BAND THEORY

Structure, Bonding, and Properties

Rule 4: Linking of polyhedra having different cations

1.10. BAND THEORY

44

CHAPTER 1

the preferential directions of the hybridized orbitals. For example, the functions

<PI

=

0.894s

+

^0.447(px

+

^Py

+

^pz)

<P2

=

0.22s

+

^{0.975(px -} Py - pz)

<P3

=

^0.22s

+

^{0.975( -Px}

+

^{Py - pz)}

<P4 = 0.22s

+

0.975( -Px - Py

+

^pz)

(1.16)

<PI defines an orbital having contributions of 80% (0.894²)s and 20%p, and the orbitals

<P2, <P3 and <P4, each with contributions of 5%s and 95%p. They are adequate to calculate the wave functions for a molecule AX3 that has a lone electron pair (<PI) with a large s contribution and two bonds with larger p orbital contributions, i.e., Sp2 hybridization. The corresponding bond angles are 96.so.

To obtain the values of the coefficients \ti' ~i' Yi' and ()i so that the bond energy is maximized and the correct molecular geometry is produced; also the mutual interactions between the electrons must be considered. This requires a great deal of computational expenditure. However, in a qualitative manner the interactions can be estimated rather well using the valence shell electron-pair repulsion theory.

k=O

An = cos (1tnIN)

k = IINa An = cos (21tnIN)

Figure 1.17. Vibration modes of a linear chain composed of N + 1 spheres linked by springs.

function of the system is

N N

\jJk

=

L An<l'n

=

L <l'n cos(2nnka) (1.18)

n=1 n=1

For k

=

^0, \jJo

=

L~=I <l'n

=

^<1'0

+

^<1'1

+

^<1'2

+

^<1'3

+ ... ,

which is the bonding case. For k= 1I2a, \jJI/2a

=

L~=I (-It<l'n

=

<1'0 - <1'1

+

<1'2 - <1'3

+ ... ,

which is the antibonding case,

Every wave function \jJk is related to a definite energy state. Taking 10⁴H atoms in the chain we have a huge number of 10⁴ energy states E(k) within the limits E(O) and E(n/a), which is an energy band. The energy states are not distributed evenly in the band.

A function of density of states (DOS) is introduced to describe the number of energy states distributed between E and E

+

^dE. The bandwidth or band dispersion exhibits the energy difference between the highest and the lowest energy level in the band. The bandwidth becomes larger if the interaction among the atoms increases; in other words the atomic orbitals are overlapped to a greater extent. A smaller interatomic distance causes a larger bandwidth.

Based on the Pauli principle, only two electrons can occupy a single spatial state and these electrons must be opposite spin, so that the N electrons of the N hydrogen atoms take the energy states in the lower half of the band, and the band is "half occupied." The highest occupied energy level, usually referred to as highest occupied molecular orbital (HOMO), is the Fermi limit. Whenever the Fermi limit is inside a band, metallic electric conduction is observed. Only a very low energy is needed to excite an electron from an occupied state under the Fermi limit to an unoccupied state above it. The easy switchover from one state to another means the electron has high mobility. Because of thermal excitation a certain fraction of the electrons is always found above the Fermi limit.

45

STRUCTURE, BONDING, AND PROPERTIES

46

CHAPTER 1

1.10.1. THE PEIERLS DISTORTION

The linear chain of H atoms may not be stable. From chemistry the H atoms may tend to be paired into groups for forming the H2 molecule. Thus, a distortion is induced in such a way that the atoms approach each other in pairs. This effect is called Peierls distortion, or strong electron-phonon coupling. To illustrate this case, we double the length of the chain to 2a, the k values only run from 0 to 1/4a. The corresponding energy band is composed of two branches (Fig. 1.18a). The first branch with a positive slope corresponds to the bonding H2 molecules. The other branch has a negative slope representing the antibonding case. The two branches meet at k

=

^{1I4a, with two}

degenerate states.

Up to now we have assumed evenly spaced H atoms. If we allow the H atoms to approach each other pairwise, a change in the band structure takes place. The corresponding movements of the atoms are marked by arrows in Fig. 1.18a. At k

=

^{0 this}

has no consequences; at the lower (or upper) end of the band an energy gain (or loss) occurs for the atoms that approach each other; it is compensated by the energy loss (or gain) of the atoms moving apart. In the central part of the band, where the hydrogen atom chain has its Fermi limit, substantial changes take place. One of the degenerate states is stabilized while the other one is destabilized. The upper branch of the curve shifts upward, and the lower one downward. As a result a gap opens up, and the bond splits (Fig. 1.18b). For the half-filled band the net result is an energy gain. Therefore, it is energetically more favorable when short and long distances between the H atoms alternate in the chain. The chain no longer is an electrical conductor, as an electron must overcome the energy gap in order to pass from one energy state to another.

