Energy Band Gap

Energy bands and band gaps

The underlying idea behind the electronic structure of solids is that the valence electrons from the atoms involved spread throughout the entire structure, ie molecular orbitals are generally extended over all the constituent atoms. A large number of overlapping atomic orbitals lead to molecular orbitals very similar in energy over a certain range. This forms an almost continuous band. These bands are separated by band gaps, which are the energy values where there are no orbitals.

This continuous band arises from the molecular orbitals being slightly different in energy, which arises from the different degrees of bonding in each. At the bottom of the band you have the lowest energy MO which has all bonding character. At the top, with highest energy is an MO with all anti-bonding character. Therefore the rest of the band is formed from all the MO’s with intermediate bonding character between the two extremes.

Different types of orbitals form separate bands. Therefore you can end up with an s band and a p band. Whether or not they form two distinct bands with a band gap, or overlap depends on the separation of the orbitals and how strong the interaction between the atoms is. Strong interaction means wide bands and a greater chance of overlap. (fig. 3)

The distinction between metallic and non-metallic solids comes from the way the orbitals are filled. Metallic behaviour arises from a partially full band as then there is no gap between the top filled level (Fermi level) and the lowest empty one. However, a non-metallic solid has a completely filled level (the valence band) and an empty one (the conduction band). These two bands are separated by a band gap. In the filled band every electron is matched by another so you get no overall net motion of electric charge. Therefore, for conduction to occur electrons have to be excited up to the conduction band by overcoming an activation energy and hence, the conduction of these compounds increases with temperature.

Insulators have a full valence band separated from the next energy band, which is empty, by a large, forbidden gap. Diamond is an excellent insulator. It has a band gap of approximately 6eV which is very large. This means very few electrons have sufficient energy to be promoted and the conductivity is negligibly small. (fig. 5) When conductivity of insulators is able to be measured it is found to increase with temperature like the non-metallic solids.

IONIC SOLIDS – bonding due to transfer of charge from one atom to another. Energy bands formed from the atomic orbitals of anions and cations.

COVALENT SOLIDS – bonding due to overlap and sharing of electrons. Bands formed from bonding molecular orbitals (filled bands) and antibonding orbitals (empty bands).

METALLIC SOLIDS – bonding due to orbital overlap forming a delocalised cloud of electrons. Overlap of atomic orbitals can be so strong that bands are formed which are much broader than the original energy separation of the orbitals. Orbitals lose their individuality and you can look at it as the electrons moving freely.

Spectroscopic techniques for studying bandstructure

The existence of bands and a map of their desities of states can be shown using either photoelectron spectroscopy or x-ray analysis.

Photoelectron spectroscopy can be used here in much the same way as when dealing with discrete molecules. The densities of states of discrete molecules are made up of a set of widely separated, narrow peaks, corresponding to energies of discrete molecular orbitals. When looking at the photoelectron spectrum the peaks exist as photoelectrons with discrete ionisation energies.

Equivalent information about solids can be obtained from X-ray emission bands. Electrons are ejected (by electron bombardment) from the inner closed shells of the atoms. As electrons from the valence band fall into the vacancies created x-rays are emitted.

Since the valence electron falling can originate from any of the occupied levels in the band, the corresponding x-ray emissions cover a range of frequencies. Where there are many states in the valence band with similar energies you get higher emission intensities, and vice-versa.

Therefore, the x-ray emission band gives an indication of the variation of the density of states across the band. It is not an exact match though, because you must make allowances for the ease with which the incoming photon may eject an electron from different types of orbital.

X-ray emission gives information about the densities of states of the occupied areas of the band and x-ray absorption information about the unoccupied areas.


Semiconductors have a similar band structure to insulators but the band gap is not very large and some electrons have sufficient thermal energy to be promoted up to the empty conduction band. There are two types of conduction mechanism in semiconductors. Electrons promoted into the conduction band are classed as negative charge carriers and would move towards a positive electrode under an applied potential. The holes these electrons leave behind are known as positive holes. These holes move when an electron enters them. Wherever the electron that filled the hole moved from is the new positive hole. The positive holes therefore move in an opposite direction to the electrons. (fig. 7)

Semiconductors can be split into two groups. INTRINSIC and EXTRINSIC semiconductors.

Intrinsic semiconductors are pure materials with the bandstructure already discussed. The number of electrons in the conduction band is determined only by the size of the band gap and the temperature (more electrons with small band gap and high temperature).

Extrinsic semiconductors are materials where the conductivity is controlled by adding dopants with different numbers of valenece electrons to that of the original material. These are discussed in the next section.

Doping of semiconductors

Doping of semiconductors is achieved by introducing atoms with more or less electrons than the parent element. Doping is substitutional, the dopant atoms directly replace the original atoms. Suprisingly low levels of dopant are required, only 1 atom in 109 of the parent atoms.

Looking at silicon; if phosphorous atoms are introduced into a silicon crystal then extra electrons will be available (one for each dopant atom introduced as P has one extra valence electron). The dopant atoms form a set of energy levels that lie in the band gap between the valence and conduction bands, but close to the conduction band. The electrons in these levels cannot move directly as there is not enough of them to form a continuous band. However, the levels themselves can act as donor levels because the electrons have enough thermal energy to get up into the conduction band where they can move freely.

