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Introduction to Solid State Physics

Introduction to Solid State Physics
Introduction to Solid State Physics PY3PO3 Prof. Igor Shvets Slide 1 Lecture 1Syllabus  Lattice structure: lattice with a basis.  Lattice structures of common chemical elements.  Concept of Bravais lattice, definition and examples.  Primitive vectors of Bravais lattice.  Coordination number.  Primitive/Conventional unit cell.  WignerSeitz primitive cell.  Examples of common crystal structures.  Bodycentered cubic lattice.  Facecentered cubic lattice.  Crystal systems  Reciprocal lattice.  First Brillouin zone.  Lattice planes and Miller indices. Slide 2 Lecture 1Syllabus  Determination of crystal structures by Xray diffraction.  Bragg formulation of Xray diffraction by a crystal.  Von Laue formulation of Xray diffraction by a crystal.  Equivalence of Bragg and Von Laue formulations.  Diffractometers Symmetry, elements of point groups  Geometrical structure factor.  Atomic form factor. Crystal defects.  Points defects  Line defects  Stress and Strain. Slide 3 Lecture 1Recommended Reading  Solid State Physics Ashcroft Mermin, HoltSaunders • A great text for anyone with an interest in the subject.  Solid State Physics Hook Hall, Wiley • Useful text. Read as a compliment to Ashcroft or Elliott.  Introduction To Solid State Physics Kittel, Wiley • Covers a huge amount in basic detail.  The Physics and Chemistry of Solids Elliott, Wiley • Lateral reading. Quite Chemistry based. Good for problem solving.  Structure of materials De Graef, McHenry, Cambridge • Covers a huge amount in basic detail, good for problem solving. Slide 4 Lecture 1Crystals Solids can be categorised into either crystalline or non crystalline solids. This course deals with the structures found in crystalline solids i.e. crystals. A Crystal is a solid in which the constituent atoms, molecules, or ions are packed in a regularly ordered, repeating pattern extending in all three spatial dimensions. Gallium Crystal Slide 5 Lecture 1Crystals Despite an underlying crystalline structure the crystal itself may not appear regular in shape. A closer look at the substance reveals a repeating pattern. Slide 6 Lecture 1Examples Silicon (Si) Principle component of most semiconductors Structure Diamond Cubic. More on that later Slide 7 Lecture 1Examples Crystalline SiO 2 Silicon Dioxide or Silica with a definite crystalline structure. Structure Each Silicon atom is surrounded by four Oxygen atoms. Silicon Dioxide is an example of Tetrahedral oxygen termination. Oxidation State Electron Configuration 2 6 O 2 He.2s .2p Si 4 Ne Slide 8 Lecture 1Examples Amorphous SiO (Silica gel) 2 Structure As with the crystalline SiO 2 Most silicon atoms have 4 bonds. Most oxygen atoms have 2 bonds. Silica gel is an example of a non crystalline solid. The local symmetry is the same as the crystalline SiO 2 However, translational symmetry is missing. Slide 9 Lecture 1Oxygen Termination The two forms of oxygen termination are Tetrahedral and Octahedral These refer to the structure of the oxide in relation to the original structure of the crystal. Tetrahedral Oxygen termination is when four oxygen atoms create a tetrahedron around the original atom e.g. Silicon Octahedral Oxygen termination is when six oxygen atoms create a octahedron around the original atom e.g. Aluminium Slide 10 Lecture 1Examples Aluminium Oxide (Al O ) 2 3 Structure Each Aluminium atom is surrounded by six Oxygen atoms. Aluminium Oxide is an example of Octahedral oxygen termination. Oxidation Electron State Configuration 2 6 O 2 He.2s .2p Al 3 Ne Slide 11 Lecture 1Examples Magnesium Oxide (MgO) Structure Structure is the same as that of Sodium Chloride. i.e. F.C.C. lattice with a two point basis Again more on that later Oxidation State Electron Configuration 2 6 O 2 He.2s .2p Mg 2 Ne Slide 12 Lecture 1Examples Magnetite ( Fe O ) 3 4 a a 32 oxygen anions form an F.C.C. lattice This is an example of Spinel structure. 8 tetrahedral interstices are occupied Lattice constant 3+ by Fe ions (Basic repeat unit) 16 octahedral interstices are occupied by a = 0.8397nm 3+ 2+ Fe and Fe ions in equal proportions Slide 13 Lecture 1Spinel Group Refers to a class of minerals with the general formulation 2+ 3+ 2 A B O 2 4 The oxide anions arrange in a cubic lattice with the A and B cations occupying the Tetrahedral and Octahedral sites. Possibilities for A and B include Magnesium, Zinc, Iron, Manganese, Aluminium, Chromium, Titanium and Silicon. In the case of Magnetite (Fe O ) the Iron is both A and B. That is, 3 4 A and B are the same metal under different charges. 3+ 3+ 2 2+ Fe (Fe Fe )O 4 Slide 14 Lecture 1High Performance Materials The majority of the worlds electricity supply is generated in power stations using steam turbines. Through the use of coal, nuclear power, etc. steam is generated and is passed through the giant steam turbines. The turbines rotate and generate electricity. Assuming the system can be considered T1 a Carnot Engine and using current 1 average values of temperatures in the T 2 system (T =35°C and T =540°C) the 1 2 efficiency is calculated to be 62. Slide 15 Lecture 1High Performance Materials As supplies of fossil fuels are diminishing, there is large interest in making the steam turbine a more efficient process for generating electricity. A way to increase this efficiency is to operate the system with a larger temperature gap, ie by making T larger. 