Registry

Module Specifications

Archived Version 2022 - 2023

Module Title	Solid State Physics
Module Code	PS402
School	School of Physical Sciences
Online Module Resources
Module Co-ordinator	Prof. Colette McDonagh	Office Number	N145
NFQ level	8	Credit Rating	5
Pre-requisite	None
Co-requisite	None
Compatibles	None
Incompatibles	None

Description

The purpose of this module is to provide students with an overview of the principal topics in solid state physics at an advanced undergraduate level. The main topic areas will be (i) Crystal Structure, reciprocal lattice, Brillouin zones, (ii) Waves in crystals, diffraction, (iii) Lattice vibrations, Thermal properties, (iv) Electronic Structure, including free electron, tight-binding, and nearly-free electron approaches, (v) Optical Properties of metals and insulators. The course is mainly knowledge-based. Other learning activities include solving numerical problems related to the topics covered.

Learning Outcomes

1. Outline and discuss the main topics in solid state physics covered in this module
2. Identify concepts and/or physical principles which are common across one or more topics
3. Relate the physical principles to topics covered in other modules
4. Apply physical principles and relevant equations in solving numerical problems.

*Type*	*Hours*	*Description*
Workload	Full-time hours per semester
Lecture	30	An average of 2.5 lectures per week (3 lectures one week and 2 the next, repeated throughout semester)
Tutorial	6	Tutorial every second week
Independent Study	89	Studying lecture material, solving numerical problems and doing background reading
Total Workload: 125

All module information is indicative and subject to change. For further information,students are advised to refer to the University's Marks and Standards and Programme Specific Regulations at: http://www.dcu.ie/registry/examinations/index.shtml

Indicative Content and Learning Activities

Structure of solids
This section of the course will introduce the structure of solids in a general way and then move quickly to the case of crystalline solids. Key concepts are introduced such as lattice, basis, lattice translation vectors, unit and primitive cells, Wigner Seitz cell, Bravais lattices, crystal symmetries, specification of crystal planes and directions, inter-planar spacing, and Miller indices. Examples will focus on SC, FCC, BCC, HCP and diamond structures. Miller Indices and the reciprocal lattice will then be introduced as well as reciprocal lattice vectors and their calculation from real space lattice vectors. Shapes of different 2D and 3D Brillouin zones will be introduced as well as mapping into the 1st Brillouin zone. The scattering of waves by periodic crystals is introduced mathematically, including the Laue condition, the relation to Bragg's Law and the use of diffraction techniques such as x-ray, electron and neutron diffraction for the determination of crystal structure. Tutorial questions will be provided on the topics above and worked solutions will be provided online after the tutorial class.

Waves in solids
This section of the course will introduce the topics of elastic properties of solids and the relation to sounds waves in a continuum model and their dispersion relation first, followed by derivation of dispersion relations for monatomic and diatomic lattices, and acoustic and optical vibration normal modes and phase and group velocities. Introduction to density of states for phonons and van Hove singularities. Introduction of quantisation of lattice vibrations and phonon modes. Discussion of anharmonic potential energy curves and phonon-phonon collisions, normal and umklapp processes and crystal momentum. Introduction to methods for measurement of phonon dispersion curves, such as neutron diffraction and Raman scattering. Introduction to the lattice contribution to the specific heat capacity, beginning with the classical Dulong-Petit model and progressing to the quantum mechanical Einstein and Debye models, introducing the concepts of the Einstein and Debye temperatures. Calculation of the mathematical forms of the specific heat capacities and low and high temperature limits for the Einstein and Debye models and comparison with experiments. Introduction of the concept of the phonon (lattice) contribution to thermal conductivity and derivation of the formula for lattice thermal conductivity by analogy to the kinetic theory of gases model. Detailed discussion of all factors in this formula and their temperature dependence, including phonon mean free path and the importance of umklapp as well as normal collisions in such processes. Tutorial questions will be provided on the topics above and worked solutions will be provided online after the tutorial class.

