five questions with answers (under 150 words each) on Solid State Physics for your Physics Honours graduation students:
1. Question: Define the terms "primitive unit cell" and "Wigner-Seitz cell" for a crystal lattice. What is the key difference between them?
Answer: A primitive unit cell is the smallest volume unit that, when translated through all the lattice vectors, can generate the entire crystal structure. It contains exactly one lattice point. The Wigner-Seitz cell is a primitive cell constructed by drawing lines from a lattice point to all its nearest neighbors, then drawing perpendicular bisectors to these lines. The smallest volume enclosed by these bisectors is the Wigner-Seitz cell. The key difference is that the Wigner-Seitz cell uniquely associates each point in space with the nearest lattice point and always reflects the point group symmetry of the lattice.
2. Question: Explain the concept of the reciprocal lattice. What is its significance in the context of X-ray diffraction from crystals?
Answer: The reciprocal lattice is a set of vectors (\mathbf{G}) such that (\mathbf{G} \cdot \mathbf{R} = 2\pi m) for all direct lattice vectors (\mathbf{R}) and integer (m). For a direct lattice with primitive vectors (\mathbf{a}_1, \mathbf{a}_2, \mathbf{a}_3), the reciprocal lattice primitive vectors are (\mathbf{b}_1 = 2\pi \frac{\mathbf{a}_2 \times \mathbf{a}_3}{\mathbf{a}_1 \cdot (\mathbf{a}_2 \times \mathbf{a}_3)}), and cyclically permuted. In X-ray diffraction, the Bragg condition in reciprocal space is (\mathbf{k}' = \mathbf{k} + \mathbf{G}), where (\mathbf{k}) and (\mathbf{k}') are the incident and scattered wave vectors. This shows that diffraction peaks occur when the momentum transfer of the X-rays matches a reciprocal lattice vector, directly probing the reciprocal lattice and thus the periodicity of the crystal.
3. Question: Describe the nearly free electron model for electronic band structure in solids. How does it lead to the formation of energy band gaps?
Answer: The nearly free electron model considers electrons in a solid as almost free particles moving in a weak periodic potential due to the lattice ions. The weak potential introduces perturbations, particularly at the boundaries of the Brillouin zones, where the electron wavevector (k) satisfies the Bragg condition ((k = n\pi/a), where (a) is the lattice constant and (n) is an integer). At these boundaries, electron waves are reflected, leading to standing waves with energies slightly higher and lower than the free electron energy. This creates energy discontinuities or band gaps at the Brillouin zone boundaries, separating allowed energy bands.
4. Question: Distinguish between direct and indirect band gap semiconductors. Why are direct band gap semiconductors generally preferred for optoelectronic devices like LEDs and laser diodes?
Answer: In a direct band gap semiconductor, the minimum of the conduction band and the maximum of the valence band occur at the same wavevector (\mathbf{k}) in the Brillouin zone. In an indirect band gap semiconductor, these extrema occur at different (\mathbf{k}) values. For optical transitions involving the band gap energy, direct band gap materials allow for efficient photon emission and absorption because momentum is conserved by the photon (which has negligible momentum). Indirect band gap transitions require the involvement of a phonon to conserve momentum, making them less probable and less efficient for light emission, hence direct band gap materials are preferred for optoelectronic devices.
5. Question: Explain the concept of doping in semiconductors. How does n-type and p-type doping alter the carrier concentration and Fermi level?
Answer: Doping is the intentional introduction of impurities into an intrinsic semiconductor to modulate its electrical conductivity. n-type doping involves adding donor impurities (e.g., phosphorus in silicon) with more valence electrons than the host atoms. These extra electrons are easily excited into the conduction band, increasing the electron concentration ((n)) and shifting the Fermi level upwards towards the conduction band. p-type doping involves adding acceptor impurities (e.g., boron in silicon) with fewer valence electrons. These create holes (vacancies of electrons) in the valence band, increasing the hole concentration ((p)) and shifting the Fermi level downwards towards the valence band. The mass action law, (np = n_i^2) (where (n_i) is the intrinsic carrier concentration), still holds.
