Investigation of single crystal germanium pn-junctions for use in tandem CdTe/Ge solar cells by James Ross Sharp B.E. (Hons.), University of Melbourne, 2006 This thesis is submitted to the Faculty of Engineering, Computing and Mathematics of the University of Western Australia in fulfillment of the requirement for the degree of Doctor of Philosophy School of Electrical, Electronic and Computer Engineering 2016
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Investigation of single crystal germanium
pn-junctions for use in tandem CdTe/Ge solar cells
by
James Ross Sharp
B.E. (Hons.), University of Melbourne, 2006
This thesis is submitted to the
Faculty of Engineering, Computing and Mathematics
of the University of Western Australia in fulfillment
of the requirement for the degree of
Doctor of Philosophy
School of Electrical, Electronic and Computer Engineering
2016
This thesis entitled:Investigation of single crystal germanium pn-junctions for use in tandem
CdTe/Ge solar cellswritten by James Ross Sharp
has been approved for the School of Electrical, Electronic and ComputerEngineering
Winthrop Professor Lorenzo Faraone
Winthrop Professor and Dean John Dell
Date
The final copy of this thesis has been examined by the signatories, and we findthat both the content and the form meet acceptable presentation standards of
scholarly work in the above mentioned discipline.
3
Sharp, James Ross (B.E. (Hons.))
Investigation of single crystal germanium pn-junctions for use in tandem CdTe/Ge
solar cells
Thesis supervised by Winthrop Professor Lorenzo Faraone
Abstract
Thin film cadmium telluride solar cells are a viable renewable energy tech-
nology, due to low manufacturing cost, fast energy pay-back times, and an energy
gap well matched to the solar spectrum. The technology is already mature and
has commercial application, with more than 10GW of installed thin film CdTe
modules world wide. As with any commercial photovoltaic technology, research
is perpetually focussed on how to boost module efficiency, improve process yield,
and lower production costs. Although startlingly rapid progress has been made
in the field of CdTe photovoltaics in recent years, with commercial CdTe module
manufacturer First Solar reporting record breaking research module efficiencies,
there will inevitably come a point where the performance of CdTe single junction
modules cannot be improved any further, as the conversion efficiency approaches
the Shockley-Queisser limit. At such time, higher efficiencies can only be achieved
by means of adopting multijunction tandem configurations.
This work aims to investigate a possible material combination, namely CdTe
and germanium (Ge), for the advancement of CdTe technology into the realm of
multijunction and concentrated photovoltaics. Germanium is a favourable material
for this purpose due to its narrow energy gap located in a suitable area of the
solar spectrum, availability of large area Ge substrates in epi-ready format, and
idealised electrical and mechanical properties. This thesis investigates germanium
4
processing in a low-cost and manufacturable manner in order to develop a process
for the formation of the lower cell of a multijunction photovoltaic device. Novel
techniques for germanium doping, passivation and contacting are expounded and
a complete methodology for germanium device fabrication is presented. This is of
interest not only to the photovoltaics sector but more generally the techniques are
applicable to a wide range of germanium opto-electronic devices.
In order to predict performance and optimise device structures, simulation
and modelling is undertaken in both a commercial device simulator (Synopsys
Sentaurus Device) as well as in a custom developed analytical/numerical simu-
lation framework. The goal of simulation is to investigate both monolithic and
mechanically stacked configurations and determine which device structure would
be optimal in terms of photocurrent matching and also in terms of optical proper-
ties to minimise optical reflection losses from device active layers. A mechanically
stacked configuration featuring CdTe grown epitaxially on sapphire is considered in
this work and its possible performance compared to a monolithic CdTe/Ge struc-
ture. It is shown that such a structure could contribute an efficiency improvement
of 5.03% absolute over a single junction CdTe solar cell, whereas a monolithic
tandem would boost single junction solar cell efficiency by a mere 3.6% absolute.
Subsequently, doped layers of single crystal germanium were prepared from
bulk germanium wafers utilising spin on dopants, either by directly spinning on a
thin film of dopant, or by vapour transport in the “proximity” doping technique, or
by the novel “sandwich-stacked diffusion” technique developed in this work. These
layers were processed into electronic and photovoltaic devices using standard pro-
cessing techniques, passivated and contacted using the technologies demonstrated
within, and finally characterised. The result is a high quality process for germanium
opto-electronic device fabrication. Optoelectronic devices are shown with surface
recombination velocities as low as 21 cm/s and with specific contact resistivities
5
as low as 1.26 ×10−7 Ω · cm2 . This highlights the quality of the passivation and
contacting procedures developed in this work.
An investigation of germanium doping for device active region formation
is undertaken. It is concluded that both proximity doping and sandwich-stacked
diffusion yield degenerate p-type doping of germanium and surface concentrations
of up to 1020 atoms cm−3 can be achieved, but degenerate n-type doping can only
be achieved by means of direct spin-on doping. The reason is most likely the high
vapour pressure of phosphorus and its oxides at the processing temperature. Direct
spin on doping gave a maximum donor concentration of 4e19 cm−3, in contrast
with a maximum concentration of 6e18 cm−3 for proximity doping.
Germanium pn-junction devices with ideality factors equal to 1 and showing
breakdown due to Zener effect are presented, as well as a 5.4% efficient solar cell.
The solar cell illustrates the complete germanium diode fabrication process includ-
ing contacting and passivation and the device is shown to be stable in efficiency
when remeasured after eight months. The solar cell was capped with a combined
passivation/anti-reflection solution shown to reduce reflection losses to 6.47%.
Finally, CdTe was grown on both germanium and sapphire substrates and
the results were characterised by a variety of methods including RHEED, XRD,
and optical transmission measurements. CdTe thin films grown on sapphire are
presented with double crystal rocking curve (DCRC) full-width at half maxima
(FWHM) as low as 59 arc seconds as rocked about the 〈111〉 diffraction plane.
CdTe grown on germanium was processed into heterojunction CdTe/Ge P/n junc-
tions and the IV and CV characteristics were measured to elucidate the electronic
properties of the heterojunction. A CdTe/Ge diode with ideality factor n=1.65 is
presented demonstrating reasonable quality material growth and device processing
utilising this novel combination of materials.
Dedication
To my father, Peter, and my mother, Angela and to the love of my life,
Hitomi.
7
Contents
Chapter
1 Introduction 13
1.1 Harnessing the power of the sun . . . . . . . . . . . . . . . . . . 13
1.1.1 Standard solar reference spectrum . . . . . . . . . . . . . 15
manium solar cells realised with ZnS/MgF2 antireflection coatings for thermopho-
tovoltaic applications. Junctions were formed by diffusion of zinc into n-type
germanium substrates. A layer of LPE grown p-type GaAs was used for surface
passivation which improved output voltage and efficiency.
Posthuma et al. [21] demonstrated a 6.7% efficient (AM1.5) stand-alone
germanium solar cell in 2003. The shallow emitter was realised using a phosphorus-
containing spin-on dopant source. Diffusion time was kept short to reduce surface
roughening and realise a shallow emitter, and in a similar fashion the diffusion
31
temperature was optimised. For passivation, a thin layer of amorphous silicon was
deposited using plasma-enhanced chemical vapour deposition (PECVD). A silver
finger pattern served as front contact and was diffused through the passivation
layer. This innovation was necessary to circumvent the lack of a uniform wet etch
with high selectivity between the amorphous silicon and germanium. Finally an
anti reflective-coating of ZnS and MgF2 was applied.
The same authors [22] reported an improved efficiency of 7.8% (AM1.5G)
in 2007. The fabrication process was essentially identical, however it featured
contacts formed from a thin layer of palladium and a thick layer of silver (as
opposed to the aluminium used in earlier studies) which helped to improve cell fill
factor reproducibility.
These results are summarised in Fig. 1.7.
1.1.10 Growth of CdTe on germanium
Cadmium telluride has previously been grown on germanium substrates using
molecular beam epitaxy (MBE). Matsumura et al. [23] prepared CdTe 〈111〉 and
CdTe 〈100〉 on 〈111〉 and 〈100〉 Ge substrates. The growth temperature ranged
between 150 C and 400 C .
It was noted that as crystallinity improved with growth temperature, the
films became milky in appearance. Two effusion cells were used during growth,
Cd at 290 C and Te at 405 C . From these nominal temperatures, flux ratio was
varied between 1/5 to 5/1. It was found that at ratios 1/2 - 1/5 (Cd:Te) the film
was polycrystalline. Twinning was observed on 〈111〉 substrates but not on 〈100〉
substrates. X-ray diffraction showed decrease of the rocking curve full-width at
half maximum (FWHM) for increasing substrate temperature, up to 350 C .
Zanatta et al. [24] presented 〈211〉B CdTe films grown on 〈211〉 Ge sub-
strates at a growth temperature of 250 C . During growth the temperature was
32
ramped to 320 C to optimize the crystal quality. The growth rate was 45 nm
per minute. Double crystal rocking curves (DCRC) for x-ray diffraction gave the
average FWHM of 89 arc seconds with a std. deviation of 6 arc seconds.
5
6
7
8
9
Effi
cie
ncy
1990
1993
1996
1999
2002
2005
2008
Venkatasu
bram
anian
(AM0)
Year
Khvostiko
v
Posthum
aPosth
uma
Figure 1.7: Ge solar cell research milestones
The efforts of Zanatta et al. came about in the context of HgCdTe growth on
CdTe buffer layers. HgCdTe is a high performance infrared detector material that,
despite its excellent characteristics for IR detector applications, has the unfortunate
disadvantage that it lacks a high quality large area lattice matched substrate.
HgCdTe is lattice matched to Cd0.95Zn0.05Te but substrates are difficult to fabricate
in large area format due to brittle mechanical characteristics [24, 25]. Hence
alternative substrates such as Si, GaAs and Ge have been considered for HgCdTe
growth. Ge has a better coefficient of thermal expansion and lattice constant
match with HgCdTe in comparison to silicon, hence the choice as a potential
substrate candidate.
33
In this work we consider the use of Ge not only as a potential substrate but
also as an active device layer with photovoltaic devices formed prior to CdTe epitax-
ial growth. This would necessitate altered growth conditions to prevent damage to
the underlying germanium device during epitaxial growth. As a substrate, Ge still
has the unfortunate disadvantage of lattice mismatch (14%) with CdTe, however
this is more favourable than silicon (19%). Germanium is chosen predominantly for
more efficient spectral splitting as opposed to lattice matching. Since Ge has the
narrower energy gap, it can absorb the longer wavelength photons out to roughly
1800 nm, whereas silicon can only absorb out to roughly 1100nm. This extra ab-
sorption region should allow for better matching of photocurrents between the two
cells of the tandem stack. The ability to absorb longer wavelength photons as well
as match the photocurrents between the two cells should allow for the creation of
an efficient two junction solar cell, with potentially higher efficiency than its single
junction CdTe counterpart.
1.2 Research outcomes and thesis outline
From the discussion in Section 1.1.8 it is evident that good progress has been
made over the past 35 years in CdTe processing technology. With the efficiency
of single junction CdTe solar cells now having been demonstrated at 21.5%, the
pathway towards the ultimate efficiency (≃30%) of single junction CdTe seems
comparatively free of encumbrance. However when that ultimate efficiency is to
all intents and purposes reached, progress can only be made by expanding into the
realm of multi junction solar cells. The aim of this work is not to advance single
junction CdTe processing any further, since this sphere of research is being under-
taken at commercial scale by manufacturers with significant research expertise and,
due to their vested interest in the development of CdTe technology, commensurate
research budgets. We assume in this work that CdTe single junction technology
34
has already developed to the point where a tandem configuration is justified, and
we will concentrate on development of the Ge bottom cell and interconnect.
In Section 1.1.9 the progress of standalone germanium solar cells was out-
lined. The conversion efficiencies at one-sun are quite low in the results hitherto
published for germanium solar cells; it begs the question how much of an efficiency
improvement can be expected from the addition of a germanium bottom cell to
form a multijunction solar cell. Device processing techniques would need to be
advanced further until single junction germanium solar cells yielded efficiencies of
10% or more to add substantial improvement to CdTe solar cell efficiency in a tan-
dem configuration. For example, while theoretically any efficiency improvement
is significant, the extra expense may not be justifiable. The research challenges
are therefore how to reduce surface recombination in Ge standalone solar cells
with adequate surface passivation, how to make low resistivity contacts to germa-
nium, and how to form device regions in a low cost manner without any unwanted
contamination which may impair device efficiency.
This investigation therefore expands upon the work of Posthuma et al. [21,
26, 22, 27, 28] to improve upon device active region formation, passivation and
contacting technologies developed by this group in order to further germanium
standalone solar cell research, as well as investigating properties of the CdTe/Ge
heterojunction. Materials growth draws on research by Zanatta et al. [24, 29] to
develop a process for growing CdTe epitaxially on Ge.
