PHS4200/4021: Condensed Matter Physics – Assignment

Question 1 (2+2+9+6+12+4+5=40)
Consider the two-dimensional (2D) crystal in the figure below, where each black circle
represents an atom, with all atoms being identical.


Figure 1. Crystal structure of a 2D crystal.
(a) Superimposed to the sketch of the 2D crystal structure, sketch an ensemble of points that
defines the 2D lattice of this crystal. Clearly distinguish atoms from lattice points.
On your sketch of the lattice, draw two vectors a1 and a2 defining a primitive unit cell of this
crystal. Consider the case where the norm of both vectors a1 and a2 is the same, that is, ‖1‖ =
‖2‖ = .
How many atoms are there in the basis for this 2D crystal? Justify your answer.
(b) Consider the coordinate system defined by the vectors ̂ and ̂ in Figure 1. Write vectors
1 and 2 as a linear combination of vectors ̂ and ̂, and as a function of a.

(c) Using 1 and 2, and 3 defined as 3 = ̂ (where ̂ = ̂ × ŷ), calculate the reciprocal
lattice vectors 1 and 2 as linear combinations of vectors ̂ and ̂, and as a function of a.
Sketch the (two-dimensional) reciprocal lattice defined by the basis vectors 1 and 2 and
sketch the first Brillouin zone.
(d) Now, let us label the atoms in this crystal as below.


̂
̂

Figure 2. Crystal structure of a 2D crystal with atom labelling

Draw and calculate (with respect to the Cartesian coordinates ̂ and ̂) the vectors that link
atoms ,, , and , to their nearest neighbours. For example, (,+1 − ,),
(+1, − ,), (, − ,) and (, − ,) link atom , to its nearest neighbours.

(e) Let us now assume that each atom of the 2D crystal contributes with one valence electron
to a valence electronic energy band of the material, with at() = 0(), where at is the
Hamiltonian of each atom composing the material (in the case where the atom isolated), ()
is the wavefunction of the valence electron in such an atom, and 0 the electron eigenenergy
when the atom is isolated.
Now, let us consider what happens to the eigenenergies of these valence electrons when the
atom is incorporated in the 2D crystal above. According to the tight-binding model, for a crystal
with a one-atom basis, the probability |,()|
2
of finding an electron at lattice site (i, j) at
time t is governed by the equation
ℏ
,()

= 1,() − ∑ ′,′
′,′
(1)
where 1 = (0 + 0) with 0 + 0 = ∫ ,
∗() ,() ) (where now is the one-
electron Hamiltonian of the 2D system, and ,() is the electron wavefunction associated
with atom with labels (i, j) when this atom is isolated), where the sum i', j’ is performed over
the nearest neighbours, and = − ∫ ,±1
∗ , = − ∫ ±1,
∗ ,.
Let us assume that the probabilities |,()|
2
, |,()|
2
and |,()|
2
of finding an electron
at atom sites i,j, i,j, i,j , respectively, are given by the ansatz functions
̂
α,
β,
γ,
β+,
γ,+
α−,
α,−
γ−,+
β+,−
̂
,() = exp [ ( ∙ , −

ℏ
)]
,() = exp [ ( ∙ , −

ℏ
)]
,() = exp [ ( ∙ , −

ℏ
)] (2)
By replacing ,(), ,() and ,() in Eq. (1) with the expression given in Eq. (2),
show that:
∙ (



) =
with
= (
1 − −2 cos( ∙ 1) −2 cos( ∙ 3)
−2 cos( ∙ 1) 1 − −2 cos( ∙ (3 − 1))
−2 cos( ∙ 3) −2 cos( ∙ (3 − 1)) 1 −
)
and 1 = , − , and 3 = , − ,.

(f) We want nontrivial solutions for ,(), ,() and ,() (i.e., , , different from
zero), such that Eq. (1) is fulfilled. What condition must the matric M fulfil?
(g) The condition in (f) is fulfilled when
() = 2 + 1
and when
() = (, )
= 1 −
± √1 + 8 cos (−

4
+
√3
4
) cos (−

4
−
√3
4
) cos (−

2
)
Using Python, MatLab or Mathematica, plot () = (, ) [that is, three dispersion
curves], for −
2

< <
2

and −
2

< <
2

.