1. Energy Band Theory 15 N Na+b) L 2n/(a+b)2n2丌 (11.32) In the case of a three-dimensional crystal, energy band calculations are, of course,much more complicated, but the essential results obtained from the one-dimensional calculation still hold. In particular, there exist permitted energy bands separated by forbidden energy gaps. The 3-D volume of the first Brillouin zone is 8IN/V. where v is the volume of the crystal, the number of wave vectors is equal to the number of elementary crystal lattice cells, N. The density of wave vectors is given by: n(k)=density of k= number of k-vectors_NV volume of the zone 8N (11.33) 1. 1. 4. Valence band and conduction band Chemical reactions originate from the exchange of electrons from the outer electronic shell of atoms. Electrons from the most inner shells do not participate in chemical reactions because of the high electrostatic attraction to the nucleus. Likewise, the bonds between atoms in a crystal as well as electric transport phenomena, are due to electrons from the outermost shell. In terms of energy bands, the electrons responsible for forming bonds between atoms are found in the last occupied band, where electrons have the highest energy levels for the ground-state atoms However, there is an infinite number of energy bands. The first (lowest) bands contain core electrons such as the ls electrons which are tightly bound to the atoms. The highest bands contain no electrons The last ground-state band which contains electrons is called the valence band, because it contains the electrons that form the -often covalent bonds The permitted energy band directly above the valence band is called the conduction band. In a semiconductor this band is empty of electrons at low temperature(T=0K). At higher temperatures, some electrons have enough thermal energy to quit their function of forming a bond between atoms and circulate in the crystal. These electrons "jump"from the valence band into the conduction band, where they are free to move. The energy difference between the bottom of the conduction band and the top of the valence band is called"forbidden gap"or "bandgap"and is noted In a more general sense, the following situations can occur depending on the location of the atom in the periodic table(Figure 1.11) A: The last(valence) energy band is only partially filled with electrons, even at T=OK
1. Energy Band Theory 15 In the case of a three-dimensional crystal, energy band calculations are, of course, much more complicated, but the essential results obtained from the one-dimensional calculation still hold. In particular, there exist permitted energy bands separated by forbidden energy gaps. The 3-D volume of the first Brillouin zone is where V is the volume of the crystal, the number of wave vectors is equal to the number of elementary crystal lattice cells, N. The density of wave vectors is given by: 1.1.4. Valence band and conduction band Chemical reactions originate from the exchange of electrons from the outer electronic shell of atoms. Electrons from the most inner shells do not participate in chemical reactions because of the high electrostatic attraction to the nucleus. Likewise, the bonds between atoms in a crystal, as well as electric transport phenomena, are due to electrons from the outermost shell. In terms of energy bands, the electrons responsible for forming bonds between atoms are found in the last occupied band, where electrons have the highest energy levels for the ground-state atoms. However, there is an infinite number of energy bands. The first (lowest) bands contain core electrons such as the 1s electrons which are tightly bound to the atoms. The highest bands contain no electrons. The last ground-state band which contains electrons is called the valence band, because it contains the electrons that form the -often covalent- bonds between atoms. The permitted energy band directly above the valence band is called the conduction band. In a semiconductor this band is empty of electrons at low temperature (T=0K). At higher temperatures, some electrons have enough thermal energy to quit their function of forming a bond between atoms and circulate in the crystal. These electrons "jump" from the valence band into the conduction band, where they are free to move. The energy difference between the bottom of the conduction band and the top of the valence band is called "forbidden gap" or "bandgap" and is noted In a more general sense, the following situations can occur depending on the location of the atom in the periodic table (Figure 1.11): A: The last (valence) energy band is only partially filled with electrons, even at T=0K
Chapter B: The last(valence) energy band is completely filled with electrons at T=OK, but the next (empty) energy band overlaps with it(i.e:an empty energy band shares a range of common energy values; Eg <0) C: The last(valence) energy band is completely filled with electrons and no empty band overlaps with it(Eg>0) cases and B, electrons with the highest energies can easily acquire an infinitesimal amount of energy and jump to a slightly higher permitted energy level, and move through the crystal. In other words, electrons can leave the atom and move in the crystal without receiving any energy. a material with such a property is a metal. In case C, a significant amount of energy (equal to Eg or higher) has to be transferred to an electron in order for it to jump"from the valence band into a permitted energy level of the conduction band this means that an electron must receive a significant amount of energy before leaving an atom and moving"freely in the crystal. A material with such properties is either an insulator or a semiconductor Figure 1. 11: Valence band(bottom) and conduction band in a metal (A and B)and in a semiconductor or an insulator(C)[] The distinction between an insulator and a semiconductor is purely quantitative and is based on the value of the energy gap. In a semiconductor Eg is typically smaller than 2 eV and room-temperature thermal energy or excitation from visible-light photons can give electrons enough energy for "jumping"from the valence into the conduction band. The energy gap of the most common semiconductors are: 1.12 ev(silicon), 0.67 ev(germanium), and 1.42 ev(gallium arsenide). Insulators have significantly wider energy bandgaps: 9.0 eV (Sio2), 5.47 eV(diamond), and 5.0 eV (Si3 N4). In these materials room- temperature thermal energy is not large enough to place electrons in the conduction band
