Semiconductor Physics and Device Foundations

Introduction

Semiconductors underpin rectifiers, transistors, integrated circuits, light-emitting diodes (LEDs), photovoltaic cells and many sensors. Their usefulness arises because carrier populations and transport can be controlled through doping, electric fields, illumination, temperature and interfaces.

This chapter connects atomic-scale band structure with engineering device behavior. Unless stated otherwise, numerical silicon examples assume thermal equilibrium at 300 K, where the thermal voltage V_T = kT/q ≈ 0.02585\ \text{V}.

Learning Objectives

After studying this chapter, students should be able to:

• distinguish conductors, semiconductors and insulators using energy bands;
• calculate intrinsic and doped carrier concentrations;
• analyze drift, diffusion and semiconductor conductivity;
• apply continuity and generation–recombination models quantitatively;
• interpret Hall-effect measurements;
• derive the electrostatics and current–voltage behavior of p–n junctions;
• compare rectifying and ohmic metal–semiconductor contacts; and
• explain the physical foundations of diodes, transistors, LEDs, solar cells and sensors.

1. Energy Bands and Charge Carriers

1.1 Band formation

When many atoms form a crystal, discrete atomic energy levels split into closely spaced allowed bands. The highest substantially occupied band at absolute zero is the valence band, while the next higher allowed band is the conduction band. The forbidden interval between their edges is the band gap:

Here E_C and E_V are the conduction- and valence-band-edge energies, respectively, and E_g is measured in electronvolts (eV).

Silicon has an indirect band gap of approximately 1.12 eV at 300 K. Gallium arsenide has a direct band gap of about 1.42 eV, making radiative electron–hole recombination more efficient.

1.2 Electrons and holes

An electron promoted into the conduction band leaves an unoccupied valence-band state. This vacancy behaves as a positively charged quasiparticle called a hole. Conduction-band electrons carry charge -q, while holes carry +q, where q = 1.602 × 10^-19\ \text{C}.

The Fermi–Dirac occupation probability is

Here E_F is the Fermi level, k is Boltzmann’s constant and T is absolute temperature. For a nondegenerate semiconductor—one in which the Fermi level lies several kT from a band edge—the Maxwell–Boltzmann approximation gives

The electron concentration n, hole concentration p, and effective densities of states N_C and N_V have units of cm⁻³ when a consistent centimetre-based system is used.

2. Intrinsic and Doped Semiconductors

2.1 Intrinsic material

An ideal intrinsic semiconductor contains no intentional impurities. Thermal generation creates electrons and holes in pairs, so

Combining the nondegenerate carrier equations gives

Thus intrinsic concentration rises strongly with temperature and falls as band gap increases. For silicon at 300 K, a commonly used value is n_i ≈ 1.0 × 10¹⁰\ \text{cm}^-3, although the exact value depends on the material parameters adopted.

At equilibrium, nondegenerate material obeys the mass-action law:

This relation is an equilibrium result and should not be applied indiscriminately to a semiconductor under nonequilibrium carrier injection.

2.2 Donors, acceptors and charge neutrality

Pentavalent dopants such as phosphorus donate electrons to silicon and produce n-type material. Trivalent dopants such as boron accept electrons, leaving holes and producing p-type material.

General charge neutrality requires

Here N_D^+ and N_A^- are ionized donor and acceptor concentrations. If dopants are fully ionized and the semiconductor is nondegenerate,

Together with np=n_i², this produces, for net donor doping N_D>N_A,

For net acceptor doping, the analogous expression is obtained by interchanging electrons and holes. The common approximations n≈N_D-N_A or p≈N_A-N_D require complete ionization and net doping much greater than n_i. They fail during dopant freeze-out, in nearly compensated material or under degenerate doping.

Worked Example 1: Compensated silicon

Silicon at 300 K has N_D=1.0×10¹⁶\ \text{cm}^-3, N_A=2.0×10¹⁵\ \text{cm}^-3 and n_i=1.0×10¹⁰\ \text{cm}^-3.

