This is the bottom level of the book. Everything above it — gates, registers, instructions, programs — is a way of organizing one physical activity: moving small amounts of electric charge from place to place, and noticing where it is. This chapter covers the basic quantities (charge, current, voltage, resistance), explains why some materials conduct and others don't, and ends with a calculation that explains a surprising amount of computer architecture: why a millimetre of wire on a chip is slower than a gate.
The numbers in this level are computed, not quoted. Each calculation is shown as a few lines of Python with its result, so you can check it or change the assumptions.
Charge and current
Matter is made of atoms: a small, positive nucleus surrounded by electrons, each carrying a negative charge of exactly e = 1.602176634 × 10⁻¹⁹ coulomb (C). Since 2019, that value is exact by definition: the SI units are built on it. Like charges repel, opposite charges attract, and charge is never created or destroyed, only moved.
Current is the rate at which charge flows past a point, measured in amperes (A): one ampere is one coulomb per second. Since each electron carries so little charge, it takes a lot of them:
e = 1.602176634e-19 # C, exact
print(1 / e) # 6.241509074460763e+18 electrons per second in 1 A
A current of 1 A is about 6.24 × 10¹⁸ electrons per second. A historical accident makes this slightly confusing: conventional current is defined as flowing from + to −, but in a metal the carriers are electrons, which move the other way. Circuit diagrams use the conventional direction; the physics doesn't care.
Voltage, resistance and Ohm's law
Charges move when something pushes them. Voltage (or potential difference), measured in volts (V), is the energy each coulomb gains or loses between two points: 1 V means 1 joule per coulomb. A battery or power supply maintains a voltage between its terminals; connect a wire, and charge flows from one to the other.
How much current flows depends on the resistance of the path, in ohms (Ω). For most conductors at a fixed temperature, current is proportional to voltage. That's Ohm's law:
V = I × R
Double the voltage across a resistor and the current doubles. The energy the charges lose along the way becomes heat, at a rate given by the power P = V × I = I²R, in watts (W, joules per second). Every watt a processor consumes ends up as heat in its resistances — the subject of the last chapter of this level.
Resistance depends on the shape of the conductor and on the material. A wire of length L and cross-section A has
R = ρ × L / A
where resistivity ρ (rho, in Ω·m) is a property of the material alone. Longer wires resist more; thicker ones less.
| Material | Resistivity at room temperature (Ω·m) |
|---|---|
| silver | 1.59 × 10⁻⁸ |
| copper | 1.68 × 10⁻⁸ |
| gold | 2.44 × 10⁻⁸ |
| aluminium | 2.65 × 10⁻⁸ |
| tungsten | 5.6 × 10⁻⁸ |
| pure silicon | about 3 × 10³ (computed below) |
| glass | 10¹¹ to 10¹⁵ |
| fused quartz (SiO₂) | about 10¹⁷ |
The range is enormous: 25 orders of magnitude between copper and quartz. It's one of the widest ranges of any physical property, and computers depend on both ends of it. A chip is copper wires separated by silicon dioxide and similar insulators, a few tens of nanometres apart.
A metre of ordinary 1 mm² copper wire has R = 1.68 × 10⁻⁸ × 1 / 10⁻⁶ = 0.0168 Ω: negligible for household wiring. We'll see that the wires on a chip are a very different story.
Fast signals, slow electrons
How fast do the electrons move? Copper has about one free electron per atom. From its density (8.96 g/cm³) and atomic mass (63.55 g/mol):
NA = 6.02214076e23
n = 8.96 / 63.546 * NA * 1e6 # free electrons per m³
print(n) # 8.49e+28
v = 1 / (n * e * 1e-6) # drift speed for 1 A in 1 mm²
print(v * 1e3, "mm/s") # 0.0735 mm/s
With 1 A flowing in a 1 mm² wire, the electrons drift at about 0.07 mm per second: they'd take almost four hours to cover a metre. Yet a light turns on as soon as you flip the switch. The reason is that the wire is already full of electrons. Pushing on one end pushes on all of them at once, like water in a full pipe, and that push — the electromagnetic field — travels at a large fraction of the speed of light. Signals are fast; the charges carrying them are slow. The chapter on signals comes back to how fast the push travels.
The electrons' own motion is actually fast but random: they bounce around at more than a million metres per second, colliding with the vibrating atoms of the lattice. The applied voltage adds only a tiny average drift on top of that. Those collisions are what resistance is, and they're why a metal's resistance rises with temperature: hotter atoms vibrate more. Copper's resistance goes up by about 0.4 % per degree Celsius.
