1. Polytetrafluoroethylene
Metals and insulators
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
Potassium
2  10-6
10-10

2. Metals, insulators & semiconductors?
1020-
At low temperatures all materials are insulators or metals.
Diamond
1010-
Resistivity (Ωm)
Germanium
Pure metals: resistivity increases rapidly with increasing temperature.
100 -
Copper
10-10-
100
200
300
Temperature (K)
Semiconductors: resistivity decreases rapidly with increasing temperature.
Semiconductors have resistivities intermediate between metals and insulators at room temperature.

3. Bound States in atoms V(r) r
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
V(r) E2 E1 E0 r
Increasing Binding Energy

4. Bound and “free” states in solids
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.
V(r) E2 E1 E0
V(r)
Solid
V(r) lower in solid (work function). r + R
Nuclear positions

5. Energy Levels and Bands
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 partially filled bands conduct
Band of allowed energy states. E
Electron level similar to that of an isolated atom + + + + +
position

6. Reasonable for “simple metals” (Alkali Li,Na,K,Cs,Rb)
Band Theory
The calculation of the allowed electron states in a solid is referred to as band theory or band structure theory.
Free electron model:
U(r)
U(r)
Neglect periodic potential & scattering (Pauli)
Reasonable for “simple metals” (Alkali Li,Na,K,Cs,Rb)

7. Energy band theory
Solid state
N~1023 atoms/cm3
2 atoms
6 atoms

8. Metal – energy band theory

9. Band theory ctd.
To obtain the full band structure, we need to solve Schrödinger’s equation for the full lattice potential. This cannot be done exactly and various approximation schemes are used. We will introduce two very different models, the nearly free electron and tight binding models.
We will continue to treat the electrons as independent, i.e. neglect the electron-electron interaction.

10. Influence of the lattice periodicity
In the free electron model, the allowed energy states are
where for periodic boundary conditions
nx , ny and ny positive or negative integers.
L- crystal dimension
Periodic potential
Exact form of potential is complicated
Has property V(r+ R) = V(r) where
R = m1a + m2b + m3c
where m1, m2, m3 are integers and a ,b ,c are the primitive lattice vectors. E

11. Tight Binding Approximation
Tight Binding Model: construct wavefunction as a linear combination of atomic orbitals of the atoms comprising the crystal.
Where f(r) is a wavefunction of the isolated atom
rj are the positions of the atom in the crystal.

12. The tight binding approximation for s states
Solution leads to the E(k) dependence!!
1D: + a
Nuclear positions

13. E(k) for a 3D lattice Simple cubic: nearest neighbour atoms at
So E(k) = - a -2g(coskxa + coskya + coskza)
Minimum E(k) = - a -6g
for kx=ky=kz=0
Maximum E(k) = - a +6g
for kx=ky=kz=+/-p/2
Bandwidth = Emav- Emin = 12g
For k << p/a
cos(kxx) ~ 1- (kxx)2/2 etc.
E(k) ~ constant + (ak)2g/2
c.f. E = (hk)2/me
k [111] direction
p/a
-p/a
= 10
g = 1
E(k)
Behave like free electrons with “effective mass” h/a2g

14. Each atomic orbital leads to a band of allowed states in the solid
Gap: no allowed states

15. Reduced Brillouin zone scheme
The only independent values of k are those in the first Brillouin zone.
Discard for
|k| > p/a
Results of tight binding calculation

16. The number of states in a band
Independent k-states in the first Brillouin zone, i.e. kx < /a etc.
Finite crystal: only discrete k-states allowed
Monatomic simple cubic crystal, lattice constant a, and volume V.
One allowed k state per volume (2)3/V in k-space.
Volume of first BZ is (2/a)3
Total number of allowed k-states in a band is therefore
Precisely N allowed k-states i.e. 2N electron states (Pauli) per band
This result is true for any lattice:
each primitive unit cell contributes exactly one k-state to each band.

17. Metals and insulators
In full band containing 2N electrons all states within the first B. Z. are occupied. The sum of all the k-vectors in the band = 0.
A partially filled band can carry current, a filled band cannot
Insulators have an even integer number
of electrons per primitive unit cell.
With an even number of electrons per
unit cell can still have metallic behaviour
due to band overlap.
Overlap in energy need not occur
in the same k direction EF
Metal due to overlapping bands

18. EF
Full Band
Empty Band
Energy Gap
Part Filled Band
Full Band
Partially Filled Band
Energy Gap EF
INSULATOR METAL METAL
or SEMICONDUCTOR or SEMI-METAL

19. Insulator -energy band theory

20. Covalent bonding Atoms in group III, IV,V,&VI tend to form
Filling factor
T. :0.34
F.C.C :0.74

21. 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 are
anti-parallel )
Example: Carbon 1S2 2S2 2p2 C C 2D 3D
Diamond: tetrahedron,
cohesive energy 7.3eV

22. 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)

23. diamond

24. semiconductors

25. Intrinsic conductivity
ln(s)
1/T

26. Extrinsic conductivity – n – type semiconductor
ln(s)
1/T

27. Extrinsic conductivity – p – type semiconductor

28. Conductivity vs temperature
ln(s)
1/T