Solid State 2 – Exercise 1
1) Why there are no linear in k terms in the dispersion?
2) What is the difference (if there is any) between the electronic density of
states of a semiconductor, and that of the free electron gas (which describes a
metal rather well)?
3) Cyclotron Resonance – Semiclassical Drude Description
a) Start with the equation of motion for an electron in constant magnetic field
B:
m
dv
v
 m  e(v  B)
dt

now add the appropriate term for an external (weak) electric field E.
b) Assume B to be parallel to z . Let v (t )  v0 e
and E (t )
the explicit relation: (i.e. find the elements of the matrix).
iwt
 vx   a xx
 
 v y   a yx
 E0 eiwt . Get
axy  Ex 
 
a yy  E y 
c) Using J   E (the conductivity tensor) and the definition of J, get the
conductivity tensor (2D) as follows:
 2D
  xx  xy 
 0 

  yx  yy 
What is  0 ? Try to replace some variables with wc 
eB
m
.
d) Can you guess the complete form of the 3D conductivity tensor? What is
special about the z direction? What can you say about  zz ?
 3D
  xx ... ... 


  0  ...  yy ... 
 ... ...  
zz 

e) Now suppose this external E is caused by radiating the sample with a light
source.The power absorbed by the sample is the time average:
P  J (t ) E (t ) t 
1
Re[ J w Ew* ]
2
Assume E is polarized linearly along the x axis E(t )  E(t ) x . Show that
when w  wc maximum absorption occurs. What is the physical meaning
of this result ?
f) What is the relation between wc and  for the measurement to be valid?
g) For m* = 0.1m0 (where m0 = 9.1 E-31 Kg) and B = 1 Tesla, what is the
wavelength for the resonance? What is the condition on  ? Does this
seem like a reasonable condition ?
4) Use the semi-classical equations of motion of an electron in an (time)
oscillating electric field.
a) Derive the frequency dependent conductivity. ( scalar, not tensor).
b) Find the dielectric constant. Write it in terms of wp - the plasma frequency.
c) Find the ion dielectric constant for ions with mass M and charge Ze.
5) Derive explicitly eq. (2.10) and (2.11) from the lecture notes.