Condensed Matter Physics

Question 1

how to plot miller index

Answer 1

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

Question 2

spacing between planes, d

Answer 2

d = a/sqrt(N)
where N = h^2 + k^2 + l^2

Question 3

describe covalent bonding. where is it present

Answer 3

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

Question 4

descirbe van-der-waals bonding. where is it present

Answer 4

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

Question 5

optical mode

Answer 5

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

Question 6

acoustic branch

Answer 6

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

Question 7

debye approximation of acoustic branch, sketch

Answer 7

acoustic branch
constant velocity (y=x line on E-R graph)

Question 8

einstein approximation of optical branch, sketch

Answer 8

optical branch
all phonons have same energy/frequency (straight horizontal line at E through optical prediction)

Question 9

Drude model
what assumptions bout electrons are made?
what is the drift velocity profile of an electron under an applied electric field?

Answer 9

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

Question 10

origin of energy band gap in crystalline solid
(between valence and conduction band)

Answer 10

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

Question 11

state bragg’s law

Answer 11

2dsinθ = nλ

d = plane spacing
θ = scattering angle

Question 12

explain diffraction of x-rays from crystal lattice using bragg’s law

Answer 12

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.

Question 13

structure factor expression

Answer 13

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

Question 14

what does structure factor equal for a bcc lattice

Answer 14

S = 2f when h+k+l = even
S = 0 when h+k+l = odd

Question 15

what does structure factor equal for an fcc lattice

Answer 15

S = 4f when h,k,l: all even or all odd
S = 0 when h,k,l: mixed parity

Question 16

what does structure factor equal for a simple cubic lattice

Answer 16

S = non-zero when h=k=l=0
therefore scattering only occurs when incident X-ray is aligned exactly with lattice planes

Question 17

describe how these functions give the total electron energy distribution:
energy-density of states, fermi-dirac, fermi-energy

Answer 17

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)

Question 18

equation relating fermi energy to electron density

Answer 18

EF = h^2/2me (3(π^2)n)^(2/3)

where n = density
& h = h_bar

Question 19

fermi velocity from fermi wave vector

Answer 19

vF = hkF/me
where, kF =(3(π^2)n^(1/3)
& h = h_bar

Question 20

what assumption can be made about free electron density

Answer 20

the same as atomic density
aka. density of a substance is the electron density x atomic mass

Question 21

how does the confinement of electrons to a 2D layer alter the density of states function

Answer 21

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

Question 22

equipartition theorem (classical method for getting speed)

Answer 22

kinectic energy
= (1/2)mv^2 = (1/2)kBT

Question 23

what is a primitive cubic lattice, body-centered, or face-centred cubic lattice

Answer 23

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)

Question 24

what is the debye approximation

Answer 24

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

Question 25

equation for scattering time/average relaxation time (tau)

Answer 25

from electron conductivity:
(sigma) = ne^2(tau)/me

from mean path:
mean free path = (tau)v
where v is from equipartition theorem

Question 26

equation for atomic spacing

Answer 26

= cube root (V)

Question 27

what is group velocity at brilluoin zone boundary

Answer 27

where k = +/- (pi)/a
vg = (1/h_bar)(dE(k)/dk) = 0

Question 28

at what value of k does the effective mass tend to infinity

Answer 28

m = (pi)/sqrt(3)a

Question 29

weidmann-franz law

Answer 29

𝜅/𝜎 = (3/2)((kB/e)^2)T
ratio of electrical to thermal conducivity

Question 30

use hall coefficient to find density of free electron carriers (n)

Answer 30

RH = -1/ne

Question 31

thermal energy in terms of boltzman constant

Answer 31

E = kBT