Semiconductor physics: energy bands, carriers and junctions

In short

Semiconductor physics explains why silicon conducts a controllable amount of current: electrons in a crystal may occupy only certain energy bands, a small forbidden gap separates the filled valence band from the empty conduction band, and doping or temperature decides how many mobile electrons and holes exist. Carrier motion by drift and diffusion, and the electrostatics of a pn junction, are the foundation on which every diode, BJT, MOSFET and IC is built. It is the first unit of the Electron Devices paper in every Indian university syllabus and carries a large share of the 8-12 marks of the Electronic Devices section in GATE EC (carrier transport, Poisson and continuity equations, pn junction) as well as in IES/ESE, BARC, ISRO and DRDO papers. Interviews at SCL Mohali, BEL, ISRO, Micron, Intel, AMD, Qualcomm, Texas Instruments, Analog Devices, Applied Materials and the fabs coming up under the India Semiconductor Mission open with exactly these questions.

Explained

From atoms to energy bands

• An isolated atom has discrete levels; bringing N atoms into a crystal splits each level into N closely spaced levels that form a band. The highest (nearly) filled band is the valence band (top edge Ev), the lowest empty band is the conduction band (bottom edge Ec).
• Forbidden gap Eg = Ec − Ev. No allowed states exist inside it in a perfect crystal; impurities and defects create levels within it.
• Classification: metal - bands overlap or the conduction band is partly filled, Eg ≈ 0; semiconductor - Eg ≈ 0.2-3 eV (Si 1.12 eV, Ge 0.66 eV, GaAs 1.42 eV at 300 K); insulator - Eg > ~4 eV (SiO₂ ≈ 9 eV).
• Si and Ge are group-IV, diamond-cubic, covalently bonded (4 bonds per atom); Si atom density ≈ 5 × 10²² cm⁻³. GaAs, InP, GaN, SiC are compound semiconductors.
• E-k diagram and effective mass: the band edge is parabolic, E = ħ²k²/2m*. The carrier responds to force as if its mass were m\* (for Si, m*_e ≈ 0.26 m₀ for conduction, m*_h ≈ 0.39 m₀); a smaller m* means higher mobility. A hole is the absence of an electron in the valence band - it behaves as a positive charge moving opposite to the electron.
• Direct gap (GaAs, InP, GaN): the conduction-band minimum and valence-band maximum occur at the same k, so an electron-hole pair recombines by emitting a photon - these make LEDs and lasers. Indirect gap (Si, Ge): a phonon must assist, so radiative efficiency is poor - silicon cannot make a practical LED.
• Gap narrows with temperature: Eg(T) = Eg(0) − αT²/(T+β) (Varshni). For Si, dEg/dT ≈ −0.27 meV/K; this is why ni rises so steeply with T.

Intrinsic and extrinsic semiconductors

• Intrinsic: pure crystal; the only carriers are thermally generated pairs, so n = p = ni. At 300 K ni(Si) is quoted as 1.5 × 10¹⁰ cm⁻³ in most Indian textbooks and GATE keys; modern measurement gives ≈1.0 × 10¹⁰ cm⁻³. Use the value your paper supplies and state it.
• n-type: add a group-V donor (P, As, Sb) - four bonds plus one loosely held electron at a donor level Ed just below Ec (ionisation energy ≈ 45 meV for P in Si), so it is fully ionised at room temperature. Electrons are majority, holes minority.
• p-type: add a group-III acceptor (B, Al, Ga, In) - a missing bond creates a level Ea just above Ev that accepts an electron, leaving a hole. Holes majority, electrons minority.
• The doped crystal stays electrically neutral: the ionised donor/acceptor cores balance the extra carriers. Doping in practice is 10¹⁴-10²⁰ cm⁻³, i.e. parts per billion to parts per thousand - which is why 11N "electronic-grade" purity is needed before doping.
• Compensation: if both are present, only the net (Nd − Na) sets the majority concentration, but mobility is degraded by the total ionised impurity (Nd + Na).
• Degenerate (n⁺/p⁺) doping (> ~10¹⁹ cm⁻³): the Fermi level moves into the band, Boltzmann statistics fail, the gap narrows, and the material behaves almost metallically - used for ohmic contacts, tunnel diodes and poly-Si gates.
• Temperature zones of an extrinsic sample: freeze-out (low T, dopants not ionised, n rises with T), extrinsic/saturation (n ≈ Nd, flat - the useful operating region), intrinsic (high T, ni > Nd, device loses its doping identity; Ge fails near 100 °C, Si near 200 °C, SiC/GaN far higher).

