11.5 Semiconductors and Doping

Semiconductors are the backbone of modern electronics, bridging the gap between conductors and insulators. Their unique properties, controlled by doping, allow for the creation of various electronic devices that power our digital world.

Doping involves adding impurities to pure semiconductors, altering their electrical properties. This process creates n-type and p-type semiconductors, which form the basis for diodes, transistors, and solar cells. Understanding doping is crucial for grasping semiconductor behavior and applications.

Electronic Structure of Semiconductors

Band Structure and Energy Levels

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• Intrinsic semiconductors consist of pure crystalline materials from groups IV, III-V, or II-VI of the periodic table (silicon, germanium)
• Electronic band structure comprises:
  • Valence band (filled with electrons at absolute zero)
  • Conduction band (empty at absolute zero)
  • Forbidden energy gap (band gap) separating the two bands
• Fermi level sits approximately in the middle of the band gap
• Band gap energy typically ranges from 0.1 eV to 4 eV
  • Silicon band gap measures approximately 1.1 eV at room temperature

Thermal Excitation and Conductivity

• Temperature increase excites electrons across the band gap
  • Creates electron-hole pairs
  • Enhances electrical conductivity
• Conductivity falls between metals and insulators
  • Range: 10^-8 to 10^3 (Ω⋅m)^-1
• Electron-hole pair generation follows the equation: $n_i = \sqrt{N_c N_v} e^{-E_g/2kT}$
  • $n_i$: intrinsic carrier concentration
  • $N_c$, $N_v$: effective density of states in conduction and valence bands
  • $E_g$: band gap energy
  • $k$: Boltzmann constant
  • $T$: absolute temperature

Doping and Semiconductor Properties

Doping Process and Impurities

• Doping introduces impurity atoms into the semiconductor crystal lattice
• Dopant atoms differ in valence electrons from host material
  • Donor impurities (n-type): one more valence electron (phosphorus in silicon)
  • Acceptor impurities (p-type): one fewer valence electron (boron in silicon)
• Dopant concentration ranges from parts per million to parts per billion
• Doping creates additional energy levels within the band gap
  • Donor levels near conduction band
  • Acceptor levels near valence band

Effects on Electrical Properties

• Doping significantly increases semiconductor conductivity
  • Adds charge carriers (electrons or holes)
• Shifts Fermi level position
  • N-type: closer to conduction band
  • P-type: closer to valence band
• Enables precise control of electrical properties
  • Facilitates creation of various electronic devices (transistors, solar cells)
• Doped semiconductor conductivity follows the equation: $\sigma = q(n\mu_n + p\mu_p)$
  • $\sigma$: conductivity
  • $q$: elementary charge
  • $n$, $p$: electron and hole concentrations
  • $\mu_n$, $\mu_p$: electron and hole mobilities

N-type vs P-type Semiconductors

N-type Semiconductors

• Created by doping with donor impurities (phosphorus, arsenic)
• Majority charge carriers electrons
• Fermi level shifts closer to conduction band
• Conductivity increases with temperature
  • Thermal excitation of electrons from donor levels to conduction band
• Electron concentration in n-type semiconductor: $n = N_D + \frac{n_i^2}{N_D}$
  • $N_D$: donor concentration
  • $n_i$: intrinsic carrier concentration

P-type Semiconductors

• Created by doping with acceptor impurities (boron, gallium)
• Majority charge carriers holes
• Fermi level shifts closer to valence band
• Conductivity increases with temperature
  • Thermal excitation of electrons from valence band to acceptor levels
• Hole concentration in p-type semiconductor: $p = N_A + \frac{n_i^2}{N_A}$
  • $N_A$: acceptor concentration
  • $n_i$: intrinsic carrier concentration

Semiconductor Junctions and Devices

• Junction between n-type and p-type semiconductors forms basis for many devices
  • Diodes: allow current flow in one direction
  • Transistors: amplify or switch electronic signals
  • Solar cells: convert light into electrical energy
• PN junction characteristics depend on doping levels and applied voltage
  • Built-in potential: $V_{bi} = \frac{kT}{q} \ln\left(\frac{N_A N_D}{n_i^2}\right)$

Temperature Dependence of Carrier Concentration

Intrinsic Semiconductors

• Carrier concentration strongly depends on temperature
• Follows Arrhenius equation: $n_i \propto e^{-E_g/2kT}$
  • $n_i$: intrinsic carrier concentration
  • $E_g$: band gap energy
  • $k$: Boltzmann constant
  • $T$: absolute temperature
• Low temperatures yield very low carrier concentration
  • Insufficient thermal energy to excite electrons across band gap
• Increasing temperature exponentially increases carrier concentration
  • Thermal generation of electron-hole pairs

Doped Semiconductors

• Carrier concentration shows less temperature dependence at low to moderate temperatures
  • Ionization of dopant atoms dominates
• High temperatures cause intrinsic carrier concentration to approach or exceed dopant concentration
  • Semiconductor behaves more like intrinsic material
• Temperature dependence affects device parameters
  • Reverse saturation current in diodes
  • Transistor gain
• Carrier freeze-out occurs at very low temperatures
  • Dopant atoms become un-ionized
  • Carrier concentration decreases rapidly

Design Implications

• Understanding temperature dependence crucial for reliable semiconductor devices
  • Operate over wide temperature ranges (automotive, aerospace applications)
• Temperature compensation techniques employed
  • Bandgap reference circuits
  • Temperature-dependent biasing
• Thermal management essential in high-power devices
  • Heat sinks, active cooling systems