Figure 1

(a) Schematic view of geometric frustration in a TQD with antiferromagnetic coupling between spins, or Ising isospins by the charge degree of freedom. QD1 and QD2 are in up and down spin (or isospin) states, respectively. Then, due to the antiferromagnetic coupling and the geometric symmetry, the spin-up (-down) state of QD1 (QD2) forces QD3 to have spin-down (-up). Hence, the spin of QD3 cannot be determined and is frustrated (as denoted by the question mark). (b) Charge stability diagram of a double QD. On the dashed line connecting electron (filled circle) and hole (open circle) triple degeneracy points, there are two degenerate ground charge states of (1,0) and (0,1). These are described by antiferromagnetic isospin coupling. (c),(d) SEM images of a symmetric TQD fabricated on a $\mathrm{GaAs}/\mathrm{AlGaAs}$ 2DEG wafer. The 2DEG is buried 77 nm below the surface of the wafer. The carrier density is $1.9\times {10}^{11}{\mathrm{cm}}^{-2}$, and the mobility is $1.1\times {10}^6{\mathrm{cm}}^2/\mathrm{V}\mathrm{s}$ at 4.2 K. The TQD is defined by $15/30\mathrm{nm}$ thick $\mathrm{Ti}/\mathrm{Au}$ metallic gates, which is patterned by electron-beam lithography.Reuse & Permissions

Figure 2

(a) The measurements are done by measuring current flow from QD1 to QD2 and QD3. (b) The conductance, measured with varying $V_{P2}$ and $V_{P3}$, constitutes the stability diagram on the $P2\mathrm{\text{−}}P3$ plane. $V_{P1}$ is fixed at the value where a sixfold degeneracy point (marked by a white arrow) appears. (c) The conductance measured on different planes of the plunger gates. (d) Stability diagram on the $P2\mathrm{\text{−}}P3$ plane, calculated from Eq. (1). It shows the charge frustration points [open and solid (red and yellow) circles], the triple degeneracy points of QD1 and QD2 [gray (blue) triangles], and the triple points of QD2 and QD3 [gray (green) squares]. The occupation numbers in the stability diagram are labeled such as (0,0,0), for clarity, by subtracting arbitrary constant numbers from the actual electron occupation numbers (which are positive) in TQD.Reuse & Permissions

Figure 3

(a) Energy diagrams of the sixfold degenerate ground-state charge configurations on the charge frustration point of a TQD. A red circle represents an electron occupying a dot, and a dark gray (blue) line shows the electrochemical potential of a dot. The six different configurations have the same electrostatic energy, resulting in omnidirectional transport via the degenerate states. (b) Measurement setup and (c) data of conductance through QD2 and QD3. The boundary lines (dotted lines) of the stability diagram are shown as guidelines.Reuse & Permissions

Figure 4

(a) Stability diagram, around sixfold degeneracy points, in a situation where electron tunneling between the TQD and the reservoirs is weaker than the case of Fig. 2b. The diagram is obtained by measuring conductance from QD1 to QD2 and QD3; see Fig. 2a. The highly conducting stripes are shown between two sixfold degeneracy points. Inset: The stability diagram inside the dashed box in the figure. The maximum value of the conductance in the stripe is higher than the values on the neighboring sixfold degeneracy point (white arrow) and triple points [gray (red) arrows]. (b) Temperature dependence of the conductance at various points in the stripes. (c) The calculated electrostatic energies of the TQD along the border between (0,0,1) and (1,0,0) [dashed dark gray (red) line in Fig. 2d].Reuse & Permissions