Close-packing: - hexagonal close packing. - Cubic close packing. MO vs. band structure: - Diodes, QD

Transcription

1 Chapter 7. The Crystalline Solid State What is the crystalline solid state?: - unitcell, crystal system, - Bravais lattice, lattice centering Close-packing: - hexagonal close packing - Cubic close packing Simple structure: - Metals - diamond Superconductor, silicates -low/high-temp. superconductor - zeolites MO vs. band structure: - Diodes, QD Binary compound: - NaCl vs. CsCl - Zinc blend vs. wurtzite

2 Chapter 7. The Crystalline Solid State Two major classifications of solid materials 1) crystals 2) amorphous 7.1 Formulas and structures Crystalline solids: contain atoms, ions, molecules packed in regular geometric arrays unit cell Simple structures unit cell: structural components resulting in a macroscopic crystal when repeated in all directions Bravais lattice: 14 possible crystal str. (Fig.7.1)

3 7.1.1 Simple structures Fig.7.1 Seven crystal classes and Fourteen Bravais Lattice.

4 7.1.1 Simple structures Atoms are shared w/ other unit cells when they are on corners, edges and faces. corners: shared by 8 unit cells contribute 1/8 to each for a single unit cell 8 X 1/8 = 1 edges: shared by 4 unit cells 1/4 contribution to each faces: shared by 2 unit cells 1/2 contribution to each Example corners = 8 X 1/8 faces = 6 X 1/2 = 3

5 7.1.1 Simple structures positions of atoms: described in lattice points : expressed as fractions of the unit cell dimensions ex) body-centered cubic origin: x = 0, y = 0, z = 0 (0, 0, 0) center: x = 1/2, y = 1/2, z = 1/2 (1/2, 1/2, 1/2) other can be generated by moving three directions in increments of one unit cell length. CUBIC: primitive cubic simple cube - 8 atoms at the corners - description: one length, one angle (90º), on lattice point (0, 0, 0) - total # atom in a unit cell: 8 X 1/8 = 1 - coordination number (CN) = 6 (surrounded by 6 atoms) - No efficient packing occupation of only 52.6 % of volume r

6 7.1.1 Simple structures BODY-CENTERD CUBIC (bcc): primitive cubic + sphere at the center - if the r(central sphere) = r(others) diagonal distance = 4r then, new unit cell 2.31r two atoms in the unit cell 2 lattice points unit cell expands (0, 0, 0), (1/2, 1/2, 1/2) CLOSE-PACKED STRUCTURES: close-packed layers each sphere is surrounded by six others in the same plane most efficient packing possible 1) hexagonal close packing (hcp): ABA structure 3 rd layers is on directly above those of the 1 st layer 2) cubic close packing (ccp) : ABC structure 3 rd layer is above the hole in the 1 st layer face-centerd cubic (fcc) - coordination # (CN) = 12

7 7.1.1 Simple structures Fig.7.2 Close-packed structures (2 X) tetrahedral holes/atom (CN = 4) (1 X) octahedral holes/atom (CN = 6)

8 7.1.1 Simple structures Fig.7.3 Hexagonal close packing - Hexagonal prism w/ the unit cell outlined in bold - 2 atoms in the unit cell - 2r X 2r X 2.83r, 120º - lattice points: (0, 0, 0), (1/3, 2/3, 1/2)

9 7.1.1 Simple structures Fig.7.4 Cubic close packing - 4 layers to complete the cube - 1 st layer: 1 atom 2 nd layer: 6 atoms 3 rd layer: 6 atoms (60º vertical rotation) 4 th layer: 1-4 atoms in the unit cell (8 X 1/8) + (6/2) = 4 - Lattice point: (0,0,0), (1/2, 1/2, 0), (1/2, 0, 1/2), (0, 1/2,1/2) - 74% occ. of the total volume

10 7.1.1 Simple structures Fig.7.5 Tetrahedral and octahedral holes in close-packed layers - tetrahedral hole X 2 /atom octahedral holes X 1 /atom can be filled by smaller ions: 0.225r tetrahedral hole 0.414r octahedral hole

11 7.1.1 Simple structures metallic crystals: most metals crystallized in body-centered cubic cubic close-packed hexagonal close-packed str. : str. can be changed by pressure or temperature size & packing of atoms are variable balance b/w attraction & repulsion simple geometric calculations are not sufficient properties of metals: 1) conductivity (heat & electricity) (= 1/resistance) - metal - nonmetal - * diamond: electrical conductivity heat conductivity

12 7.1.1 Simple structures properties of metals metallic crystals: 2) soft metal: malleable (Cu, fcc str.) hard metal: brittle (Zn, hcp str.) : most metals are shapable by hammering & bending bonding is non-directional : dislocation atoms can be slide into new str. realigned impurity atom w/ a size difference discontinuation in crystal less uniform allow gradual slippage rather than movement of entire layer hammering defect tend to group together resisting deformation

