Transient effects during laser processing of silicon for photovoltaic applications

(1)

Fabian Stefan Meyer | TRANSIENT EFFECTS DURING LASER ...

SOLAR ENERGY AND SYSTEMS RESEARCH

FRAUNHOFER VERLAG

Fabian Stefan Meyer

TRANSIENT EFFECTS DURING LASER PROCESSING OF SILICON FOR PHOTOVOLTAIC APPLICATIONS

Intense laser pulses deliver fine doses of highly concentrated energy to modify solid targets in an exceptionally well controllable manner. Two applications of this are treated herein: »laser annealing« as an approach to transfer amorphous silicon into the crystalline state and »laser ablation« to create local openings in dielectric layers for subsequent contacting. Both can be used in manufacturing of silicon solar cells as a cost-effective and environmentally friendly alternative to thermal and chemical processing.

This work presents implementations of the pump-probe approach which enable the observation of the surface during and shortly after the arrival of a laser pulse. With an automated opto- mechanical setup concurrent physical processes happening in the material are made visible on the femtosecond to nanosecond scale. Using micro- and nano-characterization, modeling, and simulations, this work shows how the absorption of intense laser pulses leads to phase transfor- mations and material removal. It demonstrates that time-resolved observations are essential to investigate the origin of undesired side-effects, such as laser induced damage or incomplete layer removal.

9 7 8 3 8 3 9 6 1 7 4 2 7 ISBN 978-3-8396-1742-7

(2)

Transient effects during laser processing of silicon for photovoltaic applications

Fabian Stefan Meyer

FRAUNHOFER VERLAG

SOLARE ENERGIE- UND SYSTEMFORSCHUNG / SOLAR ENERGY AND SYSTEMS RESEARCH

Fraunhofer Institute for Solar Energy Systems ISE

(3)

79110 Freiburg

Phone +49 761/4588-5150 info@ise.fraunhofer.de www.ise.fraunhofer.de

Cover illustration: © Fraunhofer ISE/Fabian Stefan Meyer

Bibliographic information of the German National Library:

The German National Library has listed this publication in its Deutsche Nationalbibliografie;

detailed bibliographic data is available on the internet at www.dnb.de.

ISSN: 2512-3629 ISBN: 978-3-8396-1742-7

D 25 Zugl.: Freiburg, Univ., Diss., 2021

Book Series: »Solare Energie- und Systemforschung / Solar Energy and Systems Research«

Print and finishing:

Fraunhofer Verlag, Mediendienstleistungen

The book was printed with chlorine- and acid-free paper.

70569 Stuttgart Germany

verlag@fraunhofer.de www.verlag.fraunhofer.de

is a constituent entity of the Fraunhofer-Gesellschaft, and as such has no separate legal status.

Fraunhofer-Gesellschaft zur Förderung der angewandten Forschung e.V.

Hansastrasse 27 c 80686 München Germany

www.fraunhofer.de

All rights reserved; no part of this publication may be translated, reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, recor- ding or otherwise, without the written permission of the publisher.

Many of the designations used by manufacturers and sellers to distinguish their products are claimed as trademarks. The quotation of those designations in whatever way does not imply the conclusion that the use of those designations is legal without the consent of the owner of the trademark.

(4)

Transient effects during laser processing of silicon

for photovoltaic applications

Dissertation

zur Erlangung des Doktorgrades der Technischen Fakultät der

Albert-Ludwigs-Universität Freiburg im Breisgau vorgelegt von

Fabian Stefan Meyer

(5)

Erstgutachter und Betreuer der Arbeit:Prof. Dr. Stefan W. Glunz Zweitgutachter:Prof. Dr. Ulrich T. Schwarz

(6)

Abstract

Lasers offer a very finely controllable way of depositing energy into dielectrics, semiconductors and metals to produce a variety of permanent modifications.

For the manufacture of solar cells, lasers are used to dope, anneal and structure silicon, to create and improve junctions and contacts. However, the underlying physical mechanisms are rarely considered holistically and even more rarely measured. Focused pulsed lasers in particular induce very high excitation levels in the semiconductor, which lead to extreme conditions, high pressure (gigapascals) and temperature (thousands of kelvin), confined to a very small volume. The consequent huge diversity of the possible outcomes implies that a deep understanding of the transient states of matter during processing is required, if accurate control over the result is to be achieved. This work presents direct observations of the dynamics induced by laser pulses and demonstrates how the most important microscopic effects can be measured on a timescale of femtoseconds to nanoseconds.

Pulsed lasers not only induce structural changes but are also well suited to monitor them. A short laser pulse can be both, a source of modification (pump) and a flash (probe) to take a snapshot of the surface with a very short exposure time. In the course of this work, two beams of femtosecond pulses (190 fs) have been introduced as illumination sources into a specifically designed microscope. With a green beam to probe reflectivity and an infrared beam to probe transmittance, transient micrographs of the sample can be collected, during or after it is hit by another, stronger pump pulse. This pump- probe microscope and all the peripherals to control the experiment are integrated in a hardware and software environment, which is designed for an automated acquisition and evaluation of thousands of transient images. With digital image processing techniques complete and consistent reflectivity and transmission data sets are obtained from the sequence of frames. The following two laser processes are observed with these methods: (i) pulsed laser annealing of thin amorphous layers, a purely thermal process, conducted with ultraviolet (355 nm) nanosecond pulses and (ii) laser ablation of dielectric layers from silicon substrates, induced by ultrafast pulses in the visible (515 nm) and ultraviolet (343 nm).

Pulsed laser annealing is a means to turn amorphous silicon into nano-, micro- or polycrystalline silicon, to activate dopants or to remove crystalline 3

(7)

defects. This work demonstrates how the completeness of crystallization within the area irradiated by a single pulse is dependent on the dynamics of melting and subsequent solidification. Transient micrographs show how a layer of molten silicon with metallic optical properties forms at the surface, expands and then contracts within tens to hundreds of nanoseconds. The formation of nano-crystallites, regions of low and high crystallinity could be identified and linked to variant solidification velocities reaching 380 m/s. In addition, the observation of the melt dynamics at the surface is complemented with finite- element simulations of the heat- and melt-front propagation in the bulk. A comparison of pump-probe measurements with these simulations yields very good agreement and demonstrates the relevance of the presented methods for the validation of thermal simulations. Using the shown time-resolved anal- yses, the temporal parameters, such as pulse duration and pulse shape, can be tailored to the application. This is particularly relevant for those novel solar cell concepts where conventional high-temperature processing is not possible and which could therefore benefit greatly from the selective laser treatment of thin films shown herein.

Laser ablation is a now widespread technology to locally open thin dielectric passivation layers for contacting the silicon substrate. Using second and third harmonics of an infrared 190 fs Yb:KGW laser, the ultrafast dynamics of laser ablation could be observed with sub-picosecond resolution. The excitation of the surface within the first hundreds of femtoseconds upon pulse arrival is observed and compared to an extended Drude-model which explains the modulated surface reflectivity as the optical response of an electron- hole plasma with excited carrier densities beyond 10²²cm^-3. Ultrafast melting within picoseconds is observed thereafter. The melt is found to persist around 2 ns for 343 nm pulses and around 10 ns for 515 nm pulses. The depth of the melt is measured through the transient optical attenuation of the transmitted infrared beam and is found to reach less than 20 nm into the material. More- over, the solidification dynamics are derived to yield a spatially resolved map of the local solidification velocity below the surface, with peak values of more than 25 m/s. These findings explain the distribution of amorphous zones in the peripheral regions of openings created with 515 nm and 343 nm pulses – an important observation, since the amorphization through ultrafast melting is seen as one of the primary sources of laser-induced performance loss in the device.

