2. Carrier Transport and Nonequilibrium Phenomena

Introduction and Context

The operation of every semiconductor device depends on how electrons and holes move, appear, disappear, and respond to electric fields, concentration gradients, temperature, and material interfaces. Electrostatic analysis alone cannot explain diode current, transistor action, photodetection, solar-cell operation, or the speed limits of nanoscale devices. These behaviors require a transport description.

In thermal equilibrium, carrier populations are stationary in a statistical sense: generation and recombination balance, and the Fermi level is spatially constant. Practical devices, however, are usually biased, illuminated, injected, heated, or switched. They then operate under nonequilibrium conditions. Carrier concentrations vary with position and time, currents flow, and the equilibrium Fermi-level description must be replaced or supplemented by quasi-Fermi levels.

This chapter develops the classical drift-diffusion framework and then examines its limits. It introduces mobility and conductivity, generation-recombination mechanisms, continuity equations, high-field transport, tunneling, and ballistic motion. Together, these ideas form the foundation for interpreting semiconductor device characteristics and for selecting an appropriate transport model.

Learning Objectives

After studying this chapter, the reader should be able to:

1. Distinguish drift and diffusion currents and derive their semiconductor current-density relations.

2. Relate mobility, conductivity, carrier concentration, scattering, and temperature.

3. Formulate and interpret carrier continuity equations.

4. Explain generation-recombination mechanisms and carrier lifetimes.

5. Use quasi-Fermi levels to describe nonequilibrium carrier populations.

6. Identify the physical consequences of high electric fields, velocity saturation, tunneling, and ballistic transport.

7. Select suitable transport models for bulk, short-channel, and nanoscale devices.

8. Analyze practical device situations involving transient, optical, and high-field transport.

1. Equilibrium, Nonequilibrium, and the Transport View

At equilibrium, the electron and hole concentrations are related by

\[

np=n_i^2,

\]

for a nondegenerate semiconductor, where \(n_i\) is the intrinsic carrier concentration. The equilibrium Fermi level \(E_F\) is constant throughout a homogeneous system. Individual carriers remain in motion, but their average current is zero because opposing microscopic motions cancel.

A departure from equilibrium occurs when an external voltage, optical excitation, carrier injection, temperature gradient, or time-dependent stimulus disturbs this balance. The local carrier concentrations may then differ from their equilibrium values, and the net current density is nonzero. The transport problem asks three connected questions: how carriers move, how their numbers change, and how electrical fields and carrier distributions influence one another.

The classical semiconductor model assumes that carriers undergo many scattering events over the distance of interest. Their motion can then be represented by local quantities such as mobility, diffusion coefficient, and carrier concentration. This is the drift-diffusion regime. When device dimensions become comparable with the mean free path, a particle may cross a region with few collisions; ballistic or quasi-ballistic models are then required.

2. Drift: Motion Driven by an Electric Field

An electric field \(\mathbf E\) exerts a force on a carrier. Electrons experience \(\mathbf F=-q\mathbf E\), whereas holes experience \(\mathbf F=+q\mathbf E\). Although thermal velocities are large and random, scattering prevents indefinite acceleration. The field produces a small average velocity called the drift velocity.

For moderate fields,

\[

\mathbf v_{n,dr}=-\mu_n\mathbf E, \qquad \mathbf v_{p,dr}=\mu_p\mathbf E,

\]

where \(\mu_n\) and \(\mu_p\) are electron and hole mobilities. Conventional current is in the direction of the electric field for both carrier types. Therefore,

\[

\mathbf J_{n,dr}=q n\mu_n\mathbf E,

\qquad

\mathbf J_{p,dr}=q p\mu_p\mathbf E.

\]

The total drift current is

\[

\mathbf J_{dr}=q(n\mu_n+p\mu_p)\mathbf E.

\]

Mobility is not a fundamental constant of a material under all conditions. It reflects momentum loss through scattering by lattice vibrations, ionized impurities, neutral defects, alloy disorder, interfaces, and other carriers. At low impurity concentration, lattice scattering often limits mobility; at low temperature, ionized-impurity scattering may dominate. In a real device, the effective mobility is determined by the combined scattering processes.

