Mobility & Resistivity Origin

When a battery is connected to a wire, electrons do not shoot through in a straight line. Instead, they zigzag chaotically, bouncing off vibrating metal ions, drifting only slowly in the direction the electric field pushes them. The average velocity of this slow drift — called the drift velocity — is directly controlled by how strongly the electric field accelerates electrons between collisions and how quickly collisions knock them back. The ratio of drift velocity to electric field strength is called electron mobility, and it lies at the very heart of understanding why some materials conduct electricity well and others resist it. Resistivity is the material property that tells you how strongly a substance opposes the flow of electric current, regardless of the object's shape or size. It emerges from the microscopic world: the density of free electrons available to carry charge, and how freely those electrons can move (their mobility). A material with many free electrons and high mobility has very low resistivity — like copper — while a material with few carriers or frequent scattering events has high resistivity. Understanding this link transforms resistivity from an abstract table value into a physical story about electrons navigating an obstacle course of vibrating atoms. Together, mobility and carrier density give us a microscopic formula for conductivity (the inverse of resistivity). This bridges the gap between the atomic scale — where quantum mechanics governs electron behaviour — and the macroscopic scale of Ohm's Law, circuit diagrams, and electrical engineering. Mastering this connection equips you to reason about semiconductors, superconductors, and novel materials far beyond simple copper wire.

Key takeaways
  • Drift Velocity and the Random Walk
  • Defining Mobility
  • Current Density from the Microscopic Picture
  • Resistivity as the Inverse
  • Connecting Back to Ohm's Law

Drift Velocity and the Random Walk

Electrons in a metal already move at enormous speeds (~10⁶ m/s) due to quantum-mechanical energy. However, this thermal motion is entirely random — electrons travel in all directions equally, so there is no net current. When an electric field E is applied, it superimposes a tiny but systematic push in one direction. The resulting net velocity in that direction is the drift velocity, vd, which is typically only ~10⁻⁴ m/s — far slower than thermal motion. Think of a crowded shopping centre: people rush about chaotically, but a slow evacuation drift emerges when a fire alarm sounds.

Defining Mobility

Mobility (μ) is defined as the drift velocity per unit electric field: μ = vd / E. It tells you how responsive a charge carrier is to an applied field. A high mobility means a small field produces a large drift — the electrons move relatively freely. Mobility depends on the mean free time τ (average time between collisions) and the electron's effective mass m: μ = eτ/m. In copper, τ ≈ 2.5 × 10⁻¹⁴ s, giving μ ≈ 4.4 × 10⁻³ m²V⁻¹s⁻¹. In semiconductors like gallium arsenide, smaller effective mass means much higher mobility.

Current Density from the Microscopic Picture

Current density J (current per unit cross-sectional area) is given by J = nqvd, where n is the number density of charge carriers and q is their charge. Substituting vd = μE gives J = nqμE. Since Ohm's Law states J = σE (where σ is conductivity), we can immediately identify σ = nqμ. This is a powerful result: conductivity is the product of carrier density, charge, and mobility. High n (many carriers) and high μ (freely moving carriers) both independently increase conductivity.

Resistivity as the Inverse

Resistivity ρ is simply the reciprocal of conductivity: ρ = 1/σ = 1/(nqμ). This formula reveals exactly why resistivity changes with temperature. As temperature rises, metal ions vibrate more vigorously, increasing the rate of electron–ion collisions and reducing τ. This lowers μ and therefore raises ρ. In contrast, for semiconductors, raising temperature frees more carriers (n increases), which can actually lower resistivity — the opposite behaviour to metals. The formula ρ = 1/(nqμ) encapsulates both effects simultaneously.

Connecting Back to Ohm's Law

The macroscopic resistance R of a conductor of length L and cross-sectional area A is R = ρL/A. Substituting ρ = 1/(nqμ) gives R = L/(nqμA). Every factor has physical meaning: a longer path (larger L) means more collisions; a larger cross-section (larger A) provides more parallel paths; higher carrier density or mobility reduces resistance. Ohm's Law, V = IR, is therefore not a mysterious empirical rule but a direct consequence of electrons drifting through a lattice of scattering centres.

