Semiconductor Physics

Semiconductor physics is the foundation of modern electronics. The remarkable properties of semiconductor materials, which conduct electricity better than insulators but not as well as metals, enable the transistors, diodes, and integrated circuits that power our digital world. Understanding semiconductor physics reveals why these materials behave as they do and how their properties can be controlled to create useful electronic devices.

This field bridges quantum mechanics, solid-state physics, and electrical engineering. The concepts introduced here explain how electrons move through crystalline materials, how adding impurities dramatically changes electrical properties, and how junctions between different semiconductor regions create the fundamental building blocks of electronic devices.

Crystal Structure and Atomic Bonding

Semiconductor behavior begins with atomic structure. Silicon, the most common semiconductor, has four valence electrons that form covalent bonds with neighboring atoms. In a pure silicon crystal, each atom shares its four valence electrons with four neighboring atoms, creating a tetrahedral crystal structure known as the diamond cubic lattice.

This crystalline arrangement is crucial for semiconductor properties. The regular periodic structure creates an environment where electron behavior can be described using quantum mechanical band theory. Defects in the crystal structure, whether intentional or unintentional, significantly affect electrical properties. Understanding the crystal structure helps explain why semiconductor processing must achieve extraordinary levels of purity and crystalline perfection.

Other semiconductor materials include germanium (also with four valence electrons), gallium arsenide (a compound semiconductor), and silicon carbide. Each has a specific crystal structure and bonding arrangement that determines its electrical and thermal properties, making it suitable for different applications.

Energy Bands and the Bandgap

In isolated atoms, electrons occupy discrete energy levels. When atoms combine into a solid crystal, these discrete levels broaden into continuous bands of allowed energies. The two most important bands for understanding semiconductor behavior are the valence band and the conduction band.

The valence band contains electrons that participate in covalent bonding and are relatively localized near their parent atoms. The conduction band contains electrons that are free to move through the crystal and participate in electrical conduction. Between these bands lies the bandgap, a range of energies that electrons cannot occupy in a perfect crystal.

The bandgap is what distinguishes semiconductors from conductors and insulators. In conductors like metals, the valence and conduction bands overlap, so electrons can move freely. In insulators, the bandgap is so large (typically greater than 4 eV) that very few electrons can gain enough energy to reach the conduction band. Semiconductors have moderate bandgaps, typically between 0.5 and 3 eV, allowing significant conduction under appropriate conditions.

Silicon Bandgap Characteristics

Silicon has a bandgap of approximately 1.12 eV at room temperature. This value decreases slightly as temperature increases, which explains why silicon devices become more conductive at higher temperatures. The temperature dependence follows the relationship:

E_g(T) = E_g(0) - (alpha * T²) / (T + beta)

Where E_g(0) is the bandgap at absolute zero, and alpha and beta are material-specific constants. For silicon, this means the bandgap decreases by about 0.3 meV per degree Kelvin near room temperature.

Direct and Indirect Bandgaps

Semiconductors are classified as having either direct or indirect bandgaps based on the relationship between electron energy and momentum in the crystal. In a direct bandgap material like gallium arsenide, the minimum energy state in the conduction band aligns with the maximum energy state in the valence band in momentum space. Electrons can transition between bands by absorbing or emitting photons without changing momentum.

Silicon has an indirect bandgap, meaning the conduction band minimum and valence band maximum occur at different momentum values. Transitions require both a photon and a phonon (lattice vibration) to conserve momentum. This makes silicon inefficient for light emission, which is why LEDs and laser diodes typically use direct bandgap materials like gallium arsenide or gallium nitride.

Charge Carriers: Electrons and Holes

Electrical conduction in semiconductors involves two types of charge carriers: electrons and holes. When an electron gains enough energy to move from the valence band to the conduction band, it leaves behind an empty state called a hole. This hole behaves as a positive charge carrier and can move through the crystal as neighboring electrons fill it, creating a new hole elsewhere.

In intrinsic (pure) semiconductors, electrons and holes are created in pairs and exist in equal concentrations. This intrinsic carrier concentration, denoted n_i, depends strongly on temperature and the bandgap:

n_i = sqrt(N_c * N_v) * exp(-E_g / 2kT)

Where N_c and N_v are the effective density of states in the conduction and valence bands, E_g is the bandgap, k is Boltzmann's constant, and T is absolute temperature. For silicon at room temperature (300 K), n_i is approximately 1.5 x 10¹⁰ carriers per cubic centimeter.

Effective Mass

Electrons and holes in a semiconductor crystal do not behave exactly like free particles. The periodic potential of the crystal lattice affects their motion, which is accounted for by assigning them an effective mass different from the free electron mass. This effective mass depends on the curvature of the energy bands and differs for electrons and holes, and even varies with direction in the crystal.

In silicon, the electron effective mass for conductivity calculations is about 0.26 times the free electron mass, while the hole effective mass is about 0.39 times the free electron mass. These values affect carrier mobility, diffusion rates, and the density of states in each band.

Doping and Extrinsic Semiconductors

The real power of semiconductors comes from the ability to precisely control their electrical properties through doping, the intentional introduction of impurity atoms. By adding atoms with different numbers of valence electrons, semiconductor conductivity can be increased by many orders of magnitude and made predominantly due to either electrons or holes.

