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
ƒ 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