20
Compact Models for Integrated Circuit Design
The silicon wafers used in the IC fabrication processes are cut parallel to
either the <111> or <100> crystal planes. However, the <100> material is most
commonly used due to the fact that, during IC fabrication processes, <100>
wafers produce the lowest amount of charges at the silicon/silicon-dioxide
(Si/SiO 2 ) interface and offer higher carrier mobility [14,15].
2.2.1 Energy Band Model
In a silicon crystal, each atom has four valence electrons and four nearest
neighboring atoms. Each atom shares its valence electrons with its four
neighbors in a paired configuration called covalent bond. It is predicted by
quantum mechanics (QM) that the allowed energy levels of electrons in a
solid is grouped into two bands, called the valence band (VB) and the conduction band (CB). These bands are separated by an energy range that the
electrons in a solid cannot possess and is referred to as the forbidden band
or forbidden gap. The VB is the highest energy band and its energy levels
are mostly filled with electrons forming the covalent bonds. The CB is the
next higher energy band with its energy levels nearly empty. The electrons
that occupy the energy levels in the CB are called free electrons or conduction
electrons.
Typically, the energy is a complex function of momentum in a threedimensional space and there are many allowed energy levels for a large
number of electrons in silicon, and therefore, the energy band diagram is
also complex. For the simplicity of representation, only the edge levels of
each of the allowed energy bands are shown in the energy band diagram in
Figure 2.1. In Figure 2.1, E c and E v are the bottom edge of the CB and the top
edge of the VB, respectively, and E g is the bandgap energy separating E c and
E v . And, at any ambient temperature T(K), E g is given by
E E E
g
c
v
= −
(2.1)
When a valence electron is given sufficient energy (≥E g ), it can break out of the
chemical bonding state and excite into the CB to become a free electron leaving behind a vacancy, or hole in the VB. A hole is associated with a positive
charge since a net positive charge is associated with the atom from which the
electron broke away. Note that both the electron and hole are generated simultaneously from a single event. The electrons move freely in the CB and holes
move freely in the VB. In silicon, the bandgap is small (~1.12 eV); therefore,
even at room temperature a small fraction of the valence electrons are excited
into the CB, generating electrons and holes. This allows limited conduction
to take place from the motion of the electrons in the CB and holes in the VB.
As shown in Figure 2.1, when an electron in the CB gains energy, it moves up
to an energy E > E c , while a hole in the VB gains energy, it moves down to an
energy E < E v . Thus, the energy of the electrons in the CB increases upward
while the energy of the holes in the VB increases downward.
Compact Models for Integrated Circuit Design
The silicon wafers used in the IC fabrication processes are cut parallel to
either the <111> or <100> crystal planes. However, the <100> material is most
commonly used due to the fact that, during IC fabrication processes, <100>
wafers produce the lowest amount of charges at the silicon/silicon-dioxide
(Si/SiO 2 ) interface and offer higher carrier mobility [14,15].
2.2.1 Energy Band Model
In a silicon crystal, each atom has four valence electrons and four nearest
neighboring atoms. Each atom shares its valence electrons with its four
neighbors in a paired configuration called covalent bond. It is predicted by
quantum mechanics (QM) that the allowed energy levels of electrons in a
solid is grouped into two bands, called the valence band (VB) and the conduction band (CB). These bands are separated by an energy range that the
electrons in a solid cannot possess and is referred to as the forbidden band
or forbidden gap. The VB is the highest energy band and its energy levels
are mostly filled with electrons forming the covalent bonds. The CB is the
next higher energy band with its energy levels nearly empty. The electrons
that occupy the energy levels in the CB are called free electrons or conduction
electrons.
Typically, the energy is a complex function of momentum in a threedimensional space and there are many allowed energy levels for a large
number of electrons in silicon, and therefore, the energy band diagram is
also complex. For the simplicity of representation, only the edge levels of
each of the allowed energy bands are shown in the energy band diagram in
Figure 2.1. In Figure 2.1, E c and E v are the bottom edge of the CB and the top
edge of the VB, respectively, and E g is the bandgap energy separating E c and
E v . And, at any ambient temperature T(K), E g is given by
E E E
g
c
v
= −
(2.1)
When a valence electron is given sufficient energy (≥E g ), it can break out of the
chemical bonding state and excite into the CB to become a free electron leaving behind a vacancy, or hole in the VB. A hole is associated with a positive
charge since a net positive charge is associated with the atom from which the
electron broke away. Note that both the electron and hole are generated simultaneously from a single event. The electrons move freely in the CB and holes
move freely in the VB. In silicon, the bandgap is small (~1.12 eV); therefore,
even at room temperature a small fraction of the valence electrons are excited
into the CB, generating electrons and holes. This allows limited conduction
to take place from the motion of the electrons in the CB and holes in the VB.
As shown in Figure 2.1, when an electron in the CB gains energy, it moves up
to an energy E > E c , while a hole in the VB gains energy, it moves down to an
energy E < E v . Thus, the energy of the electrons in the CB increases upward
while the energy of the holes in the VB increases downward.