Arrangement rule of stability regions and single-electron transfer in common-gate quadruple-dot devices for the real ratio of gate capacitances

For single-electron transfer in common-gate multidot devices, the arrangement of stability regions along the gate voltage (V_g) axis is important because single-electron transfer occurs around the overlap of stability regions. The stability regions along the V_g axis are well known to have periodicity when the device has an integer ratio of gate capacitances (C_g). However, the arrangement rule for the real C_g ratio is unclear. In this paper, stability regions for quadruple-dot devices with symmetric C_g are exhaustively examined. The arrangement of stability regions along the V_g axis is drawn as a map of the real C_g ratio in a newly proposed diagram. Here, the arrangement for a particular C_g ratio is drawn along a straight line that passes through the origin and has a slope depending on the C_g ratio. In the diagram, stability regions are arranged two-dimensionally, and the abovementioned periodicity for integer C_g ratios clearly appears. How neighboring stability regions interrelate with each other in the diagram is mathematically examined and described in detail. Next, the sequences of tunneling events around the overlap of stability regions are investigated, and eight kinds of tunneling sequences for single-electron transfer are determined.