ELCT363_Lecture11_Crriers_in_Semiconductor_Spring2025 -IA8

Course: ELCT 363: Introduction to Microelectronics
Topics Covered:
- Doping of semiconductors (Reference: Streetman, Ch. 3)
  - Intrinsic carriers in semiconductors
  - Carrier concentrations in n-type semiconductor
  - Carrier concentrations in p-type semiconductor
  - Fermi level in n- and p-type semiconductor
  - Compensated semiconductor
Announcements:
- 3rd Quiz (Quiz 3) scheduled for Tuesday, 2/18/25
- Midterm exam on Thursday, 3/20/25

Electrons and Holes:
- Formula for electron concentration in n-type:n0 = Nc exp[-(Ec - Ef)/(kT)]
- Formula for hole concentration in p-type:p0 = Nv exp[-(Ef - Ev)/(kT)]

In intrinsic semiconductors:
- Fermi level (EF) equals the intrinsic energy level (Ei): EF = Ei = Eg/2.
- Electron and Hole Pairing:
  - Electrons and holes appear in pairs: ni = pi
Equations:
- For intrinsic semiconductors, the relationships include:
  - n = Nc exp[-(Ec - Ef)/(kT)]
  - p = Nv exp[-(Ef - Ev)/(kT)]

Doping with Phosphorus:
- Phosphorus (P) has 5 outer shell electrons, resulting in one additional valence electron.
- Minimal energy is needed to create a free electron from phosphorous atom, resulting in each donor atom creating one free electron.
- Concentration of free electrons in n-type is calculated as: n = ND + ni, where ni is typically 1.3 × 10^10 cm-3.
- Doping concentration range: ND = 10^15 cm-3 to 10^19 cm-3.
- In n-type materials: Electrons are the majority carriers, holes are minority carriers.

Doping with Boron:
- Boron (B) has three valence electrons, creating a hole in the silicon lattice (not filling all bonds).
- This hole can participate in conduction, making boron an acceptor impurity.
- Concentration of holes is given by: p = NA + pi, where NA is the doping concentration, typically NA = 10^15 to 10^19 cm-3.
- In p-type, holes are the majority carriers while electrons are the minority carriers.

Intrinsic Semiconductor:
- n = p
Extrinsic Carrier Concentration:
- For n-type: n > p (Donor) => nn = ND
- For p-type: p > n (Acceptor) => pp = NA
- Formula for charge neutrality: n x p = ni^2
- Doping enhances conductivity.

n-Type (Majority carriers: Electrons):
- Donor impurities (Column V of Periodic Table): P, As, N.
p-Type (Majority carriers: Holes):
- Acceptor impurities (Column III of Periodic Table): B, Al, Ga.
Silicon (IV element) and III-V Compounds: GaN, GaAs, etc.

Energy Levels:
- In n-type, most mobile charges are free electrons; therefore:
  - EF_n ≈ EC.
- Phosphorus (P) provides free electrons due to its five valence electrons.

Energy Levels:
- In p-type, most mobile charges are holes, hence:
  - EF_p ≈ EV.
- Boron (B) introduces electron vacancies (holes).

Generalization:
- In n-type: EF ≈ EC (mostly free electrons).
- In p-type: EF ≈ EV (mostly holes).
- In intrinsic semiconductors: EF = Ei = Eg/2.

Formula for electron density in n-type:
- n_d = N_d - N_d +
- At temperature T, fraction of electrons in donor level:
  - n_d = N_d / [1 + 1/2 exp[(E_d - EF)/(kT)]]

Example: Determine the fraction of total electrons in donor states at 300K with:
- Nd = 10^16 cm-3 (Phosphorus).
- Calculate using E_d = Ec - Ed = 0.045ev and kT = 0.0259 eV.
- Result: 0.0041 or 0.41%, indicating most electrons are in conduction band and free to move.

Formula:
- n_a = N_a - N_a +
- Fraction of holes in donor level:
  - n_a = N_a / [1 + 1/4 exp[(EF - E_a)/(kT)]]

Comparison of intrinsic, n-type, and p-type band structures to understand energy levels and band gaps.

Increasing Doping:
- As doping increases, dopant atoms become closer; their wave functions overlap, creating bands.
- These are called degenerate semiconductors, while the course focuses on non-degenerate semiconductors.

Position calculation for n-type semiconductors:
- n0 ni Nc e^[-(Ec - EF)/(kT)] = Nc e^[-(Ec - Ei)/(kT)] = e^(EF/kT)e^(Ei/kT).
The position of the Fermi level depends on doping concentration.

Position calculation for p-type semiconductors:
- p0 pi NV e^[-(EF - Ev)/(kT)] = NV e^[-(Ei - Ev)/(kT)] = e^[-EF/kT]e^[-Ei/kT].

Note: As doping slowly increases in silicon, the Fermi level shifts toward respective bands.

Charge Carriers:
- Two types: "Free" Electrons (n) and Holes (p).
Carrier concentrations:
- At thermal equilibrium (n0 or p0).
- For intrinsic (ni or pi).
- For doped semiconductors (nn, pn, np, or pp).