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Previous: 3.4.1 Band Structure of Relaxed Si Up: 3. Strained Bulk Band Structure Next: 3.5.1 Hierarchy of systems |
Application of strain to a crystal reduces its symmetry. The Bravais lattice
basis vectors
of a crystal under homogeneous strain are obtained by
deforming the vectors
of the unstrained crystal
Generally, the strain tensor (3.5) determines only the deformation of the primitive cell as a whole, but the relative displacements of the atoms within the cell differ for the atoms forming the basis. This has no impact on the lattice symmetry but has to be accounted for in band structure calculations. In Section 3.8.2 it is shown how to take this internal displacement into account properly.
Points and lines of the BZ can be classified according to their symmetry. For
each lattice there are symmetry operations that leave the lattice and hence the
BZ invariant. These operations bring the BZ into coincidence with itself. In such
an operation a specific vector
is not necessarily projected onto itself
or
+
. For each vector
there exists a set of
symmetry operations
fulfilling
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(3.32) |
In Figure 3.5b the symmetry points (filled circles) and symmetry lines
(open circles) of the fcc lattice are shown. Strictly, the given points
and
are not symmetry points according to the definition of
(3.33), since they have the same symmetry as the points along the
symmetry lines
and
. Hence, they can be included in these symmetry
lines.
Since the center point of the BZ is mapped onto itself at any
point operation
of the crystal lattice, all symmetry operations of the
lattice are included in the point group
. Thus, this group
determines the shape and volume of the irreducible wedge. The number of
symmetry elements of
determines the volume of the irreducible wedge
as [Nowotny98]
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Previous: 3.4.1 Band Structure of Relaxed Si Up: 3. Strained Bulk Band Structure Next: 3.5.1 Hierarchy of systems |