Conduction in Semiconductors

At room temperature of 300°K, it requires an energy of E_G = 1.12 eV to break covalent bonds in Silicon material and E_G = 0.7 eV to break the covalent bonds in Germanium material and to produce some ‘electron–Hole pairs’.

Even at room temperature, a few of the covalent bonds will be broken, leading to equal number of electrons and Holes in Conduction Band and Valence Band, respectively. Electrons in the Conduction Band and Holes in the Valence Band, in an intrinsic semiconductor, are shown in Fig. 2.12. Small dashes represent free or conduction electrons. Holes are represented by circles in valence band.

Fig. 2.12 Energy-band diagram for an intrinsic semiconductor

Conductivity and Resistivity of Semiconductor Materials

The value of conductivity of a material gives us an estimate of the extent to which a material supports the flow of current through it. Electrical conductivity depends upon the number of electrons available in the conduction process. The concept of conductivity is useful in many engineering applications including medical electronics.

J = nqμE

Equation (2.17) derived in the previous section can also be written as

is called as conductivity of the material.

Thus, electrical conductivity of a material is defined as the ratio of current density J and electric field intensity E.

Conductivity of semiconductor materials increases with temperature, as an increase in temperature causes increase in conduction current. This is due to increase in broken covalent bonds that result in more charge carriers for current flow. So more electrons from Valence Band jump to Conduction Band with increase in temperature. The conductivity of semiconductors varies completely in the opposite way to that of metals.

Here it is found that current density (J) and field strength (E) are proportional to each other with σ as the constant of proportionality: J ∝ I and E ∝ v.

So σ has the dimensions of Siemens/m as shown below:

As already explained, semiconductors contain two types of mobile charge carriers, electrons and Holes. In semiconductors, the conductivity depends upon the concentrations and mobility of both electrons and Holes (Fig. 2.11).

Fig. 2.11 Electrons in a conducting medium

where n is the concentration (number) of electrons, p is the concentration (number) of Holes, μ_n is the mobility of electrons and μ_p = mobility of Holes.

In an intrinsic semiconductor n = p = n_i

If the values for the mobility and concentrations of electrons and Holes are known, the conductivity of the materials can be estimated.

Current Density in a Conducting Medium

Currents in metals are due to the movement of charge carriers ‘electrons’.

where I is the current in Amperes and A is the cross-sectional area of conducting medium in metre². Describing current density J as current per unit area has the advantage, since the dimensions of the conducting medium are not directly involved. Relation between current density and charge density ρ is described in the following:

Current density: Current I (Amperes) through a conductor by definition is Charge (in Coulombs)/Time (in seconds). Current is due to the movement of charges through a conducting medium in a given time. If, 1 C of charge moves through a conducting medium in 1 s, the resulting current is 1 A.

electrons carry 1 Coulomb of charge. So the movement of 6.25 × 10¹⁸ electrons for 1 s contributes to 1 A of current in a conductor.

where q is the charge of an electron and N is the number of electrons in a given volume. If the charge passes through a distance L (metres) in time T (seconds), through a conducting medium, then the velocity v with which the electrons move is L/T.

Substituting the value of T from Eq. (2.13) in Eq. (2.12), we get

where n = N/AL is the concentration of electrons that is the number of electrons per unit volume.

Using v = μE in Eq. (2.16), we get

where μ is the mobility of charge carriers.

Current density J_p due to the movement of Holes = pqμ_pE.

Conduction in conductors and semiconductors

Conduction in conductors and semiconductors

Mobility μ: In good conductors like metals, free electrons exist in abundance. They are supposed to be accelerated under the influence of electric or magnetic field as per ballistic (dynamics) laws. But in practice it is found that the electrons move with a constant velocity proportional to the field. The reason for this is the random nature of the electron movement involved in repeated collisions. The loss of energy during collisions is supplemented due to acceleration caused by the applied field E. Thus it is observed that the random motion of electrons when resolved in the direction of the field, the electrons acquire a constant speed called the drift speed v that is proportional to the field E (V /m) and velocity v is in metres/second.

where μ is the constant of proportionality. μ is called as mobility. It is measured as m²/V-s. Mobility of electrons and Holes due to the influence of electric field is given in Eq. (2.10). Because of the lighter mass of electrons, electrons have large values of mobility μ_n compared to Hole mobility μ_p.

