Semiconductor physics

In short

Semiconductor physics explains how materials like silicon, with a band gap of about 1.1 eV, conduct through two carrier types — electrons and holes — whose numbers can be tuned by temperature, doping and light. It gives the theory behind every diode, transistor, sensor, LED and solar cell, so it underpins both analogue and digital electronics. B.Sc and M.Sc papers examine carrier statistics, the mass-action law, drift-diffusion transport, the Hall effect and p-n junction behaviour; GATE (PH and EC), JAM, CSIR-NET and JEST ask numerical problems on exactly these. Indian employers — SCL Mohali, ISRO/SAC, BEL, CDAC, Tata Elxsi, Applied Materials, Lam Research, Micron Sanand, Tata–PSMC Dholera and semiconductor-mission fabs — test it in technical rounds for process, device-characterisation and test-engineering roles.

Explained

Bands, bonding and classification

• A crystal's allowed electron energies form bands; the highest filled band at 0 K is the valence band (VB), the next empty one the conduction band (CB), separated by the forbidden gap Eg.
• Metal: bands overlap or CB partly filled — conduction at all temperatures. Insulator: Eg large (diamond ≈ 5.5 eV). Semiconductor: Eg moderate — Ge 0.66 eV, Si 1.12 eV, GaAs 1.42 eV at 300 K.
• Si and Ge are elemental (group IV) semiconductors with diamond-cubic lattice and covalent tetrahedral bonds; compound semiconductors are III-V (GaAs, GaN, InP), II-VI (CdS, ZnO) or IV-IV (SiC).
• Direct gap (GaAs, GaN, InP): CB minimum and VB maximum at the same k, so radiative recombination is efficient — used for LEDs and lasers. Indirect gap (Si, Ge): a phonon must assist, so Si is a poor light emitter but an excellent switch and detector.
• Breaking a covalent bond creates a free electron and a hole — a vacancy that behaves as a mobile positive charge of effective mass mh*; holes move by successive bond-to-bond electron hops.
• Effective mass m* comes from band curvature, m* = ħ²/(d²E/dk²); it absorbs the lattice's periodic potential so carriers can be treated as free particles.
• Temperature dependence of Eg (Varshni form): Eg falls with T — for Si roughly Eg(T) ≈ 1.17 − 4.73×10⁻⁴T²/(T+636) eV.

Intrinsic and extrinsic semiconductors

• Intrinsic: chemically pure; n = p = ni. For Si ni ≈ 1.0–1.5×10¹⁰ cm⁻³ at 300 K (textbooks use 1.5×10¹⁰; modern measurements give ≈1.0×10¹⁰ — state which value you use), Ge ≈ 2.4×10¹³ cm⁻³, GaAs ≈ 2×10⁶ cm⁻³.
• Fermi level EF is the energy at which occupation probability is ½. In an intrinsic sample EF sits near mid-gap, shifted by (3kT/4)ln(mh*/me*).
• n-type: doping with group V (P, As, Sb) adds donors with a level ED just below EC (Si:P ≈ 45 meV); electrons are majority carriers, EF moves up toward EC.
• p-type: doping with group III (B, Al, Ga, In) adds acceptors with EA just above EV (Si:B ≈ 45 meV); holes are majority, EF moves down toward EV.
• Doped material stays electrically neutral: n + NA⁻ = p + ND⁺. With complete ionisation and ND ≫ ni, n ≈ ND and p ≈ ni²/ND.
• Mass-action law: np = ni² holds at thermal equilibrium for non-degenerate material regardless of doping — the single most used relation in numericals.
• Compensated semiconductor: both dopants present, net doping |ND − NA| decides the type.
• Degenerate doping (≳10¹⁹ cm⁻³) pushes EF into the band; Boltzmann statistics fail, the material behaves semi-metallically (used for ohmic contacts and tunnel diodes).
• Three temperature regimes of n(T) in n-type Si: *ionisation/freeze-out* (low T, carriers released from donors), *extrinsic/saturation* (wide plateau, n ≈ ND — devices are designed here), *intrinsic* (high T, ni swamps ND and the device loses its doping identity).

