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### Transcript of Photoelectric Effect in CCD (Charge Coupled Devices)

Mohsen Mirtalebi ECE491 Prof. Maher Rizkalla Final Report Photoelectric Effect in Charge Coupled Devices

Abstract This paper is to bring a mathematical model of light from Maxwell and Einsteins points of view and the effect of photon on MOSFET devices. At the beginning, there will be an introductory concept of light in Newtonian theory and also as part of electromagnetic spectrum as well as an entity in quantum mechanics. Then the emphasis would be on the mathematical model of light in electromagnetic form and photon form. Finally, It comes to adopt the photon form of light to explain the photoelectric energy conversion in charge-coupled devices. It has been always fascinating for me to write and talk about light.

From Newton to Einstein Isaac Newton recognized light as a beam of particles. His theory about light could very well answer why light travels in a straight line. In addition, his theory could comply with rules in geometric optics and the properties of mirror, lenses, and prism. In the Newtonian fashion, light is shown as a straight line.

A glass prism spreads white light out into a spectrum. (Baierlein, 16)

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But Newtons theory failed when it came to explain diffraction and interference of light. Thomas Young set up an experiment that Newtons theory could not answer the mechanism of the his experiment. He allowed sunlight to pass through a pinhole in the sheet S1 and then pass through two pinholes in sheet S2 and observing the result on the sheet S3. The result was bunch of dark and light stripes.

Youngs experiments set up. (Goldin, 91)

Youngs experiments real results. (Goldin, 90)

Mirtalebi In the wave theory of light we assume light is an infinite continues monochromatic plane wave. f(x,t) = A cos (kx - t) The constant A is the amplitude of the wave, k = 2 / and = 2 / T, T is the period, and is the wavelength. The wave is traveling along x-axis. This suggested model for light could explain the Youngs experiment until they realized that wave is a state of material and it propagates with means of matter.

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However we know that light travels in free space where there is no matter to let the wave travel. This problem was answered with Maxwells equation in electromagnetic wave where the wave was riding on electric and magnetic fields. Maxwells equations in free space: H = 0 E / t E = -0 H / t .E=0 .H=0 Where B = 0H, D = 0 E, and 00 = c-2. The first equation says if electric field changes by time, it causes the magnetic field changes in different way, which is the curl of magnetic field at that point. The second equation says that if magnetic field changes with time it similarly causes the curl of electric field at that point, which means we once more have a changing electric field but this new electric field is a bit distant from the original changing electric field. Maxwell says that the changing electric field travels with the speed of light.

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E = (Ez / y - Ey / z) i + (Ez / z - Ez / x) j + (Ey / x - Ex / y) k =-0Hz /t H=(Hz / y - Hy / z) i + (Hx / z - Hz / x) j + (Hy / x - Hx / y) k =-0Ey/t If we assume that the electromagnetic wave is one dimensional field such Ey, then the field becomes a function of x and t resulting in: Ey / x = -0 Hz /t. Hz / x = -0 Ey/t In addition we now can predict the plane waveform for the electric and magnetic fields: Ey = E0y sin (kx - t). Hz = H0z sin (kx - t). c = / k..

The Maxwells electromagnetic wave including Ey and Hz. (Goldin,120)

The energy is associated with this electromagnetic wave is the average energy of the sum of electric and magnetic fields: E = volume (0 E2 + 0 H2 ) dv.

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For the complete solution of the electromagnetic wave we need to conclude the boundary conditions and all harmonic waves. If we choose the length L somehow that E(x,t) = E(x+L,t) applies and k = 2 n/L then: E = k E0k sin (kx - kt) and Ek = 0L 0 E2k dx.

Several years after Maxwell introduced his model, which concluded light as a electromagnetic wave the black body radiation dilemma arose. In the black body radiation we learn that a radiant energy emitted from a material body is temperature dependent and is independent of the property of the material. In the other words a material emit the same amount of energy that it absorbs. An ideal black body material absorbs all energy it gets and radiates all energy it has absorbed. The total radiation of an ideal black body is expressed as its intensity and is: I(T) = 0 I(,T) d and by Stefan-Boltzmann law is, I(T) = T4, = 5.6710-8 W/m2 K4.

Intensity spectral for the different temperature T4>T1. (Goldin, 127)

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All models within the electromagnetic theory failed to be fitted into the real data in black body experiments. The best model that could get too close to the real data was Wiens law, which is: I (,T) = a1e-a2/T/5. The results could not fit into the real data as the wavelengths were increasing.

Wiens law vs. experiment. (Goldin, 127) Max Planck introduced quanta in order to explain the black body radiation. He believed that each oscillator has a discrete energy states and can absorb or emit a quantum of energy E = hf. Plancks radiation law was formed as the following equation: U(,T) = 8hc/(5(ehc/KT-1)).

Mirtalebi Heinrich Hertz discovered the photoelectric effect in 1887. Einstein in 1905 developed Plancks quanta and introduced the photon.

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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, Emax = e V0, 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:

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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 10-9. 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 Plancks 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: hf = W0 + mv2max. in the frequency form we have: eV0 = hf hf0

Mirtalebi W0 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: f0 = W0/h.

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In Einsteins relativity theory m = m0 (1- v2/c2)-1/2 where m0 is the rest mass of a particle. If E = mc2 then E2 p2c2 = E20 where p is the momentum p = mv. In case of photon m0=0 then: E = pc. Therefore: E = pc = hf p =h These are the Planck-Einstein relations. Based on observation it was discovered that an electron in an atomic structure can absorb and omit a 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: En = -22me4/n2h2 The difference in the energy level: fh=Eupper Elower

Mirtalebi 10 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:

F(x) = 1/2 G(k) e jkx dk. Where xk 1

The Gaussian wave-packet and its Fourier transform. (Goldin, 82) The G(k) function is Gaussian distribution function. The smaller k gets the more F(x) spreads out along x axis. When k 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.

Mirtalebi 11 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 type 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. I = I0 e-x 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/cm3 is GL, 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 minor