CHAPTER-25

Semiconducting Materials

Important points related to Semi conducting Materials

1. At room temperature the conductivity of semiconducting materials lies between conductors and Insulators

​

• There is variation in resistivity with temperature, in case of semiconductors.

​

• At absolute zero temperature semiconductors behave as insulator.

​

Basic theory of band structure in solids

BASIS OF THE BAND THEORY OF SOLIDS

1. On the basis of band theory, solids are classified as conductors, semiconductors, insulators.

​

• There is variation in resistivity with temperature, especially, in case of semiconductors.

​

• The potential experienced by an electron during its motion through the crystal is periodic, instead of being constant or zero.

In view of the above facts, it is clear that the motion of an electron in periodic potential needs more emphasis to establish the band theory of solids. The simplest quantum mechanical view of the potential experienced by an electron in passing through the crystal is perfectly periodic as shown in Fig below

ELECTRONS IN A PERIODIC POTENTIAL OF ION CORES�Kronig–Penney model and Bloch theorem yields the following results: �1. There are allowed energy bands separated by forbidden regions or band gaps. �2. The electronic energy functions E(k) are periodic in the wave vector k.

Bloch Theorem

Schrödinger wave equations for these regions are given as

ALLOWED ENERGY BANDS

ENERGY BANDS IN SOLIDS

• There are energy bands in a solid corresponding to the energy levels in an atom.

​

• An electron in a solid can have only those discrete energies that lie within these energy bands.

​

• These energy bands are known as the allowed energy bands, which are generally separated by some energy gap known as the forbidden energy bands or not allowed energy bands.

• Energy band occupied by the valence electrons is known as the valence band, and the energy band which is empty or occupied by the conduction electrons is known as the conduction band.

• Formation of energy band due to the interaction of outer energy levels of individual atoms is depicted in the following figure

CLASSIFICATION OF SOLIDS

1.Conductors (Metals)

In case of conductors, there is no forbidden gap between the valence band and the conduction band.

2. Insulators

A class of solids behaves as insulator if it satisfies the following two conditions:

(i) it has even number of valence electrons per atom and

​

(ii) the valence band and the conduction band are separated by a large energy gap compared with kT

Band formation in Pure Semiconductor (silicon)

• In order to illustrate the formation of band in pure semiconductor let us consider the case of silicon.
• Isolated atom of silicon have 14 electrons which has been shown in Fig.
• . Out of these 14 electrons, 10 occupy deep lying energy levels whose orbital radius is much smaller than the interatomic separation in the crystal.
• The four remaining valence electrons of outer energy level (n = 3) are loosely bound and can be involved in the process of conduction under the influence of external impetus.

​

​

Band formation in Pure Semiconductor (silicon) Contd.....

• Thus we need to consider only the outer shell (for n = 3) for the valence electrons.
• The 3s subshell (i.e. for n = 3, and l = 0) has two allowed quantum states per atom.
• This subshell contains two valence electrons at T = 0 K. The 3p subshell (for n = 3 and l = 1) has six allowed quantum state per atom and contains remaining two valence electrons.

+

Band structure in semiconducting materials with the concept of effective mass

• Only the conduction band and valence bands are of much interest to us here, because only these two bands contribute to the current

​

• Simple band structure of a semiconductor is shown in the Fig.
• The energy of the conduction band can be given as

​

Fermi distribution function

• Fermi surface is closely related to the free electron model.

​

• the Fermi-Dirac distribution function can be applied to deal with the charge carriers in the conduction and valence band.

​

• Fermi-Dirac distribution function can be defined as.

​

• Fermi-Dirac function F(E) is plotted versus E as shown in Fig.

​

Fermi Level and its Importance

The probability of occupying any electronic state E by an electron is given by Fermi Dirac distribution function is given as:

​

• Case I E < EF, i.e. the desired energy level is below the Fermi level and hence above equation reduced to F(E) = 1

​

• Thus probability of occupancy of any energy level below the Fermi level by carriers is 100% at T = 0 K, i.e. all the states below the Fermi level is completely filled by charge carriers at absolute zero temperature.

​

• Case II E > EF: At this condition above equation reduced to F(E) = 0 at T = 0 K from which it is clear that all the energy levels above the Fermi level is completely unoccupied by the charge carriers at T = 0 K.

​

• Thus, Fermi level can be defined as the reference level above which all the levels are unoccupied by the charge carriers.

Concentration of electrons in the Conduction Band

• The concentration of electrons throughout the conduction band can be obtained by the integration over the band

​

Concentration of Holes in the Valence Band

• The probability that a hole occupies a level E in valence band is equal to 1-F(E) since F(E) is the probability of electron occupation, so the probability of hole occupation Fh can be given as

Fh = 1-F(E) thus

​

Thus the hole concentration is given as

Temperature Dependence of Electron Concentration in Conduction Band

The concentration of electron can be evaluated explicitly using following equation

• The most important feature of this expression in that n increases very rapidly-exponentially with temperature, particularly by virtue of the exponential factor.

​

• Thus it may be concluded that as the temperature is raised, a vastly greater number of electrons are exited across the gap.

​

• A plot of log n versus 1/T has been shown in the Fig.

Effect of Doping on Band Structure and Carrier Concentration

• When small amount of impurities are doped in intrinsic semiconductor then its conductivity increases many fold.
• Due to ionization of donor impurities same number of electrons will be there in conduction band. In case of complete ionization electron density can be given as
• n = Nd
• Due to complete ionization of donor impurities the donor level ED is introduced just below the bottom of conduction band

​

Effect of Carrier Concentration on Band Structure

• The schematic diagram of density of states, Fermi-Dirac distribution function and carrier concentration is shown in Fig. 25.17(a).

​

• The schematic diagram for density of states Fermi-Dirac distribution function carriers concentration for extrinsic type semiconductors is shown in Fig. 25.17(b).

3. Semiconductors

TYPES OF SEMICONDUCTOES

1 .INTRINSIC

2 .EXTRINSIC

In case of semiconductors, the energy band gap (forbidden gap) between the filled valence band and the empty conduction band is small as compared to the insulators and more as compared to the conductors.

INTRINSIC SEMICONDUCTORS

Natural pure form of a semiconductor is known as intrinsic semiconductor

Current Conduction in Intrinsic Semiconductors

EXTRINSIC SEMICONDUCTORS

Depending on the added impurity elements in pure semiconductors, the extrinsic semiconductors are of the following two types

1. Donor or N-type semiconductor
2. Acceptor or P-type semiconductor

The semiconductor added to the impurity atoms is known as doped or extrinsic semiconductor. The added impurity may be pentavalent or trivalent. A few suitable pentavalent impurities are phosphorus, arsenic, antimony, etc., whereas trivalent elements are boron, aluminium, gallium, etc.

Donor or N-type Semiconductor

When a pentavalent atom of group V (having five valence electrons) such as phosphorus, arsenic, or antimony is added to a pure semiconductor, then the resulting extrinsic semiconductor is known as the donor or N-type semiconductor.

Acceptor or P-type Semiconductor

When a trivalent atom of III group (having three valence electrons) such as boron, aluminium, gallium, etc., is added to a pure semiconductor, then the resulting extrinsic semiconductor is known as the acceptor or P-type semiconductor.

Conductivity of Semiconductor Materials

Conductivity of N-type Semiconductors

Conductivity of P-type Semiconductors

where electrons (n) are the majority charge carriers and holes (p) are the minority charge carriers

In case of P-type semiconductors, majority charge carriers are due to the acceptor type impurity. Therefore,

P–N JUNCTION DIODE

• The contact surface formed between P-type and N-type semiconductors which are suitably joined together is separated by a thin junction known as P–N junction diode.

​

• P–N junction is usually named as a semiconductor diode because of its peculiar property that it can conduct in one direction only

Popular techniques used in the fabrication of P–N junction are as follows:

​

(i) Grown junction

(ii) Diffused junction method

(iii) Alloy junction

Depletion Layer

Behaviour of a P–N Junction under Biasing

Forward Biasing

When the positive terminal of a dc source or a battery is connected to P-type semiconductor and the negative terminal is connected to N-type semiconductor of a P–N junction, the junction is said to be forward biased.

reduction in the potential barrier

Reverse Biasing

When the positive terminal of a dc source or a battery is connected to N-type semiconductor, and the negative terminal is connected to P-type semiconductor of a P–N junction, the junction is said to be reverse biased.

