Basic Electronics - Semiconductor Physics Complete Study Notes

The Energy Band Theory

X-ray and other studies reveal that most metals and semiconductors are crystalline in structure. A crystal consists of a space array of atoms or molecules (strictly speaking, ions) built up by regular repetition in three dimensions of some fundamental structural unit. The electronic energy levels discussed for a single free atom (as in a gas, where the atoms are sufficiently far apart not to exert any influence on one another) do not apply to the same atom in a crystal. When atoms form crystals it is found that the energy levels of the inner-shell electrons are not affected appreciably by the presence of the neighboring atoms. However, the levels of the outer-shell electrons are changed considerably since these electrons are shared by more than one atom in the crystal. The new energy levels of the outer electrons can be determined by means of quantum mechanics, and it is found that coupling between the outer-shell electrons of the atoms results in a band of closely spaced energy states instead of the widely separated energy levels of the isolated atom as shown in figure 3. A qualitative discussion of this energy-band structure follows.

Figure 1: Illustrating how the energy levels of isolated atoms are spilt into energy bands when these atoms are brought into close proximity to form a crystal

Imagine that it is possible to vary the spacing between atoms without altering the type of fundamental crystal structure. If the atoms are so far apart that the Integration between them is negligible, the energy levels will coincide with those of the isolated atom. The outer two subshells for each element in Table 1 contain two s electrons and two p electrons. Hence, if we ignore the inner-shell levels, then, as indicated to the extreme right in figure 1(a), there are 2N electrons completely filling the 2N possible s levels, all at the same energy. Since the p atomic subshell has six possible states, our imaginary crystal of widely spaced atoms has 2N electrons, which fill only one-third of the 6N possible p states, all at the same level.

If we now decrease the interatomic spacing of our imaginary crystal (moving from right to left in figure 1(a)), an atom will exert an electric force on its neighbors. Because of this coupling between atoms, the atomic-wave functions overlap, and the crystal becomes an electronic system which must obey the Pauli exclusion principle. Hence the 2N degenerate s states must spread out in energy. The separation between levels is small, but since N is very large (~ 10²³ cm^-3), the total spread between the minimum and maximum energy may be several electron volts if the interatomic distance is decreased sufficiently. This large number of discrete but closely spaced energy levels is called an energy band and is indicated schematically by the lower shaded region in figure 1(a). The 2N states in this band are completely filed with 2N electrons. Similarly, the upper shaded region in figure 1(a) is a band of 6N states which has only 2N of its levels occupied by electrons.

Note that there is an energy gap (a forbidden band) between the two bands discussed above and that this gap decreases as the atomic spacing decreases. For small enough distance (not indicated in figure 1(a) but shown in figure 1(b)) these bands will overlap. Under such circumstances the 6N upper states merge with the 2N lower states, giving a total of 8N levels, half of which are occupied by the 2N + 2N = 4 N available electrons. At this spacing each atom has given up four electrons to the band; these electrons can no longer be said to orbit in s or p subshells of an isolated atom, but rather they belong to the crystal as a whole.Since tetravalent elements contribute four electrons each to the crystal. The band these electrons occupy is called the valence band.

If the spacing between atoms is decreased below the distance at which the bands overlap, the interaction between atoms is indeed large. The energy-band structure then depends upon the orientation of the atoms relative to one another in space (the crystal structure) and upon the atomic number, which determines the electrical constitution of each atom. Solutions of Schrodinger’s equation are complicated, and have been obtained

approximately for only relatively few crystals These solutions lead us to expect an energy-band diagram somewhat as pictured in figure 1(b). At the crystal-lattice spacing (the dashed vertical line), we find the valence band filled with 4N electrons separated by a forbidden band (no allowed energy states) of extent E_G from an empty band consisting of 4N additional states. This upper vacant band is called the conduction band.

Figure 2: Simplified energy band diagram of

(a) Insulator, (b) Semiconductor and (c) Conductor

Insulators

An insulating material has an energy band diagram as shown in figure 2(a).
It has a very wide forbidden-energy gap (~ 6 eV) separating the filled valence band from the vacant conduction band. Because of this, it is practically impossible for an electron in the valence band to jump the gap, reach the conduction band.
At room temperature, an insulator does not conduct. However, it may conduct if its temperature is very high or if a high voltage is applied across it. This is termed as the breakdown of the insulator.
Example: diamond.

Semiconductors

A semiconductor has an energy-band gap as shown in figure 2(b).
At 0°K semiconductor materials have the same structure as insulators except the difference in the size of the band gap E_G, which is much smaller in semiconductors (E_G ~ 1 eV) than in insulators.
The relatively small band gaps of semiconductors allow for excitation of electrons from the lower (valence) band to the upper (conduction) band by reasonable amount of thermal or optical energy.
The difference between semiconductors and insulators is that the conductivity of semiconductors can increase greatly by thermal or optical energy.
Example: Ge and Si.

