1.5 Intrinsic and extrinsic semiconductors

Semiconductors are the backbone of modern electronics. They come in two flavors: intrinsic and extrinsic. Intrinsic semiconductors are pure materials with equal electron and hole concentrations. Their conductivity increases with temperature as more electron-hole pairs form.

Extrinsic semiconductors are doped with impurities to control their electrical properties. N-type semiconductors have excess electrons, while p-type have excess holes. This allows precise tuning of conductivity and enables the creation of various electronic devices.

Intrinsic semiconductors

• Intrinsic semiconductors are pure semiconductor materials without any intentional doping or impurities
• At absolute zero temperature, intrinsic semiconductors behave as insulators due to the completely filled valence band and empty conduction band
• As temperature increases, electrons can gain enough thermal energy to overcome the bandgap and transition from the valence band to the conduction band, creating electron-hole pairs

Energy band structure

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• Intrinsic semiconductors have a valence band and a conduction band separated by an energy bandgap ($E_g$)
• The valence band is the highest occupied energy band at absolute zero temperature, while the conduction band is the lowest unoccupied energy band
• The bandgap determines the minimum energy required for electrons to transition from the valence band to the conduction band
• Examples of intrinsic semiconductors include silicon (Si) and germanium (Ge) with bandgaps of 1.12 eV and 0.67 eV, respectively

Electron and hole concentrations

• In intrinsic semiconductors, the concentration of electrons in the conduction band ($n_i$) is equal to the concentration of holes in the valence band ($p_i$)
• The intrinsic carrier concentration is given by: $n_i = p_i = \sqrt{N_c N_v} \exp(-E_g/2k_BT)$, where $N_c$ and $N_v$ are the effective density of states in the conduction and valence bands, $k_B$ is the Boltzmann constant, and $T$ is the absolute temperature
• At room temperature, the intrinsic carrier concentration for silicon is approximately $1.5 \times 10^{10} cm^{-3}$

Fermi level position

• The Fermi level ($E_F$) is the energy level at which the probability of an electron occupying a state is 0.5
• In intrinsic semiconductors, the Fermi level lies near the middle of the bandgap, approximately halfway between the conduction and valence band edges
• The exact position of the Fermi level depends on the effective density of states in the conduction and valence bands: $E_F = \frac{E_c + E_v}{2} + \frac{k_BT}{2} \ln(\frac{N_v}{N_c})$, where $E_c$ and $E_v$ are the conduction and valence band edges

Temperature dependence

• The intrinsic carrier concentration strongly depends on temperature due to the exponential term in the equation $n_i = p_i = \sqrt{N_c N_v} \exp(-E_g/2k_BT)$
• As temperature increases, more electrons gain sufficient thermal energy to overcome the bandgap, leading to an exponential increase in the intrinsic carrier concentration
• The temperature dependence of the intrinsic carrier concentration is given by: $n_i(T) \propto T^{3/2} \exp(-E_g/2k_BT)$
• This temperature dependence is crucial for understanding the behavior of semiconductor devices at different operating temperatures

Carrier transport mechanisms

• In intrinsic semiconductors, charge transport occurs through two main mechanisms: drift and diffusion
• Drift current is the movement of charge carriers (electrons and holes) under the influence of an applied electric field, with the drift velocity proportional to the electric field strength and carrier mobility
• Diffusion current arises from the random thermal motion of carriers, causing them to move from regions of high concentration to regions of low concentration
• The total current in a semiconductor is the sum of the drift and diffusion currents: $J_{total} = J_{drift} + J_{diffusion}$
• The Einstein relation connects the diffusion coefficient ($D$) and the mobility ($\mu$) of charge carriers: $D/\mu = k_BT/q$, where $q$ is the elementary charge