For a given excitation energy to electrons (due to applied field strength), electrons move faster in Germanium semiconductor when compared to Silicon semiconductor, because of small forbidden band-gap energy in Germanium semiconductors. So Germanium semiconductor devices find their use in high-frequency applications.

Conduction (Inverse of Resistance) in Intrinsic Semiconductors

Purest semiconductor is known as intrinsic semiconductor. At 0°K, semiconductor behaves like an insulator, because energies of the order of E_G cannot be acquired from an electric field. At room temperature, covalent bonds in the semiconductor may be broken into a few Hole–electron pairs, contributing to current flow through the material allowing the conductivity to increase.

With respect to energy, if an electron is given additional energy, it breaks away from its covalent bond. When the free electron enters a Hole in a Valence Band, this excess energy is released as a quantum of heat or light. In turn this quantum of energy may be reabsorbed by another electron to break its covalent bond and create a new Hole–electron pair. Thus Holes and electrons appear to move. The moving charge carries form current. Ohm's law governs the conduction phenomena in conductors and resistors.

Conduction by Holes is less when compared to that of electrons because of differences in freedom of movements for Holes and electrons, based on their mobility μ. The mobility of electrons μ_n is greater than the mobility of holes μ_p because of the differences in relative masses of electrons and Holes.

Typical values of mobility of electrons and Holes in semiconductors

The mobility μ of electrons and Holes is defined as the velocity acquired by these charged particles per unit-applied electric field.

Electrical conduction by electron–Hole pairs generated by thermal energy is called intrinsic conduction in pure semiconductors, of either Silicon or Germanium.

Classification of Materials

When voltages are applied, materials offer different values of electrical resistances to the passage of currents through them. On the basis of electrical resistances, materials are classified as conductors, semiconductors and insulators.

In solids, available energy states for the electrons form ‘bands of energy levels’ instead of discrete energy levels in atoms.

Conductors: Materials with adjacent or over-lapped conduction and Valence Bands with zero forbidden band-gap energy (E_G = 0) are known as conductors. EBD for a conductor material is shown in Fig. 2.7.

Fig. 2.7 Energy-band diagrams for conductors

Initially, the energy levels in the Conduction Band are empty. But, electrons enter the Conduction Band due to increase in temperature or energy acquired from an applied electric field. Then the electrons move freely inside the Conduction Band as charge carriers with each electron carrying an electron charge q_n = 1.6 × 10^–19 C. So in a conductor, electric current can flow freely. Most familiar conductors are metals such as gold, silver and copper.

Semiconductors: Materials with small forbidden band-gap energy (E_G), around 1 eV, re called semiconductors. Silicon, Germanium and gallium arsenide are semiconductor materials. They are also known as intrinsic or pure semiconductor materials.

Semiconductor materials have some of the following features:

• Typical value of resistivity is of the order 0.6 Ω-m at the room temperature.
• The material has negative temperature coefficient of resistance. Resistance of the semiconductor material decreases with increasing values of temperatures.
• The addition or doping of trivalent or pentavalent materials to the intrinsic semiconductors (Silicon or Germanium) modulates the electrical conductivity σ of the semiconductor materials. This is the important feature for the fabrication of P- and N-type semiconductors, which are the backbone materials for semiconductor devices in electronic engineering technology.
• At 0°K, E_G0 = 1.12 eV for a Silicon semiconductor material.
• For Germanium semiconductor, E_G0 = 0.785 eV.
• E_G0 = 1.41 eV for gallium arsenide.
• At room temperature (300°K), E_G = 1.1 eV for Silicon semiconductor.
• E_G = 0.72 eV for Germanium semiconductor.
• Forbidden band-gap energy
  
  
  
  
  
• At very low temperatures, the Conduction Band is practically empty. When the temperature is increased, the electrons in the top of Valence Band acquire sufficient thermal energy and move into the Conduction Band.

E_G is forbidden band-gap energy, E_C is the energy of the lower most energy level of Conduction Band and E^V is the energy of the top most energy level of Valence Band.

Silicon semiconductor has the forbidden bandgap energy E_G = 1.12 eV (Fig. 2.8). EBD for the Germanium semiconductor material is shown in Fig. 2.9. It has band-gap energy E_G = 0.72 eV.

Fig. 2.8 Energy-band diagram for silicon semiconductor

Fig. 2.9 Energy-band diagram for germanium semiconductor

• Silicon has wider forbidden band-gap energy compared with Germanium semiconductor material. This suggests that Silicon devices work up to higher temperatures with stable operation. Silicon devices are preferable for military and tropical country applications.
• Because of smaller forbidden band-gap energy, Germanium devices are limited to lower temperature applications.
• Typical band-gap energy in semiconductors is less than 2 eV.

