Electrical Conduction and Transport Models

Introduction

Electrical transport connects microscopic carrier behavior to measurable quantities such as resistance, Hall voltage and thermal conductivity. Classical electron theory explains drift and scattering intuitively, while quantum mechanics supplies the energy-band and Fermi-statistics framework required for metals, semiconductors and insulators.

This chapter develops transport models from their assumptions rather than treating conductivity as an empirical constant. It also emphasizes when common formulas fail—for example, in multicarrier semiconductors, nanostructures, strong magnetic fields and materials with substantial phonon heat conduction.

Learning Objectives

After completing this chapter, students should be able to:

• derive conductivity from carrier density, charge, mobility and relaxation time;
• distinguish drift, diffusion and conventional current directions;
• interpret Fermi level, band occupancy and effective mass;
• calculate conductivity, resistivity, mobility and diffusion coefficients;
• explain temperature-dependent transport in metals and semiconductors;
• derive and apply one-carrier and two-carrier Hall relations;
• relate electronic thermal conductivity to electrical conductivity;
• evaluate engineering materials using electrical, thermal, geometric and reliability constraints.

1. Charge Transport and the Drude Model

1.1 Current density

Electric current is the rate of charge transfer. For a uniform current normal to area A,

where J is current density in A·m⁻² and I is current in amperes. The vector current density produced by one carrier population is

Here n is carrier concentration in m⁻³, q_c is the signed carrier charge in coulombs, and \mathbf{v}_d is mean drift velocity in m·s⁻¹. Electrons have q_c=-e, where e=1.602\times10^-19\ \mathrm{C}. Electron drift is opposite the electric field, but electron charge is also negative; therefore, conventional electron current points along the field.

1.2 Relaxation-time model

In the Drude model, carriers accelerate between random scattering events. Using an effective mass m^*, their average motion obeys

The fields \mathbf{E} and \mathbf{B} have units V·m⁻¹ and tesla, while \tau is the momentum-relaxation time in seconds. The damping term assumes independent, statistically random scattering and a field-independent relaxation time.

For a steady electric field with \mathbf{B}=0,

Mobility is conventionally positive:

where \mu has units m²·V⁻¹·s⁻¹. Combining these relations gives Ohm’s law in local form:

For one isotropic carrier population, resistivity is

with \rho in Ω·m. For a homogeneous conductor of length L and constant cross-sectional area A,

This geometric resistance formula requires approximately uniform current density and negligible contact and spreading resistance.

Worked Example 1: Copper mobility and drift speed

At 20 °C, take copper resistivity as 1.68\times10^-8\ \Omega\cdot\mathrm{m}, carrier concentration as 8.47\times10²⁸\ \mathrm{m}^-3, and m^*=m_e=9.109\times10^-31\ \mathrm{kg}.

The mobility is

The corresponding relaxation time is

At J=5.00\times10⁶\ \mathrm{A\,m^-2}, the drift-speed magnitude is

This small drift velocity is compatible with rapid circuit response: the electromagnetic field propagates through the circuit much faster than individual electrons drift.

1.3 Strengths and limitations

The Drude model explains Ohm’s law and the approximate relation between conductivity and scattering. However, its classical carrier statistics give incorrect electronic heat capacities and cannot explain band gaps. Quantum theory corrects these deficiencies while retaining the useful relaxation-time structure.

2. Quantum States, Fermi Statistics and Energy Bands

2.1 Fermi–Dirac occupancy

Electrons are fermions, so no two electrons can occupy the same quantum state with identical quantum numbers. The equilibrium probability that a state of energy E is occupied is

Here \mu_ch is chemical potential, k_B=1.381\times10^-23\ \mathrm{J\,K^-1}, and T is absolute temperature. At T=0, every allowed state below the Fermi energy E_F is occupied and every state above it is empty. In ordinary metals at moderate temperatures, \mu_ch\approx E_F.

Only electrons within roughly a few k_BT of E_F can change into nearby unoccupied states and contribute effectively to transport. Thus, although metals contain many electrons, only a small fraction responds thermally to weak fields.

2.2 Formation of bands

When atoms form a crystal, interaction among their orbitals splits discrete atomic levels into closely spaced allowed energy bands. Regions without allowed states are band gaps.

