Improvement of a-Si:H devices by analysis, simulations and experiment

I

MPROVEMENT OF A

-S

I

:H

DEVICES BY

I

MPROVEMENT OF A

-S

I

:H

DEVICES BY

ANALYSIS

,

SIMULATION AND EXPERIMENT

Proefschrift

ter verkrijging van de graad van doctor aan de Technische Universiteit Delft,

op gezag van de Rector Magnificus prof. ir. K. C. A. M. Luyben, voorzitter van het College voor Promoties,

in het openbaar te verdedigen op dinsdag 28 februari 2014 om 12:30 uur

door

Bas V

ET

electrotechnisch ingenieur geboren te Zaanstad, Nederland.

Dit proefschrift is goedgekeurd door de promotor: Prof. dr. M. Zeman

Copromotor: Dr. R. A. C. M. M. van Swaaij Samenstelling promotiecommissie:

Rector Magnificus, voorzitter

Prof. dr. M. Zeman, Technische Universiteit Delft, promotor Dr. R. A. C. M. M van Swaaij Technische Universiteit Delft, copromotor Prof. Dr. R. E. I. Schropp Technische Universiteit Eindhoven Prof. Dr. C. I. M. Beenakker Technische Universiteit Delft Prof. Dr. I. Richardson Technische Universiteit Delft Prof. Dr. J. J. Smit Technische Universiteit Delft Dr. G. J. Jongerden Exergy

Prof. Dr. J. A. Ferreira Technische Universiteit Delft, reservelid

Keywords: amorphous silicon, solar cells, Printed by: Ipskamp Drukkers

Front & Back: R.D. Oudewortel

Copyright © 2013 by B. VET ISBN 000-00-0000-000-0

An electronic version of this dissertation is available at

To Jeep

C

ONTENTS

1 Introduction 1

1.1 Background . . . 1

1.1.1 A mote of dust suspended in a sunbeam. . . 1

1.1.2 The energy crisis. . . 3

1.1.3 The sun . . . 3

1.2 History of solar cell technology. . . 6

1.3 Crystalline silicon solar cells. . . 9

1.4 Thin-film silicon solar cells. . . 10

1.5 Multi-junction devices . . . 12

1.6 Modeling of a-Si:H solar cells . . . 12

1.7 Material and device characterization . . . 15

1.7.1 Reflection / Transmission measurement. . . 15

1.7.2 Fourier Transform Infrared spectroscopy . . . 16

1.7.3 Fourier Transform Photocurrent Spectroscopy . . . 17

1.7.4 Activation energy measurement. . . 18

1.7.5 Solar cell characterization . . . 19

1.8 Scope and Outline . . . 20

1.9 Contribution to the research field. . . 21

2 Performance limits of a-Si:H 23 2.1 Introduction . . . 23

2.1.1 Theory on performance of solar cells. . . 23

2.2 Structure of- and recombination in a-Si:H . . . 25

2.2.1 Density of states in a-Si:H . . . 25

2.2.2 Recombination processes. . . 26

2.3 Method. . . 28

2.3.1 Generation of charge carriers. . . 29

2.3.2 Detailed balance . . . 30

2.3.3 Lumped circuit model . . . 31

2.4 Direct recombination. . . 33

2.4.1 Analysis of radiative direct recombination . . . 33

2.4.2 The external parameters of the LC solar cell . . . 34

2.4.3 Results. . . 35

2.5 Auger Recombination . . . 37

2.5.1 Analysis of Auger recombination. . . 37

2.5.2 External parameters of the LC solar cell . . . 39

2.6 Valence band tail state recombination. . . 45

2.6.1 Analysis . . . 45

2.6.2 External parameters of the LC solar cell . . . 51

2.6.3 Simulation. . . 51

2.6.4 Results. . . 52

2.7 Conduction band tail state recombination. . . 55

2.8 Recombination through defect states . . . 60

2.8.1 Analysis . . . 60

2.8.2 Simulation. . . 62

2.8.3 Results. . . 62

2.9 Discussion. . . 64

2.10 Conclusion. . . 67

3 The p-i interface and its relation to the open-circuit voltage 69 3.1 Introduction . . . 69

3.2 Simulation. . . 70

3.3 Discussion. . . 74

3.4 Experiment. . . 83

3.5 Results and discussion . . . 85

3.6 Conclusion. . . 93

4 Development of stable a-Si:H absorber material on various substrates 97 4.1 Introduction . . . 97

4.2 Literature review on the role of hydrogen in a-Si:H. . . 99

4.2.1 Structural model . . . 99

4.2.2 Continous Random Network . . . 99

4.2.3 Disordered Network with Hydrogenated Vacancies . . . 99

4.2.4 Band gap and band tails . . . 100

4.2.5 Defects. . . 101 4.3 Material study . . . 102 4.3.1 Proto-crystalline regime . . . 102 4.3.2 Experiment. . . 103 4.3.3 Results. . . 103 4.4 Solar cells . . . 111 4.4.1 Introduction . . . 111 4.4.2 Experiment. . . 112 4.4.3 Results. . . 112 4.5 Conclusion. . . 115 5 References 119 References. . . 119

Summary 129

Samenvatting 131

Curriculum Vitæ 133

1

I

NTRODUCTION

1.1.

B

ACKGROUND

1.1.1.

A

February 14th, 1990, on the far edge of the solar system, a place on the rim of the magnetosphere of our sun, beyond the orbit of the outermost planet, at a distance of 6 054 558 968 km, Voyager 1 is traveling through space as a sole delegate of a human civilization. As ordered, Voyager 1 rotated its camera backwards to make a final obser-vation of the planets in the solar system. The 60 images it snapped, depict the family of planets in our solar system, as it would be witnessed by an alien observer. The picture of Earth, smaller than a single pixel, is displayed in Fig.1.1.

Look again at that dot. That’s here. That’s home. That’s us. On it everyone you love, everyone you know, everyone you ever heard of, every human being who ever was, lived out their lives. The aggregate of our joy and suffering, thousands of confident religions, ideologies, and economic doctrines, every hunter and forager, every hero and coward, every creator and destroyer of civilization, every king and peasant, every young couple in love, every mother and father, hopeful child, inventor and explorer, ev-ery teacher of morals, evev-ery corrupt politician, evev-ery "superstar", evev-ery "supreme leader," every saint and sinner in the history of our species lived there on a mote of dust suspended in a sunbeam.

Carl Sagan, Pale blue dot, a vision of the human future in space [1]

I would like to add that this dot embodies our environment and climate, it carries our civilization and it supplies our resources. It is here, where man invented fire, where nations and cooperations where formed, where wars were fought, where the industrial revolution took place and it is here, where we have to deal with the energy crisis.

1.1.2.

T

What is the energy crisis? Fluctuation of the oil price depending on the whim of the oil baron? Expensive heating for houses? An occasional electrical power failure? A wrong policy? No, the energy crisis is more than that.

