SOLID STATE PHYSICS

nullSOLID STATE PHYSICSSOLID STATE PHYSICSINTRODUCTIONINTRODUCTIONAIM OF SOLID STATE PHYSICS WHAT IS SOLID STATE PHYSICS AND WHY DO IT? CONTENT REFERENCEShttp://www.tjldzm.com led路灯、无极灯、led景观灯 http://www.u51688.com http://www.qiwhy.comAim of Solid State PhysicsEP364 SOLID STATE PHYSICS INTRODUCTIONAim of Solid State PhysicsSolid state physics (SSP) explains the properties of solid materials as found on earth. The properties are expected to follow from Schrödinger’s eqn. for a collection of atomic nuclei and electrons interacting with electrostatic forces. The fundamental laws governing the behaviour of solids are known and well tested. Crystalline SolidsEP364 SOLID STATE PHYSICS INTRODUCTIONCrystalline SolidsWe will deal with crystalline solids, that is solids with an atomic structure based on a regular repeated pattern. Many important solids are crystalline. More progress has been made in understanding the behaviour of crystalline solids than that of non-crystalline materials since the calculation are easier in crystalline materials. What is solid state physics?EP364 SOLID STATE PHYSICS INTRODUCTIONWhat is solid state physics?Solid state physics, also known as condensed matter physics, is the study of the behaviour of atoms when they are placed in close proximity to one another. In fact, condensed matter physics is a much better name, since many of the concepts relevant to solids are also applied to liquids, for example.What is the point?EP364 SOLID STATE PHYSICS INTRODUCTIONWhat is the point?Understanding the electrical properties of solids is right at the heart of modern society and technology. The entire computer and electronics industry relies on tuning of a special class of material, the semiconductor, which lies right at the metal-insulator boundary. Solid state physics provide a background to understand what goes on in semiconductors. Solid state physics (SSP) is the applied physicsEP364 SOLID STATE PHYSICS INTRODUCTIONSolid state physics (SSP) is the applied physicsNew technology for the future will inevitably involve developing and understanding new classes of materials. By the end of this course we will see why this is a non-trivial task. So, SSP is the applied physics associated with technology rather than interesting fundamentals. Electrical resistivity of three states of solid matterEP364 SOLID STATE PHYSICS INTRODUCTIONElectrical resistivity of three states of solid matterHow can this be? After all, they each contain a system of atoms and especially electrons of similar density. And the plot thickens: graphite is a metal, diamond is an insulator and buckminster-fullerene is a superconductor. They are all just carbon!null Among our aims - understand why one is a metal and one an insulator, and then the physical origin of the marked features. Also think about thermal properties etc. etc. CONTENTEP364 SOLID STATE PHYSICS INTRODUCTIONCONTENTChapter 1. Crystal Structure Chapter 2. X-ray Crystallography Chapter 3. Interatomic Forces Chapter 4. Crystal Dynamics Chapter 5. Free Electron TheoryCHAPTER 1. CRYSTAL STRUCTUREEP364 SOLID STATE PHYSICS INTRODUCTIONCHAPTER 1. CRYSTAL STRUCTUREElementary Crystallography Solid materials (crystalline, polycrystalline, amorphous) Crystallography Crystal Lattice Crystal Structure Types of Lattices Unit Cell Directions-Planes-Miller Indices in Cubic Unit Cell Typical Crystal Structures (3D– 14 Bravais Lattices and the Seven Crystal System) Elements of SymmetryCHAPTER 2. X-RAY CRYSTALLOGRAPHYEP364 SOLID STATE PHYSICS INTRODUCTIONCHAPTER 2. X-RAY CRYSTALLOGRAPHYX-ray Diffraction Bragg equation X-ray diffraction methods Laue Method Rotating Crystal Method Powder Method Neutron & electron diffraction CHAPTER 3. INTERATOMIC FORCESEP364 SOLID STATE PHYSICS INTRODUCTIONCHAPTER 3. INTERATOMIC FORCESEnergies of Interactions Between Atoms Ionic bonding NaCl Covalent bonding Comparision of ionic and covalent bonding Metallic bonding Van der waals bonding Hydrogen bonding CHAPTER 4. CRYSTAL DYNAMICSEP364 SOLID STATE PHYSICS INTRODUCTIONCHAPTER 4. CRYSTAL DYNAMICSSound wave Lattice vibrations of 1D cystal Chain of identical atoms Chain of two types of atoms Phonons Heat Capacity Density of States Thermal Conduction Energy of harmonic oscillator Thermal energy & Lattice Vibrations Heat Capacity of Lattice vibrationsCHAPTER 5. FREE ELECTRON THEORYEP364 SOLID STATE PHYSICS INTRODUCTIONCHAPTER 5. FREE ELECTRON THEORYFree electron model Heat capacity of free electron gas Fermi function, Fermi energy Fermi dirac distribution function Transport properties of conduction electrons REFERENCESEP364 SOLID STATE PHYSICS INTRODUCTIONREFERENCESCore book: Solid state physics, J.R.Hook and H.E.Hall, Second edition (Wiley) Other books at a similar level: Solid state physics, Kittel (Wiley) Solid state physics, Blakemore (Cambridge) Fundamentals of solid state physics, Christman (Wiley) More advanced: Solid state physics, Ashcroft and Mermin CHAPTER 1 CRYSTAL STRUCTURECHAPTER 1 CRYSTAL STRUCTUREElementary Crystallography Typical Crystal Structures Elements Of SymmetryObjectivesCrystal Structure*ObjectivesBy the end of this section you should: be able to identify a unit cell in a symmetrical pattern know that there are 7 possible unit cell shapes be able to define cubic, tetragonal, orthorhombic and hexagonal unit cell shapesnullCrystal Structure*matterGasesCrystal Structure*GasesGases have atoms or molecules that do not bond to one another in a range of pressure, temperature and volume. These molecules haven’t any particular order and move freely within a container.Liquids and Liquid CrystalsCrystal Structure*Liquids and Liquid CrystalsSimilar to gases, liquids haven’t any atomic/molecular order and they assume the shape of the containers. Applying low levels of thermal energy can easily break the existing weak bonds. Liquid crystals have mobile molecules, but a type of long range order can exist; the molecules have a permanent dipole. Applying an electric field rotates the dipole and establishes order within the collection of molecules.CrytalsCrystal Structure*CrytalsSolids consist of atoms or molecules executing thermal motion about an equilibrium position fixed at a point in space. Solids can take the form of crystalline, polycrstalline, or amorphous materials. Solids (at a given temperature, pressure, and volume) have stronger bonds between molecules and atoms than liquids. Solids require more energy to break the bonds.nullCrystal Structure*ELEMENTARY CRYSTALLOGRAPHYTypes of SolidsCrystal Structure*Types of SolidsSingle crsytal, polycrystalline, and amorphous, are the three general types of solids. Each type is characterized by the size of ordered region within the material. An ordered region is a spatial volume in which atoms or molecules have a regular geometric arrangement or periodicity.Crystalline SolidCrystal Structure*Crystalline SolidCrystalline Solid is the solid form of a substance in which the atoms or molecules are arranged in a definite, repeating pattern in three dimension. Single crystals, ideally have a high degree of order, or regular geometric periodicity, throughout the entire volume of the material.Crystalline SolidCrystal Structure*Crystalline SolidSingle CrystalSingle Pyrite CrystalAmorphous SolidSingle crystal has an atomic structure that repeats periodically across its whole volume. Even at infinite length scales, each atom is related to every other equivalent atom in the structure by translational symmetryPolycrystalline SolidCrystal Structure*Polycrystalline SolidPolycrystalline Pyrite form (Grain)Polycrystal is a material made up of an aggregate of many small single crystals (also called crystallites or grains). Polycrystalline material have a high degree of order over many atomic or molecular dimensions. These ordered regions, or single crytal regions, vary in size and orientation wrt one another. These regions are called as grains ( domain) and are separated from one another by grain boundaries. The atomic order can vary from one domain to the next. The grains are usually 100 nm - 100 microns in diameter. Polycrystals with grains that are <10 nm in diameter are called nanocrystalline Amorphous SolidCrystal Structure*Amorphous SolidAmorphous (Non-crystalline) Solid is composed of randomly orientated atoms , ions, or molecules that do not form defined patterns or lattice structures. Amorphous materials have order only within a few atomic or molecular dimensions. Amorphous materials do not have any long-range order, but they have varying degrees of short-range order. Examples to amorphous materials include amorphous silicon, plastics, and glasses. Amorphous silicon can be used in solar cells and thin film transistors. Departure From Perfect CrystalCrystal Structure*Departure From Perfect CrystalStrictly speaking, one cannot prepare a perfect crystal. For example, even the surface of a crystal is a kind of imperfection because the periodicity is interrupted there. Another example concerns the thermal vibrations of the atoms around their equilibrium positions for any temperature T>0°K.As a third example, actual crystal always contains some foreign atoms, i.e., impurities. These impurities spoils the perfect crystal structure.CRYSTALLOGRAPHYCrystal Structure*CRYSTALLOGRAPHYWhat is crystallography? The branch of science that deals with the geometric description of crystals and their internal arrangement. CrystallographyCrystal Structure*Crystallography is essential for solid state physics Symmetry of a crystal can have a profound influence on its properties. Any crystal structure should be specified completely, concisely and unambiguously. Structures should be classified into different types according to the symmetries they possess. CrystallographynullCrystal