Excerpt
Proyecto Fin De Carrera Günther Krauss
E.T.S.I.T. Universidad Politécnica de Madrid
Page 2 of 2
2.3.2 Photoconductivity ... 17
2.3.3 Photocurrent generation in pn junction solar cells... 19
2.3.3 Quantum efficiency ...
2.4 Saturation and recombination current ... 28
2.5 Equivalent circuit of a solar cell ... 31
2.6 Fundamental parameter for solar cell characterization... 33
3 Experimental issues  Hardware... 36
3.1 Global system description... 36
3.2 Laser characterization system ... 38
3.2.1 HeNe laser ... 39
3.2.2 Beam intensity monitoring... 40
3.2.3 Electrical measurements... 43
3.3 Laserscribing system... 44
3.3.1 NdYAG laser... 44
3.3.2 Optical system... 45
3.4 Position control... 47
4 Experimental issues  Software ... 49
4.1 Main program ... 49
4.2 Low level programming of microcontrollers... 53
4.3 RS232 programming of the NdYAG laser cutter ... 55
4.4 GPIB programming of digital multimeters ... 56
5 Cell characterization  results and discussion ... 58
5.1 Basic principles of photoresponse mapping ... 58
5.2 Calculations of expected photocurrents ... 59
5.3 I/V characteristics... 63
5.4 Spectral response... 69
5.5 Photoresponse mapping... 74
6 Prospective ... 80
6.1 Optical system for improved resolution ... 81
6.2 Transmittance maps ... 82
Proyecto Fin De Carrera Günther Krauss
E.T.S.I.T. Universidad Politécnica de Madrid
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6.3 UV laser beam for selective scribing... 83
6.4 Selective wavelengths for photoresponse mapping ... 83
6.5 Lockintechnique ... 84
7 Conclusions ... 85
References... 88
Acknowledgements... 90
Appendix A (Documentation of the designed programs)... 90
Appendix C (Security precautions)...108
Tables of illustrations ...112
Tables of equations...114
Proyecto Fin De Carrera Günther Krauss
E.T.S.I.T. Universidad Politécnica de Madrid
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Deutsche Zusammenfassung
Die Diplomarbeit wurde von mir in englischer Sprache verfaßt, da die Arbeit im
Zuge eines LEONARDO DA VINCI Traineeship Programmes am CIEMAT
(Centro de Investigaciones Energéticas, Medioambientales y Technológicas,
Madrid, Spain) durchgeführt wurde und auch von der politechnischen Universität
(Institut für Solarenergie) betreut wurde.
Diese Diplomarbeit beschreibt das Design und die Montage eines experimentellen
Systems zur Charakterisierung und Strukturierung von Solarzellen und anderen
Photovoltaikbauteilen. Das designte Lasersystem kombiniert ein Werkzeug zur
Strukturierung von Dünnschichtsolarzellen mit einer Charakterisierungseinheit
(,,Photoresponse Mapping"Technik)
Das Charakterisierungsteilsystem arbeitet nach dem LBICPrinzip (laser beam
induced currents). Das System zur Strukturierung von dünnen Schichten arbeitet
mit einem NdYAG Lasercutter, der gewöhnlicherweise auch zu ICReparaturen
verwendet wird.
Ziel dieser Diplomarbeit ist der ,,Set up" des Lasersystems, welches die
Anstrengungen in der Forschungsarbeit der Entwicklung neuer Materialen für
Solarzellen und deren Charakterisierung unterstützen soll.
Benutzt man Standardmethoden (IV Kurven, Spektrale Empfindlichkeit), um
optischelektrische Eigenschaften einer Solarzelle zu messen, erhält man ein
quantitatives Ergebnis, das einem keine Rückschlüsse auf lokale Defekte innerhalb
der Probe erlaubt. Kleine Defekte innerhalb eines photovoltaischen Bauelementes
können die Leistung dieses Bauteils negativ beeinflussen. Der Vorteil der
,,PhotoresponseMapping" Technik liegt in der Möglichkeit, lokale Messungen
am Bauteil vorzunehmen.
,,LaserScribing" ist eine Schlüsseltechnologie im Herstellungsprozess von
amorphen SiliziumSolarzellen. Diese Technologie deckt bereits 50% des
Proyecto Fin De Carrera Günther Krauss
E.T.S.I.T. Universidad Politécnica de Madrid
Page 5 of 5
Fabrikationsprozesses ab und hat die konventionelle Methode mittels
Photolithografie bereits komplett ersetzt [18].
Der erste Teil dieser Arbeit soll in das Thema einführen und erläutert die
vorgegebene Aufgabenstellung. Der Theorie folgend ([19], [3]) werden die
fundamentalen Prinzipien des Energiewandlungsprozesses einer Solarzelle in
Kapitel 2 erläutert, um den physikalischen Hintergrund der
Solarzellencharakterisierung zu verstehen.
Das Kapitel 3 gibt einen Überblick über die zum Design des Systems verwendete
Laborausstattung und erläutert, wie die Einzelteile, funktionell zusammengesetzt
zum System, eine nach dem ersten Set up zuverlässig funktionierende Laboreinheit
bilden.
Im Kapitel 4 wird die Wahl der Software und der unterschiedlichen verwendeten
Schnittstellen (GPIB, lowlevelprogramming, RS232) begründet. Ebenfalls wird
ein kurzer Überblick über die Leistungsfähigkeit der Programmierwerkzeuge
gegeben.
Nach dem Aufbau eines solchen Systems ist es natürlich von großem Interesse,
erste erhaltene Meßergebnisse zu verifizieren. Dies wird im Kapitel 5 getan. Hier
wird auch der Zusammenhang zu anderen optoelektrischen
Charakterisierungsmöglichkeiten aufgezeigt und die erhaltenen Messergebnisse
werden unter Einbeziehung dieser komplementären Charakteristiken der
verwendeten Solarzellen interpretiert.
Kapitel 6 sollte einige Ideen zur weiteren Entwicklung und Verbesserung des
Systems aufzeigen und einen Anreiz geben, dieses entwickelte System für weitere
Diplom oder Promotionen zu verwenden.
