Semiconductors

From atoms to energy bands

Why silicon conducts a little, and what a band gap really is.

Day 1 · 40 min

After this lesson

You can explain why silicon sits between a conductor and an insulator using energy bands.
You can quote the band gaps of Si, Ge, GaAs, SiC and GaN and say what each is used for.
You can explain why carrier density rises steeply with temperature and what that does to a circuit.

Why silicon sits between copper and glass

Silicon has atomic number 14. Ten of its electrons sit in tightly bound inner shells and four sit in the outer, or valence, shell. Like carbon and germanium in the same column of the periodic table, each silicon atom shares those four electrons with four neighbours, forming covalent bonds at 109.5° to one another. Repeat that in three dimensions and you get the diamond cubic lattice: a cube 0.543 nm on a side, 5 × 10²² atoms per cm³, and every bond holding exactly two electrons. At absolute zero every valence electron is locked into a bond, and pure silicon is a perfect insulator.

From energy levels to energy bands

An isolated atom has sharp, discrete energy levels. Bring two atoms together and each level splits in two, because the Pauli exclusion principle forbids two electrons from sharing one state. A sugar-cube-sized crystal holds about 10²² atoms, so each level splits into that many sublevels, packed so closely that they form a continuous band. In silicon the outer 3s and 3p levels mix and split into two bands with a forbidden range of energy between them:

Valence band: the bonding states. With four electrons per atom it is exactly full at 0 K.
Conduction band: the antibonding states, empty at 0 K. An electron promoted here can roam the whole crystal.
Band gap (Eg): the energy from the top of the valence band to the bottom of the conduction band, 1.12 eV for silicon at 300 K.

A completely full band cannot carry a net current: for every electron moving one way another moves the other way, and there are no empty states to shift into. Conduction needs a partly filled band. When an electron jumps the gap it becomes free in the conduction band and leaves an empty state, a hole, behind in the valence band, and the hole lets that band conduct as well. Every thermal jump therefore creates two carriers.

Metal, semiconductor and insulator band structures. The gaps are not drawn to the same scale: SiO₂'s 9 eV gap is eight times silicon's.

Metals, semiconductors and insulators

In copper the bands overlap and the highest occupied band is only partly filled, so about 8.5 × 10²² free electrons per cm³ conduct at any temperature, giving a resistivity of 1.7 µΩ·cm. In SiO₂ the gap is about 9 eV, and the thermal energy at room temperature, kT = 25.9 meV, is hopeless against it. Semiconductors sit in between with gaps of roughly 0.5 to 3.5 eV: small enough that heat, light or impurity atoms can put carriers into the conduction band in a controlled way.

Material	Eg at 300 K	Where you meet it
Germanium (Ge)	0.66 eV	The first transistors (Bell Labs, 1947), infrared detectors, SiGe in RF chips
Silicon (Si)	1.12 eV	More than 90% of all chips, solar cells, power MOSFETs and IGBTs
Gallium arsenide (GaAs)	1.42 eV, direct	Phone RF power amplifiers, laser diodes, space solar cells
Silicon carbide (4H-SiC)	3.26 eV	1200 V MOSFETs in EV traction inverters and fast chargers
Gallium nitride (GaN)	3.4 eV, direct	Blue and white LEDs, 65 W USB-C chargers, 5G radio amplifiers
Silicon dioxide (SiO₂)	about 9 eV	Insulator and gate oxide, grown directly on silicon

Temperature: the exponential that rules everything

The density of electrons that make the jump in pure, or intrinsic, material is the intrinsic carrier density, n_i ∝ T^1.5 · exp(−Eg / 2kT). The factor of 2 appears because each jump makes an electron and a hole, and they recombine in pairs. For silicon at 300 K, Eg/2kT ≈ 21.7 and n_i ≈ 1 × 10¹⁰ cm⁻³: one free electron for every five trillion atoms. Germanium, with about half the gap, has 2 × 10¹³ cm⁻³; GaAs about 2 × 10⁶ cm⁻³; 4H-SiC about 10⁻⁸ cm⁻³, which works out to one thermally freed electron per 100 cubic metres.

Because temperature sits in the exponent, n_i is brutally sensitive to it. In silicon it roughly doubles every 10 °C near room temperature. Anything that depends on thermally generated carriers, above all the reverse leakage of every p-n junction, grows about 1000× between 25 °C and 125 °C. That is why silicon parts carry temperature grades of 85 °C (industrial), 125 °C (automotive) and 150 °C (junction), and why a circuit can work perfectly on the bench and fail in an engine bay.

