Drift current and electric field: Question bank — Hardware deep dive

Definition Symbol glossary — read this first

Every symbol used below, in plain words, so nothing is assumed from memory:

$e$ = the elementary charge, a fixed positive number $e = 1.6 \times 1 0^{- 19} C$ . It is always positive — the "size" of one charge.
$q$ = a generic signed charge of whatever carrier we happen to discuss: for an electron $q = - e$ , for a hole $q = + e$ . So $q$ carries the sign; $e$ never does.
$m^{*}$ = the effective mass of a carrier (see Effective Mass) — how heavy it behaves inside the crystal.
$τ$ (tau) = the mean free time, the average time a carrier flies before its next collision.
$μ$ (mu) = the mobility, drift velocity per unit field, $μ = e τ / m^{*}$ ; subscripts $μ_{n}, μ_{p}$ for electrons and holes.
$n$ = electron density (electrons per volume); $p$ = hole density.
$E$ = the applied electric field; $v_{d}$ = drift velocity; $a$ = acceleration.
$J$ = current density (current per unit area, $J = I / A$ ); $σ$ = conductivity; $A$ = cross-section area; $I$ = current.
$D$ = the diffusion coefficient; $k_{B}$ = Boltzmann's constant; $T$ = absolute temperature — these appear only in the diffusion cross-link.

Notation convention used everywhere below: we write the generic force as $F = q E$ (so the sign of the carrier is built in), but once we collect the magnitudes into a current formula we use the positive $e$ , e.g. $J = e (n μ_{n} + p μ_{p}) E$ . Whenever you see $q$ , think "signed"; whenever you see $e$ , think "positive magnitude."

Figure — Drift current and electric field

The picture above is the mental model behind every trap on this page: left = no field (random zig-zag, no net motion); right = field on (same zig-zag, but the whole cloud slides slowly along $E$ ). Refer back to it whenever a question feels abstract.

True or false — justify

Drift velocity keeps growing forever under a constant field.

False — collisions randomize velocity every mean free time

τ

, so the systematic part saturates at the steady-state average

v_{d} = μ E

(like terminal velocity, not free fall).

Electron and hole drift currents cancel because their charges are opposite.

False — they also drift in opposite directions under the same field, so two sign flips cancel and the currents add:

J = e (n μ_{n} + p μ_{p}) E

Drift velocity is much larger than the random thermal velocity.

False — thermal speed comes from thermal energy (

\frac{1}{2} m^{*} v_{t h}^{2} \sim \frac{3}{2} k_{B} T

giving

v_{t h} \sim 1 0^{7} cm/s

in Si at room temperature), whereas drift

v_{d} = μ E \sim 1 0^{6} cm/s

even at

E = 1000 V/cm

(using

μ_{n} = 1350

); so drift is a tiny bias on top of huge random motion — see the two-panel figure above.

Higher temperature always raises the current in a doped semiconductor.

False — for phonon (lattice) scattering, more

T

means more collisions, so

τ

and

μ

drop; with

n

fixed by doping,

σ

can actually fall (see Scattering Mechanisms and Mean Free Time).

Mobility is a property of the carrier alone, not the material.

False —

μ = e τ / m^{*}

depends on the effective mass

m^{*}

(band structure) and

τ

(scattering in that specific lattice), so it is a carrier-in-a-material property.

At equilibrium with no field, individual electrons are motionless.

False — they move fast (

\sim 1 0^{5}

–

1 0^{7} cm/s

thermally) but in random directions, so only the net displacement (and net current) is zero — this is the left panel of the figure.

Ohm's law

J = σ E

is an assumption we impose, unrelated to microscopic physics.

False — it emerges from

J = e (n μ_{n} + p μ_{p}) E

with

σ = e (n μ_{n} + p μ_{p})

; linearity in

E

comes from

v_{d} = μ E

holding at low fields.

Doubling the field always doubles the drift velocity.

False — it doubles only at low fields where

v_{d} = μ E

is linear; at high fields carriers hit velocity saturation and

v_{d}

barely changes, so the "always" makes the statement false.

Spot the error

"Between collisions the carrier has no force on it, so it moves at constant velocity."

