7.1 Intrinsic semiconductors

Band structure of semiconductors

The band structure of a semiconductor describes the allowed energy states for electrons in the material. It's the starting point for understanding why semiconductors behave differently from metals and insulators.

Valence and conduction bands

The valence band is the highest energy band that's fully occupied by electrons at absolute zero. Electrons here are tightly bound to their parent atoms and don't contribute to electrical conduction.

The conduction band is the lowest energy band that's empty at absolute zero. Any electron that reaches this band is delocalized and free to carry current.

The energy separation between these two bands is what distinguishes materials:

• Metals have overlapping bands (or a partially filled band), so electrons conduct freely
• Semiconductors have a moderate gap (roughly 0.1 to 4 eV), so some electrons can be thermally promoted across it
• Insulators have a large gap (> 4 eV), making thermal excitation negligible at normal temperatures

Band gap characteristics

The band gap ($E_g$) is the energy difference between the top of the valence band and the bottom of the conduction band. It's measured in electron volts (eV) and sets the minimum energy needed to excite an electron into the conduction band.

• Typical semiconductor band gaps range from about 0.1 eV to 4 eV
• The band gap controls both optical properties (which wavelengths of light are absorbed) and electrical properties (how many carriers are thermally generated)
• Band gap engineering refers to tuning $E_g$ through alloying or heterostructures to tailor a material for specific applications like photovoltaics or optoelectronics

Direct vs. indirect band gaps

This distinction matters for how electrons transition between bands:

• In a direct band gap semiconductor, the conduction band minimum and valence band maximum occur at the same crystal momentum ($k$-value). An electron can transition directly by absorbing or emitting a photon. GaAs is the classic example.
• In an indirect band gap semiconductor, the band extrema are at different $k$-values. Transitions require a phonon (lattice vibration) to conserve momentum in addition to a photon. Silicon is the most important example.

The practical consequence: direct band gap materials absorb and emit light much more efficiently, making them the go-to choice for LEDs and laser diodes. Indirect band gap materials like Si have weaker optical transitions but tend to have longer carrier lifetimes, which is useful in certain electronic applications.

Electron-hole pair generation

When an electron gains enough energy to jump from the valence band to the conduction band, it leaves behind a vacancy called a hole. The hole behaves as a positive charge carrier. This electron-hole pair creation is the fundamental process that gives semiconductors their useful properties.

Thermal excitation process

Even without any external light source, thermal energy from the crystal lattice can promote electrons across the band gap.

• The probability of excitation follows the Boltzmann distribution and increases exponentially with temperature
• These thermally generated carriers are called intrinsic carriers
• At room temperature (300 K), silicon has approximately $10^{10}$ electron-hole pairs per cm³, which is tiny compared to the $\sim 10^{22}$ atoms per cm³ in the crystal

Optical excitation mechanisms

Photons with energy $E_{photon} \geq E_g$ can be absorbed to create electron-hole pairs.

• The electron is promoted from the valence band to the conduction band, and a hole is left behind
• Higher-energy photons create carriers deeper within the bands (with excess kinetic energy that quickly thermalizes)
• This process is the operating principle behind photovoltaic cells and photodetectors

Carrier concentration equilibrium

In steady state, electron-hole pairs are constantly being generated and recombining. Equilibrium is reached when these two rates balance.

This is described by the mass action law:

$n_0 p_0 = n_i^2$

where $n_0$ is the equilibrium electron concentration, $p_0$ is the equilibrium hole concentration, and $n_i$ is the intrinsic carrier concentration. This relation holds for both intrinsic and extrinsic semiconductors at thermal equilibrium. Any deviation from equilibrium (from light injection or applied voltage, for instance) drives current flow.

Intrinsic carrier concentration

The intrinsic carrier concentration ($n_i$) is the number of electrons (or equivalently, holes) per unit volume in a pure, undoped semiconductor at thermal equilibrium. It's one of the most important parameters in semiconductor physics because it sets the baseline for all carrier concentration calculations.

Temperature dependence

$n_i$ increases exponentially with temperature:

$n_i \propto T^{3/2} \exp\left(\frac{-E_g}{2kT}\right)$

where $E_g$ is the band gap, $k$ is Boltzmann's constant, and $T$ is absolute temperature. The $T^{3/2}$ prefactor comes from the density of states, but the exponential term dominates the behavior.

• For silicon: $n_i \approx 10^{10}$ cm$^{-3}$ at 300 K, rising to roughly $10^{14}$ cm$^{-3}$ at 150°C
• This steep temperature dependence is why semiconductor devices can fail or behave unpredictably at high temperatures

Effective mass influence

Electrons and holes in a crystal don't respond to forces like free particles. Their response is captured by the effective mass ($m_e^*$ for electrons, $m_h^*$ for holes), which reflects the band curvature.

