Bands and Semiconductors

Hook: Copper Conducts, Rubber Doesn't, Silicon Sits in the Middle

Pick up a cable. Inside: copper. Outside: plastic. Same laws of physics, same atoms (protons, neutrons, electrons), same temperature. Yet one carries current, the other does not. Why?

Because atoms don’t live alone. When billions sit together in a crystal, each atom’s sharp energy levels (the Bohr levels from Module 1.1) crowd into one another and become bands. Some bands give and take electrons — current flows through them. Some bands are either full or separated by forbidden zones — no current flows.

Silicon is in between — it behaves like an insulator, but can be made conducting with a small amount of energy. You supply it with a neighboring atom (phosphorus, boron — “doping”) or with heat/light. Transistors, diodes, memristors — all built on this “in-between” property.

The HfO₂ layer in SIDRA’s memristor, in classical terms, is an insulator. Big band gap. But we make it controllably conducting. By the end of this chapter you’ll see how.

Intuition: How Do Levels Become Bands?

A lone hydrogen atom has discrete lines $n=1, 2, 3, …$ . Bring two atoms close: Pauli’s principle forbids them from sharing the same state. The levels split in two.

Four atoms close → four-way split. When $10^{23}$ atoms sit together in a crystal, the splittings are so dense they merge into a continuous band. And between bands coming from different orbital origins, there are forbidden zones — energies an electron cannot have.

Three important bands:

Valence band: the topmost filled band. At $T=0$ , every state here is occupied.
Conduction band: the empty band above it. An electron here is “free” — it roams through the crystal and carries current.
Band gap ( $E_g$ ): the forbidden zone between them. An electron cannot exist inside the gap; it must jump $E_g$ worth of energy to cross.

Three material families:

Type	$E_g$	Electron state	Example
Metal	0 (bands overlap)	Conduction band half-filled; current always flows	Cu, Au, Al
Semiconductor	0.5–3 eV (modest)	A few electrons jump at $T>0$ ; doping controls the count	Si (1.12), Ge (0.67), GaAs (1.42)
Insulator	> 4 eV (big)	No electron jumps at room temperature	HfO₂ (5.7), SiO₂ (9), diamond (5.5)

Copper conducts because its conduction band is already half-filled. Plastic doesn’t because its gap is huge. Silicon is the sweet spot — perfect raw material for making electricity do work for us.

Formalism: Boltzmann, Fermi, and Doping

L1 · Intro

Temperature matters. As temperature rises, some electrons in the valence band gain enough energy to jump to the conduction band. Behind each jumped electron sits a hole — which behaves like a positive charge.

Simple rule: the probability of an electron jumping drops exponentially as $E_g$ grows, and rises with temperature. At room temperature (300 K), the typical “thermal energy” is $k_B T \approx 0.026$ eV. Silicon’s gap is 1.12 eV — 40× the thermal energy. So very few electrons jump, but just the right amount to be useful.

Doping: deliberately add phosphorus (extra electron) to silicon → n-type (electron conduction). Add boron (missing electron) → p-type (hole conduction). That’s the foundation of chip manufacturing.

L2 · Full

Intrinsic carrier concentration via Boltzmann:

n_i \approx N_0 \, \exp\!\left(-\frac{E_g}{2 k_B T}\right)

with $N_0 \sim 10^{19}$ cm⁻³ (density-of-states prefactor for Si). At room temperature:

Si ( $E_g = 1.12$ eV): $n_i \approx 10^{10}$ cm⁻³ — essentially no carriers; pure Si is a poor conductor.
HfO₂ ( $E_g = 5.7$ eV): $n_i \approx 10^{-30}$ cm⁻³ — none.
Metal: no gap → $\sim 10^{22}$ cm⁻³ free electrons.

To make silicon useful, we dope it. Example with phosphorus at $N_D = 10^{16}$ cm⁻³:

n \approx N_D = 10^{16} \text{ cm}^{-3}, \quad p = \frac{n_i^2}{n} \approx 10^4 \text{ cm}^{-3}

Fermi level ( $E_F$ ): the energy at which the occupation probability is 50%. In intrinsic semiconductors it sits near mid-gap; n-doping pushes it up toward the conduction band, p-doping down toward the valence band.