Eine Zusammenfassung der Arbeit ist als siebtes Kapitel angefügt und drei
angehängte Kapitel (Appendix 13) ermöglichen einen tieferen Einblick in das
Systemdesign.
Proyecto Fin De Carrera Günther Krauss
E.T.S.I.T. Universidad Politécnica de Madrid
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Abstract
This thesis, submitted for a diploma, describes the set up of an experimental
system which basically includes two lasers and one precision positioning system.
The facility designed should be used for the characterization of solar cells by
photoresponse mapping and for scribing thin films.
The photoresponsemapping equipment uses a HeNe laser, an intensitymeasuring
photodiode, a sample holder with electrical tips mounted on the xy positioning
tables and a set of electricalmeasurement facilities, such as digital voltmeters and
operational amplifiers. Photoresponse maps are obtained by laser beam induced
currents (LBIC). The scribing system consists of a NdYAG laser cutter, a special
microscope, and a sample holder mounted on the xy positioning table.
It was the objective of this work to put all parts properly together and to reach in a
first set up a reliable working facility as well as to do a number of initial
measurements with the characterization part of the system. Individual tasks
included:
Designing a global platform for the system, mountings and sample holders, fixing
the two lasers, the positioning system and the sample holders on this platform.
Installing a photodiode for monitoring the HeNe laser intensity.
Making the electrical connections for measuring the shortcircuit currents
generated by the device under test for photoresponse mapping, which includes
connecting the operational amplifiers and the digital multimeters.
Developing individual control subroutines for the positioning table, the NdYAG
laser and the voltmeters.
Measuring a number of photoresponse maps in order to establish the validity of
the experimental set up.
The system set up represents an automated system controlled by a personal
computer (type 80286) via different interfaces. The double laser facility already
works reliable and performs the characterization of solar cells by photoresponse
Proyecto Fin De Carrera Günther Krauss
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mapping and the scribing and cutting of thin films. With the experimental results
carried out by initial photoresponse mappings, it was possible to determine the
active area, to identify local defects, to generate maps of surface heterogeneity, and
to calculate local spectral response values of solar cells or other photovoltaic
devices.
After the set up, the laser scribing system is ready to be tested and should initially
be used to optimize scribing and cutting conditions. Moreover it should be used for
research and development of photovoltaic materials and devices.
Proyecto Fin De Carrera Günther Krauss
E.T.S.I.T. Universidad Politécnica de Madrid
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1 Introduction
Worldwide 80% of all energy used comes from fossil fuels like coil, oil, and natural
gas. Since these energy resources are exhaustible, low energy consumption and the
search for renewable energy resources should be the basis of the world energy
structure of the 21st century. Obviously, solar energy can not solve all energy
problems but could deliver a part of the energy consumed worldwide.
Nevertheless, it is not part of this thesis to discuss social and political questions.
They should be discussed elsewhere.
The topic of this thesis is the set up of a double laser facility which should support
the efforts of CIEMAT's laboratory staff to investigate in the field of photovoltaic
device development. CIEMAT is the main Spanish RTD (Research and
Technological Development) center working on energy resources. One of its groups
focuses on the preparation and characterization of thin film materials (aSi:H, CIS
and CdS) for photovoltaicdevice applications.
Part of the motivation for thinfilm work is its ability to use layers only a few
microns thick rather than the freestanding silicon wafers, which are hundreds of
microns thick [18].
The double laser facility designed combines one tool for the treatment with one
tool for the characterization of photovoltaic devices.
Using standard methods like currentvoltage  and spectral response
characterization to measure optical device performance, the measured quantity is
usually an average value over the entire wafer. Small defects within a wafer, that
can negatively influence the performance of the device, may not be discovered. By
scanning the whole sample with laser light one can induce short circuit currents in
the device under test, measure these currents, and present them by a
photoresponse map. The advantage of photoresponse mapping is the possibility to
obtain a lateral resolution. Using a scanning technique, the measurement is
performed locally in limited areas of the probe, whereby the relative influence of
Proyecto Fin De Carrera Günther Krauss
E.T.S.I.T. Universidad Politécnica de Madrid
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the defect on the signal is more pronounced and hence detectable. The simplicity in
equipment using photons instead of electrons for scanning a sample might be the