The net donor concentration is 8.0×10¹⁵\ \text{cm}^-3. Because this is much greater than n_i,

Electrons are the majority carriers and holes are the minority carriers.

3. Carrier Transport

3.1 Drift

An electric field causes an average carrier drift velocity. In the low-field regime,

Here \mu_n and \mu_p are electron and hole mobilities in cm² V⁻¹ s⁻¹, and E is electric field in V cm⁻¹. The signs describe particle velocities. Both carrier types contribute conventional current in the field direction.

The drift current densities are

Therefore,

Conductivity \sigma is in S cm⁻¹ and resistivity is \rho=1/\sigma in Ω·cm when centimetre-based units are used. Mobility generally decreases as lattice scattering increases with temperature or as heavy doping increases ionized-impurity scattering. At high fields, velocity saturation invalidates the linear mobility model.

Worked Example 2: Resistivity

For n-type silicon with n=1.0×10¹⁶\ \text{cm}^-3, negligible hole conduction and \mu_n=1350\ \text{cm}²\text{V}^-1\text{s}^-1,

Hence

3.2 Diffusion

Carriers also move from regions of high concentration toward regions of low concentration. In one dimension,

The opposite signs arise from the opposite carrier charges. The total one-dimensional current densities are

For nondegenerate carriers in thermal equilibrium, mobility and diffusion coefficient are related by the Einstein relation:

This simple form requires nondegenerate statistics; generalized relations are needed in degenerate semiconductors.

4. Generation, Recombination and Continuity

Generation creates electron–hole pairs, while recombination annihilates them. Relevant mechanisms include:

• band-to-band radiative recombination, important in direct-gap LEDs;
• Shockley–Read–Hall (SRH) recombination, mediated by defect levels;
• Auger recombination, important at high carrier concentrations; and
• surface recombination, caused by interface states and dangling bonds.

4.1 Continuity equations

Charge conservation for electrons and holes is expressed by

Here G and R are generation and recombination rates in cm⁻³ s⁻¹. Signs differ because electron and hole currents represent opposite charge species.

4.2 Shockley–Read–Hall kinetics

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

where

E_T is the trap energy, E_i is the intrinsic level, and \tau_n0 and \tau_p0 are trap-related electron and hole lifetime parameters. Positive U_SRH denotes net recombination; negative values denote net generation. At equilibrium, np=n_i², so the net rate is zero even though microscopic generation and recombination continue.

In uniformly doped n-type material under low-level injection, if excess minority holes satisfy \Delta p\ll n₀ and trapping conditions permit a single effective lifetime, the result reduces to

Similarly, in p-type material, U≈\Delta n/\tau_n. This lifetime model is a special low-level approximation, not a universal recombination law.

Radiative and Auger rates are commonly modeled as

where B is the radiative coefficient and C_n,C_p are Auger coefficients, with units chosen so that U is in cm⁻³ s⁻¹.

4.3 Lifetime and diffusion length

After generation is removed, a spatially uniform low-level excess population decays as

The minority-carrier diffusion lengths are

These lengths estimate how far excess carriers diffuse before recombination under the assumptions of constant diffusion coefficient and lifetime.

Worked Example 3: Recombination rate and diffusion length

In n-type silicon, let \Delta p=5.0×10¹³\ \text{cm}^-3, \tau_p=2.0\ \mu\text{s} and \mu_p=450\ \text{cm}²\text{V}^-1\text{s}^-1. Under low-level injection,

At 300 K, the Einstein relation gives

Therefore,

5. Hall Effect

A magnetic field perpendicular to a carrier current produces a transverse Lorentz force. Charge accumulates until the resulting Hall field balances the magnetic force.