Fermi-Dirac distribution:

f(E) = \frac{1}{1 + \exp\!\left(\frac{E - E_F}{k_B T}\right)}

Chemical potential = $E_F$ . In the $T \to 0$ limit it’s a step: 1 for $E < E_F$ , 0 above.

L3 · Deep

Real band structure lives in momentum space ( $k$ ): $E(k)$ . In a crystal’s periodic potential (Bloch’s theorem), wave functions take the form $\psi_{nk}(\mathbf{r}) = u_{nk}(\mathbf{r}) e^{i \mathbf{k} \cdot \mathbf{r}}$ . Bands $E_n(\mathbf{k})$ are drawn across the Brillouin zone.

Direct vs indirect gap:

Si — indirect: valence maximum at $\Gamma$ , conduction minimum at $X$ . A photon alone cannot bridge the gap; a phonon is needed to supply the momentum. That’s why silicon doesn’t emit light (no silicon LED).
GaAs — direct: both extrema at $\Gamma$ . Excellent for LEDs and lasers.
HfO₂ — amorphous in the SIDRA stack; strict $\mathbf{k}$ labels don’t apply. Transport is described via local orbital overlap.

Effective mass: band curvature sets how the electron accelerates: $m^* = \hbar^2 / (\partial^2 E / \partial k^2)$ . In Si, electron $m^* \sim 0.26 \, m_e$ , hole $\sim 0.49 \, m_e$ . This number drives transistor speed/power trade-offs.

Anderson localization: in a sufficiently disordered crystal, bands fragment and some conduction-band states remain localized. HfO₂’s memristive behavior sits right at this edge — oxygen vacancies open a conduction path; the filament closes when biased differently. This bridges us to Module 2.

Experiment: Material, Temperature, Voltage

In the band diagram below, switch between three materials (Metal / Semiconductor / Insulator). Adjust the temperature — in the semiconductor you’ll see electrons hop to the conduction band. Toggle voltage; electrons drift in the conduction band, holes drift oppositely in the valence band.

ℹ️ Animation note: For visualization only, the Boltzmann exponent is pedagogically stretched by a factor of ~6; otherwise the real $\exp(-E_g / 2 k_B T)$ at room temperature is so small that no electron would ever appear among 24 slots. The real physics shape is preserved (exponential rise with T, exponential drop with $E_g$ ); only the absolute count is rendered visible. You’ll compute the real number in the Lab task.

Try this:

Drop temperature to 0 K. Semiconductor has no conduction-band electrons. Voltage produces no current.
At 300 K (room T) — you’ll see about one electron in Si’s conduction band; a small current starts.
At 1000 K — Si shows ~8 electrons promoted. Transistors misbehave at this temperature.
Pick HfO₂, crank temperature to 1500 K. You’ll barely see one or two electrons (5.7 eV is still huge).
Pick Metal. Current starts the moment you apply voltage, regardless of temperature.

Quiz

1/5What is the 'forbidden zone' between the valence and conduction bands?

Lab Task: Si vs HfO₂ Intrinsic Density

Use $n_i \approx N_0 \exp(-E_g / 2 k_B T)$ with $N_0 \approx 10^{19}$ cm⁻³ and $k_B T$ (room T) $= 0.026$ eV.

Steps:

For silicon ( $E_g = 1.12$ eV), compute the exponent: $-E_g / (2 k_B T) = -1.12 / (2 \cdot 0.026) = -21.5$ .
$\exp(-21.5) \approx ?$ (a very small number)
$n_i = 10^{19} \cdot \exp(-21.5)$ .
Same calculation for HfO₂ ( $E_g = 5.7$ eV).
Compute the ratio $n_i(\text{Si}) / n_i(\text{HfO₂})$ .