most attractive feature. There is no need for vacuum and the scanning of light is
easy to accomplish. (Kai Wilhelm and Birger Drugge give in their doctoral
dissertations a detailed comparison of advantages and disadvantages of using
photons or electrons as excitation sources to scan photovoltaic devices ([5], [30]).
For thin film fabrication it is very important to be able to cut metal contacts and
ablate selective thin film material deposited on glass or other materials.
The above mentioned processes of thin film fabrication are usually named "laser
scribing". Laser scribing is a key technology in the manufacturing process of
amorphous silicon integrated modules. It covers about 50% of the fabrication
process steps and has completely replaced the conventional metal masking or
photolithography methods [7].
2 Physical principles of silicon solar cell characterization
2.1 Conventional structure of a silicon solar cell [19]
Solar cells require a particular pn junction design, which is depicted in figure 2.1
as a schematic representation of a conventional silicon solar cell. It consists of a
shallow junction built near the front surface, a front ohmic contact, usually in the
form of stripes and fingers, and a back ohmic contact that covers the entire back
surface. Usually, an antireflective coating is applied to the illuminated side to
increase the fraction of incident light not reflected.
An electrical field is created between two regions of a crystalline semiconductor
having opposed types of conductivity. One of these regions (ntype) is doped with
phosphorus, which has five valence electrons. This region has a much higher
concentration of electrons than holes. The other region (ptype) is doped with
boron, having three valence electrons. Here the concentration of holes is greater.
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The large difference in concentrations from one region to another causes a
permanent electric field from the
ntype region towards the ptype region. This is the field responsible for separating
the additional electrons and holes produced when light shines on the cell.
Sun rays
Front surface metal grid
Anti
reflective
coating
Back surface contact
Nregion
Pregion
Depletion region
W
W
N
W
P
X
N
X
P
X
figure 2.1: structure of a conventional silicon solar cell, W
p
is the width of the ptype base region, W is
the width of the depletion region, W
n
is the width of the ntype emitter region, x
n
is the depth of then n
side depletion edge and x
p
is the depth of the pside depletion edge, whereas x is the depth in general
Proyecto Fin De Carrera Günther Krauss
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In silicon cells the junction is obtained by diffusing a phosphorus layer into a wafer
of silicon previously doped with boron. The junction is typically very near to the
surface (about 0.2 to 0.5 µm). This shallow, diffused layer is commonly called the
emitter. The electrical contact with the illuminated side of the cell has to leave most
of the surface uncovered. Otherwise light cannot enter the cell. However, the
electrical resistance of the contact should not be too high. The compromise usually
adopted is to use contacts with the form of a comb. In contrast, the electrical
contact on the dark side of the cell covers the whole surface of the cell except in the
particular case of bifacial cells.
Proyecto Fin De Carrera Günther Krauss
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2.2 Operating principles of silicon solar cells [19]
If a solar cell is connected to a load and is illuminated, as shown in figure 2.2, a
potential difference will be produced across the load and current will flow. The
current leaves the cell from the positive terminal and returns to the negative
terminal. Under such operating conditions the cell operates as an energy
generator.
n
p
I
L
I
D
(V)
I
R
figure 2.2: internal currents in a solar cell; the unshaded circles represent holes and the black circles
represent electrons; the left arrow represents the photogenerated current I
L
caused by electron/hole
generation; the right arrow represents the diode current caused by the diffusion of electrons and holes
The processes going on inside the cell can be described as follows:
Photons that reach the interior of the cell and have an energy equal or greater than
the bandgap are absorbed in the bulk of the semiconductor, generating electron
hole pairs.
The internal electric field, or potential difference, produced by a pn junction is
responsible for separating as many carriers (electrons and holes) as possible and
producing the photocurrent I
L
in reversebias direction as shown. The total current
I produces a voltage drop across the load which forward biases the pn junction
and this forwardbias voltage produces a forwardbias current, the diode, or dark
current I
D
.
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In summary, when a load is connected to an illuminated solar cell, the current flow
is the net result of the two counteracting components of internal currents I
L
and
I
D
:
Assuming that the two currents can be superimposed linearly, the current in the
external circuit can be calculated as the difference between the two components.
Taking the photocurrent as the positive, we can write:
)
(U
I
I
I
D
L