(8)

A B S T R A C T 5

At a higher pulse energy density, different mechanisms of ablation of the silicon-nitride coating from silicon samples are observed. One beingindirect ablation, where the selective removal of the dielectric coating is achieved through an energy deposition in the underlying silicon, which in turn causes a local spallation of the coating. Contrary to that,direct ablationrefers to the ablation of the dielectric through an energy deposition in the coating itself. In addition, mixtures of indirect and direct ablation processes could be observed – presumably for the first time in a spatially resolved way. Transient inference patterns are analyzed using transfer-matrix models to extract the velocity of the removed silicon-nitride layer fragments. Finally it is demonstrated how the altered surface properties due to laser-induced amorphization lead to a more confined energy deposition. The effect discovered here can be used to create openings that are much smaller than a diffraction-limited beam at low numerical aperture would otherwise allow. This is of great relevance for solar cell manufacturing, where long focal lengths (over 200 mm) are used to achieve high processing speeds.

With the methods presented in this work, laser-induced effects can be observed on very short timescales and measured very accurately. It is demonstrated that this approach not only provides important insights for existing applications in silicon photovoltaics, but can also lead to the discovery of new and improved pulsed laser processes.

(9)

(10)

Zusammenfassung

Mit Lasern kann Energie sehr genau kontrollierbar in Dielektrika, Halbleiter und Metalle eingebracht werden, um damit eine Vielzahl dauerhafter Modifi- kationen zu erzeugen. Bei der Herstellung von Solarzellen werden Laser verwendet, um Silicium zu dotieren, auszuheilen und zu strukturieren, sowie um Kontakte und pn-Übergänge zu erzeugen. Die zugrundeliegenden physikali- schen Vorgänge werden aber selten in ihrer Gesamtheit betrachtet – und noch seltener vermessen. Insbesondere gepulste fokussierte Laser erzeugen sehr starke Anregung im Halbleiter, welche innerhalb eines kleinen Volumens zu extremen Bedingungen, hohem Druck (mehrere Gigapascal) und hohen Tem- peraturen (tausende Kelvin) führen. Um die daraus resultierende Fülle mögli- cher Ergebnisse zu nutzen und das Resultat exakt kontrollieren zu können, ist ein tiefes Verständnis der temporären Materiezustände während der Bearbei- tung erforderlich. Diese Arbeit legt dar wie die wichtigsten mikroskopischen Effekte auf der Zeitskala von Femtosekunden bis Nanosekunden vermessen werden können und zeigt welche Dynamiken ein Laserpuls dabei erzeugt.

Mit gepulsten Lasern können strukturelle Veränderungen im Kristall nicht nur herbeigeführt werden, sondern diese auch beobachtet werden. Ein kurzer Laserpuls kann zugleich die Ursache für eine solche Modifikation sein (Pump) und als „Blitzlicht” dienen, um in einer Momentaufnahme die Gegebenhei- ten an der Oberfläche mit sehr kurzer Belichtungszeit zu erfassen (Probe).

Im Zuge dieser Arbeit wurden zwei Strahlen eines gepulsten Femtosekunden- lasers (190 fs) als Beleuchtungsquelle in ein speziell dafür entworfenes Mi- kroskop eingekoppelt. Es können damit zeitaufgelöste Mikroskopaufnahmen erstellt werden, wobei ein grüner Strahl die Reflektivität und ein infraroter Strahl die Transmissivität des Substrats abfragt, während es von einem weite- ren, stärkeren Pump-Puls getroffen wird. Dieses Pump-Probe-Mikroskop und alle Gerätschaften zur Steuerung des Experiments wurden in eine Hardware- und Softwareumgebung eingebettet, die für eine automatisierte Datenerhe- bung und -auswertung konzipiert wurde. Aus der Sequenz tausender so ge- wonnener Einzelbilder werden schließlich mithilfe digitaler Bildverarbeitung vollständige und konsistente Datensätze gewonnen. Zwei Laserbearbeitungs- chritte wurden in dieser Arbeit besonders tiefgehend analysiert: erstens gepulstes Laserannealing dünner amorpher Schichten, ein rein thermischer Pro- zess, welcher mit ultravioletten (355 nm) Nanosekunden-Pulsen durchgeführt

7

(11)

wird und zweitens die Laserablation dünner dielektrischer Schichten von Sili- cium-Substraten mit Ultrakurzpulsen im Sichtbaren (515 nm) und UV (343 nm).

Gepulstes Laser-Annealing wird verwendet, um amorphes Silicium in nano-, mikro- oder polykristallines Silicium umzuwandeln, um Dotanten zu aktivie- ren oder Kristallschäden auszuheilen. Diese Arbeit legt dar, wie die Vollstän- digkeit der Kristallisation hierbei von der Dynamik des Schmelz- und Erstar- rungsvorgangs abhängt, welche sich innerhalb der von einem einzelnen Puls bestrahlten Fläche ergibt. Transiente Mikroskopaufnahmen zeigen wie sich eine Schicht von geschmolzenem Silicium an der Oberfläche bildet, sich aus- breitet und dann innerhalb von mehreren zehn bis hunderten Nanosekun- den kontrahiert. Die Entstehung von Nano-Kristalliten sowie Regionen hoher und niedriger Kristallinität können identifiziert werden und mit den gemes- senen lateralen Erstarrungsgeschwindigkeiten bis 380 m/s verknüpft werden.

Als Ergänzung zu den Beobachtungen an der Oberfläche werden Ergebnisse aus Finite-Elemente Simulationen gezeigt, welche zusätzlich die Ausbreitung der Wärme und Schmelzfront in die Tiefe des Substrats umfassen. Die vorge- fundene sehr gute Übereinstimmung der Pump-Probe Aufnahmen mit diesen Simulationen zeigt dabei auch, wie die hier vorgestellten Methoden verwendet werden können, um thermische Simulationen zu validieren. Mithilfe der gezeigten zeitaufgelösten Analysen können die zeitlichen Parameter, wie Puls- dauer und Pulsform, auf die Anwendung zugeschnitten werden. Dies ist vor allem für neuartige Solarzellenkonzepte relevant, bei denen eine herkömmliche Hochtemperaturbehandlung nicht möglich ist und welche daher besonders von der hier gezeigten selektiven Laser-Behandlung profitieren könnten.

