Discovering Perovskite Photophysics with Transient Reflectance Spectroscopy

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Discovering Perovskite Photophysics with 7UDQVLHQW5HÀHFWDQFH Spectroscopy

HANNU PASANEN

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Tampere University Dissertations 475

HANNU PASANEN

Discovering Perovskite Photophysics with Transient Reflectance Spectroscopy

ACADEMIC DISSERTATION To be presented, with the permission of the Faculty of Engineering and Natural Sciences

of Tampere University,

for public discussion in the lecture room Pieni sali 1 (FA032) of the Festia building, Korkeakoulunkatu 8, Tampere,

on 29^th of October 2021 at 12 o’clock.

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ACADEMIC DISSERTATION

Tampere University, Faculty of Engineering and Natural Sciences Finland

Responsible supervisor and Custos

Professor Nikolai Tkachenko Tampere University

Finland

Supervisor Professor Paola Vivo Tampere University Finland

Pre-examiners Professor Sergey Makarov Professor Frédéric Laquai ITMO University

Russia

King Abdullah University of Science and Technology Kingdom of Saudi Arabia Opponent Professor Ivan Scheblykin

Lund University Sweden

The originality of this thesis has been checked using the Turnitin OriginalityCheck service.

Cover design: Roihu Inc.

ISBN 978-952-03-2106-2 (print) ISBN 978-952-03-2107-9 (pdf) ISSN 2489-9860 (print) ISSN 2490-0028 (pdf)

http://urn.fi/URN:ISBN:978-952-03-2107-9

PunaMusta Oy – Yliopistopaino

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Dedicated to all the people who tried, in vain, to ﬁnd me in my ofﬁce.

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PREFACE

I thank the doctoral program of Tampere University for personal funding. Business Finland and Forschungszentrum Jülich GmbH (SolarWAVE project), and Jane &

Aatos Erkko Foundation (project ASPIRE) are acknowledged for funding the Hy- brid Solar Cells (HSC) group. Academy of Finland Flagship Programme, Photonics Research and Innovation (PREIN, 320165), is acknowledged for funding both the Photonic Compounds and Nanomaterials group and the HSC group.

I thank Nikolai Tkachenko for sharing his invaluable advice and expertise over the course of my PhD. I am forever thankful for all the support he gave me during the application process to the Tampere University of Technology, despite us being complete strangers before I contacted him out of the blue about PhD positions. I also thank Paola Vivo for her guidance and support even during her maternity leave at the start of my PhD. Other people who helped me kickstart the research were Tero- Petri Ruoko, who introduced me to the pump-probe system, and Arto Hiltunen, who showed me the basics of solar cell synthesis. Over the years I also received help from Maning Liu and the other members of the HSC group, who had to prepare the perovskite samples for me out of fear I would otherwise mess with their glovebox. I also thank Ramsha Khan for not getting too annoyed with my endless ramblings.

Finally, I thank my friends and family, especially the dog Lili, who was the best stress reliever during the few vacations I had.

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ABSTRACT

Halide perovskites have emerged as leading candidates for next-generation solar cells and other photovoltaic applications. The power conversion efﬁciency of perovskite solar cells has already surpassed 25%, exceeding that of commercial silicon-based solar cells while having several times smaller fabrication costs. Their advantages also extend to very high tunability of colour and other properties.

When we began the perovskite study in 2018, the understanding of perovskite photophysics had been lagging behind the rapid development of their synthesis and solar cell efﬁciency via trial and error. Thus, our objective was to investigate the charge carrier dynamics in perovskites by using ultrafast pump-probe techniques.

However, the traditional data analysis models were unable to explain the transient absorption signals we had obtained. It had been a common assumption in the research community that only photoinduced changes in absorption give rise to the signal, but we discovered that instead, the photoinduced changes in reflectance could also be a major factor. Whereas the samples in the past had been solutions or films with low refractive index, the new perovskites were instead thin-film samples with high refractive index, which boosts their reflectance. Therefore, to analyse the signal, we had to employ transient reflectance spectroscopy, but it was a rare technique and the available analysis tools were not intended for our case.

