Semiconductor Physics and Junction Devices

Introduction

Semiconductors occupy the electrical middle ground between conductors and insulators, but their technological importance comes from controllability rather than merely intermediate conductivity. Their carrier concentration can be modified through doping, electric fields, illumination, temperature and junction formation. These mechanisms enable diodes, transistors, integrated circuits, sensors, photovoltaic cells and power-electronic converters.

This chapter develops the physical basis of semiconductor devices. It connects energy-band theory to carrier statistics, transport, PN-junction electrostatics and practical diode behavior. Unless stated otherwise, numerical values refer to crystalline silicon near 300 K.

Learning Objectives

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

• explain valence bands, conduction bands, bandgaps and Fermi level;
• distinguish intrinsic, extrinsic, compensated and degenerate semiconductors;
• calculate equilibrium electron and hole concentrations;
• analyze drift, diffusion, conductivity and carrier continuity;
• explain PN-junction formation and depletion-region electrostatics;
• calculate built-in voltage and depletion width under stated assumptions;
• interpret forward bias, reverse bias, capacitance and breakdown;
• evaluate temperature effects and measured diode characteristics; and
• compare major semiconductor materials and fabrication technologies.

1. Energy Bands in Solids

1.1 From atomic levels to bands

An isolated atom has discrete electron-energy levels. When many atoms form a crystal, interactions between neighboring atoms split these levels into closely spaced allowed states called energy bands. Ranges containing no allowed states are forbidden energy gaps.

The highest normally occupied band at absolute zero is associated with bonding and is called the valence band. The next higher band is the conduction band. Their edges are denoted by E_V and E_C, respectively. The bandgap is

where E_g is usually expressed in electronvolts (eV). One electronvolt is the energy acquired by one electron moving through a potential difference of 1 V: 1 eV = 1.602 × 10^-19 J.

Figure: Conductors have overlapping or partially filled bands; semiconductors have a modest bandgap, while insulators have a much larger gap.

A conductor permits easy carrier acceleration because an allowed band is partially occupied or bands genuinely overlap. A semiconductor has a sufficiently small gap for thermal or optical excitation to create useful carrier populations. An insulator has a gap too large for appreciable room-temperature excitation under ordinary fields.

1.2 Electrons, holes and effective mass

An electron excited into the conduction band leaves an unoccupied valence-band state. Collective motion of neighboring valence electrons makes this vacancy behave as a positively charged mobile particle called a hole. Crystal forces are incorporated into an effective mass, allowing carriers to be treated approximately as quasiparticles while retaining the influence of the periodic lattice.

1.3 Fermi level and occupancy

At thermal equilibrium, the probability that an available state of energy E is occupied by an electron is the Fermi–Dirac function

Here E_F is the Fermi level, k is Boltzmann’s constant, and T is absolute temperature in kelvins. At E=E_F, the occupancy probability is one-half.

For a nondegenerate semiconductor—one whose Fermi level is several kT away from a band edge—the Maxwell–Boltzmann approximation gives

where n and p are electron and hole concentrations, while N_C and N_V are the effective densities of states. All four concentrations are commonly expressed in cm⁻³.

2. Intrinsic and Doped Semiconductors

2.1 Intrinsic material

A chemically pure semiconductor in thermal equilibrium is intrinsic. Electron–hole pairs are generated together, so

where n_i is the intrinsic carrier concentration. Combining the nondegenerate carrier equations gives the equilibrium mass-action law

This law applies at thermal equilibrium; under illumination, carrier injection or other nonequilibrium conditions, the product need not equal n_i².

The intrinsic concentration is

Because N_C and N_V each vary approximately as T^3/2, n_i rises rapidly with temperature. For silicon at 300 K, a commonly used approximate value is n_i ≈ 1.0 × 10¹⁰ cm^-3, although the reported value depends on material parameters and convention.

2.2 Donor and acceptor doping

Adding controlled impurities changes carrier concentration by many orders of magnitude.

• Pentavalent donor atoms in silicon provide electrons and produce n-type material.
• Trivalent acceptor atoms create holes and produce p-type material.

At ordinary temperatures and moderate doping, shallow dopants are often treated as fully ionized. Charge neutrality then requires

where N_A^- and N_D^+ are ionized acceptor and donor concentrations. With complete ionization, these become approximately N_A and N_D.