16 Chapter 1 B: The last (valence) energy band is completely filled with electrons at T=0K, but the next (empty) energy band overlaps with it (i.e.: an empty energy band shares a range of common energy values; C: The last (valence) energy band is completely filled with electrons and no empty band overlaps with it In cases A and B, electrons with the highest energies can easily acquire an infinitesimal amount of energy and jump to a slightly higher permitted energy level, and move through the crystal. In other words, electrons can leave the atom and move in the crystal without receiving any energy. A material with such a property is a metal. In case C, a significant amount of energy (equal to or higher) has to be transferred to an electron in order for it to "jump" from the valence band into a permitted energy level of the conduction band. This means that an electron must receive a significant amount of energy before leaving an atom and moving "freely" in the crystal. A material with such properties is either an insulator or a semiconductor. The distinction between an insulator and a semiconductor is purely quantitative and is based on the value of the energy gap. In a semiconductor is typically smaller than 2 eV and room-temperature thermal energy or excitation from visible-light photons can give electrons enough energy for "jumping" from the valence into the conduction band. The energy gap of the most common semiconductors are: 1.12 eV (silicon), 0.67 eV (germanium), and 1.42 eV (gallium arsenide). Insulators have significantly wider energy bandgaps: 9.0 eV 5.47 eV (diamond), and 5.0 eV In these materials roomtemperature thermal energy is not large enough to place electrons in the conduction band
1. Energy Band Theor Beside elemental semiconductors such as silicon and germanium, compound semiconductors can be synthesized by combining elements from column Iv of the periodic table (SiC and SiGe)or by combinin elements from columns III and V(GaAs, GaN, InP, AlGaAs, AlSb, GaP AlP and AlAs). Elements from other columns can sometimes be used as well(HgCdTe, Cds,... Diamond exhibits semiconducting properties at high temperature, and tin (right below germanium in column Iv of the periodic table) becomes a semiconductor at low temperatures. About 98% of all semiconductor devices are fabricated from single-crystal silicon, such as integrated circuits, microprocessors and memory chips. The remaining 2% make use of III-V compounds, such as light-emitting diodes, laser diodes and some microwave-frequency components BCN Al Si P a ge as In Sb Figure 1. 12: Main elements used in semiconductor technology (elemental semiconductors such as Si, and compound semiconductors such as gaAs) It is worthwhile mentioning that it is possible for non-crystalline materials to exhibit semiconducting properties. Some materials, such as amorphous silicon, where the distance between atoms varies in a random fashion, can behave as semiconductors. The mechanisms for the transport of electric charges in these materials are, however, quite different from those in crystalline semiconductors. [1 It is convenient to represent energy bands in real space instead of k-space By doing so one obtains a diagram such as that of Figure 1.13, where the x-axis defines a physical distance in the crystal. The maximum energy of the valence band is noted Ev, the minimum energy of the conduction band is noted EC, and the width of the energy bandgap is Eg It is also appropriate to introduce the concept of a Fermi level. The Fermi level, EF, represents the maximum energy of an electron in the
1. Energy Band Theory 17 Beside elemental semiconductors such as silicon and germanium, compound semiconductors can be synthesized by combining elements from column IV of the periodic table (SiC and SiGe) or by combining elements from columns III and V (GaAs, GaN, InP, AlGaAs, AlSb, GaP, A1P and AlAs). Elements from other columns can sometimes be used as well (HgCdTe, CdS,...). Diamond exhibits semiconducting properties at high temperature, and tin (right below germanium in column IV of the periodic table) becomes a semiconductor at low temperatures. About 98% of all semiconductor devices are fabricated from single-crystal silicon, such as integrated circuits, microprocessors and memory chips. The remaining 2% make use of III-V compounds, such as light-emitting diodes, laser diodes and some microwave-frequency components. It is worthwhile mentioning that it is possible for non-crystalline materials to exhibit semiconducting properties. Some materials, such as amorphous silicon, where the distance between atoms varies in a random fashion, can behave as semiconductors. The mechanisms for the transport of electric charges in these materials are, however, quite different from those in crystalline semiconductors.[7]. It is convenient to represent energy bands in real space instead of k-space. By doing so one obtains a diagram such as that of Figure 1.13, where the x-axis defines a physical distance in the crystal. The maximum energy of the valence band is noted the minimum energy of the conduction band is noted and the width of the energy bandgap is It is also appropriate to introduce the concept of a Fermi level. The Fermi level, represents the maximum energy of an electron in the