Voltage–Current Characteristics of P–N Junction

The graph showing the variation of voltage across the junction (along X-axis) and current through the circuit (along Y-axis) is known as voltage–current (V–I) characteristics of P–N junction diode.

Applications of P-N Junction Diode

• P-N junction diode is frequently used in rectification (conversion of ac into dc)
• It is used in amplification
• It is used for switching purposes
• It has important applications in logic circuits
• It is used in digital circuits

ZENER DIODE

• Zener diode is a specially designed P–N junction diode, which is properly doped to have a very sharp breakdown, and is optimised to operate in breakdown region.

​

• These diodes are exclusively operated under reverse bias conditions and designed to operate in breakdown region without damage

Voltage–Current Characteristics of Zener Diode

Applications of Zener Diode

• It is used as a voltage regulator.
• It is frequently used as a fixed reference voltage in a network for biasing and comparison.
• It is used for switching purposes.
• It is used as a safety device, which avoids the damage in previous instruments due to accidental application of excessive voltage.
• It is used for the calibration of voltmeters.

VARACTOR DIODE

If P–N junction diodes are made for their applications based on the voltage variable capacitance across the junction, then these are known as varactor diodes, varicaps, or voltacaps.

Applications of Varactor Diode

• It has important applications in voltage tuning of an LC resonant circuit.

​

• It is frequently used in-self-balancing bridge.

​

• It is also used in some special types of amplifiers known as parametric amplifiers.

LIGHT EMITTING DIODES (LEDs)

Light emitting diodes are specially designed forward biased P–N junctions. When an LED is energised, it emits visible light due to the electron–hole pair recombination.

If Eg is the band gap of the semiconductor, then the energy released due to recombination of electron–hole pair is given as

Applications of LEDs

Visible radiation produced by LEDs has important applications in numerical displays such as in watches, calculators, instrument panels, telephone, switch boards, a seven segment display unit etc.

SOLAR CELLS

A suitably designed P-N junction diode which converts solar energy into electrical energy is called solar cell or solar battery. It is also known as solar-energy converter. A solar cell is simply a photodiode which is operated at zero bias voltage.

Applications of Solar Cells

• Solar cells have been and will remain the best choice for providing electrical power used in the operations of satellites and space vehicles.
• Solar cells are used for domestic power supply in remote villages.
• These are frequently used in electrification of the health-care facilities, irrigation, and water supply.
• Many lighthouses and most buoys are now powered by solar cells.
• Solar cell power stations may be approaching economic viability in location where they assist the local grid during periods of peak demand and obviate the need to construct a new power station.

PHOTOVOLTAIC CELL

Photovoltaic cells are generally used for the conversion of light energy into electricity at the atomic level. They work on the principle of photovoltaic effect.

Expression for Photovoltaic emf

total reverse current across a junction diode is given as

PHOTOCONDUCTIVITY IN SEMICONDUCTORS

When a semiconductor of suitable band gap is exposed to radiations, some radiations are absorbed by it and consequently, its conductivity is increased. This process is known as photoconductivity in semiconductors.

Cut-off Wavelength

The maximum value of the wavelength of a photon required to produce electron–hole pair in an intrinsic semiconductor is known as the cut-off wavelength

For germanium and silicon, energy band gaps are 0.72 eV and 1.1 eV, respectively, at room temperature. The critical wavelength corresponding to these band gaps for Ge and Si will be 1.72 mm and 1.13 mm, respectively.

Effect of Impurity on Photoconductivity

Addition of impurities in semiconductors introduces new energy states in the forbidden energy gap region, which reduces the energy band gap. Donor and acceptor type impurities introduce donor and acceptor levels in the energy gap region

HALL EFFECT

When a current carrying conductor (or semiconductor) is placed in a transverse magnetic field, a potential difference is developed across the conductor in the direction perpendicular to both current and magnetic field. This phenomenon is called Hall effect.

Hall Voltage and Hall Coefficient

Applications of Hall Effect

• It is used to determine the sign of charge carriers.
• Carrier concentration can be determined using Hall effect.
• It is used to measure the mobility of charge carriers directly.
• It is used to determine whether the given material is a metal, a semiconductor, or an insulator.
• It is used to measure the conductivity of a given specimen.