# Band Theory of Electrical Conductivity

## Conductors

A conductor is a material that is able to conduct electricity with minimal impedance to the electrical flow. It is commonly a metal.

### Learning Objectives

Apply the concept of band theory to explain the behavior of conductors.

### Key Takeaways

#### Key Points

• A conductor is a material which contains movable electric charges.
• In metallic conductors, such as copper or aluminum, the movable charged particles are electrons, though in other cases they can be ions or other positively charged species.
• Band theory, where the molecular orbitals of a solid become a series of continuous energy levels, can be used to explain the behavior of conductors, semiconductors and insulators.
• Most familiar conductors are metallic.

#### Key Terms

• voltage: The amount of electrostatic potential between two points in space.
• molecular orbital: The quantum mechanical behavior of an electron in a molecule describing the probability of the electron's particular position and energy; approximated by a linear combination of atomic orbitals.
• metal: Any of a number of chemical elements in the periodic table that form a metallic bond with other metal atoms; generally shiny, somewhat malleable and hard, often a conductor of heat and electricity.

### Conductors vs. Insulators

A conductor is a material which contains movable electric charges. In metallic conductors such as copper or aluminum, the movable charged particles are electrons. Positive charges may also be mobile, such as the cationic electrolyte(s) of a battery or the mobile protons of the proton conductor of a fuel cell. Insulators are non-conducting materials with few mobile charges; they carry only insignificant electric currents.

In describing conductors using the concept of band theory, it is best to focus on conductors that conduct electricity using mobile electrons. According to band theory, a conductor is simply a material that has its valence band and conduction band overlapping, allowing electrons to flow through the material with minimal applied voltage.

### Band Theory

In solid-state physics, the band structure of a solid describes those ranges of energy, called energy bands, that an electron within the solid may have ("allowed bands") and ranges of energy called band gaps ("forbidden bands"), which it may not have. Band theory models the behavior of electrons in solids by postulating the existence of energy bands. It successfully uses a material's band structure to explain many physical properties of solids. Bands may also be viewed as the large-scale limit of molecular orbital theory.

The electrons of a single isolated atom occupy atomic orbitals, which form a discrete set of energy levels. If several atoms are brought together into a molecule, their atomic orbitals split into separate molecular orbitals, each with a different energy. This produces a number of molecular orbitals proportional to the number of valence electrons. When a large number of atoms (1020 or more) are brought together to form a solid, the number of orbitals becomes exceedingly large. Consequently, the difference in energy between them becomes very small. Thus, in solids the levels form continuous bands of energy rather than the discrete energy levels of the atoms in isolation. However, some intervals of energy contain no orbitals, forming band gaps. This concept becomes more important in the context of semi-conductors and insulators.

Conductors, Semiconductors and Insulators: On the left, a conductor (described as a metal here) has its empty bands and filled bands overlapping, allowing excited electrons to flow through the empty band with little push (voltage). Semiconductors and insulators have a greater and greater energetic difference between the valence band and the conduction bands, requiring a larger applied voltage in order for electrons to flow.

Within an energy band, energy levels can be regarded as a near continuum for two reasons:

1. The separation between energy levels in a solid is comparable with the energy that electrons constantly exchange with phonons (atomic vibrations).
2. This separation is comparable with the energy uncertainty due to the Heisenberg uncertainty principle for reasonably long intervals of time. As a result, the separation between energy levels is of no consequence.

### Conductors

All conductors contain electrical charges, which will move when an electric potential difference (measured in volts) is applied across separate points on the material. This flow of charge (measured in amperes) is what is referred to as electric current. In most materials, the direct current is proportional to the voltage (as determined by Ohm's law), provided the temperature remains constant and the material remains in the same shape and state.

Most familiar conductors are metallic. Copper is the most common material used for electrical wiring. Silver is the best conductor, but it is expensive. Because gold does not corrode, it is used for high-quality surface-to-surface contacts. However, there are also many non-metallic conductors, including graphite, solutions of salts, and all plasmas. There are even conductive polymers.

Thermal and electrical conductivity often go together. For instance, the sea of electrons causes most metals to act both as electrical and thermal conductors. However, some non-metallic materials are practical electrical conductors without being good thermal conductors.

## Semiconductors

Semiconductors are materials that have properties in between those of normal conductors and insulators; they are often produced by doping.

### Learning Objectives

Compare N-type and P-type semi-conductors, distinguishing them from semi-conductors and insulators using band theory.

### Key Takeaways

#### Key Points

• Intrinsic semiconductors are composed of only one kind of material.
• Extrinsic semiconductors are made of intrinsic semiconductors that have had other substances added to them to alter their properties (they have been doped with another element ).
• There are two types of extrinsic semiconductors: p-type (p for positive: a hole has been added through doping with a group -III element) and n-type (n for negative: an extra electron has been added through doping with a group-V element).

