Physics of Condensed Matter I
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1 Physics of Condensed Matter I INZ`PC Solid State 1 Faculty of Physics UW Jacek.Szczytko@fuw.edu.pl
2 Chemical bonding and molecules Born Oppenheimer approximation Max Born ( ) Jacob R. Oppenheimer ( )
3 Chemical bonding and molecules
4 W. Ibach J. Ginter From the molecule to the solid state Small distance between atoms bands (pasma) Large distance between atoms levels (poziomy) Dr hab. Darek Wasik 4
5 J. Ginter From the molecule to the solid state E g Small distance between atoms bands (pasma) Large distance between atoms levels (poziomy) Przerwa energetyczna Energy gap
6 Molecules Hybridization Carbon
7 Types of chemical bonds Hybridization Semiconductors Carbon The binding energy per atom: C (diamond) 7.30 ev Si 4.64 ev Ge 3.87 ev
8 Types of chemical bonds Covalent Semiconductors II III IV V VI Be B C N O Mg Al Si P S ionicity ionicity Zn Ga Ge As Se Cd In Sn Sb Te Group IV: diamond, Si, Ge Group III-V: GaAs, AlAs, InSb, InAs... Group II-VI: ZnSe, CdTe, ZnO, SdS
9 Types of chemical bonds Carriers: holes + electrons - Impurities (domieszki): Acceptrs (p-type) Donors (n-type) II III IV V VI Be B C N O Mg Al Si P S ionicity Semiconductors ionicity Zn Ga Ge As Se Cd In Sn Sb Te Group IV: diamond, Si, Ge Group III-V: GaAs, AlAs, InSb, InAs... Group II-VI: ZnSe, CdTe, ZnO, SdS
10 Types of chemical bonds Covalent bonding
11 Types of chemical bonds Covalent bonding Allotropes of carbon
12 Graphene Types of chemical bonds Covalent bonding
13 Types of chemical bonds Covalent bonding (+ polar covalent) Valence electrons are shared between atoms (non-polar Dc < 0,4; polar 0,4 < Dc < 1,7) Ibach. Luth
14 Types of chemical bonds Covalent bonding (+ polar covalent) Valence electrons are shared between atoms (non-polar Dc < 0,4; polar 0,4 < Dc < 1,7)
15 Types of chemical bonds Ionic bonding Electronegativity (symb. c) - the tendency of an atom to attract electrons. In the extreme case when the electronegativity of both elements is very different (eg. Li and F), it comes to a full transfer of an electron toward more electronegative atom, which leads to the formation of ionic bond (Dc 1,7). NaCl Tablica 2.4. Values of electronegativity (according Pauling) for several major elements (for H set 2,1) I II III IV V VI VII Li 1,0 Na 0,9 K 0,8 Rb 0,8 Be 1,5 Mg 1,2 Ca 1,0 B 2,0 Al 1,5 Ga 1,6 ionicity C 2,5 Si 1,8 Ge 1,7 Sn 1,7 N 3,0 P 2,1 As 2,0 O 3,5 S 2,5 Se 2,4 ionicity F 4,0 Cl 3,0 Br 2,8 J 2,4
16 Types of chemical bonds Ionic bonding Electronegativity (symb. c) - the tendency of an atom to attract electrons. In the extreme case when the electronegativity of both elements is very different (eg. Li and F), it comes to a full transfer of an electron toward more electronegative atom, which leads to the formation of ionic bond (Dc 1,7). NaCl Convention: Covalent bond Dc 0,4 Polar Covalent 0,4 Dc 1,7 Ionic Bonds Dc 1,
17 Types of chemical bonds Ionic bonding Electronegativity (symb. c) - the tendency of an atom to attract electrons. In the extreme case when the electronegativity of both elements is very different (eg. Li and F), it comes to a full transfer of an electron toward more electronegative atom, which leads to the formation of ionic bond (Dc 1,7). NaCl The binding energy per pair of ions : NaCl 7.95 ev NaI 7.10 ev KBr 6.92 ev Distribution of charge density in the NaCl plane based on X-ray results. C. Kittel
