Solid state physics part 1 Flashcards
What are solids called that are not crystalline and they are characterized by what? (not studying these)
Amorphous and an absence of long-range order
Where does the repulsion part of chemical bonding come from?
Pauli exclusion principle means that when the electron “clouds” of atoms start to overlap, they need to maintain orthogonality - this costs energy
What are the main 3 types of bonding in crystalline solids?
Ionic, covalent, and metallic
What is a covalent bond?
Electrons are shared between atoms to form electron pairs called shared pairs or bonding pairs.
What is an ionic bond?
It is the transfer of electrons between atoms (metal to a non-metal) to obtain a full valence shell for both atoms and occurs due to the electrostatic attraction between oppositely charged ions
Is it possible to have ‘clean’ ionic bonding, where one atom or molecule completely transfers an electron to another?
No, all ionic compound have some degree of covalent bonding
What is a metallic bond?
It is the sharing of free electrons among a structure of positively charged ions (cations)
Separation of variables gives radial and angular part, what are the angular parts called?
Spherical harmonics
What are the letters for the first 3 energy levels?
s, p and d
What is the angular quantum number for each energy level?
0 for s, 1 for p and 2 for d
If L is the angular quantum number of sub-shell, then what is the maximum electrons it can hold?
2(2L+1)
What is the maximum number of electrons each sublevel can hold?
s can hold 2 electrons, p can hold 6 electrons and d can hold 10 electrons
Since an electron can theoretically occupy all space, it is impossible to draw an orbital. All we can do is draw a shape that will include the electron most of the time, what is this shape called?
The 95% contour
What do s orbitals look like and how many types are there in any particular energy level?
They are spherically symmetric around the nucleus of the atom and only one type, and on higher levels you can have bigger ones like 2s, 3s, 4s etc (start at 1s)
What do p orbitals look like and how many types are there in any particular energy level?
There are 3 types that can occur after the first level (2p, 3p, 4p etc), they are called px, py and pz because they are at right angles to each other. They look like two spheres next to each other
How many types of d orbital are there in any particular energy level?
There are 5 types that can occur on the third level and later (3d, 4d, 5d etc), they are called dxy, dzx, dyz, dz^2 and dx^2-y^2.
What do d orbitals look like?
The dxy, dzx, dyz orbitals have 4 lobes and the letters say which plane they are in but they are on the diagonal, not along the axis. The dx^2-y^2 orbital has 4 lobes along the x and y axes, and the dz^2 orbital is only along the z axis.
What is the difference between a shell, subshell and an orbital?
A shell in an atom is a set of subshells of the same quantum number theory, n. Orbitals contain two electrons each and are made up of electrons with different spins that are all in the same energy level. Subshells are composed of electrons with the same angular momentum quantum number
What is a crystal lattice?
A set of points generated by multiples (m, n, o) of some primitive vectors (a1, a2, a3) in the form: R = ma1 + na2 + oa3
What is a lattice of these points called?
Bravais lattice
How many Bravais lattices are there with different symmetries (2D and 3D separate)?
5 in 2D and 14 in 3D
What is the primitive unit cell?
The volume of space which, when translated through all of the vectors of the Bravais lattice, fills space without overlapping or leaving any voids
How many lattice points does the primitive unit cell contain?
Just one
What are lattice points?
They represent the location of the atoms or ions. Lattice points that are shared by n cells are counted as 1/n of the lattice points contained in each of those cells
What is the Wigner-Seitz cell?
A type of primitive unit cell that is constructed by drawing the planes defined by the perpendicular bisector of lattice vectors and it surrounds one lattice point
What is a basis for a lattice?
It is what we actually place on each lattice site and can be a single atom, a few atoms or even a complex molecule
What is a simple cubic (sc) structure and what are the primitive vectors?
Looks like a cube and the primitive vectors are the x, y and z unit vectors that are multiplied by scalars
What is the simple cubic structure so rare among the elements?
Low packing efficiency, the packing fraction is small
What is the body-centred cubic (bcc) structure?
A cube with an atom in the middle of it
What type of structures are close-packed structures?
Face-centred cubic (fcc) and hexagonal closed packed (hcp)
What is the packing fraction or packing efficiency?