Extrinsic semiconductors

• Extrinsic semiconductors are created by intentionally introducing impurities (dopants) into intrinsic semiconductors to control their electrical properties
• Doping allows for the creation of n-type and p-type semiconductors, which are essential for the fabrication of various semiconductor devices
• The introduction of dopants significantly alters the carrier concentrations, Fermi level position, and electrical conductivity of the semiconductor

n-type vs p-type doping

• n-type doping involves introducing impurity atoms with one more valence electron than the host semiconductor material (e.g., phosphorus in silicon)
• These extra electrons can easily be excited into the conduction band, making them the majority charge carriers in n-type semiconductors
• p-type doping involves introducing impurity atoms with one fewer valence electron than the host semiconductor material (e.g., boron in silicon)
• The missing electrons, or holes, become the majority charge carriers in p-type semiconductors

Donor and acceptor impurities

• Donor impurities are atoms with one more valence electron than the host semiconductor, such as phosphorus, arsenic, or antimony in silicon
• Donor impurities introduce energy levels close to the conduction band, making it easy for electrons to be excited into the conduction band
• Acceptor impurities are atoms with one fewer valence electron than the host semiconductor, such as boron, aluminum, or gallium in silicon
• Acceptor impurities introduce energy levels close to the valence band, making it easy for electrons to be excited from the valence band, leaving behind holes

Majority and minority carriers

• In n-type semiconductors, electrons are the majority carriers, while holes are the minority carriers
• The electron concentration in n-type semiconductors is much higher than the intrinsic carrier concentration: $n \gg n_i$
• In p-type semiconductors, holes are the majority carriers, while electrons are the minority carriers
• The hole concentration in p-type semiconductors is much higher than the intrinsic carrier concentration: $p \gg n_i$

Fermi level shifts

• Doping alters the position of the Fermi level in semiconductors
• In n-type semiconductors, the Fermi level shifts towards the conduction band, as the additional electrons from donor impurities occupy energy states close to the conduction band edge
• In p-type semiconductors, the Fermi level shifts towards the valence band, as the additional holes from acceptor impurities occupy energy states close to the valence band edge
• The shift in the Fermi level depends on the doping concentration and the temperature

Carrier concentration control

• The carrier concentration in extrinsic semiconductors can be controlled by adjusting the doping concentration
• The majority carrier concentration is approximately equal to the doping concentration: $n \approx N_D$ for n-type and $p \approx N_A$ for p-type semiconductors, where $N_D$ and $N_A$ are the donor and acceptor concentrations, respectively
• The minority carrier concentration can be calculated using the law of mass action: $np = n_i^2$, where $n$ and $p$ are the electron and hole concentrations, and $n_i$ is the intrinsic carrier concentration

Compensation doping

• Compensation doping occurs when both donor and acceptor impurities are present in a semiconductor
• The presence of both types of impurities can reduce the overall effectiveness of the doping process
• If the concentration of donor and acceptor impurities is equal, the semiconductor becomes intrinsic-like, as the effects of the impurities cancel each other out
• Compensation doping can be used to fine-tune the electrical properties of semiconductors

Electrical properties

• The electrical properties of semiconductors, such as conductivity, resistivity, and carrier mobility, are crucial for understanding their behavior in electronic devices
• These properties depend on various factors, including the bandgap, carrier concentrations, temperature, and scattering mechanisms
• Measuring and controlling the electrical properties of semiconductors is essential for the design and optimization of semiconductor devices

Conductivity and resistivity

• Conductivity ($\sigma$) is a measure of a material's ability to conduct electric current, while resistivity ($\rho$) is the reciprocal of conductivity and represents the material's resistance to current flow
• In semiconductors, conductivity depends on the carrier concentrations and mobilities: $\sigma = q(n\mu_n + p\mu_p)$, where $q$ is the elementary charge, $n$ and $p$ are the electron and hole concentrations, and $\mu_n$ and $\mu_p$ are the electron and hole mobilities, respectively
• The resistivity of a semiconductor is given by: $\rho = 1/\sigma$
• The conductivity of extrinsic semiconductors is significantly higher than that of intrinsic semiconductors due to the increased carrier concentrations

