Question

Within a material, electrons exist within energy bands, often referred to as the conduction band and the valence band (and I imagine there can be other named bands as well). I understand why this occurs -- that electrons exist in orbitals around an atom and that when you bring many atoms together in a material, these orbitals aren't allowed to overlap due to the Pauli Exclusion principle and therefore form large bands of nearly continuous energy states. My question however is that these states within a band are not actually continuous but instead differ by very small energy levels, so what determines the value of these very small energy gaps within the band? For instance, the Wikipedia entry on the subject says:

Since the number of atoms in a macroscopic piece of solid is a very large number (N~1022
) the number of orbitals is very large and thus they are very closely spaced in energy (of the order of 10−22
eV)

however this does not seem very well motivated to me at all. Am I to understand the energy spacing is simply proportional to the inverse of the number of atoms in a material? If I continue to add more atoms to the material, would I expect that the energy levels just get more and more closely packed, or is there something else determining this energy spacing?

Answer 1

Energy levels within bands in a material are determined by quantum mechanics and the crystal lattice structure, which leads to very closely spaced but quantized energy levels. Electrical conductivity is related to the ease of electron transition across the energy band gap, where conductors, semiconductors, and insulators differ in their gap sizes.

Step-by-step explanation:

Understanding Energy Bands and Energy Gaps within Materials

The small energy gaps within energy bands in materials are determined by quantum mechanics. As numerous atomic orbitals combine to form molecular orbitals in a solid, the energy levels become very closely spaced. These spacings are not actually continuous but are so fine that they appear nearly continuous. The spacing between the energy levels within a band is essentially related to the inverse of the number of atoms in the material. More precisely, the quantum states of the electrons are described by wave functions that must satisfy the boundary conditions of the material's crystal lattice structure. These conditions lead to quantization of energy levels, but because of the huge number of atomic orbitals overlapping, the available states are incredibly closely packed.

When discussing electrical conductivity, we refer to the valence band and the conduction band. Electrons must transition from the valence band to the conduction band to contribute to electrical conductivity. The ease of this transition is determined by the size of the energy gap (band gap) between these bands. A small gap makes it easier for electrons to be excited into the conduction band, characteristic of conductors. Large band gaps are found in insulators, hindering electron excitation and compromising conductivity. Semiconductors exhibit intermediate band gaps, which allows for controlled conductivity useful in electronic devices.

Overall, the energy gap size is not simply a function of the number of atoms but results from the collective electronic wave functions and the crystal structure of the solid. As the number of atoms in a material increases, the available energy states become more densely packed, but they remain quantized, no matter how large the material grows.