1
A Review of the Electromagnetic Properties of
Photovoltaic Materials
Paul Sokomba, ENGR 302, Calvin College. Supervision: Prof. Ribeiro

Abstract— a solar cell is a semiconductor designed to take
advantage of the electromagnetic properties of the
material and enable it to convert solar energy to
electricity. They provide the primary power for many
terrestrial and space applications and their use is very
commonplace. To obtain the highest device performance,
optical engineering is employed through the proper
selection of a number of material properties and an
understanding of the electromagnetic properties of the
semiconductor that facilitate it to generate electricity. In
this paper, the electromagnetic characteristics and power
generation technologies of solar cells are discussed and a
model showing the physical dynamics of the photoelectric
effect by the application of these technologies is presented.
I.
INTRODUCTION
Solar cells are semiconductors with the ability to convert
solar energy into electrical energy. The sun’s spectrum is
roughly equivalent to that of a black body radiating at 5800 K.
A significant portion of the energy in this spectrum is in the
visible range (400nm-700nm).
Solar energy or flux consists of tiny particles called photons.
Photons carry a certain amount of energy that can be absorbed
by the electrons in a solar cell. These charged electrons with
their increased energy, leap out of their bonding state to an
excited state. In the presence of an electric field, these
electrons will flow in only one direction (current) and
complete an electric circuit.
The first stable solar cells were invented in Bell
Laboratories in the early 1950s. Through the ‘90s, the concept
and design of solar cells went through a major revolution and
became more popular, especially for domestic (terrestrial) use.
This paper describes the electromagnetic properties of solar
cells and power generation technology by means of the
photoelectric effect. Section two describes solar irradiance as
the source of photovoltaic (solar cell) energy, and describes
important concepts and laws associated with solar energy.
Section 3 describes the material properties of solar cells,
including the quantum mechanic theory that governs its
operation. Also discussed are the photoelectric effect and the
electromagnetic laws that are associated with semiconductors.
Section 4 describes the P-N junction and its role in device
operation. Section 5 describes the properties of the solar cell
and section and section 6 outlines the various types and
applications of solar cells. Section 7 discusses an innovative
solar cell technology. The paper concludes with a model for a
solar cell using MATHCAD that will apply some of the
concepts discussed in this paper.
II.
SOLAR RADIATION
This section discusses the properties of the solar spectrum and
how it is important to solar power generation. Discussed are
the irradiance constant, Spectral irradiance, the concept of
black-body radiation and two important laws which are SefanBoltzmann’s law and the Plank’s constant.
A.
SOLAR SPECTRUM
As earlier stated, the sun can be assumed to be a black body
radiating at 5800K. This is because the spectrum emitted by
the sun is closely related to the spectrum of a black body, as is
shown in figure 2. The sun mostly emits particles within the
visible light range i.e. from wavelengths between 400nm700nm. The solar energy reaching the earth is attenuated
considerably by the earth’s atmosphere, and part of the energy
is reflected by the constituents of the earth’s atmosphere. The
spectral irradiance is the power received by a unit surface area
within a particular wavelength and has units W/m^2um. The
solar constant or irradiance is the rate at which solar energy is
incident on a normal surface to the sun’s rays at the outer edge
of the atmosphere when the earth is at a mean distance from
the sun [11]. Two spectral distributions are defined for the
solar constant:
 AM0, which is the spectral irradiance outside the
earth’s atmosphere, or 1353 W/m^2, and
 AM1.5 which is the spectrum at sea level
AM stands for Air Mass and is 0 at outer space and 1.5 at
sea level.
FIGURE 1: Electromagnetic spectrum
2
B.
FIGURE 2: Spectral irradiance at AM0 and AM1.5 courtesy
B.
PLANCK’S CONSTANT AND ELECTROMAGNETIC SPECTRUM
Contained in the solar energy flux are particles which carry
energy as wave packets. These particles are called Photons.
This theory was presented in 1900 by Max Planck. His view
holds that each photon of frequency has energy described by:
e
h 
h c
( 1)

where v is the frequency of the photon, c is the speed of light
in a vacuum, λ is the wavelength and h is Planck’s constant
equal to 6.625x10^-34 J sec. From equation (1), it can be seen
than particles with shorter wavelength have larger photon
energies and thus will be desirable for solar cell operation.
The energy within a photon may be transferred to charges in
a static field which cause the charges to become excited and
move to a higher potential. This is called the photoelectric
effect.
Color
Violet
Blue
Green
Yellow
Orange
Red
Wavelength
Band
0.40-0.44um
0.44-0.49um
0.49-0.54um
0.54-0.60um
0.60-0.673um
0.63-0.76um
TABLE 1: Wavelength ranges of the different colors of the visible
spectrum
III.
SEMICONDUCTORS
This section presents the material properties of a
semiconductor, particularly the atomic structure and the
quantum mechanics that govern the theory of operation.
A.
QUANTUM MECHANIC THEORY OF SEMICONDUCTORS
The quantum mechanic theory is used to explain such
interesting phenomena as the photoelectric effect, blackbody
radiation and radiation from an excited hydrogen gas. It further
explains the concepts of quantization of light into photons and
the particle-wave duality of light.
PHOTOELECTRIC EFFECT
The photoelectric effect is an experiment which
demonstrates the quantized nature of light. The experiment
reveals that when a monochromatic light is incident on a metal,
electrons are released. This theory shows that electrons are
confined within the metal and if sufficient energy is delivered
to the metal, the electrons will absorb this energy and be
liberated. Albert Einstein described this phenomenon as the
quantized nature of light. According to Einstein’s theory, the
kinetic energy of the electrons equals the energy in the photon
minus the energy required to extract the electrons from the
metal. This is called the work function or Ф.
So for a metal with a work function of 4.3V, the minimum
photon energy to emit an electron through the photoelectric
effect is:
 19
Q  1.6  10
e
  4.3V
C
 19
Q 
6.89  10
J
(2)
and the frequency of the photon is determined from
(1) as:

 19
e
6.89 10
h
6.626 10
J
1040THz
 34
J s
The wavelength of the emergent photon can also be
determined from (1) as:
 34

6.626 10
h c
8 m
 J s  3 10 
 19
e
6.89 10
s
0.288m
J
C.
BLACKBODY RADIATION
A black body is a material with the ability to absorb all the
energy incident on it as well as emit all its energy. Planck
described this phenomenon based on the assumption that the
energy associated with light is quantized. This led to the
definition of the spectral density from a black body as:
u
h
2 3
 c


3
 h  


k T 
e
1
(4) [1]
Where h is Planck’s constant, ω is the radial frequency, k is
the Boltzmann’s constant and T is the temperature. The peak
black body radiation occurs at 2.82kT and increases with third
power of the temperature [1]. Thus if the sun has a peak
wavelength of 900nm and it behaves like a black body, the
estimated temperature of the sun is:
3
h c
e
 19
2.21  10
9
J
900  10
T
D.
 19
e
2.21  10
2.82k
2.82  1.38  10
J
 23
5672K
J
The energy level of an electron is described using the
Schrodinger wave function. It describes the probability that an
electron is in a particular energy level in a quantum well. The
wave function also quantizes the available energies of the
particle i.e. each electron in an atom has energy in form of
quanta or packets (Einstein’s theory). Schrodinger’s equation
for a particle is:
THE BOHR MODEL
The Bohr model of an element, proposed by Neils Bohr helps
to explain the electronic configuration of elements in the
periodic table. This is important because each element used to
develop semiconductors has certain characteristics that enable
electrons to transcend from one energy level to another and
conduct electricity. Bohr provided a model of the atom which
helps to explain the energy levels an electron could occupy in
an atom. He proposed that electrons move in a circular orbit
round the nucleus in a circular trajectory as shown in figure 3.
E ( x)

h
2

d
2
2 m d x2
 ( x)  V( x)   ( x)
(8),
and the probability density function, which gives the
probability that electron is contained within a one dimensional
infinite quantum well is:
P( x)



 ( x)    ( x)
( 9)

P( x) d x
1

FIGURE 3: Bohr model of the trajectory of an electron around the
nucleus. The figure is the model of a hydrogen atom.
The model holds that the electrostatic force on the
electron by the proton in the nucleus equals the centrifugal
force and hence:
m v
2
mrv
2
q
2
( 5)
2
2
4 q
 o  r
The probability density function is interpreted as the phi
function multiplied by its complex conjugate ψ *. In an infinite
quantum well, the potential energy is zero within the well and
infinite outside the well. Thus, the electron is less likely to be
found outside a well and more likely to be found inside.
Therefore the wave function is zero outside the well.
Following from [1], the energy corresponding to a specific
value of n (from (6)) is:
En
2

2 Lx 

2 m 
h
En
mo  q

 n ( x)
2 2 2
n
1  2...
8 o  h  n
( 6)
[1]
2 level
4 of the electron and mo is the mass
where n is the energy
mqo  q
Velectron.
( r)  The
of a free E
potential energy of the proton is equal

n
4potential
2 o  r and is calculated relative to radial
to the electrostatic
2 2
8 o  h form
 n the nucleus as:
distance of the electron
V( r)
E.

q
2
4  o  r
SCHRODINGER’S EQUATION
( 7)
n
2
(10)
(the asterisk in m* marks the complex conjugate) where Lx
marks the boundary of the well. The wave function normalized
to give a probability of unity that an electron is within an
infinite well is:
r
2
4 of
othe
 r electron is found as:
And the total energy
4
 
2
Lx
 2 m En 
 x
 h

 sin 
for 0 < x < Lx
(11)
(For a complete derivation of this formula, see
[1]) According to the Pauli exclusion theory, for a given
orbital shell around a nucleus, no two electrons can have the
same quantum number. This theory governs the distribution of
electrons into different levels if not the electrons will be all
piled up in the lowest energy level where n = 1! The two
important quantum numbers are n and s, where s denotes the
spin of the electron which could be a positive or negative half
integer spin.
4
Es
Q
2
4  o  r
(13)
for a point charge,
Es
The solutions above can be extended to 3 dimensional
quantum wells. In a simple application, if an electron is
confined within a 1 micron layer of silicon and the
semiconductor is assumed to be a one dimensional quantum
well with infinite walls, the lowest possible energy within the
material is:
En
2
n 
 