For uncompensated n-type material with N_D ≫ n_i,

For uncompensated p-type material with N_A ≫ n_i,

In compensated material, both donors and acceptors are present. If donors dominate and complete ionization is valid, n ≈ N_D-N_A only when the net doping is much larger than n_i. At very high doping, Fermi–Dirac statistics, bandgap narrowing and incomplete mobility models must be considered; simple nondegenerate formulas are then inadequate.

Worked Example 1: Carrier concentrations and resistivity

Silicon at 300 K has N_D=1.0 × 10¹⁶ cm^-3, negligible acceptors and n_i=1.0 × 10¹⁰ cm^-3. Assume complete ionization and electron mobility μ_n=1350 cm²/(V·s).

Because N_D ≫ n_i,

Neglecting the very small hole contribution, conductivity is

Therefore,

The mobility value is an assumed low-field value; actual mobility decreases with increasing dopant concentration and temperature over common operating ranges.

3. Charge Transport

3.1 Drift

An electric field causes carriers to acquire an average drift velocity. In the low-field regime,

where E is electric field in V/cm and μ_n, μ_p are mobilities in cm²/(V·s). Electron velocity is opposite to the field, but conventional electron current is in the field direction. At high fields, velocity saturation makes the linear relationship inaccurate.

The total drift conductivity is

where q=1.602 × 10^-19 C. Resistivity is ρ=1/σ for a homogeneous isotropic material.

3.2 Diffusion

Carriers also move from regions of high concentration toward regions of low concentration. In one dimension, the electron and hole current densities are

Here J_n and J_p are conventional current densities in A/cm², and D_n, D_p are diffusion coefficients in cm²/s. The opposite gradient signs follow from the opposite carrier charges.

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

The thermal voltage V_T is approximately 25.85 mV at 300 K.

3.3 Generation, recombination and continuity

Generation creates electron–hole pairs; recombination annihilates them. Carrier conservation is expressed by the continuity equations

where G and R are generation and recombination rates in cm⁻³s⁻¹. Important mechanisms include defect-assisted Shockley–Read–Hall recombination, radiative recombination and Auger recombination. Carrier lifetime and diffusion length are related, under a constant-lifetime low-injection model, by

where τ_n and τ_p are excess-carrier lifetimes.

4. Formation of a PN Junction

A PN junction is formed by adjoining p-type and n-type regions within one continuous crystal. Immediately after formation, electrons diffuse from n to p and holes diffuse from p to n. Recombination near the interface removes mobile carriers and exposes fixed ionized dopants: negatively charged acceptor ions on the p side and positively charged donor ions on the n side.

The resulting depletion region contains very few mobile carriers. Its electric field points from the positive donor ions toward the negative acceptor ions, opposing further majority-carrier diffusion. Equilibrium occurs when drift and diffusion balance and the Fermi level becomes spatially constant.

Figure: The p/n material boundary and metallurgical junction coincide at x = 450; fixed acceptor ions remain only on the p side and donor ions only on the n side.

5. Depletion Approximation and Junction Electrostatics

For an abrupt junction under the depletion approximation, mobile charge is neglected inside the depletion region and regions outside it are treated as electrically neutral. If x_p and x_n are depletion widths into the p and n sides, charge neutrality requires

Thus, depletion extends farther into the more lightly doped side.

For nondegenerate material with complete ionization, the equilibrium built-in potential is

This expression assumes an abrupt homojunction and approximately uniform doping on each side.

Let V_A be the externally applied p-side voltage relative to the n side. Positive V_A denotes forward bias. Before high injection or breakdown, the depletion width is

where ε_s is semiconductor permittivity in F/cm. The individual widths are

Reverse bias has V_A<0, so V_bi-V_A=V_bi+|V_A| and the depletion region widens.

Worked Example 2: Silicon abrupt junction

At 300 K, let N_A=1.0 × 10¹⁷ cm^-3, N_D=1.0 × 10¹⁶ cm^-3, n_i=1.0 × 10¹⁰ cm^-3, V_T=0.02585 V and ε_s=1.036 × 10^-12 F/cm.

The built-in potential is

At zero applied bias,

Therefore,

Most depletion lies in the more lightly doped n region, as expected.

6. Junction Biasing and the Diode Equation

6.1 Forward bias

Forward bias lowers the junction barrier and narrows the depletion region. Majority carriers cross the junction and become minority carriers on the opposite side. Their diffusion and recombination produce a rapidly increasing current.

Under steady state, low-level injection, negligible depletion-region recombination and sufficiently long neutral regions, the ideal diode equation is

where I is terminal current, V_D is p-to-n diode voltage, and I_S is reverse saturation current. For a junction of area A,

The equation follows from solving the minority-carrier diffusion equations with junction-edge concentrations increased by the Boltzmann factor exp(V_D/V_T).

