excitons

Lecture 3. Elementary excitations in crystals.


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 obtains an equation for the Bloch amplitude:
, (3.1.3)
where . Consideration of the operators in the parentheses as a perturbation constitutes the method of the perturbation theory which readily allows to find the shape of the electronic dispersion in the vicinity of k=0 points of all bands, which appears to be strongly different from the free electron dispersion in vacuum. Approximation
(3.1.4)
is called the effective mass approximation with being the electron effective mass in n-th band :
. (3.1.5)
The frequencies and polarization of the optical transitions in direct gap semiconductors are governed by the energies and dispersion of two bands closest to the Fermi level , referred to as the conduction band (first above the Fermi level) and the valence band (first below the Fermi level).

Figure 3.1.1 Zinc-blend (a) and wurtzite (b) crystal lattices.

Semiconductors can be divided into direct band gap and indirect band gap ones. In indirect gap semiconductors (like Si and Ge) the electron and hole occupying lowest energy states in conduction and valence bands cannot directly recombine emitting a photon due to the wave-vector conservation requirement. While a weak emission of light by these semiconductors due to phonon-assisted transitions is possible, they can hardly be used for fabrication of light-emitting devices and studies of light-matter coupling effects in microcavities. In the following, we shall only consider the direct gap semiconductor materials like GaAs, CdTe, GaN, ZnO etc. Most of them have either a zinc-blend or a wurtzite crystal lattice (see Figur