Concept of carrier mobility: Question bank — Hardware deep dive

The whole page rests on three anchors from the parent note: $μ = \frac{q τ}{m ^{*}}, v_{d} = μ E, σ = n q μ .$ Here $μ$ is mobility (drift speed per unit field), $τ$ the mean free time between collisions, $m^{*}$ the effective mass, $v_{d}$ the drift velocity, $E$ the field, $n$ the carrier count per volume, $σ$ the conductivity. Keep those three in view and every trap below unravels.

True or false — justify

Mobility depends on how strong the applied field

E

is.

False. In

μ = v_{d} / E

the field cancels because

v_{d} = μ E

grows in exact proportion to

E

;

μ

is a property of the material (via

τ, m^{*}

), temperature and doping — not of the push.

A material with more carriers automatically has higher mobility.

False. Mobility (

q τ / m^{*}

) is per-carrier ease of motion; carrier count

n

is separate. Both raise

σ = n q μ

, but they are independent knobs — doping raises

n

yet usually lowers

μ

Electron mobility

μ_{n}

is negative because the electron charge is negative.

False. We define

μ = ∣ q ∣ τ / m^{*} > 0

for both carriers. Electrons drift opposite to

E

, but that sign is tracked separately in the current, not buried inside

μ

Raising temperature always speeds up drift because carriers gain energy.

False. Extra thermal energy is random, not directed. In the lattice-limited regime more phonons mean more collisions, smaller

τ

, and thus lower

μ

and slower drift per unit field.

The drift velocity is roughly the electron's true speed inside the crystal.

False. True thermal speed is ~

1 0^{5}

m/s in random directions (zero average);

v_{d}

is only the tiny net bias (~mm–m/s) the field adds on top. See Drift and Diffusion currents.

Holes usually have lower mobility than electrons in the same material.

True (typically). Since

μ \propto 1/ m^{*}

and holes usually have larger effective mass,

μ_{p} < μ_{n}

(e.g. Si:

1350

480

cm²/V·s). See Effective mass.

Adding a second scattering mechanism can only lower the overall mobility.

True. Rates add:

1/ τ = 1/ τ_{1} + 1/ τ_{2}

, so total

τ

shrinks and

1/ μ = 1/ μ_{1} + 1/ μ_{2}

makes

μ

smaller than either alone. The worst (smallest-

μ

) mechanism dominates.

Conductivity

σ = n q μ

means doubling the field doubles the conductivity.

False.

σ

contains no

E

; it depends on

n

q

μ

. Doubling

E

doubles the current

J = σ E

, but leaves

σ

(and

μ

) unchanged. See Conductivity and Resistivity.

A perfectly rigid, perfectly pure crystal at 0 K would have infinite mobility.

Conceptually true (idealized). With no phonons and no impurities,

τ \to \infty

, so

μ = q τ / m^{*} \to \infty

. Real crystals always have some defect or boundary scattering, so

μ

stays finite.

Spot the error

"Since

μ = v_{d} / E

, a bigger field gives a bigger mobility." — find the flaw.

v_{d}

itself scales with

E

(

v_{d} = μ E

), so their ratio is constant. The equation defines

μ

; it does not make

μ

a function of

E

μ = q τ / m^{*}

, so heavier doping raises

μ

by adding carriers." — find the flaw.

Doping changes

n

, which does not appear in

μ = q τ / m^{*}

. Worse, extra ionized impurities shorten

τ

, so

μ

actually falls. See Doping and carrier concentration.

"Electrons drift opposite to

E

, so we must write

μ_{n} < 0

." — find the flaw.

The direction of drift is handled by the direction of the force

q E

; mobility is defined with

∣ q ∣

so it stays positive. Both

μ_{n}, μ_{p} > 0

σ = q (n μ_{n} + p μ_{p})

, so a metal with huge

n

must have low mobility." — find the flaw.

High

σ

can come from large

n

or large

μ

or both; the formula gives no inverse link between

n

and

μ

. They are independent factors multiplied together.

"Because

μ \propto 1/ m^{*}

, using rest mass

m_{0}

instead of

m^{*}

is fine — it's close enough." — find the flaw.

In a crystal the carrier responds to

E

with its effective mass

m^{*}

, which can differ from

m_{0}

by a large factor (e.g.

0.26 m_{0}

in Si). Using

m_{0}

can be off by several times.

"Matthiessen's rule says

μ = μ_{l a tt i ce} + μ_{im p u r i t y}

." — find the flaw.

