KAHIBARO
Table of Contents
From Single Atoms to Solids
A single isolated atom has electrons that can occupy only certain allowed energies. These energies are discrete, which means they come as separated levels rather than a continuous range. When many atoms come together to form a solid, the situation changes. The atoms are so close that their electrons influence one another. As a result, each atomic energy level splits into a very large number of closely spaced levels.
In a solid containing an enormous number of atoms, these split levels lie so close together that they form what we call energy bands. Instead of thinking about electrons as having only a few isolated energies, we now think of them as being allowed to occupy ranges of energies.
This band idea is one of the central concepts of solid-state physics because it explains why some materials conduct electricity easily, why some do not, and why semiconductors behave in their special way.
In a solid, allowed electron energies are grouped into bands, and some energy ranges are forbidden. These forbidden ranges are called band gaps.
Allowed and Forbidden Energies
An energy band is a continuous range of allowed electron energies in a solid. Between these bands there may be regions where no electron state is allowed. These regions are called forbidden bands, or band gaps.
The existence of forbidden gaps is a quantum effect. Electrons in a crystal do not behave like tiny classical particles moving freely at any energy. Their wave nature and the regular arrangement of atoms in the crystal restrict which energies are possible.
A simple picture is this. If one atom has one electron level, then two nearby atoms produce two nearby levels, three atoms produce three levels, and so on. For a huge number of atoms, the levels become an almost continuous band.
Valence Band and Conduction Band
The most important bands for electrical behavior are the valence band and the conduction band.
The valence band is the highest energy band that is normally filled with electrons at very low temperature. These electrons are often associated with bonding between atoms.
The conduction band is the next higher band. Electrons in this band can move through the solid much more easily and contribute strongly to electrical conduction.
If the valence band is full and the conduction band is empty, the material may not conduct well. For current to flow, electrons usually need access to available nearby states. A completely full band does not allow easy net motion because all nearby states are already occupied.
A full band does not conduct well. Electrical conduction requires electrons to have available empty states at nearby energies.
The Band Gap
The energy difference between the top of the valence band and the bottom of the conduction band is called the band gap, often written as $E_g$.
$$
E_g = E_{\text{conduction bottom}} - E_{\text{valence top}}
$$
This quantity is extremely important. If the band gap is very large, electrons in the valence band cannot easily reach the conduction band, so the material tends to be an insulator. If the gap is small, some electrons can move into the conduction band, especially at higher temperatures, and the material behaves as a semiconductor. If there is no gap, or if bands overlap, the material behaves like a conductor.
Why Bands Form
The detailed origin of bands comes from quantum mechanics and the periodic structure of crystals. In a crystal, atoms are arranged in a regular repeating pattern. Electrons move in the electric field created by all the atomic nuclei and other electrons. Because this environment repeats from atom to atom, only certain electron wave patterns are allowed.
These allowed wave patterns correspond to allowed energies, while other energies are not possible. The result is a band structure, which is a map of which electron energies are allowed in the solid.
For beginners, the key point is not the full mathematical theory, but the physical meaning. The regular atomic structure of a solid changes isolated atomic levels into broad bands of allowed energies.
Occupation of Bands
Not every allowed energy state is occupied. Whether a state is occupied depends on how many electrons the solid has and on the temperature.
At very low temperature, electrons fill the lowest available energy states first. As the lower bands fill up, the highest occupied energies define an important level called the Fermi level, often written $E_F$. A full treatment of the Fermi level belongs elsewhere, but here it is enough to know that the position of this level relative to the bands helps determine whether a material is metallic, semiconducting, or insulating.
If the highest occupied states lie within a partly filled band, electrons can move easily and conduction is good. If the highest occupied band is full and separated from the next band by a gap, conduction is much harder.
The electrical behavior of a solid depends not only on the existence of bands, but also on whether the bands are full, empty, or partially filled.
Band Structure and Material Type
Energy bands give a direct way to classify materials.
| Material type | Band picture | Electrical behavior |
|---|---|---|
| Conductor | Partially filled band, or overlapping bands | Electrons move easily |
| Semiconductor | Full valence band, empty conduction band, small gap | Limited conduction, increases with temperature |
| Insulator | Full valence band, empty conduction band, large gap | Very poor conduction |
In conductors, electrons already have nearby empty states in the same band, so an applied electric field can change their motion easily.
In insulators, the valence band is full and the conduction band is too far away in energy. Very few electrons can cross the large gap, so current is extremely small.
In semiconductors, the situation is in between. The gap is small enough that some electrons can be promoted to the conduction band. This creates mobile electrons in the conduction band and leaves behind holes in the valence band. The details of semiconductors are treated in later chapters, but energy bands are the foundation for understanding them.
Electrons and Holes
When an electron gains enough energy to move from the valence band to the conduction band, it leaves an empty state behind in the valence band. This empty state is called a hole.
A hole behaves in many ways like a positive charge carrier. Thus, in semiconductors, both conduction-band electrons and valence-band holes can contribute to electric current.
This idea is important because band theory does not only describe electrons. It also explains why the absence of an electron in a nearly full band can act like a moving particle.
Band Diagrams
A band diagram is a simple drawing that shows the valence band, conduction band, and the band gap. It helps us visualize the electron energy structure of a material.
These diagrams do not usually show the actual paths of electrons in space. Instead, they show energy levels. Vertical position means energy, not height in the material.
This distinction is important for beginners. A higher position on a band diagram means higher energy, not a physically higher location.
In a band diagram, the vertical axis represents energy. It does not represent the physical position of the electron in space.
Typical Band Gap Sizes
Band gaps vary widely from one material to another. Metals typically have no gap relevant to conduction. Semiconductors have moderate gaps, often around a fraction of an electron volt to a few electron volts. Insulators usually have larger gaps.
A useful unit here is the electron volt, abbreviated eV. One electron volt is the energy gained by one electron moving through a potential difference of one volt.
$$
1 \, \text{eV} = 1.602 \times 10^{-19} \, \text{J}
$$
Some rough examples are shown below.
| Material type | Typical band gap |
|---|---|
| Metal | approximately $0 \, \text{eV}$ or overlapping bands |
| Semiconductor | about $0.1$ to $3 \, \text{eV}$ |
| Insulator | often greater than $3 \, \text{eV}$ |
These values are only general guides. Real materials can be more complicated.
Importance of Energy Bands
The concept of energy bands explains many important properties of solids. It helps us understand electrical conductivity, optical absorption, semiconductors, and electronic devices.
If light shining on a material has enough energy, it can lift electrons across the band gap. This is why band gaps are also important in optics and devices such as solar cells and LEDs. The full device behavior belongs to later chapters, but the essential idea starts here, with the existence of allowed bands and forbidden gaps.
Core Ideas to Remember
Energy bands arise because many atoms in a crystal bring their electron energy levels so close together that they form continuous ranges of allowed energies. Between these allowed ranges there can be forbidden gaps. The valence band is the highest normally occupied band, and the conduction band is the next higher band that can support mobile electrons. The size of the band gap determines whether a solid behaves as a conductor, semiconductor, or insulator.
Key relations and facts:
$$
E_g = E_{\text{conduction bottom}} - E_{\text{valence top}}
$$
A conductor has a partially filled band or overlapping bands.
A semiconductor has a small band gap.
An insulator has a large band gap.
A full band by itself does not conduct well.