Figure 3.2: The first Brillouin zone of the relaxed silicon lattice is shown.
The valley positions and high symmetry points are shown as well.
Figure 3.3: The silicon band structure calculated by the pseudopotential
method (CB is the conduction band, and VB is the valence band) is described.
The VB edges are located exactly at the Γ-point and the minimum of the
lowest CB lies on the symmetry Δ line close to the X-point. The lowest two
CBs degenerate exactly at the X-point.
Figure 3.4: Left: constant energy surface of unstrained silicon (six-fold
degeneracy) is shown, right: conduction band splitting under shear tensile
strain on (001) plane is shown. The red (green) color-fill signifies high (low)
Center of the Brillouin zone (k space origin)
Middle of square faces
Middle of hexagonal faces
Middle of edge shared by two hexagons
Middle of edge shared by a hexagons and a
Middle of edge shared by two hexagons and a
Directed from Γ to X
Directed from Γ to L
Directed from Γ to K
Table 3.1: The symmetry points of the first Brillouin zone in silicon are
listed according to Figure 3.2 are listed.
The first Brillouin zone is defined as the primitive cell in the reciprocal lattice.
In silicon, the first Brillouin zone has a shape of a truncated octahedron
(c.f. Figure 3.2), and is characterized by eight hexagonal faces and six
square faces. The conduction band (CB) edge is located near the zone
boundary X points along the Δ symmetry lines. The corresponding symmetry
points and directions are tabulated in Table 3.1. The valence bands (VB)
contain the last filled energy levels at T=0K, whereas the conduction
bands are empty. The band gap Egap separates the CB from the VB. The
band structure is usually visualized by plotting En(k) along symmetry
The general band structure of unstrained silicon can be sketched as in
Figure 3.3. The principal conduction band CB minima are located along the ,
, and  directions at a distance of about 85% from the Γ-point to the
X-points (or equivalently, 15% from the X-point to the Γ-point). The energy of
the two lowest CBs are degenerate at the X-points. Close to the CB edge, the
band structure can be approximated by constant ellipsoidal energy surfaces
(c.f. Figure 3.4 left) and a parabolic energy dispersion [140, 141]. The
semi-axes of the ellipse show the direction of the longitudinal ml and the
transversal electron mass mt. The six-fold degeneracy of the valleys arise due to
the symmetry of the lattice along the , , and  directions.
The electrons occupy all of these 3 pairs equally, making the transport
3.2.2 Strained Silicon
The application of stress modifies the various symmetry properties, causes a
change in the band structure, and thus modifies the effective mass which leads to
the mobility enhancement . Th values of ml and mt change under the
influence of shear strain . The shear strain also induces a shift in the energy
levels of the conduction band and the valence band . This energy shift can be
calculated by using the deformation potential theory . The six-fold degenerate
Δ6 valleys in silicon are split into a two-fold degenerate Δ2 valley pair along 
direction, and a four-fold degenerate Δ4 valley pair located along the
other two axes. Strain results in lowering (increasing) energy of the Δ2
(Δ4) valley pair(s) [72, 143, 141]. Under this condition, the electron
population of the Δ2 (Δ4) valley pair(s) is increased (decreased), c.f.