FUDAMETAL LOSSES I SOLAR CELLS
L. C. Hirst*, N. J. Ekins-Daukes.
Department of Physics, Imperial College London, SW7 2AZ, UK
*corresponding author: [email protected]
Fundamental loss processes occurring in solar cells have been quantified. A comprehensive picture of fundamental
solar cell function is presented, with the physical origins of losses discussed in terms of photon absorption and
emission. The effects on device characteristics are calculated with losses identified as reducing current or voltage.
Fundamental losses are represented graphically, accounting for all energy received from the sun. Quantifying losses
in solar cells and presenting them graphically allows the origins of fundamental limiting efficiency to be visualised.
Potential areas for improvement to cell design can be identified, providing a guide to innovation and development.
Keywords: Fundamentals, Solar Cell Efficiencies, Thermodynamics
1
INTRODUCTION
The efficiency with which solar radiation can be
converted into useful work in a solar cell has been of
interest since the first photovoltaic devices were
developed. Fundamental limiting efficiencies have been
discussed by many authors and several analyses have
been presented. Shockley and Queisser published the
detailed balance limit [1, 2] which uses a Planck
distribution to calculate a non-empirical current-voltage
relationship for a photovoltaic device with the maximum
power point giving the fundamental limiting efficiency.
The same result can be obtained by taking a
thermodynamic approach. A solar cell can be modelled as
a heat engine, transferring heat from a hot source to a
cold sink, doing work and generating entropy in the
process. The laws of thermodynamics can be used to
describe solar cell function and calculate limiting
efficiencies through the consideration of energy and
entropy fluxes [3, 4].
Presented is a comprehensive description of solar cell
function. The effects of all fundamental loss processes
have been quantified, graphically demonstrating the
origins of solar cell theoretical limiting efficiency.
Understanding the source of fundamental loss in a solar
cell allows for potential efficiency advantages to be
identified.
2
of heat, equilibrating with the absorber in a process
known as thermalisation. Thermalised electrons have
kinetic energy described by the temperature of the lattice.
Since the heat energy in the material is not extracted as
electrical current thermalisation is a significant loss
process.
A built-in voltage across a solar cell causes
thermalised electrons to either recombine with a valence
band hole emitting a luminescent photon or to be
extracted as an electrical current. Increasing built-in
voltage increases luminescence. The point at which
luminescence is equal to absorbed solar radiation is
known as the open circuit voltage (Voc). At this voltage
all electrons recombine radiatively and no current can be
extracted. The energy per electron for a device at Voc is
significantly lower than that of the thermalised electron
distribution. The energy loss associated with this voltage
drop is described in this paper as thermodynamic loss.
Operating a device with a built-in voltage below the
open circuit limit allows for current to be extracted. The
loss process associated with operating a cell at the
maximum power point is referred to as the fill factor loss
and is described by the shape of a device’s currentvoltage characteristic.
The remaining fraction of incident solar radiation
which has not been dispersed through unavoidable loss
processes equates to the fundamental limiting efficiency
of the device.
LOSS PROCESSES IN A SOLAR CELL
3
The sun can be modelled as a blackbody, emitting
photons radially outwards in an energy distribution
described by its temperature (approximately 6000K). A
few degrees of this isotropic emission reaches the earth
and can be used to generate electricity in a photovoltaic
solar cell.
This paper considers absorption and emission in a
single threshold absorber. Every incident solar photon
with energy above the bandgap energy is absorbed. The
photon is annihilated and its energy promotes a valence
band electron into the conduction band of the absorbing
material, generating an electron hole in the valence band.
The first loss process occurs at this stage as photons with
energy below the bandgap are transmitted and hence their
energy is not used.
Excited electrons initially form an energy distribution
corresponding to that of the absorbed photons however,
they rapidly lose energy to the material lattice in the form
QUANTIFYING LOSS
This paper considers a single junction semiconductor
device under one sun illumination. The electrical power
that can be extracted from a solar cell has been calculated
as the difference between the solar radiation absorbed
and the luminescent radiation emitted (figure 1). This
method assumes particle number is conserved, all
recombination is radiative and zero series resistance.
Although effects such as reflection and cell resistances
might be significant loss processes in a real solar cell
they are theoretically avoidable and thus do not
contribute to the fundamental limiting efficiency
calculated in this paper.
