PHYS 74500 Solid State Physics
Homework #4 Due December 4, 2020
Tight binding theory, phonons and semiconductors I
1. Tight binding I: BCC lattice
Show that the tight binding band structure based on a single orbital per site for a body centred cubic lattice
(include only the hopping to the eight nearest neighbours) is:
" " "
𝐸(𝑘) = 𝐸! + 8𝑡 cos -# 𝑘$ 𝑎/ cos -# 𝑘% 𝑎/ cos -# 𝑘& 𝑎/ .
Explain the meaning of all of the symbols in the above formula.
2. Tight binding II: graphite
A single sheet of graphite has two carbon atoms in the unit cell at positions
𝒅" = 0 and 𝒅" = (a/√3)(0,1,0).
The translation vectors for the two-dimensional hexagonal lattice are:
𝒕" = (a/2)(1, √3, 0) and 𝒕" = (a/2)(−1, √3, 0).
The electronic configuration of the carbon atom is 1s22s22p2, and ignoring the 1s core states, we need to make a
band structure from the s, px, py and pz orbitals. Because s, px and py orbitals are even under reflection through
the plane, and pz odd, the two sets do not mix. The first three states hybridize to form σ−bonds with a large gap
between the bonding and anti-bonding orbitals. Within this gap lie the 𝜋-orbitals arising from the hybridised pz.
The three bonding σ orbitals will accommodate 6 electrons per cell, leaving 2 electrons per unit cell in the 𝜋-bands.
This question considers the electronic 𝜋-bands only.
Figure 3: Two dimensional structure of graphite.
(a) Construct Bloch states that consist of a linear mixture of the two pz orbitals in the unit cell, and show how this
gives rise to the secular equation to determine the eigenstate energies
𝐸' − 𝐸 𝑡𝐹(𝒌)
< <=0
𝑡𝐹 ∗ (𝒌) 𝐸' − 𝐸
where t is the two center hopping matrix element between neighbouring pz orbitals, and
1 √3
𝐹(𝒌) = 1 + 2 cos ? 𝑘$ 𝑎@ exp D−𝑖 𝑘 𝑎F
2 2 %
PHYS 74500 Solid State Physics
Show that the reciprocal lattice is also a hexagonal lattice, at an angle of 𝜋/6 to the real-space lattice. Show that
the first Brillouin zone is a hexagon centered at the point Γ= (000), whose corners are at the points 𝑃 =
#
(2𝜋/𝑎)() , 0,0).
(c) Determine a formula for the dispersion curves for the two eigenstates, and plot them in the directions ΓP , and
" √)
ΓQ. (Here 𝑄 = (2𝜋/𝑎)(# , #
, 0). is at the middle of a zone face.)
(d) Where will the 𝜋 bands lie in energy relative to the sp2 σ- orbitals? Is a single layer of graphite a metal or an
insulator?
(e) Carbon nanotubes are formed by curling a graphite sheet into a tube, connecting the atoms with periodic
boundary conditions. There are many ways to do this, and the different nanotubes can be indexed by the vector
mt" + nt # that identifies which atoms are connected periodically. Assuming the band-structure is unchanged, show
that the allowed k-states now lie on a set of lines whose direction is parallel to the tube. Discuss the situations
under which the resulting tube will be semiconducting or metallic.
3. Phonons
(a) The effective spring constant, 𝐽+,, , for vibrations along the [100] direction in argon, neon, krypton etc (which
are fcc solids at low temperature) is twice the actual spring constant, 𝐽 between nearest neighbor atoms. Now,
we will apply our work in class on phonons in fcc solids to solid krypton. The dispersion curves shown below
show experimental data (as points) and a fit using a model.
Figure 1: Phonon dispersion data for krypton at 10 K, reproduced from J. Skalyo Jr., Y. Endoh and G. Shirane,
Physical Review B 9, 1797 (1974).
(a) The spring constant, 𝐽, is the value of the second derivative of the potential energy, 𝜙 with neighbour
separation, 𝑟, evaluated at the equilibrium separation. If the potential energy is of Lennard-Jones "12-6" form:
𝜎 - 1 𝜎 "#
𝜙 = −2𝜖 D- / − - / F
𝑟 2 𝑟
then find an algebraic relation between 𝐽, 𝜖 and 𝜎. [HINT: the equilibrium separation is defined by 𝜕𝜙/𝜕𝑟 = 0.]
(b) If for krypton, 𝜎 = 3.966 Å, use the data in Figure 1 to obtain an estimate of the "12-6" model value of 𝐽 and
hence a value of 𝜖 for krypton.
PHYS 74500 Solid State Physics
(c) The fit to a 12-6 model by G.K Horton in 1968 [American Journal of Physics 36, 93 (1968)] gave a value for 𝜖
for krypton of 325x10-23 J/atom. Compare your answer in (b) to this answer.
4. Semiconductors: The free exciton
Consider the effect of the Coulomb attraction between a hole (h) and an electron (e). You can quote the results
of a Bohr-type model in your answer but you will need to adapt that model’s conclusions to suit the electron-hole
interaction.
(a) Show that a bound e-h pair state can exist whereby the electron and hole orbit each other with quantised
energy states µ 1 hcRH
Eα = − ,
me εr2 α 2
and define the symbols. State under what conditions this phenomenon might be observed.
[HINT: be aware, a is a quantization label, to avoid confusion through the use of “n”.]
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(b) Calculate the exciton Bohr radius for GaAs, for which er=12.8, me* = 0.067me and mh* = 0.2me.
Questions 1 and 2 have been adapted from a University of Cambridge course on solid state physics.
