Solid-state chemistry — band theory, semiconductors, superconductivity, magnetism in solids

1. WHY bands form (Derivation from first principles)

WHAT: A band is a near-continuous range of allowed electron energies in a solid.

WHY: Start with 2 H atoms. Two $1s$ orbitals combine to give a bonding ($\sigma$) and antibonding ($\sigma^*$) MO — two levels. Add a 3rd atom → 3 levels. Add $N$ atoms → $N$ levels.

The spread between lowest and highest level is fixed by neighbour overlap (a constant-ish bandwidth $W$), but the number of levels $= N \sim 10^{23}$. So the spacing between adjacent levels:

$\Delta E \approx \frac{W}{N} \approx \frac{1\ \text{eV}}{10^{23}} \approx 10^{-23}\ \text{eV}$

2. Metals, insulators, semiconductors

Temperature dependence — the diagnostic test:

The number of carriers thermally excited across a gap follows Boltzmann statistics: $n \propto \exp\!\left(-\frac{E_g}{2k_BT}\right)$

(The factor of 2: exciting one electron creates two carriers — an electron in CB and a hole in VB — and the chemical potential sits mid-gap, so the activation energy is shared, $E_g/2$.)

So conductivity $\sigma \propto e^{-E_g/2k_BT}$.

3. Doping — extrinsic semiconductors

• n-type: dope Si (group 14) with group 15 (P, As). Extra electron sits in a shallow donor level just below CB → easily ionised, gives electrons as majority carriers.
• p-type: dope Si with group 13 (B, Al). Missing electron creates an acceptor level just above VB → grabs a VB electron, leaving a mobile hole as majority carrier.

A p–n junction (p-type joined to n-type) lets current flow one way → the basis of diodes, LEDs, solar cells, transistors.

4. Superconductivity

WHY zero resistance (BCS idea, conceptual): Below $T_c$, an electron distorts the lattice (attracts nearby positive ions); a second electron is attracted to that distortion. The two form a Cooper pair via lattice (phonon) coupling. Cooper pairs are bosons — they all condense into a single coherent quantum state that flows without scattering off impurities or vibrations → no resistance.

• Type I: sudden loss of superconductivity above a critical field $H_c$ (most pure metals, low $T_c$).
• Type II: field penetrates as quantized vortices between $H_{c1}$ and $H_{c2}$ (alloys, cuprates).
• High-$T_c$: cuprate ceramics like $\text{YBa}_2\text{Cu}_3\text{O}_7$ ($T_c \approx 92$ K, above liquid N$_2$, 77 K).

5. Magnetism in solids

Magnetism comes from unpaired electron spins (and orbital motion). Each unpaired electron is a tiny magnet.

Spin-only magnetic moment (derived): A single electron spin contributes moment $\mu = g\sqrt{s(s+1)}\,\mu_B$ with $g\approx 2$, $s=1/2$. For $n$ unpaired electrons total spin $S = n/2$, so: $\mu = 2\sqrt{\tfrac{n}{2}\left(\tfrac{n}{2}+1\right)}\,\mu_B = \sqrt{n(n+2)}\ \mu_B$

Common mistakes

Active recall

Connections

• Molecular Orbital Theory — bands are MOs scaled to $N$ atoms.
• Crystal Field Theory — counts unpaired d-electrons for $\mu$.
• Boltzmann Distribution — gives $n\propto e^{-E_g/2k_BT}$.
• Diodes and Transistors — applications of p–n junctions.
• Meissner Effect / BCS Theory — superconductivity mechanism.
• Crystal Lattices and Unit Cells — the structural backdrop.

Concept Map

Hinglish (regional understanding)