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Cmp Pt 2: Lectures 10-19 :(((((
what is fermi energy? - ANS>at temperatures approaching T = 0, the energy of the level at which the most energetic fermion (electron) lies. ie. the difference between the lowest and highest energy states at absolute zero what is the fermi level? - ANS>the fermi level separates occupied from unoccupied states, can be at any temperature. in comparison, the fermi energy is a specific value at T=0 that defines the difference between lowest/highest energy states. the fermi level is just the highest energy level that is occupied by electrons. what happens to the momentum / kinetic energy at the fermi energy? - ANS>fermi energy states have non-vanishing momentum and kinetic energy. in fact, the maximum electron momentum and velocity occurs at the fermi energy at T = 0 what do we use fermi energy for? - ANS>the fermi energy level determines the probability of electrons to occupy different energy levels. it gives us information about whether a solid will be conducting/insulating what is wave number? - ANS>wave number is used to describe the frequency of a wave. in cmp we use k, angular wave number. k = 2π/λ
what is a wave vector? - ANS>a vector that helps to describe a wave - the wave vector's magnitude is the wave number what is the fermi surface? - ANS>an abstract surface that defines the allowable energies of electrons in a solid what is the fermi sphere? is it filled continuously? - ANS>fermi spheres are a type of fermi surface that is used to graphically represent the distribution of the momenta of electrons as the uniform filling of a sphere, using vectors that point from the centre of the sphere to the point where the electron sits. the sphere is not filled continuously what is the density of states? why do we use it? - ANS>a quantity that specifies the number of energy levels per a unit interval (eg. energy levels per unit energy) we use this because the energy levels of the fermi gas are discrete so we can compare the n° of states at a certain energy range to others. what is are fermi-dirac statistics? - ANS>fermi-dirac statistics describe a distribution of particles over energy states in systems with many indistinguishable particles that obey the pauli exclusion principle (no more than two electrons can occupy the same orbital and two electrons in the same orbital must have opposite spins) what is the mean energy of electrons belonging to the fermi gas? - ANS>at T=0, the mean energy of the electrons belonging to the fermi gas is 0.6Ef
what is heat capacity? - ANS>the amount of energy needed to raise the temperature of a body by one degree. what is wrong with the classical interpretation of electron heat capacity? - ANS>classically:
- electron thermal energy E = (3/2)kBT
- thus total electron gas thermal energy U = n(3/2)kBT
- this electron gas would then have to turn into a significant contribution to the heat capacity of the metal it is in
- in reality, this contribution is much smaller and metals don't have that much heat capacity how does the free-electron gas contribute to a metal's heat capacity? - ANS>at room temp, the free electron gas of a metal is practically fully degenerate; the fully degenerate electrons are not able to absorb any more energy and hence they cannot contribute to the heat capacity of the metal. only the few electrons close to Ef are able to absorb thermal energy and jump to higher states, and hence contribute to the heat capacity of the metal. what is the zone/band theory of solids? what does it help to define? - ANS>the band theory of metals is based on the valence band and conduction band, and it defines conductors, semiconductors and insulators very clearly. it includes a periodic electric potential that represents the potential that the electrons 'feel' as it travels across the crystal lattice of positive ions
what is the valence band? - ANS>the valence band is made up of the valence shell orbitals which have electrons in them. eg. sodium valence band is made up of 3s1 orbital (the last orbital) what is the conduction band? - ANS>the conduction band is made up of those orbitals which are unoccupied by electrons either in the valence shell or higher unoccupied shells. thus, the orbitals of the conduction band are empty. eg. sodium's valence band is 3s1, and the next orbital 3p is empty so it forms a conduction band what assumptions do we need to make in order to build the model of the wavefunction of an electrion in a periodic potential? - ANS>1. positive ions in the lattice are completely immobile, moving electrons don't transfer momentum to them and thermal vibrations of the lattice are not present
- the ions are the sources for the periodic electrostatic force that acts on the mobile, delocalised electrons
- the ions are sitting precisely on the vertices of the crystal lattice
- consider each separate electron as a single electron moving in a periodic potential
V(x) = 0 to find that not all values of energy are possible. the prohibited energy ranges are called energy gaps, and the allowed ones are energy bands. what is the reduced zone scheme? - ANS>when we only consider momentum values/energy bands within the 1st BZ for the crystal lattice. 〰 just one lol what is the periodic zone scheme? - ANS>when you extend the reduced zone scheme periodically to higher and lower momenta. it's the same because it's periodic 〰〰〰 what is the extended zone scheme? - ANS>when you extend the reduced zone scheme but only at the maxima. it allows one to understand the mechanism of the transformation of the dispersion of the electron probability wave for a free space into the dispersion of the electron probability wave for a delocalised electron in a periodic positive potential. why are there energy gaps in the kronig-penney model? - ANS>the origin of the energy gaps in the energy band structure is bragg reflection of the electron probability wave for delocalised electrons from the lattice of positive ions.