Real devices are often represented over a limited current range by

where η is an empirical ideality factor, commonly between 1 and 2. This model does not include series resistance, high-level injection, leakage mechanisms or breakdown. At large forward current, bulk and contact resistance cause the terminal voltage to rise approximately as IR_s in addition to the junction voltage.

6.2 Reverse bias

Reverse bias raises the barrier and widens the depletion region. Before breakdown, current is small and consists of thermally generated minority carriers and leakage through surfaces or defects. Practical leakage is often more temperature-sensitive and voltage-dependent than the ideal constant I_S.

6.3 Junction and diffusion capacitance

Changing reverse voltage changes depletion charge, producing junction capacitance:

This parallel-plate form applies within the depletion approximation. For a one-sided abrupt junction, C_j varies approximately as (V_bi+V_R)^-1/2, where V_R is reverse-bias magnitude. Graded junctions have a different voltage exponent.

Under forward bias, stored minority charge creates diffusion capacitance, often approximated in a suitable low-frequency operating region by

where τ_T is an effective transit or storage time. Stored charge also causes reverse-recovery current during switching.

7. Breakdown Mechanisms

If reverse voltage becomes sufficiently large, current increases sharply.

Zener breakdown arises primarily from quantum-mechanical tunneling across a narrow depletion region. It is favored by heavy doping and high electric field and commonly dominates at relatively low breakdown voltages.

Avalanche breakdown occurs when carriers gain enough kinetic energy to generate additional electron–hole pairs by impact ionization. It is favored by wider depletion regions and commonly dominates at higher voltages.

Both mechanisms may coexist. Breakdown is not inherently destructive if external circuitry limits current and junction temperature remains within ratings. Uncontrolled current can produce thermal runaway, localized heating and permanent damage. Zener and avalanche diodes intentionally exploit controlled reverse breakdown for voltage reference and transient suppression.

8. Temperature Effects and Device Characteristics

Temperature affects several interacting quantities:

• n_i rises strongly with temperature, increasing leakage and saturation current.
• Carrier mobility generally falls as lattice scattering increases, although impurity scattering complicates behavior at low temperature.
• The silicon bandgap decreases modestly as temperature rises.
• At a specified forward current, a silicon diode’s forward voltage commonly decreases by roughly 1.5–2.5 mV/°C over a practical range, but this is not a universal constant.
• Breakdown voltage may have a negative temperature coefficient when tunneling dominates and a positive coefficient when avalanche multiplication dominates.

The diode current equation also shows direct temperature dependence through V_T=kT/q, while I_S changes much more strongly through n_i². Accurate thermal design therefore requires manufacturer data rather than a single textbook coefficient.

Important datasheet characteristics include forward voltage at stated current and temperature, repetitive peak reverse voltage, leakage current, surge-current rating, reverse-recovery time or charge, junction capacitance, thermal resistance and maximum junction temperature. Ratings are conditional: for example, allowable current depends on cooling and waveform.

9. Semiconductor Material Technologies

Bandgap alone does not determine device performance. Critical electric field, carrier mobility, thermal conductivity, defect density, interface quality, substrate cost and packaging are also decisive. Silicon remains dominant because of manufacturing maturity and the exceptional quality of its silicon-dioxide interface. Wide-bandgap SiC and GaN enable thinner drift regions and faster switching in many power applications, but cost, gate reliability, parasitic inductance and thermal packaging remain significant engineering concerns.

Fabrication commonly includes crystal growth, wafer preparation, oxidation or dielectric deposition, photolithography, ion implantation or diffusion, epitaxial growth, etching, metallization, passivation, testing and packaging. Junction profiles are rarely perfectly abrupt; process simulation and measurement are needed for precision devices.

10. Applied Case: Selecting a Diode for a Fast Power Converter

Background

This is an illustrative engineering case. A 400 V converter switches at 200 kHz and carries an average diode current of 10 A. Two rated devices are considered: a silicon PN diode with a 1.1 V forward drop and reverse-recovery charge of 100 nC, and a SiC Schottky diode with a 1.6 V forward drop and negligible minority-carrier reverse recovery for this first-order comparison.

Problem

The silicon device has lower static forward loss, but its stored minority charge creates switching loss. The engineer must avoid selecting a device using forward voltage alone.

