Foundations — Understand conductors, insulators, and semiconductors

This page assumes you know nothing. We build each idea, one brick at a time, and only use a symbol after we have drawn its picture. Read top to bottom.

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1. What is electric charge? (the symbol $e$ and $q$)

Plain words. Charge is a property some tiny particles carry that makes them push and pull on each other. There are two kinds: negative (carried by electrons) and positive (carried by protons). Like charges repel; opposite charges attract.

The picture. Imagine two little balls. Two negatives shove apart; a negative and a positive snap together.

The symbols.

• $q$ = "some amount of charge", measured in coulombs (C).
• $e$ = the size (magnitude) of the charge on one electron, the smallest chunk that exists alone: $e=1.6\times 10^{-19}\text{C}(\text{a positive number}).$

Why the topic needs it. Conduction is charge moving. To count how much moves, we need a unit of charge — that unit is $e$. See Electric Charge and Current for the full story.

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2. What is current? (drifting electrons) — and which way it "flows"

Plain words. Current is charge in motion — how much charge flows past a point each second.

The picture. Picture a pipe full of electrons. When you tilt the pipe (apply a push), they slowly slide along. Current is "how many pass the checkpoint per second".

$I=\frac{q}{\Delta t}\text{measured in amperes (A)}=\text{coulombs per second}$

Here $\Delta t$ means "a chunk of time" (the little triangle $\Delta$ just reads "change in"). If the flow speeds up and slows down, we shrink that chunk to an instant and write the derivative form: $I=\frac{dq}{dt}$ which is just "$I=q/\Delta t$ measured over an instant instead of over a big interval". Read $dq/dt$ as "how fast the charge past the checkpoint is piling up right now".

Why keep this backwards-looking convention? Because the physics of how much current flows never depends on the sign of the carrier — a negative charge ($-e$) drifting left produces exactly the same current as a positive charge ($+e$) drifting right. We fixed one direction as "the current arrow" long ago and never needed to change it.

Why the topic needs it. The whole point of a conductor is that it lets current happen; an insulator stops it. Current is the thing being classified.

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3. Free carriers and their count $n$

Plain words. A free carrier is a charged particle that is loose enough to move through the material when pushed. In metals these are electrons that escaped their home atoms.

The picture. Zoom inside a solid: a grid of fixed atom-cores, with a fog of loose electrons wandering between them.

The symbol $n$. $n$ = carrier density = number of free carriers packed into one cubic metre. Units: carriers per m³, written ${\text{m}}^{-3}$.

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4. Holes — the empty-seat carrier

Plain words. When an electron leaves its spot, it leaves behind an empty positive-looking gap called a hole. Neighbouring electrons can hop into the gap, which makes the gap itself appear to move — like an empty seat drifting backwards down a row when people shuffle forward.

The picture. A row of chairs, all full but one. Person moves right into the empty seat; the empty seat effectively moves left.

Why the topic needs it. In semiconductors carriers come in pairs: freeing one electron creates one hole. Both carry current — and, importantly, they can have different counts and different mobilities. We label them separately:

• $n_e$ = electron density, ${\mu }_e$ = electron mobility (each electron carries $-e$).
• $n_h$ = hole density, ${\mu }_h$ = hole mobility (each hole carries $+e$).

This pairing is also the reason a certain "2" shows up later in the temperature formula.

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5. Energy bands and the band gap $E_g$

Plain words. Electrons in a solid can only have certain allowed energies, grouped into ranges called bands.

• Valence band = the low "sitting" range where electrons are bound to atoms.
• Conduction band = the higher "roaming" range where electrons are free to carry current.
• Band gap $E_g$ = the forbidden energy range in between — the "wall" an electron must jump to get free.

The picture. Two shelves, one low (valence) and one high (conduction), with an empty gap between them. The gap's height is $E_g$.

The unit — electron-volt (eV). $E_g$ is measured in electron-volts. One eV is the tiny energy an electron gains crossing a 1-volt push: $1\text{eV}=1.6\times 10^{-19}\text{J}$ (Notice that number is exactly $e$ — because energy = charge $\times$ voltage, and here charge $=e$, voltage $=1$ V.)

Why the topic needs it. The size of the gap decides how easily electrons reach the conduction band, which decides $n$, which decides everything. Conductor: no gap. Semiconductor: small gap ($\sim 1$ eV). Insulator: huge gap ($>5$ eV). Deeper detail lives in Energy Bands in Solids.

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6. The electric field $E$, current density $J$, and mobility $\mu$

The electric field $E$. An electric field is the "push per unit charge" — the force felt by each unit of charge. It is a vector (it points in a direction), and its units are newtons per coulomb (N/C), which is the same as volts per metre (V/m): $\vec{E}=\frac{\vec{F}}{q}[\text{N/C}=\text{V/m}]$ Sign convention: $\vec{E}$ points the way a positive test charge is pushed — i.e. the same direction as the current arrow. An electron (charge $-e$) feels force $\vec{F}=(-e)\vec{E}$, which is opposite to $\vec{E}$, which is why it drifts backwards.

