Semiconductor Physics: Atoms, Bonds, Doping, Charge Carriers

Atomic Structure and Bonding in Semiconductors

Semiconductors are crystalline solids (e.g., silicon, germanium) whose electrical properties lie between conductors (metals) and insulators. Their behavior depends on atomic structure and covalent bonding.

1.1 Atomic Structure of Semiconductors

• Silicon (Si) and Germanium (Ge) are group IV elements with 4 valence electrons in their outermost shell.
• In a pure (intrinsic) semiconductor, atoms form a crystalline lattice where each atom shares electrons with its neighbors.
• Bonding: Each Si atom forms 4 covalent bonds with neighboring atoms, creating a stable, tetrahedral structure (no free electrons at absolute zero).

graph LR
    A["Si Atom\n(4 valence e⁻)"] -->|"4 bonds"| B["Crystalline Lattice\n(Stable at 0K)"]
    B -->|"Thermal energy"| C["Free e⁻-hole pairs\n(At room temp)"

1.2 Intrinsic Semiconductors

• At absolute zero (0K), no free electrons exist (all are bound in covalent bonds).
• At room temperature (~300K), thermal energy breaks some bonds, creating:
  • Free electrons (n-type carriers) in the conduction band.
  • Holes (p-type carriers) in the valence band (absence of an electron).
• Carrier concentration in intrinsic semiconductors: $$n_i=p_i=\sqrt{N_c{N}_v}{e}^{-\frac{E_g}{2kT}}$$
  • $n_i$ = intrinsic electron concentration
  • $p_i$ = intrinsic hole concentration
  • $E_g$ = bandgap energy (~1.1 eV for Si, ~0.7 eV for Ge)
  • $k$ = Boltzmann’s constant, $T$ = temperature in Kelvin.

Worked Example: Carrier Concentration in Si at 300K Given:

• $N_c=2.8\times 10^{19}{\text{cm}}^{-3}$ (effective density of states in conduction band)
• $N_v=1.04\times 10^{19}{\text{cm}}^{-3}$ (valence band)
• $E_g=1.1\text{eV}$
• $k=8.62\times 10^{-5}\text{eV/K}$
• $T=300\text{K}$

Calculate $n_i$: $$n_i=\sqrt{(2.8\times 10^{19})(1.04\times 10^{19})}\cdot e^{-\frac{1.1}{2\times 8.62\times 10^{-5}\times 300}}$$ $$n_i\approx 1.5\times 10^{10}{\text{cm}}^{-3}$$ Answer: At 300K, intrinsic Si has $1.5\times 10^{10}$ free electrons/cm³ and the same number of holes.

Extrinsic Semiconductors: Doping

Doping introduces impurity atoms to modify carrier concentration, creating n-type or p-type semiconductors.

1.3 N-Type Semiconductors (Donor Impurities)

• Dopants: Group V elements (e.g., phosphorus (P), arsenic (As)) with 5 valence electrons.
• Mechanism:
  • 4 electrons form covalent bonds; the 5th electron is loosely bound and becomes a free electron at room temperature.
  • The dopant atom becomes a donor, donating electrons to the conduction band.
• Majority carriers: Electrons ($n$)
• Minority carriers: Holes ($p$)
• Carrier concentration: $$n\approx N_d\text{(if }{N}_d\gg n_i\text{)}$$ $$p=\frac{n_i^2}{N_d}$$

1.4 P-Type Semiconductors (Acceptor Impurities)

• Dopants: Group III elements (e.g., boron (B), aluminum (Al)) with 3 valence electrons.
• Mechanism:
  • 3 electrons form covalent bonds; the missing electron creates a hole in the valence band.
  • The dopant atom becomes an acceptor, accepting electrons from the valence band, creating holes.
• Majority carriers: Holes ($p$)
• Minority carriers: Electrons ($n$)
• Carrier concentration: $$p\approx N_a\text{(if }{N}_a\gg n_i\text{)}$$ $$n=\frac{n_i^2}{N_a}$$

Worked Example: Doping in a Solar Cell (Real-World Tie) In Nepal’s solar panels (used in rural areas like Dolpa or Mustang), p-type silicon is doped with boron to create the base layer, while n-type silicon (doped with phosphorus) forms the emitter. When sunlight hits, photons excite electron-hole pairs near the p-n junction, generating electricity. The doping levels determine the efficiency of charge separation.

