Page 53 - Semiconductor For Micro- and Nanotechnology An Introduction For Engineers

The Crystal Lattice System
The semiconductors that we consider are covalently bonded. We now
give a qualitative description of the tetrahedral covalent bond of the
atoms of an fcc crystal. Solving the Schrödinger equation (see Box 2.2
Box 2.2. The stationary Schrödinger equation and the Hamiltonian of a Solid [2.4].
In principle, all basic calculations of solid state The second term is the Coulombic potential
properties are computed with the Schrödinger energy due to the electron charges. Note that the
equation sum is primed: the sum excludes terms where
Hϕ = Eϕ (B 2.2.1) α = β .
The Hamiltonian operator H defines the dynam- For the atoms the Hamiltonian looks similar to
ϕ
ics and statics of the model, represents a state that of the electron
of the model, and is the scalar-valued energy. H = T + U ii
E
i
i
Thus the Schrödinger equation is an Eigensystem 2
P (B 2.2.4)
α
equations with the energy plying the role of an = ∑ -------- + 1 --- ∑ ′V R –( α R )
β
i
eigenvalue and the state the role of an eigen-func- α 2M 2 αβ
tion. For the electron-atom interaction we associate
only a potential energy
The Hamiltonian is usually built up of contribu-
tions from identifiable subcomponents of the sys- H ei = U ei = ∑ V ( r – R ) (B 2.2.5)
α
β
ei
tem. Thus, for a solid-state material we write that αβ
Equation (B 2.2.1) is hardly ever solved in all its
H = H + H + H + H (B 2.2.2)
i
x
ei
e
generality. The judicious use of approximations
i.e., the sum of the contributions from the elec-
and simplifications have yielded not only tremen-
trons, the atoms, their interaction and interactions
dous insight into the inner workings of solid state
with external influences (e.g., a magnetic field).
materials, but have also been tremendously suc-
Recall that the Hamiltonian is the sum of kinetic
cessful in predicting complex phenomena.
and potential energy terms. Thus, for the electrons
We will return to this topic in the next chapter,
we have
where we will calculate the valence band structure
H = T + U ee of silicon to remarkable accuracy.
e
e
2
p α 1 e 2 (B 2.2.3)
= ∑ ------- + ------------ ∑ ′-------------------
α
α 2m 8πε 0 αβ r – r β
and Chapter 3) for a single atom yields the orthogonal eigenfunctions
that correspond to the energy levels of the atom, also known as the orbit-
als. The spherical harmonic functions shown in Table 2.2 are such eigen-
functions. The tetrahedral bond structure of the Si atom can be made,
3
through a superposition of basis orbitals, to form the hybrid sp orbital
50 Semiconductors for Micro and Nanosystem Technology