SOLID STATE PHYSICS

Content

1. Crystal Structures
2. Crystal Binding
3. Interatomic Forces, Lattice Vibrations
4. Phonons
5. Free Electron Fermi Gas.
6. Energy bands
7. Semiconductor Crystals
8. Fermi Surfaces.
9. Appendix
10. Problems

 

CHAPTER 1:  Crystal Structures

 

Solid state physics is largely concerned with crystals and electrons in crystals, more specifically  with understanding the mechanical, thermal, magnetic, and optical properties of solid matter The study of solid State physics began in the early years of the 20^th century following the discovery of x—ray diffraction by crystals and the publication of a series of simple calculations and successful predictions of the properties of crystals.

        When a crystal grows in a constant environment, the form develops as if identical building blocks were added continuously .  The building blocks are atoms or groups of atoms, so that a crystal is a three—dimensional periodic array of atoms.

        This was known in the 18th century when mineralogists discovered that the index numbers of the directions of all laces of a crystal are exact integers. Only the arrangement of idenitical particles in a periodic array can account for the law of integral indices.

 

 

PERIODIC ARRAYS OF ATOMS

 

A crystal is a periodic array of atoms. An ideal crystal is constructed by the infinite repetition of identical structural units in space. In the simplest crystals the structural unit is a single atom, as in copper, silver, gold, iron, aluminum, and the alkali metals. But the smallest structural unit may comprise many atoms or molecules.

The structure of all crystals can be described in terms of a lattice, with a group of atoms attached to every lattice point. The group of atoms is called the basis; when repeated in space it forms the crystal structure.

 

Lattice Translation vectors

 

The lattice is defined by three fundamental translation vectors a₁, a₂, a₃ such that the atomic arrangement looks the same in every respect when viewed from the point r as when viewed from the point

                             r’= r + u₁a₁ + u₂a₂ + u₃a₃ ,                                                                                    (1)

where u₁ ,u₂ ,u₃ , are arbitrary integers. The set of points r’ defined by (1) for all  u₁ ,u₂ ,u₃ defines a lattice.

A lattice is a regular periodic array of points in space. (The analog in two dimensions is called a net) A lattice is a mathematical abstraction : the crystal structure is formed when a basis of atoms is attached identically to every lattice point. The logical relation is

                                    lattice + basis = crystal structure

 

Fig. 1.1

 

The lattice and the translation vectors a₁, a₂, a₃ are said to be primitive if any 2 points r,r’ from which the atomic arrangement looks the same always satisfy (1) with a suitable choice of  the integers u₁, u₂, u₃. With this definition of the primitive translation vectors, there is no cell of smaller volume that can serve as a building block for the crystal structure.

 

Fig.  1.2

 

A lattice translation operation is defined as the displacement of a crystal by a crystal translation vector

                                                                    T = u₁a₁ + u₂a₂ + u₃a₃ .                                      

Any two lattice points are connected by a vector of this fbrm.

The number of atoms in the basis may be one or more. The position of the centre of an atom j of the basis relative to the associated lattice point is:

 

r_j = x_ja₁ + y_ja₂ + z_ja₃

Primitive Lattice Cell

 

A primitive cell is a type of (unit) cell :  a minimum—volume cell.

The number of atoms in a primitive cell or primitive basis is always the same for a given crystal structure.There is always 1 lattice point per primitive cell. For a parallelepiped with lattice points at each of the 8 corners, each lattice point is shared among 8 cells, so that the total number of lattice points in the cell is 8 x 1/8 =1.  The volume of the parallelepiped with axes a₁, a₂, a₃ is:

 

V_c= |a₁.a₂ x a₃|

 

Many elements and quite a few compounds are crystalline at low enough temperatures, and many of the solid materials in our everyday life (like wood, plastics and glasses) are not crystalline. Nevertheless, typical solids state physics texts start with the discussion of crystals for a good reason: The treatment of a large number of atoms is immensely simplified if they are arranged into a periodic order.

  
Figure 1.3: Examples of crystals in two dimension. Dots, curved lines or shaded areas represent various molecules or atomic arrangements.

Figure 1.3  shows a few two-dimensional ``crystals''. All crystals have discrete translational symmetry: If displaced by a properly selected lattice vector , every atom moves to the position of an identical atom in the crystal.     Due to this translational symmetry, a crystal can be constructed by repeating the basis at every Bravais lattice point.   The basis is the ``building block" of the crystal. It may be simple, a spherical atom, or as complex as a DNA molecule. Sometimes we have to use a basis made up of two (or more) atoms, even if there is only one type of atom in the crystal (see the example in Figure 1.4).

