2.11. STRAINED HETEROSTRUCTURES 81
The built in strain causes several different effects on electronic properties: i) It can lift the
degeneracies or band edges; ii) it can change the bandgap; iii) it can alter effective masses.
To calculate the effect of strain one uses perturbation theory using equation 2.11.1. we will
summarize the relevant equations for a direct gap conduction band, an indirect gap X-valley
conduction bandedge and for the valence bands.
Case 1:Let us first examine how strain influences the bottom of the non degenerateΓ
′
2 state
which represents the conduction bandedge of direct bandgap semiconductors. This state is an
s-type state and has the full cubic symmetry associated with it. The effect of the strain is to
produce a shift in energy.
δE(000) = H
= Dxx(xx+yy+zz) (2.11.2)Conventionally we write
Dxx=Ξ(000)d (2.11.3)whereΞ(000)d represents the dilation deformation potential for the conduction band (000) valley.
Case 2:In this next case we will examine indirect gap materials like Si which have the con-
duction bandedge along the (100) and equivalent directions. The bandedges are shifted according
to the following equations.
δE(100)=Ξ(100)d (xx+yy+zz)+Ξ(100)u xx (2.11.4)By symmetry we can write
δE(010)=Ξ(100)d (xx+yy+zz)+Ξ(100)u yy (2.11.5)δE(001)=Ξ(100)d (xx+yy+zz)+Ξ(100)u zz (2.11.6)We note that if the strain tensor is such that the diagonal elements are unequal (as is the case in
strained epitaxy), the strain will split the degeneracy of the six valleys in Si. This occurs in the
SiGe/Si structures so that the 6-fold degenerate valleys split into 2-fold and 4-fold valleys. The
amount of splitting will be given later in this section.
Case 3:The triple degenerate states describing the valence bandedge.
The valence band states are defined (near the bandedge) by primarilypx,py,pz(denoted
byx,y,z) basis states. We have already discussed the strain tensor in epitaxial growth. For
(001) growth which has been the main growth direction studied because of its compatibility with
technology of processing we have
xx=yy = zz = −2 c 12
c 11(2.11.7)
The effects of the strain can be shown to be like heavy hole and light hole degeneracy at the
valence bandedge. This also causes the hole mass to become smaller. For the InyGa 1 −yAs