⚛️ Module 1 · Physics Foundation · Chapter 1.1 · 15 min read

The Atom and the Electron

Before understanding a memristor, let's look inside one of its cells.

Prerequisites

0.1 — What is SIDRACHIP?

What you'll learn here

Name the parts of an atom (nucleus, electron) and give their relative sizes
Use the Bohr model to compute orbit radii and energy levels for hydrogen
Explain photon absorption / emission as jumps between energy levels
Estimate, to an order of magnitude, the number of atoms in a SIDRA memristor cell
Distinguish the classical orbit picture from the quantum probability cloud

Hook: Seeing an Atom Is Harder Than Thinking One

Around 440 BC. Democritus reasoned that if you keep cutting a piece of rope, you’d eventually hit a piece that cannot be cut further. He called it atomos (indivisible). Nobody actually saw one.

21 centuries later, in 1897, J. J. Thomson played with cathode rays and found a negatively charged, detachable piece of the atom — the electron. Fourteen years later, Rutherford shot alpha particles at gold foil and realized the atom had a small, dense nucleus at its center. In 1913, Niels Bohr said the electron circled that nucleus on discrete energy levels.

Another 100 years passed. Today a SIDRACHIP memristor cell has a volume of only ~5 × 10⁴ nm³ — containing a few million atoms. To store each “1” or “0”, we reshuffle a specific fraction of them.

To understand the memristor, we first understand the atom.

Intuition: Four Different Pictures of the Atom

Each historical model tries to patch a hole the previous one left:

Dalton (1803) — billiard ball. Atom is an indivisible sphere. Chemical reactions rearrange the balls. Problem: no charge carriers, no explanation for light emission.
Thomson (1897) — plum pudding. Negative electrons embedded in a positive dough. Problem: Rutherford’s experiment disproved it.
Rutherford (1911) — solar system. Small, dense, positive nucleus at the center; electrons orbit in empty space. Problem: classical physics says an orbiting charge radiates, so the electron should spiral into the nucleus.
Bohr (1913) — quantized orbits. Electrons occupy only specific energy levels; transitions absorb or emit exactly one photon. Problem: fails for multi-electron atoms.

Today’s view: quantum mechanics — the electron’s location is uncertain, described as a probability cloud (Schrödinger, 1926). But the Bohr model is still the best bridge to start learning. That’s where we focus in this chapter.

Relative sizes: If the atom were a football stadium, the nucleus would be a chickpea at the center. The rest of the stadium is empty. All of it. The chair you’re sitting on holds you up through electromagnetic interactions of tiny charged particles — even though it feels like contact, no atoms actually touch.

Formalism: The Bohr Atom

L1 · Intro

In a hydrogen atom, there’s 1 proton at the center and 1 electron around it. Per Bohr:

The first allowed orbit has radius 0.529 Å (1 Å = 10⁻¹⁰ m).
As the orbit number $n$ grows, radius grows as $n^2$ . The $n = 2$ orbit is 4× farther, $n = 3$ is 9× farther.
Energy levels are fixed and negative: $E_1 = -13.6$ eV, $E_2 = -3.4$ eV, $E_3 = -1.5$ eV. The negative sign means “bound to the atom”. 0 = electron detached = ionization.

When an electron jumps between levels it absorbs or emits an exact amount of energy (a photon). No intermediate energies. That’s what “quantum” means.

L2 · Full

Bohr’s two postulates:

Angular momentum is quantized: the electron orbits with angular momentum equal to integer multiples of $\hbar$ (the reduced Planck constant):

L_n = n \hbar, \quad n = 1, 2, 3, \dots

Photon on transition: if $E_f < E_i$ , a photon is emitted with frequency:

h\nu = E_i - E_f

Balancing the central Coulomb force $\frac{1}{4\pi\varepsilon_0} \frac{e^2}{r^2}$ with the centripetal term $\frac{m_e v^2}{r}$ , plus the quantization above, gives:

r_n = \frac{4\pi\varepsilon_0 \hbar^2}{m_e e^2} \cdot n^2 = a_0 \cdot n^2

E_n = -\frac{m_e e^4}{8 \varepsilon_0^2 h^2} \cdot \frac{1}{n^2} = -\frac{13.6 \text{ eV}}{n^2}

Here $a_0 = 0.529 \text{ Å}$ is the Bohr radius and $13.6$ eV is the Rydberg energy.

Lyman, Balmer, Paschen series: $n = 2 \to 1$ is ultraviolet (Lyman-α, 121 nm), $n = 3 \to 2$ is red visible light (Balmer-α, 656 nm — hydrogen’s fingerprint in the solar spectrum). Try it in the animation below.

L3 · Deep

Bohr’s model is uniquely clean for hydrogen; beyond that you need wave functions. The time-independent Schrödinger equation:

-\frac{\hbar^2}{2m_e} \nabla^2 \psi + V(\mathbf{r}) \psi = E \psi

In a spherically symmetric Coulomb potential, it separates: $\psi(r, \theta, \phi) = R_{n\ell}(r) \, Y_{\ell}^{m}(\theta, \phi)$ . Three quantum numbers:

Principal $n = 1, 2, \dots$ (energy)
Orbital angular momentum $\ell = 0, 1, \dots, n-1$ (s, p, d, f orbitals)
Magnetic $m = -\ell, \dots, +\ell$ (orbital orientation)
plus spin $s = \pm \tfrac{1}{2}$ .

