6. crystal structurelampx.tugraz.at/~hadley/ss1/lectures18/mar22.pdf · 2018. 4. 5. · miller...
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Institute of Solid State PhysicsTechnische Universität Graz
6. Crystal structure
March 22, 2018
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513.001 Molecular and Solid State Physics
A crystal is a three dimensional periodic arrangement of atoms.
Crystal structure
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7 Crystal Systems
triclinic: a ≠ b ≠ c and ≠ ≠ ≠ 90°
monoclinic: a ≠ b ≠ c and ≠ 90° = = 90°
orthorhombic: a ≠ b ≠ c and = = = 90°
tetragonal: a = b ≠ c and = = = 90°
rhombohedral: a = b = c and ≠ ≠ ≠ 90°
hexagonal: a = b ≠ c and = = 90°, =120°
cubic a = b = c and = = = 90°
is the angle between b and c
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face centered cubic, fcc
Bravais lattice
body centered cubic, bcc
simple cubic
basis
Bravais lattice Crystal
= +
Auguste Bravais
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14 Bravais lattices
Points of a Bravais lattice do not necessarily represent atoms.
P … primitiveI … body centered F … face centeredC … centered
http://en.wikipedia.org/wiki/Bravais_lattice
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Primitive lattice vectors
Primitive lattice vectors
a1
a3
a2
Every point of a Bravais lattice can be reached from another point on the lattice by a translation vector
1 1 2 2 3 3 1 2 3 , , 2, 1,0,1,2,T n a n a n a n n n
Translation vector
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Primitive Unit Cell
1 2 3a a a
volume of a unit cell =
a1
a3
a2
There is more than one choice for a primitive unit cell
1 1 2 2 3 3 1 2 3 , , 2, 1,0,1,2,T n a n a n a n n n
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Unit Cells
There is more than one choice for a primitive unit cell
bcc fcc
Wigner-Seitz primitive unit cell
EugeneWigner
FrederickSeitz
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Wigner-Seitz cells
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is the angle between b and c is the angle between a and c is the angle between a and b
6 faces, 8 corners
http://en.wikipedia.org/wiki/Bravais_lattice
Conventional (crystallographic) unit cell
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Conventional (crystalographic) unit cell
simple cubic
zincblende
fcc
bcc
18 18
1 18 6 48 2
18 1 28
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Close packing
HCP FCC
HCP = Hexagonal close packHexagonal Bravais lattice with two atoms in the basis.
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From: Hall, Solid State Physics
Fcc
Crystalographic unit cellshowing close packed plane
Crystalographic lattice vectorsPrimitive lattice vectors
Wigner-Seitz cell
a b
c
1a
3a 2a
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[ uvw ] = vector in direction u a + v b + w c
Miller indices: Crystal direction [uvw]
[ ] specific direction< > family of equivalent directions
notation: -1 = 1
lattice vectors of the crystallographic unit cell
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Miller indices: Crystal planes
( ) specific plane{ } family of equivalent planes
always use integers for h,k,l
A plane with the intercepts 1/h,1/k,1/l is the (h,k,l) plane.
a
b
c
xyz
MOSFETs are made on <100> wafers
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Crystals
Planes with high atomic densities tend to dominate
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Silicon surfaces
(Source: Sandia Nat.Labs.)
Si(100)
Si(111)
atomic step in Si(100)
unreconstructedSi(111)
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KOH etches Si {110} > {100} > {111}, producing a characteristic anisotropic V-etch, with sidewalls that form a 54.7° angle with the surface (35.3° from the normal).
http://www.ece.uncc.edu/research/clean_room/fabprocesses/KOH-EtchingAndDecon.pdf
KOH etching of silicon
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Silicon
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simple cubicPo
Number: 221
Primitive Vectors:
Basis Vector:
1
2
3
ˆˆˆ
a axa aya az
1 (0,0,0)B
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fcc
Al, Cu, Ni, Sr, Rh, Pd, Ag, Ce, Tb, Ir, Pt, Au, Pb, Th
Primitive Vectors:
Basis Vector:
Number 225
1
2
3
ˆ ˆ2 2
ˆ ˆ2 2
ˆ ˆ2 2
a aa y z
a aa x z
a aa x y
1 (0,0,0)B A
B
C
[111]
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Rh(15,15,13) fcc
Reaction studies on nanostructured surfaces, Adolf Winkler, in The Oxford Handbook of Nanoscience and Technology, A. V. Narlikar and Y. Y. Fu ed., 2009.
Crystal planes: Miller indices
(15,15,13) fcc
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hcp
Mg, Be, Sc, Ti, Co, Zn, Y, Zr, Tc, Ru, Cd, Gd, Tb, Dy, Ho, Er, Tm, Lu, Hf, Re, Os, Tl
Space group 194
Hexagonal Bravais latticeBasis vectors:
2 1 11 2 3 3 2(0,0,0) ( , , )B B
AB
C
Hexagonal unit cellCrystallographic unit cell
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bcc
W Na K V CrFe Rb Nb Mo Cs Ba EuTa
Primitive Vectors:
Number 229
1
2
3
ˆ ˆ ˆ2 2 2
ˆ ˆ ˆ2 2 2
ˆ ˆ ˆ2 2 2
a a aa x y z
a a aa x y z
a a aa x y z
Basis Vector: 1 (0,0,0)B
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218
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http://lamp.tu-graz.ac.at/~hadley/ss2/crystalphysics/crystalclasses/crystalclasses.html
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http://it.iucr.org/A/
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Asymmetric unit
Unit cellcell 5.09000 6.74800 4.52300 90.000 90.000 90.000 natom 3 Fe1 26 0.18600 0.06300 0.32800 Fe2 26 0.03600 0.25000 0.85200 C 6 0.89000 0.25000 0.45000 rgnr 62 Cohenite (Cementite) Fe3 C
Asymmetric unitGenerated by PowderCell