3.2 Carrier concentration in semiconductors

Carrier concentration in semiconductors is a key concept that determines their electrical properties. It refers to the number of charge carriers per unit volume, which can be electrons or holes.

Understanding carrier concentration is crucial for designing and optimizing semiconductor devices. Factors like temperature, doping, and bandgap energy affect carrier concentration, influencing the behavior of transistors, solar cells, and LEDs.

Carrier concentration fundamentals

• Carrier concentration is a fundamental concept in semiconductor physics that describes the number of charge carriers (electrons and holes) per unit volume in a semiconductor material
• Understanding carrier concentration is crucial for designing and optimizing semiconductor devices used in various applications, such as transistors, solar cells, and light-emitting diodes (LEDs)
• The carrier concentration depends on several factors, including temperature, doping, and the material's bandgap energy

Intrinsic vs extrinsic semiconductors

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• Intrinsic semiconductors are pure materials without any intentional doping (silicon, germanium)
  • In intrinsic semiconductors, the number of electrons in the conduction band equals the number of holes in the valence band
  • The carrier concentration in intrinsic semiconductors is relatively low and strongly dependent on temperature
• Extrinsic semiconductors are intentionally doped with impurities to increase the carrier concentration and modify the material's electrical properties (n-type, p-type)
  • n-type semiconductors are doped with donor impurities that provide extra electrons to the conduction band (phosphorus, arsenic)
  • p-type semiconductors are doped with acceptor impurities that create extra holes in the valence band (boron, gallium)

Electrons and holes

• Electrons are negatively charged particles that occupy energy states in the conduction band of a semiconductor
  • When an electron gains enough energy to overcome the bandgap, it moves from the valence band to the conduction band, leaving behind a hole
  • Electrons in the conduction band are free to move and contribute to electrical conduction
• Holes are positively charged quasiparticles that represent the absence of electrons in the valence band
  • Holes can be thought of as "missing electrons" that can move through the valence band, contributing to electrical conduction
  • The movement of holes is equivalent to the movement of electrons in the opposite direction

Density of states

• The density of states (DOS) is a function that describes the number of available energy states per unit energy and per unit volume in a semiconductor
  • The DOS depends on the material's band structure and the effective mass of charge carriers
  • The DOS is higher near the conduction band minimum and valence band maximum, where the energy bands are more closely spaced
• The DOS is used to calculate the carrier concentration by integrating the product of the DOS and the Fermi-Dirac distribution over the relevant energy range

Fermi-Dirac distribution

• The Fermi-Dirac distribution describes the probability of an energy state being occupied by an electron at a given temperature
  • The distribution depends on the Fermi level, which is the energy level where the probability of occupation is 50% at absolute zero temperature
  • As temperature increases, the Fermi-Dirac distribution becomes smoother, and more electrons can occupy higher energy states
• The Fermi-Dirac distribution is used in conjunction with the DOS to calculate the carrier concentration in semiconductors

Intrinsic carrier concentration

• The intrinsic carrier concentration ($n_i$) is the number of electrons (or holes) per unit volume in an intrinsic semiconductor at thermal equilibrium
• $n_i$ is an important parameter that determines the electrical properties of intrinsic semiconductors and serves as a reference for extrinsic semiconductors

Temperature dependence

• The intrinsic carrier concentration strongly depends on temperature, increasing exponentially with increasing temperature
  • As temperature rises, more electrons gain enough thermal energy to overcome the bandgap and move from the valence band to the conduction band
  • The temperature dependence of $n_i$ is described by the equation: $n_i = \sqrt{N_c N_v} \exp(-E_g / 2k_B T)$, where $N_c$ and $N_v$ are the effective density of states in the conduction and valence bands, $E_g$ is the bandgap energy, $k_B$ is the Boltzmann constant, and $T$ is the absolute temperature
• The strong temperature dependence of $n_i$ has significant implications for the performance of semiconductor devices, such as increased leakage current and reduced efficiency at higher temperatures

Bandgap energy

• The bandgap energy ($E_g$) is the energy difference between the top of the valence band and the bottom of the conduction band in a semiconductor
  • The bandgap energy determines the minimum energy required for an electron to move from the valence band to the conduction band
  • Materials with larger bandgap energies (wide bandgap semiconductors) have lower intrinsic carrier concentrations and are suitable for high-temperature and high-power applications (silicon carbide, gallium nitride)
• The bandgap energy appears in the exponential term of the intrinsic carrier concentration equation, indicating its strong influence on $n_i$

Effective density of states

• The effective density of states ($N_c$ and $N_v$) represents the number of available energy states per unit volume near the conduction band minimum and valence band maximum, respectively
  • $N_c$ and $N_v$ depend on the material's effective mass of electrons and holes and the temperature
  • The effective density of states is given by: $N_c = 2(2\pi m_e^* k_B T / h^2)^{3/2}$ and $N_v = 2(2\pi m_h^* k_B T / h^2)^{3/2}$, where $m_e^*$ and $m_h^*$ are the effective masses of electrons and holes, and $h$ is Planck's constant
• $N_c$ and $N_v$ appear in the intrinsic carrier concentration equation, and their values affect the magnitude of $n_i$