Laser-Ablation wird mittlerweile häufig eingesetzt, um dünne dielektrische Passivierschichten lokal zu öffnen, damit das Silicium-Substrat darunter kon- taktiert werden kann. In dieser Arbeit werden die frequenzverdoppelten und -verdreifachten Strahlen eines 190 fs Yb:KGW Lasers zur Laserablation verwen-

det und die dabei entstehenden ultraschnellen Dynamiken mit einer Zeitauf- lösung von unter einer Pikosekunde beobachtet. Die Anregung der Oberfläche innerhalb der ersten hundert Femtosekunden nach Auftreffen des Pulses ist deutlich erkennbar. Der Vergleich mit einem erweiterten Drude-Modell erklärt die veränderte Oberflächenreflektivität als optische Antwort der angeregten Ladungsträger, deren Dichte demnach 10²²cm^-3übersteigt. Es folgt ein ultra- schnelles Schmelzen innerhalb von Pikosekunden. Im Falle von 343 nm Pulsen dauert die Schmelze 2 ns an; bei 515 nm Pulsen werden etwa 10 ns gemessen.

(12)

Z U S A M M E N FA S S U N G 9

Die Tiefe der Schmelze wird durch die optische Abschwächung eines trans- mittierten Infrarotstrahls ermittelt und beträgt in beiden Fällen weniger als 20 nm. Darüberhinaus wird die räumliche Verteilung der Erstarrungsgeschwin- digkeiten in der Tiefe abgeleitet, wobei Spitzengeschwindigkeiten von 25 m/s gemessen werden. Diese Messungen erklären die unterschiedliche Verteilung amorpher Zonen in den Randbereichen der bearbeiteten Stellen, welche bei der Verwendung von 515 nm und 343 nm Pulsen vorzufinden ist. Dies ist insbesondere deshalb eine bedeutsame Beobachtung, da die Effizienzverluste in den so bearbeiteten Zellen vor allem auf die Amorphisierung durch ultra- schnelles Schmelzen zurückgeführt werden.

Bei höherer Intensität werden schließlich verschiedene Ablationsvorgänge identifiziert. Bei derindirekten Ablationwird die dielektrische Schicht dadurch abgelöst, dass die Energie im darunterliegenden Silicium deponiert wird, wel- ches dann zu einem Abplatzen der Schicht führt. Im Gegensatz dazu bezeich- netdirekte Ablationdas Abtragen des Dielektrikums durch Absorption in der Schicht selbst. Weiterhin werden Mischzustände aus direkter und indirekter Ablation beobachtet und vermutlich zum ersten Mal räumlich aufgelöst gemessen. Unter Zuhilfenahme von Transfer-Matrix Modellen werden transiente Interferenzmuster analysiert, um die Geschwindigkeit der abgelösten Silicium- Nitrid Fragmente zu bestimmen. Schlussendlich wird dargelegt, wie die durch den Laser erzeugte Amorphisierung dazu führen kann, dass die Absorption räumlich eingeschränkt wird. Dieser Effekt kann genutzt werden, um Öffnun- gen zu erzeugen, welche viel kleiner sind als es ein beugungsbegrenzter Strahl bei niedriger numerischer Apertur sonst erlaubt. Dies ist besonders relevant für die Solarzellenfertigung, bei der große Brennweiten (über 200 mm) eingesetzt werden, um hohe Bearbeitungsgeschwindigkeiten zu erreichen.

Mit den hier gezeigten Methoden können Laser-induzierte Effekte auf sehr kurzen Zeitskalen beobachtet und sehr genau vermessen werden. Diese Ar- beit demonstriert, dass dies nicht nur wichtige Erkenntnisse für bestehende Anwendungen in der Silicium-Photovoltaik liefern kann, sondern auch zur Ent- wicklung ganz neuer Prozesse beigetragen kann.

(13)

(14)

1 Introduction

Lasers are arguably the most versatile and most precise instrument mankind has created. Lasers can be used to generate the highest pressures and the strongest electrical fields on earth [1, 2], some of the highest and some of the lowest temperatures [3–5]. They are used to produce the smallest artifi- cial structures and to cut the hardest materials [6, 7]. At the same time, they are used to measure the smallest distances and the shortest time spans [8, 9]. Because of these unique capabilities, they play a decisive role in solv- ing the most demanding questions and have become a standard tool in ad- vanced manufacturing environments. The limits of what commercially available systems can provide in terms of flexibility, power and ultimately processing speed are constantly shifting and new technical applications are in- vented nearly every day. With this in view, it seems obvious that lasers have an important role to play in technical solutions to the biggest challenges of our time, above all in the mitigation of climate change [10]. And indeed, they are now an integral part of the production lines to create silicon solar cells, which are one key approach for a decarbonized energy production [11–13]. Not unlike lasers, solar cells are constantly improving in every disci- pline – most importantly price per watt and conversion efficiency. Intensive research and rapid upscaling have made solar energy competitive with conventional energy sources [14, 15]. However, the total efficiency potential of a silicon solar cell is limited [16]; and the further developed an industrialized concept becomes, the harder it becomes for new technologies to challenge it. Experience has shown, that only those innovations are put into practice 15

(19)

on a large scale, which integrate well with the existing manufacturing conditions. The photovoltaic industry imposes very special requirements on its production tools. Given the low price of the product, material consumption and other running costs have to be held as low as possible. The production itself should be based on sustainable materials and practices. At the same time, a solar cell is a high-tech device. Its properties are tuned on the nanometer and femtoampère scale to be effective. Finding the balance between high cell efficiency and low production cost is therefore a challenge frequently encountered in the process of transferring a high-efficiency concept into large scale industrial production.

In addressing this challenge, I want to argue, laser based processes can excel. Very often they are not subject to the compromise between running cost and capability or at least to a lesser extent than conventional tools. The underlying cause is found in their unique physical properties, which are not governed by the basic laws of mechanics and thermodynamics that impose limitations elsewhere. For instance, a thermal heat source can only heat up to its own temperature, whereas lasers can heat while remaining cool. Mov- ing mechanical components within a machine requires force, creates friction, vibration and wear, whereas laser beams are not subject to inertia and can be effortlessly accelerated. The chemicals used to modify and structure semiconductors are often harmful to the environment and costly to dispose of, while direct laser processing for the same purpose requires no consumables at all.

Even simple laser systems can make use of these exceptional properties.

Still, they are easy to integrate and cost-effective to operate. Some modern laser systems can by the press of a button switch between processes that heat the workpiece to thousands of degrees to ones that modify the sample while it remains almost entirely cold. The dimensionality and magnitude of the parameter space at the disposal of the operator is huge. Val- ues range temporally from femtoseconds to hours, spectrally from x-ray to terahertz and intensities reach terawatts per square centimeter. The wide range of input parameters comes with a wide range of possible outcomes.

Small deviations in the way the process is performed can have drastic effects. A variation of only a few percent can mean a difference of thousands of kelvin or even trigger an entirely different effect. A variety of physical and chemical processes are involved, many of which show a nonlinear or

(20)

1 I N T R O D U C T I O N 17

self-enforcing behavior [17, 18]. Material properties measured as constants at equilibrium, such as the band gap or heat capacity, lose their validity. Ap- proximations fail or have only a limited scope of application. For these rea- sons, developing models that yield practicable predictions is a difficult task.

