Electronic band structure

In the field of solid-state physics, electronic band structure describes the available energy levels for electrons within a solid. It identifies the specific ranges of energy that electrons are allowed to occupy. It also identifies forbidden ranges, known as band gaps or forbidden bands, where no electron states exist.

Understanding these structures is vital for modern science. Band theory provides the foundation for all solid-state devices, including transistors and solar cells. It also explains physical properties like electrical resistivity and optical absorption.

Band formation occurs when many atoms are brought together to form a solid crystal. When atoms are isolated and far apart, their electrons occupy discrete atomic orbitals with specific energy levels. As atoms move closer to form a lattice, these atomic orbitals begin to overlap. This overlap causes the orbitals to hybridize, or split, into many new molecular orbitals.

Because a macroscopic solid contains a massive number of atoms, often around 10^22, the number of resulting orbitals is also massive. These orbitals are spaced so closely in energy that they form a continuous energy band. The Pauli exclusion principle dictates that only two electrons can occupy a single orbital. Consequently, electrons fill these bands starting from the lowest energy levels.

Scientists use two main models to describe this behavior. The first is the nearly free electron model. In this model, electrons move almost freely through the material. They are only slightly perturbed, or bumped, by the periodic lattice of atoms. This model helps explain the electronic dispersion relation. The second model views electrons as being tightly bound to individual atoms. In this view, electrons can tunnel between atoms when their orbitals overlap. This tunneling is a primary driver of the hybridization that creates bands. This process mostly involves valence electrons, which are the outermost electrons used in chemical bonding.

Band gaps arise because energy bands have finite widths. If two adjacent bands are not wide enough to touch, a gap remains between them. The width of a band depends on how much the atomic orbitals overlap. For example, core orbitals, such as those for 1s electrons, have very little overlap. This results in extremely narrow bands and large band gaps. In contrast, higher energy bands involve larger orbitals with more overlap. These higher bands become progressively wider, often resulting in no band gaps at higher energies. In a diamond crystal, for instance, two bands form with a 5.5 eV band gap.

To study these states, physicists use the concept of a wavevector, denoted as k. The single-electron Schrödinger equation is solved for an electron in a periodic potential. This provides solutions called Bloch electrons. For every value of k, there are multiple energy solutions labeled by a band index. These energy levels change smoothly as k changes, forming a band. The relationship between energy and the wavevector is called the dispersion relation.

Because of the crystal's symmetry, scientists focus on the Brillouin zone. This is a specific polyhedron in wavevector space that represents all unique physical states.

Band gaps are further classified by the wavevectors of the states surrounding them. A material has a direct band gap if the lowest-energy state above the gap has the same wavevector as the highest-energy state below it. In an indirect band gap material, these closest states do not share the same wavevector.

This distinction is important for how materials interact with light. Scientists also use the density of states function to count electronic states per unit volume and energy. This function is essential for calculating electrical conductivity and the rate of optical absorption.

Band theory relies on several key assumptions to remain accurate. First, it assumes an infinite-size system where the material is large enough to have continuous bands. Second, it assumes a homogeneous system where the chemical makeup is uniform. Third, it assumes non-interactivity, meaning electrons travel in a static potential without hitting lattice vibrations or other electrons. These assumptions can break down near surfaces or interfaces. In very small systems, like a single molecule or a quantum dot, continuous band structures do not exist. In these cases, researchers must move into the realm of mesoscopic physics.