Definitions

Question 1

What is a crystal lattice, and what are the 2D Bravais lattices?

Answer 1

A crystal lattice is a regular repetition (lattice) of a basic structural unit (basis). In 2D, there are five Bravais lattices: oblique, rectangular primitive, rectangular centered, square, and hexagonal.

Question 2

What are the different types of crystal cells?

Answer 2

Crystal cells include primitive (P), face-centered (F), body-centered (I), and base-centered (A/B/C).

Question 3

What is the mathematical representation of a crystal?

Answer 3

Mathematically, a crystal is a convolution of the lattice and basis, where the lattice is an array of delta-function-like points.

Question 4

Describe a BCC unit cell.

Answer 4

The BCC unit cell has two lattice points per cell with right nearest neighbours. It’s Wigner-Seitz cell is a truncated octahedron.

Question 5

What is close packing, and how does it occur in FCC and HCP lattices?

Answer 5

Close packing is optimal space-filling in crystals. In FCC, it occurs in an ABCABCAB… hexagonal pattern, while in HCP, it occurs in an ABABAB… hexagonal pattern.

Question 6

Describe the diamond structure, and what are Miller indices used for in crystal planes?

Answer 6

Diamond structure is an FCC structure with a 2-atom basis. Miller indices (h, k, l) are used to define crystal planes and vectors, denoting the orientation and spacing of planes.

Question 7

Describe diffraction from a crystal.

Answer 7

Crystals act as 3D gratings, scattering radiation in all directions. Diffraction occurs when the incident wavelength is comparable to the sample periodicity.

Question 8

Describe X-ray diffraction.

Answer 8

X-ray diffraction involves strong scattering from crystal electron density, using X-ray tubes or synchrotrons. X-rays are sensitive to the electron density and atomic number. Where the resulting spectrum has characteristic peaks.

Question 9

Describe neutron diffraction and how neutrons are produced.

Answer 9

Neutron diffraction involves neutrons scattering from nuclei and magnetic moments. Nuclear reactors are used to produce fast neutrons where neutron production must be moderated.

Question 10

Describe electron diffraction.

Answer 10

Electrons scatter strongly from materials and can scatter from electrostatic potential of atoms. Where electron diffraction requires thin samples.

Question 11

Describe diffraction in a lattice structure.

Answer 11

Diffraction is related to lattice structure through form factor and structure factor. Structure factors are determined by the Fourier transform of electron density.

Question 12

What is the significance of the reciprocal lattice, Brillouin Zone, and Ewald sphere in diffraction?

Answer 12

Reciprocal lattice and Brillouin Zone are concepts in reciprocal space, defining possible vectors and the smallest volume enclosed by a set of planes. The Ewald sphere visualizes diffraction patterns in reciprocal space.

Question 13

How does the Ewald sphere differ for X-rays and fast electrons in diffraction?

Answer 13

The Ewald sphere for X-rays has few diffraction spots, while for fast electrons, it produces many diffraction peaks with a mostly flat central part, intersecting other reciprocal lattice planes.

Question 14

What are Higher Order Laue Zones?

Answer 14

Higher Order Laue Zones are rings of diffraction spots formed when the Ewald sphere intersects other reciprocal lattice planes in electron diffraction.

Question 15

What is the Ewald Sphere?

Answer 15

The Ewald sphere maps out the possible elastically scattered vectors and its radius is the wavevector of the microscope electrons.

Question 16

What is the basis of the “free electron theory,” and how is it related to metallic bonding?

Answer 16

The electron is considered a classical particle, bound within a solid in a potential well. The “free electron theory” is a quantum mechanical modification of this model where metallic bonding promotes a close-packed structure.

Question 17

What is the valence number?

Answer 17

The valence is the number of outer electrons contributing to the electron sea.

Question 18

How are electrons described quantum mechanically, and what is the significance of the Pauli exclusion principle?

Answer 18

Electrons obey the Schrödinger equation, assuming constant potential and perfect confinement. The Pauli exclusion principle restricts each spatial state to two electrons with different spins.

Question 19

What is the Fermi level?

Answer 19

At T = 0, the highest energy electrons sit at the Fermi level.

Question 20

What is the Fermi surface?

Answer 20

The filled states in 3D form a spherical region in k-space, and its surface is known as the Fermi Surface. It represents the boundary between filled and empty states.

Question 21

What is the Fermi Sphere?

Answer 21

Electrons fill up to a max radius, kF forming the fermi sphere.