In practice, the distortion of certain atoms can modify the unit cell of a solid; then it induces the change of the band structure, so the physical property is changed. For example, when the V cations are distorted pairwisely in V02 , the unit cell transforms from tetragonal to monoclinic, resulting in a transition from an insulator to a metallic conductor due to the change of the band structure.

The Peierls distortion is a substantial factor influencing what structure a solid adopts.

The driving force is the tendency to maximize bonding (the same tendency that force H atoms or other radicals to bond with each other). In a solid, if the Peierls distortion makes the unit cell double or triple, the density of states at the Fermi level must be shifted certain amount toward lower energy level in bonding state or higher energy level in antibonding state. By opening up an energy gap the bands become narrower; within a band the energy levels become more crowded. The extreme case is a band that has shrunk to a single energy value (all levels have the same energy). This happens, for example, when the chain of hydrogen atoms consists of widely separated H2 molecules; then we have separated, independent H2 molecules whose energy levels all have the same value.

The bonds are localized in the molecules, by which we mean that the electron is tightly bonded to the molecule. Generally, the bandwidth is a measure for the degree of localization of the bonds: a narrow band represents a high degree of localization, and with increasing bandwidth the bonds become more delocalized. Since narrow bands can hardly overlap and are usually separated by intervening gaps, compounds with essentially localized bonds are electrical insulators.

When the atoms are forced to move closer by the exertion of pressure, their interaction increases and the bands become wider. At sufficiently high pressures the bands overlap again and the material is transformed from an insulator to a conductor. The

(a)

I~~ I ~~I ~~I

~

I

o

(b)

. . . EI

o

k ant ibonding

k

1/4a

D

- - - Ef

I

DOS

1/4a

Figure 1.18. (a) Band structure for a chain of equivalent H atoms that was built up from H2 molecules. The solid circles represent the atoms that contribute + Xn to the total sum, and the open circles for the atoms contributing - Xn. (b) Band structure of the linear H chain after the Peierls distortion to form H2 molecules.

pressure-induced transition from a nonmetal to a metal has been shown experimentally in some cases, iodine for example. Under extremely high pressures even hydrogen should become metallic. Modifying the band structure of a crystal is the key to achieving specific functionality.

1.10.2. Two-AND THREE DIMENSIONAL BONDS

The calculation of bonding in two or three dimensions is basically similar to the calculation for the one-dimensional chain. The difference is that the three-dimensional structure and symmetry come into play. The number k is replaced by a wave vector k

=

(kx' ky, kz). In the directions of a, b, c the components of k run from 0 to I12a, I12b, 1I2c, respectively. The magnitude of k corresponds to a wave number 1I''A and is

47

STRUCTURE, BONDING, AND PROPERTIES

48

CHAPTER 1

measured with a unit of reciprocal length. For this reason k usually is considered a vector in reciprocal space or "k space," which is introduced in detail in Chapter 6. A reciprocal vector is also a fundamental mathematical tool in diffraction physics.

The region within which k is considered (-1 /2a

:s

^kx

:s

¹ j2a for example) is defined as the first Brillouin zone. In the coordinate system of k space, it is a polyhedron.

The first Brillouin zone for a cubic primitive crystal lattice is shown in Fig. 1.19a. The symbols commonly given to certain points of the Brillouin zone are labeled. Figure 1.19b schematically show how different s orbitals interact with each other in a two-dimensional square lattice. Depending on the k values, i.e., for different points in the Brillouin zone,

s

Px

P

y (a)

.;' .;'

x r

I

. ; ' - - -I

0*,1t

0*,1t

(c)

i

_E

X kx=1/2a, ky=O

cr,1t

cr*, 1t*

x

kx=O, ky=1I2a X'

cr*, 1t*

cr,1t

r

M kx= 1I2a, ky= 1I2a

cr,1t*

cr, 1t*

Figure 1.19. (a) The First Brillouin zone for a cubic primitive crystal lattice. The various points represent different k vectors. (b) Combinations of s and p orbitals in a square net, and (c) the resulting band structure.

(Reprinted with permission from Wiley & Sons, Inc.).

different kinds of interactions are produced. Between adjacent atoms there are only bonding interactions at 1, and only antibonding interactions at M. The wave function corresponding to 1, therefore, is the most favorable one energetically, and the one corresponding to M the least favorable. At X every atom has two bonding and two antibonding interactions with adjacent atoms, and its energy level is intermediate between those of 1 and M. It is hardly possible to visualize the energy levels for the entire Brillouin zone, but one can plot diagrams that show how the energy values run along certain directions within the zone. This has been done in the lower part of Fig.