Carrier statistics and transport

• Occupation follows Fermi–Dirac f(E) = 1/[1+exp((E−EF)/kT)]; for E−EF > 3kT it reduces to the Boltzmann tail, giving n = Nc exp(−(Ec−EF)/kT), p = Nv exp(−(EF−Ev)/kT).
• Effective density of states Nc = 2(2πme*kT/h²)^{3/2} (×MC valleys), similarly Nv; for Si at 300 K Nc ≈ 2.8×10¹⁹ cm⁻³, Nv ≈ 1.04×10¹⁹ cm⁻³.
• Drift: an applied field gives drift velocity vd = µE; current density J = q(nµn + pµp)E, so conductivity σ = q(nµn + pµp) and resistivity ρ = 1/σ.
• Mobility µ (cm²/V·s) at 300 K, lightly doped Si: µn ≈ 1350, µp ≈ 480; Ge 3900/1900; GaAs ≈ 8500 for electrons — hence GaAs for microwave devices.
• Scattering: lattice (phonon) scattering gives µ ∝ T⁻³/2; ionised-impurity scattering gives µ ∝ T^{3/2}/Nimpurity. µ therefore peaks at intermediate T and falls with heavy doping.
• At high fields vd saturates near 10⁷ cm/s in Si; GaAs shows negative differential mobility (Gunn effect).
• Diffusion: a concentration gradient drives Jn = qDn dn/dx, Jp = −qDp dp/dx. Einstein relation D/µ = kT/q (≈0.0259 V at 300 K).
• Total current is drift + diffusion for each carrier: a bipolar picture, unlike metals.
• Generation–recombination: excess carriers decay as Δn(t) = Δn(0)e^{−t/τ}, τ = minority-carrier lifetime (µs to ms in Si). Mechanisms: band-to-band radiative, Shockley–Read–Hall trap-assisted (dominant in Si), and Auger (dominant at high injection).
• Diffusion length L = √(Dτ) — the average distance a minority carrier travels before recombining; sets base width in a BJT and collection depth in a solar cell.
• Continuity equation: ∂n/∂t = (1/q)∂Jn/∂x + Gn − Rn — the master equation from which the diode law and transistor equations are derived.
• Hall effect: with current along x and B along z, carriers deflect, building a transverse field. Its sign gives the carrier type and its magnitude gives the concentration and, with σ, the mobility — the standard characterisation experiment in Indian B.Sc labs.

The p-n junction

• Joining p and n material makes majority carriers diffuse across; the uncovered dopant ions leave a depletion (space-charge) region with a built-in field opposing further diffusion. At equilibrium EF is flat and the net current is zero.
• Built-in potential V0 = (kT/q)ln(NAND/ni²); for Si typically 0.6–0.8 V. It cannot be measured with a voltmeter — contact potentials cancel round the loop.
• Depletion width extends mostly into the lightly doped side (W1ND = W2NA); W ∝ √(V0 − V), so reverse bias widens it.
• Forward bias (p positive) lowers the barrier, narrows W and injects minority carriers — current rises exponentially beyond the cut-in voltage (≈0.7 V Si, ≈0.3 V Ge, ≈1.8–3.3 V for LEDs depending on colour).
• Reverse bias widens W; only a small reverse saturation current I0 flows, set by minority carriers and roughly doubling every 10 °C in Si.
• Junction capacitances: *transition/depletion* capacitance CT = εA/W dominates in reverse bias (used in varactors, CT ∝ V⁻¹/2 for abrupt and V⁻¹/3 for graded junctions); *diffusion* capacitance CD = τI/(ηVT) dominates in forward bias and limits switching speed.
• Breakdown: Zener (field-assisted tunnelling, heavily doped, narrow junction, VBR ≲ 5 V, *negative* temperature coefficient) and avalanche (impact ionisation, lightly doped, VBR ≳ 6 V, *positive* temperature coefficient). Both are non-destructive if current is limited.
• Other junctions: metal–semiconductor Schottky barrier (majority-carrier device — fast, low forward drop ≈0.3 V, no charge storage), ohmic contact (heavily doped interface), heterojunction (different Eg, e.g. AlGaAs/GaAs — used in HEMTs and laser diodes).