The objectives of this study became twofold: 1) establish new transient reﬂectance models for perovskite analysis, and 2) use them to acquire new information on perovskite photophysics. This approach led to several discoveries, such as evidence of non-ambipolar charge carrier diffusion and signals correlating with perovskite sample quality. It also provided knowledge of key carrier dynamics in greater detail than before, such as the charge trapping and the hot-carrier behaviour. Many of these results have practical implications for future perovskite devices, and some of the ﬁndings dispute popular theories in the literature.

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TIIVISTELMÄ

Halidiperovskiitit ovat nousseet yhdeksi johtavista seuraavan sukupolven materi- aaleista aurinkokennoihin sekä muihin valosähköisiin sovelluksiin. Perovskiittiau- rinkokennojen hyötysuhde on jo ylittänyt 25%, mikä on enemmän kuin kaupalli- sissa piikennoissa, vaikka perovskiittien valmistuskustannukset ovat niihin nähden murto-osa. Niiden etuihin lukeutuvat myös värin ja muiden ominaisuuksien säädet- tävyys.

Aloittaessamme perovskiittitutkimuksen vuonna 2018, tietämys perovskiittien fotofysiikasta oli jäljessä niiden synteesi- ja aurinkokennokehitystä. Täten meidän tavoitteenamme oli tutkia varausten käytöstä perovskiiteissa ultranopealla aikaerot- teiselle spektroskopialla. Perinteiset analyysimallit eivät kuitenkaan kyenneet selit- tämään näkemiämme aikaerotteisia absorptiosignaaleita. Tiedeyhteisössä oli tapana olettaa, että ainoastaan virityksen tuottamat absorbanssin muutokset vaikuttavat sig- naaliin, mutta me havaitsimme, että myös muutokset näytteen heijastuksessa olivat hyvin merkittävä osa signaalia. Siinä missä aiemmin näytteet olivat olleet pienen taitekertoimen liuoksia tai kalvoja, uudet perovskiittinäytteet olivat korkean taitekertoimen ohutkalvoja, mikä lisäsi merkittävästi valon heijastusta. Signaalin tulkin- taa varten meidän täytyi turvautua aikaerotteiseen heijastusspektroskopiaan, mutta se oli harvinainen metodi eivätkä saatavilla olleet mallit sopineet tilanteeseemme.

Tutkimuksellemme muodostui siten kaksi tavoitetta: 1) uusien mallien kehit- täminen aikaerotteisille heijastusmittauksille, ja 2) niiden hyödyntäminen perovskiittien fotofysiikan tutkimuksessa. Tämä lähestymistapa johti lukuisiin uusiin havain- toihin, kuten todisteisiin ambipolaarittomasta varausten diffuusiosta ja signaaleihin, jotka korreloivat perovskiitin laadun kanssa. Se myös tuotti aiempaa yksityisko- htaisempaa tietoa tärkeistä varauksiin liittyvistä tekijöistä, kuten varausansoista ja kuumien varausten käyttäytymisestä. Monilla näistä tuloksista on käytännön merk- itystä tuleville perovskiittilaitteille, ja jotkin löydöksistä haastavat kirjallisuudessa suosittuja teorioita.

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1.2 (a) Glasses are often coated with anti-reflective coatings that produce destructive thin-film interference, minimizing the reflectance. (b) The steady-state reflectance and transmittance spectra of a 500 nm thick perovskite film. The wavy pattern above 700 nm wavelength is caused by thin-film interference. . . 3

3.1 Reﬂection of light at an interface: I is the original intensity,T is the transmitted light and Ris the reﬂected light. Reprinted under CC BY-SA 3.0 licence from Wikipedia[36]. . . 12

3.2 (a) Constructive and (b) destructive interference. Reprinted under GNU Free Documentation License from Wikipedia[37]. . . 13

3.3 Light initially propagates through medium 0, presented by the ma- trixP_M₀. Then some of it is reflected at the first interface (I_M₁) while the rest continues propagation in medium 1 (P_M₁). The transfer matrix method presents these propagations and interfacial reflections as matrices and multiplies them together to acquire the total reflectance and transmittance of the system. . . 14