Why silicon won

A native oxide. Heating silicon in oxygen or steam at 900-1100 °C grows SiO₂ with a 9 eV gap, a breakdown field near 10 MV/cm and a clean interface. That oxide enabled Jean Hoerni's planar process (Fairchild, 1959) and the MOSFET (Atalla and Kahng, Bell Labs, 1959-60). Germanium's oxide dissolves in water.
Abundance and purity. Silicon is about 28% of the Earth's crust by mass. Sand is refined into 99.9999999% (nine nines) pure polysilicon, then grown into single crystals and sliced into 300 mm wafers.
A useful gap. 1.12 eV keeps leakage manageable to about 150 °C, while germanium's 0.66 eV limits it to roughly 70-85 °C.
Cost. Seventy years of investment mean a bare 300 mm silicon wafer costs around $100. A 150 mm SiC wafer, with a quarter of the area, costs several times more.

Why wide band gaps matter for EVs

A wider gap does two things. It slashes the intrinsic carrier density, so SiC and GaN devices keep low leakage at 175-200 °C, and it raises the critical electric field from about 0.3 MV/cm in silicon to about 2.8 MV/cm in 4H-SiC. Ten times the field means a 1200 V device needs a drift layer roughly ten times thinner and far more heavily doped, so its resistance per unit area can fall by over a hundred times at the material limit. Tesla's Model 3 (2017) was the first high-volume car with a SiC MOSFET traction inverter, and most 800 V platforms now use 1200 V SiC parts. GaN does the same job in 65-240 W laptop chargers, switching several times faster than silicon designs so the transformer can shrink.

Interactive: heat the crystal

Band diagram: heat the crystal

Electrons hop from the full valence band across the gap at a rate set by exp(−Eg/2kT), leaving holes behind. The animation is log-scaled (each extra electron in the conduction band means about 10× more carriers); the readouts are exact.

Temperature300 K (27 °C)

Material

Material: Silicon, Eg = 1.12 eV
Thermal energy kT: 25.9 meV
Eg / 2kT: 21.7
Free electrons n_i (= holes): 1.0 × 10¹⁰ cm⁻³
Compared with Si at 300 K: 1×
Intrinsic conductivity σ: 3.0 × 10⁻⁶ S/cm

Things to try:

Silicon from 300 K to 400 K: the conduction band gains about three electrons, which on this log scale means several hundred times more carriers.
Germanium at 300 K against silicon at 440-450 K: they have about the same n_i, so room-temperature germanium leaks like silicon at 170 °C. That is why germanium transistors failed in hot equipment.
SiC at 600 K (327 °C): at most one electron in the conduction band, and an intrinsic conductivity about 40,000 times lower than silicon's at room temperature.

In practice

Practice

How much hotter is 'hot' for silicon?

Silicon's n_i at 300 K is 1 × 10¹⁰ cm⁻³. Using n_i ∝ T^1.5 · exp(−Eg/2kT) with Eg = 1.12 eV and k = 8.617 × 10⁻⁵ eV/K, estimate n_i at 400 K (127 °C). Then explain why a SiC MOSFET can run with its junction at 200 °C while a silicon power device cannot.

text

Eg / 2k   = 1.12 / (2 × 8.617e-5)       = 6499 K
exponent  = 6499 × (1/300 − 1/400)      = 6499 × 8.33e-4 = 5.42
exp(5.42) = 225
(400/300)^1.5                           = 1.54
ratio     = 1.54 × 225                  ≈ 350
n_i(400 K) ≈ 350 × 1e10                 ≈ 3.5e12 cm^-3

SiC at 473 K: Eg/2kT = 3.26 / (2 × 8.617e-5 × 473) = 40.0
Si  at 300 K: Eg/2kT = 1.12 / (2 × 8.617e-5 × 300) = 21.7
e^-40 / e^-21.7 = e^-18.3 ≈ 1e-8

Answer: about 350× with the gap held at 1.12 eV, so n_i ≈ 3.5 × 10¹² cm⁻³. Silicon's gap actually shrinks to about 1.10 eV at 400 K, and measured data give roughly 550×. Either way it is hundreds of times. The doubling-every-10 °C rule predicts 2¹⁰ ≈ 1000×, a slight overestimate for n_i, because the doubling interval stretches from about 9 °C at room temperature to about 15 °C at 400 K. Junction leakage has a diffusion component that grows as n_i², so real leakage often rises even faster than n_i, and 1000× remains a sensible design margin.

For SiC at 200 °C the exponent Eg/2kT is still 40, far larger than silicon's 21.7 at room temperature, so hot SiC has tens of millions of times fewer intrinsic carriers than cool silicon (n_i ≈ 2 × 10² cm⁻³) and its junction leakage stays negligible. Silicon at 200 °C reaches n_i ≈ 10¹⁴ cm⁻³, comparable to the doping of the drift layer in a 1200 V device, so that layer stops behaving like doped material and the part can no longer block. For SiC the real 200 °C limits are the package (solder, gel, mould compound) and gate-oxide reliability, not the semiconductor.