Nevertheless, modeling of laser applications has seen great advances rang- ing from ab-initio molecular dynamics to mesoscopic fluid dynamics – often with remarkable agreement with experiments [19, 20]. However, results from a model for one case are usually not easily transferable to a specific condi- tion of another application. Furthermore, the impact of the laser process on the performance of an electronically active device remains unaddressed in these studies and instructions on improving them can usually not easily be derived. Since the silicon solar cell efficiency has turned out to be very sensi- tive to laser induced modifications, an experimental approach seems much needed. On these grounds, this work aims to improve the understanding of the dynamics during pulsed laser processing of samples that are typically used for silicon solar cell production. The multitude of microscopic and very fast effects, on a timescale from femtoseconds to microseconds, makes a direct observation challenging. Pump-probe techniques, however, are perfectly suited to observe laser-based processing. The pulses used for the processing can be used for a flashed illumination of the sample and thus allow for a measurement of transient optical parameters [21]. Single snapshots already reveal intermediate steps of the process, which cannot be explored with ex-situ observations alone. The approach of this work is to apply this powerful technique to laser processing applications in the field of silicon solar cell manufacture and extend its capabilities. In order to measure the fast dynamics during laser annealing on the nanosecond scale and during laser ablation on the sub-picosecond scale, a pump-probe microscope for reflectivity and transmission measurements is developed, automated and paired with a consistent set of algorithms and models. The shown techniques reveal dynamics, which have not been observed before, such as multiple curved ablation fronts during dielectric removal. Numerous ex-situ characterization results are also presented and demonstrate that these transient effects are not only scientifically interesting to be observed, but also have consequences for the application and need to be taken into consideration in order to develop a tailored laser process.

(21)

Structure of this work

Chapter 2 briefly summarizes the known physical processes that govern the interaction of an intense laser pulse with silicon samples. In chapter 3, I re- port on the experimental and mathematical methods that have been developed and applied in this work. The results of their use are presented in the two following chapters, covering many observations which are of relevance for pulsed laser processing of silicon in general: chapter 4 deals with the laser crystallization of thin deposited amorphous films on silicon; in chapter 5 the dynamics of laser ablation of thin dielectric layers from silicon substrates are investigated.

Parts of chapter 3 and chapter 5 have been published in [22]. Some of the content of chapter 4 is also found in [23] and [24].

(22)

2 Theory

Laser interaction with semiconducting substrates

In our daily lives we perceive the interaction of light with matter as a mostly reversible process. Visible light from the sun or from light bulbs is reflected, scattered, transmitted and absorbed, without inducing permanent changes in the materials it is interacting with. But already at low intensities, the abil- ity of ultraviolet (UV) light to induce permanent changes in organic materials can be observed, for instance as a sunburn or photobleaching of dyes, if a certain exposure is exceeded. Intensity, duration of exposure and photon energy are the decisive parameters to consider, if the effect of light induced modifications are to be described. This applies to both sunburn and laser material processing. Laser processing today makes use of the capabilities of modern laser systems to fine-tune any of these three parameters and of the fact that a high intensity can lead to a variety of permanent modifications.

If the effects of the photons on the material of interest are known, the out- come can be predicted. Moreover, if the underlying principles are known, the parameters of the laser can be chosen, such that a desired modification is achieved.

In this chapter, some of the most important principles that determine the interaction of a laser beam with solid matter under normal atmospheric conditions are discussed. While some of the described effects might be transferable to other materials, the treatment is focussed on semiconductors and more specifically silicon, as this is the material of primary interest in this 19

(23)

work. Other materials, dielectrics, that are used alongside with it are also discussed to the extent that they are found to be of importance to the application. Special focus is placed on the parameter space which is usually used in the context of laser processing of silicon for the manufacture of solar cells.

This means that the content of this chapter is concerned with pulsed laser beams in the range from the UV to near-infrared (NIR) and in the duration of hundreds of femtoseconds to nanoseconds at a pulse energy density below 2 J cm⁻².

2.1 Optical properties of isotropic solids

The optical properties of semiconductors are a consequence of their specific band structure and are determined by the population of the energy states.

Under laser illumination, these can vary strongly, but if the outer conditions are constant and in thermal equilibrium, the mathematical description of the optical properties is greatly simplified through the usage of measured or tabulated values of the complex refractive index. The refractive index is linked to the electric permittivity𝜖as defined by the constitutive relation

𝐃 = 𝜖𝐄, (2.1)

which describes the displacement field𝐃in the material, induced by an ex- ternal electric field𝐄. It can be factorized into the permittivity of the vacuum 𝜖₀ and the relative permittivity𝜖_𝑟, which in the absence of permeability is equal to the square of the complex refractive index𝑛̂

𝜖 = 𝜖₀𝜖_𝑟 (2.2)

𝜖_𝑟= ̂𝑛². (2.3)

The electric field vector of a plane wave with wavelength𝜆and frequency𝜔, according to the Maxwell equations, propagates in direction𝑧as

𝐄(𝑧, 𝑡) = Re [𝐄₀exp (𝑖 (2𝜋 ̂𝑛𝑧

𝜆 − 𝜔𝑡))]

= exp (−2𝜋𝜅𝑧

𝜆 ) Re [𝐄₀exp (𝑖 (𝑘𝑧 − 𝜔𝑡))] , (2.4) where the complex refractive index is split into real and imaginary part as

̂

𝑛 = 𝑛 + 𝑖𝜅and the definition of the wavenumber𝑘 = 2𝜋𝑛/𝜆is used. Under the

(24)

2 .1 O P T I C A L P R O P E R T I E S O F I S OT R O P I C S O L I D S 21

conditions of interest to this work, the electric field applied by a laser beam is described with good approximation by such a plane wave. The intensityI of the beam at depth𝑧and time𝑡in the material is then given by

I(𝑧, 𝑡) = 𝑐𝑛𝜖₀

2 |𝐄(𝑧, 𝑡)|², (2.5)

where𝑐is the speed of light in vacuum.

If the infinitesimal intensitydIwhich is absorbed in a layer of infinitesimal thicknessd𝑧is proportional to the irradiated intensityI, the absorption is said to be linear, with proportionality constant𝛼:

dI= −𝛼Id𝑧. (2.6)

The solution to this differential equation with initial intensityI₀is called the Lambert-Beer law

I(𝑧) =I₀exp (−𝛼𝑧) (2.7)

and𝛼is called attenuation coefficient or linear absorption coefficient. With equations 2.4 and 2.5 we find that𝛼is linked to the imaginary part of the refractive index𝜅as [25]

𝛼 =4𝜋𝜅

𝜆 . (2.8)

The complex refractive index incorporates the information to describe the propagation of an electromagnetic wave and consequently the linear optical properties of a homogeneous, isotropic material. Generally both𝑛and𝜅are wavelength dependent. This dependence can be measured using spectroscopic ellipsometry (SE) or taken from tabulated values. The values used for silicon throughout this work are shown in figure 2.4.

