🧗‍♀️Semiconductor Physics Unit 3 – Equilibrium Carrier Statistics in Semiconductors

Key Concepts and Definitions

• Semiconductors materials with electrical conductivity between insulators and conductors (silicon, germanium)
• Carriers charge carriers responsible for electrical conduction in semiconductors
  • Electrons negative charge carriers in the conduction band
  • Holes positive charge carriers in the valence band
• Intrinsic semiconductors pure semiconductors without any added impurities
• Extrinsic semiconductors semiconductors with added impurities (dopants) to modify electrical properties
• Fermi level energy level with a 50% probability of being occupied by an electron at thermal equilibrium
• Band gap energy difference between the top of the valence band and the bottom of the conduction band
• Density of states (DOS) number of available energy states per unit volume and energy interval in a semiconductor

Fermi-Dirac Distribution

• Describes the probability of an electron occupying an energy state at thermal equilibrium
• Represented by the equation: $f(E) = \frac{1}{1 + e^{(E - E_F)/kT}}$
  • $E$ energy of the state
  • $E_F$ Fermi level
  • $k$ Boltzmann constant
  • $T$ absolute temperature
• At $T = 0K$, the distribution is a step function, with all states below $E_F$ occupied and all states above $E_F$ empty
• As temperature increases, the distribution becomes smoother, with some states above $E_F$ being occupied and some below $E_F$ being empty
• The Fermi level determines the carrier concentrations in semiconductors
• The position of the Fermi level relative to the band edges affects the electrical properties of the semiconductor

Density of States

• Quantifies the number of available energy states per unit volume and energy interval in a semiconductor
• Depends on the band structure and the effective mass of carriers
• For a parabolic band, the density of states is proportional to the square root of energy: $DOS(E) \propto \sqrt{E}$
• The conduction band density of states is given by: $N_C = 2\left(\frac{2\pi m_e^* kT}{h^2}\right)^{3/2}$
  • $m_e^*$ effective mass of electrons
  • $h$ Planck's constant
• The valence band density of states is given by: $N_V = 2\left(\frac{2\pi m_h^* kT}{h^2}\right)^{3/2}$
  • $m_h^*$ effective mass of holes
• The density of states plays a crucial role in determining the carrier concentrations in semiconductors

Carrier Concentration Equations

• Carrier concentrations in semiconductors are determined by the Fermi-Dirac distribution and the density of states
• For intrinsic semiconductors, the electron and hole concentrations are equal: $n_i = p_i$
  • $n_i$ intrinsic electron concentration
  • $p_i$ intrinsic hole concentration
• The intrinsic carrier concentration is given by: $n_i = \sqrt{N_C N_V e^{-E_g/2kT}}$
  • $E_g$ band gap energy
• For extrinsic semiconductors, the majority carrier concentration is determined by the doping level
  • n-type semiconductors: $n \approx N_D$, where $N_D$ is the donor concentration
  • p-type semiconductors: $p \approx N_A$, where $N_A$ is the acceptor concentration
• The minority carrier concentration is calculated using the mass action law: $np = n_i^2$

Temperature Effects on Carrier Concentration

• Temperature significantly influences the carrier concentrations in semiconductors
• As temperature increases, more electrons are excited from the valence band to the conduction band
• The intrinsic carrier concentration increases exponentially with temperature: $n_i \propto e^{-E_g/2kT}$
• In extrinsic semiconductors, the majority carrier concentration is relatively insensitive to temperature changes
  • Determined by the doping level, which is fixed
• The minority carrier concentration increases with temperature, following the intrinsic carrier concentration
• At high temperatures, the semiconductor may become intrinsic, with the intrinsic carrier concentration dominating over the doping-induced carriers
• Temperature effects on carrier concentration have important implications for semiconductor device performance and reliability

Doping and Its Impact

• Doping intentional introduction of impurities into a semiconductor to modify its electrical properties
• n-type doping introduces donor impurities (phosphorus, arsenic) that provide extra electrons to the conduction band
• p-type doping introduces acceptor impurities (boron, gallium) that create holes in the valence band
• Doping shifts the Fermi level towards the corresponding band edge
  • n-type doping: Fermi level moves closer to the conduction band
  • p-type doping: Fermi level moves closer to the valence band
• The doping concentration determines the majority carrier concentration in extrinsic semiconductors
• Heavily doped semiconductors have a higher conductivity and lower resistivity compared to lightly doped or intrinsic semiconductors
• Doping enables the creation of p-n junctions, which are the building blocks of many semiconductor devices (diodes, transistors, solar cells)

Equilibrium vs. Non-Equilibrium States

• Equilibrium state: no external forces acting on the semiconductor, carrier concentrations determined by the Fermi-Dirac distribution
  • Characterized by a constant Fermi level throughout the semiconductor
  • Carrier generation and recombination rates are balanced
• Non-equilibrium state: external forces (electric field, light, temperature gradients) disturb the equilibrium carrier concentrations
  • Characterized by a non-constant Fermi level or quasi-Fermi levels for electrons and holes
  • Carrier generation and recombination rates are not balanced
• In non-equilibrium states, the carrier concentrations deviate from their equilibrium values
  • Excess carriers are generated or injected into the semiconductor
  • Carrier lifetime and diffusion play a role in the transport and recombination of excess carriers
• Understanding the difference between equilibrium and non-equilibrium states is crucial for analyzing semiconductor devices under various operating conditions

Applications and Real-World Examples

• Solar cells: p-n junctions that convert light into electrical energy
  • Carrier generation by photon absorption creates a non-equilibrium state
  • Separation of photogenerated carriers by the built-in electric field produces a photocurrent
• Light-emitting diodes (LEDs): p-n junctions that emit light when forward-biased
  • Injection of minority carriers leads to radiative recombination and photon emission
  • The wavelength of the emitted light depends on the band gap of the semiconductor material
• Transistors: three-terminal devices used for amplification and switching
  • Bipolar junction transistors (BJTs) rely on minority carrier injection and transport
  • Field-effect transistors (FETs) control the conductivity of a channel by applying an electric field
• Semiconductor lasers: p-n junctions that produce coherent light through stimulated emission
  • Population inversion achieved by injecting high current densities
  • Used in fiber-optic communication, barcode scanners, and laser pointers
• Photodetectors: devices that convert light into electrical signals
  • p-n junctions or PIN structures that generate a photocurrent when illuminated
  • Used in digital cameras, optical receivers, and light sensors