Recent figures from the US Energy Information Administration (EIA) show that the total global energy consumption1in 2006 amounted to472× 1015BTU (British thermal units;498 × 1015kJ or138 × 1012kWh) In table1.1, the figures and and projections of the EIA are depicted. One of the conclusions to draw from this table is the fact that the countries within the OECD (Organization for Economic Cooperation and Development, typically regarded as the ’developed’ countries) together consume more than 51% of the energy, while, as is shown in table1.2only 17% of the world population lives there. And in this equation Russia is not even included. Imagine the implications, if we would want to distribute the energy evenly among the world population; each person in the Western world would have to reduce his energy consumption by a factor of three. Alternatively, if every person in this world would consume energy at the level currently consumed by Europeans, the total global energy consumption will increase by a factor of three-and-a-half. Meanwhile our world population is growing and our energy demand is increasing even further. Imagine the strain this will put on our environment. Can we even produce such enormous amount of energy? Do we have sufficient oil? Even in the more realistic projections of the EIA, the global energy consumption will increase by over 50% before the year 2030, an incredible amount, of which the consequences are difficult to predict. The scenario above gives a pessimistic view of the future, in which the global society becomes increasingly unsustainable. However, I would like to go back to our voyage through space and to point at our pale blue dot, suspended in a sunbeam; a single pixel in a vast empty space illuminated by a enormous ball of light with a luminosity of

3.846× 1026Watt [2]. From this perspective the solution to the energy crisis is obvious.

All we need to do is take that, which is abundantly available: solar energy.

1.1.3.

T

The Sun is the star at the center of the Solar System. In Fig. 1.2a sketch illustrates the relative size of the Sun (diameter 1.392_×106km) compared to the Earth (diameter 12.7_×103km). By itself the sun accounts for about 99.86 of the mass of the entire solar system [3]. The sun derives its energy from nuclear fusion and releases this energy as electromagnetic radiation. Its luminosity is 3.846_×1026Watt.

The sun is Earth’s primary source of energy. The solar energy is the driving force for almost all life on Earth, as well as for the climates. In addition, the world economy runs on solar energy accumulated millions of years ago and stored in the form of fossil fuels. The solar irradiation power density (I) reaching the Earth’s atmosphere is 1365 Wm−2. [4] The total amount of solar power that reaches the Earth, therefore, amounts toπr2I = 1.74 × 1017W.2

1_{The figures include oil, coal, natural gas, nuclear power, hydro power, geothermal power, wood, biomass,}

wind and solar power

Region/Country History Projections

1990 2005 2006 2010 2015 2020

OECD

OECD North America 100.7 121.6 121.3 121.1 125.9 130.3

United States 84.7 100.5 100.0 99.9 102.9 105.4 Canada 11.0 14.2 14.0 14.6 15.6 16.5 Mexico 5.0 6.9 7.4 6.6 7.4 8.3 OECD Europe 70.0 81.4 81.6 82.2 84.8 87.9 OECD Asia 27.0 38.4 38.7 39.5 41.8 43.1 Japan 18.7 22.7 22.8 21.9 22.9 23.4 South Korea 3.8 9.2 9.4 11.0 11.6 12.0 Australia/New Zealand 4.5 6.4 6.5 6.7 7.3 7.7 Total OECD 197.7 241.3 241.7 242.8 252.4 261.3 Non-OECD

Non-OECD Europe and Eurasia 67.3 50.6 50.7 54.0 57.6 60.3

Russia 39.4 30.1 30.4 32.2 34.3 36.0

Other 28.0 20.6 20.3 21.7 23.3 24.3

Non-OECD Asia 47.4 109.4 117.6 139.2 163.2 190.3

China 27.0 66.8 73.8 90.5 105.9 124.0

India 7.9 16.3 17.7 19.1 22.9 26.8

Other Non-OECD Asia 12.5 26.3 26.1 29.6 34.4 39.5

Middle East 11.2 22.7 23.8 27.7 30.3 32.2

Africa 9.5 14.5 14.5 16.2 17.7 19.1

Central and South America 14.5 23.4 24.2 28.3 30.3 32.5

Brazil 5.8 9.4 9.6 11.4 12.9 14.5

Other Central and South America 8.8 14.0 14.6 17.0 17.5 18.0

Total Non-OECD 149.9 220.7 230.8 265.4 299.1 334.4

Total World 347.7 462.1 472.4 508.3 551.5 595.7

Region/Country History Projections

1990 2005 2006 2010 2015 2020

OECD

OECD North America 366 433 438 455 478 500

United States 254 297 300 311 327 343 Canada 28 32 33 34 35 37 Mexico 84 104 105 110 116 121 OECD Europe 497 536 538 547 555 561 OECD Asia 187 200 201 202 203 202 Japan 124 128 128 128 127 124 South Korea 43 48 48 49 49 49 Australia/New Zealand 20 24 25 26 27 28 Total OECD 1,050 1,169 1,176 1,204 1,235 1,262 Non-OECD

Non-OECD Europe and Eurasia 348 342 342 340 337 333

Russia 149 144 143 140 136 132

Other 200 198 198 199 200 201

Non-OECD Asia 2,760 3,431 3,471 3,631 3,826 4,007

China 1,149 1,313 1,321 1,352 1,389 1,421

India 860 1,134 1,152 1,220 1,303 1,379

Other Non-OECD Asia 751 984 999 1,060 1,135 1,206

Middle East 137 193 197 213 234 255

Africa 637 922 944 1,032 1,149 1,271

Central and South America 360 454 460 483 512 539

Brazil 150 187 189 199 210 220

Other Central and South America 211 267 270 284 302 319

Total Non-OECD 4,243 5,342 5,413 5,699 6,058 6,405

Total World 5,293 6,512 6,590 6,903 7,293 7,667

Figure 1.2: In scale sketch of the size of the sun compared to the size of Earth.

Due to scattering, absorption and reflection the amount of energy that reaches the Earth’s surface is slightly less. Nevertheless, the solar energy incident on the surface exceeds the total global energy consumption by more than a factor 10 000. Fig. 1.3

illustrates how much area would be required for solar cells with an efficiency of 10% to produce an yearly amount of energy equal to the total global consumption in 2006, approximately730 × 103km2. Although the area is huge, larger than the area of France

with674 × 103km2, it makes up only a small fraction of the Earth’s surface.

This perspective reduces the concept of the energy crisis to a technical challenge. To reach a sustainable future we do not need a leap of genius, or a scientific breakthrough. The technology is available, the resources are within reach. All we need to do is fill the square!

1.2.

H

ISTORY OF SOLAR CELL TECHNOLOGY

A solar cell is a device that converts solar or other electromagnetic radiation (i.e. light) directly into electric power. The technological history of solar cells began with the discovery of the photovoltaic effect in 1839 by Alexandre-Edmond Becquerel, a French scientist [5]. The photovoltaic effect refers to the observation of the generation of a potential difference at the junction of two different materials in response to visible or other radiation. Practically all solar cell technologies are based on this principle.

Figure 1.3: Rough estimate of the area required for solar cells with 10% efficiency to produce as much energy as the total global energy consumption 2006.

[6] in 1877 and an early experimental solar cell was built by Fritz[7] in 1883,3the effect remained unexplained until 1905, when Einstein published its Nobel prize winning ar-ticle On a Heuristic Viewpoint Concerning the Production and Transformation of Light [8].

Up until the development of transistors the photovoltaic effect remained a scientific curiosity. The modern era of photovoltaics dawned with the development (or rather discovery, as the researchers learned by accident that pn junction diodes demonstrated the photovoltaic effect) of the silicon solar cell by Bell Labs in 1954 [9]. It demonstrated an initial efficiency of 6% and aroused considerable interest on the prospects of power generation from photovoltaics. Within the decade an efficiency of 10% was reached [10]. However, despite this initial success, the technology for preparation of silicon ingots was not yet at sufficient level to enable a large scale industry for photovoltaics [11]. The application of solar cells was restricted to space until the early 1970’s.

In 1973 the oil crisis shocked the world. Aside the political and economic effects, the perception of how strongly the industrial world depends on fossil fuels grew globally and resulted in a political, economical and environmental demand for alternative sources of energy. As a result photovoltaics research and development institutions were set up around the world and the use of photovoltaics for terrestrial power supplies became widespread [11].

