Download Solid State - Wade Naylor

Solid State
From Chapter 12 of Modern Physics
Contents for Chapter 12
1.
2.
3.
4.
5.
6.
7.
Bonding in Solids
Classical Free Electron
Quantum Theory of Metals
Band Theory of Solids
Semiconductor Devices
Superconductors* (advanced reading)
Lasers (extra reading)
In the beginning …
• Solid state or condensed matter physics is the largest branch of
physics; both experimental and theoretically.
• The solid state has entranced humans ever since we have found
diamonds and gems thousands of years ago.
• What is really interesting is that we really do need quantum
mechanics to correctly model the behaviour of these solids.
• In particular the wave-like nature of the electrons in these solids turns
out to be very important!
12.1 Bonding in Solids
• Ionic Solids:- They form stable hard solids and are poor conductors;
they have a high melting point and are transparent to visible radiation
(in the infra-red they absorb radiation due to lattice vibrations)
• Covalent Bonds:- What properties do we have as compared to ionic
solids?
• Metallic Solids: A free electron sea?
• Molecular Crystals:- Via the Van de Waals force due to dipole
molecules
• Amorphous Solids (Glass):-
Ionic Solids:
Ionic attraction: Coulomb law
is balanced by repulsions
from inner electron orbitals
leading to following potential
The minimum leads to
Covalent Solids
These form from atoms
sharing outer orbital electrons
(remember the wave nature
of an electron)
For example in Diamond
these bonds can be stronger
than ionic bonds
*Table 12.2 should be multiplied by a factor of 2 to compare with Table 12.1
Metallic Solids
Generally weaker than ionic
or covalent bonds.
There are a large number of
mobile electrons in a metal.
A metal can be viewed of as a
lattice of positive ions
surrounded by an electron
gas.
States of matter
Fig. 12-5, p.410
12.2 Classical Free Electron Model
• We have briefly discussed this in Electricity & Magnetism (E&M)
• Thomson, Drude and Lorentz developed this model at the turn
of the 20th Century
• Ohm’s Law: 𝐽 = 𝜎𝐸 is observed experimentally in many metals
and semiconductors and can be derived as in (E&M)
𝑛𝑒 2 𝜏
𝑚𝑒
𝑛𝑒 2 𝐿
𝑚𝑒 𝑣𝑟𝑚𝑠
• Recall that 𝜎 =
=
where
𝜏 = 𝐿/𝑣𝑟𝑚𝑠 implies that
𝑛𝑒 2 𝜏
𝜎=
1/2 , but this leads to a too
(3𝑘𝐵 𝑇𝑚𝑒 )
small for the conductivity (see Ex 12.1)
Wiedeman-Franz Law
• Thermal Conductivity K:
defined in
much same way as conductivity 𝐽 = −𝜎Δ𝑉/Δ𝑥,
Δ𝑉
where E = −
Δ𝑥
• The W-F law shows that
and this
ratio as a constant is predicted by classical theory
although wrong magnitude
• So we need to move onto a QM theory
12.3 Quantum Theory of Metals
• We need to use Fermi-Dirac (FD) distribution not Maxwell-Boltzmann
• In a semi-classical approximation
we just replace 𝑣𝑟𝑚𝑠 → 𝑣𝐹 : use the
2
𝑛𝑒 𝐿
1
Fermi velocity for 𝜎 =
and 𝐾 = 𝐶𝑣 𝑣𝐹 𝐿 (see next slide)
𝑚𝑒 𝑣𝐹
3
• Using now the fact the FD only has a small fraction of electrons near
𝐸𝐹 (see next-next slide) leads a modified Wiedemann-Franz Law (see
pg 422 & cf. Table 12.7):
• Note the quantum mean free path is ~100 times larger than the
classical one: Electrons have infinite path length in a perfect ion
lattice
Replacement of 𝑣𝑟𝑚𝑠 → 𝑣𝐹
Fig. 12-14, p.421
Electron fraction near Fermi surface
Fig. 10-12, p.358
12.4 Band Theory of Solids
• Much like the 𝐻2± bonding/antibonding orbitals in a molecule the
same thing can happen with solids
Fig. 12-16, p.426
Fig. 12-17, p.426
Fig. 12-18, p.426
Fig. 12-19, p.427
Fig. 12-20, p.428
Table 12-8, p.428
Fig. 12-21, p.428
Fig. 12-22, p.429
Energy Bands from Electron Wave Reflections
• This is almost real quantum mechanics and very elegant
Fig. 12-23, p.430
Fig. 12-24, p.430
Fig. 12-25, p.432
Fig. 12-26, p.432
12.5 Semiconductor Devices
• In progress
Fig. 12-27, p.433
Fig. 12-28, p.434
Fig. 12-28a, p.434
Fig. 12-29, p.435
Fig. 12-30, p.436
Fig. 12-31, p.437
p.437
Fig. 12-32, p.438
Fig. 12-33, p.438
p.439
Fig. 12-34, p.440
Fig. 12-35, p.441
Fig. 12-36, p.442
Fig. 12-37, p.442
Fig. 12-38, p.443
12.6 Superconductivity
• A little out of the scope of this book, but let’s note that we have
• A critical temp 𝑇𝑐 (𝐾) where material super conducts (see Table 12.9)
• The Meissner effect: magnetic fields are expelled from a
superconductor
• Industrial applications range from transport to computing!
• Origin of basic superconductivity comes from the work of Bardeen,
Cooper and Scheiffer (BCS) theory:• Two electrons can actually interact with each other attractively …
Table 12-9, p.443
Fig. 12-39, p.444
Fig. 12-40, p.445
12.7 Lasers: Interaction of light with atoms
• Absorption, Spontaneous Emission & Stimulated Emission
Fig. 12-41, p.448
Fig. 12-42, p.450
Fig. 12-43, p.451
Table 12-10, p.451
Semiconductor Lasers
• Extra reading ….
Fig. 12-44, p.452
Fig. 12-45, p.453
Fig. 12-46, p.453