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 Einstein’s
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 Newton’s theory failed when it came to explain diffraction and interference of
light. Thomas Young set up an experiment that Newton’s 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.
Young’s experiment’s set up. (Goldin, 91)
Young’s experiment’s real results. (Goldin, 90)
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In the wave theory of light we assume light is an infinite continues
monochromatic plane wave.
f(x,t) = A cos (kx - ωt)
The constant A is the amplitude of the wave, k = 2π / λ and ω = 2π / T, T is the
period, and λ is the wavelength. The wave is traveling along x-axis. This suggested
model for light could explain the Young’s experiment until they realized that wave is a
state of material and it propagates with means of matter.
However we know that light travels in free space where there is no matter to let
the wave travel. This problem was answered with Maxwell’s equation in electromagnetic
wave where the wave was riding on electric and magnetic fields.
Maxwell’s equations in free space:
∇ × H = ε0 ∂E / ∂t
∇ × E = -µ0 ∂H / ∂t
∇ . E = 0
∇ . H = 0
Where B = µ0H, D = ε0 E, and µ0ε0 = 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 =-µ0∂Hz /∂t
∇ × H=(∂Hz / ∂y - ∂Hy / ∂z) i + (∂Hx / ∂z - ∂Hz / ∂x) j + (∂Hy / ∂x - ∂Hx / ∂y) k =-ε0∂Ey/∂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 = -µ0 ∂Hz /∂t.
∂Hz / ∂x = -ε0 ∂Ey/∂t
In addition we now can predict the plane waveform for the electric and magnetic fields:
Ey = E0y sin (kx - ωt).
Hz = H0z sin (kx - ωt).
c = ω / k..
The Maxwell’s 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 = ½ ∫volume (ε0 E2 + µ0 H
2 ) 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 = 2π n/L then:
E = Σk E0k sin (kx - ωkt) and Ek = ∫0L ε0 E
2k 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(λ,T) dλ and by Stefan-Boltzmann law is, I(T) = σ T4, σ= 5.67×10
-8 W/m
2 °K
4.
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 Wien’s
law, which is:
I (λ,T) = a1e-a2/λT
/λ5.
The results could not fit into the real data as the wavelengths were increasing.
Wien’s 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.
Planck’s radiation law was formed as the following equation:
U(λ,T) = 8πhc/(λ5(ehc
/λKT-1)).
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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, 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 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:
hf = W0 + ½ mv2
max.
in the frequency form we have:
eV0 = hf – hf0
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“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.
In Einstein’s relativity theory m = m0 (1- v2/c
2)-1/2
where m0 is the rest mass of a particle.
If E = mc2 then E
2 – p
2c
2 = E
20 where p is the momentum p = mv. In case of photon m0=0
then:
E = pc.
Therefore:
E = pc = hf
p =hλ
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 = -2π2me
4/n
2h
2
The difference in the energy level:
fh=Eupper – Elower
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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) = 1/2π ∫∫∫∫ G(k) e jkx
dk.
Where ∆x∆k ≈ 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.
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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-αx
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 minority carrier, electrons and holes, in the see of
then: GL ( x,λ) = ∂n/∂t = ∂p/∂t ; GL = GL0 e-αx
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GL0 is the rate of photogeneration/cm3 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 ∆p is the exceeded minority carriers as a
result of light then:
∆p = GLτp
τp 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.
∆p = GLτp (1-e-t/τp
)
The current that is produced as a result of photogeneration process can be calculated as
follows:
IL = -qGL AL
A is the area, L is the length and q is the charge as a result of generated electron-hole
pairs.
Also λ = 1.24/E ; E = hf
MOSFET Devices
The two main advantages of Metal Oxide Semiconductor Field Effect Transistors
have made them the most widely used transistors in the new digital world. They have a
much smaller size and faster switching time.
