1. Impact of Restricted Radiative Emission from
Compressively Strained Quantum Wells
on the Photonic Coupling
Panagiota Theodoulou
A thesis submitted in partial fulfilment of the requirements for the degree of
Master of Physics and the Diploma of Imperial College London.
September 15, 2014
2. Abstract
One of the approaches to reduce the radiative losses in a solar cell is to suppress the
Transverse Magnetic (TM) mode associated with the in-plane emission, which can be
achieved by growing quantum wells (QWs) under compressive strain in the solar cell.
The compressive strain leads to a suppression of the light hole (LH) transitions and
consequently the suppression of the TM mode, which only couples to LH transitions.
The Transverse Electric (TE) polarized light is emitted both in and out of the plane
of the structure and is therefore associated with the coupling of light emitted from
one junction to the next in a multijunction device, which is called photonic coupling
and results in an enhanced device efficiency. However, the TE mode has also a small
contribution from LH transitions, consequently the key research question of this study
is whether the restricted in-plane emission due to the suppression of the LH transitions
can limit the photonic coupling. Photoluminescence (PL) measurements from the
front surface were performed on samples containing AlGaAs/AlInGaAs test QWs in
the top region and GaAs/InGaAs probe QWs in the bottom region. Since an increase
in the Indium fraction leads to further suppression of the LH transition, the Indium
fraction in the test QWs was varied for each sample to observe how the photonic
coupling from the test to the probe QWs changes as the emission is further restricted.
It was found that the samples had a thin top region, allowing direct laser excitation of
the probe QWs, so no coupling of light was observed from the test to the probe QWs.
In an identical sample with a thicker top region and unstrained AlGaAs/GaAs test
QWs, photonic coupling was demonstrated by Photoluminescence Excitation (PLE)
measurements. In addition, calculations for a “virtual” QW suggest that the current
density in the bottom region due to coupling of light from the test QWs decreases
by 35% when the LH transitions are completely suppressed compared to the case of
isotropic emission. Finally, in PL spectra obtained from the edge of the samples to
measure the restricted emission, an unexpected polarization pattern was observed,
which was mainly attributed to Heavy Hole (HH)-LH band mixing.
1
5. List of Figures
1.0.1 Layer structure of a pin solar cell with GaAs/InGaAs QWs in the
intrinsic region. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.0.2 Emission from a typical solar cell upon light absorption. . . . . . . . 10
1.0.3 Band structure for an unstrained and compressively strained QW. . . 11
1.0.4 Propagation of TE and TM polarized light with respect to the QW
plane. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.1.1 Schematic diagram of the layer structure of a multijunction solar cell. 14
2.1.2 Band diagram and IV curve of a tunnel diode at different values of
forward bias. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.2.1 Simplified p-i-n layer structure for the QWSC. . . . . . . . . . . . . . 17
2.2.2 Band diagram of the p-i-n QWSC. . . . . . . . . . . . . . . . . . . . 18
2.2.3 DoS for a QW (g2D) and a bulk semiconductor (g3D) . . . . . . . . . . 19
2.2.4 Schematic diagram illustrating the transition from the first hole state
in the valence band to the first electron state in the conduction band
of a QW, for a nonzero in-plane wavevector. . . . . . . . . . . . . . . 20
2.2.5 Band structure of an unstrained, direct bandgap bulk semiconductor. 21
3.1.1 Schematic diagram showing the four processes that can take place upon
photon emission in a solar cell. . . . . . . . . . . . . . . . . . . . . . . 24
3.1.2 IV curves for a two-junction solar cell in which the bottom cell is
current-limited. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
4.2.1 Calculated absorption coefficients, αTE(E) and αTM (E) for the QWs
in the sample QSOL. . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
4
6. 5.1.1 Double-structure sample designed by Dr. Diego Alonso-Alvarez to
study the impact of restricted radiative emission on the photonic cou-
pling. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
5.2.1 Schematic diagram of the set-up used to perform the front PL mea-
surements. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
5.2.2 Simulated and experimental emission from single-structure samples
QSOL, QT1657 and T6 at 300K. . . . . . . . . . . . . . . . . . . . . 44
5.2.3 Schematic diagram of the experimental set-up used to perform the edge
PL measurements. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
5.2.4 Edge PL spectrum of T6 at 300K. . . . . . . . . . . . . . . . . . . . . 47
5.3.1 PL spectra of double-structure, SBQW samples, obtained at 300 K. . 49
5.3.2 PL spectra of double-structure, SBQW samples, obtained at 30 K. . . 50
5.3.3 PL spectrum at 300 K, of the sample U34101, with a 1.5 µm thick top
region and 0% Indium (no strain) in the test QWs. . . . . . . . . . . 52
5.3.4 PL spectrum at 30 K, of the sample U34101, with a 1.5 µm thick top
region and 0% Indium (no strain) in the test QWs. . . . . . . . . . . 53
5.3.5 PLE spectrum of the U34101 sample at 30K. . . . . . . . . . . . . . . 54
5.3.6 PL spectrum of sample U34101 as a function of temperature and power. 56
5.3.7 Edge PL spectrum of U34101 at 300K. . . . . . . . . . . . . . . . . . 59
6.0.1 Calculated dark IV for the sample U34097 at 300 and 30 K. . . . . . 62
6.0.2 Calculated dark IV for a “virtual” QW at 300 K, for isotropic and
suppressed in-plane radiative emission. . . . . . . . . . . . . . . . . . 64
6.0.3 Calculated absorption coefficients, αTE(E) and αTM (E), for the QWs
in the sample QT1657. . . . . . . . . . . . . . . . . . . . . . . . . . . 69
6.0.4 Calculated absorption coefficients, αTE(E) and αTM (E), for the QWs
in the sample T6. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
6.0.5 Calculated absorption coefficients, αTE(E) and αTM (E), for the test
QWs in U34097, at 300 K. . . . . . . . . . . . . . . . . . . . . . . . . 70
6.0.6 Calculated absorption coefficients, αTE(E) and αTM (E), for the test
QWs in U34097, at 30 K. . . . . . . . . . . . . . . . . . . . . . . . . . 71
5
7. Declaration
Declaration of work undertaken
The work described in this thesis was undertaken at Imperial College London, under
the supervision of Dr. Diego Alonso-Alvarez. The experimental work in this project,
including the equipment set-up and data acquisition, was carried out in collaboration
with Dr. Diego Alonso-Alvarez, with additional support from Dr. Markus Fuhrer.
The samples used in the experiments were fabricated by Dr. Peter Spencer. As
stated in the report, the parameters used in the simulations were calculated by Dr.
Diego-Alonso Alvarez, using a program developed by Dr. Markus Fuhrer. The code
for the simulations, the analysis of the presented experimental and simulated data,
the planning and production of this thesis are entirely my own work.
Declaration of Originality
I hereby declare that the composition of this thesis is the result of my own, inde-
pendent research and work and that this paper has not been previously published or
submitted at another University for any other academic award. Where ideas have
been drawn from the published work of others, complete references have been given.
Panagiota Theodoulou
London, September 2014
6
8. Acknowledgments
I owe my deepest gratitude to my supervisor, Dr. Diego Alonso-Alvarez, for his
support, guidance, patience and enthusiasm at all stages of this project. I truly
appreciate the time he has taken to help me with the experimental work and to
understand complex theories related to this study. I would also like to thank him for
the long, inspiring discussions that made this project so interesting and enjoyable!
My special thanks go to Dr. Peter Spencer, for fabricating the samples used in this
work and for taking time to demonstrate the fabrication process.
I am particularly grateful to Dr. Markus Fuhrer for developing the program providing
the parameters used in my simulations, as well as for additional support with the
experimental set-up.
My appreciation is extended to Dr. Paul Stavrinou and Dr. Ned-Ekins Daukes for
their useful comments regarding the findings of my research.
I would also like to thank the members of the Quantum Photovoltaics Group for the
interesting discussions during the weekly group meetings.
Finally, I thank my parents for giving me the opportunity to study this course and
for their continuous support and encouragement throughout the academic year.
7
9. Chapter 1
Introduction
Due to the high cost and CO2 emissions associated with the production of energy from
fossil fuels, the generation of electric power has shifted to alternative sources, such as
photovoltaics. Photovoltaic solar cells convert light light from the sun to electricity
and hold the promise for environmentally friendly and sustainable energy production
[1, 2]. Therefore, there has been ongoing research on the development of methods to
deliver the maximum power from these devices. One of the approaches that have been
adopted to achieve this, is the fabrication of multijunction solar cells. Multijunction
solar cells are comprised of several semiconductor subcells, each having a different
bandgap energy, so that different portions of the incident solar light can be harvested
more efficiently [3, 4]. A four junction GaInP/GaAs//GaInAsP/GaInAs solar cell
developed by researchers in the Fraunhofer Institute for Solar Energy Systems holds
a world record efficiency of 44.7% [5], whereas the highest efficiency achieved for single-
junction solar cells is only 29.1% [6]. An advantage of the multijunction configuration
is that the light emitted from one junction can be absorbed in the underlying junction
and contribute to its photocurrent, which is known as photonic coupling [7]. As
opposed to the photocurrent induced by direct sunlight absorption, the photocurrent
induced by photonic coupling does not depend on the spectrum of the incident light,
making the device less sensitive to fluctuations in the incident spectrum and enhancing
the device efficiency [8, 9].
Another approach to increase the efficiency of solar cells is by incorporating QWs in
the solar cell. Due to their low bandgap, QWs enable the absorption of lower energy
light, increasing the photocurrent [10]. The simplest structure for a p-i-n QW solar
cell consists of GaAs p and n regions, with InxGa1-xAs quantum wells in the intrinsic
region, with the GaAs layers acting as the barriers, as shown in Figure 1.0.1.
8
10. Figure 1.0.1: Layer structure of a pin solar cell with GaAs/InGaAs QWs in the intrinsic region.[11]
The addition of Indium lowers the bandgap, forming the QW, and with increased
amount of Indium the QW can become deeper and the bandgap even smaller. In
this way, the bandgap of the conventional GaAs semiconductor, used in single and
multijunction solar cells, can be brought from a value of 1.42 eV close to the ideal
single-junction value of 1.1 eV [12, 10]. At the same time, as shown in the diagram,
the InxGa1-xAs layer has a larger lattice constant than the bulk GaAs, therefore it
has to be compressively strained to be lattice matched to the GaAs.
Obviously, increasing the number of QWs in the intrinsic region can enhance the ab-
sorption and increasing the Indium content shifts the bandgap closer to the ideal, im-
proving the photocurrent. However, both the increase in the number of QWs and the
Indium fraction contribute to a strain built-up in the structure [10, 11].The approach
to overcome this problem is to grow QW layers in compressive strain, surrounded by
higher bandgap barrier layers in tensile strain. Due to the alternate compressive and
tensile strain, the net strain in this double-heterostructure1
is zero, preventing the
built-up of strain, hence the name strain-balanced QW solar cell (SBQWSC) [11].