Current density $\vec{J}$. Before we can link "push" to "flow", we need the local flow quantity. Current density $\vec{J}$ = current per unit cross-sectional area, units amperes per square metre (A/m²). It is a vector that points in the direction of conventional current (the way positive charge effectively flows). Think of it as "how thick the stream of charge is, per square metre of pipe": big pipe carrying $I$ has small $J$; thin pipe carrying the same $I$ has large $J$.

Mobility $\mu$. $\mu$ answers: for a given push, how fast do the carriers actually drift? Its units are metres-squared per volt-second, ${\text{m}}^2/(\text{V⋅s})$ — because ${\vec{v}}_d=\mu \vec{E}$ (for a positive carrier) means

$\underset{v_d}{\underbrace{\text{m/s}}}=\mu \cdot \underset{E}{\underbrace{\text{V/m}}}\Rightarrow \mu =\frac{\text{m/s}}{\text{V/m}}=\frac{{\text{m}}^2}{\text{V⋅s}}.$

A big $\mu$ = slippery material where carriers glide fast; a small $\mu$ = carriers keep bumping into things. $\mu$ is defined as a positive number; the drift direction is set by the carrier's sign — a positive carrier drifts along $\vec{E}$, an electron drifts opposite to $\vec{E}$.

${\vec{v}}_d=\mu \vec{E}(\text{positive carrier}),{\vec{v}}_d=-\mu \vec{E}(\text{electron}).$

The picture. A ball rolling downhill through pegs (atoms): the field is the downhill tilt; every peg-collision slows it; the steady average speed it settles at is $v_d$.

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7. Building $\sigma =ne\mu$ from scratch (the derivation)

We now have every piece: $n$, $e$, ${\vec{v}}_d$, $\mu$, $\vec{J}$, $\vec{E}$. Watch them snap together. To keep signs honest we do the electron case explicitly, then read off the magnitude.

Step A — count the charge that crosses an area (get $\vec{J}=ne{\vec{v}}_d$). What we do: We count how much charge crosses a checkpoint of area $A$ in a short time $\Delta t$. Why we count a slab: In time $\Delta t$ every carrier moves forward by $v_d\Delta t$. Only the carriers within that distance behind the checkpoint can reach it in time — carriers further back haven't arrived yet, carriers already ahead have left. So the charge that crosses is exactly the charge sitting in a slab of area $A$ and length $v_d\Delta t$. That is why the slab, and only the slab, is the right thing to count. The picture: the slab has volume $A{v}_d\Delta t$; it holds $n\cdot (A{v}_d\Delta t)$ carriers. Signs: each electron carries $-e$ and its velocity ${\vec{v}}_d=-\mu \vec{E}$ points opposite to $\vec{E}$. The current-density vector is charge $\times$ (carriers per volume) $\times$ velocity: $\vec{J}=(-e)n{\vec{v}}_d=(-e)n(-\mu \vec{E})=ne\mu \vec{E},$ so the two minus signs cancel and $\vec{J}$ points along $\vec{E}$ — matching the current arrow, exactly as the convention promised. In magnitude: $J=ne{v}_d.$

Step B — replace $v_d$ using the field (plug in $v_d=\mu E$). Why: Step A left the mystery speed $v_d$ in the answer; Section 6 told us $v_d=\mu E$, tying that speed to the push $E$ we actually control. $J=ne(\mu E)=(ne\mu )E.$

Step C — read off conductivity. Why: The whole bracket $(ne\mu )$ is a constant of the material multiplying $E$. Compare with the macroscopic law $J=\sigma E$ ("current density is proportional to push, with $\sigma$ the proportionality"). Matching the two forms term-by-term: $\boxed{\sigma =ne\mu }\rho =\frac{1}{\sigma }=\frac{1}{ne\mu }.$

So $\sigma =ne\mu$ is not a definition dropped from the sky — it is exactly what you get when you count carriers ($n$), give each its charge magnitude ($e$), and let the field set their speed ($\mu E$).

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8. Resistivity $\rho$ and conductivity $\sigma$ — the material numbers

Plain words.

• Resistivity $\rho$ = how strongly a material opposes current, independent of its shape. Units: ohm-metre (Ω·m).
• Conductivity $\sigma =1/\rho$ = how easily the material conducts — the same information flipped over.

The picture. Two identical-shaped cubes: one of copper (low $\rho$, current gushes) and one of rubber (high $\rho$, current trickles).

Why the topic needs it. $\rho$ is the single number that most sharply sorts the three classes (copper $\approx 10^{-8}$, rubber $\approx 10^{14}$). Every earlier symbol shows up in it: $n$ (how many carriers), $e$ (charge each), $\mu$ (how freely they move). More in Ohm's Law and Resistance.

