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Lecture VII
Why Solids?
most elements are solid at room temperature
atoms in ~fixed position
“simple” case - crystalline solid
Crystal Structure
Why study crystal structures?
description of solid
comparison with other similar materials - classification
correlation with physical properties
Metals, insulators & semiconductors?
At low temperatures all materials are insulators or metals.
Semiconductors: resistivity decreases rapidly with increasing temperature. Semiconductors have resistivities intermediate between metals and insulators at room temperature.
Pure metals: resistivity increases rapidly with increasing temperature.
1020-
1010-
100 -
10-10-R
esis
tivi
ty (
Ωm
)
100 200 3000Temperature (K)
Diamond
Germanium
Copper
Measured resistivities range over more than 30 orders of magnitude
Material Resistivity (Ωm) (295K)
Resistivity (Ωm) (4K)
10-12
“Pure”Metals
Copper
10-5
Semi-Conductors
Ge (pure) 5 102 1012
Insulators Diamond 1014
Polytetrafluoroethylene (P.T.F.E)
1020
1014
1020
Potassium
2 10-6 10-10
Metals and insulators
Early ideas• Crystals are solid - but solids are not
necessarily crystalline• Crystals have symmetry (Kepler) and long
range order• Spheres and small shapes can be packed to
produce regular shapes (Hooke, Hauy)
?
Kepler wondered why snowflakes have 6 corners, never 5 or 7. By considering the packing of polygons in 2 dimensions, demonstrate why pentagons and heptagons shouldn’t occur.
Empty space not allowed
CRYSTAL TYPES
Three types of solids, classified according to atomic arrangement: (a) crystalline and (b) amorphous materials are illustrated by microscopic views of the atoms, whereas (c) polycrystalline structure is illustrated by a more macroscopic view of adjacent single-crystalline regions, such as (a).
quartz
Crystal structure
Amorphous structure
Definitions1. The unit cell
“The smallest repeat unit of a crystal structure, in 3D, which shows the full symmetry of the structure”
The unit cell is a box with:
• 3 sides - a, b, c
• 3 angles - , ,
14 possible crystal structures (Bravais lattices)
3D crystal lattice
cubica = b = c = =
tetragonala = b c = = = 90o
monoclinica b c = = 90o
90o
orthorhombica b c = = = 90o
hexagonala = b c = = 90o; = 120o
triclinica b c 90o
trigonal (rhombohedral)a = b = c = = 90o
Chemical bonding
Types:
Ionic bonding
Covalent bonding
Metallic bonding
Van der Walls bonding + -+ -
Bonding in SolidsMelting point(K)
Molecularcrystals
Metals Ioniccrystals
Covalentcrystals
3000
2000
1000
0
organiccrystals
W(3683)
Mo(2883)
Pt(2034)Fe(1808)
Cu(1336)Al(933)Pb(600)Na(371)Hg(234)
LiF(1143)KCl(1063)
C(<3500)
BN(3270)
SiO2(2001)
Si(1683)Ge(1240)
Electrons in metals
P. Drude: 1900 kinetic gas theory of electrons, classicalMaxwell-Boltzmann distributionindependent electronsfree electronsscattering from ion cores (relaxation time approx.)
A. Sommerfeld: 1928Fermi-Dirac statistics
F. Bloch’s theorem: 1928Bloch electrons
L.D. Landau: 1957Interacting electrons (Fermi liquid theory)
Metallic bond
Atoms in group IA-IIB let electrons to roam ina crystal. Free electrons glue the crystal
Na+ Na+
e-
e-
Attract
Attract
Attract
AttractRepelRepel
Additional binding due to interaction of partially filled d – electron shells takes place in transitional metals: IIIB - VIIIB
Core and Valence Electrons
Simple picture. Metal have CORE electrons that are bound to the nuclei, and VALENCE electrons that can move through the metal.
Most metals are formed from atoms with partially filled atomic orbitals.
e.g. Na, and Cu which have the electronic structure
Na 1s2 2s2 2p6 3s1
Cu 1s2 2s2 2p6 3s23p63d104s1
Insulators are formed from atoms with closed (totally filled) shells e.g. Solid inert gases
He 1s2 Ne 1s2 2s2 2p6
Or form close shells by covalent bonding i.e. Diamond
Note orbital filling in Cu does not follow normal rule
Why are metals good conductors?Consider a metallic Sodium crystal to comprise of a lattice of Na+ ions, containing the 10 electrons which occupy the 1s, 2s and 2p shells, while the 3s valence electrons move throughout the crystal.