13 7.1.1 Simple structures diamond (Fig.7.6): same as Zinc Blend str. w/ all atoms identical : fcc divided into 8 smaller cubes add atoms in the centers of 4 of the smaller cubes (none of them are adjacent) diamond str. : tetrahedral coordi. (covalent bond) : Si, Ge, α-sn Fig.7.6

14 7.1.2 Structure of Binary Compounds Fig.7.7: two simple examples of binary compds. Fig.7.7 larger ions (usually anions) close-packed str. smaller ions w/ the opposite change occupy the holes (tetrahedral-hole, octahedral-hole) two factors: 1) the relative sizes radius ratio (r + /r - ) small cation tetra-holes or octa-holes larger cations only octa-holes 2) the relative # of cation and anion ex) M 2 X: too many cation no close-packing of anion

15 7.1.2 Structure of Binary Compounds Common structure type (named after the most common compound w/ the str.) NaCl: (Fig.7.7(a)): two face-centered cubes of Na & Cl interpenetrating each other offset by 1/2 unit cell length : many alkalihalides : quite diff. sizes of ions (r(cation) < r(anion)) : 1 st nearest-neighbor per each ion (X 6) Fig.7.7 CsCl (Fig.7.7(b)): Cl - - simple cubes : Cs in the center of cubes (vice versa) : 0.73r will fit exactly in the center : intermediate distance (d(cs-cl) = 356 pm < (r(cs) = 167 pm + r(cl) = 202 pm) 3.5 % smaller : CsCl, CsBr, CsI, TlCl, TlBr, TlI, ordinary conditions

16 7.1.2 Structure of Binary Compounds Zinc blend (Fig.7.8(a)): one of two common crystal form of ZnS : diamond str.-type (CN = 4) : face-centered lattice of S + 1/2 occupied T d hole by Zn same result,, if Zn & S reversed Wurtzite (Fig.7.8(b),(c)): formed at higher temperature : hexagonal close-packed lattice + 1/2 occupied T d hole by other same result,, if Zn & S reversed Fig.7.8

17 7.1.2 Structure of Binary Compounds Fluorite, CaF 2 (Fig.7.9): 1) Ca + cubic close-pack lattice F - occupy all tetra-holes 2) alternative description (Fig.7.9(b)) : F - simple cube array Ca + alternative body centers - antifluorite: cation-anion stoichiometry is reversed e.g.) Li 2 Te, Be 2 C : unlike ZnS,, all tetrahedral sites are occupied by cations Fig.7.9

18 7.1.4 Radius Ratio radius ratio: r + /r - simple, but best approximation to predict coordination # treat the ions as if they were hard spheres possible crystal str. Example 7.3) NaCl and ZnS

19 7.1.4 Radius Ratio All ratio predictions should be used w/ caution!! they are not hard spheres incorrect ratio predictions when r (cation) > r (anion) radius ratio r - /r + ex) CsF: r - /r + = 119/181 = (CN = 6) when nearly equal size cubic of anions center of cation other stoichiometry than 1:1 e.g.) CaF 2, Na 2 S cation & anion have diff, CN or only fraction of possible sites are occupied

20 7.2 Thermodynamics of Ionic Crystal Formation Born-Harber cycle: consider the series of components reactions that can be imagined as the individual steps in compds. formation ex) LiF Li(s) Li(g) 1/2 F 2 (g) F(g) Li(g) Li + (g) + e - F(g) + e - F - (g) Li + (g) + F - (g) LiF(g) Li(s) + 1/2F 2 (g) LiF(s) H sub = 161 KJ/mol Sublimation H dis = 79 KJ/mol Dissociation H ion = 531 KJ/mol Ionization E H ion = -328 KJ/mol -Electron affinity H xtal = KJ/mol Lattice Enthalpy H form = -796 KJ/mol Formation historically: used to determine electron affinities currently: used to determine more accurate lattice enthalpies

21 7.2.1 Lattice Energy and Madelung Constant lattice E of a crystal: the sum of electronic E b/w each pair, U = Z i Z j /r 0 (e 2 /4πε 0 ) Z i, Z j = ionic charges in electron units r 0 = distance b/w ion centers e = e - charge = X C 4πε 0 = permittivity of a vacuum = 1.11 X C 2 /Jm e 2 /4πε 0 = X Jm However, this doesn t include long-range interactions!! e.g.) in NaCl for Na 1 st nn: 6 Cl (1/2a) 2 nd nn: 12 Na (0.707a) Madelung constant: sum of all these geometric factors U = NMZ + Z - /r 0 (e 2 /4πε 0 ) N = Avogadro s # M = Madelung const.