Analysis

Approximate conduction losses, assuming continuous 10 A conduction during the interval being compared, are

A first-order estimate of silicon reverse-recovery switching loss is

Thus, the simplified totals are approximately 19 W for silicon and 16 W for SiC.

Approach and outcome

The preliminary analysis favors SiC despite its greater forward drop. Final selection must use measured switching waveforms because Q_rr depends on current, current slope and temperature. SiC junction capacitance, electromagnetic interference, package inductance, heat sinking, cost and reliability must also be evaluated.

Learning points

• Static and dynamic losses must be considered together.
• A material advantage becomes useful only with appropriate circuit layout and packaging.
• Simplified equations support screening, not final qualification.
• Datasheet conditions must match the intended operating region.

11. Engineering Practice, Limitations and Governance

Semiconductor models are hierarchical. Constant mobility, complete ionization, abrupt junctions, low-level injection and uniform temperature simplify analysis, but each has a limited domain. Professional practice requires documenting assumptions, checking electric-field and thermal limits, and validating models against measurements or calibrated device simulation.

Automated parameter extraction, machine-learning-assisted process control and digital twins are increasingly used in fabrication and reliability analysis. Such tools can identify patterns but do not replace physical constraints, traceable calibration or independent safety review.

Semiconductor production also has environmental and ethical dimensions. Crystal growth and cleanrooms consume substantial energy and water; fabrication may involve toxic, corrosive or high-global-warming-potential chemicals. Responsible governance includes worker protection, emissions control, conflict-mineral due diligence, product traceability, cybersecurity for connected fabrication equipment and compliance with hazardous-substance and electronic-waste regulations. Engineers should design for energy efficiency, reliability, repairability and responsible end-of-life recovery.

Chapter Summary

Energy-band theory explains why conductors possess overlapping or partially occupied bands, while semiconductors have a controllable bandgap. Thermal excitation creates electron–hole pairs, and doping establishes majority and minority carriers. Equilibrium concentrations follow charge neutrality and, for nondegenerate material, the mass-action law.

Carrier transport consists primarily of drift and diffusion, linked through the Einstein relation. Generation, recombination and continuity determine time-dependent carrier behavior. When p-type and n-type regions meet, carrier diffusion exposes fixed dopant ions and creates a depletion region, electric field and built-in potential. The depletion approximation enables calculation of depletion width and its bias dependence.

Forward bias lowers the junction barrier and produces minority-carrier injection, leading under restricted assumptions to the diode equation. Reverse bias widens the depletion region until leakage or breakdown becomes important. Capacitance, reverse recovery, temperature, series resistance and thermal limits distinguish practical devices from ideal models. Silicon dominates mainstream electronics, while GaAs, SiC and GaN offer advantages in optoelectronic, high-frequency and power applications.

Key Terms

Energy band: A range of allowed electron energies in a solid.

Bandgap: The forbidden energy interval between valence and conduction bands.

Fermi level: The equilibrium energy at which the electron occupancy probability is one-half.

Electron: A negatively charged mobile carrier, especially important in the conduction band.

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

Intrinsic semiconductor: A semiconductor whose carriers arise primarily from thermal generation rather than doping.

Donor: An impurity that supplies electrons to a semiconductor.

Acceptor: An impurity that produces holes by accepting valence electrons.

Majority carrier: The more abundant mobile carrier type in doped material.

Minority carrier: The less abundant mobile carrier type in doped material.

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

Diffusion: Carrier motion driven by a concentration gradient.

Recombination: A process that removes an electron–hole pair.

Depletion region: A junction region depleted of most mobile carriers and containing fixed ionized dopants.

Built-in potential: The equilibrium electrostatic potential across a junction depletion region.

Forward bias: Applied voltage that reduces a PN-junction barrier.

Reverse bias: Applied voltage that increases a PN-junction barrier.

Reverse recovery: Transient removal of stored charge when a conducting diode is switched toward reverse bias.

Zener breakdown: Reverse breakdown dominated by field-induced tunneling.

Avalanche breakdown: Reverse breakdown caused by impact-ionization multiplication.

Review Questions

Short Answer Questions

1. Distinguish a conduction-band electron from a valence-band hole.
2. State the conditions under which np=n_i² applies.
3. Why does depletion extend farther into the more lightly doped side of a PN junction?
4. What physical processes produce drift and diffusion currents?
5. Distinguish Zener breakdown from avalanche breakdown.