We now turn our attention to investigating these research challenges in the
following chapters. In Chapter 2, we explore analytical and numerical simulation of
photovoltaic devices and create a simple simulation framework for the evaluation
of single junction solar cells. This framework consists of both well-known closed
form solutions of the drift-diffusion equations as well as a compact numerical drift-
diffusion solver that discretises and solves these equations over a small mesh. The
35
purpose of developing this framework is to add insight into numerical simulation
and complement more complicated commercial device simulators.
In chapter 3 we consider, simulate and optimise the device structures. We
do so by employing a commercial simulator application and our self-developed
simulation framework to simulate devices by solving the drift-diffusion equations for
solar cells illuminated by the solar spectrum. Of particular importance is matching
the photocurrent between the two cells, particularly in a monolithic configuration.
A comparison is made between monolithic and mechanically stacked tandem solar
cells. For the case of a mechanically stacked tandem, photocurrents are matched
so that the individual sub-cells can be series connected to form a two terminal
device.
Chapter 4 considers low cost techniques for junction formation for germa-
nium optoelectronic devices. Doping of bulk germanium is considered using spin-on
dopants, silica or polymer films which are spun onto a target wafer to deliberately
introduce impurity atoms into device regions. Three techniques are presented for
device active region formation, these being “sandwich-stacked” diffusion, proximity
doping, and direct spin-on doping. The three methods give the three possible con-
figurations for doping germanium in a low cost manner by either direct contact or
vapour transport. Hence a proof-of-concept of all three techniques demonstrates
possible avenues for low cost manufacture of germanium sub-cells for multijunction
tandem application.
Chapter 5 examines passivation, antireflection and contacting techniques for
germanium optoelectronic devices. We consider a range of chemical pre-treatments
and passivation layers and compare their efficacy by measuring the lifetime using
a simple photoconductive decay apparatus. Wet and dry chemical pre-treatments
are necessary to prepare the sensitive germanium surface prior to passivation layer
deposition. This is done so by terminating dangling bonds at the highly chemi-
36
cally reactive germanium surface so that when the passivation layer is deposited
the complete structure is comparatively free of recombination centers and defects.
This is critical to high performance photovoltaic devices. Investigated passiva-
tion layers are inductively coupled plasma enhanced chemical vapour deposition
(ICPECVD) grown thin films. The advantage of using an ICPECVD reactor to
deposit the passivation layer is the potential for in-situ dry pre-treatments with re-
active gases, and it is found that in-situ ammonia pretreatment increases minority
carrier lifetime.
In Chapter 6, we investigate materials growth of CdTe and ZnTe on sapphire
and germanium, and investigate the properties of a CdTe/Ge heterojunction. CdTe
and ZnTe thin films on germanium and sapphire are prepared using molecular
beam epitaxy (MBE) and thermal evaporation, respectively, and are characterised
by a variety of methods including RHEED, optical transmission methods, and
X-ray diffraction (XRD). A sample of CdTe grown on a germanium substrate is
processed into a heterojunction device and the electronic properties are investigated
by measuring the IV and CV characteristics.
Chapter 7 summarizes and makes conclusions about the work as a whole.
Chapter 2
Analytical and numerical techniques for optoelectronic device modelling
2.1 Elementary theory of solar cells
2.1.0.1 Theory of pn junctions
A pn junction is a semiconducting device that consists of two doped regions,
one doped p-type, that is to say, doped with acceptors, and the other doped n-
type, that is, with donors. By bringing these two semiconducting regions together,
a potential barrier known as the built-in potential is formed at the metallurgical
junction, which can be used to separate charge carriers, and hence produce a
photocurrent when illuminated.
Fig. 2.1 shows the pn junction at equilibrium. The symbols used are ex-
plained in Tab. 2.1. Diagram a) shows the space-charge distribution, b) the
electric field distribution, c) the potential and d) the energy band diagram. At
equilibrium, the differing signs of the charge on either side of the junction brings
about the space-charge region. Here, local electric fields cause carriers to cross
the metallurgical junction at x = 0 to cancel out the imbalance of carriers. The
space-charge or depletion region then becomes depleted of carriers, and an electric
field develops across the junction, with a built-in potential ψbi . This forces the
Fermi-level to sit level within the device.
It is this built-in potential that sweeps carriers out of the junction if they
enter it, and in particular separates electron-hole pairs into their constituent parts,
38
i.e. electron and hole. In this way, minority carriers can be swept across the
junction to become majority carriers and be collected at the contact. This is the
basic principle by which pn-junction solar cells operate.
d)
c)
b)Area = built in potential,
a)
+
-
0
0
depletion region
Depletion charge
p-region n-region
Donor density
Acceptor density
Figure 2.1: pn junction at equilibrium, after [30]
39
ND the donor doping density
NA the acceptor doping density
WDp the width of the depletion region in the p-type material
WDn the width of the depletion region in the n-type region
E the electric field
Em the maximum electric field
ψbi the built-in potential
ψp the potential in the p-type region
ψn the potential in the n-type region
ψBp
the energy difference between the
intrinsic level and the Fermi level
in the p type region
ψBn
the energy difference between the
intrinsic level and the Fermi level in
the n type region
φp
the energy difference between the
valence band and the Fermi level in
the p-type region
φn
the energy difference between the
conduction band and the Fermi level
in the n-type region
Table 2.1: Symbols for Fig. 2.1
2.2 Derivation of an analytical model
Whilst solar cells are becoming increasingly complicated in terms of device
structure, necessitating complex numerical simulation techniques for their analysis,
simplified analytical models can be derived subject to certain assumptions. An
40
analytical model can be used as an adjunct to a more thorough numerical model,
and has the following advantages:
• An analytical model will yield results much faster than a numerical simu-
lation (which may take hours to converge), facilitating experimentation.
• An analytical model may help to verify results from numerical simulation.
• Analytical models give rise to closed form expressions for determining the
effect of parameter variation on key device metrics.
Analytical solutions can be obtained for the light and dark currents in the
three regions of a p-n junction solar cell, subject to certain simplifying assumptions.
These are well known and can be found in many references, such as [31]. These
solutions are derived in the following section to lay the foundation for the analytical
model. The pn-junction structure under consideration is depicted in Fig 2.2. Here,
x is defined to be the position within the cell, with x = 0 set to be at the
metallurgical junction. H is the total width of the cell, and Hp and Hn are the
widths of the p and n type regions respectively. Wn and Wp are the widths of
the depletion region in the n and p-type regions respectively and W is the total
depletion region width.
Figure 2.2: Cell considered for derivation of an analytical model[31]
41
In deriving an analytical model, we begin with the following assumptions
[31]:-
• The analysis is restricted to one dimension.
• Light is incident normal to the surface, and we neglect scattering and
internal reflection.
• Both regions of the pn junction are non-degenerate and donors/acceptors
are fully ionized.
• There are no hot carrier effects and a photon excites a single electron-hole
pair.
• Minority carrier recombination is pseudo-first order in the bulk and at the
surfaces.
• Low level minority carrier injection/diffusion is the operative transport
mechanism.
• Device parasitics are ignored (we will consider them later using a circuit
analysis approach).
By assuming that minority carrier injection and diffusion are the only opera-
tive modes of transport, that we can ignore device parasitics, and further that the
cell remains in low injection throughout the bias/optical excitement range [31],
we can appeal to the principle of superposition and essentially decouple the light
and dark current densities, so that they can be modelled separately and the results
superposed, greatly simplifying the results.
When modelling the electrical properties of semiconductors, we solve the
following set of equations:-
42
∇2φ(x) = −∇E(x) = − 1
ǫǫ0ρ(x) (2.1)
Je(x) = q · (n(x) · µe · E(x) + Dn · ∇n(x)) (2.2)
Jh(x) = q · (p(x) · µh · E(x)− Dp · ∇p(x)) (2.3)
1
q∇ · Je(x)− re(x) + ge(x) = 0 (2.4)
−1
q∇ · Jh(x)− rh(x) + gh(x) = 0 (2.5)
Where φ(x) is the potential at x, E(x) is the electric field at x, ρ is the
space charge, ǫ is the relative permittivity of the material, ǫ0 is the vacuum per-
mittivity, Je and Jh are the electron and hole current densities, respectively, q is
the elementary charge, n(x) and p(x) are the electron and hole concentrations
respectively, re and rh are the electron and hole recombination rates respectively,
and ge and gh are the electron and hole generation rates, respectively. Equation
2.1 is the Poisson equation, Eqns. 2.2, 2.3 are the electron and hole drift-diffusion
equations, and Eqns. 2.4/2.5 are the electron and hole continuity equations.
By restricting ourselves to one dimension, we can differentiate Eqn. 2.2 and
substitute it into Eqn. 2.4. The procedure is then repeated for holes, yielding the
following set of equations which can be solved to yield the carrier concentrations
in the device [31]:-
Dn
d2n
dx+ µeE
dn
dx+ nµe
dE
dx− re(x) + ge(x) = 0 (2.6)
Dp
d2p
dx− µhE
dp
dx− pµh
dE
dx− rh(x) + gh(x) = 0 (2.7)
43
2.2.1 Recombination
For the recombination terms in Eqns. 2.6 and 2.7, we assume low injection
conditions and hence that recombination in the semiconductor is pseudo-first order
[31]. Hence the recombination rates may be written as [31]:-
re(x) =np − n0p
τe=
De(np − n0p)
L2e0 ≤ x ≤ Hp (2.8)
rh(x) =pn − p0
n
τh=
Dh(pn − p0n)
L2h−Hn ≤ x ≤ 0 (2.9)
Where np is the electron concentration in the p region, pn is the electron
concentration in the n region, De and Dh are the electron and hole diffusivities
respectively, Le , Lh are the electron and hole diffusion lengths respectively, and n0p
and p0n are the dark minority carrier densities.
The diffusion length for electrons and holes is given by [32]:-
Le =√
Deτ (2.10)
Lh =√
Dhτ (2.11)
where τ is the bulk lifetime for electrons and holes.
2.2.1.1 Surface recombination velocity
Surface recombination velocity is the rate at which carriers recombine at sur-
faces. These surfaces include the front and back surfaces of a solar cell as well as
any grain boundaries within the cell if it consists of polycrystalline material. Grain
boundaries act as recombination centers because the lattice is unterminated and
there are a great many defects and dangling bonds at such sites. The interfaces
between layers within the solar cell also act as surfaces, with a certain surface re-
combination velocity used to express the recombination rate due to surface effects.
44
Passivation is necessary to adequately terminate the crystal lattice at such
sites to reduce recombination rates and hence lower surface recombination velocity.
By passivating defects such as dangling bonds at surfaces and grain boundaries,
that is, rendering them inert, recombination can be prevented in such areas. This
will serve to increase charge collection probability, since carriers now have a lower
probability of recombining at interfaces and grain boundaries and hence more
chance of being collected at contacts.
In the context of analytical models, surface recombination velocity is usually
a boundary condition imposed on surfaces, i.e.
Dh
dp
dx
∣
∣
∣
∣
x=surface
= Sp · p|x=surface (2.12)
In analytical simulations of a single dimension, this parameter is used to
model the effectiveness of contacts to allow majority carriers to recombine in
preference to minority carriers, which determines the charge collection probability.
Hence in a single dimension, where the surface beyond the contact region cannot
be accounted for, surface recombination velocity at the contact itself is used to
encompass surface effects at cell front and back surfaces. Although Ohmic con-
tacts are usually considered to be sites of infinite recombination, the use of this
parameter in a one-dimensional simulation serves to factor in surface effects and
model the quality of device passivation for a particular cell.
2.2.2 Carrier absorption/generation
We now consider the generation of carriers in the semiconductor. These are
given by the so called Beer-Lambert expression, as follows [31]:
45
g ne (x) =g n
h (x) = αnλφ
emitterλ exp[−αn
λ(Hn + x)] − Hn ≤ x ≤ 0 (2.13)
g pe (x) =g
ph (x) = αp
λφbaseλ exp[−αp
λ(x −Wp)] 0 ≤ x ≤ Hp (2.14)
where αnλ and αp
λ are the absorption coefficients in the n and p regions,
respectively, and φemitterλ and φbase
λ are the photon fluxes into the emitter and base
region, respectively, as given by [31]:
φemitterλ = φ0
λ(1− rλ) photons ·m−2s−1 (2.15)
φbaseλ = φemitter
λ exp(−αnλHn)exp(−αp
λWp) photons ·m−2s−1 (2.16)
where φ0λ is the illumination, and rλ is the reflectivity for the wavelength
under consideration.
The absorption coefficients ανλ where ν ∈ (n, p) can be taken from tables
of the complex refractive index of the material, since the absorption coefficient is
related to the imaginary part (or k-value) of the complex refractive index [30]:
α =4πkrλ
(2.17)
2.2.3 Reflection
When two media of differing refractive index meet, light incident on the
interface will be partially reflected, partially absorbed, and partially transmitted.
This leads to a decrease in efficiency in solar cell devices, since any light reflected
from the air-semiconductor interface cannot contribute to the photocurrent.