Conductivity and Resistivity

For a uniform semiconductor, the conductivity is

\[

\sigma=q(n\mu_n+p\mu_p),

\]

and the resistivity is \(\rho=1/\sigma\). In an n-type material, if \(n\gg p\), then \(\sigma\approx qn\mu_n\). In a p-type material, \(\sigma\approx qp\mu_p\). Both carriers must nevertheless be included in compensated, intrinsic, illuminated, or strongly injected material.

Conductivity is a local property. In a nonuniform device, carrier concentration and mobility can vary with position, so the current-density relation must be applied locally rather than by assigning one resistance to the entire structure.

3. Diffusion: Transport Down a Concentration Gradient

Random thermal motion alone produces no net current in a uniform material. If carrier concentration varies spatially, however, more carriers cross from a high-concentration region toward a low-concentration region than in the reverse direction. This net motion is diffusion.

For one-dimensional transport, the diffusion current densities are

\[

J_{n,diff}=qD_n\frac{dn}{dx},

\qquad

J_{p,diff}=-qD_p\frac{dp}{dx},

\]

where \(D_n\) and \(D_p\) are diffusion coefficients. The opposite signs arise because electron motion and conventional electron current have opposite directions.

Combining drift and diffusion gives the drift-diffusion equations:

\[

J_n=qn\mu_nE+qD_n\frac{dn}{dx},

\]

\[

J_p=qp\mu_pE-qD_p\frac{dp}{dx}.

\]

In vector form, replace the spatial derivatives by gradients. These equations are central to semiconductor-device analysis, but they presume local thermal equilibrium and fields that are not sufficiently strong to produce velocity saturation or significant hot-carrier effects.

Einstein Relation

For nondegenerate carriers in thermal equilibrium with the lattice,

\[

\frac{D_n}{\mu_n}=\frac{D_p}{\mu_p}=\frac{kT}{q}=V_T,

\]

where \(V_T\) is the thermal voltage. At 300 K, \(V_T\) is approximately 25.9 mV. The relation links random thermal motion to field-driven motion and is a consequence of detailed balance. It must be modified in degenerate semiconductors or under strongly nonequilibrium conditions.

Equilibrium Cancellation

Consider a graded semiconductor with no externally applied current. A concentration gradient creates diffusion current, while the associated electrostatic field creates drift current. At equilibrium these terms cancel. For electrons, setting \(J_n=0\) and using the nondegenerate relation gives

\[

E=-\frac{kT}{q}\frac{1}{n}\frac{dn}{dx}.

\]

This built-in field is not an externally imposed voltage; it is a self-consistent consequence of charge redistribution and electrostatics. The same principle explains the built-in potential of a p-n junction.

4. Generation, Recombination, and Carrier Lifetime

Carrier transport cannot be understood without accounting for changes in carrier number. Generation creates electron-hole pairs, while recombination removes them. The net rate is commonly written as \(G-R\), with units of carriers per unit volume per unit time.

Important generation mechanisms include thermal generation, optical absorption, impact ionization, and carrier injection from contacts. Recombination mechanisms include radiative recombination, Auger recombination, and Shockley-Read-Hall (SRH) recombination through defects or traps.

Shockley-Read-Hall Recombination

For a single trap level, the SRH net recombination rate is

\[

U_{SRH}=\frac{np-n_i^2}{\tau_p(n+n_1)+\tau_n(p+p_1)},

\]

where \(\tau_n\) and \(\tau_p\) are capture-related lifetimes, and \(n_1\) and \(p_1\) depend on trap energy. SRH recombination is especially important in depletion regions and materials containing defects. It can reduce minority-carrier lifetime and strongly affect diode leakage, photodiode response, and solar-cell efficiency.

Radiative recombination is important in direct-band-gap materials used for light emitters and lasers. Auger recombination involves three carriers and becomes significant at high carrier concentrations, limiting the performance of heavily injected devices and some semiconductor lasers.

Lifetime and Diffusion Length

The minority-carrier lifetime describes the characteristic time over which an excess population decays. In a simple low-level-injection approximation, excess concentration decays approximately exponentially:

\[

\Delta n(t)=\Delta n(0)e^{-t/\tau}.

\]

The diffusion length is

\[

L=\sqrt{D\tau}.

\]

It represents the approximate distance a minority carrier diffuses before recombining. Lifetime and diffusion length are therefore essential design parameters in bipolar transistors, photodetectors, and solar cells.