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Cross-section view of n-channel MOSFET. (Pierret, 612)
In these devices the voltage that is applied to the gate (G) controls the width or
ON/OFF states of the channel, consequently this voltage controls the flow of charges
from source (S) to drain (D) (in conventional form, from drain to source). There are three
types of current flowing inside of a MOSFET. First one is the drift current, which is
generated by an applied electric field. The diffusion current exists because of the holes
and electrons potential difference. Charges like to travel from a higher potential to a
lower potential level. The third one is the current as a result of thermal or light effect on
semiconductor. This last type of current is used to measure light wavelength and
intensity. The most advanced use of MOSFET devices is in CCD camera. A CCD is an
area array of MOSFETS that acts similar to a human eye’s retina. It converts the photon
energy to electric energy.
In MOSFETs if the transistor is of type npn it means that the substrate is a p-type
semiconductor and the channel will be a n-channel. In order to create an n-channel we
have to apply a positive voltage to the gate in order to attract electrons to the surface. We
could use the same analogy for a pnp p-channel MOSFET to see that the voltage on the
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ate should be negative. There many different MOSFET technologies available but they
are out of scope of this paper.
Charge Coupled Devices (CCD)
A side view of an array of transistors in a CCD. (Pierret, 240)
The charge coupled devices is a an image detector that was developed in 1970.
There are two types of CCD’s, linear array and area array. The linear array is typically
used in scanners but area array CCD’s are good as an advanced photodetectors in digital
cameras.
After a period of time is being exposed to light, photocharges will be transferred
one after the other to the output stage. The exposure time is named the integration time.
The readout electric signal at the output stage is directly proportional to the charge was
generated by the incident illumination falling on the CCD.
GL= -q AL / IL
A is the area, L is length, GL is photo generation rate/cm3, and q is the charge.
There is always a constant voltage applied to input and output stages. The light
illumination causes variation in this constant voltage. Therefore by measuring the output
signal after integration time we should get the photogeneration rate GL . Of course,
Mirtalebi 15
because GL is function of light wavelength λ therefore we can approximate the
wavelength of each light incident.
Let’s see how really a MOSFET forms a light trap. A MOSFET is an actually a
capacitor. The oxide layer works as an insulator. The substrate material can be n or p-
type semiconductor. The gate is made of metal. The voltage applied to the gate absorbs
the minority carriers of the substrate forming a conducting channel to let the charges pass
through. This channel in CCD is called the charge well.
Side view of a MOS capacitor. (Buil, 2)
Side of a MOS Capacitor after applying a positive voltage to the gate G. (Buil, 3)
As one can see in the last picture the gate voltage has pushed the positive charges
(holes) of the substrate down and has opened a channel of electrons. This channel will
eventually direct the current of charges to the output stage. The channel is named charge
well and is the space that charges are not in equilibarium state. It means that because of
the forced voltage, applied gate voltage, the electron-hole pairs are separated, electrons
became absorbed to the surface and holes are pushed back to the bottom of the device.
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Because Semiconductor in general and MOSFETs specificly are very temperature
dependent there is a phenamenon known as dark current. This current is the result of
thermal energy absorbed by the semiconductor during a period known as thermal
relaxation time without the presence of light. In CCD technology we should always avoid
dark current because the only thing we need to impact our CCD is light not the heat. To
get around this problem we need to cool down the CCD in order to minimize the process
of electron-hole pairs generation by the temperature. Sdlkjfnbs
On the other hand the depth of the charge well is directly related to the voltage applied to
the gate. This means the stronger gate voltage the deeper charge well. As you see in the
following picture the two identical MOSs have different volume of well. This is because
V2 has a higher voltage that V1. Notice that there is a barrier between two wells this is
because of the gap between two gate leads put them apart.
Two identical MOS with different gate voltage. (Buil, 4)
The charge transfer machanisim is based on the arrangement of MOS and their
gate voltages. If two considered MOSs were so close to each other then there would not
be any barrier in between them.
