Condensed Matter Physics Flashcards
how to plot miller index
axes go x-z-y clockwise round the triangle
- axes extend backwards through origin
line above indes means negative
0 means parallel to that axis
1 means at a on that axis
2 means at a/2
etc
spacing between planes, d
d = a/sqrt(N)
where N = h^2 + k^2 + l^2
describe covalent bonding. where is it present
in C02
occurs in systems where identiical or similar atoms bond
sharing of one electron from each atom
in covalent bond electrons have opposite spins; produces a spatially symmetric wavefunction with a high density of electrons between the atoms
descirbe van-der-waals bonding. where is it present
in C02
olds together individual molecules
weak type of bonding
from coulomb interaction between electric dipoles, varies as 1/r^6
repulisve component from pauli exclusion principle at close distances, increasing energy as the spatial components of the wavefunctions overlap
optical mode
atoms vibrate out of phase
Ka < < 1
w^2 ~ 2C(1/M1 + 1/M2)
w indpendent of K
sketch on E-R graph: negatove shallow curve from high E going horizontal at R=pi/a
acoustic branch
atoms vibrate in phase
Ka ~ 1 - (1/2)K^2a^2
w^2 ~ (1/2)C/(M1 + M2) K^2a^2
w is proportional to K
sketch on E_R graph: straight diagonal that curves to horizontal by R=pi/a
debye approximation of acoustic branch, sketch
acoustic branch
constant velocity (y=x line on E-R graph)
einstein approximation of optical branch, sketch
optical branch
all phonons have same energy/frequency (straight horizontal line at E through optical prediction)
Drude model
what assumptions bout electrons are made?
what is the drift velocity profile of an electron under an applied electric field?
assumption: electrons are classical particles with mass me, charge -e
drift velocity profile: under constant E electrons undergo constant acceleration. Drude assumes that electron drift velocity increases linearly (with const’ a) until the electron undergoes a collision then it resets to zero.
times between collisions varies randomly, scattering time (tau) = mean time between collisions
origin of energy band gap in crystalline solid
(between valence and conduction band)
bragg scattering of electrons from planes of atoms leadin to regions where there are no energy states
braggs arises from destructive interference between the electron wavefunctions and the fourier components of the periodic potential in the crystal
state bragg’s law
2dsinθ = nλ
d = plane spacing
θ = scattering angle
explain diffraction of x-rays from crystal lattice using bragg’s law
models crystal as parrallel planes.
braggs law arises from path difference in relections from adjacent planes.
bragg scattering involves the partial reflection of incident X-ray waves from parallel planes of atoms in a crystal. each plane of atoms reflects a proportion or the intensity of the incident x-ray wave - related to the electron density.
structure factor expression
S(h,k,l) = sum over cell (f exp-2πi(hxj + kyj + lzj )
h,k,l from bragg peak
x,y,z from fractional coords of atoms in unit cell
S describes interference within a unit cell which is not considered by bragg’s law
what does structure factor equal for a bcc lattice
S = 2f when h+k+l = even
S = 0 when h+k+l = odd
what does structure factor equal for an fcc lattice
S = 4f when h,k,l: all even or all odd
S = 0 when h,k,l: mixed parity
what does structure factor equal for a simple cubic lattice
S = non-zero when h=k=l=0
therefore scattering only occurs when incident X-ray is aligned exactly with lattice planes
describe how these functions give the total electron energy distribution:
energy-density of states, fermi-dirac, fermi-energy
energy density of states function = number of available electron energy states per unit energy range which can be occupied
fermi-dirac function = probability of these states being occupied
fermi energy = hgihest occupied electron state in a system when it is in the ground state (below fermi energy all states are filled, above none are)
equation relating fermi energy to electron density
EF = h^2/2me (3(π^2)n)^(2/3)
where n = density
& h = h_bar
fermi velocity from fermi wave vector
vF = hkF/me
where, kF =(3(π^2)n^(1/3)
& h = h_bar
what assumption can be made about free electron density
the same as atomic density
aka. density of a substance is the electron density x atomic mass
how does the confinement of electrons to a 2D layer alter the density of states function
function is then constant with respect to energy instead of sqrt(E).
this alters the availabe electron energy states at band edge
gives favourabl electronic properties for some devices
equipartition theorem (classical method for getting speed)
kinectic energy
= (1/2)mv^2 = (1/2)kBT
what is a primitive cubic lattice, body-centered, or face-centred cubic lattice
primitve: only one lattice point at each corner of unit cell
bcc: atoms at corners & in centre of cube
fcc: atoms at corners (shared between 8 points) & centre of each face (shared between 2 points)
what is the debye approximation
assumes that the dispersion relation is linear w=vk,
that the velocity is constant throughout the brillouin zone & given by the long wavelength limit approximation
equation for scattering time/average relaxation time (tau)
from electron conductivity:
(sigma) = ne^2(tau)/me
from mean path:
mean free path = (tau)v
where v is from equipartition theorem
equation for atomic spacing
= cube root (V)
what is group velocity at brilluoin zone boundary
where k = +/- (pi)/a
vg = (1/h_bar)(dE(k)/dk) = 0
at what value of k does the effective mass tend to infinity
m = (pi)/sqrt(3)a
weidmann-franz law
𝜅/𝜎 = (3/2)((kB/e)^2)T
ratio of electrical to thermal conducivity
use hall coefficient to find density of free electron carriers (n)
RH = -1/ne
thermal energy in terms of boltzman constant
E = kBT
