Foundations — Drift current and electric field

This page assumes you have seen none of the notation in the parent note. We build every symbol from the ground up, in an order where each one only uses ideas already earned. By the end you will be able to read $J_{drift}=e(n{\mu }_n+p{\mu }_p)E$ and know what every letter means and why it is there.

0. What is a semiconductor "carrier"? (before any symbol)

Picture a nearly-full parking lot. A few cars can move into the empty spaces — those moving cars are like electrons. But you can also describe the motion by watching the empty space itself shuffle backwards — that empty space is a hole, and we treat it as a positive particle. Both descriptions are the same reality; both let charge flow.

Why do we need both? Because in real doped material, one type usually dominates, and the current formula has one term for each. If you only ever tracked electrons you'd get the wrong answer in p-type material.

1. Charge $q$ and the elementary charge $e$

• An electron has charge $q=-e$ (negative).
• A hole has charge $q=+e$ (positive).

Why split $e$ off from $q$? Because $e$ is a fixed number of nature, while $q$ carries the sign (direction of push). Keeping them separate lets us write clean formulas and track signs on purpose instead of by accident.

2. The electric field $\vec{E}$

The picture: imagine a tilted floor. The steeper the tilt, the stronger the field; the direction of steepest descent is the field's direction.

Why do we need it? Because "apply a voltage" really means "create a field inside the material," and it is the field — not the voltage directly — that pushes each carrier. Units: volts per centimetre, written $\text{V/cm}$.

3. Force $\vec{F}=q\vec{E}$

• Positive charge ($+e$): force points along $\vec{E}$ (downhill).
• Negative charge ($-e$): force points opposite to $\vec{E}$ (uphill).

Why is this the very first physics we need? Because drift current is nothing but "what happens when a steady push acts on jiggling charges." No push, no drift.

4. Effective mass $m^{\ast }$

The picture: pushing a shopping trolley through thick mud feels heavier than through open air, even though the trolley's real mass is unchanged. The crystal's atomic lattice is that "mud," and $m^{\ast }$ is the felt heaviness.

Why the star ${}^{\ast }$? To remind us it is not the free-electron mass — it can be larger or smaller. We need it because acceleration is $a=F/m^{\ast }$: a lighter effective mass means the same push produces more speed. See Effective Mass for the full story.

5. Acceleration $a$ and Newton's law $F=m^{\ast }a$

The picture: a ball rolling down our tilted floor speeds up steadily; that steady speeding-up is $a$. Between two bumps, a carrier does exactly this.

Why do we need it? A carrier isn't at constant speed while the field acts on it — it is speeding up. To find how much velocity it gains, we must know how fast it gains it ($a$) and for how long ($\tau$, next).

6. Mean free time $\tau$

The picture: a pinball rolling, hitting a bumper, rolling, hitting another. The average roll-time between bumpers is $\tau$. Each collision erases the direction the carrier had built up — this is the crucial reset.

Why do we need it? Because it answers "for how long does the field get to speed up a carrier before the reset?" That interval is exactly $\tau$. Details of what causes collisions live in Scattering Mechanisms and Mean Free Time.

7. Drift velocity $v_d$

Between collisions a carrier gains velocity $a\tau$; each collision throws away the direction; averaging over the crowd leaves a steady

Why "average"? Because any single carrier is mostly doing wild thermal loops; only when you average millions of them does the tiny biased slide survive. That surviving slide is $v_d$.

8. Mobility $\mu$

Big $\tau$ (few collisions) and small $m^{\ast }$ (light carrier) both make $\mu$ big — a nimble carrier. Units: ${\text{cm}}^2/(\text{V⋅s})$.

Why invent it? Because in a real problem you never re-derive $\tau$ and $m^{\ast }$ each time — you look up one number, $\mu$, that already contains all the messy scattering physics. Electrons and holes have their own values, written ${\mu }_n$ and ${\mu }_p$.

9. Carrier densities $n$ and $p$

The picture: count the moving charges inside one little cube and divide by the cube's volume. More carriers packed in ⇒ more charge available to flow.

Why do we need both? A general semiconductor conducts through electrons and holes at once; the total current adds one term per type — which is why $n$ and $p$ each get their own place in the formula.

10. Current density $J$

The picture: watch a window of area $A$ inside the wire; count the charge sailing through it each second, then divide by the window's size.

Why use $J$ instead of the current $I$ we actually measure? Because $J$ doesn't care how fat the sample is — it is a property of the material and field, not the sample shape. We compute $J$ from physics, then multiply by area at the very end to get the measurable $I$.

Combining everything: a slab of electrons with density $n$, each drifting at $v_d={\mu }_{n}E$, gives

and holes give $J_p=ep{\mu }_{p}E$. Both point along $\vec{E}$ (electrons are negative but drift backwards — two sign flips cancel), so they add:

11. Conductivity $\sigma$ and Ohm's law

Why bundle again? Just as $\mu$ hid the scattering physics, $\sigma$ hides both the carrier counts and the mobilities in one measurable material constant. See Conductivity and Resistivity of Semiconductors.

How the foundations feed the topic

Read it top-down: charge and field make a force; force and effective mass make an acceleration; acceleration acting for one mean free time makes a drift velocity; folding the constants gives mobility; mobility plus carrier density gives current density; and that packages into conductivity.

Equipment checklist

Test yourself — cover the right side and see if you can state each from memory.

Connections

• Effective Mass — the $m^{\ast }$ we treated as "felt heaviness."
• Scattering Mechanisms and Mean Free Time — what sets $\tau$.
• Diffusion Current and Carrier Gradients — the other way carriers move.
• Drift-Diffusion Equation — drift and diffusion combined.
• Einstein Relation — links $\mu$ and diffusion.
• Conductivity and Resistivity of Semiconductors — $\sigma$ and $\rho$.
• Velocity Saturation and High-Field Effects — where $v_d=\mu E$ breaks down.
• Parent: Drift current and electric field