To this end, solar cells front surfaces are usually capped with an antireflection
coating, which may consist of multiple layers of different materials. Antireflection
46
coatings are designed to give the lowest possible reflectance for the widest possible
region of the solar spectrum, in order to maximize cell efficiency.
The propagation of light in the system can be modelled using the direct
matrix method [33], [34]:-
Meq =
n∏
j=1
cos(φj)iηjsin(φj)
iηj sin(φj) cos(φj)
(2.18)
where Meq is the characterisation matrix of the thin film stack, nj is the
refractive index of the j th layer, φj =2πλneffj dj ,ηj = Y0njcos(θ) for parallel polari-
sation, ηj = Y0nj
cos(θ)for perpendicular polarisation, θj is the angle of incidence in
layer j , Y0 is the admittance of free space, and dj is the thickness of the j th layer.
From this, the characteristic matrix of the assembly can be written down:
B
C
= Meq
1
ηs
(2.19)
where ηs is the effective complex refractive index of the substrate similarly
defined as above.
The reflectance, R , transmittance, T , and absorption, A of the assembly of
j thin film layers can be obtained as follows [34]:
R =
(
η0B − C
η0B + C
)(
η0B − C
η0B + C
)∗(2.20)
T =4η0Re(ηs)
(η0B + C )(η0B + C )∗(2.21)
A = 1− R − T =4η0Re(BC
∗ − ηs)
(η0B + C )(η0B + C )∗(2.22)
To compute the reflectance at the top active layer (usually emitter) of a
solar cell, a matrix stack of all thin films from the illumination source to the
active layer is assembled for all wavelengths of the spectrum. The transmission
47
and reflectance can then be computed using the above relations. This yields the
incident light for all wavelengths of the spectrum for the topmost active region of
the solar cell. From there, the solar cell’s efficiency can be calculated. Note: this
does not incorporate reflection from the cell’s bottom contact and hence multiple
passes through the device. This is because the analytical models presented in this
work only compute photocurrents for a single pass of illumination.
2.2.3.1 Emitter (n-type) quasi-neutral region
In the emitter region we consider the current from minority carrier holes.
To obtain the hole concentration throughout the emitter (−Hn ≤ x ≤ −Wn), we
first substitute Eqn. 2.9 into Eqn. 2.7, then based on our assumption that the
carrier concentrations obey the principle of superposition, we subtract the terms
involving the dark hole concentration [31], yielding a differential equation solely
for the photo-generated holes in the n region (=pphn ) [31]:
Dh
d2pphn
dx2− Dhp
phn
L2h+ g n
λ (x) = 0 (2.23)
Note that the terms involving E are set to zero since there is no electric field
in the quasi-neutral region. Solutions to this equation will yield the carrier gener-
ation profile as a function of x for a particular wavelength of light, λ. Solutions
can be obtained subject to the following boundary conditions [31]:
pphn (−Wn) = 0 (2.24)
Dh
dpphn
dx
∣
∣
∣
∣
x=−Hn
= Sppphn (−Hp) (2.25)
These boundary conditions state that the hole concentration is zero at the
edge of the space charge region, and that the rate at which holes leave through
48
the front contact is equal to the front hole surface recombination velocity Sp times
the hole concentration at that point.
Equation 2.23 is an inhomogeneous second order differential equation, hence
its solution is of the form [31]:-
pphn (x ,λ) = CF + PI (2.26)
CF, the complementary function, is the solution of 2.23 with g nλ(x) set to
zero, and is of the form [31]:-
CF = Aphin cosh
(
x
Le
)
+ Bphin sinh
(
x
Le
)
(2.27)
The particular integral PI is some constant C times Eqn. 2.14. To find the
constant C, we simple set the CF to zero and substitute Eqn. 2.26 into Eqn. 2.23
[31]:-
PI =Cφemitterλ exp[−αn
λ(Hn + x)] = C · y (x)
⇒Dh
d2C · y (x)dx2
− DhC · y (x)L2h
+ y (x) = 0
⇒C
(
Dh(αnλ)
2 − Dh
L2h
)
+ 1 = 0
∴ C =− L2hDh[(αn
λ)2L2h − 1]
With some algebra, the solution for the hole density can be obtained [31]:-
49
pphn (x ,λ) =
φemitterλ αn
λL2h
Dh[(αnλ)
2L2h − 1]exp(−αn
λQn)
×
cosh[(Hn + x)/Lh] + (ShLh/Dh) sinh[(Hn + x)/Lh]
+(αnλLh + ShLh/Dh) sinh[−(Wn + x)/Lh]exp(α
nλQn)
cosh(Qn/Lh) + ShLh/Dh sinh (Qh/Lh)− exp[−αn
λ(Wn + x)]
(2.28)
where Qn is the width of the emitter region, Hn −Wn.
Differentiation gives the hole current profile in the emitter region [31]:-
jphemitter (x ,λ) =qDh
dpphn (x)
dx
=− qφemitterλ αn
λLh
[(αnλ)
2L2h − 1]exp(−αn
λQn)
×
− sinh[(Hn + x)/Lh]− (ShLh/Dh) cosh[(Hn + x)/Lh]
+(αnλLh + ShLh/Dh) cosh[−(Wn + x)/Lh]exp(α
nλQn)
cosh(Qn/Lh) + ShLh/Dh sinh (Qh/Lh)
−αnλLh exp[−αn
λ(Wn + x)]
(2.29)
The total current entering the junction from the emitter is given by jphemitter (−Wn,λ)
[31]:-
50
jphemitter (−Wn,λ) =− qφemitter
λ αnλLh
[(αnλ)
2L2h − 1]exp(−αn
λQn)
×
− sinh[Qn/Lh]− (ShLh/Dh) cosh[Qn/Lh]
+(αnλLh + ShLh/Dh) exp(α
nλQn)
cosh(Qn/Lh) + ShLh/Dh sinh (Qh/Lh)
−αnλLh
(2.30)
2.2.3.2 Base (p-type) quasi-neutral region
The solution for the photocurrent in the base quasi-neutral region is found
in a similar way, by solving electron continuity equation for the minority carrier
electron concentration in the region Wp ≤ x ≤ Hp [31]:-
De
d2nphp
dx2−
Denphp
L2p+ g
pλ (x) = 0 (2.31)
The solution takes the following form [31]:-
jphbase(x ,λ) =− qφbase
λ αpλLe
(αpλ)
2L2e − 1
×
− sinh[(Hp − x)/Le]− (SeLe/De) cosh[(Hp − x)/Le]
−(αnλLe − SeLe/De) cosh[(x −Wp)/Le ]exp(−αp
λQp)
cosh(Qp/Le) + SeLe/De sinh (Qp/Le)
+αpλLe exp[−α
pλ(x −Wp)]
(2.32)
where Qp is the width of the base region, Hp−Wp. The photocurrent density
flowing into the junction is given by jphbase(Wp,λ) [31]:-
51
jphbase(Wp,λ) =− qφbase
λ αpλLe
(αpλ)
2L2e − 1
×
− sinh[Qp/Le ]− (SeLe/De) cosh[Qp/Le]
−(αnλLe − SeLe/De) exp(−αp
λQp)
cosh(Qp/Le) + SeLe/De sinh (Qp/Le)
+αpλLe
(2.33)
2.2.3.3 Space-charge region
In deriving the current in the space charge region, we may consider either
electrons or holes; here a choice is made in favour of electrons. The assumption in
the space-charge region is that carriers are swept out by the built-in electric field
sufficiently quickly that no recombination occurs [31]. Hence, Eqn. 2.4 reduces
to:-
1
q
dJe
dx+ ge(x) = 0 (2.34)
This can be solved using simple integration to yield jphscr . To simplify the
integration, the problem can be made symmetric about the origin, x = 0. Defining
the illumination at this point as:-
φjnλ = φemitter
λ exp(−αnλHn) (2.35)
and defining the generation of carriers in the SCR:-
gλ(x) =
αnλφ
jnλ exp(−αn
λx) −Wn ≤ x ≤ 0
αpλφ
jnλ exp(−α
pλx) 0 ≤ x ≤ Wp
(2.36)
We can now integrate the carrier generation in the SCR to find the photocurrent:-
52
jphscr =− q
∫ Wp
−Wn
gλ(x)dx
=− q
(∫ 0
−Wn
αnλφ
jnλ exp(−αn
λx)dx +
∫ Wp
0
αpλφ
jnλ exp(−α
pλx)dx
)
=− qφjn [exp(αnλWn)− exp(−αn
λWp)] (2.37)
2.2.4 Total photocurrent
The total photocurrent for a single wavelength λ is simply the sum of the
previously derived photocurrents for the emitter, base, and space-charge regions
[31]:-
jph(λ) = jphemitter (−Wn,λ) + jphscr (λ) + j
phbase(Wp,λ) (2.38)
The total photocurrent is then the integral over all wavelengths of the illu-
mination spectrum [31]:
jph =
∫
AM1.5G
jph(λ) dλ (2.39)
2.2.5 Depletion region width
The depletion region width for a homojunction can be found by adopting the
exhaustion layer (Schottky) approximation. This allows the following expressions
to be derived [31]:-
Wn =NA
NA + ND
W (2.40)
Wp =ND
NA + ND
W (2.41)
W =
[
2ǫ0ǫps ǫ
nsψbi(NA + ND)
2
qNAND(ǫpsNA + ǫnsND)
]1/2
(2.42)
53
The built-in potential ψbi is given by [30]:-
ψbi = kT lnNDNA
n2i(2.43)
For a heterojunction, the depletion region width is given by [30]:-
WD1 =
[
2NA2ǫs1ǫs2(ψbi − V )
qND1(ǫs1ND1 + ǫs2NA2)
]1/2
(2.44)
WD2 =
[
2ND1ǫs1ǫs2(ψbi − V )
qNA2(ǫs1ND1 + ǫs2NA2)
]1/2
(2.45)
where WDn is the width of depletion region n, ǫsn is the dielectric constant
of region n, NDn is the donor concentration in region n, NAn is the acceptor
concentration in region n, ψbi is the built-in potential, and V the applied potential.
The built-in potential of the heterojunction is given by [35]:-
qψbi =∆EC −∆EV
2+ kTln
NdNa
ni ,n + ni ,p(2.46)
where ∆EC and ∆EV are the conduction and valence band discontinuities
respectively, Nd and Na are the doping densities either side of the heterojunction,
and ni ,p and ni ,n are the intrinsic carrier concentrations in the p and n regions,
respectively.
2.2.6 Dark current
2.2.6.1 Emitter dark current
Minority carrier holes are considered when computing the emitter dark cur-
rent in the n-type emitter region, in an analogous manner to that of section
2.2.3.1. Substituting Eqn. 2.9 into Eqn. 2.7 and dropping the electric field and
carrier generation terms, yields [31]:-
54
Dp
d2pDKn
dx− Dh(p
DKn − p0
n)
L2h= 0 (2.47)
which can be solved subject to the following boundary conditions [31]:-
De
dpDKn
dx
∣
∣
∣
∣
∣
x=−Hn
= Sh[pDKn (−Hn)− p0
n] (2.48)
pn(−Wn) = p0n exp(qVj/kT) (2.49)
where Eqn. 2.48 states that the flux of minority carriers at the emitter surface
(i.e. the front contact) is equal to the surface recombination velocity times the
number of carriers present at the surface. Equation 2.49 arises from the so-called
low-level injection conditions [31] and gives the minority carrier concentration at
the edge of the junction for some junction bias Vj .
The solution can be shown to take the following form [31]:-
pDKn (x) =p0
n + p0n[exp(qVj/kT)− 1]
×
cosh[(Hn + x)/Lh] + (ShLh/Dh) sinh[(Hn + x)/Lh]
cosh(Qn/Lh) + (ShLh/Dh) sinh(Qn/Lh)
−Hn ≤ x ≤ −Wn
(2.50)
Substituting into Eqn. 2.3 gives the hole current density profile in the emitter
[31]:-
jDKemitter (x) =
qDhp0n
Lh[exp(qVj/kT)− 1]
×
sinh[(Hn + x)/Lh] + (ShLh/Dh) cosh[(Hn + x)/Lh]
cosh(Qn/Lh) + (ShLh/Dh) sinh(Qn/Lh)
(2.51)
At the edge of the junction, i.e. at x = −Wn, the current flowing into the
emitter can be expressed as [31]:-
55
jDKemitter (−Wn) =
qDhp0n
Lh[exp(qVj/kT)− 1]Ξn (2.52)
where Ξn is the emitter width factor, given by [31]
Ξn =sinh[Qn/Lh] + (ShLh/Dh) cosh[Qn/Lh]
cosh(Qn/Lh) + (ShLh/Dh) sinh(Qn/Lh)(2.53)
2.2.6.2 Base dark current
The expressions for the dark electron density and electron current density
profile are derived in an identical manner to that of section 2.2.6.1. They are as
follows [31]:-
nDKp (x) =n0p + n0p[exp(qVj/kT)− 1]
×
cosh[(Hp − x)/Le ] + (SeLe/De) sinh[(He − x)/Le ]
cosh(Qp/Le) + (SeLe/De) sinh(Qp/Le)
Wp ≤ x ≤ Hp
(2.54)
jDKbase(x) =
qDen0p
Le[exp(qVj/kT)− 1]
×
sinh[(Hp − x)/Le] + (SeLe/De) cosh[(Hp − x)/Le]
cosh(Qp/Le) + (SeLe/De) sinh(Qp/Le)
(2.55)
As in section 2.2.6.1, the current flowing into the base from the junction at
x = Wp can be expressed as [31]:-
jDKbase(Wp) =
qDen0p
Le[exp(qVj/kT)− 1]Ξp (2.56)
where Ξp is the base width factor, given by [31]:-
Ξp =sinh[Qp/Le ] + (SeLe/De) cosh[Qp/Le ]
cosh(Qp/Le) + (SeLe/De) sinh(Qp/Le)(2.57)
56
2.2.6.3 Space-charge region dark current
In the dark space charge region Eqn. 2.6, and Eqn. 2.7 become [31]:-
Dn
d2n
dx+ µeE
dn
dx+ nµe
dE
dx= 0 (2.58)
Dp
d2p
dx− µhE
dp
dx− pµh
dE
dx= 0 (2.59)
Their solution is quite involved and can be found in [31]. However, since in
the dark space charge region there is (effectively) no generation or recombination,
it follows that the dark current density in the space charge region is constant.