2 Lx 

2 m 
2
6.626  10 242
1


 31 
6 
2 0.26  9.11  10
k  2  10 m 
2
 25
2.32  10
J
Es
H.
G.
2
I.
4 r
2
(12)
The point of Gauss's Law is that it uses the symmetry of
special cases to simplify the calculation of electric field
properties for those cases:
2
x
V
2

y
2
V
2

2
z
V
 v

SEMICONDUCTOR PROPERTIES
The electronic properties of a semiconductor material are
intimately related to their lattice structures. The basic structure
of a semiconductor crystal is presented in this section. The
material studied is the crystal structure of silicon and how the
atoms are arranged in its lattice.
J.
o
2

which the solution for a three dimensional structure in
Cartesian coordinates.
Gauss’ law is one of Maxwell’s equations and relates the
charge density to the electric field. It states that “the electric
flux passing through any closed surface is equal to the total
charge enclosed by the surface” or:
Q
(15)
(16)
GUASS’ LAW
Q
2 o
POISSON’S LAW
 V
ELECTROMAGNETICS OF SEMICONDUCTORS

 E dS
 s

s
Poisson’s equation extends from the point form of Gauss’s
law. This law presents the relationship between charge density
and potential. It is outlined in [8] for (16) below and gives the
scalar potential of a charge distribution.
1.45meV
This section presents the use of electromagnetic laws to
calculate the electrostatic potential within a semiconductor
device as a function of the existing charge distribution. The
two important laws outlined here are Gauss’ law and Poisson’s
law.
(14)
for a sheet charge.[8]
Where m* is evaluated by considering the effective mass of
electrons in silicon is 0.26mo, and mo is equal to 9.11 x 10^31.
F.
2  o  r
for a line charge and
FIGURE 4: Potential of an infinite well, with width Lx. Also shown
are the 5 lowest energy levels.
h
L
BRAVAIS LATTICES
The Atoms of elements organize into an array of different
types. These structures formed are called a crystal lattice.
There are altogether 14 Bravais lattices that can be formed by
materials.
Silicon is element number 14 on the periodic table. Thus, it
has 14 positively protons charged in its nucleus and 14
negatively charged electrons orbiting the nucleus. Of the 14,
10 are filled in the 2, and 8 orbital shells of electrons, leaving
4 available for conduction of electricity. These are called the
valence electrons. These electrons bond with 4 other atoms to
make up a crystal structure.
5
Hexagonal
Packed
Wurtzite
Close
Lithium (Li), Cadmium (Cd)
Galluim
Nitride
(GaN),
Induim
Nitride
(InN),
Cadmuim Sulphide (CdS)
TABLE 2: Summary of some crystal structures and examples of
semiconductors that have them.[1]
L.
FIGURE5: Crystal structure or lattice of silicon showing the bond
with
neighboring
silicon
atoms
courtesy
http://www2.ece.jhu.edu/faculty/andreou/487/2003/LectureNotes/
Doping8b.pdf
ENERGY BANDS
In this section, energy bands are studied and the concepts
discussed in earlier sections are combined together. Energy
bands are levels of spaced energy that exist in a crystalline
material. Wave functions of electrons overlap with those of
neighboring atoms and owing to Pauli’s Exclusion Principle,
there are only certain levels electrons can exits in, hence
energy bands are created.
The double line in between each bond indicates that each
shares its four electrons with four from four neighboring
atoms, thus having eight electrons in its outer shell. The
sharing of electrons keeps the crystal structure intact and this is M. ENERGY LEVELS
called a tetrahedral bond because each atom is connected to
It was discussed in section 3.3 that in an atom, electrons
four other atoms. Most of the electrons in this bond are
orbit around the nucleus. It was also considered that electrons
trapped in the bond and are not available for conduction. So a
can exist in only certain energy levels and never in between.
pure silicon crystal conducts only very minimal current.
The electrons further away from the nucleus have higher
Other elements form other bonds for example trigonal, cubic energy that those closer to the nucleus and as such since every
and hexagonal. Most semiconductors are manufactured in such atom will always tend towards the lowest energy possible
a way that they have more electrons to conduct, i.e. one free (Silicon forms four other bonds to attain a stable state), the
electron in the bond with four other electrons. This concept is first place to cut the energy is from the valence electrons. If
two atoms come form bonds and the electrons are at the same
called doping and is studied in a later section.
energy level, the electrons will separate minimally and from
bands (Pauli’s exclusion principle). When billions of atoms
K.
SEMICONDUCTOR CRYSTAL STRUCTURES
with similar energy levels come together, each level must shift
There are three basic vectors that govern the crystal slightly and be different from the next. This explains the theory
structure of a semi conductor and give it its name [1]. Based of a quantum well, a region where an electron can exist
on these three vectors, the packing of the atoms in a lattice virtually any where within the region. Each of these new
structure can either be simple cubic, body-centered cubic, face energy bands is separated by energy band gaps.
centered cubic, diamond, zinc blend, hexagonal close packed
and Wurtzite. A lot of semiconductors either have a diamond N. VALENCE BAND AND CONDUCTION BAND
or zinc blend structure, which have tetrahedral basis. Silicon
There are two very important types of bands than exist. The
forms the diamond structure while compounds like Gallium
valence
band, which is the most filled band at temperature T=
Arsenide (GaAs) from zinc blend crystal. Most III-V
0
K,
and
the conduction band which is the first empty band at
compounds (like GaAs) from zinc blend structures. A
T=
0K,
or
the first unoccupied energy level. Electrons can
summary of structures and examples is presented in table 2.
exist
in
either
of these bands. Electrons in the valence band do
Depending on the crystal structure of a semiconductor, a
not
conduct
electricity
and are tightly bonded. However,
suitable dopant must be found.
electrons in the conduction band are free to move around the
material and thus conduct electricity. The valence and
Crystal Structure
Example Semiconductore
conduction band are separated by a band gap and is similar to
Diamond
Carbon (C), Germanium the work function earlier described. The band gap is thus the
energy required to promote an electron from the conduction
(Ge), Silicon (Si),
band to a valence band. This involves breaking free of the
Zinc Blend
