5
Introduction to Compact Models
assumptions and derived I ds equations for circuit analysis [26]. In these models,
the device is considered to be turned on above a certain applied input voltage, referred to as the threshold voltage (V th ), and turned off at the input bias
below V th . This approach is known as threshold voltage–based or, V th -based compact modeling.
With the great potential of MOSFET devices in ICs during 1960s, a detailed
understanding of MOSFET device physics became critical. In 1966, Pao and
Sah [22] reported an I ds equation to describe MOSFET device characteristics
under varying biasing conditions in terms of a physical parameter called the
surface potential (f s ), where f s describes the mode of operation of MOSFET
devices under the applied biasing conditions. It is to be noted that V th is
defined at a particular value of f s above which the device starts conducting
whereas f s defines the entire range of operation of MOSFETs from off-state to
on-state, depending on the applied biasing conditions. The value of f s is calculated, iteratively, from an implicit expression derived from Poisson’s equation and Gauss’s law. This I ds model is a double integral equation, commonly
known as the Pao-Sah model, that can only be solved numerically. Inherently,
it takes into account both the drift and diffusion components of I ds , and is
valid in all regions of device operation: from the subthreshold (below V th )
to strong inversion region (above V th ). This method is now known as surface potential–based or, f s -based compact modeling. Sah’s f s -based modeling
requires iterations and integration and is computationally demanding for
circuit CAD. Thus, the Pao-Sah model is inefficient for circuit CAD due to
its complexities involving integration and iterations to get I ds at each value
of applied voltage. Thus, the search for simplified models for circuit CAD
began in the late 1960s.
In the late 1970s, SPICE emerged as an essential circuit CAD tool to perform
accurate and efficient design and analysis of ICs under the EDA environment [27]. In order to use SPICE, accurate and efficient compact models are
required to describe the behavior of the devices used in the circuits. Thus,
the explicit development of MOSFET compact models for circuit CAD started
with the widespread usage of SPICE and continues today as the mainstream
MOSFET devices rapidly approach their fundamental scaling limit near the
10-nm regime [1,28–33].
The first approach used in developing I ds model is to circumvent the iterative computation of f s from the implicit relation [22] using V th as the boundary between the off-state or weakly conducting state, referred to as the weak
inversion region, and on-state, called the strong inversion region, of MOSFET
devices, that is, use V th -based compact modeling. This approach results in two
current equations, one for the weak inversion and the other for strong inversion [25,34]. In V th -based modeling, a linear approximation is made between
f s and the applied input voltage to eliminate f s and relate the input voltage
to the output current I ds . This approach results in a simple I–V equation in
the parabolic form and was first used for circuit simulation in 1968 [34]. This
is the first known compact MOSFET model for circuit CAD and is referred
Introduction to Compact Models
assumptions and derived I ds equations for circuit analysis [26]. In these models,
the device is considered to be turned on above a certain applied input voltage, referred to as the threshold voltage (V th ), and turned off at the input bias
below V th . This approach is known as threshold voltage–based or, V th -based compact modeling.
With the great potential of MOSFET devices in ICs during 1960s, a detailed
understanding of MOSFET device physics became critical. In 1966, Pao and
Sah [22] reported an I ds equation to describe MOSFET device characteristics
under varying biasing conditions in terms of a physical parameter called the
surface potential (f s ), where f s describes the mode of operation of MOSFET
devices under the applied biasing conditions. It is to be noted that V th is
defined at a particular value of f s above which the device starts conducting
whereas f s defines the entire range of operation of MOSFETs from off-state to
on-state, depending on the applied biasing conditions. The value of f s is calculated, iteratively, from an implicit expression derived from Poisson’s equation and Gauss’s law. This I ds model is a double integral equation, commonly
known as the Pao-Sah model, that can only be solved numerically. Inherently,
it takes into account both the drift and diffusion components of I ds , and is
valid in all regions of device operation: from the subthreshold (below V th )
to strong inversion region (above V th ). This method is now known as surface potential–based or, f s -based compact modeling. Sah’s f s -based modeling
requires iterations and integration and is computationally demanding for
circuit CAD. Thus, the Pao-Sah model is inefficient for circuit CAD due to
its complexities involving integration and iterations to get I ds at each value
of applied voltage. Thus, the search for simplified models for circuit CAD
began in the late 1960s.
In the late 1970s, SPICE emerged as an essential circuit CAD tool to perform
accurate and efficient design and analysis of ICs under the EDA environment [27]. In order to use SPICE, accurate and efficient compact models are
required to describe the behavior of the devices used in the circuits. Thus,
the explicit development of MOSFET compact models for circuit CAD started
with the widespread usage of SPICE and continues today as the mainstream
MOSFET devices rapidly approach their fundamental scaling limit near the
10-nm regime [1,28–33].
The first approach used in developing I ds model is to circumvent the iterative computation of f s from the implicit relation [22] using V th as the boundary between the off-state or weakly conducting state, referred to as the weak
inversion region, and on-state, called the strong inversion region, of MOSFET
devices, that is, use V th -based compact modeling. This approach results in two
current equations, one for the weak inversion and the other for strong inversion [25,34]. In V th -based modeling, a linear approximation is made between
f s and the applied input voltage to eliminate f s and relate the input voltage
to the output current I ds . This approach results in a simple I–V equation in
the parabolic form and was first used for circuit simulation in 1968 [34]. This
is the first known compact MOSFET model for circuit CAD and is referred