Metals

There is no forbidden energy gap between the valence and conduction bands. The two bands actually overlap as shown in figure 2(c).
Without supplying any additional energy such as heat or light, a metal already contains a large number of free electrons and that is why it works as a good conductor.
Example: Al. Cu etc.

Semiconductor Materials: Ge, Si AND GaAs

Semiconductors are a special class of elements having a conductivity between that of a good conductor and that of an insulator.
Single crystal and compound crystal semiconductor are two ramifications of semiconductor depending upon, number of constitutional elements. Examples of single crystal semiconductors are germanium (Ge) and silicon (Si) whereas compound semiconductors are gallium arsenide (GaAs), cadmium sulphide (CdS), gallium nitride (GaN) and gallium arsenide phosphide (GaAsP) etc.

Intrinsic Materials & Covalent Bonding

Semiconductor in its purest form (without any impurity) is known as intrinsic semiconductor.
An intrinsic semiconductor (such as pure Ge or Si), has only four electrons in the outermost orbit of its atoms. When atoms bond together to form molecules of matter, each atom attempts to acquire eight electrons in its outermost shell. This is done by sharing one electron from each of the four neighboring atoms. This sharing of electrons in semiconductors is known as covalent bonding. Below figure shows covalent bonding of the silicon atom.

Figure 3: Covalent bonding of the silicon atom

Charge Carriers in Intrinsic Semiconductor

At room temperature (say 300°K) sufficient thermal energy is supplied to make a valence electron of a semiconductor atom to move away from the influence of its of its nucleus. Thus, a covalent bond is broken. When this happens, the electron becomes free to move in the crystal. This is shown in figure below.

Figure 4: Generation of electron- hole pair in an intrinsic semiconductor

(a) Crystal structure, (b) Energy-band diagram

When an electron breaks a covalent bond and moves away, a vacancy is created in the broken covalent bond. This vacancy is called a hole.
Free electrons and holes are always generating in pairs. Therefore, the concentration of free electrons and holes will always be equal in an intrinsic semiconductor

n = p = ղ_i

Where ղ_i is called the intrinsic concentration

Effect of Temperature on Conductivity of Intrinsic Semiconductor

A semiconductor (Ge or Si) at absolute zero, behaves as a perfect insulator. At room temperature, some electron-hole pairs are generated. Now, if we raise the temperature further, more electron hole pairs are generated. The higher the temperature, the higher is the concentration of charge carriers. As more charge carriers are made available, the conductivity of intrinsic semiconductor increases with temperature. In other words, the resistivity (inverse of conductivity) decreases as the temperature increases. That is; semiconductor have negative temperature coefficient of resistance.

For Intrinsic concentration

Where, E_G0: Energy gap at 0°K in eVs

k: Boltzman’s constant in eV/°K

A₀: Material constant independent of temperature

Extrinsic Materials

In addition to the intrinsic carriers generated thermally, it is possible to create carriers in semiconductors by purposely introducing impurities into the crystal. This process is called doping. It is the most common technique for varying the conductivity of semiconductors.
When a crystal is doped such that the equilibrium carrier concentrations n₀ and p₀ are different from the intrinsic carrier concentration ղ_i the material is said to be extrinsic.

n-type Semiconductor

An n-type semiconductor is created by introducing impurity elements that have five valence electrons (pentavalent), such as antimony, arsenic and phosphorus.
The effect of such impurity elements is indicated in figure below. Note that the four covalent bonds are still present. There is, however an additional fifth electron due to the impurity atom, which is unassociated with any particular covalent bond. This remaining electron loosely bound to its parent atom (antimony) atom, is relatively free to move within the newly formed n-type material.

· Since the inserted impurity atom has donated a relatively “free electron to the structure; so diffused impurities with five valence electrons are called donor atoms.

Figure 5: Antimony impurity in n-type material

When impurities or lattice defects are introduced into an otherwise perfect crystal, additional levels are created in the energy band structure, usually within the band gap. For example, an impurity from column V of the periodic table (P, As and Sb) introduces an energy level very near the conduction band in Ge or Si. Such an impurity level is called a donor level. In case of germanium, the distance of new discrete allowable energy level is only 0.01 eV ( 0.05 eV in silicon) below the conduction band, and therefore at room temperature almost all the “fifth” electrons of the donor material are raised into the conduction band.

Figure 6: Energy-band diagram of n-type semiconductor

p-type Semiconductor

The p-type semiconductor is formed by doping a pure germanium or silicon crystal with impurity atoms having three valence electrons (trivalent). The elements most frequently used for this purpose are boron, gallium and indium.

