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What is the Band Theory of Solids?

Answer:

Band Theory of Solids

In the case of solid material, all the atoms are close to each other, so the neighboring atoms affect the energy levels of outermost orbit electrons. When the isolated atom or two single atoms are brought close to each other, then the outermost orbit electrons of the two atoms are interacting with each other. i.e., the electrons in the outermost orbit of one atom experience an attractive force from the nearest or neighboring atomic nucleus.  Because of this, the energies of the electrons will not be at the same level, the energy levels of electrons are changed to a value that is higher or lower than that of the original energy level of the electron. The electrons in the same orbit exhibit different energy levels. The grouping of these different energy levels is called an energy band.

Band overlap will not occur in all substances, no matter how many atoms are close to each other. In some substances, a substantial gap remains between the highest band containing electrons is known as the Valence band and the next band, which is empty is known as the conduction band. As a result, valence electrons are “bound” to their constituent atoms and cannot become mobile within the substance without a significant amount of imparting energy. So, the band-gap is the minimum amount of energy required for an electron to break free of its bound state. When the band-gap energy is met, the electron is excited into a free state, and can therefore take part in conduction. Similarly, in the electronic band structure of solids, the band gap refers to the energy difference between the top of the valence band and the bottom of the conduction band.

In terms of Band Gap energy solids are in 3 categories: Metal, Insulator and Semiconductor. Band Theory of Solids

The electrical properties of a given material depend on the electronic populations of the different allowed energy bands. Electrical conduction is the result of electron motion within each band. When an electric field is applied to the material, electrons start to move in the direction opposed to the direction of the electric field. An empty energy band (in which there is no free electron) does not of course participate in the formation of an electric current. It is also the case for a fully occupied band.

Conductor:

According to the band theory of solids this implies that there is a no energy gap between the energies of the valence electrons i.e valance band and the energy at which the electrons can swing through the material i.e the conduction band. For a conductor, conduction bands and valence bands are not separated and there is therefore no energy gap. The conduction band is then partially occupied (even at low temperatures), resulting in a “high” electrical conductivity.

Electrons in metals are also arranged in bands, but in a metal the electron distribution is different – electrons are not localized on individual atoms or individual bonds. In a simple metal with one valence electron per atom, such as sodium, the valence band is not full, and so the highest occupied electron states lie some distance from the top of the valence band. Such materials are good electrical conductors, because there are empty energy states available just above the highest occupied states, so that electrons can easily gain energy from an applied electric field and jump into these empty energy states. Band Theory of Solids

Insulator:

In terms of the band theory of solids this implies that there is a large gap between the energies of valance band and conduction band. Glass is an insulating material which may be transparent to visible light for reasons closely correlated with its nature as an electrical insulator. The visible light photons do not have enough quantum energy to bridge the band gap and get the electrons up to an available energy level in the conduction band.

Semiconductor:

Materials that fall within the category of semiconductors have a narrow gap between the valence and conduction bands. Thus, the amount of energy required to motivate a valence electron into the conduction band where it becomes mobile is quite modest. For Pure semiconductors like silicon and germanium, the Fermi level is essentially halfway between the valence and conduction bands. Although no conduction occurs at 0 K, at higher temperatures a finite number of electrons can reach the conduction band and provide some current. In doped semiconductors, extra energy levels are added. The increase in conductivity with temperature can be modeled in terms of the Fermi function, which allows one to calculate the population of the conduction band

Key Features:

• Metals have free electrons and partially filled valence bands, therefore they are highly conductive; at last conductors are materials with high conductivities (like silver: 106S/cm)
• Insulators have filled valence bands and empty conduction bands, separated by a large band gap Eg (typically >4eV), they have high resistivity; Insulators are materials having an electrical conductivity order of 10^-8 s/cm (like diamond: 10^-14S/cm)
• Semiconductors have similar band structure as insulators but with a much smaller band gap. Some electrons can jump to the empty conduction band by thermal or optical excitation  Eg=1.12 eV for Si, 0.67 eV for Ge and 1.43 eV for GaAs; semiconductors have a conductivity order of 10^-8 to 10^3 s/cm (for silicon it can range from 10^-5S/cm to 10^3S/cm). Band Theory of Solids

Problem 1: The mobility of free electron & holes in pure Ge are 0.38 & 0.18 m²/Vs. The corresponding values for pure Si are 0.13 & 0.05 m²/Vs. Determine the value of intrinsic resistivity for both Ge an Si. If n_i for Ge = 2.5 x 10¹⁹ /m³ & for Si = 1.5 x 10¹⁶ /m³ at room temperature.

Solution: Given That : n_i for Ge = 2.5 x 10¹⁹ /m³ ; n_i for Si = 1.5 x 10¹⁶ /m³;

For Ge : mobility of e^– (µ_e)=0.38 m²/Vs ; mobility of Holes (µ_h)= 0.18 m²/Vs

The intrinsic conductivity of Ge semiconductor is:

s_I = q n_i (µ_e+ µ_h )

s_I = 1.6 x 10^-19 x 2.5 x 10¹⁹ (0.38 + 0.18)

s_I = 2.243 Ω^-1m^-1

So the intrinsic resistivity of Ge is

r_I =1/  s_I

r_I  = 1/ 2.243

r_I   = 0.4458 Ωm

For Si: mobility of e^– (µ_e)= 0.13 m²/Vs ; mobility of Holes (µ_h)= 0.05 m²/Vs ;

The intrinsic conductivity of Si semiconductor is:

n_i = 1.5 x 10¹⁶ /m³

s_I = q n_i (µ_e+ µ_h )

s_I = 1.6 x 10^-19 x 1.5 x 10¹⁶ (0.13 + 0.05)

s_I = 0.4325 x 10^-3 Ω^-1m^-1

So the intrinsic resistivity of Si is

r_I =1/  s_I

r_I  = 1/ 0.4325 x 10^-3

r_I   = 2.312 x 10³ Ωm

Problem 2: A peace of Ge semiconductor has a length of 1 cm and 1 mm² cross section area; at 300K the intrinsic concentration of Ge is 2.5 x 10¹⁹ m^-3. The free electron & holes mobility are 0.38 m³/ V.s & 0.18 m²/V.s respectively. Given that there are 4.4 x 10²⁸ Ge atoms /m³ , calculate parts per 10⁹ of p-type doping that results in a peace resistance of 1kΩ.

Solution Given that: 4.4 x 10²⁸ Ge atoms /m³ in G; n_i =  2.5 x 10¹⁹ m^-3; T = 300K

 Mobility of e^– (µ_e)=0.38 m²/Vs ; mobility of Holes (µ_h)= 0.18 m²/Vs

L = 10^-2m; A = 10^-6m²; R = 10³ Ω

s = L/RA

s = 10^-2/10³x 10^-6

s = 10 /Ωm

For a p-type semiconductor

P = N_A>>n=n_i²/N_A

s = p µ_hq = N_A µ_pq

N_A = 3.47 x 10²⁰ m^-3

N_A = 4.4 x 10²⁸ (x/10⁹)

x = 3.47x 10²⁰/ 4.4×10¹⁹

x = 7.69 parts per 10⁹