52
Compact Models for Integrated Circuit Design
profile along the cutline of the active device is shown in Figure 2.14b and c.
The metallurgical junction depth X j is indicated as the point where the net
impurity concentrations of donors and acceptors are equal. For compact
modeling, the actual impurity profile is approximated by a step or abrupt
(high–low) shallow junctions, Figure 2.14b or a linearly graded (deep) junctions, Figure 2.14c, so that a tractable circuit model can be developed. A step
doping profile is characterized by constant p-type dopant concentration N a
that changes with position in a stepwise fashion to a constant n-type dopant
concentration N d .
From the 1D impurity profiles in Figure 2.14b and c, we find that there is a
large carrier concentration gradient at the junction resulting in carrier diffusion. Holes from the p-side diffuse into the n-side, leaving behind negatively
charged acceptor ions N a
−
( ) and electrons from the n-side diffuse into the
p-side leaving behind positively charged donor ions N d
+
( ) . Consequently, a
space charge region is formed (negative charge on the p-side and positive
charge on the n-side), creating thereby an electric field E, and, hence, a potential difference as shown in Figure 2.15. The direction of the field (n-region to
p-region) is such that it opposes further diffusion of carriers so that, in thermal equilibrium, the net flow of carriers is zero; that is, an electric field is set
up, which tends to pull electrons and holes back to the original positions. The
internal potential difference between the two sides of the junction is called
the built-in potential or barrier height, f bi . The space charge region on two sides
of the metallurgical junction is often called the depletion region, because the
region is depleted of the free carriers.
Figure 2.16a shows the energy-band diagram of a p-type silicon and
n-type silicon physically separated from each other. As discussed in Section
2.2.4, the Fermi level for an n-type silicon lies close to its CB, and for a p-type
silicon lies close to its VB. Also, as we will show later, the Fermi level of a
semiconductor is flat, that is, spatially constant, when there is no current
flow in it. Therefore, as the p-type region and the n-type region are brought
Depletion
region
E
n
p
Holes
Electrons
FIGURE 2.15
Formation of built-in electric field due to the space charges left behind by mobile carriers after
diffusion from the high- to the low-concentration region on either side of the junction.
Compact Models for Integrated Circuit Design
profile along the cutline of the active device is shown in Figure 2.14b and c.
The metallurgical junction depth X j is indicated as the point where the net
impurity concentrations of donors and acceptors are equal. For compact
modeling, the actual impurity profile is approximated by a step or abrupt
(high–low) shallow junctions, Figure 2.14b or a linearly graded (deep) junctions, Figure 2.14c, so that a tractable circuit model can be developed. A step
doping profile is characterized by constant p-type dopant concentration N a
that changes with position in a stepwise fashion to a constant n-type dopant
concentration N d .
From the 1D impurity profiles in Figure 2.14b and c, we find that there is a
large carrier concentration gradient at the junction resulting in carrier diffusion. Holes from the p-side diffuse into the n-side, leaving behind negatively
charged acceptor ions N a
−
( ) and electrons from the n-side diffuse into the
p-side leaving behind positively charged donor ions N d
+
( ) . Consequently, a
space charge region is formed (negative charge on the p-side and positive
charge on the n-side), creating thereby an electric field E, and, hence, a potential difference as shown in Figure 2.15. The direction of the field (n-region to
p-region) is such that it opposes further diffusion of carriers so that, in thermal equilibrium, the net flow of carriers is zero; that is, an electric field is set
up, which tends to pull electrons and holes back to the original positions. The
internal potential difference between the two sides of the junction is called
the built-in potential or barrier height, f bi . The space charge region on two sides
of the metallurgical junction is often called the depletion region, because the
region is depleted of the free carriers.
Figure 2.16a shows the energy-band diagram of a p-type silicon and
n-type silicon physically separated from each other. As discussed in Section
2.2.4, the Fermi level for an n-type silicon lies close to its CB, and for a p-type
silicon lies close to its VB. Also, as we will show later, the Fermi level of a
semiconductor is flat, that is, spatially constant, when there is no current
flow in it. Therefore, as the p-type region and the n-type region are brought
Depletion
region
E
n
p
Holes
Electrons
FIGURE 2.15
Formation of built-in electric field due to the space charges left behind by mobile carriers after
diffusion from the high- to the low-concentration region on either side of the junction.