18 Types of chemical bonds Ionic bonding Electronegativity (symb. c) - the tendency of an atom to attract electrons. In the extreme case when the electronegativity of both elements is very different (eg. Li and F), it comes to a full transfer of an electron toward more electronegative atom, which leads to the formation of ionic bond (Dc 1,7). NaCl V Ԧr ij = ± e2 4πε 0 r ij Potential energy between two singly charged ions i, j r ij = a p ij separation of nearest neighbours depends on crystal structure V r = 4πε 0 a i j e2 ±1 p ij = Ibach. Luth 25/01/ e2 4πε 0 a i j The Madelung constant A depends on the structure e.g. A NaCl = ±1 = e2 p ij 4πε 0 a A
19 Types of chemical bonds Covalent bonding (+ polar covalent) Valence electrons are shared between atoms (non-polar Dc < 0,4; polar 0,4 < Dc < 1,7) Ionic bonding electrons are tranfered between atoms (Dc 1,7). An essential contribution to bonds energy of ionic crystals comes from the electrostatic interaction (Madelung energy): U r = N e2 4πε 0 r i j r the distance between atoms rp ij - the distance between pair of ions i, j B, n repulsive potential parameters (n = 6 12) ±1 + B p ij r n i j 1 p ij n A = σ i j ±1 p ij - the Madelung constant (for NaCl structure A = 1,748, for CsCl A = 1,763)
20 Types of chemical bonds Metalic bonding The chemical bond in metals, formed by the electrodynamic interaction between the positively charged atom cores, which are located in nodes of the lattice, and negatively charged plasma electrons (delocalized electrons, electron gas). Similar to a covalent bond, but electrons forming a bond are common to a large number of atoms. Na + e Na + e Na + e Na + e Na + e Na + e Na + e Na + e Na + e e e e Na + Na + Na + Na + Na + e e e e e Na + Na + Na + Na + Na + e e e e Na + Na + Na + Na + Na + e e e e Na + Na + Na + Na + Na + e e Electron gas
21 Types of chemical bonds Metalic bonding The chemical bond in metals, formed by the electrodynamic interaction between the positively charged atom cores, which are located in nodes of the lattice, and negatively charged plasma electrons (delocalized electrons, electron gas). Similar to a covalent bond, but electrons forming a bond are common to a large number of atoms. In the alkali metals only delocalized electrons of the last shell ns contribute to bonding. In these metals the length of the bonds can be easily changed (high compressibility) The metals of further columns of the Periodic Table also deeper shells give an important contribution to bonding (in particular, transition metals and rare earths d and f shells). In these metals the length of the bonds is much harder to change (small compressibility) The bonds in metals are usually not very strong, but there are also metals with quite strong bonding - eg. Tungsten (wolfram) 25/01/
22 Types of chemical bonds Metalic bonding The chemical bond in metals, formed by the electrodynamic interaction between the positively charged atom cores, which are located in nodes of the lattice, and negatively charged plasma electrons (delocalized electrons, electron gas). Similar to a covalent bond, but electrons forming a bond are common to a large number of atoms
23 Types of chemical bonds Hydrogen bonding Hydrogen is shared between atoms Celulose
24 Wiązania Hydrogen bonding Hydrogen is shared between atoms HF 2 C. Kittel
25 Types of chemical bonds Hydrogen bonding Hydrogen is shared between atoms (H 2 0)
26 Types of chemical bonds Van der Waals bonds Ne, Ar, Kr, Xe interaction of induced dipole moments
27 Types of chemical bonds Dipole bonding (also intermolecular interaction) attractive forces between the positive end of one polar molecule and the negative end of another polar molecule - intermolecular interaction (e.g. ICl)
