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Electronic Devices - Energy Bands in Solids

Grade 12ICSEPhysics

Review the key concepts, formulae, and examples before starting your quiz.

🔑Concepts

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Energy Bands: In a solid, the discrete energy levels of isolated atoms broaden into bands due to the proximity of atoms and the Pauli Exclusion Principle. The two most important bands are the Valence Band (VBVB) and the Conduction Band (CBCB).

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Valence Band (VBVB): This is the range of energy levels occupied by valence electrons. It is the highest occupied energy band and can be completely or partially filled.

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Conduction Band (CBCB): This is the energy band above the valence band. Electrons in this band are free to move and contribute to the electrical conductivity of the solid.

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Forbidden Energy Gap (EgE_g): The energy difference between the bottom of the conduction band (ECE_C) and the top of the valence band (EVE_V). No electron can exist in this gap: Eg=EC−EVE_g = E_C - E_V.

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Conductors: In conductors, the VBVB and CBCB overlap, or the CBCB is partially filled even at 0 K0\text{ K}. There is no forbidden gap (Eg≈0E_g \approx 0), allowing electrons to move freely.

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Insulators: In insulators, the forbidden energy gap is very wide (typically Eg>3 eVE_g > 3\text{ eV}). At room temperature, electrons do not have enough thermal energy to jump from VBVB to CBCB.

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Semiconductors: In semiconductors, the forbidden energy gap is small (typically Eg<3 eVE_g < 3\text{ eV}). For Silicon (SiSi), Eg≈1.1 eVE_g \approx 1.1\text{ eV}, and for Germanium (GeGe), Eg≈0.7 eVE_g \approx 0.7\text{ eV}.

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Effect of Temperature: As temperature increases, more electrons in a semiconductor gain thermal energy to jump from VBVB to CBCB, thereby increasing conductivity and decreasing resistivity.

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Intrinsic and Extrinsic Semiconductors: Pure semiconductors are 'Intrinsic'. Adding impurities (doping) creates 'Extrinsic' semiconductors (nn-type with pentavalent impurities or pp-type with trivalent impurities).

📐Formulae

Eg=EC−EVE_g = E_C - E_V

nenh=ni2n_e n_h = n_i^2

σ=e(neμe+nhμh)\sigma = e(n_e \mu_e + n_h \mu_h)

ρ=1σ\rho = \frac{1}{\sigma}

E=hν=hcλE = h\nu = \frac{hc}{\lambda}

💡Examples

Problem 1:

The energy gap of a semiconductor is Eg=1.12 eVE_g = 1.12\text{ eV}. Calculate the maximum wavelength (λ\lambda) of a photon that can excite an electron from the valence band to the conduction band. (Use h=6.63×10−34 J sh = 6.63 \times 10^{-34}\text{ J s}, c=3×108 m/sc = 3 \times 10^8\text{ m/s}, and 1 eV=1.6×10−19 J1\text{ eV} = 1.6 \times 10^{-19}\text{ J})

Solution:

Given Eg=1.12 eV=1.12×1.6×10−19 JE_g = 1.12\text{ eV} = 1.12 \times 1.6 \times 10^{-19}\text{ J}. The energy of the photon must be at least equal to the band gap energy: Eg=hcλE_g = \frac{hc}{\lambda}. Rearranging for λ\lambda: λ=hcEg=6.63×10−34×3×1081.12×1.6×10−19\lambda = \frac{hc}{E_g} = \frac{6.63 \times 10^{-34} \times 3 \times 10^8}{1.12 \times 1.6 \times 10^{-19}} λ≈1.11×10−6 m=1110 nm\lambda \approx 1.11 \times 10^{-6}\text{ m} = 1110\text{ nm}

Explanation:

To transition an electron across the forbidden gap, the incident photon must provide energy E≥EgE \geq E_g. The maximum wavelength corresponds to the minimum energy required.

Problem 2:

In an intrinsic semiconductor, the concentration of electrons is ni=1.5×1016 m−3n_i = 1.5 \times 10^{16}\text{ m}^{-3}. If it is doped with a pentavalent impurity such that the electron concentration becomes ne=4.5×1022 m−3n_e = 4.5 \times 10^{22}\text{ m}^{-3}, calculate the new hole concentration nhn_h.

Solution:

According to the Law of Mass Action: nenh=ni2n_e n_h = n_i^2 Substituting the given values: nh=ni2ne=(1.5×1016)24.5×1022n_h = \frac{n_i^2}{n_e} = \frac{(1.5 \times 10^{16})^2}{4.5 \times 10^{22}} nh=2.25×10324.5×1022=0.5×1010 m−3=5×109 m−3n_h = \frac{2.25 \times 10^{32}}{4.5 \times 10^{22}} = 0.5 \times 10^{10}\text{ m}^{-3} = 5 \times 10^9\text{ m}^{-3}

Explanation:

Doping an intrinsic semiconductor increases the majority carrier concentration (electrons in this nn-type case) while significantly decreasing the minority carrier concentration (holes) to maintain the equilibrium constant ni2n_i^2.

Energy Bands in Solids - Revision Notes & Key Formulas | ICSE Class 12 Physics