Carrier statistics and the Fermi level

• Fermi-Dirac: f(E) = 1/[1 + exp((E − EF)/kT)] is the probability that a state at E is occupied. At E = EF, f = 0.5 at every temperature. EF is a chemical potential, not necessarily an allowed state.
• For E − EF > 3kT the Boltzmann approximation f ≈ exp(−(E − EF)/kT) applies - the non-degenerate case assumed in exams.
• Carrier density = ∫ (density of states) × (occupancy) dE, giving n = Nc exp(−(Ec − EF)/kT) and p = Nv exp(−(EF − Ev)/kT), where Nc, Nv are the effective densities of states (Si at 300 K: Nc ≈ 2.8 × 10¹⁹, Nv ≈ 1.04 × 10¹⁹ cm⁻³; values differ slightly between textbooks). Both scale as T³/2.
• Law of mass action: np = ni² = Nc Nv exp(−Eg/kT), independent of doping. Doping one carrier up pushes the other down by the same factor.
• Position of EF: mid-gap (Ei) for intrinsic; close to Ec for n-type; close to Ev for p-type; it moves back towards mid-gap as temperature rises. EF − Ei = kT ln(n/ni).
• Quasi-Fermi levels Fn, Fp are used under bias or illumination, when n and p are no longer in equilibrium with one another; the splitting Fn − Fp equals the applied junction voltage and np = ni² exp(qV/kT) > ni².
• In equilibrium the Fermi level is flat across any junction or contact - this single rule lets you draw every band diagram correctly.

Drift, diffusion and transport

• Drift: an electric field E gives a net average velocity vd = μE (μ = mobility, cm²/V·s), so Jdrift = q(nμn + pμp)E = σE. At high fields vd saturates near 10⁷ cm/s in Si (velocity saturation); at very high fields carriers gain enough energy for impact ionisation.
• Scattering limits mobility: lattice (phonon) scattering ∝ T⁻³/2 dominates at high T and low doping; ionised-impurity scattering ∝ T³/2/N dominates at high doping and low T. Hence μ falls as doping rises (Si electrons: ≈1350 cm²/V·s undoped, ≈1200 at 10¹⁶, ≈300 at 10¹⁸ cm⁻³). Electrons are ~3× more mobile than holes in Si - the reason NMOS beats PMOS.
• Resistivity ρ = 1/σ. A semiconductor has a negative temperature coefficient when carrier generation dominates (intrinsic / thermistor behaviour) but a positive one in the extrinsic region where only μ changes - this is why MOSFET on-resistance rises with temperature and allows safe paralleling.
• Diffusion: carriers move down a concentration gradient. Jn,diff = +qDn (dn/dx), Jp,diff = −qDp (dp/dx) (signs follow from the opposite charges). D is the diffusion coefficient, cm²/s.
• Einstein relation: D/μ = kT/q = VT = 25.9 mV at 300 K - links the two transport mechanisms because both arise from the same random thermal motion.
• The total current of each species is drift + diffusion; inside a bipolar device minority-carrier diffusion dominates, inside a MOSFET channel drift dominates (diffusion dominates in sub-threshold).
• Poisson's equation relates field and charge: dE/dx = ρ_charge/εs. Continuity equations account for carriers: ∂n/∂t = (1/q)(∂Jn/∂x) + Gn − Rn. Together they are the complete device-physics set, and GATE lists them by name.
• Hall effect: a transverse magnetic field deflects carriers, producing a Hall voltage whose sign gives the carrier type and magnitude gives the carrier concentration; with a resistivity measurement it gives mobility. Used for doping metrology, and Hall sensors for current/position sensing.