Figure: Schematic band occupancy. In the metal, the occupied-state shading terminates exactly at the depicted Fermi level at approximately zero temperature.

A metal has a partially filled band or overlapping bands, so nearby empty states are available. A semiconductor has a relatively small gap E_g, while an insulator has a larger gap. The distinction is not determined by a universal gap threshold alone; defects, temperature and electric-field strength also matter.

2.3 Effective mass and semiclassical dynamics

Within a band of dispersion E(\mathbf{k}), an electron wave packet has group velocity

where \hbar is the reduced Planck constant and \mathbf{k} is the wave vector. In one dimension,

This scalar formula applies near a band extremum with approximately parabolic dispersion. In anisotropic crystals, effective mass is a tensor. Near a valence-band maximum, describing missing electrons as positively charged holes yields a convenient positive effective mass.

3. Drift, Diffusion and Semiconductor Conductivity

3.1 Multiple carrier populations

For electrons and holes in an isotropic semiconductor under low electric field,

Here n and p are electron and hole concentrations, while \mu_n and \mu_p are their positive mobilities. Contributions add because both carrier types produce conventional current along the applied field.

Mobility is not generally constant. It may decrease through phonon scattering at elevated temperature and through ionized-impurity scattering at high doping. Strong fields can also cause velocity saturation, invalidating v_d=\mu E.

Worked Example 2: Two-carrier conductivity

A silicon sample at 300 K has n=1.00\times10²²\ \mathrm{m^-3}, p=2.00\times10²⁰\ \mathrm{m^-3}, \mu_n=0.135\ \mathrm{m²V^-1s^-1} and \mu_p=0.048\ \mathrm{m²V^-1s^-1}. Then

Therefore,

The electron term dominates, but omitting holes would still be an approximation rather than an identity.

3.2 Diffusion

A carrier-concentration gradient produces net motion from high to low concentration. For particle concentration c, the drift–diffusion particle flux is

where \boldsymbol{\Gamma} is in m⁻²·s⁻¹ and D is diffusion coefficient in m²·s⁻¹. Its conservation equation is

Here G and R_c are volumetric generation and recombination rates in m⁻³·s⁻¹. Boundary conditions at contacts and surfaces are required to solve practical devices.

For nondegenerate carriers in thermal equilibrium, the Einstein relation is

It assumes Maxwell–Boltzmann carrier statistics and a consistent scattering model. At 300 K, k_BT/e=0.02585\ \mathrm{V}. Thus, for the electron mobility in Worked Example 2,

Degenerate semiconductors require a generalized Einstein relation involving the carrier chemical potential.

4. Temperature Dependence of Resistivity

4.1 Metals

Near a reference temperature T₀, many pure metals can be approximated by

where \alpha is the temperature coefficient in K⁻¹. This is a local linear approximation, not a universal law. Over wider ranges, electron–phonon scattering is nonlinear, especially at cryogenic temperatures.

Matthiessen’s rule is often written

where \rho_res represents temperature-independent scattering from impurities and defects, and \rho_ph represents phonon scattering. The rule is approximate because scattering mechanisms need not be independent.

The residual-resistivity ratio RRR=\rho(300\,\mathrm{K})/\rho(T_low) is used to assess metallic purity, provided the specified low temperature and measurement procedure are reported.

4.2 Semiconductors

In an intrinsic, nondegenerate semiconductor,

where N_C and N_V are effective densities of states. Conductivity commonly rises strongly with temperature because carrier generation can outweigh mobility reduction.

Doped semiconductors exhibit three broad regimes:

1. Freeze-out: dopants are incompletely ionized.
2. Extrinsic regime: dopant-generated carriers dominate.
3. Intrinsic regime: thermally generated electron–hole pairs dominate.

Consequently, a single constant temperature coefficient is usually inadequate for semiconductors.

5. Hall Effect

5.1 Physical mechanism and geometry

Consider current along +x and magnetic field along -z, into the page. Moving carriers experience the Lorentz force q_c\mathbf{v}\times\mathbf{B}, causing charge accumulation across the y direction. The resulting Hall electric field is therefore transverse to both current and magnetic field.

Figure: Hall voltage is measured between transverse contacts along y, perpendicular to current along x and magnetic field along z.