2.1.1 Thin film optics

If electromagnetic waves hit an interface between two materials of different refractive indices, they are split up into reflected and transmitted parts. The amplitudes of the reflected and transmitted partial waves can be calculated from the refractive indices of the two materials𝑛₁ and𝑛₂. The reflection coefficient𝑟is the ratio of the reflected wave’s complex electric field amplitude to that of the incident wave and𝑡 the ratio of transmitted to incident

(25)

𝜃_𝑖

𝜃_𝑡

̂

𝑛_𝑙

̂𝑛_𝑙+1

𝑟_𝑙,𝑙+1

𝑟_{𝑙+1,𝑙+2} 𝑡_𝑙,𝑙+1

̂

𝑛_𝑙+2

𝑑_𝑙+1

𝑡_{𝑙+1,𝑙+2}

Figure 2.1 On any interface of two materials with different refractive indices𝑛̂_𝑙and thicknesses𝑑_𝑙, partial waves are reflected with complex amplitude𝑟_𝑙,𝑙+1, which can be calculated from Fresnel’s equations. The transmitted part, with complex amplitude 𝑡_𝑙,𝑙+1is in turn partially reflected at the next interface with amplitude𝑟_{𝑙+1,𝑙+2}and so on.

The resulting reflectivity𝑅 = |𝑟|²and transmittanceΘ = |𝑡|²of the complete stack can be conveniently calculated using the transfer-matrix method.

one. Fresnel’s equations then give these ratios as 𝑟_s= 𝑛₁cos 𝜃_i− 𝑛₂cos 𝜃_t

𝑛₁cos 𝜃_i+ 𝑛₂cos 𝜃_t, (2.9) 𝑡_s= 2𝑛₁cos 𝜃_i

𝑛₁cos 𝜃_i+ 𝑛₂cos 𝜃_t, (2.10) 𝑟_p = 𝑛₂cos 𝜃_i− 𝑛₁cos 𝜃_t

𝑛₂cos 𝜃_i+ 𝑛₁cos 𝜃_t, (2.11) 𝑡_p = 2𝑛₁cos 𝜃_i

𝑛₂cos 𝜃_i+ 𝑛₁cos 𝜃_t, (2.12) where the angle of the transmitted wave vector𝜃_tcan be calculated from the angle of incidence𝜃_iusing Snell’s law

𝑛₁sin 𝜃_i= 𝑛₂sin 𝜃_t. (2.13) If a partially transmitting thin layer of a material is applied onto a material of another refractive index, an additional interface is formed and the partial waves that are reflected from either of them interfere. To facilitate the

(26)

calculations for stacks of different materials it is advisable to use a matrix formalism, called transfer-matrix method (TMM) [26, 27]. With the nomencla- ture in figure 2.1, the properties of every layer𝑙 are expressed as a matrix 𝑀_𝑙

𝑀_𝑙 = 1

𝑡_𝑙,𝑙+1(𝑒^−𝑖𝛿^𝑙 0

0 𝑒^𝑖𝛿^𝑙) ( 1 𝑟_𝑙,𝑙+1

𝑟_𝑙,𝑙+1 1 ) (2.14)

where the partial reflection amplitudes𝑟_𝑙,𝑙+1are to be calculated using the Fresnel formulae (equations 2.9 to 2.12). The complex phase contribution of each layer𝛿_𝑙is given by

𝛿_𝑙= {

𝜋 𝑙 = 1

0 𝑙 = 𝑁

2𝜋 ̂𝑛_𝑙𝑑_𝑙/𝜆 else

(2.15)

where the first (𝑙 = 1) and last (𝑙 = 𝑁) layer represent the two infinite half spaces surrounding the layer stack and𝑑_𝑙 is the thickness of intermediate layer𝑙. The matrices are then multiplied in the order of interaction with the incident light:

𝑀 =

𝑁

∏

𝑙=1

𝑀_𝑙 (2.16)

The complex field amplitude ratios of the complete stack, for reflection𝑟and transmission𝑡can then be calculated from the matrix elements of𝑀:

𝑟 = 𝑀₂₁

𝑀₁₁ 𝑡 = 1

𝑀₁₁. (2.17)

By tuning the thickness or refractive index of the thin layers, the reflectivity of the surface can be strongly increased or suppressed. This technique has numerous applications, for example for the fabrication of highly reflective laser mirrors or to increase absorption of solar cells. In air, a flat silicon surface has a reflectance of over 30%. For an uncoated solar cell, the reflected power would be lost for electric conversion and an anti-reflection coating (ARC) is therefore usually applied on silicon solar cells. This coating is optimized in thickness and refractive index to reduce the reflectance of the encapsulated cell under sunlight [28, 29]. The optical requirements of such a coating (high𝑛and low 𝜅), are fulfilled by dielectrics like silicon dioxide

(27)

Newton’s rings

curved interface flat interface

Figure 2.2 If the partial waves reflected from a curved surface interfere with those reflected of a flat surface below, bright and dark rings are formed where the conditions for constructive and destructive interference are met. A symmetrically curved surface creates concentric rings, which are widely called Newton’s rings

or silicon nitride. Additionally the layer can serve electrical purposes: it can passivate the silicon surface and significantly reduce surface recombination [30]. On the front-side of silicon solar cells, hydrogenated silicon nitride SiN_X, deposited with plasma-enhanced chemical vapor deposition (PECVD) is most widely used [31]. On a textured surface, for a single layer coating, a refractive index of around 2.05 and a coating thickness of around 75 nm is chosen [12].

Newton’s rings If the plane partial waves reflected by two non-parallel interfaces interfere, dark and bright fringes can be observed. This is because alternatingly, the conditions for destructive and constructive interference of the two waves are met. In the special case, that one of the interfaces is spherical and the other planar, these fringes form concentric rings, which are called Newton’s rings (fig. 2.2). The fringes can be used to detect slight- est deviations of a perfectly symmetrically curved surface, as it causes them to be locally bent and deviate from the circle. In this work, the term Netwon’s rings is used in a more widely sense for the interograms caused by curved interfaces in general, although they are mostly elliptic here and although the shape of the surface is more parabolic than spherical.

Fabry-Pérot interference filters A configuration where two reflecting interfaces face each other with a non-absorbing medium of thickness𝑑 in between them is called Fabry-Pérot interferometer (fig. 2.3). Even though both

(28)

𝑅

d cavity

Θ

a) Fabry-Pérot filter 0.0

0.5

1.0 b) reflectivity𝑅

0 200 400 600 800

𝑑(nm) 0.0

0.5 1.0

c) transmittanceΘ

Figure 2.3 Two facing reflecting surfaces (a) form an interferometric filter, with well defined reflectance valleys (b) and transmittance peaks (c).

interfaces may be strongly reflecting, this setup is transmissive for coherent light if the first transmitted wave interferes constructively with those after one or more round trips within the cavity. That is, when two successive transmitted partial waves have an optical path length difference of

2𝑛_𝑐𝑑 = 𝑞𝜆 with 𝑞inℕ,

where𝑛_𝑐is the refractive index of the cavity. The sharpness of the transmission peaks is dependent on the absorption in the cavity and the reflectivity of the interfaces. If the transmission is monitored, while the wavelength is gradually changed, the recurring transmission peaks can act as a scale for the linearity of wavelength sweep for spectroscopic applications. In this work, the wavelength is generally constant, but configurations are found, where two facing and reflective interfaces move away from each other. The grad- ual increase of the cavity length then creates a similar pattern. Instead of transmittanceΘ, reflectance is monitored in this case, which shows the com- plementary shape1 − Θif absorption can be neglected.

