Integrative lecture on Condensed Matter Physics José A. Flores Livas Fall -2019 1 Calendar of supporting lectures on CMP 2019 (Part I) Lectures colored in blue are in PART_I_integrative-CMP_fall_2019.pdf 1st Midterm exam Lectures colored in green are in PART_II_integrative-CMP_fall_2019.pdf 2nd Midterm exam 2 Outline Basis for Crystal structures (direct lattice) Reciprocal Lattice, Brillouin Zone in Crystals Wave Scattering by Crystals (XRD) Lattice vibrations (Quantum Modes: Phonons) Lattice vibration and specific heat of Solids 3 Who I am? Why am I here? Why are you here? Jose A. Flores-Livas Bio Contact: ➔ Researcher (RTD-a) at University of Rome Email: jose.fl[email protected] ➔ 2 postdocs Building Fermi, room 101 ➔ Phd in Physics Available all Mondays from 14:00 to 16:00. ➔ Undergrad and master Office phone: +39 (0)649913443 ➔ Interests: Design of quantum materials and climbing My website: http://www.fysik-aztek.net/ ● Progress shifted by 2-3 lectures from Prof. Caprara’s main lecture ● Depending on progress, problems will be introduced, and if necessary, more supporting material will be offered. ● Objective: is that you reach a clear understanding of solid-state physics and eventually get perfect records! 4 Start: “Basis for crystals” ● Introduction ● Motivation to CMP For today’s lecture ● Catch up with some concepts: Crystals, Bravais, lattice, basis Series of question to gradually test progress and set up exercises. I made these slides from different sources, including books: Ashcroft, Kittel, Simon (Oxford) and public websites. Some of the figures were done by myself and others adapted from the internet. You can share, mutilate or remove slides from this pdf [in case let me know of typos or corrections]. Bear in mind: This material is intended as a guide for you to assimilate concepts. In no circumstance, it covers all the content that was given during the main lecture. 5 Introduction: What is Condensed Matter Physics Condensed matter physics is the field of physics that deals with the macroscopic and microscopic physical properties of matter. In particular, it is concerned with the “condensed” phases that appear whenever the number of constituents in a system is extremely large, and the interactions between the constituents are strong. The most familiar examples of condensed phases are solids and liquids, which arise from the electromagnetic forces between atoms. The term CMP was coined by Nobel Laureate P. W. Anderson Why Do We Study Condensed Matter Physics? ● Because it is the world around us (why a metal is metal? Why glass transparent? etc ● Because it is useful: iphone laptops, etc ● Because it is a Laboratory: best laboratory to study quantum physics. ● During the CMP course we will discuss the work of no fewer than 50 Nobel laureates! 6 Crystal Structure Definition: A Lattice is an infinite set of points defined by integer sums of a set of linearly independent primitive lattice vectors OR simply: “Pattern of repeats” IMPORTANT: in Ashcroft-Mermin it refer to this as a Bravais Lattice. This enables them to use the term Lattice to describe other things that we would not call a lattice (cf, the honeycomb). Lattice is more common, and more correct mathematically as well. A lattice is defined as integer sums of a set of In 2D, 3D, 4D.. : The choice of primitive lattice 7 primitive lattice vectors. vectors for a lattice is not unique It turns out that any periodic structure can be expressed as a lattice of repeating motifs. A cartoon of this statement is shown: Definition: A unit cell is a region of space such that when many identical units are stacked together it tiles (completely fills) all of space and reconstructs the full structure. One should be cautious however, that not all periodic arrangements of points are lattices. The honeycomb is not a lattice. Any takers why? < Do not to call this a hexagonal lattice. But a triangular lattice “a lattice of triangles” where each point has six nearest neighbor points. 