In the eighties and nineties the solar cell industry began to mature. In the USA, Japan, and Europe manufacturing lines for the production of crystalline silicon solar

Figure 1.4: Mapping of the first, second and third generation photovoltaics with respect to conversion effi-ciency and cost per area. (Adapted with permission from [12])

(c-Si ) modules were built. Different solar cell technologies emerged from research labs and entered pilot production, including "second generation" photovoltaics, i.e. thin film technologies. The first generation of solar cells is characterized by a conversion efficiency of over 12%, but rather high cost per area of over $200 m−2as can be seen in Fig. 1.4. The thin film solar cells offer the benefit of very low cost per area, at the ex-pense of lower conversion efficiency. The cost per Watt are slightly lower for the second generation of solar cells. Notable thin film technologies are hydrogenated amorphous silicon(a-Si:H), cadmium telluride (CdTe) and copper-indium-diselenide (CuISe2).

Due to lack of government and private investments in the early eighties large solar cell companies were forced to cut on their research budget. Nevertheless, the market for photovoltaics grew exponentially throughout these two decades.4

In the first decade of the 21stcentury an important trend in photovoltaics research is the investigation of new concepts that would fall under the category of 3r d generation photovoltaics, which is illustrated in Fig. 1.4. These photovoltaic devices promise both high conversion efficiency and low price per square meter. Today the most mature third generation technology is the multi-junction thin film silicon technology. Several facilities around the world manufacture this type of solar cells.

From a perspective of the solar cell market the German restructuring of the feed-in tariff for renewable electricity was a major milestone. The feed-in tariff is a subsidy construction where the owner of a solar cell project (e.g. a solar farm or a homeowner with a couple of modules on his roof) receives a guaranteed price for the energy he

delivers to the grid. Although the system existed in Germany since 1990, it only became a major success after the restructuring of the feed-in tariff law in 2000. From 2000 the feed-in tariff was methodologically based on the cost of generation from renewable energy sources. Additionally the feed-in tariff was guaranteed for a period of 20 years. Due to these guarantees it became profitable for investors to invest in large scale solar farms. The German feed in tariff became a big success and other countries like Spain (2008) enacted a feed-in tariff system based on the German policy framework.

In 2010 the global photovoltaic market reached more than 17 GW, while the cumu-lative installed PV power at the end of that year exceeded 37 GW. In 2011 the market grew by 60%, reaching 27 GW. Experts believe the PV market can grow to reach 40 GW in 2013 [13–15].

1.3.

C

RYSTALLINE SILICON SOLAR CELLS

The dominant solar cell technology on the market is based on crystalline silicon. Its market share in 2010 amounted to approximately 90% . This technology is the most mature technology as it has been in the market since 1960.

The operation principle of c-Si solar cells is based on the photovoltaic effect. The photovoltaic effect is the generation of a potential difference over a junction of two materials in response to radiation. In case of c-Si solar cells the junction of two mate-rials consists of p-type and n-type silicon, i.e. a pn-junction. The pn-junction converts light into electricity by three operations on charge carriers, detailed below: generation, separation and collection.

Illumination of the silicon induces photo-excitation, i.e. generation of electron-hole pairs. The incident photon is absorbed and its energy is transferred to an electron in the valence band, which is consequently excited to a state in the conduction band. The "hole" is the vacany it leaves in the valence band.

In c-Si these electron-hole pairs have a lifetime in the order of 1 ms before they recombine. During their lifetime they diffuse through the material. Upon reaching the p-n junction, the electric field of the junction seperates the electron and hole spatially.

The holes are collected at the p-side contact and the electrons at the n-side contact, both contributing to the electrical current through the external circuit.

A schematic structure of a c-Si solar cell is depicted in Fig. 1.5. The solar cell is prepared from a wafer. This is typically a p-type wafer, i.e. a wafer doped with accep-tor dopants, such as Boron atoms. Subsequently, donor dopants, such as Phosphorous atoms are applied by a diffusion process. The P-dopants diffuse up to a certain depth into the wafer as such forming a (thin) n-type top layer. Next, the anti-reflective win-dow layer is deposited on top of the n-type layer. This layer is often silicon nitride, deposited by chemical vapour deposition. Finally the metal contacts are deposited. The back sheet is usually fully covered with aluminum, while at the front side the H-pattern is printed with silver paste to minimize shadow losses.

Crystalline silicon solar cells have relatively high conversion efficiency, up to 25% [16], however, they are also relatively expensive to manufacture. One of the reasons is that the base material, i.e. the wafer, is expensive. Preparing silicon with high purity

N-type c-Si

P-type c-Si

Metal back contact

Anti-reflective coating

Figure 1.5: Schematic structure of a PN c-Si solar cell

(> 99.9%at) is costly. In addition, the wafers are prepared from a molten silicon at

temperature exceeding1400 ◦C, which requires a lot of energy. One of the ways to reduce the cost of crystalline silicon solar cells is to use thinner wafers, but the material is brittle become increasingly difficult to handle when made thinner.

1.4.

T

HIN

-

FILM SILICON SOLAR CELLS

The interest in thin film silicon was aroused in 1970’s by the discovery that amorphous silicon exhibits semiconducting properties [17] and can be doped [18,19]. In 1976 Carlson and Wronski reported on the first solar cell based on amorphous silicon with a conversion efficiency of 2.4% [20]. They had discovered that films of amorphous silicon produced by glow-discharge have superior electrical properties compared to amorphous silicon films prepared by, for example, evaporation.5

This discovery received a lot of attention as enabled manufacturing of a-Si:H de-vices, more specifically solar cells. Thin film silicon solar cells have the potential to be a low-cost alternative for crystalline silicon solar cells. They are thin ( 200nm) com-pared to crystalline silicon wafers ( 300µm) and therefore require less raw materials. In addition, they require less energy as they are processed at low temperature (150◦_C-300 ◦_{C). With low cost for energy and raw materials, a-Si:H solar cells can be manufactured}

cheaply.

However, a-Si:H solar cells have lower efficiency than crystalline silicon solar cells. Even though a-Si:H features semiconducting properties, designing an efficient device is challenging. Due to the disordered atomic structure (see Fig. 2.1) , a-Si:H has a much lower charge carrier lifetime than crystalline silicon. In addition, the mobility of the charge carriers is lower. Consequently, a solar cell design as depicted in Fig.

5_{Glow discharge is deposition of films from a plasma (which has a typical glow), nowadays the term plasma}

Al (300 nm) ~ ~ ~ ~ ~ ~ ~ ~ Glass (0.3 mm) Front TCO (650 nm) ~_~ ~ ~ Intrinsic a-Si:H (300 nm) P-type a-Si:H (10 nm) N-type a-Si:H (20 nm)

Figure 1.6: Schematic structure of a a-Si:H solar cell

1.5is ineffective for a-Si:H. Most of the charge carriers generated by photo-excitation would recombine long before they reach the p-n junction. Alternatively, if the p-layer thickness would be reduced to match the diffusion length,6the absorption would be too low effectively absorb photons and generate charge carriers.

To overcome this problem, an intrinsic absorber layer is placed in between the p-n junction of a-Si:H solar cells, forming a p-i-n structure, see Fig. 1.6. The absorber layer is of intrinsic (undoped) a-Si:H as the charge carrier lifetime of intrinsic a-Si:H is much higher than that of doped a-Si:H. In this design, the absorber layer is placed inside the electric field of the p-n junction. Consequently, photogenerated charge cariers are driven towards the doped layers by the electrical force. Electrons drift to the n-layer and holes to the p-layer. That is the reason why a-Si:H are refered to as drift devices, whereas c-Si solar cells are diffusion devices.