Structure*A basic knowledge of crystallography is essential for solid state physicists; to specify any crystal structure and to classify the solids into different types according to the symmetries they possess. Symmetry of a crystal can have a profound influence on its properties. We will concern in this course with solids with simple structures.ELEMENTARY CRYSTALLOGRAPHYCRYSTAL LATTICECrystal Structure*CRYSTAL LATTICE What is crystal (space) lattice? In crystallography, only the geometrical properties of the crystal are of interest, therefore one replaces each atom by a geometrical point located at the equilibrium position of that atom.PlatinumPlatinum surfaceCrystal lattice and structure of Platinum(scanning tunneling microscope)nullCrystal Structure*An infinite array of points in space, Each point has identical surroundings to all others. Arrays are arranged exactly in a periodic manner.Crystal LatticeCrystal StructureCrystal Structure*Crystal StructureCrystal structure can be obtained by attaching atoms, groups of atoms or molecules which are called basis (motif) to the lattice sides of the lattice point.Crystal Structure = Crystal Lattice + BasisA two-dimensional Bravais lattice with different choices for the basisA two-dimensional Bravais lattice with different choices for the basisBasisCrystal Structure*EHb) Crystal lattice obtained by identifying all the atoms in (a)a) Situation of atoms at the corners of regular hexagonsBasis A group of atoms which describe crystal structureCrystal structureCrystal Structure*Crystal structureDon't mix up atoms with lattice points Lattice points are infinitesimal points in space Lattice points do not necessarily lie at the centre of atoms Crystal Structure = Crystal Lattice + BasisnullCrystal Structure*Types Of Crystal LatticesCrystal Structure*Types Of Crystal Lattices1) Bravais lattice is an infinite array of discrete points with an arrangement and orientation that appears exactly the same, from whichever of the points the array is viewed. Lattice is invariant under a translation. Types Of Crystal Lattices Crystal Structure* Types Of Crystal Lattices The red side has a neighbour to its immediate left, the blue one instead has a neighbour to its right. Red (and blue) sides are equivalent and have the same appearance Red and blue sides are not equivalent. Same appearance can be obtained rotating blue side 180º.2) Non-Bravais Lattice Not only the arrangement but also the orientation must appear exactly the same from every point in a bravais lattice.Translational Lattice Vectors – 2DCrystal Structure*Translational Lattice Vectors – 2D A space lattice is a set of points such that a translation from any point in the lattice by a vector; Rn = n1 a + n2 b locates an exactly equivalent point, i.e. a point with the same environment as P . This is translational symmetry. The vectors a, b are known as lattice vectors and (n1, n2) is a pair of integers whose values depend on the lattice point. PPoint D(n1, n2) = (0,2) Point F (n1, n2) = (0,-1)Lattice Vectors – 2DCrystal Structure*The two vectors a and b form a set of lattice vectors for the lattice. The choice of lattice vectors is not unique. Thus one could equally well take the vectors a and b’ as a lattice vectors.Lattice Vectors – 2DnullCrystal Structure*Lattice Vectors – 3D An ideal three dimensional crystal is described by 3 fundamental translation vectors a, b and c. If there is a lattice point represented by the position vector r, there is then also a lattice point represented by the position vector where u, v and w are arbitrary integers.   r’ = r + u a + v b + w c      (1) Five Bravais Lattices in 2DCrystal Structure*Five Bravais Lattices in 2DUnit Cell in 2DCrystal Structure*Unit Cell in 2DThe smallest component of the crystal (group of atoms, ions or molecules), which when stacked together with pure translational repetition reproduces the whole crystal.SSUnit Cell in 2DCrystal Structure*Unit Cell in 2DThe smallest component of the crystal (group of atoms, ions or molecules), which when stacked together with pure translational repetition reproduces the whole crystal.The choice of unit cell is not unique.ab2D Unit Cell example -(NaCl)Crystal Structure*2D Unit Cell example -(NaCl)We define lattice points ; these are points with identical environmentsChoice of origin is arbitrary - lattice points need not be atoms - but unit cell size should always be the same.Crystal Structure*Choice of origin is arbitrary - lattice points need not be atoms - but unit cell size should always be the same.This is also a unit cell - it doesn’t matter if you start from Na or ClCrystal Structure*This is also a unit cell - it doesn’t matter if you start from Na or Cl- or if you don’t start from an atomCrystal Structure*- or if you don’t start from an atomThis is NOT a unit cell even though they are all the same - empty space is not allowed!Crystal