

Theory
Elimination of spurious solutions in 8band Hamiltonian
The number of bulk bands needed for such calculation depends on the value of band gap and the required accuracy. Thus for relatively large gap materials such as GaAs, it is a quite good approximation to consider conduction band and hole bands separately. But for narrow band material such as InAs and InSb it is necessary to treat conduction and hole bands simultaneously [8]. The resulting eightband model leads to the wellrecognized problem of spurious solutions [912] in the envelope function calculation of confined states. The origin of spurious solution is the incompleteness of the set of basis functions in the k•p approach, which makes it impossible for energy E to be a periodic function of wave vector k when it moves through the various Brillouin zones [10]. The spurious solutions with large imaginary k are related to the wingband solutions discussed in [13]. The wingband solutions are rapidly decaying in nature and therefore considered to be harmless in contrast to the oscillatory solutions with large real k. If the dispersion curve crosses the constant energy line twice, the two sets of confinement states appears. The spurious solution corresponds to the large k.
Bulk dispersion for InAs from the eightband k•p theory.


The two approaches are used to eliminate the spurious solutions. The first one is to introduce a small correction to the
Hamiltonian [14]. The small correction is a new αk_{z}^{2} term, which is added to √2/3Pk_{z} term (interaction between light hole and conduction bands) in the original Hamiltonian. As an example, after this modification the Hamiltonian with k_{x}=k_{y}=0 is
The coefficients s, β, t and μ can be derived from the previous expressions (5)(27) for the Hamiltonian. The α parameter has a critical value, above which the Hamiltonian does not have fast oscillating spurious states. This value is
if we use eightband model and
for six or fourband model without splitoff band. The second approach is to modify Kane P parameter in Eq. (47) and put s value equal to zero [15]:
where P can be extracted from Eq. (13).
