UNIT I SOLID STATE PHYSICS

UNIT I SOLID STATE PHYSICS

CHAPTER 1 CRYSTAL STRUCTURE 1.1 INTRODUCTION When two atoms are brought together, two kinds of forces: attraction and repulsion come into play. The force of attraction increases rapidly as the separation between atoms, decreases. At small distances the force of repulsion comes into being and increases more rapidly with further decrease in interatomic distance. At distance r = r 0, attractive force becomes equal to the repulsive force and at this distance the interaction energy acquires minimum value and the atoms form a bound state. The minimum energy corresponds to the equilibrium-state. When a large number of atoms are brought together, the atoms in equilibrium-state, arrange themselves to form a closed pack structure, called solid. Thus, the forces of interaction between the structural particles (atoms, ions, group of atoms) are responsible for the existence of solids. The force of attraction between the particles prevents them from flying apart and the force of repulsion prevents them from merging. Fig. 1.1.1 Variation of inter-atomic force and potential energy with distance Solids occur in two forms: Crystalline and amorphous. A crystalline solid is one in which atoms or other structural units are arranged in an orderly repetitive array. That is, a crystalline solid exhibits long-range orders.

4 Introduction to Modern Physics II Amorphous (non-crystalline) solids do not show long range periodicity. An amorphous substance, such as glass, is actually regarded as a super-cooled liquid that differs from ordinary liquid in its physical properties. Usually viscosity is used to differentiate a liquid from amorphous solid. If the coefficient of viscosity is less than 10 15 poise, it is called liquid, if it is greater than this, it is a solid. The transition from liquid to amorphous-state and vice-versa is gradual whereas the transition between a liquid and crystalline solid is sharp. 1.2 SPACE LATTICE OR CRYSTAL LATTICE Space lattice is a geometrical abstraction. To have its meaning, imagine a network of straight lines constructed in such a way that it divides the entire space into identical volumes. The points of intersection of these lines are called lattice points. The network of points in three dimensions, in which the surrounding of each point is identical with the surrounding of other points, is called space lattice. By associating each lattice point with a single atom or group of atoms, called basis, crystal structure results. Thus, for every crystal there is a network of lattice points, which are occupied by either a single atom or group of atoms. Fig. 1.2.1 Space lattice and fundamental vectors a, b and c In 1848, a Russian mathematician Bravais showed that there are just 14 ways of arranging points in space lattice such that all the lattice points have exactly the same surroundings. Such lattices are called Bravais lattices. From these 14 space lattices, an unlimited number of different crystal structures can be made. Fig. 1.2.2 Two dimensional space lattice spanned by basis vectors a and b

Crystal Structure 5 For any type of lattice, there exists three fundamental translation vectors a, b, and c, not lying in a plane, in terms of which any lattice point can be specified. A translation operation T is defined by T = n 1 a + n 2 b + n 3 c...(1.2.1) where n 1, n 2, n 3 are integers. Thus means that by applying the operation T to a point r we reach another point r', which has the same environment as point r. r' = r + T = r + n 1 a + n 2 b + n 3 c...(1.2.2) The set of points r' given by Eqn. (2) for all possible values the integers n 1, n 2, and n 3 defines the space lattice. 1.3 UNIT CELL A parallelepiped formed by fundamental vectors a, b and c of space lattice is called unit cell. Each crystal is built up of a repetitive stacking of unit cells each identical in size, shape and orientation. The unit cell with basis is thus the fundamental building block of the crystal. A unit cell is always drawn with lattice point at each corner, but there may also be lattice points at the center of some of the faces or at the body center of the cell. Although there are infinite number of ways of choosing the unit cell, but the simplest way is conventionally accepted. Primitive Cell A unit cell with lattice points at its corners only is called primitive cell. Thus, there is only one lattice point to each primitive cell. It is also defined as the smallest unit cell in volume of space lattice. Basis vectors: A set of linearly independent vectors a, b, c, which can be used to define a unit cell, are called basis vectors. Primitive basis vectors: A set of linearly independent vectors that define a primitive cell is called primitive basis vectors. 1.4 PARAMETER OF A UNIT CELL A unit cell is specified by six quantities: three edges a, b, c and three angles a, b, g defined in the figure. These quantities are termed the parameters of the unit cell. In crystallography, the distances are expressed in terms of a, b, c. The 14 space lattices with conventional unit cell, some by their primitive cells and others by non-primitive cells are shown in the Fig. (1.4.2). The 14 lattices may be grouped into 7 crystal systems, each of which has in common certain characteristic symmetry elements. Fig. 1.4.1 Parameters of a unit cell