Generation, recombination and excess carriers

• Thermal generation creates pairs at rate G; recombination destroys them at rate R. In equilibrium G = R and np = ni².
• Mechanisms: band-to-band radiative (strong only in direct-gap materials), Shockley-Read-Hall (SRH) trap-assisted via deep levels from gold, copper, defects - dominant in Si, and the reason fabs are obsessive about metal contamination; Auger (three-carrier, matters at very high injection/doping); surface recombination at unpassivated surfaces, which SiO₂ passivation suppresses.
• Minority-carrier lifetime τ: excess carriers decay as Δp(t) = Δp(0) e^−t/τ. τ ranges from nanoseconds (gold-doped fast-recovery diodes) to milliseconds (solar-grade float-zone Si). Short τ means fast switching but more leakage and lower solar-cell efficiency.
• Diffusion length L = √(Dτ) - the average distance a minority carrier travels before recombining. It fixes transistor base width (base must be ≪ Ln) and solar-cell thickness.
• Low-level injection: Δn = Δp ≪ majority concentration; the majority density and hence conductivity are hardly changed, and small-signal device models assume it. High-level injection flattens the diode's exponential and causes β fall-off in BJTs.
• Photoconductivity: photons with hν ≥ Eg create pairs, raising σ - the basis of photodiodes, LDRs and solar cells. The cut-off wavelength is λc(µm) ≈ 1.24/Eg(eV); for Si, 1.1 µm.
• Haynes-Shockley experiment measures minority-carrier mobility, lifetime and diffusion coefficient from a drifting, spreading injected pulse.

The pn junction

• Formation: joining p and n regions makes holes diffuse into n and electrons into p; they leave behind immobile ionised acceptors (−) and donors (+). This space-charge / depletion region creates a field from n to p that opposes further diffusion. Equilibrium = zero net current, with drift and diffusion of each species exactly cancelling, and one flat Fermi level.
• Built-in potential Vbi = VT ln(NaNd/ni²) - about 0.6-0.8 V for typical Si doping, higher for larger Eg. It cannot be measured with a voltmeter (contact potentials cancel it) and cannot drive a current in an unbiased junction.
• The depletion layer extends further into the lightly doped side (charge balance: Na·xp = Nd·xn). In a one-sided (p⁺n) junction, essentially all of W lies in the n region.
• The field is triangular for an abrupt junction, peaking at the junction: Emax = 2Vbi/W (zero bias). Band diagram: both bands bend upward by qVbi going from n to p.
• Reverse bias (p negative): adds to Vbi, widens W, raises Emax, and only a small saturation current flows (minority carriers thermally generated within a diffusion length, plus generation inside W). Forward bias (p positive) shrinks W and the barrier, injecting minority carriers exponentially - leading to the ideal diode law I = I₀[exp(qV/ηkT) − 1].
• Junction (transition/depletion) capacitance Cj = εsA/W ∝ (Vbi + VR)⁻¹/2 for an abrupt junction and ∝ (Vbi + VR)⁻¹/3 for a linearly graded one; it dominates in reverse bias. Diffusion capacitance Cd = τ·I/VT dominates in forward bias. A plot of 1/Cj² against VR is linear - its slope gives the doping and its intercept gives Vbi.
• Breakdown mechanisms: Zener (tunnelling) at high doping and thin W, below about 5 V, with a *negative* temperature coefficient of breakdown voltage; avalanche (impact ionisation) at light doping and wide W, above about 7 V, with a *positive* coefficient. Between 5 and 7 V both act and the coefficient is near zero (used for reference diodes). Punch-through occurs when W reaches the far contact in a thin, lightly doped region.
• Junction types: abrupt/step (ion implantation + anneal), graded (diffusion), one-sided p⁺n or n⁺p, isotype n-n⁺ high-low junction (used in solar-cell back-surface fields).

