Thermal effects on conductivity: Foundations — Hardware deep dive

This page assumes you know nothing. Every letter, arrow, and word the parent note uses is built here, in an order where each idea rests on the one before it. If a symbol appears in the parent topic and you can't picture it, this is where you fix that.

0. What even is "current"? (the starting picture)

Imagine a pipe full of tiny marbles. Tilt the pipe and the marbles all start rolling the same direction. Electric current is exactly this: a crowd of charged particles all drifting the same way through a material.

What to see in Fig 1: a horizontal channel (the material) holds many amber dots (the charge carriers). Each dot has a small white arrow — they all point the same way. A big white arrow underneath labels the overall flow direction: this shared movement of the whole crowd is the current.

Two things obviously control how big the flow is:

How many marbles are in the pipe.
How fast each marble is willing to roll for a given push.

Everything on this page is just naming those two ideas precisely and adding the ones needed to measure them. Keep the marble picture in your head — we return to it constantly.

1. Charge, and the symbol $q$ (with its sign)

Picture: the "handle" on each marble that an electric field can grab. No handle → no push → no current. The sign says which way the handle is pulled.

Why the topic needs it: conductivity asks how charge moves. If a carrier had zero charge it could never carry current no matter how fast it moved. $q$ is the very first factor.

2. Carrier density, and the symbol $n$

Picture: count the marbles inside a $1 m \times 1 m \times 1 m$ box. That count is $n$ .

3. Electric field, and the symbol $E$

Picture: the tilt of the pipe. A steeper tilt = stronger field = harder push on every marble.

Why the topic needs it: without a push the marbles just jiggle randomly and go nowhere on average. $E$ is what turns random jiggling into net flow.

4. Drift velocity, and the symbol $v_{d}$

What to see in Fig 2: one carrier's cyan path zig-zags wildly across the frame — that is its random thermal motion. But the two amber dots (start and end) sit at the same height while the end is well to the right: an amber arrow along the bottom marks that small rightward net displacement. Divide that net displacement by the elapsed time and you get $v_{d}$ , the shared creep hiding under the chaos.

Picture: a swarm of bees buzzing chaotically inside a slowly-moving bus. Each bee's own motion is wild, but the whole swarm creeps forward at the bus's pace. That creep is $v_{d}$ .

Why the topic needs it: current flow is only about the shared drift, not the wild random speeds (which cancel out). $v_{d}$ isolates exactly the part that matters. More in Mobility and Drift Velocity.

5. Cross-sectional area $A$ , current $I$ , current density $J$ , and the relation $J = n q v_{d}$

Now we can connect the drift picture to something you can measure on a meter.

Picture: the size of the circular opening you'd see looking down the barrel of the wire.

6. Mean free time $τ$ , carrier mass $m$ , and deriving $v_{d} = q E τ / m$

Before the derivation we need one more actor — the carrier's mass — so let's name it here where it is first used.

Picture (mass): a heavy marble vs a light one on the same tilt — the heavy one speeds up more slowly.

What to see in Fig 3: cyan star-shaped obstacles are the vibrating lattice atoms. An amber path threads between them in straight segments, kinking sharply at each star — every kink is a collision that wipes out the drift. Each straight run is labelled $τ$ : it is the average length of one of those runs before a smash.

Picture ( $τ$ ): running across a room full of poles. $τ$ is how long, on average, you run before smacking a pole. Sparse poles → long $τ$ . Dense/wobbling poles → short $τ$ .

Formula Deriving drift velocity from Newton

Step 1 — acceleration. The field pushes each carrier with force $F = q E$ (charge times field-push). Newton's second law $F = ma$ gives acceleration $a = \frac{F}{m} = \frac{q E}{m} .$ Why this step? It converts "how hard the push is" into "how fast the carrier speeds up," using the mass $m$ we just defined.

Step 2 — how long the push acts. Each collision randomizes the carrier's motion, so on average it accelerates freely for a time $τ$ , then gets reset. The extra drift speed built up in that window is (velocity gained = acceleration × time): $v_{d} = a τ = \frac{q E τ}{m} .$ Why this step? $τ$ is the carrier's "memory" — how long it keeps the speed the field gave it before a collision erases it. What it looks like: in Fig 3, each straight amber segment is one such free-flight of duration $τ$ ; the drift is the small forward lean built up during each segment before the next kink resets it.

7. Mobility $μ$ — reading the formula

Picture: how slippery the pipe is. A smooth pipe → same tilt gives more roll → high mobility. A rough, obstacle-filled pipe → same tilt barely moves the marbles → low mobility.

Why the topic needs it: $μ$ bundles together everything about how easily an individual carrier moves, separate from how many there are. That clean split (many vs easily) is the whole game. More in Mobility and Drift Velocity.

8. Recap of mass $m$ — why it sits in the denominator

We defined $m$ in section 6 because the derivation needed it there. It's worth restating why it belongs where it does.

9. The band gap, and the symbol $E_{g}$

What to see in Fig 4: a lower cyan band (bound carriers, not conducting) and an upper amber band (free, conducting), separated by a vertical white double-arrow labelled $E_{g}$ — the wall height. One amber carrier is shown being kicked from the lower band up over the wall into the free band; the dashed circle it leaves behind in the lower band is the hole (its positive partner). Metals have essentially no wall; semiconductors have a real one that heat helps carriers clear.