=
equation 2.1: fundamental characteristic equation of a solar cell
This is the fundamental characteristic equation of a solar cell. It is valid over all
operating conditions, even when the device consumes rather than generates
electricity, in which case the recombination outweighs the photogeneration.
Proyecto Fin De Carrera Günther Krauss
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2.3.1 Absorption of light [21]
The light generated current I
L
is partly determined by the absorption behavior of
the semiconductor. The fraction of the incident light D = 1  R that actually
penetrates the absorbing material can be calculated from the complex refraction
index n
c
= n  i, where is the extinction coefficient and R is the reflectivity, given
by
(
)
(
)
2
2
2
2
1
1
+
+
+

=
n
n
R
equation 2.2: reflectivity
where n and are functions of the wavelength of the incident light (perpendicular
light penetration is assumed). Substituting the appropriate values shows that for
many semiconductors a considerable fraction of light is reflected. It is therefore
desirable from the perspective of making an efficient solar cell to decrease R. This
can be achieved by means of an antireflection coating or a textured surface
structure.
The important process for photovoltaic conversion is the excitation of electrons
from the valence into empty states of the conduction band, which can occur if the
energy of the incident photons is larger than the band gap energy. The light
passing through the material is absorbed then, and the number of generated
electronhole pairs depends on the number of incident photons S
0
() per unit area,
unit time and unit energy, that can be calculated from the spectral distribution of
the sunlight in figure 2.3.
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figure 2.3 : spectral distribution of sunlight, referring to different air mass factors (AM)
The frequency , or the photon energy h, is related to the wavelength by the
relation [µm] = c/ = 1.24/h[eV] where c is the velocity of light. Inside the crystal
the photon flux S(x,) decreases exponentially according to
( )
( )
( )
(
) (
)
x
R
S
x
S