N-Type Semiconductors

When silicon is doped with atoms having five valence electrons, such as phosphorus, arsenic, or antimony, the extra electron from each dopant atom is loosely bound and easily ionized into the conduction band. These dopants are called donors because they donate electrons. The resulting material is n-type, with electrons as the majority carriers and holes as minority carriers.

Donor atoms introduce energy levels just below the conduction band, typically 0.01 to 0.05 eV below the band edge. At room temperature, virtually all donors are ionized, so the electron concentration approximately equals the donor concentration for typical doping levels.

P-Type Semiconductors

Doping with atoms having three valence electrons, such as boron, aluminum, or gallium, creates acceptor sites that can capture electrons from the valence band, leaving behind holes. These materials are p-type, with holes as majority carriers. Acceptor levels lie just above the valence band, and at room temperature, acceptors are essentially fully ionized.

The ability to create both n-type and p-type regions in the same crystal is fundamental to semiconductor device operation. Transistors, diodes, and integrated circuits all depend on controlled junctions between differently doped regions.

Compensation and Carrier Concentration

When both donors and acceptors are present, they partially compensate each other. The net carrier type depends on which dopant has higher concentration. The carrier concentration is reduced by compensation, and the mobility may also be affected due to increased scattering from ionized impurities.

The relationship between carrier concentrations in thermal equilibrium is governed by the mass action law:

n * p = n_i²

This relationship holds regardless of doping. If electrons are the majority carrier with concentration n = N_D (the donor concentration), then the hole concentration is p = n_i² / N_D, which can be many orders of magnitude smaller than the electron concentration.

Carrier Transport Mechanisms

Charge carriers in semiconductors move through two primary mechanisms: drift under the influence of electric fields and diffusion due to concentration gradients. Understanding both is essential for analyzing device behavior.

Drift Current

When an electric field is applied to a semiconductor, charge carriers accelerate in response but are frequently scattered by lattice vibrations (phonons) and ionized impurities. This results in a constant average velocity called the drift velocity, proportional to the electric field:

v_d = mu * E

Where mu is the carrier mobility and E is the electric field. Mobility has units of cm²/(V*s) and depends on temperature, doping concentration, and the semiconductor material. For silicon at room temperature with moderate doping, electron mobility is approximately 1400 cm²/(V*s) and hole mobility is approximately 450 cm²/(V*s).

The drift current density is:

J_drift = q * (n * mu_n + p * mu_p) * E = sigma * E

Where sigma is the conductivity. This is the semiconductor form of Ohm's law.

Diffusion Current

Carriers also move from regions of high concentration to low concentration through diffusion, even without an applied electric field. The diffusion current density for electrons is:

J_diff,n = q * D_n * (dn/dx)

Where D_n is the electron diffusion coefficient. The diffusion coefficient is related to mobility through the Einstein relation:

D = (kT/q) * mu

At room temperature, kT/q is approximately 26 mV, called the thermal voltage. This relation reflects the fundamental connection between random thermal motion (diffusion) and directed motion under a field (drift).

High-Field Effects

At high electric fields, carrier velocity no longer increases linearly with field. The drift velocity saturates at approximately 10⁷ cm/s in silicon as carriers gain enough energy between collisions that additional energy goes into lattice heating rather than increasing velocity. Modern nanoscale transistors routinely operate in this velocity saturation regime.

At even higher fields, carriers may gain enough energy to ionize other atoms, creating electron-hole pairs in a process called impact ionization or avalanche multiplication. This effect is exploited in avalanche photodiodes but must be avoided in most other devices.

P-N Junction Physics

The p-n junction, formed at the boundary between p-type and n-type semiconductor regions, is the fundamental building block of semiconductor devices. Understanding junction physics is essential for analyzing diodes, transistors, solar cells, and LEDs.

Junction Formation and the Depletion Region

When p-type and n-type materials are brought together, electrons diffuse from the n-side (where they are plentiful) to the p-side, while holes diffuse in the opposite direction. This diffusion leaves behind ionized dopant atoms: positive donor ions on the n-side and negative acceptor ions on the p-side. These fixed charges create an electric field that opposes further diffusion.

The region depleted of mobile carriers is called the depletion region or space charge region. In equilibrium, the drift current due to the built-in field exactly balances the diffusion current, resulting in zero net current. The built-in potential for a silicon p-n junction at room temperature is typically 0.6 to 0.7 V, depending on doping levels.

Forward and Reverse Bias

Applying a voltage across the junction changes the balance between drift and diffusion. Forward bias (positive voltage on p-side) reduces the barrier, allowing more carriers to diffuse across the junction and producing significant current. The current increases exponentially with voltage according to the ideal diode equation:

I = I_s * (exp(qV/nkT) - 1)

Where I_s is the saturation current (typically 10^-12 to 10^-15 A for silicon diodes) and n is the ideality factor (1 for ideal diodes, 1-2 in practice).