In addition to moving the bandgap closer to the optimum, compressively strained
QWs offer the advantage of reducing the entropy associated with the difference be-
tween the angular spread of absorbed and emitted light in a conventional solar cell.
1
A double-heterostructure refers to a structure in which a low bandgap semiconductor is grown in the
middle of two higher bandgap semiconductors [13].
9
11. As shown in Figure 1.0.2, a typical bulk semiconductor solar cell, absorbs light cov-
ering a narrow angular range, indicated by Ωabs and re-emits it isotropically, as a
result of the LH and HH band degeneracy. The broad angular range over which light
is emitted is indicated by Ωemit. The entropy generated as a result of this difference
is given by kBTln( Ωabs
Ωemit
), which in turn reduces the voltage of the solar cell. A zero
entropy could therefore be achieved by matching the angular spread of absorbed and
emitted light, so that Ωabs = Ωemit [14].
Figure 1.0.2: Emission from a typical solar cell upon light absorption. The range of angles over
which light is absorbed from the solar cell is much smaller than the angular range over which light
is emitted, which constitutes a radiative loss mechanism for the solar cell [14].
In QWs, the LH and HH bands are split to some extent because of quantum confine-
ment effects (Section 2.2). As illustrated in Figure 1.0.3, if the QW is compressively
strained, the splitting increases because the LH band shifted is further away from the
conduction band and the LH transitions can be suppressed. This in turn, leads to a
suppression of the TM mode associated with the in-plane emission of light, since this
mode has only contributions from LH transitions. As the TM polarized light is not
associated with the absorption of sunlight incident perpendicular to the QW plane,
the TM mode constitutes an unnecessary radiative loss and it can be suppressed [15],
leading to restricted radiative emission so that Ωabs = Ωemit , as required.
10
12. Figure 1.0.3: Band structure for an unstrained and compressively strained QW. As opposed to
a bulk semiconductor, the HH and LH bands are already non-degenerate due to the quantum
confinement in the z direction. For a compressively strained QW, the LH band is shifted downwards
with respect to the conduction band, leading to the suppression of the LH transition. [15].
However, as shown in Figure 1.0.4, the TE polarized light, which has mostly contribu-
tions from HH transitions but also a small contribution from LH transitions, can be
emitted/absorbed both parallel and perpendicular to QW plane [12, 16]. Returning
to the photonic coupling effect introduced previously, it is reasonable that it is light
emitted in the normal direction from a junction that can be absorbed in the next
junction, rather than light emitted in the plane, in other words, the TE mode is the
one associated with the photonic coupling.
Figure 1.0.4: Propagation of TE and TM polarized light with respect to the quantum well plane.
Since the polarization direction of the electric field is perpendicular to the direction at which the
light travels, TE polarized light, for which the electric field is polarized in the x or y direction, can
be emitted both parallel and perpendicular to the QW plane. TM polarized light, on the other
hand, can only be emitted in the QW plane since the electric field in that case is polarized in the
z direction [16].
It follows that, by suppressing the LH transition to achieve restricted in-plane emission
and minimize the radiative losses, a part of the out-of plane emission (TE polarized
11
13. light) associated with the photonic coupling will also be suppressed. Consequently,
the central question this thesis seeks to answer, is whether the restricted radiative
emission from a compressively strained QW can limit the photonic coupling.
The overall structure of this study takes the form of six chapters, including this intro-
ductory chapter. Since photonic coupling and restricted emission are associated with
multijunction and QW solar cells, respectively, Chapter 2 will introduce basic prin-
ciples, as well as some of the design aspects of these devices. Chapter 3 gives a more
detailed discussion of the photonic coupling. Chapter 4 begins with a brief demon-
stration of why the emission/absorption in QWs, as opposed to bulk semiconductors,
depends on the polarization of light and the particular transition taking place, by
considering the optical momentum matrices for bulk semiconductors and QWs. This
chapter continues by laying out the theory behind the simulation of the compressively
strained QW emission profile and the calculation of the current density induced in the
bottom junction of a tandem device by coupling of light from a compressively strained
QW in the top junction. Chapter 5 describes the experimental work in this project:
Section 5.1 gives details about the structure of the samples used in this project. Sec-
tion 5.2 presents PL spectra of samples containing compressively strained QWs. Since
each of these samples comprises a single region of a double-heterostructure, they will
be simply referred to as “single-structure” samples. Section 5.3 presents PL mea-
surements on samples with test QWs on the top double-heterostructure region and
probe QWs on the bottom double-heterostructure region. As each of these samples
has two double-heterostructure regions (top and bottom), they will be called “double-
structure” samples. The degree of compressive strain (restricted emission) on the test
QWs is different for each sample, to study how the photonic coupling from the test to
the probe QWs changes as the LH transition in the test QWs is further suppressed,
which is the primary aim of this work. Chapter 6 presents a calculation to examine
the impact of restricted emission on the photonic coupling, based on the theory in
Chapter 4. Finally, a brief summary and critique of the main findings of this work is
given and recommendations for further work are made.
12
14. Chapter 2
Multijunction & Quantum Well solar
cells
2.1 Multijunction solar cells
Multijunction solar cells are devices in which, solar cells made of different bandgap
materials, namely III and V semiconductor elements in the periodic table, are stacked
on top of each other and are essentially connected in series. The cells can be stacked
mechanically, meaning that the different semiconductor materials of which they are
made of need not be lattice matched. Alternatively, multijunction solar cells can be
fabricated as monolithic devices, where the different cells are grown on top of each
other by epitaxial techniques and are separated by tunnel diodes, as shown in Figure
2.1.1. This requires the materials to have the same lattice constant in order to avoid
defects [2]. The advantage brought by multijunction solar cells is that thermalisation
losses can be decreased: If light has lower energy than the bandgap of the semicon-
ductor comprising the solar cell, it will not be absorbed, and if it has higher energy
than the bandgap, only the amount of energy that is equal to the bandgap will be
absorbed and the remainder will be converted to heat. In multijunction devices, this
problem is solved, because the cells are arranged in order of decreasing bandgap, so
that each of them can absorb more efficiently a particular portion of the solar spec-
trum and convert it to photocurrent, depending on its bandgap [17]. A multijunction
solar cell can therefore deliver a much higher efficiency compared to a conventional,
single-junction solar cell. The bandgap of each cell will not only affect the spectral
range of light it will absorb, but also the current and voltage it will produce [18].
In particular, a cell with large bandgap will have large open-circuit voltage, Voc but
13
15. limited short-circuit current Jsc [19].
Figure 2.1.1: Schematic diagram of the layer structure of a monolithic multijunction device, in
which the top cell is composed of GaInP, the middle cell is composed of GaInAs, whereas the
bottom subcell is made of Ge. The cells are made as thin n doped emitters on thick p doped bases
and are connected by tunnel diodes. Also shown is the portion of the solar spectrum absorbed by
each cell [20].
The fact that the cells are connected in series implies, on the one hand, by Kirchoff’s
law, that their voltages will add [2], and on the other hand, that the cell produc-
ing the least current determines the current flowing through the whole device [18].
Current matching, meaning that the same current is flowing through each junction,
is consequently essential in achieving the maximum efficiency in these devices. As
the thickness of a given cell determines the amount of light it absorbs and therefore
the amount of current it produces, current matching can be achieved by tuning the
thickness of each cell [21].
Nevertheless, current matching is not the only requirement for maximizing the effi-
ciency of these devices. The design of the tunnel diodes used to separate the different
junctions also plays an important role in the multijunction solar cell performance.
As illustrated in Figure 2.1.1, the different cells in a monolithic multijunction device
are made as thin n-doped emitters on thick p-doped bases, since minority holes (in
the n-doped emitter) have much shorter lifetimes than minority electrons (in the p-
doped base) in semiconductors that belong in the III-V groups. When the cells are
connected in series, the pn junction formed at their interfaces does not allow current
to flow from one cell to the next, which is the reason why tunnel diodes are needed to
connect the different cells. A tunnel diode is a highly doped p-on-n diode which, as
its name implies, is based on the principle of quantum tunneling, where particles can
14
16. tunnel through a barrier, provided it is sufficiently thin. In the case of electrons or
holes being the particles that can tunnel through the barrier with a high probability,
a large current can flow, through the tunnel junction, from one cell to the next and
the tunnel diode is said to have an ohmic response [20].
Tunnel diodes have a peak tunneling current at low positive voltages, where the band
alignment is such that electrons from the filled states on the n side can flow to the
empty states on the p side, giving a large current. As forward bias is further increased,
the band alignment is such that the number of available empty states on the p side
decreases and the current is reduced, since the electrons have less states to move into
[22]. The band diagram and the current-voltage (IV) curve of a tunnel diode for these
two situations is shown in Figure 2.1.2. Tunnel diodes should be substantially doped,
as the peak tunneling current increases with doping. However, in order to avoid ab-
sorption losses in multijunction solar cells, a tunnel diode should have a high bandgap
so that it does not absorb the light emitted from the cell above it, allowing it to be
transmitted to the next cells in the device, and a high bandgap material cannot be
doped to very high levels. Therefore, there is a trade off between the transparency of
a tunnel diode to the light emitted from the overlying junction and the peak tunneling
current [20, 2]. In addition to high transparency and peak tunneling current, tunnel
diodes should have low resistivity, so that they permit current produced by one cell
to flow to the next without losses [18].
15
17. Figure 2.1.2: Band diagram and IV curve of a tunnel diode at different values of forward bias. Left:
At low forward bias, there is a large number of available states in the p side for electrons to move
into, resulting in peak tunneling current. Right: For higher forward bias, the band alignment is
modified in a way such that the number of available states in the p side is reduced, resulting in a
decrease in the tunneling current [22].
Another factor that affects the efficiency of multijunction solar cells is their extremely
high sensitivity in fluctuations in the incident solar spectrum, which becomes partic-
ularly important in terrestrial applications. The incident solar spectrum exhibits
variations throughout the day and it depends on the weather conditions as well as on
the chosen location for solar cell operation. As multijunction solar cells are designed
in a way such that current matching between cells is satisfied for a given spectrum,
a change in the spectral distibution of incident light results in a current mismatch
between the different junctions, leading to a loss in efficiency [23].
It is clear from the above that, optimizing the efficiency of multijunction solar cells
presents a challenge and is gaining special interest in the scientific community. In
general, multijunction solar cells have received a lot of attention, because in addition
to their ability of delivering much higher efficiencies compared to conventional single
junction solar cells, they exhibit an interesting effect known as radiative coupling.
Radiative coupling refers to the process in which the light emitted from one junction
is absorbed in the adjacent junction, contributing to its photocurrent [8]. This effect
16
18. will be introduced in more detail in Chapter 3.
