محاضره مهمه عن exciton

Solutions of the problems to Lecture 2.

1. a) ;
b) ;
c) .

2. For the coherent distribution:
;
(reminder: )
For the thermal distribution:
Let .
.
Thus,
.

3. For the coherent distribution:
.


4.
Let .
.
.
Note that and

For the thermal distribution:











3.1 Optical transitions in semiconductors.


We remind here the most essential features of the structure of optical transitions in semiconductors.
Fermi level Energy gap width Conductivity (S m-1)
metals Inside the band any Up to 6.3 107 (silver)
semiconductors Inside the gap < 4 eV Varies in large limits
dielectric Inside the gap 4 eV
Can be as low as 10-10
Table 3.1 Classification of solids.
It is well-known that the discrete electronic levels of individual atoms form large bands in crystals where thousands of atoms are assembled in a periodic structure. There are also gaps between the allowed bands where no electronic states exist in an ideal infinite crystal. Those crystals which have a Fermi level inside one of the allowed bands are metals, while the crystals having a Fermi level inside the gap are semiconductors or dielectrics. The difference between semiconductors and dielectrics is quantitative: the materials where the band gap containing the Fermi level is narrower than about 4 eV are usually called semiconductors, the materials with wider band gaps are dielectrics. In this Chapter we consider only semiconductor crystals.
The eigen-functions of electrons inside the bands have a form of so-called Bloch waves.
The concept of the Bloch waves was developed by a Swiss physicist Felix Bloch in 1928, to describe the conduction of electrons in crystalline solids. The Bloch theorem states that a wave-function of an electronic eigen-state in an infinite periodic crystal potential can be written in form:
, (3.1.1)
where (called Bloch amplitude) has the same periodicity as the crystal potential, k is so-called pseudo-wave vector of an electron (further we shall omit “pseudo” while speaking about this quantity), n is the index of the band.
Substitution of the wave-function (3.1.1) into the Schroedinger equation for an electron propagating in crystal
, (3.1.2)
with being the free electron mass, one obtain