Semiconductors

Section 02

The Idea, Step by Step

Start simple: a material caught between "yes" and "no"

A copper wire always carries current; a dry rubber band never does. Silicon is the interesting in‑between — a "maybe" material. Cold and perfectly pure, it barely conducts. But warm it up, shine light on it, or sprinkle in a few foreign atoms, and it suddenly carries current. That controllability — a switch you can flip with heat, light, or a whisper of voltage — is exactly why silicon, not copper, runs every computer.

Build it up: electrons need a running start

The electrons that bind a silicon crystal together sit in a packed lower level called the valence band. To carry current, an electron has to jump up into the nearly empty conduction band above it. The height of that jump is the band gap $E_g$ — about $1.12$ eV for silicon. Ordinary room‑temperature heat can lift only a tiny fraction of electrons across, so pure silicon conducts only weakly. Every electron that jumps leaves an empty seat behind — a hole — which drifts like a positive charge moving the opposite way.

Rather than rely on heat, engineers dope the crystal: swap about one‑in‑a‑million silicon atoms for phosphorus (one spare electron, making n‑type) or boron (one missing electron, making p‑type). Press an n block and a p block together and you get a p‑n junction: electrons and holes meet at the seam, cancel, and leave a thin charged "depletion" zone with a built‑in voltage. Current now flows easily one way and is blocked the other — that is a diode.

Go deeper (AP / intro‑college)

The number of free carriers in pure silicon climbs exponentially with temperature, which is why a hot chip leaks more current:

Intrinsic carriers grow with heat

$$n_i=\sqrt{N_cN_v}\,e^{-E_g/(2k_BT)}$$

Forward‑bias the junction and the current rises along the Shockley curve, where the thermal voltage $V_T=k_BT/e\approx26$ mV at room temperature — so the current roughly doubles every $\sim18$ mV of extra bias:

Diode current versus voltage

$$I_D=I_s\!\left(e^{V_D/V_T}-1\right)$$

Stack a third layer onto the junction and it becomes a transistor: a tiny base current steers a roughly $100\times$ larger collector current, $I_C=\beta I_B$, packing amplification and switching into a single speck of silicon. In the sim, the Band gap slider sets $E_g$ (watch $n_i$ collapse as the gap widens), Temperature kicks carriers across it, Doping fixes the junction's built‑in voltage $V_{bi}$, and Diode voltage walks you along the I‑V curve.

Try this in the sim above

In Band Theory, slide $E_g$ from $1.12$ eV down toward $0.1$ eV and watch $n_i$ explode — a tiny gap behaves almost like a metal. Still in Band Theory, raise Temperature from $77$ K to $600$ K and watch thermal electrons fill the conduction band. Then switch to Diode I‑V and sweep $V_D$ from $-2$ V up to $+1$ V to find the sharp "knee" near $0.6$ V where silicon abruptly begins to conduct.

Section 03

Equations & Derivation

Intrinsic Carrier Concentration

$$n_i=\sqrt{N_cN_v}\,e^{-E_g/(2k_BT)},\quad n_i(\text{Si, 300K})\approx1.5\times10^{10}\;\text{cm}^{-3}$$

Diode IV (Shockley Equation)

$$I_D=I_s(e^{V_D/nV_T}-1),\quad V_T=k_BT/e\approx25.9\;\text{mV at 300K},\quad n\approx1\text{–}2$$

Built-in Voltage & Depletion Width

$$V_{bi}=\frac{k_BT}{e}\ln\frac{N_aN_d}{n_i^2},\quad W=\sqrt{\frac{2\varepsilon(V_{bi}-V)}{e}\left(\frac{1}{N_a}+\frac{1}{N_d}\right)}$$

BJT Transistor

$$I_C=\beta I_B,\quad I_E=I_C+I_B=(1+\beta)I_B,\quad V_{BE}\approx0.6\;\text{V (Si)}$$

Common Semiconductors

Symbol	Quantity	Unit/Value
$\text{Si}$	Band gap 1.12 eV, dominant in electronics	n_i=1.5×10¹⁰ cm⁻³ at 300K
$\text{Ge}$	0.67 eV, early transistors, infrared	n_i=2.4×10¹³ cm⁻³
$\text{GaAs}$	1.42 eV, direct bandgap, LEDs/lasers	High electron mobility
$\text{GaN}$	3.4 eV, blue LEDs/lasers, power devices	Nobel Prize 2014

Band theory. In solids, atomic energy levels broaden into bands due to periodic crystal potential. Conductors: conduction band partially filled or overlaps valence band. Insulators: $E_g>5$ eV, thermal energy insufficient. Semiconductors: $E_g\sim0.1$–3 eV — at room temperature, some electrons thermally excited across the gap.

p-n junction. Doping: $n$-type adds electrons (donor atoms like P in Si); $p$-type adds holes (acceptor atoms like B). At the junction, diffusion creates a depletion region and built-in electric field $V_{bi}$. Forward bias reduces $V_{bi}$, current flows; reverse bias increases $V_{bi}$, current blocked (until breakdown).