It's the rates

1/ τ

(hence

1/ μ

) that add, not the mobilities:

1/ μ = 1/ μ_{l a tt i ce} + 1/ μ_{im p u r i t y}

. Adding mobilities would wrongly make more scattering help. See Scattering mechanisms in semiconductors.

"At high

T

mobility falls, at low

T

it also falls — so the curve has no peak." — find the flaw.

The two regimes fight: lattice scattering (

μ \propto T^{- 3/2}

) hurts at high

T

, ionized-impurity scattering (

μ \propto T^{+ 3/2}

) hurts at low

T

. Between them

μ

reaches a maximum, so there is a peak.

Why questions

Why does the field

E

cancel out of the definition of mobility?

Because drift is linear in the push: doubling

E

doubles both the acceleration and

v_{d}

, so

v_{d} / E

is a fixed ratio set by

τ

and

m^{*}

— a material fingerprint, not a field property.

Why do we use the effective mass

m^{*}

instead of the true electron mass in

μ

Inside the periodic lattice the carrier's response to a force is modified by the crystal's internal forces;

m^{*}

packages all of that so Newton's law

a = q E / m^{*}

works with the applied field alone.

Why does more scattering (smaller

τ

) mean lower mobility?

τ

is how long a carrier accelerates before a collision randomizes it. Shorter

τ

means less time to build up drift, so

v_{d} = q E τ / m^{*}

and hence

μ

shrink.

Why is drift velocity so tiny compared to thermal velocity yet still carries all the current?

The huge thermal motion averages to zero (random directions), contributing no net charge flow. Only the small systematic drift

v_{d}

survives averaging, so it alone makes the current.

Why can mobility relate to diffusion at all (Einstein relation)?

Both drift and diffusion are limited by the same collisions (

τ

), so the same microphysics ties them:

D = μ k T / q

. See Einstein relation.

Why does heavy doping usually lower mobility even though it boosts conductivity?

Dopants become ionized impurities that scatter carriers, cutting

τ

and

μ

. But they add so many carriers (

n

) that

σ = n q μ

still rises — the

n

gain outweighs the

μ

loss.

Why must

τ

be an average (mean free time) rather than a single fixed value?

Collisions happen at random moments, so different carriers accelerate for different durations. Averaging over that spread gives one

τ

that reproduces the crowd's mean drift.

Edge cases

What is

v_{d}

when

E = 0

Zero net drift —

v_{d} = μ \cdot 0 = 0

. Carriers still move thermally at ~

1 0^{5}

m/s, but with no field their average velocity vanishes and no current flows.

What happens to

μ

τ \to 0

(extremely dirty or hot crystal)?

μ = q τ / m^{*} \to 0

: carriers are randomized before they can drift, so conductivity collapses toward that of an insulator-like behaviour.

What happens to

μ

for a very heavy carrier,

m^{*} \to \infty

μ \to 0

: an infinitely heavy carrier can't be accelerated by the field, so it acquires no drift and contributes no current.

In an intrinsic semiconductor with equal

n = p

, which term dominates

σ = q (n μ_{n} + p μ_{p})

The electron term, because

μ_{n} > μ_{p}

typically. With

n = p

, the larger mobility carries proportionally more current.

If mobility doubles but carrier count halves, what happens to conductivity?

It stays the same.

σ = n q μ

is a product; halving

n

and doubling

μ

cancel, giving identical

σ

(and

ρ

At very low

T

in a lightly doped sample, why can mobility fall instead of rising?

Phonons freeze out (helping), but slow carriers spend longer near ionized impurities, so impurity scattering (

μ \propto T^{+ 3/2}

) worsens and dominates, pulling

μ

down.

Can two materials have equal conductivity but very different mobilities?

Yes — one with few high-mobility carriers, another with many low-mobility carriers can share the same

σ = n q μ

. Conductivity alone can't separate

n

from

μ

; a Hall measurement can.

Recall One-line survival kit

$E$ cancels in $μ = v_{d} / E$ ::: so $μ$ is a material number, not a field number. Rates add, not mobilities ::: $1/ μ = 1/ μ_{1} + 1/ μ_{2}$ . $n$ and $μ$ are independent ::: doping raises $n$ but lowers $μ$ . $μ > 0$ always ::: direction lives in the force $q E$ , not in $μ$ .

Connections

Concept of carrier mobility — the parent note these traps drill.
Scattering mechanisms in semiconductors — sets $τ$ ; source of the temperature traps.
Doping and carrier concentration — the $n$ -vs- $μ$ independence trap.
Effective mass — why $m^{*}$ , not $m_{0}$ .
Conductivity and Resistivity — $σ = n q μ$ misreadings.
Einstein relation — the drift–diffusion link.