(IV-IV) Silicon Carbide (SiC) bonds and leaping to the conduction band. An electron must
(III-IV)
Aluminum find somewhere to source this energy from, and this is the
Antimonide (AlSb), Galluim theory behind the solar cell operation. As the discussion
Arsenide (GaAs), Induim progresses, it will become more apparent how the band gap
Phosphide
(InP) plays a big role in the conduction process. The figure below
(II-VI) Zinc Sulphide (ZnS)
shows a sample of band gaps for some semi conductors, where
the graphs are plotted for energy against wave vector, which is
6
the symmetrical constant of the lattice structure. The odd
shapes of the graphs are as a result of the crystal structure of
the material.
The material band gap of a semiconductor ranges between
1.5 to 4eV, however band gaps as low as 0.7eV have been
achieved for the manufacture of blue light emitting diodes.
For a semiconductor with the top of the valence band at say
3eV and the band gap between the conduction and valence
band at say 2eV, an electron will need approximately 5eV or
more to reside in the conduction band, or at the bottom of the
conduction band.
P.
DIRECT AND INDIRECT BAND GAP
Other important features of band gaps especially in
optoelectronic devices (like solar cells) are the direct and
indirect band gap nature of the material. If the conduction
band minimum and the valence band maximum occur at the
same wave vector number, then the material is said to have an
indirect band gap. From figure 6, GaAs has a direct band gap.
If this is not the case, as is for the other two Ge and Si in figure
6, the material is said to have an indirect band gap. The
distinction is of interest because direct band gap materials
provide more efficient absorption and emission of light.
Q.
FIGURE 6: Energy band diagrams for Ge, Si, and GaAs. Eg is the
Energy band gap and the (+) indicate holes in the valence bands and
(-) indicate electrons in the conduction band. Courtesy: Physics of
semiconductors [ref physic_semi_1].
From the figure above, it can be seen that for Ge, the bottom
of the conduction band can either be at the axis <111> (L) or
at the axis <100> (X) or at Г.
O.
TEMPERATURE DEPENDENCE ON BANDGAP
As the temperature increases, the band gap of semi
conductors tends to decrease. This is due to the fact that the
interatomic spaces increase as the interatomic vibrations
increase due to an increase in thermal energy
[phys_semi_cond]. The temperature dependence on band gap
is given by (17) below [1].
METALS, INSULATORS AND SEMICONDUCTORS
Eg ( T)
Eg ( 0) 
T 
( 17)
The previous section introduced the topic of energy bands.
T
Using this theory, the conduction of electricity through metals,
insulators and semiconductors will now be explored. Figure 7
where  and  are fitting parameters of the
shows the energy band diagrams for metals, insulators and
semiconductor, and T is the temperature
semiconductors. Since the band gap is the energy required to
promote an electron from the valence band to the conduction
R.
ELECTRONS AND HOLES
band, the length of this band is what determines and materials
conductivity. Metals have no band gap i.e. their valence and
This section describes the current generation mechanism of
conduction bands overlap and as such readily conduct the solar cell. It introduces the concepts of charge carriers and
electricity. Insulators on the other hand have a very wide band current density and elucidates further the concepts of energy
gap and the energy required to overcome this gap is usually bands.
much larger than is practical. Semiconductors are in between
and have band gaps that can adequately be crossed to the next S.
ELECTRON EXCITATION
level.
As discussed earlier, a certain amount of energy is required
to promote an electron from the valence band to the
conduction band. The source of this energy could be from
heat, potential field or a light source. In our case, it is a light
source. The use of light to excite electrons is called optical
FIGURE 7: Energy
generation.
band gap analogy for
metals, insulators and
semiconductors.
In optical generation, a photon travels and strikes an
electron in a semiconductor material, like silicon. The
electron, trapped in a bond will be accelerated by the new
found energy and if the energy is large enough, the electron
7
will break its bond and move to the conduction band. Once
there, it is free to float and conduct electricity.
T.
electrons). The sum over all the states is zero and so equation
(20) shows the current is due to the positively charged
particles in the almost filled band.
CHARGE CARRIERS
When an electron is free from the valence band and leaps to
the conduction band, a vacancy is created within the now
empty bond. This vacancy is called a hole. Holes are positive
charge carriers while electrons are obviously negatively
charged carriers. Holes are conceptual and only stand for
missing electrons in a bond.
An interesting concept that governs the movement of charge
carriers is Space Charge Neutrality. This theory says that a
sample material will always contain the same total amount of
charge in it, or in other words the same number of electrons.
So if a piece of silicon were connected to a battery that sends
one electron into the silicon material, one electron must move
out. This chain reaction of electrons is the concept behind
electric conduction in semiconductors. If there are no electrons
in the conduction band, then no electron will be allowed in
because for every one electron that goes in, one must come
out. This is why insulators do not conduct electricity.
Holes, described earlier move around the valence band just
as electrons move around the conduction band. But it must be
noted that holes are an abstraction. When one electron moves
from the valence to the conduction band, a nearby electron fills
its place and creates another vacancy or hole that will be again
filled by another nearby electron and so on. Thus it is said that
the hole has moved. Space charge neutrality still holds for
holes in the valence band. However the theory is that if an
electron was introduced in to a material with one hole in its
valence band (and no free electrons in the conduction band),
the electrons will shift around till the number of holes and
electrons are neutralized (back to its original number).
U.
CURRENT DENSITY
When current is passed through a material, the electrons and
holes go in opposite directions. The electrons go in the
direction opposing the current flow and the holes move in the
direction of the current.
The total current due to the electrons in the valence band is
calculated as:
Jvb
1
V