Metal-semiconductor and hetero-junctions

• Compare the metal work function φm with the semiconductor work function φs. For n-type: φm > φs gives a rectifying Schottky barrier of height φB = φm − χ (χ = electron affinity); φm < φs gives an ohmic contact. The rules reverse for p-type.
• Schottky diodes conduct by majority-carrier thermionic emission, so there is no stored minority charge: very fast switching and low forward drop (0.2-0.4 V) but higher reverse leakage. Common metals on Si: Pt, Ni, Ti silicides.
• Real contacts are usually made ohmic by degenerate n⁺/p⁺ doping under the metal, making the barrier thin enough to tunnel through - surface states also "pin" the Fermi level, so measured barriers deviate from the ideal work-function prediction.
• Heterojunctions (two different materials, e.g. AlGaAs/GaAs, AlGaN/GaN, SiGe) have conduction- and valence-band offsets that let designers confine electrons and block holes independently - the basis of HBTs, HEMTs (GaN RF power) and quantum-well lasers.
• MOS structure: metal-oxide-semiconductor with accumulation, depletion and inversion regimes - built on the same band-bending ideas but treated in the MOSFET topic.
• Indian context: SCL Mohali runs a 180 nm fab for ISRO/strategic use; under the India Semiconductor Mission (2021) and the Chips-to-Startup (C2S) programme, Micron's ATMP plant at Sanand (commercial production from February 2026), the Tata Electronics-PSMC fab at Dholera, CG Power-Renesas and Kaynes units are in various stages of build-out, with timelines and capacities still shifting - check current announcements before quoting figures.

Key points

• Energy band - a continuum of allowed electron energies formed when atomic levels split in a crystal; conduction and valence bands are separated by the forbidden gap Eg.
• Eg at 300 K - Si 1.12 eV, Ge 0.66 eV, GaAs 1.42 eV, 4H-SiC ≈ 3.26 eV, GaN ≈ 3.4 eV; Eg falls as T rises.
• Hole - a vacancy in the valence band acting as a mobile positive charge, with lower mobility than an electron in Si.
• Direct vs indirect gap - direct (GaAs, GaN) emits light efficiently; indirect (Si, Ge) needs a phonon, so Si is a poor emitter but a fine detector.
• Donor/acceptor - group-V dopant gives a free electron (n-type); group-III gives a hole (p-type). Both keep the crystal neutral.
• Mass action law - np = ni² always holds in thermal equilibrium, whatever the doping.
• Majority concentration - n ≈ Nd (n-type) or p ≈ Na (p-type) in the extrinsic range; minority = ni²/majority.
• Fermi level - the energy at which occupancy is ½; flat across any structure in equilibrium; near Ec for n-type, near Ev for p-type.
• VT = kT/q - 25.9 mV at 300 K (≈26 mV); appears in every exponential of device physics.
• Einstein relation - D/μ = kT/q, so Dn(Si) ≈ 31 cm²/s and Dp(Si) ≈ 12 cm²/s at typical doping.
• Conductivity - σ = q(nμn + pμp); doping raises n or p by orders of magnitude while cutting μ only modestly, so ρ is set by doping.
• Mobility vs temperature - lattice scattering (∝T⁻³/2) dominates at high T, impurity scattering (∝T³/2) at low T; μ peaks in between.
• Lifetime and diffusion length - τ (SRH-dominated in Si) and L = √(Dτ) fix switching speed, leakage and transistor base width.
• Built-in potential - Vbi = VT ln(NaNd/ni²); it exists without bias but delivers no current and cannot be metered.
• Depletion width - W ∝ √(Vbi + VR); it spreads mainly into the lightly doped side, and Cj = εsA/W.
• Breakdown - Zener/tunnelling below ~5 V (negative tempco), avalanche above ~7 V (positive tempco).
• Schottky contact - φm > φs on n-type gives a fast majority-carrier rectifier; heavy n⁺ doping makes a tunnelling ohmic contact instead.
• Reverse leakage - rises steeply with T: ~2× per 10 K where generation current (∝ ni) dominates, ~2× per 5 K where diffusion current (∝ ni²) does.