Devices built on the physics

• Rectifier diode, Zener regulator, varactor, tunnel diode (negative resistance from band-to-band tunnelling), Schottky diode (mixers, fast clamps).
• Optoelectronic: LED (direct-gap radiative recombination; λ ≈ 1240/Eg nm with Eg in eV), photodiode (reverse-biased, photocurrent ∝ illumination), solar cell (fourth-quadrant operation; characterised by Voc, Isc, fill factor and efficiency), photoconductor (LDR), laser diode.
• BJT: two junctions sharing a thin, lightly doped base; minority-carrier injection and diffusion across the base give current gain β = IC/IB, α = IC/IE, β = α/(1−α). Needs EB forward- and CB reverse-biased for active mode.
• FET/MOSFET: field-controlled majority-carrier device; the gate field creates an inversion channel beyond the threshold voltage VT. Enhancement/depletion, n/p channel; CMOS pairs them for near-zero static power — the basis of all VLSI.
• Thyristors (SCR, TRIAC, IGBT): four-layer and hybrid structures for power control in drives, inverters and HVDC.
• Processing chain (what a fab interview probes): Czochralski crystal growth → wafer slicing and polishing → oxidation → photolithography → ion implantation or diffusion doping → annealing → CVD/PVD deposition → etching → metallisation → packaging and testing.

Indian standards, policy and practice

• Measurements use SI units; concentrations are conventionally quoted in cm⁻³ (1 cm⁻³ = 10⁶ m⁻³) — convert carefully.
• Indian device standards largely adopt IS/IEC semiconductor-device series (IEC 60747 family) and IS/IEC 60749 for mechanical/climatic test methods; quote the subject rather than a number if unsure of the current edition.
• India Semiconductor Mission (ISM) under MeitY, announced with ₹76,000 crore outlay in 2021, supports fabs, OSAT/ATMP and compound-semiconductor units — Micron's ATMP at Sanand, Tata–PSMC's fab at Dholera, Tata's Assam OSAT, CG Power–Renesas and Kaynes Semicon OSATs at Sanand. Project statuses and incentive terms change; verify current figures before quoting.
• SCL Mohali (Semi-Conductor Laboratory, moved from the Department of Space to MeitY) is the legacy government fab (180 nm) used for strategic ICs; BEL, ISRO and DRDO labs are major users.
• E-waste and hazardous handling: E-Waste (Management) Rules, 2022 and the Hazardous and Other Wastes Rules govern fab chemicals and discarded electronics; BIS compulsory registration (CRS) applies to many electronic products.
• Electronics goods and components fall under GST slabs of 5%/18%/40% as revised from 22 September 2025; most electronic components attract 18%.
• Fab and lab employment conditions now sit under the four Labour Codes in force from 21 November 2025, which replaced the Factories, Minimum Wages, Industrial Disputes and BOCW Acts.

Key points

• Band gap Eg - energy to free a bonded electron: Ge 0.66, Si 1.12, GaAs 1.42 eV at 300 K; Eg falls as T rises.
• Hole - mobile vacancy in the valence band acting as a positive carrier with its own effective mass and mobility.
• Mass-action law - np = ni² at equilibrium, for any non-degenerate doping.
• ni dependence - ni² = NcNv exp(−Eg/kT), so ni rises steeply with T and falls with Eg.
• Donor/acceptor - group V gives n-type (EF near EC), group III gives p-type (EF near EV); both ionise almost fully at 300 K.
• Charge neutrality - n + NA = p + ND fixes carrier numbers together with np = ni².
• Conductivity - σ = q(nµn + pµp); in doped material one term dominates, so σ ≈ qNDµn or qNAµp.
• Einstein relation - D/µ = kT/q = 25.9 mV at 300 K, linking diffusion and drift.
• Mobility vs T - lattice scattering T⁻³/2 at high T, impurity scattering T^{3/2} at low T; heavy doping always reduces µ.
• Semiconductor resistance has a negative temperature coefficient - more carriers overwhelm reduced mobility, unlike metals.
• Diffusion length - L = √(Dτ); the base of a BJT must be much thinner than L.
• Built-in potential - V0 = (kT/q)ln(NAND/ni²), 0.6–0.8 V in Si; depletion width ∝ √(V0 − V).
• Diode equation - I = I0[exp(V/ηVT) − 1], VT = kT/q, η ≈ 1–2.
• Zener vs avalanche - tunnelling below ~5 V with negative temperature coefficient; impact ionisation above ~6 V with positive coefficient.
• Direct vs indirect gap - GaAs/GaN emit light efficiently; Si and Ge do not, but Si dominates ICs for its stable oxide and cost.
• Hall effect - sign of VH identifies carrier type; RH = 1/(nq) gives concentration, µ = σ|RH|.
• Junction capacitance - depletion CT (reverse bias, varactors) and diffusion CD (forward bias, limits switching).
• Degenerate doping - above ~10¹⁹ cm⁻³ EF enters the band; used for ohmic contacts and tunnel diodes.