Reverse bias (negative voltage on p-side) increases the barrier, reducing current to the small saturation current caused by thermally generated minority carriers. The depletion region widens under reverse bias, a property exploited in varactor diodes for voltage-controlled capacitance.

Junction Capacitance

The depletion region acts as a parallel-plate capacitor, with the ionized dopants forming the "plates" and the depleted semiconductor as the dielectric. The depletion capacitance depends on voltage:

C_j = C_j0 / (1 - V/V_bi)^m

Where C_j0 is the zero-bias capacitance, V_bi is the built-in potential, and m is approximately 0.5 for abrupt junctions. This voltage-dependent capacitance is fundamental to varactor operation and must be considered in high-frequency circuit design.

Quantum Mechanical Foundations

Classical physics cannot explain semiconductor behavior. Quantum mechanics provides the theoretical framework for understanding why energy bands exist, why electrons can tunnel through thin barriers, and why semiconductor properties are so sensitive to crystal structure and purity.

Wave-Particle Duality

Electrons exhibit both wave and particle properties. In a semiconductor crystal, electron waves interact with the periodic potential of the lattice, leading to allowed and forbidden energy bands. The electron wavelength, called the de Broglie wavelength, becomes comparable to atomic dimensions at semiconductor carrier energies, making quantum effects essential.

Quantum Tunneling

Quantum mechanics allows electrons to penetrate through potential barriers that would be insurmountable according to classical physics. This tunneling effect is important in thin oxide layers, heavily doped p-n junctions (tunnel diodes), and flash memory operation. Tunneling current increases exponentially as barrier thickness decreases, which becomes significant when oxide layers are only a few nanometers thick in modern transistors.

Fermi-Dirac Statistics

Electrons are fermions and obey Fermi-Dirac statistics, which determines the probability that an energy state is occupied. The Fermi level is the energy at which the occupation probability is exactly 50%. In semiconductors, the position of the Fermi level relative to the band edges determines carrier concentrations and is shifted by doping. Understanding Fermi level behavior is essential for analyzing junctions, contacts, and device operation.

Generation and Recombination

Carriers are continuously being generated (moving from valence to conduction band) and recombining (electrons falling back into holes). In thermal equilibrium, these rates are equal, maintaining constant carrier concentrations. Under non-equilibrium conditions such as illumination or current flow, generation and recombination dynamics determine device behavior.

Recombination Mechanisms

Three main recombination mechanisms occur in semiconductors:

Radiative recombination: The electron releases energy as a photon. This is the dominant mechanism in direct bandgap materials and is exploited in LEDs and laser diodes.
Auger recombination: The energy is transferred to another carrier rather than emitted as light. This becomes significant at high carrier concentrations.
Shockley-Read-Hall (SRH) recombination: Recombination occurs through defect states in the bandgap. This is often the dominant mechanism in indirect bandgap materials like silicon and is why crystal quality affects device performance.

Carrier Lifetime and Diffusion Length

The average time a carrier survives before recombining is called the carrier lifetime, typically ranging from nanoseconds to milliseconds depending on material quality and type. The diffusion length is the average distance a carrier diffuses before recombining:

L = sqrt(D * tau)

Where D is the diffusion coefficient and tau is the lifetime. The diffusion length is critical for devices like solar cells, where carriers generated by light must reach a junction to be collected.

Temperature Effects

Semiconductor properties vary significantly with temperature, affecting device performance across operating conditions.

Intrinsic carrier concentration: Increases exponentially with temperature, approximately doubling every 11 degrees Celsius for silicon near room temperature.
Mobility: Generally decreases with increasing temperature due to increased phonon scattering, though at low temperatures ionized impurity scattering may dominate.
Bandgap: Decreases slightly with increasing temperature.
Junction characteristics: Forward voltage decreases approximately 2 mV per degree Celsius, and leakage current increases with temperature.

These temperature dependencies must be considered in circuit design and are exploited in temperature sensors while presenting challenges for precision circuits.

Applications of Semiconductor Physics

Understanding semiconductor physics enables the design and optimization of countless devices:

Transistors: MOSFETs and bipolar transistors rely on controlled carrier injection and transport in semiconductors.
Diodes: P-n junction physics explains rectification, zener breakdown, and LED light emission.
Solar cells: Light generates electron-hole pairs that are separated by the junction field and collected as current.
Photodetectors: Similar to solar cells but optimized for sensitivity and speed rather than power generation.
Integrated circuits: Combine many transistors, diodes, resistors, and capacitors on a single semiconductor chip.

Summary

Semiconductor physics provides the foundation for understanding how electronic devices work at a fundamental level. The key concepts include the band structure of crystalline materials, the behavior of electrons and holes as charge carriers, how doping controls conductivity, carrier transport through drift and diffusion, and p-n junction behavior. These principles, grounded in quantum mechanics and solid-state physics, explain why semiconductors have such versatile and controllable electrical properties.

As devices shrink and new materials emerge, semiconductor physics continues to evolve. Quantum effects become more prominent, new materials like gallium nitride and silicon carbide enable high-power and high-frequency applications, and the fundamental limits of semiconductor scaling drive innovation in device architecture. A solid understanding of semiconductor physics prepares engineers to work with current technology and adapt to future developments.