Inorganic chemistry

Semiconductors and superconductors

Both semiconductors and superconductors are electronic solids, but in opposite regimes: a small band gap limits carrier excitation in a semiconductor, whereas electron pairing opens a superconducting gap that enables resistance-free transport.

1 Lower secondary 2 Upper secondary 3 Undergraduate 4 Advanced 5 Research

IntuitionIntuition: gaps control charge motion

In a filled band, electrons cannot gain energy without crossing to an empty band. An insulator’s gap is large, a semiconductor’s small, and a metal’s bands overlap. Superconductivity is different: a tiny gap below the Fermi level protects paired electrons from ordinary scattering.

Choose an insulator, intrinsic or doped semiconductor, metal or superconductor and compare band filling and gaps.

SchoolSchool level: semiconductors as controlled conductors

Definition: Intrinsic semiconductor

An intrinsic semiconductor has a modest band gap (for Si, about 1.1 eV at room temperature). Thermal energy excites only a small fraction of electrons from valence to conduction band, leaving equal numbers of electrons and holes.

n_i\propto\exp\!\left(-\frac{E_g}{2k_BT}\right)

Because intrinsic carrier concentration depends exponentially on gap and temperature, a semiconductor can switch from almost insulating to conducting. That sensitivity enables diodes, transistors, sensors and solar cells.

Three electrical regimes
Material	Electronic feature	Effect
Insulator	Large gap between valence and conduction bands	Very few carriers
Semiconductor	Small gap; carriers thermally or optically excited	Tunable conductivity
Metal	Partly filled band or overlapping bands	High conductivity

Example: Estimate intrinsic carriers in silicon

Use the exponential form ni ∝ exp(−Eg/2kBT). At T = 300 K, kBT ≈ 0.0259 eV; compare Eg = 1.1 eV with Eg = 5.5 eV.

Solution

The exponents are roughly −21 and −106. The larger gap therefore makes thermal carrier production vanishingly small even though it is only five times the smaller gap.

UndergraduateUndergraduate: doping, junctions and superconductivity

Substitutional doping adds carriers deliberately: P provides electrons (n-type), B creates acceptor states and holes (p-type). The Fermi level shifts toward the band from which carriers originate, and contacts between differently doped regions create built-in electric fields.

In a normal metal, scattering by phonons and defects causes resistance. A superconductor instead develops a correlated state of Cooper pairs below a critical temperature Tc. In conventional BCS theory, weak electron–phonon coupling lets pairs condense with a gap of order kBTc.

2\Delta(0)\approx3.53\,k_BT_c\qquad(\text{BCS weak coupling})

Example: Estimate a BCS gap

For MgB₂, take Tc ≈ 39 K and kB = 8.617×10⁻⁵ eV K⁻¹. Estimate the weak-coupling 2Δ.

Solution

2Δ ≈ 3.53 × 8.617×10⁻⁵ × 39 ≈ 1.19×10⁻² eV ≈ 11.9 meV. Actual MgB₂ is multiband, so its gap structure is more complex than a single weak-coupling value.

The Meissner effect, not merely zero resistance, is defining: a superconductor expels magnetic flux when cooled through Tc (within its penetration length). Type-II materials admit flux as quantized vortices between lower and upper critical fields.

AdvancedPhysics and materials perspectives

Semiconductor behaviour is described by band dispersion, carrier statistics, mobility, recombination and contact effects; the idealized band picture hides interfaces, traps and defects. Superconductivity raises additional questions about pairing symmetry, flux pinning, coherence and competing electronic orders.

ResearchResearch: high-temperature superconductors and quantum semiconductors

References

Solid State Physics · N. W. Ashcroft, N. D. Mermin, 1976
Possible high-Tc superconductivity in the Ba–La–Cu–O system · J. G. Bednorz, K. A. Müller, 1986