=
exp
1
,
0
with
( )
c
4
=
equation 2.3: photon flux and absorption coefficient
The absorption coefficient () is determined by the absorption process in the
semiconductor and can be used to calculate the generation rate G(x,) of electron
hole pairs (per unit time, volume and energy) at a distance x from the
semiconductor surface. The fraction of photons that penetrate into the crystal is
given by S
0
()*(1R); therefore, the number of electronhole pairs generated per
unit time in the volume between x and
x + x can be calculated from the negative derivative of equation 2.3 with respect
to x:
( )
( ) ( ) ( )(
)
( )
(
)
x
R
S
x
G
qi


=
exp
)
(
1
,
0
equation 2.4: generation rate
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The internal quantum efficiency
qi
() takes into account that only a fraction of the
absorbed photons generates electronhole pairs. For many compound
semiconductors it is observed that
qi
() 1 near the absorption edge. This fact is
due to the formation of excitons or bound electronhole pairs, which carry no
charge and do not contribute to the conductivity. However, for the elemental
semiconductor germanium and particularly silicon, and some IIIV compounds,
values of
qi
() 1 are measured at room temperature even near the absorption
edge.
Proyecto Fin De Carrera Günther Krauss
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2.3.2 Photoconductivity [21]
First we assume a homogeneous semiconductor under illumination and after that
we consider the photocurrent generation in a pn junction cell (chapter 2.3.3).
If p
0
and n
0
are the carrier concentrations in thermodynamic equilibrium, which
are determined by the doping concentrations of donors and acceptors, the changes
in the electron n and hole concentrations p can be expressed by p = p  p
0
and n
= n  n
0.
The basic equations describing the flux of electrons and holes in a
semiconductor under illumination are the currentdensity equations and the
continuity equations:
)
grad
(
p
D
p
q
p
p
P

=
E
J
µ
)
grad
(
n
D
n
q
n
n
n
+
=
E
J
µ
equation 2.5: flux of electrons and holes in a semiconductor
p
p
p
G
div
q
t
p
p

+

=
J
1
with
p
: minority carrier lifetime in the ntype region
n
n
n
n
G
div
q
t
n

+
=
J
1
with
n
: minority carrier lifetime in the ptype region
equation 2.6: continuity equations
The diffusion coefficients D
n
and D
p
are related to the mobility of the carriers µ
n
,
µ
p
by the general Einstein relationship D
n/p
= (KT/q) µ
n/p
. For lightgenerated
carriers, the generation rates for electrons and holes are equal to G
n
= G
p
= G. If
the excess concentrations are low, one can assume additionally that n = p and
combine equation 2.5 and equation 2.6. For instance, for holes, one obtains
Proyecto Fin De Carrera Günther Krauss
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µ
p
G
p
divgrad
D
p
t
p

+
+

=
grad
E
equation 2.7: time derivation of hole concentration
where the ambipolar diffusion coefficient D and the ambipolar mobility µ are
given by
(
)
(
)
p
n
n
p
p
n
p
n
D
p
D
n
D
µ
µ
µ
µ
+
+
=
(
)
p
n
p
n
p
n
p
n
µ
µ
µ
µ
µ
+