2.2 Quantum Well Solar Cells (QWSCs)
A quantum well solar cell (QWSC) is a p-i-n structure in which lower bandgap layers-
the QWs-are incorporated in the intrinsic region [14]. A simplified p-i-n layer structure
for the QWSC is shown in Figure 2.2.1. The first p-i-n photodiode with multiple GaAs
well layers surrounded by higher bandgap AlGaAs barrier layers, in the intrinsic re-
gion, was demonstrated in 1991 by Barnham et al. [24], who proposed, at the same
time, the use of this structure as a solar cell. The advantage offered by this structure
is that, the QWs can increase the photocurrent by harvesting more efficiently the
lower energy parts of the spectrum, compensating for the reduction in the Voc that
results from the low bandgap of the QW material [15].
Figure 2.2.1: Simplified p-i-n layer structure for the QWSC[14].
In order, however, for a QW to produce photocurrent, the electrons and holes gener-
ated after light absorption in the QW must escape from it and be swept in opposite
directions for collection in the external circuit, as shown in the band diagram in Figure
2.2.2. This happends provided, firstly, carriers are supplied with sufficient thermal
energy to escape, which is usually the case at room temperature and secondly, there
is a strong electric field, which pulls the carriers away from the intrinsic region once
they are able to escape [10]. In fact, the reason why QWs are grown in the intrinsic
17
19. region at the first place, is because of the very low background doping in the intrinsic
region which helps to sustain such a strong electric field, as opposed to the p and n
layers where the background density is high, reducing the electric field [25].
Figure 2.2.2: Band diagram of the p-i-n QWSC. The diagram shows electrons and holes moving
in opposite directions after their escape from the QW. They can be then collected in the external
circuit and contribute to the photocurrent or be captured in a different QW and recombine as
illustrated in this diagram [14].
QWs are made as extremely thin layers, with widths of the order of a few nanome-
tres, which gives rise to quantum confinement of carriers. In particular, carriers
are confined in the growth direction, z, but are able to move freely in the remain-
ing two, which define the quantum well plane. Heisenberg’s uncertainty principle,
∆x∆p ≥ /2, shows that, an increase in the momentum uncertainty, ∆p, follows
when the width of the QW is reduced, since then, the uncertainty in the position of
the carrier, ∆x is reduced. The increase in ∆p is the reason why the ground level in
the QW corresponds to a particular energy value rather than lying at zero energy as
it would be expected. The well width, therefore, determines the absorption threshold
of the QW and consequently its bandgap [25]. Therefore, the width of the QW can
be adjusted so that its bandgap is closer to the optimum bandgap corresponding to
a particular incident spectrum [26] and this constitutes an additional motivation for
incorporating quantum wells in a solar cell. The well width, however, does not only
affect the bandgap of the QW, but also the carrier recombination taking place in the
QW. As the well width increases, it becomes more difficult for carriers to escape,
which results in higher recombination rates and a reduction in the Voc [27]. Conse-
18
20. quently, the QW width must be tuned to satisfy the requirement for both an optimum
bandgap and low recombination rates.
In addition to determining the absorption threshold/bandgap of a QW, quantum
confinement has also an effect in the density of states (DoS). Since the carriers are
confined in one dimension, the DoS for a QW is 2D rather than 3D as for a bulk
semiconductor. The 2D and 3D DoS are given by [16]:
g2D(E) =
m∗
π 2
n
H(E − En) (2.2.1)
and [25]:
g3D(E) =
1
2π2
2m∗
2
3
2 √
E (2.2.2)
In the above expressions, m∗ is the electron or hole effective mass. The DoS for both
the QW and the bulk semiconductor are shown in Figure 2.2.3, as a function of the
energy measured from the bandgap.
Figure 2.2.3: DoS for a QW (g2D) and a bulk semiconductor (g3D).The energy is measured from
the bandgap [25].
In the case of the QW, H(E − En) in the expression for the DoS is the Heaviside
function, which has a value of 1 for E > En and a value of 0 for E < En and is
responsible for the staircase shape of the DoS. The DoS is independent of energy,
except when a different transition energy, En is reached [16]. The DoS for the bulk
19
21. semiconductor, on the other hand, increases proportionally to
√
E and takes a finite
value at the bandgap energy, as opposed to the DoS for the QW, which becomes
nonzero at a higher energy. This can be understood by looking at the valence to
conduction band transitions in a QW, as illustrated in Figure 2.2.4.
Figure 2.2.4: Schematic diagram illustrating the transition from the first hole state in the valence
band to the first electron state in the conduction band of a QW, for a nonzero in-plane wavevector
[28].
The vertical arrow in the diagram shows in particular, that in order for a photon
to excite an electron from the first hole state (quantum number n=1) in the valence
band to the first electron state in the conduction band, the photon must have an
energy equal to the sum of the energies of the QW bandgap, ground hole and ground
electron states. This can be expressed as [28] :
ω = Eg +
2
k2
xy
2m∗
h
+ Eh1
+
2
k2
xy
2m∗
e
+ Ee1
(2.2.3)
The second and third terms in brackets represent the total energy of the ground hole
and electron states, respectively. The terms enclosed in brackets show that the total
energy of an electron (hole) in the ground state is the sum of the quantized energy in
the z direction, which, here is represented by Ee1 (Eh1) and the kinetic energy of the
electron (hole) in the (x,y) plane,
2
k2
xy
2m∗
e
2
k2
xy
2m∗
h
. The kinetic energy term comes from
the fact that the carriers are free to move in the x,y directions and is defined by the
wavevector in the quantum well plane, kxy and the effective mass of the carrier in the
corresponding band, which is m∗
h for holes in the valence band and m∗
e for electrons
20
22. in the conduction band. For a bulk semiconductor, the absorption edge is simply at
ω = Eg, hence the finite value of the DoS for the bulk semiconductor at the bangap
energy.
In the infinite well approximation, where the potential V (z) is infinite outside the
well and zero inside the well, the general expression for the quantized hole or electron
energies in the z direction, in eq. 2.2.3, is given by [28]:
En =
2
k2
n
2m∗
=
2
π2
n2
2m∗d2
(2.2.4)
where kn is the quantized wavevector in the z direction and d is the QW width. Since
the quantized energy in the growth direction is inversely proportional to the hole or
electron effective mass, the first LH level lies at a greater energy than the first HH
level for a QW. For a bulk semiconductor, this quantization of energy does not take
place, so the first light and heavy hole bands have the same energy, as shown in Figure
2.2.5.
Figure 2.2.5: Band structure of an unstrained, direct bandgap bulk semiconductor. The HH and
LH bands have the same energy at k=0 [29].
Due to the effect of quantum confinement and the behavior of the 2D DoS, QWs are
useful for a variety of applications for optoelectronic devices. For example, QW lasers
have considerably lower threshold current density for lasing operation compared to
conventional semiconductor laser diodes, due to the sharp rise in the DoS at the onset
of each transition [30]. Another important advantage brought by these structures is
the restriction of the radiative emission in solar cells, upon incorporation of QWs,
which can result in a reduction in the radiative losses and consequently to an increase
21
23. in the solar cell efficiency. This effect, known as anisotropic or restricted radiative
emission has been explained in the Introduction and is heavily studied in this project.
22
24. Chapter 3
Radiative coupling
3.1 Theory of radiative coupling
3.1.1 Introduction to the concept of radiative coupling
When electron-hole pairs recombine radiatively in a high bandgap junction of a multi-
junction solar cell, the photons emitted can be reabsorbed in the active layer, leading
to the generation of new electron-hole pairs (“photon recycling”) or absorbed para-
sitically in layers of the structure other than the active layer, such as the substrate,
which constitutes a loss mechanism. In addition, the emitted photons can escape
from the front surface of the solar cell, or absorbed in the adjacent junction of lower
bandgap. The latter process is termed radiative coupling. Other expessions in the lit-
erature for this process include luminescent, optical or photonic coupling. Figure 3.1.1
demonstrates the aforementioned processes. The last process results in an increase in
the photocurrent of the lower bandgap junction and therefore in an enhancement in
the solar cell efficiency [7]. The greatest improvement in performance from radiative
coupling is achieved when the junction to which light is coupled is the current-limited
junction, that is, the one to which the least current is flowing [31].
23
25. Figure 3.1.1: Schematic diagram showing the four processes that can take place upon photon
emission in a solar cell : Reabsorption of the emitted photons (red), escape from the front surface
(dark blue), absorption to the next junction(light blue) and parasitic absorption (yellow) [7].
Then, the photocurrent of the current-limited junction will be increased to the point
where the series-connected junctions of the solar cell become current matched. The
IV curves for a two-junction GaInP/GaAs solar cell in Figure 3.1.2 serve to illustrate
this, where the bottom GaAs junction is the current-limited one. The Jsc of the device
is measured when the voltage applied to it is zero. Due to the requirement that at
equilibrium the currents of the top, bottom junctions and full device must be equal,
the Jsc is obtained when a forward bias is applied to the top cell to induce radiative
recombination and force it to act as an LED and a reverse bias to the bottom cell,
as shown in the diagram. Radiative recombination in the top cell induces radiative
coupling from the top to the bottom junction, increasing the photocurrent of the
latter. At some point radiative coupling will be reduced, due to the decrease in
the recombination current and equilibrium will be reached [32], where the current-
matched junctions are at their maximum operating points and the efficiency of the
overall device is maximised [33].
24
26. Figure 3.1.2: IV curves for a two-junction solar cell in which the bottom cell is current-limited.
The green, blue and black lines correspond to the IV curves of the bottom, top junctions and full
device, respectively. More importantly, the increase in the photocurrent of the bottom junction, in
the presence of radiative coupling, is shown. The current density added to the bottom cell due to
radiative coupling is represented by JLC12. The dashed lines represent the new IV curves, in the
presence of radiative coupling, when equilibrium has been established. The circles represent the
maximum operating points of each junction and show that when the short circuit current of the
device is measured in this case, the top and bottom cells are operated in forward and reverse bias,
respectively [32].
