F7360 Characterization of thin films and surfaces Lenka Zajíčková Faculty of Science & CEITEC, Masaryk University, Brno lenkaz@physics.muni.cz 2. Chapter - Structure of Condensed Matter spring semester 2018 Central European Institute of Technology BRNO I CZECH REPUBLIC • 2. Structure of Condensed Matter • 2.1 Amorphous and Crystalline Materials • 2.2 Bonds in solids • 2.2.1 Ionic Bonds o 2.2.2 Van der Waals bonds • 2.2.3 Covalent bonds • 2.2.4 Metallic Bonds o 2.2.5 Summary of Bonds in Solids o 2.3 Types of Materials F7360 Characterization of thin films and surfaces: 2.1 Amorphous and Crystalline Materials ka Zajíčková 3/57 When the temperature of a melt is lowered to a certain point, the liquid will form either a crystalline or amorphous solid. ► Crystalls are periodic arrays of long-range ordered atoms. A real crystal is never perfect - contains defects (vacancies, dislocations, impurities, and other imperfections). ► Amorphous materials posses only short-range ordering. Si02 demonstrates the difference between crystalline and amorphous materials: ► Short-range ordering: silicon atoms are surrounded by three oxygen atoms. ► Long-range ordering in quartz: hexagonal structure. F7360 Characterization of thin films and surfaces: 2.1 Amorphous and Crystalline Materials Lenka Zajíčková 4/57 An ideal crystal is constructed by the infinite repetition of identical groups of atoms (a motif): ► A group is called the basis. ► The set of mathematical points to which the basis is attached is called the lattice. o o o o o o o o o O O o 0 0 o o o o o o o O O O O O © fj © o o o o o o o o O o O o © 0 © rj o o o o o o o O O o o O © o © <& o o o o o o o O O o o O o = Motif © = l Motif O o o o o o o O O O O O O O O O O O O O O O O O O O O O O O O O = Motif r i Square lattice The lattice in 3D is defined by three translation vectors a\, a2, a3 - the arrangement of atoms in the crystal have to look the same when viewed from the points r and ?' f f+ u-\ aA + u2a2 + u3a3 where u<\, u2 and u3 are arbitrary integers. F7360 Characterization of thin films and surfaces: 2.1 Amorphous and Crystalline Materials Lenka Zajíčková 5/57 The lattice is primitive if any two points from which the atomic arrangement looks the same always satisfy 7 = ~r+ l/i<3i + u2a2 + u3a3 with a suitable choice of the integers , u2 and u3. Then, the vectors a\, a2 and a3 are primitive translation vectors. Lattice points in 2D - all pairs of a\, a2 are translational vectors but a\\ a^" are not primitive. F7360 Characterization of thin films and surfaces: 2.1 Amorphous and Crystalline Materials ka Zajíčková 6/57 The parallelepiped defined by the primitive axes a\, a2 and a3 is a primitive cell. A primitive cell is type of unit cell (or just cell). It is the smallest cell that can serve as a building block for the crystal structure. Its volume is V = |a-i .a2 x a3| Primitive translation vectors a, are often used to define the crystal axes - three adjacent edges of the primitive parallelepiped. Nonprimitive axes are used as crystal axes when they have a simple relation to the symmetry of the structure. 2D centered rectangular lattice with ► primitive translation vectors a\ and a2 and ► nonprimitive translational vectors C\ and c2. F7360 Characterization of thin films and surfaces: 2.1 Amorphous and Crystalline Materials ka Zajíčková 7/57 In order to describe the crystal structure it is necessary to answeer three important questions: 1. what lattice we have (for a particular structure can be more than one), 2. what translational vectors a\, a2, a3 are we using to describe the lattice (more sets of translational vectors can be selected for a given lattice) and 3. what is the basis (which is chosen after the lattice and translational vectors are selected). F7360 Characterization of thin films and surfaces: 2.1 Amorphous and Crystalline Materials ka Zajíčková 8/57 Crystal lattice can be transformed into themselves by the lattice translation f T = u-\3-\ + u2a2 + 1/3 a3 and by various other symmetry point operations. A typical symmetry point operation - rotation about an axis that passes through a lattice point Lattices can be found such that one-, two-, three-, four- and six-fold rotation axes carry the lattice into itself (corresponding rotations by 2n, 2n/2, 2n/3, 2n/4 and 2n/6 and their integral multiples). Another symmetry