Carrier mobility

• Carrier mobility ($\mu$) is a measure of how easily charge carriers (electrons and holes) can move through a semiconductor under the influence of an electric field
• Mobility depends on various factors, such as the effective mass of the carriers, the temperature, and the scattering mechanisms
• The electron mobility is generally higher than the hole mobility in semiconductors due to the lower effective mass of electrons
• In silicon at room temperature, the electron mobility is approximately 1,400 cm²/V·s, while the hole mobility is around 450 cm²/V·s

Hall effect measurements

• The Hall effect is a powerful technique for measuring the carrier concentration, mobility, and conductivity type (n-type or p-type) of semiconductors
• When a magnetic field is applied perpendicular to the current flow in a semiconductor, a transverse voltage (Hall voltage) develops due to the deflection of charge carriers
• The Hall coefficient ($R_H$) is given by: $R_H = \frac{E_y}{J_xB_z} = \frac{1}{nq}$ for n-type semiconductors and $R_H = \frac{E_y}{J_xB_z} = \frac{-1}{pq}$ for p-type semiconductors, where $E_y$ is the Hall electric field, $J_x$ is the current density, and $B_z$ is the magnetic field
• By measuring the Hall voltage and knowing the current, magnetic field, and sample geometry, the carrier concentration and mobility can be determined

Temperature dependence

• The electrical properties of semiconductors exhibit a strong temperature dependence due to the variation in carrier concentrations and mobilities with temperature
• As temperature increases, the intrinsic carrier concentration increases exponentially, leading to higher conductivity in intrinsic semiconductors
• In extrinsic semiconductors, the carrier concentration is relatively temperature-independent at low temperatures (freeze-out regime) and high temperatures (exhaustion regime), but it increases in the intermediate temperature range (extrinsic regime) due to the ionization of dopants
• Carrier mobility generally decreases with increasing temperature due to increased scattering from lattice vibrations (phonons)

Scattering mechanisms

• Scattering mechanisms are processes that impede the motion of charge carriers in semiconductors, limiting their mobility
• The main scattering mechanisms in semiconductors are lattice (phonon) scattering, ionized impurity scattering, and neutral impurity scattering
• Lattice scattering arises from the interaction of carriers with lattice vibrations (phonons) and is the dominant scattering mechanism at high temperatures
• Ionized impurity scattering occurs when carriers interact with the Coulomb potential of ionized dopant atoms and is more significant at low temperatures and high doping concentrations
• Neutral impurity scattering is caused by the interaction of carriers with neutral impurity atoms and is generally less significant than the other scattering mechanisms

Optical properties

• The optical properties of semiconductors, such as absorption, emission, and luminescence, are closely related to their electronic band structure and carrier dynamics
• Understanding and controlling the optical properties of semiconductors is crucial for the development of optoelectronic devices, such as solar cells, light-emitting diodes (LEDs), and lasers
• The interaction of light with semiconductors can lead to various phenomena, including photoconductivity and photoluminescence

Absorption and emission

• Absorption occurs when a photon with energy greater than the bandgap is absorbed by a semiconductor, exciting an electron from the valence band to the conduction band and creating an electron-hole pair
• The absorption coefficient ($\alpha$) depends on the photon energy and the band structure of the semiconductor
• Emission occurs when an electron in the conduction band recombines with a hole in the valence band, releasing a photon with energy equal to the bandgap
• The emission process can be spontaneous (random) or stimulated (coherent) and is the basis for light-emitting devices like LEDs and lasers

Direct vs indirect bandgaps

• Semiconductors can have direct or indirect bandgaps, depending on the relative positions of the conduction band minimum and valence band maximum in the crystal momentum (k-space) diagram
• In direct bandgap semiconductors (e.g., GaAs), the conduction band minimum and valence band maximum occur at the same crystal momentum, allowing for efficient absorption and emission of photons
• In indirect bandgap semiconductors (e.g., Si), the conduction band minimum and valence band maximum occur at different crystal momenta, requiring the involvement of phonons (lattice vibrations) for absorption and emission processes
• Direct bandgap semiconductors are generally more efficient for optoelectronic devices due to their higher absorption and emission coefficients