The valence electrons form a very dense ‘electron gas’.
_
_
_
_
_ _ __
_
_
__
_
_
__
+ +
+
+
+
+ +
+
+ + +
+
+
+
+ +
+
+
+ + + + + +
Na+ ions:Nucleus plus 10 core electrons
We might expect the negatively charged electrons to interact very strongly with the lattice of negative ions and with each other.
In fact the valence electrons interact weakly with each other & electrons in a perfect lattice are not scattered by the positive ions.
Free classical electrons:AssumptionsWe will first consider a gas of free classical electrons subject to external electric and magnetic fields. Expressions obtained will be useful when considering real conductors
(i) FREE ELECTRONS: The valance electrons are not affected by the electron-ion interaction. That is their dynamical behaviour is as if they are not acted on by any forces internal to the conductor. (ii) NON-INTERACTING ELECTRONS: The valence electrons from a `gas' of non-interacting electrons. They behave as INDEPENDENT ELECTRONS; they do not show any `collective' behaviour. (iii) ELECTRONS ARE CLASSICAL PARTICLES: (iv) ELECTRONS ARE SCATTERED BY DEFECTS IN THE LATTICE: ‘Collisions’ with defects limit the electrical conductivity. This is considered in the relaxation time approximation.
Ohms law and electron drift
V = E/L = IR (Volts)
Resistance R = rL/A (Ohms)
Resistivity r = AR/L (Ohm m)
E = V/L = / rI A = rj (Volts m-1)
Conductivity = 1/ s r (low magnetic field)
j = sE (Amps m-2)
I = dQ/dt (Coulomb s-1)
Area A
dx
vd
denvdt
dxen
dt
dQ
A
1j
L
Area A
Electric field E
Force on electron F
Drift velocity vd
Current density j = I/A
n free electrons per m3 with charge –e ( e = +1.6x10-19 Coulombs )
Force on electrons F = -eE results in a constant electron drift velocity, vd.
Charge in volume element dQ = -enAdx
Electrical ConductivityIn the absence of collisions, the average momentum of free electrons subject to an electric field E would be given by
The rate of change of the momentum due to collisions is
At equilibrium
Now j = -nevd = -nep/me = (ne2tp /me) E So the conductivity is s = j/E = ne2tp /me
Ep
eFdt
d
Field
p/dt
d
Collisions
p
p
Εppp
e- So0dt
d
dt
d
CollisionsField
The electron mobility, , m is defined as the drift velocity per unit applied electric field
m = vd / E = etp /me (units m2V-1s-1)
sodium ion (Na+)
Examples of ionic bonding• Metal atoms with 1 electron to lose can form
ionic bonds with non-metal atoms which need to gain 1 electron:– Eg. sodium reacts with fluorine to form sodium
fluoride:
sodium atom
(Na)
fluoride ion (F-)
fluorine atom
(F)
So the formula for
sodium fluoride is
NaF
CsCl Crystal Structure
• Chloride ions form simple cubes with cesium ions in the center
Examples of ionic bonding
Examples of ionic bonding:NaCl• Each sodium atom is surrounded by
its six nearest neighbor chlorine atoms (and vice versa)
• Electronically – sodium has one electron in its outer shell: [Ne]3s1 and Chlorine has 7 (out of 8 “available” electron positions filled in its outer shell) [Ne]3s23p5
• Sodium “gives up” one of its electrons to the chlorine atom to fill the shells resulting in [Ne] [Ar] cores with Na+ and Cl- ions
• Coulombic attraction with tightly bound electron cores
NaCl
mr
B
r
eU
2
04
1
• Potential energy:
a - Madelung constant, m – integer number
for ro, the equilibrium position between the ions:
mr
eU
11
4 0
2
00
U0 is the cohesive energy, i.e. the energy per ion to remove the ion out of the crystal.
Repulsive potential 1/rm
Attractive potential -1/r
Total potential
Ionic bonding
Properties of the ionic crystals
• medium cohesive energy (2-4 eV/ atom).– low melting and boiling temp. .