Mass action law

• The mass action law states that the product of the electron and hole concentrations in an intrinsic semiconductor is equal to the square of the intrinsic carrier concentration
  • Mathematically, the mass action law is expressed as: $n \times p = n_i^2$, where $n$ and $p$ are the electron and hole concentrations, respectively
  • The mass action law holds true for both intrinsic and extrinsic semiconductors at thermal equilibrium
• The mass action law is useful for calculating the minority carrier concentration in extrinsic semiconductors, given the majority carrier concentration and the intrinsic carrier concentration

Extrinsic carrier concentration

• Extrinsic carrier concentration refers to the number of charge carriers per unit volume in a semiconductor that has been intentionally doped with impurities
• Doping allows for the control of carrier concentration and the creation of n-type and p-type semiconductors, which are essential for fabricating semiconductor devices

n-type vs p-type doping

• n-type doping involves the introduction of donor impurities that have one more valence electron than the host semiconductor material (phosphorus, arsenic in silicon)
  • Donor impurities easily donate their extra electron to the conduction band, increasing the electron concentration and making electrons the majority carriers in n-type semiconductors
  • The Fermi level in n-type semiconductors is shifted closer to the conduction band compared to intrinsic semiconductors
• p-type doping involves the introduction of acceptor impurities that have one fewer valence electron than the host semiconductor material (boron, gallium in silicon)
  • Acceptor impurities readily accept an electron from the valence band, creating holes and increasing the hole concentration, making holes the majority carriers in p-type semiconductors
  • The Fermi level in p-type semiconductors is shifted closer to the valence band compared to intrinsic semiconductors

Donor and acceptor impurities

• Donor impurities are atoms that have one more valence electron than the host semiconductor material (group V elements in silicon or germanium)
  • When a donor impurity is incorporated into the semiconductor lattice, it can easily donate its extra electron to the conduction band, increasing the electron concentration
  • Common donor impurities in silicon include phosphorus, arsenic, and antimony
• Acceptor impurities are atoms that have one fewer valence electron than the host semiconductor material (group III elements in silicon or germanium)
  • When an acceptor impurity is incorporated into the semiconductor lattice, it can readily accept an electron from the valence band, creating a hole and increasing the hole concentration
  • Common acceptor impurities in silicon include boron, gallium, and indium

Ionization energy

• Ionization energy is the energy required to ionize a donor or acceptor impurity, i.e., to remove an electron from a donor atom or add an electron to an acceptor atom
  • Donor ionization energy ($E_D$) is the energy difference between the donor energy level and the conduction band minimum
  • Acceptor ionization energy ($E_A$) is the energy difference between the valence band maximum and the acceptor energy level
• Ionization energies are typically much smaller than the bandgap energy, allowing for efficient ionization of impurities at room temperature
  • For example, the ionization energies of phosphorus and boron in silicon are approximately 45 meV and 45 meV, respectively, compared to silicon's bandgap energy of 1.12 eV at room temperature

Charge neutrality

• The charge neutrality principle states that in a semiconductor at thermal equilibrium, the total charge of the positive and negative carriers must balance each other
  • In an intrinsic semiconductor, the electron and hole concentrations are equal, maintaining charge neutrality: $n = p = n_i$
  • In an n-type semiconductor, the electron concentration (majority carriers) is much higher than the hole concentration (minority carriers), and the charge neutrality equation becomes: $n + N_A^- = p + N_D^+$, where $N_A^-$ and $N_D^+$ are the ionized acceptor and donor concentrations, respectively
  • In a p-type semiconductor, the hole concentration (majority carriers) is much higher than the electron concentration (minority carriers), and the charge neutrality equation becomes: $p + N_D^- = n + N_A^+$
• The charge neutrality principle is essential for understanding the behavior of semiconductor devices and is used in the derivation of the Fermi level position and carrier concentration calculations

Carrier concentration calculations

• Carrier concentration calculations involve determining the electron and hole concentrations in a semiconductor based on the doping levels, temperature, and other material properties
• These calculations are crucial for designing semiconductor devices and predicting their electrical behavior

Fermi level position

• The Fermi level position is a key parameter in determining the carrier concentrations in a semiconductor
  • In an intrinsic semiconductor, the Fermi level lies close to the middle of the bandgap
  • In an n-type semiconductor, the Fermi level shifts towards the conduction band, while in a p-type semiconductor, it shifts towards the valence band
• The Fermi level position can be calculated using the charge neutrality equation and the expressions for electron and hole concentrations
  • For example, in an n-type semiconductor, the Fermi level position ($E_F$) can be approximated as: $E_F - E_C = k_B T \ln(N_D / N_C)$, where $E_C$ is the conduction band minimum, $N_D$ is the donor concentration, and $N_C$ is the effective density of states in the conduction band

