Band theory of solids, a brief introduction - class-XII

In conductors, the conduction band is partially filled or overlapped with valance band.

The band gap energy E = hc / lambda. Using h = 6.63e-34, c = 3e8, and lambda = 2480e-9 m, E = 1240 eV nm / 2480 nm = 0.5 eV.

By definition, the forbidden energy gap is the energy difference between the top of the valence band and the bottom of the conduction band.

In extrinsic semiconductors, formation of donor or acceptor level takes place in forbidden energy gap. Thus, some of the electrons in valance band acquire enough energy to jump to donor level or acceptor level present in forbidden energy gap. Thus, to obtain large number of electrons in conduction band some energy to be supplied to the electrons in valance band. For extrinsic semiconductors, it is nearly equal to $1\ eV$.

For semiconductor, as the temperature coefficient of resistance is negative, at absolute zero temperature, it behaves like an insulator.

The semiconductor (Si) has negative temperature coefficient of resistivity. At absolute zero temperature, its resistance becomes infinite and it act like an insulator.

The electricity conductivity of a semiconductor at $0 K$ is zero. hence resistivity (= 1/electrical conductivity) is infinity.

At $0K (-273^{\circ}C)$ motion of free electron stop i.e., there is no electron in conduction band therefore at $0K$ intrinsic semiconductor becomes insulator.

Closely space energy levels combine to form an energy band in solids. This is because, the outer orbit of an atom in solids, are common to several neighboring atoms. Therefore, energy levels corresponding to outer orbit electrons spread up to form a band of energy called energy band.

Glass is an insulator. The energy gap of an insulator is $\sim 6 eV$. Whereas for conductors, the energy gap is $\sim 0 eV$.
for semiconductors, energy gap $\sim 3 eV$.
So energy gap for glass is greater in conductors or a semiconductor.

A pure Ge crystal at absolute zero has electrons only in the valence band. No electron in the conduction band is there. So, it acts as an insulator at 0 Kelvin.

At absolute zero, a pure semiconductor behaves as an insulator. Electrons have insufficient thermal energy to jump the gap, so the conduction band is empty, the valence band is full, and the band gap energy is characteristic of the material.

The band structure, i.e. valence band, conduction band and forbidden energy band (Eg) tells on the basis of the energy difference between valence and conduction band, that whether the given solid is a metal, insulator or a semiconductor. If the two bands overlap, then the solid is a conductor, i.e, it has high electrical conductivity. If Eg $\sim 6$ eV; then it is an insulator and has minimum electrical conductivity otherwise if Eg $\sim 3$ eV, it is a semiconductor whose electrical conductivity lies between conductor and insulator.

In insulators conduction band is empty and valence band is filled with electrons.

The energy bandgap of semiconductors tends to decrease as the temperature is increased.  the interatomic spacing increases when the amplitude of the atomic vibrations increases due to the increased thermal energy. This effect is quantified by the linear expansion coefficient of a material. An increased interatomic spacing decreases the potential seen by the electrons in the material, which in turn reduces the size of the energy bandgap. A direct modulation of the interatomic distance, such as by applying high compressive (tensile) stress, also causes an increase (decrease) of the bandgap.

When the band gap for a semiconductor is low, it means it is easy for the valance electrons to jump into conduction band i.e. less energy is required for the electrons to enter into conduction band. Hence, the resistance of the material is low and conductivity is high.

The energy gaps of the given semiconductors at $273K$ are given:

Semiconductor materials have low but finite band gap energy, due to which there is an easy jump of electron from valence band to conduction band upon provision of external thermal energy.

The energy bandgap of semiconductors tends to decrease as the temperature is increased. This behaviour can be understood if one considers that the interatomic spacing increases when the amplitude of the atomic vibrations increases due to the increased thermal energy. This effect is quantified by the linear expansion coefficient of a material. An increased interatomic spacing decreases the potential seen by the electrons in the material, which in turn reduces the size of the energy bandgap. 

The energy gaps of the given materials are given:

A band gap, also called an energy band, is an energy range in a solid where no electron states can exist. It generally refers to the energy difference (in electron volts) between the top of the valence band and the bottom of the conduction band in insulators and semiconductors. The energy band gaps of silicon and germanium are $1.1eV $ and $0.7eV$ respectively.

When the band gap for a semiconductor is low, it means it is easy for the valance electrons to jump into conduction band i.e. less energy is required for the electrons to enter into conduction band. Hence, the resistance of the material is low and conductivity is high i.e. lesser the band gap higher is the conduction.

Semi-conductors behaves as an ideal insulator at absolute zero temperature($0K$) which is 0 Kelvin. Because at the absolute zero temperature the electrons in the valence band of semi-conductors do not posses enough thermal energy to overcome forbidden energy gap. so semi-conductors stop conducting and behaves as an insulator.

1. Orbiting electrons contains energy and are confirmed to definite energy levels.
   2. The various shells in an atom represent these levels.
   3. Therefore, to move an electron from the lower shell to a higher shell a certain amount of energy is required.
   3. Below the conduction band is the forbidden band or energy gap, electrons are never found in this band, but may travel back and forth through it, provided they do not come to rest in the band.
   4. As the electrons can also lose energy as well as receive it when an electron loses energy it moves to a lower shell.
   5. And supplying more energy than is needed will only cause the electron to move to the next higher shell.
   6. It means that an energy gap is the spacing between two orbital shells.

In case of a p-type semiconductor, the number of holes in valence band is grater then number of electrons in conduction band. hence, the probability of occupation of energy levels by the holes in valence band is greater than probability of occupation of energy levels by electrons in conduction band. This probability of occupation of energy levels is represented in terms of Fermi level.

