3C25 Solid State Physics Equations

Memorized equations Below are a series of equations from the lecture notes that you should memorize. It is important to recognise that this does not define all the material you need to know for your exam: that is defined by the complete set of notes. The other equations within the notes you will be expected to understand, but not memorize. Note that no explanation for the equations is given below. You will need to refer to your lecture notes for that.

Crystal Structure V c =∣a1⋅ a2× a3 ∣

 

E n ≃exp − V N kBT

 c =G

a 2d

Diffraction methods and structure determination 2 d sin =n  r  n  r =∑ nG exp  i G⋅  G

 ⋅r  n G=V cell −1 ∫ d V n r exp i G V cell

 T =2 n G⋅

bi⋅ a j=2 ij

 

 

 

a × a a ×a a ×a b1=2  2 3 ; b2 =2  3 1 ; b3=2 1 2 V cell V cell V cell

  k =G   d 3 r f G =∫ natom  r exp i G⋅r  rj  S G =∑ f j expi G⋅ j

Cohesion U  R=4 

[    ]  R

12

 − R

6

=

A B − 6 12 R R

Vibrations d 2 us F s= M =C u s−1−u s C u s 1−u s=C u s1u s−1−2u s  d t2

u s t = u s 0 e i K s a e−i  t= u s 0 ei  K s a −t 

K ZB =±

v p=

v g=

 a

 K

d dK

 

= n

1 ℏ 2

 pK =ℏ K

2

ℏ 1 m v 2=  kK2  E2' =E±ℏ 2m

  K  k '=k G±

Thermal Properties U =U lattice =∑ ∑ 〈 n K , p 〉 ℏ  p  K  K

〈n〉=

U=

p

1 exp ℏ /k B T −1

ℏ ∑ ∫ D p  exp ℏ / k

p all bands

CV =

B

T −1

d

∂U ∂T

C V =3N k B K=

m  m = N a L

 

3

dN L V K2 D K = =4  K 2 = dK 2 2 2 2

D=

 

2

dN dK VK dK VK 1 = D K  = = 2 2 d d  2 d  2  vg

 

3

4 L  K 3D =N cells =N DOF 3 2   V k−sphere

1 /V state

C V =constant ×N k B

  T D

3

1 = C V v s l 3

Free Electron Gas k =

ℏ2 2 ℏ2 2 2 k =  k k k 2  2m 2m x y z 4 V sphere =  k 3F 3

 

2 V state= L

f =

3

1 exp [−/ k B T ]1 ∞

U =∫ D  f  d  0

C V =constant ×T

 =q   F E v ×  B

  = d p =m d v =ℏ d k F dt dt dt 2

j=n q v = n e  E  m 2

=

ne  m

1 1 1 =   i  p

j V =

dT dx

 =L T 

Electron Bands  x=u x e i k x

k = 〈  k ∣H∣ k 〉=−− ∑ exp− k⋅m  m

Semiconductors v g=

1 d ℏdk

  =ℏ d k F dt 2

1 1 d  = 2 * m ℏ d k2 2

k = E c 

ℏ k2 2 me

 

3

 

kBT E n p=n i  =4 me mh3 /2 exp − g 2 kBT 2ℏ 2

Semiconductor Devices n p= n i 2

=n e

   

e e e h pe =n e e  p e h me mh

=

vg E

1 1 1 =   L i

−∇ 2 =

 r 0

n D wn =n A w p

j = j e

e kB T

s

I D =I DSS



−1



V 1− GS Vp

=

IC IE

=

IC IB



2

Magnetic Materials  E =−m⋅  B

  M  =0  r H  B =0  H

 = H  M

〈 〉=

B e  e

B

B/ k B T

 B B/ k B T

− B e− e

B

B /k B T

−B B /k B T

=

C T

Beff = M =

C T −T C

= B tanh

  B B kBT