• Lighter effective masses produce higher densities of states near the band edges and higher carrier mobilities
• GaAs has a much lower electron effective mass than Si, which is why GaAs has higher electron mobility
• Some semiconductors have anisotropic effective masses, meaning carrier properties depend on crystal direction

Density of states calculation

The density of states (DOS) gives the number of available energy states per unit energy per unit volume. Near the band edges of a 3D semiconductor:

$DOS(E) \propto E^{1/2}$

To find $n_i$, you integrate the product of the DOS and the Fermi-Dirac occupation probability across the conduction band (for electrons) or valence band (for holes). The result is what gives you the full expression for $n_i$ including the effective masses and temperature dependence.

Fermi level in intrinsic semiconductors

The Fermi level ($E_F$) is the energy at which the probability of electron occupation is exactly 50%. It doesn't need to correspond to an actual allowed state. In intrinsic semiconductors, its position tells you that electron and hole concentrations are equal.

Fermi-Dirac distribution

The probability that a state at energy $E$ is occupied by an electron is:

$f(E) = \frac{1}{1 + \exp\left(\frac{E - E_F}{kT}\right)}$

• At $T = 0$ K, this is a sharp step function: all states below $E_F$ are filled, all above are empty
• At finite temperature, the step broadens. States within a few $kT$ of $E_F$ have partial occupation
• This distribution respects the Pauli exclusion principle and is the correct statistics for fermions

Intrinsic Fermi level position

For an intrinsic semiconductor, charge neutrality requires $n = p = n_i$. Solving this condition gives:

$E_F = \frac{E_c + E_v}{2} + \frac{3}{4}kT \ln\left(\frac{m_h^*}{m_e^*}\right)$

where $E_c$ and $E_v$ are the conduction and valence band edges.

• The first term places $E_F$ at the exact midgap
• The second term is a correction due to the difference in electron and hole effective masses
• If $m_h^* = m_e^*$, the Fermi level sits exactly at midgap. In practice, the correction is small (on the order of $kT$), so the intrinsic Fermi level is always close to the center of the gap.

Temperature effects on Fermi level

As temperature rises, the intrinsic Fermi level shifts slightly toward the true midgap. The rate of this shift depends on the band gap and the ratio of effective masses. In silicon, the shift is approximately 0.3 meV/K. While small, this matters when you're calculating precise carrier concentrations at elevated temperatures.

Electrical conductivity

Conductivity ($\sigma$) quantifies how easily current flows through a material. In intrinsic semiconductors, both electrons and holes contribute.

Mobility of carriers

Mobility ($\mu$) measures how quickly a carrier moves in response to an electric field:

$\mu = \frac{v_d}{E}$

where $v_d$ is the drift velocity and $E$ is the applied electric field.

• Mobility is limited by scattering: lattice vibrations (phonons), impurities, and carrier-carrier interactions all slow carriers down
• In silicon at 300 K: electron mobility $\approx 1400$ cm²/(V·s), hole mobility $\approx 450$ cm²/(V·s)
• Electrons are generally more mobile than holes because the conduction band typically has lighter effective masses

Drift and diffusion currents

Two mechanisms drive current in semiconductors:

• Drift current: carriers move in response to an applied electric field. $J_{drift} = \sigma E$
• Diffusion current: carriers move from regions of high concentration to low concentration, driven by concentration gradients

The total current density combines both:

$J = J_{drift} + J_{diffusion}$

The drift-diffusion model is the workhorse framework for most semiconductor device simulations.

Conductivity vs. temperature

For intrinsic semiconductors, conductivity follows an Arrhenius-like relationship:

$\sigma \propto \exp\left(\frac{-E_g}{2kT}\right)$

The exponential increase in carrier concentration with temperature dominates, so conductivity rises steeply with heating. Silicon's conductivity increases by roughly an order of magnitude for every 50°C increase near room temperature.

At very high temperatures, increased phonon scattering starts to reduce mobility, which can partially offset the rising carrier concentration. But for most practical temperature ranges in intrinsic material, the carrier concentration effect wins.

Optical properties

How semiconductors interact with light is governed by their band structure. These optical properties are the foundation of devices like solar cells, LEDs, and photodetectors.

Absorption spectrum

• The absorption coefficient rises sharply for photon energies above $E_g$. Below the band gap, the material is largely transparent.
• Direct band gap materials (GaAs) have a steep absorption edge and strong absorption near $E_g$
• Indirect band gap materials (Si) have a more gradual absorption onset because transitions require phonon assistance
• Measuring the absorption spectrum is a standard way to determine a semiconductor's band gap experimentally

Photoluminescence

Photoluminescence (PL) occurs when a semiconductor absorbs photons and then re-emits light as excited carriers recombine.

• The emitted photon energy is typically close to $E_g$
• PL spectra reveal information about electronic states, defects, and impurities
• Spectrum shape and intensity are sensitive to temperature and material quality
• PL is widely used as a non-destructive characterization tool in research and manufacturing

Radiative recombination

This is the process where an electron in the conduction band recombines with a hole in the valence band by emitting a photon.

