bonds, then a more global approach is required(using Huckels theories for chemists or Floquet's theories for physicists) 1.3.2. Weak bonds This approach involves improving the potential box model. This is done by the electrons interacting with a periodical internal potential generated by a crystal lattice (of Coulumbic potential varying 1/r with respect to the ions placed at nodes of the lattice). In Figure 1.1, we can see atoms periodically spaced a distance"a"apart Each of the atoms has a radius denoted"R"(Figure 1.la). A ID representation of the potential energy of the electrons is given in Figure 1. 1b. The condition a< 2R has been imposed Depending on the direction defined by the line Ox that joins the nuclei of the atoms, when an electron goes towards the nuclei, the potentials diverge. In fact, the study of the potential strictly in terms of Ox has no physical reality as the electrons here are conduction electrons in the external layers. According to the line(D)that does not traverse the nuclei. the electron-nuclei distance no longer reaches zero and potentials that tend towards finite values join together. In addition, the condition a<2R decreases the barrier that is midway between adjacent nuclei by giving rise a strong overlapping of potential curves. This results in a solid with a periodic slightly fluctuating potential. The first representation of the potential as a flat bottomed bowl(zero order approximation for the electrons) is now replaced with a periodically varying bowl. As a first approximation, and in one dimension(r= x), the potential can be described as V(x)=wo cos The term wo, and the associated perturbation of the crystalline lattice, decrease in size as the relation a 2R becomes increasingly valid. In practical terms, the smaller a"is with respect to 2R, then the smaller the perturbation becomes, and the more justifiable the use of the perturbation method to treat the problem becomes. The corresponding approximation( first order approximation with the Hamiltonia perturbation being given by H=wo cos -x)is that of a semi-free electron and is an improvement over that for the free electron(which ignores H ). The theory that results from this for the weak bond can equally be applied to the metallic bond where there is an easily delocalized electron in a lattice with a low value of wo(see Chapter 3)
Representations of Electron-Lattice Bonds 7 bonds, then a more global approach is required (using Hückels theories for chemists or Floquet’s theories for physicists). 1.3.2. Weak bonds This approach involves improving the potential box model. This is done by the electrons interacting with a periodical internal potential generated by a crystal lattice (of Coulumbic potential varying 1/r with respect to the ions placed at nodes of the lattice). In Figure 1.1, we can see atoms periodically spaced a distance “a” apart. Each of the atoms has a radius denoted “R” (Figure 1.1a). A 1D representation of the potential energy of the electrons is given in Figure 1.1b. The condition a < 2R has been imposed. Depending on the direction defined by the line Ox that joins the nuclei of the atoms, when an electron goes towards the nuclei, the potentials diverge. In fact, the study of the potential strictly in terms of Ox has no physical reality as the electrons here are conduction electrons in the external layers. According to the line (D) that does not traverse the nuclei, the electron-nuclei distance no longer reaches zero and potentials that tend towards finite values join together. In addition, the condition a < 2R decreases the barrier that is midway between adjacent nuclei by giving rise to a strong overlapping of potential curves. This results in a solid with a periodic, slightly fluctuating potential. The first representation of the potential as a flatbottomed bowl (zero order approximation for the electrons) is now replaced with a periodically varying bowl. As a first approximation, and in one dimension (r { x), the potential can be described as: V(x) = w0 cos 2 x . a S The term w0, and the associated perturbation of the crystalline lattice, decrease in size as the relation a < 2R becomes increasingly valid. In practical terms, the smaller “a” is with respect to 2R, then the smaller the perturbation becomes, and the more justifiable the use of the perturbation method to treat the problem becomes. The corresponding approximation (first order approximation with the Hamiltonian perturbation being given by H(1) = w0 cos 2 ) a x S is that of a semi-free electron and is an improvement over that for the free electron (which ignores H(1)). The theory that results from this for the weak bond can equally be applied to the metallic bond, where there is an easily delocalized electron in a lattice with a low value of w0 (see Chapter 3)