q   i
1
V


q   i 
filled_states
1
Jvb
V


1
V


q   i
( 19)
empty_states
q  i
empty_states
( 20)
[1]
Thus current density is the amount of charge passing through a
unit area, perpendicular to the direction of the flow of current,
per unit time.
V.
CARRIRER TRANSPORT
The motion of free electrons and holes leads to the current
in a semiconductor. This motion is caused by an electric field
due to an externally applied voltage.
There are two transport theories for electrons and holes one
is Drift transport, which is the motion of a carrier drifting in a
semiconductor due to an applied electric field. This field
causes the carrier to move with a velocity, v. The carriers
collide with one another as they move along the path. When an
electric field is applied across a semiconductor, negatively
charged electrons will accelerate in a direction opposite the
field and positively charged holes will accelerated in a
direction parallel to the field.
The other is the Diffusion current (gradient driven transport
process) and is the process in which electrons diffuse down
concentration gradient, and carry a negative charge. The
diffusion current points in the direction of the gradient. Holes
diffuse down the gradient and carry a positive charge and
diffusion is opposite the current. In summary, the charge
carriers move from an area where the carrier concentration is
high to areas where the carrier concentration is lower.
The applied electric field causes electrons to move with a
velocity v. Assuming all charges move at the same velocity,
the total current in a semiconductor can be expressed as a ratio
of the total charge within the semiconductor to the time it takes
to travel from one electrode to the other, as shown in (21):
( 18)
filled_states
where V is the volume of the semiconductor
q is the electronic charge and v is the
electron velocity.
Jvb
I
[1]
Q
Q
tr
L
( 21)
v
The sum is taken over all occupied states in the valence band.
This equation can also be estimated by solving for the sum of
all the states in the valence band and subtracting it from the
states in the empty states in the valence band (missing
where tr is the time constant which is a function of L, the
length of the wire carrying the charges and v the velocity of
the charges. The current density can then be expressed as a
8
function of either charge density or the density of carriers in
the semiconductor as:
J
Q
AL
v
 v
( 22)
q n v
Charges however do not move at a constant velocity but
move randomly throughout the material, even in the presence
of an electric field. The random motion is due to the influence
of the thermal energy of the electrons. Thus, in the presence of
an electric field, the charge carriers are said to have an average
net motion in the direction of the field. The motion of
electrons is visualized in Figure 8 below.
FIGURE 8: Random motion of electrons due to thermal energy.
The mobility of charge carriers in the presence of an electric
field can be calculated using Newton’s laws. The carriers’
acceleration is proportional to the applied electrostatic force
and this yields:
F
ma
d
m v
dt
(22)
charge carriers in it. For example, silicon can be doped with
phosphorous. Phosphorous is an element next to silicon on the
periodic table with five valence electrons in its outermost
shell. This means when Phosphorous is bonded to silicon, it
has one more valence electron that is free to float around the
material in the conduction band. Adding more phosphorous
electrons will increase the number of electrons in the
conduction band available for conduction. This material is
now called extrinsic and has more electrons than holes (the
electrons did not form from bond breaking or excitation). To
reinforce the concept, consider an intrinsic silicon
semiconductor with 1010 charge carriers per cubic centimeter
at room temperature. When an excess of 1017 phosphorous
atoms per cubic centimeter are implanted into the silicon bond,
the number of charge carriers are increased to 10 17 charge
carriers per cubic centimeter.
The same process can be applied to provide excess holes in
a semiconductor. Boron has three electrons in its outer most
shell. When bonded to a silicon atom, the result is that one
bond does not get an electron from Boron and is left
incomplete.
When a semiconductor has excess electrons, it is called an
N-type semiconductor because the majority charge carriers are
electrons (negative charge). Likewise, when a semiconductor
has excess number of holes, it is called a P-type
semiconductor.
PN junctions are formed when p-type semiconductors and ntype semiconductors materials intersect.
For complicated reasons beyond the scope of this study, the
mean acceleration for most metals is zero and the numerical
value of the velocity of the charge carriers is proportional to
the electric field:

v
q 
E
m
( 23)
where t is the collision time constant.
The drift current can also be expressed as a function of the
mobility:
J
q n  E
(24)
where n is the concentration of charge carriers.
IV.
PN JUNCTIONS
In this section, semiconductor pn-junctions are introduced.
The discussion will begin with a description of the pn-junction,
then following the basics of intrinsic and doped
semiconductors. Finally, the parameters of the semiconductor
junction will be analyzed of performance specific information.
A.
FIGURE 9: Electrons and Holes in a P and N type semiconductor
I.
RECOMBINATION
Recombination is the process where electrons in an intrinsic
semiconductor fill the available holes. The recombination rate
is proportional to the number of free electrons and holes,
which in turn is proportional to the ionization rate. The
ionization rate itself is a strong function of temperature. In
thermal equilibrium, the recombination rate is equal to the
ionization energy and this relation can be used to calculate the
number of free electrons or holes, which should be equal. This
can be calculated using the equation:
DOPING SEMICONDUCTORS
Intrinsic semiconductor materials are materials with the same
number of electrons and holes. Electrical conductivity is
increased in the material if the number of charged carriers is
increased. This method is called doping. A silicon material can
be doped with another element to increase the number of
 EG
2
ni
3
B T  e
k T
(25)
where ni is the intrinsic carrier concentration (holes or
electrons), B is a material dependent parameter which is 5.4 x
9
1031 for silicon, EG is the band gap of the material which is
1.12 eV for silicon, and k is the Boltzmann’s constant = 8.62 x
10-5 eV/K.
J.
DEPLETION REGION
As mentioned earlier, diffusion is the process where
electrons and holes diffuse across the junction pn-junction.
Hole that diffuse across the junction into the n region quickly
recombine with some of the majority electrons present there.
This is called the carrier depletion region where electrons are
depleted by diffused holes. The same goes for electrons that
diffuse along the junction. They too get lost in the diffusion
region by recombining with holes in the p-type material. The
depletion region is best explained by space charge theory
presented in previous sections.
The charges on both sides of the depletion region create an
electric field. This field results in a potential between these
two materials (n and p type), where the n-type semiconductor
becomes positive relative to the p-type. This potential barrier
makes it difficult for holes to diffuse to the n-type
semiconductor and this in turn reduces the diffusion current
through the region. The greater the potential, the less the
current through the junction.
In another case, the drift current caused by minority carriers
in either material can be swept across the junction. For
example, the minority holes generated in the n-type
semiconductor by thermal generation, move toward the
depletion region, where they get swept to the p-type
semiconductor. The value of the drift current it strongly
dependent on temperature and it is not affected by the potential
in the depletion region.
Under open circuit conditions, the drift current is equal to
the diffusion current since there is no external current driving
the circuit. This is because the potential in the depletion region
balances the two. The theory is that if the drift current exceeds
the diffusion current, there will be an imbalance of electrons or
holes in either material which will result in a bond change
were more carriers will be uncovered and the depletion region
will widen
The potential that exist across the depletion region is found
using the relation:
Vo
 NA  ND 

 n2 
 i  (27)
VT ln 
[3]
Where NA is the acceptor concentration and ND is the donor
concentration, VT is the thermal voltage of the material.
The width of the depletion region is given by:
q xp A NA
q xn A ND
Where x.n and xp are the width of the p side and n side
depletion regions respectively and A is the cross sectional area
of the junction. Under open circuit conditions, the width of the
depletion region could be found as:
W dep
xn  xp
2 s
q
 

1
NA

V
ND  o

1
(28)
[3]
Where εs is the electrical permittivity of the material.
V.
THE SOLAR CELL
The basic parameters for understanding how solar cells work
have been outlined in the previous sections. In the following
section, the characteristic parameters will be discussed.
A.
PHOTO GENETATION OF LIGHT
As presented in the section 2, solar cells work as a result
of electrons being liberated from their bonds. Presented here
are the important semiconductor properties that show how
effective the conduction process is for a given solar cell.
B.
ABSORPTION COEFFICIENT
The absorption coefficient depends on the semiconductor
material, in particular the band gap of the material and whether
it is direct or indirect. If this parameter is very high for a given
semiconductor, the photons are absorbed within a short
distance from the surface of the cell. However, if this
parameter is small, the photons travel longer distances. In
some cases, this value is zero and the material is said to be
transparent to the particular wavelength.

REFLECTANCE
The reflectance of the surface of the semiconductor depends
on the finishing type and the antireflection coating applied to
the cell.

DRIFT-DIFFUSION
These parameters control the migration of charge across the
junction and dictate carrier lifetimes and motilities of electrons
and holes.

SURFACE RECOMBINATION
Recombination that occurs at the surface of the cell where the
minority carriers recombine also affect the performance of the
solar cell.
C.
SHORT CIRCUIT CURRENT DENSITY
The short circuit current density is a function of the
wavelength, absorption coefficient, reflectance and the spectral
irradiance. The equation for the spectral short circuit current is
presented in the model at the end of this study. The spectral
short circuit current density must be calculated for both n-type
and p-type semiconductors. The total short circuit spectral
current density is calculated as:
Jsc
Jscn  Jscp
(29)
The short circuit current density is the integral of the short
circuit spectral current density and is found as follows:
10
Jsc




Jsc d 
0




Jscn  Jscp d 
0
(30)
The units for this equation are A/cm2
[4]
K.
It is important to note however, that the open circuit voltage
is independent on area. Thus regardless of the cell area, the
open circuit voltage stays the same.
DARK CURRENT DENSITY
The dark current operation of a solar cells us similar to that
of a diode. The resulting curve also has two equations for the
n-type and the p-type semiconductors, both presented in the
model. The total dark current density for the n and p-type
semiconductor is given as:
 V

V
 T  1
Jdark Jdarkn  Jdarkp Jo  e

(31)
where Jo is the saturation current density [4].
L.
SHORT CIRCUIT CURRENT
A simplified model governing the current of the solar cell
can be written as:
 V