operations are mirror reflections about a plane through a lattice point. The collection of symmetry point operations which, applied about a lattice point, carry the latice into itself is called lattice point group. F7360 Characterization of thin films and surfaces: 2.1 Amorphous and Crystalline Materials ka Zajíčková 9/57 Bravais lattices in two dimensions: ► general lattice known as oblique lattice - invariant only under rotation of n and 2n about any lattice point ► four special lattices (rectangular, centered rectangular or rhombic, hexagonal and square) - can be invariant under rotation 2n/3, 2n/4 and 2n/6 or under mirror reflection Q Two-fold rotation axis /\ I l>uv told roljtion axis ] Fourfold rotation axis Sixfold rotation axis | Mirror symmetry plane -f- Orthogonal mirror planes 3|£ Mirror planes every 4V Oblique I alticc ii ' b . .1 ' sH> řx. Rectangular I ait ice |a|*lhl- a - w *.......-----1 1 1 } '■X / / / Centered Rectangular o - cos '(aj2b) I he symmetry is the same as that for any other rectangular lattice, in addition to the minimum symmetries ot any oblique lattice Square Lattice |a = b . <» = 90° Hexagonal I uttice la|=|b|. u = 120* F7360 Characterization of thin films and surfaces: 2.1 Amorphous and Crystalline Materials Lenka Zajíčková 10/57 In 3D - 7 distinguishable point groups of unit cells (7 crystal systems) that can fill the space (triclinic, monoclinic, orthorhombic, tetragonal, rhombohedral, hexagonal and cubic). The Bravais lattices are obtained by combining one of the 7 lattice systems with one of the lattice centerings: ► simple - lattice points on the cell corners only. ► body-centered ► face-centered ► base-centered in total 7 x 6 = 42 combinations but from the full symmetries (point operations and translations) 14 different space groups (14 Bravais lattices) have been found. 47' m Simple cubic Face-centered cubic Ä7I Body-centered cubic Simple tetragonal Body-centered tetragonal Hexagonal V3r I. V Simple orthorhombic Body-centered orthorhombic Base-centered orthorhombic Face-centered orthorhombic Rhombohedral Simple Monoclinic Base-centered monoclinic Triclinic Crystal Centering Axial Distances Axial Angles Examples System (edge lengths) Cubic simple body-centred face-centred a = b = c a = p = 7 = 90° NaCI, Zinc Blende, Cu Tetragonal simple body-centered a = b / c a = (3 = 7= 90° White tin, Sn02, Ti02, CaS04 Orthorhombic simple body-centered face-centered base-centered a/ c a = (3 = 7 = 90° Allotropes of sulfur, KN03, BaS04 Hexagonal simple a = b / c a = /3 = 90°,7 = 120° Graphite, ZnO, CdS Rhombohedral simple a = b = c a = 0 = 7 /90° CaC03, (trigonal) HgS Monoclinic simple base-centered a/ c a = 7 = 90°, p /90° Monoclinic Sulphu Na2S04 *10H2O Triclinic simple a / p / 7 ^90° K2Cr207, CuS04 *5H20, H3BO3 F7360 Characterization of thin films and surfaces: 2.1 Amorphous and Crystalline Materials nka Zajíčková 12/57 Primitive translation vectors of the body-centered cubic (bcc) lattice (in units of lattice parameter a) ► a, =(1/2,1/2,-1/2); ► a2 = (-1/2,1/2,1/2); ► a3 = (1/2,-1/2,1/2) The primitive cell is the rhombohedron. The packing ratio is 0.68, defined as the maximum volume which can be filled by touching hard spheres in atomic positions. Each atom has 8 nearest neighbors. The conventional unit cell is a cube based on vectors a\ = (0,0,1); a2 = (0,1,0); a3 = (0,0,1). It is twice big compared to the primitive unit cell and has two atoms in it with coordinates rj = (0,0,0) and r2 = (1 /2,1 /2,1 /2). The bcc lattice have alkali metals such as Na, Li, K, Rb, Cs, magnetic metals such as Cr and Fe, and refractory metals such as Nb, W, Mo, Ta. F7360 Characterization of thin films and surfaces: 2.1 Amorphous and Crystalline Materials ka Zajíčková 13/57 Primitive translation vectors of the face-centered cubic (fee) lattice (in units of lattice parameter a) ► ai =(1/2,1/2,0); ► a2 = (0,1/2,1/2); ► a3 = (1/2,0,1/2). The primitive cell is the rhombohedron. The packing ratio is 0.74. Each atom has 12 nearest neighbors. The conventional unit cell is a cube based on vectors a\ = (0,0,1); a2 = (0,1,0); a3 = (0,0,1). It is 4 times bigger than the primitive unit cell and has 4 atoms in it with coordinates 7A = (0,0,0); r2 = (1 /2,1 /2,0); r3 = (0,1 /2,1 /2); r4 = (1 /2,0,1 /2). z The fee lattice have noble metals such as Cu, Ag, Au, common metals such