Such semiconductors are known as n-type semiconductors, representing the negative charge carriers or electrons.

What if, instead of doping with phosphorous, we doped silicon with an element with one less valence electron such as gallium. Now for every dopant atom there is an electron missing, and the atoms form a narrow, empty band consisting of acceptor levels which lie just above the valence band. Electrons from the valence band may have enough thermal energy to be promoted into the acceptor levels, which are discrete levels if the concentration of gallium atoms is small. Therefore, electrons in the acceptor levels cannot contribute to the conductivity of the material. However, the positive holes in the valence band left behind by the promoted electrons are able to move.

These type of semiconductors are known as a p-type semiconductors, representing the positive holes.


There are two fundamental differences between extrinsic and intrinsic semiconductors:

1) At standard temperatures extrinsic semiconductors tend to have significantly greater conductivities than comparable intrinsic ones.

2) The conductivity of an extrinsic semiconductor can easily and accurately be controlled simply by controlling the amount of dopant which is introduced. Therefore materials can be manufactured to exact specifications of conductivity.


http://www.chem.ox.ac.uk/vrchemistry/solid/Page09.htm







The electrical, optical, and other properties of semiconductors depend strongly on how the energy of the delocalized electrons involves the wavevector k in reciprocal or k-space, with the electron momentum p given by p=mv=hk. We will consider three dimensional crystal; in particular, we are interested in the properties of III-V and II-VI semiconducting compounds that have a cubic structure, so their three lattice constants are the same, namely, a=b=c….

The various bands have prominent maxima and minima at the center point Γ of the Brillouin zone. The energy gap or region where no band appears extends from the zero of energy at one point to the other(eg. Γ8 to Γ6) directly above the gap at the energy Eg = 1.35 eV. Hence Γ6 is the lowest energy point of the conduction band, and Γ8 is the highest point of the valence band. At the absolute zero all of the energy bands below the gap are filled with electrons, and all the bands above the gap are empty, so at T=0 K the material is an insulator. At room temperature the gap is sufficiently small so that some electrons are thermally excited from the valence band to the conduction band, and these relatively few excited electrons gather in the region of the conduction band immediately above its minimum at Γ6, a region that is referred to as a “valley.” These electrons carry some electric current; hence the material is a semiconductor.

The physics and chemistry of Nanosolids, Frank Owens and Charles Poole, Wiley


Fermion

In particle physics, fermions are particles which obey Fermi-Dirac statistics; they are named after Enrico Fermi. In contrast to bosons, which have Bose-Einstein statistics, only one fermion can occupy a quantum state at a given time; this is the Pauli Exclusion Principle. Thus if more than one fermion occupies the same place in space, the properties of each fermion (e.g. its spin) must be different from the rest. Therefore fermions are usually associated with matter while bosons are often force carrier particles, though the distinction between the two concepts is not clear cut in quantum physics.

Fermions can be elementary, like the electron, or composite, like the proton. All observed fermions have half-integer spin, as opposed to bosons, which have integer spin. This is in accordance with the spin-statistics theorem which states that in any reasonable relativistic quantum field theory, particles with integer spin are bosons, while particles with half-integer spin are fermions.




In the Standard Model there are two types of elementary fermions: quarks and leptons. The 24 fundamental fermionic flavours are:

* 12 quarks - 6 particles (u · d · s · c · b · t) with 6 corresponding antiparticles (u · d · s · c · b · t);

* 12 leptons - 6 particles (e− · μ− · τ− · νe · νμ · ντ) with 6 corresponding antiparticles (e+ · μ+ · τ+ · νe · νμ · ντ).

Composite fermions, such as protons and neutrons, are essential building blocks of matter. Weakly interacting fermions can also display bosonic behaviour, as in superconductivity.


Spin-statistics theorem

The spin-statistics theorem in quantum mechanics relates the spin of a particle to the statistics obeyed by that particle. The spin of a particle is the angular momentum that the particle has when it is not moving, and in quantum mechanics, it gives the change in wavefunction phase when you do certain rotations. All particles have either integer spin or half-integer spin (in multiples of \ \hbar (Dirac's constant)). Integer spin means that the phase change for a 360 degree rotation is 1, while half integer spin means that the phase change for a 360 degree rotation is -1.

The theorem states that:

* The wave functions of a system of identical integer-spin particles, spin 0, 1, 2, 3, has the same value when the positions of any two particles are exchanged. Particles with wavefunctions symmetric under exchange are called bosons.

* The wave functions of a system of identical half-integer-spin s = 1/2, 3/2, 5/2, are anti-symmetric under exchange, meaning that the wavefunction changes sign when the positions of any pair of particles are swapped. Particles whose wavefunction changes sign are called fermions.

So the spin statistics theorem states that integer spin particles are bosons while half-integer spin particles are fermions.

WIKIPEDIA





ENERGY
There are various forms of energy : chemical energy, heat, electromagnetic radiation, potential energy (gravitational, electric, elastic, etc.), nuclear energy, rest energy. These can be categorized in two main classes: potential energy and kinetic energy.