Formulas and examples

Example 1 - doped silicon: carriers, resistivity and Fermi level. Si doped with Nd = 10¹⁶ cm⁻³ phosphorus at 300 K; ni = 1.5 × 10¹⁰ cm⁻³, μn = 1200 cm²/V·s at this doping.

• Donors are fully ionised and Nd ≫ ni, so n ≈ 10¹⁶ cm⁻³.
• p = ni²/n = (1.5 × 10¹⁰)²/10¹⁶ = 2.25 × 10²⁰/10¹⁶ = 2.25 × 10⁴ cm⁻³ - minority holes are 12 orders of magnitude fewer, so their current is negligible.
• σ = qnμn = 1.6 × 10⁻¹⁹ × 10¹⁶ × 1200 = 1.92 S/cm → ρ = 0.52 Ω·cm.
• EF − Ei = 0.0259 × ln(10¹⁶/1.5 × 10¹⁰) = 0.0259 × 13.41 = 0.347 eV above mid-gap (so Ec − EF = 0.56 − 0.347 = 0.213 eV).
• For comparison, intrinsic Si: σi = q ni(μn+μp) = 1.6 × 10⁻¹⁹ × 1.5 × 10¹⁰ × 1830 = 4.39 × 10⁻⁶ S/cm → ρ ≈ 2.3 × 10⁵ Ω·cm. Doping at 0.2 ppm has cut resistivity by a factor of ~4 × 10⁵.

Example 2 - compensated sample. Si with Nd = 5 × 10¹⁶ and Na = 2 × 10¹⁶ cm⁻³; at a total impurity of 7 × 10¹⁶ cm⁻³, μn ≈ 1000 cm²/V·s.

• Net donors = 3 × 10¹⁶ cm⁻³ ≫ ni, so n ≈ 3 × 10¹⁶ cm⁻³ (still n-type), p = 2.25 × 10²⁰/3 × 10¹⁶ = 7.5 × 10³ cm⁻³.
• σ = 1.6 × 10⁻¹⁹ × 3 × 10¹⁶ × 1000 = 4.8 S/cm → ρ = 0.208 Ω·cm.
• Teaching point: carrier count follows the net doping (3 × 10¹⁶), mobility follows the total ionised impurity (7 × 10¹⁶) - a favourite trap.

Example 3 - temperature and minority carriers. Same n-Si (Nd = 10¹⁶) heated from 300 K to 400 K; take Eg = 1.12 eV constant.

• ni(400)/ni(300) = (400/300)^1.5 × exp[(Eg/2)(1/kT₃₀₀ − 1/kT₄₀₀)]; kT = 0.02585 and 0.03447 eV.
• Exponent = 0.56/0.0259 − 0.56/0.0345 = 21.66 − 16.25 = 5.41 → e^5.41 = 224; prefactor = 1.54. Ratio ≈ 346.
• ni(400) ≈ 1.5 × 10¹⁰ × 346 = 5.2 × 10¹² cm⁻³ (band-gap narrowing makes it somewhat larger still).
• n stays ≈ 10¹⁶, but p = ni²/n = (5.2 × 10¹²)²/10¹⁶ = 2.7 × 10⁹ cm⁻³ - up by ~1.2 × 10⁵. Reverse leakage follows this, which is why Ge devices are useless above ~100 °C and Si above ~175-200 °C.

Example 4 - pn junction electrostatics. Abrupt Si junction, Na = 10¹⁷ cm⁻³ (p-side), Nd = 10¹⁶ cm⁻³ (n-side), 300 K, area A = 10⁻⁴ cm².

