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In the case of degenerate bands strain does not only shift the band as a whole, but may also split bands as a result of partial or complete removal of degeneracy upon the reduction of symmetry. In the diamond crystal structure, the lowest two conduction bands and touch at the zone boundary due to a special symmetry of the diamond structure, namely the presence of three glide reflection planes, given by , and [Yu03]. For example, the plane is a glide plane since diamond is invariant under a translation by followed by a reflection on this plane. Whenever the strain tensor in the crystal system contains a shear component (for example, as a result from stress along the [110] direction), the strained lattice belongs to an orthorombic crystal system (see Section 3.5.5). The shear component removes the glide reflection plane and consequently the degeneracy of the two lowest conduction bands and at the symmetry points is lifted [Hensel65,Bir74] (compare Section 3.5.5). It should be noted that the glide reflection symmetry is preserved in biaxially strained Si layers grown on {001} SiGe substrate, as well as in Si uniaxially strained/stressed along a fourfold rotation axis .
From kp theory (see Section 3.7) including terms of third order Bir and Pikus found that by lifting the degeneracy at a zone boundary point a comparatively large change in the energy dispersion of the conduction band minimum located close to this point is induced [Bir74]. In Si this effect was verified experimentally by Hensel and Hasegawa [Hensel65], who measured the change in effective mass for stress along , and by Laude [Laude71] who measured the indirect exciton spectrum.
Equation (3.47) has to be modified as follows when taking into account the lifting of the degeneracy of the two lowest conduction bands and at the points [Hensel65]
Tr | (3.50) | |
(3.51) |
Figure 3.12 shows the splitting for three levels of strain . The splitting is very pronounced even for relatively small strain (). One can also observe that the conduction band is deformed in the vicinity of the symmetry points due to the lifting of the degeneracy.
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In unstrained Si the constant energy surfaces of the six conduction band valleys have a prolate ellipsoidal shape, where the semi-axes are characterized by and , denoting the longitudinal and transverse electron masses, respectively. The minima of the three valley pairs are located along the three equivalent directions and have the same energies (see Section 3.4.1).
From Figure 3.12 it can be observed that a non-vanishing shear component in the strain tensor affects the energy dispersion of the lowest conduction band in three ways:
If the splitting of the conduction bands is different at the different zone boundaries (for example, ), the conduction band minima along the axes have different energies. This may result in a repopulation between the six conduction band valleys. Note that such an effect cannot be explained via equation (3.47) alone, where a possible lifting of the degeneracy at the point induced by shear strain is neglected and application of shear strain yields no valley repopulation.
An analytical expression for the valley shift along the direction can be derived using a degenerate kp theory at the zone boundary point [Bir74,Hensel65]. A shear strain causes an energy shift between the conduction band valleys along [001] with respect to the valleys along [100] and [010]. The shift is given by
(3.55) |
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