Introduction to Quantum Mechanics 2

Appendix 1Quantum Well of Semiconductor Materials

A1.1. 2D, 1D and 0D confinement

At temperature T = 0 K, the energy bands of a crystal are filled with valence electrons. The last of these bands is known as a valence band (VB) of energy E_v separated from a conduction band (CB) of energy E_c by a forbidden band (FB) corresponding to the material gap of positive energy E_g = E_c − E_v (Figure A1.1). In these energy bands, Schrödinger’s equation has no solution.

Figure A1.1. Relative arrangement of the energy bands in a material

Three categories of materials can be distinguished, according to the gap value: metals, insulators and semiconductors. As an example, let us consider several values of the gap at 300 K [SAK 15]. Metal: tin (Sn): 0.0 eV; insulator: diamond (C): 6.0 eV; semiconductor: silicon (Si): 1.12 eV; germanium: 0.67 eV; gallium arsenide (GaAs): 1.40 eV.

• – Metals have low resistivity at ambient temperature (of about 10^–5 Ωcm). Conduction is due to the free electrons in the CB (density: 10²² to 10²⁸ cm^–3). An increase in temperature drives only a small increase in resistivity, because the motion of free electrons is hindered by the vibrations of metal atoms.
• – The resistivity of insulators is above 10⁸ Ω cm. This is the case for glass, mica, silica (SiO₂), carbon, etc. In insulators, the release of electrons is triggered by an increase in temperature. This drives a ...