Determining the most probable charge state of the island of a single electron transistor

When a current is flowing through a single electron transistor, the charge on its island is not constant. In dynamic equilibrium, the probability P(n) of a charge state n being occupied is constant and depends on the temperature and the voltages being applied. It is useful to know which of the charge states has the highest propability of being occupied. Here a formula for n_opt is derived that estimates the most probable charge state.

The relationship between the rates and the probabilities is,

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The maximum of the probability distribution will be between the two states where P(n^*-1) = P(n^*). The following equation can then be solved for n^*.

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At low temperature, the tunnel rates can be approximated by

for ,

and for .

For V₁ > V₂, current will only flow right through the junctions thus,

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This can be written as,

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Here, V(n) is the voltage on the island for the nth charge state. Finally, solving for n^*,

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A similar calculation for V₁ < V₂ yields the same result. The optimal charge state is halfway between n^* - 1 and n^*. The estimate for the most probable charge state is thus,

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