Introduction and Bravais Lattices

8/10/2019 Introduction and Bravais Lattices

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PHY 527 Solid State Physics IIntroduction

Solids encompass tremendous variety of properties andenormous number of elements and compounds.

Examples:! Elec. conductivities from infinity (supercond.) to 10 -16 (! -cm) -1 (insulators likeSiO

2); piezoelectric, ferroelectric " .

! Optical properties (visible)" reflecting" transparent (of a multitude of colors)" Absorbing

! Non-magnetic, diamagnetic, paramagnetic, ferromagnetic, antiferromagnetic,

Ferrimagnetic (examples in metals, semiconductors and insulators).! Can be prepared with remarkable purity (example is Ge; foreign atoms at alevel of 10 10 imp./cm 3! mechanically very hard (diamond); mechanically very soft and malleable (Pb)! Tremendous range of applications

What ties all these materials and properties together?


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Crystal Structure

Answer: NOT MUCH! 1. AMOST ALL SOLIDS EXIST IN CRYSTALLINE FORM2. FUNDAMENTAL PROPERTY THAT UNDERLIES MOST OF OUR

UNDERSTANDING OF PROPERTIES OF SOLIDS (AMORPHOUSSOLIDS LIKE GLASSES ARE EXCEPTIONS)

CRYSTAL STRUCTURE

For this course well consider all solids to be crystals, i.e., there is anunderlying xtal lattice. This structure is periodic some fundamental unit isrepeated ad infinitum. Simple geometric regularities in macroscopicstructures are examples of this. e.g. facets on surfaces of large crystals.

Crystal structures are understood from purely geometric solid considerations(symmetries).

Very important to understand.

Continually appears and reappears in SSP.


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Crystal StructureBravais Lattice

Fundamental concept -- specifies periodic arrayof points in 3D (or 2D or 1 D)

Definition:Bravais lattice is an infinite array of points (doesnt specify whats on orat the points, e.g., atoms, molecules) that appears the same fromwhichever of the points the array is observed.

Operational or mathematical definition: A 3D Bravais lattice consists of all points with position vectors of theform

are any threevectors not all in the same plane, and the are integers (pos., neg. and0), Every point in the lattice must be able to be generated in this manner.The are called primitive vectors and span the lattice.

See next slide for examples.

!

R

!

R = n1

!

a1

+ n!

a2

+ n!

a3 , where

!

a1,

!

a2 , and

!

a3

n i'

s

!

ai


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2D examples (oblique net)

Crystal Structure

P

!

a 3

!

b3

!

a 2

!

b 2

!

a 1

!

b1

!

P1

= 3!

a1

+!

b1

!

a 4

!

b4 Not possible,

nor is P

P

!

P2

= ! !a2

+ 2!

b2

!

P3

= 6!

a3

+ 3!

b3

The a i and b i for 1 3 are primitive vectors; a 4 and b 4 are not

A Set ofprimitive

vectors is notunique


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Bravais Lattices


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2D honeycomb lattice

Crystal StructureImportant material

GRAPHENE (monolayer ofGraphite)

! Not a Bravais lattice! Two sublattices (O, )! Points P and Q are

inequivalent

(consider first definitionof a BL)

b 1

b 2

P

Q2

R

Vector like R = n 1 ! 1+ n 2 ! 2 generates points not on either sublattice; ( 1+ 2 is a vector R to the center of the hexagons . A Centered honeycomblattice IS a Bravais lattice.


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3D ExamplesBody centered cubic (bcc) two interpenetrating simple cubiclattices (or simple cubic lattice with a two atom basis).

Primitive vectors:

These primitive vectors are not unique; could also pick centers ofthree adjacent cubes as primitive vectors

y

x

z

3a

!

Crystal Structure

a a

a

1a!

2a!

!

a1

= a x !

a 2 = a y

!

a 3 =a

2 x + y + z( )


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Crystal StructureFace Centered Cubic: take simple cubic lattice and add point to

center of each face (6 faces). Closest pt. to any point lies in planes definedby the cube faces at 45 degree angles to cube axes (four ineach plane). Each pt. has same surroundings # BL

Primitive vectors

xy

z

!

a 1 =a2

x + y( ); !

a 2 =a2

y + z( ); !

a 3 =a

2 x + z( )

aa

a

1a!

2a! 3a

!

What is another set ofprimitive vectors?

Very important structure; enormousvariety of solids crystallize in fcc;simple cubic is very rare ( $ phase

of Po)


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Crystal StructureCoordination Number:

Points in lattice closest to a given point are nearest neighbors (nn),etc., nnn, nnnn. Every BL point has the same number of nns.This is the coordination number.

