7.1 Intrinsic semiconductors

Semiconductors are the backbone of modern electronics, with their unique band structure enabling precise control of electrical properties. This section explores intrinsic semiconductors, focusing on their band gaps, carrier generation, and Fermi levels.

Understanding intrinsic semiconductors is crucial for grasping more complex semiconductor physics. We'll examine how temperature affects carrier concentration, electrical conductivity, and optical properties, laying the groundwork for applications in solar cells, LEDs, and photodetectors.

Band structure of semiconductors

• Condensed matter physics explores the electronic properties of materials, with semiconductors playing a crucial role in modern technology
• Band structure forms the foundation for understanding semiconductor behavior, describing allowed energy states for electrons within the material

Valence and conduction bands

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• Valence band represents the highest occupied energy level in semiconductors at absolute zero temperature
• Conduction band denotes the lowest unoccupied energy level where electrons can freely move and contribute to electrical conduction
• Energy separation between these bands determines the electrical properties of the material (metals, semiconductors, insulators)
• Electrons in the valence band are tightly bound to atoms, while those in the conduction band are delocalized

Band gap characteristics

• Band gap defines the energy difference between the top of the valence band and the bottom of the conduction band
• Measured in electron volts (eV), typical semiconductor band gaps range from 0.1 to 4 eV
• Determines the minimum energy required for electronic excitation and influences optical and electrical properties
• Band gap engineering allows for tailoring semiconductor properties for specific applications (optoelectronics, photovoltaics)

Direct vs indirect band gaps

• Direct band gap materials have the conduction band minimum aligned with the valence band maximum in momentum space
• Indirect band gap semiconductors have misaligned band extrema, requiring phonon assistance for transitions
• Direct band gap materials (GaAs) exhibit stronger light absorption and emission, making them ideal for optoelectronic devices
• Indirect band gap semiconductors (Si) generally have longer carrier lifetimes, beneficial for certain electronic applications

Electron-hole pair generation

• Electron-hole pair generation is a fundamental process in semiconductor physics, driving the operation of many electronic devices
• Understanding this mechanism is crucial for designing and optimizing semiconductor-based technologies in condensed matter systems

Thermal excitation process

• Occurs spontaneously due to thermal energy in the crystal lattice
• Probability of thermal excitation increases exponentially with temperature, following the Boltzmann distribution
• Generates intrinsic carriers in semiconductors, contributing to the material's conductivity
• At room temperature, silicon has approximately $10^{10}$ thermally generated electron-hole pairs per cubic centimeter

Optical excitation mechanisms

• Involves absorption of photons with energy greater than or equal to the band gap
• Creates electron-hole pairs by promoting electrons from the valence to the conduction band
• Photon energy determines the depth of carrier generation within the semiconductor
• Forms the basis for photovoltaic and photodetector devices

Carrier concentration equilibrium

• Steady-state condition where generation and recombination rates of electron-hole pairs balance
• Influenced by temperature, material properties, and external factors (light, electric fields)
• Described by the mass action law: $n_0p_0 = n_i^2$, where $n_0$ and $p_0$ are equilibrium electron and hole concentrations
• Deviations from equilibrium drive current flow in semiconductor devices

Intrinsic carrier concentration

• Intrinsic carrier concentration represents the number of electrons and holes present in a pure semiconductor at thermal equilibrium
• This parameter is fundamental to understanding semiconductor behavior and device performance in condensed matter physics

Temperature dependence

• Intrinsic carrier concentration increases exponentially with temperature
• Follows the relationship $n_i \propto T^{3/2} \exp(-E_g/2kT)$, where $E_g$ is the band gap and $k$ is Boltzmann's constant
• Higher temperatures lead to more thermal excitations across the band gap
• Silicon's intrinsic carrier concentration increases from $10^{10}$ cm⁻³ at room temperature to $10^{14}$ cm⁻³ at 150°C

