Figure 5: Calculation of the electron density and estimation of the Wigner crystal size.

(a) Geometry of the metallic gates and SGM tip defined in the Comsol simulation software and example of electrostatic potential map computed for a given gate voltage and tip position. The Fermi energy E_F is 8 meV in the 2DEG. (b) Map of the two-dimensional electron density n_2D in the 2DEG computed classically but self-consistently with the potential, when the tip is at 1 μm from the QPC. (c) One-dimensional electron density n_1D obtained by integration of n_2D along the y axis, when the tip is at 1 μm and 400 nm from the QPC. Choosing a critical density determines the expected size L_crystal of the 1D Wigner crystal. (d) Computed size of the Wigner crystal as a function of tip position in an horizontal plane 30 nm above the surface (V_gate=−1 V). The region in red corresponds to a closed contact (the electron density is zero at the QPC center for these tip positions). Black lines indicate tip positions for which L_crystal is enlarged by , corresponding at first order to the addition of one charge to the crystal. (e) Computed size of the Wigner crystal for tip positions in a vertical plane (above line X₁ at 35° from QPC axis).