Photoconductivity

• Photoconductivity is the increase in electrical conductivity of a semiconductor when exposed to light
• When photons with energy greater than the bandgap are absorbed, electron-hole pairs are generated, increasing the concentration of free carriers and, consequently, the conductivity
• The photoconductivity $\Delta\sigma$ is proportional to the incident light intensity $I$ and the carrier generation rate $G$: $\Delta\sigma = q(\mu_n + \mu_p)G\tau$, where $\tau$ is the carrier lifetime
• Photoconductivity is the basis for various optoelectronic devices, such as photodetectors and solar cells

Luminescence mechanisms

• Luminescence is the emission of light from a semiconductor due to the recombination of electron-hole pairs
• There are several luminescence mechanisms, including band-to-band recombination, excitonic recombination, and impurity-related recombination
• Band-to-band recombination occurs when an electron in the conduction band directly recombines with a hole in the valence band, emitting a photon with energy equal to the bandgap
• Excitonic recombination involves the recombination of bound electron-hole pairs (excitons), which have a slightly lower energy than the bandgap due to the attractive Coulomb interaction between the electron and hole
• Impurity-related recombination occurs when electrons and holes recombine through energy levels introduced by impurities or defects in the semiconductor

Quantum efficiency

• Quantum efficiency is a measure of the effectiveness of a semiconductor in converting incident photons into electron-hole pairs (for photodetectors) or converting electron-hole pairs into emitted photons (for light-emitting devices)
• The external quantum efficiency (EQE) is the ratio of the number of charge carriers collected (or photons emitted) to the number of incident photons
• The internal quantum efficiency (IQE) is the ratio of the number of charge carriers collected (or photons emitted) to the number of absorbed photons
• Factors that can reduce the quantum efficiency include reflection losses, recombination losses, and carrier trapping
• Improving the quantum efficiency is a key goal in the design and optimization of optoelectronic devices

Applications

• Semiconductors' unique electrical and optical properties make them essential materials for a wide range of electronic and optoelectronic applications
• The ability to control the conductivity, carrier concentrations, and bandgap of semiconductors through doping and material engineering has led to the development of numerous devices that have revolutionized modern technology
• Some of the most important applications of semiconductors include solar cells, LEDs, lasers, and thermoelectric devices

Semiconductor devices

• Semiconductor devices are electronic components that exploit the electrical and optical properties of semiconductors to perform various functions
• Examples of semiconductor devices include diodes, transistors, solar cells, LEDs, and lasers
• Diodes are two-terminal devices that allow current to flow in one direction but block it in the reverse direction, making them useful for rectification, voltage regulation, and protection
• Transistors are three-terminal devices that can amplify or switch electronic signals, forming the basis for modern electronic circuits and computing devices
• Solar cells, LEDs, and lasers are optoelectronic devices that convert light into electricity or electricity into light

Photovoltaic cells

• Photovoltaic cells, also known as solar cells, are devices that convert sunlight directly into electricity through the photovoltaic effect
• When photons with energy greater than the bandgap are absorbed by a semiconductor (usually silicon), electron-hole pairs are generated and separated by an internal electric field, creating a photocurrent and photovoltage
• The internal electric field is typically created by forming a p-n junction, which is the interface between n-type and p-type regions of the semiconductor
• The efficiency of solar cells depends on factors such as the bandgap, absorption coefficient, carrier mobility, and recombination rates
• Improving solar cell efficiency and reducing manufacturing costs are key goals in the development of photovoltaic technology

Light-emitting diodes (LEDs)

• Light-emitting diodes (LEDs) are semiconductor devices that