6 Introduction to Modern Physics II Fig. 1.4.2 14 Bravais space lattices with conventional unit cell

Crystal Structure 7 Table 1.4.1: Seven Crystal Systems System Bravais lattice Symbol Unit cell characteristics 1 Triclinic Simple P a ¹ b ¹ c, a ¹ b ¹ g ¹ 90 2 Monoclinic Simple P a ¹ b ¹ c, a = b = 90 0 ¹ g Base centered C 3 Orthorhombic Simple P a ¹ b ¹ c, a = b = g = 90 Base centered C Body centered I Face centered F 4 Tetragonal Simple P a = b ¹ c, a = b = g = 90 Body centered I 5 Cubic Simple P a = b = c, a = b = g = 90 Body centered I Face centered F 6 Hexagonal Simple P a = b ¹ c, a = 120, b = g = 90 7 Trigonal Simple P a = b = c, a = b = g ¹ 90 (Rhombohedral) Fig. 1.4.3 Primitive (rhombohedral) and non-primitive (cubic) unit cells for f.c.c. lattice The crystallographic description of crystal requires specification of shape and size of unit cell and distribution of matter in it. The choice of unit cell is not unique. While choosing the unit cell, emphasis is given on the symmetry exhibited by it. If a non-primitive cell exhibits more symmetry

8 Introduction to Modern Physics II than a primitive cell, the former is preferred. For example, in F.C.C. lattice the simplest primitive cell is rhombohedral with three fundamental translation vectors, each of length L/Ö2, which make angles of 60 with respect to each other Fig. (1.4.3). A non-primitive cubic unit cell of length L exhibits strong symmetry and therefore, it preferred to primitive cell. The rhombohedral unit cell has volume (L 3 )/4 whereas the cubic unit cell has volume L 3. Fig. 1.4.4 Primitive and non-primitive (conventional) unit cell for b.c.c. lattice Similarly for B.C.C. lattice, because of strong symmetry shown by cubic unit cell, it is conventionally accepted in comparison to primitive rhombohedral unit cell, which can be constructed with primitive vectors of length ( LÖ3)/2, The primitive cell has volume (L 3 )/2 and non-primitive cubic unit cell has volume L 3. 1.5 SYMMETRY An object is said to have symmetry if it coincides with itself as a result of some kinds of its spatial movement or rotation through some angle about an axis. The larger the number of ways by which the object can be made to coincide with itself, the more symmetric it is said to be. By symmetry element we mean an operation by which equivalent points of the object or space lattice are brought into coincidence. All the possible symmetry operations on an object can be expressed as a linear combination of following four symmetry elements. (i) n-fold rotation axis (Symmetry axes): If a rotation of a lattice about an axis through an angle 2p/n brings a lattice into a position indistinguishable from its initial position, then the axis is a n-fold rotation symmetry axis. The possible values of n are 1, 2, 3, 4 and 6. Five-fold rotational symmetry in a crystal lattice is impossible. Obviously any axis of a lattice is a one-fold symmetry axis.

Crystal Structure 9 (ii) Plane of symmetry: This type of symmetry exists if one half of the lattice coincides with itself as a result of the mirror reflection of its points in a certain plane. This plane is called the symmetry plane of the lattice. (iii) Inversion center (center of symmetry): A crystal lattice is said to possess the center of symmetry if it coincides with itself upon inversion through a certain point. This point is called the center of symmetry. By inversion operation the position vector r of a lattice point is converted into r. (iv) Rotation-inversion axis: If a lattice remains unchanged upon rotation about an axis through angle 2p/n, n = 1, 2, 3, 4 and 6, followed by inversion about a lattice point through which the rotation axis passes, it is said to have rotation-inversion axis. This kind of symmetry is indicated in the figure. The point A takes the position A' upon rotation through angle p. Upon inversion about point O, the point A' goes to A". Thus, A and A'' are related by two-fold rotation-inversion axis. Fig. 1.5.1 Two-fold rotation-inversion axis 1.6 MILLER INDICES A plane in a crystal is specified in terms of Miller indices. To find the Miller indices of a plane, following procedure is used. (i) Determine the intercepts of the plane on crystal axes in terms of fundamental vectors a, b and c. (ii) Take the reciprocal of these numbers in order. Reduce them to three smallest integers having the same ratio. (iii) Enclose these numbers in parentheses as (h k l). The Miller indices (h k l) denote a set of parallel planes. For example consider a plane with intercepts 4a, 6b and 3c on the crystal axes. In terms of axial units, the intercepts are 4, 6, 3. The reciprocals of these numbers are 1, 1, 1. By multiplying 4 6 3

10 Introduction to Modern Physics II each of these numbers with lowest common multiple 12, these are reduced to smallest integers 3, 2, 4. The Miller indices of the plane are (324). Fig. 1.6.1 Miller indices of a plane cutting intercepts 4a, 6b, 3c on crystallographic axes If a plane intersects the crystal axes on the negative side of origin, the corresponding index is negative. By convention a minus sign is placed above the corresponding index. A plane intercepting the crystal axes at 4a, 2b, c, has Miller indices (12 4). Miller indices of some important planes of cubic lattices are shown in the Fig. (1.6.2). Fig. 1.6.2 Miller indices of some important planes of cubic lattices