# PART 1 Introduction to Theory of Solids

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1 Elsevier UK Job code: MIOC Ch01-I :03p.m. Page:1 Trim: MM TS: Integra, India PART 1 Introduction to Theory of Solids

2 Elsevier UK Job code: MIOC Ch01-I :03p.m. Page:2 Trim: MM TS: Integra, India

3 Elsevier UK Job code: MIOC Ch01-I :03p.m. Page:3 Trim: MM TS: Integra, India CHAPTER 1 Periodic Structures Contents 1.1 Fundamental Types of Lattices Diffraction of Waves by a Crystal and the Reciprocal Lattice Reciprocal lattice vectors Brillouin Zones 8 References FUNDAMENTAL TYPES OF LATTICES In a solid crystal, the atoms are located in the same positions with respect to each other. Their relative location depends on the character of chemical bonding and the conditions for minimum energy. For crystals built of identical atoms the energy minimum is reached when all atoms have the same surrounding. Atomic positions are called lattice points and the whole lattice is called the crystal structure. In the case of a compound, the lattice points of the crystal structure are formed by molecules of the compound. The smallest part of the crystal structure is the primitive (elementary) cell. It can be created in many ways and is repeated translationally in each direction. The translational symmetry allows the description of the whole crystal by defining primitive axes a 1 a 2 a 3. Every lattice point can be described by the multiplicity of these axes: r n = n 1 a 1 + n 2 a 2 + n 3 a 3 (1.1) The choice of primitive basis vectors generating a given lattice is to some extent arbitrary but they have been selected to have the smallest possible length. 3

4 Elsevier UK Job code: MIOC Ch01-I :03p.m. Page:4 Trim: MM TS: Integra, India 4 Introduction to Theory of Solids a 2 a 1 (a) (b) FIGURE 1.1 Primitive cells: (a) defined by the primitive axes a 1 a 2 ; (b) Wigner Seitz primitive cell. This is the ideal crystal lattice. Every real lattice has limited dimensions and defects caused by impurities of other elements, dislocations and vacancies which influence the mechanical, electric and thermal properties of the lattice. For example, in good conductors the current carriers can be scattered on impurities, decreasing the conductivity, but in semiconductors impurities donating charges (donors) can increase the conductivity. In a real lattice, three primitive basis vectors create the primitive cell, which is the smallest of all elementary cells. Its volume is V = a 1 a 2 a 3. A primitive cell, containing only one lattice site, may also be chosen by drawing lines connecting a given lattice point with all nearby lattice points and then, at the midpoint of all these lines, drawing planes normal to these lines (see Fig. 1.1). The smallest volume enclosed in this way is the Wigner Seitz primitive cell. The most popular structures are: sc: simple cubic; : body-centred cubic; : face-centred cubic; : hexagonal close packed. The simple cubic cells are characterized by a 1 = a 2 = a 3 and a 1 a 2 = a 2 a 3 = a 1 a 3 = 0. These cells are shown in Fig In Table 1.1 are listed the most common crystal structures and lattice structures of the elements. For a crystal structure of different elements it is advised to consult Wyckoff [1.1]. 1.2 DIFFRACTION OF WAVES BY A CRYSTAL AND THE RECIPROCAL LATTICE Diffraction is the main method of investigating crystal structures. On the other hand the diffraction of electron waves of electrons belonging to the crystal

5 Elsevier UK Job code: MIOC Ch01-I :03p.m. Page:5 Trim: MM TS: Integra, India Periodic Structures 5 sc FIGURE 1.2 The cubic space lattices (sc,, ) and the hexagonal lattice (). is the origin of the Brillouin zones (see Section 1.3) and of electron bands in crystals (see Chapter 4). The diffraction of a beam on a crystal is the reflection of waves on the periodic structure of the crystal followed by their interference. In diffractional analysis, one uses radiation with a wavelength comparable with inter-atomic distances or, in other words, with the lattice constant. Most commonly, X-ray radiation, electron and neutron beams are used. Neutron radiation, being only weakly absorbed, is used for larger samples, while the electron beams, which are strongly absorbed, are used mostly for surface analysis. W.L. Bragg described the diffraction of beams from a crystal. The crystal is a periodic set of parallel atomic planes, each of which reflects a very small fraction of the incident beam (see Fig. 1.3). The distance of the planes is d. The incident angle is defined as in Fig For coherent diffraction the extra path 2d sin must be an integral number of wavelengths: This is Bragg s law. 2d sin = n (1.2)

6 Elsevier UK Job code: MIOC Ch01-I :03p.m. Page:6 Trim: MM TS: Integra, India 6 Introduction to Theory of Solids Table 1.1 Crystal structure of the elements H He Li Be B rhom. C diam. N cubic (N 2 ) O (O 2 ) F mon. Ne Na Mg Al Si diam. P S Cl (Cl 2 ) Ar K Ca Sc Ti V Cr Mn cubic Fe Co Ni Cu Zn Ga Ge diam. As rhom. Se hex. chains Br (Br 2) Kr Rb Sr Y Zr Nb Mo Tc Ru Rh Pd Ag Cd In tetr. Sn diam. Sb rhom. Te hex. chains I (I 2) Xe Cs Ba La Lu Hf Ta W Re Os Ir Pt Au Hg rhom. Tl Pb Bi rhom. Po sc At Rn Fr Ra Ac Lr La hex Ce Pr hex Nd hex Pm Sm Eu Gd Tb Dy Ho Er Tm Yb Lu Ac Th Pa tetr. U Np Pu Am hex Cm Bk Cf Es Fm Md No Lr sc, simple cubic;, body-centred cubic;, face-centred cubic; hex, hexagonal;, hexagonal close packed; diam., diamond; rhom., rhombic; tetr., tetragonal;, complex; mon., monoclinic.