Exponential I-V curve. Shockley equation: $I_D=I_s(e^{V_D/nV_T}-1)$. At room temperature $V_T=26$ mV. Forward bias: current doubles every ~18 mV (Si). Reverse bias: $I\approx-I_s$ (leakage). Ideal diode: rectifier. LED: emits photon of energy $\approx E_g$. Solar cell: photon generates electron-hole pair → voltage.

Transistor amplification. BJT: base current $I_B$ controls collector current $I_C=\beta I_B$ ($\beta\sim100$). Small input signal $\to$ large output power. MOSFET: gate voltage controls drain current (essentially infinite input impedance). CMOS (complementary MOS): pairs of n-MOSFET and p-MOSFET, near-zero static power — basis of all modern digital logic.

Ref: Kittel — Introduction to Solid State Physics (8th Ed.), Ch. 8; Sedra & Smith — Microelectronic Circuits (7th Ed.); Neamen — Semiconductor Physics and Devices (4th Ed.).

Section 04

Frequently Asked Questions

At higher $T$, more electrons have enough thermal energy to cross the band gap $E_g$ to the conduction band: $n_i\propto T^{3/2}e^{-E_g/(2k_BT)}$. This gives exponentially more carriers. In contrast, metals become less conductive at higher $T$ because increased lattice vibrations scatter electrons more. This opposite behaviour distinguishes semiconductors from metals.

Key takeaway: Semiconductor: more carriers at higher T (exponential). Metal: more scattering at higher T (less conductive).

No transistors → no computers, smartphones, or the internet. No solar cells → no photovoltaic power. No LEDs → inefficient lighting (incandescent bulbs only). No diode rectifiers → no AC-DC adapters. No laser diodes → no optical fibres, no DVDs. Semiconductor devices power every aspect of modern civilisation.

Key takeaway: Semiconductors enable computing, solar power, LEDs, telecommunications, and virtually all electronics.

The BJT operates near the edge of the p-n junction barrier. A small base current slightly forward-biases the emitter-base junction, injecting a stream of minority carriers into the base region. These diffuse across the thin base and are swept into the collector by the large reverse-biased collector junction. Small changes in base injection control a large collector current — current amplification $\beta=I_C/I_B\sim100$.

Key takeaway: BJT: small base current modulates minority carrier injection; collector current follows with gain β≈100.

$n_i=10^{10}e^{(T/300-1)\cdot E_g/(2k_B\times300)}$... More precisely: $n_i\propto T^{3/2}e^{-E_g/(2k_BT)}$. Setting $n_i=10^{16}$ and solving numerically (Si, $E_g=1.12$ eV) gives $T\approx650$ K. Above this, intrinsic carriers dominate over doping — devices fail. This sets upper temperature limits for Si devices (~150°C practical, ~300°C absolute).

Key takeaway: Si becomes intrinsic (~10¹⁶ cm⁻³) at ~650 K. This sets upper temperature limit for Si devices.

Resources: Khan Academy; HyperPhysics; MIT OCW.

Section 05

Common Misconceptions

❌ Semiconductors are poor conductors at all temperatures.

✅ Semiconductors have conductivity that varies exponentially with temperature. At low $T$: near insulator. At room temperature: intermediate. At high $T$: approaches metal conductivity as $n_i$ grows. Silicon at 650 K is nearly metallic. Heavily doped semiconductors (>10¹⁸ cm⁻³) behave like metals even at room temperature.

📖 Kittel — Introduction to Solid State Physics (8th Ed.), Ch. 8.

❌ Holes in semiconductors are actual positively charged particles.

✅ Holes are missing electrons in the valence band — a collective property of many electrons. When an electron moves left, the hole appears to move right. The hole behaves like a particle with positive charge $+e$ and effective mass $m_h^*$ (different from $m_e$). In GaAs, holes have $m_h^*\approx0.5m_e$.

📖 Kittel — Introduction to Solid State Physics, Ch. 8; Ashcroft & Mermin — Solid State Physics, Ch. 12.

❌ A transistor amplifies power because it creates energy from nothing.

✅ A transistor is not an energy source. The output power comes from the power supply $V_{CC}$. The transistor acts as a valve: the small input signal controls how much power is drawn from the supply and delivered to the load. Power gain is real but limited by conservation of energy — transistors always dissipate some supply power as heat.

📖 HRW 10th Ed., §38-4; Sedra & Smith — Microelectronic Circuits (7th Ed.), Ch. 5.

Misconception research: Driver et al. — Making Sense of Secondary Science; Chabay & Sherwood — Matter & Interactions.