Long Answer Questions

1. Explain the formation of energy bands and compare conductors, semiconductors and insulators.
2. Develop the charge-neutrality relationships for intrinsic, n-type, p-type and compensated semiconductors.
3. Explain PN-junction formation from initial carrier diffusion to thermal equilibrium.
4. Derive the abrupt-junction depletion-width expression, stating its assumptions and sign convention.
5. Discuss how temperature influences carrier concentration, mobility, leakage, forward voltage and breakdown.

Analytical / Application Questions

1. A silicon sample at 300 K has N_A=3 × 10¹⁶ cm^-3, N_D=1 × 10¹⁶ cm^-3 and n_i=10¹⁰ cm^-3. Assuming complete ionization and nondegenerate behavior, calculate its majority and minority carrier concentrations.
2. For the junction in Worked Example 2, calculate the depletion width at V_A=-5 V and determine the portions on the p and n sides.
3. Compare a silicon PN diode and a SiC diode for a high-frequency rectifier using forward loss, switching loss, thermal limits, capacitance, cost and reliability as decision criteria.

Multiple Choice Questions with Answers

Question 1

Which energy-band description correctly characterizes an ordinary electrical conductor?

A. Its valence and conduction bands are separated by a large forbidden gap.

B. It has an overlapping or partially occupied allowed band.

C. Its conduction band must be completely empty.

D. Its Fermi level can exist only at the center of a bandgap.

Correct Answer: B. It has an overlapping or partially occupied allowed band.

Question 2

At equilibrium, silicon has n_i=1.0 × 10¹⁰ cm^-3 and N_D=1.0 × 10¹⁶ cm^-3, with negligible acceptors and complete ionization. What is the approximate hole concentration?

A. 1.0 × 10⁴ cm^-3

B. 1.0 × 10⁶ cm^-3

C. 1.0 × 10¹⁰ cm^-3

D. 1.0 × 10¹⁶ cm^-3

Correct Answer: A. 1.0 × 10⁴ cm^-3

Question 3

An n-type sample has n=1.0 × 10¹⁶ cm^-3 and μ_n=1350 cm²/(V·s). Neglecting holes, what is its conductivity?

A. 0.462 S/cm

B. 1.35 S/cm

C. 2.16 S/cm

D. 21.6 S/cm

Correct Answer: C. 2.16 S/cm

Question 4

Which expression gives one-dimensional electron current density in a nondegenerate semiconductor under the usual drift–diffusion model?

A. J_n=qnμ_nE+qD_n(dn/dx)

B. J_n=qnμ_nE-qD_n(dn/dx)

C. J_n=-qnμ_nE-qD_n(dn/dx)

D. J_n=qpμ_pE-qD_p(dp/dx)

Correct Answer: A. J_n=qnμ_nE+qD_n(dn/dx)

Question 5

For an abrupt PN junction with N_A=10¹⁷ cm^-3, N_D=10¹⁶ cm^-3, n_i=10¹⁰ cm^-3 and V_T=0.02585 V, what is the approximate built-in potential?

A. 0.238 V

B. 0.417 V

C. 0.595 V

D. 0.774 V

Correct Answer: D. 0.774 V

Question 6

In an abrupt PN junction where N_A is ten times N_D, where does most of the depletion region lie?

A. In the p region because it has more acceptors

B. In the n region because it is more lightly doped

C. Equally on both sides regardless of doping

D. Outside both semiconductor regions

Correct Answer: B. In the n region because it is more lightly doped

Question 7

What is the primary electrostatic effect of applying forward bias to a PN junction before high-level injection?

A. The barrier rises and depletion width increases.

B. The barrier falls and depletion width decreases.

C. The barrier and depletion width both remain fixed.

D. The junction immediately enters avalanche breakdown.

Correct Answer: B. The barrier falls and depletion width decreases.

Question 8

Which statement correctly describes avalanche breakdown?

A. It is caused principally by majority-carrier diffusion at zero field.

B. It results from impact ionization and carrier multiplication.

C. It requires the semiconductor bandgap to become zero.

D. It can occur only in forward bias.

Correct Answer: B. It results from impact ionization and carrier multiplication.

Question 9

At a fixed moderate forward current, increasing the temperature of a typical silicon diode usually causes its forward voltage to:

A. decrease

B. increase without limit

C. remain exactly constant

D. become equal to the reverse breakdown voltage

Correct Answer: A. decrease

Question 10

Which semiconductor is widely used for high-voltage power devices because of its wide bandgap, high critical electric field and strong thermal capability?

A. Germanium

B. Silicon carbide

C. Intrinsic silicon only

D. Copper

Correct Answer: B. Silicon carbide