Universitat Bonn 22.10.2014Physikalisches InstitutProf. Dr. Johann Kroha, Johannes Rentrop and Ammar Nejatihttp://www.kroha.uni-bonn.de/teaching/lectures-and-seminars/wt2014/tcmp

Condensed Matter Theory I — WS14/15

Exercise 2

(Please return your solutions before 12:00 h Tue. 28.10.2014 [room PI 1.055].)

2.1: Quasicrystals (10 points)In exercise 1.1 you proved that lattices never have a 5-fold symmetry. Neverthelessquasicrystals with such symmetries can be observed in nature. These objects havea local rotational symmetry but no translational symmetry and are built from atleast two different unit cells. We will construct an example of such a one-dimensionalquasicrystal in the following.Consider a 2-dimensional cubic lattice and choose a coordinate system (e‖, e⊥), whichis rotated by an angle α = 30 (Fig. 1). We will project now all lattice points ina stripe around the e‖ axis onto this axis. As we will see, the result will be a 1-dimensional quasicrystal.To perform the projection we have to work a little bit more formally: For each latticepoint ~n = (n1, n2) ∈ Z2 its unit cell is given by C(~n) := (x1, x2) ∈ R2 |xi ∈[ni, ni + 1) , i = 1, 2 (Fig. 2). Denote further the e‖ axis by l and define S := ~n ∈Z2 | l ∩ C(~n) 6= ∅. S is the set of all lower left-handed vertices of all square cells cutby l (open dots in Fig. 2).

a) Project all points of S onto l and show that the result is a non-periodic inter-section of l. What does happen for α = 45? Which condition must α fulfil toensure non-periodicity?

b) Show that the intersection of l is built up by two different unit cells (Fig. 2),i.e. the distance between two projected, neighbouring points on l is either a orb. Express a and b as functions of α.

e

e

30°

Figure 1: projection scheme

aa b b ba a ba

Figure 2: 1d quasicrystal

2.2: The Empty-Lattice and Nearly-Free-Electrons Approximations (10 points)

a) The Schrodinger Equation for Bloch wave-functions. Derive explicitly the time-independent Schrodinger equation, Hψ = εψ, for the normalized Bloch waves,ψnk(r) = 1√

Veik·r unk(r), including a lattice-periodic potential U(r), U(r+ t) =

U(r) for t ∈ B. Use the Fourier series expansions,

unk(r) =∑Gj∈R

cnk(Gj)eiGj ·r and U(r) =

∑Gj∈R

U(Gj)eiGj ·r ; U(Gj = 0)

!= 0 ,

and show that the Schrodinger equation is recast as

[ ~2

2me

(k + Gi)2 − εnk

]cnk(Gi) +

∑Gj 6=Gi

U(Gi −Gj)cnk(Gj) = 0 . (1)

b) The Empty-Lattice Approximation. Consider Eq. (1) in the asymptotic limitU = 0. Show that the solution is

cnk(Gi) = 0 or εnk = E(0)k+Gi

:=~2

2me

(k + Gi)2 ,

with a normalized Bloch wave, ψnk(r) = 1√Veik·r eiGi·r. This procedure estab-

lishes a one-to-one correspondence between the band indices n and reciprocallattice vectors Gi.

c) The One-dimensional Lattice. As a concrete example, consider a 1d chain oflength L = Na (a is the lattice constant and N number of lattice sites). Im-posing periodic boundary conditions, show that the wave-vectors k can onlytake values k = 2π

Lm = 2π

Nam,m ∈ Z. Derive also the reciprocal lattice vectors

as G = 2πan, n ∈ Z. Then, by taking the k-vectors in the first Brillouin zone,

−πa≤ k ≤ π

a, show that when U = 0, the energy eigenvalues are given by

εnk =~2

2me

(k +

2π

an)2

=~2

2me

( 2π

Na

)2(m+ nN)2 .