what is bragg reflection? - ANS>the reflection of the electron probability wave from the lattice of positive ions, such that it satisfies the bragg condition. what is the bragg condition? - ANS>2dsinΘ = nλ, or we use q(sub B) = πn/T qB = wavenumber of the wave where bragg reflection is maximum T = lattice parameter n = ±1, ±2, ±3... how do we find the edges of the 1st BZ using the bragg condition? - ANS>we obtain the edges of the first BZ by setting n=-1 and +1. what wavelength does the bragg wavenumber correspond to? - ANS>the bragg wave number qB = πn/T, corresponds to a wavelength λ=2T/n. eg. for the first energy gap, we have n=1 and λ=2T, or λ/2=T what happens to the bragg reflection / energy gap size when E < Vo? - ANS>when the energy is less than the potential, the only way for the particle to cross the barrier is by tunnelling, which is not very efficient, so the probability for the electron to reflect is high. this will lead to a relatively large energy gap what happens to the bragg reflection / energy gap size when E > Vo? - ANS>when the kinetic energy is greater than the potential, it can pass the barrier as an electron travelling wave, and the probability for it to be
the internal orbitals of atoms do not overlap noticeably = those bands are narrower why are there discrete energy levels within an energy band? how many electrons states are there in an energy band? - ANS>there are discrete energy levels because of the countable number of atoms in a crystal. the number of electron states in an energy band is 2N·n(v), where N is the number of atoms in a solid and nv is the number of valence electrons in an atom. how do we determine whether a solid is a metal or not? - ANS>by counting the number of valence electrons per atom. nv = odd: the valence band is half-full, and fermi level sits in the middle of the band = this solid is a metal. nv = even: the valence band is full, and the solid is an insulator. exclusions: two energy bands overlap without forming an energy gap. then, fermi level will sit (somewhere) in the middle of the combined energy band, and the solid will be a metal. eg. alkaline earth metals what is reciprocal space? - ANS>"reciprocal space" is an imaginary space where planes of atoms are represented by reciprocal points, and all lengths are the inverse of their length in real space.
what is the reciprocal lattice? - ANS>an alternative lattice defined by reciprocal vectors. it is convenient to work in the reciprocal lattice than the 'real' spatial lattice. we define the reciprocal lattice as the set of wave vectors or which the corresponding plane waves have the periodicity of the original lattice. what is a semiconductor in terms of energy gaps? - ANS>semiconductors are physically insulators (valence band is full), but with the tiny width of the energy gap = kBT, separating its valence and conduction bands how do we label semiconductors? - ANS>eg. A4 = the first element (A) is in group 10+4= eg. A3B5 = the first element (A) is in group 13, the second element is in group 15. the power is also the number of valence electrons that the element has, eg. A4 has a 4 valence electrons what is an intrinsic semiconductor? what are the two types of charge carriers? - ANS>it is a solid in which the band gap is so small that some electrons from the valence band will occupy energy levels in the conduction band. there are two types of charge carriers: electrons of the conduction band and holes of the valence band. what is the relationship between semiconductor conductivity and temperature? - ANS>semiconductor conductivity is zero at T=0, and grows exponentially with temperature; exp[-Eg/(kBT)], where Eg is the gap width
mobility. if the mobility is low, the concentration is probably low and if the mobility is high, the concentration is probably high where is the fermi level in an intrinsic superconductor? what happens as you increase temperature to room temp? - ANS>at precisely mid-height of the energy gap separating the valence and conduction bands. with an increase in temperature, it slowly moves up in energy, but the difference from Ef for T=0 remains practically negligible for room temperature. what is a donor impurity? - ANS>donor impurity means that the doping element has an extra electron which it can donate. it will increase the concentration of electrons and decrease the hole concentration what is an acceptor impurity? - ANS>acceptor impurity means that the doping element has fewer electrons which it can donate. it can 'accept' electrons to decrease the concentration of electrons and increase the hole concentration. how do we obtain a p-type semiconductor? - ANS>the p-type semiconductor is obtained by doping A4 semiconductors with a valence 3 impurity, so that there is an excess hole (p = positive) how do we obtain an n-type semiconductor? - ANS>the n-type semiconductor is obtained by doping A4 semiconductors with a valence 5 impurity, so that there is an excess electron (n=negative) how does doping effect the band structure? - ANS>doping adds an extra single energy level to the band structure. the level sits within the energy