3.1.2 Quantifying the radiative coupling: the coupling efficiency
As mentioned in 3.1.1, the increase in the solar cell efficiency in the presence of
luminescent coupling comes from the increase in the photocurrent of the current-
limited junction. In order to obtain expressions for both the added current density
and the efficiency of this process, we first start by simply considering the total current
flowing through each of the junctions under illumination:
Jtot(V, φ) = −Jsc(φ) + Jrec(V ) (3.1.1)
This is the superposition approximation, where Jsc is the short circuit current and
Jrec is the recombination current [33]. Each junction is modelled as a two-diode cir-
cuit, in which one diode stands for the radiative recombination current, arising from
conduction band-to-valence band recombination and the other for the non-radiative
recombination current, due to the non-radiative Shockley-Read-Hall (SRH) recombi-
nation mechanism. The SRH mechanism refers to the non-radiative recombination
of electrons with holes in defect states, which lie in the bandgap [19]. The ideality
25
27. factors for the radiative and non-radiative processes are 1 and 2, respectively1
. The
dark recombination current for each subcell is then written as follows:
Jrec(V ) = J01 exp
eV
KBT
− 1 + J02 exp
eV
2KBT
− 1 (3.1.2)
where J01 and J02 represent the radiative and non-radiative recombination current
densities, respectively [31, 32]. When luminescent coupling is taken into account,
another term needs to be added in eq. 3.1.1 [34]:
JLCij = ηijJ01,i exp(
eVi
KBT
) − 1 (3.1.3)
where JLCij is the extra photocurrent density in the jth junction due to the coupling
of light from the overlying ith junction and ηij is the coupling efficiency, which,
as can be seen from the above equation, is the ratio of the current density due to
radiative coupling to the radiative recombination current in the ith junction. In other
words, it is the fraction of light from radiative recombination in the ith subcell that
contributes to the photocurrent of the current-limited jth junction. Eq. 3.1.1 can
then be modified to express the total current in the jth junction, in the presence of
radiative coupling, as:
Jtot,j(Vj, φ) = −Jsc,j(φ) + J01,j exp
eVj
KBT
− 1 + J02,j exp
eVj
2KBT
− 1
+ηijJ01,i exp(
eVi
KBT
) − 1 (3.1.4)
The coupling efficiency is always < 1, since a small fraction of light, of about 1
4n2
escapes from the front surface of the ith junction rather than being absorbed in the
subsequent junction to increase its photocurrent [34, 9]. For forward bias V > 0, the
exponential terms dominate and eqs. 3.1.2 and 3.1.3 can be combined to give [32]:
Jrec =
JLCij
ηij
+ J02
JLCij
ηijJ01
1
2
(3.1.5)
Which gives the following solution for JLCij:
1
Actually, an ideality factor of 1 can also include non-radiative processes, but this is not accounted for in
most of the literature. In addition, Auger non-radiative recombination is neglected in eq. 3.1.2 [32].
26
28. JLCij = ηij φ2
i + Jrec − φi
2
(3.1.6)
where φi = J02,i
2
√
J01,i
is the coupling linearity [35] and in the above equation indicates
a non linear relationship between the luminescent coupling current and the radiative
recombination current. This is termed non-linear luminescent coupling. If the non-
radiative SRH recombination mechanism in the ith junction can be neglected, then
J02,i = φi = 0 and a linear relationship can be obtained, leading to linear luminescent
coupling [32]:
JLCij = ηijJrec (3.1.7)
Various expressions can be found in the literature for the coupling efficiency ηij.
Considering the processes that can occur following photon emission, as shown in
Figure 3.1.1, the coupling efficiency for a tandem solar cell is [7]:
η12 =
ηintPLC
1 − ηintPabs
(3.1.8)
where ηint is the internal luminescent efficiency or the probability of radiative recom-
bination in the first junction, PLC is the probability that light will be coupled to the
second junction and Pabs is the probability that light emitted in the first junction will
be reabsorbed in the same junction and generate more electron-hole pairs.
In summary, photonic coupling is a useful effect in a solar cell having multiple junc-
tions, simply because the light emitted by one junction that is not lost by parasitic
absorption or does not escape from the solar cell is effectively recycled, contributing
to the photocurrent of the next junction and the overall solar cell efficiency. There-
fore, it is important to investigate any processes that can limit this effect, which is
the primary aim of this project.
27
29. Chapter 4
Emission in bulk semiconductors and
QWs
4.1 The optical matrix element for bulk semiconduc-
tors and QWs
In the Introduction, it was mentioned that the TE polarized light is mostly associated
with HH transitions, with a small contribution from LH transitions, whereas the TM
polarized light couples only to LH transitions. The aim of this section is to briefly
demonstrate, starting by the Fermi Golden rule, how this conclusion is drawn. The
Fermi Golden rule forms the basis for the derivation of the absorption coefficients for
both bulk semiconductors and QW, which will be used later to calculate the emission
profile of a QW and most importantly, the fraction of emitted light from the strained
QW that can be coupled to the next cell in a tandem structure. The Fermi Golden rule
is derived from the time-dependent perturbation theory, which considers the evolution
of a system upon its interaction with light and it is an expression of the probability
per unit second (transition rate) that an electron will transit from an initial state
|i of energy Ei to a final state |f of energy Ef . This transition is induced by the
absorption or emission of a photon. In the case of photon absorption, the electron
undergoes, in the presence of a perturbed Hamiltonian, H (r), a transition from a
lower energy state to a higher energy state so Ef > Ei and the transition rate is given
by [36]:
Wabs =
2π
| f| H (r)|i |
2
δ(Ef − Ei − ω) (4.1.1)
28
30. where ω is the photon energy. The transition rate Wems for the emission of a photon
when the electron drops from |f to |i is:
Wems =
2π
| i| H (r)|f |
2
δ(Ei − Ef + ω) (4.1.2)
To calculate the total rate of upward transitions upon photon absorption, we need
not only to sum over all initial and final states, but also to take into account their
occupancies. For example, absorption can only take place when the initial state is
occupied and the final is empty, whereas the opposite applies for emission. The total
rate of upward electron transitions per unit volume is then [36, 37]:
Ri→f =
2
V ki kf
2π
| f| H (r)|i |
2
δ(Ef − Ei − ω)fi(1 − ff ) (4.1.3)
where f is the Fermi Dirac distribution, which gives the occupancy of a given state and
ki, kf are the wavevectors of the initial and final states, respectively.The occupancy
of the initial state is given by [37]:
fi =
1
1 + exp(Ei−EF
KBT )
(4.1.4)
In eq. 4.1.3, the factor (1 − ff ) indicates that the final state is empty, whereas V
denotes the volume of the sample and the factor of 2 accounts for the two possible
electron spin directions. The total rate of downward electron transitions, Rf→i is
given by a similar expression, using eq. 4.1.2 and ff (1−fi) for the state occupancies.
The net rate of upward electron transitions per unit volume is then:
R = Ri→f − Rf→i =
2
V ki kf
2π
| f| H (r)|i |
2
δ(Ef − Ei − ω)(fi − ff ) (4.1.5)
The above expression is equivalent to the number of absorbed photons per unit time
per unit volume, which, when divided by the number of input photons in the sample
per unit time per unit area yields the absorption coefficient as a function of photon
energy a( ω) (cm−1
) [37]:
a( ω) =
2 µc
nωA2
o
2
V ki kf
2π
| f| H (r)|i |
2
δ(Ef − Ei − ω)(fi − ff ) (4.1.6)
29
31. where µ represents the magnetic permeability, ω and c are the frequency and speed
of light, respectively, n stands for the refractive index of the sample and H (r) =
−e
mo
A f |p| i = −eAo
2mo
ˆe · pfi. In the latter expression, A is a vector potential associated
with the electromagnetic field of light in the system, ˆe is the polarization vector of
the electric field of light and pfi is the momentum matrix [37]. It is obvious then
that, the absorption coefficient is proportional to |ˆe · pfi|
2
.
To calculate a( ω), the momentum matrix elements need to be evaluted according to
the transition taking place. The general expression for the momentum matrix is:
pfi = f |p| i (4.1.7)
The wavefunctions of the initial and final states, |i and |f are given by [36]:
Fi(r) = uvi (r)fi(r) =
1
√
S
uvi (r)exp(ik⊥ · r⊥)χi(z) (4.1.8)
Ff (r) = uvf
(r)ff (r) =
1
√
S
uvf
(r)exp(ik⊥ · r⊥)χf (z) (4.1.9)
In the above equations, the subscripts i and f stand for initial and final, respectively,
uv(r) is the Bloch function having the lattice periodicity, v denotes a band (light,
heavy hole or conduction band), whereas k⊥ and r⊥ are the wavevector and position
vector, respectively, in the (x,y) plane (QW plane). The carrier motion in the z
direction is described by the envelope function, χi(z) and S is the area of the sample.
Inserting 4.1.8 and 4.1.9 into 4.1.7 gives:
e · pfi = e · uvf
|p| uvi ff | fi + e · uvf
| uvi ff |p| fi (4.1.10)
The intraband transitions involve the second part of the right hand side of eq. 4.1.10,
where uvf
| uvi is non zero only when vf = vi and therefore forces the electrons to
remain in the same band. Interband transitions involve the first part of the right
hand side of eq.4.1.10, where uvf
|p| uvi depends on the polarization of light and
ff | fi determines the quantum number of the initial and final electron and hole
states for which transitions are allowed, as well as the strength of the transitions and
defines the conservation of k⊥. In particular, for an infinite QW, the quantum num-
bers of the electron and hole levels must be the same for the transition to be allowed,
because this is the only way for ff | fi to be equal to 1. For example, a transition
30
32. from the first electron level to the first hole level would be allowed, but a transition
from the first electron level to the second hole level would be forbidden. However,
for a finite QW, the quantum numbers do not necessarily have to be the same, since
the wavefunctions are not exactly orthogonal. It is sufficient, in that case, that the
difference between the quantum numbers of the hole and electron levels is an even
number, since then the parities of these levels would be the same. Of course, such
a transition would be weaker compared to an interband transition where the initial
and final states have the same quantum number. Finally, a transition for a finite QW
would be forbidden if the difference between the quantum numbers is an odd number,
as this implies that the parities of the levels are opposite and their overlap is zero
[28, 38, 36] .
Since we are considering transitions between the valence (LH or HH) and the conduc-
tion band, the optical momentum matrix will be uc |p| uv = M, where uc represents
either the spin up or spin down wavefunctions for the conduction band and uv repre-
sents the wavefunctions for either the heavy hole , uhh or light hole, ulh , bands [37].
As the optical momentum matrix depends on the polarization of light, |ˆe · M|
2
can be
evaluted for ˆe = ˆx (or ˆe = ˆy) and ˆe = ˆz if the transition involves TE or TM polarized
light, respectively. Table 4.1, adopted from [37] summarizes the values of |ˆe · M|
2
for
each transition, for both bulk semiconductors and QWs, which are calculated from
the explicit expressions for the conduction and valence band wavefunctions.
Table 4.1: Momentum matrix elements for bulk semiconductors and QWs [37].
In this table, the subscripts c, lh and hh denote conduction, LH and HH bands,
respectively, whereas Ep is an energy parameter (constant) depending on the material.
The results here show that the |ˆe · M|
2
has a constant value, independent of the
polarization of light, for a bulk semiconductor, equal to M2
b = mo
6 Ep, whereas in the
case of the QW, M2
b is multiplied by an anisotropy factor, which is different for each
31
33. transition and polarization of light. This is due to the fact that, the structure of the
bulk semiconductor is symmetric/isotropic, as opposed to the asymmetric structure of
the QW [39, 40]. Since |ˆe · M|
2
is a quantity that enters into the absorption coefficient,
a( ω), this results in the polarization dependent absorption/emission of light in QWs.