• Low electrical conductivity.– (the lack of the free electrons).
• Transparent for VIS light– ( energy separation between neighbouring levels > 3 eV)
• Easily dissolved in water.– Electrical dipoles of water molecules attract the ions
Covalent bonding: molecular orbitalsConsider an electron in the ground, 1s, state of a hydrogen atom
The Hamiltonian is
The expectation value of the electron energy is
This gives <E> = E1s = -13.6eV
o
2
4e = where
RadiusBohr theis a where a 1
= (r) i.e. oo e ar/-3/2 o
r
- 2m
- = H
22
(r)dV H (r) = > E <
+
E1s
V(r)
F(r)
Hydrogen Molecular Ion
Consider the H2+ molecular ion in which
one electron experiences the potential
of two protons. The Hamiltonian is
We approximate the electron wavefunctions as
and
|R - r|-
r -
2m
- = )rU( +
2m
- = H
2222
] + A[ |)] R - r(| + )r([ A = )r( 21
] B[ |)]R - r(| )r([ B = )r( 21
p+ p+
e-
R
r
Bonding andanti-bonding states
Expectation values of the energy are:
E = E1s – g(R) for
E = E1s + g(R) for
g(R) - a positive function
Two atoms: original 1s stateleads to two allowed electron states in molecule.
Find for N atoms in a solid have N allowed energy states
)r(
)r(
)r(
-6 -4 -2 0 2 4 6
-1.4
-1.2
-1.0
-0.8
-0.6
-0.4
-0.2
0.0
0.2
0.4
0.6
0.8
1.0
1.2
r
-6 -4 -2 0 2 4 6
-1.4
-1.2
-1.0
-0.8
-0.6
-0.4
-0.2
0.0
0.2
0.4
0.6
0.8
1.0
1.2
r
V(r)
2)r(
)r(
1s
2s
bonding
Anti-bonding
Anti-bonding
bonding
covalent bonding – H2 molecule
• 8
6
4
2
0
-2
-4
-6
R00.1 0.2 0.3 0.4
nuclear separation (nm)
ener
gy(e
V)
parallel spin
antiparallel spin
system energy (H2)
Covalent bonding
Crystals: C, Si, Ge
Covalent bond is formed by two electrons, one from each atom, localised in the region between the atoms (spins of electrons areanti-parallel )
Example: Carbon 1S2 2S2 2p2
C C
Diamond: tetrahedron, cohesive energy 7.3eV
3D 2D
Covalent Bonding in Silicon
• Silicon [Ne]3s23p2 has four electrons in its outermost shell
• Outer electrons are shared with the surrounding nearest neighbor atoms in a silicon crystalline lattice
• Sharing results from quantum mechanical bonding – same QM state except for paired, opposite spins (+/- ½ ħ)
Covalent bond
Atoms in group III, IV,V,&VI tend to form covalent bond
Filling factor
T. :0.34 F.C.C :0.74
ionic – covalent mixed
diamond lattice
zinc blend crystals (ZnS, GaAs)
As
Properties of the covalent crystals
• Strong, localized bonding.
• High cohesive energy (4-7 eV/atom).– High melting and boiling temperature.
• Low conductivity.
The Hall Effect
An electric field Ex causes a current jx to flow.
Ex, jxEyBz
vd = vx
The Hall coefficient is RH = Ey/jxBz = -1/ne
The Hall resistivity is rH = Ey/jx = -B/ne
jx = -nevx so Ey = -jxBz/ne
Therefore Ey = +vxBz
F = -e (E + v B). In equilibrium jy = 0 so Fy = -e (Ey - vxBz) = 0
A magnetic field Bz produces a Lorentz force in the y-direction on
the electrons. Electrons accumulate on one face and positive charge on the other producing a field Ey .
j
For a general vx.
vx+ve or -ve
The Hall Effect
The Hall coefficient RH = Ey/jxBz = -1/ne
The Hall angle is given by tan f = Ey/Ex = rH/r
For many metals RH is quiet well described by this expression which is useful for obtaining the electron density, in some cases.
However, the value of n obtained differs from the number of valence electrons in most cases and in some cases the Hall coefficient of ordinary metals, like Pb and Zn, is positive seeming to indicate conduction by positive particles!
