PHY2009 Physics of Crystals

2010-2011

Code: PHY2009
Level: 2
Title: Physics of Crystals
Instructors: Prof. R.J. Hicken
CATS Credit Value: 10
ECTS Credit Value: 5
Pre-requisites: N/A
Co-requisites: N/A
Duration: M1-M11
Availability: unrestricted
Background Assumed: Waves and Oscillators (PHY1106)
Directed Study Time: 22 lectures
Private Study Time: 66 hours
Assessment Tasks Time: 12 hours

Aims

This module will develop the students' understanding of how electrons, and other waves, propagate within crystalline materials. The first section involves learning about the crystal structures common in nature. The fundamental properties of periodic structures are discussed, which are central to our understanding of solid-state physics, particularly the relationship between real space and reciprocal space and the representation of elastic and inelastic scattering in both spaces. The second section involves learning about the vibrational excitations of the crystal lattice - phonons. These are of central importance to the properties of insulators and will be further discussed in PHY3102 (Solid State Physics I). Both phonons and electrons are profoundly influenced by the crystal structure in which they propagate. The last section of this module considers the transport of electrons in the free-electron and nearly-free-electron approximations, which give a good description of the behaviour of electrons in metals and semiconductors.

Intended Learning Outcomes

Students should be able to:

Module Specific Skills

• Describe the elementary models for bonding of atoms and molecules and the consequential classifications used in solid state physics; relate the general properties (electrical, thermal and optical) for each class, including details of the expected crystal structures, to the mechanical properties.
• Describe, and perform simple calculations involving, the hexagonal close-packed structure and various cubic structures, which are commonly found in nature.
• Explain how the problem of elastic scattering by a crystal is treated using the concept of the reciprocal lattice and how calculations separate factors which depend on the lattice and on the basis, i.e. yield the Laue equations and a structure factor; solve problems relating to representative solid state materials.
• Give a detailed description of the features of the vibrations of monatomic and of diatomic linear chains and explain the significance of dispersion curves in three dimensions.
• Discuss in an informed manner the scattering of phonons, and in particular the occurrence of Umklapp scattering of phonons near the Brillouin zone edge.
• Describe the free electron model and apply it in calculations involving: the dispersion relation, the effective mass, the density of states, the Fermi distribution.
• Use the nearly free electron model to account for the occurrence of energy gaps at the Brillouin zone edges, and the consequent behaviour of the group velocity and effective mass of the electrons.

Discipline Specific Skills

• Use mathematical abstraction to represent and solve problems involving periodic structures.

Personal and Key Skills

• Solve problems requiring spatial reasoning.

Learning / Teaching Methods

Lectures, tutorials, problems classes, and e-learning resources.

Assignments

Regular problems sheets for self-study and prepartion for problems classes.

Assessment

Problems-class assignments (10%), 30-minute mid-term test in Week M7 (20%) and one 90-minute examination (70%).

Syllabus Plan and Content

1. Review of Bonding in Solids
   1. Interatomic bonding in solids
      1. Ionic, covalent, metallic, van de Waals and Hydrogen bonds Central core repulsion.
2. Crystal Lattices
   1. Concept of crystal structure as lattice plus basis. Lattice symmetries.
   2. Two-dimensional and three-dimensional lattices.
      1. General features, Bravais lattices, crystal systems.
      2. Positions, directions and planes in crystals.
   3. Typical Crystal Structures
      1. Cubic and hexagonal close-packed.
      2. Body-centred cubic; Rock-salt.
      3. Ionic solids.
      4. Diamond and zincblende.
3. Elastic Scattering of Waves
   1. General features of scattering by solids.
   2. Scattered-wave amplitude.
   3. Laue conditions for diffraction.
   4. Reciprocal lattice and Brillouin zones.
   5. Structure factor.
4. Atomic Vibrations
   1. Lattice vibrations of the monatomic linear chain.
   2. Diatomic linear chain.
   3. Lattice vibrations of three-dimensional crystals
      1. Longitudinal and transverse phonons;
      2. Plotting of dispersion relations.
5. Electrons in crystals
   1. The free-electron model
      1. Free electron Fermi gas.
      2. Dispersion relation, group velocity and effective mass.
      3. Density of states in 1-, 2- and 3-dimensions, and the Fermi distribution.
      4. Electrical conductivity
   2. The nearly-free-electron model
      1. Qualitative discussion of the dispersion curves: energy gaps, group velocity and effective mass.
      2. Consequences for electrical conductivity: Band picture classification of metals insulators and semiconductors.

Core Text

Kittel C. , Introduction to Solid State Physics, Wiley (UL: 530.41 KIT)

Supplementary Text(s)

Christman J.R. (1988), Fundamentals of Solid State Physics, Wiley, ISBN 0-471-81095-9 (UL: 530.41 CHR)
Hook J.R. and Hall H.E. (1991), Solid State Physics (2^nd edition), Wiley, ISBN 0-471-928054 (UL: 530.41 HOO)

Formative Mechanisms

This module is supported by problems classes and tutorials. Students are able to monitor their own progress by attempting problems sheets provided in the lectures. The graded mid-term test scripts are discussed by tutors. Students with specific problems should first approach their tutor, and if the problem is not resolved, the lecturer.

Evaluation Mechanisms

The module will be evaluated using information gathered via the student representation mechanisms, the staff peer appraisal scheme, and measures of student attainment based on summative assessment.