5. Continuity Equations: Conservation Applied to Carriers

The continuity equation expresses conservation of particles. For electrons,

\[

\frac{\partial n}{\partial t}=\frac{1}{q}\nabla\cdot\mathbf J_n+G_n-R_n,

\]

and for holes,

\[

\frac{\partial p}{\partial t}=-\frac{1}{q}\nabla\cdot\mathbf J_p+G_p-R_p.

\]

The signs differ because electron current is opposite to electron particle flux, while hole current follows hole motion. In a one-dimensional steady state with no generation or recombination, the appropriate current density is constant with position. If recombination is present, current changes spatially because carriers are being removed.

The continuity equations must be solved together with Poisson's equation,

\[

\nabla\cdot(\epsilon\nabla\phi)=-\rho,

\]

where \(\rho=q(p-n+N_D^+-N_A^-+\rho_{trap}/q)\), and with the drift-diffusion equations. This coupled set is the basic drift-diffusion device model. It explains why carrier transport and electrostatics cannot generally be treated independently.

Boundary Conditions and Contacts

A mathematically correct transport model requires physically meaningful boundary conditions. Ohmic contacts may impose carrier concentrations close to equilibrium values and allow current flow. Schottky or insulating contacts impose different injection and transmission conditions. Surface recombination velocity is often used to describe carrier loss at a semiconductor surface:

\[

J_{surface}=qS\Delta n.

\]

Incorrect contact assumptions can produce apparently precise but physically invalid simulation results.

6. Quasi-Fermi Levels and Nonequilibrium Populations

Under nonequilibrium conditions, a single Fermi level cannot generally describe both electrons and holes. Instead, separate quasi-Fermi levels are introduced. For a nondegenerate semiconductor,

\[

n=N_C\exp\left[-\frac{E_C-E_{Fn}}{kT}\right],

\]

\[

p=N_V\exp\left[-\frac{E_{Fp}-E_V}{kT}\right].

\]

At equilibrium, \(E_{Fn}=E_{Fp}=E_F\). Under illumination or electrical injection, the quasi-Fermi levels split. The separation \(E_{Fn}-E_{Fp}\) is directly related to the carrier population and, in suitable devices, to the maximum attainable voltage.

In a spatially varying system, gradients in quasi-Fermi levels are closely related to current. A constant quasi-Fermi level corresponds to zero current under the relevant assumptions. This makes quasi-Fermi-level diagrams particularly useful for interpreting p-n junctions, LEDs, solar cells, heterostructures, and contacts.

Quasi-Fermi levels are not additional energy bands and do not imply that the system is in global equilibrium. They are effective statistical potentials that describe separate carrier populations when each population is sufficiently close to local thermal equilibrium.

7. High-Field Transport

The low-field relation \(v_d=\mu E\) fails when the field becomes large. Carriers gain energy between scattering events and may heat the carrier distribution above the lattice temperature. Mobility becomes field-dependent, and drift velocity may approach a limiting saturation velocity. A simple engineering model is

\[

v_d=\frac{\mu E}{[1+(\mu E/v_{sat})^\beta]^{1/\beta}},

\]

where \(v_{sat}\) and \(\beta\) are empirical parameters. The exact behavior depends on material, crystal direction, temperature, and carrier density.

High-field effects include velocity saturation, intervalley transfer, impact ionization, hot-carrier injection, and, in some materials, negative differential mobility. Velocity saturation limits the current and switching speed of short-channel field-effect transistors. Impact ionization can cause avalanche multiplication but also device breakdown and reliability degradation.

A simulation that uses constant mobility at high field may substantially overestimate current and underestimate power dissipation. Energy-balance or hydrodynamic models are more appropriate when carrier temperature and energy relaxation are important.

8. Tunneling and Ballistic Effects

Tunneling

Tunneling is a quantum-mechanical process in which a carrier passes through a classically forbidden energy barrier. Its probability depends strongly on barrier width, height, effective mass, and electric field. In a sufficiently strong field, the barrier becomes narrow enough for appreciable band-to-band tunneling, as in tunnel diodes and some breakdown mechanisms.

Trap-assisted tunneling proceeds through defect states and can dominate leakage in imperfect materials or thin dielectrics. Direct tunneling is important in ultrathin gate oxides and semiconductor barriers. Because tunneling depends exponentially on barrier properties, small changes in thickness or field can produce large changes in current.

Tunneling models must distinguish between direct and indirect transitions, conserve momentum when appropriate, and include the actual band structure. A classical drift-diffusion model alone cannot describe tunneling accurately.