=
equation 2.8: ambipolar diffusion coefficient and ambipolar mobility
A corresponding equation is obtained for electrons if p is replaced by n. The
lifetimes for electrons and holes are assumed equal in this case,
n
=
p
= , and can
be calculated, for instance, by the ShockleyReadHall statistics.
Under steadystate conditions, a homogeneous generation of electrons and holes
and for a homogeneous doped crystal, the time derivative and the gradient of p (or
n) are zero, and equation 2.7 leads to the condition p = G. For negligible electric
fields, the quasiFermi energies under illumination then can be calculated, and
yield:
+

=
1
ln
0
p
G
kT
E
E
F
Fp
+
=
+
1
ln
0
n
G
kT
E
E
F
Fn
equation 2.9: quasiFermi energies under illumination
Under normal illumination conditions for solar cells (AM1.5), the product G is
usually smaller than the majority carrier concentration; therefore, only the quasi
Fermi energy of the minority carriers is essentially changed.
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When an external electric field E is applied to the semiconductor, under the above
mentioned assumptions, a current flows. The total current density J derived from
equation 2.5 is given by
E
J
J
J
=
+
=
n
p
equation 2.10: total current density under illumination
with the conductivity
(
)
n
p
n
p
q
µ
µ
+
=
equation 2.11: conductivity
One can separate the conductivity into two terms =
0
+
ph
with the following
expressions for the dark conductivity
0
and the photoconductivity
ph
:
(
)
n
p
n
p
q
µ
µ
0
0
0
+
=
(
)
µ
µ
G
q
n
p
ph
+
=
equation 2.12: dark conductivity and photoconductivity
In doped crystals, the dark conductivity depends on the concentration and
mobility of the majority carriers, whereas the photoconductivity is mainly
determined by the lifetime of the minority carriers.
2.3.3 Photocurrent generation in pn junction solar cells [21]
Photovoltaic energy conversion requires the separation of electrons and holes by
an internal electric field. This means solar cells require a particular pn junction
design, which is depicted in the schematic representation of figure 2.2.
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The internal electric field E =  grad leads to an inhomogeneous distribution of
electrons and holes, and the calculation of the currents requires the solution of the
complete currentdensity equation 2.5 and continuity equation 2.6.
The potential is determined from Poisson's equation,
)
(
0
p
n
N
N
q
A
D
+



=
equation 2.13: Poisson´s equation
where N
D
and N
A
are in general, functions of the position and are equal to the
concentrations of the ionized acceptors and donors. is the dielectric constant of
the material and
0
is the permittivity of the vacuum.
For an abrupt pn junction with uniform doping concentrations on each side of the
junction, the usual approximation is that within a certain width W, the
semiconductor is completely free of charge carriers. The depletion width W, derived
from the upper equation is then given by
A
D
A
D
bi
N
N
N
N
q
V
W
+
=
0
2
equation 2.14: depletion width
where the built in voltage V
bi
is determined by the doping concentrations N
D
and N
A
on either side of the junction
=
2
ln
i
A
D
bi
n
N
N
q
kT
V
equation 2.15: built in voltage
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When light is incident on the front surface and penetrates the crystal, the number
of electrons and holes generated at a distance x from the surface is given by the
generation rate (equation 2.4), which was determined in the previous section.
In thermodynamic equilibrium, when no current flows, minority carriers reaching
the edges of the depletion region are immediately accelerated by the electric field
to the opposite side of the junction. One can approximately assume in this case that
the boundary conditions for the minority carrier concentrations at the depletion
edges are
p 0 at x = x
n
and n 0 at x = x
p.
(see figure 2.4)
A current flows under illumination when the two sides of the pn junction are
connected externally. The external load resistance determines the corresponding
voltage drop U across the junction in forward bias direction. If the current is
small, the solution of Poisson's equation can easily be derived in the depletion
approximation which is depicted schematically in figure 2.4.
figure 2.4: schematic representation of the pn junction band diagram. Under low illumination, the quasi
Fermi energies of majority carriers E
Fn
and E
Fnp
in the n or pdoped region, respectively, remain almost
constant and equal to the Fermi energy in thermal equlibrium E
F
, and extend horizontally into the space
charge region.
The electric field outside the depletion region remains small compared to the field
across the junction and can be ignored; thus E = 0.
Under low illumination the minority carrier concentrations increase noticeable
with respect to the thermal equilibrium whereas the majority carrier
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concentrations remain almost constant. In figure 2.4 this fact is illustrated by
means of the quasiFermi energies in the n or pdoped region, respectively.
The quasiFermi energies vary only slowly as long as the carrier concentrations
are large. Therefore, if the recombination in the depletion region is small, the
quasiFermi energies of the majority carriers can be extended approximately
horizontally into the depletion region. Considering the situation depicted in figure
2.4, the quasiFermi energies of the minority carriers at the depletion edges are
then given by
Proyecto Fin De Carrera Günther Krauss
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qU
E
E
F
Fp

