equation units
dQ = I(V,T) dt aC = uA ps
V = Q/C mV = aC/fF
dT = f(T-Te) dt uK
First compute the current thru each transistor, adjusted for
temperature. Add the resulting charge to input and output nets. Change
the temperature depending on the amount it differs from power
equilibrium. Then compute the voltage on each net from its charge and
capacitance. Repeat indefinitely.
To speed net identification, the program first traces the largest nets, core power and ground. It starts at the input pad and uses a recursive algorithm to follow the trace through metal and diffusion, branching as required. It marks each tile with a flag:
01 - power
10 - ground
00 - neither
It then scans the poly layer and locates transistors where poly crosses
diffusion. It measures their size by following the poly trace. It then
identifies the nets for source, gate and drain. It can distinguish
source and drain only when source is core power or ground. The result
is 2 tables:
TRAN: +0 Source net index
+2 Gate net
+4 Drain net
+6 Drive (uA)
LOCATION: +0 Lower-left gate tile
To identify a net that is not power or ground, it follows the trace,
recursively when it forks, changing layers as needed. It does 2 things:
1 - Computes capacitance (to substrate) depending upon the layer:
diffusion 3 fF/tile
gate 5
poly .4
metal .3
metal-2 .5
These numbers can be calculated from process parameters, but are best measured by an on-chip test circuit.
2 - Looks for a tile bit indicating the 'owner' of the net. If it finds an owner bit while on a plausible layer, it searches an 4-byte table to identify the net:
ANCHOR: +0 Location of owner
NET: +0 Capacitance (fF)
+4 Coupled charge (aC)
The net is the one whose location matches the current position.
Lacking an owner indicates a new net. An entry is created, with the owner being the location the search was started. The layer is important. For example, a metal trace might cross a poly owner, yet be on a different net. Since the owner bit does not identify the layer, it is determined by these rules:
Not metal-2
Not metal if over poly or diffusion
Transistor gates are owned by poly, source or drain by diffusion. Pads are owned by metal.
A table of coupling capacitance is constructed:
G_D: +0 Gate net
+2 Drain or source net
+4 Capacitance
Transistors with the same gate and drain are combined (inverters) and only significant coupling retained.
Clearly this process takes some time (10 s). A scan of the entire layout looks for transistors. Another scan looks for resistors. These are drawn as a 1-tile wide, vertical n-well.
IV: +0 Voltage (mV)
+2 Previous voltage
+4 Charge (aC)
For each transistor:
TK: +0 Temperature (uK, actually 2*20)
+4 kS (um)
(Ta/T)*1.5
Chip temperature is only 2 K above ambient, which may be specified (298
K). This current times 7.8 ps is the charge added to the drain net and
subtracted from the source net. The current thru a p-transistor is
negative.
Current is multiplied by voltage across the transistor to get power (p). This is summed for all transistors to get total chip power. It is also used to compute an equilibrium temperature proportional to energy density
Te = (p dt)/kS
If temperature is below equilibrium, it is approached exponentially, increasing by
a (Te-T)
If above, it is decreased by
b (T-Te)*3
This approximates diffusion decay of t*.5 (square-root time).
Coefficients a and b must be measured. Temperature rises rapidly (ps)
and falls slowly (ns).
The effect of coupling capacitance is to change the charge and thus the voltage on both nets. The equations for drain to gate are:
Qd = Vd Cd + (Vd-Vg)Cgd
Vd(Cd+Cgd) = Qd + Vg Cgd
So to the drain charge we add the gate voltage times the coupling
capacitance. And conversely. Scanning the coupling table applies all
significant effects. Coupling charge must be stored separately from net
charge since it changes independently. Several gates may contribute to
the same drain.
Finally, a voltage is computed for each net by dividing charge (plus coupling charge) by capacitance. This is extrapolated by using previous voltage, to the time for the next current calculation. In fact, this extrapolation leads to instability in small-capacitance nets and is neglected.
The time step of 7.8 ps resulted from the plotting convenience of 4x32 steps to 1 ns. It can't be larger because of small-capacitance voltage instability. To avoid most of these, series transistors are specially treated. There remain the internal nets of Schmitt inverters and some others.
Ids = f(Vd,Vs,Vg,Vb)
variations of the Ebers-Moll model are usually used by SPICE
simulators. But this model is too elaborate to calculate for 10,000
transistors. We use a 3-parameter empirical model adjusted to fit the
measured Ids curves for a particular process. Even this model is best
computed by table-lookup, inorder to minimize computation time.
The first approximation is to write the current as a product
Ids = D(Vd,Vs)G(Vg)This
separation is not strictly accurate, but close enough. Body voltage is
+0 for n-transistors and +5 for p-. It has no other function.
The IV curves
look like this for a 1-tile n-transistor. The green curves are D vs Vd
for Vg = 1, 2, 3, 4 and 5 V. The red curve is G vs Vg. Although the
green curves should run to 5 V, we can't measure them. All available
information is provided and the leaf-like symmetry is attractive.
The black dots are measured values. The green curves have been corrected for temperature and represent cold a transistor. The difficulty in relating dots to curves emphasizes the huge difference temperature makes.
The
IV curves for a 2-tile p-transistor. It suggests that 3-tile
transistors would better match the n-transistor curve. The dots show
the p-transistors less affected by temperature, because of their lower
current density.
G is computed by a clumsy exponential converging to linear. It is best considered a table look-up of the red curve. There are 300 values between 0-6 V at 20 mV increments.
D is computed as the parallel combination of 2 linear functions. Again, it is a table look-up with the same characteristics as G. Vs is +0 (or +5).
When Vs is not the same as Vb, SPICE includes a "body effect" correction. A more accurate treatment of this effect is to observe that
Ids = Id0 - Is0
Thus the form used becomes
Ids = (D(Vd)-D(Vs))G(Vg)
This product form can be effectively used to compute the current thru series transistors.
Polysilicon 80 Ohms/tile
Polysilicide 8
n-diffusion 35
p-diffusion 70
metal .02
n-well 700
transistor 5000
Non-metal traces must be kept short enough that this assumption is valid.
There are several interesting features in this display. Note the output is delayed from the input, perhaps 200 ps. The rise/fall time of the output is faster than the input. The output load is small enough that the inverter has some gain (in this sense).
The output vs input plot shows considerable hysteresis. This is typical of these relatively small gates. It's due to the fast transition and the time required to charge the output capacitance. The curve is not symmetrical because of the difference between n- and p-transistors. Textbooks don't include these effects.
The slight rise just before output falls is due to gate-drain capacitance. It is so small as to be negligible, as is its effect on rise/fall time. SPICE models claim a larger effect.