Title: Summary Using Bandstructures to Understand and Explain the Properties of Solids

Choose the correct statement.

- The properties of a material are strongly coupled to its band structure. It lets us distinguish between metals (the Fermi energy lies within the band) and insulators or semiconductors (some bands are completely filled and others are completely empty). (correct)
- The properties of a material are strongly coupled to its band structure. It lets us distinguish between insulators (the Fermi energy lies within the band) and metals or semiconductors (some bands are completely filled and others are completely empty).

- The conductivity of a metal is reflected in the band structure. Broad bands present a large ß, which is a sign of strongly overlapping orbitals, and therefore of good conductivity. Analogously, narrow bands are a sign of poor conductivity. (correct)
- The conductivity of a metal is reflected in the band structure. Broad bands present a small ß, which is a sign of poorly overlapping orbitals, and therefore of poor conductivity. Analogously, narrow bands are a sign of good conductivity.

- The effective mass of an electron is inversely proportional to the curvature of the band. For large curvatures, the effective mass is small, and therefore conductivity is high. (correct)
- The effective mass of an electron is proportional to the curvature of the band. For large curvatures, the effective mass is large, and therefore conductivity is low.

- In some regions of the band structure the curvature of the band can be negative, and so the effective mass is also negative. Instead of considering a negative mass, we think of the current in these regions as being carried by positively charged holes having positive mass. (correct)
- In some regions of the band structure the curvature of the band can be positive, and so the effective mass is negative. Then, we think of the current in these regions as being carried by positively charged holes having negative mass.

- In optical absorption, an incoming photon excites an electron from the valence band to the conduction band and is absorbed during the process, which requires the conservation of energy and momentum. (correct)
- In optical absorption, an incoming photon excites an electron from the conduction band to the valence band and is absorbed during the process, which requires the conservation of energy and momentum.

- In direct gap semiconductors, the k value of the electron is the same in its ground and excited states, and therefore the transition is vertical within the the band structure. On the other hand, in indirect gap semiconductors the transition takes place in 2 steps: the absorption of a photon is followed by the emission or absorption of a phonon, allowing for the conservation of momentum. (correct)
- In indirect gap semiconductors, the k value of the electron is the same in its ground and excited states, and therefore the transition is vertical within the the band structure. On the other hand, in direct gap semiconductors the transition takes place in 2 steps: the absorption of a photon is followed by the emission or absorption of a phonon, allowing for the conservation of momentum.

- The transition probability amplitude (TPA) is the probability of an electron in a particular valence band to make the transition into a particular conduction band state. (correct)
- The transition probability amplitude (TPA) is the probability of an electron in a particular valence band to be excited into the conduction band by means of the absorption of a photon. It takes into account the joint density of states, a quantity which depends both on how many electrons are available to be excited and how many empty states are available to accept them.

- The transition probability amplitude can be evaluated using symmetry arguments. As the transition operator presents odd parity, the probability will only be different from zero (and therefore the transition possible) when the initial and the final states have different parity, such as between an s orbital (even parity) and a p orbital (odd parity). (correct)
- The transition probability amplitude can be evaluated using symmetry arguments. As the transition operator presents odd parity, the probability will only be different from zero (and therefore the transition possible) when the initial and the final states have the same parity, such as between an s orbital (even parity) and another s orbital (even parity).

- In summary, optical transitions must conserve energy, momentum and parity. (correct)
- In summary, optical transitions must conserve energy and parity.

- In semiconductor quantum dots, the bandgap depends on the size: as the size decreases, the bandgap becomes larger. (correct)
- In semiconductor quantum dots, the bandgap depends on the size: as the size increases, the bandgap becomes larger.

- An exciton is an electron-hole pair bound by a Coulomb interaction, with a binding energy inversely proportional to the relative permittivity. (correct)
- An exciton is an electron-hole pair bound by a Coulomb interaction, with a binding energy proportional to the relative permittivity.

- According to the Peierl's theorem, a one-dimensional metal is always unstable with respect to a structural distortion that removes a degeneracy at the Fermi level, with a periodicity equal to π over the k at the Fermi level. It consists of a balance between the strain energy cost and electronic energy lowering. (correct)
- According to the Peierl's theorem, a one-dimensional metal is always stable with respect to a structural distortion that removes a degeneracy at the Fermi level, with a periodicity equal to 2π over the k at the Fermi level. It consists of a balance between the strain energy cost and electronic energy lowering.

- Crystal field splitting is a consequence of the coupling between the electronic structure and the physical structure. For example, in the case of d orbitals, depending on the coordination (tetrahedral, octahedral...) the energy of some orbitals is raised and that of others is lowered. (correct)
- Crystal field splitting is a consequence of the coupling between the electronic structure and the physical structure. For example, in the case of s orbitals, depending on the coordination (tetrahedral, octahedral...) the energy of some orbitals is raised and that of others is lowered.

- The Jahn-Teller distortion is an electronically-driven structural distortion, which consists of the elongation along an axis and a shortening along the other two axes. It will take place when the saving in electronic energy is higher than the cost in strain energy. (correct)
- The Jahn-Teller distortion is an electronically-driven structural distortion, which consists of the elongation along an axis and a shortening along the other two axes. It will take place when the saving in electronic energy is lower than the cost in strain energy.