as AI, Pb, Ni and inert gas solids such as Ne, Ar, Kr, Xe. NaCI is fee with the basis consisting of two atoms (Na and CI) - the closest neighbours are 6 atoms of different type. Diamond is fee, the basis two same atoms at 000, \ tetraedral bonds (4 closest neighbours). F7360 Characterization of thin films and surfaces: 2.1 Amorphous and Crystalline Materials ka Zajíčková 14/57 Index systems for crystal directions and planes (Miller indices) F7360 Characterization of thin films and surfaces: 2.2 Bonds in solids ka Zajíčková 15/57 Interatomic bonds in solids are ► ionic (a) covalent (b) ► metalic (c) Van der Waals (d), (e) bond energy: 1 kJ/mol = 0.010364 eV/atom DIOIO DI3IO la) lb) ••••• •■ é • m id) F7360 Characterization of thin films and surfaces: 2.2.1 Ionic Bonds ka Zajíčková 16/57 Ionic bonds - between particles which have a net electrical charge ► positive ions - cations, atoms with low ionization energy (lose electrons easily) - alkaline metals (only 1 s electron in outer shell) ► negative ions - anions, atoms with high electron affinity (easily accept electrons) - halogens (missing 1 p electron) Interaction force - Coulomb. Repulsive forces of similarly charge ions and attractive forces of differently charged ions are equilibrated. Pauli exclusion principle does not allow ions to come too close. Periodic Table of the Elements 1 H Hydrogen 1.0(179 2 tIA 2A 13 IMA 3A 14 IVA 4A 15 VA BA 16 VIA 6A 17 2 Wo VIIA 1 1C -,A Helium 'A 4.0D26U 3 Li Lilliium 6.941 4 Be Barvili urn 9.01218 5 El iijjjjj] c Carbon 12.011 N Nitrogen 14.00674 8 0 Oxysen 15.9994 9 10 F Ne Fluorine Neon IE.0934113 20.1797 11 Na Sodium 23.9bb7g6 12 Mg Magnesium 24-305 3 1MB 3B 4 IVB 4B 5 VB 5B 6 VIB 6B 7 VIIB 7B s f 9 — VIII — 8 10 ) 11 IB 1B 12 IIB 2B 13 Al Aluminum ÍG.981539 14 Si Silicon 2B.ÜS55 15 p Phosphorus 30-973762 16 s Sulfur 32.066 17 18 CI Ar Chlorine Argon 36.4627 364*6. 19 K Potassium 39.09(S3 20 Ca Calcium 40.078 21 Sc Scandium 44.95591 22 Ti Titanium 47 Bfl 23 V Vanadium 50.9415 24 Cr 51.9951 25 Mn Manganese 54.938 26 Fe 55 847 27 Co Cobalt 58.9332 28 Ni Nickel 5B.6934 29 Cu Copper 63 546 30 Zn Zinc 65.39 31 Ga Gallium 69.732 32 Ge Germanium 79.64 33 As 74.92159 34 Se Selenium 7B.9G 35 36 Br Kr Bromine Krypton 79.904 £3.80 37 Rb Rubidium be.467b 38 Sr Sirů fitl urn B7.62 3d Y Yttrium assoEss 40 Zr ZirtonuT 91.224 41 Nb Niobium 92.90638 42 Mo Molybdenum 95.94 43 Tc Technetium 3 S. 00 72 44 Ru 101.07 45 Rh Rhodium 102.9055 46 Pd Palladium 106.42 47 AJ 107 .«682 46 Cd Cadmium 112.411 49 In Indium 114.31 S 50 Sn Tin 118.71 51 Sb Antimony 121.7SQ 52 Te Tellurium 127.6 53 54 1 Xe Iodine Xenon 123.90447 131.20 55 Cs Cesium 132.90643 56 Ba Barium 137.327 57-71 72 Hf Hafnium 176.49 73 Ta 130-9479 74 W Tungsten 183-Í5 75 Re Rhanium 1Í6.2Q7 76 Os Osmium 190.23 77 lr Iridium 192.22 78 Pt Platinum 10508 79 Au Gott 1 96.9665 80 Hg Mercury 2Ů0Ě9 81 Tl Thalii jih 2Ů4.3A3Í 82 Pb 207 5 83 Bi Bismuth 84 Po Polonium [2ÖB.9&24J 85 86 At Rn Astatine Radon 209.9*71 J33.9176 87 JFr 2230137 88 Ra Radium 220.0254 89-103 104 Rf Ruthe rtordium 105 Db 106 [2SG] 107 Bh Bob Hum [264] 108 Hs Hssslum [26*1 109 Mt Meitnerium [26*] 110 Ds Darmsudtium [2691 111 Rg Roentgenium [2721 112 Cn Copernlclum [277] 113 Uut 114 115 116 Uuq Uup Uuh Lnunquadium UnunpenUum Uiunheiium [28S] unknown (2SB] 117 118 Uus Uuo li r i u r l s s p Li u n i u -i l r o c i i j rn 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 La Ce Pr Nd Pm Sm Eu Gd Tb Dy Ho Er Tm Yb Lu Lanthanum Cerium Praseodymium NeedymUm Prumeihlum Saw sin m Europium Gadolinium Terbium Dy^presíum Holmlum Erbium Thulium YtlarDUím Luteitlum 13B.9055 140.115 140.80785 144.24 144.0127 150.3B 151.9655 157.25 158/92534 162.50 164.93032 167.26 168.03421 173.04 174.867 €3 94 95 96 97 98 99 100 101 102 103 Np Pu Am Cm Bk Cf Es Fm Md No Lr Neptunium Plutonium Amerlclum Curium Berka Hum Californium Einsteinium Formlum Mandele vlum N obeli urn Lew rend urn 237.04ÍI 244.084? 