5.2 One-carrier derivation

At steady state, transverse electric and magnetic forces balance:

For one isotropic carrier type in the weak-field regime, the Hall coefficient is

Thus, electron conduction gives a negative Hall coefficient and hole conduction gives a positive one under the stated sign convention. For a rectangular plate of thickness t parallel to B,

The measured sign also depends on lead polarity and field direction, so experimental conventions must be documented.

Worked Example 3: Hall voltage

An n-type plate has n=5.00\times10²²\ \mathrm{m^-3}, thickness t=0.500\ \mathrm{mm}, current I=10.0\ \mathrm{mA} and perpendicular field magnitude B=0.500\ \mathrm{T}. For electrons,

Hence, for the stated reference orientation,

The magnitude is 1.25 mV. Reversing either current or magnetic field reverses the Hall voltage and helps remove offset voltages.

5.3 Two-carrier Hall coefficient

When both electrons and holes contribute, R_H=1/(nq_c) is invalid. In the low-field limit where \mu_nB\ll1 and \mu_pB\ll1,

The squared mobilities mean that a minority carrier with high mobility can substantially influence the Hall sign. Strong fields, anisotropic bands and anomalous Hall phenomena require tensor or quantum treatments.

Hall measurements are used to estimate carrier type, concentration and mobility. If one-carrier behavior is justified, \mu=|R_H|\sigma.

6. Thermal–Electrical Transport Relationships

Electrons can transport both charge and thermal energy. Fourier’s law is

where \mathbf{q}_th is heat flux in W·m⁻² and \kappa is thermal conductivity in W·m⁻¹·K⁻¹. The minus sign indicates heat flow toward lower temperature.

For a simple degenerate metal dominated by elastic scattering, the Wiedemann–Franz relation is

where \kappa_e is the electronic part of thermal conductivity and L is the Lorenz number. The Sommerfeld value is

For copper at 300 K with \sigma=5.95\times10⁷\ \mathrm{S\,m^-1},

The relation is most reliable for ordinary metals where electrons dominate heat conduction and scattering is nearly elastic. In semiconductors and insulators, phonons may dominate total thermal conductivity, so \kappa\neq L\sigma T. Correlated materials and nanostructures may also have non-Sommerfeld Lorenz numbers.

Thermoelectric performance is commonly described by

where S is Seebeck coefficient in V·K⁻¹ and \kappa_ph is lattice thermal conductivity. The numerator contains the power factor S²\sigma. Improving ZT is difficult because electrical and electronic thermal transport are coupled.

7. Applied Engineering Case: Elevated-Temperature Busbar Selection

Background

This is an illustrative case. A power-electronics manufacturer requires a 0.50 m busbar carrying 100 A. The available cross-sectional area is initially 20 mm², and the conductor may reach 100 °C. Copper and aluminum are being compared.

Problem

The engineer must estimate conductor losses, assess whether aluminum can match copper resistance, and consider implications not captured by room-temperature resistivity alone.

Analysis

Use the local linear temperature model from 20 °C to 100 °C. Take:

For copper,

With A=20\ \mathrm{mm²}=2.00\times10^-5\ \mathrm{m²},

and

For aluminum,

Therefore, R_Al=9.32\times10^-4\ \Omega and P_Al=9.32\ \mathrm{W} at the same geometry.

To match copper resistance at 100 °C,

The mass ratio for equal length and equal hot resistance is

Approach and Outcome

Aluminum can provide approximately equal hot resistance at about 51% of the copper mass, but it requires 69% more cross-sectional area. Copper is preferable where volume, joint size and low contact resistance dominate. Aluminum may be preferable where mass and material cost dominate.

Final selection must also account for oxide films, galvanic compatibility, creep, thermal expansion, connector qualification, short-circuit forces, fire safety and applicable electrical codes. A coupled electrothermal simulation should use temperature-dependent properties and validated convection and contact-resistance data.

Learning Points

• Material selection cannot be based on room-temperature conductivity alone.
• Geometry can compensate for bulk resistivity but changes mass and packaging.
• Contact behavior and long-term reliability may control performance.
• Electrical loss and temperature are coupled because resistivity changes with temperature.