Figure 7: Boron impurity in p- type material

Note that there is now an insufficient number of electrons to complete the covalent bonds of the newly formed lattice. The resulting vacancy is called a hole and is represented by a small circle or a plus sign, indicating the absence of a negative charge.
Since the resulting vacancy will readily accept a free electron; so, the diffused impurities with three valence electrons are called acceptor atoms.
The resulting p-type material is electrically neutral for the same reasons described for the n-type material.
Atoms from Column-III (B, Al, Ga arid in) introduce impurity levels in Ge or Si near the valence band. These levels are empty of electrons at 0K. At low temperatures, enough thermal energy is available to excite electrons from the valence band into the impurity level, leaving behind holes in the valence band. Since this type of impurity level “accepts” electrons from the valence band, it is called an acceptor level.

Figure 8: Energy-band diagram of p-type semiconductor

The Mass-Action Law

In a semiconductor under thermal equilibrium (constant temperature) the product of electrons and holes concentrations is always a constant and is equal to the square of intrinsic concentration.

np=(n_i)²

The intrinsic concentration ղ_i is a function of temperature.

For a p-type semiconductor,

For an n-type semiconductor,

Direct & Indirect Semiconductors

Let us assume that a single electron travel through a perfectly periodic lattice. The wave function of the electron is assumed to be in the form of a plane wave moving, for example, in the x-direction with propagation constant k, also called a wave vector as shown below in the figure.

Figure 9: Direct Transition with accompanying photon

Figure 10: Indirect transition via a defect level

The band structure of GaAs has a minimum in the conduction band a maximum in the valence band for the same k value (k = 0). On the other hand, Si has its valence band maximum at different value of k than its Conduction band minimum. Thus 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 band gap semiconductor and
Indirect band gap semiconductor.

Comparison between these two classes of semiconductors is given below:

Direct Band Gap Semiconductor

During recombination’s energy is dissipated in the form of light.
When electron is falling from conduction band to valence band, the falling electron directly dissipates energy in the form of light.
Used for microwave devices and in fabrication of LEDs and LASERs.

Indirect Band Gap Semiconductor

During recombination’s energy is dissipated the form of heat.
When free electron is falling from conduction band to valence band, with the crystals of the atom and the crystals will be absorbing the energy from the falling electron and they become heated up and the energy is released in the form of heat.
Used in all other applications.

Transport Phenomena In Semiconductors

The only current contributing particle in case of metals are free electrons. Electrically, semiconductors on the other hand have both holes and electrons giving rise to the net current. Where we can enhance one’s (holes/electrons) contribution in comparison to another (electrons/holes) choosing a trivalent or pentavalent impurity atom for doping purpose. The transport of the charges in a crystal under the influence of an electric field (a drift current), and also as a result of a nonuniform concentration gradient (a diffusion current), is investigated.

Mobility

If a constant electric field, say E(Volts/m) is applied. As a result of electrostatic force, the electrons would be accelerated and the velocity would increase indefinite with time, were it not for the collisions with the ions. However, at each inelastic collision with an ion, an electron losses energy, and a steady state condition is reached where finite value of drift speed v is attained. This drift velocity is in the direction opposite to that of the electric field. The speed at time t between collision is at, where a = is the acceleration. Hence the average speed v is proportional to E.

Thus, v = μE

where μ (square meters per volt-second) is called the mobility of the electrons.

as v = at

Where, q is electronic charge (1.6 × 10^–19C) and m is the mass of the electron (9.1 × 10^–31 kg).

Current Density

If N electrons are contained in a length L of a conductor as shown in figure below. If T is time taken to traverse distance L, the total number of electrons passing through any cross-section of wire in per unit time is N/T.

Figure 11

Therefore,

∴ Current density =

⇒ J = qv [Unit of J = amp/m²]

since, = n(electron concentration in electrons per cubic meter).

∴ J = nqv = ρv

where, ρ = nq is the charge density in coulombs/m³ and v in m/s.

Conductivity

From the above discussion

J = nq v = nqμE = σE

The above equation is recognized as ohm’s law.

Where, σ = nqμ is the conductivity of the metal in (hm-meter)^–1.

With the effect of applied electric-field, as a result of collisions of electrons with the lattice ions, electron power is dissipated within the metal and is given by

JE = σE² Watts/m³

Conductivity is the reciprocal of resistivity.
Conductivity denotes current carrying capacity of the material or device.

For semiconductors conductivity

σ = nqμ_n + pqμ_p

For intrinsic semiconductor

σ=n_i(μ_n+μ_p)q

For n-type semiconductor

n >> p

∴σ=nμ_nq

but, n ~ N_D

so, σ=N_dμ_nq

For p-type semiconductor

p >> n

σ=nμ_nq

but, p ~ N_A

so, σ=N_A μ_nq

It is clear that conductivity of semiconductor increases with increase in doping concentration.

Conductivity Vs Temperature

As we know that in metals, resistivity of metal increases with increase in temperature. So, conductivity of metals decreases with increase in temperature.
In pure semiconductors conductivity mainly depends upon number of charge carriers. So, in a semiconductor conductivity increases with temperature.
For 1°C increase in temperature, conductivity of Ge increases by 6% while in Si it increases by 8%.
Conductivity of extrinsic semiconductor decreases above normal temperature with temperature.

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