Problem sheet 2

21.06.2017

1. Considering a lattice in three dimensions with the crystal potential

czbyaxUzyxU /2cos/2cos/2cos,,

With the help of central equation, please try to find approximately the band gap at the corner point

(π/a,-π/b,π/c) of the Brillouin zone.

2. The carrier density in an intrinsic Silicon semiconductor at room temperature (300K) is

6.6*109cm-3. The effective mass of electrons is me* = 1.08me, the effective mass of holes is

mh* = 0.59 mh.. Please calculate:

A) The band gap energy.

B) The position of the Fermi level.

3. To produce a silicon diode, a boron doped (n(Boron) = 1016cm-3) silicon crystal (ni =

1.5*1010cm-3, Eg = 1.12eV) was radiated by phosphorous atoms (radiation density = 16nA*cm-2)

for 33s, which resulted in a constant concentration profile and the range of Phosphorous atoms’

distribution is 300nm.

At a certain temperature, the concentration of electrons is n = 8*1016cm-3, the concentration of

holes is p = 2.5*10-19cm-3. (n0(T) = 2.8*1019cm-3, p0(T) = 1019cm-3)

Please calculate:

(For simplification, here we assume that the energy gap is temperature-independent)

A) The ‘certain Temperature’.

B) The position of the fermi level.

C) The ionization energy of the Phosphorous dopant at the certain temperature.

4. A two dimensional metal has one atom with valency one in a simple rectangular primitive cell a

= 0.6nm, b = 0.9nm. Please:

A) Calculate the radius of the Fermi energy in cm-1.

B) Draw the Fermi sphere to first, second and third Brillouin zones.

C) Draw another scheme to show the first few periods of the free electron band in the periodic

zone scheme, for both first and second energy bands. Assume that there is a small band gap at the

zone boundary.

5. What is polaron and how is it connected with the temperature dependency of electrical

conductivities?

6. Please scetch and describe the following relations:

A) Dielectric function of a free electron gas versus frequency. (Hint: in units of plasma frequency

ωp)

B) Dielectric function of coupled Phonon-photon in an ionic crystal versus frequency.