1. Metallic Solids
Metallic bond: The valence electrons are loosely bound. Free valence electrons may be shared by the lattice. The common structures for metals are: fcc, bcc and hcp The cohesive energy can be low, 1-4eV/atom – metals are less strongly bound than ionic or covalent solids

2. Band Theory of Solids
To understand the electronic states of a solid we must consider the presence of many N~1023 atoms
The band theory of solids describes the interaction between the electrons and the lattice ions that comprise a solid

3. Band Theory: “Bound” Electron Approach
For the total number N of atoms in a solid (1023 cm–3), N energy levels split apart within a width E.
Leads to a band of energies for each initial atomic energy level (e.g. 1s energy band for 1s energy level).
Two atoms
Six atoms
Solid of N atoms
Electrons must occupy different energies due to Pauli Exclusion principle.

4. Band Theory of Solids
The task is to compute the quantum states and associated energy levels of this simplified model by solving the Schrödinger equation 1 2 3

5. Band Theory of Solids For periodic potentials, Felix Bloch showed that
the solution of the Schrödinger equation must
be of the form:
and the wavefunction must
reflect the periodicity of
the lattice: 1 2 3

6. Band Theory of Solids By requiring the wavefunction and its derivative
to be continuous everywhere, one finds energy
levels that are grouped into bands separated by
energy gaps. The gaps occur at
The energy gaps
are basically energy
levels that cannot
occur in the solid 1 2 3

7. Band Theory of Solids
Completely free
electron
electron in a
lattice

8. Band Theory of Solids
The theory can explain why some substances are conductors, some insulators and others semi conductors

9. Fermi-Dirac “Filling” Function
Probability of electrons to be found at various energy levels.
For E – EF = 0.05 eV  f(E) = 0.12 For E – EF = 7.5 eV  f(E) = 10 –129
Exponential dependence has HUGE effect!
Temperature dependence of Fermi-Dirac function shown as follows:

10. Conductors, Insulators, Semiconductors
NaCl is an insulator, with a band gap of 2 eV,
which is much larger than the thermal energy at
T=300K
Therefore, only a tiny fraction of electrons are
in the conduction band

11. Conductors, Insulators, Semiconductors
Silicon and germanium have band gaps of 1 eV and
0.7 eV, respectively. At room temperature, a small
fraction of the electrons are in the conduction
band. Si and Ge are intrinsic semiconductors

12. Free Electron Gas in Metals
Solid metals are bonded by the metallic bond
One or two of the valence electrons from each atom are free to move throughout the solid
All atoms share all the electrons. A metal is a lattice of positive ions immersed in a gas of electrons. The binding between the electrons and the lattice is what holds the solid together

13. Free Electron Gas in Metals
The electron has two spin states, so W = 2.
If the electron’s speed << c, we can take its
energy to be
So the number of states in (E, E+dE) is given by

14. Free Electron Gas in Metals
The number of electrons in the interval
E to E+dE is therefore
The first term is the Fermi-Dirac distribution
and the second is the density of states g(E)dE

15. Free Electron Gas in Metals
The total number of electrons N is given by
The average energy of a free electron is given by

16. Free Electron Gas in Metals
At T = 0, the integrals are easy to do.
For example, the total number of electrons is

17. Free Electron Gas in Metals
The average energy of an electron is
This implies
EF = k TF defines
the Fermi
temperature

18. Fermi Energy for T>0
In metals this change is small at room temperature
kT=0.025eV while for most metals EF(0)~ several eV
Change is 1 part in 104

19. Free Electron Gas in Metals
At T > 0, only
the electrons
near the
Fermi energy
can be excited
to higher
energy states

20. Heat Capacity of Electron Gas
Therefore, the total energy can be written as
where a = p2/4
The heat capacity of the
electron gas is predicted to be

21. Heat Capacity of Electron Gas
Consider 1 mole of copper. In this case NAk = R
For copper, TF = 89,000 K. Therefore, even at room temperature, T = 300 K,
the contribution of the electron gas to the heat capacity of copper is
small: CV = R
C=3R for lattice

22. Summary of metallic state
The ions in solids form regular lattices
A metal is a lattice of positive ions immersed in a gas of electrons. All ions share all electrons
The attraction between the electrons and the lattice is called a metallic bond
At T = 0, all energy levels up to the Fermi energy are filled

23. Band Diagram: Insulator
Conduction band
(Empty)
T > 0 EC
Egap EF EV
Valence band
(Filled)
At T = 0, lower valence band is filled with electrons and upper conduction band is empty, leading to zero conductivity.
Fermi energy: EF is at midpoint of large energy gap (2-10 eV) between conduction and valence bands.

24. Band Diagram: Intrinsic Semiconductor
Conduction band
(Partially Filled) EC EF EV
Valence band
(Partially Empty)
At T = 0, lower valence band is filled with electrons and upper conduction band is empty, leading to zero conductivity.
Fermi energy EF is at midpoint of small energy gap (<1 eV) between conduction and valence bands.

25. Donor Dopant in a Semiconductor
For group IV Si, add a group V element to “donate” an electron and make n-type Si (more negative electrons!).
“Extra” electron is weakly bound, with donor energy level ED just below conduction band EC.
Dopant electrons easily promoted to conduction band, increasing electrical conductivity by increasing carrier density n.
Fermi level EF moves up towards EC. EC EV EF ED
Egap~ 1 eV
n-type Si

26. Band Diagram: Acceptor Dopant in Semiconductor
For Si, add a group III element to “accept” an electron and make p-type Si (more positive “holes”).
“Missing” electron results in an extra “hole”, with an acceptor energy level EA just above the valence band EV.
Holes easily formed in valence band, greatly increasing the electrical conductivity.
Fermi level EF moves down towards EV. EA EC EV EF
p-type Si

27. pn Junction: Band Diagram
pn regions “touch” & free carriers move
Due to diffusion, electrons move from n to p-side and holes from p to n-side.
Causes depletion zone at junction where immobile charged ion cores remain.
Results in a built-in electric field (103 to 105 V/cm), which opposes further diffusion.
Note: EF levels are aligned across pn junction under equilibrium.
n-type
electrons EC EF EF EV
holes
p-type
pn regions in equilibrium – – EC – – + – + – EF + + – – – + + + – – + + – + + + EV
Depletion Zone