From Atomic Orbitals to Energy Bands and Semiconductor Behaviour

Atomic Energy Levels vs. Many-Atom Systems

• A single isolated atom has discrete, well-defined atomic energy levels (e.g., $1s, 2s, 2p$ …).
• When billions of atoms are brought together, their individual energy levels begin to interact and split.
  • Result: formation of closely-spaced levels that merge into energy bands.
• Key conceptual leap: move from "one electron in one orbital" thinking to collective, crystal-wide descriptions.

Fundamentals of Quantum Description

• Electrons are quantized (Planck/Boltzmann hypothesis) and obey the Pauli exclusion principle.
  • No two electrons in the same system can share an identical set of quantum numbers ${n,\;l,\;m,\;s}$.
• Wave function $\psi (\vec r, t)$
  • Complex function containing all measurable information about the electron.
  • Specified uniquely by the four quantum numbers.
  • Written generally as $\psi = \Re{\psi} + i\,\Im{\psi}$.
• Probability density
  • $|\psi|^2 = \psi \psi^*$ (modulus squared) gives the probability of finding an electron at a position $\vec r$ at time $t$.

Schrödinger Equation: The Core Theory

• Time-independent form used in the lecture: $\hat H\,\psi = E\,\psi$
  • $\hat H$ (Hamiltonian operator) represents the total energy (kinetic + potential) acting on the wave function.
  • RHS is a simple multiplication by the scalar energy $E$; LHS is an operation (cannot be cancelled algebraically with $\psi$).
• Goal in atomic/solid-state problems: choose or construct an appropriate $\psi$, apply $\hat H$, and solve for allowed energies $E$.

Linear Combination of Atomic Orbitals (LCAO)

1. Two-Atom Example: $\text{H} + \text{H} \to \text{H}_2$

• Start with two isolated $1s$ orbitals: $\psi<em>1,\;\psi</em>2$.
• Bring atoms together ⇒ orbitals overlap ⇒ form two new molecular orbitals:
  • Bonding: $\psi<em>{12} = \psi</em>1 + \psi_2$ (constructive interference).
  • No node between nuclei; high probability density between atoms ⇒ electron sharing ⇒ covalent bond.
  • Energy $E_{12}$ lower than atomic $1s$.
  • Antibonding: $\psi<em>{12}^* = \psi</em>1 - \psi_2$ (destructive interference).
  • Has a node (probability $|\psi|^2 = 0$) between nuclei.
  • Energy $E_{12}^*$ higher than atomic $1s$.
• Electron filling (H has one valence electron):
  • Two electrons (spin up/down) populate only the lower bonding MO ⇒ net energy decrease ⇒ stable $\text{H}_2$ molecule.

2. Why He$_2$ Does Not Form

• $\text{He}$ supplies four electrons.
  • Bonding MO accommodates 2; antibonding takes the next 2.
  • Energy gain from bonding is cancelled by energy cost of filling antibonding ⇒ no net stabilization.

From Molecules to Solids: Birth of Bands

n Hydrogen Atoms (Conceptual Metallic Hydrogen)

• Each added atom contributes one atomic orbital + one electron.
• $n$ atoms ⇒ $n$ bonding-type + $n$ antibonding-type orbitals ⇒ 2$n$ total states.
• Only $n$ electrons available ⇒ band half-filled.
• Resulting solid behaves like a metal:
  • Highest occupied states sit at the Fermi level $E_F$ (top of filled portion).
  • Electrons at $E_F$ can acquire kinetic energy and conduct when an electric field is applied.
• Real-world link: under extreme pressure (e.g., Jupiter’s core) hydrogen becomes a metallic conductor generating planetary magnetic fields.

Silicon: Valence Structure and sp$^3$ Hybridization

1. Isolated Si Atom

• Electron configuration beyond neon core: $3s^2\,3p^2$ (4 valence electrons).
• Upon bonding, $3s$ and $3p$ mix to form four equivalent sp$^3$ hybrids pointing toward the corners of a tetrahedron.

2. Building a Silicon Crystal (LCAO with $sp^3$)

• Each Si contributes four orbitals (sp$^3$) and four valence electrons.
• $n$ Si atoms ⇒ 8$n$ total sp$^3$-derived states (because each orbital splits into a bonding-like and antibonding-like set).
  • Lower set (Valence Band): $4n$ states – filled by the $4n$ electrons.
  • Upper set (Conduction Band): $4n$ states – completely empty at 0 K.
• The energy gap $E_g$ separating the two sets defines silicon as a semiconductor.
  • No available states for electrons to move unless thermal/photon energy $\ge E_g$ promotes them across the gap.

Key Quantities & Terminology

• $|\psi|^2$ – probability density; always real and non-negative.
• Node – spatial point where $\psi = 0$; characteristic of antibonding orbitals.
• Fermi Level $E_F$ – highest occupied energy at 0 K; governs metallic conduction.
• Bonding vs. Antibonding States – lower- vs. higher-energy combinations arising from orbital overlap.
• Valence Band (VB) – fully occupied band in a semiconductor.
• Conduction Band (CB) – empty band above $E_g$; carriers promoted here enable conduction.
• Band Gap $E_g$ – energy difference between CB minimum and VB maximum.

Practical, Technological & Cosmic Relevance

• Understanding band formation underpins transistor, diode, and photovoltaic design.
• Metallic hydrogen theory explains the strong magnetic field of gas giants (Jupiter, Saturn).
• Silicon’s specific band structure (moderate $E_g$, tetrahedral bonds) makes it the workhorse material for modern electronics.

Study Checklist

• [ ] Derive bonding/antibonding energies for $\text{H}_2$ via simple overlap integrals.
• [ ] Practice interpreting $|\psi|^2$ plots: identify nodes and bonding regions.
• [ ] Sketch band diagrams for: metal (half-filled), semiconductor (VB + CB + $E<em>g$), insulator (large $E</em>g$).
• [ ] Memorize typical silicon parameters: $E_g \approx 1.12\,\text{eV}$ at 300 K, lattice constant $a \approx 5.43\,\text{Å}$.
• [ ] Be able to explain why He does not form a stable molecular bond in terms of orbital filling.