18 Chapter I material at zero degree Kelvin(OK). At that temperature, all the allowed energy levels below the Fermi level are occupied, and all the energy levels bove it are empty. Alternatively, the Fermi level is defined as an energy level that has a 50% probability of being filled with electrons, even though it may reside in the bandgap. In an insulator or a semiconductor we know that the valence band is full of electrons and the conduction band is empty at 0 K. Therefore, the Fermi level lies somewhere in the bandgap, between Ey and Ec. In a metal, the Fermi level lies within an energy ba Conduction band Fermi level, EF Ey Valence band Figure 1. 13: Valence and conduction band in real space B A Eg TB k[Ill Figure 1.14: Examples of energy band extrema(minimum of the conduction band and maximum of the valence band in two crystals). In crystal A, Eg is the bandgap energy. There is no bandgap in crystal B because the conduction and the valence bands overlap It is impossible to represent the energy bands as a function of k k(kx, ky, k a three-dimensional crystal in a drawing made on a two- dimensional sheet of paper. One can, however, represent E(k) along main crystal directions in k-space and place them on a single graph. For
18 Chapter 1 material at zero degree Kelvin (0 K). At that temperature, all the allowed energy levels below the Fermi level are occupied, and all the energy levels above it are empty. Alternatively, the Fermi level is defined as an energy level that has a 50% probability of being filled with electrons, even though it may reside in the bandgap. In an insulator or a semiconductor, we know that the valence band is full of electrons, and the conduction band is empty at 0 K. Therefore, the Fermi level lies somewhere in the bandgap, between and In a metal, the Fermi level lies within an energy band. It is impossible to represent the energy bands as a function of k = for a three-dimensional crystal in a drawing made on a twodimensional sheet of paper. One can, however, represent E(k) along main crystal directions in k-space and place them on a single graph. For
1. Energy Band Theory example, Figure 1. 14 represents the maximum of the valence band and the minimum of the conduction band as function of k in the [100 and the [lll] directions for two crystals. Crystal A is an insulator or a semiconductor(Eg>0); crystal B is a metal (Eg <0 The energy band diagrams, plotted along the main crystal directions allow us to analyze some properties of semiconductors. For instance, in Figure 1. 15. B the minimum energy in the conduction band and the maximum energy in the valence band occur at the same k-values (k=0). A semiconductor exhibiting this property is called a direct-band semiconductor. Examples of direct-bandgap semiconductors include most compound elements such as gallium arsenide (gaAs). In such a semiconductor an electron can "fall"from the conduction band into the valence band without violating the conservation of momentum law, ie an electron can fall from the conduction band to the valence band without a change in momentum. This process has a high probability of occurrence and the energy lost in that jump"can be emitted in the form of a photon with an energy hv=Eg. In Figure 1. 15.A, the minimum energy in the conduction band and the maximum energy in the valence band occur at different k-values. A semiconductor exhibiting this property is called an indirect bandgap semiconductor. Silicon and germanium are indirect bandgap semiconductors. In such a semiconductor, an electron cannot fall" from the conduction band into the valence band without a change in momentum. This tremendously reduces the probability of a direct"fall of an electron from the conduction band into the valence band as will be discussed in Chapter 3. B A Eg k[llll k [100] k [Illl Figure 1. 15: A: Indirect bandgap semiconductor, B: Direct bandgap semiconductor. [8 1.1.5. Parabolic band approximation For electrical phenomena, only the electrons located near the maximum of the valence band and the minimum of the conduction band
1. Energy Band Theory 19 example, Figure 1.14 represents the maximum of the valence band and the minimum of the conduction band as function of k in the [100] and the [111] directions for two crystals. Crystal A is an insulator or a semiconductor crystal B is a metal The energy band diagrams, plotted along the main crystal directions, allow us to analyze some properties of semiconductors. For instance, in Figure 1.15.B the minimum energy in the conduction band and the maximum energy in the valence band occur at the same k-values (k=0). A semiconductor exhibiting this property is called a direct-band semiconductor. Examples of direct-bandgap semiconductors include most compound elements such as gallium arsenide (GaAs). In such a semiconductor, an electron can "fall" from the conduction band into the valence band without violating the conservation of momentum law, i.e. an electron can fall from the conduction band to the valence band without a change in momentum. This process has a high probability of occurrence and the energy lost in that "jump" can be emitted in the form of a photon with an energy In Figure 1.15.A, the minimum energy in the conduction band and the maximum energy in the valence band occur at different k-values. A semiconductor exhibiting this property is called an indirect bandgap semiconductor. Silicon and germanium are indirectbandgap semiconductors. In such a semiconductor, an electron cannot "fall" from the conduction band into the valence band without a change in momentum. This tremendously reduces the probability of a direct "fall" of an electron from the conduction band into the valence band, as will be discussed in Chapter 3. 1.1.5. Parabolic band approximation For electrical phenomena, only the electrons located near the maximum of the valence band and the minimum of the conduction band