Question 22

Explain the concept of Density of States (DOS) and its application to a hypothetical 1D metal.

Answer 22

DOS, represented by D(E), is the number of electron states per unit volume in an energy range (E, E + dE). In a hypothetical 1D metal, atoms contribute electrons, and DOS varies with electron density.

Question 23

How does the 3D Density of States vary with temperature?

Answer 23

At T = 0, states with E > EF are empty, and E < EF are filled. At finite temperatures, only electrons within kBT of EF can be thermally excited.

Question 24

What is the chemical potential (μ) in Fermi-Dirac statistics?

Answer 24

μ is the chemical potential, the Gibbs Free energy per particle, and is EF at T = 0.

Question 25

What is the magnetic moment?

Answer 25

Magnetic moment quantifies the strength of magnetic effects in a material.

Question 26

How does the magnetic field affect the energy populations?

Answer 26

For no magnetic field and 0K there is no energy difference between populations and spins are balanced. In an applied magnetic field there is an unstable situation resolved by ‘spin flips’ to leave a spin imbalance.

Question 27

What can we conclude about from the density of states under an applied magnetic field?

Answer 27

Although energies change, only a small fraction of electron spins change in an applied magnetic field.

Question 28

What happens to the electrons under an applied electric field?

Answer 28

An applied electric field results in the net movement of electrons, the whole sea shifts in the opposite direction by momentum k, leading to conduction. For no electric field electrons move randomly, and scattering from ion cores results in no net motion.

Question 29

What happens when you turn off the electric field when a steady state has developed?

Answer 29

Once a steady-state (delta k) has developed, turning off the electric field results in an exponential decay in current.

Question 30

Describe the Hall bar.

Answer 30

Charges accumulate on one side of the hall bar and produces an electric field measured as the hall voltage. The hall bar exploits the magnetoresistance results. Producing a force which balances the lorentz force.

Question 31

Explain the transport of thermal energy and the Wiedemann-Franz law.

Answer 31

Thermal conductivity (Kel) is estimated for a free electron sea. The Wiedemann-Franz law links thermal and electrical conductivity it works well at extremely low temperatures but it fails at ~10K due to different scattering mechanisms.

Question 32

What are the successes and failures of the free electron theory?

Answer 32

Successes include qualitative agreement on electronical specific heat, spin susceptibility, electrical conduction, and the Hall effect. Failures include specific heat predictions, Hall effect measurements, and the assumption that everything is a metal.

Question 33

How does the presence of ion cores affect the electron wave in a 1D crystal lattice?

Answer 33

Ion cores create deep potential wells, forming a periodic potential with sharp variations. Electrons traveling through the lattice experience back-scattering wavelets due to ion cores.

Question 34

Describe the Bragg condition.

Answer 34

The Bragg condition occurs when back wavelets are in phase, resulting in coherent back-scattered wavelets. It is defined as a travelling wave producing coherent back-scattered wavelets equal in amplitude to the forward-traveling wave. Resulting in a standing wave.

Question 35

Describe the ion cores effect on the electron density.

Answer 35

electron density is determined by |Ψ|^2. Electrons pile up either at (Ψ+) or between (Ψ-) ion cores. Electrostatic energy is lower for Ψ+ than Ψ-.

Question 36

Explain the formation of gaps in the allowed energies in the nearly-free electron model (NFE).

Answer 36

A split into two evergy levels occurs at the Bragg condition. Where gaps are entirely due to the periodic potential produced by the lattice. Away from the Bragg condition, free electron behaviour persists.

Question 37

What is Bloch’s theorem?

Answer 37

Bloch’s theorem states that the electron wavefunction in a periodic potential has the periodicity of the crystal lattice.

Question 38

How does Bloch’s theorem simplify the central equations of the NFE model?

Answer 38

Bloch’s theorem simplifies the central equations by showing that strong coupling occurs only near the Brillouin zone boundary. The electron can only change its wavevector by G in a periodic structure.

Question 39

What are Bloch functions?

Answer 39

Bloch functions are the solutions of Schrodinger’s equation in a periodic potential. They contain plane wave components at all k - G.

Question 40

Describe Brillouin zones.

Answer 40

Brillouin zones are regions in reciprocal space between successive points satisfying the Bragg condition. The zones are constructed based on the periodicity of the lattice.

Question 41

What are Bloch waves?

Answer 41

Bloch waves are plane waves modulated by a function with the periodicity of the lattice.

Question 42

Explain the concept of Fermi surfaces and how they relate to the Fermi energy.

