As mentioned earlier, an enhancement mode MOSFET can be modeled as a simple switch, through which current can flow in either direction. A slightly more complex model could be to consider the device to act as a resistor on its output, and a capacitor at its input. More sophisticated models can be readily derived, but the two mentioned above are useful for logic and approximate timing simulations of the behavior of a MOS integrated circuit.
A number of circuit simulation programs have been written which allow the simulation of MOS integrated circuits with a wide range of transistor models. Some of the more sophisticated models have many parameters which can be varied, and can produce quite accurate simulation results. Most of these circuit simulators have been derived from the SPICE circuit simulation program, developed at UCB. SPICE has 3 sophisticated models for MOS transistors, and is generally considered to be quite an accurate circuit simulator.
We can derive the simplest (and least accurate) of the SPICE models from simple physics; this exercise is useful because it provides some ``rules of thumb'' for the design of VLSI devices. In particular, it allows us to see how devices ``scale''; that is, how important properties such as the switching speed of a transistor vary with the linear dimensions of the transistor.
One of the most important parameters for a transistor is its ``switching time''. We might expect this to be related to the transit time, , for a charge carrier (an electron, for an N channel MOSFET or a hole, for a P channel MOSFET) to cross the channel region from the source to the drain region. If we consider a transistor as shown in Figure , the transit time, , is simply where v is the average speed of the charge carrier.
Under simple conditions, we can calculate the average speed, v, for a charge carrier. If no nuclei were present in the channel region, and there was no voltage applied between the source and the drain ( i.e., no applied electric field) across the channel, then the average velocity would be zero. If there was an electric field in the channel region, the charge carriers would accelerate with an acceleration , where e is the charge of the charge carrier, E is the magnitude of the applied electric field, and m is the mass of the charge carrier. (The quantity e is equal in magnitude to the charge of an electron). For free charge carriers, the average speed would be , where t is the time for the charge carrier to travel across the channel region. In reality, however, the charge carriers will collide with the nuclei in the channel region quite frequently. This collision is inelastic, the charge carrier gives up some of its energy to the nucleus. We will assume that the collisions are totally inelastic, and that each collision brings the charge carrier to a stop. The electron, therefore, accelerates only in the interval between collisions with nuclei. There will be an ``average time between collisions'', , and the average speed of the charge carrier will be given by
The parameter is called the ``mobility''; there is an ``electron mobility'', , and a corresponding ``hole mobility``, . For silicon, and, . (Actually, there are two types of mobility; a ``bulk'' mobility and a ``surface'' mobility. For MOSFET's, the surface mobility is important, and corresponds to the values quoted. For bipolar transistors, the bulk mobility is important.)
So, the characteristic time, , for the transistor is
Although this is not a rigorous derivation, there are two important things to note:
These are ``scaling rules'' for MOS devices, and are believed to be approximately valid for device sizes (channel dimensions) down to about 0.5 microns.
We can also calculate, approximately, some of the important electrical properties of the MOS transistor; e.g., the current which can flow through the channel region, and the impedance of the channel. These will be functions of both the size of the channel region and of the voltages applied to the gate (), drain () and source () of the device.
Referring to Figure , the charge, Q, which will be available at the oxide/channel boundary is simply where C is the capacitance of the capacitor formed between the gate and the substrate, is the gate-to-source voltage, and is the threshold voltage. (The quantity is the ``effective'' voltage applied to the gate). For a parallel plate capacitor, where A = LW is the area of the gate over the channel of width W and length L, D is the distance between the gate and the substrate ( i.e., the oxide thickness) and is the permittivity of the material (for .) So,
The current flowing in the channel is
(Note that this analysis neglects some important effects; for example, it implicitly assumes that there is a uniform electric field in the entire channel region.) A more sophisticated analysis gives slightly different values for the current in the channel region:
in the linear (ohmic) region, and
in the saturation region. These expressions are often written in terms of the ``process transconductance parameter'', , where where is the capacitance per unit area of the oxide. The gate capacitance . The quantity is a SPICE parameter for MOS transistors.
We can now calculate the effective impedance of the channel region as:
if the transistor is operated in the linear region, and
in the saturation region.
The fact that the impedance, Z, for a MOS transistor is proportional to the ratio in both their linear and saturated regions means that we can use these transistors as resistors; the relative resistances scale with the device size. This is true for both P- and N- channel devices, as well as for both enhancement and depletion mode transistors. In fact, a ``permanently turned on'' transistor is often used as a resistor, for example as a ``pull-up resistor'' for a logic gate, to provide a current limit for the gate output.
We can also calculate the time constant for a transistor to charge the gate of another identical transistor. This time is very important, since it will be related to the ``gate delay'' for a logic device.
assuming that the transistor providing the charge is operating in its saturation region. Although this result is not rigorously correct, it is worth noting that the delay time for a transistor to switch on a second transistor has a simple relationship to the transit time .