Thin film silicon solar cells require a substrate for mechanical support, unlike c-Si solar cells that acquire their structure from the crystalline wafer itself. Usually glass is used, which is cheap, rigid and highly transparant. A transparant conduction oxide (TCO) is deposited on the glass, which serves as the front contact, instead of an metal H-pattern. The TCO is required as the conductivity of the doped a-Si:H layers is too low to conduct the generated current. Next the active a-Si:H layers are deposited in p-i-n order. Finally the metal back contact is deposited.

Solar cells with a structure similar to above can reach an efficiency of 11% initially. However, a drawback of a-Si:H is the fact that the material is not stable when exposed

to light. a-Si:H exhibits the Staebler-Wronski effect [21]. The Staebler-Wronski effects is the light-induced changes in the dark- and photoconductivity. The effect is caused by light-induced generation of defects within a-Si:H. For solar cells the Staebler-Wronski effect results in a degrading efficiency. The efficiency can drop 5%-20% with respect to initial efficiency (relatively).

1.5.

M

ULTI

-

JUNCTION DEVICES

A fundamental loss factor in photovoltaic conversion of electromagnetic radiation is thermalization. To generate an electron hole pair, a photon must be absorbed with sufficient energy to excite an electron from the valence band into the conduction band. When a photon’s energy exceeds this required energy, the excess energy is lost as heat. This is called thermalization.

Multi-junction solar cells can reduce the thermalization losses. By using materials with different band gaps,7the solar spectrum can be used more effectively. For instance, a tandem configuration consists of a high band gap top cell and a low band gap bottom cell. The high band gap top cell absorbs high energy photons, but is transparent for low energy photons. Due to the high band gap, the thermalization is minimal. In the low band gap bottom cell, thermalization of high energy photons would be high, however, most high energy photons are allready absorbed by the top cell. The remaining low energy photons are absorbed by the bottom cell.

Multi-junction structures are commonly applied in thin film silicon devices. a-Si:H solar cells are used at the top of the multi-junction device, as it has a high band gap ( 1.7-1.8 eV). a-Si:H-alloys, such as hydrogenated amorphous silicon-germanium (a-SiGe:H, 1.3-1.4 eV) or hydrogenated microcrystaline silicon (µc-Si:H, 1.1-1.2 eV), are often used for the bottom cell. µc-Si:H is a mixed phase material, consisting of small ( 50-500 nm) crystalline particles embedded in an amorphous matrix. Using a triple juction (a-Si:H-_µc-Si:H-_µc-Si:H) configuration an efficiency of 13% was reached [22].

1.6.

M

ODELING OF A

-S

I

:H

SOLAR CELLS

The complexity of the optical and electrical interaction between the layers in single-and multi-junction a-Si:H solar cells is so high that analysis of these solar cells without the aid of a computer is difficult. For this reason computer modeling assumes an im-portant role in studying and designing these solar cells. Imim-portant physical features that state-of-the-art simulators must have include: optical simulation of textured interfaces; advanced recombination models; and models for tunnel recombination junctions. Ad-ditionally, technical features like numerical stability, computational speed, and choice of a sufficient large stack of layers are essential. The electro-optical device simulator ASA(Advanced Semiconductor Analysis) supports all the above features.

The ASA program is designed for the simulation of devices based on amorphous and

7_{For now: energy offset between valence and conduction band. In chapter 2 the band gap is defined more}

Multiple layers

Band discontinuities in the CB and VB Large band gap materials: Eg >2.0 - 3.7 eV

Grading of material parameters

Recombination and charge in the localized states

Simulation of non-routine measurements: J(V), QE, C(V), ... as a function of T Fast and easy to use

Table 1.3: Model requirements for a thin-film solar cell simulation program

crystalline semiconductors. The ASA program solves the basic semiconductor equa-tions in one dimension (the Poisson equation and two continuity equaequa-tions for electrons and holes) and uses the free electron concentration, the hole concentration, and the elec-trostatic potential, as variables. Further it uses several advanced physical models which describe specific device operation and material opto-electronic properties.

ASAis highly suitable for the simulation of (thin-film) silicon solar cells. The gram meets the standard requirements for a suitable thin-film solar cell simulation pro-gram which are listed in table1.3[23,24]. Modeling of thin-film silicon devices re-quires one to take the electronic structure of a-Si:H andµc-Si:H into account. The spa-tial disorder in the atomic structure of a-Si:H results in a continuous density of states (DOS) in the energy band gap with no well defined conduction-band (CB) and valence-band (VB) edges. When considering the transport properties of charge carriers in a-Si:H one has to distinguish between the extended states and the localized states in the DOS distribution.

The localized states within the mobility gap strongly influence the trapping and recombination processes and therefore the trapped charge in the localized states cannot be ignored as is often the case in modeling of crystalline semiconductor devices. The localized states in the mobility gap of a-Si:H can be of different nature which requires different approaches for the calculation of recombination-generation (R-G) statistics through these states. The models that are commonly used to describe the localized states in a-Si:H and their corresponding R-G statistics are described in detail in [25]. The Shockley-Read-Hall (SRH) recombination through the states introduced by dopants and/or impurities is negligible in a-Si:H compared to the recombination via the tail states or dangling bond states and therefore is not used for amorphous films in device structures. However, the ASA program allows the SRH recombination based on the carrier lifetimes to be used for crystalline materials.

From the optical point of view, both the efficient use of the solar spectrum and the light management inside solar cells are important to obtain high conversion efficiencies. In today’s thin-film solar cells light management is accomplished by implementing light trapping techniques. The light trapping techniques are based on the introduction of surface-textured substrates and the use of special reflector layers. The surface-textured substrates introduce rough interfaces into the solar cell. The incident light is scattered

at rough interfaces and the scattering processes at rough interfaces must be taken into account in order to determine accurately the generation profile of charge carriers inside the solar cell. This requires the development of optical models that take both coherent non-scattered (specular) and incoherent scattered (diffused) light propagation through a device into consideration.

The efficient use of solar spectrum requires a multi-junction approach to thin-film silicon solar cells. The tunneling assisted recombination at an interface between two adjoining junctions is responsible for charge carrier transfer through a multi-junction solar cell. This interface is described as the tunnel-recombination junction (TRJ). There are two approaches that can be used in the ASA program to model TRJ. The Delft approach is based on the trap-assisted tunneling model and enhanced carrier transport in the high-field region of the TRJ. The Pennsylvania approach is based on the introduction of a highly defective layer with strongly reduced band gap at the n/p interface and grading of the mobility gap of the n-layer and p-layer in the regions adjacent to the defective layer.

The main features of the ASA program are summarized:

• Modeling of multilayer amorphous and/or crystalline semiconductor devices

• Optical simulation of scattering at 2-D randomly textured surfaces using a model that takes into account both the specular and the scattered part of the incident light.

• Models describing a complete DOS as function of energy, which include both the extended and localized (tail and defect) states

• Calculation of the defect-states distribution in a-Si:H using the defect-pool mod-els

• Correct recombination-generation statistics for the acceptor- and donor-like states and for ambipolar states

• Change (grading) of all input parameters as a function of position in the device or energy level in the gap

• Model for the tunnel-recombination junction

• Modeling of degradation of a-Si:H solar cells

• High computational speed due to the native ANSI C implementation

• User-friendly graphical interface.

1.7.