Wrong — between collisions the electric force

F = q E

still acts (there is no scattering obstacle, but the field is always on), so the carrier accelerates with

a = q E / m^{*}

"Since

v_{d} = a τ

and

a = q E / m^{*}

, mobility must depend on the applied field

E

Wrong — the

E

cancels:

v_{d} = (q E τ / m^{*})

μ = v_{d} / E = e τ / m^{*}

(magnitude) has no

E

in it. Mobility is field-independent in the linear regime.

"Because holes are just missing electrons, hole current is the same thing as electron current counted twice."

Wrong — holes are a distinct positive carrier with their own density

p

and mobility

μ_{p}

; they contribute a separate term

e p μ_{p} E

that adds to the electron term.

"In n-type silicon we drop the

p μ_{p}

term because holes have zero mobility."

Wrong — holes still have finite

μ_{p}

; we drop the term because

p ≪ n

, so

p μ_{p} ≪ n μ_{n}

— an 80/20 approximation, not a zero.

"Current density

J

depends on the sample's cross-section, so bigger samples have bigger

J

Wrong —

J = I / A

is defined per unit area precisely so it is size-independent; the current

I = J A

scales with area, but

J

does not.

"Electrons drift along

E

because the field pushes charges forward."

Wrong — electrons carry charge

q = - e

, so

F = - e E

points opposite to

E

; the electrons drift against

E

, yet conventional current still points along

E

Why questions

Why do we average acceleration over the mean free time

τ

instead of over the whole sample's transit time?

Because each collision resets the drift velocity to random (net-zero); only the velocity accumulated within one free flight of duration

τ

survives as systematic drift.

Why is

J = q n v_{d}

the right way to count current, rather than just

q n

Current is charge crossing an area per second; in time

d t

only carriers within

v_{d} d t

of the surface cross, so the rate is proportional to

v_{d}

, giving

J = q n v_{d}

Why does mobility rise when effective mass falls?

A lighter carrier (

m^{*}

small) accelerates more for the same force (

a = q E / m^{*}

), so it builds more drift velocity per free time —

μ = e τ / m^{*}

is inversely proportional to

m^{*}

Why is drift current only one of two current mechanisms?

Drift needs a field to bias motion; carriers can also flow down a concentration gradient with no field at all — that is diffusion current, and both combine in the drift-diffusion equation.

Why does the same

τ

that controls mobility also link drift to diffusion?

The scattering time sets both how far carriers drift and how far they random-walk; the Einstein relation

D = μ k_{B} T / e

ties the two together through this shared microscopic origin.

Why does conductivity depend on the product

n μ

, not

n

μ

separately?

Current needs both many carriers (large

n

) and each responding to the field (large

μ

); either being zero kills the current, so they multiply — see Conductivity and Resistivity of Semiconductors.

Edge cases

What is the drift current when

E = 0

Exactly zero — with no field there is no drift velocity (

v_{d} = μ \cdot 0 = 0

), so despite fast thermal motion the net charge flow vanishes.

What happens to

v_{d} = μ E

at very large

E

It breaks down — carriers can't gain unlimited speed because they dump energy to the lattice (optical phonons), so

v_{d}

flattens to a saturation velocity (

\sim 1 0^{7} cm/s

in Si); see Velocity Saturation and High-Field Effects.

What is the drift current in an intrinsic semiconductor where

n = p

Nonzero and it still adds:

J = e n (μ_{n} + μ_{p}) E

; even though densities are equal, electrons and holes both push current along

E

τ \to 0

(carriers scatter constantly), what happens to mobility and current?

μ = e τ / m^{*} \to 0

, so drift velocity and drift current vanish — an infinitely-collided carrier can never build up any net bias.

What is the drift velocity of a carrier immediately after a collision, before any acceleration?

Its systematic drift is zero — the collision randomized its direction, so the biased velocity must be rebuilt over the next free flight.

If a material had

m^{*} \to \infty

(infinitely heavy carrier), what current would a field drive?

None —

μ = e τ / m^{*} \to 0

, so the carrier is too inert to accelerate; heavy carriers respond weakly to the field regardless of how many there are.

Connections

Drift current and electric field — the parent this bank stress-tests.
Effective Mass — why light carriers drift faster.
Scattering Mechanisms and Mean Free Time — the temperature and $τ$ traps.
Velocity Saturation and High-Field Effects — the high-field edge cases.
Diffusion Current and Carrier Gradients · Drift-Diffusion Equation · Einstein Relation — the "other mechanism" why-questions.
Conductivity and Resistivity of Semiconductors — the $n μ$ product trap.