On the Farsightedness (hyperopia) of the Standard k � p Model

Abstract

The use of a small number of bands in conventional k Á p treatment of nanostructures leads to "farsightedness" (hyperopia), whereby the correct, detailed atomistic symmetry is not seen by the model, but only the global landscape symmetry is noted. Consequently, the real symmetry is confused with a higher symmetry. As a result, a number of important symmetry-mandated physical couplings are unwittingly set to zero in the k Á p approach. These are often introduced, after-thefact, "by hand", via an ansatz. Sometimes physical effects (e.g., piezoelectricity) are invoked to fix the otherwise incorrect symmetry. Thus, whereas in atomistic theories of nanostructures (tightbinding, pseudopotentials) the physically correct symmetry is naturally forced upon us by the structure itself, in the standard k Á p model it is accommodated ex post facto once it is known from sources outside the model itself.

Anisotropic optical response of GaN and AlN nanowires

“…[18,20,21,19] This last methodology could omit the truth symmetry of the system, missing important effects. [22,23] This problem related to symmetry is automatically resolved in the atomistic approaches. Thus, the most accurate atomistic methodology are those funded in ab initio, and already implemented for nanowires of GaN, ZnO, [24] AlN [25] and ZnO/ZnS.…”

Symmetry reduction in multiband Hamiltonians for semiconductor quantum dots: The role of interfaces and higher energy bands

“…When one is not interested in very fine details of the electronic structure, such a shortcoming can be converted into a strength from the computational point of view, as the existing symmetry (not necessarily the correct one) can be exploited to block-diagonalize the Hamiltonian and therefore reduce the computational time. [57][58][59][60] There is however a widespread belief 61 that the mentioned deficiency is a general feature of multiband kÁp Hamiltonians and that these cannot be used if one needs to capture the true symmetry of the nanostructure. If this were true, this would be indeed a serious issue.…”

Reliable k⋅p band structure calculation for nanostructures using finite elements

“…(11) in [10]) in that particular basis. All diagonal terms are treated in the standard way Mk 2 x → k x Mk x while the offdiagonal terms Nk i k j in the valence band are split asymmetrically Nk i k j → k i N + k j + k j N − k i (6) and N − is estimated as [10] …”

“…Various simplifications during the perturbative treatment of remote bands obfuscate physical details, such as intervalley-and interface mixing [1,2]. This leads to an ongoing dispute about the precision and validity of the obtained results [3][4][5][6] where the raised arguments apply to the commonly used k·p standard model. It was shown recently [1,2] that an exact derivation of the envelope equations from first principles does allow for a consistent incorporation of interface mixing effects and an exact description of the band structure.…”