V
 T  1  J
J Jo   e
 sc
(32)
These values can be replaced by the value of the current
generated by the solar cell using the relation that:
Isc
Io
Jsc A
Jo  A
with the short circuit current, and this also results in a
dependence of the open circuit voltage on the short circuit
current as discussed in the previous section of the depletion
region.
N.
MAXIMUM POWER POINT
The output power of a solar cell is the product of the output
current delivered to the electric load and the voltage across the
cell. Thus using equation (34), the power could be expressed
as:

 V
 


V
T



P V I V IL  Io   e
 1 

 (36) [4]
This equation goes to zero during short circuit operation since
there is no voltage drop. The power is also zero at open circuit
conditions.
The maximum power point is the maximum power generated
by the solar cell and basically occurs at maximum voltage and
maximum current. The relationship can be found using:
Vmax
 Vmax 


VT
 VT
 Vmax
d
P 0 IL  Io   e
 1 
 Io  e
VT
dV
 Vmax 


 VT

Imax IL  Io   e
 1
(33)
to give the equation:
 V

V
 T  1  I
I Io   e
 sc
Vmax
(34) [4]
Vmax 

Voc  VT ln  1 

VT


(37)
[4]
M.
OPEN CIRCUIT VOLTAGE
The open circuit voltage has already been visited when
discussing the built in potential of a junction. The open circuit
voltage is given as:
Vo
O.
The fill factor is the ratio between the maximum power Pmax
and the open circuit voltage and short circuit current. This
value can be expressed as:
Isc 

VT ln  1 

Io


(35)
The equation reveals that the open circuit voltage depends on
the Isc/Io ratio. This implies that under constant temperature the
value of the open circuit voltage will change logarithmically
FILL FACTOR
FF
P.
Vmax Imax
Voc Ioc
(38)
[4]
POWER CONVERSION EFFICIENCY
11
The power conversion efficiency η, is the ratio between the
solar cell power and the solar power incident on the solar cell
surface. This input power is the irradiance multiplied by the
cell area.

