In Chapter 2 we found that electrons are restricted to sets of discrete ener-gy levels within atoms. Large gaps exist in the energy scale in which no en-ergy states are available. In a similar fashion, electrons in solids are restricted to certain energies and are not allowed at other energies. The basic differ-ence between the case of an electron in a solid and that of an electron in an isolated atom is that in the solid the electron has a range, or band, of avail-able energies. The discrete energy levels of the isolated atom spread into bands of energies in the solid because in the solid the wave functions of electrons in neighboring atoms overlap, and an electron is not necessarily lo-calized at a particular atom. Thus, for example, an electron in the outer orbit of one atom feels the influence of neighboring atoms, and its overall wave function is altered. Naturally, this influence affects the potential energy term and the boundary conditions in the Schrodinger equation, and we would expect to obtain different energies in the solution. Usually, the influence of neighboring atoms on the energy levels of a particular atom can be treated as a small perturbation, giving rise to shifting and splitting of energy states into energy bands.
3.1.1 Bonding Forces in Solids
The interaction of electrons in neighboring atoms of a solid serves the very important function of holding the crystal together. For example, alkali halides such as NaCl are typified by ionic bonding. In the NaCl lattice, each Na atom
is surrounded by six nearest neighbor CI atoms, and vice versa. Four of the nearest neighbors are evident in the two-dimensional representation shown in Fig. 3-la.The electronic structure of Na (Z = 11) is [Ne] 3s1, and CI (Z = 17) has the structure [Ne]3s23p5. In the lattice each Na atom gives up its outer 3s electron to a CI atom, so that the crystal is made up of ions with the electronic structures of the inert atoms Ne and Ar (Ar has the electronic structure [Ne]3s23/?6). However, the ions have net electric charges after the electron ex-change. The Na+ ion has a net positive charge, having lost an electron, and the Cl~ ion has a net negative charge, having gained an electron. Each Na+ ion exerts an electrostatic attractive force upon its six Cl~ neighbors, and vice versa. These coulombic forces pull the lattice together until a balance is reached with repulsive forces. A reasonably accurate cal-culation of the atomic spacing can be made by considering the ions as hard spheres being attracted together (Example 1-1). An important observation in the NaCl structure is that all electrons are tightly bound to atoms. Once the electron exchanges have been made between the Na and CI atoms to form the Na+ and CI" ions, the outer orbits of all atoms are completely filled. Since the ions have the closed-shell con-figurations of the inert atoms Ne and Ar, there are no loosely bound electrons to participate in current flow; as a result, NaCl is a good insulator. In a metal atom the outer electronic shell is only partially filled, usual-ly by no more than three electrons. We have already noted that the alkali met-als (e.g., Na) have only one electron in the outer orbit. This electron is loosely bound and is given up easily in ion formation. This accounts for the great chemical activity in the alkali metals, as well as for their high electrical con-ductivity. In the metal the outer electron of each alkali atom is contributed to the crystal as a whole, so that the solid is made up of ions with closed shells immersed in a sea of free electrons. The forces holding the lattice together arise from an interaction between the positive ion cores and the surrounding free electrons. This is one type of metallic bonding. Obviously, there are com-plicated differences in the bonding forces for various metals, as evidenced by the wide range of melting temperatures (234 K for Hg, 3643 K for W). Howev-er, the metals have the sea of electrons in common, and these electrons are free to move about the crystal under the influence of an electric field. A third type of bonding is exhibited by the diamond lattice semicon-ductors. We recall that each atom in the Ge, Si, or C diamond lattice is sur-rounded by four nearest neighbors, each with four electrons in the outer orbit. In these crystals each atom shares its valence electrons with its four neighbors (Fig. 3-lb). Bonding between nearest neighbor atoms is illustrated in the di-amond lattice diagram of Fig. 1-9. The bonding forces arise from a quantum mechanical interaction between the shared electrons. This is known as cova-lent bonding; each electron pair constitutes a covalent bond. In the sharing process it is no longer relevant to ask which electron belongs to a particular atom—both belong to the bond. The two electrons are indistinguishable, ex-cept that they must have opposite spin to satisfy the Pauli exclusion principle. Covalent bonding is also found in certain molecules, such as H2.