1.19b for three directions (1 ---+ X, X ---+ M, and 1 ---+ M). The Px orbitals oriented perpendicular to the square lattice interact in the same way as the s orbitals, but the 1t- type interactions are inferior, so the bandwidth is smaller. For Px and py orbitals the situation is somewhat more complicated, because ^0' and 1t interactions have to be considered between adjacent atoms (Fig. 1.19b). For instance at 1 the Px orbitals are ^0'- antibonding, but 1t-bonding. At X Px and py are different, one being 0'- and 1t-bonding, and the other ^0'- and 1t-antibonding.

In a three-dimensional structure (for example a cubic primitive lattice) the situation is similar. To use stacking square nets and to consider how the orbitals interact at different points of the Brillouin zone, we can obtain a qualitative picture of the band structure.

1.11.

MIXED VALENT COMPOUNDS AND FUNCTIONAL MATERIALS The band structure of a compound is the key for determining its properties. The band structure of a crystal can be modified by varying the lattice constants, symmetry, and/or ligand bonding. Changing oxidation state of a portion of metal ions to form a mixed valence system may be a significant factor to induce the variation of a band structure and property. For example, W03 (or W^V10 3) and LiW03 (or LiW^V0 3) are insulators, while LixW03 (or LixW~W~x03' with 0.1 <x < 0.4) is a conductor. LaMn03 (or LaMnIl103) is antiferromagenetic but LaxMn03 (LaxMn;vMf~x03' with 0.16 < x < 0.3) is ferromag- netic. The most obvious feature of many mixed valent compounds is the existence of an intensive absorption characteristics of visible light, so they usually have colors. But the compounds containing a single metallic valent state are usually colorless. This is a simple feature for distinguishing the compounds with mixed valences from those with a single valent state.

The mixed valence usually describes inorganic or metal-organic compounds in which an element exhibits more than one oxidation state as determined by the number of valence shell electrons associated with each atom. Sodium, for example, has one electron in valence shell (2s22p⁶3s1), but manganese has seven electrons in valence shell (3s23p⁶3d⁵4s2). The oxidation state of manganese can vary from 2 to 7 under different conditions. In the period table of fundamental elements the d- and [shell elements generally have more than one oxidation state. Therefore, more than 40 elements have the ability to form oxides with mixed valence states. The oxidation state depends on the chemical environment of the element. In solids the ligand radical plays a significant role in bonding. In some compounds covalency or electron delocalization between the constituent atoms is so great that we cannot judge, even approximately, how many valence shell electrons should be assigned to each kind of atom. Determination and creation of mixed valence system is an important field particularly for functional

49

STRUCTURE, BONDING, AND PROPERTIES

so

CHAPTER 1

materials. The analysis of valence state using electron energy loss spectroscopy will be discussed in Chapter 8.

The mixed valence system can be classified into three groups (Robin and Day, 1967). The basis for this classification relies on the simplicity and unambiguity for distinguishing the two sites occupied by elements with different valence states. If we use wave function and energy level to illustrate the mixed valence system, a valence delocalization coefficient av is introduced. We consider a compound with the octahedron and tetrahedron coordinated polyhedra. Each site is occupied by an ion with two valences: the A site, for example, is occupied by IV valence ion, A(IV), the B site by III valence ion, B(IIl). Then the valence bond configuration A(IV)B(III) should have an energy different from A(III)B(IV). If there exists a suitable perturbation matrix element to mix the two configurations, the ground state wave function is a linear combination of the two:

(1.19) where 0 ::: av ::: 1. The following classifications are given based on the value of avo 1.11.1. CLASS I COMPOUNDS: av

=

0

The valence state is firmly trapped in the state of A(III)B(IV). Metal ions in the ligand field have very different symmetry and/or strength. The different valence ions can be easily distinguished by coordination polyhedra. There is no valence electron transition between valence-different ions in the visible light region. This type of compound usually is magnetically dilute, paramagnetic, or diamagnetic at very low temperatures and an insulator with a resistivity of 10^1O0hmcm or greater. GaCl2 is a typical example. The environments of the two Ga sites in GaCh can be described as follows: Half of the Ga atoms are surrounded by tetrahedra of Cl atoms at distance about O.219nm and the other half by an irregular dodecahedron of Cl at distances ranging from 0.32 to 0.33 nm. From the known coordination preferences of Ga(I) and Ga(ill) it is clear that the tetrahedral site is Ga(IlI) and the dodecahedral one is Ga(l). GaCl2 is colorless, diamagnetic, and insulative. Another example is orthorhombic cervantite a-Sb204• There are two coordinated sites: one pentavalent antimony bonded to six oxygen atoms at the corners of a distorted octahedron forms a corrugated sheet by sharing parallel edges, and the other trivalent antimony has fourfold coordination of oxygen atoms joined with the octahedron sheet. It is also an insulator.