Effective mass influence

• Effective mass of electrons and holes affects the density of states and carrier mobility
• Lighter effective masses result in higher intrinsic carrier concentrations
• Anisotropic effective masses in some semiconductors lead to direction-dependent electronic properties
• GaAs has a lower electron effective mass than Si, contributing to its higher electron mobility

Density of states calculation

• Density of states (DOS) describes the number of available energy states per unit energy and volume
• For a 3D semiconductor, DOS is proportional to $E^{1/2}$ near band edges
• Intrinsic carrier concentration depends on the integral of DOS multiplied by the Fermi-Dirac distribution
• Accurate DOS calculations are crucial for predicting carrier concentrations and device performance

Fermi level in intrinsic semiconductors

• Fermi level concept is central to understanding electron distribution in semiconductors and other condensed matter systems
• In intrinsic semiconductors, the Fermi level position provides insights into carrier concentrations and material properties

Fermi-Dirac distribution

• Describes the probability of electron occupancy for a given energy state at thermal equilibrium
• Given by $f(E) = \frac{1}{1 + \exp((E-E_F)/kT)}$, where $E_F$ is the Fermi energy
• Accounts for the Pauli exclusion principle and quantum mechanical effects
• At T = 0 K, the distribution becomes a step function with all states below $E_F$ occupied and all above empty

Intrinsic Fermi level position

• Located near the middle of the band gap in intrinsic semiconductors
• Determined by the condition of charge neutrality: $n = p = n_i$
• Expressed as $E_F = \frac{E_c + E_v}{2} + \frac{3}{4}kT\ln(\frac{m_h^*}{m_e^*})$, where $m_h^*$ and $m_e^*$ are effective masses
• Slight deviation from exact midgap due to differences in effective masses and density of states

Temperature effects on Fermi level

• Fermi level shifts towards the middle of the band gap as temperature increases
• Rate of shift depends on the band gap and effective masses of the semiconductor
• Influences the temperature dependence of carrier concentrations and device characteristics
• In silicon, the intrinsic Fermi level moves by approximately 0.3 meV/K towards midgap with increasing temperature

Electrical conductivity

• Electrical conductivity is a key property in condensed matter physics, determining how materials respond to applied electric fields
• Understanding conductivity mechanisms in semiconductors is crucial for designing and optimizing electronic devices

Mobility of carriers

• Describes the ease with which charge carriers move through the semiconductor lattice
• Defined as the ratio of drift velocity to applied electric field: $\mu = v_d / E$
• Influenced by various scattering mechanisms (phonon, impurity, carrier-carrier)
• Electron mobility in silicon at room temperature is approximately 1400 cm²/(V·s), while hole mobility is about 450 cm²/(V·s)

Drift and diffusion currents

• Drift current results from carrier motion due to an applied electric field
• Diffusion current arises from carrier concentration gradients within the material
• Total current density is the sum of drift and diffusion components: $J = J_{drift} + J_{diffusion}$
• Drift-diffusion model forms the basis for many semiconductor device simulations

Conductivity vs temperature

• Conductivity in intrinsic semiconductors increases with temperature due to increased carrier concentration
• Follows an Arrhenius-like relationship: $\sigma \propto \exp(-E_g/2kT)$
• At very high temperatures, conductivity may decrease due to increased phonon scattering
• Silicon's conductivity increases by about an order of magnitude for every 50°C increase in temperature near room temperature

Optical properties

• Optical properties of semiconductors are essential for understanding light-matter interactions in condensed matter systems
• These properties form the basis for various optoelectronic devices and applications in modern technology

Absorption spectrum

• Describes how light of different wavelengths is absorbed by the semiconductor
• Absorption coefficient increases sharply for photon energies above the band gap
• Indirect band gap materials (Si) have weaker absorption near the band edge compared to direct band gap materials (GaAs)
• Absorption spectrum can be used to determine the band gap and electronic structure of semiconductors

Photoluminescence

• Process where a semiconductor emits light after absorbing photons
• Provides information about electronic states, impurities, and recombination mechanisms
• Spectrum shape and intensity depend on material quality and temperature
• Used as a non-destructive characterization technique in semiconductor research and quality control