M

ATERIAL AND DEVICE CHARACTERIZATION

One general note with respect to material and device characterization is that the interpretation of results is difficult due to:

• the inhomogeneities in the material

• the small thickness of the films (_<200 nm)

• the role of interfaces

• run-to-run variation

• general measurement uncertainty

To correctly interpret results, these factors must be kept in mind.

1.7.1.

R

/ T

A versatile means of analyzing films is using optical techniques. The interaction of light with the film results in scattering, absorption and interference patterns, which are reflected in the reflection and transmission spectra. Scattering, i.e. the propagation and reflection of light in other than the specular direction, at the surface of the film offers information about the interface roughness. Absorption imparts information about the optical band gap of the material. From the interference pattern the thickness of the film can be calculated.

The most important optical measurement technique for films is reflection / trans-mission (R/T) measurement. The R/T data is used to calculate the thickness, the refrac-tive index and the absorption coefficient as a function of the wavelength. In turn the latter is used to derive the optical band gap.

For R/T measurements presented in this thesis, the Lambda 950 (Perkin Elmer) spectrometer is used. It is an advanced multipurpose optical measurement system that has a special accessory, the total integrating sphere, which can be used for the purpose of R/T measurements. The total integrating sphere responds to light that is reflected or transmitted from the sample in any direction. The intensity measured at the specular direction can selectively be substracted from the measurement.

From the R/T data the absorption coefficient as a function of the wavelength is determined. From the absorption coefficient the optical band gap is determined by fitting: a q Ephα¡Eph¢ n ¡Eph ¢ ∝ ¡Eph− Eg¢ (1.1)

whereEph is the photon energy,α¡Eph

¢

is the absorption coefficient andn¡Eph

¢ is the refractive index. Fora = 2 the fitted optical band gap (Eg) is the Tauc optical band

gap [26,27]. The Klazes optical gap uses cubic root (a = 3). The parameter is derived from the assumption that the energetic distribution of the density of states near the band edges is quadratic (a = 2) or linear (a = 3).

In some samples, the absorption coefficient cannot be determined for the entire wavelength range. The reason for this is that, due to interference effects, for part of the wavelength range the absorption was too high to calculate a reliable value forα. In some cases this impedes the calculation of the Tauc band gap.

Therefore within this thesis both the Tauc optical band gap and the Klazes optical band gap are used. Within a series or experiment the same band gap is used consistently. The Klazes optical band gap is typically 0.15 eV lower than the Tauc optical band gap [28].

The R/T measurement technique is useful to determine the absorption coefficient in the wavelength range of 1.8 eV to 3.5 eV. For a-Si:H films we are also interested in the subband gap absorption, i.e. the absorption coefficient in the range of 0.5 eV to 1.8 eV. for this purpose we use the FTPS measurement technique described in section1.7.3.

1.7.2.

F

T

I

Infrared absorption spectroscopy is used to detect and characterize the chemical bonds between silicon and hydrogen atoms in a-Si:H. The vibrational modes of Si-Hx bonds

result in characteristic absorption peaks. In this work we have used:

• the 640 cm−1_{peak, associated with the wagging mode of bonded hydrogen;} • the 1980-2030 cm−1_{, associated with the stretching mode of monohydrides}

lo-cated in divacancies, in this work refered to as low stretching mode (LSM)

• the 2060-2160 cm−1, associated with the stretching mode of mainly monohy-drides located at the surface of nano-sized voids, in this work referred to as high stretching mode (HSM)

The integrated intensity of the peaks relates directly to concentration of the associ-ated bonds:

Nx= Ax

Z α(ω)

ω dω (1.2)

whereα(ω)is the absorption coefficient andAx is a proportionality constant.

The total hydrogen concentration of a-Si:H is determined using the 640 cm−1

wag-ging mode, since all bonded hydrogen contributes equally to the corresponding absorp-tion peak. The proporabsorp-tionality constant for this mode has been calibrated by hydro-gen effusion and nuclear measurement techniques. The value amounts to1.6 × 10−19

-2.1 × 10−19cm−2[29–32]. In this work we take the value of1.6 × 10−19cm−2[33].

Similarly the constants for the LSM and HSM modes have been determined. In this work we use the constants as determined by Smets,ALSM= AH SM= 9.1 × 10−19cm−2

[33]. Infrared absorption spectra were recorded using a Thermo Electron Nicolet 5700 spectrometer. The films measured using this technique were deposited on crystalline silicon wafers, since glass substrates absorb the infrared light.

0 0.5 1 1.5 2 2.5 3 3.5 4 10−4 10−2 100 102 104 106

Photon energy (eV)

α

(cm

−1 )

Figure 1.7: Typical absorption spectrum of a-Si:H from RT and FTPS measurements.

1.7.3.

F

T

P

S

Several methods exist to determine the subband gap absorption of a-Si:H, such as con-stant photocurrent measurement (CPM), dual beam photocurrent spectroscopy (DBP) and Photothermal Deflection Spectroscopy (PDS) [34]. For this work we have applied a recently developed method to measure the subband gap absorption, Fourier Transform Photocurrent Spectroscopy (FTPS) [35].

The measurement setup consists of the same Thermo Elektron Nicolet 5700 spec-trometer as used for the FTIR measurements and an external optical system. The oper-ation of the FTPS is very similar to FTIR spectrometry:

• A light source generates infrared light

• The light is guided through an optical system to a modulator.

• The modulator, consisting of a rigid and a vibrating mirror, modulates the infrared light.

• The modulated light is guided to the sample.

The difference between FTPS and FTIR is that the sample itself is utilized as detector. The absorbed light is detected as a current by the application of a voltage over the sample. The measured current is amplified and demodulated by the spectrometer.

The processing of the measured data involves the application of corrections for the source spectrum, the frequency behavior of the sample and the frequency behavior of the

0.0 0.5 1.0 -150 -100 -50 0 50 C ur re nt d en si ty (A m -2 ) Voltage (V) Jsc voc FF

Figure 1.8: Current-voltage characteristic of a a-Si:H solar cell under illumination

amplifier. Also to improve the signal to noise ratio several optical filters can be applied, in which case the measured data must be merged. The result of the data processing is a absorption coefficient curve in arbitrary units in the photon energy range of 0.8 eV to 1.7 eV. By scaling the curve to the RT data in the overlapping energy range (1.65 eV -1.75 eV) an absolute absorption spectrum in the range of 0.8 eV to 2.5 eV is obtained, as is shown in Fig.1.7.

FTPS has several advantages over DBP, such as high optical-through put and high resolution [35]. Also, the method allows the measurement of subband gap absorption in the absorber layer of solar cells, however, the interpretation of the results is not yet fully understood [36].

1.7.4.

A

In general the conductivity (σ) of a semiconductor is proportional to the electron and hole concentration,nandp:

σ = q ¡µnn + µpp

¢

(1.3)

with_µnandµpthe electron and hole mobilities, respectively andqthe electron charge.

In thermal equilibrium the charge carrier concentrations constitute a balance be-tween thermal generation and recombination. Using the Maxwell-Boltzmann

approxi-mation the charge carrier concentration is described as: n ₌ Nce µ_{E f −Ec} kT ¶ (1.4) p = Nve µ Ev −E f kT ¶ (1.5) whereNv andNcare the effective density of states in the valence and conduction band,

respectively, Ef is the Fermi-level, Ev andEc are the valence and conduction band

mobility edges, respectively,kis Boltzmann’s constant andT is the temperature. The above equations show that the carrier concentration has an exponential relation to the temperature. The activation energy (Ea) is determined by measuring the

conduc-tivity of the sample as a function of the temperature and fitting this relation:

σ(T ) = σ0e ³ −Ea kT ´ (1.6) The activation energy in this equation is used as a measure for the separation between the Fermi-level and the mobility edge corresponding to the majority carriers.