(29)

0.2 0.4 0.6 0.8 1.0 1.2 wavelength (𝜇𝑚)

1 2 3 4 5 6 7

refractive index𝑛

0.2 0.4 0.6 0.8 1.0 1.2

wavelength (𝜇𝑚) 10⁻¹

10⁰ 10¹ 10² 10³ 10⁴ 10⁵ 10⁶

10⁷ absorption coefficient𝛼(cm⁻¹)

a-Si l-Si c-Si

Figure 2.4 Tabulated values for the real part of the refractive index of silicon (amorphous, crystalline and liquid) and the absorption coefficients as calculated from the literature values of κ from [32–34]. For an extensive review of optical and electrical properties of crystalline silicon see [35].

2.2 Absorption and excitation

The first step in the sequence of events that leads to a structural modification of a semiconductor through a laser pulse is the excitation of carriers and generation of electron-hole pairs. Depending on the density of states and their population levels, photon energy and intensity, different absorption mechanisms dominate.

2.2.1 Linear absorption in silicon

If the photon energy matches the energy difference between its current and a higher unpopulated state and the transition is not suppressed by a selection rule, an incoming photon can transfer its energy to an electron and excite it to the state of higher energy in a process called stimulated absorption.

In a semiconductor the excitation of an electron from the valence band to the conduction band is likely, if the energy of the photon exceeds the band gap. Crystalline silicon is an indirect semiconductor and absorption of a photon above the indirect band gap of 1.14 eV but below 3.4 eV, can only happen

(30)

2 . 2 A B S O R P T I O N A N D E X C I TAT I O N 27

under absorption or emission of a phonon, which carries the difference in momentum between initial and final state. The momentum of the photon itself is negligible compared to that of the phonons. As it is a three parti- cle scattering process, absorption in the visible range in silicon is weaker than for semiconductors with a smaller direct band gap. Photons with a wavelength shorter than 364 nm (≥3.4 eV) can be absorbed without phonon participation. Consequently absorption is very strong for UV radiation and absorption depth is only around 10 nm at𝜆 =350 nm [36].

In amorphous silicon (a-Si), a long range order is missing and the band theory based on periodic boundary conditions can strictly speaking not be applied. The disorder leads to additional localized states in an energy range which is forbidden in crystalline silicon, which are called Urbach tails [37].

Through this smoothing of the band edges, the effective band gap is reduced, turning it into a direct semiconductor, with stronger absorption in the visible and a sudden drop of absorption below the band gap (see fig. 2.4).

Intermediate optical absorption spectra between those of amorphous and crystalline are found, as the degree of non-amorphous content can vary depending on the origin [38, 39]. Artificially, a-Si can be deposited (e.g. with low pressure chemical vapor deposition (LPCVD)) or created from crystalline substrates through self-implantation. Deposited amorphous silicon is usually passivated with hydrogen (denoted a-Si:H) to saturate dangling bonds, as opposed to the amorphous silicon created through laser impact (to be discussed in section 2.5.2), which can thus be considered highly recombination active.

2.2.2 Nonlinear absorption

Photons well below the band gap can still be contribute to interband absorption, if their density is high enough that nonlinear effects become relevant.

This is the case, if linear absorption is weak or if ultrashort pulses are used.

two-photon absorption (TPA) is the most dominant multi-photon absorption mechanism in silicon, where two photons excite one electron. Because the probability of two photons to be available for absorption is dependent on their density, the contribution of the TPA is proportional to the intensity.

Thus, a contribution𝛽Iis added to the linear absorption𝛼in equation 2.6

𝜕

𝜕𝑧I= −(𝛼 + 𝛽I)I. (2.18)

(31)

0 1.1 eV 3.2 eV

ℏ𝜔 Energy

Γ 𝑋

valence band conduction band

A. interband absorption B. FCA C. impact

ionsation

D. Auger re- combination

band-gap

Figure 2.5 Transitions of carrier energy can be induced optically (dashed lines) or through collisions with other carriers (solid lines). Silicon as an indirect semiconductor is absorbing in the visible (ħω < 3.4 eV) only under assistance of phonons (dotted green), which bridge the gap in momentum between the Γ and X point [49]. Direct absorption is possible for UV light beyond 3.2 eV [50]. Already excited carriers within the conduction band can contribute to broadband absorption as free carrier absorption (FCA) (B), especially if their density is high. In a process called impact ionization (C) highly excited carriers of the conduction band transfer their energy to unexcited carriers to lift them into the conduction band. The inverse process, Auger recombination (D), leads to a reduction of the number density of excited electron-hole pairs, while increasing the energy of electrons in the conduction band. Their energy is then in turn mostly transferred to heat through scattering with phonons.

Literature values for the two-photon absorption coefficient𝛽are subject to high uncertainty. In so called z-scan experiments, it is found to be in the order of 2 cm GW⁻¹for silicon [40, 41]. Other publications state higher values (38 cm GW⁻¹ [42] at 620 nm, 6.8 cm GW⁻¹at 800 nm [43]). TPA and other nonlinear effects in silicon nitride are a topic of current research because of its importance for silicon photonics [44–48]. Values in the range of 10 cm GW⁻¹ to 80 cm GW⁻¹are reported for silicon-rich nitride [44].

2.2.3 Generation of the electron-hole plasma

For every photon absorbed through interband absorption, one electron-hole pair is generated (fig. 2.5A). In the case of TPA, half as many carriers are created per photon. Combining the rates of the two mechanisms, a carrier den-

(32)

2 . 2 A B S O R P T I O N A N D E X C I TAT I O N 29

sity generation rate of

𝜕

𝜕𝑡𝑁(𝑧, 𝑡) = (𝛼 +1

2𝛽I(𝑧, 𝑡))I(𝑧, 𝑡)

ℏ𝜔 , (2.19)

is concluded [51]. A gaussian temporal pulse shape is often a suitable approximation, especially for ultrashort pulses [52]. Using a pulse duration of 𝜏_𝑃 and

I(𝑡) =I₀exp (− (𝑡/𝜏_𝑝)²) , (2.20) equation 2.19 can be integrated to receive the total density of generated car- riers𝑁_𝑒−ℎ. Due to reflectance𝑅, only a fraction of1 − 𝑅photons are absorbed and lead to the excitation of carriers in a density of

𝑁_𝑒−ℎ = √𝜋𝜏_𝑝(1 − 𝑅)I₀

ℏ𝜔 (𝛼 + 𝛽(1 − 𝑅)I₀

2√2 ) . (2.21)

In general𝑅is not constant and neglecting its time-dependence is only valid at the beginning of the pulse [51]. Afterwards, the optical properties can be strongly affected by the excited carriers.

2.2.4 Optical response of the excited semiconductor

Interband transitions are not the only mechanism to lead to energy deposition. Already excited carriers can also significantly contribute to the absorption if their density is high (fig. 2.5B). Because of their high mobility, they are treated as free carriers and their contribution is called free carrier response.

