Triple-dot single-electron devices with a single common gate have been studied. The overall stability diagram of the single-electron device with a homogeneous tunnel capacitance Cj and a homogeneous gate capacitance Cg is derived algebraically. If the set of excess electron numbers in the three islands (n1, n2, n3) is (n, n, n), (n, n + 1, n), or (n, n - 1, n), where n is an arbitrary integer, the corresponding stability region S(n1, n2, n3) exists for any Cg/Cj. S(n, n, n), S(n, n + 1, n), and S(n, n - 1, n) for all n are arranged along the Vg axis in the order of n1 + n2 + n3, where Vg is gate voltage, and neighboring stability regions overlap for any Cg/Cj ratio. Overlaps between S(n, n, n) and S(n, n ±1, n) for all n have identical kite-like shapes. Overlaps between S(n, n + 1, n) and S(n + 1, n, n + 1) for all n have identical rhombus shapes. Turnstile operations are possible by alternating gate voltage around overlaps between S(n, n, n) and S(n, n ±1, n) and around overlaps between S(n, n + 1, n) and S(n + 1, n, n + 1), though the sequences of single-electron transfers are different. The range of drain voltage and the swing of gate voltage for turnstile operation are estimated. The overlap between S(n, n + 1, n) and S(n + 1, n, n + 1) seems superior to the overlap between S(n, n, n) and S(n, n ±1, n) because of the larger ratio of the drain voltage range to the gate voltage swing, though the overlap between S(n, n + 1, n) and S(n + 1, n, n + 1) might require higher reliability against the nonuniformity of gate capacitances.
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