Solid State Theory Physics 545
Structure of Matter • Properties of materials are a function of their: Atomic Structure Bonding Structure Crystal Structure Imperfections • If these various structures are known then the properties of the material can be determined. • We can tailor these properties to achieve the material needed for a particular product.
¾ Properties of a material z
z
Depends on bonding of atoms that form the material Dictates how it interacts and responds to the world around it. These include: • Physical Properties z
Form (Gas, Liquid, Solid) ,Hardness (Rigid, Ductile, Strength), Reactivity
• Thermal Properties z
Heat capacity, Thermal conductivity,
• Electrical Properties z
Conductivity, Dielectric Strength, Polarizability
• Magnetic Properties z
Magnetic permittivity,
• Optical Properties z
Spectral absorption, Refractive index, Birefringence, Polarization,
Atomic Theory • Nucleus of the atom is made up of protons (+) and neutrons (-) • Number of electrons surrounding the nucleus must equal the number of protons (free state-electrically neutral). • Atomic Number - the number of protons in the nucleus • Atomic Number determines the properties and characteristics of materials
Atomic Structures And Atomic Bonding • gives chemical identification Nucleus • consists of protons and neutrons • # of protons = atomic number • # of neutrons gives isotope number Electrons • participate in chemical bounding • described by orbitals
Electron Configurations Valence electrons: occupy the outermost filled shell. Na+
22s2in 63s electron the 3rd1 Example: Sodium atom:Only Na:1 1s 2p
shell, it is readily released.
+11e
Once this electron is released, it becomes Sodium ion (Na+). Cation: positive charge Anion: negative charge
Electron Behavior • Elements cannot be fully explained by nucleus alone requires understanding electron behavior as well. • When shells are filled, the atom is stable. • Electrons in unfilled shells are know as valence electrons. • Valence electrons are largely responsible for element behavior. • Partially filled shells mean that electrons may be given up, accepted from other atoms, or may share them with other atoms. • Manner in which this stabilization occurs determines the type of bonding.
Bonding Forces and Energies FA
FR
r
FN = FA + FR Where FN: Net force between the two atoms FA: Attractive force FR: Repulsive force
Bonding Energy Energy and force relation: E = ∫ Fdr r
E N = ∫ FN dr ∞
=
When
∫
r
∞
r
FA dr + ∫ FR dr ∞
= E A + ER → Bonding energy
dE N = 0 , E N = E0 dr
Bonding Forces And Energies FA + FR = 0
r0 r
E N = ∫ ( FA + FR )dr = E A + E R ∞
Primary bondings: Ionic, Covalent, Metallic bonds
Secondary bondings: Van der Waals bond, Hydrogen bond
Classification of Bonds Primary bonds Ionic Bonding Covalent Bonding Metallic Bonding Secondary bonds van der Waals bonds Hydrogen bonds
•
Van de Waals
Bonds
– Low temperature(~0K), Noble gases (ie Ar, He, Kr, Ne) – Electrostatic attraction due to electron orbit variations – results in charge polarization of atoms – Weak bond
•
•
•
Ionic – Unequal sharing of outer electrons – eg NaCl Na loses electron to Cl -> ions Na+, Cl– Bond due to net electrostatic attraction between ions – Electrons tightly bound to atoms Covalent – Outer valance electrons shared equally between atoms – Strong bond, electrons tightly bound to atoms – No net charge - No Electrostatic forces – Eg C, Ge, Si Metallic – Variation of Covalent – Valance electrons stripped from nucleus – Shared equally between all atoms as sea of community e’s – Bonding due to electrostatic attraction between sea of -ve electrons and sea of +ve nuclei – High thermal and electrical conductivity due to free electron gas – Eg: Na, Cu, Al, Mg, Fe
Van Der Waals Bond
• Formed when an atom or Hydrogen bonds: permanent dipole bonds molecule is asymmetric, creating a net polar moment in the charges. • The bond is weak and is found in neutral atoms such as inert gases. •No electron transfer or Van der Waals bonds: sharing Based upon the fluctuating dipole bonds, attraction of dipoles Ar Ar •Bonding energy: ~0.01 eV (weak) •Compared to thermal vibration energy kBT ~ 0.026 eV at T = 300 K •Examples: inert gases
+ Ar -
+ Ar -
Dipole-dipole interaction
Secondary Atomic & Molecular Bonds [Van der Waals Bonds] Permanent Dipole Bonds • Weak intermolecular bonds are formed between molecules which possess permanent dipoles. • A dipole exists in a molecule if there is asymmetry in its electron density distribution.
Fluctuating Dipole Bonds • Weak electric dipole bonding can take place among atoms due to an instantaneous asymmetrical distribution of electron densities around their nuclei. • This type of bonding is termed fluctuation since the electron density is continuously changing.
