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Solid State Physics by Heesang Kim at SSU

Solid State Physics

Text: Introduction to Solid State Physics

by Charles Kittel

Why do we study Solid State Physics?


Chapter 1. Crystal Structure

Niobium #41 Nb

copper #29 Cu


Iron-sulfides, FeS2


rock salt, NaCl

diamond, #6 C



Periodic array of atoms


Lattice: regular periodic arrangement of points in

space

Lattice sites:

( : 정수)

: Basis vector ( primitive vectors )

332211 auauauu

321 ,, uuu

321 ,, aaa


Translational Symmetry:

Basis vectors

Primitive lattice translation vector

: 정수

원자 구조가 만큼 이동해도 변화 없음

crystal(lattice) translation vector :

321 ,, aaa

332211 anananT

321 ,, nnn

T

T


Crystal structure = lattice + basis


Primitive unit cell:

A volume of space that, when translated through all the vectors in a Bravais

lattice, just fills all of space without either overlapping itself or leaving voids.

인 체적.

There are various possible choices of primitive unit cell.

332211 axaxax

1,,0 321 xxx


Wigner-seitz primitive cell:

Primitive unit cell 중에서 가장 기본이 되는 것.

정해진 한 점에서 가장 가까운 lattice sites들을 잇는 vector를 수직 이등분한

선으로 둘러싸인 cell.


Bravais lattice

Crystal 구조를 나누는 기본 단위의 lattice

14개(3차원) 5개(2차원)


90,,, baba

Two-dimensional lattice

1. Oblique lattice

2. Special lattice (4)

Hexagonal lattice

Square lattice

Rectangular lattice

Centered rectangular lattice


Crystal structure: 3차원에서 crystal의 기본 구조 7개와 14 Bravaus lattices

Triclinic ( ) 1가지

Monoclinic ( ) 2 가지

Orthorhombic ( ) 4가지

Tetragonal ( ) 2가지

Cubic ( ) 3가지

Trigonal ( ) 1가지

Hexagonal ( ) 1가지

,cba 90,cba 90, cba 90, cba 90, cba

90,120, cba 120,90, cba


3a

3a3a3a

3

2

1a

3

4

1a

a a2

3 a2

1

Simple

Body-centered

Face-centered

Volume(conventional cell)

Lattice points per cell

1

2

4

Volume of primitive cell

No. of nearest neighbors

6

8

12

Nearest neighbor distance

Packing fraction

0.524

0.68

0.740

Characteristics of Cubic Lattices


Simple cubic lattice (sc)

zaayaaxaa ˆ,ˆ,ˆ 321


: A cubic lattice with two atoms per unit cell.

and

: integers

Primitive vectors

I) , ,

II) , ,

zalyalxall zyxˆˆˆ

zalyalxall zyxˆ)

2

1(ˆ)

2

1(ˆ)

2

1(

zyx lll ,,

)ˆˆˆ(2

11 zayaxaa

)ˆˆˆ(2

12 zayaxaa )ˆˆˆ(

2

13 zayaxaa

xaa ˆ1

yaa ˆ2

)ˆˆˆ(

23 zyx

aa

Body centered cubic structure (bcc)


Face-centered cubic lattice (fcc)

Primitive vectors

)ˆˆ(2

),ˆˆ(2

),ˆˆ(2

321 yxa

axza

azya

a


Hexagonal structure

Primitive vectors

yaxa

a ˆ2

3ˆ

21

,ˆ2

3ˆ

22 yax

aa zca ˆ

3

312 aaa

2

120


Miller index

Plane : (100)

Direction : [100]

Set of planes : {100} = {(100), (010), (001), (100), (010), (001)}

)3,3,2(

)6

3,

6

3,

6

2()

2

1,

2

1,

3

1()2,2,3(


Simple Crystal Structures

Sodium Chloride Structure : fcc with basis(Na, Cl)

ex) LiH, MgO, MnO, NaCl, AgBr, PbS, KCl, KBr


Cesium Chloride Structure : sc with basis(Cs, Cl)

ex) BeCu, AlNi, CuZn, CuPd, AgMg, LiHg, NH4Cl, TlBr, CsCl, TlI


Close-packed Structures : packing ratio 0.74, infinite number of ways

random stacking – ABABCABCABABC……

hcp – ABABABAB…

fcc – ABCABCABC… (111) surface

Polytypism – (ABA…CAB)(ABA…CAB)….. ZnS, SiC


Hexagonal Close-packed Structure : hexagonal with basis of two atoms

ex) He, Be, Mg, Ti, Zn, Cd, Co, Y, Zr, Gd, Lu


Diamond structure : carbon, silicon, germanium, tin

fcc with basis at 000, ¼ ¼ ¼

packing ratio 0.34

Inversion symmetry at 1/8 1/8 1/8


Cubic Zinc Sulfide structure : CuF, SiC, CuCl, ZnS, AlP, GaP, ZnSe, GaAs, AlAs

CdS, InSb, AgI

fcc with basis of different atoms at 000, ¼ ¼ ¼

packing ratio N/A

No inversion symmetry


Direct Imaging of Atomic Structure : Scanning tunneling microscopy (STM)

STM image of Pt (platinum)

fcc (111) surface at 4K

nn spacing is 2.78 o

A


Homework

1. Chapter 1-problems 1, 2, 3.

2. Show that a honeycomb structure is not Bravais. Identify its lattice structure and basis.

solid state physics - ssu.ac.krphys-in.ssu.ac.kr/~course/ssp/chap1-crystal_structure.pdf ·...

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