1.11.2. CLASS II COMPOUNDS: av > 0 BUT SMALL

The valences of ions are distinguishable, but with a slight delocalization. Metal ions are located in ligand fields of nearly identical symmetry. The difference between the valence-different ions is the distortion of the coordination polyhedron by a few tenths of an angstrom. It can have one or more mixed valence transition in the visible light region.

The compounds are magnetically dilute with both ferromagnetic and antiferromagnetic interactions at low temperatures. Usually this kind of compound is a semiconductor with resistivity in the range 10_10⁷ ohm em. A typical example is Prussian blue, Fe4[Fe(CN)6h ·xH20 (x= 14-16) which has a fcc unit cell. The iron ions have two valence states: Fe(Il) and Fe(IIl). In the fcc lattice the metal sites are occupied alternatively by Fe(I!) and Fe(III) bridged with cyanide in a regular array. In a compound

with 25% of Fe (II) sites vacant, an inherent structural disorder is formed. The Fe(II) has octahedron coordination, but Fe(III) is in a mixed nitrogen (NC)-oxygen(H20) environment with an average composition of Fe(III)N4.s01.5. It is a semiconductor, and the blue color is due to the valence electron transition between Fe (II) and Fe(III).

EU3S4 and Ti407 are other examples.

1.11.3. CLASS III COMPOUNDS: r:lv

=

^r:lmax

There are two different cases: in case IlIa r:lv is maximal locally, and in case Illb r:lv is maximal but completely delocalized over the cation sublattice. In IlIa the metal ions are indistinguishable but grouped into polynuclear clusters. It is magnetically dilute and has valence electron transition between the valence-different ions in the visible light region.

This kind of compound is probably an insulator but changeable. In case IIIb, all metal ions are indistinguishable. It is either ferromagnetic with a high Curie temperature or diamagnetic, depending upon the presence or absence of local magnetic moments. The Curie temperature Tc is a critical temperature above which a ferromagnetic material becomes paramagnetic since thermal motion inhibits the orientations of the magnetic moments. This type of compound has metallic conductivity with resistivity in the range 1O-2-10-60hmcm and optical absorption edge in the infrared, opaque with metallic reflectivity in the visible light region. A typical example is Nax W03 (0.4 < x < 0.9). The W03 matrix has the Re03 structure, with W06 octahedra sharing vertices, while Na occupies some ofthe cube centers. Nuclear magnetic resonance (NMR) measurements on 23Na nucleus show that Na is present as Na+; thus, electrons have been donated to the t2g

orbitals ofW. But the W sites are all crystallographic ally equivalent so that no distinction between W(VI) and W(V) is made. If the ^r:lv value is the maximum, the valence electrons are delocalized. The bronzes are, therefore, metallic, their specific conductivity is proportional to x, and their optical and magnetic properties are typical of metals.

Another series of class III compounds has chains of mixed valence metal atoms. For example, K2Pt(CN)4Bro.33H20 (KCP) is a molecular metal. In this compound the Br- are nonstoichiometrically distributed in channels between the chains of square Pt(CN) groups and stacked plane to plane. At room temperature the Pt atoms are equally spaced, but their average oxidation state is

+

2.3. In becoming oxidized,

+

^2(d8) electrons are removed from the dz² orbitals so that KCP is a metallic conductor along the Pt chains.

When the electric vector of the incident light is parallel to the chairts the optical properties are similar to those of a metal, but when perpendicular to the chains the crystal is transparent. The one-dimensional conductivity is unstable due to lattice distortions, which introduce a long-wavelength periodicity into the chains. Based on band theory this has the effect of opening a gap at the Fermi surface, rendering the material semiconducting.

The valence delocalization coefficient r:lv is related to bridging ligands and molecular vibration. Changing a ligand can affect the site potential of the cation with different valences and the strength of interaction between the cation sites as well. The oxygen vacancy, for example, changes the coordination situation, causing an increase or decrease in the distance between the cations and possibly leading to a modification in the band structure and the electron transition between the mixed valence cations. As a result, the compound may change from an insulator to a conductor.

In summary, compounds with mixed valences are classified into three groups. In class I compounds, the coordination sites are quite different from each other and the

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STRUCTURE, BONDING, AND PROPERTIES