Van der Waals Bonding Energy of the van der Waals bond
U=
A r6
B + rn
(n ≅ 12)
3 ways to lead to a rather weak bonding
- Fluctuating induced dipole bonds - Polar molecule-induced dipole bonds - Permanent dipole bonds Pauli exclusion principle:
Ionic Bond • When elements donate or receive an electrons in its outer shell a charged particle or an ION is formed. • If the element gives up an electron, it is then left with at net + 1 charge, and is called a POSITIVE ION. • Charged particles are attracted to each other.
Ionic Bonding NaCl
Formation of ionic bond 1. Mutual ionization occurs by electron transfer 2. Ions are attracted by coulombic forces
EA = − A / r ER = B / r n , n ~ 8
~ 640 KJ/mole or 3.3eV/atom, Tm~ 801oC Bonding energy: 1-10 eV (strong)
An ionic bond is non-directional (Ions can be attracted to one another in any direction)
Ionic Bonding Some aspects to remember: 1. Electronegative atoms will generally gain enough electrons to fill their valence shell and more electropositive atoms will lose enough electrons to empty their valence shell. e.g.
Na: [Ne]3s1 → Na+: [Ne] Cl: [Ne]3s2 3p5 → Cl-: [Ar]
Ca: [Ar]4s2 → Ca+2: [Ar] O: [He]2s2 2p4 → O-2: [Ne]
2. Ions are considered to be spherical and their size is given by the ionic radii that have been defined for most elements (there is a table in the notes on Atomic Structure). The structures of the salts formed from ions is based on the close packing of spheres. 3. The cations and anions are held together by electrostatic attraction.
Ionic Bonding Because electrostatic attraction is not directional in the same way as is covalent bonding, there are many more possible structural types. However, in the solid state, all ionic structures are based on infinite lattices of cations and anions. There are some important classes that are common and that you should be able to identify, including:
NaCl
CsCl
Fluorite
Zinc Blende
Wurtzite
And others…Fortunately, we can use the size of the ions to find out what kind of structure an ionic solid should adopt and we will use the structural arrangement to determine the energy that holds the solid together - the crystal lattice energy, U0.
Ionic Bonding The “cation” has a + charge & the “anion” has the - charge. The cation is much smaller than the anion.
Ionic Bonding Most ionic (and metal) structures are based on the “close packing” of spheres - meaning that the spheres are packed together so as to leave as little empty space as possible - this is because nature tries to avoid empty space. The two most common close packed arrangements are hexagonal close-packed (hcp) and cubic close packed (ccp). Both of these arrangements are composed of layers of close packed spheres however hcp differs from ccp in how the layers repeat (ABA vs. ABC). In both cases, the spheres occupy 74% of the available space. Because anions are usually bigger than cations, it is generally the anions that dominate the packing arrangement.
hcp
ccp
Usually, the smaller cations will be found in the holes in the anionic lattice, which are named after the local symmetry of the hole (i.e. six equivalent anions around the hole makes it octahedral, four equivalent anions makes the hole tetrahedral).
Ionic Bonding Some common arrangements for simple ionic salts:
Cesium chloride structure 8:8 coordination Primitive Cubic (52% filled) e.g. CsCl, CsBr, CsI, CaS
Rock Salt structure 6:6 coordination Face-centered cubic (fcc) e.g. NaCl, LiCl, MgO, AgCl
Zinc Blende structure 4:4 coordination fcc e.g. ZnS, CuCl, GaP, InAs
Wurtzite structure 4:4 coordination hcp e.g. ZnS, AlN, SiC, BeO
Ionic Bonding Fluorite structure 4:8 coordination fcc e.g. CaF2, BaCl2, UO2, SrF2
Rutile structure 6:3 coordination Body-centered cubic (bcc) (68% filled) e.g. TiO2, GeO2, SnO2, NiF2 Nickel arsenide structure 6:6 coordination hcp e.g. NiAs, NiS, FeS, PtSn
Anti-fluorite structure 8:4 coordination e.g. Li2O, Na2Se, K2S, Na2S You can determine empirical formula for a structure by counting the atoms and partial atoms within the boundary of the unit cell (the box). E.g. in the rutile structure, two of the O ions (green) are fully within the box and there are four half atoms on the faces for a total of 4 O ions. Ti (orange) one ion is completely in the box and there are 8 eighth ions at the corners; this gives a total of 2 Ti ions in the cell. This means the empirical formula is TiO2; the 6:3 ratio is determined by looking at the number of closest neighbours around each cation and anion.
There are many other common forms of ionic structures but it is more important to be able to understand the reason that a salt adopts the particular structure that it does and to be able to predict the type structure a salt might have.
