Physics PY4118 Physics of Semiconductor Devices Crystalography Coláiste na hOllscoile Corcaigh, Éire University College Cork, Ireland 2.1ROINN NA FISICE Department of Physics Taken mostly from: Crystal2.ppt 0xue%20-%20course%20materials/class%20slides/

PY4118 Physics of Semiconductor Devices 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 ROINN NA FISICE Department of Physics

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 OperationElement InversionPoint ReflectionPlane RotationAxis RotoinversionAxes 4

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° 6

Stereographic projections of symmetry groups Types of pure rotation symmetry Draw point group diagrams (stereographic projections) Rotation1, 2, 3, 4, 6 Rotoinversion 1 (= i), 2 (= m), 3, 4, 6 Rotation1, 2, 3, 4, 6 Rotoinversion 1 (= i), 2 (= m), 3, 4, 6 symmetry elements equivalent points 7

Stereographic projections of symmetry groups Types of pure rotation symmetry Draw point group diagrams (stereographic projections) Rotation1, 2, 3, 4, 6 Rotoinversion 1 (= i), 2 (= m), 3, 4, 6 Rotation1, 2, 3, 4, 6 Rotoinversion 1 (= i), 2 (= m), 3, 4, 6 symmetry elements equivalent points 8

Stereographic projections of symmetry groups Types of pure rotation symmetry Draw point group diagrams (stereographic projections) Rotation1, 2, 3, 4, 6 Rotoinversion 1 (= i), 2 (= m), 3, 4, 6 Rotation1, 2, 3, 4, 6 Rotoinversion 1 (= i), 2 (= m), 3, 4, 6 symmetry elements equivalent points 9

Stereographic projections of symmetry groups Types of pure rotation symmetry Draw point group diagrams (stereographic projections) Rotation1, 2, 3, 4, 6 Rotoinversion 1 (= i), 2 (= m), 3, 4, 6 Rotation1, 2, 3, 4, 6 Rotoinversion 1 (= i), 2 (= m), 3, 4, 6 symmetry elements equivalent points All objects, structures with i symmetry are centric All objects, structures with i symmetry are centric 10

Rotation1, 2, 3, 4, 6 Rotoinversion 1 (= i), 2 (= m), 3, 4, 6 Rotation1, 2, 3, 4, 6 Rotoinversion 1 (= i), 2 (= m), 3, 4, 6 Stereographic projections of symmetry groups Types of pure rotation symmetry Draw point group diagrams (stereographic projections) symmetry elements equivalent points 11

Rotation1, 2, 3, 4, 6 Rotoinversion 1 (= i), 2 (= m), 3, 4, 6 Rotation1, 2, 3, 4, 6 Rotoinversion 1 (= i), 2 (= m), 3, 4, 6 Stereographic projections of symmetry groups Types of pure rotation symmetry Draw point group diagrams (stereographic projections) symmetry elements equivalent points 12

Stereographic projections of symmetry groups More than one rotation axis - point group 222 symmetry elements equivalent points 13

Stereographic projections of symmetry groups More than one rotation axis - point group 222 symmetry elements equivalent points 14

Stereographic projections of symmetry groups More than one rotation axis - point group 222 symmetry elements equivalent points orthorhombic 15

Stereographic projections of symmetry groups More than one rotation axis - point group 222 [100] 16

Stereographic projections of symmetry groups More than one rotation axis - point group 222 [010] [100] 17

Stereographic projections of symmetry groups More than one rotation axis - point group 222 [010] [001] [100] [001] [010] [100] 18

Stereographic projections of symmetry groups Rotation + mirrors - point group 4mm [001] 19

Stereographic projections of symmetry groups Rotation + mirrors - point group 4mm [100] 20

Stereographic projections of symmetry groups Rotation + mirrors - point group 4mm [110] [001] [010] [100] [110] 21

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 [010] 23

Stereographic projections of symmetry groups Rotation + mirrors - point group 2/m symmetry elements equivalent points monoclinic 24

And here are the 32 point groups System Point groups Triclinic 1, 1 Monoclinic 2, m, 2/m Orthorhombic 222, mm2, 2/m 2/m 2/m Tetragonal 4, 4, 4/m, 422, 42m, 4mm, 4/m 2/m 2/m Cubic 23, 2/m 3, 432, 43m, 4/m 3 2/m Hexagonal 6, 6, 6/m, 622, 62m, 6mm, 6/m 2/m 2/m Trigonal 3, 3, 32, 3m, 3 2/m System Point groups Triclinic 1, 1 Monoclinic 2, m, 2/m Orthorhombic 222, mm2, 2/m 2/m 2/m Tetragonal 4, 4, 4/m, 422, 42m, 4mm, 4/m 2/m 2/m Cubic 23, 2/m 3, 432, 43m, 4/m 3 2/m Hexagonal 6, 6, 6/m, 622, 62m, 6mm, 6/m 2/m 2/m Trigonal 3, 3, 32, 3m, 3 2/m 25