1.7.5.

S

The illuminated current-voltage measurements are carried out under standard condi-tions. The Oriel solar simulator uses a He-Xe arc lamp to illuminate the samples. The light bundle is filtered in order to approximate the AM1.5 spectrum (1000 Wm−2). The voltage is swept from -0.05 V to 1.0 V,8while the current is measured. The measure-ment is performed at room temperature, however, the temperature is not controlled.

Solar cell samples are produced according to the layout depicted in Fig. 1.9. The solar cells are deposited on an Asahi U-type substrate (glass with a TCO layer of SnO2).

An aluminum bar is deposited on the substrate to contact the TCO layer. The a-Si:H layers are deposited on the substrate. Consequently, the solar cell areas are defined by the deposition of metal back contacts. In total 30 solar cell dots are deposited on each sample.

Measurements are performed at maximum speed in order to avoid the effect of heat-ing in response to the intense illumination. The current-voltage measurements are used to determine the external parameters, as indicated in Fig.1.8:

• Open-circuit voltage (Voc) • Short-circuit current (Jsc)

• Fill factor (ff), derived from the maximum power point(mpp):ff =Jmpp_JscV_Vmpp_oc

• Efficiency (η):η =Jmpp_PVmpp

i (withPi the incident power, 1000 Wm −2₎

Figure 1.9: Schematic layout of a typical solar cell sample (not in scale)

Spectral response measurements are carried out using a custom-built setup. This setup uses a halogen light source and a monochromator to illuminate the sample. The current is measured at 0 V or by applying an external voltage over the sample. By calibrating this setup to a reference diode the external quantum efficiency (EQE) is obtained, which is the ratio of collected photo-generated current over the flux of incident photons:

EQE (_{λ) =}JL(λ,V )

qΦ(λ) (1.7)

In case of multi junction solar cells,9bias light is applied. When measuring EQE of the top cell, the bottom cell is made conductive by applying an intense (infra-)red bias light (λ =850 nm). The bias light only affects the bottom cell as the a-Si:H top cell is insensitive to illumination at that wavelength. When measuring the EQE of the bottom cell, the top cell is short-circuited with green bias light (λ =545 nm). The bottom cell is not affected, since over 99% of the illumination at that wavelength is absorbed before it reaches the bottom cell.

1.8.

S

COPE AND

O

UTLINE

As mentioned before, the challenge for massive implementation of photovoltaics lies with the reduction of the price of PV. Two paths lead to cost reduction: increasing the conversion efficiency and lowering the production cost [37]. Thin film silicon solar cells have the potential to fulfill this objective, because they use little raw materials and little energy for their production and consequently can be produced at low cost. The

9_{of which the individual unit cells have distinct band gaps, like micromorph tandem solar cells (top cell a-Si:H}

draw-back of this technology is the low conversion efficiency compared to wafer-based silicon solar cells.

To enhance the performance of a-Si:H solar cells, research efforts are mainly di-rected to improving the short circuit current by designing improved light-trapping schemes [38]. In this work we focus on a different approach to improve the performance i.e. im-proving the electrical characteristics of the solar cell in order to increase theVoc.

However, despite many theoretical [39,40], experimental [41] and simulation [42,

43] studies, a firm theoretical understanding of theVoc still lacks. In chapter 2 the

theoretic limits of theVocand the efficiency of a-Si:H based solar cells are studied by a

thorough analysis of recombination processes.

Simulations facilitate further understanding of the behavior of a-Si:H and a-Si:H devices. Simulations allow studying the behavior of physical parameters inside devices that are difficult to measure experimentally, such as the concentration gradient of charge carriers throughout the device. In second part of chapter2and in chapter3simulations are employed to study recombination processes and the effect of p-i interface parameters on the charge carrier concentration profile, respectively.

We observe a gap between theVoclimits derived in chapter2and theVocmeasured

in practical solar cells. This may largely be explained by the fact that practical devices are not as ideal as the model used in chapter2assumes. In practice theVocis limited by

the doped layers. Also, simulations and experiments [41,44–46] have shown that the

Vocof a-Si:H solar cells is sensitive to the p-i interface region, due to high recombination

at the heterogeneous interface.

One strategy to reduce recombination in the p-i region is optimizing the material properties of the p- and intrinsic layer. Hydrogenated amorphous silicon carbide (a-SiC:H), fluorized microcrystalline silicon p-layers, and other materials have been in-vestigated as p-type window layers [39,47,48].

Another common strategy to enhance theVoc is the use of a thin, wide band gap

p-i interface layer. Implementing wide band gap a-Si:H or a-SiC:H as interface layer results in an enhancedVoc[41,42,44,46,49]. In chapter3simulations are employed to

study the p-i interface region and the effect of implementing a p-i interface layer. The results are used in the second part of chapter3, to study different materials as candidates to improve the p-i interface layer.10

In chapter 4 the experimental results of an optimization study are presented. The a-Si:H material is optimized for use in a flexible micromorph tandem solar cell. The experimental results are discussed in relation to the continuous random network model and a novel model: the disordered network with hydrogenated vacancies.

1.9.

C

ONTRIBUTION TO THE RESEARCH FIELD

The work described in this thesis has contributed with the following new results to the research field:

10_{Although implementation of a p-i interface layer helps to improve the initial efficiency, several groups have}

reported a stronger degradation during light exposure [49]. The scope of this chapter is limited to study the effect of interface layers on solar cells before light degradation.

• A comprehensive analysis is presented of the recombination losses in a-Si:H and the effect these losses have on the open circuit voltage and the efficiency limit of photovoltaic devices.

• This analysis is applied to calculate the optimal band gap combination for tandem solar cells

• Simulations are applied to demonstrate that open circuit voltage of a-Si:H solar cells is sensitive to the properties of the p-i interface because the p-i interface features a large concentration of photo-generated electrons and, as a result, a high recombination rate.

• This thesis demonstrates how modeling can be applied to aid the process of de-signing improved solar cells. However, the process involves extensive experi-mental work to search for materials with appropriate properties.

• We show that a-SiN:H has a good potential for use as a p-i interface layer in solar cells. However, the nitride atoms act as n-type doping, which interferes with the built-in voltage of the cell.

• The samples deposited with a hydrogen dilution ratio ofR = 10have lower Ur-bach energy than those deposited with a hydrogen dilution ratio of R = 0 and

R = 20.

• Experimental results on the relation between band gap and Urbach energy versus the concentration of hydrogen in LSM (associated with divacancies), are well explained in terms of the disordered network with hydrogenated vacancies model.

• Decreasing the deposition temperature from 180◦_{C to 160}◦_{C increases the}

effi-ciency of a-Si:H solar cells deposited from a hydrogen diluted (R = 10) silane source gas by 5% relatively.

2

P

ERFORMANCE LIMITS OF

A

-S

I

:H

2.1.

I

NTRODUCTION

Ever since the report on the first a-Si:H based solar cell in 1976 [19] scientists have studied methods on how to improve the performance of such solar cells, out of human curiosity, but especially, because they believe in the potential of a-Si:H solar cells for large scale generation of solar electricity. The first question that comes to a scientists’ mind is, is how big is this potential? How much can we improve this solar cell? What is the performance limit of these devices?

In this chapter a systematic approach is applied to examine this question from a theoretical viewpoint.

2.1.1.

