Solid State Physics
Lecture 5 – Brillouin Zones and Bonding
Professor Stephen Sweeney
Advanced Technology Institute and Department of Physics
University of Surrey, Guildford, GU2 7XH, UK
[email protected]
Solid State Physics - Lecture 5
Recap from Lecture 4
• The reciprocal lattice is a useful construction when considering
interaction of photons with a real lattice, e.g. diffraction
• Reciprocal lattice points represent real space planes
• The vector joining reciprocal lattice points has a magnitude
inversely proportional to the real space lattice inter-planar spacing
• The Ewald construction can be used to derive the diffraction
condition
Some very useful on-line material here:
http://www.matter.org.uk/diffraction/geometry/geometry_of_diffraction_braggs_law_1.htm
Solid State Physics - Lecture 5
Brillouin Zones
From Ewald construction, diffraction occurs when
k’
k'  k  G
k
If we square both sides of the equation:
k 'k '  k  G   k  G  k  G 

2
G
 k cos 
2
 k '  k  G  2k  G
2
2
2
If scattering is inelastic, then
Hence:
G
k'  k
G 2  2k  G  2k  G  2kG cos 
(If G is a lattice vector then so is –G !)
G
 k cos  
2
i.e. diffraction condition is satisfied if k lies in the
plane that perpendicularly bisects G
Solid State Physics - Lecture 5
Brillouin Zones
• We define the “Brillouin Zone” as the
boundary in the reciprocal lattice where
scattering will occur
1st Brillouin Zone for a square lattice
2
a
a
• A Brillouin Zone is a primitive unit cell in
reciprocal space (c.f. Wigner-Seitz cell in
real space)
• The Brillouin zone describes the behaviour
of the entire crystal, e.g. how electrons are
influenced by the periodicity of the crystal
(Bloch Waves) with implications for:
1st Brillouin Zone for a fcc lattice (3D)
• electrical conduction
• light emission/absorption
• etc.
• We will return to this later…
Solid State Physics - Lecture 5

a
Léon Nicolas Brillouin (1889-1969)
• Founder of modern solid-state physics
• Studied in Paris and Munich
• PhD supervisor was Paul Langevin
examined by Marie Curie
• Developed theories of interactions of light with phonons
• Directed research on radio and radar for French
government during WWII
• Moved to the USA and worked, amongst other places
at IBM and Harvard
• 1953: Elected to US National Academy of Sciences
Solid State Physics - Lecture 5
Bonding in Solids
- Interatomic/intermolecular forces
• Gases turn into liquids when slower moving atoms/molecules are
able to interact for long enough that long-range forces cause them to
coalesce
• Below a certain interatomic spacing atoms can no longer be attracted
together since short range forces become significant and are
repulsive
• Such interactions are described semi-empirically, e.g. the LennardJones potential.
Solid State Physics - Lecture 5
Bonding in Solids
- Attractive hard sphere model
• Consider each atom as an attractive hard sphere:
a
• If K.E. of freely moving atoms can be overcome by force F then they will coalesce
• The closer the atoms get, the stronger the interaction up until they touch (hard
spheres)
Solid State Physics - Lecture 5
Bonding in Solids
- Force-separation curves
Hard Spheres
Real system
Force
Force
0
0
a
Separation (r)
No possibility of separation < a
Solid State Physics - Lecture 5
a
Separation (r)
Large energy cost in pushing
atoms too closely together
(repulsion in overlapping electron
clouds)
Bonding in Solids
- Repulsive forces
As atoms are forced together their electron
clouds overlap:
• Pauli exclusion principle demands that two
electrons with same spin state cannot occupy
the same state
 Energy is required to promote electrons into
higher orbitals
• Repulsive force varies quickly with distance,
represented by a high power law:
1
F  13
r
(short-range force)
Solid State Physics - Lecture 5
Bonding in Solids
dU
F 
dr
At equilibrium separation, F=0
U minimised = -
Zero energy defined at r =
Potential Energy (au)
As with all physical systems, aim is to minimise energy:
repulsive
0
bond energy
()
attractive
a
Solid State Physics - Lecture 5
separation, r (au)
Bonding in Solids
- Attractive forces
Attractive forces depend on the type of chemical bond between the atoms
Primary bonds
• Covalent bonds: electrons are shared between neighbouring atoms
• Ionic bonds: complete transfer of an electron from one atom to a
neighbouring atom
• Metallic bonds: electrons are delocalised
Secondary bonds
• Van der Waals bonds: no electron transfer – long range forces bond
atoms
• Hydrogen bonds: electrostatic attraction
Solid State Physics - Lecture 5
Bonding in Solids
Covalent Bond
Ionic Bond
• Strong and very hard
• Highly directional
• Not easy to draw on a F-r
curve
• Examples: Diamond and
Silicon
• Purely Coulombic:
F 
q1q2
r2
and
U
q1q2
r
• Exist between permanently
charged ions (e.g. Na+Cl-)
Solid State Physics - Lecture 5
Bonding in Solids
Metallic Bond
Molecular/Hydrogen Bond
• Delocalised electrons
• Valence electrons are weakly
bound
• Electrostatic interactions
between electrons and ions
• Strong and non-directional
• H atoms in molecules often
have a net positive charge
• Can easily bond
electrostatically to
electronegative atoms, e.g.
N, F or O
• Bond is directional with
intermediate stength
• Example – bonding between
water molecules
Solid State Physics - Lecture 5
Bonding in Solids
Van der Waals bonding
•
Due to an alignment of dipoles or induced dipoles, e.g.
1. Two molecules with permanent dipoles (Keesom
energy)
-
+
+
Occur between atoms and molecules with no net charge
-
•
-
+
+
-
2. One molecule with a permanent dipole induces a
temporary dipole in a neighbour (Debye energy)
3. Fluctuations in charge induce temporary dipoles
(London energy), e.g. in a noble gas
•
Force falls off rapidly with distance:
1
1
F   7 and U  6
r
r
Solid State Physics - Lecture 5
Van der Waals Bonds
Examples:
Graphite
(graphene sheets interacting
via Van der Waals bonds)
Solid Helium
Solid State Physics - Lecture 5
Johannes Diderik van der Waals
(1837-1923)
• Physicist and thermodynamicist
• Born in Leyden, The Netherlands
• Studied in his spare time, becoming a secondary
school teacher
• First Professor of Physics at University of Amsterdam
• Worked on thermodynamics, developed The law of
corresponding states
• Work allowed Kamerlingh Onnes to liquify Helium in
1908
• Received Nobel Prize for Physics in 1910
Solid State Physics - Lecture 5
Bonding in Solids
Solid State Physics - Lecture 5