Ionic Bonding The ratio of the radii of the ions in a salt can allow us to predict the type of arrangement that will be adopted. The underlying theory can be attributed to the problem of trying to pack spheres of different sizes together while leaving the least amount of empty space. The maximum possible size of a cation (the smaller sphere) that can fit into the hole between close packed anions (the larger spheres) can be calculated using simple geometry. The ratio of the radii can thus suggest the coordination number of the ions which can then be used to predict the structural arrangement of the salt. E.g. for a 3-coordinate arrangement where A is at the center of the hole (of radius r+) and B is at the center of the large sphere (of radius r-), one can define the right triangle ABC where the angle CAB must be 60°. Sin(60°) = 0.866 = BC/AB = r-/r++rA C
0.866 (r++r-) = r-
0.866 r+ + 0.866 r- = r-
0.866 r+ = r- - 0.866 r-
0.866 r+ = (1 - 0.866) r- = (0.134) r-
B
So: r+/r- = 0.134/0.866 = 0.155 This means that the largest cation that will fit in the hole can only have a radius that is 15.5% of the radii of the anions. Coordination number 3
4
6
r+/r-
0.155
0.255 0.414 0.732
structure
covalent ZnS
NaCl
8 CsCl
Limiting and optimal radius ratios for specific coordinations not „in touch“
in touch
optimal radius ratio for given CN („ions are in touch“): CN 8 6 4 3
ratio > 0.7 0.4 – 0.7 0.2 – 0.4 0.1 – 0.2
Variation of ionic radii with coordination number
The radius of one ion has to be fixed to a reasonable value (r(O2-) = 140 pm) → Linus Pauling. That value is then used to compile a set of self consistent values for all other ions.
Ionic Bonding The energy that holds the arrangement of ions together is called the lattice energy, Uo, and this may be determined experimentally or calculated. Uo is a measure of the energy released as the gas phase ions are assembled into a crystalline lattice. A lattice energy must always be exothermic. E.g.: Na+(g) + Cl-(g) → NaCl(s) Uo = 788 kJ/mol
Ionic Bonding Coulombic forces: the attractive forces Attractive energy:
EA=
A
where A=
z1z2 q2
r 4πε0 where z1, z2 are the valences of ions, q is the charge of an electron, and ε0 is the permittivity of vacuum (=8.85×10-12 C2/Nm2) Repulsive energy
ER =
B rm
Ionic Bonding The origin of the equations for lattice energies.
U0 = Ecoul + Erep The lattice energy U0 is composed of both coulombic (electrostatic) energies and an additional close-range repulsion term - there is some repulsion even between cations and anions because of the electrons on these ions. Let us first consider the coulombic energy term: For an Infinite Chain of Alternating Cations and Anions: In this case the energy of coulombic forces (electrostatic attraction and repulsion) are: Ecoul = (e2 / 4 π ε0) * (zA zB / d) * [+2(1/1) - 2(1/2) + 2(1/3) - 2(1/4) + ....] because for any given ion, the two adjacent ions are each a distance of d away, the next two ions are 2×d, then 3×d, then 4×d etc. The series in the square brackets can be summarized to give the expression: Ecoul = (e2 / 4 π ε0) * (zA zB / d) * (2 ln 2) where (2 ln 2) is a geometric factor that is adeqate for describing the 1-D nature of the infinite alternating chain of cations and anions.
Ionic Bonding For a 3-dimensional arrangement, the geometric factor will be different for each different arrangement of ions. For example, in a NaCl-type structure:
Ecoul = (e2 / 4 π ε0) * (zA zB / d) * [6(1/1) - 12(1/√2) + 8(1/√3) - 6(1/√4) + 24(1/√5) ....] The geometric factor in the square brackets only works for the NaCl-type structure, but people have calculated these series for a large number of different types of structures and the value of the series for a given structural type is given by the Madelung constant, A. This means that the general equation of coulombic energy for any 3-D ionic solids is: Ecoul = (e2 / 4 π ε0) * (zA zB / d) * A Note that the value of Ecoul must be negative for a stable crystal lattice.
Calculation of the Madelung constant Cl
Na
typical for 3D ionic solids: Coulomb attraction and repulsion
Madelung constants: CsCl: 1.763 NaCl: 1.748 ZnS: 1.641 (wurtzite) ZnS: 1.638 (sphalerite) ion pair: 1.0000 (!)
