows the stability regions for each state (we characterize the states by the number of electrons on the islands). The line above the number indicates a minus sign.

Let us assume that tunneling of one electron, i.e., transition between neighboring states (across any boundary on the graph), requires time $\tau_0$.

Let the voltages across the gates depend on time according to the equations
$$\begin{cases}
U_1 (t) = U_1^{\text{d.c.}} + u\exp\left[ 2\pi i f t\right],\\
U_2 (t) = U_2^{\text{d.c.}} + u\exp\left[ 2\pi i f t + i\varphi\right].
\end{cases}$$Let us assume that $f\ll \tau_0^{-1}$. Then, after averaging over the period, the current value will be determined precisely by the frequency $f$ of the signal at the gates.

B1 At what value of $\varphi$ will the system in coordinates $U_1$ — $U_2$ move counterclockwise in a circle around point $P$ (see Fig. 3b)? What will be the direction of the current through the left ammeter?

B2 How will the answers to the questions in the previous paragraph change if the center of the circle is moved to point $N$?

The graph below shows the dependence of the current flowing through the left ammeter on the DC component of the voltage at the first gate, with a constant DC voltage at the second gate and $u = 0.3~\text{mV}$.

B3 Using the data in the graph, estimate the frequency $f$ of the signal at the gates.

Thus, by setting constant voltage components on the gates and changing the frequency of the variable component, it is possible to regulate the current flowing in the circuit.