Which of the following options correctly represents the relative order of band gap energies for Diamond, Silic

Question

Which of the following options correctly represents the relative order of band gap energies for Diamond, Silicon, and Germanium?

Examoogle · Accepted Answer

The correct answer is Option A: E(Diamond) > E(Silicon) > E(Germanium). The band gap energy ($E_g$) of a semiconductor is determined by the strength of the atomic bonds and the crystal structure. Diamond, being a wide-bandgap insulator with extremely strong covalent bonds, possesses the highest energy gap, followed by Silicon and Germanium.

Step-by-Step Solution & Concepts:
The band gap energy ($E_g$) of a semiconductor is determined by the strength of the atomic bonds and the crystal structure. Diamond, being a wide-bandgap insulator with extremely strong covalent bonds, possesses the highest energy gap, followed by Silicon and Germanium.

Working Principle: The band gap energy is inversely proportional to the interatomic distance and directly proportional to the strength of the orbital overlap. In Diamond (carbon), the atoms are smaller and form shorter, stronger $sp^3$ hybrid bonds compared to Silicon and Germanium, requiring higher energy to excite an electron from the valence band to the conduction band.

Key Formulas: $E_g \approx E_c - E_v$ — where $E_c$ is the conduction band energy and $E_v$ is the valence band energy.; $E_g(T) = E_g(0) - \frac{\alpha T^2}{T + \beta}$ — Varshni empirical relation representing temperature dependence.

Key Points: Diamond is a wide-bandgap material with $E_g \approx 5.5 	ext{ eV}$.; Silicon is the standard semiconductor with $E_g \approx 1.1 	ext{ eV}$ at $300 	ext{ K}$.; Germanium has a smaller band gap of $E_g \approx 0.67 	ext{ eV}$ at $300 	ext{ K}$.; Smaller band gaps make materials more sensitive to thermal excitation at room temperature.

Advantages: Diamond's high band gap allows for high-power and high-temperature operation.; Silicon's moderate band gap is ideal for stable integrated circuit operation at ambient temperatures.

Disadvantages: Germanium's low band gap causes high leakage currents at room temperature.; Diamond is difficult to dope and process for standard semiconductor devices.

Applications: Silicon is used in standard logic and memory microchips.; Germanium is used in high-speed transistors and infrared optics.; Diamond is used in radiation detectors and specialized high-power electronics.

Additional Info: Values at $300 	ext{ K}$: $E_g(	ext{Diamond}) = 5.5 	ext{ eV}$, $E_g(	ext{Silicon}) = 1.12 	ext{ eV}$, $E_g(	ext{Germanium}) = 0.66 	ext{ eV}$.. Option B is the reverse of the true order; Option C and D are incorrect as they misorder the materials.

Answer: A is correct — Diamond has the widest band gap, followed by Silicon, and then Germanium, which has the smallest.

Exam Tip: Remember that band gap is inversely related to thermal conductivity; materials with wider band gaps (like Diamond) often exhibit better thermal management properties.

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Option B: E(Germanium) > E(Silicon) > E(Diamond)

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Option C: E(Silicon) > E(Diamond) > E(Germanium)

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Option D: E(Diamond) > E(Germanium) > E(Silicon)

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