Visual walkthrough — Drift current and electric field

Before we begin, one promise: every symbol gets a picture and a plain-word meaning before it is used. If a word looks scary, look at the figure beside it first.

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Step 1 — One electron, no field: going nowhere fast

WHAT. Inside a piece of silicon, a free electron is already moving — and moving fast, about $10^5$ metres every second. But it changes direction constantly because it keeps hitting things (vibrating atoms, impurities). We call each hit a collision (or "scattering event").

WHY start here. If we don't understand the motion without a field, we can't see what the field actually changes. The key fact: the fast motion is random, so if you add up where the electron went over many collisions, it ends up roughly back where it started. Net travel = zero. No current.

PICTURE. The zig-zag path below. Each straight segment is free flight; each dot is a collision that kicks the direction to something new. The green circle marks the finish — practically on top of the start.

Let the average time a carrier flies between two collisions be $\tau$ (the mean free time, studied in Scattering Mechanisms and Mean Free Time).

$\tau =\text{average seconds of free flight between collisions}$

• $\tau$ small ⇒ the electron is interrupted often (dirty or hot material).
• $\tau$ large ⇒ long clean runs between hits.

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Step 2 — Switch on the field: a gentle sideways push

WHAT. We apply an electric field $\vec{E}$ — imagine tilting the whole chalkboard so everything feels a steady pull in one direction. A charge $q$ in a field feels a force

$\underset{\text{push on the charge}}{\underbrace{\vec{F}}}=\underset{\text{how much charge}}{\underbrace{q}}\underset{\text{field strength \& direction}}{\underbrace{\vec{E}}}$

WHY this matters. The field does not stop the random bouncing — the electron is still staggering fast in all directions. The field only adds a tiny extra nudge the same way every free flight. That consistent nudge is the whole story of drift.

PICTURE. Same zig-zag as Step 1, but now every free-flight segment curves slightly toward the field (the pale-yellow arrow). The finish point (pink) has crept a little downfield of the start.

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Step 3 — Between collisions, Newton takes over

WHAT. During one free flight there is nothing in the way, so the only force is the electric one. Newton's law $F=m^{\ast }a$ gives the acceleration. Here $m^{\ast }$ is the effective mass — how heavy the carrier behaves inside the crystal (see Effective Mass).

$a=\frac{F}{m^{\ast }}=\frac{\stackrel{\text{charge}}{\overbrace{q}}\stackrel{\text{field}}{\overbrace{E}}}{\underset{\text{effective mass}}{\underbrace{m^{\ast }}}}$

WHY $a$ and not just a velocity? A steady force produces steady acceleration, meaning the extra downfield speed grows during the flight — it starts at zero right after a collision and builds up until the next collision cuts it off.

PICTURE. A single free-flight segment magnified. The downfield speed starts at $0$ (just after a collision) and ramps up linearly to $a\tau$ (just before the next). The slope of that ramp is $a$.

Reading the graph: the horizontal axis is time since the last collision; the vertical axis is the extra velocity the field has added so far. It is a straight line from $(0,0)$ to $(\tau ,a\tau )$.

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Step 4 — Averaging the ramp: the drift velocity

WHAT. A collision randomizes direction, so the extra downfield speed is thrown away at every collision and rebuilt from zero. The typical carrier is somewhere in the middle of a ramp. To get the average extra speed we average the straight line from $0$ to $a\tau$.

WHY average, and why does it equal $a\tau$ in the parent note? The average of a straight ramp from $0$ to $a\tau$ is exactly $\frac{1}{2}a\tau$. The parent note writes the characteristic value $v_d=a\tau$ (velocity gained over one full free time $\tau$); the factor of $\frac{1}{2}$ is a bookkeeping constant that a more careful average introduces. Either way the structure is the same, and we fold every such constant into one measured symbol in the next step. Using the parent's convention:

$\underset{\text{average downfield speed}}{\underbrace{v_d}}=a\tau =\frac{q\tau }{m^{\ast }}E$

PICTURE. The saw-tooth: extra velocity climbs, collision drops it to zero, climbs again — over and over. The dashed horizontal line is the steady average, the drift velocity $v_d$.

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Step 5 — Naming the material's response: mobility

WHAT. Look at $v_d=\frac{q\tau }{m^{\ast }}E$. Everything except the field $E$ is a property of the material ($q$, $\tau$, $m^{\ast }$). Bundle them into one symbol, mobility $\mu$:

$\boxed{\mu =\frac{q\tau }{m^{\ast }}}⟹\underset{\text{speed}}{\underbrace{v_d}}=\underset{\text{how responsive}}{\underbrace{\mu }}\underset{\text{how hard we push}}{\underbrace{E}}$

WHY bundle them. In the lab you don't measure $\tau$ or $m^{\ast }$ directly — you measure how fast carriers drift for a given field. That single number is $\mu$. It answers: how easily does this carrier respond to a push?

• Big $\tau$ (few collisions) ⇒ big $\mu$.
• Small $m^{\ast }$ (light carrier) ⇒ big $\mu$.

PICTURE. $v_d$ versus $E$ is a straight line through the origin whose slope is $\mu$. Two materials, two slopes: the steeper line is the higher-mobility material.

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Step 6 — Counting how much charge crosses a wall

WHAT. Now turn one drifting carrier into a current. Put an imaginary wall (cross-section area $A$) across the material. In a time $dt$, every carrier within a distance $v_{d}dt$ of the wall will cross it. Those carriers fill a slab of volume $A\times v_{d}dt$.