Charge Carriers and Current Flow

1.5 Electron and Hole Movement

• Electrons move from n-type to p-type (conventional current flows p → n).
• Holes move in the opposite direction (diffusion due to concentration gradient).
• Drift current: Due to an electric field (e.g., applied voltage).
• Diffusion current: Due to concentration gradient (e.g., at a p-n junction).

stateDiagram-v2
  [*] --> p_region
  p_region --> depletion_region : Diffusion (holes)
  n_region --> depletion_region : Diffusion (electrons)
  depletion_region --> [*] : Recombination
  state n_region {
    [*] --> majority_carriers : n >> p
  }
  state p_region {
    [*] --> majority_carriers : p >> n
  }

Mermaid Diagram: Carrier Movement in a p-n Junction

graph LR
    A["n-type (high e⁻)"] -->|"Diffusion"| B["Depletion Region"]
    C["p-type (high h⁺)"] -->|"Diffusion"| B
    B -->|"Electric Field"| D["Drift Current\n(e⁻: n→p, h⁺: p→n)"]
    B -->|"No Bias"| E["Equilibrium\n(No net current)"

1.6 Law of Mass Action

In thermal equilibrium, the product of electron and hole concentrations is constant: $$n\cdot p=n_i^2$$

• Example: If $n_i=1.5\times 10^{10}{\text{cm}}^{-3}$ and $N_d=10^{15}{\text{cm}}^{-3}$ (n-type), then: $$p=\frac{(1.5\times 10^{10}{)}^2}{10^{15}}=2.25\times 10^5{\text{cm}}^{-3}$$

Comparison: Intrinsic vs. Extrinsic Semiconductors

## In the Real World

1. eSewa and Khalti (Digital Payments)

   • Idea Used: Doped semiconductors in microcontrollers
   • How? The ARM Cortex-M processors in payment terminals use n-type and p-type silicon in CMOS (Complementary Metal-Oxide-Semiconductor) logic. When you scan a QR code for eSewa, the processor’s transistors (doped silicon) switch rapidly to authenticate and process transactions. Without precise doping, these chips would fail.

2. Nepal Electricity Authority (NEA) Power Grids

   • Idea Used: Semiconductor diodes in rectifiers
   • How? NEA uses silicon diodes (doped p-n junctions) to convert AC power from transmission lines into DC power for battery storage or electrolysis (e.g., in hydropower plants like West Seti). The forward bias of diodes allows current to flow only one way, preventing backflow that could damage equipment.

3. Pathao and Daraz Delivery Logistics

   • Idea Used: Semiconductor sensors in GPS and route optimization
   • How? Pathao’s delivery drivers use GPS modules with silicon-based sensors (doped for high sensitivity). These sensors detect minority carriers (electrons/holes) generated by weak signals, ensuring accurate location tracking. Similarly, Daraz’s warehouse RFID tags use semiconductor diodes to transmit product data wirelessly.

## Exam Tip

1. Memorize Key Equations:

   • Intrinsic carrier concentration: $n_i=\sqrt{N_c{N}_v}{e}^{-\frac{E_g}{2kT}}$
   • Law of mass action: $n\cdot p=n_i^2$
   • Doping approximations: $n\approx N_d$ (n-type), $p\approx N_a$ (p-type)

2. Differentiate n-type and p-type:

   • n-type: Extra electrons from donor impurities (P, As).
   • p-type: Extra holes from acceptor impurities (B, Al).
   • Always draw the crystal lattice when explaining doping.

3. Real-World Applications:

   • Exams often ask: "Why is silicon preferred over germanium in modern electronics?" Answer: Silicon has a larger bandgap (1.1 eV vs. 0.7 eV), making it less temperature-sensitive and more stable for high-speed devices.

4. Graphical Questions:

   • Expect carrier concentration vs. temperature graphs. Sketch:
     • Intrinsic: Exponential rise with temperature.
     • Extrinsic: Almost flat (doping dominates) until very high temps.

5. Common Mistakes to Avoid:

   • Confusing majority/minority carriers (e.g., saying holes are majority in n-type).
   • Ignoring temperature effects (carrier concentration changes with $T$).
   • Forgetting the law of mass action in equilibrium problems.

Final Note: Semiconductor physics is the foundation of all electronic devices. Master doping, carrier behavior, and the bandgap model—these concepts appear in every unit of this course. Practice sketching energy band diagrams and calculating carrier densities under different conditions.

In the real world

• Nepal’s NTC’s fiber-optic network uses doped germanium (Ge) in photodetectors to convert light signals (from fiber) into electrical current. The p-type Ge (doped with gallium) absorbs photons, generating electron-hole pairs that create the current needed to decode data (e.g., your Ncell 4G signal). Without precise doping, the detector’s sensitivity would drop, increasing errors in your calls or internet.

• eSewa’s QR code scanners rely on CMOS sensors (complementary n-type/p-type silicon transistors). When you scan a QR code, the p-n junctions in the sensor’s pixels convert light into voltage signals. The doping levels (e.g., 10¹⁵ cm⁻³ boron for p-type) determine how fast the sensor responds—critical for quick transaction processing.

• NEA’s solar inverters use doped silicon diodes (e.g., p-type base + n-type emitter) to convert DC from solar panels into AC for the grid. The law of mass action (n·p = n_i²) ensures that even under varying sunlight, the diode maintains a stable current flow, preventing power loss during Nepal’s frequent load-shedding periods.