Insulators: The materials with large forbidden band-gap energy E_G > 6 eV do not support conduction at all. Large forbidden band-gap energy between the Valence Band and the Conduction Band shown in Fig. 2.10 suggests that no electron can reach the Conduction Band. Such materials are known as insulators.

Fig. 2.10 Energy-band diagram for insulator materials

Insulators practically have no free electrons to act as charge carriers to support electrical conduction. Non-metallic solids such as glass, porcelain and mica behave as insulators. Their resistivity is very high, while conductivity is very low.

Energy-band Concepts of Materials

Energy-band Concepts of Materials

• The electron energy levels for a single free atom in a gaseous medium are discrete, since the atoms are sufficiently far apart. So the energy levels of individual atoms are not perturbed.
• The proximity of neighbouring atoms in solid media such as crystals does not appreciably affect the energy levels of inner shell electrons. But, groups of energy levels of outer shell electrons are changed due to the influence of electrons in the neighbouring atoms. They allow sharing of electrons among them to form covalent bonds between neighbouring atoms in the process of getting on to stable ‘8-electron configuration’ in Silicon and Germanium semiconductors.

Sharing of outer shell electrons to form covalent bonds is shown in Fig. 2.5.

• Valence band The coupling between the outer shell electrons of the atoms results in a group or a band of closely spaced energy levels or states instead of the widely spaced energy levels of the isolated atoms. Because of the coupling between atoms in the crystals (As the inter-atomic distance is quite small in solid materials.) completely filled and partially filled energy levels are merged into an ‘energy band’ known as Valence Band.
• Top most energy level of Valence Band is E_V.
• Merging of empty energy levels in atoms form Conduction Band (top energy band).
• Lower most energy level of Conduction Band is E_C.
• Region between Conduction Band and Valence Band is known as forbidden band gap E_G, or band gap equal to (E_C – E_V). It decreases with temperature.
• The magnitude of the band-gap energy E_G predicts the type of the materials, such as conductors, semiconductors and insulators, which is discussed later.
• Energy-band diagrams (EBD) show the energy of electrons (in electron volts) associated with the energy levels on the y axis and the momentum (P) on the x axis. Energy of electrons is measured in eV (electron volts).
• The unit of electron volt (eV) is the energy acquired by an electron while falling through a potential difference of 1 V.

Fig. 2.5 Covalent bonds about silicon atoms

According to quantum-mechanical theory, when the energy band has all filled energy levels; electron there cannot contribute to electrical conduction. There is no open energy level to which they can move after absorbing any energy from the applied electric field. Therefore they do not absorb energy and do not become conduction electrons. Only the band containing the unfilled or empty energy levels is the Conduction band, to which electrons enter to contribute electrical conduction.

Conductivity of a pure semiconductor at ‘Absolute-Zero temperature’ is zero, since lower Valence Band is filled and there are no electrons in the upper Conduction Band.

At the ambient temperature, some electrons may acquire sufficient energy equal to or greater than the forbidden band-gap energy E_G and they will move to energy levels in the upper band. These electrons will be in an incompletely filled band and they can contribute to electrical conduction. While the electrons move to the Conduction Band, they leave Holes in the Valence Band (Holes were formed due to the formation of Hole–electron pairs during the process of breakage of covalent bands in Valence Band). Hole will have positive charge. Formation of Hole–electron pairs is shown in Fig. 2.6.

Fig. 2.6 Formation of hole–electron pair

Conductivity of ‘intrinsic semiconductor’ is due to the Hole–electron pairs formed during broken covalent bonds or due to supply of energy to free electrons to cross the forbidden band gap to enter the Conduction Band.

Resistivity of semiconductor material can be expressed as

where A is a coefficient that varies slightly with temperature, and ρ is the resistivity of the emiconductor material. It is a function of temperature T and forbidden band-gap energy ΔE_G.

Resistance across a standard mass and shape of a material at a given temperature is called the resistivity of the material. The reciprocal of resistivity is conductivity (σ).

Electronic configuration of a Germanium atom

Fig. 2.3 Electron configuration of germanium atom

Germanium semiconductor atom has ‘atomic number’ Z = 32. It has 32 positive charges in the nucleus and 32 electrons in various shells containing 2, 8, 18 and 4 electrons. Germanium atom is electrically neutral. Germanium semiconductor as a whole is electrically neutral.