Boltzmann approximation

• The Boltzmann approximation is a simplification of the Fermi-Dirac distribution that is valid when the Fermi level is several $k_B T$ away from the conduction or valence band edges
  • Under the Boltzmann approximation, the electron and hole concentrations can be expressed as: $n = N_C \exp((E_F - E_C) / k_B T)$ and $p = N_V \exp((E_V - E_F) / k_B T)$, where $E_V$ is the valence band maximum, and $N_V$ is the effective density of states in the valence band
  • The Boltzmann approximation simplifies carrier concentration calculations and is widely used in semiconductor device modeling
• The Boltzmann approximation breaks down when the Fermi level is close to the band edges, such as in heavily doped semiconductors or at low temperatures

Degenerate semiconductors

• Degenerate semiconductors are heavily doped semiconductors in which the Fermi level lies within the conduction band (n-type) or valence band (p-type)
  • In degenerate semiconductors, the Boltzmann approximation is no longer valid, and the Fermi-Dirac distribution must be used to calculate the carrier concentrations
  • The electron concentration in a degenerate n-type semiconductor can be approximated using the Joyce-Dixon approximation: $n = N_C F_{1/2}(\eta)$, where $F_{1/2}$ is the Fermi-Dirac integral of order 1/2, and $\eta = (E_F - E_C) / k_B T$ is the reduced Fermi level
• Degenerate semiconductors exhibit distinct electrical properties, such as reduced carrier mobility and increased optical absorption, which are important for certain applications (tunnel diodes, solar cells)

Numerical solutions

• In some cases, the carrier concentration calculations involve complex equations that cannot be solved analytically, requiring numerical solutions
  • Numerical methods, such as the Newton-Raphson method or the bisection method, can be used to solve the charge neutrality equation and determine the Fermi level position and carrier concentrations
  • Semiconductor device simulation software, such as Silvaco Atlas or Synopsys Sentaurus, often employs numerical techniques to model carrier transport and calculate carrier concentrations in complex device structures
• Numerical solutions are particularly useful when dealing with non-uniform doping profiles, complex band structures, or high-level injection conditions

Carrier concentration measurements

• Measuring carrier concentration is essential for characterizing semiconductor materials and devices, as well as validating theoretical models and simulations
• Several techniques are available for measuring carrier concentration, each with its advantages and limitations

Hall effect

• The Hall effect is a widely used method for measuring carrier concentration and mobility in 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 voltage is proportional to the applied magnetic field and the current, and inversely proportional to the carrier concentration and the sample thickness
• By measuring the Hall voltage, the carrier concentration can be determined using the equation: $n = I B / (q V_H t)$, where $I$ is the current, $B$ is the magnetic field, $q$ is the elementary charge, $V_H$ is the Hall voltage, and $t$ is the sample thickness
  • The sign of the Hall voltage indicates the type of majority carriers (electrons or holes)
  • The Hall effect can also be used to measure the carrier mobility by combining the carrier concentration with resistivity measurements

Resistivity vs Hall measurements

• Resistivity measurements provide information about the overall electrical conductivity of a semiconductor, which depends on both the carrier concentration and mobility
  • The resistivity ($\rho$) is related to the carrier concentration ($n$) and mobility ($\mu$) by the equation: $\rho = 1 / (q n \mu)$
  • Resistivity measurements alone cannot distinguish between the contributions of carrier concentration and mobility
• Hall effect measurements, on the other hand, directly provide information about the carrier concentration and can be combined with resistivity measurements to determine the carrier mobility
  • By measuring both the resistivity and the Hall coefficient ($R_H = 1 / (q n)$), the carrier mobility can be calculated using the equation: $\mu = R_H / \rho$
• Combining resistivity and Hall measurements provides a comprehensive characterization of the electrical properties of semiconductors

Majority and minority carriers

• Carrier concentration measurements typically focus on determining the majority carrier concentration, as it dominates the electrical conductivity in extrinsic semiconductors
  • In n-type semiconductors, electrons are the majority carriers, while in p-type semiconductors, holes are the majority carriers
  • The Hall effect measurements directly provide the majority carrier concentration and type
• Measuring the minority carrier concentration is more challenging, as it is usually several orders of magnitude lower than the majority carrier concentration
  • Techniques such as photoconductivity, photoluminescence, or capacitance-voltage measurements can be used to estimate the minority carrier concentration
  • The minority carrier concentration can also be calculated using the majority carrier concentration and the intrinsic carrier concentration, based on the mass action law: $n_{\text{minority}} = n_i^2 / n_{\text{majority}}$

Carrier mobility

• Carrier mobility is a measure of how easily charge carriers can move through a semiconductor under the influence of an electric field
  • Electron mobility ($\mu_e$) and hole mobility ($\mu_h$) are important parameters that affect the performance of semiconductor devices, such as the speed of transistors and the efficiency of solar cells
  • Carrier mobility depends on various factors, including temperature, doping concentration, and scattering mechanisms (lattice vibrations, ionized impurities, defects)
• Carrier mobility can be measured using the Hall effect, as discussed earlier, by combining the Hall coefficient and resistivity measurements
  • Other techniques, such as time-of-flight, charge extraction by linearly increasing voltage (CELIV), or field-effect transistor (FET) measurements, can also be used to determine carrier mobility
• Understanding and optimizing carrier mobility is crucial for developing high-performance semiconductor