Structure*This is NOT a unit cell even though they are all the same - empty space is not allowed!In 2D, this IS a unit cell In 3D, it is NOTCrystal Structure*In 2D, this IS a unit cell In 3D, it is NOTWhy can't the blue triangle be a unit cell?Crystal Structure*Why can't the blue triangle be a unit cell?Unit Cell in 3DCrystal Structure*Unit Cell in 3DnullCrystal Structure*Unit Cell in 3DThree common Unit Cell in 3D Crystal Structure*Three common Unit Cell in 3D nullCrystal Structure*Body centered cubic(bcc) Conventional ≠ Primitive cellSimple cubic(sc) Conventional = Primitive cell The Conventional Unit CellCrystal Structure*The Conventional Unit CellA unit cell just fills space when translated through a subset of Bravais lattice vectors. The conventional unit cell is chosen to be larger than the primitive cell, but with the full symmetry of the Bravais lattice. The size of the conventional cell is given by the lattice constant a.nullCrystal Structure*Primitive and conventional cells of FCCnullPrimitive and conventional cells of BCCPrimitive Translation Vectors:Primitive and conventional cellsCrystal Structure*Simple cubic (sc): primitive cell=conventional cell Fractional coordinates of lattice points: 000, 100, 010, 001, 110,101, 011, 111Primitive and conventional cellsBody centered cubic (bcc): conventional ¹primitive cell Fractional coordinates of lattice points in conventional cell: 000,100, 010, 001, 110,101, 011, 111, ½ ½ ½ Primitive and conventional cellsCrystal Structure*Body centered cubic (bcc): primitive (rombohedron) ¹conventional cell Face centered cubic (fcc): conventional ¹ primitive cell Fractional coordinates: 000,100, 010, 001, 110,101, 011,111, ½ ½ 0, ½ 0 ½, 0 ½ ½ ,½1 ½ , 1 ½ ½ , ½ ½ 1Primitive and conventional cellsPrimitive and conventional cells-hcpCrystal Structure*Hexagonal close packed cell (hcp): conventional =primitive cell Fractional coordinates: 100, 010, 110, 101,011, 111,000, 001Primitive and conventional cells-hcpnullCrystal Structure* The unit cell and, consequently, the entire lattice, is uniquely determined by the six lattice constants: a, b, c, α, β and γ. Only 1/8 of each lattice point in a unit cell can actually be assigned to that cell. Each unit cell in the figure can be associated with 8 x 1/8 = 1 lattice point. Unit CellPrimitive Unit Cell and vectorsCrystal Structure*A primitive unit cell is made of primitive translation vectors a1 ,a2, and a3 such that there is no cell of smaller volume that can be used as a building block for crystal structures. A primitive unit cell will fill space by repetition of suitable crystal translation vectors. This defined by the parallelpiped a1, a2 and a3. The volume of a primitive unit cell can be found by V = a1.(a2 x a3) (vector products) Cubic cell volume = a3Primitive Unit Cell and vectorsPrimitive Unit CellCrystal Structure*The primitive unit cell must have only one lattice point. There can be different choices for lattice vectors , but the volumes of these primitive cells are all the same. P = Primitive Unit Cell NP = Non-Primitive Unit CellPrimitive Unit CellWigner-Seitz MethodCrystal Structure*Wigner-Seitz MethodA simply way to find the primitive cell which is called Wigner-Seitz cell can be done as follows; Choose a lattice point. Draw lines to connect these lattice point to its neighbours. At the mid-point and normal to these lines draw new lines. The volume enclosed is called as a Wigner-Seitz cell.Wigner-Seitz Cell - 3DCrystal Structure*Wigner-Seitz Cell - 3DLattice Sites in Cubic Unit CellCrystal Structure*Lattice Sites in Cubic Unit CellCrystal DirectionsCrystal Structure*Crystal DirectionsFig. Shows [111] direction We choose one lattice point on the line as an origin, say the point O. Choice of origin is completely arbitrary, since every lattice point is identical. Then we choose the lattice vector joining O to any point on the line, say point T. This vector can be written as; R = n1 a + n2 b + n3c To distinguish a lattice direction from a lattice point, the triple is enclosed in square brackets [ ...] is used.[n1n2n3] [n1n2n3] is the smallest integer of the same relative ratios.ExamplesCrystal Structure*X = ½ , Y = ½ , Z = 1 [½ ½ 1] [1 1 2]ExamplesNegative directionsCrystal Structure*Negative directionsWhen we write the direction [n1n2n3] depend on the origin, negative directions can be written as R = n1 a + n2 b + n3c Direction must be smallest integers. Y directionExamples of crystal directionsCrystal Structure*X = -1 , Y = -1 , Z = 0 [110]Examples of crystal directionsX = 1 , Y = 0 , Z = 0 [1 0 0]ExamplesCrystal Structure*ExamplesX =-1 , Y = 1 , Z = -1/6 [-1 1 -1/6] [6 6 1]We can move vector to the origin.Crystal Planes Crystal Structure*Crystal Planes Within a crystal lattice it is possible to identify sets of equally spaced parallel planes. These are called lattice planes. In the figure density of lattice points on e