Drift current and electric field

WHY do we even have drift current?

WHAT is the problem? In a semiconductor at equilibrium (no field), electrons and holes are in constant thermal motion. They fly around at high speed (~$10^5$ m/s at room temperature) but in random directions. Averaged over many carriers, the net displacement is zero — no current.

WHY does a field change this? An electric field $\vec{E}$ exerts a force $\vec{F} = q\vec{E}$ on each charge. This force adds a tiny systematic velocity on top of the huge random thermal velocity. That small biased velocity is the drift velocity $v_d$, and it is what produces net current.

HOW does drift velocity relate to the field? (Derivation from scratch)

Step 1 — Newton between collisions. Between two scattering events, a carrier of charge $q$ and effective mass $m^*$ accelerates: $a = \frac{qE}{m^*}$ Why this step? Between collisions there is no obstacle, so the only force is the electric force; $F = m^*a$ gives the acceleration.

Step 2 — Carriers randomize at each collision. After a collision the velocity direction is randomized, so the drift velocity gained is lost. On average a carrier travels for the mean free time $\tau$ before colliding.

Step 3 — Average the gained velocity. Velocity gained just before a collision is $a\tau$; averaged from $0$ up to $\tau$ the mean drift velocity is: $v_d = a\tau = \frac{qE\tau}{m^*}$ Why this step? Each carrier restarts from random (net-zero) velocity after a collision, so the systematic part is what was accumulated over one free time $\tau$.

Step 4 — Define mobility. We bundle all the material constants into one symbol: $\boxed{\mu = \frac{q\tau}{m^*}} \qquad \Rightarrow \qquad v_d = \mu E$

HOW do we get the current density?

Step 1 — Count charge crossing an area. Consider electrons of density $n$ (per volume) each drifting at $v_d$. In time $dt$, all electrons within distance $v_d\,dt$ of a cross-section $A$ cross it. Charge crossing: $dQ = q\,n\,(A\,v_d\,dt)$

Step 2 — Current = charge per time, density = per area. $J = \frac{1}{A}\frac{dQ}{dt} = q\,n\,v_d$ Why this step? Current density $J = I/A$ is the natural quantity because it doesn't depend on sample size.

Step 3 — Substitute drift velocity. For electrons ($q = -e$, but they drift opposite to $E$, so the conventional current is along $E$): $J_n = e\,n\,\mu_n\,E$ and for holes: $J_p = e\,p\,\mu_p\,E$

Why the two terms add (not subtract)? Electrons carry negative charge moving one way; holes carry positive charge moving the other way. Both give current in the same direction (along $E$), so their contributions add.

Worked Examples

Common Mistakes (Steel-manned)

Flashcards

Connections

• Effective Mass — the $m^*$ inside $\mu = q\tau/m^*$.
• Scattering Mechanisms and Mean Free Time — sets $\tau$ and temperature dependence of $\mu$.
• Diffusion Current and Carrier Gradients — the other current mechanism; combined via Drift-Diffusion Equation.
• Einstein Relation — links $\mu$ (drift) and $D$ (diffusion).
• Conductivity and Resistivity of Semiconductors — $\sigma = e(n\mu_n+p\mu_p)$.
• Velocity Saturation and High-Field Effects — where $v_d = \mu E$ breaks down.

Concept Map

Hinglish (regional understanding)