The fraction of the crystal or the unit cell actually occupied by the atoms
How do you calculate the packing fraction?
The number of particles in the unit cell multiplied by the volume of each particle divided by the volume occupied by the unit cell
What is the packing fraction of simple cubic (sc), body-centred cubic (bcc), face-centred cubic (fcc) and hexagonal closed packed (hcp)?
SC = 0.52, BCC = 0.68, FCC & HCP = 0.74
What does a face-centred cubic (fcc) structure look like?
A cube with another atom in the middle of each face on each side
Out of sc, bcc, fcc and hcp structures, which ones are Bravais lattices?
sc, bcc and fcc
Why is a hexagonal close-packed (hcp) structure not a Bravais lattice?
Two atoms are needed with each point
What is the most common way of measuring crystal structures?
X-ray diffraction
X-rays interact with the charge distribution in a solid but only in a weak way, what does this mean we can assume?
They only scatter once, we call this the kinematic approximation
If the source of x-rays and the detector are sufficiently far away from the sample being studied, what can we assume about the ingoing and outgoing waves?
We can treat them like plane waves
What is the Bragg condition for constructive interference between two rays that are reflected by successive planes?
An integer multiplied by the wavelength of the ways is equal to 2 multiplied by the distance between planes multiplied by the sin of the angle between the rays and the plane
What is a lattice plane?
It contains at least 3 non-colinear points of a given Bravais lattice
How do you find the Miller indices of a plane?
Find where the plane intercepts the crystallographic axes in lattice vector units, take the reciprocal value of the 3 numbers and reduce them to the smallest set of integers with the same ratio
How are Miller indices presented for a specific plane?
(hkl)
How do you express a set of planes that are related by symmetry of the crystal to each other?
{U,V,W} planes
What if we cross the axis at a negative number when calculating the Miller indices?
Put a little bar above the number
How are directions in the crystal labelled?
With square brackets eg []
A set of crystallographic directions which are symmetrically equivalent, how are these labelled?
With triangular brackets eg <>
What causes the scattering of x-rays?
Its electromagnetic field causes the electrons in the material to oscillate at the same frequency as that of the field and the electrons emit new x-rays that give rise to an interference pattern
For a given Bravais lattice, R, and its reciprocal lattice, G, what is the dot product G.R equal to?
2 pi multiplied by an integer
How are the reciprocal lattice vectors calculated?
2 pi multiplied by a fraction involving the primitive vectors
What is the dot product between one of the primitive vectors and one of the reciprocal lattice vectors equal to?
2 pi multiplied by the Kronecker delta
What plane do the Miller indices (hkl) define in relation to the reciprocal lattice vector hb1+kb2+Lb3?
A plane which is perpendicular to it
What is the Laue condition?
It is that constructive interference occurs when K (scattering vector) = G (reciprocal lattice vector set)
What is the integral equal to for an oscillating function over a whole number of wavelengths?
Zero
What is the reciprocal lattice of a bcc real space lattice with sides of length a?
FCC lattice with sides of length 4 pi over a
What is the intensity of the scattering proportional to when the Laue condition it satisfied at a particular scattering vector, G?
The absolute value of the particular complex Fourier coefficient of the charge density squared
What is the phase problem of x-ray diffraction?
We can only know the modulus and not the phase of the intensity through measurements (the intensity is the modulus of the complex Fourier coefficients of the charge density squared)
What is the denominator when using the primitive vectors to calculate the reciprocal lattice vectors?
It’s a triple product. The first primitive vector dot product with the cross product between the second and third primitive vectors
What does the Ewald construction allow you to visualise?
The Laue condition
Is the Ewald construction on real lattice points or on reciprocal lattice points?
Reciprocal lattice points
How do you draw the Ewald construction?
Draw the incoming wave vector, k, and draw a circle (or sphere in 3D) with the vector as the radius and wherever the circle intersect a reciprocal lattice point, the Laue condition is satisfied and there will be diffraction intensity in that direction
When the Laue condition is satisfied, what is G equal to?
k-k’. This is the incoming wave vector minus the outgoing wave vector
Why is the Laue condition fulfilled for two of the three direction components in the Bragg picture of specular reflection?