For a material dominated by one carrier type,

Under the ideal single-carrier model,

For a rectangular sample of thickness t,

The Hall-voltage sign identifies the dominant carrier type; its magnitude can estimate carrier concentration. Mobility may then be estimated from \mu≈|R_H|\sigma. These simple equations require predominantly one carrier type and neglect Hall scattering factors, geometric errors and contact misalignment.

Worked Example 4: Hall measurement

An n-type specimen has n=1.0×10¹⁶\ \text{cm}^-3=1.0×10²²\ \text{m}^-3. Thus

For I=5.0\ \text{mA}, B=0.50\ \text{T} and t=0.50\ \text{mm},

The negative sign confirms electron-dominated conduction for the stated terminal convention.

6. The p–n Junction

6.1 Depletion-region formation

When p-type and n-type materials are joined, majority carriers diffuse across the interface and recombine. Their departure exposes fixed negative acceptor ions on the p side and fixed positive donor ions on the n side. This space charge creates an electric field that opposes further diffusion. Equilibrium is reached when drift and diffusion balance and the Fermi level becomes spatially constant.

Under the abrupt-junction depletion approximation, mobile carriers are neglected inside the depletion region. If the metallurgical junction is at x=0, the charge density is

Figure: In an abrupt junction, fixed acceptor charge is negative on the p side and fixed donor charge is positive on the n side; both SVG rectangles use positive dimensions and are placed relative to the zero-charge axis.

Charge neutrality of the depletion region requires equal charge magnitudes:

The built-in potential for a nondegenerate abrupt homojunction is

For applied junction voltage V_a, taken positive for forward bias, the total depletion width is

This equation assumes an abrupt junction, depletion approximation, uniform doping, and V_a<V_bi. Reverse bias is represented by negative V_a, which increases W. The individual widths are

The depletion region extends farther into the more lightly doped side.

Worked Example 5: Silicon depletion width

Let N_A=1.0×10¹⁷\ \text{cm}^-3, N_D=1.0×10¹⁶\ \text{cm}^-3, n_i=1.0×10¹⁰\ \text{cm}^-3 and \varepsilon_s=11.7\varepsilon₀=1.036×10^-12\ \text{F cm}^-1. At zero applied bias,

Therefore,

Most depletion occurs in the more lightly doped n region.

6.2 Junction current

Forward bias lowers the carrier-injection barrier; reverse bias raises it. For an ideal diode under low-level injection, negligible depletion-region recombination, steady state and negligible series resistance,

A practical empirical form introduces ideality factor n_d:

Typically n_d lies between 1 and 2 over relevant operating ranges, but it is not a universal constant. Series resistance, leakage, high-level injection and self-heating cause further departures.

Reverse breakdown occurs through Zener tunneling in highly doped, narrow junctions or avalanche multiplication in wider, more lightly doped junctions. Breakdown is nondestructive only when current and power are externally limited.

Junction capacitance under reverse bias results from changing depletion charge. Forward-biased devices also exhibit diffusion capacitance due to stored minority carriers, affecting switching speed.

7. Metal–Semiconductor Interfaces

A metal–semiconductor contact may be rectifying or approximately ohmic. For an ideal n-type interface without interfacial states, the Schottky barrier height is

Here \Phi_M is the metal work function and \chi is the semiconductor electron affinity, both in eV. Real contacts often depart from this Schottky–Mott relation because of interface states, chemical reactions, image-force lowering and Fermi-level pinning.

A Schottky diode conducts mainly through majority carriers and therefore switches rapidly with little minority-carrier storage. An ohmic contact should have a nearly linear, low-resistance current–voltage characteristic. It is commonly achieved through suitable metal selection and heavy semiconductor doping, which narrows the barrier sufficiently for tunneling.

8. Physical Foundations of Semiconductor Devices

8.1 Bipolar junction transistor

A bipolar junction transistor (BJT) contains two p–n junctions. In forward-active operation, the emitter–base junction is forward biased and injects minority carriers into the thin base. Most diffuse across the base before recombining and are collected by the reverse-biased collector junction. Good current gain requires strong emitter injection, a thin base and a minority-carrier diffusion length exceeding the effective base width.