243.0614 247.0703 247.0703 251.07» \254] 257 .OSM 255.1 259.100S |262J 89 ^ctinide Series Actinium 227.027S 90 91 92 Th Pa U Thorium Pjr>i actinium Uranium 232.0381 231.03586 238.028» F7360 Characterization of thin films and surfaces: 2.2.1 Ionic Bonds ka Zajíčková 17/57 Ionization energy (energy necessary to release electron) is periodic function of atomic number, large atoms or molecules have lower ionization energy. First Ionization Energies He 25 20 - > Ä S) L. = = o "ní 15 10 - TT Ne D Noble gases O Alkali metals Third transition series -t- -t- -+- 10 20 30 40 50 60 Atomic number 70 80 90 100 Electron affinity (energy released if electron is added): Fluorine Chlorine Bromine Iodine 3.45 eV 3.61 eV 3.36 eV 3.06 eV Directionality of ionic bond is low - electron configuration of ions resemble filled shells of inert gases, i. e. electron density is sherically symmetric. High coordination - cation (anion) is surrounded by as many anions (cations) as possible. F7360 Characterization of thin films and surfaces: 2.2.1 Ionic Bonds Lenka Zajíčková 18/57 Relative size of cations and anions determines the lattice type. The most frequent types ► fee - typical example NaCI (6 neighbors of different type) ► bcc - typical example CsCI (8 neighbors of different type) F7360 Characterization of thin films and surfaces: 2.2.1 Ionic Bonds ka Zajíčková 19/57 Cohesive energy of the crystal - energy released if ionic crystal is formed. The most important contribution - Coulomb interactions between ions, long range interaction Consider Na+ in NaCI. It is surrounded by six Cl~ at the distance r: V,=- 6e< 47re0r Another neighbors are 12 Na+ each at the distance V2r V2 = + 12e< 47T£0V2r Summed for the entire crystal: e2 ^Coulomb — — ~A ~ 4-Treor n 12 6"71 + (1) = -1.748 47T£0r = —a 47T£0r The constant a is called Madelung crystal constant, values 1.6-1.8 for simple crystals. F7360 Characterization of thin films and surfaces: 2.2.1 Ionic Bonds nka Zajíčková 20/57 The contribution of the quantum-mechanical repulsive force to the total potential energy can be written as: B repulsive — n Total potential energy in the crystal is: V — ^Coulomb "I" ^repulsive — ae 2 B + — 47T£0r rn where n is about 9. o E 01 c c i. 589 Repulsive interactions Attractive interactions In the steady case the separation of ions at a distance r0 must have minimum: dV_ dr = r0 = 0. so B =-1 47T£0n 0 Total potential energy: V = - aec 47T£0r0 n F7360 Characterization of thin films and surfaces: 2.2.1 Ionic Bonds ka Zajíčková 21 157 Total potential energy (from previous slide): In case of NaCI ► r0 = 2.81, thus V=-1.27x 10-18 J =-7.97 eV we have take into account energy for electron transfer between Na and CI, i. e. the difference between the ionization energy 5.14 eV for Na and the electron affinity of -3.61 eVforCI 1.53 eV ► each atom is contributing with half of the value, so the overall cohesive energy per atom EcoheSive = (-3.99 + 0.77) eV/atom = -3.22 eV/atom. =^ Ionic crystals are hard and they have high melting point. They conduct electricity when molten or in solution, but not as a solid. They tend to be soluble in water. F7360 Characterization of thin films and surfaces: 2.2.2 Van der Waals bonds ka Zajíčková 22/57 All atoms and molecules, even inert-gas atoms, exhibit weak, short-range attractions for one another (proportional to r~7) due to van der Waals forces (0.01-0.1 eV/molecule). polar-polar attraction polar-nonpolar attraction nonpolar-nonpolar The different types of van der Waals forces were first explained by different people at different times different names ► London dispersion forces between non-polar atoms or molecules were described by Fritz London in 1930. He suggested that the motion of electrons within an atom or non-polar molecule can result in a transient dipole moment. Dispersion forces are the weakest of the van der Waals forces. They are stronger for larger atoms and molecules (higher polarizability). ► dipole-dipole interactions explained by Keesom in 1912 as interaction between permanent electrical dipole moments of molecules (depends on the value of electrical dipole). F7360 Characterization of thin films and surfaces: 2.2.2 Van der Waals bonds ka Zajíčková 23/57 Hydrogen bonds - a special type of attractive dipole-dipole interaction between an electronegative atom and a hydrogen atom bonded to another electronegative atom (e.g. for H-F, H-0 or H-N). ► It is strong type of van der Waals forces (0.04-0.26 eV/molecule) because H atom has only 1 electron that is "donated" almost whole to the electronegative atom, leaving the small effective size of proton unshielded (electric forces vary as r~2). ► Hydrogen bonds can occur between molecules or within parts of a single molecule. Typical