Why some materials conduct: bands
Why does copper conduct and quartz doesn't, when both are packed with electrons? The answer comes from quantum mechanics. In an isolated atom, electrons can only have certain energies, the atom's energy levels. In a solid, the levels of billions of neighbouring atoms merge into continuous bands of allowed energies, separated by band gaps: ranges of energy that no electron can have.
At low temperature, electrons fill the lowest states first. Whether a material conducts depends on where the filling stops:
- In a metal, the highest occupied band is only partly filled. Just above the most energetic electrons there are empty states at almost the same energy, so the smallest push from a voltage can move electrons into them: they're free to carry current.
- In an insulator, the electrons exactly fill a band, the valence band, and the next band up, the conduction band, is empty and far away. A full band carries no net current: every electron moving one way is matched by one moving the other way, and there's no free state to shift into. To conduct, an electron must jump the whole gap, and in SiO₂ that gap is about 9 electron-volts (eV). One electron-volt is the energy an electron gains across 1 V: 1.602 × 10⁻¹⁹ J.
- A semiconductor is an insulator with a small gap: 1.12 eV for silicon at room temperature, 0.66 eV for germanium.
The only energy available to push electrons across the gap is heat. At temperature T, the typical thermal energy is kT, where k is Boltzmann's constant:
k = 1.380649e-23 # J/K, exact
kT = k * 300 # room temperature, 300 K
print(kT, kT / e) # 4.14e-21 J = 0.02585 eV (25.9 meV)
At 300 K, kT is 25.9 meV — about one fortieth of an electron-volt. The fraction of electrons thermally excited across a gap Eg varies roughly as exp(−Eg / 2kT), and that exponential is ferocious:
import math
for name, Eg in [("Ge", 0.66), ("Si", 1.12), ("diamond", 5.5), ("SiO2", 9.0)]:
print(name, math.exp(-Eg / (2 * 0.025852)))
# Ge 2.9e-06 Si 3.9e-10 diamond 6.3e-47 SiO2 2.5e-76
Going from silicon's 1.12 eV to silicon dioxide's 9 eV changes the factor by 66 orders of magnitude. That's why quartz is one of the best insulators known, and why silicon, with its small gap, is interesting: a little heat, light or a few impurity atoms make it conduct, in amounts we can control. The next chapter is about exactly that control. For pure silicon, the thermally excited carriers give the resistivity in the table above:
ni = 1e10 # intrinsic carriers per cm³ in Si at 300 K
mu_n, mu_p = 1400, 450 # electron and hole mobilities, cm²/(V·s)
rho = 1 / (e * ni * (mu_n + mu_p))
print(rho, "ohm·cm") # 3.4e+05 ohm·cm = 3.4e+03 ohm·m
That's about 10¹¹ times worse than copper, and about 10¹³ times better than quartz.
Capacitance: where bits live
The other key component is the capacitor: two conductors separated by an insulator. Put a voltage V across it and charge +Q collects on one side, −Q on the other, in proportion:
Q = C × V
The constant C is the capacitance, in farads (F). For two parallel plates of area A a distance d apart, C = εA/d, where ε is the insulator's permittivity: ε₀ = 8.854 × 10⁻¹² F/m for a vacuum, about 3.9 times that for SiO₂. Big plates close together make a big capacitor. The capacitors in a chip are tiny, measured in femtofarads (1 fF = 10⁻¹⁵ F).
Capacitance is everywhere in a computer, wanted or not:
- every gate input is a capacitor: the transistor's gate is a conductor sitting on a thin insulator above the channel, as the MOSFET chapter shows;
- every wire is a capacitor with its neighbours and the layers above and below;
- a DRAM bit is literally a capacitor holding charge or not, as the chapter on storing a bit calculates.
So changing a logic value from 0 to 1 means charging a capacitor through some resistance, and changing it back means discharging it. That takes time, and it costs energy. Charging a capacitance C to voltage V from a supply stores ½CV² in the capacitor and — whatever the resistance — dissipates another ½CV² as heat on the way. Discharging burns the stored half. This is the origin of the dynamic power formula of the last chapter.
The RC time constant
Charge a capacitor C through a resistor R and the voltage doesn't jump; it approaches its final value exponentially, with a time constant τ = RC (ohms times farads give seconds). After one τ the voltage has covered 63 % of the way; after about 0.69 τ, half of it.
A wire on a chip isn't a single resistor and a single capacitor: its resistance and capacitance are spread along its whole length. For such a distributed RC line, the delay to reach 50 % at the far end is about 0.38 RC, where R and C are the whole wire's totals. And here's the important part: both R and C grow with the length L, so the delay grows as L².