Semiconductors absorb visible light at room temperature primarily through interband absorption because the conduction band is not as populated as the valence band. However, free carrier absorption (FCA) becomes relevant for photons below the band gap energy, where it can be the only absorption mechanism besides multi-photon absorption. In the case of strongly excited semiconductors, they are the primary source of some variant optical properties that are of particular interest to this work. The contribution to the dielectric constantΔ𝜖_FCRof free carriers is described by the Drude model [51, 53]

Δ𝜖_FCR= − (𝜔_𝑃 𝜔)

2 1

1 + 𝑖 ¹

𝜔𝜏_𝐷

, (2.22)

(33)

where the definition of the plasma frequency is used:

𝜔_𝑃= √ 𝑁_𝑒−ℎ𝑒² 𝜖₀𝑚^∗𝑚_𝑒.

𝜏_𝐷is the Drude damping time,𝑒and𝑚_𝑒are charge and mass of an electron and𝑚^∗is the optical effective mass of the carriers.

At low excitation levels, it can be assumed that(𝜔_𝑃, 𝜏_𝐷⁻¹) ≪ 𝜔and a linear absorption coefficient can be derived using equations 2.3 and 2.8

𝛼_FCR= 𝑒²𝑁_𝑒−ℎ

4𝜋²𝜖₀𝑚_𝑒𝑚^∗𝜏_𝐷√𝜖_BG𝜆²,

where𝜖_BGis the background dielectric function of the semiconductor without the contribution of excited carriers [54]. The total absorbed intensity from equation 2.18 then reads

𝜕

𝜕𝑧I(𝑧, 𝑡) = − (𝛼 + 𝛼_FCR(𝑧, 𝑡))I(𝑧, 𝑡) − 𝛽I(𝑧, 𝑡)²

[19]. For solar cells, the absorption of the free carriers is considered parasitic because it does not generate new electron-hole pairs. It is important to note that it grows with𝑁_𝑒−ℎand consequently with temperature and doping level [54, 55]. The𝜆²-dependency is an often quoted rule of thumb and is of particular importance for semiconductors irradiated below the band gap, where interband absorption is low. Tabulated values for𝑚^∗ and𝜏_𝐷 exist for low- density plasmas (𝑚^∗ = 0.15,𝜏_𝐷≈100 fs [51]) but are also not applicable in the case of high-density plasmas or liquid layers as generated by laser pulses [51, 56]. When irradiated with short laser pulses, the excitation levels are much higher and𝜔_𝑃approaches𝜔. Hence the assumption(𝜔_𝑃, 𝜏_𝐷⁻¹) ≪ 𝜔does not hold and FCA increases strongly [54]. Through free carrier absorption the energy of the excited carriers is increased, while their number density remains constant. But indirectly additional carriers are created through collisions of hot carriers with bound electrons in a process called impact ionization (see fig. 2.5C and fig. 2.6). This increases absorption of the electron gas further and can create an avalanche effect. To account for the additional source of excitation, equation 2.19 is extended

𝜕

𝜕𝑡𝑁(𝑧, 𝑡) = (𝛼 + 1

ℏ𝜔 + 𝛿_imp𝑁(𝑧, 𝑡),

(34)

2 . 3 C A R R I E R R E L A X AT I O N 31

where 𝛿_imp is called impact ionization coefficient [19, 57]. While they find the Drude model alone to describe the transient reflectivity of the excited surface already surprisingly well, Sokolowski-Tinten and von der Linde [58]

list other transient contributions to the dielectric constant, namely band- fillingΔ𝜖_BF and band-structure renormalization Δ𝜖_BGS. The total dielectric function is then given by

𝜖^∗= 𝜖_𝑟+ Δ𝜖_BF+ Δ𝜖_BGS+ Δ𝜖_FCR, (2.23) where𝜖_𝑟is the dielectric function of the unexcited material. As the states to be excited become filled, the number of free target states is reduced and the probability of further excitation decreases [59]. Neglecting spectral effects

Δ𝜖_BF= −(𝜖_𝑟− 1) 𝑁_𝑒−ℎ 𝑁_𝑒−ℎ,0

is concluded, where𝑁_𝑒−ℎ,0 is the number density of valence-band carriers of the unexcited semiconductor [58]. In the excited plasma, the effective band structure and size of the band gap, that an individual carrier encoun- ters, is affected by the many-body interactions. Sokolowski-Tinten and von der Linde treat this with the contributionΔ𝜖_BGS using a spectrally shifted dielectric function [42, 58].

2.3 Carrier relaxation

Absorption of light in a semiconductor means that at first, all energy is deposited as excitation of electron-hole pairs or increasing the energy of already excited ones (fig. 2.6). If this deposition takes longer than the relaxation mechanisms, carriers and lattice have the same temperature. However, especially if ultrashort laser pulses are used, the deposition of energy in the electron-hole plasma can precede its transfer into heat of the lattice leading to a non-equilibrium state. To describe this highly dynamic process, a two- temperature model is applied, which regards the carriers and the lattice as independent systems of different temperature that interact to reach thermal equilibrium (fig. 2.7) [19, 60, 61]. After the local deposition of energy into the electron-hole plasma, multiple processes restore thermal equilibrium within itself and with its environment. The excited carriers’ energy is reduced in two

(35)

photon energy ℏ𝜔 carrier excess

energy electron-hole

pairs impact

ionization

Auger recombination heats carrier

gas to𝑇_𝑒> 𝑇_𝑙 optical phonons acoustic phonons

Figure 2.6 The storage of photon energy into the carrier subsystem leads to its heating to temperature T_e. Depending on the rates of these transitions, which depend on the carrier concentration itself, their temperature T_ecan be initially higher than that of the lattice T_l. Thermal equilibrium is restored through phonon emission. Schematic adapted from [57].

T

time 𝜏_𝐸 Laser

pulse T_e

T_l

Figure 2.7 The fast deposition of energy into the electron subsystem by an ultrashort laser pulse, leads to a tempo- rary difference between temperatures of the electrons T_e and and the lattice T_l. Equilibrium between the two is restored through carrier-phonon scattering within the time τ_e[65].

processes that happen concurrently during the first few hundred femtoseconds: carrier-carrier and carrier-phonon scattering [62, 63]. Carrier-carrier scattering is an electrostatic interaction, which restores the Fermi-Dirac distribution, but does not change the total energy of the carrier subsystem [63].

Actual cooling of the carriers is performed through carrier-phonon scattering, where carriers emit or absorb phonons. In a single scattering process only little energy can be transferred to a phonon and it thus takes many scattering events and several picoseconds for the lattice and the electron-hole plasma to reach a common well-defined temperature [63, 64].

