Phys 446 Solid State Physics Last Lecture Dec 7th, 2007Course reviewI. Crystal structure, and symmetry Crystal: atoms are arranged so that their positions are periodic in all three dimensions. Crystal lattice - the periodic array of points.An atom or a group of atoms associated with every lattice point is called basis of the lattice In addition to periodicity, each lattice can have other symmetryproperties (inversion, mirror planes, rotation axes) 7 crystal systems and 14 Bravais lattices A particular combination of symmetry operations determined by symmetry of the basis and the symmetry of the Bravais lattice defines a point symmetry group. There are 32 point groups -crystal classes. Notation for crystallographic directions and planes: Miller indicesII. Interatomic forces and types of chemical bondsAttractive electrostatic interaction between electrons and nuclei –the force responsible for cohesion of solidsRepulsive interaction between atoms is primarily due to electrostatic repulsion of overlapping charge distributions and Pauli principleSeveral types of attractive forces: • Ionic crystals – electrostatic forces between "+" and "-" ions• Covalent bond: overlap of charge distributions with antiparallelspin• Metals: reduction of kinetic energy of electrons in free state compared to the localized state of a single atom• Secondary forces (Van der Waals, hydrogen) become significant when the other bonds are impossible, e.g. in inert gasesPhysical properties are closely related to the type of bondingDiffraction of waves by crystal lattice Most methods for determining the atomic structure of crystals are based on scattering of particles/radiation (X-rays, electrons, neutrons). The wavelength of the radiation should be comparable to a typical interatomic distance (few Å=10-8cm) The Bragg law: condition for a sharp peak in the intensity of the scattered radiation. Various statements of the Bragg condition: 2d·sinθ = mλ ; ∆k = G ;2k·G = G2 Constricting the reciprocal lattice from the direct lattice. Reciprocal lattice is defined by primitive vectors:  A reciprocal lattice vector has the form G = hb1 + kb2+ lb3It is normal to (hkl) planes of direct lattice The set of reciprocal lattice vectors determines the possible scattering wave vectors for diffractionDiffraction, reciprocal lattices and Brillouin zones First Brillouin zone is the Wigner-Seitz primitive cell of the reciprocal lattice Only waves whose wave vector drawn from the origin terminates on a surface of the Brillouin zone can be diffracted by the crystal  Reciprocal lattices for common structures (e.g. sc ↔ sc; bcc ↔ fcc)Brillouin zones for these common structures Diffraction amplitude is determined by a product of several factors: atomic form factor, structural factor, Debye-Waller factor- Atomic scattering factor (form factor) reflects distribution of electronic cloud.- Structure factor – for lattices with a basis of several atoms- Debye-Waller factor – atomic vibrations are taken into account ∑++=jlwkvhuiajjjjefF)(2πElastic propertiesElastic properties are determined by forces acting on atoms when they are displaced from the equilibrium positionsFor small displacements, the potential is harmonic, and the restoring force is linear with respect to the displacement of atoms.Elastic properties are described by considering a crystal as a homogeneous continuum medium rather than a periodic array of atoms •Applied forces are described in terms of stressσ, •Displacements of atoms are described in terms of strainε.•Elastic constants C relate stressσand strainε, so thatklklijklijCεσ∑=(Hooke's law) Elastic waves sound velocity All crystal vibrational waves can be described by wave vectors within the first Brillouin zone in reciprocal space More than one atom in a unit cell – acoustic and optical vibrations.In general three-dimensional lattice with s atoms per unit cell there are 3s vibrational branches: 3 acoustic, 3s - 3 optical Phonon - the quantum of lattice vibration. Energy ħω; momentum ħq Density of states is important characteristic of lattice vibrations; It is related to the dispersionω = ω(q). Simplest case of isotropic solid, for one branch: Real density of vibrational states is more complicated2222xuCtuxeff∂∂=∂∂ρρeffCv =Lattice vibrationsdqdVqDωπω12)(22= Heat capacity is related to the density of states.  Debye model – good when acoustic phonon contribution dominates.At low temperatures givesCv∝T3 Einstein model - simple model for optical phonons (ω(q) = constant, DOS is approximated by a δ-function ) At high T both models lead to the Dulong-Petit law: Cv= 3NkB Phonon thermal conductivity  Anharmonism of potential energy is responsible for such effects as: • phonon-phonon interaction• thermal expansion• Pressure and temperature dependence of elastic constantsPhonon heat capacity, thermal conductivity, anharmonismCvlK31=Free electron model of metals  Free electron model – simplest way to describe electronic properties of metals: the valence electrons of free atoms become conduction electrons in crystal and move freely throughout the crystal.  Fermi energy - the energy of the highest occupied electronic level at T = 0 K; Density of states of 3D free electron gas:  Heat capacity of free electron gas at low temperatures kBT << EF :322232⎟⎟⎠⎞⎜⎜⎝⎛=VNmEFπ=3123⎟⎟⎠⎞⎜⎜⎝⎛=VNkFπ3123⎟⎟⎠⎞⎜⎜⎝⎛=VNmvFπ=ENEmVdEdNED2322)(212322=⎟⎠⎞⎜⎝⎛===π*2mneτσ−=Electrical conductivity:Thermal conductivity:Wiedemann-Franz law TLKσ=223⎟⎠⎞⎜⎝⎛=ekLBπρ= ρi+ρph(T)The free electron model gives a good insight into many properties of metals, such as the heat capacity, thermal conductivity and electrical conductivity. However, it fails to explain a number of importantproperties and experimental facts, for example:• the difference between metals, semiconductors and insulators•It does not explain the occurrence of positive values of the Hall coefficient. •Also the relation between conduction electrons in the metal and the number of valence electrons in free atoms is not always correct.Bivalent and trivalent metals are consistently less conductive than the monovalent metals (Cu, Ag, Au)⇒ need a more accurate theory, which would be able to answer thesequestions – the band