Formulas and examples

Example 1 — carrier concentrations in doped Si. Si at 300 K with ni = 1.5×10¹⁰ cm⁻³ is doped with ND = 5×10¹⁶ cm⁻³ phosphorus.

• ND ≫ ni, so n ≈ 5×10¹⁶ cm⁻³.
• p = ni²/n = (1.5×10¹⁰)²/(5×10¹⁶) = 2.25×10²⁰/5×10¹⁶ = 4.5×10³ cm⁻³.
• σ ≈ qnµn = 1.602×10⁻¹⁹ × 5×10¹⁶ × 1350 ≈ 10.8 S/cm, so ρ ≈ 0.0925 Ω·cm.
• Fermi-level position: Ec − EF = kT ln(Nc/n) = 0.0259 × ln(2.8×10¹⁹/5×10¹⁶) = 0.0259 × ln(560) = 0.0259 × 6.328 ≈ 0.164 eV below Ec.

Example 2 — intrinsic conductivity of germanium. Ge at 300 K: ni = 2.4×10¹³ cm⁻³, µn = 3900, µp = 1900 cm²/V·s.

• σ = q ni (µn + µp) = 1.602×10⁻¹⁹ × 2.4×10¹³ × 5800.
• 1.602×10⁻¹⁹ × 2.4×10¹³ = 3.845×10⁻⁶; × 5800 = 0.0223 S/cm → ρ ≈ 44.8 Ω·cm.
• A bar 1 cm long, cross-section 1 mm² = 0.01 cm²: R = ρL/A = 44.8 × 1/0.01 ≈ 4.5 kΩ.

Example 3 — Hall effect measurement. A semiconductor slab of thickness t = 0.5 mm carries I = 10 mA in B = 0.2 T; the measured Hall voltage is VH = 1.25 mV, positive at the expected p-type face.

• Use SI: t = 5×10⁻⁴ m. n = IB/(qtVH) = (10×10⁻³ × 0.2)/(1.602×10⁻¹⁹ × 5×10⁻⁴ × 1.25×10⁻³).
• Numerator = 2×10⁻³. Denominator = 1.602×10⁻¹⁹ × 6.25×10⁻⁷ = 1.00125×10⁻²⁵.
• n = 2×10⁻³/1.00125×10⁻²⁵ ≈ 2.0×10²² m⁻³ = 2.0×10¹⁶ cm⁻³, and the sign of VH shows holes are the majority carriers.
• RH = 1/(nq) = 1/(2.0×10²² × 1.602×10⁻¹⁹) ≈ 3.12×10⁻⁴ m³/C; if σ = 50 S/m then µ = σRH ≈ 1.56×10⁻² m²/V·s = 156 cm²/V·s.

Example 4 — built-in potential and diffusion length. Si p-n junction, NA = 10¹⁷ cm⁻³, ND = 10¹⁵ cm⁻³, ni = 1.5×10¹⁰ cm⁻³, T = 300 K.

• NAND/ni² = 10³²/2.25×10²⁰ = 4.44×10¹¹; ln(4.44×10¹¹) = 26.82.
• V0 = 0.0259 × 26.82 ≈ 0.695 V.
• Minority electrons in the p-side with Dn = µnkT/q = 1350 × 0.0259 ≈ 35 cm²/s and τn = 1 µs: Ln = √(35 × 10⁻⁶) = √(3.5×10⁻⁵) ≈ 5.9×10⁻³ cm = 59 µm.

Example 5 — diode current and dynamic resistance. A Si diode has I0 = 10 nA, η = 1, VT = 0.0259 V, forward bias V = 0.6 V.

• V/VT = 0.6/0.0259 = 23.17; exp(23.17) ≈ 1.16×10¹⁰.
• I ≈ 10×10⁻⁹ × 1.16×10¹⁰ ≈ 116 A (unrealistically large — a real diode's series resistance and high-injection effects hold the drop near 0.7 V at ampere currents; this is why the ideal law is used only at low currents).
• At I = 1 mA instead, rd = ηVT/I = 0.0259/10⁻³ ≈ 25.9 Ω.