As in the case of the ionic crystals, no free electrons are available to the lattice in the covalent diamond structure of Fig. 3-lb. By this reasoning Ge and Si should also be insulators. However, we have pictured an idealized lattice at 0 K in this figure. As we shall see in subsequent sections, an elec-tron can be thermally or optically excited out of a covalent bond and there-by become free to participate in conduction. This is an important feature of semiconductors. Compound semiconductors such as GaAs have mixed bonding, in which both ionic and covalent bonding forces participate. Some ionic bonding is to be expected in a crystal such as GaAs because of the difference in place-ment of the Ga and As atoms in the periodic table. The ionic character of the bonding becomes more important as the atoms of the compound become further separated in the periodic table, as in the II-VI compounds. Such elec-tronic structure, and specifically the idea that the outermost valence shell is complete if it has a stable set of eight electrons (Ne, Ar, Kr), is the basis of most of chemistry and many of the semiconducting properties.
3.1.1 Bonding Forces in Solids
The interaction of electrons in neighboring atoms of a solid serves the very important function of holding the crystal together. For example, alkali halides such as NaCl are typified by ionic bonding. In the NaCl lattice, each Na atom
is surrounded by six nearest neighbor CI atoms, and vice versa. Four of the nearest neighbors are evident in the two-dimensional representation shown in Fig. 3-la.The electronic structure of Na (Z = 11) is [Ne] 3s1, and CI (Z = 17) has the structure [Ne]3s23p5. In the lattice each Na atom gives up its outer 3s electron to a CI atom, so that the crystal is made up of ions with the electronic structures of the inert atoms Ne and Ar (Ar has the electronic structure [Ne]3s23/?6). However, the ions have net electric charges after the electron ex-change. The Na+ ion has a net positive charge, having lost an electron, and the Cl~ ion has a net negative charge, having gained an electron. Each Na+ ion exerts an electrostatic attractive force upon its six Cl~ neighbors, and vice versa. These coulombic forces pull the lattice together until a balance is reached with repulsive forces. A reasonably accurate cal-culation of the atomic spacing can be made by considering the ions as hard spheres being attracted together (Example 1-1). An important observation in the NaCl structure is that all electrons are tightly bound to atoms. Once the electron exchanges have been made between the Na and CI atoms to form the Na+ and CI" ions, the outer orbits of all atoms are completely filled. Since the ions have the closed-shell con-figurations of the inert atoms Ne and Ar, there are no loosely bound electrons to participate in current flow; as a result, NaCl is a good insulator. In a metal atom the outer electronic shell is only partially filled, usual-ly by no more than three electrons. We have already noted that the alkali met-als (e.g., Na) have only one electron in the outer orbit. This electron is loosely bound and is given up easily in ion formation. This accounts for the great chemical activity in the alkali metals, as well as for their high electrical con-ductivity. In the metal the outer electron of each alkali atom is contributed to the crystal as a whole, so that the solid is made up of ions with closed shells immersed in a sea of free electrons. The forces holding the lattice together arise from an interaction between the positive ion cores and the surrounding free electrons. This is one type of metallic bonding. Obviously, there are com-plicated differences in the bonding forces for various metals, as evidenced by the wide range of melting temperatures (234 K for Hg, 3643 K for W). Howev-er, the metals have the sea of electrons in common, and these electrons are free to move about the crystal under the influence of an electric field. A third type of bonding is exhibited by the diamond lattice semicon-ductors. We recall that each atom in the Ge, Si, or C diamond lattice is sur-rounded by four nearest neighbors, each with four electrons in the outer orbit. In these crystals each atom shares its valence electrons with its four neighbors (Fig. 3-lb). Bonding between nearest neighbor atoms is illustrated in the di-amond lattice diagram of Fig. 1-9. The bonding forces arise from a quantum mechanical interaction between the shared electrons. This is known as cova-lent bonding; each electron pair constitutes a covalent bond. In the sharing process it is no longer relevant to ask which electron belongs to a particular atom—both belong to the bond. The two electrons are indistinguishable, ex-cept that they must have opposite spin to satisfy the Pauli exclusion principle. Covalent bonding is also found in certain molecules, such as H2.
As in the case of the ionic crystals, no free electrons are available to the lattice in the covalent diamond structure of Fig. 3-lb. By this reasoning Ge and Si should also be insulators. However, we have pictured an idealized lattice at 0 K in this figure. As we shall see in subsequent sections, an elec-tron can be thermally or optically excited out of a covalent bond and there-by become free to participate in conduction. This is an important feature of semiconductors. Compound semiconductors such as GaAs have mixed bonding, in which both ionic and covalent bonding forces participate. Some ionic bonding is to be expected in a crystal such as GaAs because of the difference in place-ment of the Ga and As atoms in the periodic table. The ionic character of the bonding becomes more important as the atoms of the compound become further separated in the periodic table, as in the II-VI compounds. Such elec-tronic structure, and specifically the idea that the outermost valence shell is complete if it has a stable set of eight electrons (Ne, Ar, Kr), is the basis of most of chemistry and many of the semiconducting properties.
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