Introduction to Solid State Physics

1. EDAYATHANKUDY G S PILLAY ARTS AND SCIENCE COLLAGE Prof. K.SENTHILARASAN SOLID STATE PHYSICS

2. INTRODUCTION • AIM OF SOLID STATE PHYSICS • WHAT IS SOLID STATE PHYSICS AND WHY DO IT? • CONTENT • REFERENCES EGSP

3. Aim of Solid State Physics • Solid 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.

4. Crystalline Solids • We 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.

5. What 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. EP364 SOLID STATE PHYSICS INTRODUCTION

6. What 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.

7. Solid state physics(SSP) is the applied physics • New 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.

8. Electrical resistivity of three states of solid matter • How can this be? After all, they each contain a system of atoms andespecially electrons of similar density.And the plot thickens: graphite is a metal, diamond is an insulatorand buckminster-fullerene is a superconductor. They are all just carbon!

9. Among our aims - understand whyone is a metal and one an insulator,and then the physical origin of themarked features. • Also think about thermal propertiesetc. etc.

10. CONTENT • Chapter 1. Crystal Structure • Chapter 2. X-ray Crystallography • Chapter 3. Interatomic Forces • Chapter 4. Crystal Dynamics • Chapter 5. Free Electron Theory

11. CHAPTER 1.CRYSTAL STRUCTURE • Elementary 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 Symmetry

12. CHAPTER 2. X-RAY CRYSTALLOGRAPHY • X-ray • Diffraction • Bragg equation • X-ray diffraction methods • Laue Method • Rotating Crystal Method • Powder Method • Neutron & electron diffraction

13. CHAPTER 3. INTERATOMIC FORCES • Energies of Interactions Between Atoms • Ionic bonding • NaCl • Covalent bonding • Comparision of ionic and covalent bonding • Metallic bonding • Van der waals bonding • Hydrogen bonding

14. CHAPTER 4. CRYSTAL DYNAMICS • Sound 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 vibrations

15. CHAPTER 5. FREE ELECTRON THEORY • Free electron model • Heat capacity of free electron gas • Fermi function, Fermi energy • Fermi dirac distribution function • Transport properties of conduction electrons

16. REFERENCES • Core 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

17. CHAPTER 1CRYSTAL STRUCTURE Elementary Crystallography Typical Crystal Structures Elements Of Symmetry

18. Objectives • By 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 shapes Crystal Structure

19. matter

20. Gases • Gases 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.

21. Liquids and Liquid Crystals + - + - + - + - + - + - + - • Similar 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. Crystal Structure

22. Crytals • Solids 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. Crystal Structure 22

23. SOLID MATERIALS CRYSTALLINE POLYCRYSTALLINE AMORPHOUS (Non-crystalline) Single Crystal ELEMENTARY CRYSTALLOGRAPHY Crystal Structure 23

24. Types of Solids • Single 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. Crystal Structure 24

25. Crystalline Solid • Crystalline 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. Crystal Structure 25

26. Crystalline Solid • Single crystalhas 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 symmetry Single Pyrite Crystal Amorphous Solid Single Crystal Crystal Structure 26

27. Polycrystalline Solid • Polycrystalis 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 Polycrystalline Pyrite form (Grain) Crystal Structure 27

28. Amorphous Solid • Amorphous (Non-crystalline) Solidis 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. Crystal Structure 28

29. Departure From Perfect Crystal • Strictly 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. Crystal Structure 29

30. CRYSTALLOGRAPHY What is crystallography? The branch of science that deals with the geometric description of crystals and their internal arrangement. Crystal Structure 30

31. Crystallography • 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. Crystal Structure 31

32. ELEMENTARY CRYSTALLOGRAPHY • A basic knowledge of crystallography is essential for solid state physicists; • to specify any crystal structure and • to classify the solids into different types accordingto 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. Crystal Structure 32

33. CRYSTAL LATTICE What is crystal (space) lattice? In crystallography, only the geometrical properties of the crystalare of interest, therefore one replaces each atom by a geometricalpoint located at the equilibrium position of that atom. Crystal lattice and structure of Platinum Platinum surface Platinum (scanning tunneling microscope) Crystal Structure 33

34. y B C D E α b O x a A Crystal Lattice • An infinite array of points in space, • Each point has identical surroundings to all others. • Arrays are arranged exactly in a periodic manner. Crystal Structure 34

35. Crystal Structure • Crystal 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 + Basis Crystal Structure 35

36. A two-dimensional Bravais lattice with different choices for the basis

37. y y B C D E B C D α b F G b O x x a A O a A Basis • A group of atoms which describe crystal structure E H a) Situation of atoms at the corners of regular hexagons b) Crystal lattice obtained by identifying all the atoms in (a) Crystal Structure 37

38. Crystal structure • Don'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 + Basis Crystal Structure 38

39. Crystal Lattice Bravais Lattice (BL) Non-Bravais Lattice (non-BL) • All atoms are of the same kind • All lattice points are equivalent • Atoms can be of different kind • Some lattice points are not equivalent • A combination of two or more BL Crystal Structure 39

40. Nb film Types Of Crystal Lattices 1) 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. Crystal Structure 40

41. Honeycomb Types Of Crystal Lattices 2) Non-Bravais Lattice Not only the arrangement but also the orientation must appearexactly the same from every point in a bravais lattice. • 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. Crystal Structure 41

42. 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, bare known as lattice vectors and (n1, n2) is a pair of integers whose values depend on the lattice point. P Point D(n1, n2) = (0,2) Point F (n1, n2) = (0,-1) Crystal Structure 42

43. Lattice Vectors – 2D • 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. Crystal Structure 43

44. 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 ware arbitrary integers. r’ = r + ua + vb + wc      (1) Crystal Structure 44

45. Five Bravais Lattices in 2D Crystal Structure 45

46. S S S S S b S S S S a S S S S Unit Cell in 2D • The smallest component of the crystal (group of atoms, ions or molecules), which when stacked together with pure translational repetition reproduces the whole crystal. 2D-Crystal S S Unit Cell Crystal Structure 46

47. S S S S Unit Cell in 2D • The smallest component of the crystal (group of atoms, ions or molecules), which when stacked together with pure translational repetition reproduces the whole crystal. 2D-Crystal The choice of unit cell is not unique. b a Crystal Structure 47

48. 2D Unit Cell example -(NaCl) We define lattice points ; these are points with identical environments Crystal Structure 48

49. Choice of origin is arbitrary - lattice points need not be atoms - but unit cell size should always be the same. Crystal Structure 49

50. This is also a unit cell - it doesn’t matter if you start from Na or Cl Crystal Structure 50