Electronic Properties & Superconductivity

How bands form. An isolated atom has sharp energy levels. When a huge number (N) of atoms pack together into a crystal, each atomic orbital splits into N very closely spaced levels. These merge into a continuous energy band. Inner orbitals overlap little and give narrow bands; the outer valence orbitals overlap strongly and give broad bands.

Key terms:

• Valence band: the highest band containing electrons (full at 0 K).
• Conduction band: the next higher band — empty at 0 K; electrons here move freely and conduct.
• Forbidden gap (E_g): the gap between the bands where no electron can stay.
• Fermi level (E_F): the energy of the highest filled state at 0 K.

Why metals conduct (band picture). In a metal the valence and conduction bands overlap (or the top band is only half-filled), so the Fermi level lies inside a band. Electrons just below E_F can move into empty states just above it with almost no energy — so even a tiny electric field makes them flow. This is why metals conduct even at 0 K.

Contrast (one line each): in a semiconductor a small gap (E_g ≈ 0.5–3 eV) separates full and empty bands — heating pushes electrons across, so conductivity rises with T. In an insulator the gap is huge (E_g > 3–4 eV) — electrons cannot cross it, so there is practically no conduction.

Semiconductors are solids whose electrical conductivity lies between conductors and insulators. In band terms they have a small forbidden gap (E_g ≈ 0.5–3 eV; Si: 1.1 eV, Ge: 0.66 eV) between a full valence band and an empty conduction band. At 0 K they behave like insulators, but heating gives some electrons enough energy to jump the gap, leaving holes behind — so their conductivity rises sharply with temperature (opposite to metals).

n-type semiconductor: pure Si or Ge doped with a tiny amount of pentavalent impurity (P, As, Sb). Four of the impurity’s five valence electrons form bonds with the host; the fifth electron stays loosely bound and is easily promoted into the conduction band. This creates a donor level just below the conduction band. Because current is carried mainly by negative charges (electrons), it is called n-type.

Example: phosphorus-doped silicon (Si:P).

Metals — conductivity decreases on heating. In a metal the conduction band already contains a huge number of free electrons, and heating hardly changes this number. But heating makes the lattice ions vibrate more strongly, and these vibrations scatter the moving electrons — the electrons collide more often and their drift slows down. Resistivity ρ rises (roughly linearly with T), so conductivity σ = 1/ρ falls.

Semiconductors — conductivity increases on heating. At low temperature almost all electrons sit in the full valence band and very few reach the conduction band. Heating gives exponentially more electrons enough thermal energy to jump the small band gap, creating electron–hole pairs. The number of charge carriers grows exponentially with T:

This flood of new carriers completely outweighs the extra scattering, so conductivity rises sharply with temperature — the exact opposite of a metal.

Density of states D(E): the number of allowed quantum states per unit energy interval at energy E — D(E)·dE is the number of states between E and E + dE. It decides how many electron states are available to be filled, so it controls heat capacity, conductivity and optical properties of solids.

Derivation (1D — this gives the E^−1/2 law). Confine a free electron to a wire of length L. Allowed wavevectors are quantised (k = nπ/L), so the number of k-states (counting both spins) up to wavevector k is

For a free electron E = ℏ²k²/2m, so k = (2mE/ℏ²)^1/2 and dk/dE = m/(ℏ²k). Differentiating,

Hence D(E) ∝ E^−1/2 in one dimension — it diverges (spikes) at low energy. These spikes are the van Hove singularities.

Expressions asked in the question:

• 1D quantum wire: D(E) = (L/π)(2m/ℏ²)^1/2E^−1/2 ∝ 1/√E
• 2D thin film (area A, per subband): D(E) = mA/(πℏ²) = constant — each subband adds a flat step
• 0D solid / quantum dot: D(E) = 2Σ_nδ(E − E_n) — sharp delta-function lines, fully discrete atom-like levels

(For reference — 3D bulk: D(E) = (V/2π²)(2m/ℏ²)^3/2E^1/2 ∝ √E.)

