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168 Cards in this Set

  • Front
  • Back

define crystal lattice

Construcyed by infinite repetition of identical group of atoms (basis)

define primative lattice cell

a parallelogram with the axes of primative vectors

define primative vectors

for any two points r and r prime look the same

how many atoms are in the 3 types of cubic lattice

simple cubic = 1 atom


BCC = 2 atoms


FCC = 4 atoms

translation vector

T = U1a1 + u2a2 + u3a3

r prime

r' = r + T

distance between basis

rj = xja1 + yja2 + zja3

r prime with a basis of 2 atoms

r' = r + T + rj

bands of an insulator

bands of a semiconductor

bands of a metal

directions of r

characterised by u, v, w that have the ratio of u1 u2 u3

planes of lattices

defined by the intercepts of plane with crystal lattice.


find the reciprocal and reduce the fractions to integers

Fermi Dirac distribution

dispersion relationship

total number of states inside a sphere

Fermi k vector

Fermi energy

density of states

momentum

Fermi temperature

Fermi level at small temperatures

Fermi level = Fermi energy

proof that Y(x+L) = Y(x)

principles of free electron model

ignored crystal lattice



valence electrons move almost freely through volume

cons of free electron model

fails to account for:


metallic


colour

define scattering time tau

time taken for an electrone velocity to relax to zero

mobility mu

a steady state electron velocity acquired in the electric field with a value of 1 V/m

electron motion

velocity in terms of scattering times

v = - eEtau/m

mobility

v = - mu E



mu = - e tau/m

current density

J = - nev = sigma E

conductivity

sigma = ne^2 tau/ m

resistivity

ro = 1/ne mu

free mean path

L = v tau

drude model

ignores ion cores and interactions. of electrons with periodic crystal lattice

electrons in Fermi sphere in absense of E field

electrons in Fermi sphere with application of E field

phonon scattering

defect scattering

collisions are elastic


scattering with large delta k causes resistance

matthiesens rule

Bragg diffraction

2asin(theta) = N lamda

reciprocal lattice vector

G = 2pi/a = delta k

group velocity

vg = dw/dk = 1/h bar dE/dk

electrons experiencing Bragg diffraction

have a k vector of k = ±npi/a

K vectir in 1st brouillion zone

±pi/a

when are standing waves formed in a 1D chain

when K = ±pi/a

positive and negative wavefunctions. of standing waves

Y+ = 2cos(pix/a)


Y- = 2isin(pix/a)

probability of wavefunctions

Y+*Y- piles up ON ions


Y-*Y+ piles up INBETWEEN ions

application of E field in a half filled Conduction band

enables shift of electron distribution with respect to zero


non zero current

application of E field in a filled Conduction band

current allows some electrons to jump the band gap

overlapping bands

energy of first state can be higher than that of the second so the states in second band are filled first

efficient diffraction

delta k = 2ksin(theta)

Bragg plane

perpendicular bisector planes that divide boundries of brouillion zones

reduced zone schemes

high order BZ are folded back into 1st BZ by changing k vectors by a multiple of G

Construction of Bragg planes

define Fermi surface

surface of a sphere separating full and empty states in K space

diagram of Fermi surface

deformation of Fermi surface

modification of Fermi surface rules

how to measure Conduction band distribution of metals

find Fermi energy by emission of xrays

define tight binding model

constructs electron bands and wavefunctions

how does tight binding model work

how are energy bands formed

from different electron orbitals depending on the distance between atoms

derivation of effective mass

define a hole

an electron with positive mass and change

creation of a hole in a semiconductor

do holes create current

yes because they have positive charge

how to make a semiconductor more conductive

add elements to make more holes or higher electron concentration

radius of an atom

donors

what group will be donors for group IV

group V

acceptors

what group will be acceptors for group IV

group III

binging energy

number of electrons

electron/hole density

dependance of Fermi level on temperature

Fermi level. at low temperature

concentration of electrons at low. temp

entrinsic semiconductors

doped semiconductors where electron concentrationd are given by dopants

intrinsic material

when Na = Nd then both will be ionised and no holes or electrons will be in bands