T

In 1955, quickly after the report on the first silicon based solar cell, the first report on the efficiency limit was published [50]. The article studies the relation between the band gap of a semiconductor and the maximum possible efficiency of a solar cell based on that material. Until 1960 several review papers appeared in literature [51,52]. These papers described theoretical efficiencies, however they were founded on assumptions derived from experimental results.

A different line of thinking was introduced by Shockley and Queisser [53]. They proposed the detailed balance limit,1which is a theoretical limit that is a consequence of, in their words, the nature of atomic processes required by the basic laws of physics. In principle they consider a solar cell as a heat engine and apply the fundamental laws of thermodynamics to it. Würfel provides a clear, descriptive elaboration on how the laws of thermodynamics impose a limit on the efficiency of a photovoltaic heat engine

[54]. The detailed balance limit, popularly known as the Shockley-Queisser limit, in fact describes the curve of maximal achievable solar cell efficiency versus the band gap, but is often cited as 30%, the efficiency limit for semiconductors with a band gap of 1.1 eV, which is also the maximum efficiency in this curve.

For solar cell types such as monocrystalline silicon and GaAs the record efficien-cies (25% and 26.1% respectively) approach the detailed balance limit closely. How-ever, a-Si:H based solar cells cannot achieve a performance close to the detailed bal-ance limit. The amorphous nature of the material results in a continuous destribution of states throughout the band gap of the material. As a consequence different recom-bination processes take place which impose constraints on the conversion efficiency. The next section gives a brief description of the distribution of states and the different recombination processes.

Finding a theoretical description on how the amorphous nature of a-Si:H affects the maximum achievable efficiency for photovoltaic devices based on this material proves difficult. To approach this problem the efficiency (η) is often studied by splitting it up into parameters:

•The short circuit current density (Jsc).

•The open circuit voltage (Voc).

•The fill factor (ff).

 



η = JscVocf f

Pi HerePi is the incident

ir-radiance (i.e. the power of the incident light).

Optimization of the short circuit current density is a technical issue. In theory it is possible to have each photon with energy above the band gap converted to an electron hole pair. In practice there are optical losses, but there is no fundamental reason why these losses cannot be minimized.

The next parameter that determines the efficiency is the open circuit voltage. This parameter is studied extensively theoretically [39,40], by experiments [41] and by sim-ulations [42,43]. Despite this effort the relation between structural and material prop-erties of a-Si:H solar cells and the open circuit voltage is still not well understood.

The works of Tiedje [39] and Lagos et al. [40] study the relation between theVocof

a-Si:H solar cells and the recombination through band tail states. They both consider the "lumped circuit" limit.2Tiedje uses non-equilibrium steady-state statistics [55] to derive expressions for recombination rate as a function of the electron and hole quasi-Fermi levels. These equations introduce an integration constant and can be solved numerically. For the case where the recombination rate through conduction band tails is negligible, an approximate solution for the quasi-Fermi level splitting is obtained, which is used to determine the performance limit. The calculated efficiency limit amounts to16 %3.

Lagos et al. provide a general theoretical framework called "electron-hole kinetics", which deals with an arbitrary distribution of states within the band gap. For a material with symmetric conduction and valence band tails, they derive an equation for the cur-rent density as a function of the voltage. However, this equations can only be solved numerically, resulting in a efficiency limit of12 %.4

2_{In section2.3.3the lumped circuit model is discussed} 3_{Based on an short-circuit current of181 Am}−2

Tiedje and Lagos et al. both examined the recombination rate in a-Si:H and how these steady state statistics impose a constraint on the maximum efficiency. Schiff, instead, studies the dynamics and transport processes within the structure of an a-Si:H device [56]. His work concludes that the low mobility of holes in a-Si:H is the limiting factor to the performance of photovoltaic devices based on this material. He proposes a model that can be applied to photovoltaic devices based on low mobility semiconductors to calculate a maximal performance.

In this chapter the performance of a-Si:H solar cells is studied in relation to the recombination rate. This approach is in line with the work of Tiedje and Lagos et al. as we also employ the lumped circuit model. However our approach starts off by con-sidering only direct recombination, so that we obtain the Shockley-Queisser limit and than systematically increase the complexity of the model by considering the different forms of recombination that occur in a-Si:H, direct recombination, Auger recombina-tion, recombination through valence and conduction band tail states and recombination through defect states..

2.2.

S

TRUCTURE OF

-

AND RECOMBINATION IN A

-S

I

:H

2.2.1.

D

-S

:H

Contrary to crystalline silicon, where the silicon atoms are perfectly ordered in a pe-riodically repeating crystal lattice, as shown in Fig. 2.1a, the lattice of a-Si:H is not perfectly repeating over long ranges. The atomic network features small variations in the bond-length and orientation. In this lattice defects are present, where an atom is only covalently bonded to 3 instead of 4 neighbouring atoms. The fourth bonding electron is not bonded to a neighbouring atom and forms a so-called "dangling bond". 56 The differences between atomic structure of c-Si and a-Si:H are illustrated in Fig.2.1.

The periodic structure of c-Si yields an energy band of free electron states, the con-duction band, and an energy band of free hole states, the valence band. The concon-duction band states and valence band states are called the extended states. The electron and hole wavefunctions in these bands are non-localized, i.e. the wavefunctions extend over the crystal lattice. Consequently, charge carriers (electrons or holes respectively) in these states have high mobility and can be considered free charge carriers. The seperation between the bottom of the conduction band and he top of the valence band is the band gap, i.e. an energy band that does not contain energy states that can be occupied by charge carriers. For c-Si the width of the valence band is 1.12 eV.

In a-Si:H the same three energy regions (i.e. valence band, conduction band and band gap) can be distinguished as for c-Si. However, besides the extended states of the valence band and conduction band, the disordered lattice of a-Si:H gives rise to states whose wavefunctions are localized. The variations in bond length and -angle alter the wavefunctions of electrons in proximity to these bonds. The result is that the edges of the valence- and conduction band are no longer sharply defined. Instead these

5_{Chapter 4 discussed defects in a-Si:H in more detail}

6_{In hydrogenated amorphous silicon (as opposed to amorphous silicon) many of these dangling bonds are}

Figure 2.1: a) The perfectly ordered crystal lattice of c-Si b) The disordered atomic network of a-Si:H (used with permission from [57]

Ex tended v alenc e band sta tes Ex tended c onduc tion band sta tes Tail sta tes Tail sta tes Defect states E log(N DOS )

Figure 2.2: Density of states diagram of a-Si:H

bands feature band tails that exponentially decline into the band gap as depicted in Fig.

2.2. In addition the defects in the a-Si:H lattice introduce defect states, with localized wavefunctions around the defect. The energies associated with these states lie deep within the band gap.

As a consequence of both the band tail states and the defect states, a-Si:H features a continuous density of states throughout the band gap. Nevertheless, the density of states in the middle of the band gap is about 5-7 orders of magnitude lower than the density of states in the valence and conduction band. Besides, the the band tail and defect states are localized. Consequently the mobility charge carriers occupying these states is 2-4 orders of magnitude lower than the mobility of charge carriers in the extended states. This is called the mobility band gap.

2.2.2.

R

Recombination is the transition of an electron from a state in the conduction band into an empty valence band state associated with a hole. Both free charge carriers (i.e. the

E

a)

b)

c)

E

CB

E

VB

E

T

Figure 2.3: Charge carrier recombination paths in semiconductors: a) direct recombination, b) trap-assisted recombination, c) Auger recombination.

free electrons and holes) are eliminated in the process. The energy difference between the initial and final state of the electron is released in the form of a photon (radiative recombination), one or more phonons (vibrations in the atomic lattice) or in the form of kinetic energy that is transferred to another electron.