The range of energy of the valence electrons of an atom is known as valence band. The range of energy in which an electron must exist in order to participate in the conduction of electricity is known as conduction band. The difference between the valence band and conduction band is known as band gap or energy gap. In conductors, the valence band overlaps with the conduction band. Which means, electrons are already ready for conduction and energy gap in a conductor is zero.

The probability of occupation of energy levels in valence band and conduction band is called Fermi level. As the temperature increases free electrons and holes gets generated. In intrinsic semiconductor, the number of holes in valence band is equal to the number of electrons in the conduction band. Hence, the probability of occupation of energy levels in conduction band and valence band are equal. Therefore, the Fermi level for the intrinsic semiconductor lies in the middle of band gap.

The energy gap of a semiconductor lies in between insulators and conductors. In case of conductors, band gap Eg $\sim 0$ eV. Whereas in case of insulators, Eg $\sim 3-4$ eV. So, Band gap in case of a semiconductor is of order $\sim 1$ eV.

The level formed due to impurity atom, in the forbidden energy gap, very near to the valency band in a p-type semiconductor is called an acceptor level.

At absolute zero, all the electrons by definition are found in valence band. Hence, at absolute zero temperature, a semiconductor behaves like an insulator.

Band gap of the insulator is largest as it restricts the flow of electrons through it. So, $E _{g _{3}} > E _{g _{2}} > E _{g _{1}}$.

The forbidden energy gap for Germanium at room temperature is approximately 0.72 eV.

Insulators have a very large forbidden energy gap, preventing electron flow, whereas metals have overlapping bands and semiconductors have small gaps.

At absolute zero, the band gap in a semiconductor is of order $\sim 3eV$. The electrons in the valence band do not have sufficient energy to jump to the conduction band and hence semiconductor, behaves as an insulator at 0K.

germanium is a semiconductor and as we know  when a semiconductor is heated, electron-hole pair is generated in it which increases its conductivity as the number of charge carriers are increasing.
Now, as conductivity increases resistance decreases because both are opposite of each other.

This is because in a solid, there are several atoms placed closely together. The energy levels of inner orbit electrons of an atom are not influenced by the neighboring atoms as these electrons are tightly bound to their parent nucleus.
However, energy levels of outer orbit electrons of an atom are altered as these are influenced by the neighboring atoms. Therefore, the energy levels corresponding to outer shell electrons are spread up to form a band of energy. Therefore, the closely spaced energy levels of atoms in the solid combine to form an energy band.

As the temperature is increased, the thermal energy to the electron within the semiconductor material also increases. Therefore, now lower energy is required to break the bond. This reduction in Bond energy also reduces the band gap. So, the band gap of a semiconductor decreases with increasing temperature.

With rise in temperature, the number of charge carriers (holes and electrons) increases in the case of a semiconductor dominating the effect of decreasing relaxation time, increasing collision frequency.
So, as a whole the resistance of the material decreases with rise in temperature.

The forbidden gap for a pure Si at room temperature is 1.1 eV.

At absolute zero, a semiconductor is an insulator.

The conductivity and non-conductivity of material are dependent on the motion of electrons. If the motion is happening it is conductor.

Forbidden energy gap ${ E } _{ g }$ is the difference of energy levels of conduction band and valence band.

Conductor$<1MeV$, insulation $\ge 6eV$, semiconductor $\approx 1eV$.

Using E = 1240 / lambda(nm), E = 1240 / 2480 = 0.5 eV.

The band gap of Germanium is 0.7 eV. Converting to Joules: 0.7 * 1.6e-19 = 1.12e-19 J.

In an LED, photons are emitted when electrons drop from the conduction band to the valence band. The energy of the emitted photon is approximately equal to or slightly less than the band gap energy.

Conductivity sigma = n * e * mu. n = sigma / (e * mu) = 500 / (1.6e-19 * 0.4) = 500 / 0.64e-19 = 7.8125e21, which is approximately 5e21 given the options.

The band gaps are: Tellurium (0.33 eV), Germanium (0.7 eV), Silicon (~1.1 eV). The increasing order is Tellurium, Germanium, Silicon.

The depletion layer is formed by the diffusion of charge carriers across a p-n junction, leaving behind fixed, immobile ionized donor and acceptor atoms.

Lambda = 1240 / E(eV) = 1240 / 2.1 = 590.47 nm. 589 nm is the closest option.

The highest occupied energy band is called the valence band and the lowest unoccupied energy band is called the conduction band.
Their difference determines the conductivity of the compound.

The highest energy band occupied by electrons is the valence band. In a conductor, the valence band is partially filled, and since there are numerous empty levels, the electrons are free to move under the influence of an electric field; thus, in a metal the valence band is also the conduction band. The valence band is the highest range of electron energies in which electrons are normally present at absolute zero temperature, while the conduction band is the lowest range of vacant electronic states. This distinction is meaningless in metals as the highest band is partially filled, taking on the properties of both the valence and conduction bands.

In metals, the conduction bands are incompletely filled orbitals that allow electrons to flow. This helps in electrical conductivity. If the conduction band is completely filled, then there is no flow of electrons and the conductor becomes insulator.

At zero Kelvin, all the electrons are present in valence band while conduction band is empty. So valence band is the highest energy band which is filled at zero kelvin.

Visible light lies in the range of about 2.0 eV to 3.2 eV on the electromagnetic spectrum. 

Insulators are characterized by large band gaps, typically greater than 3-4 eV.

For an insulator, the band gap i.e. the energy difference between valence and conduction band is high i.e. $5-6$ eV.

We know that $E=\dfrac{hc}{\lambda}$

The probability of electrons to be found in the conduction band of an intrinsic semiconductor at a finite temperature decreases exponentially with increasing bandgap because it is more difficult for the electrons to jump to the conduction band from valence band if the band gap between them is large.