Ballistic and Quasi-Ballistic Transport

The mean free path is the average distance traveled between significant scattering events. If the device length is much larger than this distance, transport is diffusive. If it is comparable to or smaller than the mean free path, transport becomes quasi-ballistic or ballistic.

In a ballistic channel, current is determined by carrier injection from contacts and the transmission of available modes rather than by a local conductivity alone. The Landauer viewpoint expresses conductance in terms of transmission probability and quantum modes. Contact resistance and injection velocity may then dominate the measured device resistance.

Ballistic transport does not mean that scattering is absent everywhere. Scattering may occur in reservoirs, at contacts, or at interfaces. It means that the active transport region does not establish the local equilibrium assumed by drift-diffusion theory. Quantum confinement, source-drain injection, and mode occupation must be considered in nanoscale devices.

9. Choosing a Transport Model

Model selection should follow the physical regime rather than the convenience of a familiar equation. Drift-diffusion is usually suitable when carriers are locally thermalized, fields are moderate, and dimensions exceed the mean free path. A mobility model with velocity saturation is useful for many conventional device simulations.

Energy-balance or hydrodynamic models are appropriate when carrier heating, nonlocal energy transport, or hot-carrier effects influence results. Monte Carlo methods resolve stochastic carrier motion and scattering in high-field or nanoscale structures but are computationally intensive. Quantum-corrected drift-diffusion, Schrödinger-Poisson, NEGF, or related quantum-transport methods are required when tunneling, confinement, interference, or phase-coherent effects are central.

The choice should be validated by comparing predicted currents, lifetimes, temperature dependence, and length-scale effects with measurements. Increasing model complexity without reliable material parameters does not necessarily increase predictive accuracy.

Applications in Devices and Industry

Carrier transport concepts appear throughout semiconductor engineering. The forward current of a p-n diode results from minority-carrier injection, diffusion, recombination, and contact transport. In a bipolar transistor, emitter injection and base transport depend on minority-carrier diffusion length and recombination. In a MOSFET, channel drift, mobility degradation, velocity saturation, contact resistance, and short-channel effects determine the on-current and switching speed.

Photodiodes and solar cells rely on optical generation followed by drift and diffusion toward selective contacts. Their performance is governed by absorption depth, diffusion length, depletion-region field, surface recombination, and collection probability. LEDs and laser diodes depend on carrier injection, quasi-Fermi-level separation, radiative recombination, and nonradiative loss.

In manufacturing, defects, contamination, interface roughness, and process variation modify mobility, lifetime, trap density, and tunneling leakage. Device engineers therefore use transport analysis not only to predict nominal performance but also to diagnose yield loss, aging, breakdown, and reliability failures.

Illustrative Case Study: Photodiode Response Limited by Transport

Background

Consider a silicon p-i-n photodiode used in an optical receiver. Light is absorbed mainly in the intrinsic region, generating electron-hole pairs. The device is reverse biased so that the electric field sweeps carriers toward the contacts.

Situation and Problem

The measured responsivity is acceptable, but the bandwidth is lower than expected. Two possible causes are excessive carrier transit time and recombination before collection. A third possibility is excessive junction capacitance, which is electrical rather than transport-limited.

Analysis

The generated carriers in the depleted intrinsic region experience drift. If the field is approximately uniform, their transit time is roughly the depletion width divided by the drift velocity. Increasing reverse bias initially increases velocity and reduces transit time. At sufficiently high field, velocity saturation limits further improvement. Carriers generated outside the depletion region must diffuse, so their collection depends on diffusion length \(L=\sqrt{D\tau}\). A low lifetime caused by defects reduces collection efficiency and produces a slow diffusion tail in the impulse response.

Approach and Solution

The design team measures bandwidth as a function of reverse bias and temperature, then compares the results with a transport model containing depletion width, mobility, saturation velocity, lifetime, and capacitance. If bandwidth improves with bias until a plateau, transit-time saturation is implicated. If the response contains a long tail and is strongly temperature- or material-quality-dependent, diffusion and recombination are implicated. The design may be improved by reducing absorption in neutral regions, increasing material lifetime, optimizing intrinsic-layer thickness, and reducing parasitic capacitance.

Outcome

The case illustrates that simply increasing electric field cannot indefinitely improve speed. Once velocity saturation is reached, material quality, geometry, contact design, and capacitance become decisive.

