The Bravais lattice vectors for Si are given by
(3.32) |
(3.33) |
(3.34) |
The first Brillouin zone has the shape of a truncated octahedron. It can be
visualized as a set of eight hexagonal planes halfway between the centre of the
cell and the lattice points at the corner, and six square planes halfway to the
lattice points in the center of the next cell. The Brillouin zone is shown in
Fig. 3.4b with the points and directions of high-symmetry marked
using Greek letters and Roman letters for points on the
surface. Table 3.3 summarizes these symmetry points and
directions. These points and directions are of importance for interpreting the
band structure plots.
Symmetric Points/ | ||
Directions | Coordinate | Remark |
() | Origin of k space | |
() | Middle of square faces | |
Middle of hexagonal faces | ||
Middle of edge shared by two hexagons | ||
Middle of edge shared by a hexagons and a square | ||
Middle of edge shared by two hexagons and a square | ||
Directed from to | ||
Directed from to | ||
Directed from to |
Several symmetry operations can be performed on the diamond structure which leave the structure unchanged. Apart from translation, such operations can be categorized as rotation or inversion, or a combination, and are collectively known as symmetry operations. Due to the fcc lattice of the diamond structure there exist a total of 48 such independent symmetry operations. This symmetry of the lattice in real space is also reflected in the first BZ. The lowering of symmetry of the fcc structure due to strain can distort it to some of the 14 Bravais lattice forms, such as the tetragonal, orthorhombic or trigonal lattice. The new lattice structure formed has different symmetry properties which can have pronounced consequences on the band structure. Numerical methods for calculating the band structure allow the prediction of several important electronic and optical properties.
The numerical band structure of unstrained Si calculated using the empirical pseudopotential method [Chelikowsky76] is shown in Fig. 3.5. At , the highest energy band that is completely filled is denoted as the valence band. The next higher band is completely empty at and called the conduction band. The two bands are separated by a forbidden energy gap, known as the bandgap .
The principal conduction band minima of Si are located along the , and directions at a distance of about 85% from the -point to the X-points. The minima of the second conduction band touch the first conduction band at the X-points. This degeneracy of the two conduction bands can produce interesting effects on applying shear strain as discussed in Section 3.3.4. Close to the principal minima the band structure can be represented by ellipsoidal constant energy surfaces by assuming a parabolic energy dispersion relation, as shown in Fig. 3.6a. For valleys located along the [100] direction, the energy dispersion reads
Unlike the conduction band, the valence band structure looks highly anisotropic even in the unstrained case. The valence band maximum in Si is located at the point. Due to the degeneracy of the bands at the point, the energy surfaces develop into quartic surfaces of the form [Dresselhaus55]
(3.36) | |
(3.37) |
In the method, the single-electron Schroedinger equation is expanded using Bloch's theorem, to yield a term and an additional term parabolic in to the original equation.
A very powerful tool for obtaining the complete band structure is the empirical pseudopotential method. In this method, the actual hard core ionic potential, , experienced by an electron is replaced by a soft pseudopotential, . This also results in a modification of the true wave function, , to a pseudo-function, , as shown in Fig. 3.7. The pseudopotential is found to depend on pseudopotential form factors which can be obtained by matching experimental results and are available for a large set of material systems [Yu03]. Knowing this potential, the Schroedinger equation gets modified to a so called pseudo wave equation, the solution of which delivers the energy dispersion relation.