Lattice and basis (Solid state physics)

1
Lattice and basis (Solid state physics)

1. ??(vector)???? ?? ??
2. ??(lattice)? basis
3. ???(reciprocal lattice)
4. Bloch vector, Bloch function

????? ??? ???? Halliday Resnick Walker?
Aschroft Mermin, ??? Kittel??? scan??? ????.
2
Vector length(magnitude) and direction
B

• (?) ?? ??? ??(??)? ???? ???? 10m ? ? ???? 10m???
  ? ??? ??? ? ??? ?
• ???? ?? 20 m ??? ???? ?? (X).
• ???? ?? ? 14.12m ??? ???, ?? ??? ?? (O).

O
A
20m 10m 10m (??? ??? ??, ????)
3
??? ??? ? ??
4
??? ??(inner product), ????(scalar product)
5
?? ?? (?) ??? 1?? ?? ??? ??? ??
?? ??
6
(?) 3??? ??? ????
7
???(vector product), ??(outer product)
? ???
??? ? ? ??? ??? ?? ?????
???? ????? ???? ????? ?? ??? ??? ?????.
8
??
? ??? ???? ?????? ??? ??? ?? ???? ????.
???? ????? ???? ????? ?? ??? ??? ?????.
9
??
? ??? ???? ?????? ??? ??? ?? ???? ????.
10
?? (Lattice) An infinite repeation
?? ?? ? ?? ??? ?? ??? ?? Lattice ? ?? ? ??.
11
?? (Lattice) ?? ??? ??? ?? ??? ???? ??? ?????
?? ? ? ?
12
primitive lattice vector? Lattice Translation
vector
Lattice point? ?? 1?? ??? ?? lattice
13
?? Lattice point (?? lattice vector)? ? ?? ???
?????? ??? ? ?, ? ? ?? ??? primitive vector? ??.

Primitive unit cell ?? lattice translation? ???
???? ????, ?? ?? ?? ? ?? ?? ??.
Figure 4.10 scan.
Unit cell? ??? ???? ??? ??? ???? ?? ??.
14
Lattice ? ??(basis)
Lattice point? ?? 1?? ??? ?? lattice ?? 1?? ???
????
15
Lattice ? ??(basis)
16
????(primitive lattice vector)? Lattice
Translation vector
17
????(basis) Lattice point? ???? ?

• Basis ? ?? 1? ? ??
• Basis ? ?? 1?? ?? 1?? ? ??
• Basis ? ?? 2?,??1??, ??? 1? ? ??
• Basis ? ?? 1?? ??
• Basis ? ?? ? ?? ??

18
?? Lattice? ??? ??? ?? ?? 1?? ???? !!!

• ??(basis)? spherical symmetric object ? ? ?
  Bravais Lattice
• 7?? point group ?? ??(??, unit cell ??)
• 14?? space group ?? ???Crystallography
• (Ashcroft Mermin, Ch.7, Kittel, Ch.2)

19
Cubic Lattice
(a) (b)
(c)
(a)? unit cell ? ? Simple
Cubic Bravais lattice ??.
20
Unit cell ?? ??? ? ? ?
(a) (b)
(c)
(c)? unit cell?
? Simple Cubic Lattice ? ??, unit cell??
??? 4? ??? ???. (Bravais lattice ? ??) ? FCC?
simple cubic lattice ? ???.basis ( ?) ?? ??????
21
Face-Centered Cubic
(a)
Unit cell ?? ?? 1? ??. ? FCC Bravais lattice
22
Face-Centered Cubic
Unit cell ?? ( )? ( )? ??. ? SC with ( ?
) basis
(a)
23
???

• ???(x,y,z?) ? ??? ??
• ??? ??? ??
• Lattice vector? basis ? ?? ????? ?? ? ?? ? (??)
• ??? ??? ??. ??? ??? ??? ????.
• ??? ???? ?? ???.

24
Body-Centered Cubic
25
Triangular Lattice
26
Honey Comb ??
Aschcroft Figure 4.17
Honeycomb ??
27
Honey Comb ??
Graphene, Graphite,Carbon Nanotube
Lattice Vector ? ?
Basis ???
28
(No Transcript)
29
IV? ????? III-V? ??? ???
???? ??
30
Diamond ??
31
FCC Two basis
???? ??? ?? ?? ?, Basis atom? ??? ? ? ???
???????? ???? FCC ?? ???.
(a)
32
??
Diamond ??? ??? ?? simple cubic conventional
cell ? ?? ? ?, ? conventional cell??? ? ?? ?????
basis ? ?? ??? ? ?? ??? ?? ?? ??? ? basis atom?
????? ???
33
?? ??? ?? ?? ??? ? basis atom? ????? ???
34
Tetrahedral bond ??? ??
35
Simple Hexagonal Lattice
Figure 4.19, Figure 4.21, Figure 4.20
36
???? ?? ??
37
Hexagonal Closed Pack(HCP)
38
??
?4?? ??

???? ?? ??? ????? ??? ? ?? ?? ?? ?? ?? ????.
??(???)? ??? a ? ? ? ?? h ? ??? ????. HCP ????
?? ?? ???? ?? c ? a? bond
length ? ??????
39
Hexagonal Closed Pack(HCP)
40
FCC??? Hexagonal lattice ? ??(111)?? Z ?? ???
??.