1 BONDING FORCES IN SOLIDS

1 Bonding Forces in solids

Solids are made up of atoms which are held together by intramolecular bonds. The interaction of
electrons in neighbouring atoms of a solid helps in holding the crystal together called as bonding.
Types of bonding
ˆ Ionic Bonding

ˆ Metallic Bonding

ˆ Covalent Bonding

1.1 Ionic bonding

Ionic bonding is found in compounds that are composed of both metallic and non-metallic elements.
Atoms of a metallic element easily give up their valence electrons to the non-metallic atoms. In
the process all the atoms acquire stable or inert gas configurations. By accepting or donating an
electron, atoms become ions. Example: NaCl. Na (Z = 11) gives up its outermost shell electron to
Cl (Z=17) atom, thus the crystal is made up of ions with the electronic structures of the inert atoms
Ne and Ar. An electrostatic attractive force is established, and the balance is reached when this
equals the net repulsive force. This pulls the lattice together. All the electrons are tightly bound to
the atom. Since there are no loosely bound electrons to participate in current flow, NaCl is a good
insulator.

Figure 1: Ionic bonding in NaCl

1.2 Metallic Bonding

In a metal atom the outer electronic shell is only partially filled, usually by no more than three
electrons. Example: Na atoms have only one electron in the outer orbit. This electron is loosely
bound and is given up easily in ion formation. This accounts in high electrical conductivity. The
electrons in metal atoms are loosely held by nucleus in the outermost energy levels. The metal is
considered as a cluster of free electrons, forming electron cloud. These electrons are free to move
about the lattice under the influence of an electric field.

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1.3 Covalent Bonding 2 PAULI EXCLUSION PRINCIPLE

Figure 2: Metallic bonding

1.3 Covalent Bonding

The sharing of valence electrons between two or more atoms produces covalent bonding. This
type of bonding is exhibited in column IV elements in periodic table. Column IV elements have
four electrons in the valence shell. Example: Silicon has fewer than the required number of eight
electrons needed in the outer shell. Its atoms are united with other atoms until eight electrons are
shared. This gives each atom a total of eight electrons in its valance shell; four of its own and
four borrowed from surrounding atoms. Due to this sharing process, the valance eelctrons are held
tightly together. In covalent bonding, No free electrons are available to the lattice. Therefore, pure
Si and Ge are considered as insulators at 0 K. By proper application of temperature or electrical
field, these electrons can break the covalent bond and become free.

Figure 3: Covalent Bonding

2 Pauli Exclusion principle

ˆ Electrons in a single atom occupy discrete levels of energy

ˆ No two energy levels or states in an atom can have same energy.

ˆ Each energy level can consist of atmost two electrons.

ˆ If two or more atoms brought together, their outer energy levels(valance) levels must shift
slightly. So, they are different from one another.

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 3 ENERGY LEVELS IN SI AS A FUNCTION OF INTERATOMIC SPACING

3 Energy levels in Si as a function of interatomic spacing

Figure 4: Energy levels in Si as a function of interatomic spacing.

ˆ In a solid, when many atoms brought together, the energy levels will split to form continuous
band of energies. Each isolated silicon atom has an electronic structure 1s2 2s2 2p6 3s2 3p2
ˆ Each atom has available two 1s states, two 2s states, six 2p states, two 3s states, six 3p states.

ˆ If we consider N atoms, there will be 2N, 2N, 6N, 2N, and 6N states of type 1s, 2s, 2p, 3s, and
3p, respectively. As the interatomic spacing decreases, these energy levels split into bands,
beginning with the outer (n = 3) shell.
ˆ As the 3s and 3p bands grow, they merge into a single band composed of a mixture of energy
levels. This band of 3s3p levels contains 8N available states. As the distance between atoms
approaches the equilibrium interatomic spacing(At equilibrium the net forces are zero) of
silicon, this band splits into two bands separated by an energy gap Eg.

ˆ The upper band (called the conduction band) contains 4N states, as does the lower (valence)
band.
ˆ 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 wide, 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.
ˆ There were 4N electrons in the original isolated n = 3 shells (2N in 3s states and 2N in 3p
states).

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 4 METALS, SEMICONDUCTORS, AND INSULATORS

ˆ These 4N electrons must occupy states in the valence band or the conduction band in the
crystal. At 0 K the electrons will occupy the lowest energy states available to them.
ˆ In the case of the Si crystal, there are exactly 4N states in the valence band available to the
4N electrons. Thus at 0 K, every state in the valence band will be filled, while the conduction
band will be completely empty of electrons.

4 Metals, Semiconductors, and Insulators

4.1 Metals
In metals the bands either overlap or are only partially filled. Thus electrons and empty energy
states are intermixed within the bands so that electrons can move freely under the influence of an
electric field. Metals have high electrical conductivity.