Worked example: a millimetre of wire on a chip
Take a thin copper wire in the lower metal layers of a modern chip: 50 nm wide, 100 nm tall, 1 mm long. For capacitance, use the common rule of thumb for on-chip wires, about 0.2 fF per micrometre of length.
rho = 1.68e-8 # copper, ohm·m (bulk value: optimistic)
def wire(L, W=50e-9, H=100e-9, c_per_m=0.2e-15 / 1e-6):
R = rho * L / (W * H)
C = c_per_m * L
return R, C, 0.38 * R * C # 50% delay of a distributed RC line
R, C, t = wire(1e-3)
print(R, C, t) # 3360 ohm, 2e-13 F (200 fF), 2.55e-10 s = 255 ps
| Length | R | C | RC delay |
|---|---|---|---|
| 20 µm | 67 Ω | 4 fF | 0.1 ps |
| 0.1 mm | 336 Ω | 20 fF | 2.6 ps |
| 1 mm | 3.4 kΩ | 200 fF | 255 ps |
| 2 mm | 6.7 kΩ | 400 fF | 1.0 ns |
| 10 mm | 34 kΩ | 2 pF | 25.5 ns |
A millimetre of this wire has 3,360 Ω of resistance — 200,000 times more than a metre of household cable — and a delay of about 255 ps. At 4 GHz, a clock cycle is 250 ps. Twice the length, four times the delay: an unbroken 10 mm wire across a large die would take 25 ns, about a hundred clock cycles.
Compare with the speed of light. In SiO₂ (relative permittivity 3.9), light travels at c/√3.9, about half its vacuum speed, and crosses 1 mm in 6.6 ps. The RC delay is almost 40 times longer. On a chip, wires are limited by resistance and capacitance, not by the speed of light.
The calculation is optimistic, too. At widths of a few tens of nanometres, electrons scatter off the wire's surfaces and grain boundaries, and a thin barrier layer that keeps copper from diffusing into the silicon takes up part of the cross-section, so the real resistance is noticeably higher than the bulk value predicts.
Tanenbaum makes a related point in his discussion of pulse generators: a 20-micron path adds only about 0.0001 ns, because the signal travels "at the speed of light". The conclusion holds — the table gives about 0.1 ps for 20 µm of this wire, just as negligible — but the reason is different. Short on-chip wires are fast because L² is tiny, not because signals move at light speed; at a millimetre, the RC delay is already 40 times slower than light.
What chip designers do about it
The L² law shapes how chips are built:
- A stack of wiring layers. A modern chip has more than a dozen metal layers. The lowest are thin and dense, for short local connections. The top ones are thick and wide — much lower resistance — and carry long-distance signals, the clock and power.
- Repeaters. A long wire is cut into segments with an inverter (a buffer) between each. Ten 1 mm segments cost 10 × 255 ps plus ten small buffer delays, instead of 25.5 ns for the whole: the delay becomes linear in length again.
- Better materials. IBM replaced aluminium wires with copper in 1997, cutting resistivity by more than a third. Insulators with a lower permittivity than SiO₂ (low-k dielectrics, some of them porous) reduce C.
- Architecture. Since a signal can't cross the chip in one cycle, designs keep communication local. Pipelining, the split of big caches into banks, and multicore chips that talk over on-chip networks are all partly answers to wire delay.
Shrinking makes the problem worse, not better. Scale a wire's width and height down by a factor s, and its resistance per unit length goes up by s²; its capacitance per unit length stays roughly the same. Transistors get faster with each generation; long wires don't. That's why the MOSFET chapter will say that transistors are no longer the only thing that matters.
Takeaways
- Charge comes in units of e = 1.602 × 10⁻¹⁹ C; current is charge per second (1 A ≈ 6.24 × 10¹⁸ electrons/s); voltage is energy per charge. Ohm's law: V = IR, and power P = VI.
- Resistivity spans 25 orders of magnitude, from copper (1.68 × 10⁻⁸ Ω·m) to quartz (about 10¹⁷ Ω·m). Chips use both ends.
- In a wire, signals move fast but electrons drift slowly — about 0.07 mm/s for 1 A in 1 mm² of copper.
- In the band picture, metals have a partly filled band; insulators and semiconductors have a band gap (9 eV for SiO₂, 1.12 eV for silicon) that heat (kT = 25.9 meV at 300 K) can only rarely cross.
- Every gate input and wire is a capacitor. Switching a bit means charging or discharging it, which costs ½CV² of heat each way.
- On-chip wires are RC-limited: delay grows as length². A 1 mm, 50 × 100 nm copper wire takes about 255 ps — a whole clock cycle — while light would take 6.6 ps. Chips answer with thick upper layers, repeaters, copper, low-k insulators, and architectures that keep signals local.