The k.p method can be derived from the one-electron Schrödinger equation as follows:
The wave function
can be expressed as the product of a plane wave and the function
, which reflects the periodicity of the lattice.
denotes the band index and
represents a wave vector. If the given potential
only depends on one spatial coordinate (also called local), (3.27) can be substituted in (3.26).
Luttinger [173] showed that it is possible to use the eigenfunctions of the ground states as a complete set of eigenfunctions and that the wave function can be expanded by
This way, for any fixed wave vector
, (3.29) for the unperturbed system, delivers a complete set of eigenfunctions
, which completely cover the space of the lattice periodic functions in real space. Therefore, the wave function
at
, for the full system, can be expressed via
![]() |
(3.30) |
Due to the
term in (3.29) this method is also known as the k.p method. Provided that the energies at
and that the matrix elements of
between the wave functions, or the wave functions themselves, are known, the band structure for small
's around
can be calculated. The entire first Brillouin zone can be calculated by diagonalizing (3.29) numerically, provided a sufficiently large set of
to approximate the complete set of basis functions is used [172].
The following subsections will explain the effective masses for the non-degenerate conduction band of silicon and the energy dispersion utilizing a non-degenerate k.p theory. In order to analyze the effects of shear strain on the two lowest conduction bands
and
, the k.p method is adapted to enable degeneracy, due to the coincidence of the
and
bands at the
point.
The conduction band minima of silicon reside on the
axes at a distance of
from the
symmetry points. By means of non-degenerate perturbation theory and the knowledge of the eigenenergies
and the wave functions
at the conduction band minima
, the eigenvalues
at neighboring points
can be expanded to second order terms in
.
![]() |
(3.32) |
![]() |
(3.34) |
![]() |
(3.35) |
From the derived equations follows that due to the coupling between electronic states in different bands (via k.p term), an electron in a solid has a different mass than a free electron. The coupling terms are related to the following criteria:
It is possible to calculate numerically all matrix elements and subsequently the effective masses from (3.33) via the empirical pseudo potential method [177].
However, there is a clear experimental proof that shear strain changes the effective masses of electrons in the lowest conduction band[170] and the exciton spectrum of silicon[171]. In order to explain this behavior one has to take the splitting of the lowest two conduction bands at the symmetry point by shear strain into account. The lifting of the degeneracy can be calculated with the deformation potential constant
via (3.20). (3.20) is only valid at the
symmetry point and cannot be used to predict the effect of strain on the valley minima
. In order to circumvent this obstacle a degenerate k.p theory has to be applied around the
symmetry point.
A different approach was adapted in [161]. The Hamiltonian at the
points can be described via the theory of invariants:
![]() |
(3.38) |
![]() ![]() |
(3.39) |
![]() |
(3.40) |
Up to now it has been assumed that the conduction band minima are located at
. This is only valid for small shear strain.
The minimum of the conduction band moves towards the
point in conjunction with an increasing splitting between the conduction bands, when the shear strain rises (as can be seen in Fig. 3.2). This causes a change in the shape of the conduction bands and the assumption that the minima lie fixed at
does not hold anymore.
Therefore, a model which is able to cover the effects of shear strain on the effective masses has to take the movement of the conduction band as a function of strain into account. In the following a model will be derived that takes this movement of
into account.
Starting with (3.41) and setting
the minimum can be found from the dispersion relation
The strain dependent longitudinal mass
can be calculated from (3.44) with
![]() |
(3.46) |
![]() |
(3.47) |
![]() |
(3.48) |
![]() |
(3.52) |
![]() |
(3.53) |
![]() |
(3.54) |
As can be seen along the
direction the effective mass is reduced (mobility is enhanced) for
, while for the
direction the effective mass is increased (the mobility is reduced) for increasing shear strain (
). For shear strain above
the effective mass is a constant which depends on the sign of the strain.
The analytical valley shift induced by shear strain
(given in (3.22)) can now be calculated. Substituting the expression for
from (3.45) into equation (3.44) delivers the equation for shear strain.The shift between the valley pair along
and the valley pairs
or
due to
can be obtained in the form of