3.4 Reflections from the substrate. If coated with a thin film, the respec- tiveRandT need to be replaced with the reflectance and transmittance of the film either from the air-film interface or substrate-film interface. . . 15

3.5 Relation betweenΔnandΔkaccording to KK relations: a photoinduced change in absorption also changes the refractive index. . . 16

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4.1 Pump-probe schematics: The sapphire laser produces ultrashort (about 100 fs) pulses with a wavelength of 800 nm. 1000 of these pulses are generated each second, but the optical modulator reduces this by half. The Topas-C unit converts the original 800 nm pump beam to the desired pump wavelength. The probe beam is directed to a tunable delay setup which adjusts the travel distance of the probe pulses:

it adjusts the delay time, in other words the time difference between the pump and the probe pulses hitting the sample. Before hitting the sample, the probe pulse passes through a medium that turns it into white light: a cuvette ﬁlled with heavy water for producing visible light or a sapphire crystal for producing stronger NIR light. The probe beam is split twice to acquire one reference measurement before the white light conversion and another after the conversion. The pump spot is between 1-2 mm², while the probe spot is much smaller, a fraction of a square millimeter. . . 18

4.2 Δn and Δk ﬁt done with two different ﬁlm thicknesses, 520 and 525 nm, using linear approximation. TheΔk has been multiplied by 5 for clarity. . . 21

4.3 A 300 nm ﬁlm withn=1.7 deposited either on pure glass (n=1.5) or glass coated with a 50 nm thick TiO₂layer (n = 2.75): (a) the steady-state reﬂectance spectra and (b) transient transmittance spectra withΔñ=0.001+0.001iapplied only to the low-refractive index layer. . . 23

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4.4 ApproximatedΔA_OD based on Equation (4.15) versus the original ΔA_OD when the|Δk/Δn|ratio is (a) 1, (b) 0.1, and (c) 0.01. (a-c) present the visible range while (d-f) present the NIR under similar conditions. More precisely, theΔkis ﬂipped from negative to positive at 600 nm wavelength and back to negative at 1000 nm, while the Δnis maintained at constant -0.001. (g) The fullΔR_OD,ΔT_ODand ΔA_ODspectra over the entire wavelength range with|Δk/Δn|=0.1.

The modelled system was a 600 nm GaAs ﬁlm on glass, but theΔn andΔkspectra used here were for modelling purposes only and do not represent the real photoinducedΔnandΔk of a GaAs ﬁlm. . . . 26 4.5 TFI based TR model which accounts for the inhomogeneous carrier

distribution. The solid black line is the refractive index across the model, the dashed line and the blue line are the refractive index of the film before and after the excitation, respectively. An inhomogeneous carrier distribution causes a gradient in the refractive index inside the film (n_f), which is modelled by splitting the film into multiple layers, each with theΔñmultiplied by the carrier concentration in that layer (a>b>c). . . . 28 4.6 The TFI based diffusion fitting requires three preliminary datasets:

1) the complex refractive index and the thickness of the ﬁlm (including the roughness if necessary), 2) the photoinduced Δn and Δk spectra, and 3) the recombination rates and surface recombination velocity. . . 30 4.7 The distribution of charge carriers immediately after excitation (a)

without roughness and (b) with roughness. The latter distribution is stretched at the perovskite surface to have the same total number of charge carriers in both cases. . . 32

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4.8 Effect of surface roughness on transient reﬂectance and transmittance simulations at 880 nm[41]. The roughness thickness was 90 nm while the rest of the perovskite ﬁlm on a glass substrate was 496.5 nm thick. The transmittance signal was multiplied by 4 for clarity. . . 33 4.9 Some parts of the TFI pattern are more sensitive to the differences in

the carrier distribution than the others. (a) The carrier distributions as a function of time and (b) the TR spectra in NIR at 0 and 1000 ps delay times. (c) The TR response to the diffusion is sensitive up to 1 ns at 875 nm wavelength whereas (d) at 1050 nm wavelength the TR loses responsiveness to the carrier distribution after 200 ps. A 540 nm perovskite ﬁlm (ñ≈2.4+i0.013 andΔñ=−0.001+i0.0001) on glass was used in this simulation. . . 34 4.10 Comparison between the TFI model and Equation (4.23) by mod-