Obtain the band structure by plotting the energy eigenvalues (in appropriateunits) in the first Brillouin zone.

Universitat Bonn 29.11.2014Physikalisches InstitutProf. Dr. Johann Kroha, Johannes Rentrop and Ammar Nejatihttp://www.kroha.uni-bonn.de/teaching/lectures-and-seminars/wt2014/tcmp

Condensed Matter Theory I — WS14/15

Exercise 3

(Please return your solutions before 12:00 h Tue. 4.11.2014 [room PI 1.055].)

3.1: The Empty-Lattice and Nearly-Free-Electrons Approximations (10 points)[continued from exercise sheet 2]

The Nearly-Free-Electrons Approximation. Consider a finite but relatively weak pe-riodic potential U (U Ekin). This finite potential (due to ions and other electrons)modifies the energy spectrum. The weakness of the potential allows one to obtain thephysical quantities as perturbative series in orders of U , keeping only the lowest non-vanishing corrections. This approximation provides qualitatively correct informationabout the bands and Fermi surfaces in metals with s- and p-electrons. We considertwo major cases in the following.

i) Non-degenerate case: Consider a state for which the unperturbed energy in the

empty-lattice approximation, E(0)k+Gi

, is relatively far from the energies E(0)k+Gj

of states with the same k in all other bands, i.e., for Gi 6= Gj,

|E(0)k+Gi

− E(0)k+Gj| U .

Write down the Schrodinger equation for cnk(Gi) and cnk(Gj 6=i) as

[ ~2

2me

(k + Gi)2 − εnk

]cnk(Gi) +

∑Gj 6=Gi

U(Gi −Gj)cnk(Gj) = 0 ,

[ ~2

2me

(k + Gj)2 − εnk

]cnk(Gj) + U(Gj −Gi)cnk(Gi) +

∑Gk 6=Gi,j

U(Gj −Gk)cnk(Gk) = 0 .

Now, cnk(Gi) depends on all other coefficients cnk(Gl). Reminding that cnk(Gl 6=i) ∼ O(U)and keeping the first non-vanishing correction, determine cnk(Gj) as

cnk(Gj) ≈U(Gj −Gi)

εnk − E(0)k+Gj

cnk(Gi)

and

εnk ≈ E(0)k+Gi

+∑

Gj 6=Gi

|U(Gj −Gi)|2

E(0)k+Gi

− E(0)k+Gj

.

ii) Degenerate case: Consider a state with an energy close to that of another state

with the same vector k, i.e., E(0)k+Gi

≈ E(0)k+Gj

. This can occur especially at the

center and boundaries of the Brillouin zone. Show that the latter condition isequivalent to 2k · (Gi − Gj) + G2

i − G2j ≈ 0. The calculation of the energy

shifts requires degenerate perturbation theory. Due to the mixing of degeneratestates, the coefficients associated with vectors Gi and Gj can be large; therefore,by neglecting the terms associated with other vectors of the reciprocal lattice,show that the following equations are obtained for the energies:[

εnk − E(0)k+Gi

]cnk(Gi) ≈ U(Gi −Gj) cnk(Gj) ,[

εnk − E(0)k+Gj

]cnk(Gj) ≈ U(Gj −Gi) cnk(Gi) .

Solve the set of two linear equations and obtain the peturbed energies as

εnk = 12

(E

(0)k+Gi

+ E(0)k+Gj

)±

14

(E

(0)k+Gi

− E(0)k+Gj

)2+ |U(Gj −Gi)|2

12

.

Show the new energy eigenvalues schematically at the edge of the Brillouin zone.

3.2: The Wannier Basis (10 points)Based on the fact that the Bloch wave-functions ψnk(r) provide a complete and or-thonormal basis for the Hilbert space of lattice electrons, prove that the set of Wannierfunctions,

WnR(r) =1

ΩBZ

∑k∈BZ

ψnk(r)eik·R , R ∈ B,

provides also a complete and orthonormal basis for the same Hilbert space.