Picture: carriers sit in a low valley (bound, not conducting). To join the current they must be kicked up over a wall of height $E_{g}$ into the upper "free" region. In a metal there is no wall ( $E_{g} \approx 0$ ) — carriers are already free.

10. Temperature $T$ , Boltzmann's constant $k_{B}$ , and the exponential $e^{- E_{g} / (2 k_{B} T)}$

Picture: a dial from $0 K$ (everything frozen still) upward; the higher the dial, the more violently everything shakes.

Picture: a thermostat that translates "how hot" ( $T$ ) into "how big a random kick each carrier typically gets" ( $k_{B} T$ ).

Reading the fraction $\frac{E _{g}}{2 k _{B} T}$ : it's (price ÷ budget). Small $T$ → tiny budget → giant fraction → $e^{- giant} \approx 0$ (almost no carriers). Large $T$ → fatter budget → smaller fraction → exponential climbs toward 1.

11. Conductivity $σ$ , resistivity $ρ$ , and resistance $R$

Now every ingredient is defined, we can assemble the parent's master formula — and connect it to a resistance you can actually measure.

Why a derivative and not just a ratio? Because we want the local sensitivity at a temperature, not just endpoints — the slope right where you are.

Prerequisite map

Recall Plaintext walkthrough of the map (in case the diagram doesn't render)

The signed charge $q$ , the carrier density $n$ , the cross-section area $A$ , and the drift velocity $v_{d}$ combine into the current density $J = n q v_{d}$ (section 5).
Drift velocity $v_{d}$ is itself built from the field $E$ , the mean free time $τ$ , and the mass $m$ via $v_{d} = q E τ / m$ ; the same three give mobility $μ = q τ / m$ .
$J$ and $μ$ together yield the conductivity $σ = n q μ$ (with $J = σ E$ ).
Separately, the band gap $E_{g}$ plus the temperature $T$ (through $k_{B} T$ ) set the carrier density $n$ via the Boltzmann factor $e^{- E_{g} / (2 k_{B} T)}$ — that arrow feeds back into $n$ .
Conductivity flips to resistivity $ρ = 1/ σ$ , which combines with wire geometry to give measurable resistance $R = ρ L / A$ , whose temperature slope is the coefficient $α$ .
The two temperature stories split off: the metal law (via shrinking $τ$ ) and the semiconductor law (via exploding $n$ ).

Equipment checklist

Cover the answer and test yourself before moving to the derivations.

What does $\frac{d R}{d T}$ mean and what is $α$ ? :::

What does

q

stand for, and is it signed?

The charge each carrier carries, in coulombs; it is signed — electrons have

q = - e

, holes

q = + e

e = 1.602 \times 1 0^{- 19}

For an electron, which way does its drift

v_{d}

point relative to

E

, and why is

J

still along

E

v_{d}

points against

E

(negative charge), but

q

is also negative, so the product

q v_{d}

(hence

J

) points along

E

— the two minus signs cancel.

What does

n

measure?

The number of mobile charge carriers per unit volume (

m^{- 3}

) — how crowded the pipe is.

What is the electric field

E

physically?

The push per unit charge — the "tilt of the pipe"; units V/m. (Not energy!)

What is the cross-sectional area

A

The area of the flat face exposed by slicing the wire straight across, perpendicular to flow; units

m^{2}

— how many lanes wide the pipe is.

Define drift velocity

v_{d}

The average shared forward speed the field adds on top of the carriers' random motion.

State

J = n q v_{d}

and where it comes from.

Current density = carriers-per-volume × charge each × drift speed; derived by counting the charge in a slab of length

v_{d} Δ t

and area

A

that crosses a face in time

Δ t

Derive

v_{d}

from Newton.

Force

q E

gives acceleration

a = q E / m

; acting for the free-flight time

τ

builds drift

v_{d} = a τ = q E τ / m

Define mobility

μ

two ways.

μ = v_{d} / E

(drift per field)

= q τ / m

(from Newton).

What is

τ

and why does heat shrink it?

The average time between collisions; heat makes atoms vibrate harder → bigger scattering targets → collisions sooner → smaller

τ

What role does

m

play in mobility?

Carrier sluggishness; bigger

m

→ less acceleration for the same push → lower

μ

(denominator of

μ = q τ / m

What is

E_{g}

, and where do holes come from?

The band-gap energy — the wall height an electron must clear to conduct (≈0 in metals); kicking an electron over it leaves a positive hole (

q = + e

) behind, so carriers are born in pairs.

What is

T

and why kelvin?

Absolute temperature — the random thermal energy scale; use kelvin because the formula needs a scale starting at true-zero energy (

0

K).

What does

k_{B}

do?

Converts temperature

T

(kelvin) into a typical thermal energy budget

k_{B} T

Why an exponential

e^{- E_{g} / (2 k_{B} T)}

and not a line?

Overcoming a fixed energy wall is a lottery whose winners explode as budget approaches price — only an exponential captures that.

State the master equation and

J = σ E

σ = n q μ

(how many × charge each × how easily each moves); substituting

v_{d} = μ E

into

J = n q v_{d}

gives

J = σ E

Why is

σ

always positive despite negative electrons?

Because

q < 0

and

v_{d}

-against-

E

flip together,

J

ends up along

E

; with

J = σ E

pointing the same way,

σ

is positive.

How does

σ

connect to a measurable resistance

R

ρ = 1/ σ

, and

R = ρ L / A = L / (σ A)

— longer wire raises

R

, fatter wire lowers it.