The different forms in which the energy is released allow for the distinction be-tween radiative and non-radiative recombination. The difference is significant since emitted photons arising from radiative recombination have sufficient energy to gener-ate an electron-hole pair when they are absorbed again. Non-radiative recombination generally leads to thermalization. The energy is transformed into heat, i.e. in terms of photovoltaics the energy is lost, it cannot be converted to electricity.

The recombination processes can also be classified by considering the individual charge carriers and energy states involved. Fig.2.3illustrates this classification.

Direct recombination is typically a radiative process in which the potential energy of the electron in the conduction band is released in the form of a photon.

In the trap-assisted recombination process an energy state, or "trap-state", within the band gap aids the recombination process. For example a hole from the valence band gets trapped in the trap-state within the band gap. The trapped hole then recombines with an electron from the conduction band. The released energy can by either radiative or non-radiative. However, in the radiative case the release photon has an energy lower than the band gap and cannot be reabsorbed.

The Auger recombination process involves a third particle besides the free electron and hole that recombine. The energy released during the recombination process is trans-ferred to a third particle, which can be either an electron or a hole. Typically, the third particle thermalizes before it is able to ionize an electron from the valence band into the conduction band.

2.3.

M

ETHOD

In this chapter the open circuit voltage and the performance of solar cells, in particular solar cells based on a-Si:H, are systematically studied in relation to different recombi-nation processes. We start with the simplest recombirecombi-nation, direct recombirecombi-nation, and systematically add more complex recombination processes until our model adequately represents the recombination processes in a-Si:H.7Our approach is to analyse the re-combination rate using statistical methods and comparing it to simulations.

The statistics of recombination processes in actual solar cell devices is too com-plex to study with an analytical model, as most parameters vary throughout the device. Therefore we depart from the following simplifying assumptions, which are detailed later in this chapter:

• Spatial dependence of parameters is disregarded by using the lumped circuit model as proposed by Tiedje [39] and Lagos et al.

• The quantum efficiency is 1, meaning that all photons with energy larger than the band gap are absorbed, in line with the assumptions of Shockley and Queisser [53]

• The principle of detailed balance, also applied by Shockley and Queisser. [40] This chapter starts out with the analysis of the simplest form of recombination, ra-diative recombination. By mathematical analysis the relation between rara-diative recom-bination and the external parameters of solar cells is described, which is in line with the work of Shockley and Queisser. Simulations are used to verify these results. The similarities and difference between this work and theirs is discussed.

In the next step we add Auger recombination to the model, and study how the exter-nal parameters and the maximum performance of solar cells is effected by this process. In the third step we add recombination through valence band tail states to the model. Recombination through valence band tail states is expected to be dominant over recom-bination through the condunction band tail states, as experiments have indicated that the valence band tail is broader and contains more localized states than the conduction band tail. It is logical to assume that recombination through valence band tail states has more influence on the external parameters of a solar cell than recombination through conduction band tail states.

By mathematical analysis a closed form expression for the recombination rate through valence band tail states is obtained, which is to our knowledge a premier in literature. The effect of the recombination through valence band tail states on the external param-eters is analyzed. The results are verified by simulations.

In the last two sections conduction band tail state recombination and defect recombi-nation are added to the model. Mathematical analysis of these two recombirecombi-nation paths are given. However, we were unable to find a similar closed form expression for these recombination paths as for recombination through valence band tail states alone. With

simulations the effects of the recombination through conduction band tail and defect states on the external parameters of solar cell are studied.

2.3.1.

G

An important process in the operation of solar cells is the absorption of light and the consequent generation of charge carriers in the absorber layer of the solar cell. The amount of absorption depends on the optical characteristics of the solar cell, the ab-sorption coefficient of the absorber layer, the thickness of the absorber layer and the intensity of incident light. Because the focus of this study lies in investigating the elec-trical properties of solar cells, a simple model of the absorption of incident light and the consequent generation of charge carriers is applied for evaluation of the performance of the solar cell: total absorption.

The total absorption model is used by Shockley and Queisser [53] and assumes that all photons with energy above the band gap are absorbed, i.e. the optical quantum efficiency is a stepwise function of the photon energy. This model neglects multiple excitation, where a single high energy photon generates multiple electron and hole pairs, and sub-band-gap absorption. It serves as an upper limit for the absorption in a solar cell. In the rest of this chapter the total absorption model is used unless otherwise mentioned.

Practical solar cells have optical characteristics that are far from optimal. Many light trapping techniques, like anti-reflective front side coating, rough transparent front-contacts that scatter the incident light and back reflectors, are implemented. But still a part of the light is reflected before it enters the solar cell or leaves the solar cell without having been absorbed. Also part of the light is absorbed outside of the absorber layer, where it does not contribute to the generation of charge carriers.

Unlike Shockley and Queisser,8 the irradiation source used in this article has an AM1.5 spectrum with a total power density of1000 W m−2_.9 _{Both AM0 and AM1.5}

spectra are illustrated in Fig.2.4together with the irradiation spectrum of a black body at 6000 K.

The incident solar irradiation is described by the incident (spectral) photon flux densityΦi, the amount of incident photons (per wavelength) per square meter. The part

of the photon flux that is absorbed within the absorber layer is described by the absorbed photon flux densityΦa:

Φa⊂ Φi (2.1)

The absorbed photon flux relates to the absorption rate Aby integration over the thickness of the absorber layerd:

Φa=

Z d

0

A(x)d x (2.2)

8_{Shockley and Queisser use a black-body irradiation source at a temperature of 6000K}

9_{Air-mass 1.5, means that this is the spectrum of solar radiation that has passed through a layer of air}

corre-sponding to 1.5 times the thickness of the earth’s atmosphere. This is representative for the spectrum of solar irradiation measured on earth at 60olattitude. AM0 refers to the solar spectrum outside earth’s atmosphere.

0 5 0 0 1 0 0 0 1 5 0 0 2 0 0 0 2 5 0 0 3 0 0 0 3 5 0 0 4 0 0 0 0 1 x 1 0 2 7 2 x 1 0 2 7 3 x 1 0 2 7 4 x 1 0 2 7 5 x 1 0 2 7 6 x 1 0 2 7 Ph ot on F lu x De ns ity (m -3 s -1 ) W a v e l e n g t h ( n m ) A M 1 . 5 A M 0 B l a c k b o d y a t 6 0 0 0 K

Figure 2.4: The 6000 K blackbody spectrum, the AM1.5 spectrum and the AM0 spectrum

The quantum efficiency (ηg), i.e. the amount of charge carriers generated by an

absorbed photon, relates the generation rate of charge carriers to the absorption rate. High energy photons can in principle generate multiple electron hole pairs, however, in this article multiple excitation is neglected. Therefore the internal quantum efficiency is unity:

ηg= 1 ⇒ G = A (2.3)

2.3.2.

D

In the many different fields of physics, including mechanics, dynamics, quantum physics, particle physics, we observe that all processes are reversible. That means that, if in a conservative system all particles are reversed with equal velocity, the system will retrace their former paths, reversing the entire succession of configurations.10 A consequence of this principle is that in the long run each kind of process occurs with equal probability as its reverse [58].

For molecular systems the same principle of dynamical reversibility holds. If a con-servative molecular system has three states, as depicted in Fig. 2.5a), the reversibility principle must be applied to each transition. This is called the principle of detailed balance [58–60]. Fig. 2.5b) shows a conservative system in an equilibrium, which

10_{a conservative system is a system in which the number of particles remains constant, i.e. no particles go in}