#### Key Terms

• semiconductor: a substance with electrical properties between those of a good conductor and those of a good insulator
• conductor: something that can transmit electricity, heat, light, or sound
• doped: describing a semiconductor that has had small amounts of elements added to create charge carriers

Semiconductors are materials that have properties of both normal conductors and insulators. Semiconductors fall into two broad categories:

• Intrinsic semiconductors are composed of only one kind of material; silicon and germanium are two examples. These are also called "undoped semiconductors" or "i-type semiconductors. "
• Extrinsic semiconductors, on the other hand, are intrinsic semiconductors with other substances added to alter their properties -- that is to say, they have been doped with another element.

### Intrinsic Semiconductors

In the classic crystalline semiconductors, electrons can have energies only within certain bands (ranges of energy levels). The energy of these bands is between the energy of the ground state and the free electron energy (the energy required for an electron to escape entirely from the material). The energy bands correspond to a large number of discrete quantum states of the electrons. Most of the states with low energy (closer to the nucleus ) are occupied, up to a particular band called the valence band.

Semiconductors and insulators are distinguished from metals by the population of electrons in each band. The valence band in any given metal is nearly filled with electrons under usual conditions. In semiconductors, only a few electrons exist in the conduction band just above the valence band, and an insulator has almost no free electrons.

An Illustration of the Electronic Band Structure of a Semiconductor: This is a comprehensive illustration of the molecular orbitals in a bulk material. As the energy in the system increases, electrons leave the valence band and enter the conduction band.

Semiconductors and insulators are further distinguished by the relative band gap. In semiconductors, the band gap is small, allowing electrons to populate the conduction band. In insulators, it is large, making it difficult for electrons to flow through the conduction band.

### Extrinsic Semiconductors

The name "extrinsic semiconductor" can be a bit misleading. While insulating materials may be doped to become semiconductors, intrinsic semiconductors can also be doped, resulting in an extrinsic semiconductor. There are two types of extrinsic semiconductors that result from doping: atoms that have an extra electron (n-type for negative, from group V, such as phosphorus) and atoms that have one fewer electron (p-type for positive, from group III, such as boron).

In semiconductor production, doping intentionally introduces impurities into an extremely pure, or intrinsic, semiconductor for the purpose of changing its electrical properties. The impurities depend on the type of semiconductor. Lightly and moderately doped semiconductors are referred to as extrinsic. When a semiconductor is doped to such a high level that it acts more like a conductor than a semiconductor, it is referred to as degenerate.

### N-Type Semiconductors

N-type semiconductors are a type of extrinsic semiconductor in which the dopant atoms are capable of providing extra conduction electrons to the host material (e.g. phosphorus in silicon). This creates an excess of negative (n-type) electron charge carriers.

N-type Semiconductor: After the material has been doped with phosphorus, an extra electron is present.

Doping atom usually have one more valence electron than one type of the host atoms. The most common example is atomic substitution in group-IV solids by group-V elements. The situation is more uncertain when the host contains more than one type of atom. For example, in III-V semiconductors such as gallium arsenide, silicon can be a donor when it substitutes for gallium or an acceptor when it replaces arsenic. Some donors have fewer valence electrons than the host, such as alkali metals, which are donors in most solids.

### P-Type Semiconductors

A p-type (p for "positive") semiconductor is created by adding a certain type of atom to the semiconductor in order to increase the number of free charge carriers. When the doping material is added, it takes away (accepts) weakly bound outer electrons from the semiconductor atoms. This type of doping agent is also known as an acceptor material, and the vacancy left behind by the electron is known as a hole. The purpose of p-type doping is to create an abundance of holes.

P-type Semiconductor: After the material has been doped with boron, an electron is missing from the structure, leaving a hole. This allows for easier electron flow.

In the case of silicon, a trivalent atom is substituted into the crystal lattice. The result is that one electron is missing from one of the four covalent bonds normally part of the silicon lattice. Therefore the dopant atom can accept an electron from a neighboring atom's covalent bond to complete the fourth bond. This is why these dopants are called acceptors.

When the dopant atom accepts an electron, this causes the loss of half of one bond from the neighboring atom, resulting in the formation of a hole. Each hole is associated with a nearby negatively charged dopant ion, and the semiconductor remains electrically neutral overall. However, once each hole has wandered away into the lattice, one proton in the atom at the hole's location will be "exposed" and no longer cancelled by an electron. This atom will have three electrons and one hole surrounding a particular nucleus with four protons.

For this reason a hole behaves as a positive charge. When a sufficiently large number of acceptor atoms are added, the holes greatly outnumber thermally excited electrons. Thus, holes are the majority carriers, while electrons become minority carriers in p-type materials.