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Materials Science on CD-ROM User Guide

Introduction to Electrons in Crystals

Version 2.1

Peter Goodhew, University of Liverpool
Andrew Green, MATTER

Assumed Pre-knowledge

Atomic structure in terms of protons, neutrons and electrons, the periodic table, the concepts of potential and kinetic energy, the idea of electron energy levels, spdf orbital and KLM shell terminologies (although these are also defined in the glossary and compared in the module), quantum numbers.

Module Structure

The module covers five topics:

Introduction to energy levels

This section is intended to ensure that the terminology is understood. It, and the other sections, are supported by more than 100 electron-related terms in the MATTER glossary. The section starts with a consideration of the two principal conventions for defining the energy of an electron bound to an atom. Potential energy compared to the vacuum level (defined to be negative) is contrasted with binding energy (defined to be positive). The concept of localised electrons is introduced, in contrast to the formation of electron energy bands. The term delocalised is defined in the glossary entry for localised. The need for the valence and conduction bands is explained very briefly, but only qualitatively, in terms of the Pauli exclusion principle.

A table shows the comparison between the KLM, spdf and quantum number descriptions of localised states. This introduces the terminologies commonly used to describe characteristic X-ray emission and Auger transitions. The selection rules are mentioned briefly and a simple animation illustrates the forbidden 2s to 1s transition and the permitted 2p to 1s. The section ends with a comparison of the widely-used but non-intuitive Ka, Kb convention and the logical but under-used convention preferred by IUPAC.

After completing this section, the student should be able to explain:

  • The conventions for describing the energy levels of localised electrons in terms of 'spdf ' terminology and quantum numbers
  • The reason for the formation of energy bands for conduction and valence electrons
  • The terminology used to describe core level transitions leading to X-ray and Auger electron emission.

Electron-atom interactions

This section introduces the interaction of high energy electrons with atoms of aluminium. It is directed towards the use of high-energy electron beams in materials analysis, for example in electron microscopes and electron spectrometry. Electron microscopy is covered in a series of companion MATTER modules. Aluminium is chosen for the simulation because it has electrons in three shells, and therefore allows for a range of transitions, but is not too complex to simulate on the screen. Plasmon, phonon, inner-shell and valence electron excitations are considered, and subsequent relaxation to give Auger electrons and characteristic X-rays are dealt with. The relative frequencies of each excitation are mentioned and further study in this direction is stimulated by a question on page E:4. In order that a student can collect sufficient data to comment on these frequencies a button has been provided on page E:4 which will run 20 interactions quickly. Each press of this button will accumulate a further set of 20 data. The teacher should note, and the student should discover, that all the scattering processes occur with equal probability in the simulation, which is far from the situation in reality. It is clearly impractical to simulate the real probabilities of occurrence, since some processes would almost never be seen by the student.

After studying these simulations, the student should be able to:

  • List the major inelastic scattering processes for high energy electrons;
  • Name (using the IUPAC terminology) the X-rays and Auger electrons emitted after specific electron transitions.

This section is primarily intended to support the teaching of physical techniques such as electron microscopy and spectroscopy. The interaction between high energy electrons and atoms is particularly important in electron microscopy. The material in this section forms the basis for understanding the use of electron microscopy for elemental analysis.


The treatment begins with the ideas of electron excitation to a higher energy state and subsequent decay to the lower state. The difference between spontaneous and stimulated emission is illustrated and the idea of pumping is introduced via a simulation of a two-state system. The fundamental concepts of population inversion and confinement are introduced, and stimulated emission is illustrated. The first major simulation in this section relates to the pumping of a two-state system. The system is pumped as long as the mouse button is held down, but decays rapidly once pumping is stopped by release of the button. The student should note the relative occupancies of the two levels.

A system with three states is then simulated, to show that population inversion can be achieved. The user can change the decay time of the middle energy level in order to explore the conditions necessary for inversion. A second way of achieving population inversion by the use of a forward-biased p-n junction is then animated. The idea of confinement is then incorporated and the section ends with an animation of a semiconductor heterojunction laser.

After completing this section, the student should be able to:

  • Distinguish between spontaneous and stimulated emission;
  • Explain the need for population inversion and how this can be achieved either in a three-state system or in a p-n junction;
  • Explain the operation of a double heterojunction semiconductor laser.