Learning Points

Transport analysis separates drift-limited, diffusion-limited, recombination-limited, and circuit-limited behavior. It also demonstrates why a single measured time constant should not automatically be attributed to one physical mechanism.

Limitations, Risks, and Common Errors

Several errors recur in transport analysis. First, mobility is sometimes treated as constant over all fields, temperatures, and carrier densities. Second, the signs of electron diffusion current and particle motion are confused. Third, equilibrium relations such as \(np=n_i^2\) are applied to strongly injected or illuminated devices without qualification. Fourth, generation-recombination is omitted from regions where minority-carrier lifetime controls performance.

Other errors include using an inappropriate contact boundary condition, neglecting surface recombination, interpreting a quasi-Fermi level as a true equilibrium Fermi level, and applying drift-diffusion to a ballistic structure. Tunneling leakage may also be incorrectly modeled as thermionic emission. Numerical solutions can suffer from convergence problems when abrupt doping, thin barriers, or strongly nonlinear recombination are present.

Professional analysis should state assumptions, identify parameter sources, perform dimensional and limiting checks, and compare predictions with independent measurements. Reliability decisions must consider hot-carrier damage, oxide charging, electromigration, avalanche stress, and thermal feedback.

Current Developments and Best Practices

Modern device research increasingly combines multiscale models. Atomistic calculations provide band structure and scattering information; Monte Carlo or quantum transport describes nanoscale motion; compact models translate the results into circuit simulation. Machine-learning methods are also used to calibrate mobility, lifetime, and process-dependent parameters, but predictions remain dependent on the quality and representativeness of the training data.

Best practice is to begin with conservation laws and the simplest valid model, add physical effects only when justified by characteristic length, time, field, and energy scales, and verify each addition against experiment. Particular attention should be given to contacts, interfaces, traps, temperature rise, and uncertainty in material parameters. These practices improve both physical insight and engineering reliability.

Chapter Summary

Carrier transport in semiconductors arises primarily from drift in an electric field and diffusion caused by concentration gradients. Their combined current relations, together with the Einstein relation, provide the drift-diffusion framework. Mobility determines low-field response but depends on scattering, temperature, impurities, and interfaces. Conductivity follows from the contributions of electrons and holes.

Generation and recombination change carrier populations, while continuity equations impose conservation. These equations must be solved with electrostatics and appropriate contact conditions. Under nonequilibrium, separate quasi-Fermi levels describe electron and hole populations. At high fields, velocity saturation, carrier heating, impact ionization, and other non-ohmic effects require refined transport models. Tunneling requires quantum mechanics, and ballistic devices require contact- and mode-based descriptions rather than local conductivity alone.

The practical lesson is that transport models must be matched to physical scale and operating conditions. Accurate device analysis depends on recognizing which mechanism—drift, diffusion, recombination, tunneling, ballistic injection, or parasitic capacitance—controls the observed behavior.

Key Terms

Drift: Carrier motion and current produced by an electric field.

Diffusion: Net carrier transport caused by a spatial concentration gradient.

Mobility: Proportionality between low-field drift velocity and electric field.

Conductivity: Local ability of a semiconductor to carry electrical current.

Diffusion coefficient: Parameter describing carrier spreading due to random thermal motion.

Einstein relation: Relation \(D/\mu=kT/q\) for nondegenerate carriers in local equilibrium.

Generation: Creation of electron-hole pairs.

Recombination: Removal of an electron-hole pair through a physical process.

Carrier lifetime: Characteristic time for excess carriers to decay.

Diffusion length: Approximate distance a carrier diffuses during its lifetime.

Continuity equation: Conservation equation describing local carrier-number change.

Quasi-Fermi level: Effective chemical potential describing an individual nonequilibrium carrier population.

Shockley-Read-Hall recombination: Trap-assisted generation-recombination through defect states.

Velocity saturation: Limiting of drift velocity at high electric field.

Hot carrier: Carrier whose energy distribution is substantially above lattice equilibrium.

Impact ionization: Carrier-energy process that creates an additional electron-hole pair.

Tunneling: Quantum transmission through a classically forbidden barrier.

Mean free path: Average distance between significant scattering events.

Ballistic transport: Transport through a region with negligible internal scattering.

Drift-diffusion model: Local transport model combining electric-field and concentration-gradient effects.