At the band edges, where the wavevector is zero, we obtain, for transitions involving
TE polarized light [37]:
|ˆx · Mc−hh|
2
=
3
2
M2
b (4.1.11)
|ˆx · Mc−lh|
2
=
1
2
M2
b (4.1.12)
and for transitions associated with TM polarized light [37]:
|ˆz · Mc−hh|
2
= 0 (4.1.13)
|ˆz · Mc−lh|
2
= 2M2
b (4.1.14)
The above equations are physically significant, since they provide important infor-
mation about the coupling of light to a particular transition, depending on its polar-
ization. In particular, it can be deduced from equations 4.1.11 and 4.1.12 that the
coupling of TE polarized light to the conduction-LH band transition is three times
weeker than its coupling to the conduction-HH transition. Likewise, equations 4.1.13
and 4.1.14 show that TM polarized light cannot couple to the HH transition.
A different way of understanding the difference between the optical momentum ma-
trix of bulk semiconductors and QWs, is by looking at Figure 2.2.5 in Section 2.2.
Since for an unstrained, bulk semiconductor, the first LH and HH bands have exactly
the same energy at the maximum of the valence band, the emission of light following
radiative recombination of an electron from the conduction band with a hole in the
valence band will not depend on whether the hole comes from the LH or HH band, so
it will be isotropic. However, in the case of a QW, the expression for the quantized
energy in the z direction, given by eq. 2.2.4, shows that the first LH band lies at a
higher energy than the first HH band, due to the difference in the effective masses
of the light and heavy holes. Since the thermal occupation of the HH band is higher
than that of the LH band, the HH transition will be stronger than the LH transition
for light propagating in the z direction [10]. This is consistent with the values of
32
34. the momentum matrices for a QW, given by equations 4.1.11 and 4.1.12, which show
that the HH transition is three times stronger than the LH transition for TE polarized
light propagating in the z direction.
4.2 Simulation of the radiative emission in strained
QWs
4.2.1 Modelling the strained QW emission profile
Since the momentum matrix elements |ˆe · M|
2
for QWs have different values depend-
ing on the polarization of light and a( ω)∝|ˆe · M|
2
, then there will be different
absorption coefficients for TE and TM polarized light, indicated by αTE and αTM ,
respectively, rather than a single, isotropic absorption coefficient, as for bulk semi-
conductors. In addition, the value of αTM can be varied depending on the amount
of strain on the QW structure. In particular, an increase in the QW compressive
strain, achieved by increasing the Indium fraction in a QW, leads to a decrease in
αTM . Since the absorption coefficient for QWs is not isotropic, the fraction of light
absorbed or emitted from the QW structure will have not only spectral, but also an-
gular dependence. For a compressively strained QW, where the absorption/emission
of TM polarized light is suppressed, the fraction of light absorbed in the structure is
given by [16]:
αsp(E, θ) = 1 − e(−βαn(E)z)
(4.2.1)
where z is the thickness of the QW layer, αn is the absorption coefficient for light
incident in a direction perpendicular to the plane of the QW and β is a factor that
accounts for the suppression of the TM mode, which can be written as [16]:
β = cos2
θ +
αTE(E) + αTM (E)
2αTE(E)
sin2
θ (4.2.2)
where θ is the angle at which light from the QW is emitted with respect to the normal
to the QW plane. Assuming that αn ≈ αTE , since the TM mode is not associated
with absorption/emission in the normal, the factor βαn(E) in eq. 4.2.1 is rewritten
as:
33
35. αn(E, θ) = αTE(E) cos2
θ +
sin2
θ
2
+
αTM (E)
2
sin2
θ (4.2.3)
Then, eq. 4.2.1 can be multiplied with the Planck function for non-blackbody radia-
tion, which can be used to decribe the rate of light emission from radiative recombi-
nation by a semiconductor, to give the emission profile of the QW [16]:
˙N =
2πn2
semi
c2h3
E2
e
E−qV
KBT
− 1
αsp(E, θ) (4.2.4)
Inserting equations 4.2.1 and 4.2.3 into eq. 4.2.4 gives:
˙N =
2πn2
semi
c2h3
E2
e
E−qV
KBT
− 1
1 − exp − αTE(E) cos2
θ +
sin2
θ
2
+
αTM (E)
2
sin2
θ z
(4.2.5)
Here, nsemi is the refractive index of the semiconductor , E is the energy of the emitted
light, KB is the Boltzmann constant, h is Planck’s constant, T is the temperature
and V is the amount of forward bias applied to the solar cell. Here, it is assumed that
the forward bias matches the difference between the electron and hole quasi-Fermi
levels [16]. The coefficients αTE(E) and αTM (E) for a given sample were calculated
by Dr Diego Alonso-Alvarez, using a program, based on k p theory, developed by
Dr Markus Fuhrer. As an example, the profile of the calculated coefficients for the
GaAs0.9P0.1/GaIn0.109As0.891 QWs in the sample QSOL ( Table 5.1) is shown in Figure
4.2.1.
34
36. Figure 4.2.1: The absorption coefficients αTE(E) and αTM (E) for the
GaAs0.9P0.1/GaIn0.109As0.891 QWs in the sample QSOL (Table 5.1). These were calculated
by Dr Diego Alonso-Alvarez, using a program developed by Dr Markus Fuhrer. The first and
second steps in αTE(E) correspond to the first HH and LH transitions, respectively, whereas the
single step in αTM (E) corresponds to the first LH transition.
It can be seen, that αTE(E) has two sharp peaks at 1.33 and 1.38 eV, both followed
by plateaus, whereas αTM (E) shows a single sharp peak at 1.38 eV, followed by a
single plateau. From the staircase shape of the DoS for a QW, it is clear that the first
step in αTE(E) corresponds to the first HH transition, whereas the second step to
the first LH transition. Only the step corresponding to the LH transition is seen for
αTM (E) , as expected, since the TM mode has no contributions from HH transitions.
The sharp peaks at 1.33 and 1.38 eV correspond to the excitons associated with the
first HH and LH transitions, respectively, which are bound electron-hole states [40].
A MATLAB code, based on eq.4.2.5, was written to calculate, using αTE(E) and
αTM (E), the QW emission profile for the samples in Table 5.1. The simulated emis-
sion profile is compared with the experimental emission profile for these samples in
Section 5.2.1.
4.2.2 Theoretical investigation of the impact of restricted radia-
tive emission on the radiative coupling
In order to calculate the fraction of the QW emission that can be coupled to the
adjacent junction of a multijunction solar cell, eq. 4.2.1 is modified to account for the
35
37. fact that light emitted from the QW can reach the next junction at a range of angles
and thus will travel a longer distance compared to light emitted in the normal [16]:
αsp(E, θ) = 1 − e(−αn(E,θ)z
cosθ ) (4.2.6)
where θ is the angle at which light travels in the structure. In that case, eq.4.2.4
must be integrated over solid angle and over the whole energy range at which QW
emission occurs [31]:
˙N =
2n2
semi
c2h3
ˆ ∞
Eg
E2
e
E−qV
KBT
− 1
ˆ
S
αsp(E, θ)cosθdΩ · dS dE (4.2.7)
where dΩ = sinθdθdϕ is the solid angle element and
´
S αsp(E, θ)cosθdΩ · dS can be
explicitly written as [31]:
ˆ
S
αsp(E, θ)cosθdΩ · dS = 2π
ˆ 1
cosθc
(1 − r(E, θ)) 1 − e−αn(E,θ)z
cosθ cosθd(cosθ)+
2π
ˆ 1
cosθc
1 − r(E, θ)e−αn(E,θ)z
cosθ 1 − e−αn(E,θ)z
cosθ cosθd(cosθ)+
2π
ˆ cosθc
0
1 − e−2αn(E,θ)z
cosθ cosθd(cosθ)
(4.2.8)
where
´
S
´
dΩ · dS =
´ 2π
0 dφ
´
S
´ θc
0 sinθdθ. In the above equation, θc is the critical
angle for total internal reflection and r(E, θ) is the reflectivity of the top surface.
The first integral corresponds to light propagating in the junction at angles in the
range 0 ≤ θ ≤ θc, which means that it can escape from the top surface. The second
integral corresponds to photons travelling at the same range of angles but instead of
escaping from the top surface are absorbed in the subsequent junction. Finally, the
third integral corresponds to light propagating in the angular range θc ≤ θ ≤ π
2 . This
means that T.I.R occurs and light undergoes a second pass through the QW layer
before being coupled to the next cell, hence the factor of 2 in the exponential. For
the sake of simplicity, one can assume r(E, θ) = 0 so that there is no refractive index
difference between the semiconductor and air. This allows to set nsemi = nair = 1
and θc = π/2.Since cos(π/2) = 0, the third integral vanishes and the first and second
integrals are the same. Multiplying eq. 4.2.7 with the electron charge,q, allows to
36
38. calculate the current density generated by the light coupled to the adjacent junction,
in analogy to eq. 3.1.3, giving:
Jcoupled =
8πq
c2h3
ˆ
E2
e
E−qV
KBT
− 1
ˆ 1
0
1 − e−αn(E,θ)z
cosθ cosθd(cosθ)
dE (4.2.9)
with αn(E, θ) as defined in eq. 4.2.3. Finally, repeating this calculation for different
values of forward bias gives the radiative dark IV curve of the QWSC.
37
39. Chapter 5
Experiment
5.1 Samples
5.1.1 Sample Fabrication
The samples were fabricated by Dr. Peter Spencer at Imperial College London,
using Molecular Beam Epitaxy (MBE). This technique can be briefly described as
follows: Different types of atoms are heated in different furnaces, also known as
effusion cells, to generate atomic or molecular beams. The beams are then aimed
towards a substrate wafer which has been previously heated to a sufficiently high
temperature in order to produce a clean surface. The chemical reaction of these beams
with the substrate results in the formation of extremely thin compound semiconductor
layers. Since the beams are directed towards the wafer at different angles, the wafer
holder is rotated to ensure uniform deposition of species. In order to control which
layers will be grown on the wafer, or to terminate the deposition of a given species,
shutters are used in front of the effusion cells, which can be opened or closed to
turn the beam on or off very fast, resulting in abrupt changes in the composition of
the layer formed on the substrate. The growth process is performed in ultra high
vacuum conditions to achieve clean surfaces and pure, high quality layers. Detailed
information about the crystal structure grown on the substrate can be obtained during
the growth process, by diffraction of electron beams by the atoms that constitute the
layer of interest [41, 42, 43].
38
40. 5.1.2 Single-structure, strain-balanced QW (SBQW) samples
The first set of photoluminesce (PL) measurements was performed on three single-
structure, strain-balanced QW samples, having different compositions and different
degrees of compressive strain. The properties of the samples are summarised in Table
5.1.
