Formulas Flashcards
Density
bello
M/V = M/a^3
where a is the lattice constant
packing fraction
Packing fraction = Vsphere/Vlattice
lattice constant of an FCC
a = 4r/sqrt(2)
lattice constant of a BCC
a = [4r/sqrt(3)]^3
Monoclinic
|a1| ≠ |a2| ≠ |a3| and alpha = gamma = 90 but beta ≠ 90
Orthorhombic
|a1| ≠|a2| ≠|a3| alpha = beta = gamma = 90
Tetragonal
|a1| = |a2| ≠ |a3| alpha = beta = gamma = 90
Hexagonal
|a1|=|a2| ≠|a3| alpha = beta = 90, gamma = 120
Structure factor
F(K) = Σj fj exp(iK.rj)
where G is the reciprocal lattice vector
and r is (x,y,z)
Intensities of diffraction peaks
I ∝ Ψ(K)^2 = F(K)^2
where F(K) is the structure factor
number density / number of free electrons per unit volume
n = zp(Na)/A
where Na is Avogadro’s number and only used if p is in mol.
or
n = n.atoms z/a^3
Fermi temperature
EF = kB TF
Fermi velocity
vF = pF/m
where pF = ℏkF
electronic specific heat constant
cel,V = gamma T
where gamma = π^2nkB^2/2EF
Volume
V = M/p
First brillouin zone
kF = |G|/2
cyclotron frequency
ω = eB/m
Resistivity
p = 1/σ
where σ is conductivity
conductivity
σ = 1/p
where p is resistivity
scattering rate
1/τ
Hall voltage
VH = -IB/dne
where d is the thickness
Resistance
R = L/σA
fermi sphere from the origin to the Brillouin zone
kF/kBZB
estimating the temperature of the intrinsic carrier concentration
use ni formula and where mc and mv are omitted use me for both
effective mass
m* formula on the formula sheet
The probability of a vacancy at energy Ev is equivalent to the probability of finding a hole at energy Eh if
Eh - μ = μ - Ev
where μ is the chemical potential
Band gap energy
E = hf
intrinsic limit
~ ni
conductivity in terms of pv
σ = pv e μ
where μ is the mobility
when the sample is intrinsic
We can ignore donor contributions
pv = ni
ni = (pvnc)^1/2
area of a fermi circle
A = πkF^2
similary a sphere would be A = 4πkF^2
Wiedeman-Franz law
L0 = κ/σT = π^2/3 (kb/c)^2
where L0 is the Lorenz number which is independent of temperature and material.
derive the fermi wavevector in 3D kF = (3π^2n)^(1/3)
Periodic potential psi (x+L)
k space separated by lambda ~ 2π/L so volume of (2π/L)^3
nL^3 = number of states x Vfermi sphere/Vk space
states filled with two electrons in each state
nL^3 = 2 Vfermi sphere/Vk space
fermi sphere has radius kF
Ewald sphere has wavevector
k = 2π/λ
Radius of Ewald Sphere
r = 2pi/lambda
To find the form of the Ewald sphere
[direction vector] . x = 0
whatever vector dotted with the direction vector gives 0.
It must obey the rules of FCC / BCC etc.
Debye Scherrer radius
L = tan2 theta
Steady state conditions
d/dt vd = 0
Current density
J = -nevd
also J = I/A
A = wd
where w is the width and d is the thickness
Minimum energy for photon absorption
Ec - Ev = Eg
Carrier concentration
ni = (ncpv)^1/2
Mean free path
L = vF tau
where vF is the fermi velocity
Show that the heat capacity contribution of the free electrons is ∝ T
Only electrons with kBT of EF can contribute to the heat capacity.
∆U/V = D(EF) kbT x kbT
∆U/V = D(EF) (kbT)^2
cV = d/dT (∆U/V)
cV = 2D(EF)kb^2T
cV ∝ T
what does the term vd/τ represent
the damping term from the material.
how are n and vd related?
through current density
J = -nevd
Relationship between donors and acceptors
nc = ND - NA
or
pv = NA - ND
f value in structure factor
f is proportional to the atomic number, Z.
Hall voltage
Ey = VH/w
where w is the width
Electric field in the x direction across the hall bar.
Ex = Vx / L
where L is the length
derive the conductivity
d/dt vd + vd/τ = -e/m (E + v x B)
assume v x B = 0
assume a steady state and J = -nevd = σE
rearranging gives as required
What are the boundary conditions for a periodic potential
ψ(x) = ψ(x+L)
hence exp[±ik(x+L)] = e±ikx
eikL = 1
k = 2πnx/L
L = nλ
Derive the density of states
N = spin states x shell volume/volume per kstate
shell volume = 4πk^2 dk
volume per k state = (2π/L)^3
N = VD(k)dk , V = L^3
D(k)dk = k^2/π^2 dk
E = ħ^2k^2/2m
rearrange for k and take the derivative hence substitute into D(k) and dk for D(E) dE
Show that the average energy is 3/5 EF
<E> = (∞ ∫ 0) D(E) E / (∞ ∫ 0) D(E)
</E>
number density of electrons
n = (∞ ∫ 0) D(E) f(E) dE
totel energy per volume
U/V = (∞ ∫ 0) D(E) f(E) EdE
Derive the electronic susceptibility
M = gain in spin up - loss of spin down
M = 1/2 D(EF) μB B x μB - 1/2 D(EF) μB B x (-μB)
M = D(Ef) μB^2 B
B = μ0H
M = Χel H
rearrange for Χel
derive Matthiessen’s Rule
1/τ = 1/τi + 1/τL
p = m/ne^2 (1/τi + 1/τL) = pi + pL
giving pi and pL
such that
μe = eτ/m giving σ = neμe
Show that we get the matrix form (1 -ωcτ ωcτ 1) (vx vy) = -eτ/m(Ex Ey)
Start from d/dt vd + vd/τ = -e/m (E + vd x B)
steady state
find the cross product term
express vd as vx and vy
which gives the two components of using the cyclotron frequency ωc = eBz/m
derive Ey = RH Jx Bz
starting from Jx Jy matrix
Jy = 0 such that Ey = -ωcτ Ex
Jx = σEx
Ey = -1/ne Jx Bz
the number of states in the 1st BZ
BZ width/ width per state = 2π/a / 2π/L = L/a
Show that we must introduce the k momentum for the effective mass
F = m* d/dt vg
d/dt vg = 1/ħ d/dt(dE/dk)
dE/dk becomes dE/dt
use chain rule dE/dk dk/dt
F = ħ dk/dt equating the two gives
m* = ħ^2 / (d^2E/dk^2) on formula sheet.
Hall coefficients for semiconductors
RH = -r/nce and RH = +r/pve
Optical absorption
I = I0 exp(-alphad)
Bloch’s theorem
Ψk(r) on the formula sheet under NFE
the free electron energy for wavevector k
λk = ħ^2k^2/2m
where this is not a wavelength
The central equations
(λk-E)c(k) + (ΣG) UG c(k-G) = 0
The schrodinger equation written in momentum space where the periodic potential imposes a restriction on the allowed k terms.
wave vector electron vs hole
electron kj
hole -kj
effective mass electron vs hole
electron me = ℏ^2/(d^2E/dk^2)
mh = -me
effect of an electric field electron vs hole
electron F = -eE
hole F = +eE
velocity of electron vs hole
electron ve = 1/ℏ (dE/dk)
hole vh = ve