The difference is in the band structure, not the geometry:

• Graphite — conductor. Every carbon is identical and sp² bonded; the fourth valence electron of each C sits in a delocalised π-band spread over the whole layer. The valence and conduction π-bands touch at the Fermi level (no effective gap), so electrons move freely within the layers — graphite conducts (strongly anisotropically: along the layers, not between them).
• Boron nitride — insulator. B and N have very different electronegativities, so the B–N bonds are strongly polar: the π-electrons are localised on the nitrogen atoms instead of being shared. This opens a huge band gap of about 6 eV — far too large for thermal excitation — so there are no free carriers and BN is an insulator (used as a ceramic precisely because it conducts heat but not electricity).

One line: identical atoms give graphite a delocalised π-band with no gap (conductor); polar B–N bonds localise the electrons and open a ~6 eV gap in BN (insulator).

Einstein model: each atom of the solid is an independent 3D quantum harmonic oscillator, and all oscillators vibrate with the same single frequency ν_E.

Step 1 — partition function of one 1D Planck oscillator. Energy levels E_n = nhν (n = 0, 1, 2, …; the constant zero-point energy hν/2 drops out of the heat capacity):

Each atom vibrates in three independent directions, so for one atom z_3D = z³.

Step 2 — mean energy. From ε̄ = kT²(∂ln z/∂T):

Step 3 — total energy of one mole. One mole has 3N_A oscillators:

θ_E is the Einstein temperature.

Step 4 — heat capacity. C_V = (∂U/∂T)_V:

Step 5 — the two limits (2 marks):

• High T (T ≫ θ_E): θ_E/T is small, e^θ_E/T ≈ 1 + θ_E/T, so C_V → 3R — the Dulong–Petit law. ✓ Success.
• Low T (T ≪ θ_E): C_V ≈ 3R(θ_E/T)²e^−θ_E/T → 0 exponentially. Going to zero is qualitatively right (third law), but experiment shows C_V ∝ T³ (Debye T³ law) — the exponential fall is too fast. ✗ Failure.

Debye T³ law: at very low temperatures (T ≪ θ_D, the Debye temperature), the molar heat capacity of a solid is proportional to the cube of the absolute temperature:

Why T³? At low T only the low-frequency acoustic vibrations (phonons) are thermally excited. Their density of states grows as ν², and only modes with hν ≲ kT are active — the number of excited modes ∝ T³, so the vibrational energy U ∝ T⁴ and C_V = dU/dT ∝ T³.

Utility:

• It correctly explains the low-temperature heat capacity where Einstein’s model fails (Einstein predicts an exponential fall, experiment gives T³).
• Measuring the low-T heat capacity lets us determine the Debye temperature θ_D of a solid — a characteristic elastic property.
• In metals, writing C_V = γT + AT³ lets us separate the electronic (γT) and lattice (AT³) contributions by plotting C_V/T against T².

Drawback of Einstein’s equation. Einstein assumes every atom vibrates independently with one single frequency ν_E. This predicts that at low temperature the heat capacity falls exponentially, C_V ∝ e^−θ_E/T — but experiments show it falls as T³ (Debye T³ law). The single-frequency assumption is too crude: in a real solid the atoms are coupled and vibrate with a whole range of frequencies. (The fitted θ_E also drifts with temperature instead of staying constant.)

Einstein vs Debye — the key difference in assumptions:

One line: Einstein = independent oscillators, one frequency; Debye = coupled lattice, continuous spectrum up to ν_D — and that spectrum is what gives the correct T³ law at low temperature.

Polythiazyl, (SN)_x, shows metallic electrical conductivity — it was the first known conducting inorganic polymer (a “synthetic metal”). Its chains of alternating sulphur and nitrogen atoms carry a delocalised π-electron system along the backbone, so electrons move freely down the chains (conductivity of the order of 10³ S cm⁻¹ along the chains). It even becomes superconducting below about 0.26 K.

One-line answer for 1 mark: polythiazyl is an inorganic polymer with metallic electrical conductivity along its (S–N)_x chains.