Introduction to solid state physic

7/30/2019 Introduction to solid state physic

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Ph sics

Goan Hsi-Shen

Department of Physics


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Condensed matter physics, the largest branch

of modern physics, mostly concerns the study of,substances, glasses, and liquids.

conditions of temperature and pressure.

What makes a material be a metal, insulator, or

New physics?

.


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1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18

1

H

2per o c a e

He

3 4 5 6 7 8 9 10

e

e

11Na

12M --- --- --- ---

--- --- --- ---

13Al

14Si

15P

16S

17Cl

18Ar

19K

20Ca

21Sc

22Ti

23V

24Cr

25Mn

26Fe

27Co

28Ni

29Cu

30Zn

31Ga

32Ge

33As

34Se

35Br

36Kr

37Rb

38Sr

39Y

40Zr

41Nb

42Mo

43Tc

44Ru

45Rh

46Pd

47Ag

48Cd

49In

50Sn

51Sb

52Te

53I

54Xe

55Cs

56Ba

57La

72Hf

73Ta

74W

75Re

76Os

77Ir

78Pt

79Au

80Hg

81Tl

82Pb

83Bi

84Po

85At

86Rn

87

Fr

88

Ra

89

Ac

104

Rf

105Db

106Sg

107Bh

108Hs

109Mt

110

Uun

111

Uuu

112

Uub


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Crystal: constructed by the infinite

re etition of identical rou s of atoms. Basis: a group of atoms.

a ce: e se o ma ema ca po n s owhich the basis is attached.An infinitearray of points in space, in which each

others.


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Tr n l i n l mm r

rans a ona y symme r c or per o c: r ng ng a po nback to an equivalent point in the crystal.

One can reach an oint in s ace b addin an inte ernumber of translation vectors.

If the lattice was infinite, you could not tell if you moved.


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Don't mix up

atoms with latticepoints

Lattice points are

in space Atoms are h sical

objects

Lattice points donot necessarily lieat the centre of


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Lattice

+

Basis

=

Crystal structure


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Even if there is onl one t e of atoms in the cr stal wehave to use a basis made up with two atoms.


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We need to

identify thesymmetry att cevectors) andlattice contents

(basis) to fullydescribe a crystalstructure.

How do wechoose lattice

lead us tothinking about

primitive cell.


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crystal structure (it fills space).

.

The choice of the origin for unit cells is

arbitrary as well. The only requirement is that the lattice must

look the same when translated by a crystal

translational vector:1 1 1 2 3 3T u a u a u a= + +

G

G G G

1 2 3where , and are integers.u u u


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Different crystal facesare developed evenwith identical buildinblocks.

Rock salt


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Primitive cell A primitive cell is a type of unit cell.

primitive translation vectors (axes)

1 2 3, anda a a

G G G

can be used as a building block for crystalstructure.

A primitive cell is a minimum-volume cell.

( )V a a a=

G G G

A primitive cell will fill space by the repetition of.


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A, B and C are primitive.

D, E and F are not. Why? ,

C are the same and the

different. But this does

not matter. There is only one lattice

oint er rimitive cell.


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-

Another way of choosing a

primitive cell. Procedure:

1. Draw lines to connect aiven lattice oint to all

nearby lattice points2. At the midpoint and

,draw new lines (or

planes in 3D).enclosed in this way isthe Wigner-Seitzpr m t ve ce .


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Unit cell is primitive

(1 lattice point) but

the basis. Atoms at the corner

contribute only 1/4 tounit cell count

the 2D unit cellcontribute only 1/2 to

unit cell count . Atoms within the 2D

unit cell contribute 1(i.e. uniquely) to thatunit cell

A single layer of graphite


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mapped into themselves by thelattice translations Tand by variousother s mmetr o erations.

Physicists use the symmetry of theunit cells to classify crystalstructures and how the fill s ace.

This involves the use ofpoint groupoperations and translationoperations.

By lattice point group we mean thecollection ofsymmetry operations

which, applied about a lattice point,carry the lattice into itself. Examples: mirror reflection,

inversion, n-fold rotational axisThe symmetry planes andaxes of a cube

symme ry


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- There are only five distinct

lattices in 2D. These are

defined by how you canrotate the cell contents( and get the same cell

Bravais lattice: commonphrase for a distinct lattice

. Unit cells made of these 5

Bravais lattices in 2D canspace. o er onescannot.

oblique lattice


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Penta onsdo not fittogether tofill all s ace

Penrosetiling in2D

2/3, 2/4, and 2/6 radians.

But rotation of 2/5, and 2/7 do not fill space. -

distinct designs of tiles.


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P=primitive (1

I=Body-centered(2 lattice pts)

= ace-cen ere(4 lattice pts)

C=Side-centerd (2lattice pts)

In this course, we

will concentrateupon only a few ofthese types.


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Cubic s ace lattices

For cubic systems, wehave three Bravais lattice:

the simple cubic (SC),o y-cen ere cu c ,

and face-centered cubic(FCC).