8.2 Field-effect transistor

A metal–oxide–semiconductor field-effect transistor (MOSFET) controls a conducting surface channel electrostatically. In an n-channel enhancement MOSFET, positive gate voltage depletes holes and then creates an inversion layer of electrons. Gate oxide quality, interface traps, threshold voltage, mobility, short-channel effects and leakage determine performance.

8.3 LEDs and photodetectors

An LED converts injected electrical carriers into photons through radiative recombination. Efficient LEDs generally use direct-band-gap compounds and heterostructures that confine carriers and light. The emitted photon energy is approximately related to band gap by hν≈E_g, although spectra broaden through thermal and band-structure effects.

Photodiodes operate by the reverse process: absorbed photons with adequate energy create electron–hole pairs. The junction field separates carriers and produces photocurrent. Important specifications include responsivity, quantum efficiency, dark current, bandwidth and noise.

8.4 Solar cells

A solar cell is an illuminated junction that converts optical power into electrical power. Its idealized current relation is

The power-conversion efficiency is

Here V_OC is open-circuit voltage, I_SC is short-circuit current, and FF is fill factor. Recombination, optical reflection, resistive loss and temperature reduce efficiency.

8.5 Semiconductor sensors

Semiconductor properties are sensitive to environmental inputs. Thermistors exploit temperature-dependent carrier concentration or mobility; Hall sensors measure magnetic field; photodiodes detect radiation; piezoresistive sensors use stress-induced resistance changes; and chemical sensors use surface-charge modulation. Calibration must address temperature cross-sensitivity, drift, packaging stress and aging.

9. Applied Case: Selecting a Photovoltaic Module

Background: This is an illustrative engineering case. A design team evaluates a silicon module for a remote monitoring station. Under standard test conditions, the proposed module has area 100 cm², V_OC=0.68\ \text{V}, current density J_SC=36\ \text{mA cm}^-2, fill factor 0.80 and incident irradiance 100\ \text{mW cm}^-2.

Problem: Determine the standard-condition efficiency and estimate output at a cell temperature of 60 °C if power decreases by 0.35% per degree Celsius relative to its 25 °C value.

Analysis:

The incident power is (0.100\ \text{W cm}^-2)(100\ \text{cm}²)=10.0\ \text{W}. Therefore,

The temperature rise is 35 °C. Applying the specified relative coefficient,

Approach and solution: The team should size the power system using hot-condition output rather than nameplate efficiency alone. It should also consider irradiance variation, battery charging losses, partial shading, soiling and long-term degradation.

Outcome: The module supplies approximately 1.72 W at the stated hot condition, about 12.25% less than its standard-condition maximum power.

Learning points: Device performance follows carrier generation, separation and recombination physics, but system design must also incorporate temperature, optical, electrical and environmental losses.

10. Engineering Limitations, Responsibility and Contemporary Developments

Semiconductor models are hierarchical. Simple formulas are valuable only within their assumptions. Heavy doping may require Fermi–Dirac statistics and band-gap narrowing; nanoscale devices require quantum and ballistic transport models; high-field devices require impact-ionization and velocity-saturation treatment; and surfaces may dominate microscale sensors.

Best engineering practice includes documenting temperature, doping profile, geometry, contact resistance, uncertainty and model validity. Device simulation should be validated against measurements rather than treated as an unquestionable prediction. AI-assisted process optimization can identify relationships in fabrication data, but biased datasets and poorly constrained models can recommend physically implausible operating conditions.

Environmental and ethical responsibilities include safe handling of toxic compounds, responsible sourcing of gallium, indium and other critical materials, worker protection during fabrication, energy and water use, export-control compliance, and end-of-life recovery. Reliability standards and electrical safety rules are especially important for power semiconductors and photovoltaic installations.