Review Questions

Short Answer Questions

1. Why can a semiconductor have large microscopic carrier velocities but zero net current at equilibrium?

2. State the drift-diffusion current equations for electrons and holes.

3. What physical processes determine carrier mobility?

4. Define carrier lifetime and diffusion length.

5. Why are two quasi-Fermi levels needed under nonequilibrium conditions?

Long Answer Questions

1. Derive the conductivity of a semiconductor containing both electrons and holes, and discuss limiting cases for intrinsic and doped material.

2. Explain the continuity equations for electrons and holes, including the significance of each term and the difference in signs.

3. Describe SRH, radiative, and Auger recombination and identify situations in which each becomes important.

4. Explain the breakdown of low-field transport at high electric fields and discuss suitable advanced models.

5. Compare drift-diffusion, hydrodynamic, Monte Carlo, and quantum-transport approaches in terms of assumptions and applications.

Analytical / Application Questions

1. A photodiode has a carrier lifetime of 10 ns and diffusion coefficient of 25 cm²/s. Estimate its diffusion length and discuss whether a neutral-region thickness of 20 μm is likely to permit efficient diffusion collection.

2. A semiconductor region has \(n=10^{16}\) cm⁻³, \(p=10^{15}\) cm⁻³, \(\mu_n=1200\) cm²/V·s, and \(\mu_p=400\) cm²/V·s. Calculate its conductivity and identify the dominant carrier contribution.

3. A short-channel transistor shows little increase in current when the electric field is increased. Determine which transport mechanism may be responsible and propose two experiments or simulations to distinguish velocity saturation from contact resistance.

22. Multiple Choice Questions with Answers

Question 1

Which expression gives the total low-field drift current density in a semiconductor?

A. q(nμn − pμp)E

B. q(nμn + pμp)E

C. q(nDn + pDp)E

D. q(n + p)E/(μn + μp)

Correct Answer: B. q(nμn + pμp)E

Question 2

For a nondegenerate semiconductor in local thermal equilibrium, the Einstein relation is

A. D/μ = q/kT

B. Dμ = kT/q

C. D/μ = kT/q

D. D/μ = T/qk

Correct Answer: C. D/μ = kT/q

Question 3

In a uniformly illuminated semiconductor under open-circuit nonequilibrium conditions, which statement is generally correct?

A. The electron and hole quasi-Fermi levels may split

B. The equilibrium relation np = ni² must always hold

C. Both quasi-Fermi levels disappear

D. The lattice temperature must become zero

Correct Answer: A. The electron and hole quasi-Fermi levels may split

Question 4

The diffusion length of a minority carrier is primarily given by

A. Dτ

B. D/τ

C. √(Dτ)

D. √(τ/D)

Correct Answer: C. √(Dτ)

Question 5

Which mechanism is especially associated with recombination through defect or trap states?

A. Radiative recombination

B. Shockley-Read-Hall recombination

C. Elastic scattering

D. Direct tunneling

Correct Answer: B. Shockley-Read-Hall recombination

Question 6

At sufficiently high electric field, a major reason that drift current ceases to increase linearly with field is

A. Carrier charge becomes zero

B. Carrier velocity approaches saturation

C. The diffusion coefficient becomes infinite

D. The band gap vanishes in every semiconductor

Correct Answer: B. Carrier velocity approaches saturation

Question 7

A device whose active length is shorter than the carrier mean free path is most appropriately described initially as

A. Purely diffusive

B. Ballistic or quasi-ballistic

C. In thermal equilibrium only

D. A recombination-only system

Correct Answer: B. Ballistic or quasi-ballistic

Question 8

Which phenomenon cannot generally be described accurately by a classical drift-diffusion model alone?

A. Moderate-field ohmic conduction

B. Low-field diffusion

C. Quantum tunneling through a thin barrier

D. Conductivity in a uniform bulk sample

Correct Answer: C. Quantum tunneling through a thin barrier

Question 9

In a one-dimensional steady state with no generation or recombination, the carrier current density is

A. Constant with position

B. Always zero

C. Exponential with position

D. Equal to the carrier concentration

Correct Answer: A. Constant with position

Question 10

For an n-type semiconductor in which n is much greater than p, the conductivity is approximately

A. q p μp

B. q n μn

C. q(n + p)/(μn + μp)

D. q ni² μn μp

Correct Answer: B. q n μn