example of molecules with permanent electric dipole moments is H20: F7360 Characterization of thin films and surfaces: 2.2.2 Van der Waals bonds ka Zajíčková 24/57 Characteristics of attractive force between polar and nonpolar molecules: The electric field Ě at a distance r from a induced electric dipole moment p' in polar molecule having dipole moment p the other, normally nonpolar molecule, E = 1 pr = prcos6 {0 is angle between pand r). pf = aĚ where a is a constant called polarizability of the molecule. The energy of the induced dipole in the electrical field E is S = -p/E = - iA a x9(1 +cos26>)P (47T£0)' 6 The mutual energy of the molecules that arises from their interaction is thus negative, signifying that the force between them is attractive, and is proportional to r~6. The force itself is equal to d8/dr and so proportional to r~7, which means that it drops rapidly with increasing separation. Doubling the distance between two molecules reduces the attractive force between them to only 0.8 % of its original value. http: //chemwiki.ucdavis.edu/Physical_Chemistry/Physical_Properties_of_Matter/ Atomic_and_Molecular_Properties/Intermolecular_Forces/Intermolecular_Forces F7360 Characterization of thin films and surfaces: 2.2.3 Covalent bonds ka Zajíčková 25/57 Explanation of covalent bonds - quantum mechanics is necessary. Two theories ► valence bond (VB) theory or local electron model: chemical bonds are formed by overlapping of atomic orbitals. This overlap of orbitals causes localization of the electrons in the bond region. ► molecular orbital (MO) theory: construction of new orbitals called molecular orbitals, electrons are redistributed throughout the molecules. VB theory provides an excellent agreement with observed molecular geometries (bond angles and bond lengths) but physical properies cannot be explained =^ MO theory. In discussing chemical bonds it is helpful to Schrodinger's equation for hydrogen atom: visualize the various atomic orbitals qualitatively resembling those of hydrogen: H^(T) = E^(T) h2 p2 H =--V2 + V(r) and V =-- 2/7? 47r£0r Solution for lim -0 = 0 is ^,/,m(r) = Rni{r)P™{cosO)eim* 2fr2(47T£o)2 n2 n = 1,2,3,... principal quantum number / = 0,1,2,... n — 1 orbital quantum number m =—/,..., 0,/ magnetic quantum number F7360 Characterization of thin films and surfaces: 2.2.3 Covalent bonds ka Zajíčková 26/57 3s 3p □□□□□ nDPD 3d 4s 4P □ 2s 2p 1s H Is 3 Li 2s 11 Na 3s 37 Rb 5s 55 Cs 6s 87 Fr 7s 4 Be 12 Me 20 Ca 38 Sr 56 Ba —> 88 Ka —> ► Carbon, C - 1s22s22p2 ► Nitrogen, N - 1s22s22p3 ► Oxygen, 0 - 1s22s22p4 ► Fluorine, F - 1s22s22p5 ► Titanium, Ti -1s22s22p63s23p63d24s2 Vi-H-i^* / /t.tt.tt.t ř\ r\ r*\\-v r\TTYi i n-f n /n3(Ton7 /AQAOn+^Vil tltffi Electron Configurations in the Perodic Table 21 Sc 39 Y 57 La 89 Ac 22 Ti 40 Zr 72 Hf 104 Rf 23 V 41 Nb 73 la 105 Db 24 Cr 42 Mo 74 W 106 25 Mn 26 Fe 3d 43 44 Ru 4d 75 Re 76 Os 5d 107 Bh 108 Hs 6d 27 Co 45 Rh 77 Ir 109 Mt 28 Ni 46 Pd 78 Pt 110 29 Cu 47 Ag 79 Au 111 30 Zn 48 Cd 80 Hg 112 5 B 13 Al 31 Ga 4r- 49 In 81 II 113 6 C 14 Si 32 Ge 50 Sn 82 Pb 114 7 N 8 O 15 P 16 s 3p 33 As 34 Se 4p 51 Sb 52 Te 5p 83 Bi 84 Po 6p 9 F 17 CI 35 Br 53 I 85 At 2 He Is 10 Ne —> 18 Ar 36 Kr —^ 54 Xe 86 Rn by: SaraJi Faizi 58 Ce 59 Pr 60 INd 61 Pm 62 Sm 63 Lu 64 Gd A If 65 lb 66 Dy 67 Ho 68 Er 69 Tin 70 Yb 71 Lu 90 Th 91 Pa 92 U 93 Np 94 Pu 95 Am 96 Cn 1 if 97 Bk 98 Cf 99 Es 100 Fm 101 Md 102 No 103 Lr F7360 Characterization of thin films and surfaces: 2.2.3 Covalent bonds Lenka Zajíčková 27 / 57 H+ molecule - the easiest quantum mechanical calculation i i i -200 -100 0 100 200 r/pm For separated cores h2 V J V 2/7? and similarly for 0Ď; E° = E° = E° If the cores come closer electron from the core a will be influenced by the core b. Additionally, Coulomb repulsive force occurs (shifts energies by constant value up, i. e. omitted for now) (- ft V2- 2/7? 