The Nature of Light - Part 5 (Wave Packet and Semiconductors)

The last concept that I want to talk about is the wave-packet. The wave-packet is a superposition of waves. This concept brings the particle and wave properties of light together. A wave-packet consisting of a plane wave in one dimension is:

The Gaussian wave-packet and its Fourier transform. (Goldin, 82)

The G(k) function is Gaussian distribution function. The smaller gets the more F(x) spreads out along x axis. When becomes a delta function, F(x) becomes a sinusoidal function, in which it will comply with the classic light wave laws. F(x) and G(k) are Fourier transform pairs.

So far we have just reviewed a very brief history of optics from Newton to Einstein. In the next part I like to review some materials on MOSFET devices technology and design.

Photoelectric Effect in Semiconductors

There are two major types of semiconductors that use and work with the photoelectric effect, diodes and photocells. The transistor with the GaAs substances will do very well with the photoelectric effect. These types of transistors are called direct transistors because in their recombination process photon emits as result of electrons moved to a less energy level.

The photon emitted in recombination process in a direct semiconductor. (Pierret, 109)

As soon as monochromatic light strike the surface of the semiconductor some of it reflects and the rest of it has intensity of I₀. This intensity decays as far the photon penetrates the semiconductor.

The intensity of light at the distance x from the surface is I. After a photon transfer its energy to the semiconductor, pairs of electron-hole will be created. The rate of photogeneration / is , which is the essence of created current in the material and is function of depth of light penetration and the frequency of the light.

If n and p are number of generated minority carrier, holes in the sea of electrons then:

;

is the rate of photogeneration/ at the surface of semiconductor where the light strikes first.

Therefore if there is not any other effect than light on the semiconductor the generated current is directly proportional to the photogeneration rate. For example in a N-type semiconductor if the minority carrier is p and Dp is the exceeded minority carriers as a result of light then:

is the minority lifetime in from generation to recombination process. As a result of quantitative solution and boundary conditions to the photogeneration problem we can formulate the following equation for the excess minority carriers, in here holes.

The current that is produced as a result of photogeneration process can be calculated as follows:

A is the area, L is the length and q is the charge as a result of generated electron-hole pairs. Also:

The Nature of Light - Part 4 (Einstein and the Photoelectric Effect)

Heinrich Hertz discovered the photoelectric effect in 1887. Einstein in 1905 developed Planck’s quanta and introduced the photon.

Photoelectric-effect circuit. (Peleg, 1)

In photoelectric effect a piece of metal sheet is biased above a threshold voltage V₀ and exposed to the light. A galvanometer measures the current upon light incident. When monochromatic light with high enough frequency falls on a metal electrons eject form the sheet to the anode pole, this happens instantaneously even for a very weak light intensity. This means that a change in the frequency of the radiation changes the maximum kinetic energy of electrons, , while a change in the light intensity does not affect this energy. However the current read by the Galvanometer is intensity dependent.

The following figures illustrates the properties of photoelectric effect:

Properties of photoelectric effect. (Peleg, 2)

a) If the light intensity stays constant the current proceed to its steady-state position. The transition time is about .

b) The relationship between light intensity and the photoelectric current is linear.

c) The photocurrent stops at potential that reaches the maximum energy of electrons.

d) For different frequency of light there is a different maximum energy.

But in classical explanation of light the intensity of light determines the maximum energy absorbed by the electrons. However we just saw that based on Planck’s quantum theory the maximum absorbed by the electron is frequency dependent not intensity.

If E = hf is absorbed energy by the electron through light incident, then:

in the frequency form we have:

“” is the minimum energy to overcome atomic binding energy to generate a free electron.

“v” is the speed of the electron proportional to the frequency of the incident light.

Therefore the minimum threshold frequency is:

In Einstein’s relativity theory where is the rest mass of a particle.

If then where p is the momentum p = mv. In case of photon then:

E = pc.

Therefore:

E = pc = hf

p =hl

These are the Planck-Einstein relations.

Based on observation it was discovered that an electron in an atomic structure could absorb and cancel discrete frequencies. This formed the Niels Bohr model of atomic structure in terms of wrapping orbits around the core of the atom. His mathematical model of orbits is:

The difference in the energy level: