Applicability of the $\mathbf{k}\cdot \mathbf{p}$ method to the electronic structure of quantum dots

The $\mathbf{k}\cdot \mathbf{p}$ method has become the “standard model” for describing the electronic structure of nanometer-size quantum dots. In this paper we perform parallel $\mathbf{k}\cdot \mathbf{p}$ $(6\mathrm{}\times 6$ and $8\times 8)$ and direct-diagonalization pseudopotential studies on spherical quantum dots of an ionic material—CdSe, and a covalent material—InP. By using an equivalent input in both approaches, i.e., starting from a given atomic pseudopotential and deriving from it the Luttinger parameters in $\mathbf{k}\cdot \mathbf{p}$ calculation, we investigate the effect of the different underlying wave-function representations used in $\mathbf{k}\cdot \mathbf{p}$ and in the more exact pseudopotential direct diagonalization. We find that (i) the $6\times 6\mathbf{k}\cdot \mathbf{p}$ envelope function has a distinct (odd or even) parity, while atomistic wave function is parity-mixed. The $6\times 6\mathbf{k}\cdot \mathbf{p}$ approach produces an incorrect order of the highest valence states for both InP and CdSe dots: the $p$-like level is above the $s$-like level. (ii) It fails to reveal that the second conduction state in small InP dots is folded from the $L$ point in the Brillouin zone. Instead, all states in $\mathbf{k}\cdot \mathbf{p}$ are described as Γ-like. (iii) The $\mathbf{k}\cdot \mathbf{p}$ overestimates the confinement energies of both valence states and conduction states. A wave-function projection analysis shows that the principal reasons for these $\mathbf{k}\cdot \mathbf{p}$ errors in dots are (a) use of restricted basis set, and (b) incorrect bulk dispersion relation. Error (a) can be reduced only by increasing the number of basis functions. Error (b) can be reduced by altering the $\mathbf{k}\cdot \mathbf{p}$ implementation so as to bend upwards the second lowest bulk band, and to couple the conduction band into the $s$-like dot valence state. Our direct diagonalization approach provides an accurate and practical replacement to the standard model in that it is rather general, and can be performed simply on a standard workstation.