Packin fraction

atomic volume

cell volume=


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Simple cubic lattice

Counting atoms (lattice1

;a x=G

Atoms in different positionsin a cell are shared b a

2

3

;

.

a y

a z

=

=

G

G

differing numbers of unitcells

e.g., Po(Polonium)

er ex a om s are ycells 1/8 atom per cell

One lattice point per unit cell,with a total volume ofa3

cells 1/4 atom per cell

Face atom shared by 2

Number ofnearest neighbors: 6 Nearest neighbor distance: a

Number of next-nearestcells 1/2 atom per cell

Body unique to 1 cell1

neighbors: 12 Next-nearest neighbor distance:

2a.

Packing fraction:/ 6


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Body-Center cubic lattice

( )11

;2

a a x y z= + G

( )21

;

2

a a x y z= + +G

( )3 .2

a a x y z= +G

2 lattice points per cell, with a

. ., , , , , , , , ,

pr m ve ce vo ume o a Number ofnearest neighbors: 8

Nearest neighbor distance: 3 / 2a

rhombohedron

Number of next-nearest neighbors:6

Next-nearest neighbor distance: a Packing fraction: 3 / 8


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Wi ner-Seitz cells in 3D

http://www.chembio.uoguelph.ca/educmat/chm729/wscells/construction.htm

SC lattice

Procedure: Choose any lattice site as the origin. Starting at the origin draw vectors to all neighbouring lattice points. Construct a plane perpendicular to and passing through the midpoint

of each vector.

The area enclosed by these planes is the Wigner-Seitz cell.


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Unit cells of BCC lattice

defined by translation vectors.

Primitive cell of a truncatedoctahedron constructed bythe Wigner-Seitz cell.

The Wigner-Seitz cell may look as range, u as e u po nsymmetry of its lattice point andwill be useful when we look at the

.

movie


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Face-centered cubic (FCC) lattice

4 lattice points per cell, with aprimitive cell volume ofa3/4

Number ofnearest nei hbors: 12

Primitive cells of FCC latice

1

1

;a a x y= +

G

Nearest neighbor distance: Number of next-nearest neighbors:

/ 2a

( )21

;2

a a y z= +G

Next-nearest neighbor distance: a Packing fraction: 2 / 6 ( )3 1 .2a a x z= +

G

Movies


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Primitive cell is a right

rism based onrhombus with anincluded angle of 120o.

Later we will look at thehexagonal close-

,which is this structure

with a basis.1 2 3

a a a= G G G


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Indexin s stem for cr stal lanes

Since cr stal structuresare obtained from

diffraction experimentsdiffract from planes ofatoms), it is useful to

develop a system forindexing lattice planes.

constants a1, a2, a3, but

it turns out to be moreuseful to use what arecalled Miller Indices. Index


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Rules for determining Miller indices

n e n ercep s onthe axes in terms of the An example:

, , . (2) Take the reciprocals

of these numbers and

then reduce to threeintegers having the sameratio, usually the smallestof the three integers. The Intercepts: a, ,

Reciprocals: a/a, a/

, a/

, ,called the index of thelane.

= 1, 0, 0Miller index for this plane : (1 0 0)(note: this is the normal vector for

s p ane


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Examples of Miller indices

Intercepts: a, a,Reciprocals: a/a, a/a, a/

= 1, 1, 0

Intercepts: a,a,aReciprocals: a/a, a/a, a/a

= 1, 1, 1

Miller index for this plane : (1 1 0) er n ex or s p ane :


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Intercepts: 3a1, 2a2, 2a3Intercepts: 1/2a, a,Reciprocals: 2a/a, a/a, a/

= 2, 1, 0

Reciprocals: a1/3a1, a2/2a2, a3/2a3= 1/3, 1/2, 1/2

The smallest three integers having

Miller index for this plane : (2 1 0) the same ratio: 2, 3, 3

Miller index for this plane : (2 3 3)


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m g mean a s ng e p ane, oraset of planes

If a plane cuts a negative axis, welace a minus si ns above the index: (001) face

(hkl) Planes equivalent by symmetry are

denoted with curly brackets {hkl} e se o cu e aces are eno e

{100} which include (100), (010),(001), (100), (010), and (001) faces.

[100]direction

___

with [uvw] (for example, The [100]direction for a cubic crystal is alongx axis )

n cu c crys a s, e rec on sperpendicular to the plane (hkl)having the same indices, but thisisnt necessarily true for other crystal

rec on

sys ems


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Im ortant lanes in a cubic cr stal

e p ane means a p ane para e o u cu ng e a1 ax s a a .

E l f t t


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Examples of common structures:

o um or e a s ruc ure e space a ce o a

structure is FCC.

The basis consists ofoneNa+ ion and one Cl- ion,separated by one-half of thebod dia onal of a unit cube.

There are four units of NaClin each unit cube. om pos ons:

Cl: 000 ; 0; 0; 0

Each atom has 6 nearest

neighbors of the oppositeOften described as2 interpenetrating

.


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-

Crystal a ( )

LiH 4.08 +MgO 4.20 a

.

NaCl 5.63

AgBr 5.77

PbS 5.92KCl 6.29

.movie


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structure is SC.