If there are $n$ carriers per unit volume, the charge that crosses is:

$dQ=\underset{\text{charge each}}{\underbrace{q}}\underset{\text{per volume}}{\underbrace{n}}\underset{\text{slab volume that crosses}}{\underbrace{(A{v}_{d}dt)}}$

WHY a slab. Only carriers close enough to reach the wall within $dt$ actually cross it — "close enough" means within one drift-step $v_{d}dt$. That defines exactly the shaded slab.

PICTURE. The wall (blue), the shaded slab of thickness $v_{d}dt$ behind it, and arrows showing every carrier in the slab sweeping across.

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Step 7 — From charge to current density

WHAT. Current is charge per time; current density $J$ is current per area (so the answer doesn't depend on how wide our sample is):

$J=\frac{1}{A}\frac{dQ}{dt}=\frac{1}{A}\cdot \frac{qnA{v}_{d}dt}{dt}=\underset{\text{charge}}{\underbrace{q}}\underset{\text{density}}{\underbrace{n}}\underset{\text{drift speed}}{\underbrace{v_d}}$

The $A$ and $dt$ cancel — that is why we use density: it is a property of the material, not of the block's size.

WHY substitute $v_d=\mu E$ next. We want current in terms of things we control ($E$) and material constants ($\mu$). Plugging in:

$J_n=en{\mu }_{n}E\text{(electrons)},J_p=ep{\mu }_{p}E\text{(holes)}$

PICTURE. A funnel: three inputs ($q$, $n$, $v_d$) flow into the box $J=qn{v}_d$, and $v_d=\mu E$ feeds in to give $J=en\mu E$.

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Step 8 — Two carriers, one direction: why they ADD

WHAT. Semiconductors have two kinds of mobile charge: electrons ($-e$) and holes ($+e$). Under the same field $\vec{E}$ they drift in opposite directions. You might expect their currents to cancel — they don't.

WHY they add. Two sign flips happen at once:

$\text{electron: }(-e)\times (\text{drifts against }E)=+\text{(current along }E)$ $\text{hole: }(+e)\times (\text{drifts along }E)=+\text{(current along }E)$

Both point the same way, so:

$\boxed{J_{\text{drift}}=e(\underset{\text{electron term}}{\underbrace{n{\mu }_n}}+\underset{\text{hole term}}{\underbrace{p{\mu }_p}})E=\sigma E}$

with conductivity $\sigma =e(n{\mu }_n+p{\mu }_p)$ — see Conductivity and Resistivity of Semiconductors.

PICTURE. Electrons drift left (blue), holes drift right (pink); both current arrows (yellow) point the same way — rightward, along $E$.

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Step 9 — Degenerate & edge cases (never leave a gap)

Every scenario the reader might hit, shown once:

The figure overlays all four edge cases on the $v_d$–$E$ line: the origin ($E=0$), the reversed branch ($E<0$), the low-field straight region, and where the real curve bends away (saturation).

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The one-picture summary

Everything above, compressed: random motion → tilt (field) → ramp between collisions → averaged drift → mobility slope → charge slab → current density → two carriers adding to $\sigma E$.

Recall Feynman retelling — the whole walk in plain words

A tiny electron zooms around inside the crystal, fast but aimless, bumping into things every $\tau$ seconds and forgetting where it was headed — so as a group it goes nowhere (Step 1). Tilt the world with a field (Step 2): now, during each little free flight, the field steadily speeds the electron up in one direction (Step 3). But every bump throws that gained speed away and it must build up again, so what you see on average is a small steady slide — the drift velocity (Step 4). Bottle up all the material's stubbornness into one number, mobility $\mu$, and the slide is just $v_d=\mu E$ (Step 5). To turn a sliding electron into a current, count how many cross an imaginary wall in a moment — a thin slab's worth (Step 6) — and divide by area to get current density $J=qn{v}_d=en\mu E$ (Step 7). Do it for holes too; they drift the opposite way but carry the opposite charge, so both currents point the same way and add up to $\sigma E$ (Step 8). And when there's no field, or the field flips, or it grows huge, the same little picture still tells you exactly what happens (Step 9).

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Recall

What does a collision do to the drift speed an electron has built up? ::: It randomizes direction, throwing the built-up drift speed back to zero — then it rebuilds over the next $\tau$. Why does the average of the velocity ramp give a steady drift velocity rather than a growing one? ::: Because the ramp restarts at zero after every collision, so the average sits at a fixed level instead of climbing. In $J=qn{v}_d$, why do $A$ and $dt$ disappear? ::: They cancel between "charge in the slab" and "per area per time" — leaving a size-independent material quantity. Why do electron and hole drift currents add? ::: Opposite charge AND opposite drift direction = two sign flips = both currents along $E$. What happens to $v_d=\mu E$ at very high fields? ::: It breaks — real drift saturates at a ceiling (Velocity Saturation and High-Field Effects); the straight line is a low-field approximation.

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Connections

• Effective Mass — the $m^{\ast }$ that set the acceleration in Step 3.
• Scattering Mechanisms and Mean Free Time — the $\tau$ that chops the ramp in Step 4.
• Conductivity and Resistivity of Semiconductors — the $\sigma$ assembled in Step 8.
• Velocity Saturation and High-Field Effects — where Step 5's straight line finally bends.
• Diffusion Current and Carrier Gradients · Drift-Diffusion Equation · Einstein Relation — the companion current mechanism and how the two combine.