Unconventional Nodal Wavefunctions in Quantum Dots

In a single band model such as one electron in a box, it is well known that the ground state wavefunction has no node maximizing its spatial symmetry. However, the ordering of eigenstates in a multiband system e.g., p-doped semiconductor quantum dots (QDs) can be very different due to spin-orbit interaction, symmetry of the underlying lattice and geometry of the confinement. Such unconventional ordering of states has appeared in the literature {[}1, 2{]} but it is often ignored or merely considered a shortcoming of $k\cdot p$ model {[}3{]}. We investigate spatial structure of hole envelope-wavefunctions in QDs with a focus on its symmetry. Our calculation shows a counter-intuitive ordering of eigenstates where a single hole ``ground-state'' has a node at the center. For simplicity, we start with a 2D QD tight-binding model and extend the discussion to 3D QD tight-binding and $k\cdot p$ models. We also discuss experimental implications of the wavefunction ordering described above. {[}1{]} K. V\'{y}born\'{y} et al., PRB \textbf{85}, 155312 (2012) {[}2{]} A. Bagga et al., PRB \textbf{71}, 115327 (2005); P. Horodysk\'{a} et al., PRB \textbf{81}, 045301 (2010); J. Xia and J. Li, PRB \textbf{60}, 11540 (1999) {[}3{]} L. W. Wang et al., APL \textbf{76}, 339 (2000)