Moore's Law and scaling trends: Foundations — Hardware deep dive

Before you can read the parent note, you must be able to read every squiggle in it. Below is a strict, build-from-zero tour of each symbol and idea the parent uses. Nothing later depends on something not yet defined. If you know all of these, jump back to Moore's Law and scaling trends.

1. The transistor — the "brick" everything counts

The picture: think of a water tap. The gate is the handle; turning it lets water (current) through or blocks it. That's all a transistor is at this level.

Why the topic needs it: Moore's Law counts these. Everything else — area, power, delay — is a property of one of these switches.

Figure 1 — A transistor drawn as a switch: source and drain (mint) joined by a channel (butter), with the gate (lavender) on top acting like a tap handle that opens or blocks the current flow (coral arrows).

2. The letters used for size: $L$ , $W$ , $t_{o x}$ , and $A$

The picture: look at the transistor as a little rectangle on the chip. $L$ and $W$ are its two side lengths; multiply them and you get the floor space $A$ it eats up.

Why we need it: "make transistors smaller" literally means shrink $L$ , $W$ , $t_{o x}$ . If area $A$ halves, twice as many fit in the same silicon — that is the mechanical reason Moore's Law can happen.

Figure 2 — Left: top view of the transistor rectangle showing length $L$ (coral) and width $W$ (lavender), with area $A = L \times W$ . Right: side view showing the thin oxide thickness $t_{o x}$ between the gate (lavender) and the silicon channel (mint).

3. Powers and the " $2^{n}$ " notation — the heart of doubling

The picture: a tree that splits in two at every step. After 1 split there are 2 branches, after 2 splits 4, after 5 splits 32. The exponent is "how many split-levels down you are".

Why this tool and not plain multiplication? Because the growth is repeated multiplication, not repeated addition. If you only added a fixed number of transistors each year you'd get a straight line. Real chips multiply, so you must use an exponent to capture it. That is why the topic is "exponential", not "linear".

4. The symbols $N$ , $N_{0}$ , $t$ , $t_{0}$ , $T_{d}$ — the growth formula's cast

The picture: a stopwatch that started at year $t_{0}$ . Every time $T_{d}$ years tick by, the transistor count doubles. $N_{0}$ is the reading at the start; $N (t)$ is the reading now.

Building the formula from §3: in §3 we saw " $n$ doublings" gives a multiplier $2^{n}$ . Here the number of doublings is just "years passed divided by doubling time", $n = (t - t_{0}) / T_{d}$ . Multiply the starting count $N_{0}$ by that $2^{n}$ and you get exactly $N (t) = N_{0} 2^{(t - t_{0}) / T_{d}}$ .

Why the subscript "0"? The little $0$ just means "the reference value we start from". It is not a special zero — it labels the anchor point of the stopwatch.

5. Logarithms — the tool that "undoes" an exponent

Why we need it here (two reasons):

To solve for an unknown exponent. In Example 2 the parent must solve $2^{22/ T_{d}} = 1 0^{3}$ . The unknown $T_{d}$ is stuck up in the exponent. Taking $lo g_{10}$ of both sides drops it down where algebra can reach it: $\frac{22}{T _{d}} lo g_{10} 2 = 3.$
To make the graph a straight line. Taking $lo g_{10}$ of the growth formula gives $lo g_{10} N (t) = lo g_{10} N_{0} + \frac{t - t _{0}}{T _{d}} lo g_{10} 2,$ which has the shape "constant + (slope) $\times t$ " — a straight line. That is why Moore's Law is drawn on a semi-log graph: the exponential curve becomes a diagonal you can read with a ruler.

Figure 3 — The same growth curve twice. Left (coral, linear axis): the exponential shoots up and is hard to read. Right (lavender, semi-log axis): the very same data becomes a straight line whose slope is $(lo g_{10} 2) / T_{d}$ .

6. The scaling factor $κ$ and the words "scales as"

The picture: a photocopy set to "shrink to $1/ κ$ ". Every length on the transistor comes out $κ$ times smaller.

Why $\propto$ and not $=$ ? Take capacitance, which the parent writes as $C = ε \frac{W L}{t _{o x}}$ . Here $ε$ (epsilon) is just a fixed material constant — a property of the oxide that never changes as we shrink. Because $ε$ is the same before and after, it cancels when we compare "after / before". We don't care about its exact value — only about whether shrinking makes $C$ bigger or smaller and by how much. $\propto$ strips away that constant clutter so the trend is visible.

7. The physical quantities: $C$ , $V$ , $I$ , $f$ , $τ$ , $P$

Each of these is a property of a switching transistor. Define them once, then the whole Dennard table reads easily.

Why "power density" $P / A$ ? A big cool chip and a tiny hot chip can burn the same total watts but the tiny one melts because the heat is squeezed into less area. So the quantity that decides "does it melt?" is $P / A$ — power per unit area — which is why the parent tracks it, not $P$ alone.

Figure 4 — The bucket analogy: capacitance $C$ is the bucket size, current $I$ is the fill-flow (coral), voltage $V$ is the pressure it fills to (lavender). Delay $τ = C V / I$ is the time to fill; power $P \propto C V^{2} f$ is filling and emptying $f$ times a second.

8. Two ideas that appear as walls: threshold voltage & leakage

Why this ends the free lunch: To keep the "cool" benefit, voltage $V$ had to keep shrinking. But $V$ can't drop below what $V_{t h}$ allows without leakage exploding — so $V$ froze around 2005, and the constant-power-density magic stopped. That is the power wall that pushed everyone toward Multi-core Architecture.

How these foundations feed the topic

The map below shows the build order: each box is one section above, and an arrow means "you need this before that". The two spines — the exponential/log spine and the physical-quantity spine — meet at the power wall.

Equipment checklist

Test yourself — cover the right side. If any answer is fuzzy, re-read its section above before opening the parent note.

What is a transistor, in one phrase?

A tiny electrical switch whose gate controls current flow.

What are

L

W

, and

t_{o x}

Channel length, channel width, and gate-oxide thickness — the sizes you shrink.

Why does area scale as

1/ κ^{2}

when each side shrinks by

κ

Area is length

\times

width, so both shrinking gives

κ \times κ = κ^{2}

What does

2^{n}

mean and what does

n

count?

2

multiplied by itself

n

times;

n

counts how many doublings happened.

N (t) = N_{0} 2^{(t - t_{0}) / T_{d}}

, what is

N_{0}

The transistor count at the reference start year

t_{0}

What question does

lo g_{10}

answer?

"What power of 10 gives this number?" — it pulls an exponent down.

Give the two reasons the topic uses logarithms.

To solve for an exponent (like

T_{d}

), and to turn the exponential into a straight semi-log line.

What does

\propto

mean and why use it here?

"Proportional to" — it shows the factor of change while ignoring constants like

ε

, revealing the trend.

What is

κ

The factor (

> 1

) by which every dimension and voltage shrinks each node.

Write and explain

τ = C V / I

Delay = charge to fill the gate (

C V

) divided by flow rate (

I

) = time to switch.

Why does the parent track power density

P / A

, not just

P

Because heat squeezed into less area melts the chip;

P / A

decides whether it overheats.

What is leakage and why did it freeze voltage?

Trickle current when the switch is "off"; lowering

V_{t h}

makes it grow exponentially (

I_{l e ak} \propto 1 0^{- V_{t h} / S}

), so

V

couldn't drop further.