Hint:

Rough answer

(1-2) $\exp(-21.5) \approx 4.6 \times 10^{-10}$ .
(3) $n_i(\text{Si}) \approx 10^{19} \times 4.6 \times 10^{-10} = 4.6 \times 10^9$ cm⁻³ ≈ 10¹⁰ cm⁻³.
(4) HfO₂: $-5.7 / (2 \cdot 0.026) = -109.6$ , $\exp(-109.6) \approx 10^{-48}$ . $n_i \approx 10^{-29}$ cm⁻³. That’s less than one electron in the entire universe.
(5) Ratio ~ $10^{40}$ . Feel the weight of material choice: a small change in gap produces a 40-order-of-magnitude change in carrier density.

Takeaway: silicon is friendly — you can dope it. HfO₂ is not a conduction medium but a control medium. Module 2 (Chemistry) goes into the how.

Cheat Sheet

Birth of bands: when $10^{23}$ atoms crowd in a crystal, single-atom levels fan into bands.
Three regions:
- Valence band — filled, bound electrons.
- Conduction band — empty, free electrons.
- Band gap ( $E_g$ ) — the forbidden zone.
Family types: Metal ( $E_g = 0$ ) · Semiconductor ( $E_g \sim 1$ eV) · Insulator ( $E_g > 4$ eV).
Thermal excitation: $n_i \approx N_0 \exp(-E_g / 2 k_B T)$ . $k_B T|_{300K} = 0.026$ eV.
Doping: n-type (P, As — donates electron), p-type (B, In — donates hole). The essence of transistor design.
Fermi level: the 50%-occupied energy. Intrinsic → mid-gap; n-type → near conduction band; p-type → near valence band.
SIDRA context: HfO₂ is insulating in bulk; memristive behavior emerges from oxygen-vacancy filament control.

Vision: Beyond Bands

The classical band model has been the language of circuits for 100 years, but modern material science is hitting its edges. Advanced paradigms:

Topological insulators: conductive surface, insulating interior — Bi₂Se₃, HgTe. Edge states are Majorana qubit candidates.
Dirac & Weyl semimetals: graphene, Cd₃As₂. Electrons behave as massless fermions — ultra-high mobility.
2D heterostructure band engineering: WSe₂/MoS₂ bilayers — engineered bands; Moiré superlattices yielding entirely new material classes (2018+).
Ferroelectric FET (FeFET): HZO gate oxide — polarization carries data; non-volatile CMOS.
Superlattice AI accelerators: analog weight cells in Moiré flat-bands — tunable G across a wide range.
Excitonic devices: information carried by electron-hole pairs; stable at room T in 2D materials.
Hot-carrier FETs: hot-electron injection gives sub-thermionic (< 60 mV/dec) subthreshold slope.
Quantum well & dot lasers: III-V heterojunctions — on-chip light source for photonic interconnect.

Biggest lever for post-Y10 SIDRA: pairing a 2D material channel with an HZO ferroelectric gate — one transistor, both logic and memory. Weights and inputs can multiply in the same device, removing the separate memristor layer. 2027–2030 horizon.

Bands and Semiconductors

Prerequisites

What you'll learn here

Hook: Copper Conducts, Rubber Doesn't, Silicon Sits in the Middle

Intuition: How Do Levels Become Bands?

Formalism: Boltzmann, Fermi, and Doping

Experiment: Material, Temperature, Voltage

Quiz

Lab Task: Si vs HfO₂ Intrinsic Density

Cheat Sheet

Vision: Beyond Bands

Further Reading

Prerequisites

What you'll learn here

🪝 Hook: Copper Conducts, Rubber Doesn't, Silicon Sits in the Middle

🧭 Intuition: How Do Levels Become Bands?

📐 Formalism: Boltzmann, Fermi, and Doping

🧪 Experiment: Material, Temperature, Voltage

📝 Quiz

🛠️ Lab Task: Si vs HfO₂ Intrinsic Density

🗂️ Cheat Sheet

🔮 Vision: Beyond Bands

📚 Further Reading

Hook: Copper Conducts, Rubber Doesn't, Silicon Sits in the Middle

Intuition: How Do Levels Become Bands?

Formalism: Boltzmann, Fermi, and Doping

Experiment: Material, Temperature, Voltage

Quiz

Lab Task: Si vs HfO₂ Intrinsic Density

Cheat Sheet

Vision: Beyond Bands

Further Reading