293
Modeling Process Variability in Scaled MOSFETs
Note that the fluctuations on one transistor of the pair cannot induce fluctuations on the second one (i.e., σV th1 and σV th2 are independent); thus, ρ = 0.
In addition, σV th1 = σV th2 ≡ σV th (i.e., V th1 – V th2 = V th2 – V th1 ). Therefore, defining σV th1 = σV th2 ≡ σV th from Equation 8.8, we can show that
σ
σ
V
V
th
th
=
∆
2
(8.9)
Equation 8.9 describes σV th of individual transistors of the closely spaced
pair. Equation 8.9 can be experimentally verified by comparing the local V th
variability obtained either in paired transistors (which give a value of σ(ΔV th ))
or in dense transistor arrays (which give a value of (σV th )) [44]. Note that the
A vt factor is defined historically from σ(ΔV th ). Therefore, in order to develop
compact variability model to simulate mismatch between identical devices,
we get σV th from Equations 8.7 and 8.9, as
σV
A
W L
th
vt
eff eff
=
⋅
2
1
(8.10)
Comparing Equations 8.3 and 8.10 we get, C
A
vt
vt
= / 2; Thus, we can estimate the mismatch coefficient A vt for any technology from Equation 8.4
using the technology parameters T ox and N CH . However, A vt is extracted
from the measured data from a set of closely spaced identical paired
transistors.
The same procedure is used to determine the mismatch σP of any parameter P between closely spaced identical devices with mismatch coefficient A p
such that
σP
A
W L
p
eff eff
= 2
1
(8.11)
8.3.2 Systematic Variability
As shown in Figure 8.1, the systematic or global variability is the shift of the
mean value of a parameter. Therefore, global variability is obtained simply
by calculating the standard deviation (σ) of any parameter P causing systematic variability. Thus, the systematic variability of V th is characterized by
calculating σV th of the total V th population, that is, V th data from the target
MOSFET test structures distributed across the wafer. The total V th population could include devices from several wafers of a lot or from several lots
collected over a period of time. The same procedure is used to determine
the systematic variation σP of any parameter causing global device performance variability.
Modeling Process Variability in Scaled MOSFETs
Note that the fluctuations on one transistor of the pair cannot induce fluctuations on the second one (i.e., σV th1 and σV th2 are independent); thus, ρ = 0.
In addition, σV th1 = σV th2 ≡ σV th (i.e., V th1 – V th2 = V th2 – V th1 ). Therefore, defining σV th1 = σV th2 ≡ σV th from Equation 8.8, we can show that
σ
σ
V
V
th
th
=
∆
2
(8.9)
Equation 8.9 describes σV th of individual transistors of the closely spaced
pair. Equation 8.9 can be experimentally verified by comparing the local V th
variability obtained either in paired transistors (which give a value of σ(ΔV th ))
or in dense transistor arrays (which give a value of (σV th )) [44]. Note that the
A vt factor is defined historically from σ(ΔV th ). Therefore, in order to develop
compact variability model to simulate mismatch between identical devices,
we get σV th from Equations 8.7 and 8.9, as
σV
A
W L
th
vt
eff eff
=
⋅
2
1
(8.10)
Comparing Equations 8.3 and 8.10 we get, C
A
vt
vt
= / 2; Thus, we can estimate the mismatch coefficient A vt for any technology from Equation 8.4
using the technology parameters T ox and N CH . However, A vt is extracted
from the measured data from a set of closely spaced identical paired
transistors.
The same procedure is used to determine the mismatch σP of any parameter P between closely spaced identical devices with mismatch coefficient A p
such that
σP
A
W L
p
eff eff
= 2
1
(8.11)
8.3.2 Systematic Variability
As shown in Figure 8.1, the systematic or global variability is the shift of the
mean value of a parameter. Therefore, global variability is obtained simply
by calculating the standard deviation (σ) of any parameter P causing systematic variability. Thus, the systematic variability of V th is characterized by
calculating σV th of the total V th population, that is, V th data from the target
MOSFET test structures distributed across the wafer. The total V th population could include devices from several wafers of a lot or from several lots
collected over a period of time. The same procedure is used to determine
the systematic variation σP of any parameter causing global device performance variability.