2.5 Minimum Tunnel Resistance for Single Electron Charging

    The formulation of the  Coulomb blockade model is only valid, if electron states are localized on islands. In a classical picture it is clear, that an electron is either on an island or not. That is the localization is implicit assumed in a classical treatment. However a preciser quantum mechanical analyses describes the number of electrons localized on an island N in terms of an average value $\langle N\rangle$ which is not necessarily an integer. The Coulomb blockade model requires $\vert N-\langle N\rangle\vert^2 \ll 1$. Clearly, if the tunnel barriers are not present, or are insufficiently opaque, one can not speak of charging an island or localizing electrons on a quantum dot, because nothing will constrain an electron to be confined within a certain volume.

A qualitative argument is to consider the energy uncertainty of an electron
$$\begin{gather}\Delta E\Delta t > h. \end{gather}$$
The characteristic time for charge fluctuations is
 $$\begin{gather} \Delta t \simeq R_T C, \end{gather}$$
the time constant for charging capacitance C through tunnel resistor R_T, and the energy gap associated with a single electron is
 $$\begin{gather} \Delta E = \frac{e^2}{C}. \end{gather}$$
Combining (2.26) and (2.27) gives the condition for the  tunnel resistance
$$\begin{gather}R_T > \frac{h}{e^2} = R_Q \doteq \text{25813}\ \mathsf{\Omega}. \end{gather}$$

Another line of thought proceeds as follows. The condition $\vert N-\langle N\rangle\vert^2 \ll 1$ requires that the time t which an electron resides on the island be much greater than $\Delta t$, the quantum uncertainty in this time.
 $$\begin{gather} t \gg \Delta t \geq \frac{h}{\Delta E} \end{gather}$$
The current I cannot exceed e/t since for moderate bias and temperature at most one extra electron resides on the island at any time. The energy uncertainty of the electron, $\Delta E$, is no larger than the applied voltage V_b.
 $$\begin{gather} \Delta E < eV_b \end{gather}$$
Inserting t=e/I and (2.30) into (2.29) results in
$$\begin{gather}R_T = \frac{V_b}{I} \gg \frac{h}{e^2}. \end{gather}$$

In fact, more rigorous theoretical studies of this issue have supported this conclusion [111]. Experimental tests have also shown this to be a necessary condition for observing single-electron charging effects [39].