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9. Resistance $R$ vs resistivity $\rho$ (don't confuse them)

Plain words. Resistance $R$ is what an actual shaped object opposes — it depends on both the material and the object's length $L$ and cross-sectional area $A$: $R=\rho \frac{L}{A}$

The picture. A garden hose: a long thin hose ($L$ big, $A$ small) resists flow; a short fat hose lets it gush. Same rubber, different resistance.

Recall Which one is a material-only property?

Resistivity $\rho$ is material-only; resistance $R$ also depends on shape ($L$ and $A$) ::: correct — same material, different $R$ for different shapes.

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10. Temperature $T$, $k_B$, and the exponential $e^{-E_g/2k_{B}T}$

Plain words. Heat makes atoms jiggle. That jiggle can hand an electron enough energy to jump the band gap. Temperature $T$ is measured in kelvin (K).

Why an exponential? This is a Boltzmann-factor argument from statistical physics. At temperature $T$, the probability that a random thermal kick supplies at least energy $\Delta$ falls off like $e^{-\Delta /(k_{B}T)}$ — the famous Boltzmann factor. Bigger wall or colder day → exponentially fewer kicks make it. So the number of electrons that reach the conduction band scales like this factor.

Why the $2$ in the exponent? Two reasons stack:

1. Pairing. Freeing one electron always creates one hole (Section 4). The energy $E_g$ is shared to make the pair, so each carrier effectively "pays" half of $E_g$.
2. Fermi-level position. In a pure (intrinsic) semiconductor the occupancy is measured from the middle of the gap (the Fermi level sits roughly mid-gap once you account for the available states — the density of states — on both sides). An electron therefore sits $E_g/2$ above that reference, not the full $E_g$.

Both point to the same result: the exponent carries $E_g/2$, giving $n\propto e^{-E_g/(2k_{B}T)}.$

Consequence. Warm the semiconductor → $n$ climbs fast → $\sigma =ne\mu$ rises → $\rho$ falls. A metal does the opposite: no gap to help, but hotter atoms scatter carriers more (collision time $\tau$ drops, so $\mu$ and hence $\sigma$ drop), so $\rho$ rises. See Temperature Coefficient of Resistance.

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11. How it all wires together

Read an arrow "A → B" as "you need A to understand B". Every foundation on this page feeds the single classification at the bottom: the size and controllability of the carrier count $n$.

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Equipment checklist

Cover the right side and test yourself. If any line is fuzzy, reread its section above.

• What is $e$ and its value, and what charge does an electron carry? ::: $e$ is the magnitude of the elementary charge, $1.6\times 10^{-19}$ C (positive); an electron carries $-e$, a hole/proton $+e$.
• Which way does conventional current point vs electron drift? ::: Current points the way positive charge moves ($+$ to $-$ outside the battery); electrons drift the opposite way.
• What is $I=dq/dt$ the limiting form of? ::: Of $I=q/\Delta t$ — the instantaneous rate of charge crossing a point.
• What does $n$ count, and in what units? ::: Free carriers per cubic metre, ${\text{m}}^{-3}$; it is the master dial separating the three classes.
• What is a hole and what charge does it act like? ::: A missing electron; it behaves as a movable positive ($+e$) charge, drifting along the current arrow.
• Name the three energy regions and what $E_g$ is. ::: Valence band (bound), conduction band (free), and $E_g$ = the forbidden gap between them.
• Convert: what is 1 eV in joules? ::: $1.6\times 10^{-19}$ J (charge $e$ times 1 volt).
• What are the units and vector direction of the field $\vec{E}$? ::: N/C = V/m; a vector pointing the way a positive charge is pushed.
• What is current density $\vec{J}$, its units, and its direction? ::: Current per unit area, A/m²; a vector along the conventional-current direction; it bridges drift to $J=\sigma E$.
• What are the units of mobility $\mu$, and how does carrier sign enter? ::: ${\text{m}}^2/(\text{V⋅s})$; $\mu$ is positive, and the carrier's sign sets whether drift is along ($+$) or against ($-$) $\vec{E}$.
• Derive $\sigma =ne\mu$ in one line, keeping signs. ::: $\vec{J}=(-e)n(-\mu \vec{E})=ne\mu \vec{E}\equiv \sigma \vec{E}$, so $\sigma =ne\mu$.
• Write the general two-carrier conductivity and say why the terms add. ::: $\sigma =e(n_e{\mu }_e+n_h{\mu }_h)$; electrons and holes push current the same way, so contributions add.
• Value and units of $k_B$? ::: $1.38\times 10^{-23}$ J/K; $k_{B}T$ is the thermal energy scale.
• Why the factor of 2 in $n\propto e^{-E_g/2k_{B}T}$? ::: Electron–hole pairing plus the mid-gap Fermi level put each carrier $E_g/2$ from the reference.
• Metal vs semiconductor when heated? ::: Metal $\rho$ rises (more scattering, smaller $\mu$); semiconductor $\rho$ falls (more carriers $n$).