22 7.2.1 Lattice Energy and Madelung Constant Born-Mayer equation: corrects repulsion b/w neighbors U = NMZ + Z - /r 0 (e 2 /4πε 0 )(1-ρ/r 0 ) * ρ = 30 pm, for simple compd. lattice enthalpy: H xtal = U + (PV) = U + nrt * n = the change in # of gas-phase particles on formation of the crystal ex) AB -2 AB 2-3 nrt = KJ/mol for AB = KJ/mol for AB 2 Small!! H xtal U for approximate calculations!!

23 7.2.2 Solubility, Ion Size & HSAB thermodynamic calculations for the effect of solvation & solubility ex) AgCl(s) + H 2 O Ag + (aq) + Cl - (aq) AgCl(g) Ag + (g) + Cl - (g) Ag + (g) + H 2 O Ag + (aq) Cl - (g) + H 2 O Cl - (aq) AgCl(s) + H 2 O Ag + (aq) + Cl - (aq) H = 917 KJ/mol -Lattice enthalpy H = -475 KJ/mol Solvation H = -369 KJ/mol Solvation H = -73 KJ/mol Dissociation if only three are known fourth can be calculated!! factors influencing the thermodynamics of solubility ionic size & charge,, HSAB,, crystal str.,, electronic str. small ions: strong electrostatic force attraction for each other for water larger ions: weaker attraction for each other for water can accumulate more water molecules

24 7.2.2 Solubility, Ion Size & HSAB factors influencing the thermodynamics of solubility soft-soft hard-hard less soluble than soft-hard combination ex) LiF, CsI LiI, CsF less soluble more soluble smaller ions: larger lattice E overcomes larger hydration enthalpies larger ions: smaller hydration enthalpies allow the lattice E to dominate LiI(s) + CsF(s) CsI(s) + LiF(s); exothermic ( H = -146 KJ/mol)

25 7.3 Molecular Orbitals and Band Structures Fermi Level intrinsic semiconductor doped semiconductor: p-type vs. n-type metal insulator semiconductor p-n junctions

26 7.3 Molecular Orbitals and Band Structures AO of 2 atoms 2MO (σ 2s & σ 2s *) AO of n atom n MO in solid n is very large!! # orbitals & # E levels w/ closely spaced E is also large band of orbitals (rather than the discrete E levels) Valence band (highest E band containing e - ) Conduction band (next higher, empty band) - band gap: E difference b/w VB & CB prevent motion of e - Fig.7.14 insulator (Fig.7.14(a))

27 7.3 Molecular Orbitals and Band Structures - Figure: Chemist s representation of the band for the polymer - except the lowest level (sometimes the highest level),, most of the orbitals are paired. - as # nodes E - infinite polymer on the right: the lowest level node-less the highest level max. # nodes

28 7.3 Molecular Orbitals and Band Structures conductor: 1) conductor of electricity e - & holes are free to move - if partially filled orbitals a little E promote e - to higher E levels holes are left behind 2) conductor of heat: free e - also transmit E density of states (N(E)): the concentration of E levels within band (a) Insulator w/ a filled VB (b) Metal w/ a partially filled VB if voltage is applied,, conducting electricity Fig.7.15

29 7.3 Molecular Orbitals and Band Structures conductivity (conductance): 1) metals temp conductivity vibration of the atom interface w/ the motion of e - resistance of the e - flow 2) semiconductor (Si and Ge) temp conductivity more e - are excited into the upper band more holes left behind conductance VB is completely filled & CB is completely empty Fig.7.16, but bands are very close in E ( 2eV)

30 7.3 Molecular Orbitals and Band Structures intrinsic semiconductors: e.g.) Si, Ge pure elements w/ a semiconducting property doped semiconductors: replace a few atoms of the original elements w/atoms having either more or fewer e -. 1) n-type semiconductor: Fig.7.16 (b) (negative) : adding material w/ more e - : e.g.) P(5e - ) Si (4e - ) e - just below CB of Si w/ E added, e - moves from added E level to empty level Fig.7.16

31 7.3 Molecular Orbitals and Band Structures doped semiconductors: 2) p-type semiconductor: Fig.7.16 (c) (positive) : adding material w/ less e - : e.g.) Al (3e - ) Si (4e - ) a band close to VB of Si w/ E applied, e - from VB of host to new level Fig.7.16 conductance by controlling the voltage applied control conductance of the device Fermi level (E F, Figure 7.16): the E level where an e - is equally likely to be in each of the CB & VB moved depending upon the dopant e - type

32 7.3.1 Diodes, the Photovoltaic Effect, and Light-Emitting Diodes p-n junction: put layers of p-type & n-type semiconductor together (Fig.7.17) (a) at equilibrium - E F are at the same E - E level of n-type is lower Fig.7.17 (b) if forward bias is applied negative n-type positive p-type 1) excess e n-type 2) E level of n-type CB 3) enough E move into the p-type holes move toward the junction from the left e - move toward the junction from the right current flow readily!!