47T£0ra 47T£0rb Solution by Linear Combination of Atomic Orbitals (LCAO) -0 = Ci^a + C2(ßb results in equation (E° - E - —^—)ci 0a + (E° — E — -^)c20Ď = 0 that will be multiplied by 0* or 0* (but 0a, 0^ are real - ground state of H) F7360 Characterization of thin films and surfaces: 2.2.3 Covalent bonds ka Zajíčková 28/57 Wave functions are normalized / ^a^a^dV = 1 Wave functions are not orthogonal overlap integral Interaction of e~ with separated cores (charge density —ecfil with core b or — ec^ with core a) Interaction of electron exchanged density -e0a0b with core - exchange integral S = f fatbdV c = f -e%b dV D= / -ecj)a(j)b 47re0ra,b dV It gives algebraic set of equations (AE + C)Ci + (AE.S + D)c2 = 0 (AE.S + D)d + (AE + C)c2 = 0 which determinant has to be equal to zero bonding state (symmetric wave functions) = c(0a + b) =^ c = Cz = ±Ci i. e. C+D ■binding + 1+S 4716 o Rab antibonding state (antisymmetric wave function) = c{(j)a — b) ^binding C-D + 1 - S 47re0Rab Bonding Anti-bonding F7360 Characterization of thin films and surfaces: 2.2.3 Covalent bonds ka Zajíčková 29/57 Valence Bond Theory Valence bond theory concentrates on direction of bonds given by the overlap of atomic orbitals (s, p) - pairing of unpaired electrons from vallence shell atomic orbitals. In VB we name the bond according to its direction: ► g) g) N2p N 2s 1 H. heteronuclear molecule 02 and F2 0 2p " 0 2s O 4ct* g) g) 2 TC* 1 R 3a 2 a* 1 a ■ 0 2p 0 2s F7360 Characterization of thin films and surfaces: 2.2.3 Covalent bonds ka Zajíčková 41 157 Molecules with more atoms, e. g. CH4 Energy LUMO HOMO 2a, a It, 1a. C - — I < s ► Highest Occupied Molecular Orbital (HOMO): The highest-energy molecular orbital in the energy ground state of a molecule occupied by at least one electron. ► Lowest Unoccupied Molecular Orbital (LUMO): The lowest-energy molecular orbital that is unoccupied in the ground state. F7360 Characterization of thin films and surfaces: 2.2.4 Metallic Bonds ka Zajíčková 42/57 H2 molecule - two 1s electrons with opposite spins (maximum electrons in K shell) saturated covalent bonds Li2 (6 unfilled 2p states with energy similar to 2s): Li + Li2 —>> Li3 (without violating exclusion principle - all valence electrons remain in L shells) unsaturated covalent bonds Li forms bcc crystals (8 nearest neighbors, i. e. each bond = 1/4 of electron instead 2 for covalent bond) - electrons participating in unsaturated bonds are not localized. ► Cloud of free electrons - atoms "lose" outermost, s or p, electrons while the positively charged ions are left over. In transition metals (partially filled d-shells under the outermost shell) further electrons may participate in metallic bonding. Energy 4 LUMO Excitation HOMO a 3 u O C Energy Conduction Band Valence Band Insulator Metal Semiconductor a 3 w S F7360 Characterization of thin films and surfaces: 2.2.4 Metallic Bonds Lenka Zajíčková 43/57 Electrons in metals Quantum mechanically solved by the one electron approximation (precise only if electrons do not interact) and ► approximation of free electrons - can correctly explain many properties of metals, such as specific heat, thermal conductivity, electrical conductance. ► approximation of weakly bound electrons - explain other important phenomena such as the difference between metals, insulators and metalloid, the relationship between conductivity and valence electrons in the metal System of electrons - ideal gas of fermions (max 1 particle in a given state). distribution function n(e) Fermi-Dirac distribution function fFD (for 7 = 0 step-wise Heavyside function) 1.0 0.9- 0.6 --k-T-jifldQ ■ '— kT=íí/10 — kT=p/2 frf - 2 3 electrochemical potential Fermi energy energy distribution of states g(e) n(e)de = fFD g(t)de f _ i 'FD — eF =/z(7 = 0) c7(e)de=^(2A77)3/V/2d( F7360 Characterization of thin films and surfaces: 2.2.4 Metallic Bonds Lenka Zajíčková 44/57 Electrons in metals (contin.) electrochemical potential is given by the normalization condition (N electrons) For highly degenerated gas (low T) its temperature dependence can be approximated as oo 0 ji = ep I 1 — 7T2 ÍWT 12 \ eF The solution for Schrodinger equation in electric field for periodic ionic crystals shows the existence of a separate area of the energy bands - forbidden band (band gap). The position of Fermi level (electrochemical potential) with respect to band gap is important for behaviour of materials. F7360 Characterization of thin films and surfaces: 2.2.5 Summary of Bonds in Solids ka Zajíčková 45/57 Property Ionic ('ovalení Met allic Ván der Waals Mechanical Thermal Electrical Opt ical Non-directional: St rue hires of high coordinal ion Strong, hard • i \ si als High melt ing IK)int. low expansion coellicient Weak insulator, conduct ion by ion transport when liquid Absorption and other properties mainly