Carrier Mobility and Conductivity

Introduction

Carrier concentration tells how many electrons or holes can conduct; mobility tells how effectively they respond to an electric field. Two semiconductor samples can have the same electron count and different resistivities because their crystals scatter carriers differently. Conversely, a heavily doped sample may contain more electrons but show lower mobility. Separating concentration from motion prevents the common mistake of equating a high dopant count with high conductivity.

Core explanation

Define drift mobility μ as the magnitude of average drift velocity divided by electric-field magnitude in the low-field linear regime: μ = v d / E . Its SI unit is m² V⁻¹ s⁻¹, often reported as cm² V⁻¹ s⁻¹. An electron drifts opposite the electric field but its negative charge makes its contribution to conventional current point with the field. Holes drift along the field and likewise give conventional current along it. For one electronic carrier, σ = neμ . With both types present, σ = e(nμ n + pμ p) , assuming a homogeneous, isotropic, low-field sample. Current density then obeys J = σE and resistivity is ρ = 1/σ. These are transport relations, not statements that the lattice atoms move with the current.

Mobility is affected by scattering. Vibrating lattice atoms or phonons scatter carriers more strongly in many ordinary temperature ranges. Ionised dopants also perturb carrier trajectories; adding dopants can increase n while decreasing μ. Grain boundaries, dislocations, neutral defects and surfaces may matter in particular samples. The observed temperature trend results from the competition among mechanisms and carrier concentration, so “conductivity must decrease as temperature rises” is not a general semiconductor rule. MIT's compound-semiconductor lecture collection includes scattering mechanisms and semiconductor mobility, while NIST's mobility measurement study compares contact and contactless measurements.

In a simple Drude-like picture, μ ≈ eτ/m , where τ is a momentum-relaxation time and m is an appropriate band effective mass. This is a useful guide, but a real semiconductor can have direction-dependent bands, multiple valleys, energy-dependent scattering and distinct electron and hole effective masses. Mobility measured by a Hall effect method is not always identical to drift mobility without a Hall factor. Quoting one number without temperature, doping level and measurement method hides important context.

At very high electric fields, carrier speed may saturate and the low-field law v d = μE no longer holds with one constant μ. In a thin film, contact resistance can dominate a two-terminal measurement. A four-terminal geometry or contactless optical method can help isolate the material's transport response. The conductivity equation is therefore a model for a specified regime, not a universal calibration for every device configuration.

Step-by-step reasoning

1. Identify the carrier type or types and their number densities. 2. Convert mobilities to the same units, remembering 1 cm² V⁻¹ s⁻¹ = 10⁻⁴ m² V⁻¹ s⁻¹. 3. Calculate each contribution neμ or peμ, then add them. 4. Interpret a conductivity trend by separating carrier concentration from mobility. 5. Check field strength, contacts, temperature and sample geometry before comparing measurements.

Visual explanation

Draw an electron zigzagging through a crystal under an applied field. Short random segments depict scattering, while a small net displacement opposite E depicts drift. Draw a hole moving effectively along E. Beside them, show two bars for sample A and B: B has more carriers but shorter scattering time, illustrating why total σ cannot be inferred from n alone.

Real-world analogy

Traffic flow depends on how many cars are available and how fast they can traverse the road. Extra cars may increase flow at first, but congestion can lower average speed. This captures the product of carrier density and mobility, although actual carrier scattering is quantum-mechanical and holes are electronic vacancies rather than cars.

Real-world example

In a doped silicon resistor, raising donor concentration usually increases electron density. At sufficiently high doping, ionised-impurity scattering reduces mobility. Designers choose a concentration that meets a target sheet resistance while considering temperature dependence and junction properties. NIST's work on mobility measurements notes that even the carrier density used for an optical measurement can influence scattering and mobility.

Why?

Why do both electrons and holes contribute positive terms to σ despite opposite charges? Electrons move opposite E while holes move with E. Multiplying each carrier velocity by its charge gives conventional current in the same direction as E for both. Conductivity is a positive proportionality in a passive material's ordinary linear-response regime.

Common misconception

“A larger electron concentration guarantees proportionally larger conductivity” ignores mobility changes. “Electron drift follows the electric field” reverses its actual motion. “Mobility is speed” omits the field: mobility is speed per field , and its value can depend on temperature and scattering regime.

Worked example

A sample has n = 1.0 × 10²¹ m⁻³, p = 2.0 × 10²⁰ m⁻³, μ n = 0.10 m² V⁻¹ s⁻¹ and μ p = 0.040 m² V⁻¹ s⁻¹. Its conductivity is e(nμ n + pμ p) = (1.602 × 10⁻¹⁹)[(1.0 × 10²¹)(0.10) + (2.0 × 10²⁰)(0.040)] = 17.3 S m⁻¹ to three significant figures. Electron transport supplies 16.0 S m⁻¹ and holes about 1.28 S m⁻¹. Thus ignoring holes here would be a modest but identifiable approximation, not an exact result.

Quick check

1. If n doubles and μ halves while all other contributions remain negligible, what happens to σ? Answer: It stays the same because σ = neμ has an unchanged product nμ.

Exam focus

Show both terms when electrons and holes matter. Keep density and mobility units compatible. Distinguish drift direction from conventional-current direction for electrons. When asked about a temperature or doping trend, discuss both carrier production and scattering instead of giving a one-factor answer.

Advanced insight

The Hall effect can estimate carrier sign and density in a simple single-carrier material, while conductivity then helps infer a mobility. With multiple carrier types, Hall coefficients require a weighted two-carrier analysis and can be misleading if interpreted by the one-carrier formula. Frequency-dependent conductivity reveals additional information about scattering times, which is why terahertz probes complement direct-current measurements.

Summary

Mobility quantifies low-field drift response, while conductivity combines mobility with carrier number. Electrons and holes contribute additively to ordinary conductivity. Scattering from lattice vibrations, dopants and defects changes mobility, so composition alone cannot predict transport performance.

Practice questions

1. Convert 500 cm² V⁻¹ s⁻¹ to SI mobility units. Answer: 500 × 10⁻⁴ = 0.050 m² V⁻¹ s⁻¹. 2. Which way do electrons drift relative to the applied electric field? Answer: Opposite the field, while their conventional-current contribution points with the field. 3. A one-carrier sample has n = 10²² m⁻³ and μ = 0.01 m² V⁻¹ s⁻¹. Estimate σ. Answer: neμ = (10²²)(1.602 × 10⁻¹⁹)(0.01) = 16.0 S m⁻¹. 4. Name two processes that can lower mobility without removing carriers. Answer: Scattering by ionised dopants and by lattice vibrations are two examples.