Semiconductor theory

Atomic nuclei and electrons are the most basic building blocks of matter. In an atom (N = 1), electrons orbit around the nucleus which is made up of protons and neutrons. The number of electrons depends on the element. For the simplest case – hydrogen – one electron orbits around one proton. Each electron can be ascribed to an individual orbit known as the atomic orbital (AO) which has a discrete energy level. When an AO from one atom overlaps with the AO from another atom (N = 2) two molecular orbitals (MOs) are formed and the electrons are delocalised over both atoms. One of the MOs is bonding and this has a lower energy than that of the original AO. The other MO is anti-bonding and has an energy that is higher than the initial AO. Figure 1.5 shows the energy level diagram for a H2 molecule. The

formation of a H2 molecule involves the overlap of two single 1s orbitals, one from

each atom. The Pauli exclusion principle limits the number or electrons that can occupy a MO to two and requires that the two electrons must be paired. For H2 both

15 Figure 1.5 – The MO energy level diagram for a H2 molecule. σ denotes the bonding and σ* the anti- bonding MO.

For three atoms there are three orbitals; one bonding and one antibonding with a nonbonding orbital in between. As more atoms are added, each one contributes one more AO and hence one more MO is formed. For N atoms in a line there are N

MOs. The orbital with the highest energy is the most antibonding and the orbital with the lowest energy in the most bonding. The remainder of the orbitals are spread between the two extremes. The total width of the band remains finite as N

approaches infinity and so there can only be a finite spread of orbital energies, regardless of N. As N approaches infinity the separation between neighbouring orbitals approaches zero. The discrete energy levels of the orbitals merge to form a band of near continuum energy levels for conductors. Figure 1.6 shows the build-up of MO for a metal.49

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Figure 1.6 – The MO energy levels for N atoms for a metal. Adapted from [49].

For conductors the bonding and antibonding orbitals are not separated by an energy gap, as shown in Figure 1.6. In the case of non-metals such as insulators and semiconductors two bands are formed, the valence band (VB) and the conduction band (CB). The VB and CB are separated by an energy gap known as the band gap (Eg). This is due to the conjugated system undergoing geometrical distortion from a

structure with equal band length to one with alternating long and short bonds. The geometrical distortion leads to the opening up of a band gap. This is known as Peirels Distortion.

The VB is normally fully occupied and the CB is generally unoccupied at 0 K. A schematic depicting the difference between metals (conductors), semiconductors and insulators is shown in Figure 1.7. Insulators have a large Eg, typically > 4 eV. At

17 the CB. On the other extreme, the electrons in conductors can be excited at room temperature are essentially free because there is no Eg. Semiconductors fall between

the two. They have an Eg of ~ 1 – 4 eV; at sufficiently low temperatures the CB is

completely empty and the VB is fully occupied. However, the Eg is sufficiently

small that the electrons have the potential to reach the CB upon the appropriate input of energy.

Figure 1.7 – The energy level diagrams of a) an insulator, b) semiconductor and c) conductor.

The Fermi-Dirac distribution (Equation 1.1) defines the probability of an electron existing at a particular energy level,

𝑓(𝐸) = 1

𝑒𝐸−𝐸𝑓𝐾𝑇 +1

Equation 1.1

where K is Boltzmann’s constant, Ef is the Fermi level energy and T is the

temperature. Ef is defined as the theoretical energy level where the population

18 remove an electron from the Fermi level to the vacuum level (EVac). The ionisation

potential (EIP) is the amount of energy required to remove an electron from the edge

of the VB to the EVac and the electron affinity (EEA) is the energy from the CB to the

EVac. These are indicated in Figure 1.8.

The conductivity of semiconductors can be altered by doping. Doping is where a few atoms (< 0.1 %) of the original element are replaced by atoms having either more or less electrons. An n-type semiconductor is where the added (donor) atoms have more electrons in the valence shell than the host atom. If the donor atoms are spread out from each other, their electrons will be localised and a thin donor band will form. The donor atoms energy levels lie at a higher energy than the valence electrons of the host lattice, usually just below the empty CB of the lattice. Some of the electrons from the donor band can be promoted to the CB which are then able to move throughout the lattice. If the added atoms have fewer electrons than the host, positive holes are added and the semiconductor is said to be p-type. The dopant atoms form a thin, empty acceptor band that lies just above the full VB of the host lattice. Small amounts of energy can promote electrons from the VB to the acceptor band generating holes in the host VB which allows the remaining electrons to become mobile.

The number of electrons that can be promoted depends on the temperature and the energy gap between the bands. The Fermi level in an intrinsic semiconductor is likely to be near the middle of the band gap. For n-type semiconductors the Fermi levels rise to an energy level near the middle of the new donor band and the CB band of the host. The Fermi level drops to a point near the middle of new acceptor band

19 and valence band of the host for p-type semiconductors. Figure 1.8 shows the band structure of intrinsic, n- and p-type semiconductors.

Figure 1.8 – Energy level diagram for a) an intrinsic, b) n-type and c) p-type semiconductors with the Φ, EIP and EEA labelled.

The terms CB and VB are typically used for inorganic materials. Organic semiconductors are made up of molecules rather than atoms and it is the MO that defines the energy structure. So instead of referring to the VB and CB, highest occupied molecular orbital (HOMO) and lowest unoccupied molecular orbital (LUMO) are used instead. Generally, the π-bonding orbitals combine to form the HOMO and the π-antibonding orbitals form the LUMO with the energy gap between the two defining the Eg.

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