程序代写案例-PHYC30024-Assignment 2

PHYC30024 Condensed Matter Physics Assignment 2
Due: 11:59pm, Friday 15/10/21
1 Tight binding 1: Kagome lattice
A sktech of the Kagome lattice is sh
own below. “Kagome”
is not french, but the japanese word for a traditional bas-
ket with such a pattern, see figure on the right. There are
several materials which have a crystal structure consisting
of (very weakly coupled) two-dimensional planes with such
a lattice. Even a few minerals, naturally occurring on earth,
were found with a kagome structure (e.g. Herbertsmithite
or kapellasite).
a) Is the kagome lattice a Bravais lattice? If not, what is the underlying Bravais lattice and how many basis
atoms are there? Choose two primitive lattice vectors and find the Wigner–Seitz cell (not just any unit
cell – it must be the Wigner–Seitz cell). What is the coordination number of the kagome lattice? Calculate
the reciprocal lattice vectors bi and draw the first Brillouin zone.
b) Use the tight-binding method and calculate the energy bands (i.e., the band structure) of the kagome
lattice below. Use the primitive vectors ai as indicated by the black arrows and formulate the Bloch
matrix in terms of the ai rather than the nearest-neighbor vectors.
A B
C
1 2
3 64
5
(n,m)(n-1,m)
(n-1,m+1) (n,m+1)
(n,m-1) (n+1,m-1)
~a2
~a1
Guideline: The Hamiltonian can be written as HK = −t

〈ij〉(c

icj + H.c.). The primitive lattice vectors
are given by a1 = a(1, 0) and a2 = a(1/2,

3/2). The basis atoms are located at r1 = 0 (A/red),
r2 = a1/2 (B/green) und r3 = a2/2 (C/blue). A unit cell is indicated by the yellow rhomboid. In the
following, each unit cell will be labeled with x and y coordinates (n,m), see above. Instead of c
(†)
j we are
using the operators A
(†)
(n,m), B
(†)
(n,m), C
(†)
(n,m), which annihilate (create) an electron on the corresponding
sublattice A, B, C within the (n,m)-th unit cell. Thus we can write all nearest-neighbor hopping processes
(see purple arrows above plus hermitian conjugation) as
HK =−t

n,m
[
C†n,mAn,m+B

n,mCn,m +A

n,mBn,m + C

n,m−1An,m +B

n−1,m+1Cn,m +A

n+1,mBn,m + H.c.
]
(1)
The Fourier transform of these terms can be conveniently arranged into a 3×3 matrix, the Bloch matrix,
and its eigenvalues are the energy bands. Calculate and sketch these bands in the Brillouin zone.
Hint: Determine the characteristic equation of the 3× 3 matrix and use the identity
cos2
(
k·a1
2
)
+ cos2
(
k·a2
2
)
+ cos2
(
k·(a1 − a2)
2
)
= 2 cos
(
k·a1
2
)
cos
(
k·a2
2
)
cos
(
k·(a1 − a2)
2
)
+ 1 .
One of the eigenvalues has a simple form which you might guess.
Clarification: Feel free to use any computer program to plot/sketch the bands of the Kagome lattice.
However, you are expected to diagonalize the Bloch matrix by hand, i.e., it must be evident from your
assignment that you calculated the energy bands.
2 Tight binding 2: boron nitride
In the lecture, we derived the tight-binding band structure for the honeycomb lattice (“graphene”) forming
a Dirac semi-metal. In the following we want to adopt these bands and modify them in order to describe
a single layer of hexagonal boron nitride (h-BN). h-BN is an insulator with an impressive band gap of
5.2 eV.
a) h-BN forms a honeycomb net just like graphene, but as you might guess from the name, it consists of
boron and nitrogen. How many atoms and which atoms do you expect in the unit cell? What is the
corresponding Bravais lattice?
b) In order to account for the difference of N and B atoms, modify the graphene tight-binding model from
the lecture and add a term which introduces a sublattice imbalance between the two types of atoms. B
atoms are subject to an onsite potential of energy +M while N atoms of energy −M (compare with the
subsection “the simplest insulator” from the lecture notes). Write down the resulting Bloch matrix.
Hint: You can use the Bloch matrix of graphene as a starting point.
c) Write the Bloch matrix in a clever way which allows you to read off the resulting energy eigenvalues (i.e.,
the band structure) immediately.
d) Perform a Taylor expansion of the Bloch matrix elements around K, one of the corner points of the
Brillouin zone. Find the energy eigenvalues in the vicinity of K and show that the energy bands are
gapped for finite M . What is the gap size?
3 Phonons
Consider a chain of silicon atoms with an interatomic spacing a, a nearest-neighbour interatomic force of
γ1 = 600 N/m and a next-nearest-neighbour interatomic force of γ2 = 200 N/m.
a) Show that
ω2 =
4
m
2∑
p
γp sin
2(pqxa/2) . (2)
b) Calculate and plot the phonon bands in the entire first Brillouin zone (i.e. −pi/a ≤ qx < pi/a) with
(p = 1, 2) and without (p = 1 only) the next nearest neighbour interaction.
c) What is the speed of sound in the material in the long wavelength limit with and without the next nearest
neighbour interaction (i.e. the group velocity at low frequencies)?

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