1.4 Crystal structure
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1 1.4 Crystal structure
2 (a) crystalline vs. (b) amorphous configurations short and long range order only short range order Abbildungen: S. Hunklinger, Festkörperphysik, Oldenbourg Verlag
3 represenatives of different types of crystals noble gas crystal ionic crystal covalent crystal metal Abbildungen: S. Hunklinger, Festkörperphysik, Oldenbourg Verlag
4 The world of crystalline matter: crystal / bonding types A-A examples A-B interaction (attractive term) dependence on the interatomic distance E bin (ev) R 0 (Å) T melt (K) 1) noble gas crystals molecular crystals vander Waals crystals noble gases: Ar, Ne CH 4 induced dipole dipole interaction ~ 1/r Ne: 3.13 Xe: 4.35 Ne: 24 Xe: 161 2) hydrogen bonds ice: H 2 O dipole dipole interaction (permanent dipoles) ~ 1/r Ice Ih: 4.5/ ) ionic crystals salts: NaCl, KCl Coulomb interaction; direct terms dominate: complete e transfer from A to B ~ 1/r 5-10 NaCl: 5.62 NaCl: ) covalent crystals homopolar Si diamond graphen heteropolar: SiGe GaAs Coulomb interaction direct and exchange terms: partial e transfer; (A and A or A and B share valence electrons) ~ 1/r 5-10 Si: 5.43 GaAs: 5.65 GaAs: ) metallic bonds Na, Li (alkali-) transition metals: Mo, W, Au, Pt, etc alloys CuSn (bronze)... screened Coulomb interaction: screening by completely delocalized electrons additional covalent bonding screened Coulomb potential (jellium model) Na: 371 CuSn: W: 3695
5 contour plot of the electron density distribution in NaCl Abbildungen: S. Hunklinger, Festkörperphysik, Oldenbourg Verlag
6 contour plots of valence electron density distribution in Germanium Abbildungen: S. Hunklinger, Festkörperphysik, Oldenbourg Verlag
7 The world of crystalline matter: crystal / bonding types A-A examples A-B interaction (attractive term) dependence on the interatomic distance E bin (ev) R 0 (Å) T melt (K) 1) noble gas crystals molecular crystals vander Waals crystals noble gases: Ar, Ne CH 4 induced dipole dipole interaction ~ 1/r Ne: 3.13 Xe: 4.35 Ne: 24 Xe: 161 2) hydrogen bonds ice: H 2 O dipole dipole interaction (permanent dipoles) ~ 1/r Ice Ih: 4.5/ ) ionic crystals salts: NaCl, KCl Coulomb interaction; direct terms dominate: complete e transfer from A to B ~ 1/r 5-10 NaCl: 5.62 NaCl: ) covalent crystals homopolar Si diamond graphen heteropolar: SiGe GaAs Coulomb interaction direct and exchange terms: partial e transfer; (A and A or A and B share valence electrons) ~ 1/r 5-10 Si: 5.43 GaAs: 5.65 GaAs: ) metallic bonds Na, Li (alkali-) transition metals: Mo, W, Au, Pt, etc alloys CuSn (bronze)... screened Coulomb interaction: screening by completely delocalized electrons additional covalent bonding screened Coulomb potential (jellium model) Na: 371 CuSn: W: 3695
8 group IV semiconductors group I II III IV V VI VII VIII
9 III-V semiconductors group I II III IV V VI VII VIII
10 II-VI semiconductors group I II III IV V VI VII VIII
11 covalent bonding: formation of molecular orbitals bonding / antibonding states of the H 2 molecule Abbildungen: S. Hunklinger, Festkörperphysik, Oldenbourg Verlag
12 electron density distribution for different separations between bonding H atoms of the H 2 -molecule
13 the role of hybridization: example of sp-hybridized orbitals sp 2 : s + p X sp 3 X Abbildungen: S. Hunklinger, Festkörperphysik, Oldenbourg Verlag
14 carbon allotropes 2005: graphene Tabelle: R. Saito, G. Dresselhaus, M. Dresselhaus, Physical Properties of Carbon Nanotubes, 1998
15 sp2 carbon allotropes (a) graphite and (b) C 60 -molecule Abbildungen: S. Hunklinger, Festkörperphysik, Oldenbourg Verlag
16 Ri
17 nomenclature: basis 1 atom 2 atoms or groups of atoms
18 nomenclature: lattice vectors a v b v basis
19 nomenclature: lattice vectors span the unit cell a v b v basis
20 nomenclature: the unit cell is not unique a v b v basis unique unit cell: the Wigner-Seitz cell constructed by setting perpendicular boundaries in the middle interconnects to next neighbors
21 entire world of crystals: total of 14 Bravais lattices in 3D Abbildungen: S. Hunklinger, Festkörperphysik, Oldenbourg Verlag
22 7 crystal systems Abbildungen: S. Hunklinger, Festkörperphysik, Oldenbourg Verlag
23 7 crystal systems reduce to 6 primitive Bravais lattices equivalent Abbildungen: S. Hunklinger, Festkörperphysik, Oldenbourg Verlag
24 further Bravais lattices are obtained by adding lattice points Abbildungen: S. Hunklinger, Festkörperphysik, Oldenbourg Verlag
25 cubic unit cells: unit cell simple cubic sc body-centered cubic bcc face-centered cubic fcc G. Lisensky - Beloit College
26 the Wigner-Seitz cell in real space sc fcc bcc hcp Abbildung: O. Madelung, Introduction to Solid-State Theory, Springer
27 different crystal phases of the same material example: bulk boron nitride (BN) fcc structure: β-bn hexagonal: α-bn, also h-bn hcp (wurtzite) structure: w-bn wikipedia
28 representatives of the fcc crystal structure: sodium chloride structure: NaCl, salts diamond structure: C, Si, Ge As C Ga C zinkblende structure: GaAs, CdTe (Zinkblende = ZnS) S. Hunklinger, Festkörperphysik, Oldenbourg Verlag
29 1.5 Reciprocal space
30 periodically structured diffraction pattern of a periodically structured real space lattice TEM electron diffraction D. Fox et al, Beilstein J. Nanotechnol. 2012, 3, / doi: /bjnano.3.67
31 diffraction criterion for constructive interference (Bragg or von Laue): periodically structured real space corresponds to periodically structured reciprocal space Bragg condition for constructive interference: diffraction pattern 2 d sinθ = nλ k = 2π λ lattice Wikipedia
32 diffraction criterion for constructive interference (Bragg or von Laue): periodically structured real space corresponds to periodically structured reciprocal space
33 diffraction pattern of graphene:
34 the 1st Brillouin zone (unique) is the Wigner-Seitz cell of the reciprocal lattice Bravais lattice: fcc bcc hcp Brillouin zone: bcc fcc hcp Γ, L, X, K etc: points of high symmetry S. Hunklinger, Festkörperphysik, Oldenbourg Verlag
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