Band Structures of Metals, Semiconductors and Insulators at 0 K: 21-Mat-B10 Properties and Processing of Micro- and Nanomaterials, December 2018, Q4

21-Mat-B10 Properties and Processing of Micro- and Nanomaterials · December 2018

Question 4: Band Structures of Metals, Semiconductors and Insulators at 0 K (15 marks)

Question text not reproduced: the examination questions are © Engineers and Geoscientists BC. Open the official past paper (linked at the top of this page) to read the question, then follow the worked solution below.

Approach. At 0 K, all three classes have completely filled lower bands; the classification hinges entirely on whether the highest filled band overlaps the next empty band (metal) or is separated from it by a gap, and if so, how wide that gap is relative to what an applied field or modest thermal energy can bridge.

(a) Metal

Valence and conduction bands overlap (or the band is only partially full): occupied states (blue) sit right up to the Fermi level E_F with no forbidden gap above it.

In a metal the highest band is either partially filled or overlaps the next band, so electrons immediately above E_F can move into empty states under an arbitrarily small applied field — there is no gap to overcome. Approximate gap: E_g ≈ 0 eV.

(b) Semiconductor

A completely filled valence band is separated from a completely empty conduction band by a narrow gap E_g, small enough for thermal excitation or doping to bridge at ordinary temperatures.

At 0 K a semiconductor conducts no better than an insulator (the valence band is full, the conduction band is empty), but its gap is narrow enough that modest thermal energy or dopant levels populate the conduction band at working temperatures. Approximate gaps: Si ≈ 1.1 eV, Ge ≈ 0.7 eV, GaAs ≈ 1.4 eV.

(c) Dielectric / insulator

The same picture as a semiconductor, but the gap is far wider — too wide for thermal energy or ordinary fields to promote a meaningful number of electrons into the conduction band.

An insulator has the identical filled-valence/empty-conduction-band picture as a semiconductor, differing only in the size of E_g: it is wide enough that essentially no electrons are thermally promoted across it at any accessible temperature, so the material stays non-conducting. Approximate gaps: diamond ≈ 5.5 eV, SiO₂ ≈ 9 eV, Al₂O₃ ≈ 8–9 eV.

Final results — Question 4
Material class	Band picture at 0 K	Approximate E_g
Metal	Overlapping / partially filled bands	≈ 0 eV
Semiconductor (Si)	Filled VB, empty CB, narrow gap	≈ 1.1 eV
Insulator (SiO₂)	Filled VB, empty CB, wide gap	≈ 9 eV

Topic: Band-theory classification of conductors, semiconductors and insulators.

Key relations: Occupation set by the Fermi–Dirac distribution at 0 K (a step function at E_F); conduction requires an empty state adjacent in energy to a filled one.

Why this works: All three material classes obey the identical quantum-mechanical band structure; the ONLY distinguishing feature at 0 K is whether the highest occupied band is full and, if so, how wide the gap to the next band is. This single parameter (E_g) explains why silicon can be doped into a useful conductor while quartz cannot — the gap is simply too wide for practical carrier populations.

Common pitfall: Drawing a semiconductor's conduction band as partially filled at 0 K (it is completely empty — conduction only appears once T > 0 or the material is doped), or treating "insulator" and "semiconductor" as different physics rather than the same picture with a different gap width.

Source: Callister, Materials Science and Engineering, Ch. 18 (electron band structures); Kasap, Principles of Electronic Materials and Devices, Ch. 4.

Question 4 of 6: Band Structures of Metals, Semiconductors and Insulators at 0 K

Question 4: Band Structures of Metals, Semiconductors and Insulators at 0 K (15 marks)

(a) Metal

(b) Semiconductor

(c) Dielectric / insulator