8 Definition: A primitive unit cell for a periodic crystal is a unit cell containing only a single lattice point Wigner-Seitz cell: choose a lattice point and draw lines to all of its possible near neighbors (not just its nearest neighbors). Then draw perpendicular bisectors of all of these lines. The perpendicular bisectors bound the Wigner-Seitz cell The definition of the unit cell is never unique: All of these unit cells reconstruct the same crystal Sometimes it is useful to define a unit cell which is not primitive in order to have it simpler to work with. This is known as a conventional unit cell. Almost always these conventional unit cells are chosen so as to have orthogonal axes. 9 Some unit cells for the triangular lattice Eugene Wigner was yet another Nobel laureate who was another one of the truly great minds of the last century of physics. Frederick Seitz was far less famous, but gained notoriety in his later years by being a consultant for the tobacco industry and a prominent sceptic of global warming. He passed away in 2007. This Wigner-Seitz construction can be done on an irregular collection of points as well as on a periodic lattice. For such an irregular set of point the resulting construction is known as a Voronoi cell. A Voronoi diagram is a partitioning of a plane into regions based on distance to points in a specific subset of the plane. That set of points (called seeds, sites, or generators) is specified beforehand, and for each seed there is a corresponding region consisting of all points closer to that seed than to any other. These regions are called Voronoi cells. 10 Scheme for constructing such a Wigner-Seitz cell: “Choose a lattice point and draw lines to all of its possible near neighbors (not just its nearest neighbors). Then draw perpendicular bisectors of all of these lines. The perpendicular bisectors bound the Wigner-Seitz cell” > It is always true that the Wigner-Seitz construction for a lattice gives a primitive unit cell. Definition: The description of objects in the unit cell in terms of the reference point in the unit cell is known as a “basis” Left: A periodic structure in two dimensions. A unit cell is marked with the dotted lines. Right: A blow-up of the unit cell with the coordinates of the particles (possibly atoms) in the unit cell with respect to the reference point in the lower left hand corner. The basis is the description of the 11 atoms along with these positions. 11 of 14 Let us recap Lattice points in 2D: Is this a periodic pattern? Is this a Lattice? Is this a triangular Lattice? < Honeycomb pattern is made of triangular lattice 12 Basis for crystal Triangular Lattice Square Lattice In this way you can say that the positions of the atoms in the crystal are “the lattice plus the basis” 13 Lattices in Three dimensions The simplest lattice in three dimensions is the simple cubic lattice It is known as cubic “P” or cubic-primitive lattice Two further lattice systems that you will need to know are the “Face Centered Cubic (fcc)” and “Body Centered Cubic (bcc)” lattices. As a side note: In fact, real crystals of atoms are In terms of atoms as being like small spheres, packing spheres in either a bcc or fcc rarely simple cubic. lattice leaves much less open space between the spheres than packing the spheres in a simple cubic lattice. Correspondingly, these two lattices are realized much > To understand why this is so, think of an atom as a more frequently in nature. 14 small sphere. 14 of 14 The Body Centered Cubic (bcc) Lattice The body centered cubic (bcc) lattice is a simple cubic lattice where there is an additional point in the very center of the cube (is known as cubic-I ) ”Innenzentriert” (inner centered) Q: How many lattice points have the unit cell? Where are they located? Packing together these unit cells to fill space, we see that the lattice points of a full bcc lattice can be described as being points having coordinates [x, y, z] where either all three coordinates are integers [n1,n2,n3] times the lattice constant a, or all three are odd-half-integers times the lattice. Any integer that can be divided exactly by 2 is an even number --last digits are: 0, 2,4,6 or 8. Any integer that cannot be divided exactly by 2 is an odd number --last digits are: 1,3,5,7 or 9. 15 It is often convenient to think of the bcc lattice as a simple cubic lattice with a basis of two atoms per conventional cell. Let’s think of a simple cubic “decorated” with another “similar” point. For convenience we took, d1=[0, 0, 0] and d2=[a/2, a/2, a/2], in units of the lattice constant. Thus the points of the bcc lattice are written as: “A Lattice is an infinite set of vectors where addition of any two vectors in the set gives a third vector in the set. Now, we may ask why it is that this set of points forms a lattice? we can write the primitive lattice vectors of the bcc lattice as: with n1, n2 and n3 integers gives a point within our definition of the bcc lattice. (that the three coordinates are either all integer or all half-odd integer times the lattice constant). 16 Let’s look the local environment, every point in the lattice should be the same. (is it?) Examining the point in the center of the unit cell, we see that it has precisely 8 nearest neighbors in each of the possible diagonal directions.