Consider a weak periodic potential of the form V (x) = V

PHY401 Exercise Sheet 2

HS2020 Prof. Dr. Johan Chang

Discussion on September 22 Due on September 28

Exercise 1 Energy Gap in 1D

Consider a weak periodic potential of the form V (x) = V

cos(Gx) created in a one dimensional electronic system, with 2k

= G, where G is the primitive unit vector of reciprocal space. The aim is to calculate the total energy of the system with and without this weak potential.

(a) The new bands after this perturbation will be given by E

(k) = E

(k) + E

(k − G)

2 ±

s

E

(k) − E

(k − G) 2

+ V

, (1)

where E

is the unperturbed energy and V

is the potential in reciprocal space, i.e. the Fourier component. Use this to express the change in energy in the lower band with respect to the unperturbed case as

∆E

(k) = ¯ h

2m

Gk

−

q

(Gk

)

+ (2mV

/¯ h

)

, (2)

where k

:= G/2 − k.

(b) Integrate equation (2) over k-space and show that for a weak potential the total change in energy is proportional to V

ln V

.

(c) Does the energy increase or decrease for this gapped system compared to free electrons?

Exercise 2 Gap in the Density of States

Consider a simplified model of a system with a constant density of states (DOS) d

= 1/W , where W ≈ E

is the bandwidth. Introducing a gap 2∆ in the spectrum, the DOS gets modified to

d(E) = d

|E − E

|

p (E − E

)

− ∆

. (3) (a) Sketch the DOS before and after the introduction of the gap. Shade the occupied energy regions

in each case, assuming that the minimum energy is 0.

(b) What is the total energy of the system before the introduction of the gap?

(c) Evaluate the energy of the system after the introduction of the gap.

(d) Show that the change in energy is

δE ∝ ∆

W ln

∆ W

. (4)

1