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Solid State Physics > Formulas > Flashcards

Formulas Flashcards

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1
Q

Density

bello

A

M/V = M/a^3

where a is the lattice constant

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2
Q

packing fraction

A

Packing fraction = Vsphere/Vlattice

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3
Q

lattice constant of an FCC

A

a = 4r/sqrt(2)

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4
Q

lattice constant of a BCC

A

a = [4r/sqrt(3)]^3

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5
Q

Monoclinic

A

|a1| ≠ |a2| ≠ |a3| and alpha = gamma = 90 but beta ≠ 90

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6
Q

Orthorhombic

A

|a1| ≠|a2| ≠|a3| alpha = beta = gamma = 90

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7
Q

Tetragonal

A

|a1| = |a2| ≠ |a3| alpha = beta = gamma = 90

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8
Q

Hexagonal

A

|a1|=|a2| ≠|a3| alpha = beta = 90, gamma = 120

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9
Q

Structure factor

A

F(K) = Σj fj exp(iK.rj)

where G is the reciprocal lattice vector
and r is (x,y,z)

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10
Q

Intensities of diffraction peaks

A

I ∝ Ψ(K)^2 = F(K)^2

where F(K) is the structure factor

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11
Q

number density / number of free electrons per unit volume

A

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

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12
Q

Fermi temperature

A

EF = kB TF

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13
Q

Fermi velocity

A

vF = pF/m

where pF = ℏkF

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14
Q

electronic specific heat constant

A

cel,V = gamma T

where gamma = π^2nkB^2/2EF

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15
Q

Volume

A

V = M/p

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16
Q

First brillouin zone

A

kF = |G|/2

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17
Q

cyclotron frequency

A

ω = eB/m

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18
Q

Resistivity

A

p = 1/σ

where σ is conductivity

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19
Q

conductivity

A

σ = 1/p

where p is resistivity

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20
Q

scattering rate

A

1/τ

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21
Q

Hall voltage

A

VH = -IB/dne

where d is the thickness

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22
Q

Resistance

A

R = L/σA

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23
Q

fermi sphere from the origin to the Brillouin zone

A

kF/kBZB

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24
Q

estimating the temperature of the intrinsic carrier concentration

A

use ni formula and where mc and mv are omitted use me for both

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25
effective mass
m* formula on the formula sheet
26
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
27
Band gap energy
E = hf
28
intrinsic limit
~ ni
29
conductivity in terms of pv
σ = pv e μ where μ is the mobility
30
when the sample is intrinsic
We can ignore donor contributions pv = ni ni = (pvnc)^1/2
31
area of a fermi circle
A = πkF^2 similary a sphere would be A = 4πkF^2
32
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.
33
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
34
Ewald sphere has wavevector
k = 2π/λ
35
Radius of Ewald Sphere
r = 2pi/lambda
36
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.
37
Debye Scherrer radius
L = tan2 theta
38
Steady state conditions
d/dt vd = 0
39
Current density
J = -nevd also J = I/A A = wd where w is the width and d is the thickness
40
Minimum energy for photon absorption
Ec - Ev = Eg
41
Carrier concentration
ni = (ncpv)^1/2
42
Mean free path
L = vF tau where vF is the fermi velocity
43
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
44
what does the term vd/τ represent
the damping term from the material.
45
how are n and vd related?
through current density J = -nevd
46
Relationship between donors and acceptors
nc = ND - NA or pv = NA - ND
47
f value in structure factor
f is proportional to the atomic number, Z.
48
Hall voltage
Ey = VH/w where w is the width
49
Electric field in the x direction across the hall bar.
Ex = Vx / L where L is the length
50
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
51
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
52
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
53
Show that the average energy is 3/5 EF
= (∞ ∫ 0) D(E) E / (∞ ∫ 0) D(E)
54
number density of electrons
n = (∞ ∫ 0) D(E) f(E) dE
55
totel energy per volume
U/V = (∞ ∫ 0) D(E) f(E) EdE
56
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
57
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
58
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
59
derive Ey = RH Jx Bz
starting from Jx Jy matrix Jy = 0 such that Ey = -ωcτ Ex Jx = σEx Ey = -1/ne Jx Bz
60
the number of states in the 1st BZ
BZ width/ width per state = 2π/a / 2π/L = L/a
61
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.
62
Hall coefficients for semiconductors
RH = -r/nce and RH = +r/pve
63
Optical absorption
I = I0 exp(-alphad)
64
Bloch's theorem
Ψk(r) on the formula sheet under NFE
65
the free electron energy for wavevector k
λk = ħ^2k^2/2m | where this is not a wavelength
66
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.
67
wave vector electron vs hole
electron kj hole -kj
68
effective mass electron vs hole
electron me = ℏ^2/(d^2E/dk^2) mh = -me
69
effect of an electric field electron vs hole
electron F = -eE hole F = +eE
70
velocity of electron vs hole
electron ve = 1/ℏ (dE/dk) hole vh = ve

Solid State Physics (2 decks)

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