28 Types of chemical bonds Van der Waals bonds (also intermolecular interaction) interaction between permanent dipoles (Keesom interaction) interaction between permanent and induced dipoles (Debye interaction) London interaction London dispersion forces (interaction between induced dipoles) Lennard-Jones potential The potential energy of N atoms U r = 4ε Van der Waals bonds Ne, Ar, Kr, Xe interaction of induced dipole moments. Responsible for the possibility of condensation and solidification of noble gases (London ineraction) σ r U tot r = 2Nε 12 σ r i j 6 σ p ij r 12 i j σ p ij r
29 Types of chemical bonds Covalent bond Ionic bond Metallic bond Directional bond (hybrydization) Isolators or semiconductors (charge between atoms) Many of the covalent compounds dissolved in non-polar solvents, and are insoluble in water non-directional bond Isolators (charge in ions) Many of the ionic compounds dissolved in a polar solvent (water) and not soluble in non-polar non-directional bond, delocalised electrons The more electrons, the stronger the bond Conductors (free charge) Metals crystallize preferentially in closed packed structures (fcc, hcp, bcc) Plastic (metal ions can easily move under the influence of an external force)
30 Types of chemical bonds Covalent bond Ionic bond Metallic bond Directional bond (hybrydization) Isolators or semiconductors (charge between atoms) Many of the covalent compounds dissolved in non-polar solvents, and are insoluble in water non-directional bond Isolators (charge in ions) Many of the ionic compounds dissolved in a polar solvent (water) and not soluble in non-polar non-directional bond, delocalised electrons The more electrons, the stronger the bond Conductors (free charge) Metals crystallize preferentially in closed packed structures (fcc, hcp, bcc) Plastic (metal ions can easily move under the influence of an external force)
31 Crystals T = n 1 Ԧt 1 + n 2 Ԧt 2 + n 3 Ԧt 3 primitive translation vectors V Ԧr = V Ԧr + T Kryształ Lattice is a regular and periodic arrangement of points in space (lattice sites or lattice points). It is a mathematical abstraction; the crystal structure there is only when the base is uniquely assigned to each network node Ciało amorficzne
32 Crystals T = n 1 Ԧt 1 + n 2 Ԧt 2 + n 3 Ԧt 3 primitive translation vectors
33 Crystals T = n 1 Ԧt 1 + n 2 Ԧt 2 + n 3 Ԧt 3 primitive translation vectors Primitive translation vectors are not selected unambiguously!
34 Crystals T = n 1 Ԧt 1 + n 2 Ԧt 2 + n 3 Ԧt 3 primitive translation vectors Primitive translation vectors are not selected unambiguously!
35 Crystals T = n 1 Ԧt 1 + n 2 Ԧt 2 + n 3 Ԧt 3 primitive translation vectors primitive lattice cell
36 Crystals T = n 1 Ԧt 1 + n 2 Ԧt 2 + n 3 Ԧt 3 primitive translation vectors Wigner-Seitz primitive cell C. Kittel
37 Crystals T = n 1 Ԧt 1 + n 2 Ԧt 2 + n 3 Ԧt 3 primitive translation vectors The basis (baza) may be a single atom, ion, a set of atoms, eg. proteins 10 5, positioned around each and every lattice point
38 Crystals T = n 1 Ԧt 1 + n 2 Ԧt 2 + n 3 Ԧt 3 R 0j Basis R nj = R 0j + T primitive translation vectors The basis (baza) may be a single atom, ion, a set of atoms, eg. proteins 10 5, positioned around each and every lattice point
39 Crystals B n t 1 A B A = CD = nt 1 = t cos φ j -j cos φ = n C A t 1 B D
40 Crystals T = n 1 Ԧt 1 + n 2 Ԧt 2 + n 3 Ԧt 3 primitive translation vectors High symmetry cell Primitive cell
41 Bravais lattice Regularna In three-dimensional space, there are 14 Bravais lattices. They form 7 lattice systems Tetragonalna Rombowa Heksagonalna Auguste Bravais Romboedryczna Jednoskośna a b c a b c Trójskośna
42 Bravais lattice Regularna In three-dimensional space, there are 14 Bravais lattices. a b c 90 They form 7 lattice systems a b c 90 Tetragonalna a b c Rombowa Heksagonalna a b c 90 Romboedryczna a b c Jednoskośna a b c a b c Trójskośna
