Condensed Matter Physics

Hall electric field

(1/ne) J x B (due to magnetic field applied) If electric field E = E_x \hat{x} and magnetic field B = B_z \hat{z}, then hall field E_H = R_H j_x B_z

Atomic potential

(e^2/(4 pi e_0 a)) \sum_k ((-1)^k n_k)/d_k) + repulsion Sum of the interactions between an atom and its nearest neighbours, in terms of shells of atoms with identical separation

Electron wavefunctions in a periodic potential (far from BZ boundary)

2nd order correction to energy levels is small, so they are like free e- states

Dispersion relation branches

3 for each atom in the unit cell Branches can be degenerate, especially in high symmetry directions 3 acoustic modes 3p - 3 optical modes for p distinct basis atoms

Amorphous material

A material that lacks the long-range (translational) order that is characteristic of a crystalline solid. Amorphous materials are either rubbers or glasses, and have only short-range order (Amorphous and crystalline phases can exist for the same material)

Drude-Sommerfeld corrections to Drude model

Accounts for density of electron states and only counts e- sufficiently close to E_F Makes electronic heat capacity depend on (T/T_F) so that it only matters at low temps MFP much larger as electrons don't scatter from individual ions, only from defects & impurities - anything that disturbs the lattice periodicity (phonons, other e-)

Diffraction approximations

All scattering is elastic (no energy losses, Compton scattering or change in wavelength when photon and e- interact) x-rays interact weakly and only scatter once (each) (kinematic)

Controlling crystal grain size

Annealing (larger grains) Smaller (large grains) Dispersing impurities to pin grain boundaries Cold working (introduces additional defects)

electrical conductivity and Fermi surface

Applying constant electric field along x-direction means uncompensated electrons arise and cause a current density j_x e- can't scatter into Fermi surface as all states filled so they have to scatter across it to the other side, which acts like damping to keep the e- acceleration due to the electric field finite

Einstein heat capacity model

Assumes all oscillators have same frequency, w_E Good assumption for high energy optical phonons, which dominate at high temperatures C_V -> 3 N_a k_B at high temperatures (agrees with Dulong-Petit) C_V -> 0 exponentially as T -> 0, not as T^3 as experimentally observed

Debye model heat capacity

Assumes linear dispersion w(k) = vk and velocity of sound is the same for all polarisations Recovers C_V \propto T^3 at low temperatures as observed experimentally C_V \propto constant at high temperatures as expected Better representation of low energy vibrations than Einstein model, which matter most at low temps (but breaks down at intermediate temps)

Single crystal

Atoms are in a repeating or periodic array over the entire extent of the material, with long range order.

Drift velocity v_d

Average velocity of electrons induced by electric field

Phonon

Behave like photons, but exist in materials, not in a vacuum, so they behave somewhat differently Represent discrete energy levels in material structures, but as number of atoms gets large, the energy level distribution approaches a continuum so quantum effects are insignificant

Model of a solid

Bound e- localised at ion cores (bound energy states) Nearly free (delocalised) e- free to move through the solid (lattice states) Wavefunctions in bound states are modelled as those of an infinite square well

Gamma point

Centre of Wigner Seitz cell

Bravais Lattice

Collection of all positions that can be reached in space from the origin with a position vector r = na_1 + m a_2 + o a_3, with a_1, a_2 and a_3 all in distinct planes. Only one basis atom present.

Single band approximation

Consider just one energy band that derives from atomic energy level E_i with a single atom/unit cell Then we can only consider the orbitals associated with this energy eigenstate, so the wavefunction can be approximated by a linear combination of just these atomic orbitals

Classical stat mech theory of heat capacity

Constant and independent of material and temperature (C_v = 3R - Dulong-Petit Law) Based on 1D harmonic oscillator, works for high T

Ewald Sphere (diameter of 1/wavelength)

Constructive interference occurs when a reciprocal lattice point lies on the Ewald sphere Equiv: distance from transmitted x-ray to scattered x-ray is 1/d_{hkl} (Rotating crystal moves more points through Ewald sphere allowing more diffraction spots to be observed as different reciprocal lattice points line up with the sphere)

Laue condition

Constructive interference when scattering vector = reciprocal lattice vector (i.e. q = k' - k = G and |k'| = |k| = 2 pi / wavelength) Or equivalently when dot product of (r_j) and G = 2\pi m for integer m (r_j = position of jth atom)