On the assumption that all majority carriers injected at one end of the junction
become minority carriers when emerging from the space-charge region, the dark
current in the space-charge region can be expressed as [31]:-
jDKscr = jDK
emitter (−Wn) + jDKbase(Wp)
=
[
qDhp0n
LhΞn +
qDen0p
LeΞp
]
[exp(qVj/kT)− 1] (2.60)
Which reduces to the Shockley equation for Ξn = Ξp = 1 [31].
2.2.7 Device parasitics
Device parasitics can be conveniently modelled as two parasitic resistors, one
in series and one in parallel. The series resistance is the lumped total of contact
resistance, which itself can be resolved into contributions from the semiconduc-
tor/metal interface as well as the metallisation itself, and any bondout resistance.
The shunt resistance encompasses a variety of processes that act like a simple
resistance in parallel with the device. This could be due to pin holes and defects
that shunt out the emitter, or lateral defects causing junction shunting current to
flow at the edges of the mesa.
57
Figure 2.3: Two diode model showing shunt (RP) and seriesresistance (RS), effective diodes (D1 and D2), load resistance(RL), and the current source representing the short circuit currentdensity (JSC)
The model includes two diodes, having different ideality factors to account
for two regimes in the device characteristics. These are space-charge region (SCR)
generation and recombination (GR) currents, and quasi-neutral region GR currents.
The quasi-neutral region GR currents dominate at higher forward biases and the
diode which models this behaviour usually has an ideality factor close to one. The
space-charge region GR currents dominate at low forward bias and, depending
on material qualities, this part of the diode model usually has an ideality factor
closer to 2. The key parameters determining the ideality factor are the radiative,
Shockley-Read-Hall (SRH) and Auger parameters (see Section 2.3).
The SRH formalism describes recombination through a trap level. In regions
where the lattice is unterminated, such as grain boundaries, dislocations and bulk
defects, and surfaces, the SRH lifetime is significantly shorter. These defect sites
are accounted for by the higher ideality factor of the second diode. A device with
comparably fewer grain boundaries, bulk defects, and well passivated surfaces may
have a lower ideality factor for this second diode or the device may be well-behaved
enough to be described by a single diode model.
We can analyze the circuit in Fig. 2.3 fairly simply and write out the circuit
58
equations for it as follows:-
IL(Vj) = Id1(Vj) + Id2(Vj) + Ishunt(Vj) (2.61)
= J0,1A
(
exp
(
Vj
n1kT
)
− 1
)
+ J0,2A
(
exp
(
Vj
n2kT
)
− 1
)
+Vj
RP
(2.62)
VL(Vj) = Vj + RS IL(Vj) (2.63)
where IL, VL are the load current and voltage, J0,i , ni are the saturation
current densities of the ith diode, kT is the thermal voltage, and Vj is the junction
voltage.
We note that the factor[
qDhp0n
LhΞn +
qDen0p
LeΞp
]
has been encapsulated into
J0,1, diode 1’s saturation current density. J0,2 on the other hand may be either set
to zero or determined empirically.
2.2.8 Summary analytical model for solar cells
The complete analytical model, including photocurrent, dark current, and
device parasitics is summarized below for convenience.
2.2.8.1 Photocurrent
jph =
∫
AM1.5G
jph(λ) dλ (2.64)
jph(λ) = jphemitter (−Wn,λ) + jphscr (λ) + j
phbase(Wp,λ) (2.65)
59
jphemitter (−Wn,λ) =− qφemitter
λ αnλLh
[(αnλ)
2L2h − 1]exp(−αn
λQn)
×
− sinh[Qn/Lh]− (ShLh/Dh) cosh[Qn/Lh]
+(αnλLh + ShLh/Dh) exp(α
nλQn)
cosh(Qn/Lh) + ShLh/Dh sinh (Qh/Lh)
−αnλLh
(2.66)
jphscr =− q
∫ Wp
−Wn
gλ(x)dx
=− q
(∫ 0
−Wn
αnλφ
jnλ exp(−αn
λx)dx +
∫ Wp
0
αpλφ
jnλ exp(−α
pλx)dx
)
=− qφjn [exp(αnλWn)− exp(−αn
λWp)] (2.67)
jphbase(Wp,λ) =− qφbase
λ αpλLe
(αpλ)
2L2e − 1
×
− sinh[Qp/Le ]− (SeLe/De) cosh[Qp/Le]
−(αnλLe − SeLe/De) exp(−αp
λQp)
cosh(Qp/Le) + SeLe/De sinh (Qp/Le)
+αpλLe
(2.68)
2.2.8.2 Darkcurrent
jDK (Vj) =
[
qDhp0n
LhΞn +
qDen0p
LeΞp
]
[exp(qVj/kT)− 1] (2.69)
jDK ,parasitics(Vj) = J0,1
(
exp
(
Vj
n1kT
)
− 1
)
+ J0,2
(
exp
(
Vj
n2kT
)
− 1
)
+vj
RP
(2.70)
VL(Vj) = Vj + RS IL(Vj) (2.71)
60
2.2.8.3 Total current
jtotal(Vj) =
∫
AM1.5G
jph(λ) dλ− jDK ,parasitics(Vj) (2.72)
2.2.9 Key device characteristics
There are several key parameters that can be extracted from solar cell device
characteristics that allow the performance of a cell to evaluated and compared with
other devices. These can be summarised as follows:-
• Short Circuit Current
• Open Circuit Voltage
• Fill Factor
• Efficiency
2.2.9.1 Short circuit current
The short circuit current (i.e. when the load is effectively a short circuit) is
roughly equivalent to the photocurrent, since at zero bias the dark current should
be zero. In practice, due to recombination effects which are not included in the
idealised diode model (Auger recombination, GR currents, trap-assisted and band
to band tunneling, shunting effects due to process related defects etc.) this may
not be the case and a very small leakage current may exist, as evidenced in the
dark IV characteristics. In general, however, at zero bias these effects are much
smaller than the photocurrent and can be neglected.
Short circuit current is therefore related to the number of photons absorbed
by the device. This necessitates adequate collection of incoming photons by en-
suring they are absorbed in the device and not lost due to reflection or inefficient
61
absorption. Hence the absorber layer must account for reflections at the front
surface and prevent photons from being lost due to optical mismatch between the
front surface and the atmosphere, as well as ensuring that any unabsorbed photons
do not exit from the rear surface, but are, in fact, reflected at the rear surface and
hence absorbed on a second or further pass through the device. Better yet would
be a light trapping design that traps photons and keeps them confined within the
body of the device due to total internal reflection.
From Eqn. 2.66 and Eqn. 2.68, it is evident that diffusion length, diffusivity,
and surface recombination velocity play a vital role in determining the probability
of photo-generated electron-hole pairs reaching a contact for collection. In order
to maximise photocurrent, surface recombination velocities must be minimised,
and diffusivity (i.e. mobility) must be maximised, in so doing improving diffusion
length. To lower surface recombination velocities, front and rear surfaces must
be adequately passivated. This necessitates combined passivation/antireflection
coatings that terminate the semiconductor surface as well as providing light trap-
ping properties. To improve diffusion lengths, mobilities must be optimised. This
necessitates high quality materials with good electronic properties. Single crystal
semiconducting devices will always outperform polycrystalline and amorphous thin
films in this regard, at the expense of increased material cost. The same can
be said of lifetimes, also a component in determining diffusion lengths. Hence
there is a trade off between material cost and electronic properties when choosing
materials to optimise photocurrents.
2.2.9.2 Open circuit voltage
Open circuit voltage conditions occur when the photocurrent exactly cancels
the dark current, yielding no net current. For this case we can write:-
62
jph =
[
qDhp0n
LhΞn +
qDen0p
LeΞp
]
[exp(qVoc/kT)− 1]
⇒ exp(qVoc/kT) = 1 +jph
qDhp0n
LhΞn +
qDen0pLe
Ξp
⇒ Voc =kT
qln
1 +jph
qDhp0n
LhΞn +
qDen0pLe
Ξp
(2.73)
giving the open circuit voltage of the cell, Voc . Since the open circuit voltage
is essentially determined by the ratio of the light and dark currents, it is necessary to
maximise the photocurrent whilst minimising the dark current to optimise the open
circuit voltage. This means that many of the the material parameters affecting
the photocurrent also determine the open circuit voltage, i.e. diffusion length,
diffusivity, and surface recombination velocity. In addition, open circuit voltage is
determined by the emitter and base doping densities. By increasing the doping
densities, dark current can be reduced, improving open circuit voltage. It can be
seen from equation Eqn. 2.73 that if the photocurrent were directly scaled, for
example by using a lens to concentrate the incident radiation, the open circuit
voltage would increase. In this way, the power increases with concentration of the
incoming solar radiation. This power increase is the principle by which concentrated
photovoltaics (CPV) is feasible.
2.2.9.3 Efficiency
Efficiency is defined as the ratio of the output power to the incident power,
or
η =Pout
Pin
× 100% (2.74)
Pin for AM1.5G illumination is 100 mW/cm2. Efficiency is determined by
the maximum output power of the device, at the device maximum power point,
63
since power output is determined by the load. To optimise efficiency therefore, the
position of the maximum power point must be optimised. This means optimising
not only short circuit current and open circuit voltage, but also fill factor, as
discussed in the next section. This is where device parasitics come into play as the
series and shunt resistance of the device shift the position of the maximum power
point, affecting the efficiency.
2.2.9.4 Fill factor
Voltage, Volts
Cell PV
Curr
ent
Area b
Area a
Pow
er
Cell JV
FF=Area b / Area a x 100%
Figure 2.4: Graphical depiction of maximum power point, opencircuit voltage, short circuit current, and fill factor and theirrelationship.
Fill factor is the geometric ratio of the area bounded by the maximum power
point and the open circuit voltage/short circuit current. This relationship is de-
picted graphically in Fig. 2.4.
FF =Jmpp · Vmpp
Jsc · Voc
× 100% =η × Pin
Jsc × Voc
(2.75)
64
Factors determining the fill factor include recombinative losses, since both
current and voltage are determined by diffusion length and surface recombination
velocity, and series resistance. Series resistance shifts the position of the maximum
power point back toward the current axis. Shunt resistance shifts the position of
the maximum power point down towards the voltage axis. Hence we need low
recombination losses, low series resistance and high shunt resistance to optimise
fill factor and hence efficiency.
2.2.10 Summary
A review of analytical solutions to the drift-diffusion equations has been
presented. From this, an analytical model for solar cells has been built up including
a discussion of device parasitics. The closed form nature of the analytical model
and governing equations means that a device can be modelled quickly. This allows
back-of-the-envelope calculations for novel device structures to be carried out
quickly, as a preparation for more detailed numerical calculations, which solve
the drift-diffusion equations numerically using finite element analysis rather than
rely on closed-form solutions allowing more complicated models to be evaluated.
We turn our attention to such solutions in the next section.
65
2.3 Numerical simulation
2.3.1 Introduction
In this section, numerical solution of the drift-diffusion equations is pre-
sented. This involves discretisation and solution of the drift-diffusion equations,
as well as discussion of various models of interest. The purpose of this section is
to build up a stand-alone unidimensional drift-diffusion solver for numerical simu-
lation of solar cells. This is a useful exercise that allows one to gain insight into
the function of numerical simulators, which are very complex and quite difficult to
understand at first.
2.3.2 Equation set in continuous form
The drift-diffusion set of equations belongs to a class of problems collectively
known as diffusion-advection-reaction problems. Their numerical solution is known
to be problematic, and this is particularly the case when dealing with semiconductor
problems, as the tight coupling between potential and carrier concentrations intro-
duces non-linearities which impair numerical stability. In this section we present the
unidimensional discretisation and possible solution methods for the drift-diffusion
equations.