• Much more efficient in direct band gap materials (GaAs, GaN) where momentum conservation is easily satisfied
• In indirect band gap materials (Si), non-radiative recombination pathways (Auger, Shockley-Read-Hall) tend to dominate
• Radiative recombination is the operating mechanism behind LEDs and semiconductor lasers

Common intrinsic semiconductors

Different semiconductor materials offer different trade-offs in band gap, mobility, thermal stability, and ease of fabrication.

Silicon vs. germanium

Silicon dominates the industry largely because SiO₂ forms a high-quality, stable insulating layer, which is essential for MOSFET gate oxides. Silicon is also far more abundant. Germanium's smaller band gap gives it higher intrinsic carrier concentration, making it more temperature-sensitive but also useful in infrared detectors and high-speed transistors.

Compound semiconductors

These are formed by combining elements from different groups of the periodic table:

• III-V compounds (GaAs, InP, GaN): GaAs has a 1.42 eV direct band gap, making it excellent for high-speed electronics and optoelectronics. InP-based materials are standard in long-wavelength fiber-optic communication.
• II-VI compounds (CdTe, ZnSe): CdTe is used in thin-film solar cells. ZnSe has applications in blue-green optoelectronics.

Compound semiconductors offer a much wider range of band gaps and can be alloyed to tune properties continuously (e.g., $\text{In}_x\text{Ga}_{1-x}\text{As}$).

Wide band gap materials

Semiconductors with $E_g > 2$ eV are classified as wide band gap:

• GaN (3.4 eV): the material behind blue/white LEDs and high-power RF transistors
• SiC (3.3 eV): used in high-voltage power electronics and harsh-environment sensors
• Diamond (5.5 eV): the ultimate wide band gap material, with extreme thermal conductivity and hardness

These materials excel in high-power, high-temperature, and high-frequency applications because their large band gaps mean very few intrinsic carriers at operating temperatures, and their strong chemical bonds provide radiation hardness and chemical stability.

Intrinsic vs. extrinsic comparison

Intrinsic semiconductors are pure and undoped. Extrinsic semiconductors have been intentionally doped with impurity atoms to control carrier concentrations. This comparison highlights why doping is so central to device design.

Carrier concentration differences

• Intrinsic: $n = p = n_i$. Electron and hole concentrations are equal and relatively low.
• n-type (doped with donors like P or As in Si): $n \gg p$. Electrons are the majority carriers.
• p-type (doped with acceptors like B in Si): $p \gg n$. Holes are the majority carriers.
• The mass action law ($np = n_i^2$) still holds at equilibrium, so increasing one carrier type suppresses the other.

Fermi level position

• Intrinsic: $E_F$ sits near midgap
• n-type: $E_F$ shifts toward the conduction band (higher electron occupation)
• p-type: $E_F$ shifts toward the valence band (higher hole occupation)

The Fermi level position directly determines the equilibrium carrier concentrations through the Fermi-Dirac distribution, so knowing where $E_F$ sits tells you almost everything about the material's electrical behavior.

Electrical characteristics

• Intrinsic semiconductors have low conductivity at room temperature and a steep temperature dependence
• Extrinsic semiconductors have much higher conductivity that's controllable through doping concentration
• Extrinsic materials show a more stable conductivity over moderate temperature ranges because the dopant-supplied carriers dominate until temperatures get high enough for intrinsic carriers to become significant (the "intrinsic regime")

Applications and devices

The physics of intrinsic semiconductors underpins several major device categories. Even though most real devices use doped (extrinsic) material, the intrinsic properties set fundamental limits on performance.

Solar cells

• Convert photon energy into electrical energy via the photovoltaic effect
• Efficiency depends on band gap (optimal around 1.1–1.4 eV for single-junction cells), material quality, and cell architecture
• Crystalline silicon cells dominate the market, with record single-junction efficiencies around 26%
• Multi-junction cells using III-V compounds stack materials with different band gaps and achieve over 47% efficiency under concentrated sunlight

Photodetectors

• Convert optical signals to electrical signals for sensing and communication
• The semiconductor band gap determines the spectral response: smaller gaps detect longer wavelengths
• Types include photodiodes, phototransistors, and avalanche photodiodes (which provide internal gain)
• $\text{In}_{0.53}\text{Ga}_{0.47}\text{As}$ photodetectors are the standard for 1.3–1.55 μm fiber-optic communication

Light-emitting diodes

• Emit light when carriers recombine radiatively across a forward-biased p-n junction
• The emitted photon energy (and therefore color) is set by the band gap of the active material
• GaN-based LEDs produce blue and ultraviolet light; adding a phosphor coating converts this to white light for solid-state lighting
• Organic LEDs (OLEDs) use organic semiconductor layers and enable flexible, thin displays