elling TR signals of a MAPbI₃perovskite film with different thicknesses. With a thickness of 480 nm, the perovskite film can produce a very similar-looking signal as the Equation (4.23) would suggest (after normalization to match the signal strength). However, a 540 nm thick perovskite film would produce an entirely opposite signal, and the Equation (4.23) only truly matches the 3000 nm thick perovskite layer near the bandgap. . . 35 4.11 ΔR response of (a) MAPbI₃perovskite and (b) GaAs according to

the TFI model. Different baseΔñ(multiplied by the carrier distribution in the TFI model) were used in order to demonstrate how the SCC compares to the TFI model under different photoinducedΔn and Δk. TheΔRresponses were normalized to match the SCC at 5 ns delay time. The ﬁlm thickness in both cases was 3000 nm, the diffusion constant was 1 cm²/s and the probing wavelength 700 nm. 36

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6.1 (a) Absorption (solid lines) and PL spectra (dashed lines) of CsSnI₃ and CsSn_0.6Ge_0.4I₃NCs in suspension. b) Deconvolution of the PL spectrum of CsSn_0.6Ge_0.4I₃NCs[84]. . . 47

6.2 ΔT spectra as function of time of (a) CsSnI₃and (b) CsSn_0.6Ge_0.4I₃ NC ﬁlms encapsulated between two glass plates. The excitation wavelength was 500 nm and intensity 10μJ cm⁻²[84]. (c) The NIR TR (blue) and TT (red) spectra of CsSnI₃NC ﬁlms at 10 ps delay time.

The wave pattern of the TR spectra is caused by the encapsulation glass plates acting as a Fabry–Pérot interferometer. The reﬂectance effects were relatively minor in these samples compared to the photoinduced absorption changes. . . 49

6.3 (a) PL decays of CsSnI₃and CsSn_0.6Ge_0.4I₃NCs in suspension, excited at 405 nm. (b) TT decays of the encapsulated CsSnI₃and CsSn_0.6Ge_0.4I₃ NC films at 700 nm probe wavelength. The excitation wavelength was 500 nm and intensity 10μJ cm⁻². The solid lines represent the fitting results, and the the insets show magnified decays in early timescale.

In (b) the bleaching of CsSn_0.6Ge_0.4I₃NCs begins its decay 1 ps later than in the case of CsSnI₃NCs[84]. . . 50

6.4 Measured and simulated steady-state (a) transmittance and reflectance, and (b) absorption of a 510 nm thick CsMAFA perovskite film. The measuredAis not corrected for reflectance, and the perceived peaks in NIR are a product of reflectance. . . 52

6.5 TT and TR spectra of the CsMAFA perovskite sample in (a-b) visible range and (c-d) NIR[89]. The excitation wavelength was 600 nm. . . 53

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6.6 The photoinducedΔn andΔk based on the TT and TR spectra of Figure 6.5 at 100 ps delay time in (a) the visible range and (b) NIR [89]. The squares connected by fine dashed lines are the directly calculated spectra, and the solid lines are the smoothed spectra. The modelled film thicknesses for the fitting were 520.6 for the visible range and 522.4 nm for the NIR. . . 54

6.7 (a) TR and TT spectra of the 70 nm thick perovskite ﬁlm in the wavelength range from 420 nm to 600 nm, and (b) the respectiveΔñﬁt (normalized and smoothed) from 450 nm to 600 nm. The excitation wavelength was 400 nm. . . 54

6.8 Δk spectra differences between different CsMAFA perovskite samples. The almost negative Δk at 900 nm hints towards more traps being present in sample 2 than in sample 1. . . 55

6.9 (a) TR spectra of different CsMAFA perovskite samples (approximately 500 nm thick) compared to their simulated TR in the 450 – 600 nm wavelength range and 2 ps delay time, excited at 400 nm wavelength. (b) The decay of these TR signals at 503 nm. The ﬁg- ures were normalized to 1 at 503 nm wavelength and 1 ps delay time.

