Universitat Bonn 5.11.2014Physikalisches 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 4

(Please return your solutions before 12:00 h Tue. 11.11.2014 [room PI 1.055].)

4.1: Band Structure: The Tight-Binding Method (10 points)We consider the Hamiltonian for an electron on a lattice,

H = − ~2

2m∇2 + U(r) , (1)

with the periodic potential U(r) =∑

Ri∈B V (r − Ri) and Ri ∈ B, the lattic vec-tors of the Bravais lattice B. We denote the complete basis of Wannier states byW = {|n, i〉 |Ri ∈ B, n = 0, 1, 2, · · · }; the wave-function of |n, i〉 in position represen-tation is the Wannier function,

WnRi(r) ≡ 〈r|n, i〉 =

∑k∈R

Bnk(r) e−ik·Ri ,

where Bnk(r) are the Bloch functions.

a) Give the representation of H in the Wannier basis W ; i.e., write down thematrix elements of H in the Wannier basis,

εnm,i = 〈n, i|H|m, i〉 , tnm,ij = 〈n, i|H|m, j〉, i 6= j .

Are εnm,i and tnm,ij diagonal in the band indices n,m? Why or why not?

b) We assume now that the Wannier functions WnRiare strongly localized around

the lattice site Ri (“tight-binding”), so that their overlap is non-vanishing onlyfor the on-site and nearest-neighbour matrix elements. Write down the Hamil-tonian for this tight-binding assumption in terms of the parameters En = εn,iand tn = tn,ij, where i, j are nearest neighbours. Use the operator notation|n, i〉〈m, j|.

c) Show that H is diagonalized by the transformation

|n, i〉 =∑k

e−ik·Ri |n,k〉, for any n = 0, 1, 2, · · · .

What is the meaning of the new basis states |n,k〉? Obtain the dispersion(eigenenergies) of the tight-binding Hamiltonian εn,k as

εn,k = En + 2td∑

i=1

cos(ki a)

for a cubic lattice in d dimensions, d = 1, 2, 3.

In practice, the Bloch functions and, therefore, the Wannier functions are not ex-actly known (otherwise the problem of diagonalizing the full Hamiltonian (1) wouldbe solved!). Therefore, one takes appropriately chosen approximations for WnRi

tocalculate the matrix elements of 4.1.a), like the atomic orbitals, possibly amended bythe proper lattice symmetries.

4.2: Density of States in the Tight-Binding Approximation (10 points)We restrict ourselves now to a single band, say n = 0 and E0 = 0 (without restrictionof generality). Use the dispersion relation obtained in 4.1.c) for a cubic lattice.

a) Calculate the group velocity vk in d = 1, 2, 3 for the tight-binding model.

b) For d = 1, sketch the dispersion relation εk and the group velocity vk. Wheredoes vk vanish?

c) For d = 2, sketch the lines of constant energy, εk = const., in the first Brillouinzone. In particular, determine and draw the line εk = 0 (band center).

d) For d = 1, calculate the density of states N (ε) of the tight-binding model forall energies −D < E < +D, D = 2|t| (including proper normalization).

e) For d = 2, 3, calculate the density of states N (ε) for E ≈ ∓D = ∓2|t|d (lowerand upper band edge, i.e. long-wavelength limit). Why is N (ε) symmetricabout E = 0? Why is the behaviour of N (ε) near the band edges independentof the lattice structure and, therefore, dependent only on the spatial dimension-ality?

f) Show that in d = 2, the density of states has a logarithmic divergence (vanHove singularity) at E = 0.

Remark: Because of the behaviour calculated in sections e) and f), the density ofstates (per spin) is often approximated by

d = 2 : N (ε) =1

2DΘ(D − |E|) ,

d = 3 : N (ε) =2

πD

√1− (

E

D)2 .

-1.5

-1

-0.5 0

0.5

0.5

0.50.5

1 1

11

1.51.5

1.51.5

-π -π

2

0 π

2

π

-π

-π

2

0

π

2

π

a kx

aky