8 Solid-State Physics for Electronics (a)(1 x (D) Potential energy (b) Resultant (D) Potential generated Potential generated Resultant potential by atom(1) by atom(2) with respect to Ox Figure 1. 1. Weak bonds and: (a) atomic orbitals(s orbitals with radius R of a periodic latticeperiod =a) respecting the condition a< 2R;(b)in ID, the resultant potential energy(thick line) seen by electrons 1.3.3. Strong bonds sofi The approach used here is more"chemical"in its nature. The properties of the id are deduced from chemical bonding expressed as a linear combination of atomic orbitals of the constituent atoms. This reasoning is all the more acceptable when the electrons remain localized around a specific atom. This approximation of a strong bond is moreover justified when the condition a 2 2R is true(Figure 1.2a) and is generally used for covalent solids where valence electrons remain localized around the two atoms that they are bonding. Q Once again, analysis of the potential curve drawn with respect to Ox gives a function which diverges as the distance between the electrons and the nuclei reduced. With respect to the line D, this discontinuity of the valence electrons can be two situations, namely(see also Figure 1. 2b) If a>> 2R, then very deep potential wells appear, as there is no longer any real overlap between the generated potentials by two adjacent nuclei. In the case, if a chain of N atoms with N valence electrons is so long that we that we have a system of N independent electrons( with N independent dee then the energy levels are degenerate N times. In this case they are ind from one another as they are all the same, and are denoted Eloc in the figure
8 Solid-State Physics for Electronics Figure 1.1. Weak bonds and: (a) atomic orbitals (s orbitals with radius R) of a periodic lattice (period = a) respecting the condition a < 2R; (b) in 1D, the resultant potential energy (thick line) seen by electrons 1.3.3. Strong bonds The approach used here is more “chemical” in its nature. The properties of the solid are deduced from chemical bonding expressed as a linear combination of atomic orbitals of the constituent atoms. This reasoning is all the more acceptable when the electrons remain localized around a specific atom. This approximation of a strong bond is moreover justified when the condition a t 2R is true (Figure 1.2a), and is generally used for covalent solids where valence electrons remain localized around the two atoms that they are bonding. Once again, analysis of the potential curve drawn with respect to Ox gives a function which diverges as the distance between the electrons and the nuclei is reduced. With respect to the line D, this discontinuity of the valence electrons can be suppressed in two situations, namely (see also Figure 1.2b): – If a >> 2R, then very deep potential wells appear, as there is no longer any real overlap between the generated potentials by two adjacent nuclei. In the limiting case, if a chain of N atoms with N valence electrons is so long that we can assume that we have a system of N independent electrons (with N independent deep wells), then the energy levels are degenerate N times. In this case they are indiscernible from one another as they are all the same, and are denoted Eloc in the figure. e - (a) (1) (2) R R O Potential a energy r (b) Potential generated by atom (1) Potential generated by atom (2) Resultant potential with respect to (D) Resultant potential with respect to Ox (D) x
(1) (2 Potential Resultant to d when a≥2R (strong bond) wells where generated Potential generated potential withwith respect Eloc level is by atom 2 D wher degenerate Figure 1. 2. Strong bonds and: (a)atomic orbitals(s orbitals with radius R) in a periodic lattice (of period denoted a) where a2 2R;(b) in ID, the resulting potential energy(thick curve) seen by electrons If a 2 2R, the closeness of neighboring atoms induces a slight overlap of nuclei generated potentials. This means that the potential wells are no longer independent and their degeneration is increased. Electrons from one bond can interact with those of another bond, giving rise to a spread in the band energy levels. It is worth noting hat the resulting potential wells are nevertheless considerably deeper than those weak bonds(where a <2R), so that the electrons remain more localized around their base atom. Given these well depths, the perturbation method that was used for weak bonds is no longer viable. Instead, in order to treat this system we will have to turn to the Huckel method or use Floquet's theorem(see Chapter 7) 1.3.4. Choosing between approximations for weak and strong bonds The electrical behavior of metals is essentially determined by that of the conduction electrons. As detailed in section 1.3.2. these electrons are delocalized throughout the whole lattice and should be treated as weak bonds
Representations of Electron-Lattice Bonds 9 Figure 1.2. Strong bonds and: (a) atomic orbitals (s orbitals with radius R) in a periodic lattice (of period denoted a) where a t 2R; (b) in 1D, the resulting potential energy (thick curve) seen by electrons – If a t 2R, the closeness of neighboring atoms induces a slight overlap of nuclei generated potentials. This means that the potential wells are no longer independent and their degeneration is increased. Electrons from one bond can interact with those of another bond, giving rise to a spread in the band energy levels. It is worth noting that the resulting potential wells are nevertheless considerably deeper than those in weak bonds (where a < 2R), so that the electrons remain more localized around their base atom. Given these well depths, the perturbation method that was used for weak bonds is no longer viable. Instead, in order to treat this system we will have to turn to the Hückel method or use Floquet’s theorem (see Chapter 7). 1.3.4. Choosing between approximations for weak and strong bonds The electrical behavior of metals is essentially determined by that of the conduction electrons. As detailed in section 1.3.2, these electrons are delocalized throughout the whole lattice and should be treated as weak bonds. e - (a) (1) (2) R (D) a Potential r energy (b) Eloc Eloc Eloc Potential generated by atom 1 Resultant potential with respect to Ox Potential generated by atom 2 Resultant potential with respect to D when a t 2R (strong bond) Deep independent wells where Eloc level is degenerate N times R x Potential wells with respect to D where a >> 2R