The generalised Planck photon distribution has been
used to calculate absorption and emission in the cell
(equation 1).
GP( E , Ω, T , µ ) =
2Ω
⋅
c2h3
E2
E−µ
exp
 −1
 kT 
Eg
(1)
c is the speed of light, h is Planck’s constant, k is
Boltzmann’s constant. The distribution is dependent on
four variables. E is photon energy, Ω is the solid angle of
absorption or emission, T is the temperature of the
emitting body and µ is the chemical potential of the
emitted radiation. Solar radiation is thermal and thus has
µ=0. Emission from the solar cell is luminescent because
of the voltage (V) across the device. Luminescent
photons have µ=eV [5].
3.1 Below Eg loss
Not all incident solar radiation is absorbed in a
photovoltaic solar cell. Photons with energy below the
bandgap energy of the absorbing material are transmitted
(figure 2). The energy lost through non-absorption of
below bandgap photons is calculated by multiplying
equation 1 for the incident photon distribution by photon
energy and integrating with respect to photon energy over
the region 0 → Eg (equation 2).
Below Eg loss =
∫ E ⋅ GP(E, Ω
A
,TS , µ = 0)dE
(2)
0
3.2 Thermalisation Loss
Electrons are initially excited into the conduction
band with the energy of the incident photon. Electrons
form a thermal distribution at a temperature initially
much higher than the lattice temperature (figure 3).
Electrons then interact with lattice modes allowing
energy and momentum to be transferred to the absorbing
material until the electron energy distribution is in
thermal equilibrium with the lattice. The electron energy
distribution is calculated by considering the density of
electron states in the conduction band and then using
Fermi-Dirac statistics to calculate the probability of
electrons occupying these states. A detailed balance
approach can be used to calculate thermalisation loss
(equation 3).
Figure 2: Photons with energy below the bandgap energy
of the absorbing material are transmitted through the cell
and consequently do not contribute to the useful
electrical power output of the device.
Figure 1: Absorption and emission in a solar cell. The
sun emits thermal radiation defined by its temperature. A
solid angle ΩA of this isotropic emission is absorbed in a
solar cell. The cell emits luminescent radiation over solid
angle ΩE, described by the cell temperature and the
voltage across it. Only photons with energy above the
bandgap of the absorbing material can be absorbed or
emitted.
Figure 3: Electron distribution before and after
thermalisation calculated using Fermi-Dirac statistics.
The dashed lines show the mean energy of each
distribution. The energy difference between the two
dashed lines gives the mean energy lost through
thermalisation per electron.
Thermalisation loss =
(3)
∞
∫ E ⋅ GP ( E , Ω
A
, TS , µ = 0)dE
Eg
Fill factor loss =
∞
3


−  Eg + kTC  ∫ GP ( E , Ω A , TS , µ = 0) dE
2

 Eg
3.3 Thermodynamic losses
The maximum chemical energy which can be
extracted from the thermalised electron distribution is
given by the free energy available in the system [6].
Several processes contribute to this thermodynamic
factor. The non-zero temperature of the lattice results in
thermalised electrons forming an energy distribution with
mean energy above the bandgap energy. The kinetic
energy of the electrons cannot be extracted as useful
work. In this paper this loss is referred to as electron
kinetic loss and is calculated by considering an electron
gas with mean kinetic energy per particle kT/2, per degree
of freedom.
Electron kinetic loss per electron =
operated at zero volts. Fill factor loss is a result of
operating at the maximum power point and has been
calculated as shown in equation 5.
3
kTc
2
I scVoc − I mpVmp
(5)
Fill factor loss is determined by the curvature of the
current-voltage characteristic of the device. In a real solar
cell this is affected by cell resistances however, in the
ideal model used here the fill factor loss tends to zero for
a cell at 0K.
(4)
The temperature of the lattice also results in a voltage
drop process associated with the entropy of carriers at
non-zero temperature. The large temperature gradient
between the sun and the cell makes this loss only small
but it must be considered to gain a complete picture of
loss processes in a solar cell. Here this loss is referred to
as Fermi level loss. In a cell under maximum
concentration Voc (i.e. the maximum energy which can be
extracted per electron) is displaced from the bandgap
energy. The energy associated with this voltage drop is
calculated to give Fermi level loss in this paper. This loss
process has been discussed and quantified by several
authors and is sometimes referred to as the Carnot factor
[7, 8].
