ROINN NA FISICE Department of Physics

Physics PY4118Physics of Semiconductor Devices

Crystalography

Coláiste na hOllscoile Corcaigh, Éire University College Cork, Ireland

2.1

Taken mostly from: Crystal2.ppt http://www.ems.psu.edu/~ryba/coursework/zhong%20shan%20da%20xue%20-%20course%20materials/class%20slides/

PY4118 Physics of Semiconductor DevicesROINN NA FISICE Depa

rtment of Physics

Symmetry?

This is actually really important for some semiconductor devices, especially:

Inversion Symmetry:

This is (not) required for: Second harmonic generation The electro-optic effect Piezo-electric effect etc.

Coláiste na hOllscoile Corcaigh, Éire University College Cork, Ireland 3.2

Each of the unit cells of the 14 Bravais lattices has one or more types of symmetry properties, such as inversion, reflection or rotation,etc.

SYMMETRY

INVERSION REFLECTION ROTATION

ELEMENTS OF SYMMETRY

3

Lattice goes into itself through Symmetry without translation

Operation Element

Inversion Point

Reflection Plane

Rotation Axis

Rotoinversion Axes4

Reflection Plane

A plane in a cell such that, when a mirror reflection in this plane is performed, the cell remains invariant.

5

Rotation Axis

This is an axis such that, if the cell is rotated around it through some angles, the cell remains invariant.

The axis is called n-fold if the angle of rotation is 2π/n.

90°

120° 180°

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Stereographic projections of symmetry groups

Types of pure rotation symmetry

Stereographic projections of symmetry groups

Types of pure rotation symmetry

Draw point group diagrams (stereographic projections)Draw point group diagrams (stereographic projections)

Rotation 1, 2, 3, 4, 6

Rotoinversion 1 (= i), 2 (= m), 3, 4, 6

Rotation 1, 2, 3, 4, 6

Rotoinversion 1 (= i), 2 (= m), 3, 4, 6

symmetry elements equivalent pointssymmetry elements equivalent points 7

Stereographic projections of symmetry groups

Types of pure rotation symmetry

Stereographic projections of symmetry groups

Types of pure rotation symmetry

Draw point group diagrams (stereographic projections)Draw point group diagrams (stereographic projections)

Rotation 1, 2, 3, 4, 6

Rotoinversion 1 (= i), 2 (= m), 3, 4, 6

Rotation 1, 2, 3, 4, 6

Rotoinversion 1 (= i), 2 (= m), 3, 4, 6

symmetry elements equivalent pointssymmetry elements equivalent points 8

Stereographic projections of symmetry groups

Types of pure rotation symmetry

Stereographic projections of symmetry groups

Types of pure rotation symmetry

Draw point group diagrams (stereographic projections)Draw point group diagrams (stereographic projections)

Rotation 1, 2, 3, 4, 6

Rotoinversion 1 (= i), 2 (= m), 3, 4, 6

Rotation 1, 2, 3, 4, 6

Rotoinversion 1 (= i), 2 (= m), 3, 4, 6

symmetry elements equivalent pointssymmetry elements equivalent points 9

Stereographic projections of symmetry groups

Types of pure rotation symmetry

Stereographic projections of symmetry groups

Types of pure rotation symmetry

Draw point group diagrams (stereographic projections)Draw point group diagrams (stereographic projections)

Rotation 1, 2, 3, 4, 6

Rotoinversion 1 (= i), 2 (= m), 3, 4, 6

Rotation 1, 2, 3, 4, 6

Rotoinversion 1 (= i), 2 (= m), 3, 4, 6

symmetry elements equivalent pointssymmetry elements equivalent points

All objects,structures withi symmetry arecentric

All objects,structures withi symmetry arecentric

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Rotation 1, 2, 3, 4, 6

Rotoinversion 1 (= i), 2 (= m), 3, 4, 6

Rotation 1, 2, 3, 4, 6

Rotoinversion 1 (= i), 2 (= m), 3, 4, 6

Stereographic projections of symmetry groups

Types of pure rotation symmetry

Stereographic projections of symmetry groups

Types of pure rotation symmetry

Draw point group diagrams (stereographic projections)Draw point group diagrams (stereographic projections)