of t he individual ions Direcional; St met ures of low coordination and low density St rong. hard crystals High melting point, low expansion coefficient Insulator in solid and liquid st ate High refractive index, absorption different in solid and gas Non-direct ional; Struct ures of high eoordinat ion and high density Variable crystals Range of melting point8 extended liquidus range Conduction by elect ron t ransport Opaque, with similar properties in liquid st ate Analogous to metallic bonds Weak, soft crystals Low melt ing point large expansion coefficient Insulator Propert ies of individual molecules F7360 Characterization of thin films and surfaces: 2.3 Types of Materials ka Zajíčková 46/57 Classification of materials based on nature and applications by Bever (1986): by nature: ceramics, glasses, metals and alloys, other inorganic materials, polymers, elastomers, fibres, composite materials, wood, paper and paperboard, other biological materials ► by application: electrical materials, electronic materials, superconductors, magnetic materials, materials for nuclear applications, materials for other energetic applications, optical materials, biomaterials, building materials, materials for textile and packaging industry (modified) M. B. Bever (ed.): Encyclopedia of Materials, Science and Engineering, 1986, sv. 1 ed. R. W. Cahn (Oxford: Pergamon) other references - material science conferences ► Spring and Fall Meetings of Material Research Society (MRS) in U.S. ► Spring and Fall Meeting of European Material Research Society (EMRS) ► TechCon of Society of Vacuum Coaters (SVC) F7360 Characterization of thin films and surfaces: 2.3 Types of Materials ka Zajickova 47 / 57 eramic Materials A combination of one or more metals with a non-metallic element (usually oxygen but others include nitrogen, carbon ...). ► May be crystalline or partially crystalline. ► The atoms are linked by ionic/covalent bonds - ionic bond character occurs especially for oxygen that effectively borrows two electrons from the neighbouring metal atoms Types of ceramics ► traditional ceramic materials: natural stone, clay minerals such as kaolinite ► modern ceramic materials, classified as advanced ceramics: aluminium oxide (alumina), silicon carbide, tungsten carbide, ... Ceramic Si3N4 bearing Fine ceramic components parts from alumina F7360 Characterization of thin films and surfaces: 2.3 Types of Materials ka Zajíčková 48/57 Ceramic Materials (contin. Physical and mechanical properties are controlled by the crystal structure and chemical composition. It can be demonstrated for Si02 silicon-oxygen tetrahedron silicon atom 4 oxygen atoms layered minerals as mica sheet of tetratierira 3D structure of quartz JLJLjLjjLJL fibrous asbestos single chain of tetrahedra [001] Pyroxene (Enstatlt) -> [010] Wollastonit F7360 Characterization of thin films and surfaces: 2.3 Types of Materials Lenka Zajíčková 49/57 Ceramic Materials (contin.) Ceramic materials ► are brittle, hard, strong in compression, weak in shearing and tension. ► withstand, in many cases, erosion that occurs in an acidic or basic environment. ► withstand very high temperatures such as temperatures that range from 1000 °C to 1600 °C, exceptions include inorganic materials that do not have oxygen such as silicon carbide. Crystal lattice imperfections (vacancies, dislocations) and microstructural defects (inclusions, pores, voids and distribution of irregular size grain) influence the properties ► mechanical failure occurs from pre-existing flaws - high mechanical stresses which exceed the local tensile strength effect crack propagation from flaws followed by rupture ► deffect is weak point for electrical load and aggressive environment ► Class of materials that does not crystallise when cooled from the molten state, no long range periodicity ► The major constituents of glasses are in two separated regions of the periodic table ► Group VI (O, Si, Se and Te) plus some neighbouring elements (B) and ► Groups I and II that are used primarily as fluxes. The addition of fluxing atoms such as sodium reduces the number of bond cross links. Calcium Sodium Oxygen Silicon F7360 Characterization of thin films and surfaces: 2.3 Types of Materials Len ka Zajickova 51/57 Metals an d Alloys Properties of metal and alloys are a consequence of the metallic