Density of States

The ideal parabolic N(E) curve is developed quantitatively and the concept of effective mass is introduced from the second differential of the energy-momentum expression. The appearance of the parabolas for light and heavy electrons and holes is animated and the student is directed to think about the origin of the energy scale for electrons and holes. A one-dimensional wave-in-a-box argument is used, with the de Broglie equation, to develop the concept of the quantised state. This is generalised to three dimensions without a rigorous treatment of the mathematics and the idea of the Fermi sphere is introduced qualitatively.

A simple geometrical k-space argument is used to derive the ideal density of states parabola. The distinction between an electron and an electron state (capable of containing two electrons) needs to be kept clearly in mind here. The filling of the lowest energy states is illustrated, leading to the concept of the Fermi energy Ef. The effect of temperature is illustrated in a simplified manner on the next screen, which refers to the glossary for a short treatment of Fermi-Dirac statistics.

The importance of diffraction in creating band gaps and zone boundaries is now introduced qualitatively. The first and second Brillouin zones for a simple cubic material are shown and the filling of the Fermi sphere within these zones is simulated. Overlap of states into the second zone is shown as the e/a ratio is increased. On the next screen the bending of the ideal parabola at zone boundaries is shown and interpreted in terms of diffraction and a change in effective mass. Its effect on the creation of a band gap is illustrated.

The section ends with a few selected band diagrams and DOS curves to illustrate the non-ideally-free nature of real materials. Band diagrams for Si and GaAs are used to show the conventional [100] and [111] plotting method. Density of states curves for g-brass and chromium are displayed to show the difference between a nearly-ideal metal and a complex transition metal.

After completing this section, the student should be familiar with:

  • Simple parabolic energy-momentum diagrams (E-k curves);
  • The concept of effective mass;
  • The Fermi sphere;
  • The ideal parabolic N(E) vs E curve;
  • The effect of diffraction leading to zone boundaries and band gaps;
  • The conventional presentation of band diagrams for Si and GaAs;
  • The reasons for the non-parabolic appearance of DOS curves for transition metals and brass.


Conduction in metals and semiconductors is covered at an introductory level.

The section starts by considering carriers and the first screen contains reminders about the nature of electrons and holes and their relationship to band diagrams. The difference between thermal motion and drift (accounting for conduction) is then treated quantitatively. A simulation shows the effect of drift superimposed on the thermal motion of a single electron. The idea of scattering, leading to constant resistivity, is then introduced and the effect of temperature is considered briefly.

The free electron wave model is then used to introduce the idea that scattering can be much reduced in a crystalline array of scatterers. Note that no mention has been made, up to this point, of the nature of the scatterers. A simulation shows that scattering will be strong either if the wavelength of the electron is not the same as the spacing of the scatterers or if the scatterers are not distributed periodically.

The nature of the scatterers is introduced via Matthiessen’s rule and the student is asked to consider the types of possible scatterer. Following this, the relationship between electrical and thermal conductivity is touched on briefly and the Wiedermann-Franz law is stated.

The treatment now moves on specifically to semiconductors and the simultaneous movement of electrons and holes. The student is led to the conclusion hat the product of n and p should be constant (at a given temperature). Finally the effect of temperature on the conductivity of a semiconductor is treated and the effective densities of states in the conduction and valance bands are presented.

After completing this section, the student should understand:

  • The difference between thermal and drift velocities of carriers
  • The reasons for the low resistivity of perfect crystals, and constant resistivity in terms of electron scattering
  • The existence of many scattering mechanisms in crystals, the effect of which can be added using Mattheissen's Rule
  • The Wiedermann - Franz Law
  • The nature of current carriers in semiconductors and the concept of mobility
  • The semiconductor equation np = constant and its independence of doping.


The student is referred to the following resources in this module:

Jones, I.P., Chemical Microanalysis using Electron Beams, Institute of Materials, 1992

Goodhew, P.J. and Humphreys, F.J., Electron Microscopy and Analysis, 2nd Edition, Taylor & Francis, 1988

Reed, S.J.B., Electron Microprobe Analysis, Cambridge University Press, 1975

Solymar, L. and Walsh, D., Lectures on the Electrical Properties of Materials, Oxford Science Pubs. 4th Ed 1988

Sze, S.M., Semiconductor Devices: Physics and Technology, John Wiley, 1985

Cottrell, A.H., Introduction to the Modern Theory of Metals, Institute of Metals, 1988

Reed-Hill, R.E., Physical Metallurgy Principles, 2nd Edition, van Nostrand, 1973

Rosenberg, H.M., The Solid State, Oxford Physics Series, 1978 or later


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Copyright of The University of Liverpool 2000