8. Measurement Practice, Limitations and Contemporary Developments

Four-terminal sensing separates lead resistance from specimen resistance and is preferred for low-resistance materials. Van der Pauw measurements can determine sheet resistance and Hall coefficient for thin, homogeneous, simply connected samples with small perimeter contacts. Thickness uncertainty directly affects inferred bulk resistivity and carrier concentration.

Recommended practice includes current reversal, magnetic-field reversal, temperature stabilization, uncertainty reporting and verification of Ohmic contacts. Joule self-heating must be limited or included in the analysis.

Classical local transport becomes inadequate when device dimensions approach carrier mean free paths or phase-coherence lengths. Ballistic and mesoscopic systems are better described by transmission-based models such as Landauer transport. At high magnetic fields and low temperatures, Landau quantization produces quantum Hall behavior. Wide-band-gap semiconductors, two-dimensional materials and topological materials introduce strongly anisotropic, interfacial or protected transport channels.

Machine-learning models are increasingly used to predict mobility and discover thermoelectric materials. Such models should preserve dimensional consistency, identify training-domain limits and be validated experimentally. Proprietary datasets, measurement bias and irreproducible processing histories are governance concerns.

Transport characterization also carries ethical and regulatory responsibilities. Engineers should report conditions and uncertainty rather than selectively presenting favorable values. Material choices should consider mining impacts, recyclability, hazardous substances, conflict-mineral requirements and lifecycle energy use.

Chapter Summary

Electrical conduction results from the directed motion of charge carriers under fields and concentration gradients. The Drude relaxation-time model leads to mobility, conductivity and resistivity, while quantum band theory explains why metals, semiconductors and insulators have different carrier populations. Fermi–Dirac statistics govern state occupancy, and band curvature determines effective mass.

Drift and diffusion jointly determine semiconductor transport, with the Einstein relation connecting mobility and diffusion for nondegenerate carriers. Metallic resistivity commonly rises with temperature because of phonon scattering, whereas semiconductor conductivity may rise because thermal generation increases carrier concentration.

The Hall effect produces a voltage transverse to current and magnetic field. Its simple inverse-density relation applies only to a justified one-carrier model; multicarrier systems require mobility-weighted analysis. The Wiedemann–Franz law links electronic thermal and electrical conductivity under restricted metallic conditions. Reliable engineering decisions must combine these models with geometry, temperature, contacts, uncertainty, reliability and lifecycle considerations.

Key Terms

Band gap: Energy interval containing no allowed electronic states in an ideal crystal.

Carrier concentration: Number of mobile carriers per unit volume.

Conductivity: Proportionality between current density and electric field.

Diffusion: Carrier transport driven by a concentration gradient.

Drift velocity: Average carrier velocity produced by an applied field.

Effective mass: Band-structure parameter describing carrier acceleration in a crystal.

Einstein relation: Equilibrium relationship between diffusion coefficient and mobility.

Fermi energy: Highest occupied energy at absolute zero.

Fermi–Dirac distribution: Occupancy probability for fermionic energy states.

Hall coefficient: Ratio of transverse Hall field to current density times magnetic field.

Hole: Positively charged quasiparticle representing a missing valence-band electron.

Lorenz number: Ratio \kappa_e/(\sigma T) in the Wiedemann–Franz relation.

Matthiessen’s rule: Approximate addition of resistivity contributions from independent scattering mechanisms.

Mobility: Drift-speed magnitude per unit electric-field magnitude in the linear regime.

Relaxation time: Characteristic time over which carrier momentum decays through scattering.

Resistivity: Reciprocal of conductivity for an isotropic material.

Seebeck coefficient: Thermoelectric voltage developed per unit temperature difference under open circuit.

Thermal conductivity: Proportionality between heat flux and negative temperature gradient.

Review Questions

Short Answer Questions

1. Why does conventional electron current point opposite to electron drift velocity?
2. State the assumptions used to derive \sigma=n e\mu.
3. What physical information is represented by effective mass?
4. Why does current reversal improve a Hall-voltage measurement?
5. Under what conditions is the Wiedemann–Franz law most reliable?

Long Answer Questions

1. Derive the Drude conductivity from the relaxation-time equation and discuss the model’s limitations.
2. Explain how Fermi statistics and energy bands distinguish metals, semiconductors and insulators.
3. Develop the drift–diffusion flux equation and explain the role of generation, recombination and boundary conditions.
4. Compare temperature-dependent conduction in pure metals, doped semiconductors and intrinsic semiconductors.
5. Derive the one-carrier Hall coefficient and explain why the result changes in a two-carrier material.

Analytical / Application Questions

1. A wire’s current density, carrier concentration and resistivity are known. Determine its drift velocity, mobility and relaxation time, clearly stating any effective-mass assumption.
2. Design a Hall experiment that identifies carrier sign while removing contact-misalignment voltage. Specify current, field and voltage directions.
3. Compare two candidate interconnect materials at operating temperature using resistive loss, mass, required area, contact behavior and thermal reliability.

Multiple Choice Questions with Answers

Question 1

For a homogeneous one-carrier conductor in the low-field regime, which expression gives conductivity?

A. \sigma=n|q_c|\mu

B. \sigma=\mu/(n|q_c|)

C. \sigma=n\mu/|q_c|

D. \sigma=|q_c|/(n\mu)

Correct Answer: A. \sigma=n|q_c|\mu

Question 2

An electron concentration is 2.0\times10²²\ \mathrm{m^-3} and electron mobility is 0.10\ \mathrm{m²V^-1s^-1}. Neglecting holes, what is the conductivity?

A. 32.0 S·m⁻¹

B. 320 S·m⁻¹

C. 3.20 × 10³ S·m⁻¹

D. 1.25 × 10⁻³ S·m⁻¹

Correct Answer: B. 320 S·m⁻¹

Question 3

At approximately zero temperature, how are electron states occupied in a metal?

A. All states in every band are occupied

B. Only states above the Fermi level are occupied

C. Allowed states below the Fermi level are occupied, while those above are empty

D. Every state has an occupancy probability of one-half

Correct Answer: C. Allowed states below the Fermi level are occupied, while those above are empty

Question 4

The ordinary Einstein relation D/\mu=k_BT/e applies directly to which situation?

A. Nondegenerate carriers in thermal equilibrium

B. Any conductor at any electric field

C. Only superconductors below their critical temperature

D. Degenerate carriers without modification

Correct Answer: A. Nondegenerate carriers in thermal equilibrium

Question 5

In a Hall plate, current is along x and magnetic field is along z. Along which direction is the Hall voltage measured?

A. Along x

B. Along y

C. Along z

D. Parallel to both x and z

Correct Answer: B. Along y

Question 6

For a one-carrier n-type material under the chapter’s sign convention, the ordinary Hall coefficient is:

A. Positive and equal to 1/(ne)

B. Zero because electrons have no transverse force

C. Negative and equal to -1/(ne)

D. Equal to resistivity divided by thickness

Correct Answer: C. Negative and equal to -1/(ne)

Question 7

An n-type Hall plate has n=5.00\times10²²\ \mathrm{m^-3}, I=10.0\ \mathrm{mA}, B=0.500\ \mathrm{T} and t=0.500\ \mathrm{mm}. What is the Hall-voltage magnitude in the one-carrier model?

A. 0.125 μV

B. 12.5 μV

C. 0.125 mV

D. 1.25 mV

Correct Answer: D. 1.25 mV

Question 8

Why can a high-mobility minority carrier strongly affect a two-carrier Hall coefficient?

A. The Hall numerator contains mobility squared

B. Minority carriers always outnumber majority carriers

C. The Hall effect is independent of concentration

D. Mobility cancels from the two-carrier expression

Correct Answer: A. The Hall numerator contains mobility squared

Question 9

Using L₀=2.44\times10^-8\ \mathrm{W\,\Omega\,K^-2}, \sigma=5.95\times10⁷\ \mathrm{S\,m^-1} and T=300\ \mathrm{K}, the Wiedemann–Franz estimate of electronic thermal conductivity is approximately:

A. 4.36 W·m⁻¹·K⁻¹

B. 43.6 W·m⁻¹·K⁻¹

C. 436 W·m⁻¹·K⁻¹

D. 4360 W·m⁻¹·K⁻¹

Correct Answer: C. 436 W·m⁻¹·K⁻¹

Question 10

Which measurement method best removes lead resistance when determining a very small specimen resistance?

A. Two-terminal sensing with long leads

B. Four-terminal sensing

C. Open-circuit Hall sensing

D. A single voltage probe without current injection

Correct Answer: B. Four-terminal sensing