An excess of excited carriers is reduced through carrier diffusion and recombination of electrons with holes [63]. The latter happens under emission of a photon (photoluminescence), while exciting another carrier (Auger recombination [66]), a defect (Shockley-Read-Hall [67, 68]) or a surface state [69, 70]. The rates associated with these recombination channels are of ut- most importance for the conversion efficiency of solar cells, as they define

(36)

2 .4 T H E R M A L E F F E C T S 33

the average lifetime of a carrier-hole pair. For the dynamics induced by a strong laser pulse, usually only Auger recombination is taken into account (2.5D), as the others are many orders of magnitude slower (typically microseconds to milliseconds) than the timescales of interest. Auger recombination is a three-body process, where the excess energy of an electron-hole recombination is transferred to excite an electron (𝑒𝑒ℎ) or hole (ℎℎ𝑒) [71]. As such, the recombination rate is proportional to𝑁_𝑒−ℎ³ and contributes to a fast reduction of the number of excited carriers if its density is high. Using the Auger coefficient𝛾_𝐴it can be included in equation 2.2.4 [19, 57]

𝜕

𝜕𝑡𝑁(𝑧, 𝑡) = (𝛼 + 1

ℏ𝜔 + 𝛿_imp𝑁(𝑧, 𝑡) − 𝛾_𝐴𝑁(𝑧, 𝑡)³.

Additionally to recombination, carrier diffusion is another mechanism to reduce the number density in the excited volume. However, the considerations of Brown et al. and van Driel et al. suggest, that induced energy band gap gradients lead to a confinement of the carriers [63, 72, 73] and van Driel et al.

conclude that carrier diffusion can be omitted for the modelling of plasma excitation during intense laser pulses [57].

2.4 Thermal effects

After the carriers have given their energy to the lattice, the system is locally in thermal equilibrium. However, because strong absorption yields a high intensity gradient, steep spatial temperature gradients prevails. If the energy is stored within a short time, most of the sample will stay cold, while locally the temperatures can reach supercritical values. The volume which is subject to laser induced modifications is therefore a strong function of the optical penetration depth of the beam, its spatial shape, the temporal pulse shape and material properties. The electronic effects, which have been discussed in the previous sections, temporarily overlap the thermal processes and have great impact on them. For example, the reflectance of the surface determines how much energy is available in the material and varies during the absorption process. The high value range of excited carrier density and temperature create a system of very high complexity, which can only be the- oretically treated, if approximations are made that allow a separation of the effects. A line can be drawn between ultrashort processes and pure thermal

(37)

processes as distinguished by the pulse duration𝜏_𝑃 relative to the duration of thermalization of the carriers𝜏_𝐸 of 1 ps to 10 ps [74, 75]. If𝜏_𝑃 ≫ 𝜏_𝐸, electrons and phonons are in thermal equilibrium during the complete process and a treatment with purely thermal models is appropriate. In the ultrashort regime𝜏_𝑃 ≤ 1 ps, a thermal description of slow processes, which happen at a time𝜏_𝑀 ≫ 𝜏_𝐸, is justified. However, phase changes and modifications on the timescale of the thermalization𝜏_𝑀 ≈ 𝜏_𝐸can lead to nonequilibrium states and a metastable solid or liquid [75].

2.4.1 Heat diffusion

In an absorptive medium, the exponentially decaying intensity distribution in the material according to the Lambert-Beer law leads to a strong gradient of the absorbed energy in the𝑧direction perpendicular to the surface. The deposited heat will redistribute in the direction of the temperature gradient.

This thermal diffusion process is described by the heat diffusion equation 𝜌𝑐_𝑃𝜕𝑇

𝜕𝑡 − ∇ ⋅ (𝐾∇𝑇) = 𝑄, (2.24)

where𝑐_𝑃 is the specific heat at constant pressure,𝑇is the temperature,𝐾is the thermal conductivity and𝜌is the mass density (for values see table 2.1).

𝑄is a volumetric heat source, which in this case is given by the absorbed energy. These quantities are generally functions of temperature and approximations are to be made to solve the equation for a specific case. Heat con- ductance can be approximated to be one dimensional (1-D), if the assumption is justified, that the depth of thermal impact is small compared to the lateral size of the beam. In this case solutions can be found, for example for a pulse of constant intensityI₀for a duration𝜏_𝑃, dropping to zero intensity afterwards:

I(𝑡) = { I₀ 𝑡 < 𝜏_𝑃 0 𝑡 ≥ 𝜏_𝑃

If we assume the heat source to be purely superficial, the heat diffusion equation is solved by the temperature𝑇(𝑧, 𝑡)as a function of the depth𝑧and time

(38)

2 .4 T H E R M A L E F F E C T S 35

𝑡after the pulse according to

𝑇(𝑧, 𝑡) = 2𝛼I₀(1 − 𝑅)

𝐾 [𝜁 (𝑧, 𝑡) − 𝜁 (𝑧, 𝑡 − 𝜏_𝑃)]

𝜁 (𝑧, 𝑡) ≔ √𝐷𝑡 ierfc ( 𝑧 2√𝐷𝑡

) ,

where𝐷 = 𝐾/𝜌𝑐_𝑃 is the thermal diffusivity andierfcis the integral comple- mentary error function [76, 77]. If the thermal penetration depth𝑙_𝐷is defined through

𝑇(𝑙_𝐷, 𝜏_𝑃)=^! 1

𝑒𝑇(0, 𝜏_𝑃),

𝑙_𝐷≈ √𝐷𝜏_𝑃 (2.25)

is received. This is an often quoted rule of thumb for the depth of the heat affected zone (HAZ) in the case of a superficial heat source, when the optical attenuation depth𝛼⁻¹is small compared to𝑙_𝐷. Bechtel shows in detail how the calculations can be extended for volumetric heating and for other pulse shapes [76]. However, the number of cases where exact solutions can be found is very limited, especially if the 1-D approximation is not applicable or if it has to be taken into account, that the optical or thermal parameters vary with time. For accurate predictions the heat diffusion equation can be solved numerically using measured or tabulated values for the constants. A finite element simulation, which incorporates some of the transient effects has been developed by Andreas Fell and is used in this work as comparison for experimental results [78].

2.4.2 Melting and solidification

High interest in laser melting of silicon emerged in the 1980s, because of the promising technical possibilities of pulsed laser annealing (PLA) [87–89]. The phase change of melting temporarily removes the structure of the solid and allows the atoms to settle to a different structure after solidification. As it turns out, this can work both ways: amorphous silicon can be turned into liquid and then solidify to the crystalline state (crystallization) or crystalline silicon can be molten and solidify to the amorphous state (amorphization).

To which state the melt solidifies, as well as the crystallites’ size and orien- tation is a function of the solidification velocity and the driving mechanism.

Transient effects during laser processing of silicon for photovoltaic applications

TRANSIENT EFFECTS DURING LASER PROCESSING OF SILICON FOR PHOTOVOLTAIC APPLICATIONS

Transient effects during laser processing of silicon for photovoltaic applications

FRAUNHOFER VERLAG

Fraunhofer Institute for Solar Energy Systems ISE

Transient effects during laser processing of silicon

for photovoltaic applications

Dissertation

Fabian Stefan Meyer

Abstract

Zusammenfassung

Contents

1 Introduction

2 Theory

Laser interaction with semiconducting substrates

2.1 Optical properties of isotropic solids

2.1.1 Thin film optics

2.2 Absorption and excitation

2.2.1 Linear absorption in silicon

2.2.2 Nonlinear absorption

2.2.3 Generation of the electron-hole plasma

2.2.4 Optical response of the excited semiconductor

2.3 Carrier relaxation

2.4 Thermal effects

2.4.1 Heat diffusion

2.4.2 Melting and solidification