Sample Barrier
composi-
tion
Barrier
width
(nm)
QW com-
position
QW
width
(nm)
# of
QWs
T6 GaAsP0.15 17 In0.2Ga0.8As 8 40
QT1657 Al0.04GaAsP0.0611.6 In0.1Ga0.9As 7.5 35
QSOL GaAs0.9P0.1 15.8 GaIn0.109As0.891 7.8 50
Table 5.1: Single-structure samples with different degrees of compressive strain.
5.1.3 Double-structure, SBQW samples
The structure of the sample designed by Dr. Diego Alonso-Alvarez to study the im-
pact of restricted radiative emission on the photonic coupling is shown in Figure 5.1.1.
The growth direction and energy are indicated in the diagram. The sample consists
of two structures grown on a GaAs substrate. The top structure contains three test
AlGaInAs QWs surrounded by AlGaAs barriers with 40% Al, whereas the bottom
structure contains four probe InGaAs QWs surrounded by GaAs barriers. Each of
the QWs is 8 nm thick. The light emitted by the test wells can be absorbed in the
bottom structure, generating carriers that diffuse towards the probe QWs and recom-
bine, resulting in the emission of lower energy light. The intensity of the light emitted
by the probe wells can then provide information about the efficiency of photonic cou-
pling between the test and probe wells. In order to determine how the restricted
radiative emission affects the radiative coupling, samples with different degrees of
strain on the test QWs were fabricated, by varying the Indium content, whereas the
strain on the probe wells was kept constant, by fixing the Indium concentration in
the InGaAs QWs to 20%. A higher Indium content leads to a higher compressive
strain and consequently greater suppression of the TM mode and a greater restric-
tion of radiative emission from the test wells. As test QWs of different strain must
emit at the same wavelength, the Aluminium fraction in the AlGaInAs QWs was also
increased as the Indium fraction was increased, since the Aluminium has the opposite
39
41. effect on the QW bandgap to the Indium. In particular, the Indium, as mentioned
previously, decreases the bandgap, whereas the Aluminium increases it. The sample
also contains three 80 nm thick AlGaAs barriers with 60% Al. The bottom, middle
and upper barriers have been grown between the substrate and the bottom cell, the
bottom and the top cell and the top cell and cap layers, respectively. The purpose
of these barriers is to prevent the carriers generated in either the test or probe QW
regions from diffusing to other areas of the structure. The probe well region has an
overall thickness of 500 nm , while the test well region is 400 nm thick. The structure
ends with a GaAs cap layer. The samples fabricated for the purpose of this study
are U34097, U34099 and U34100 with 0% (no strain-reference sample), 20% and 10%
Indium, respectively, in the test wells. The test and probe barrier/QW compositions
for these samples are shown in Table 5.2.
Figure 5.1.1: Double-structure sample designed by Dr. Diego Alonso-Alvarez to study the impact
of restricted radiative emission on the photonic coupling.
Sample Test Barrier/QW composition Probe Barrier/QW composition
U34097 Al0.4Ga0.6As/GaAs GaAs/In0.2Ga0.8As
U34099 Al0.4Ga0.6As/Al0.15Ga0.65In0.2As GaAs/In0.2Ga0.8As
U34100 Al0.4Ga0.6As/Al0.07Ga0.83In0.1As GaAs/In0.2Ga0.8As
Table 5.2: Test and probe barrier/QW compositions of the double-structure samples to study the
impact of restricted radiative emission on the photonic coupling.
40
42. 5.2 PL measurements on single-structure, SBQW sam-
ples
5.2.1 Front Surface PL measurements
Experimental Set-up
The QW emission from each sample was characterised, at room temperature, by ex-
tracting the PL from the front surface of the sample. In this set-up, an Nd:YVO4 green
laser light of 532 nm wavelength and 0.2W power from a Millenia-V Spectra Physics
Source was directed towards the front surface of the sample after being reflected from
a series of mirrors and passing through a frequency chopper. The frequency chop-
per was connected to a Bentham 218 variable frequency instrument and was used
to ensure that only the signal from the sample is measured, rather than background
signals related to ambient light. A filter was attached at the lens which focuses the
light into the sample in order to attenuate the laser power. This was done in order
to prevent a large increase in temperature due to sample heating, that would, on
the one hand, increase the excited carrier densities and on the other hand, lead to
carrier escape from the barriers, which means that no radiative recombination would
take place and the PL from the QW would be lost. In general, a high increase in
temperature due to a high power would change the optical properties of the sample.
The light emitted from the sample was collected by a lens and focused by another
lens to the slits of a Princeton Instruments monochromator. A filter was attached to
the lens focusing the light to the monochromator to block any laser light that could
reach the monochromator, for example by reflections.
After passing through the monochromator, the PL from the QWs was collected by a
Si detector. The Si detector was connected to a Stanford Research Systems lock-in
amplifier, which indicated the strength of the PL signal. The software SpectraSense
was used to acquire data from the Si detector. A schematic diagram of the set-up
used to perform the front PL measurements is shown in Figure 5.2.1.
41
43. Figure 5.2.1: Schematic diagram of the set-up used to perform the front PL measurements.
Results and Discussion
The normalized simulated and experimental data for the samples QSOL, QT1657
and T6 are shown in Figure 5.2.2. Assuming that the highest contribution to the
PL signal in this experiment is light emitted in the normal with respect to the front
surface of the samples, the simulation of the QW emission profile for each sample,
was performed for θ = 0, for comparison. The voltage V in the simulation was chosen
to be as close as possible to the energy at which the emission from each sample is
maximum. A value of nsemi = 3.6 was used in the QW emission profile for each
sample, since this is the typical value for semiconductors that belong in the III-V
groups[31] and z was set to be equal to the QW width in each sample. In the case of
the sample QSOL, the main emission peak in the experimental data is around 1.325
eV, with additional peaks around 1.345 , 1.37, 1.38 and 1.42 eV. The peak at 1.42 eV
corresponds to the bandgap of the GaAs substrate. At first, it may seem that the
peaks at 1.325 eV and 1.345 eV correspond to the transition from the first electron
level (e1) to the first HH (hh1) and first LH (lh1) levels, respectively. However, the
calculated profile of the absorption coeffiecients αTE(E) and αTM (E) for this sample,
illustrated in Figure 4.2.1, shows that the first step in αTM (E), which is associated
with the first LH transition, occurs around 1.38 eV. Therefore, it may be possible that
the peaks at 1.325 and 1.345 eV correspond to the e1-hh1 and e1-hh3 (where hh3 is
the third HH level) transitions, respectively, and that the peaks at 1.37 and 1.38 eV
correspond to the e1-lh1 and e1-lh3 (where lh3 is the third LH level) transitions, re-
spectively. Although for infinite QWs only interband transitions between levels of the
42
44. same quantum number are allowed, it was explained in Section 4.1 that for realistic,
finite QWs, such as the ones contained in this sample, transitions from electron to
hole levels where the difference in the quantum numbers is an even number can also
be observed. Then, the experimental data suggest that, for this sample, the hh1-lh1
splitting is around 45 meV and that the hh1 transition is 1.6 times stronger than the
lh1 transition. Comparing the experimental with the simulated data, it is likely that
the peak at 1.33 eV in the simulated data corresponds to the e1-hh1 transition and the
barely visible peak at around 1.38 eV to the e1-lh1 transition, giving a hh1-lh1 split-
ting of 50 meV and a hh1 transition 20 times stronger than the lh1 transition. The
above differences between the experimental and simulated data could be attributed
to several factors. The calculation of αTE(E) and αTM (E) assumes a QW with a
perfect square shape, which is not the case for the QWs in the sample. In addition,
small variations in the QW thickness and Indium fraction of each QW in a given
sample, during the growth process, can affect the QW emission in the experiment.
In particular, different QWs in a given sample emit at slightly different wavelengths,
resulting in broader emission peaks in the experimental data. Finally, the substrate
and impurities in the sample contribute to the PL, which is not accounted for in
the calculation of αTE(E) and αTM (E) and consequently in the QW emission profile
calculation.
Similarly, for the sample QT1657, the first step in αTM (E) (shown in the Appendix)
occurs at around 1.4 eV, suggesting that the e1-hh1, e1-hh3, e1-lh1 and e1-lh3 tran-
sitions correspond to the 1.35, 1.37, 1.38 and 1.40 eV peaks, respectively, in the
experimental data. In the modelled QW emission profile for this sample, the e1-hh1
transition corresponds to 1.35 eV, in excellent agreement with the experimental data,
whereas the small e1-lh1 transition peak can be seen at 1.4 eV. In the case of T6,
the LH transition is suppressed in both in the modelled and experimental emission,
since T6 has twice the Indium fraction compared to QSOL and QT1657 and a single
emission peak corresponding to the e1-hh1 transition is seen at 1.23 and 1.25 eV, for
the simulated and experimental data, respectively.
43
45. Figure 5.2.2: Simulated (red) and experimental (black) emission from the single-structure samples
QSOL (top), QT1657 (middle) and T6 (bottom) at 300K. The emission profile of the QWs in each
sample was modelled based on eq.4.2.5, with nsemi = 3.6 and z=7.8, 7.5 and 8 nm for the QWs
in the samples QSOL, QT1657 and T6, respectively. A forward bias of 1.3, 1.3 and 1.2 V was
used in the simulation for QSOL, QT1657 and T6, respectively, to match the main emission in the
experimental data.
5.2.2 Edge PL measurements
Experimental set-up
In order to collect and analyze the TE and TM polarized light from the samples and
hence measure the restricted emission, the set-up described in the previous section
was initially slightly modified by using a Glan-Thompson polarizer in front of the lens
focusing the light to the monochromator. A waveplate was placed in front of the po-
larizer, whose purpose is to rotate the plane of polarization, allowing to compare the
relative strengths of the two modes at different angles. As an example, only the TM
polarized emission from the QWs would be recorded when the plane of polarization
44
46. was at 0◦
and only the TE polarized emission would be observed when the plane of
polarization was rotated by 90◦
(which corresponds to a rotation of 45◦
in the wave-
plate).The sample was mounted in the holder in a way such that the laser spot hits
the edge rather than the front surface, to enable collection of both polarization modes.
However, when the plane of rotation was set at 0◦
and 90◦
to obtain the TM and TE
emission respectively, it was observed that, even for the T6 sample with the highest
strain (and therefore the greatest suppression of the TM mode) the two polarizations
had the same signal strength. This indicates that emission was collected not from the
QWs, as required, but from the bulk substrate, which has an emission independent of
polarization. This occurs because in these samples, the multi-QW region is around
1-3 μm thick, much thinner than the substrate having a thickness around 300 μm
and the laser spot diameter lies in the range of 50-200 μm. Consequently, the main
disadvantage of this method is that the excitation of the substrate rather than the
QW region from the laser spot is unavoidable and leads to measurement artefacts. A
different approach was therefore adopted to overcome this.