Figure shows a plot of the current in the channel region of a MOS transistor , the current between the source and drain) against the potential difference across the channel for various values of the applied gate current, for a typical MOS transistor.
Note that three different regions are distinguished; the ``cutoff'' region, in which the transistor effectively passes no current; the ``linear'' region, in which the behavior is ohmic, and the ``saturation'' region, in which the current is nearly constant.
Effectively, then, we have modelled the current in the channel region of a MOS transistor (either enhancement mode or depletion mode, P- or N- channel) as
The quantity is often written as , and is called the ``gain factor'' for a MOS transistor. Note that is proportional to .
Actually, in the saturation region, the drain current, is not completely independent of , partly because ``depletes'' charge carriers from the vicinity of the well as shown in Figure . This effect both shortens the effective channel length and adds more charge carriers to the channel region.
These effects, which increase the drain current are usually modelled empirically by the parameter , the ``channel length modulation factor'', giving the following expression for the saturation current
is typically quite small, . It is a Level 1 SPICE parameter. The term is often included with the expression for in the linear region as well, to ensure that is continuous from the linear to the saturation region.
One remaining parameter for MOS devices which can readily be derived using simple physical reasoning is , the threshold voltage.
We start by looking again at a schematic of the cross-section of an NMOS enhancement mode transistor, as in Figure (similar arguments apply for a PMOS transistor.)
Note that if no charge is applied to the gate, the source and drain regions are separated by reverse biased PN junctions, and no current can flow between the source and drain of the transistor. (The ``off'' impedance is normally of the order of thousands of megohms.)
When a positive charge is applied to the gate of the transistor, however, mobile charge carriers are attracted from the bulk of the P-doped substrate to the surface of the channel region immediately below the insulating oxide under the gate. These mobile charge carriers leave behind fixed (immobile) charges in the ``depletion'' region in the channel, as shown in Figure . The thickness, of the depletion region below the gate can be calculated as a function of , the electrostatic potential just inside the depletion layer at the oxide-semiconductor interface. In P-type semiconductors, this depletion layer is made by ``pushing back'' mobile holes.
The number of holes, dQ, originally contained in a thin horizontal layer of thickness is
The change in surface potential is
where is the constant of integration. Solving for ,
where is the permittivity of silicon, q is the charge of a single charge carrier (|q| = e, the charge of an electron), and is the density of P-type ions in the substrate.
The quantity of charge per unit area in the channel region due to the immobile ions which have been stripped of their charge carriers (in this case, ions which have been stripped of their mobile holes) is
The threshold is defined as , i.e., the induced surface potential is equal in magnitude to the original, unbiased surface potential; the density of mobile negative charge carriers (electrons) at the surface is equal to the density of holes in the original, unbiased substrate. In effect, the channel region has been ``induced'' to become as strongly N-doped as it was originally P-doped.
In order to calculate the gate voltage required to attain threshold, we need to know the concentration of charge carriers N in the unbiased substrate. This is normally equal to the dopant concentration of the substrate.
We can define the equilibrium (electrostatic) potential inside a semiconductor as
where or are the equilibrium concentrations of P or N type mobile charge carriers ( i.e. the dopant concentration) in the substrate, and is the equilibrium mobile charge concentration for pure (or ``intrinsic'') silicon, at room temperature. Note that the potential is a surface potential, possibly arising from the diffusion of charge carriers across a surface. This potential is, of course, present independent of the gate voltage . Moreover, it must be ``overcome'' by to allow conduction in the channel.
If the substrate is not biased by any ``substrate body bias'' then the immobile charge in the depletion region at threshold is
If the substrate is biased by a voltage between the source and body, the surface potential required to produce inversion is , and
We can now calculate the gate voltage required to produce inversion, the threshold voltage . The threshold voltage consists of several components:
Therefore the threshold voltage is given by
Since the last term in the previous expression is difficult to evaluate it is usual to express the threshold voltage as a function of the substrate bias voltage, and measure the unbiased threshold voltage:
where . The parameter is the zero bias threshold voltage and is called the ``body effect coefficient factor''.
We have now derived the most important relationships used in the simplest of the SPICE models (Level 1) of the MOS transistor. The following table lists estimates of these parameters for the CMOS3DLM and CMOS4S processes available to us:
Circuit simulators include the capacitance between the various elements of the transistor as components of the models. SPICE calculates the nonlinear capacitance between the gate and channel region. SPICE uses a set of parasitic capacitances modeled as constant capacitors as shown in Figure , in which the parasitic capacitors are named as in the SPICE parameter list. Also shown are the capacitances between the source and body, and the drain and body. These are each considered as two separate nonlinear capacitors, a bottom capacitance and a sidewall capacitance, calculated from the perimeters of the source and drain areas.