=
at x = x
n
qU
E
E
F
Fn
+
=
at x = x
p
equation 2.16: quasiFermi energies of the minority carriers at the depletion edges
The excess carrier concentrations p and n at the depletion edges under
illumination are given by the following expressions:

=
1
exp
2
kT
qU
N
n
p
D
i
at x = x
n

=
1
exp
2
kT
qU
N
n
n
A
i
at x = x
p
equation 2.17: excess carrier concentrations at the depletion edges
The increased carrier concentrations at the edges of the depletion region lead to
diffusion currents of the minority carriers (electrons and holes) into the p and n
doped region, respectively. If recombination in the depletion region can be
ignored, these currents flow unchanged through the depletion region. The total
current can thus be determined by calculating the minority currents at the edges of
the depletion region.
Calculation of the photocurrent densities collected from the base, emitter and
depletion region [21]:
For n or ptype semiconductors, the expressions in equation 2.5 for the minority
currents can be simplified if the electrical field outside the depletion region is
negligible. Under steadystate conditions, the time derivatives of p and n in the
continuity equations are zero and equation 2.7 outside the depletion region on the
n and p side, respectively, reduces to
Proyecto Fin De Carrera Günther Krauss
E.T.S.I.T. Universidad Politécnica de Madrid
Page 24 of 24
p
p
p
G
x
p
D
+

=
2
2
n
n
n
G
x
n
D
+

=
2
2
equation 2.18: differential equations of electrons and holes
The upper and further equations are valid in case of a onedimensional
consideration of the pn junction. Inserting equation 2.4 for the generation rate G
one obtains, for instance, the following differential equation for the minority holes
in the ntype front region:
(
)
0
1
0
2
2
=


+

p
x
qi
p
p
e
R
S
x
p
D
equation 2.19: differential equation for minority holes in the ntype front region
The corresponding equation for electrons in the pbase region is easily obtained
from the second equation of equation 2.18. The general solution of the differential
equation is:
(
)
( )
( )
x
f
L
R
S
q
L
x
c
L
x
c
p
p
qi
p
p
p
1
1
sinh
cosh
2
0
2
1



+
=
with
( )
(
)
(
)
(
)
(
)
{
}
p
p
L
x
c
p
L
x
c
p
x
e
L
e
L
e
x
f
/
/
3
3
1
1
2
1




+


+
=
equation 2.20: general solution of the differential equation
c
1
, c
2,
and c
3
are constants which have to be determined from the boundary
conditions. A corresponding equation is obtained for the electron concentrations in
the pdoped base region.
In thermodynamic equilibrium without an applied bias (U = 0), the boundary
conditions at the depletion edges are p 0 at x = x
n
and n 0 at x = x
p
, as
Excerpt out of 115 pages
 Quote paper
 Günther Krauß (Author), 1998, Setup of a laser facility for characterization and treatment of photovoltaic devices, Munich, GRIN Verlag, https://www.grin.com/document/78
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