43 Bravais lattice
44 Bravais lattice Example: close packed structure 1 layer A
45 Bravais lattice Example: close packed structure 1 layer A 2 layer B
46 Bravais lattice Example: close packed structure 1 layer A 2 layer B 3 layer A B A B
47 Bravais lattice Example: close packed structure 1 layer A 2 layer B 3 layer C A B C
48 Bravais lattice Example: close packed structure hexagonal close-packed (HCP) Hexagonal lattice with basis fcc lattice
49 Bravais lattice Example: close packed structure hexagonal close-packed (HCP) Hexagonal lattice with basis fcc lattice
50 Bravais lattice Example: close packed structure hexagonal close-packed (HCP) Hexagonal lattice with basis
51 Bravais lattice Example: close packed structure fcc lattice
52 Bravais lattice Example: close packed structure fcc lattice
53 Bravais lattice Example: close packed structure fcc lattice
54 Lattice points T = n 1 Ԧt 1 + n 2 Ԧt 2 + n 3 Ԧt 3 primitive translation vectors fcc lattice ½ ½ 1 Translation vectors of lattice n 1 Ԧa 1, n 2 Ԧa 2, n 3 Ԧa 3 Coordinates of a cell ½ ½ ½ 0½ 1½ ½ 000 ½1 ½ ½ ½
55 Lattice points T = n 1 Ԧt 1 + n 2 Ԧt 2 + n 3 Ԧt 3 primitive translation vectors fcc lattice ½ ½ 1 Translation vectors of lattice n 1 Ԧa 1, n 2 Ԧa 2, n 3 Ԧa 3 Coordinates of a cell n 1, n 2, n ½ ½ ½ 0½ 1½ ½ 000 ½1 ½ ½ ½
56 Lattice directions The set of any of integers relatively prime (coprime, względnie pierwsze) which are to each other as the projections of a vector parallel to the direction of crystal axes.. u vҧ w Negative values denoted by the bar over the number fcc lattice [001] 011 ½ ½ 1 [112] [101] [111] 0½ ½ ½ 0½ ½1 ½ 1½ ½ ത1 [100] 000 ½ ½0 [110] [010]
57 Planes in the crystal A family of lattice planes are written (hkl), and denote the family of planes that intercepts the three points: Ԧa 1 h, Ԧa 2 k, Ԧa 3 l If one of the indices is zero, it means that the planes do not intersect that axis (1/0 = infinity) C Also: the family of planes orthogonal to: h Ԧg 1 + k Ԧg 2 + l Ԧg 3 Where Ԧg 1, Ԧg 2, Ԧg 3 are reciprocal lattice vectors E.g.: A=2, B=3, C=6, plane (3,2,1) hkl plane hkl set of planes hkl diections hkl set of directions A 000 B
58 Planes in the crystal A family of lattice planes are written (hkl), and denote the family of planes that intercepts the three points: Ԧa 1 h, Ԧa 2 k, Ԧa 3 l If one of the indices is zero, it means that the planes do not intersect that axis (1/0 = infinity) Also: the family of planes orthogonal to: h Ԧg 1 + k Ԧg 2 + l Ԧg 3 Where Ԧg 1, Ԧg 2, Ԧg 3 are reciprocal lattice vectors C [321] E.g.: A=2, B=3, C=6, plane (3,2,1) hkl plane hkl set of planes hkl diections hkl set of directions A 000 B
59 Planes in the crystal A family of lattice planes are written (hkl), and denote the family of planes that intercepts the three points: Ԧa 1 h, Ԧa 2 k, Ԧa 3 l If one of the indices is zero, it means that the planes do not intersect that axis (1/0 = infinity) Also: the family of planes orthogonal to: h Ԧg 1 + k Ԧg 2 + l Ԧg 3 Where Ԧg 1, Ԧg 2, Ԧg 3 are reciprocal lattice vectors E.g.: A=2, B=3, C=6, plane (3,2,1) hkl plane hkl set of planes hkl diections hkl set of directions
60 Planes in the crystal A family of lattice planes are written (hkl), and denote the family of planes that intercepts the three points: Ԧa 1 h, Ԧa 2 k, Ԧa 3 l If one of the indices is zero, it means that the planes do not intersect that axis (1/0 = infinity) (100) (110) (111)
61 Planes in the crystal (110) (120) (212) (100) (110) (111)
62 Planes in the crystal
63 Crystalography Planes in the crystal The crystalline structure is studied by means of the diffraction of photons, neutrons, electrons or other light particles
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