The wave vector parallel to the planes is not changed
How do you find the Bragg condition from the Laue condition?
Use the perpendicular component of the wave vectors and G
What are the complex Fourier coefficients of the charge density also called?
Structure factors
How are the structure factors expressed?
The sum over all atoms in the unit cell of the atomic form factor multiplied by e to the i times G dot r
What is the atomic form factor, f?
It describes the scattering power of the particular atom
What will the coefficients of the G vector be for the beam if it has been diffracted from a plane in the crystal with Miller indices (hkl)?
h, k and l, because the direction [hkl] is perpendicular to the plane (hkl)
What is the r vector in the structure factor formula?
The position vector of the atom within the cell
If the structure factor is zero, what does this mean?
There is destructive interference for this reflection
What is the Wigner-Seitz unit cell of reciprocal space?
The first Brillouin zone
What is the formula that is the Brillouin condition?
2k dotted with G is equal to G squared
Where does the k vector need to end to satisfy the Brillouin condition?
The Brillouin zone boundary
How do you make the unit cell of the Brillouin zone in reciprocal space?
The planes are the perpendicular bisectors of G vectors and the planes satisfy the diffraction condition
What do the Brillouin zone boundaries physically represent?
They are Bragg planes which reflect (diffract) waves with particular wave vectors so that they cause constructive interference
What is the Drude model that he proposed in 1900?
The gas of electrons moved freely in the space between the positively charged ion cores
What are the 5 assumptions of the Drude model?
Electrons scatter only through collisions with the ion cores, and between collisions, they do not interact with each other or with the ion cores. The collisions are instantaneous and result in the electron’s velocity changing. The probability of an electron collision per unit time is one over the inverse scattering rate. Thermal equilibrium is reached through collisions
What does the Drude model predict and is it correct?
The specific heat is independent of temperature, which is wrong
What do energy levels that electrons can occupy in a crystal form?
A band
What happens to the energy levels of two isolated atoms when they are brought together and allow the wavefunctions of the two separate valence electrons (sodium is the example here) to overlap?
A new molecular wavefunction is made as a result of the interference between the two original wavefunctions and this can be constructive or destructive
What are the two types of states that lead from constructive and destructive interference when two atoms are brought together?
Bonding and anti-bonding states
What is the bonding state?
It comes from constructive interference, which puts extra electron density between the nuclei
What is the anti-bonding state?
It comes from destructive interference, which suppresses the electron density between the nuclei
What happens to the energy levels when the separation between two atoms decrease and the energy levels increase?
The original separate energies split into separate bonding and anti-bonding levels
If we imagine solids as being a large molecule, what happens to the energy levels when more and more atoms are added?
There will be more and more non-degenerate levels
The Sommerfeld ‘free electron’ model is a quantum mechanical version of what?
The Drude model
What are the approximations of the Sommerfeld ‘free electron’ model?
The motion of the ions are ignored (Born-Oppenheimer approximation is that electronic and ionic motion can be separated) and all the interactions between an electron and ion-cores and with other electrons can be encapsulated into an effective potential, U(r)
What does it mean in the Sommerfeld model that we treat the electrons as being ‘free’ and what does this mean for the potential?
They do not interact with either the ion cores or the other electrons and this is equivalent to saying U(r) = 0
What are the boundary conditions on the wavefunction for the Sommerfeld model?
Born-von Karman (or periodic), which is boxes with side of length L
What is the normalised solution to the Schrodinger equation with Born-von Karman boundary conditions?
1 over square root the volume multiplied by e to the power of i times the wavevector dotted with the position vector. These are plane waves
What are the allowed values of the wavevector, k, for each direction for the Sommerfeld model?
2 pi over L multiplied by n (nx, ny or nz depending on direction), which are integers (importantly, this includes zero)
What are the discrete set of of energies in the Sommerfeld model called?
Energy eigenvalues
What are the energy eigenvalues equal to in the Sommerfeld model?
h bar squared multiplied by k squared divided by 2 times the mass of an electron
What does it mean that we have a quasi-continuous band of energies in the Sommerfeld model?
The energy eigenvalues are actually discrete but the separation between them is very small
How do the electrons populate the energy levels in the Sommerfeld model?