Where it is used

• Fabs, ATMP/OSAT and equipment firms: Micron Sanand, Tata–PSMC Dholera, CG Power–Renesas, Kaynes Semicon, Applied Materials and Lam Research India — process, yield, metrology and failure-analysis engineers use doping profiles, resistivity and lifetime measurements daily.
• Strategic and space electronics: SCL Mohali, ISRO/SAC, BEL, DRDO labs — radiation-hard device design where carrier lifetime and leakage physics decide reliability.
• Power electronics and EVs: IGBT and SiC/GaN modules in Exide, Delta, ABB India, Tata Motors and BHEL drives — breakdown voltage and thermal limits come straight from this physics.
• Solar PV manufacturing: Adani Solar, Waaree, Vikram Solar, Tata Power Solar — cell efficiency depends on diffusion length, surface recombination and junction quality under the PM Surya Ghar and ALMM framework.
• Characterisation labs and academia: Hall-effect, four-probe resistivity, C–V and deep-level transient spectroscopy at IITs, CSIR labs (CEERI Pilani, NPL), and university M.Sc projects.
• Job roles for physics graduates: semiconductor process technician, device characterisation engineer, test/ATE engineer, applications engineer, photovoltaics QA, and research-fellow positions funded under the India Semiconductor Mission.

Interview questions

• Q: Why does a semiconductor's resistance fall with temperature while a metal's rises? A: Heating creates far more carriers across the gap (exponential in Eg/2kT), which outweighs the modest drop in mobility; a metal has a fixed carrier count, so only the increased scattering shows.
• Q: Does doping change the mass-action law np = ni²? A: No — at thermal equilibrium np = ni² always holds for non-degenerate material; doping only redistributes n and p between themselves.
• Q: Why is silicon preferred over germanium even though Ge has higher mobility? A: Si has a wider gap, so much lower leakage and a higher usable temperature, plus a stable native oxide (SiO₂) for gates and masking, and abundant cheap raw material.
• Q: Why can't silicon make an efficient LED? A: It has an indirect band gap, so electron–hole recombination needs a phonon; the process is slow and mostly non-radiative.
• Q: Can the built-in potential of a p-n junction be measured with a voltmeter? A: No — the metal–semiconductor contact potentials at the probes exactly cancel it around the loop, so the meter reads zero at equilibrium.
• Q: Distinguish Zener and avalanche breakdown. A: Zener is band-to-band tunnelling in a heavily doped narrow junction below about 5 V with a negative temperature coefficient; avalanche is impact ionisation in a lightly doped wide junction above about 6 V with a positive coefficient.
• Q: What does the Hall effect tell you that a resistivity measurement cannot? A: It gives the sign of the majority carrier and the carrier concentration separately; combining it with σ then yields mobility.
• Q: Why must a BJT's base be thin and lightly doped? A: Thin (≪ diffusion length) so injected minority carriers reach the collector without recombining, and lightly doped so emitter injection efficiency and hence β stay high.
• Q: Why does a Schottky diode switch faster than a p-n diode? A: Conduction is by majority carriers only, so there is no minority-carrier charge storage and no reverse-recovery delay.
• Q: What is the Einstein relation used for? A: It converts between diffusion coefficient and mobility, D = (kT/q)µ, letting one measured parameter supply both in drift-diffusion calculations.
• Q: What happens if the operating temperature pushes a doped device into its intrinsic region? A: ni exceeds the doping, so the junction loses its p-n character — leakage soars and the device stops rectifying; this fixes the maximum junction temperature.
• Q: Why does the depletion region extend further into the lightly doped side? A: The total exposed charge on both sides must be equal, so fewer dopant ions per unit volume means a greater width: NAW_p = NDW_n.

Common mistakes

• Mixing cm⁻³ and m⁻³, or using eV for k while T-terms are in joules — the commonest source of wrong carrier numbers and Hall results.
• Writing n = ND without checking whether ND is comparable to ni (fails at high temperature or very light doping) or whether the sample is compensated.
• Thinking doping makes the crystal charged — it stays neutral; only the depletion region has net space charge.
• Treating holes as physical positive particles rather than a bookkeeping description of valence-band vacancies.
• Using ni = √(NcNv)exp(−Eg/2kT) but forgetting the factor of 2 in the denominator, or confusing Eg with Eg/2.
• Applying the ideal diode equation at high forward current and reporting absurd currents — series resistance and high-injection dominate there.
• Swapping the temperature coefficients and doping levels of Zener versus avalanche breakdown.
• Assuming mobility only falls with temperature — at low temperature impurity scattering makes it rise with T.

Related topics

Solid state physics · Operational amplifiers · Number systems and Boolean algebra · Statistical mechanics and quantum statistics · Quantum mechanics: particle in a potential well · Optoelectronics and photovoltaic devices