Answer 42

Fermi surfaces represent constant energy surfaces at the Fermi energy. The Fermi energy determines the occupation of states by electrons, affecting the material’s electrical and thermal properties.

Question 43

What are extended and reduced zone schemes?

Answer 43

Extended zone schemes are plotting E(k) with k. Reduced zone schemes involve picking a G vector such that k - G maps back to a unique k in the 1st Brillouin zone. This folds the E(k) diagram back on itself giving the reduced zone scheme.

Question 44

What is the periodic extended zone scheme?

Answer 44

A plot of all the reduced zone bands at all k - G, which is a hybrid of the extended and reduced zone schemes.

Question 45

How do Brillouin zone boundaries and Fermi surfaces influence the Hall coefficient in electronic transport?

Answer 45

Brillouin zone boundaries and Fermi surfaces impact the Hall coefficient. States within the Fermi line have a negative Hall coefficient, resembling free electrons, while states outside the Fermi line have a positive Hall coefficient, a failure of the free electron model.

Question 46

How does the Fermi surface change as the potential increases?

Answer 46

As the potential increases the bandgap widens and the lower BZ fill such that the upper BZ will empty. Where Bragg scattering distorts the Fermi line to meet BZBs at right angles.

Question 47

How are band structures and Fermi surfaces visualized for different types of materials?

Answer 47

Band structures and Fermi surfaces are visualized by plotting E vs. k for different directions and by considering lines or surfaces of constant energy. The nature of the band structure and position of the fermi energy determines whether a material is metallic, semiconducting, or insulating.

Question 48

Describe an optical absorption experiment.

Answer 48

An optical absorption experiment shines light at varying wavelength through a sample of the semiconductor of thickness d.

Question 49

Describe the structure of both crystalline Si and crystalline GaAs in terms of their atomic arrangements.

Answer 49

Both c-Si and c-GaAs have an FCC structure with a 2-atom basis, forming a tetrahedral network of covalent bonds.

Question 50

What is the difference between direct and indirect band gaps, and how do they relate to materials like Si and GaAs?

Answer 50

Si has an indirect band gap, where promotion from the valence to the conduction band requires the energy of a photon plus a change of k provided by a phonon. GaAs has a direct band gap, electrons in the valence band readily absorb photons and are promoted to the conduction band.

Question 51

Explain the process of optical absorption in semiconductors.

Answer 51

Optical absorption involves the excitation of electrons from the filled valence band to the empty conduction band.

Question 52

Describe the Law of Mass Action.

Answer 52

The Law of Mass Action describes semiconductor carrier density by relating the product of electron and hole densities to a temperature-dependent material property. It depends on the bandgap, chemical potential and effective masses of carriers.

Question 53

What does effective mass depend on?

Answer 53

The effective mass depends on band curvature and is constant in a parabolic band.

Question 54

Describe doping in semiconductors.

Answer 54

Doping introduces intentional impurities to adjust carrier density, where n-type doping adds excess electrons and p-type doping adds excess holes. Doping can increase carrier concentration and conductivity.

Question 55

Define mobility and how it varies with temperature?

Answer 55

Mobility is the drift velocity per unit electric field. It varies with temperature, exhibiting power-law dependence. At low temperatures, impurity scattering dominates, while at high temperatures, phonon scattering becomes dominant.

Question 56

What information does the Hall effect provide in semiconductor measurements?

Answer 56

The Hall effect reveals the dominant carrier type (excess electrons or holes) in a semiconductor. The sign of the Hall coefficient indicates the carrier type, and it is influenced by factors like temperature and doping.

Question 57

What are the successes of the Nearly Free Electron (NFE) model in describing semiconductor properties?

Answer 57

The NFE model successfully describes semiconductor band structures using weak potentials and explains simple metal Fermi surfaces. It is particularly effective considering the simplicity of its additions to the free electron model.

Question 58

What challenges or limitations does the NFE model face?

Answer 58

The NFE model faces challenges in explaining strong and long-range Coulomb forces, requiring the concept of pseudopotentials. Additionally, it struggles to account for non-crystalline solids like amorphous Si or glass, where the tight binding approximation provides some insights.

Question 59

How is the hall voltage measured?

Answer 59

The hall voltage is measured transverse to the field and current.

Question 60

What is the drift velocity?

Answer 60

The drift velocity is the average velocity of charge carriers under an applied electric field.

Question 61

What is the relaxation time?

Answer 61

The relaxation time is the average time between electronic scattering events

Question 62

What is the mobility?