Thermal conduction

Contributions from e- and phonons => electrical insulators can have very high thermal conductance

N-process (normal scattering)

Crystal momentum conserved Emergent wavevector inside first Brillouin zone

U-process (Umklapp scattering)

Crystal momentum not conserved Emergent wavevector outside first Brillouin zone, but translated back inside 1st BZ by subtracting an integer multiple of a reciprocal lattice vector Number of umklapp collisions that occur depends on temperature Need large incident wavevector and high angular frequency (w) to get a resulting wavevector outside the 1st BZ

Polycrystalline

Crystalline materials composed of more than one crystal or grain. Periodic structure, long-range order interrupted by boundaries between highly ordered regions with differing orientations.

Interatomic spacing in amorphous materials

Described by pair correlation function: probability of finding an atom somewhere given a starting location (goes to 1 as distance -> infinity)(1 = no correlation, >1 = positive correlation)

Lever Rule

Determining composition of intermediate mixed phases X_{alpha} = (c-b)/(a-b) X_{alpha} is weight percentage of phase alpha a = concentration of element B in alpha phase (e.g. solid) b = concentration of element B in beta phase (e.g. liquid) c = total concentration of element B

Drude Sommerfeld model limitations

Doesn't explain conductivity that varies with direction, positive R_H, difference between metals, semiconductors & insulators

Dispersion relation

E = E(k), represents electronic bandstructure for a material

Bloch Theorem

Eigenstates of the one electron Schrodinger equation for a periodic potential have the form of a plane wave multiplied by a function which has the periodicity of the Bravais lattice

Localised electrons

Electron states associated with a specific atom (below level of atomic potential)

Nearly free electron bandgap

Electrons travelling in a weak potential Bandgap in bandstructure related to Fourier coefficients of potential, as the difference between the potential energies of the two standing waves formed during Bragg diffraction

Consequence of Bloch Theorem

Electrons, even in strong potentials, propagate almost like free e-, with plane wave eigenstates, that are modulated by the potential (also relationship between wavevector and momentum modified)

Semiconductor/insulator criteria

Filled band and band gap present, as electrons can't (always) jump between valence band (highest filled band) and conduction band (lowest unfilled band)

XRD and grains

Finite crystallite size => imperfect destructive interference, broadening of diffraction peaks (greater for smaller peaks)

Harmonic oscillator approximation

For small displacements from the equilibrium atomic spacing (R - R_0), the atomic potential is approximately quadratic

Eutectic alloys

Form when solubility of two elements is limited Can have two distinct phases

Reciprocal space lattice

Fourier transform of real space lattice

Amorphous material types

Glasses, liquids, thin films, metallic glasses, ion-damaged materials Generally liquids, where atomic kinetic energies are too high to remain bound in a crystal

Thermodynamic theory of heat capacity

Goes to zero as T -> 0 as finite entropy at T = 0

(Semi)metal criteria

Have nearly full valence band and partially filled conduction band

Liquid temperature and order (amorphous)

Higher temperature => amplitude of pair correlation function decreases Due to thermal expansion and effect on degree of packing of atoms

Debye frequency (w_D)

Highest frequency that would be reached if we continued the acoustic dispersion regime w_D^3 = 6 pi^2 (N/V) v^3

Quasicrystal

Highly ordered but no translational symmetry

Structure factor

How materials scatter incident radiation/what diffraction pattern to expect

Decrease of bandgap with temperature

Interatomic spacing increases with temperature as amplitude of atomic vibrations increases. Increased interatomic spacing => potential seen by electrons decreases, which leads to smaller bandgap

Electron density in bonding

Ionic: much higher around atomic nuclei Covalent: much higher between the atoms, lower near the nuclei

Born von Karman boundary conditions

Join the ends of a chain of atoms for continuity (despite materials having finite length)

Tight binding approach

Looking at localised e- (states assoc. with atoms) treat single e- states as a superposition of atomic wavefunctions e- move between atoms by tunnelling bandstructure results from spacing of atomic energy levels and formation of energy bands (and bandgaps) due to overlap of atomic wavefunctions Assume only nearest neighbour interactions as hopping rate of e- depends on proximity of neighbours

