Page 45 - Semiconductor For Micro- and Nanotechnology An Introduction For Engineers

The Crystal Lattice System
c
ab
assume that the vectors , and are perpendicular. The Bravais lattice
is inherently symmetric with respect to translations q : this is the way we
j
construct it. The remainder of the symmetries are related to rotations and
reflections. There are crystal systems: hexagonal, trigonal, triclinic,
7
monoclinic, orthorhombic, tetragonal and cubic. In addition, there are
14 Bravais lattice types, see Figure 2.12. These are grouped into the fol-
lowing six lattice systems in decreasing order of geometric generality (or
increasing order of symmetry): triclinic, monoclinic, orthorhombic, tet-
ragonal, Hexagonal and cubic. The face-centered cubic (fcc) diamond-
like lattice structure of silicon is described by the symmetric arrangement
of vectors shown in Figure 2.13 (I). The fcc lattice is symmetric w.r.t. 90 o
rotations about all three coordinate axes. Gallium arsenide´s zinc-blende
bcc-structure is similarly described as for silicon. Because of the pres-
ence of two constituent atoms, GaAs does not allow the same transla-
tional symmetries as Si.

Primitive We associate with a lattice one or more primitive unit cells. A primitive
Unit Cell unit cell is a geometric shape that, for single-atom crystals, effectively
contains one lattice point. If the lattice point is not in the interior of the
primitive cell, then more than one lattice point will lie on the boundary of
the primitive cell. If, for the purpose of illustration, we associate a sphere
with the lattice point, then those parts of the spheres that overlap with the
inside of the primitive cell will all add up to the volume of a single
sphere, and hence we say that a single lattice point is enclosed. Primitive
cells seamlessly tile the space that the lattice occupies, see Figure 2.14.

Wigner-Seitz The most important of the possible primitive cells is the Wigner-Seitz
Unit Cell cell. It has the merit that it contains all the symmetries of the underlying
Bravais lattice. Its definition is straightforward: The Wigner-Seitz cell of
lattice point p contains all spatial points that are closer to p than to any
i i
other lattice point q . Its construction is also straight-forward: Consider-
j
ing lattice point p , connect p with its neighbor lattice points q . On
i i j
each connection line, construct a plane perpendicular to the connecting
line at a position halfway along the line. The planes intersect each other

42 Semiconductors for Micro and Nanosystem Technology