Effect of island length on the Coulomb modulation in single-electron transistors

We have studied single-electron transistors with island lengths of 2, 10, 20, 30, and 40 μm and with high-resistance tunnel junctions to minimize the effects of cotunneling and electron self-heating. With longer islands, there is a marked reduction of the gate modulation of the Coulomb blockade width. According to orthodox theory, the width of the Coulomb blockade at $T=0\mathrm{}$ is equal to ${e/C}_{\Sigma },$ and it falls approximately exponentially with $k_B{T/E}_c,$ where $E_c{=e}^2{/2C}_{\Sigma },$ and $C_{\Sigma }$ is the total capacitance. Based on numerical calculations and analytic estimates, we conclude that the modulation reduction is mainly due to the large increase in stray capacitance between the island and the leads as the island length is increased, and not to some more subtle nonequilibrium effect. When the increased capacitance is combined with the effect of electron heating, the Coulomb modulation is rapidly reduced. This work also demonstrates the need to take stray capacitance into account in addition to intrinsic junction capacitance even in structures that are only a few μm long.