2 Slide 16 Lecture 1High Performance Materials This is not as simple as it sounds. To T1 increase the efficiency by just 5 would 1 require an average operating temperature T 2 increase from 540°C to 660°C. The turbine blades themselves must be able to withstand these high temperatures without melting or buckling. This is where knowledge of the crystal structure of the materials used in their production is invaluable. Without a detailed analysis of the structure engineers and scientists would not be able to combat the problem of creating a more efficient system. Slide 17 Lecture 1Lattice When dealing with crystal structures it is best to firstly consider the mathematical idea of a lattice, without the notion of atoms or molecules at this stage. A Lattice is a regular, periodic array of points throughout an area (2D) or a space (3D). This picture shows one of many possible lattice types. Example of a 2D Lattice IMPORTANT NOTE: The lattice is the underlying pattern of the crystal. The crystal is being described by the lattice that can contain more than one atom/ion assigned to each point of the lattice. This is called a Basis. Slide 18 Lecture 1Basis You have a lattice with more than one atom/ion assigned to each lattice point. Atoms can be the same or different. Complex structures will have larger/more complex bases. All crystal structure consists of identical Consider a point of the lattice copies of the same Then apply to the rest of the lattice physical unit, called Now introduce a arbitrary basis of two atoms the basis, assigned to all the points of the lattice. Lattice + Basis = Crystal Slide 19 Lecture 1Basis While the basis is assigned to each point of the lattice there is nothing to say that it is anywhere near this point. Consider the same lattice and basis as before. Increase the separation between the basis and the lattice point. We discover that the same lattice is revealed when the new position of the basis is applied to each lattice point. Slide 20 Lecture 1Basis Even the most complex crystal structures can be broken down into a lattice and a basis. Lattice + Basis = Crystal However, there is no unique choice of Basis Each of these Basis, when combined with this lattice, will yield the same crystal. Slide 21 Lecture 1Symmetry of a Basis Consider a Basis that is itself symmetric and combine with the lattice to create a simple crystal structure; Crystal properties considered along … they are also the xaxis are independent of independent of direction along the direction (left or yaxis (up or right)… down). Slide 22 Lecture 1Symmetry of a Basis Now consider a basis that is not completely symmetric; … However the Crystal properties structure now has a considered along lower symmetry the xaxis are than before and is independent of not the same from direction (left or the top and right)… bottom. Slide 23 Lecture 1Symmetry of a Basis What if the basis is completely asymmetric The properties of the substance may no longer be independent of direction they are measured. Slide 24 Lecture 1Bravais Lattice The foundation of ANY Crystal structure is the Bravais lattice. Definitions of Bravais Lattice 1. Infinite array of discrete points that appear exactly the same from whichever of the points the array is viewed. 2. All the points with position vectors R = n a + n a + n a 1 1 2 2 3 3 Where; a , a , a are three vectors not all in the same plane 1 2 3 and n , n , n are integer values. 1 2 3 3. A discrete set of vectors, not all in the same plane, closed under vector addition or subtraction. The vectors used to define a Bravais lattice, a , a , a are called the primitive vectors. 1 2 3 , Slide 25 Lecture 1Bravais Lattice Pictured: A general twodimensional Bravais lattice of no particular symmetry. The vectors a and a are primitive 1 2 vectors. One of the most common three dimensional Cubic Bravais a 3 a lattices, the Simple Cubic lattice. 1 a 2 All a , a , a are of equal length 1 2 3 , and orthogonal. Slide 26 Lecture 1NonBravais Lattices Using the first definition of a Bravais lattice it is clear that the 2D Honeycomb Structure is not a Bravais lattice. Q The lattice does not appear exactly same P when viewed from P, Q and R R In 3D an example of a non Bravais lattice is the Hexagonal ClosePacked structure. The points of the middle layer are above the centres of the triangles in the layer below. E.g. if you double the vector from corner point to centre point, you will NOT arrive to another point of the lattice. Slide 27 Lecture 12D NonBravais Lattice with a Basis All NonBravais lattices can be created from a Bravais lattice with a basis. The red points form a simple 2D bravais lattice. Adding a twopoint basis to each lattice point… and it starts to look familiar. So if we go back to the case of Honeycomb structure. While it itself is not a Bravais lattice, it can be considered to be a Bravais lattice with a twopoint basis. Slide 28 Lecture 12D NonBravais Lattice with a Basis As mentioned before, the choice of basis is not unique, there are infinitely many bases to choose from. However, that does not mean all combinations of two points will work… Consider a lattice point and the basis shown. Apply this basis to the same lattice as before. If we superimpose the picture of the honeycomb over ours it is obvious that they do not match. Slide 29 Lecture 1Exercise  In each of the following cases indicate whether the structure is a Bravais lattice. a) Basecentered Cubic (Simple cubic with additional points in the centres of the horizontal faces 2) b) Sidecentered Cubic (Simple cubic with additional points in the centres of the vertical faces 4) c) Edgecentered Cubic (Simple cubic with additional points at the midpoints of the lines joining the simple cubic points 12) c) a) b) If it is a Bravais lattice give three primitive vectors. If not describe it as Bravais with the smallest possible basis. Slide 30 Lecture 1Questions/Problems  What is a Crystal  What is a Bravais Lattice  What is a Basis  What is meant by the Symmetry of a basis I would urge you to know the answers to these questions before next time. Good resources  Solid State Physics Ashcroft, Ch. 4  Introduction to Solid State Physics Kittel, Ch. 1  The Physics and Chemistry of Solids Elliott, Ch. 2  Solid State Physics Hook Hall, Ch. 1 Slide 31 Lecture 1
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