$|\psi|^2$ is the probability density — the electron cloud’s “where are you likely to find it” distribution. For example, 1s is spherical, 2p has two lobes, 3d has four. Conduction properties of materials like HfO₂ arise from overlap of these orbitals with neighboring atoms — the subject of the next chapter (1.2 — Bands and Semiconductors).

Experiment: Walk an Electron Through the Levels

Below is a hydrogen atom. Click the n buttons to jump the electron to different orbits. Each jump shows a photon:

Outward (higher n) → absorption (photon enters the atom).
Inward (lower n) → emission (photon leaves the atom).
The photon’s color maps to its energy — UV blue, IR red. ΔE is shown on top.

The ladder on the right shows each level’s exact energy. As $n \to \infty$ energy approaches 0 — that’s the ionization threshold, where the electron leaves the atom.

Experiment idea: jump $n=1$ → $n=3$ → $n=2$ → $n=1$ . The sum of emitted-photon energies should equal a single direct jump’s energy. Conservation of energy.

Quiz

1/5In the Bohr model of hydrogen, the n = 2 orbit radius is how many times the n = 1 radius?

Lab Task: Atom Count in a Memristor Cell

Work out quiz question 5 in detail yourself. Use:

Cell: 100 nm × 100 nm × 5 nm (SIDRA 1S1R HfO₂ layer)
HfO₂ density: 9.7 g/cm³
HfO₂ molar mass: $M = 210$ g/mol (Hf: 178.5 + 2×O: 32)
Avogadro: $N_A = 6.022 \times 10^{23}$ mol⁻¹

Steps:

Convert volume V to cm³ (1 nm³ = 10⁻²¹ cm³).
Mass = V · ρ.
Moles = mass / M.
Formula units = moles · N_A.
Atoms = formula units × 3 (3 atoms per HfO₂).

Then consider:

Thermal noise: the energy needed to jostle an atom is $k_B T$ ≈ 0.026 eV at room temperature. This drives conductance fluctuations in the memristor.
If the cell shrinks to ~50 nm, atom count drops 8× — the same absolute fluctuation becomes 8× more visible electrically. This is the scaling wall.
Elementary charge $e = 1.6 \times 10^{-19}$ C. How many electrons does a single memristor state store? Make your own estimate.

Getting this number into your head sets up later chapters (especially Module 2 — Chemistry and Module 5 — Hardware) when we discuss memristor noise, endurance, and variability.

Cheat Sheet

Atomic structure: tiny dense positive nucleus (~10⁻¹⁵ m) + negative electrons (~10⁻¹⁰ m diameter cloud). 99.99999…% of an atom is empty.
Bohr (hydrogen):
- Orbit radius: $r_n = n^2 \cdot a_0$ , $a_0 = 0.529$ Å.
- Energy: $E_n = -13.6/n^2$ eV.
- Transition: $h\nu = E_i - E_f$ .
Historical chain: Dalton → Thomson → Rutherford → Bohr → Schrödinger. Each fixes the previous model’s gap.
Quantum: the electron has no definite location; an orbital is a probability cloud ( $|\psi|^2$ ).
SIDRA context: A memristor cell ≈ 10⁶ atoms. Atomic thermal jitter = memristor noise. As cells shrink, noise dominates — a design limit.

Memorize: $a_0 = 0.529$ Å · 13.6 eV · hydrogen visible (Balmer) lines: 656 / 486 / 434 / 410 nm.

Vision: Beyond the Atomic Scale

The Bohr atom was born in 1913; today we manipulate single atoms. Tomorrow’s gains:

Single-atom transistors: Simmons group (UNSW, 2012) placed a phosphorus atom on silicon for a single-electron transistor. Post-SIDRA density floor: one bit per dopant atom.
Neutral-atom qubits: QuEra, Atom Computing — Rb/Cs in optical traps as compute units. Quantum AI accelerators.
Electron spin (spintronics): spin polarization carries information; MRAM and SOT-MRAM parallel SIDRA memristors.
Atomic 2D materials: MoS₂, hBN single-atom-thick transistors (IBM 2024). The next density jump.
NV-centers (nitrogen-vacancy): a defect point in diamond — room-temperature quantum memory.
Rydberg atom arrays: many-qubit quantum simulation, atoms placed one-by-one with optical tweezers.
On-chip atomic clocks: Sr/Yb optical clocks → 10⁻¹⁸ timing stability, critical for synchronized AI clusters.
Molecular electronics: single-molecule diodes / memristors via DNA-origami self-assembly.

Biggest lever for post-Y10 SIDRA: a donor-atom (P) memristor — toggling a single phosphorus atom between two states as a weight bit. In theory, sub-attojoule per cell, near-infinite endurance. 2030+ horizon.

Prerequisites

What you'll learn here

🪝 Hook: Seeing an Atom Is Harder Than Thinking One

🧭 Intuition: Four Different Pictures of the Atom

📐 Formalism: The Bohr Atom

🧪 Experiment: Walk an Electron Through the Levels

📝 Quiz

🛠️ Lab Task: Atom Count in a Memristor Cell

🗂️ Cheat Sheet

🔮 Vision: Beyond the Atomic Scale

📚 Further Reading