Solid State Physics PHYS 40352 by Mike Godfrey Spring 2012 Last changed on May 22, 2017 ii Contents Preface v 1 Crystal structure 1 1.1 Lattice and basis . .1 1.1.1 Unit cells . .2 1.1.2 Crystal symmetry . .3 1.1.3 Two-dimensional lattices . .4 1.1.4 Three-dimensional lattices . .7 1.1.5 Some cubic crystal structures ................................ 10 1.2 X-ray crystallography . 11 1.2.1 Diffraction by a crystal . 11 1.2.2 The reciprocal lattice . 12 1.2.3 Reciprocal lattice vectors and lattice planes . 13 1.2.4 The Bragg construction . 14 1.2.5 Structure factor . 15 1.2.6 Further geometry of diffraction . 17 2 Electrons in crystals 19 2.1 Summary of free-electron theory, etc. 19 2.2 Electrons in a periodic potential . 19 2.2.1 Bloch’s theorem . 19 2.2.2 Brillouin zones . 21 2.2.3 Schrodinger’s¨ equation in k-space . 22 2.2.4 Weak periodic potential: Nearly-free electrons . 23 2.2.5 Metals and insulators . 25 2.2.6 Band overlap in a nearly-free-electron divalent metal . 26 2.2.7 Tight-binding method . 29 2.3 Semiclassical dynamics of Bloch electrons . 32 2.3.1 Electron velocities . 33 2.3.2 Motion in an applied field . 33 2.3.3 Effective mass of an electron . 34 2.4 Free-electron bands and crystal structure . 35 2.4.1 Construction of the reciprocal lattice for FCC . 35 2.4.2 Group IV elements: Jones theory . 36 2.4.3 Binding energy of metals . 37 2.4.4 Jones theory of Group V elements . 39 2.4.5 Structure of alloys . 40 2.5 Cyclotron resonance . 41 2.5.1 Effective mass tensor . 41 2.5.2 Calculation of the cyclotron frequency . 43 2.5.3 Cyclotron resonance in metals . 45 iii iv CONTENTS 2.5.4 Magnetic breakthrough: failure of the semiclassical approximation . 45 3 Magnetism 47 3.1 Electrons in a magnetic field . 47 3.1.1 Hamiltonians in classical mechanics . 47 3.1.2 Classical Hamiltonian of a charge in a magnetic field . 48 3.1.3 No magnetism in classical physics . 49 3.1.4 Quantum Hamiltonian of an electron in a magnetic field . 51 3.2 Magnetic quantities in thermodynamics . 51 3.3 Magnetism of a gas of free electrons . 52 3.3.1 Pauli spin paramagnetism of an electron gas . 53 3.3.2 Landau orbital diamagnetism of an electron gas . 53 3.3.3 Total magnetic response of the electron gas . 54 3.4 Magnetism of ions . 54 3.4.1 Hund’s rules . 55 3.4.2 Diamagnetism of closed-shell systems . 56 3.4.3 Paramagnetism of ions with partially filled shells . 56 3.5 Ordered magnetic states . 58 3.5.1 Dipolar interaction between spins . 59 3.5.2 Exchange interaction . 60 3.5.3 Exchange interaction between ions . 62 3.5.4 Why aren’t all magnets FM? . 62 3.5.5 The Heisenberg Hamiltonian . 63 3.6 Ferromagnetic groundstate and excitations . 63 3.6.1 Groundstate energy . 63 3.6.2 Spin-flip excitations and magnons . 64 3.7 Mean-field theory of the critical point . 66 Preface This document will eventually be a summary of the material taught in the course. In a few places you may find that derivations and examples of applying the results are not given, or are very much abbreviated. Conversely, it is often convenient to present some material in a different way from in the lectures. A little common sense should therefore be used when reading the notes and a good textbook consulted from time to time. I would suggest you come back to these notes from time to time, as they are (and are likely to remain) a work in progress. Please let me know if you find any typos or other slips, so I can correct them. v vi CONTENTS Chapter 1 Crystal structure In preparation: Much of the material in this chapter has been adapted, with permission, from notes and diagrams made by Monique Henson in 2013. You are strongly recommended to make sure that you understand and (where appropriate) can solve problems that involve: • the meaning of the terms lattice and motif [or basis] • simple cubic, face-centered cubic, and body-centered cubic lattices • lattice vectors and primitive lattice vectors; unit cells and primitive unit cells • diffraction of X rays by a crystal in terms of the Bragg equation and the reciprocal lattice vectors • the relation between lattice planes and reciprocal lattice vectors • be sure you know (and can derive) the reciprocal lattices for the simple cubic, FCC, and BCC lattices [these are useful for the kinds of problems that can be set on nearly-free electron theory and X-ray diffraction] • the “indexing” of X-ray diffraction patterns (i.e., given the Bragg angles q, find plausible G