symmetry elements equivalent pointssymmetry elements equivalent points 11

Rotation 1, 2, 3, 4, 6

Rotoinversion 1 (= i), 2 (= m), 3, 4, 6

Rotation 1, 2, 3, 4, 6

Rotoinversion 1 (= i), 2 (= m), 3, 4, 6

Stereographic projections of symmetry groups

Types of pure rotation symmetry

Stereographic projections of symmetry groups

Types of pure rotation symmetry

Draw point group diagrams (stereographic projections)Draw point group diagrams (stereographic projections)

symmetry elements equivalent pointssymmetry elements equivalent points 12

Stereographic projections of symmetry groups

More than one rotation axis - point group 222

Stereographic projections of symmetry groups

More than one rotation axis - point group 222

symmetry elements equivalent pointssymmetry elements equivalent points

13

Stereographic projections of symmetry groups

More than one rotation axis - point group 222

Stereographic projections of symmetry groups

More than one rotation axis - point group 222

symmetry elements equivalent pointssymmetry elements equivalent points

14

Stereographic projections of symmetry groups

More than one rotation axis - point group 222

Stereographic projections of symmetry groups

More than one rotation axis - point group 222

symmetry elements equivalent pointssymmetry elements equivalent points

orthorhombicorthorhombic

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Stereographic projections of symmetry groups

More than one rotation axis - point group 222

Stereographic projections of symmetry groups

More than one rotation axis - point group 222

[100][100]

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Stereographic projections of symmetry groups

More than one rotation axis - point group 222

Stereographic projections of symmetry groups

More than one rotation axis - point group 222

[010][010]

[100][100]

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Stereographic projections of symmetry groups

More than one rotation axis - point group 222

Stereographic projections of symmetry groups

More than one rotation axis - point group 222

[010][010]

[001][001]

[100][100][001][001]

[010][010]

[100][100]18

Stereographic projections of symmetry groups

Rotation + mirrors - point group 4mm

Stereographic projections of symmetry groups

Rotation + mirrors - point group 4mm

[001][001]

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Stereographic projections of symmetry groups

Rotation + mirrors - point group 4mm

Stereographic projections of symmetry groups

Rotation + mirrors - point group 4mm

[100][100]

20

Stereographic projections of symmetry groups

Rotation + mirrors - point group 4mm

Stereographic projections of symmetry groups

Rotation + mirrors - point group 4mm

[110][110][001][001]

[010][010]

[100][100] [110][110]21

Stereographic projections of symmetry groups

Rotation + mirrors - point group 4mm

Stereographic projections of symmetry groups

Rotation + mirrors - point group 4mm

symmetry elements equivalent points

tetragonal

22

Stereographic projections of symmetry groups

Rotation + mirrors - point group 2/m

Stereographic projections of symmetry groups

Rotation + mirrors - point group 2/m

[010][010]

23

Stereographic projections of symmetry groups

Rotation + mirrors - point group 2/m

Stereographic projections of symmetry groups

Rotation + mirrors - point group 2/m

symmetry elements equivalent points

monoclinic

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And here are the 32 point groupsAnd here are the 32 point groups

System Point groups

Triclinic 1, 1Monoclinic 2, m, 2/mOrthorhombic 222, mm2, 2/m 2/m 2/mTetragonal 4, 4, 4/m, 422, 42m, 4mm, 4/m 2/m 2/mCubic 23, 2/m 3, 432, 43m, 4/m 3 2/m Hexagonal 6, 6, 6/m, 622, 62m, 6mm, 6/m 2/m 2/mTrigonal 3, 3, 32, 3m, 3 2/m

System Point groups

Triclinic 1, 1Monoclinic 2, m, 2/mOrthorhombic 222, mm2, 2/m 2/m 2/mTetragonal 4, 4, 4/m, 422, 42m, 4mm, 4/m 2/m 2/mCubic 23, 2/m 3, 432, 43m, 4/m 3 2/m Hexagonal 6, 6, 6/m, 622, 62m, 6mm, 6/m 2/m 2/mTrigonal 3, 3, 32, 3m, 3 2/m

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