Contemporary developments include wide-band-gap SiC and GaN power devices, perovskite–silicon tandem solar cells, micro-LED displays, flexible sensors, silicon photonics and two-dimensional semiconductors. These technologies promise higher efficiency or new form factors, but face challenges involving defects, interfaces, manufacturability, stability and cost.

Chapter Summary

Semiconductor behavior emerges from energy bands, carrier statistics and controllable electron and hole populations. Intrinsic material has equal electron and hole concentrations, while donors and acceptors establish majority and minority carriers subject to charge neutrality. Carriers move through drift and diffusion, connected in nondegenerate material by the Einstein relation.

Continuity equations enforce carrier conservation. SRH, radiative and Auger models quantify important recombination mechanisms, while lifetime and diffusion length govern carrier collection and switching behavior. The Hall effect identifies carrier type and can determine concentration and mobility.

A p–n junction develops fixed negative acceptor charge on the p side and positive donor charge on the n side. Its built-in field, depletion width and carrier injection produce diode behavior. Metal–semiconductor interfaces form Schottky or ohmic contacts depending on barrier formation and tunneling. These principles explain the operation of BJTs, MOSFETs, LEDs, photodiodes, solar cells and semiconductor sensors.

Key Terms

Band gap: Forbidden energy interval between conduction and valence bands.

Conduction band: Allowed energy band in which electrons can contribute readily to conduction.

Hole: Positively charged quasiparticle representing an unoccupied valence-band state.

Fermi level: Energy associated with the statistical occupation of electronic states.

Intrinsic semiconductor: Semiconductor without intentional doping and with equal electron and hole concentrations.

Donor: Impurity that supplies electrons to the conduction system.

Acceptor: Impurity that creates holes by accepting valence electrons.

Majority carrier: More abundant mobile carrier type in a doped semiconductor.

Mobility: Proportionality between low-field drift speed and electric-field magnitude.

Diffusion: Carrier transport driven by a concentration gradient.

Generation: Creation of electron–hole pairs.

Recombination: Annihilation of electrons and holes.

Carrier lifetime: Characteristic time over which an excess carrier population decays.

Diffusion length: Characteristic distance an excess carrier diffuses before recombination.

Hall effect: Transverse voltage produced by magnetic deflection of moving charge carriers.

Depletion region: Junction region depleted of mobile majority carriers and containing fixed ionized dopants.

Built-in potential: Equilibrium electrostatic potential across a semiconductor junction.

Schottky contact: Rectifying metal–semiconductor interface dominated by a carrier barrier.

Ohmic contact: Low-resistance, approximately linear electrical contact.

Fill factor: Ratio of photovoltaic maximum power to V_OCI_SC.

Review Questions

Short Answer Questions

1. Distinguish intrinsic, n-type and p-type semiconductors.
2. State the assumptions required for the mass-action law and complete-ionization approximation.
3. Explain why electron and hole diffusion-current equations have opposite signs.
4. What information can be obtained from a Hall-effect measurement?
5. Why does a Schottky diode generally switch faster than a p–n diode?

Long Answer Questions

1. Derive the equilibrium electron and hole concentrations in a compensated, nondegenerate semiconductor.
2. Explain drift and diffusion transport, including current-density equations and the Einstein relation.
3. Develop the SRH recombination model and show how it reduces to a minority-carrier lifetime model under low-level injection.
4. Derive the built-in potential and depletion width of an abrupt p–n homojunction, stating all assumptions.
5. Compare the physical operation and design requirements of BJTs, MOSFETs, LEDs, photodiodes and solar cells.

Analytical / Application Questions

1. A silicon sample has N_D=5×10¹⁵\ \text{cm}^-3, N_A=1×10¹⁵\ \text{cm}^-3 and n_i=10¹⁰\ \text{cm}^-3. Calculate the equilibrium carrier concentrations and resistivity using stated mobility data.
2. For a specified abrupt silicon junction, calculate built-in voltage, depletion widths and maximum electric field at zero and reverse bias; then assess whether the depletion approximation is reasonable.
3. A photodiode exhibits excessive dark current. Analyze how bulk defects, surface states, temperature, reverse bias and contact leakage could contribute, and propose a measurement plan to separate the mechanisms.