Center point in top plane is

equidistant from corner pointsand center points 12 nns

Simple cubic = 6Body centered cubic = 8


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Crystal StructurePrimitive Unit Cell:Definition :

Volume of space that, when translated through ALL vectors of the BL, just fills all space without overlapping or leaving voids (like stacking buildingblocks of odd shapes. There is NO UNIQUE CHOICE!

Example (2D oblique lattice)

B

A

C

D

IV

III

II

Primitive cell contains ONE lattice point;n = density of points in lattice (#/vol. or area)

v = vol. of unit cell; then nv = 1 (useful later)

I

Obvious prim. cell is the oblique oneshown in red . (each corner point is shared

Among 4 identical cells so each cell hasone atom (4 x # ).

The hexagonal cells are also primitive cells:one atom in each and exactly the same area.Show by translating I through CD, III through

AD, and II through BD --- generates thehexagonal cell


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Crystal Structure

x

z

y

!

a 1

3a!

2a

!

( ) ( ) ( )

( ) ( )

( ) ( )

4

1000108

0)(8

)(8

iscell primitiveof vol.

2

;2

;2

3

33

3

321

321

a

a y x z y x

a

z x z y y xa

aaav

z xa

a z ya

a y xa

a

=

+++++=+++!+=

+"++="=

+=+=+=

!!!

!!!

1/4 vol. ofcubic cell

Primitive Unit cell (3D example):

Face Centered Cubic (fcc)

Primitive Vectors


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2D example

Draw lines connecting centerpoint to all nearby lattice points

Draw perpendicular bisectors(lines or planes) Smallest volume enclosed isW-S unit cell

3D examples

bcc truncated octahedron

Wigner-Seitz unit cell

fcc rhombicdodecahedron


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Lattice with a basis

Real crystal described by an underlying BL together with a

particular arrangement of atoms, molecules, etc.!

within anindividual Primitive CellIdentical copies of the same physical unit (the Basis) translatedthrough all vectors of the BL reproduce the crystal

With a single atom or ion as basis the lattice is called a monatomicBL

Simple 2D example

x

y

Basis

Lattice + Basis = crystal

Lattice Points


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Non-primitive Unit Cell

Can also describe monatomic BL as a lattice with a basis bychoosing non-primitive Unit cell

Example: Conventional (cubic) unit cell for fcc or bcc

( ) z y xr r z aa yaa xaa

a ,0 basis

,,

221

321

++==

===

!!

!!!

y

x

z

bcc is simple cubic cell with two-point basis

fcc is simple cubic cell with four-point basis(same cubic primitive cell as above) y

x

z

!

r 3

( ) ( ) ( ) x z a

r z ya

r y xa

r r 2,2,2,0 4321 +=+=+==!!!!

!

r 1

!

r 2

!

r 4

!

r 1

!

r 2


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Described as fcc BL with a two-point basis:

( ) z y xar r 4

,0 21 ++==!!

2r !

W

Two interpenetrating fcc latticesDisplaced along the body diagonal by" the length of the diagonal

Important ExamplesDiamond ( and Zinc-blende ) structure

Diamond: (carbon, Silicon, Germanium, grey-tin)

Zinc- blende:

Two interpenetrating fcc lattices

that make up the diamond structureare occupied by different atoms ( a nd )

Most III-V compound semiconductors(GaAs,GaP, GaSb, InP, InSb, InAs, ! )and many II-VI s.ZnSe, CdTe, HgTe, ! .)

yx

z

!

r 1


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Hexagonal Close-Packed Structure

!

a 3

!

a1 !

a 2

Top ViewHCP structure not BL, but very important.(30 elements). Underlying structure is simplecentered hexagonal (BL); stack 2D triangularnets directly above one another three prim.vectors

z ca ya

xa

a xaa ,2

3

2, 321 =+==

!!!

a

b

a

Stacking sequence:ababab

x

y

xaa 1 =!

ya

xa

a 2

3

22

+=!

60

Every other triangle in a hexagon in ana- layer has a b-layer atom underneath itDisplaced along z-direction by c/2.

The ideal hexagonalClose-packedstructure has a

c/a ratio = ! ! # "

To get hcp: take two interpenetrating

simple hexagonal BLs and displace onefrom the other by!

a 1

3+

!

a 2

3+

!

a 3

2=

a

2 x +

a

3

3

2 y +

c

2 z


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NaCl Structure

y

x

z

Again, two interpenetrating fcc latticesdisplaced along body diagonal by half thediagonal lengthfcc BL with two atom basis

( ) z y xar r 2

,0 21 ++==!!

2r !

CsCl StructureEach atom has 6 nn of the other kind (along cube edges)

Two interpenetrating simple cubic latticesdisplaced along body diagonal by half thediagonal length

Simple cubic BL with two atom basis

( ) z y xar r 2

,0 21 ++==!!

Each atom has 8 nn of the other kind (along body diagonals)

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Introduction and Bravais Lattices