In the new, so called micro-PL setup, the laser light passes through a series of mirrors
before reaching a beamsplitter, which is placed in front of the waveplate. The beam-
splitter transmits 70% of the incident laser light, permitting 30% of the light to be
reflected towards a microscope objective. The microscope objective focuses the laser
light into a very small spot, with a diameter of about 2µms to the edge of the sample,
so that the desirable multi-QW region can be excited. The sample was mounted on a
differential adjuster, which allows to adjust as precisely as possible, in steps of µms,
the position of the sample in the upward/downward, left/right and forward/backward
directions. 70% of the light emitted from the sample, after excitation, can then be
transmitted through the beamsplitter and directed through the monochromator to an
InGaAsP photomultiplier tube detector. For the purpose of ensuring that the laser
spot hits the edge of the sample, a mirror was placed between the waveplate and the
beamsplitter at an angle of about 45◦
with respect to the path of the light emitted
from the sample. Since 70% of the laser light reflected from the edge of the sample
can also be transmitted through the beamsplitter, the mirror reflects the laser light
into a camera connected to the computer, allowing to check the position of the laser
spot with respect to the sample and make fine adjustments. The mirror was removed
when data were acquired to permit light emitted from the sample to pass through the
monochromator. The photomultiplier tube detector is extremely sensitive and gives
a higher signal than the Si detector used in the previous setup, so measurements had
to be taken in the dark. A schematic diagram of the experimental set-up used to
45
47. perform the edge PL measurements is shown in Figure 5.2.3.
Figure 5.2.3: Schematic diagram of the experimental set-up used to perform the edge PL measure-
ments.
Results and Discussion
Measurements were performed, at 300K, only for the T6 sample, as the emission
from the other samples did not coincide with the detection range of the InGaAsP pho-
tomultiplier tube. The PL spectrum obtained for both the TE and TM components
is shown in Figure 5.2.4.
46
48. Figure 5.2.4: Edge PL of T6 at 300 K. The black and red lines represent the emission patterns of
the TE and TM polarized light, respectively.
The black line corresponds to the TE emission, whereas the red line to the emission
from the TM mode. The peak at 1.245 eV in the spectrum of the TE mode represents
the first HH transition (e1-hh1), whereas the lower peak at 1.31 eV represents the
first LH transition (e1-lh1), suggesting a hh1-lh1 splitting of 65 meV. According to
equations 4.1.11 and 4.1.12, the LH transition should be three times weaker than the
HH transition for TE polarized light, and for compressively strained QWs such as
those in T6 the LH transition should be even weaker. Therefore, we would expect
the intensity of the peak corresponding to the e1-lh1 transition to be more than three
times smaller than the intensity of the peak corresponding to the e1-hh1 transition.
Instead, it is obvious from this spectrum that the e1-lh1 transition is only 1.6 times
weaker than the e1-hh1 transition, which could be attributed to contributions to the
e1-lh1 transition from other layers in the structure.
Looking at the spectrum of the TM mode, on the other hand, we would expect
to observe a small peak at 1.31 eV since, as deduced from equations 4.1.13 and
4.1.14, TM polarized light is only associated with LH transitions and here the LH
transition should be mostly suppressed. A possible explanation for the unexpected
47
49. presence of the peak at 1.245 eV is that the light emitted in the structure, associated
with the HH transition, undergoes multiple internal reflections and escapes with a
random polarization from the edge [15]. This problem can be overcome by growing
an antireflection coating in the internal edge of the sample. This result could also be
explained by the fact that, away from the Brillouin zone centre, that is for k = 0,
the HH and LH bands are mixed, which means that, even though a HH transition is
forbidden in the case of the TM mode, it could still be observed due to its mixing
with the allowed LH transition [44, 45].
5.3 PL measurements on double-structure, SBQW sam-
ples
5.3.1 Front surface PL measurements
Experimental set-up
To extract the PL from the front surface of the double-structure SBQW samples
introduced in Section 5.1.3, the same method as in Section 5.2.1 was adopted. Data
were acquired both at room and low temperatures, by enclosing the samples in a
cryostat. The temperature was monitored and controlled using a Lakeshore 321 Au-
totuning controller.
Results and Discussion
The PL spectra obtained for the samples U34097, U34099 and U34100 (Table 5.2)
at a temperature of about 300 K, are shown in Figure 5.3.1. At 300 K, no emission
from the test QWs is observed, except for the sample U34097, which shows a small
peak around 1.5 eV. The probe QWs, however, emit strongly at around 1.23 eV for
each sample. A possible explanation for the absence of emission from the test QWs
could be that, at room temperature, the carriers can escape from the test QWs, and
diffuse towards the probe QWs, where they can recombine, resulting in emission of
light from the probe QWs instead.
The PL spectra obtained at 30 K for these samples are shown in Figure 5.3.2. It can
be seen that the emission energy for both the test and probe QWs is higher at low
temperatures, in particular it is 1.3 eV for the probe QWs and 1.57 eV for the test
QWs in U34097. This is a consequence of the Varshni relation, according to which
48
50. the bandgap of a semiconductor, and consequently the energy of emitted photons,
decreases as the temperature increases 1
[46].
Figure 5.3.1: PL spectra of double-structure, SBQW samples, obtained at 300K. The samples
U34097, U34099 and U34100 contain 0%, 20% and 10% Indium in the test QWs, respectively. No
emission from the test QWs is observed,except for the sample U34097, which shows a small peak
around 1.5 eV. The probe QWs show a strong emission around 1.23 eV.
1
The Varshni relation is given by [46]:
Eg(T) = Eg(0) −
αT2
β + T
where Eg(0) denotes the bandgap at 0 K, whereas α and β represent the Varshni parameters. The
above equation shows that at low temperatures the bandgap of the semiconductor increases. This happens
because at low temperatures the lattice contracts, on the one hand, reducing the distance between atoms
and strengthening the interatomic bonds and on the other hand, the lattice vibrations become much weaker,
reducing the interactions between electrons and phonons [47].
49
51. Figure 5.3.2: PL spectra of double-structure, SBQW samples, obtained at 30K. The samples
U34097, U34099 and U34100 contain 0%, 20% and 10% Indium in the test QWs, respectively.
The emission energy from the probe QWs increases to 1.3 eV for the probe QWs and 1.57 eV for
the test QWs in U34097, due to the increase in the semiconductor bandgap at lower temperatures.
However, the most surprising aspect of the results in Figure 5.3.2 is that, even at
such a low temperature, no PL signal can be detected from the test QWs in U34099
and U34100, whereas the PL from the probe QWs in the case of U34097 is 14 times
stronger than the PL from the test QWs. This discrepancy could be attributed to
several factors. Since performing the experiment at low temperatures increases the
bandgap, the top region becomes more transparent to laser light and this increases
the likelyhood of light being directly absorbed in the bottom region, resulting in a
higher emission from the probe QWs. Nevertheless, the carrier escape rate from the
QWs decreases exponentially with decreasing temperature [48], which means that this
process dominates over any increase in the top region bandgap. Therefore, bandgap
effects should not be important in this case, as they cannot be responsible for the
high emission from the probe QWs. Another possible explanation for this result is
the presence of defects in the top region, which can lead to increased non-radiative
recombination upon carrier trapping in these states and low emission from the test
QWs. Measuring the decay lifetimes for radiative recombination in the top and bot-
50
52. tom region could resolve this, since a high radiative recombination lifetime in the top
region would be an indication of a low quality material with a high defect density.
Another likely cause could be that light emitted from the probe QWs undergoes mul-
tiple internal reflections at the lowest diffusion barrier, before escaping from the front
surface. Since light emitted from the probe QWs cannot be absorbed in the top cell
on its way to the front surface, it is not attenuated and adds to the intensity of the
emitted light. Yet, the added intensity from multiple reflections saturates to a very
small value, which implies that the strong emission from the probe QWs cannot be
the result of multiple reflections at the lowest diffusion barrier.
These findings suggest that, even at low temperatures, the carriers from the test QWs
can still escape and diffuse towards the probe QWs, where the carrier concentration
is smaller, recombining and resulting in high emission from that area of the sample.
It could also be possible that, the top region of these samples is not sufficiently thick
to absorb laser light efficiently, resulting in direct excitation of the bottom region,
which would explain the high PL from the probe QWs. To investigate this possibility,
measurements were also performed on the sample U34101, which is identical to the
reference sample U34097, but with a 1.5 µm thick top region 2
. The PL from this
sample at 300 and 30 K is shown in Figures 5.3.3 and 5.3.4, respectively.
2
It should be noted that only the thickness of the surrounding layers was increased in the top region of
the sample U34101 and not the thicknesses of the test QWs, since this would change the energy at which
the test QWs emit.
51
53. Figure 5.3.3: PL spectrum at 300 K, of the sample U34101, with a 1.5 µm thick top region and
0% Indium (no strain) in the test QWs. The emission from the probe and test QWs can be seen
around 1.23 and 1.47 eV, respectively.
52
54. Figure 5.3.4: PL spectrum at 30 K, of the sample U34101, with a 1.5 µm thick top region and 0%
Indium (no strain) in the test QWs. The emission from the probe and test QWs can be seen around
1.3 and 1.54 eV, respectively.
At 300 and 30K, the emission from the probe QWs in this sample is observed around
1.23 eV and 1.3 eV, respectively. The emission from the test QWs is observed around
1.47 and 1.54 eV at 300 and 30 K, respectively. The fact that the test QWs in this
sample emit at a slightly lower energy than the test QWs in U34097 could be due
to an error in the growth process. Nevertheless, the emission from the test QWs in
U34101 is significantly higher than the emission from the probe QWs, which could be
an indication of photonic coupling from the test to the probe QWs.
To determine whether photonic coupling takes place in this sample, PL Excitation
(PLE) measurements were performed at 30 K, which involve exciting the sample at a
range of wavelengths and detecting the emitted PL at a fixed wavelength [49].In this
method, the green laser was used to illuminate a Ti:Sapphire crystal and the emission
from the crystal was directed towards the front surface of the sample. Using an
Ocean Optics Spectrograph and an Oriel Instruments Mike Controller, the excitation
wavelength was varied from 710 to 805 nm (1.54-1.75 eV), to include the emission
wavelength range of the test wells, whereas the detection wavelength was fixed at 948
53
55. nm (1.3 eV), which corresponds to the emission wavelength of the probe QWs. The
PLE spectrum of the sample is illustrated in Figure 5.3.5 by the black line. The peaks
at 1.54 eV and 1.62 eV correspond to the e1-hh1 and e1-lh1 transitions, respectively,
suggesting a HH-LH splitting of 80 meV. Such a large value is not expected, as the
HH-LH splitting for an unstrained QW (such as the unstrained test QWs in this
sample) is typically around 10-15 meV [50]. The reason for this observation is not
clear, but it may be related to a mixing effect between the HH and LH bands. It may
also be that the peak at 1.54 eV corresponds to the e1-hh1 transition, whereas the
peak at 1.62 eV to a higher order LH transition (for example, e2-lh2), hence the large
separation between the peaks. Nevertheless, the fact that the LH and HH transitions
taking place in the test wells of the sample can be detected at the emission wavelength
of the probe wells provides solid evidence for the coupling of emission from the test
to the probe wells. The laser power was also measured at each wavelengh and the
emission of the sample was divided by the power to obtain a more realistic spectrum,
which is indicated by the red line.