vectors, or lattice planes—you can get additional practice from past paper questions) 1.1 Lattice and basis A fundamental property of a crystalline solid is its periodicity: a crystal consists of a regular array of iden- tical “structural units”. The structural unit, which is called the basis [or motif ] can be simple, consisting of just one atom (as in sodium or ion), or complex, consisting of two or more atoms (as in diamond or in haemoglobin); see Fig. 1.1. The positions in space of these structural units define the points of a lattice.1 Although any real crystal has only a finite number of atoms, this number can be very large indeed (1023, say), so that it is often useful to imagine the crystal and its corresponding lattice to be infinite, extending through all space. The environment of every lattice point is identical in all respects, including orientation, so that we can get from one lattice point to any other by a simple translation. A vector connecting any two points of the lattice [and hence a possible translation vector] is called a lattice vector, and can be expressed in the form R = n1a1 + n2a2 + n3a3 , (1.1) where n1, n2, n3 can take any of the integer values 0, ±1, ±2, . 1Usually named after Bravais, who made a systematic study [ca. 1845] of the lattices possible in two and three dimensions. 1 2 CHAPTER 1. CRYSTAL STRUCTURE Figure 1.1: An example of a crystal structure. Atoms are represented by grey circles. Three atoms (shaded green) make up the motif (or basis) of the structure. Lattice points are indicated by blue dots. Any direction in the lattice can be specified by the coefficients [n1, n2, n3]. A set of non-coplanar lattice vectors a1, a2, a3 that can be used to generate all of the lattice vectors in accordance with (1.1) is said to be primitive. The choice of these vectors is not unique; in particular, they need not be the shortest possible lattice vectors. This is illustrated in Fig. 1.2. Figure 1.2: Two choices of primitive vectors for a 2D lattice. Primitive unit cells are also shown. The shaded region is a non-primitive cell with twice the area of a primitive cell. 1.1.1 Unit cells Any region of space that contains only one lattice point and can be translated by lattice vectors R to fill the whole of space without leaving gaps or forming overlaps is called a primitive unit cell. For example, 2 the parallelepiped whose edges are the primitive vectors a1, a2, a3 is always a primitive unit cell. There are many possible choices of unit cell. In Fig. 1.2 a1 and a2 are primitive vectors and the rectangle they span is a primitive unit cell. The 0 0 choice of vectors is not unique: we could equally well choose a1 and a2. They are the edges of a primitive cell (a parallelogram) of the same area as the rectangle with edges a1 and a2. These are just two simple examples – they don’t necessarily have to be that simple. Consider the shaded rectangle in Fig. 1.2. It is still a unit cell, as it could be repeated to fill the whole of space, but it is not a primitive cell as it does not have the smallest possible area. It is not always convenient to work with a primitive unit cell. For example, when discussing the lattices of the cubic system we generally use a unit cell that has the shape of a cube, even though for BCC and FCC this conventional cubic unit cell is non-primitive; see Figs. 1.13 and 1.14 later. 2A parallelepiped is illustrated in Fig. 1.11. 1.1. LATTICE AND BASIS 3 1.1.2 Crystal symmetry Lattice symmetries include translation by a lattice vector, discrete rotations (discussed below), and reflec- tions. Mirror symmetry (reflections) Mirror symmetry should be familiar enough not to need discussion. Rotational symmetry Figure 1.3: Conventional symbols for axes of pure rotational symmetry. These are shown shaded in this figure, as is usual in textbooks of crystallography, but elsewhere in this chapter (and in the lectures) the symbols have been left unfilled. The notation for axes of rotational symmetry is shown in Fig. 1.3. These are the only four possible rotational symmetries that are consistent with the periodicity of a crystal. Why? Consider an n-fold rotation axis A, in two dimensions, such that a rotation through an angle f = 2p/n about A maps the crystal onto itself. This is illustrated in Fig. 1.4. Now consider a second n-fold axis, B, that is related to axis A by a lattice translation, which we suppose to be the shortest possible.