Multiple Choice Questions with Answers

Question 1

In nondegenerate silicon at thermal equilibrium, which relationship is the mass-action law?

A. n+p=n_i

B. np=n_i²

C. n-p=n_i²

D. n/p=n_i

Correct Answer: B. np=n_i²

Question 2

Silicon has N_D=9×10¹⁵\ \text{cm}^-3 and N_A=2×10¹⁵\ \text{cm}^-3. Assuming complete ionization and negligible intrinsic concentration, what is the majority-carrier concentration?

A. 2×10¹⁵\ \text{cm}^-3

B. 7×10¹⁵\ \text{cm}^-3

C. 9×10¹⁵\ \text{cm}^-3

D. 11×10¹⁵\ \text{cm}^-3

Correct Answer: B. 7×10¹⁵\ \text{cm}^-3

Question 3

Which equation gives semiconductor conductivity in the low-field regime?

A. \sigma=q(n\mu_n+p\mu_p)

B. \sigma=q(n/\mu_n+p/\mu_p)

C. \sigma=(n+p)/(qE)

D. \sigma=qD_nD_p

Correct Answer: A. \sigma=q(n\mu_n+p\mu_p)

Question 4

At 300 K, a nondegenerate carrier has mobility 400\ \text{cm}²\text{V}^-1\text{s}^-1. Using V_T=0.02585\ \text{V}, its diffusion coefficient is approximately:

A. 0.0646\ \text{cm}²\text{s}^-1

B. 10.34\ \text{cm}²\text{s}^-1

C. 15.47\ \text{cm}²\text{s}^-1

D. 400\ \text{cm}²\text{s}^-1

Correct Answer: B. 10.34\ \text{cm}²\text{s}^-1

Question 5

Under low-level injection, an excess minority-carrier concentration of 6×10¹²\ \text{cm}^-3 has a lifetime of 3\ \mu\text{s}. What is the net recombination rate?

A. 2×10⁶\ \text{cm}^-3\text{s}^-1

B. 1.8×10⁷\ \text{cm}^-3\text{s}^-1

C. 2×10¹⁸\ \text{cm}^-3\text{s}^-1

D. 1.8×10¹⁹\ \text{cm}^-3\text{s}^-1

Correct Answer: C. 2×10¹⁸\ \text{cm}^-3\text{s}^-1

Question 6

A negative Hall coefficient under the stated conventional terminal orientation usually indicates that the dominant carriers are:

A. Electrons

B. Holes

C. Ionized acceptors

D. Photons

Correct Answer: A. Electrons

Question 7

In the depletion region of an abrupt p–n junction, the fixed charge is:

A. Positive on both sides

B. Negative on both sides

C. Negative on the p side and positive on the n side

D. Positive on the p side and negative on the n side

Correct Answer: C. Negative on the p side and positive on the n side

Question 8

For an abrupt junction with N_A=10N_D, which statement is correct?

A. The depletion region extends equally into both sides.

B. The depletion region extends primarily into the p side.

C. The depletion region extends primarily into the n side.

D. No depletion region forms at equilibrium.

Correct Answer: C. The depletion region extends primarily into the n side.

Question 9

Which recombination mechanism is typically dominant in a well-designed direct-band-gap LED active region?

A. Radiative electron–hole recombination

B. Mechanical recombination

C. Hall recombination

D. Purely resistive recombination

Correct Answer: A. Radiative electron–hole recombination

Question 10

A solar cell has V_OC=0.60\ \text{V}, I_SC=2.0\ \text{A} and fill factor 0.75. What is its maximum electrical power?

A. 0.45 W

B. 0.90 W

C. 1.20 W

D. 1.60 W

Correct Answer: B. 0.90 W