Good thermal insulators

Low heat capacity at temperature of operation, short MFP, low velocity of sound Short MFP: small particles or many defects in the crystal structure Low speed of sound: weak interatomic forces & heavy atomic masses

Mean free path (MFP, for phonons)

Mean distance between collisions Constrained by phonon collisions with crystal impurities, boundary surfaces and other phonons

Effect of neglecting higher order terms in potential coefficients

Means bandgaps at higher energies are not present/accounted for, as band gap between bands n and n+1 is related to nth Fourier coefficient of potential

Atomic form/scattering factor

Measure of scattering power of an isolated atom

Free e- (Drude) model assumptions

Metals consist of immobile positive ions and high density of mobile e- e- don't interact with each other but scatter from positive ions - instantaneous collisions/no long range interactions - mean time between collisions tau e- can be treated as an ideal gas and reach thermal eqm with lattice through collisions - their energy distribution reset to the the MB distribution after the collision In between collisions e- respond to external fields (e.g. EM fields) (Builds on kinetic theory of gases to develop model of e- transport in solids)

Alloys

Mixtures of elements - for when there is non-zero solubility between two different solid materials

Zone melting (high purity single crystals)

Moving a warm zone upward and out, leaving the rest to solidify as a high purity single crystal Impurity atoms will diffuse to the liquid regions of the crystal and thus can be separated

Bravais lattice vectors

Must form a basis for a given basis atom, and all lie in distinct planes - any two cannot lie in the same plane

Maxwell Boltzmann ideal gas

N_t distinguishable gas atoms/molecules in thermal eqm at temp T avg KE per atom: sqrt((3 k_B T)/m) total KE of gas: (3/2) N_T k_B T total specific heat of gas: C_V = (3/2) N_T k_B thermal conductivity k = (1/3) <v^2> tau C_V

Mass chain dispersion relation (long wavelength limit)

Near k = 0, wavelengths are long so sound waves are present (acoustic phonons) that travel through the lattice

Drude Sommerfeld (free e-) model assumptions

Neglect interactions between ions and valence e- except in boundary conditions ions are not necessarily source of collisions The interactions between electrons are ignored. The electrostatic fields in metals are weak because of the screening effect. There is a relaxation time that doesn't depend on the electronic configuration Pauli exclusion principle (Fermi gas based model)

Neutron diffraction

Neutrons scatter from nuclei so can distinguish different isotopes and discern magnetic structures given spins of neutrons

Reciprocal lattice vector

Normal to planes they represent, and correspond to the spatial frequency of the planes magnitude of b_i = spatial freq. of planes in a_i direction

Fermi trap effect (Drude-Sommerfeld model)

Only a small fraction of e- can participate in energy exchange (those within ~3 k_B T of E_F) Drude model says that all e- can participate

Fermi Dirac distribution

Only e- near E_F can participate in energy exchange "Probability that a given energy level is filled" - changes with temperature All states up to E_F filled at T = 0 K

Covalent bonding in H2

Parallel e- spin: no bonding, energy = 2 E_0 + ΔE↑↑

Strongest perturbation to energy of free e-

Periodic potential couples free e- wavefunctions with k values that differ by a reciprocal lattice vector (At BZ boundary, when the Bragg diffraction condition is satisfied (k = ± n pi/a) - need to consider at least two dominant terms, c_k and c_{k-G}, assuming all others are zero)

Bragg plane

Perpendicular bisector to a line passing from the origin of reciprocal space to a reciprocal lattice point (plane of atoms that x-rays reflect off in XRD)

Phonon collisions

Phonon momentum means energy of lattice vibrations (phonons don't exist) Harmonic wavepackets don't scatter - they just pass through each other Anharmonic forces only can produce phonon "collisions"

Form of Bloch functions

Plane wave times function with periodicity of the Bravais lattice, which causes periodic modulation of the plane wave

Drude model limitations

Predicts resistivity proportional to sqrt(T), but experimentally it's proportional to T States resistivity of alloys should be between those of its components - actually resistivity can be much higher than that of either component Gives anomalous R_H values for some metals (due to presence of holes which it doesn't account for)

Hall coefficient

R_H = E_y/(j_x B_z) if electric field applied along x-direction and magnetic field applied along z-direction In steady state R_H = -1/(ne) - Lorentz and electric forces balance so e- pass straight through the material