An additional voltage drop is observed for a cell with
Tc>0K, under one sun illumination. This loss process is
referred to as etendue loss and has been considered by
several authors [9]. Etendue loss can be thought of
thermodynamically as the expansion of photon modes
resulting in entropy generation. The physical origins of
this loss can be appreciated by simply considering
absorption and emission in a cell. The solid angle of
absorption (ΩA) can be increased using lenses and
mirrors to focus light onto a cell. The maximum chemical
energy per electron which can be extracted from a cell is
Voc and occurs when absorption and emission equate.
Increasing ΩA increases absorption but has no effect on
emission resulting in an increased Voc. The difference in
Voc for one sun and maximum concentration is calculated
to give etendue loss (Figure 4). All thermodynamic loss
process are zero at a lattice temperature of 0K.
3.4 Fill factor loss
Operating a device at Voc results in all absorbed
photons recombining radiatively and no current being
extracted. All absorbed photons will be extracted at short
circuit current (Jsc) however, this will only occur in the
absence of luminescent radiation when the device is
Figure 4: Absorption and emission with varying
concentration, voltage and Tc for a material with bandgap
1.32eV. Voc occurs when lines of emission and
absorption intersect. This intersection occurs at a lower
voltage for one sun illumination than for maximum
concentration resulting in an efficiency loss at one sun.
At a lower lattice temperature Voc increases and the effect
of etendue loss is reduced.
4
LOSSES GRAPHICALLY REPRESENTED
Once quantified, loss processes can be graphically
represented allowing the origins of fundamental limiting
efficiency to easily be visualised.
Every loss process can be described in terms of a
reduction in current or voltage. Absorbed current (photon
number above the bandgap energy x electron charge) is
plotted against varying bandgap energy for a device
under one sun illumination (figure 5). Integrating under
this plot gives the total power available from the solar
spectrum. The current-voltage characteristic for a device
at 300K, under one sun illumination with a bandgap
energy of 1.32eV (optimal bandgap) is superimposed.
The power available in the solar spectrum has been
divided up into regions showing the current and voltage
drops associated with loss processes. Below bandgap loss
is effectively a current drop as it limits the number of
electrons available to be extracted. Thermalisation and
thermodynamic losses result in a reduction in electron
chemical energy which is effectively a voltage drop.
Operating a device at the maximum power point along
the current-voltage characteristic results in both current
and voltage reduction which equates to the fill factor
loss.
Figure 5: An I-V plot showing loss processes as current and voltage drops for a device with bandgap 1.32eV.
Figure 6: Efficiency with varying bandgap. All energy received from the sun is accounted for. Maximum efficiency is shown
to be 0.31 for a bandgap of 1.32eV.
All loss processes in a solar cell are a function of
device bandgap (figure 6). Increasing the bandgap
reduces the number of absorbed photons and also reduces
thermalisation loss. This current-voltage trade off is
optimized for a device with bandgap energy 1.32eV
giving 31% efficiency under one sun illumination. The
reason for this Shockley Queisser limit is easily
visualised as all energy received from the sun is
accounted for in figure 6.
5
CONCLUSIONS
Technological developments have created practical
devices which are well described by the idealized model
presented here. Device efficiency has incremented
towards the fundamental limit achievable with this device
design. Fundamental losses are unavoidable in the single
junction semiconductor device under one sun
illumination modelled in this paper, however, changing
device design enables some of these losses to be avoided.
The fundamental limiting efficiency of this cell design is
far from the ultimate limiting efficiency of solar energy
conversion.
A simple design enhancement is to concentrate
incident solar radiation onto a solar cell using lenses or
mirrors. This reduces the voltage drop associated with
etendue loss, increasing efficiency. Third generation cell
designs such as multi-junction cells and hot carrier cells
offer the potential for substantial efficiency
improvements by limiting below Eg losses and
thermalisation losses. The remaining loss processes can
only be eliminated by operating the device at 0K and as
such are ultimately unavoidable in any practical device.
Understanding the origins of loss in solar cells is
essential to improving efficiency. Quantifying loss allows
for potential areas of improvement to cell design to be
easily identified, providing a guide to innovation and
development.
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