12 8 6 24 A= 6− + − + ... = 1.748... (NaCl) 2 3 2 5 (infinite summation)
2. Born repulsion (VBorn)
(Repulsion arising from overlap of electron clouds, atoms do not behave as point charges)
Because the electron density of atoms decreases exponentially towards zero at large distances from the nucleus the Born repulsion shows the same behaviour
VAB
r0 r
approximation:
V Born = VAB
B
r
n
B and n are constants for a given atom type; n can be derived from compressibility measurements (~8)
Ionic Bonding The numerical values of Madelung constants for a variety of different structures are listed in the following table. CN is the coordination number (cation,anion) and n is the total number of ions in the empirical formula e.g. in fluorite (CaF2) there is one cation and two anions so n = 1 + 2 = 3. lattice
A
CN
stoich
A/n
CsCl
1.763
(8,8)
AB
0.882
NaCl
1.748
(6,6)
AB
0.874
Zinc blende
1.638
(4,4)
AB
0.819
wurtzite
1.641
(4,4)
AB
0.821
fluorite
2.519
(8,4)
AB2
0.840
rutile
2.408
(6,3)
AB2
0.803
CdI2
2.355
(6,3)
AB2
0.785
Al2O3
4.172
(6,4)
A2B3
0.834
Notice that the value of A is fairly constant for each given stoichiometry and that the value of A/n is very similar regardless of the type of lattice.
Covalent Bonding • Occurs when valence electrons are shared • Form between elements that have too many or require too many electrons for Ionic Bond to form. •Covalent bonds form in compounds composed of electronegative elements, especially those with 4 or more valence electrons • The nuclei is POSITIVE (+), therefore, if electrons (-) are shared by adjacent nuclei, the result is a VERY strong bond.
The atoms share their outer s and p electrons so that each atom attains the noblegas electron configuration.
Covalent bond • Two atoms share a pair of electrons • Bonding energy: ~1-10 eV (strong) • Examples: C, Ge, Si, H2
A covalent bond is directional (Bonds form in the direction of greatest orbital overlap)
H + H
H H
C
C C
C
C
Covalent Bonding in Carbon A carbon atom can form form sp3 orbitals directed symmetrically toward the corners of a tetrahedron. [Note the examples below.]
Bond angle: 109.5o
Metallic Bond • Metallic elements – have only 1, 2, or 3 electrons in their outer shell. • Since fewer electrons, bond is relative loose to the nucleus. • When valence electrons approach adjacent atoms orbit, electrons may be "forced out of natural orbit". • Results in positive ions being formed. • These floating electrons form a "cloud" of shared valence electrons, and electron movement can occur freely. •Solids composed primarily of electropositive elements containing 3 or fewer valence electrons are generally held together by metallic bonds.
Metallic Bond Large interatomic forces are created by the sharing of electrons in a delocalized manner to form strong nondirectional bonding. Positive ions in a sea of electrons Na+
Na+
Electron sea
Na+ Na+
Na+
•Bonding energy: ~1-10 eV (strong)
• It is convenient for many purposes to regard an atom in a metal as having a definite size, which may be defined by the distance between its center and that of its neighbor. • This distance is that at which the various forces acting on the atom are in equilibrium. • In a metal, the forces can be considered as – (a) the attractive forces between electrons & positive ions, – (b) the repulsion between the complete electron shells of the positive ions, & – (c) the repulsion between the positive ions as a result of their similar positive charges.
The Hard Sphere Model • This approach can be called the "hard sphere" model of an atom, however the radius of an atom (or ion) determined for a particular crystal structure is not a real characteristic of that atom, because when the same atom appears in different crystal structure it displays different radii. • The radius of an atom (or ion) can be determined for a particular metal by using the dimensions of the unit cell of the crystal structure it forms.
Different types of atomic radii
(!! atoms can be treated as hard spheres !!)
element or compounds
compounds only
elements or compounds („alloys“)
Mixed Bonding Metallic-Covalent Mixed Bonding: The Transition Metals are an example where dsp bonding orbitals lead to high melting points. Ionic-Covalent Mixed Bonding: Many oxides and nitrides are examples of this kind of bonding.
Influence of Bond Type on engineering Properties Ductility
metallic bonds → atoms are easy to slip → ductile ionic bonds → difficult to slip → brittle Conductivity metallic bonds → free valence electrons → good electrical and thermal conductivity ionic and covalent bonds → no free valence electrons → good electrical and thermal insulators
Summary: Bonding Directional Bonds Covalent Permanent dipole
Non-Directional Bonds Metallic Ionic Fluctuating dipole
Metallic
M
Examples: Metals: Metallic bonding Secondary Ceramics: Ionic/covalent Polymers: Covalent and secondary Semiconductors: Covalent or covalent/ionic
Ionic
C P
S Covalent
Bonding Energy vs. Inter-atomic Distance Energy Parabolic Potential of Harmonic Oscillator ro
Distance
1-D Array of Spring Mass System
Eb
Spring constant, g Equilibrium Position Deformed Position
Mass, m
a
x n-1
xn
x n+1