Many perovskite samples exhibited a sharp positive peak at 505 nm, while according to the simulation based on theΔñfrom Figure 6.7, the perovskite should have no such peak. This signal at 500 nm can be attributed to either unreacted PbI₂or undesired yellow-phase perovskite, which have their bandgap near 500 nm, and therefore we can estimate the sample quality based on the relative strength of this signal. 56

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6.10 Perovskite TT spectrum changes if any HTL or ETL layers are added to the sample structure despite theΔnandΔk staying the same. In this example, we took theΔñfrom Figure 6.6 (b) and modelled the reﬂectance of a 530 nm perovskite ﬁlm with either 50 nm TiO₂at the bottom or 50 nm spiro-OMeTAD on top. . . 57

6.11 Normalized bandgap TT response of a CsMAFA perovskite ﬁlm with 400 nm excitation and different pump energies. (a) The solid lines and the dashed lines are the TT spectra at 2 ps and 1 ns delay times, respectively. The bandgap bleaching is initially blueshifted with high pump energy, and remains partially blueshifted up to nanoseconds if the pump energy is high. (b) TT response at 705 nm normalized to have matching maximum bleaching at 2 ps delay time, also showing how the blueshift can last up to nanoseconds with high pump power.

TR was also measured to ensure these signals were not signiﬁcantly affected by any TR effects. . . 58

6.12 Initial carrier concentration in csMAFA perovskite with 400 nm excitation. Majority of the carriers are generated within 50 nm from the ﬁlm surface. This 50 nm layer of perovskite has absorbance of only 0.05 at the bandgap at 750 nm probe before the excitation, and after the excitation and initial carrier cooling the bandgap bleaching is approximately -0.05. Therefore, the bandgap of the surface layer is fully or almost fully saturated. . . 59

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6.13 Semilogarithmic and normalized TT response of a 70 nm thin perovskite ﬁlm probed at 750 nm wavelength. Apart from the pristine case, the perovskite was either deposited on C₆₀or coated with spiro- OMeTAD. The C₆₀layer reduced the charge carrier lifetime whereas spiro-OMeTAD did not, implying that photoexcited holes do not impact the TT or TR signals. The excitation wavelength was 610 nm and energy 2.3μJ cm⁻². . . 60

6.14 (a) TT and (b) TR spectra as a function of time. Diffusion causes a change in the TR response, which also appears in the TT response but in the opposite direction[41]. . . 61

6.15 Simulated (a) charge carrier distributions and (b-d) TT and TR signals at wavelengths 880, 936, and 1040 nm, respectively [41]. ΔT was multiplied by 4 for clarity. A 90 nm thick roughness layer was included in the simulation, and the diffusion constant was approximately 1.64 cm²s⁻¹. Carrier lifetime was estimated as 14.7 ns, and the higher-order recombination rates were presumed negligible. The excitation wavelength was 530 nm. . . 62

6.16 TR data of the perovskite sample 2 in Table 6.2 compared to simulation with D = (0.20+1.40e^−t/^40ps) cm²s⁻¹: (a) carrier distribution over time, and (b) TR signal compared to the TR simulation at 477 nm wavelength. The ﬁgure shows both our TFI model (sim.ΔR) and the SCC model from the literature. The excitation wavelength was 400 nm and energy 7.7μJ cm⁻². . . 64

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6.17 Diffusion dependence on excitation energy with 510 nm excitation wavelength: (a) Normalized ΔT decays of the bandgap bleaching with solid and dashed lines representing the measured and the modelled decays, respectively. The higher-order recombination becomes more prevalent with increasing excitation density. (b) Normalized ΔRdecays with the same excitation densities, where the solid and dashed lines represent measured and modelled decays, respectively, at the 1159 nm wavelength. The diffusion fitting values are given in Table 6.2 sample 3, but overall the increase in excitation energy or density had little impact on the diffusion fitting result once the increase in carrier recombination was taken into account. . . 65 6.18 The influence of the excitation wavelength on the TR signal of per-

ovskite sample 6 in Table 6.2: (a) TR spectra at 5 ns delay time at 1000-1200 nm wavelengths, normalized to 1 at 1070 nm. (b)ΔRde- cays from 1 ps to 5 ns at 1124 nm, solid and dashed lines are the measured data and the diffusion ﬁts, respectively, withD= (0.20+ 1.20e^−t^/^{60 ps})cm²s⁻¹. . . 65 6.19 Sum mobility of photo-excited electrons and holes measured by time-

