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Theory

Elimination of spurious solutions in 8-band 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 eight-band model leads to the well-recognized problem of spurious solutions [9-12] 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 wing-band solutions discussed in [13]. The wing-band 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 eight-band 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 αkz2 term, which is added to √2/3Pkz term (interaction between light hole and conduction bands) in the original Hamiltonian. As an example, after this modification the Hamiltonian with kx=ky=0 is

(47)
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
(48)
if we use eight-band model and
(49)
for six or four-band model without split-off band. The second approach is to modify Kane P parameter in Eq. (47) and put s value equal to zero [15]:
(50)

where P can be extracted from Eq. (13).

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