Starting at the lowest energy, each level can be populated with two electrons and the levels are filled until all electrons have been assigned
If we have N electrons to accommodate, how many levels do we need in the Sommerfeld model?
N/2
How do we find out the value of n max in the Sommerfeld model?
Equate the number of states that we need to accommodate with the volume of the sphere of radius n max
What is the Fermi energy?
The maximum energy level that electrons occupy for single-particle states
What is the Fermi wavevector?
The maximum wavevector that occurs at the Fermi energy and is equal to 2 pi over L multiplied n max or the square root of 3 pi squared times the number density of electrons
What is one unit of volume in k-space (in 3D)?
(2 pi over L) all cubed
What is the density of levels (number of allowed levels per unit volume) in k-space in d dimensions?
(L over 2 pi) to the power of d
How do you calculate the number of allowed states between k and k + dk, eg D(k)dk?
The volume of the shell between spheres of radii k and k+dk (4 pi k squared dk) multiplied by the density of levels in k-space multiplied by 2 (as each level can have 2 electrons)
The density of states for 3D free electrons increases with what power of energy?
Half
How do you work out the number of electrons at zero temperature using the density of states?
It is the integral between 0 and the Fermi energy of the energy density of states dE
For electrons at finite temperature, what will they be distributed according to?
The Fermi-Dirac distribution
For metals at zero temperature, what can we put the chemical potential equal to?
The Fermi energy
At zero temperature, what does the Fermi-Dirac distribution look like?
A step function that is 1 until the Fermi energy and then zero
What happens to the Fermi-Dirac distribution at finite temperature?
It smears at the chemical potential, so some states below it are unoccupied and some above it are occupied
What is the ‘soft zone’ of the Fermi-Dirac distribution?
The smearing of the function around the chemical potential
What happens to the probability of occupation above and below the chemical potential for hotter temperatures?
The probability below the chemical potential reduces whilst the probability above increases
What is the energy width of the ‘soft zone’ of the Fermi-Dirac distribution?
4 multiplied by Boltzmann’s constant multiplied by the temperature
What is the equation for the number of electrons when there is a finite temperature?
The integral between zero and infinity of the density of states multiplied by the Fermi-Dirac distribution dE
What is the free-electron Fermi surface?
The sphere of occupied states in k-space with radius of the Fermi wavevector (max)
What is a general Fermi surface?
The surface of constant energy in reciprocal space which separates occupied states from unoccupied states
What are 2 other ways of saying reciprocal space?
Wavevector space or k-space
Why is the existence of a Fermi surface due to the Pauli principle?
Because it forbids electrons to be in the same quantum state, forcing electrons to occupy higher and higher energies until all have been accomodated
Why are most of the electrons below the Fermi level effectively inert?
They cannot reach (scatter into) unoccupied states if their energy changes by just a small amount
What are the only electrons that matter in a Fermi-Dirac distribution and why?
Those in the ‘soft zone’ around the Fermi energy, which is the electrons on or very close to the Fermi surface. Because they are the only ones that are not inert/restricted
What happens to the Fermi surface when it is at zero temperature versus when it is at finite temperature?
It is sharp at zero temperature but blurs at finite temperature
Does the bonding or anti-bonding state have less energy and why?
Bonding because the increased electron density acts as a screen between the positive ions and reduces the effect of coulomb repulsion
What is the linear term of the temperature in the heat capacity due to?
Electrons
When we raise the temperature, a number of electrons will be excited from below the Fermi energy to above the Fermi energy, what is the order of the energy they will gain?
Boltzmann’s constant multiplied by the temperature
Why does the coefficient to the linear temperature term of the specific heat vary away from the free electron value for real materials?
Mass enhancement, which is the electrons not behaving like they have free electron mass but a different one, because in reality they are not free and they have an ‘effective mass’ that is different from their rest mass
What are 2 shortcomings of the Sommerfeld model?
We do not know why some solids are insulators and others are metals. Also, Fermi surfaces of real metals are never spherical
What is the independent electron approximation?
The electrons are not interacting with each other
For the nearly free electron model, what can is the same and what is different from the free electron model?
We keep the independent electron approximation but electrons are no longer free, but nearly free. The potential must have the periodicity of the lattice U(r) = U(r+R). We still use the Born-von Karman boundary conditions
Every solution of the Schrodinger equation with periodic boundary conditions can be written as what?
