Semiconductors

Conductors, insulators, semiconductors

Materials' electrical conductivity depends on availability of free charge carriers:

Material type	Resistivity (Ω·m)	Carriers at room T
Conductor (metal)	~10⁻⁸	Lots of free electrons (overlapping bands)
Semiconductor	10⁻⁵ to 10⁹	Few free; bandgap ~1-2 eV
Insulator	> 10¹⁰	Almost none; bandgap > 4 eV

Band theory

Atoms' electrons occupy energy levels. In a solid, levels broaden into bands:

• Valence band — filled with electrons in covalent bonds.
• Conduction band — empty (or partially filled), where electrons can move freely.
• Bandgap — energy region between valence and conduction; electrons can't have these energies.

Material	Bandgap (eV)	Application
Diamond	5.5	Insulator
SiC (silicon carbide)	2.2-3.3	High-power, high-T electronics
GaN (gallium nitride)	3.4	Blue LEDs, RF power
GaP (gallium phosphide)	2.3	Green/red LEDs
GaAs (gallium arsenide)	1.4	Solar cells, RF amplifiers
Silicon	1.1	Computers, most electronics
Germanium	0.67	Old transistors; IR detectors
InSb (indium antimonide)	0.17	IR sensors
HgCdTe	0 to 1.5 (tunable)	Multi-band IR cameras

Doping creates n- and p-type

Type	Dopant atoms	Carrier	Examples
n-type	Group V (P, As, Sb)	Electrons (negative)	5 outer electrons; 1 extra → conduction
p-type	Group III (B, Al, Ga)	Holes (positive)	3 outer electrons; missing 1 → "hole"

Doping levels typical: 10¹⁵-10²⁰ atoms/cm³ (1 dopant per 10⁶ to 10⁹ Si atoms).

Common semiconductor devices

Device	Structure	Function
Diode	Single p-n junction	One-way current valve
BJT (bipolar transistor)	NPN or PNP	Amplifier or switch
MOSFET	Metal-Oxide-Semiconductor FET	Voltage-controlled current; basis of CPUs
Solar cell	p-n junction with light absorption	Light → electricity
LED	p-n junction in direct-bandgap material	Electricity → light
Photodetector	Reverse-biased p-n	Light → electrical signal
Thermistor	Doped semiconductor	Temperature sensor (R varies with T)

JavaScript — semiconductor calculations

// Photon wavelength corresponding to bandgap
function bandgapToWavelength(E_eV) {
  // E = hc/λ → λ = hc/E. h = 6.626e-34, c = 3e8, e = 1.602e-19
  return 6.626e-34 * 3e8 / (E_eV * 1.602e-19);
}

console.log(`Si (1.1 eV): ${(bandgapToWavelength(1.1) * 1e9).toFixed(0)} nm`);  // ~1127 nm (IR)
console.log(`GaAs (1.4 eV): ${(bandgapToWavelength(1.4) * 1e9).toFixed(0)} nm`); // 886 (near IR)
console.log(`GaN (3.4 eV): ${(bandgapToWavelength(3.4) * 1e9).toFixed(0)} nm`);  // 365 (UV)

// LED color from semiconductor bandgap
function ledColor(bandgap_eV) {
  const lambda_nm = bandgapToWavelength(bandgap_eV) * 1e9;
  if (lambda_nm < 380) return 'UV';
  if (lambda_nm < 450) return 'violet';
  if (lambda_nm < 495) return 'blue';
  if (lambda_nm < 570) return 'green';
  if (lambda_nm < 590) return 'yellow';
  if (lambda_nm < 620) return 'orange';
  if (lambda_nm < 750) return 'red';
  return 'IR';
}

console.log(`AlGaInP (2.0 eV): ${ledColor(2.0)}`); // red

// Diode I-V relation (Shockley equation)
function diodeCurrent(V_diode, I_s = 1e-12, T_K = 300) {
  // I = I_s · (exp(qV/kT) - 1)
  const k = 1.380649e-23;
  const e = 1.602e-19;
  return I_s * (Math.exp(e * V_diode / (k * T_K)) - 1);
}

console.log(`Diode at 0.6V: ${(diodeCurrent(0.6) * 1000).toFixed(2)} mA`);  // significant
console.log(`Diode at 0.5V: ${(diodeCurrent(0.5) * 1e6).toFixed(2)} µA`);  // small
console.log(`Diode at -0.5V: ${diodeCurrent(-0.5).toExponential(2)} A`);   // tiny (reverse)

// Thermal voltage at T
function thermalVoltage(T_K = 300) {
  const k = 1.380649e-23;
  const e = 1.602e-19;
  return k * T_K / e;
}

console.log(`V_T at 300K: ${thermalVoltage(300).toFixed(4)} V`);  // 0.0259 V (~26 mV)

// Carrier concentration ratio (n × p = n_i² for intrinsic)
function intrinsicCarrierConcentration(bandgap_eV, T_K = 300, m_eff_factor = 1) {
  // Rough estimate: n_i ~ T^(3/2) · exp(-E_g/(2kT))
  // For Si at 300K, n_i ≈ 1.5 × 10¹⁰ /cm³
  const k = 1.380649e-23;
  const E_g = bandgap_eV * 1.602e-19;
  return 4.83e15 * m_eff_factor * Math.pow(T_K, 1.5) * Math.exp(-E_g / (2 * k * T_K));
}

console.log(`Intrinsic Si at 300K: ${intrinsicCarrierConcentration(1.1).toExponential(2)} /cm³`);

Where semiconductors matter

• Computing. Every CPU, GPU, RAM, storage uses billions of MOSFETs.
• Power electronics. Inverters, converters in solar/wind/EV/grid use IGBTs, MOSFETs, SiC, GaN.
• Sensors. Photodiodes (cameras), thermistors (T sensors), pressure sensors, accelerometers.
• Lighting. LEDs replacing incandescent (efficiency 5× to 10×).
• Solar cells. All photovoltaic panels are semiconductor based.
• Communications. RF transistors in radios, cell phones, WiFi.
• Quantum. Quantum computers being developed using semiconductor qubits (silicon, GaAs).

Common mistakes

• Treating semiconductors as ohmic. Diodes and transistors have non-linear I-V; don't apply V=IR directly.
• Ignoring temperature. Semiconductor conductivity strongly T-dependent (more carriers at higher T). Different from metals (less conductivity at higher T).
• Forgetting bandgap drives behavior. Almost everything (LED color, solar wavelength, transistor speed) traces to bandgap.
• Confusing electron and hole motion. "Holes" are absences of electrons that move opposite to electrons; treated as positive carriers.
• Treating doping as free. Excess doping creates defects, reduces carrier mobility. Optimal doping density chosen carefully.
• Mixing direct and indirect bandgap. Direct bandgap (GaAs) — efficient for light emission/absorption (LEDs, lasers). Indirect (Si) — needs phonon for transition; less efficient for light. Affects device choice.