Figure 5.3.5: PLE spectrum from the front surface of the U34101 sample at 30K .The black line
represents the emission spectrum of the sample, whereas the red line represents the emission divided
by the laser power. The peaks at 1.54 and 1.62 eV represent the e1-hh1 and e1-lh1 transitions,
respectively, in the test QWs.
54
56. Since no other double-structure samples with a thick top region, but different degrees
of strain on the test QWs, were available, to study the impact of restricted emission
on the photonic coupling, PL measurements were performed, as a function of tem-
perature (T) and power (P), for the sample U34101. Increasing the temperature or
power should increase the population of the LH band and consequently minimize the
suppression of the LH transition in the test QWs, which is equivalent to restricting
the emission from the test QWs to a lesser extent. The PL spectrum as a function of
temperature and power is shown in Figure 5.3.6.
55
57. Figure 5.3.6: Top: PL spectrum of U34101 as a function of temperature, at a fixed power of 116
mW. Bottom: PL spectrum of U34101 as a function of power, at a fixed temperature of 30K.
It can be observed that, the PL from both the test and probe QWs shifts to a slightly
56
58. lower energy as the temperature is increased, as a consequence of the Varshni relation.
The decreased intensity of the PL as the temperature is increased can be attributed
to two factors: The first is the increased carrier escape rate from the test QWs with
temperature, which leads to decreased recombination and consequently PL signal and
the other is the thermal activation of defect states. The latter leads to trapping of
electrons and holes in these states, causing them to recombine non-radiatively.
If the population of the LH band in the test QWs is increased at a certain temperature,
then more light from the test QWs would be emitted and coupled to the bottom
region and that would be observed as a higher signal from the probe QWs at that
particular temperature. The results, however, show that the emission from the test
QWs continues to decrease, even at 74 K, which suggests that this temperature is
not sufficiently high to populate the LH band. Furthermore, no additional peak
corresponding to the LH transition can be observed in emission from the test QWs.
Indeed, the expression for the HH to LH band occupation ratio, given by fhh
flh
= e
Elh−Ehh
KBT
[16], where Elh and Ehh correspond to the energies of the LH and HH transitions,
respectively, can explain this: For a HH-LH splitting of 80 meV, as deduced from the
PLE experiment for this sample, the LH band is far away from the HH band and
the occupation of the HH band at 74 K is about 12 times larger than that of the LH
band.
Turning now to the PL measurements as a function of power, it can be seen that the
intensity of the PL signal increases with power, which is expected, since more carriers
can be excited across the test QWs and recombine. Then, more light could also be
coupled to the bottom region, resulting in a higher emission from the probe QWs,
regardless of whether the LH band population increases or not. It is not clear, in that
case, if the increased emission from the probe QWs is related to a higher occupation
of the LH band or is merely associated with the increased excitation of carriers across
the test QWs. Given, however, that the temperature was fixed to only 30 K and no
additional peak from the LH transition was observed in the PL from the test QWs,
it is likely that the LH band was not populated under these conditions and that it
is the increased rate of carrier excitation in the test QWs that is responsible for the
higher PL from the probe QWs.
5.3.2 Edge PL measurements
Experimental set-up
Edge PL measurements were performed at 300 K for the sample U34101, using the
57
59. same method as for single-structure samples, but with the InGaAsP photomultiplier
tube detector replaced by a Si detector.
Results and Discussion
The polarized emission pattern obtained for the sample U34101 is illustrated in
Figure 5.3.7. In the case of the TE mode, two peaks are observed at 1.23 and 1.47
eV, whereas in the spectrum of the TM mode only the peak at 1.47 eV is seen.
Comparing the positions of these peaks with those corresponding to the emission
from the test and probe QWs at 300K, for this sample (Figure 5.3.3), it is obvious
that the emission at 1.23 eV comes from the probe QWs, whereas the emission at 1.47
eV comes from the test QWs. The small peak observed at 1.42 eV in both the TE and
TM spectra is likely due to contribution from the GaAs substrate. Given that the
test QWs are unstrained, we would expect, in the case of the TE mode, to observe a
peak corresponding to the LH transition and a peak, at a slightly lower energy, three
times stronger, corresponding in the HH transition, whereas in the case of the TM
mode a single peak corresponding to the LH transition would be expected (according
to equations 4.1.11-4.1.14). In the case of the compressively strained probe QWs, a
similar polarization pattern would be expected, but the LH transition and hence the
TM mode would be mostly suppressed. Since the TM mode is suppressed at 1.23
eV, whereas the TE mode is not, it is likely that the peak at 1.23 eV corresponds to
the HH transition in the probe QWs. At 1.47 eV, the TE and TM modes have equal
strength, suggesting that the signal is unpolarized. The reasons for this have been
explained in Section 5.2.2, where we have seen how mixing between LH and HH bands
or multiple internal reflections in the structure can lead to a random polarization.
The difference in the emission intensity from the probe and test QWs, on the other
hand, could be related to the DoS for unstrained and strained QWs. For unstrained
QWs, such as the test QWs in this sample, the mixing effect dominates, due to the
fact that the HH and LH bands are closer to each other and this in turn leads to
an increased DoS. For compressively strained QWs, such as the probe QWs in this
sample, the mixing effect is minimized 3
, since the LH band is shifted away from the
HH band, which results in a reduction in the DoS. A reduction in the DoS would imply
that less states are available for electrons to be excited to, which could explain why
the emission from the strained probe QWs is so weak compared to the emission from
the unstrained test QWs [38]. In addition, it is possible that there are contributions
3
It should be noted that the reduced mixing of the valence bands for compressively strained QWs could
explain why the emission from the probe QWs is polarized, as opposed the emission from the test QWs.
58
60. from the substrate emission to the test QW emission.Finally, the response of the Si
detector reduces significantly at energies lower than 1.24 eV, which implies that the
HH transition is considerably stronger than it appears in the TE mode spectrum.
Figure 5.3.7: Edge PL spectrum of U34101 at 300K. The black and red lines represent the emission
patterns of the TM and TE modes, respectively.
59
61. Chapter 6
Impact of restricted emission on the
radiative coupling current density
In Section 5.3.1, it was observed that the double-structure samples used to study the
effect of restricted emission on the photonic coupling, had a thin top region, resulting
in direct laser excitation of the bottom region. The high emission from the probe
QWs was therefore due to absorption of laser light, rather than absorption of light
emitted by the test QWs. The intensity of the PL from unstrained test QWs in a
thicker top region sample (U34101), was shown to be significantly higher than the
emission from the probe QWs, suggesting that photonic coupling from the test to the
probe QWs takes place, which was confirmed by PLE measurements. Since no other
thicker top region samples were available, with strained test QWs to study how the
photonic coupling is affected by the restricted emission, a calculation was performed
to investigate this.
Returning to Section 4.2.2, when light emitted from radiative recombination in a
QW region is coupled in the subsequent junction of a multijunction solar cell, the
contribution to the current density of the latter, Jcoupled is:
Jcoupled =
8πq
c2h3
ˆ
E2
e
E−qV
KBT
− 1
ˆ 1
0
1 − e−αn(E,θ)z
cosθ cosθd(cosθ)
dE (6.0.1)
where all the terms have been previously defined and Jcoupled will be referred to as
the radiative coupling current density. Although the double-structure samples used
in this project are not devices, but merely heterostructures, the main idea here is
that, given the emission from the test QWs in the top region is known, the fraction of
60
62. light coupled in the bottom region, and consequently the radiative coupling current
density in the latter can be estimated from eq. 6.0.1. More importantly, changes in
the radiative coupling current density can be calculated as the emission from the test
QWs is restricted to a greater or lesser extent.
The particular structure of these samples, however, requires eq. 6.0.1 to be reformu-
lated for a more accurate calculation of the radiative coupling current. The present
form of eq. 6.0.1, in particular, assumes that the bottom region is sufficiently thick
to absorb all the light emitted from the the test QWs in the top region, but as can
be seen in Figure 5.1.1, the total thickness of the bottom region is only 500 nm. To
account, therefore, for the fact that not all of the emitted light can be absorbed and
contribute to the current density of the bottom region, eq. 6.0.1 needs to be multi-
plied by the correction factor 1−e−α (λ)t
cosθ , where t is the thickness of the bottom region
and hence is rewritten as:
Jcoupled =
8πq
c2h3
ˆ
E2
e
E−qV
KBT
− 1
ˆ 1
0
1 − e−α (λ)t
cosθ 1 − e−αn(E,θ)z
cosθ cosθd(cosθ)
dE
(6.0.2)
Since the bottom region is mostly comprised by GaAs barriers, the absorption of
emitted light takes place mainly in these layers so, in that case,α (λ) in the correction
factor corresponds to the absorption coefficient of the GaAs at the emission wave-
length λ of the test QWs. These can be determined, using λ(nm) = 1240
E(eV ), by the
emission energies of the test QWs in the PL spectra (Section 5.3.1) . The cosθ in the
correction factor accounts for the fact that light can reach the GaAs at a range of an-
gles. Then, the absorption coefficients αTE(E) and αTM (E) that enter into αn(E, θ)
correspond to the absorption coefficients of the test QWs and z is equal to the total
thickness of the emitter. Given that each of the double-structure samples has three
test QWs, each 8 nm thick, z is equal to 24 nm.
Figure 6.0.1 illustrates the radiative coupling current density, calculated based on eq.
6.0.2, using MATLAB, as a function of voltage (dark IV curve) at 300 and 30 K for
the sample U34097, which contains unstrained Al0.4Ga0.6As/GaAs test QWs. The
absorption coefficients for the test QWs in U34097, at 300 and 30 K are shown in the
Appendix.
61
63. Figure 6.0.1: Dark IV for the sample U34097 at 300K (Top) and 30K (Bottom) calculated using
eq.6.0.2. The radiative coupling current density is shown in logarithmic scale, whereas in the insets
is shown in linear scale. As can be seen from Figures 5.3.1 and 5.3.2 the emission energies of the
test QWs in the sample U34097 at 300 and 30 K are about 1.5 and 1.57 eV, respectively, which
correspond, by λ(nm) = 1240
E(eV ) , to emission wavelengths of 828 and 792 nm, respectively. The
absorption coefficient of GaAs used in these calculations was therefore α (828nm) = 1.216·106m−1
and α (792nm) = 1.386 · 106m−1 .
62