Vmax Imax
Pin
FF
Voc Isc
Pin
FF
Voc Isc
G Area
FF
Voc Jsc
G
(39) [4]
Where G is the solar constant in W/m^2. the power conversion
efficiency is proportional to the three important parameters of
a solar cell: the short circuit current, the open circuit voltage
and the fill factor. The figure below shows the relationship
between the Ioc,, Voc, Imax, Vmax and FF.
ISC
PMAX
The shunt resistances are more uncontrollable by the end
user as they are a result of the manufacturing process. They
occur during fabrication and are identified as localized shorts
at the n-type layer or perimeter shunts along the cell borders.
These also are lumped to one resistance model parallel to the
device called Rsh.
R.
Recombination has the effect of degrading the open circuit
voltage of the cell. The extent to the degradation is
investigated in the model that will follow at the end of this
study. The results show that there is significant degradation of
the open circuit voltage and the fill factor when recombination
dominates. The short circuit current however is not affected by
recombination.
Recombination result in the loss of minority carriers. Electrons
drop from the conduction band back into the valence band and
recombine with the holes. This is known and band
recombination
S.
VOC
FIGURE 9: Relationship between Voc and Isc with Pmax
ISC
EFFECTS OF RECOMBINATION
EFFECTS OF TEMPERATURE
The operating temperature has significant effects on the
performance of solar cells. The effects of temperature are also
investigated and presented in the solar cell model. As seen
before, a lot of parameters of a semiconductor depend on
temperature: band gap, saturation current density etc.
The effect of temperature to the saturation current density is
realized with the following equation:
Eg
Jo
VOC
FIGURE 10: The fill factor is the fitness for the I-V curve between
the ideal (sharp corner) and the actual I-V relationship (bent knee
corner)
Q.
RESISTANCE LOSSES IN SOLAR CELLS
There are some factors that affect a solar cell’s response.
These losses can be modeled as series shunt resistances.
The series resistances are resistances that exist in the
contacts that are connected to the solar cell material where the
generated current passes. Also the resistance of the metal grid
contacts and current collecting bus also contribute to resistance
losses in the generated current. All these losses can be lumped
into one resistance model called the series resistance of the
solar cell Rs.
T.
3 k T
B T  e
(40)
CHARACTERISTICS OF SOLAR CELLS
The main operating characteristics of solar cells are the
beginning of life (BOL) and end of life (EOL) efficiency. They
mainly depict the degradation of minority carrier lifetimes in
semiconductors. The result is a degradation of photocurrent
absorbed by the solar cell. The effect also reduces the
maximum power of the solar cell significantly.
BOL and EOL efficiencies are more pertinent to space
applications where the performance of the solar cell should not
degrade below a certain level to ensure power is constantly
generated through out the mission.
VI.
TYPES OF SOLAR CELLS
12
Solar cells are specified by the elements used in fabrication, as
well as the number of junctions in the solar cell. These factors
are investigated in this section.
A.
SINGLE JUNCTION SOLAR CELLS
To design solar cells, the designer optimizes current and
voltage in order to maximize power. To maximize the current,
it is obvious that a large number of photons must be captured
from the spectrum of solar radiation. Small band gaps may be
chosen to capture the lower energy photons, but this has the
effect of lowering the voltage in the cell. Increasing the band
gap will not necessarily improve the situation as less photons
will be captured by the cell. To optimize this, engineers select
band select material combinations with band gaps near the
middle of the energy spectrum of solar radiation. Silicon and
GaAs are two popular materials with band gaps of 1.1eV and
1.4 eV respectively.
Figure 12: Splitting different fractions of the spectrum [9]
Multi-junctions are created by mechanically stacking or
growing multiple layers of semiconductors with decreasing
band gaps so that the top layers absorb higher energy photons
and the lower layers absorb the transmitted lower energy
photons.
FIGURE 11: Figure shows optimal band gaps of popular solar cell materials
[9]
The figure above shows that designs of single junction solar
cells are inherently limited to 25% under 1000W/m^2 of solar
radiation [9].
B.
MULTI-JUNCTION SOLAR CELLS
With multi-junction solar cells, it is possible to capture a larger
range of photons. This is because there are more junctions that
can be tuned to different wavelengths. This is done without
sacrificing the voltage.
A different semiconductor is selected that would best match a
particular portion of sunlight as shown in the figure below:
Group III-V elements of the periodic table are popular
choices for multi-junction solar cells (tandem solar cells). By
carefully adjusting the compositions of the alloys, the band gap
can be tuned to a particular wavelength. For example
Ga0.5In0.5P is one of NREL’s record setting triple junction
tandem solar cells with a band gap of 1.85 eV and a lattice
constant of 5.65 angstroms. By reducing the ratios of Gallium
and Indium, it is possible to achieve lower or higher band gap
energies.
One important factor earlier discussed must be taken into
account when building tandem solar cells. This is the lattice
constant. The lattice constant must be the same in all layers to
produce optical transparency and maximum current
conductivity. Again, the lattice constant is the measure of the
distance between the atom locations in the crystal structure. A
mismatch in the crystal lattice constant creates defects or
dislocations in the lattice where recombination can occur,
which leads to a degradation in performance as discussed
earlier.
VII.
INOVATIVE SOLAR CELL RESEARCH
On the 22nd of November 2002, Berkley National Laboratories
announced a solar cell with the capability of converting
approximately 50% of the suns light to electricity. The highest
efficiencies currently attained by multi junction solar cells are
about 28% under 1 sun or 45% under concentrator systems [9].
13
This achievement could revolutionize the solar cell industry
and its applications.
The solar cell is composed of InxGayN (Indium Gallium
Nitride) and the crystal is grown using molecular beam
empitaxy (MBE). This method is said to produce pure indium
with a band gap of 0.7eV which is pertinent to the operation of
the device. The earlier stated value of the band gap for InN
was 2.0 eV, which the scientist believe the error was a result of
the method used to grow the crystal. The method used was
metal-organic chemical vapor deposition (MOVCD), which
unfortunately produces samples that are less free of impurities.
VIII.
1
2
At this present time, the research team is still researching
better ways to produce the cell to establish the best efficiency
possible for all layers of the cell. Most of the problems that
must be overcome are lattice mismatching and also monolithic
growth of the crystal. It is desirable that all the crystals be
grown together so that the cell is series connected and has only
two terminals. This is important so that each layer of the
device have matched currents, i.e. the rate at which they
absorb photons is the same.
FIGURE 13: Innovative solar cell research discovery by LBNL
capturing 50% of the suns light. Courtesy:
http://www.lbl.gov/Science-Articles/Archive/MSD-full-spectrumsolar-cell.html
Van Zeghbroeck B, “Principles of Semiconductor Devices” University
of Colorado http://ece-www.colorado.edu/~bart/book/
Solar Electricity – How Solar Cells Work
http://www.soton.ac.uk/~solar/intro/tech0.htm
3
4
5
6
7
8
According to the leading scientist Wladek Walukiewicz,
two layers of indium gallium nitride, one tuned to a band gap
of 1.7 eV and the other to 1.1 eV, could attain the theoretical
50 percent maximum efficiency for a two-layer multijunction
cell.
REFERENCES
9
10
Sedra, Smith, “Microelectronic Circuits” Fourth Edition, Oxford Press
Castañer, Silvestre, “Modeling Photovoltaic Systems with PSPICE”
Wiley Press
Decher, “Direct Energy Conversion” Oxford Press, 1997
Sribar, Divkovic-Puksec, “Physics of Semiconductor Devices”
Markvart, “Photovoltaic Solar Energy Conversion” School of
Engineering Sciences, University of Southampton. July 2002
Hayt, Buck, “Engineering Electromagnetics” Sixth Edition, Mac Graw
Hill Press.
NREL, Basic Physics and design of III-V Multijuction Solar Cells
http://www.nrel.gov/ncpv/pdfs/11_20_dga_basics_9-13.pdf
Wladek Walukiewicz, Full Spectrum Solar Cell, NBNL 2002
http://www.lbl.gov/Science-Articles/Archive/MSD-full-spectrum-solarcell.html
http://www.lbl.gov/msd/PIs/Walukiewicz/02/02_8_Full_Solar_Spe
ctrum.html
11
Cengel, Fundamentals of Thermal Fluid Sciences
Paul Tama Sokomba is a senior Electrical and Computer
Engineering student at Calvin College. He will be
graduation in May 2004 with a
BSE in Engineering. His home
country is Nigeria, where he
hopes someday to go back to
and
help
support
the
development of technology. He
hopes to find a full time job
after graduation and then further
his education with a Master’s
degree in Digital Signal
Processing or Control Systems later in the future.
Contact email: psokom40@yahoo.com