resolved terahertz spectroscopy[41]. The real part is extrapolated to the DC-value. . . 68 6.20 GaAs thin ﬁlm (a) TT and (b) TR visible range spectra with 600 nm

pump, and (c) TT and (d) TR NIR spectra with 700 nm pump. The spectra show slow carrier relaxation to the lowest conduction band up to 7 ps, after which the response is very stable due to the slow recombination rate. . . 69 6.21 PhotoinducedΔn and Δk in GaAs thin ﬁlms above the bandgap

after 10 ps delay time. . . 70

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6.22 MeasuredΔRdecay of the GaAs wafer compared to simulation with D=120 cm²s⁻¹similarly to Figure 6.16: (a) Carrier distribution as a function of time, (b) measuredΔRsignal at 1060 nm wavelength compared to both the TFI and SCC models. Beginning around 50 – 100 ps the measuredΔRdeviates from the models and becomes negative at longer delay times likely due to charge trapping at the surface. 71 6.23 NIR TR spectra of the GaAs wafer with 500 nm excitation in the

later delay times, showing how the TR signal drops to negative at most wavelengths after 100 ps. . . 71

List of Tables

6.1 NC PL and TT decay fitting results . . . 51 6.2 Diffusion fitting results . . . 66 6.3 Additional diffusion fitting settings . . . 66 6.4 Observed phenomena and their timescales for CsMAFA perovskite . 72 6.5 Observed phenomena and their timescales for GaAs . . . 72

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ABBREVIATIONS

CsMAFA (CsI)_0.05[(MAPbBr3)_0.15(FAPbI₃)_0.85)]_0.95 DMF dimethylformamide

DMSO dimethyl sulfoxide

EDS energy-dispersive X-ray spectroscopy ETL electron transport layer

FA formamidinium

FAMACs alternative name to CsMAFA

HC hot carrier

HTL hole transport layer

KK Kramers-Kronig

LiTFSI lithium bis(triﬂuoromethanesulfonyl)imide

MA methylammonium

NC nanocrystal

NIR near-infrared

PCE power conversion efﬁciency

PL photoluminescence

QY quantum yield

SCC surface-carrier concentration SnGe-NC CsSn_xGe₁₋_xI₃nanocrystal TA transient absorption

TCSPC time-correlated single-photon counting

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TFI thin-ﬁlm interference TR transient reﬂectance

TRMC time-resolved microwave conductivity TRPL time-resolved photoluminescence TRTS time-resolved terahertz spectrocopy TT transient transmittance

XPS X-ray photoelectron spectroscopy

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SYMBOLS

A absorbance

D diffusion constant E electric ﬁeld

I intensity

I_M interface matrix

L thickness

M full transfer matrix OD optical density

P Cauchy principal value P_M propagation matrix R reﬂectance

S surface recombination velocity T transmittance

T_a temperature

V_{f rac} relative volume fraction

Δn_S photoexcited sheet carrier concentration α attenuation coefﬁcient

ε

ˆ complex dielectric function

λ wavelength

μ carrier mobility μ sum mobility

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ω angular frequency τ lifetime

ñ complex refractive index θ angle of incidence ₀ vacuum permittivity c speed of light

c_P relative defect concentration

f frequency

k extinction coefﬁcient k_B Boltzmann’s constant n refractive index

n_{e f f} effective refractive index

r₁ 1st order recombination rate constant r₂ 2nd order recombination rate constant r₃ 3rd order recombination rate constant r_p reflectance coefficient for p-polarized light r_s reflectance coefficient for s-polarized light

t time

t_p transmittance coefﬁcient for p-polarized light t_s transmittance coefﬁcient for s-polarized light u carrier density

x location

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ORIGINAL PUBLICATIONS

Publication I H. P. Pasanen, P. Vivo, L. Canil, A. Abate and N. Tkachenko.

Refractive Index Change Dominates the Transient Absorption Response of Metal Halide Perovskite Thin Films in the near In- frared.Phys. Chem. Chem. Phys.21.1 (2019), 14663–14670. DOI:

10.1039/C9CP02291K.