The drift-diffusion set of equations for the steady state behaviour of an
electronic device were presented in Section 2.2.
The discretisation of the electron and hole drift-diffusion and continuity equa-
tions (equations 2.2 - 2.5) over a mesh is itself problematic since a non-linear in-
terpolation is usually necessary to determine mesh points on the odd grid (for first
differences) as required for solution of the electron and hole continuity equations.
66
2.3.3 Dependent variables
We will mainly consider the set of dependent variables (φ, n, p) for electron
and hole concentration and potential, respectively. However, it is possible to
work with other sets of dependent variables. One particular example is the set of
variables (φ, u, v ) where u and v are given by:-
n = ni · exp(
φ
kT
)
· u (2.76)
p = ni · exp(
− φ
kT
)
· v (2.77)
Although it is fairly obvious that the dynamic range required by this variable
set is very large, to the extent that practical implementation even on modern
computers is very difficult [36].
Another such variable set is (φ, φn, φp), where raw carrier concentrations
are replaced by quasi-Fermi potentials according to the following relation:-
n = ni · exp(
φ− φn
kT
)
(2.78)
p = ni · exp(
φp − φ
kT
)
(2.79)
Finally another set of dependent variables are the Slotboom variables, (φ,
Φn, Φp) where:-
Φn = ni · exp(
− φn
kT
)
(2.80)
Φp = ni · exp(
φp
kT
)
(2.81)
however, it is noted that this formulation is difficult to extend to degenerate
conditions. Although the variables (φ, n, p) seem the most natural choice for
67
solving semiconductor problems, a change of variables is often advantageous to
circumvent the problems associated with the non-linearity and tight coupling of
the drift-diffusion model.
2.3.4 Discretisation of Poisson equation in 1D
The Poisson equation is comparatively easily discretised, since the Laplacian
operator requires only physical mesh points (not odd grid, or interpolated, mesh
points) since it is discretised by a second difference, and so the Poisson equation
(2.1) becomes (see e.g. [36]):-
Fφ(φ, n, p)|i = ǫǫ0
(
φxi−1− φxi
xi − xi−1
+φxi+1
− φxi
xi+1 − xi
)
2
xi+1 − xi−1
−e·(n − p − ND + NA) = 0
(2.82)
where the subscripts i , i − 1 etc. refer to the mesh point.
2.3.5 Discretisation of electron and hole drift-diffusion and continuity
equations in 1D
After combining Eqns. 2.2 and 2.4, the following expression can be discre-
Figure 3.2: CdTe/Ge heterojunction tunnel diode current-voltagecharacteristics. Peak tunneling current density more than satis-fies the requirements for cells operating under 1 sun illumination(≃ 26mA/cm−2) [58].
Parameter Description Valueµe,CdTe CdTe electron mobility 1260 cm2/Vs [2]µh,CdTe CdTe hole mobility 104 cm2/Vs [2]Eg,CdTe CdTe band gap 1.5 eV [2]χ0,CdTe CdTe electron affinity 4.05 eV [58]τSRH,CdTe CdTe SRH lifetime 1 ns [66]µe,Ge Ge electron mobility 3800 cm2/Vs [67]µh,Ge Ge hole mobility 1800 cm2/Vs [67]Eg,Ge Ge band gap 0.67 eV [42]χ0,Ge Ge electron affinity 4.0 eV [58]τSRH,Ge Ge SRH lifetime 1 ms [42]
Table 3.1: Simulation parameters for crystalline CdTe homojunc-tions on crystalline Ge substrates
since the peak tunneling current density (≃ 1200 mA/cm−2) is much greater than
the device short circuit current density (≃ 24− 26 mA/cm−2).
The full device structure and energy band diagram are presented in Fig.
108
Figure 3.3: Proposed energy band diagram for a CdTe homo-junction on Ge substrate. Note desirable overlap of valence andconduction bands in the tunnelling / recombination junction.Not to scale.
Table 3.2: Efficiency vs. Ge FSF/emitter doping, under 1-sun,AM1.5G illumination. Note efficiency improvement with increas-ing front surface field (FSF:n+germanium tunnel diode region)doping.
3.1 and Fig. 3.3, respectively. The device is modelled using optical databases
109
(n and k values) from Tousek [68] for CdTe and Palik [69] for Ge. Minority
carrier lifetimes are set to 1 ns [66] and 1 ms [42] for CdTe and germanium
respectively. Front and back surface recombination velocities are set to 5e5 cm/s
and 1e3 cm/s [58], respectively. Surface recombination at the interface is set
to 1e2 cm/s; however simulation results are relatively insensitive to the value
assumed (since recombination in the tunnel junction is desirable). Simulation
takes into account band gap narrowing and doping dependent mobility in the
germanium cell. The doping dependent mobility data is from Palankovski [67].
Such effects are neglected for the CdTe cell, due to a lack of experimental data.
Incomplete ionization is also included. TAT is simulated using the Hurkx model
[41]. Simulation parameters for this cell are given in Tab. 3.1.
As noted above, simulation has shown that majority carrier flow across the
tunnel junction is also possible and this imposes limits on ultimate device efficiency.
Electron flow from the CdTe absorber region to the Ge emitter is negligible; this
is owing to the tunnel junction p+ region acting as a back surface field (BSF).
The same cannot be said for hole flow from the Ge emitter to the CdTe absorber;
however, this may be mitigated by reducing the Ge emitter doping and doping
the tunnel diode n+ region more heavily, producing a front surface field (FSF).
The emitter doping ought not to be reduced too drastically, as this will result in
lower open circuit voltage for the lower cell (and hence the tandem structure). By
reducing minority carrier flow accross the tunnel diode we may make some gains
on open circuit voltage due to reduced recombination but there will inevitably be
a cross over point where any potential advantages are out weighed by diminishing
bottom cell Voc . Results of overall cell efficiency vs. tunnel junction n+ region
doping are presented in Tab. 3.2 for devices with a 2.5 µm thick CdTe absorber
layer and a 15 µm thick Ge absorber layer. The device structure is simulated under
1 sun, AM1.5G illumination.
110
0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8Wavelength, m
0.2
0.0
0.2
0.4
0.6
0.8
1.0a.u
EQE and Reflection
EQE CdTeEQE GeReflection
Figure 3.4: CdTe homojunction on Ge substrate: external quan-tum efficiency and reflectance plot versus wavelength. Note thereflection losses caused at the CdTe/Ge interface. Data obtainedfrom Sentaurus simulation using transmission matrix method.
Although the heterostructure tunnel junction performs well in simulation,
there are definite concerns about physical realizability, since p type doping of CdTe
to the levels required is non-trivial. Although hole concentrations of 5× 1019 have
been reported using ion-implantation and pulsed electron beam annealing [70], ion
implantation is not particularly amenable to commercial manufacture of large-scale
modules. Nevertheless, this approach can be used as a starting point to investigate
whether or not such a structure can be physically realised.
Optical modelling using the transfer matrix method (TMM) shows significant
111
reflection at the interface. The external quantum efficiency (EQE) and reflectance
plot for the two cells of the tandem is shown in Fig. 3.4 as a function of wavelength.
The optical losses at the CdTe/Ge interface, combined with the non-idealities of
the tunnel junction, present difficulties in matching photocurrents between the two
cells of the tandem. Note that the the bi-layer antireflection coating of the device
structure consists of 60 nm of TiOx covered by 100 nm of MgF2 and serves to
minimize reflection losses at the front surface.
3.1.1.2 All germanium tunnel junction
In order to relax the doping requirements for p-type CdTe somewhat, and cir-
cumvent the problem of potentially unfavourable band alignment at the CdTe/Ge
interface, an alternative structure is presented in Fig. 3.5, which features a tunnel
diode realised entirely in germanium. The energy band diagram for this structure
is depicted in Fig. 3.6. This is the second approach to the problem of intercon-
nection of the two tandem cells. In this configuration, the underlying Ge substrate
wholly contains the tunnel diode which is defined using e.g. spin on diffusants.
In order to improve majority carrier injection while reducing minority carrier flow
across the heterointerface, a back surface field is realised in the CdTe absorber.
The doping density required for the back surface field is considerably less than for
the above heterojunction tunnel diode p+ region. Results are presented in Tab.
3.3.
The disadvantages of this structure are the limitations imposed on the doped
substrate in terms of high temperature processing. For example, it is usual during
MBE growth on Ge to outgas the native oxide at 650C [24] which would cause
significant diffusion of the dopants in the germanium substrate [22]. The time for
oxide outgassing therefore must be kept to a minimum.
Another disadvantage is that there is now a p+-CdTe / p+-Ge region formed,
112
Figure 3.5: CdTe homojunction on Ge substrate featuring all-germanium tunnel junction. Tunnel diode is formed entirelywithin the underlying germanium substrate.
Table 3.3: Efficiency vs. CdTe BSF doping for structure withall-Ge tunnel diode under 1 sun, AM1.5G illumination. Notethe efficiency improvement with increasing BSF doping. Theshort circuit current density is limited by the germanium bottomcell and hence is unaffected by CdTe BSF doping. This can beimproved slightly by optimising CdTe thickness, see Tab. 3.5.
and interdiffusion of dopants from either side of this junction will cause parasitic
doping which will affect the net doping of each region detrimentally (for example
Sb is a p-type dopant in CdTe but an n-type dopant in Ge).
113
Figure 3.6: Proposed energy band diagram for all-Ge-TRJ con-taining structure
Parameter Description Valueµe,CdTe CdTe electron mobility 320 cm2/Vs [?]µh,CdTe CdTe hole mobility 40 cm2/Vs [?]Eg,CdTe CdTe band gap 1.5 eV [?]τSRH,CdTe CdTe SRH lifetime 1 ns [66]µe,Ge Ge electron mobility 350 cm2/Vsµh,Ge Ge hole mobility 150 cm2/VsEg,Ge Ge band gap 0.88 eV [57]τSRH,Ge Ge SRH lifetime 1 µs
Table 3.4: Simulation parameters for thin film CdS/poly-CdTe/poly-Ge heterostructures
3.1.2 Conventional CdTe heterojunction
Unlike the previous structures, which are intended to serve as proof of con-
cept cells to be grown on single crystal germanium substrates by MBE, in this
114
Figure 3.7: CdS/CdTe/Ge tandem solar cell structure. 100 nmthick MgF2 is deposited on the glass front surface as an anti-reflection coating.
by cleaning (soak in warm trichloroethylene, acetone, methanol plus ultrasonic
bath), pre-treatment and passivation layer deposition, and finally depositing Cr/Au
Schottky contacts after stripping the passivation layer using reactive ion etching
(RIE) from the contact pad regions.
It has been noted that surface pre-treatment prior to passivation is important
to reducing SRVs and reducing interface trap densities. Two main wet treatments
are investigated in this work: HF/H2O, and HCl/HBr. HF/H2O treatment has
been shown to strip native oxides and terminate the Ge surface with hydrogen
[109]. HCl/HBr has been shown to passivate dangling bonds and leave the surface
terminated with Cl [110] or potentially Br. This treatment is carried out at room
temperature, since at elevated temperatures, this solution can etch the germanium
surface and potentially consume device active regions. Each treatment is preceded
with a 30s dip in 1:1:5 NH4OH : H2O2 : DI to remove approx 100nm of the surface
damaged region of the wafer (i.e. due to sawing) and any diffused impurities at
the surface. The chemical treatments are tabulated in Tab. 5.1.
5.2.1.2 Dry pre-treatment
Two main plasma treatments are considered in this work, namely H2, and
NH3 plasmas. The purpose of the dry plasma treatment is much the same as the
wet treatment, i.e. to desorb any native oxides and passivate dangling bonds. It
has been suggested that ammonia plasma treatment leaves the surface terminated
with nitrogen [4]. Whilst the ammonia pre-treatment can be carried out in-situ
159
Treatment Steps
A - Hydrogenation
1. Dip 1:1:5 NH4OH:H2O2:DI 30s
2. HF dip (1:3 BOE), 1 min
3. DI soak, 3 min
B - Halide Passivation1. Dip 1:1:5 NH4OH:H2O2:DI 30s
2. 1:1:5 HCl:HBr:DI soak, 1 min
Table 5.1: Wet pre-treatments for Germanium passivation
Treatment Gas flow rate Plasma Power
A. H2 plasma (ex situ) 10 sccm RF, 15W 30s
B. NH3 plasma (in situ) 10 sccm ICP, 200W 30s
Table 5.2: Dry pre-treatments for Germanium passivation
within the ICPEVCD reactor, the hydrogen plasma treatment was performed in a
separate tool, a Plasmalab System 100 RIE from Oxford Instruments [Austin, TX,
USA]. It must be noted that the brief exposure to atmosphere when transferring
to the ICPECVD after H2 plasma treatment may negate some of its beneficial
effects. The basic recipes are described in Tab. 5.2. The ammonia treatment
took place at 250 C , whereas the H2 plasma treatment was performed at room
temperature.