10 Solic cs for Electr Dielectrics(insulators), however, have electrons which are highly localized around one or two atoms. These materials can therefore only be described using strong-bond theory Semiconductors have carriers which are less localized. The external electrons can delocalize over the whole lattice, and can be thought of as semi-free. Thus, it can be more appropriate to use the strong-bond approximation for valence electrons from the internal layers, and the weak-bond approximation for conduction electrons 1.4. Complementary material: basic evidence for the appearance of bands in solids This section will be of use to those who have a basic understanding of wave nechanics or more notably experience in dealing with potential wells. For others, it is recommended that they read the complementary sections at the end of Chapters 2 and 3 beforehand This section shows how the bringing together of two atoms results in a splitting of he atoms'energy levels First, we associate each atom with a straight-walled potential well in which the electrons of each atom are localized Second we recall the solutions for the straight-walled potential wells, and then analyze their change as the atoms move closer to one another. It is then possible to imagine without difficulty the effect of moving N potential wells, together representing N atoms making up a solid. 1. 4.1. Basic solutions for narrow potential wells In Figure 1.3, we have w>0, and this gives potential wells at intervals such that -a/2,+al2] where-w<0 We can thus state that w=. and the energy E is the sum of kinetic energy(Ec =5=0) and potential energy As the related states are carry electrons then E<0, and we can therefore write E=y+E <0 By making a2=(r2-k2)>0, a is real
10 Solid-State Physics for Electronics Dielectrics (insulators), however, have electrons which are highly localized around one or two atoms. These materials can therefore only be described using strong-bond theory. Semiconductors have carriers which are less localized. The external electrons can delocalize over the whole lattice, and can be thought of as semi-free. Thus, it can be more appropriate to use the strong-bond approximation for valence electrons from the internal layers, and the weak-bond approximation for conduction electrons. 1.4. Complementary material: basic evidence for the appearance of bands in solids This section will be of use to those who have a basic understanding of wave mechanics or more notably experience in dealing with potential wells. For others, it is recommended that they read the complementary sections at the end of Chapters 2 and 3 beforehand. This section shows how the bringing together of two atoms results in a splitting of the atoms’ energy levels. First, we associate each atom with a straight-walled potential well in which the electrons of each atom are localized. Second, we recall the solutions for the straight-walled potential wells, and then analyze their change as the atoms move closer to one another. It is then possible to imagine without difficulty the effect of moving N potential wells, together representing N atoms making up a solid. 1.4.1. Basic solutions for narrow potential wells In Figure 1.3, we have W > 0, and this gives potential wells at intervals such that [– a /2, + a/2] where – W < 0. We can thus state that ² ² 2 0, m W J ! = and the energy E is the sum of kinetic energy ² ²² 2 2 ( ) c p k m m E = and potential energy. As the related states are carry electrons then E < 0, and we can therefore write that: ² ² ² ² ² 0. 2 2 E WE k c m m D J = = By making D J ² ²² k > 0, D is real
Schrodinger's equation d=+ h(E-v)v=0(where V is the potential energy such that V=-wbetween-ah2 and al2 but v=0 outside of the well) can be written for the two regions -region I for x)>a: dy_2m E=0, d - region II for、4xs 2a小(E+W)=0, [19 The solutions for equation [1.8 are(with the limiting conditions of y(x) being finite when x→)±oo) for x The solution to equation [1.9] must be stationary because the potential wells are narrow(which forbids propagation solutions). There are two types of solution a symmetric solution in the form vu(x )=B cos kr, for which the conditions of continuity with the solutions of region I give
Representations of Electron-Lattice Bonds 11 Schrödinger’s equation ² 2 ² ² 0 d m dx E V \ \ = (where V is the potential energy such that V = – W between –a/2 and a/2 but V = 0 outside of the well) can be written for the two regions: 2 2 2 2 2 2 2 – region I for x : 0, 2 so that 0 ad m E dx d dx \ ! \ \ D \ = [1.8] 2 2 2 2 2 2 a 2 – region II for – : ( ) 0, 2 2 so that 0. ad m x E W dx d k dx \ dd \ \ = \ [1.9] The solutions for equation [1.8] are (with the limiting conditions of \(x) being finite when x o rf ): – for 2 : () x I a x x Ae D !\ – for 2 : () . x I a x xAe D \ The solution to equation [1.9] must be stationary because the potential wells are narrow (which forbids propagation solutions). There are two types of solution: – a symmetric solution in the form II \ ( ) cos , x B kx for which the conditions of continuity with the solutions of region I give: Figure 1.3. Straight potential wells of width a Ep 0 – W – a/2 + a/2 x Region II Region I Region I