Bravais Lattice

A Bravais lattice is an infinite array of discrete points with an arrangement and orientation that appears exactly the same viewed from any point of the array. A three-dimensional Bravais lattice consists of all points with position vectors R (lattice vectors , ) of the form:

such that every is an integer. The 's are the three primitive vectors of the Bravais lattice; in three dimensions they must have a nonzero product.       There are an infinite number of different choices for the primitive vectors of a given lattice. For example, ; ; and will describe the same lattice.   The lattice spacings are the lengths of the shortest possible set of primitive vectors. The primitive lattice vectors can be used to generate the lattice by varying the integers. The smallest parallelepiped with a lattice point at each corner is called the primitive unit cell of the crystal. There is only one lattice point per unit cell since the points at the eight corners are shared by eight adjacent unit cells. Crystallographers often describe a crystal in terms of a non-primitive unit cell which is larger than the primitive cell. For instance, a body-centered cubic crystal has a non-cubic primitive cell, but it is often described in terms of a cubic conventional cell which is twice the size of the primitive cell.  The 14 Bravais Lattices  are:

Fig.1.3a. The 14 Bravias lattices: Triclinic (P) :a ¹ b ¹ c, a ¹ ß ¹ g ¹ 90^o ; Monoclinic (P): a ¹ b ¹ c, a = g = 90^o, ß ¹ 90^o ; Monoclinic (C) ; a ¹ b ¹ c, a = g = 90^o, ß ¹ 90^o ; Orthorhombic (P) ; a ¹ b ¹ c, a = ß = g = 90^o; Orthorhombic (C) : a ¹ b ¹ c, a = ß = g = 90^o; Orthorhombic (I) :a ¹ b ¹ c, a = ß = g = 90^o; Orthorhombic (F) :a ¹ b ¹ c, a = ß = g = 90^o; Hexagonal (P) :a = b ¹ c, a = ß = 90^o, g = 120^o;  Rhombohedral (R): a = b = c, a = ß = g ¹ 90^o; Tetragonal (P) : a = b ¹ c, a = ß = g = 90^o; Tetragonal (I) :a = b ¹ c, a = ß = g = 90^o; Cubic (P) : a = b = c, a = ß = g = 90^o;  Cubic (I) :a = b = c, a = ß = g = 90^o; Cubic (F) :a = b = c, a = ß = g = 90^o

TABLE 1.1. Seven Lattice Systems and Fourteen Bravais Lattices

Lattice systems	Angles and basis vectors	Bravais lattices
Triclinic	b1 ≠ b2 ≠ b3	Simple
	α ≠ β ≠ γ
Monoclinic	b1 ≠ b2 ≠ b3	Simple, base-centered
	α = β = 90° ≠ γ
Orthorhombic	b1 ≠ b2 ≠ b3	Simple, base-centered, body-centered, face-centered
Tetragonal	b1 = b2 ≠ b3	Simple, body-centered
	α = β = γ = 90°
Trigonal	b1 = b2 = b3	Simple
	α = β = γ ≠ 90°
Hexagonal	b1 = b2 ≠ b3	Simple
	α = β = 90°, γ = 120°
Cubic	b1 = b2 = b3	Simple, body-centered, face-centered.
	α = β = γ = 90°

 

All three crystals in Figure 1.3 have the same Bravais lattice. Note that not all symmetric arrays of points are Bravais lattices! For example, Figure 1.4 shows a honeycomb lattice and a choice for its Bravais lattice and basis.  

  
Figure 1.4: Example of a regular array of points that is not a Bravais lattice (honeycomb lattice).

  In addition to the translational symmetry, most crystals also have other symmetries, including reflection, rotation, or inversion symmetry, or more complicated symmetry operations, like the combination of rotation and translation by a fraction of the lattice vector. In Figure 1.3 the three-fold rotational symmetry around the point P is common to all three crystals. The honeycomb lattice (Figure 1.3c) also has a ``mirror line'' m, while the other two crystals do not have this symmetry. A less trivial symmetry operation is mirroring the honeycomb lattice with respect of the line m', and then shifting it parallel to the line, until it overlaps with itself.

   The collection of symmetry operations forms a symmetry group. The important property that defines a symmetry group is the relationship between the symmetry elements - i.e., what happens if two symmetry operations are applied subsequently. In the language of group theory, this relationship is described by the multiplication (direct product) table.The symmetry group can be represented in many ways (collections of matrices, symmetry operations of a simple geometric object, and so on). As long as the multiplication table is the same, we are dealing with the same group. The crystals in Figures 1.3a and 1.3b have equivalent symmetry groups, while some of the symmetries of the honeycomb lattice are different.

  When all possible symmetry operations are taken into account we talk about crystallographic space groups. Any given three-dimensional crystal belongs to one of the 230 possible crystallographic space groups. (Two-dimensional crystals are much simpler; there are only 17 inequivalent ``crystallographic plane groups''.) The symmetries are often identified by the name of a representative material, like ``sodium chloride structure'', ``diamond structure'', ``wurzite (or zincblende, zinc sulfide) structure'', and so on. More sophisticated group theoretical notations are used by crystallographers. For a complex structure the identification of the symmetry group may be a rather nontrivial task.    

A subset of symmetry operations that leaves at least one point invariant makes up the crystallographic point group.     There are 32 different crystallographic point groups in three dimensions, and 10 in two dimensions. Considering the examples in Figure 1.3, the rotations around P and the mirror line m are point group symmetries, but the combination of mirroring around m' and the subsequent shift is not a point group operation.

      Sorting out the symmetries of the Bravais lattices is much simpler. There are 14 different space groups for three-dimensional Bravais lattices, including the simple cubic (sc), face centered cubic (fcc), body centered cubic (bcc), simple tetragonal, body centered tetragonal, and others. Figure 1.5 shows all possible Bravais lattices in two dimensions. It is important to emphasize that the symmetries of the Bravais lattice are intimately related to the symmetries of the original lattice. For example, the three-fold rotational symmetry of the honeycomb lattice results in the requirement that its Bravais lattice must have three-fold rotational invariance (which leaves the hexagonal lattice as the only choice, see Figure 1.4).