5.2.1.3 Passivation layer deposition
In this work we consider SiNx, a-Si:H/SiNx, and SiO2 as potential passivation
layers. These are deposited in a Sentech SI500D ICPECVD [Sentech Instruments
GmbH, Berlin, Germany]. The recipes are shown in Tab. 5.3. It was noted in
previous experiments that low ICP power favours higher quality SiO2 and a-Si:H
160
MaterialTable
temp. ( C)Flow rates ICP Power
SiO2 120 SiH4 = 6.5 sccm 450W
He = 123 sccm
Ar = 126 sccm
O = 70 sccm
a-Si:H 300 SiH4 = 5 sccm 35W
Ar = 95 sccm
SiNx 100 SiH4 = 7.3 sccm 800W
Ar = 139 sccm
NH3 = 10 sccm
Table 5.3: Passivation layer film deposition parameters
with good optical and mechanical properties and resilience to high temperature
processing, however for SiNx, the reverse is true. Here SiNx is deposited at very
high (∼ 800 W) ICP power, which furnishes very high quality, chemically resistant
SiNx with wide area uniformity (> 5” diameter uniformity area).
5.2.1.4 Anti-reflection coatings
In order to investigate the potential for combining a-Si:H, SiNx and MgF2
as both passivation and anti-reflection (AR) coating, films were first deposited
individually on germanium in order to obtain optical constants using spectroscopic
ellipsometry. This data was then used to optimise AR film parameters. Whereas
a-Si:H and SiNx were deposited in the Sentech ICPECVD, MgF2 was evaporated
in a bell-jar thermal evaporator at a rate of 0.5-1 A/S. Optical constants (complex
refractive index) and thicknesses were extracted by spectroscopic ellipsometry using
a Woollam M2000D [J. A. Woollam Co., Lincoln, NE, USA].
161
5.2.1.5 Contact deposition
Contacts were deposited using thermal evaporation of 5nm Cr and 50nm Au
at a rate of 0.5 A/s (Cr) and 10 A/s (Au), respectively, in a bell jar evaporator
at a vacuum of 1e-6 Torr. The contacts were patterned using photo-lithography
and lift-off patterning, and lift-off was performed in boiling acetone, with residual
resist removed using a swab dipped in hot acetone.
5.2.2 Characterisation
5.2.2.1 Photoconductive decay
PMOS
White LED
Sample under test
To power supply
To square
wave generator
probes
To parameter
analyser
To parameter
analyser
Test structure viewed from above
Contact
pads
Mesa
Bulk
+ -Applied bias
Figure 5.1: Measurement setup for photoconductive decay mea-surements
Photoconductive decay measurements were undertaken by illuminating the
device with a white LED broadband source pulsed at 100 Hz. The test setup is
shown in Fig. 5.1, which also shows a photoconductor viewed from above. The
162
mesa region of the photoconductors was 40 µm wide and between 96 and 900
µm long. Devices were biased at 30V, in order to ensure significant photocurrent.
The white LED was switched using a p-channel MOSFET with switching times
several orders of magnitude shorter than the minority carrier lifetime. Since the
surface recombination velocity (SRV) is essentially a boundary condition on the
drift diffusion equations (i.e. for holes dp
dx|surface = S ·p where S is the SRV, p is the
hole concentration) a useful expression can be derived by solution of the diffusion
equation [111]. If we make the assumption that S is much less than the diffusivity
of electrons divided by half the wafer thickness (= 500 µm ), in this case roughly
4000 cm/s, then we can make use of the following expression relating the effective
lifetime to the bulk lifetime and S [26]:
1
τeff=
1
τbulk+
1
τSRV=
1
τbulk+
2S
d(5.1)
where τeff is the effective lifetime, τSRV is the surface recombination dom-
inated lifetime, τbulk is the bulk lifetime, S is the surface recombination velocity
(total for both sides) and d is the wafer thickness. The effective lifetime is taken
to be the dominant decay curve of a potentially multi-exponential curve [112].
The bulk lifetime was assumed to be roughly 3.8 ms, a value taken from
the literature [26]. In [26] this value was determined for wafers of comparable
quality from the same manufacturer as were used in these experiments so should
be approximately valid. The usual method for determining the bulk lifetime and
surface recombination velocity is to perform measurements on samples of varying
thickness. A plot of 1/τeff vs. 1/d is then constructed; the y intercept is equal to
1/τbulk and the slope is equal to 2S [26, 112]. Due to budget and time constraints,
wafers of only a single thickness were used in these experiments, hence a value for
the bulk lifetime is assumed, rather than experimentally determined.
163
EntryWet
treatment
Dry
treatmentPassivation Layer τSRV S (cm/s)
1 A A 40nm SiO2 150 µs 173
2 A A40nm a-Si:H
75nm SiNx171 µs 150
3 A A 75nm SiNx 60 µs 423
4 B B 40nm SiO2 150 µs 173
5 B B40nm a-Si:H
75nm SiNx1.7 ms 21
6 B B 75nm SiNx 60 µs 423
7 B -40nm a-Si:H
75nm SiNx78 µs 327
8 B - 75nm SiNx 40 µs 631
9 B - Unpassivated 200 µs 131
Table 5.4: Passivation layer film deposition parameters, showingwet pretreatment (see table Tab. 5.1) and dry pretreatment(see table Tab. 5.2), measured lifetime and extracted surfacerecombination velocity.
The photoconductive decay results are shown in table Tab. 5.4.
From the photoconductive decay measurements, we may draw a few infer-
ences about the various passivation processes. Firstly, the HCl/HBr pretreatment,
which has been shown previously to reduce interface state density can successfully
terminate the lattice so as to reduce surface recombination velocity, which is evi-
denced by entry 9 in table Tab. 5.4. Secondly, in situ dry treatment is preferable
to ex situ or wet treatment alone, which may be attributed to native oxide re-
moval. Finally, insulating wide band gap (optical band gap ≃ 2.2eV) amorphous
silicon is a superior passivation layer in combination with SiNx, reducing interfacial
164
recombination/tunneling, which leads to longer surface recombination dominated
lifetimes and hence reduced surface recombination velocities. The photoconduc-
tive decay measurement of the NH3 plasma treated sample is depicted in Fig. 5.2
showing extraction of the dominant τeff .
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0Time,ms
10-2
10-1
100
101
Photocurrent, m
A
Photoconductive decay of NH3 treated Ge:a-Si:H:SiNx
Fit, (τeff = 1.7 ms)
Meas.
Figure 5.2: Photoconductive decay of NH3 plasma treated Gecapped with ICPECVD a-Si:H/SiNx
5.2.2.2 Summary
It is insightful to compare these results with published values for comparable
technologies in germanium. This compares the results obtained in this work to
two references from the literature which employed the microwave reflected pho-
toconductive decay (µPCD) method as opposed to the simple constant voltage
photoconductive decay method employed herein. In the µPCD method, the pho-
165
toconductivity is measured by microwave reflection after carriers are injected by
optical pulses from a pulsed laser [85].
These results are set out in Tab. 5.5. Note that the results of Posthuma et
al. [26], while impressive, are derived from extrapolation from curves of 1/τ vs.
1/d, which gives very optimistic values for the surface recombination velocity and
bulk lifetime. Indeed the bulk lifetime seems quite high, even for very pure CZ
germanium, which tend to contradict lifetime studies as published in the literature
[113, 114], which indicate a maximum lifetime of ∼ 2ms for the bulk lifetime.
Nevertheless, this might be attributable to advances in Ge material processing,
since the first measurements of lifetime were made many decades ago. On the
other hand, the results of Swain et al. [115] seem a great deal more realistic
and serve to highlight the affinity the germanium surface has for both hydrogen
and chlorine. However, we must concede that wet treatment alone is simply not
enough to adequately passivate germanium, as it does not deal with the issues
of necessary A/R coatings nor environmental passivation. Hence, we can surmise
that wet treatment (with its reasonable stability in air) can be used to prevent
oxidation of the sample prior to loading into an ICPECVD tool to deposit capping
and A/R layers. Finally, we must point out that since the PCD technique as used in
this work involves making contacts, and that contacts are recombination centers,
the PCD results will always give shorter lifetimes than the µPCD technique, which
is a contactless microwave-reflected photoconducive decay method. Nevertheless,
we can conclude that the SRVs presented in this thesis are comparable to those
reported in the literature.
5.2.3 Anti-reflection coatings
In addition to passivation (which lowers surface recombination velocities,
thus improving photocurrent in optoelectronic devices), anti-reflection coatings are
166
Ref.Passivation
Layer stackMaterial
τeff
τbulkS (cm/s)
Meas.
techn.
Posthuma
et al. [26]
a-Si:H
∼ 100nm
p-type
CZ Ge
550µs
3.8ms17 µPCD
Swain
et al. [115]
Wet treatment
(HF
HCl)
p-type
CZ Ge
∼ 475, 370µs
1.17ms23, 37 µPCD
This
work
a-Si:H
∼ 40nm :
SiNx
100nm
n-type
CZ Ge
1.7ms
3.8ms21 PCD
Table 5.5: Summary of Germanium passivation technologies
necessary to further maximize photoresponse or efficiency of power transfer. In
this work, we consider the use of a-Si:H/SiNx/MgF2 as combined passivation/anti-
reflection (A/R) coating. This is depicted in Fig. 5.3.
Figure 5.3: Schematic diagram of proposed 3 layer A/R coating.Ei is the incident wave, Er the reflected wave. dj is the layerthickness; ηj is the refractive index of each layer.
It is informative to calculate what the optimal refractive indices ought to be
in order to totally cancel any reflections at the interfaces between layers, in order
167
to compare them with the chosen materials’ properties. In this case, we must
satisfy [33, 34]:-
η1η0
=η2η1
=η3η2
=ηSη3
(5.2)
with dj set to a quarter wavelength. A possible solution is to have η3 : η2 : η1
= 2.8 : 2 : 1.4, which is a reasonable match for a-Si:H (n ∼ 3.5) : SiNx (n ∼ 2)
: MgF2 (n ∼ 1.4).
In order to optimise this AR structure, the optical constants were first ex-
tracted for each material used in the coating. In the case of SiNx, it is known
that the refractive index can be tuned by adjusting the ammonia flow rate. For
the SiNx recipe in Tab. 5.3, refractive index as a function ammonia flow rate is
depicted in Fig. 5.4.
Figure 5.4: Refractive index (@ 632nm) vs. ammonia flow rate
Since the higher refractive index is required, the lowest ammonia flow rate
(10 SCCM) is used; lower flow rates were not considered since it was assumed
that this would disturb the governing reactions adversely and give rise to poor
quality films. Another property that is necessary to tailor is the hydrogen content
of the SiNx film; high hydrogen content will cause absorption throughout the
168
visible and infrared regions such that any anti-reflection properties would be more
than negated. High ICP power and low growth rate were seen to ensure low
hydrogen content and hence low absorption, as evidenced by the imaginary part
of the complex refractive index (i.e. k) being less than 10−5 for all wavelengths
longer than 400 nm.
This is in contrast to a-Si:H, which is deposited with low ICP power and
higher substrate temperatures to ensure high quality films. Under these limit-
ing conditions, the deposition technology approaches chemical vapour deposition
proper. These low hydrogen content films are quite robust and well suited to high
temperature processing, surviving 10-15 minute treatment at 750C without any
change in optical thickness, and without evidence of bubbling (which can occur if
H content is substantial). Optical constants of all 3 films are shown in Fig. 5.5.
Using the optical constants, 3 layer anti-reflection coatings may be designed
and optimised for a particular wavelength or spectrum. In this work, we consider
optimisation of the AR coatings for the solar (AM1.5G) spectrum for use in pho-
tovoltaic applications. The AM1.5G spectrum was chosen in order to improve the
efficiency of stand-alone Ge solar cells, as presented in chapter 6. In this work,
we do not choose to optimise for longer wavelengths, such as for the irradiation
expected in a four terminal tandem cell configuration with CdTe top cell. This is
because we the precise thickness of the CdTe top cell was not firmly determined at
time of writing. However, an identical procedure can be followed to optimise the
structure for longer wavelengths. As we shall see, the below structure shows rea-
sonable response at longer wavelengths so we wouldn’t expect much would change
if we were to optimise for the infra-red portion of the spectrum. Note that no anti-
reflection coating is required for germanium in the case of a monolithic tandem
cell; instead, we would like a reflective back contact to reflect any light reaching
the back surface back through the device. Nonetheless, adequate passivation of
169
0 500 1000 1500 2000
Wavelength (nm)
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5R
efr
acti
ve index, n
SiNx a-Si:H MgF2
Optical properties of films
Figure 5.5: Refractive indices of films used for passivation/ARcoating
the back surface is crucial to reducing back surface recombination velocities so
the structure presented below is still applicable to a monolithic tandem to improve
electronic properties of the back surface.