gap (between valence/conduction bands), either closer to the valence band (p-type, because there is an excess hole that it wants to fill), or conduction band (n-type, because there is an excess electron that it wants to get rid of). what charge carriers contribute to the conductivity of intrinsic semiconductors at room temp? - ANS>both holes and electrons what charge carriers contribute to the conductivity of extrinsic semiconductors at room temp? - ANS>either electron or hole conductivity dominates, depending on the type of doping n-type: electron dominates p-type: hole dominates what is a p-n junction? what is the depletion region? - ANS>a p-n junction is a piece of semiconductor that has one half doped n-type and the other half is doped p-type. when we put a p-type and an n-type semiconductor together a depletion layer is formed. this depletion layer is the area that contains no majority charge carriers, so it behaves as an insulator. how is the depletion region formed? what does it do? - ANS>a depleted layer is formed at the interface due to the spread of holes to the n-side of the junction and electrons to the p-side with subsequent recombination of the charge carriers. this process leaves behind non-compensated charges of the crystal lattice and creates an electric field that opposes further diffusion of the free charge carriers.
what is the exchange interaction? - ANS>the effect of the 3d spin co- alignment for the 3d-shell spins of the neighbouring atoms in a ferromagnetic solid is called "the exchange interaction" how does ferromagnetism occur? - ANS>transition metals such as iron/nickel/cobalt have unpaired electrons in their 3d-shells, as shown in the diagram. every single electron with unpaired spin acts as an elementary bar magnet that creates a stray magnetic field around the electron. due to the exchange interaction, the spins of nearest-neighbour atoms co- align which leads to a very large magnetic moment for ferromagnets. list the ferromagnetic metals in order from strongest to weakest. why? - ANS>iron, cobalt, nickel. iron as 4 single electrons, cobalt has 3 and nickel has 2. what is the heisenberg hamiltonian? - ANS>it describes ferromagnetic order of spins. A simple solution to the Hamiltonian can be obtained in the mean-field approximation. when is the heisenberg hamiltonian minimised? - ANS>the hamiltonian is minimised when spins at all spin sites are co-aligned to each other and the applied magnetic field. this is the ground state of the hamiltonian how does temperature affect ferromagnetism? - ANS>at finite temperatures, changes in temp start to affect the perfect ferromagnetic order and reduces specimen magnetisation.
what is the curie temperature? - ANS>the temperature above which a metal is no longer magnetised. the curie temperature is unique for different types of metals what is the magnetic domain? - ANS>magnetic domains are long range alignments of magnetic moments what is magnetostatic energy? - ANS>the magnetic potential energy generated when a magnetic body is present in a magnetic field. what is the 'single-domain' state of a ferromagnet, why is it not energetically favourable? - ANS>the single domain state is when the co- alignment of ferromagnetic spins creates a large magnetic field in/around the ferromagnet. it is not energetically favourable because it lots of energy is stored within the magnetic field that overcompensates the gain in exchange energy due to the spin co-alignment. how are multi-domain states formed? - ANS>the exchange and magnetostatic interactions of the spins act on very dissimilar scales. this allows a system of spin pairs to minimise magnetic energy by creating a structure of magnetic domains. what is the magnetisation curve? - ANS> what is the hysteresis loop? - ANS>
what is a spin wave? - ANS>spin waves are a special type of magnetisation precession whereby spins at different spin sites precess with different initial phases, such that a travelling wave of precession phase is formed what is a magnon? - ANS>quanta of spin waves are called magnons.