The same MOSs but now closer together. (Buil, 5)
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There are numerous techniques for transfering charges mainly based on the phase
difference between voltages for each gate. For example in three-phase charge transfer as
in the following figure is shown a deep voltage well is produced under electrod 1 by
appling an strong gate voltage. The charge accordingly will be stored under this lead.
Three-phase transfer technique. (Buil, 6)
Then when electrod 2 is prograssively asserted the charge will be transferred from
under the previous electrod to under this present electrod which meand from lead 1 to the
lead 2. If we constantly continue asserting and disasserting consecetive electrodes we can
transfer the charge from under the first electrod to the last electrod and then output stage
to read the signal. This is because the charge acts as a flowing water. As water like to
travel from a higher potential level to a lower potential level, charges also want to get to
the lower energy level. These techniques are for preventing charges to be mixed together.
We need to transfer each packet of charge separately to measure it in a voltage scale.
Each amount of charge transfers valuable information about the picture.
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There are also different number of phase that can be used to transfer charges. The
advantage of using phases more than three-phase is speed. For example in four-phase
clock timing sequence the structure four consecutive MOS are in charge of one packet of
charge. This reduces greatly the chance of charge backflow.
After all transfer quality is the most important part of transfering charges. During
switching the phases there would be certain amount of charge that are left behind. These
charges either can be recombined or mixed with the next packet of charge. This problem
is known as transfer inefficiency. The main reson behind this problem is the impurity in
semiconductor that holds some charges still. To reduce this phenomenon we should use a
burried channel to facilitate charge transformation through the device. If we are using a
npn transistor we need to add a n-channel between two ends of the transistor.
The last and most complicated part of a CCD is the output stage. Usually
engineers should design this part because the CCD manufacturers would not add this part
to their CCD array. It is simply because the output stage will form the final shape of the
video signal and there are different type of video signals for different standards such as
PAL, NTSC, SECAM, and so on.
The next figure shows the output register of a CCD’s output pin. This satge use a
floating diode technique. This technique is based on precharging a diod which acts as a
capacitor at a refrence level. The capacitor is then partly discharged by a packet of
incoming charge. The difference between the refrence level voltage and the varied level
voltage is the encountoured photon energy at that particular x and y coordination system
in a CCD area array.
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A typical output stage of a CCD array. (Buil, 9)
At time t=t1 a signal 0R is send to the transistor Tr1 to turn it on. Consequently
the capacitor Cs is being precharged to the value of Vdr voltge. At time t2 the switch Tr1
is off which isolates the Cs. At time t3 electrod 02 releases the carrying charge into the
output stage. Remember that the upper side of the capacitor is alrady positively charged.
The charge is beinging transferred is negative therefore as soon as the carried charge
arrives at the output satge it reduces the potential level of the capacitor making a
deviation in the refrence level. Then the transistor Tr2 reads this new voltage.
∆V = q N G / Cs
where ∆V = Vrefrence - Vread ; q = electron charge; N= Number of electron per packet
G = gain at the amplifier; Cs = capacitance of the output diode.
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Bibliographic Refrences
Baierlein, Ralph. Newton to Einstei: The Trail of Light. Great Britain. Cambridge
University Press. 1992.
Buil, Christian. CCD Astronomy: Construction and Use of an Astronomical CCD
Camera. Richmond, Virginia. Willmann-Bell Inc. 1991.
Goldin, Edwin. Waves and Photons: An introduction to Quantum Optics. United Sates.
John Wiley & Sons, Inc. 1982.
Hayat, William H. Engineering Electromagnetics. United States. McGraw-Hill. 2001.
Peleg,Yoav. Schaum’s outline of Quantum Mechanics. United States. McGraw-Hill.
1998.
Pierret, Robert F. Semiconductor Device Fundamentals. United States. Addison-Wesley.
1996.
Ziemer, R.E. Principle of Communications: System, Modulation, and Noise. United
States. John Wiley & Sons, Inc. 1995.
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