Solid State Physics

This is a topic from Higher Physics 1B

Introduction

This segment of the course explores some of the characteristics and behaviours of solids on the quantum level

Remodelling Metallic Solids

The bonds between molecules in metals are generally weaker than ionic or covalent bonds, such that electrons in the outer shell (as in the Bohr model of the atom) are relatively free to move. In a metal the number of such electrons is large, and so the metal lattice structure can be thought of as a grid of ions surrounded by a 'cloud' or 'gas' of electrons. It is this sea of electrons that cause metals to be shiny and to conduct electricity so well. This cloud model of accounts for many of the shortcomings of the classical model of metallic solids by considering the wave nature of the electrons and the probabilistic phenomenon that accompany this.

Fermi-Dirac Distribution Function

The electrons in the cloud vary in energy. The probability that a particular energy level E is occupied by an electron in the solid is:

Where E represents the energy level in question

e represents in exponential constant
EF is a constant known as the Fermi energy which is the maximum energy of an electron in a substance at absolute zero (T = 0 Kelvin)
kB is Boltzmann's constant
T is temperature in Kelvin

• At T = 0 all states with less energy than the Fermi energy are occupied (remember the allowed energy levels are discrete and not continuous so it is possible to occupy them all). There are no electrons of an energy greater than the Fermi energy
• As temperature increases the probability of lower energy levels being occupied is still 1, however towards the Fermi energy the probability decreases slightly to allow for a small probability that the energy level is higher (the total probability must equal 1.0 x EF)

The Density-of-States Function

The allowed energy levels for electrons is discrete, and so it is possible to conceptualise the number of allowed states per unit volume of energy between E and dE:

Energy Bands in Solids

Atoms have been seen to have discrete energy levels. When a huge number of atoms are combined to form a solid however, these discrete energy levels are replaced by discrete ranges of energy, or energy bands, within which there are so many individual allowed energy values that within the bands the distribution can be considered to be continuous. This idea is seen in the following figure:

• In between energy bands are ranges of energy which are entirely impossible known as band gaps
• Different substances have different band structures which in many ways dictate the characteristics of that substance in terms of electrical conduction, as there exists an energy band called the conduction band, in which electrons can be propagated as current. The valence band on the other hand is the energy level of valence electrons which are bound into the atomic structure of the substance
• When a substance is placed in an electric field its electrons gain potential energy. For a metal (conductor) the conduction and valence bands overlap, and so the additional energy is enough to free electrons and cause current as there are allowable energy values directly above the initial state. For insulators however, there is often a band gap between the initial energy state and the next possible value, and so a huge amount of energy is required to cause the electron to be freed and accelerated as current. For semiconductors there is still a band gap however it is a lot smaller. This is all seen in the following band diagram:

Semiconductors

• When an electron moves from the valence band to the conduction band it leaves behind a vacancy called a hole which moves around as electrons fill it and leaves a hole in its original place
• Holes act like positively charges particles
• Thus the carriers of charge in a semiconductors are positive and negative
• Electrons can be excited from the valence band to the conduction band due to temperature, electric field, or the photoelectric effect

Direct and Indirect Gap Semiconductors

It is important to be aware of the distinction between two types of semiconductors based on the characteristics of their behaviours in the photoelectric effect. For direct gap semiconductors such as Gallium Arsenide (GaAs) light absorption involves a change in electron energy only, where as for indirect gap semiconductors such as Silicon (Si) light absorption involves a change in electron momentum as well as energy.

Doping

• A pure semiconductor containing only one element is called an intrinsic semiconductor and contains an equal number of conduction electrons and holes (electron-hole pairs)
• Adding other elements to a semiconductor is called doping, and it can affect band structure and resistivity. A doped semiconductor is called an extrinsic semiconductor
• N-type semiconductors involve the addition of donor atoms which have an extra electron in the valence band which does not fit into the structure and so is free to be a charge carrier:

• P-type semiconductors involve the addition of acceptor atoms which have one less electron in the valence band, creating a hole to act as a charge carrier: