TIME-DEPENDENT PERFORMANCE AND MOORE’S LAW
If the voltages applied to the MOSFET are not stationary (DC), but time dependent (AC), so is the current in the MOSFET. In principle, one would have to solve the time dependent version of the equations we have used so far, especially the time dependent continuity equation, but fortunately other less computationally expensive methods exist which belong more to the regime of Electronic Engineering than to the theory of charge transport. A short introduction, however, will be given here. This introduction will serve only to extract Moore’s law that we have qualitatively described in the beginning of this chapter.
Consider the simplified circuit diagram shown in figure 7.22. At the gate a DC and AC voltage generator are applied. This will create DC and AC components in all major quantities associated with the MOSFET. We adopt the following convention.
DC part of quantities: capital letter, capital subscript AC part of quantities: small letter, small subscript Total quantities: capital letter, small subscript
We then write for DC and AC currents and voltages
If the magnitude of the AC components is small (which normally is the case), a linear Taylor expansion of the above time dependent drain current can be made. Therefore
FIGURE 7.22 A simplified circuit diagram for the derivation of the small-signal equivalent circuits.
If the variation in vgs(t) and vds(t) is slow, that is if the frequencies involved are low, then the derivatives in 7.60 can be equated to their DC values, that is
It is important to note that equations 7.61 hold only for low frequencies. When DC measurements of an I-V characteristic are made, it is always assumed that the system has enough time to relax between one measurement and the next so the DC derivatives of ID can not in principle be used when the system is continuously varying. In essence “low” frequencies means that the system (MOSFET) can follow the input signal with no time delay. At higher frequencies capacitive effects do delay the MOSFET and these effects have to be taken into account.
The methodology of equation 7.60 and 7.61 proceeds therefore in two stages: first the approximation of equation 7.61 is used and a simple circuit, mathematically equivalent to equations 7.60 and 7.61, is derived for low frequencies; then additional capacitors are added to it to account for the actual delays in the MOSFET. All the required information is contained in the AC parts of the quantities of interest, so we only need a circuit that relates these components only. This is called “small signal equivalent circuit” and is just a topological representation of the relations between the AC components of 7.59. The small signal, low frequency circuit equivalent to equations 7.60 and 7.61 is shown in figure 7.23a. Note that the derivative
is represented as a current generator whereas the derivate
FIGURE 7.23 Small-signal equivalent circuits for (a) low frequencies and (b) high frequencies.
as a resistor. It can easily be seen that the gate is open-circuited because there is no gate current. However, as the input frequencies are increased, AC electric currents can cross dielectrics and so for higher frequencies a capacitor labelled Cgs is added between the source and the gate electrodes. So, the small signal equivalent circuit at high frequencies is given in figure 7.23b. Additional capacitances are needed to have a proper representation of the MOSFET time dependent behaviour, but we will stop at this stage because figure 7.23b is adequate in order to extract Moores Law. But first, we have to find expressions for the elements appearing in figure 7.23b.
From equation 7.19 we have
We also have from 7.19
Note that the above parameters gm and r0 depend on the DC values of the current and gate voltage.
Next, for the high frequency small signal circuit we have to calculate Cgs, i.e. the capacitance between the gate and the source. In the linear region this is easy because we can imagine that the charge of the channel is shared equally between the following two pairs of electrodes: a) gate with source and b) gate with drain. Hence, we can write
In the saturation region, however, where most of the interest lies, the drain has very little contact with the channel and a more detailed calculation is needed. The total charge Q,ol below the gate can be written, making use of equation 7.7, as
where, we remind the reader, V(x) is the part of Vos up to the point x, measured from the source. Using also equation 7.12 and transforming variables from у to У we get
Performing the integration, we get
so that the required capacitance is
We note that in both equations 7.65 and 7.66 the upper limit of integration is the point of pinch-off. Substituting the above values of gm, r0 and Cgs in the small signal equivalent circuit of figure 7.23 gives a very approximate picture of the performance of a MOSFET. More capacitances are needed for an accurate description. However, for the purposes of obtaining Moore’s Law it is adequate.
From figure 7.23b we have
Furthermore, omitting r„ which is small
A MOSFET is useful for digital circuits if given an input signal (i.e. i,„) of digital “1” we get a response (i.e. iout) of also “1”, that is expression 7.71 should not become less than one. Given that 7.71 is a decreasing function of со, this condition sets the maximum allowed frequency for the MOSFET usually denoted by /;■. Substituting the expressions for gm and Cgs we have derived, we get
This led to the famous Moore’s Law. Gordon Moore predicted, based on 7.72, that the number of MOSFETs in a chip would double every one or two years and the frequency of operation would increase accordingly. The law proved correct for several decades, but the corresponding increase in frequency came to a stagnation because of heat dissipations: as the frequency is increased, heat is dissipated faster and there is a danger of chip malfunction or burning. Furthermore, as the length of the channel is decreased to nanometric lengths fT scales as 1/L and not as 1/L2. The primary reason for that is that the drift velocity saturates and is no longer proportional to the electric field, see more in the next chapter.
As the length of the channel was decreased the oxide thickness was also decreased to preserve the gate control on the channel. At one point in time the oxide thickness was between lnm and 2nm. Appreciable tunnelling currents were developing that ruined the performance of the MOSFET. The obvious solution was the use of oxides of higher dielectric constant. Hence the traditional Si02 was replaced by НЮ2 which has a relative dielectric constant of approximately 25.