bonds. They ► have good mechanical strength, high thermal and electrical conductivity, ► are opaque, lustrous and relatively heavy, ► are easily fabricated and shaped. In general, they form one of the face centred cubic (fee), body centred cubic (bec) or hexagonal close packed (hep) structures. Changes in the strength of metallic bond cause differences in optical, electrical, thermal and mechanical properties. The overall mechanical properties of metals and alloys are controlled by the crystal lattice defects, such as dislocations and vacancies. Mechanical and chemical properties can be modified by the addition of alloying elements in varying proportion. F7360 Characterization of thin films and surfaces: 2.3 Types of Materials ka Zajíčková 52/57 Polymers are by definition materials composed of long-chain molecules, typically 10 to 20 nm, that have been developed as a consequence of the linking of many smaller molecules, monomers. H H H H H H H H I I 1 1 1 1 1 1 C = C + C= C —^- —C —C— C- C l I 1 1 1 l l 1 H H H H H H H H Ethylene mer units by opening of double bonds j j j j j * j\j j \^ Cartoon Polyetby lene C hai n Hy d rog en The combination of tensile strength and flexibility make these materials attractive. ~7~-~L. polymer cross-linking F7360 Characterization of thin films and surfaces: 2.3 Types of Materials Lenka Zajíčková 53/57 If the molecular chains are packed side by side, the molecules form an array with a crystalline structure. Natural polymers have complex microstructure comprising a mixture of crystalline and amorphous material. The interatomic bonds between molecular chains are the weak van der Waals forces, but in the crystalline structures, the chains are closer =^ more rigid material. CHjOH ch2oh • beta-glucose Cellulose OH Non ivduciiiĽ mil Jn.; Reducing ond R-C1-to-C4 bonds hydrogen bonds To develop stronger, more rigid, polymers: 1. production of a crystalline structure (polyethylene, nylon), 2. formation of a strong covalent bond between the molecular chains by cross linking (vulcanising raw rubber by heating with the controlled addition of sulphur atoms). F7360 Characterization of thin fill ms and surfaces: 2.3 Types of Materials ka Zajickova 54 / 57 I composites A composite material ► was originally considered to be a combination of two materials ► now it is regarded as any combination of various materials or their polymorphs. Composites have particular physical, mechanical and other properties that are not found in their constituents: ► natural composites: wood - cellulose fibres provide tensile strength and flexibility and lignin provides the matrix for binding and adds the property of stiffness; bone - strong, but soft, protein collagen and the hard, brittle mineral apatite, ► synthetic composites: combining individual properties such as strong fibres of a material (for example carbon) in a soft matrix (such as an epoxy resin). The concept of composite materials has led to the design and manufacture of a new range of structural materials that are generally lighter, stiffer and stronger than anything previously manufactured. F7360 Characterization of thin films and surfaces: 2.3 Types of Materials ka Zajickova 55 / 57 1 composites (contin. Exnuded Minenal/Potymer Composit* Billflt Finished Profile Pulled Through Die (b) Intercalated Nanocomposite (a) Exfoliated Nanocomposite F7360 Characterization of thin films and surfaces: 2.3 Types of Materials ka Zajickova 56 / 57 1 Composites (contin. Alloys can have properties superior to each component 5000 NDC Multilayer structures can combine properties of different compounds 0 20 40 60 AO 100 WOt % Figure 12-5. Mitrohardncss of rmied car hides due to solid solution and precipitation hardening (From Ret*. 5). M. Ohring, The Materials Science of Thin Films SJE-M. k3W Flguri 134. SEM imj^cv uf CVD muJlilavrr iijiijn£i fur LuCjnu !ix>l irtytru. (a) Ctrbidc whrtrawr/TiC/TiCN^TiN (5500 x ). (0 H—» c o CL ~o c o -Q c o -Q i_ (0 o 2 4 6 8 10 12 14 16 18 20 22 oxygen percentage [%] Changes of carbon bonding (by XPS) after plasma funct. in low pressure RF discharges =>- significantly improved hardness and elastic modulus of polyurethane/CNTs composites for optimum plasma conditions (Ar/H20, 02/C2H60) L. Zajíčková et al. Plasma Process. Polym. 6 (2009) S864-S869 L. Zajíčková et al. Thin Solid Films 538 (2013) 7-15