Figure 1

(a) Conduction and valence band edges of the triple quantum well schematically shown in the bottom. The widths \(d_0\) of the HgTe wells and the thickness t of the Hg\(_{1-x}\)Cd\(_x\)Te barriers (\(x=0.3\)) are indicated. Single well states located on each of left (L), central (C) and right (R) wells are illustrated in purple (electron-like) and blue (hole-like). As t decreases their hybridization leads to a large energy splitting of the E-like states at \(k=0\), while the H-like remain nearly degenerate due to the larger effective mass. (b,c,d) Band structures for \(t=3\) nm and varying \(d_0\). (b) For \(d_0 = 5\) nm, in the trivial regime, all E subbands are above the H subbands. \(\Delta E_I\) indicates the indirect band gap. Phase transitions occur as each E subband crosses the H subbands with increasing (c) \(d_0 = 6\) nm, (d) \(d_0 = 7\) nm, and (e) \(d_0 = 8.5\) nm. For large \(d_0\) the indirect gap \(\Delta E_I\) closes and the system becomes semi-metallic (shown in the Supplementary information which contains the \(8\times 8\) Kane Hamiltonian^53,54,55,56, the table of material parameters, the expressions for the matrix elements of the 2D model^42,43, and extra figures complementing the ones in the main text). All of the plotted subbands above present degeneracy in spin.