COEFFICIENT OF OPTIC ABSORPTION. PART 2

Hello friends of steemit, here I present the second part of the coefficient of optical absorption.

If you have not seen the first part, I invite you to see it here.

Fundamental or intrinsic absorption in a direct gap semiconductor.

In the second type of materials the ends of the conduction band and the valence band are in different k, (), being called indirect gap, as shown in the following figure:

Optical transitions in an indirect gap semiconductor.

Most semiconductors, including silicon and germanium, correspond to this type of material.

The wave vector of a photon is very small compared to the maximum value of the wave vector of an electron in the crystal, since the minimum wavelength of an electron in a crystal is approximately , while is of the order of . Because of this we have the following:

eq. (1) and (2)

The above correlations, called selection rules for electronic transitions, indicate that during the interaction of the semiconductor electron with the radiation field only such transitions are possible, for which the electron wave vector is conserved.

Let us examine the fundamental absorption for direct transitions between bands of a semiconductor whose valence and conduction bands have spherical symmetry. Suppose that transitions of electrons are produced from a state of the valence band with wave vector corresponding to the interval between , as shown in the following figure:

Direct transitions

In this case, the energy of the absorbed photon, h, can be determined, based on the law of conservation of energy

eq.(3)

Where:

is the width of the band gap

is the reduced effective mass of the electron and the gap:

eq. (4)

The absorption coefficient ![15.png]() is inversely proportional to the length of the free path ![16.png](), whose magnitude in turn is determined by the speed of movement ![17.png]() and the free travel time of the photon ![18.png](), that is:

eq.(5)

Where:

g (v) is the probability of absorption of the photon in the unit of time.

is the speed of movement of the photon in the material.

n is the refractive index.

c is the speed of light.

eq. (6)

Here the coefficient 2 takes into account two possible directions of polarization of the light. The density of the quantum states near the maximum of the valence band is determined by the relationship:

eq. (7)

so:

eq.(8)

Taking into consideration equation 3 we obtain:

eq. (9)

Based on equations 5 and 9, we can write:

eq. (10)

Where we are going to consider that B is a constant number.

Taking into account that P (v) is proportional to 1 / v, the above equation can be written as follows:

eq. (11)

Where A is a new constant.

Once the absorption coefficient is known, the energy gap can be determined through the graph of ![26.png]() The approximation is only valid in the region of abrupt increase of ![27.png](), called edge of absorption. The slope in the fundamental absorption region is ![28.png]() and the cutoff point is ![29.png]() Therefore ![30.png]() is given by:

eq. (12)

The following figure shows this linear dependence of ![32.png]() with respect to hv in the region of fundamental absorption or absorption edge.

.

Dependence of with respect to hv.

REFERENCES:

Pankove J, (1971),Optical Processes in Semiconductors, New York,Dover Publications.

Hecht E, Zajac A, (1977),Óptica, Fondo Educativo Interamericano.

Díaz R, Merino J. M., Martín T, Rueda F, León M,(1998),An approach to the energygap determination from the reflectance measurements.

McKelvey J. P, (1993),Física del Estado Solido y de Semiconductores, 1º Edición,Mexico, Editorial Limusa.

D.B Gadkari, K. B Lal y B M Arora. (1999). Growth of undoped and Te doped InSb crystals by vertical directional solidification technique, Indian Academy of Sciences.

FurdynaJ.K, (1988), Diluted magnetic semiconductors, Journal of Applied Physcs, Vol 64, Nº 4.