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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
Q

effective mass

A

m* formula on the formula sheet

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26
Q

The probability of a vacancy at energy Ev is equivalent to the probability of finding a hole at energy Eh if

A

Eh - μ = μ - Ev

where μ is the chemical potential

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27
Q

Band gap energy

A

E = hf

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28
Q

intrinsic limit

A

~ ni

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29
Q

conductivity in terms of pv

A

σ = pv e μ
where μ is the mobility

30
Q

when the sample is intrinsic

A

We can ignore donor contributions
pv = ni

ni = (pvnc)^1/2

31
Q

area of a fermi circle

A

A = πkF^2

similary a sphere would be A = 4πkF^2

32
Q

Wiedeman-Franz law

A

L0 = κ/σT = π^2/3 (kb/c)^2

where L0 is the Lorenz number which is independent of temperature and material.

33
Q

derive the fermi wavevector in 3D kF = (3π^2n)^(1/3)

A

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
Q

Ewald sphere has wavevector

A

k = 2π/λ

35
Q

Radius of Ewald Sphere

A

r = 2pi/lambda

36
Q

To find the form of the Ewald sphere

A

[direction vector] . x = 0

whatever vector dotted with the direction vector gives 0.
It must obey the rules of FCC / BCC etc.

37
Q

Debye Scherrer radius

A

L = tan2 theta

38
Q

Steady state conditions

A

d/dt vd = 0

39
Q

Current density

A

J = -nevd

also J = I/A

A = wd

where w is the width and d is the thickness

40
Q

Minimum energy for photon absorption

A

Ec - Ev = Eg

41
Q

Carrier concentration

A

ni = (ncpv)^1/2

42
Q

Mean free path

A

L = vF tau

where vF is the fermi velocity

43
Q

Show that the heat capacity contribution of the free electrons is ∝ T

A

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
Q

what does the term vd/τ represent

A

the damping term from the material.

45
Q

how are n and vd related?

A

through current density

J = -nevd

46
Q

Relationship between donors and acceptors

A

nc = ND - NA
or
pv = NA - ND

47
Q

f value in structure factor

A

f is proportional to the atomic number, Z.

48
Q

Hall voltage

A

Ey = VH/w

where w is the width

49
Q

Electric field in the x direction across the hall bar.

A

Ex = Vx / L

where L is the length

50
Q

derive the conductivity

A

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
Q

What are the boundary conditions for a periodic potential

A

ψ(x) = ψ(x+L)

hence exp[±ik(x+L)] = e±ikx
eikL = 1
k = 2πnx/L
L = nλ

52
Q

Derive the density of states

A

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
Q

Show that the average energy is 3/5 EF

A

<E> = (∞ ∫ 0) D(E) E / (∞ ∫ 0) D(E)
</E>

54
Q

number density of electrons

A

n = (∞ ∫ 0) D(E) f(E) dE

55
Q

totel energy per volume

A

U/V = (∞ ∫ 0) D(E) f(E) EdE

56
Q

Derive the electronic susceptibility

A

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
Q

derive Matthiessen’s Rule

A

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
Q

Show that we get the matrix form (1 -ωcτ ωcτ 1) (vx vy) = -eτ/m(Ex Ey)

A

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
Q

derive Ey = RH Jx Bz

A

starting from Jx Jy matrix

Jy = 0 such that Ey = -ωcτ Ex

Jx = σEx

Ey = -1/ne Jx Bz

60
Q

the number of states in the 1st BZ

A

BZ width/ width per state = 2π/a / 2π/L = L/a

61
Q

Show that we must introduce the k momentum for the effective mass

A

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
Q

Hall coefficients for semiconductors

A

RH = -r/nce and RH = +r/pve

63
Q

Optical absorption

A

I = I0 exp(-alphad)

64
Q

Bloch’s theorem

A

Ψk(r) on the formula sheet under NFE

65
Q

the free electron energy for wavevector k

A

λk = ħ^2k^2/2m

where this is not a wavelength

66
Q

The central equations

A

(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
Q

wave vector electron vs hole

A

electron kj
hole -kj

68
Q

effective mass electron vs hole

A

electron me = ℏ^2/(d^2E/dk^2)

mh = -me

69
Q

effect of an electric field electron vs hole

A

electron F = -eE
hole F = +eE

70
Q

velocity of electron vs hole

A

electron ve = 1/ℏ (dE/dk)
hole vh = ve

Solid State Physics (2 decks)

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