The basis consists ofoneCs atom and one Cl atom,with each atom at thecenter of a cube of atoms

of the opposite kind. There is on unit of CsCl in

Atom positions:

Cs : 000Cl : (or vice-versa) Each atom has 8 nearest

Often described as 2interpenetrating SC lattices

opposite kind.


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rys a a

BeCu 2.70

.

CuZn 2.94a

.

AgMg 3.28

.

NH4Cl 3.87

.

CsCl 4.11Why are the avaluessmaller for the CsCl

. s ruc ures an or e

NaCl (in general)?


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-

What does stacking fruit have to do with solid state physics?

Cl d k d t t


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Closed-packed structures There are an infinite number of ways to arrange identical

spheres in a regular array that maximize the packingfraction.

The centersof spheresat A, B, and

pos ons

There are different ways you can pack spheres together. Thisshows two ways, one by putting the spheres in anABAB

arrangement, and the other withABCABC.

H l l d k d (HCP) t t


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Hexagonal closed-packed (HCP) structure

The HCP structure is made up of stacking spheres in aABABAB configuration. The HCP structure has the primitive cell of the hexagonal lattice, with a basis oftwo identical atoms

Atom positions: 000, 2/3 1/3 1/2 (remember, the unit axes are not allperpendicular)

The number of nearest-neighbors is 12.e.g., Be, Sc, Te, Co, Zn, Y, Zr, Tc, Ru, Gd,

Tb, Py, Ho, Er, Tm, Lu, Hf, Re, Os, Tl

Rotatethree times

To get the fullstructure

Primitive cell has a1= a2 Conventional HCP unit cell

FCC (also called the cubic


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FCC (also called the cubic

c ose -pac e s ruc ure, or

s ruc ure as e s ac ng arrangemen o

ABCABCABC sequence along the [111] direction.

HCP and FCC stacking arrangement


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HCP and FCC stacking arrangement

z

y

FCC lattice A B CA B CA B C

HCP and CCP (FCC) stacking


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HCP and CCP (FCC) stacking

arrangemen

HCP FCC

[1 1 1][0 0 1]

(CCP)(looking

along [111]rec on

ABABsequence

ABCABCsequence

movies


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-packed (HCP) and FCC

structures both have the

rys a c a ra o

He(hcp) 1.633

ea pac ng ract on o0.74.

.

Mg 1.623

packing is (8/3)1/2 = 1.633(see Problem 3 in Kittel)Zn 1.861Cd 1.886

Why arent these valuesperfect for real materials?

Co 1.622

Y 1.570 y wou rea ma er a s

pick HCP or FCC (whatdoes it matter if the ack

Zr 1.594

Gd 1.592

the same?)? Lu 1.586

Diamond structure


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Diamond structure

The space lattice is FCC. The primitive basis has two

identical atoms at the 000 and positions.

There are 8 atoms in each Number of the nearest

neighbors: 4

2 interpenetrating FCC lattices

neighbors: 12 Packing fraction: 0.34 (show

Crystal a ()

C 3.567

Si 5.430

a

e .

Sn 6.49

Diamond structure


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Diamond structure

FCC lattice + basis of 2 atoms at (0,0,0) and (1/4,1/4,1/4)

1/43/4

01/2 1/2

1/4 3/4

0 01/2

12 next nearest neighbors

two FCC displaced from eachother by of a body diagonal

Maximum packing fraction = 0.34

Cubic Zinc Sulfide (ZnS) structure


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Cubic Zinc Sulfide (ZnS) structure

The space lattice is FCC. The primitive basis has two different

positions (i.e., Zn and S) There are 4 units ofZnS in each

conventional unit cube. About each atom, there are 4

nearest neighbors atoms of theopposite kind arranged at thecorners o a e ra e ron.

Cr stal a

SiC 4.35

ZnS 5.41AlP 5.45

GaP 5.45

GaAs 5.65

AlAs 5.66movie

Periodic table of the elements


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1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18


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1H

2per o c a e

He

3 4 5 6 7 8 9 10

e

e

11Na

12M --- --- --- ---

--- --- --- ---

13Al

14Si

15P

16S

17Cl

18Ar

19K

20Ca

21Sc

22Ti

23V

24Cr

25Mn

26Fe

27Co

28Ni

29Cu

30Zn

31Ga

32Ge

33As

34Se

35Br

36Kr

37Rb

38Sr

39Y

40Zr

41Nb

42Mo

43Tc

44Ru

45Rh

46Pd

47Ag

48Cd

49In

50Sn

51Sb

52Te

53I

54Xe

55Cs

56Ba

57La

72Hf

73Ta

74W

75Re

76Os

77Ir

78Pt

79Au

80Hg

81Tl

82Pb

83Bi

84Po

85At

86Rn

87Fr

88Ra

89Ac

104Rf

105

Db

106

Sg

107

Bh

108

Hs

109

Mt

110Uu

n

111Uu

u

112Uu

b


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Many crystals undergo structural changes with T, P:For example:

LiquidBCCFCCBCC

-ferrite -ferrite

empera uree910oC 1400oC 2100oC

TemperatureNa

LiquidFCCBCC


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Density and atomic concentration

Documents

Introduction to solid state physic