4.2 Insulators
A material with fully occupied or empty energy bands is then an insulator. This is the case when
the gap energy exceeds 9eV, because for such gaps, the thermal energy at 300K ( 25 meV) is clearly
insufficient to allow electrons from the valence band to be promoted to the conduction band. Eg >
9eV in insulators

4.3 semiconductors
ˆ Electrons to experience acceleration in an applied electric field, they must be able to move into
new energy states. This implies there must be empty states (allowed energy states which are
not already occupied by electrons) available to the electrons.
ˆ In a silicon band structure at 0 K, the valence band is completely filled with electrons and the
conduction band is empty.
ˆ There can be no charge transport within the valence band, since no empty states are available
into which electrons can move. There are no electrons in the conduction band, so no charge

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 6 DIRECT AND INDIRECT SEMICONDUCTORS

transport can take place there either. Thus silicon at 0 K has a high resistivity typical of
insulators
ˆ Semiconductor materials at 0 K have basically the same structure as, a filled valence band
separated from an empty conduction band by a band gap containing no allowed energy states.

ˆ The difference lies in the size of the band gap Eg , which is much smaller in semiconductors
than in insulators.
ˆ In a silicon band structure at 0 K, the valence band is completely filled with electrons and the
conduction band is empty.

ˆ There can be no charge transport within the valence band, since no empty states are available
into which electrons can move. There are no electrons in the conduction band, so no charge
transport can take place there either. Thus silicon at 0 K has a high resistivity typical of
insulators.
ˆ The relatively small band gaps of semiconductors allow for excitation of electrons from the
lower (valence) band to the upper (conduction) band by reasonable amounts of thermal or
optical energy.
ˆ An important difference between semiconductors and insulators is that the number of electrons
available for conduction can be increased greatly in semiconductors by thermal or optical
energy

5 E-k diagram
Note: Derivation done in class. Also E-k diagrams of conduction and valance band is also discussed.
Refer class notes.

6 Direct and Indirect semiconductors

The distinguishing feature of semiconductors is the location of the conduction band energy minimum
with respect to the valence band maximum on the E-k diagrams.

ˆ E= energy of an electron, k= 2π
λ wave vector or propagation constant.

In silicon and germanium, the valence band maximum does not occur at the conduction band
minimum. The valence band maximum in all semiconductors occurs at k=0, where as the conduction
band minimum for Si and Ge occurs at a different k, indicating the difference in momentum between
these two points. In gallium arsenide, the conduction band minimum and the valence band maximum
occur at k=0. Hence, GaAs is known as a direct band gap semiconductor, and Si and Ge are known
as indirect band gap semiconductors.
An electron making a smallest-energy transition from the conduction band to the valence band
in GaAs can do so without a change in k value; on the other hand, a transition from the minimum
point in the Si conduction band to the maximum point of the valence band requires some change in
k. Thus there are two classes of semiconductor energy bands: direct and indirect.

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 6 DIRECT AND INDIRECT SEMICONDUCTORS

In a direct semiconductor such as GaAs, an electron in the conduction band can fall to an empty
state in the valence band, giving off the energy difference Eg as a photon of light. Ex: Ga As, InAs,
GaSb, InSb.
An electron in the conduction band minimum of an indirect semiconductor such as Si cannot fall
directly to the valence band maximum but must undergo a momentum change as well as changing
its energy. Electron may go through some defect state (Et) within the band gap. In an indirect
transition which involves a change in k, part of the energy is generally given up as heat to the lattice.
Ex: Si, Ge This difference between direct and indirect band structures is very important for deciding
which semiconductors can be used in devices requiring light output.

Figure 5: Direct and indirect electron transitions in semiconductors: (a) direct transition with
accompanying photon emission; (b) indirect transition via a defect level.

he application to which the direct semiconductors become important is the optical device. In
this case, GaAs is a principal semiconductor used in semiconductor lasers and light-emitting diodes.

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 7 CURRENT CARRIERS—ELECTRONS AND HOLES

7 Current carriers—Electrons and Holes

When a pure semiconductor, such as silicon, which is initially at = 0K, acquires thermal energy
equal to or greater than the band gap energy , electrons are excited from the top of the valence
band and into the conduction band. They become free electrons. For every electron that leaves
the valence band, a vacancy in the covalent bonding is left behind into which another electron in
that valence band may move. When an electric field is applied to the silicon, the electrons in the
conduction band acquire velocity in a direction opposite to that of the field. Similarly, the electrons
in the valence band, which moved to fill the vacancies, gain velocity. By having the vacancy occupied
by another electron, the vacancy moves in the direction of the field. This vacancy is the hole. Thus,
both the electron and the hole cause electric current in the same direction with the hole moving in
the direction of the field and the negatively charged electron moving opposite to the direction of the
electric field.
If the conduction band electron and the hole are created by the excitation of a valence band electron
to the conduction band, they are called an electron-hole pair (abbreviated EHP).
Electrons are thermally excited from the valence band to the conduction band, leaving empty states
in the valence band. When an electric field is applied, electrons in the conduction band are acceler-
ated, and so are the electrons in the valence band as they move into the empty states. The current
density (amps/m2 ) of electrons in the valence band, JVB, can be determined by a summation of the
motion of all the electrons in the valence band as

[H]Jvb = (1)

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