Radiative recombination

• Direct process where an electron in the conduction band recombines with a hole in the valence band, emitting a photon
• Efficiency depends on the band structure and competing non-radiative processes
• More prominent in direct band gap semiconductors (GaAs) compared to indirect band gap materials (Si)
• Forms the basis for light-emitting diodes (LEDs) and semiconductor lasers

Common intrinsic semiconductors

• Intrinsic semiconductors play a crucial role in condensed matter physics research and technological applications
• Understanding the properties of various semiconductor materials enables the design of advanced electronic and optoelectronic devices

Silicon vs germanium

• Silicon dominates the semiconductor industry due to its abundance and suitable band gap (1.12 eV at 300 K)
• Germanium has a smaller band gap (0.66 eV at 300 K) and higher carrier mobilities
• Si forms a stable oxide (SiO₂), crucial for MOSFET technology, while Ge lacks a stable native oxide
• Ge has higher intrinsic carrier concentration, making it more sensitive to temperature variations

Compound semiconductors

• Formed by combining elements from groups III and V (GaAs, InP) or II and VI (CdTe, ZnSe) of the periodic table
• Offer a wide range of band gaps and electronic properties not available in elemental semiconductors
• GaAs (1.42 eV band gap) is widely used in high-speed electronics and optoelectronics
• InP-based compounds find applications in long-wavelength optical communication systems

Wide band gap materials

• Semiconductors with band gaps significantly larger than silicon (typically > 2 eV)
• Include materials like GaN (3.4 eV), SiC (3.3 eV), and diamond (5.5 eV)
• Offer advantages in high-power, high-temperature, and high-frequency applications
• Exhibit strong chemical bonds, making them resistant to radiation damage and chemical degradation

Intrinsic vs extrinsic comparison

• Comparing intrinsic and extrinsic semiconductors is essential for understanding how doping affects material properties
• This comparison forms the basis for designing and optimizing various semiconductor devices in condensed matter physics

Carrier concentration differences

• Intrinsic semiconductors have equal electron and hole concentrations (n = p = nᵢ)
• Extrinsic semiconductors have unequal carrier concentrations due to intentional doping
• n-type materials have electron concentrations much higher than hole concentrations (n >> p)
• p-type materials have hole concentrations much higher than electron concentrations (p >> n)

Fermi level position

• Intrinsic Fermi level lies near the middle of the band gap
• In n-type semiconductors, Fermi level shifts towards the conduction band
• p-type materials have Fermi levels closer to the valence band
• Fermi level position determines the equilibrium carrier concentrations and influences device behavior

Electrical characteristics

• Intrinsic semiconductors have lower conductivity compared to extrinsic ones at room temperature
• Extrinsic semiconductors offer better control over electrical properties through doping
• Temperature dependence of conductivity is stronger in intrinsic semiconductors
• Extrinsic semiconductors maintain more stable electrical characteristics over a wider temperature range

Applications and devices

• Semiconductor applications form the backbone of modern electronics and optoelectronics industries
• Understanding device physics is crucial for advancing condensed matter research and developing new technologies

Solar cells

• Convert light energy into electrical energy using the photovoltaic effect
• Efficiency depends on material properties, cell design, and light spectrum
• Silicon dominates the market, with single-junction cell efficiencies reaching 26%
• Multi-junction cells using III-V compounds achieve efficiencies over 40% under concentrated sunlight

Photodetectors

• Convert optical signals into electrical signals for various sensing applications
• Types include photodiodes, phototransistors, and avalanche photodiodes
• Spectral response depends on the semiconductor band gap and device structure
• InGaAs photodetectors are widely used in optical communication systems for near-infrared detection

Light-emitting diodes

• Emit light when an electric current passes through a forward-biased p-n junction
• Color of emitted light depends on the semiconductor band gap
• GaN-based LEDs have revolutionized solid-state lighting with high-efficiency blue and white light emission
• Organic LEDs (OLEDs) offer flexible, thin displays for mobile devices and televisions