• First, second and third orbits are completely filled.
• Fourth orbit (shell) is partially filled.
• Energy levels from fifth orbit onwards are empty energy levels.

Germanium atom representation is shown in Fig. 2.4. It is a basis to know the formation of covalent bonds and so on. Germanium is also considered as a ‘tetravalent’ material, as it has 4 valence electrons in its outer incomplete shell. Thus, Silicon and Germanium materials are referred as tetravalent materials with similar electrical and chemical properties.

Fig. 2.4 Representation of germanium atom with four valence electrons

Electron Configurations of Silicon and Germanium Atoms

REVIEW OF SEMICONDUCTOR PHYSICS

• The electronics subject begins from the concepts of behaviour of charge carriers in electron devices and Integrated Circuits (ICs) under influence of electric fields.
• A model of an atom is shown in Fig. 2.1. The aspect of electron motion is analogous to the planetary motion in which the planets rotate round the sun. On similar lines, electrons move in closed stationary orbits around the positive nucleus in an atom.

Fig. 2.1 Electron configuration of silicon atom

2.1.1 Electron Configurations of Silicon and Germanium Atoms

• The shell structure and states occupied by electrons depend on the valence of material and its atomic number Z. Silicon and Germanium semiconductor materials are used for the manufacture of semiconductor devices.

The distribution of electrons in the various orbits for Silicon and Germanium atoms is shown in Table 2.1 and in Figs. 2.1 and 2.3.

Table 2.1

Element	Atomic number (Z)	Configuration
Silicon (Si)	14	1s2 2s2 2p6 3s2 3p2
Germanium (Ge)	32	1s2 2s2 2p6 3s2 3p6 3d10 4s2 4p2

Electron configuration of Silicon atom (Fig. 2.1)

• The atomic number of Silicon atom is Z = 14. It contains 14 positive charges in the nucleus and 14 electrons that move about the nucleus in closed stationary orbits. The orbits are assumed to be concentric circles. Thus, each atom is electrically neutral (Zero charge for the atom as a whole). Hence, the Silicon material is an ‘Electrically Neutral material’.
• The planetary model for the atom is considered only from the classical model. Each ‘Silicon atom’ has its electrons arranged in groups of energy levels or shells as the following:
  1. First orbit, the inner most energy level has 2 electrons (completely filled).
  2. Second orbit has 8 electrons (completely filled).
  3. Third orbit has the balance of 4 electrons (partially filled).
  4. Energy levels starting from the fourth level are empty energy levels.
  5. This last partially filled shell (third orbit) is called valence shell.
  6. The 4 electrons in the third orbit (shell) are known as valence electrons.
  7. Valence electrons are responsible for the chemical and electrical properties of the material.
  8. Electrons extracted from valence shell and not subject to force of attraction of nucleus on them are called free electrons.

Silicon atom representation as a tetravalent material is shown in Fig. 2.2 as a basis to understand the concept of covalent bond formation etc. Silicon semiconductor, as a ‘Tetravalent’ material, has ‘four valence electrons’. The force of attraction between the nucleus (core) and the electron inside the atom is given by

where electronic charge q is in coulombs, r is the separation distance between electrons and nucleus (in an atom) in metres, the force F is in Newtons, ɛ₀ is the permittivity of free space in farads/metre, and permittivity of the free space ɛ₀ = 8.849 × 10^–12 farads/metre.

Fig. 2.2 Representation of silicon atom with its valence electrons

This force of attraction F between the nucleus and the electron is counter balanced by

where m is the electronic mass, m = 9.109 × 10^–31 kg, v is the speed of the electron in the orbit, acceleleration a = v²/r and is directed towards the nucleus.

Then according to Newton's second law of motion

The potential energy PE of the electron at a distance ‘r’ from the nucleus .

According to the conversation of energy, energy associated with the electrons

W = Kinetic energy + Potential energy:

where the energy W is in joules.

Substituting the value from Eq. (2.2A) into Eq. (2.3), we get

Equation (2.4) shows the relation between the radius r (distance of electron in the circular orbit from the nucleus) and the energy W of the electrons. It also shows that the energy of the electron becomes less (i.e., more negative) as it approaches closer to the nucleus. The relation is given by Eq. (2.4A):

where n = 1, 2, 3 and so on.