The optical stack is optimised by evaluating the usual expressions for the
reflectance (R), transmittance (T) and absorption (A) of the multilayer film [33,
34]:
R =
(
η0B − C
η0B + C
)(
η0B − C
η0B + C
)∗(5.3)
T =4η0Re(ηS)
(η0B + C )(η0B + C )∗(5.4)
A = 1− R − T =4η0Re(BC
∗ − ηS)
(η0B + C )(η0B + C )∗(5.5)
170
where:
B
C
= Meq
1
ηS
(5.6)
Meq =n∏
j=1
cos(φj)iηjsin(φj)
iηj sin(φj) cos(φj)
(5.7)
η0 = Y0 = ǫ0 · c0 (5.8)
ηS = Y0(nS − ikS) (5.9)
ηj = Y0(nj − ikj) (5.10)
(5.11)
φj =2πηj · dj
λ(5.12)
where Y0 is the permittivity of free space and nl , kl are the refractive index
and extinction coefficient of medium l respectively (l = 0 = air, l = S = substrate,
l = j = layer j, etc.) and λ is the wavelength.
Although dispersion in the films is comparably small, it cannot be neglected
entirely, necessitating solution for the optimal structure via numerical (i.e. global
solution), rather than analytical means (i.e. for one particular wavelength). This
allows one to generate a 3 dimensional dataset to optimize the optical properties
of the film stack, as weighted over the AM1.5G spectrum, in order to maximize
energy transfer to the underlying device.
Fig. 5.7 shows the slice through the data cube for d1 = 90 nm, showing
reflection, transmission, and absorption characteristics as a function of the two
remaining free parameters, d2 and d3.
The amorphous silicon films are highly absorptive. This is due in part to the
above band gap absorption of the films (the optical bandgap of the amorphous
171
400 800 1200 1600
Wavelength, nm
0
20
40
60
80
100
%
Reflectance, RTransmission, T
Reflectance and Transmission for A/R coating
Figure 5.6: Transmission/reflectance of as-deposited a-Si:H/SiNx/MgF2 A/R stack
silicon as deposited in this work is roughly 2.2eV) and partly due to absorption
due to excess hydrogen. By neglecting the absorption of the amorphous silicon,
we may optimise the AR stack and minimise reflection. This yields a minimum
reflectivity of the stack of 6.47%, with d1:d2:d3 = 89:58:76 nm, as shown in Fig.
5.8. However, this does not maximise the transmission. The optical transmission
of the coating is maximised for d1:d2:d3 = 89:61:10 nm, where the amorphous
silicon is made as thin as possible, as shown in Fig. 5.7. Figure 5.6 shows the
transmission and reflectance of the latter, transmission-optimised, AR stack.
As part of future investigations, annealing should be experimented with to
reduce the absorption of these films and maximise transmission. However, an-
The capacitance of the heterojunction is given by the following relation [30]:-
Chj = A
√
q · ǫnǫp2
NaNd
(Naǫp + Ndǫn)(φbi − VA)(7.3)
A more useful quantity to consider is the inverse square of the capacitance,
since this transforms the complicated relation in Eqn. 7.3 into a simple linear
relationship.
1
C 2hj
=1
A2
2
q · ǫnǫp(Naǫp + Ndǫn)(φbi − VA)
NaNd
(7.4)
By inspection of Eqn. 7.4, it is evident that the x-intercept is equal to the
built in potential of the device. Furthermore, it is possible to determine the doping
profile as a function of depth in an unknown region of the device from the slope
of the 1C2hj
versus VA plot.
242
d
dV
(
1
C 2hj
)
= − 1
A2
2
q · ǫnǫp(Naǫp + Ndǫn)
NaNd
(7.5)
If the doping in one region of the Pn heterojunction is significantly higher
than in the other region, it is possible to solve Eqn. 7.5 analytically. In other cases,
graphical methods can be employed to determine the carrier concentration from
the transcendental equation. Since in this case the doping concentrations on both
sides of the heterointerface are comparable, this technique must be employed.
The capacitance-voltage profiling of the CdTe/Ge heterojunction is shown
in Fig. 7.23. Doping densities some distance from the junction are extracted
using the above techniques. The doping density for the germanium substrate is
computed from wafer resistivity (40 Ω · cm ≃ 1e14 cm−3). The values of the
dielectric constants for CdTe and Ge are taken to be 10.4 and 16 ǫ0 respectively
[2]. The built in potential is extracted by extrapolation of the 1C2hj
versus VA plot,
and is around 1.42V. This built in potential has components attributable to the
doping densities either side of the interface and the band offsets themselves, as
illustrated by equation Eqn. 2.46. If one band offset were known, the other
could be computed from this equation. The doping concentration extracted is
the acceptor concentration in the p-type CdTe and is fairly constant. From the
literature [127] we would expect a Gaussian profile of Cu acceptor ions at the
surface of the CdTe (a distance 6 µm from the junction as indicated in Fig. 7.23).
We do not see such a feature. This could be attributable to the limited accuracy
of the CV profiling at such a distance from the junction. The doping density
extracted, however, may not be completely due to the introduction of copper ion
but may be due to a native defect that arises during the diffusion process, which
serves to anneal and activate the sample in air, a common step in CdTe solar cell
processing.
243
−10 −8 −6 −4 −2 0 2Bias, V
0.00
0.01
0.02
0.03
0.04
0.05
0.06
0.071
/ C
apaci
tance
square
d, 1/pF2
1/C2 −V Plot of CdTe/Ge heterojunction
0
5
10
15
20
25
30
35
40
Capaci
tance
, pF
Built in potential = 1.42±0.1 V
−6 −5 −4 −3 −2 −1 0Distance from junction, µm
1013
1014
1015
Dopin
g c
once
ntr
ati
on, cm
−3
Emitter doping profile of CdTe/Ge heterojunction
Figure 7.23: CdTe / Ge heterojunction capacitance profiling
7.6 Summary and conclusions
In this chapter we computed the true orientation of substrates used in this
work using the stereographic projection. We noted the misorientation of wafers
244
can lead to a dramatic change in wafer orientation, for instance nominally <2 1
1> wafers were in fact oriented <5 3 3>. This is important to take notice of,
particularly when performing XRD on these samples. This also effects the growth
orientation for crystals grown on such wafers.
ZnTe thin films were prepared using a simple thermal evaporation method,
based on preparing stoichiometric ZnTe from Zn and Te and mixing in a crucible
to form a stoichiometric source. The ZnTe was deposited at temperatures ranging
from 200 to 400C . Optically transparent and reasonably uniform films were
deposited within this temperature range.
Molecular Beam Epitaxy was employed to deposit high quality CdTe films
on both germanium and sapphire substrates. MBE is the growth technology of
choice for high quality semiconducting films with good control over interfaces
and material composition. This growth technology yielded high quality layers as
compared to thermal evaporation. The success of the sapphire grown samples
illustrates that MBE is a viable technology for experimentation with mechanically
stacked multijunction CdTe solar cells utilising sapphire substrates.
Optical contrast microscopy allowed the surface morphology of the as-deposited
epilayer to be examined. The surface morphology was smooth and did not feature
any noticeable defects. The surface was slightly corrugated with small hillocks of
1 micron in size. The overall appearance was seen to indicate a reasonable quality
film.
In situ RHEED was used during growth to characterise the film in situ and
showed high quality nucleation followed by a transition to smooth morphology with
a streaky appearance of the RHEED pattern. RHEED was also used during initial
preparations to determine the growth temperature by adjusting the temperature
prior to nucleation iteratively according to the RHEED pattern. RHEED is a
useful characterisation tool and one of the many advantages of ultra-high vacuum
245
technologies such as MBE.
Optical transmission measurements showed high quality films with good op-
tical transparency. The films did not match up with the models particularly well
with differing optical bandgaps and optical properties. Nonetheless, the films
themselves showed good properties highlighting the quality of the material.
X-ray diffraction was used to investigate the crystallinity of samples of CdTe
and ZnTe prepared on Ge and Sapphire. The samples prepared on sapphire had the
best crystallinity as shown by double crystal rocking curves (DCRC) full width at
half maximum (FWHM) as low as 59 arc seconds about the < 1 1 1 > diffraction
plane, indicating a very high quality crystal. The samples prepared on < 5 3 3 >
Ge was less crystalline owing to suboptimal growth conditions. The ZnTe samples
were polycrystalline due to the simpler, lower quality deposition technology.
A CdTe/Ge heterojunction was fabricated using copper spin-on dopants to
dope the CdTe region p-type with Cu acceptors. Current-voltage (IV) and capac-
itance voltage (CV) measurements were made of the heterojunction diode. The
diode showed rectifying characteristics, a series resistance of approximately 10kΩ,
a shunt resistance of 1.25MΩ, and an ideality factor of 1.65. This indicated a
reasonable quality diode had been formed. The CV profiling showed a built-in
potential of 1.42V and the extracted doping concentration indicated the doping
profile was fairly uniform throughout the epilayer. This may be attributable to
doping due to a native defect in the CdTe which was activated during the diffusion
process.
The CdTe/Ge heterojunction shows that the electronic properties of the
CdTe/Ge interface are suitable to create a heterojunction device such as a tandem
solar cell, since the device showed good characteristics over a certain voltage
range and obeyed theory well enough to admit a simple curve fit. The fitted
curve had ideality factor 1.65, showing the presence of defects in the device most
246
likely due to the lattice mismatch, the skewed orientation of the substrate and the
suboptimal growth conditions. Future investigations of CdTe/Ge heterojunctions
could include deep-level transient spectroscopy (DLTS) to determine the energy
levels of the defects at the CdTe/Ge interface. This technique can yield a wealth
of information about the electronic properties of CdTe and Ge which can feed back
into device models. Knowledge of the trap levels in the interface could help design
an optimised tunneling/recombination junction and improve solar cell efficiency.
In summary, it was shown that it is possible to grow high quality II-VI films
on sapphire, an optically transparent, mechanically hard, and insulating substrate.
This shows that mechanically stacked multijunction solar cells featuring high qual-
ity II-VI thin films grown on sapphire are physically realisable and should be consid-
ered. Future work should include in situ doping of the II-VI films during epitaxial
growth to realise complete tandem structures on sapphire. These structures can
then easily be combined with the germanium pn-junction diode and solar cell pro-
cess developed in this work and presented in Chapter 6. It was also shown that the
electronic properties of the CdTe/Ge heterojunction were favourable and a device
was fabricated to show this. This tends to indicate that a monolithic device should
be possible, however further work is needed to devise processes for heavily doping
cadmium telluride, since the doping density achieved with copper spin on doping
was not particularly high.
Chapter 8
Conclusions
8.1 Summary
Throughout this work we have been concerned with demonstrating the feasi-
bility of germanium as a bottom cell for monolithic or mechanically stacked tandem
solar cells with cadmium telluride. We have found that monolithic growth of the
materials is possible but troublesome and plagued with difficulties, particularly re-
lating to high temperature processing. Since Ge and CdTe form a eutectic alloy
at around 319 C , there is a restriction on the temperatures that can be used
on CdTe devices deposited on Ge. Also, the lattice mismatch makes it difficult to
grow high quality CdTe epilayers on top of Ge, even using a high-cost, low deposi-
tion rate technology like MBE. Nevertheless, a CdTe/Ge heterojunction device was
created that allowed electronic properties of the heterostructure to be explored.
This device was fabricated using MBE grown CdTe on Ge grown at the University
of Illinois at Chicago, and processed at the UWA clean room (A. G. Nassibian
Nanofabrication Facility). The device was a P/n heterojunction and exhibited
good rectifying characteristics, with ideality factor of 1.65 indicating reasonable
quality material and a shunt resistance of 1.25 MΩ suggesting very few shunt like
defects at the material interface.
Work was undertaken to develop low cost technologies for device active re-
gion formation. The techniques investigated were direct spin-on doping, proximity
248
doping, and sandwich stacked diffusion. Direct spin-on doping is seen to give the
best profile for phosphorus doped devices in terms of abruptness, however, direct
spin on doping was to be avoided due to issues with inhomogeneities and defects
causing poor yield. Proximity doping was shown to give favourable results but
lacked the surface concentration for P and Sb necessary for degenerate doping.
Proximity doping was shown to work quite well for Ga as well, and degenerate
doping could be achieved. Finally, sandwich stacked diffusion, the novelty intro-
duced in this thesis, was used to degenerately dope Ge with Ga acceptors. These
low-cost techniques are all amenable to mass manufacture and adaptable to the
mass manufacturing environment. These techniques could one day be used to
form the Ge junction part of a CdTe/Ge tandem solar cell.
To demonstrate the viability of the low cost doping technologies, Ge diodes
were fabricated using Ge single crystal wafers from Sylarus inc. [St. George,
Utah, USA] and University Wafer [Boston, MA, USA]. These were seen to be high
quality diodes with ideality factors close to 1. These devices did suffer from Zener
breakdown in reverse bias. However, on the lower doped substrate, the reverse
breakdown occurred at -25V which is acceptable reverse breakdown performance.
These devices illustrated the use of the low cost doping techniques.
