SEMICONDUCTOR DEVICE PHYSICS AND DESIGN

2.3. ELECTRONS IN CRYSTALLINE SOLIDS 37

The wavefunction is spread in the entire sample and has equal probability (ψ∗ψ) at every point
in space. In the periodic crystalelectronprobabilityisthesameinallunitcellsofthecrystal
becauseeachcellisidentical. This is shown schematically in figure 2.4.

PERIODIC POTENTIAL

U(r)

Wavefuntions |ψ|^2 have the same periodicity as the potential

ψ(r) =u(r)eik•r

r

r

|ψ(r)|^2

Figure 2.4: A periodic potential,|ψ|^2 has the same spatial periodicity as the potential.

Bloch’s theorem states the eigenfunctions of the Schrodinger equation for a periodic potential ̈
are the product of a plane waveeik·rand a functionuk(r), which has thesameperiodicityasthe
periodicpotential. Thus
ψk(r)=eik·ruk(r) (2.3.10)

is the form of the electronic function. The periodic partuk(r)has the same periodicity as the
crystal, i.e.
uk(r)=uk(r+R) (2.3.11)
The wavefunction has the property

ψk(r+R)=eik·(r+R)uk(r+R)=eik·ruk(r)eik·R

= eik·Rψk(r) (2.3.12)

To obtain the allowed energies, i.e. the band structure, computer techniques are used to solve the
Schrodinger equation. One obtains a series of allowed energy bands separated by bandgaps as ̈
shown schematically in figure 2.5. Each band has anEvs.krelation Examples of such relations
called bandstructure will be shown later in section 2.6. The product ofand thek-vector behaves
like an effective momentum for the electron inside the crystal.
The smallestk-values lie in ak-space called the Brillouin zone (see figure 2.6). If thek-value
is chosen beyond the Brillouin zone values, the energy values are simply repeated. The concept