This is totally inexplicable within the free electron model.
j=jx
Ey
Bzvd = vx
Ex
E
The (Quantum)Free Electron model: Assumptions
(i) FREE ELECTRONS: The valance electrons are not affected by the electron-ion interaction. That is their dynamical behaviour is as if they are not acted on by any forces internal to the conductor. (ii) NON-INTERACTING ELECTRONS: The valence electron from a `gas' of non-interacting electrons. That is they behave as INDEPENDENT ELECTRONS that do not show any `collective' behaviour. (iii) ELECTRONS ARE FERMIONS: The electrons obey Fermi-Dirac statistics. (iv) ‘Collisions’ with imperfections in the lattice limit the electrical conductivity. This is considered in the relaxation time approximation.
Free electron approximation
U(r)U(r)
Neglect periodic potential & scattering (Pauli)
Reasonable for “simple metals” (Alkali Li,Na,K,Cs,Rb)
Eigenstates & energies
dtd
iUm2
22
)L/n,L/n,L/n(2k
km2
E
e)t,r(
zzyyxx
22
k
)rkt(i0k
Ek
|k|
Unit volume in k-space: 1/(2p3)
Density of states
G(E) - this is the number of allowed states within a band:
g(E)dE=2g(k)dk
with g(k) equal to the density of states within the k-space. 2 is due to two possible spin values.
dkdE
kgEg
/
)(2)(
g(k) is equal to the number of states within the space between two spheras of radii k and k+dk, which is equal to the number of states per unit volume (1/(2p)3) multiplied by the volume between the spheras (4pk2dk). Thus :
2
22
3 2)(4
)2(
1)(
kkgdkkdkkg
m
kE
2
22
2
2
2)(
k
kg
mEmmk
k
mk
dkdE
kEg 2
2
2
/
1)(
322222
2
2
2
For:
dkdE
k
dkdE
kgEg
/
1
/
)(2)(
2
2
g(E)
E
Statistics & DOS
Density of states:
mEm
dE
dkkgEg 2)(2)(
32
Fermi-Dirac statistics:
1e
1)E(f
kT
EFD
T=0.1m
0.5 1 2
1
m=EF
g(E)
E
2kBT
Occupation of states
|kF|
FermiSurface
D(E)
E
2kBT
EF
Bound States in atoms
r4
qe = )r(V
o
2
Electrons in isolated atoms occupy discrete allowed energy levels E0, E1, E2 etc. .
The potential energy of an electron a distance r from a positively charge nucleus of charge q is
-8 -6 -4 -2 0 2 4 6 8-5
-4
-3
-2
-1
0
F6 F7 F8 F9
r
V(r)E2
E1
E0
r
0
Increasing Binding Energy
Bound and “free” states in solids
-8 -6 -4 -2 0 2 4 6 8-5
-4
-3
-2
-1
0
F6 F7 F8 F9
r
-8 -6 -4 -2 0 2 4 6 8-5
-4
-3
-2
-1
0
F6 F7 F8 F9
r
-8 -6 -4 -2 0 2 4 6 8-5
-4
-3
-2
-1
0
F6 F7 F8 F9
r
V(r)E2
E1
E0
The 1D potential energy of an electron due to an array of nuclei of charge q separated by a distance R is
Where n = 0, +/-1, +/-2 etc.
This is shown as the black line in the figure.
n o
2
nRr4
qe = )r(V
r
0
0
+ + + + +RNuclear positions
V(r) lower in solid (work function).
Naive picture: lowest binding energy states can become free to move throughout crystal
V(r)Solid
Energy Levels and Bands
+E
+ + + +position Electron level similar to
that of an isolated atom
Band of allowed energy states.
In solids the electron states of tightly bound (high binding energy) electrons are very similar to those of the isolated atoms.
Lower binding electron states become bands of allowed states.
We will find that only partial filled band conduct
Solid stateN~1023 atoms/cm32 atoms 6 atoms
Energy band theory
Metal – energy band theory
Insulator -energy band theory
diamond
semiconductors
Intrinsic conductivity
kTEss
ge2/
0
ln(s)
1/T
1/T
ln(s)
kTEdd
de /0
Extrinsic conductivity – n – type semiconductor
Extrinsic conductivity – p – type semiconductor
Conductivity vs temperature
kTEss
ge2/
0
ln(s)
kTEdd
de /0
1/T