Wigner-Seitz cell

Region of space closer to a chosen atom than to any of its nearest neighbours (It has the point symmetry of the lattice)

Periodicity of reciprocal lattice

Repeats with period G e- properties must reflect this symmetry so we can use reduced zone scheme to represent the energy states through k-space

Pairwise atomic interaction

Repulsion at short distances, attraction at long distances (Pauli exclusion principle)

Primitive unit cell

Smallest volume cells that reflect symmetry of crystal lattice, and contain one lattice point per cell Can have a basis of more than one atom

Phonons in liquids

Sound propagates in liquids therefore possible to have phonons, but fluids cannot support the low frequency transverse modes High frequency transverse modes can exist as liquids take a finite amount of time to relax Min transverse frequency = Fermi frequency, w_F = 2pi/(relaxation time) No BZs in dispersion curves for amorphous materials as the dispersion relations are not periodic

2D Bravais lattices

Square, rectangular, hexagonal, oblique, centered rectangular

Quenching of alloys

Sufficiently rapid cooling of an alloy allows freezing a higher temperature phase into place, resulting in a "metastable state" occurs as atoms need to overcome a potential barrier to change arrangement, that requires a certain amount of thermal energy

Pairwise potential

Sum of Coulomb repulsion and nearest neighbour repulsion

Fermi surface

Surface with radius k_F that contains all the e- in a given volume at T = 0 K

Space group

Symmetry group of a three-(two-) dimensional (crystal) pattern. Each group has an infinite set of translations. Also related to the Bravais lattices (shapes of unit cells) combined with the locations of the basis atoms.

Electron wavefunctions in a periodic potential (at BZ boundary)

System is degenerate in energies so use degenerate perturbation theory to get E = e_0 (k) ± |V_G| for e_0 energies of unperturbed states

First Brillouin Zone

The Wigner-Seitz cell of the origin of the reciprocal lattice (centered on an atom, unlike FCC, BCC, simple cubic)

Thermal conduction by phonons

Treat phonons as particles - their interactions are described by thermal vibrations in the crystal, as if they are particles in an ideal gas Wavepacket size >> lattice spacing, but << crystal size k_p = (1/3) C_V (MFP) v_p v_p = phonon (i.e. sound) speed

Debye density of states (D_P (w))

V w^2/(2 pi^2 v^3)

Types of defects

Vacancy, interstitial, substitution, self-interstitial Common because they increase entropy and lower Gibbs free energy

XRD vs neutron diffraction

X-rays scatter from electrons so hard to distinguish atoms in XRD Neutrons scatter from nuclei so can distinguish different isotopes and discern magnetic structures given spins of neutrons

Distance between lattice planes

d_{hkl} = 2 pi / (|G|) = 2 pi / sqrt(h^2 + k^2 + l^2)

Delocalised electrons

e- states associated with the crystal as a whole (below the crystal potential level but above that of the atoms)

e- motion in electric field (Drude)

electrons drift between collisions in response to electric field at drift velocity v_d statistically looks at average response of e- to an external force F to get p'(t) = F(t) - (1/tau) p(t)

e- velocity (Drude)

given by thermal velocity from MB distribution, <v> = sqrt(3 k_B T /m)

Mass chain dispersion relation (k = pi/a)

group velocity here is zero and thus standing waves occur - no energy propagation, wavelength = 2a

Ohmic electric field

m/(ne^2 tau) J (due to current density from applied field)

Bragg's law

n wavelength = 2 d sin (theta)

e- motion in absence of field (Drude)

net displacement = 0, random motion, average velocity = 0

k-space

points in k-space represent quantum state of the system (each can hold two e-) Volume of one state is (2pi/L)^3 for material of length L constant radius = constant energy surface in k-space

Substitutional alloys

produced by mutually soluble (miscible) elements

Weidemann Franz Law

ratio of thermal to electrical conductivity constant at a given temperature for all metals (k/sigma = LT)

electronic heat capacity

small at high temperatures, dominates at low temperatures estimated as constant by Drude (3/2 n k_B) and too large - overestimated as it assumes all electrons contribute to thermal energy exchange C_V = y T + B T^3 in reality (yT from e-, BT^3 from lattice)

Fermi velocity, thermal velocity, drift velocity

v_F ~ 10^6 m/s, v_d ~ 10^{-2} m/s, v_t ~ 10^5 m/s