types of semiconductor

lorentz force

F = - ev x B

Hall electric field

eEh=ev x B

velocity in direction of Ex

what direction is hall field. in

Y direction

Hall coefficient

Rh = 1/ne = EyB/j

current density in terms of width and thickness

J = I/wt

Hall velocity

Vh=Ey w

Hall effect with electrons and holes

direction of electrons around B

will move in anticlockwise direction

cyclotron resonance

absorption coefficient

complex velocity

circularly polarised E field

FWHM of cyclotron resonance

define electrical conductivity

depends on carrier concentration and their distribution of thermal velocities

where is Fermi level

somewhere in band gap

kinetic energy of electrons in Conduction band

1/2 mv^2 = 3/2 kT

charge defects

free mean path wrt T at low T

proportional to T^2

scattering time with respect to T at low T

tau is proportional to T^3/2

free mean path wrt T at high T

proportional to T^-1

scattering time wrt T at high T

proportional to T^-3/2

optical absorption

what is an exciton

moss-burstein effect

derive dielectric function

what happens when w>wp

wave will propagate

what happens when w<wp

wave will be reflected

electrons under electric field - electrostatic screening

dielectric function of semiconductors

dispersion relation wrt dielectric function

define plasmon

a quantum or longitudinal plasma oscillation

plasma frequency

wp

n^2/epsilon naught me*

where is electronic paramagnetic found

atoms, molecules and lattice defects with odd number of electrons



free atoms and ions with partially. filled inner shell



metals

total angular momentum

J = angular momentum + spin

hunds rules

magnetic dipole of orbital motion

magnetic moment for spin

equation for bohr magneton

resultant magnetic momentum

precesses around J



component of mu R that lies parallel to J gives actual value of dipole moment



perpendicular component averages to zero

magnetic moment of J

landé splitting factor g

what are the values of the splitting factor g

when S=0 g=1


when L=0 g=2

magnetic energy of parallel and antiparallel dipoles

probability of finding an electron in parallel and antiparallel states

what happens to probability when thermal energy is larger than the magnetic energy

probability of parallel is roughly equal to probability of perpendicular



magnetic dipole is roughly zero

what happens to probability when thermal energy is smaller than magnetic energy

probability of parallel is larger than probability of perpendicular



brillouin function

this gives the average dipole component in the field direction as a function it B and T and takes a value between 1 and 0

derive an expression for the magnetic susceptibility in the limit of small. applied fields and high temperatures

equations for magnetic susceptibility

bulk properties of a ferromagnetic

magnetisation is large and positive



its contribution to total B field is significant



magnetism is a complex function of the applied magnetic H field



magnetism depends on history of sample

what causes the increasing magnetism in a paramagnet

increasing alignment of magnetic dipoles

how to produce large values of magnetisation in a ferromagnet

very small H field



direct alignment of magnetic dipoles

edges of domains

called walls



width of a few 100 atoms wide

two main mechanisms by which a ferromagnet is magnetised

growth of domains with favourably. orientated magnetisation vectors



rotation of magnetisation vectors

hysteresis effects

irreversible domain wall motions give rise to these effects

domain model for the magnetisation of a ferromagnet

displacement of domain walls eat away the unfavourable domains whose direction is not aligned with the applied magnetic field

what makes a domain wall motion irreversible

motion of walls is strongly influenced by defects or impurities in the sample

the total energy of a ferromagnet crystal has a contribution from 3 mechanisms

magnetostatic energy



anisotropy energy



exchange energy



sum of these must be minimised

temperature dependance of a ferromagnet

if you heat a permanent magnet past the curie temperature and then cool it, it will destroy domain structure

sources of an magnetic field

electric current that flows in a conductor



from intrinsic magnetic properties of particles having a spin

define H field

can only be produced by a free current

define B field

related to total current, bound and free

what is a magnetic dipole ina in wire

current times area

define magnetisation

where a non magnetic material is converted to a magnetic material



magnetic dipole moment per unit volume

relationship of B and H

B = mu naught H

relationship between B and H when there's magnetisation

B = mu naught (H+M)



B = mu naught mu R H

2 measures of susceptibility

force on a conductor in a magnetic field


circulating negative charge

magnetic dipole moment of a circulating electron

energy when placing a magnetic dipole in a B field

torque when placing a magnetic dipole in a B field

magnetisation and H field

theory of diamagnetism

all materials are diamagnetic



have a small negative values of susceptibility



magnetism will oppose an applied magnetic field

precession

the motion of a magnetic dipole in a constant magnetic field

precessional motion

langeins theory of diamagnetism

what does the precessional motion mean.

what can cause spontaneous magnetisation

need alignment of neighbouring magnetic dipoles

pauli exclusion principle

for 2 identical fermions, total wavefunction is antisymmetic under exchange of particles

4 ways of writing a wavefunction

one singlet = spins are antiparallel = symmetric


three triplets = spins are parallel = antisymmetic

overlap in distributions

smaller for two electrons in an antisymmetic spatial wavefunction



therefore a smaller coulombic repulsion in a triplet wavefunction

energy comparison for states

triplet state is lower than the singlet state

energy difference between singlet and triplet

exchange energy



best condition for ferromagnetiv state is when J is positive