Bonding in solids

The interaction of electrons in neighboring atoms of a solid serves the very important function of holding the crystal together.

For example Nacl

In the Nacl lattice, each Na atom is surrounded by 6 nearest neighbor Cl atoms and vice versa. The electronic structure of Na ( Z=11) is [Ne] 3S1, Cl ( z=17) has the structure [Ne] 3S2 3P5. In the lattice each Na atom gives up its outer 3S electron to a Cl atom, so that the crystal is made up of ions with the electronic structure of the inert atoms Ne & Ar ([Ne] 3S23P6).

However, the ions have net electric charges after the electron exchange. The Na+ ion has a net +ve charge, having lost an electron, and the Cl- ion has a net –ve charge, having gained an electron.

Each Na+ ion exerts an electrostatic attractive force upon its ‘6’ Cl- neighbors and vice versa. These columbic forces pull the lattice together until a balance is reached with repulsive forces.

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 & Cl atoms to form the Na+ and Cl- ions, the outer orbits of all atoms are completely filled. Since the ions have the closed shell configuration of the inert atoms Ne & 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, usually by no more than ‘3’ electrons.

For example the alkali metal ( 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 conductivity. 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 +ve ion cores and the surrounding free electrons. This is metallic bonding. The metals have the sea of electrons 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 semiconductors. Each atom in the Ge, Si or C diamond lattice is surrounded by four nearest neighbors, each with ‘4’ neighbors. The bonding forces arise from a quantum mechanical interaction between the shared electrons. This is known as covalent bonding.

As in the case of the ionic crystals, no free electrons are available to the lattice in the covalent diamond structure. By this reasoning Ge & Si should also be insulators. But we shall see an electron can be thermally or optically excited out of a covalent bond and thereby become free to participate in conduction. This is an important feature of semiconductors.

Energy bands

As isolated atoms are brought together to form a solid, various interactions occur between neighboring atoms. The forces of attraction and repulsion between atoms will find a balance at the proper inter-

atomic spacing for the crystal. In the process, important changes occur in the electron energy level configurations and these changes result in the varied electrical properties of solids.

Si14:-1s2 2s2 2p6 3s2 3p2

In the outermost shell or valence shell, n=3, two 3s and two 3p electrons interact to form the four ‘hybridized’ sp3 electrons when the atoms are brought close together.

Solving the Schrödinger equation for such an interacting system, the composite two electron wave functions are linear combinations of the individual atomic orbital (LCAO).

When the atoms are brought together, the application of the Pauli Exclusion Principle becomes important. When two atoms are completely isolated from each other so that there is no interaction of electron wave functions between them, they can have identical electronic structures. As the spacing between the two atoms becomes smaller, electron wave functions begin to overlap.

The exclusion principle says that no two electrons in a given interacting system may have the same quantum state. Thus there must be at most one electron per level after there is a splitting of the discrete energy levels of the isolated atoms into new levels belonging to the pair rather than to individual atoms.

In a solid, many atoms are brought together so that the split energy levels form continuous bands of energies. Each isolated silicon atom has an electronic structure 1s2 2s2 2p6 3s2 3p2 in the ground state. If we consider N atoms, there will be 2N, 2N, 6N, 2N and 6N states of type 1s, 2s, 2p, 3s, 3p respectively. As the inter-atomic space decreases, these energy levels split into bands, beginning with the outer (n=3) shell. The 3s and 3p bands merge into a single band composed of a mixture of energy levels. This band of 3s-3p levels contains 8N available states. As the distance between atoms approaches the equilibrium inter-atomic spacing of silicon, this band splits into two bands separated by an energy gap Eg.

The upper band called the conduction band contains 4N states, as does the lower band called the valence band. Thus apart from the low lying and tightly bound core levels, the silicon crystal has two bands of available energy levels separated by an energy gap Eg which contains no allowed energy levels for electrons to occupy. This gap is sometimes called a ‘forbidden band’, since in a perfect crystal it contains no electron energy states. The lower 1s band is filled with the 2N electrons. 2s and 2p bands will have 2N and 6N electrons in them respectively. However, there are 4N electrons in the original isolated (n=3) shells (2N in 3s states and 2N in 3p states). These 4N electrons must occupy states in the valence band or the conduction band in the crystal. At 0K, the electrons will occupy the lowest energy states available to them.

In the case of the Si crystal, there are exactly 4N states in the valence band available to the 4N electrons. Thus at 0K, every state in the

valence band will be filled, while the conduction band will be completely empty of electrons.

The Bloch theorem

The one dimensional Schrödinger equation for an electron moving in a constant potential is

) ψ=0

The solutions to this equation are plane waves of the type

Ψ(x) = where ( )=

For an electron moving in one dimensional periodic potential , the Schrodinger equation is written as

ψ=0 ------------------------ (1)

Since the potential is periodic with period equal to the lattice constant “a”, we have

There is an important theorem known as Bloch theorem or the Floquet’s theorem concerned with the solutions to the wave equation

ψ=0

According to Bloch theorem the solutions to this equation are plane

waves of the type Ψ(x) = which are modulated by a function having the same periodicity as that of the lattice.

Thus the solutions are of the form Ψ(x) = with

Ψ(x) = ---- The wave functions of this type are called Bloch functions.

The Kronig-Penney model

This model illustrates the behavior of electrons in a periodic potential by assuming a relatively simple one dimensional model of periodic potential.

The potential energy of an electron in a linear array of positive nuclei is assumed to have the form of a periodic array of square well with period of (a+b). At the bottom of a well i.e. for , the electron is assumed to be in the vicinity of a nucleus and the potential energy is taken as zero where as outside a wel l i.e. , the potential energy is assumed to be .

The wave functions are obtained by writing the Schrodinger equations for the two regions as

=0,

) ψ=0, )

Assuming that the energy E of the electrons is less than , we define two real quantities as

= , =

Putting these values we can write

Since the potential is periodic, the wave functions must be of the form of Bloch function

Ψ(x) = where is the periodic function in x with periodicity of (a+b).

i.e.

+

= -

Where A, B, C, and D are constants which can be determined from the following boundary conditions –

,

,

From these boundary conditions, we will obtain 4 linear homogeneous equations.

These equations are used to determine the constant A,B,C &D . A non zero solution to these equations exists only if the determinant of the coefficients of A,B,C & D vanishes. i.e

On solving this determinant-

To simplify this equation, Kronig & Penney considered the case when tends to infinity and approaches to zero but the product

remains finite i.e. the potential barriers become delta functions.

Under these circumstances, the model is modified to one of a series of wells separated by infinitely thin potential barriers. The quantity

is called the barrier strength.

As

Let’s define a quantity which is a measure of the area of

the potential barrier.

Thus increasing has the physical meaning of binding an electron more strongly to a particular potential well.

This is the condition which must be satisfied for the solutions to the wave equation to exist.

Since lies between , the left hand side should assume only those values of for which its value lies between . Such values of , therefore represent wave like solutions of the type of

equation and are allowed. The other values of are not allowed.

This is a plot of vs

The vertical axis lying between -1 and +1 as indicated by the horizontal lines represents the values acceptable to the left hand side i.e.

.

Following conclusions may be drawn-

1. The energy spectrum of the electrons consists of alternate regions of allowed energy bands and forbidden energy bands.

2. The width of the allowed energy bands increases with or the energy.

3. The width of a particular allowed energy band decreases with increase in the value of ‘p’ i.e. with increase in the binding energy of the electrons.

As p , the allowed energy bands are compressed into energy levels and a line spectrum is resulted.