• Vbi = 0.0259 × ln(10¹⁷ × 10¹⁶/2.25 × 10²⁰) = 0.0259 × ln(4.44 × 10¹²) = 0.0259 × 29.12 = 0.754 V.
• (1/Na + 1/Nd) = 10⁻¹⁷ + 10⁻¹⁶ = 1.1 × 10⁻¹⁶ cm³. W = √[2 × 1.036 × 10⁻¹² × 0.754 × 1.1 × 10⁻¹⁶/1.6 × 10⁻¹⁹] = √(1.074 × 10⁻⁹) = 3.28 × 10⁻⁵ cm = 0.328 µm.
• xn = W·Na/(Na+Nd) = 0.328 × (10¹⁷/1.1 × 10¹⁷) = 0.298 µm; xp = 0.030 µm - 91% of the depletion layer sits in the lightly doped n-side.
• Emax = 2 × 0.754/3.28 × 10⁻⁵ = 4.6 × 10⁴ V/cm (well below the ~3 × 10⁵ V/cm avalanche field of Si).
• Cj = εsA/W = 1.036 × 10⁻¹² × 10⁻⁴/3.28 × 10⁻⁵ = 3.16 pF (31.6 nF/cm²).
• At VR = 5 V: W scales by √(5.754/0.754) = 2.76 → W = 0.905 µm, Cj = 1.14 pF. A varactor exploits exactly this.

Example 5 - diffusion and Hall measurement.

• Minority electrons in p-Si with μn = 1200 cm²/V·s, τn = 1 µs: Dn = 1200 × 0.0259 = 31.1 cm²/s, Ln = √(31.1 × 10⁻⁶) = 5.58 × 10⁻³ cm = 55.8 µm. A BJT base must be far thinner than this.
• A hole excess falling linearly by Δp = 10¹⁵ cm⁻³ over 10 µm, with Dp = 450 × 0.0259 = 11.7 cm²/s: gradient = 10¹⁵/10 × 10⁻⁴ = 10¹⁸ cm⁻⁴, so Jp = qDp × gradient = 1.6 × 10⁻¹⁹ × 11.7 × 10¹⁸ = 1.87 A/cm² - with zero electric field.
• Hall bar, thickness t = 100 µm = 10⁻⁴ m, I = 1 mA, B = 0.1 T, measured VH = 0.625 mV, polarity indicating electrons: n = IB/(q t VH) = (10⁻³ × 0.1)/(1.6 × 10⁻¹⁹ × 10⁻⁴ × 6.25 × 10⁻⁴) = 10²² m⁻³ = 10¹⁶ cm⁻³. With ρ = 0.52 Ω·cm, RH = 1/qn = 625 cm³/C and μH = σRH = 1.92 × 625 = 1200 cm²/V·s.

Where it is used

• Wafer fab and process engineering: doping by ion implantation and diffusion, four-point-probe sheet resistance, Hall and C-V profiling of doping - SCL Mohali, Micron Sanand, Tata-PSMC Dholera, Tower/Analog Devices design-and-test teams.
• Device modelling and TCAD roles: solving Poisson plus continuity equations in Sentaurus/Silvaco; SPICE parameter extraction (Is, η, Cj0, Vbi, τ) for foundry PDKs.
• Power electronics: choosing Si vs SiC vs GaN for EV traction inverters, solar string inverters and railway traction (CDAC/CPRI work, Exicom, Delta, Tata Power Solar) - the choice is an Eg, critical-field and leakage argument.
• Photovoltaics: cell efficiency is governed by τ, L and surface recombination; Indian PLI-backed lines (Adani, Reliance, Waaree, Tata Power Solar) run lifetime and quantum-efficiency measurements daily.
• Optoelectronics and sensing: direct-gap LEDs and laser diodes, Si photodiodes and image sensors, Hall-effect current and position sensors in motor drives and smart meters.
• Space and strategic electronics: radiation effects (displacement damage raising SRH recombination, leakage rise) in ISRO/DRDO/BEL components; this is standard interview ground for those PSUs.