Passivation technologies were investigated and it was found that a pre-
treatment using ammonia followed by a-Si:H/SiNx passivation layer deposition
gave the most favourable surface recombination velocities, with a surface recom-
bination velocity as low as 21 cm/s reported in this work. This was as compared
with unpassivated but chemically treated control samples as well as a variety of
other passivation techniques, including hydrogen pretreatement and SiO2 passiva-
tion layers. These passivation technologies are again low cost and amenable to
mass-manufacture.
Antireflection coatings to Ge were optimised and a final design of a-Si:H,
249
SiNx and MgF2 of thicknesses 10-80 nm, 50-60 nm, and 80-90 nm, respectively,
were found to minimise reflectivity. Specifically, for thicknesses 76nm (a-Si:H),
58nm (SiNx) and 89nm (MgF2) the reflectivity was reduced to 6.47%. However,
thicker a-Si:H implies more absorption and hence less transmission to the underly-
ing Ge layer. Hence it is favourable to trade off reflectivity for transmissivity and
thus incorporate a thinner amorphous silicon layer.
Contacts were fabricated and optimised to germanium. It was found that
Ni had the best specific contact resistivity but was hard to measure with the test
structures in question since the resistances involved were so low the measurements
had impaired accuracy. A specific contact resistivity of 1.26 ×10−7 Ω · cm2 is
reported. Other metals investigated included Al, Cr/Au and Ti. With the exception
of Al, all metals made good quality ohmic contacts to the germanium.
Ge solar cells were fabricated, with a peak efficiency of 5.4%. Although this
is not world changing in and of itself, it nevertheless demonstrates the technology.
These cells were fabricated from single crystal germanium by using the techniques
developed in this work: that is, diffusion by low cost proximity doping technique,
passivation and anti-reflection coating, and contacts. Hence it is fair to say that
the solar cell symbolises the culmination of all the thesis’ work in one single device.
The device was stable 8-9 months after initial testing which proved the potential
efficacy of the passivation layer as an environmental passivant. However, a lack of
environmental testing means the conclusions which may be drawn are limited.
Single crystal CdTe was grown of very good quality on R-plane sapphire.
The best X-ray DCRC FWHM was 59 arc seconds, indicating very high quality
material. Since this is a viable option, it may be possible to create mechanically
stacked tandem solar cells with top cell featuring CdTe grown on sapphire. This
could be a possible future research direction.
250
8.2 Research outcomes
The key research aims of this work were to reduce surface recombination in
Ge standalone solar cells with adequate surface passivation, to make low resistivity
contacts to germanium, and form device regions in a low cost manner without
any unwanted contamination which may impair device efficiency. These aims
were addressed sufficiently by the research in this work. Further research aims
included to investigate the feasibility of combining CdTe and Ge together to form
an electronic device and determine the electronic properties of the heterostructure.
This was also addressed and a viable heterojunction was demonstrated.
The following can be concluded about the combination of CdTe with Ge:-
• MBE growth yields good results, but heteroeptiaxy is difficult due to lattice
mismatch
• The optimum growth MBE temperature lies somewhere between 300 and
305C , and the optimum fluxes are beam equivalent pressure (BEP) 1.1e-
6 Torr for CdTe, and 1.5e-6 Torr for Te.
• Care must be taken to always keep the temperature of the sample below
the eutectic point (319 C ) to prevent alloying of CdTe and Ge. This is
particularly necessary if annealing in the presence of Cd vapour.
Provided that low temperature processing is used, monolithic CdTe and Ge
devices can be adequately fabricated with good electronic properties. If elevated
processing temperatures are required, a mechanically stacked tandem should be
pursued, and for this case, sapphire can be used as an optically transparent sub-
strate.
251
8.3 Future work
Due to time and budget constraints, sandwich-stacked diffusion was only
investigated for Ga spin-on dopants and only with diffusion followed by drive-in.
It would be highly recommended to research all dopants investigated in this work
using this technique to determine its applicability to a wide range of dopant sources.
Furthermore, models for dopant profiles should be compiled so that devices can
be designed. It would be interesting to see if higher surface concentrations can be
achieved with sandwich-stacked diffusion for the high vapour pressure phosphorus
dopant sources. If so, this will overcome some of the limitations of phosphorus
proximity doping which cannot achieve a high surface concentration due to the
high vapour pressure and volatility of phosphorus.
Process models need to be focussed on to improve the prospects of realising
a tunnel-diode. In particular, co-doping with phosphorus and gallium needs to
be investigated to determine the effects one dopant has on the diffusivity of the
other. Such a study should make extensive use of secondary ion mass spectrometry
(SIMS) depth profiles to adequately ascertain the secondary effects of codoping.
These models then need to be utilised to design a tunnel diode, which can then be
used to realise the structure introduced in Chapter 3, which consisted of an all-Ge
tunnel/recombination junction.
Work needs to be undertaken to improve the efficiency of the standalone
germanium solar cells. Their efficiency was hampered by two main factors: poor
quality VGF wafers from JMP technology Sarl, which contained a great many
defects and dislocations, and poor fill factor attributed to high series resistance due
to insufficiently thick front contacts. A technology needs to be developed for thick
plating of contacts to micron scale thickness. One possible technique would be
a simple nanoparticle paste which is spread on the contact material and annealed
252
to form a thick contact. Waste material should not adhere to the passivation
layer and should instead “ball up” or migrate to the contacts. An example nano-
particle paste would be simple Sn/Pb solder paste. Other nanoparticle pastes
might be suitable. Another possible technology could be electroplating. Thick
layers of contact material could be deposited by passing a current through the
device itself in a chemical bath. Finally, inkjet printing with nanoparticle inks is
a novel technology that has only recently entered the research arena. This could
be used combined with annealing with a nanoparticle solution of, for example, Ni,
to make thick low resistivity contacts to the Ge. These techniques should lower
series resistance and improve fill factor.
Open circuit voltage can be improved in the solar cells by using slightly
more heavily doped wafers. The wafers from JMP Technology Sarl were nominally
1e17 cm−3 doped. By raising the substrate doping to 5e17 or 1e18 cm−3 a few
tens of millivolts open circuit voltage can be gained. In addition, open circuit
voltage is limited by shunt resistance. The device presented in chapter 6 had a
suboptimal shunt resistance which can be attributed to defects during the doping
process which are more prominent in this device due to its larger area. This means
the proximity doping technique is still affected by defects and inhomogeneities
present in the spin-on dopant matrix. Or alternatively, the shunt resistance may
be attributable to lateral defects due to the dislocations on the wafer itself as
evidenced by SEM on a similarly fabricated diode. Again, due to large area, the
perimeter is considerable and will contain a great many of these defects, some
located within the junction. These may be the cause of the shunt resistance, in
which case simply changing wafer manufacturers should solve this problem.
It was seen that the optical transparency of the a-Si:H films was not partic-
ularly good. There are two reasons for this, one is that the optical band gap of
the film lies within the visible range, and hence absorbs energy (a-Si:H has a rea-
253
sonably high absorption coefficient, hence its use in thin film solar cells); the other
is the presence of excess hydrogen. It should be possible to anneal out some of
the excess hydrogen without adversely affecting the device. Of critical importance
is that the passivating interactions at the surface when prepared with ammonia
pretreatment are not interfered with during the annealing process. For example, it
is thought that nitrogen plays a role in terminating the lattice and passivating Ge
dangling bonds. If so, the passivation should be relatively unaffected by thermal
treatment. If however hydrogen itself is responsible in part for the passivation of
the surface, then it is unlikely that the structure will be temperature-stable and
hence may degrade during the process. This warrants further investigation.
Future work should encompass doping studies of CdTe. Since spin-on Cu
solution worked quite well to form a simple heterojunction diode, this avenue may
be followed. To increase doping density, higher temperatures need to be consid-
ered. This means the experiments need to take place in saturated Cd environment
with Cd overpressure supplied from a solid charge to prevent Cd out-diffusion and
damage to the epilayer structure. This necessitates annealing in sealed ampoules.
This technique would be limited to sapphire substrates since Ge will dissolve in
the Cd environment forming a eutectic alloy. A much better alternative would be
to develop in situ doping in the MBE system itself. As for choice of dopants, it
would be difficult indeed to find suitable materials since in situ doping of CdTe is
quite difficult. It may be possible to use iodine or indium as donors, and antimony
or arsenic as acceptors but would require a great amount of work to perfect this
technology.
Growth of CdTe on Ge needs to be improved to reduce the full width at half
maximum of double crystal X-ray rocking curves. Perhaps a better orientation of
wafer should be chosen since growth on <5 3 3> germanium was not particularly
successful. A better orientation is <1 0 0> but if miscut for epitaxy then X-ray
254
peaks can be difficult to find.
8.4 Final Conclusions
This thesis has demonstrated a low cost fabrication process for germanium
optoelectronic devices including solar cells, that features optimised passivation and
contacting technologies. Low cost techniques for device active region formation
were demonstrated. The process as a whole was amenable to mass manufacture.
The devices were stable over time indicating that the passivation was adequate.
These key elements were satisfied in the context of developing the technology
for monolithic and mechanically stacked tandem solar cells. It was shown that
the doping densities demanded in the modelling and design of the tandem solar
cells could be achieved using the low cost technologies developed in this work, at
least in the case of Ge. Heavy doping of CdTe could not be demonstrated, but
moderate doping was achieved using a Cu spin-on dopant source. A solar cell
of moderate efficiency was demonstrated using the diode process. This solar cell
could be improved by many of the techniques discussed in the previous section.
Having satisfied the main research aims, this thesis has been moderately
successful. It is perhaps lamentable that a full tandem solar cell could not be
fabricated, however the amount of work involved in fabricating such a device and
developing from scratch the technology to do so is well outside the realm of a single
PhD thesis. Several recommendations for future work in the previous section tackle
the work involved and break it down into smaller pieces.
Since the author chose to carry out this study at a research group with a
background in infrared detectors, it is not surprising that the main outcome has
been a high-quality process for photodiodes as opposed to large area solar modules.
This is simply a question of research know-how within a research group. Since this
was the first solar project that this group has ever undertaken, the outcome of
255
working solar cells and high quality diodes can be seen as a major success. Should
the research into solar cells at the Microelectronics Research Group continue, there
should be no reason why efficiencies cannot match those of world leading research
groups, as intellectual property is developed over time.
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Appendix A
Decoupled solution - Gummel’s method
1 # minimalist implementation of Gummel ’s method for Silicon pn junction
IV curve trace
2 impor t math
3 impor t scipy . linalg
4 impor t numpy as np
5 impor t matplotlib . pyplot as plt
6 from pylab impor t ∗7 impor t time
8 c l a s s units: # Units and scaling
9 cm = 1e8 ; s = 1e8 ; V = 1.0 / 0 . 02585 ;10 C = 1.0 /1 . 602176462 e−19; K = 1.0 /300 ; m = 1e2∗cm ;
11 um = 1e−4∗cm ; J = C∗V ; eV = 1.602176462e−19∗J ;12 A = C/s ; mA = 1e−3∗A ; kb = 1.3806503 e−23∗J/K ;
13 e = 1 ; eps0 = 8.854187818e−12∗C/V/m ;
14 eps = 11.7∗ units . eps0 ; affinity = 4.17∗ units . V ; Eg = 1.12∗ units . V; ni = 1.45 e10 ∗pow ( units . cm, −3.0) ;
15 mun = 100∗ units . cm∗units . cm/units . V/units . s ; mup = 40∗ units . cm∗units . cm/units . V/units . s
16 tau = 1e−8∗units . s ; kT = 1 ; Nd = 1e16 ∗ pow ( units . cm,−3) ; Na = 5e19
# if we didn’t break due to res increase update old mesh
92 N . set_ydata ( old_mesh_n ∗ pow ( units . cm,3 ) ) ; P . set_ydata ( old_mesh_p ∗pow ( units . cm,3 ) ) ; # Plot carrier concentrations
93 Ec . set_ydata (−affinity/units . V − old_mesh_phi/units . V ) ; Ev . set_ydata (−affinity/units . V − old_mesh_phi/units . V − Eg/units . V ) # Plot
155 b2 = [ j f o r k i n zip ( [ f_phi_RHS ( i ) f o r i i n range (1 ,npoints−1) ] ,\156 [ f_n_RHS ( i ) f o r i i n range (1 ,npoints−1) ] , \157 [ f_p_RHS ( i ) f o r i i n range (1 ,npoints−1) ] ) f o r j i n k ]
− mesh_phi [ 0 ] ) ] + \258 [−f_phi_RHS_0 ( i ) f o r i i n range (1 ,npoints−1) ]\259 + [ ( f_phi_0 ( mesh_n [ npoints −1] ,mesh_p [ npoints −1],doping (
299 p r i n t "Bias, V\t\tJ, mA/cm2 \t\t||RHS ||\ t\tIterations \t\tTime (s)"
300
301
302 f o r Vapp i n np . array ( [ 0 . 0 ∗ units . V,0 . 2∗ units . V,0 . 4∗ units . V,0 . 6∗ units . V,0 . 7∗units . V,0 . 8∗ units . V ] ) : # Solver loop