The Key Equation

ρ=1nqμ=m∗nq2τ\rho = \frac{1}{nq\mu} = \frac{m^*}{nq^2\tau}
  • \rho — Resistivity (Ω·m): Material property describing resistance to current flow, independent of shape.
  • n — Carrier number density (m⁻³): Number of free charge carriers (electrons or holes) per unit volume.
  • q — Carrier charge (C): Magnitude of charge on each carrier; for electrons q = 1.6 × 10⁻¹⁹ C.
  • \mu — Carrier mobility (m²·V⁻¹·s⁻¹): Drift velocity produced per unit electric field; measures how freely carriers move.
  • m^* — Effective mass (kg): Quantum-corrected mass of the electron in the crystal lattice; differs from rest mass.
  • \tau — Mean free time (s): Average time between successive collisions of an electron with lattice ions.

Worked Example

Problem. Copper has a carrier density of n = 8.5 × 10²⁸ m⁻³ and a measured resistivity of ρ = 1.7 × 10⁻⁸ Ω·m at room temperature. (a) Calculate the electron mobility in copper. (b) If the electron's effective mass in copper is approximately equal to its rest mass (9.11 × 10⁻³¹ kg), estimate the mean free time τ between collisions.

μ≈4.3×10−3 m2V−1s−1,τ≈2.4×10−14 s\mu \approx 4.3 \times 10^{-3}\,\text{m}^2\text{V}^{-1}\text{s}^{-1},\quad \tau \approx 2.4 \times 10^{-14}\,\text{s}

Answer. Electron mobility μ ≈ 4.3 × 10⁻³ m²V⁻¹s⁻¹; Mean free time τ ≈ 2.4 × 10⁻¹⁴ s

Figure
Diagram
Microscopic Current Model: Drift in a MetalCu²⁺Cu²⁺Cu²⁺Cu²⁺Cu²⁺Cu²⁺Cu²⁺Cu²⁺Cu²⁺Cu²⁺— electron path (random collisions + slow drift →)Electron (e⁻) zigzag pathMetal ion (Cu²⁺) lattice siteElectric field E →Drift velocity vd = μE (net rightward, ~10⁻⁴ m/s)σ = nqμ → ρ = 1/(nqμ)
Origins · 1 minPaul Karl Ludwig Drude and how this was discovered
PK
Paul Karl Ludwig Drude
1863–1906 · German

Drude developed the classical free-electron model of electrical conduction in metals in 1900, just three years after the electron's discovery. His model treated conduction electrons as a gas of classical particles drifting through a lattice of positive ions, deriving expressions for conductivity, resistivity, and the Wiedemann–Franz Law. The concepts of drift velocity, mean free time, and mobility all flow directly from his framework. Although quantum mechanics later refined the model, Drude's equations remain the starting point for understanding electrical conduction. Drude published his landmark paper on electron conduction in 1900 — the same year Max Planck introduced the quantum of action — yet his entirely classical theory remained impressively accurate for metals for decades, until quantum mechanics explained why it worked so unexpectedly well.

  • Drift Velocity and Current — Drift velocity is the central quantity linking mobility to measurable current; understanding vd = μE is prerequisite to this concept.
  • Ohm's Law and Resistance — Macroscopic Ohm's Law (V = IR) is the large-scale consequence of the microscopic mobility and resistivity model developed here.
  • Resistivity and Temperature — Temperature affects mean free time τ and hence mobility, directly changing resistivity — the next step beyond this foundational model.
  • Semiconductors and Band Theory — In semiconductors both carrier density n and mobility μ vary with conditions; understanding ρ = 1/(nqμ) is essential for analysing semiconductor behaviour.