A sum of plane waves
What is the central equation?
The Schrodinger equation in a periodic lattice written as a set of algebraic equations (rather than differential as we usually see it)
What is Bloch’s theorem?
The solutions (eigenfunctions) to the Schrodinger equation with periodic boundary conditions take the form of a plane wave modulated by a function with the same periodicity as the lattice
Bloch waves that differ by a reciprocal lattice vector are what?
They are identical
What do we know about the eigenvalues of Bloch waves?
The eigenvalues are periodic functions of k so if we know the eigenvalues within the first Brillouin zone then we know them for all possible k
What is the electronic band structure of a solid?
The range of energy levels that electrons may have within it
Do Bloch waves have periodicity in real space and why?
No because it is the product of two functions and only one of which are periodic
Do Bloch waves have periodicity in reciprocal space and why?
Yes because Bloch waves that differ by a reciprocal lattice vector are identical
When trying to solve the central equation for the first time, how do we simplify it?
Use one dimension, set all Fourier coefficients of the potential equal to zero, except U1 and U-1 and have them both real and both called U. We also assume the wavefunction has only 3 non-zero components
What are the eigenfunctions and eigenvalues of the central equation?
The coefficients of the wavefunction are the eigenfunctions and the energies are the eigenvalues
When U=0 (empty lattice approximation, electrons more free) for the central equation with 3 coefficients of the wavefunction, what does the plot of energy against k look like?
3 parabolas centred on k=g, k=g and k=-g, with the parabolas crossing each other at the first Brillouin zone
When U is not 0 for the central equation with 3 coefficients of the wavefunction, what changes from the graph from when U=0?
There are some energy gaps at the first Brillouin zone where the parabolas have crossed
What happens to the energy against k plot for the central equation when a second Fourier coefficient is is added to the potential?
More energy gaps show higher up, which is on the second Brillouin zone boundary
Using the central equation to find the bandstructure, what are the regions called where there are no allowed states and what are these due to?
Band gaps and the potential
How big is the energy gap at the first Brillouin zone boundary when solving the central equation?
2U (eg 2 times the potential)
How many dominant coefficients are there to the wavefunction?
2
What does the + and - charge densities for the Sommerfeld model look like?
proportional to cos squared and sin squared respectively
When the potential (U) is attractive, is it positive or negative?
Negative
What is the crystal momentum and is it conserved?
h bar times k. It is conserved
Are Bloch states simultaneous eigenstates of the Hamiltonian and momentum operator? (free electron states are)
No
How many independent values of k to each energy band can each primitive cell contribute?
Exactly one
How many electrons can each band contain per primitive unit cell and why?
2 (because 2 spin states)
When do we have enough electrons to completely fill a band (so all independent k-points are occupied)?
An atom contributes two electrons per primitive cell
When do we have enough electrons to half fill a band?
An atom contributes a single electron per primitive cell
How are an electron’s options for finding a state to scatter into limited by the Pauli principle?
It has to be an empty state and it has to be within an infinitesimally small energy jump of where it came from
What is an insulator in terms of energy bands?
It has a completely full energy band and because the energy gap between the bands is large, it is impossible for the electron have enough energy to move into the higher band so it can’t carry a current
What is the case in terms of energy bands for an intrinsic semiconductor?
The band is full but the energy gap between bands is small and there may be enough thermal energy in the system for electrons to jump across the gap to find an unoccupied state
What is a conductor in terms of energy bands?
Partially filled band so electrons can move and it can carry a current.
Do metals and insulators have a Fermi surface?
Metals do, insulators don’t
What is the reduced zone scheme?
Only calculating and plotting the bands within the first Brillouin zone (we usually use this scheme)
What is the extended zone scheme?
Plotting each band in a separate zone (1st band in 1st zone, 2nd band in 2nd zone etc)
What is the repeated zone scheme?
Plotting all bands in all zones (repeat reduced zone scheme in every Brillouin zone)
In the tight binding approximation, what are we going assume about the wavefunctions of the electrons?