Interview questions

• Q: Why does a semiconductor's resistance fall with temperature while a metal's rises? A: In a semiconductor, extra electron-hole pairs are generated exponentially with temperature and swamp the modest fall in mobility; a metal already has a fixed carrier count, so increased phonon scattering alone raises its resistance.
• Q: Does np = ni² hold in a forward-biased diode? A: No - it holds only in thermal equilibrium. Under bias the quasi-Fermi levels split and np = ni² exp(qV/kT) in the injected regions.
• Q: If doping raises the carrier count by 10⁶ times, why doesn't conductivity rise by exactly 10⁶? A: Ionised dopants scatter carriers, so mobility falls as doping rises; conductivity rises slightly less than proportionally.
• Q: Why can't you measure the built-in potential of a pn junction with a voltmeter? A: The contact potentials at the two metal-semiconductor probes exactly cancel it around the loop; with no net emf there is no current, consistent with equilibrium.
• Q: Why is silicon used for ICs but GaAs for lasers? A: Si is indirect-gap (poor light emission) but has a stable native oxide, high purity and low cost; GaAs is direct-gap, so electron-hole recombination emits photons efficiently.
• Q: In a p⁺n junction, where does the depletion region lie and why? A: Almost entirely in the lightly doped n-side, because the charge on each side must be equal (Na·xp = Nd·xn), so the low-doping side must be much wider.
• Q: How do you distinguish Zener from avalanche breakdown experimentally? A: By the temperature coefficient - a Zener (tunnelling) breakdown voltage decreases with temperature, an avalanche breakdown voltage increases; Zener also dominates below about 5 V with heavy doping.
• Q: What does the Einstein relation tell you physically? A: That drift and diffusion both stem from the same random thermal motion, so the diffusion coefficient and mobility are locked together by kT/q.
• Q: What is the Fermi level if there is no allowed state at that energy inside the gap? A: It is a statistical reference (chemical potential) that fixes occupancy probabilities, not an occupied state; its flatness across a structure in equilibrium is what makes band diagrams solvable.
• Q: Why is the Hall effect preferred for finding carrier concentration? A: Its sign identifies the carrier type and its magnitude gives concentration directly from VH = IB/qnt, without assuming a mobility; combining it with resistivity then yields mobility.
• Q: Why must a BJT's base be much thinner than the minority-carrier diffusion length? A: So injected minority carriers cross to the collector before recombining, which keeps the transport factor and hence β high.
• Q: Why does a Schottky diode switch faster than a pn diode? A: Conduction is by majority carriers, so there is no stored minority-carrier charge to remove - no reverse-recovery delay, only junction capacitance.

Common mistakes

• Using the net doping instead of the total impurity when picking mobility in a compensated sample (and vice versa for carrier count).
• Mixing units: ni in cm⁻³ with εs in F/m, or µm with cm in W and gradient calculations - convert everything to one system first.
• Applying np = ni² under bias or illumination, where it is invalid.
• Saying the built-in potential can drive a current, or treating Vbi as the diode's "cut-in" voltage (cut-in is set by the exponential I-V, not by Vbi).
• Assuming donors are always fully ionised - at cryogenic temperatures carriers freeze out onto the dopant levels.
• Forgetting that holes are not actual positive particles and double-counting valence-band electrons along with holes.
• Dropping the sign convention in diffusion current, so the electron and hole diffusion currents cancel instead of adding when both gradients point the same way.
• Quoting ni(Si) = 1.5 × 10¹⁰ cm⁻³ without noting the temperature, or mixing it with a Nc·Nv product from a different textbook (which yields ≈10¹⁰) in the same calculation.

Related topics

• Semiconductor diodes and applications · Special diodes and optoelectronic devices · BJT and MOSFET · FET amplifiers and biasing · Solid-state physics: crystal structure and quantum mechanics · VLSI fabrication and CMOS technology