7 Elsevier UK Job code: MIOC Ch01-I :03p.m. Page:7 Trim: MM TS: Integra, India Periodic Structures 7 θ θ θ d d sin θ FIGURE 1.3 The reflection of an incident beam (wave) on atomic planes in crystal Reciprocal lattice vectors The Bragg condition can be expressed in general terms. Let k and k be the wave vectors of an incident and reflected beam, respectively k =2 /. Vector k = k k is normal to the reflection plane, and its length (see Fig. 1.4) is k = k 4 sin k = (1.3) The Bragg s law (1.2) (for n = 1) can be written as k = k k = 2 d (1.4) To proceed further with the Bragg s law one has to introduce the reciprocal lattice. One defines the axis vectors b 1 b 2 b 3 of the reciprocal lattice as being orthogonal to the axis vectors a 1 a 2 a 3 of the real lattice: b 1 = 2 a 2 a 3 a 1 a 2 a 3 b 2 = 2 a 3 a 1 a 1 a 2 a 3 b 3 = 2 a 1 a 2 a 1 a 2 a 3 (1.5) Δk k θ θ k k FIGURE 1.4 The relation between incident k, reflected k and the wave vector k = k k.

8 Elsevier UK Job code: MIOC Ch01-I :03p.m. Page:8 Trim: MM TS: Integra, India 8 Introduction to Theory of Solids where a 1 a 2 a 3 = V is the volume of the elementary cell of the real lattice. Given the primitive (basic) vectors of the reciprocal lattice b 1 b 2 b 3 one can write each vector of reciprocal lattice as G = hb 1 + kb 2 + lb 3 (1.6) where h k l are integers. The scalar product of an arbitrary vector in a real lattice, r, and an arbitrary vector in the reciprocal lattice, G, is the multiplicity of the factor 2 : therefore their product fulfils the relation r G = n 1 h + n 2 k + n 3 l 2 (1.7) exp ir G = 1 (1.8) Vectors in the reciprocal lattice have the dimension of [1/length], and the volume of the elementary reciprocal lattice cell is 2 3 /V. Having defined the reciprocal lattice one can return to the Bragg s law. It can be proved that the spacing between parallel lattice planes that are normal to the direction G = hb 1 + kb 2 + lb 3 is d hkl = 2 / G (see [1.2]). Thus one can write the Bragg condition (1.4) as k = k k = G (1.9) Condition (1.9) is the basic condition for the reflection of scattered waves by the crystal. In elastic scattering of electrons, the magnitudes k and k are equal, and k 2 = k 2. Therefore we have k k = G k G 2 = k 2 2k G = G 2 (1.10) 1.3 BRILLOUIN ZONES The primitive cell of the reciprocal lattice can be spanned on the primitive axes b 1 b 2 b 3. It can also be created by the Wigner Seitz method explained above. The Wigner Seitz primitive cell is bound by planes normal to the vectors connecting the origin with the nearest-neighbour points of the reciprocal lattice and drawn at their midpoints. This cell is called a Brillouin zone. An example of the first Brillouin zone for the two-dimensional (2D) rectangular lattice is shown in Fig For complicated structures the shape of the first Brillouin zone becomes spherical. The second Brillouin zone is the space between the first zone and the planes drawn at the midpoints of vectors pointing to the second neighbours and so on

9 Elsevier UK Job code: MIOC Ch01-I :03p.m. Page:9 Trim: MM TS: Integra, India Periodic Structures 9 b 2 First Brillouin zone b 1 Second Brillouin zone FIGURE 1.5 First and second Brillouin zones for a two-dimensional rectangular lattice. k z 0,0,2 1, 1,1 1,1,1 1, 1,1 2,0,0 0,2,0 1,1,1 Γ 0,0,0 1, 1, 1 0, 2,0 2,0,0 k x k y 1,1, 1 1,1, 1 1, 1, 1 0,0, 2 FIGURE 1.6 First Brillouin zone for the face-centred cubic () lattice. The square and hexagonal limiting planes come from the points (2,0,0) and (1,1,1) of the reciprocal lattice, respectively. for subsequent Brillouin zones, see Fig In Fig. 1.6, the first Brillouin zone is shown for the lattice, which by itself is the lattice in the reciprocal space. The Brillouin zone gives a physical interpretation of diffraction condition (1.10). After dividing both sides of (1.10) by 4 we obtain ( ) ( ) k 2 G = 2 G (1.11)

10 Elsevier UK Job code: MIOC Ch01-I :03p.m. Page:10 Trim: MM TS: Integra, India 10 Introduction to Theory of Solids This relation states that the reflected wave with the wave vector on the boundary of the Brillouin zone fulfils the Bragg condition. In the process of interference with incoming wave, it forms the standing wave (see Chapter 4) and, in consequence, generates an energy gap on the Brillouin zone boundary. We can see how the internal diffraction of crystal electrons by obeying the Bragg s law creates the Brillouin zones and in effect the electron bands in crystals. REFERENCES [1.1] W.G. Wyckoff, Crystal Structures, Krieger, Florida (1981). [1.2] H. Ibach and H. Lüth, Solid-State Physics. An Introduction to Principles of Materials Science, Springer, Berlin (1995).

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