Answer 62

The mobility is the measure of the response of charge carriers to an applied electric field. Or the drift velocity per unit electric field.

Question 63

Compare classical energy to Fermi energy.

Answer 63

Classical thermal energy is much less than fermi energy.

Question 64

Describe an FCC unit cell.

Answer 64

The BCC unit cell has four points per cell with twelve nearest neighbours. It’s Wigner-Seitz cell is a rhombic dodecahedron.

Question 65

What are the four crystal diffractive techniques?

Answer 65

X-ray diffraction, thermal energy neutrons, electron beams and thermal energy atoms and ions.

Question 66

Describe powder diffraction.

Answer 66

In X-ray diffraction, we can use a powdered crystalline sample where each Bragg angle produces a diffraction cone of semi-angle 2θ. Where an image of Debye-Scherrer rings can be collected.

Question 67

How do we know the electron gas is not classical?

Answer 67

The heat capacity is dominated by lattice vibrations where the electronic contributions are much smaller than the lattice. Hence they only show up strongly at low T.

Question 68

How does heat capacity vary with insulators and metals?

Answer 68

For insulators without free electrons, the electronic heat capacity is zero. For metals the electronic heat capacity is small. The lattice heat capacity varies with lattice vibrations.

Question 69

What scattering effects do we need to consider?

Answer 69

Scattering effects such as defects and lattice vibrations randomise the k vectors and velocities of conduction electrons.

Question 70

Describe the importance of the Brillouin zone.

Answer 70

It represents regions in reciprocal space between successive points where the Bragg condition is met. This is where the E ∝ k^2 free electron result breaks down.

Question 71

Why do we see a difference in optical absorption?

Answer 71

Differences in optical absorption are due to differences in band structure between materials.

Question 72

How does semiconductor conductivity vary with temperature?

Answer 72

For T = 0, the conduction bands are empty and the valence bands are full therefore we have zero conductivity. For T > 0, electrons can be thermally excited into the conduction band contributing to electrical conductivity.

Question 73

Describe a quantum gas.

Answer 73

In a quantum gas electrons are fermions which are governed by Schrodinger’s equation where energy states are quantised by boundary conditions. Where they can only move from filled to empty states.

Question 74

Describe a classical gas.

Answer 74

In a classical gas, atoms have translational KE and collide with walls where there are no boundary conditions. Conduction electrons are like a gas in a box that are free to move around.

Question 75

Describe how the fermi sphere varies with the BZB?

Answer 75

If the fermi sphere is inside the 1st BZB then free electron behaviour persists. Near the BZB the fermi sphere is distorted and exhibits behaviour between NFE and FE. If the fermi sphere crosses the BZB then we get energy gaps and energy bands - semiconductors, insulators etc.

Question 76

What happens to the fermi circle/sphere close to the zone boundaries?

Answer 76

The fermi circle will become distorted by the BZB where the surface will intersect the BZB at right angles. Resulting in electron and hole like states in the periodic zone scheme.

Question 77

Describe a poly-crystalline materials diffraction pattern.

Answer 77

A polycrystalline material has a ring pattern due to the superposition of reciprocal lattice planes in different orientations.

Question 78

Why do indirect gaps in semiconductors appear in 2D but not in 1D?

Answer 78

In 1D we have simple energy splitting at k value of BZB. Hence a gap occurs at that k value. In 2D the distance to the BZB depends on direction so it occurs at different values of k.

Question 79

Why is the conductivity of semiconductors smaller than that of metals?

Answer 79

The density of charge carriers is much lower.

Question 80

Describe the concept of effective mass.

Answer 80

The effective mass is used to account for electron transport behaviour in solid materials. It uses the effective mass in place of the electronic mass where there is no direct relationship between the electronic rest mass and the effective mass of an electron or its hole.

Question 81

Describe alloyed atoms.

Answer 81

Alloyed atoms act as defect sites that increase the scattering rate for conduction electrons.

Question 82

Describe the group velocity.

Answer 82

The group velocity depends on band gradient where |vg| reduces as the BZB is approached and is zero at the BZB.

Question 83

Describe an electron-like and hole-like orbit.

Answer 83

An electron-like orbit occurs when we get an anticlockwise cyclotron orbit and a hole like orbit occurs if the cyclotron orbit is clockwise.

Question 84

Describe a-Si (amorphous siliicon) which has no lattice.

Answer 84

The momentum conservation conditions are different and the absorption edge moves to lower wavelengths making it a good photovoltaic. This is not considered in the NFE, a failure of the model.