Real-world scenario

Sensor board reads garbage above 125 °C

An engine-bay pressure sensor module passed every bench test and went into AEC-Q100 Grade 1 qualification (−40 to +125 °C ambient). The bridge amplifier drives a 1 MΩ series resistor and a 10 nF capacitor, a 16 Hz anti-alias filter, and a BAV99 dual switching diode clamps the filtered node to the 3.3 V and ground rails to protect the ADC input. At 25 °C the reading is accurate to 0.1%. In the thermal chamber it starts drifting above 85 °C and is useless at 125 °C.

Symptoms

Within 0.1% at 25 °C, about 2% high at 85 °C, and nearly 1 V (30% of full scale) off at 125 °C
The error is smooth and repeatable, and it disappears completely when the chamber cools; nothing is damaged
Swapping the pressure sensor for a precision resistor bridge does not change the drift
A second board built with a 10 kΩ filter resistor for a different customer drifts far less

Interview questions

screenWhy is silicon a semiconductor while copper is a conductor and glass is an insulator?

It comes down to how full the bands are and how big the gap is. In copper the highest occupied band is only partly full, so roughly 10²³ electrons per cm³ have empty states right next to them and conduct at any temperature. In glass the valence band is full and the gap is about 9 eV, hundreds of times the 26 meV of thermal energy at room temperature, so essentially no electrons get across. Silicon also has a full valence band, but its gap is only 1.12 eV, so about 10¹⁰ electrons per cm³ are thermally excited at 300 K, and doping can then set its conductivity over many orders of magnitude.

screenWhat are the band gaps of Si, SiC and GaN, and why do EV and charger makers care?

Silicon is 1.12 eV, 4H silicon carbide is 3.26 eV and gallium nitride is 3.4 eV. A wider gap means far fewer thermally generated carriers, so leakage stays low at 175-200 °C, and the critical electric field is roughly ten times silicon's. That lets a 1200 V device use a drift layer about ten times thinner and much more heavily doped, which cuts on-resistance and switching loss. That is why 800 V traction inverters use SiC MOSFETs and compact USB-C chargers use GaN transistors.

deepWhy does leakage rise so quickly with temperature, and how would you design an analog front end that must work at 150 °C?

Reverse leakage comes from thermally generated carriers, and the intrinsic carrier density goes as T^1.5 × exp(−Eg/2kT), so in silicon it roughly doubles every 10 °C. From 25 °C to 150 °C that is more than three orders of magnitude, so a 1 nA leak at room temperature becomes several microamps. I would keep every node that touches a junction at low impedance, budget leakage times impedance at the maximum temperature, and choose low-leakage clamps and CMOS- or JFET-input amplifiers characterized at 150 °C. For harsher jobs, such as downhole tools at 200 °C, I would move to SOI or SiC parts, where the buried oxide or the wider gap keeps leakage down.

deepWhy did silicon beat germanium and gallium arsenide for mainstream integrated circuits?

Germanium came first, but its 0.66 eV gap makes it leaky above about 70 °C and its oxide dissolves in water, so it cannot protect or insulate a surface. Silicon grows a stable, high-quality SiO₂ simply by heating it in oxygen, and that native oxide made the planar process and the MOSFET possible around 1959-1960. Silicon is also abundant, about 28% of the Earth's crust, strong enough to be grown into 300 mm wafers, and its 1.12 eV gap keeps leakage manageable to about 150 °C. GaAs has roughly six times the electron mobility, but it has no usable native oxide and comes on smaller, costlier wafers, so it stays in RF amplifiers, lasers and LEDs.

screenWhat is the difference between a direct and an indirect band gap?

In a direct-gap material such as GaAs or GaN, the bottom of the conduction band and the top of the valence band sit at the same crystal momentum, so an electron can fall across the gap and emit a photon on its own. In an indirect-gap material such as silicon, the electron must also trade momentum with a lattice vibration (a phonon), which makes light emission tens of thousands of times less likely. That is why LEDs and laser diodes use GaAs, InGaN and similar compounds. Silicon still absorbs light above its gap well enough to make good solar cells and camera sensors.

References and further reading

Where the ideas in this lesson come from, and what to read next. Books are the standard texts; papers are the original sources; standards are what employers work to.

Book
Ben G. Streetman and Sanjay K. Banerjee (2015). Solid State Electronic Devices. Pearson, 7th edition.
Read: Chapters 1-3: crystal structure, energy bands and carrier concentration.
Book
Donald A. Neamen (2012). Semiconductor Physics and Devices: Basic Principles. McGraw-Hill, 4th edition.
Read: The chapter on energy bands, for a second explanation with worked numbers.
Book
S. M. Sze and Kwok K. Ng (2007). Physics of Semiconductor Devices. Wiley, 3rd edition.
Read: The appendix tables for the band gaps and intrinsic carrier densities quoted in the lesson.

Next: Doping, electrons and holes →