33 7.3.1 Diodes, the Photovoltaic Effect, and Light-Emitting Diodes p-n junction: (c) reverse bias: E level of n-type (Fig.7.17) : hole & e - move away from the junction : little current flows diode: current flow only one direction Fig.7.18 Diode Behavior

34 7.3.1 Diodes, the Photovoltaic Effect, and Light-Emitting Diodes photovoltaic effect: e.g.) can be used as light-sensitive switch (Fig.7.19) : reverse bias no current flow But,, E diff. b/w VB and CB of semiconductor is small enough. visible light is energetic enough to promote e - from VB into CB current flow despite the reverse bias e.g.) photovoltaic cell, solar device Fig.7.19

35 7.3.1 Diodes, the Photovoltaic Effect, and Light-Emitting Diodes light-emitting diode (LED): with the forward-biased junction : e - on the n-type + hole on the p-type if E change is of the right magnitude can be released as visible light (luminescence) can be used as LED example) GaP x As 1-x (0.4 x 1.00): red (b.g. = 1.88 ev) GaAs b.g. = 1.4 ev GaP b.g. = 2.25 ev green (b.g. = 2.23 ev) as a fraction of P b.g.

36 7.3.2 Quantum Dots quantum confinement effect: as the size of semiconductors become smaller no longer continuum states (bulk properties) quantized E states at limiting case as if a simple molecule w/ MO discrete E level quantum dots: nanoparticles showing quantum confinement effect : < 10 nm as the particle size bandgap more E needed to excite e - exciton: excited electron-hole combination radiation of specific E light of a particular color can be obtained by adjusting a particle size

37 7.3.2 Quantum Dots example) size of QD vs. electronic emission spectra : ZnSe : as size b.g. emission max. moves from UV to Vis. other examples) ZnSe: UV & violet PbS: near IR & visible CdSe: through the range of visible light applications: conversion of solar E to electricity : data processing & recording : biosensor : medical applications - labeling cell surface - tracking tumor : LED

38 7.4 Superconductivity superconductor: near liquid helium temperature (< 10K),,, metals show no resistance to the flow of e - &, current will continue to flow Fig.7.20

39 7.4.1 Low-temperature Superconducting Alloys most common superconducting material: Nb-Ti alloys typeⅠ superconductors: expelling all magnetic flux cooled below T c - Meissner effect : highest T c for Nb-alloys 23.3 K for Nb 3 Ge typeⅡ superconductors: two T c : below 1 st T c excluded the magnetic field completely 1 st < b/w < 2 nd partial penetration 2 nd < loose superconductivity Meissner effect: e.g.) magnetic levitation train : demonstration superconducting material cooled blow T c + small magnet above it superconductor repels the magnetic flux of the magnet suspended magnet above the material

40 7.4.1 Low-temperature Superconducting Alloys levitation effect: only typeⅡ superconductor magnetic field line partially do enter the superconductor balance the repulsion float the magnet above the superconductor major goal of superconductor research: high temp. superconductor then,, no cooling is required!!

41 7.7 Silicates Silica (SiO 2 ) in three forms: 1) quartz T < 870 ºC 2) tridymite 870 ºC < T < 1470 ºC 3) cristobalite 1470 ºC < T < 1710 ºC Molten silica frequently forms a glass. high viscosity too slow crystallization quartz: most common form of silica : helical chains of SiO 4 tetrahedra chiral w/ clockwise or counterclockwise twists : Fig.7.23 crystal str. of β-quartz Fig.7.23

42 7.7 Silicates Fig Common Silicate Structure

43 7.7 Silicates Al 3+ can substitute for Si 4+ require the addition of another cation to maintain charge balance zeolite: mixed aluminosilicates : (Si, Al) n O 2n frameworks + cations synthesized zeolite: contain cavities large enough for other molecule can be tailored (4 12 silicon to form the cavity entrance) sodalite unit (β-cage): 1) 4- & 6-rings linked together 2) 24 silica/alumina tetrahedra linked together 3) basket-like shape A truncated octahedron

44 7.7 Silicates zeolite A (Linde A, Fig. 7.26) - synthetic zeolite - sodalite units are stacked & linked by oxygen bridges b/w 4-rings - 3-D network of linked cavities each w/ a truncated cuboctahedron shape space-filling run parallel to the three axes B Truncated cuboctahedron Fig.7.26

45 7.7 Silicates synthesized zeolite applications 1) ion exchange properties: for alkali & alkali-earth metal water softener to remove excess Ca 2+, Mg 2+ 2) molecular sieves: absorb oil, water, other molecules e.g) cat litter, oil absorbent 3) catalyst & supports for other catalysts 4) surface can be prepared w/ reactive metal atoms surface-catalyzed reactions