Electronic properties of solids
Detailed discussion of the free electron theory of solids, including the application of quantum mechanics to free electrons and the associated states, as well as the use of Fermi-Dirac statistics in the filling of such states and the concepts of Fermi level, Fermi temperature, Fermi velocity and Fermi wavevector. The discussion refers to the Drude model covered in a previous module. The concept of density of states is revisited for the case of free electrons. The heat capacity of the free electron gas is calculated using both an approximate and an exact approach to yield the Sommerfeld constant and the need for an effective mass correction is introduced. Comparisons with experimental data are given. The electrical conductivity of the free electron gas is discussed and calculated in terms of collision times and mean free paths and examples including DC and AC fields are given, as well as for crossed E and B fields, for the case of the Hall effect. The Wiedemann-Franz Law and the Lorenz constant are discussed and the failures of the Drude model considered. Bloch's theorem is introduced and proved for a periodic potential, and the Bloch function is introduced, along with the crystal momentum eigenfunction. The concept of bands arising due to continuous variation of energy with k is introduced. The empty lattice model is introduced as the first step towards the introduction of a periodic potential and the form of the Brillouin zones is discussed in detail along with the form of the empty lattice "bands". The Nearly Free Electron Model and the central equation are then introduced and derived and the opening up of bandgaps related to the relevant Fourier coefficients is explored, and the effects on the effect mass close to the band edges. Some examples of simple potential types are explored and their Fourier coefficients calculated. The tight binding approximation is introduced, including all key assumptions and the concepts of molecular orbitals, bonding and anti-bonding states, hybridisation etc. are introduced and discussed. The simplified Schroedinger equation for the tight binding model is introduced and solved for a variation of 2D and 3D lattices and the material properties such as effective masses etc. are introduced and calculated. The concepts of density of states for electrons in 1D, 2D and 3D are introduced, as well as the form of the density of states for electrons in bands in terms of the gradient of the E versus k surface and sample calculations for e.g. a tight binding model are given. The conducting nature of the material (metal, insulator, semiconductor) is discussed in light of the material's Fermi surface and the bending and warping of Fermi surfaces in real materials is discussed and explained with worked examples. The detailed effects of bandstructure on electron transport, including effective mass, negative effective mass and the behaviour of holes and why they yield a positive Hall coefficient are explored in depth. Tutorial questions will be provided on the topics above and worked solutions will be provided online after the tutorial class.

Optical properties of solids
The optical properties of dielectrics are introduced using Maxwell's equations and the complex dielectric constant and refractive index, leading to the concepts of extinction coefficient etc. The relation of the material polarisability to its optical properties is introduced, including orientational and ionic polarisation. The electronic polarisation is introduced using the Drude-Lorentz oscillating dipole model and its consequences explored. Multiple resonances and continuum absorption are treated briefly. Ionic polarisation is then revisited in the context of phonon modes and the Lyddane-Sachs-Teller (LST) relationship and the reflectance behaviour close to phonon resonances and restrahlen are explored. The optical properties of metals are then explored by using the Drude-Lorentz approach but without the restoring force assumed in the case of the oscillating dipole (as used for dielectrics), and this connection is made clear. The reflectivity of metals is discussed and the plasma frequency derived, and the case of heavily doped semiconductors is also introduced. Inter-band transitions, the Fermi-Golden rule and the effects on the coloration of real metals such as Ag, Al, Au and Cu are discussed in detail. The optical properties of metals are also related back to the free carrier AC conductivity discussed earlier in the module, and the concept of skin depth is discussed based on absorption by the metal. Finally the concept of plasmons as quantised modes of electron gas vibrations are described, and methods for plasmon measurement briefly mentioned, as are some applications of plasmons in terms of tuning optical properties. Tutorial questions will be provided on the topics above and worked solutions will be provided online after the tutorial class.

Selected current topics in solid state/condensed matter physics
If time permits, a brief introduction (single lecture in seminar format) to one or more selected topics of strong current interest will be given (e.g. topological insulators, graphene etc.). The choice will be informed by considerations such as recent Nobel prize awards in the area. There will be no tutorial questions on this section of the module and it will not be examinable.

Learning activities
Students are expected to attend lectures and tutorials and to prepare for both by appropriate study. This will include using books and online loop and other resources.

Assessment Breakdown
Continuous Assessment	20%	Examination Weight	80%

Type	Description	% of total	Assessment Date
Course Work Breakdown

Reassessment Requirement
Resit arrangements are explained by the following categories; 1 = A resit is available for all components of the module 2 = No resit is available for 100% continuous assessment module 3 = No resit is available for the continuous assessment component
Unavailable

Indicative Reading List

M. Ali Omar: 0, Introductory Solid State Physics,
A. M. Fox: 0, Optical Properties of Solids, Oxford University Press,
J. Singleton: 0, Band Theory and Electronic Properties of Solids, Oxford University Press,
M. T. Dove: 0, Structure and Dynamics, Oxford University Press,
H. M. Rosenberg: 0, The Solid State,
J.R. Hook and H.E. Hall: 0, Introductory Solid State Physics,
C. Kittel: 0, Introduction to Solid State Physics,

Other Resources

None

Programme or List of Programmes

AP	BSc in Applied Physics

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