They are similar to free electrons, where the electrons are completely localised on the atomic sites (as isolated atoms). We also assume that the true wavefunction is made from a making a linear combination of the atomic wavefunctions
To a first approximation of the tight binding model, do the wavefunctions on neighbouring sites overlap?
No
What is the Hamiltonian in the crystal a sum of? (tight binding approximation)
The Hamiltonian of isolated atom and the small perturbation potential coming from other atoms
The energies for the tight binding model is made from the energy level in the free atom minus two other parts (A and B multiplied by a function that shapes the band) , what do these represent?
A is the correction due to other atoms and B represents the overlap that would actually be there
What part of the energy equation for the tight binding approximation leads to band formation and what does it represent?
B. It is called a transfer (or hopping) integral because it represents the probability of an electron hopping from one site to another
In the tight binding approximation, if there is not much overlap (small B) then is the energy more or less flat?
More flat
What Brillouin zones does the free electron Fermi surface cover in its construction?
Fully the 1st, lots of the 2nd, small amount of the 3rd and tiny amount of the 4th
Is there a Fermi surface in the 1st Brillouin zone for free electrons and why?
No because it is bigger than the 1st zone and therefore the first zone is full
When a magnetic field is applied, what force occurs on the charged particles and what does this cause them to do?
Lorentz force and they will execute orbits in real and reciprocal space (perpendicular to the field for real space)
What is the momentum equal to in a magnetic field?
h bar times k plus the charge multiplied by the magnetic vector potential
What is the quantised magnetic flux for an electron in a magnetic field?
(n + 1/2) multiplied by h over the electron charge (where n is an integer)
The quantisation condition on the allowed orbits in a magnetic field is also a quantisation condition on what?
The enclosed magnetic flux
What is the normal magnetic flux equation when the magnetic field is perpendicular to the motion?
The magnetic field multiplied by the area in real space (this is quantised too, so its the area of the nth allowed level)
What is a line element in reciprocal space squared divided by the a line element in real space squared in a magnetic field and this ratio also applies to what?
The square of all the charge multiplied by the magnetic field divided by h bar. The ratio of the quantised area in reciprocal space to real space
In a magnetic field, what increments are needed to produce orbits of the same area? (what is the period?)
1 over B (the magnetic field)
What are Landau levels?
The circles in 2D or cylinders in 3D that the allowed values of k in the kx-ky plane are confined to when there is a magnetic field
What is the separation of the Landau levels in energy?
h bar multiplied by the cyclotron frequency, which is the charge multiplied by B over the mass
When a Landau level is at the same energy as the Fermi energy, will the total energy be a maximum or lower than maximum?
Maximum
When a Landau level passes through the Fermi energy (not at the same energy), will the total energy be a maximum or lower than maximum?
Lower
As the magnetic field is changed, what values oscillate and what is this concept called?
The total energy and any other thermodynamic quantities that depend on the total energy and they are called quantum oscillations
What is the change in one over the magnetic field equal to?
2 pi multiplied by electron’s charge divided by h bar times S (area in reciprocal space)
For a 3D Fermi surface, there are lots of areas perpendicular to B, which ones dominate the experimental signal?
Extremal orbits (largest or smallest orbits)
Is there any restrictions in the kz direction with Landau tubes (cylinders) if the magnetic field is applied is the z direction?
No
Why do we need low temperatures when measuring Fermi surfaces?
At higher temperatures, the Fermi surface with smear and there will be increased disorder which will scatter electrons
What 3 conditions are needed to accurately obtain Fermi shapes?
High-quality samples, low temperatures and high magnetic fields
What does ARPES stand for?
Angle-resolved photoemission
What is the general idea of ARPES?
UV photons hit a sample being studied and photoelectrons are emitted and their energy and angle are analysed
What are the strengths of ARPES and positron annihilation and Compton scattering techniques?
Direct visualisation of Fermi surfaces and electrons mean-free path doesn’t matter
What are the weakness of ARPES?
Surface sensitive and works best in 2D systems
What is similar in the way positron annihilation and Compton scattering techniques measure the Fermi surface?
They measure it via the momentum distribution of electrons
What are the weaknesses of the positron annihilation and Compton scattering techniques?
Momentum resolution not as good as other techniques, slow in comparison to others, and a synchrotron is needed