Publication II H. P. Pasanen, P. Vivo, L. Canil, H. Hempel, T. Unold, A.

Abate and N. V. Tkachenko. Monitoring Charge Carrier Dif- fusion across a Perovskite Film with Transient Absorption Spectroscopy. J. Phys. Chem. Lett. 11.2 (2020), 445–450. DOI:

10.1021/acs.jpclett.9b03427.

Publication III M. Liu, H. Pasanen, H. Ali-Löytty, A. Hiltunen, K. Lahtonen, S.

Qudsia, J.-H. Smått, M. Valden, N. V. Tkachenko and P. Vivo. B- Site Co-Alloying with Germanium Improves the Efﬁciency and Stability of All-Inorganic Tin-Based Perovskite Nanocrystal Solar Cells.Angew. Chem. Int. Ed.59 (2020), 22117–22125. DOI:^10.

1002/anie.202008724.

Publication IV H. P. Pasanen, M. Liu, H. Kahle, P. Vivo and N. V. Tkachenko.

Fast non-ambipolar diffusion of charge carriers and the impact of traps and hot carriers on it in CsMAFA perovskite and GaAs.

Materials Advances(2021). DOI:10.1039/d1ma00650a. Author’s contribution

Publication I I performed the measurements, some of the data analysis, and wrote parts of the manuscript such as where the Δn and Δk

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are ﬁtted without the linear approximation. Nikolai Tkachenko established the linear approximation method for calculating the Δn andΔk from the measuredΔRandΔT. Laura Canil and Antonio Abate prepared the perovskite samples and wrote the corresponding section.

Publication II I performed the transient reﬂectance and absorption measurements and wrote the matlab-based ﬁtting programs except parts of the transfer matrix code and the diffusion simulation, which were acquired from elsewhere. I then performed the data analysis and wrote majority of the manuscript. Laura Canil and An- tonio Abate prepared the perovskite samples and wrote the corresponding section. Hannes Hempel and Thomas Unold performed the time-resolved terahertz spectroscopy measurements and data analysis, and wrote the corresponding section.

Publication III I performed the transient absorption and reﬂectance measurements, participated in the photophysics analysis of the photoluminescence and pump-probe results, and wrote parts of the corresponding analysis. Maning Liu synthesized the nanocrystals, performed the time-correlated single-photon counting measurements, and wrote the majority of the manuscript as the main author.

Publication IV Similarly to Publication 2, I performed the measurements, data analysis and wrote the majority of the manuscript. Maning Liu prepared the perovskite samples and wrote the corresponding ex- perimental section. Hermann Kahle provided the GaAs samples and wrote the corresponding sample preparation section.

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1 INTRODUCTION

Metal halide perovskites are one of the most promising materials for next-generation solar cells. Despite their relatively recent debut in solar cell research in 2009 [1], perovskite-based solar cells quickly caught up to the traditional silicon solar cells in power conversion efﬁciency (PCE)[2]. However, the main advantage of perovskite over the Si, GaAs and other common semiconductors is not the PCE, despite the fairly high record value of over 25%[2], but the very cheap fabrication process: perovskites can be synthesized by solution processing methods in ambient conditions [3], whereas other high PCE devices have to be prepared with expensive equipment and methods, such as growing the solar cell in very high temperature over several hours[4]. Perovskite solar cells can be printed from start to ﬁnish in a roll-to-roll process, which makes them highly desirable from industrial and economic perspec- tives[5].

Perovskites are a class of minerals with a speciﬁc crystal structure (Figure 1.1 (a)), and one of their subsets, halide perovskites, can be utilized in various photovoltaic applications[6]. The rich chemistry of halide perovskites enables an endless number of different combinations. Aside from synthesizing bulk ﬁlms, perovskites can also be turned into nanocrystals (NCs) and quantum dots[7], further expanding the pos- sibilities. This wide tunability allows for relatively easy optimization of the optical properties of perovskite materials for each application, such as light-emitting diodes [8, 9], lasers[10], photodetectors[11, 12], and multi-junction solar cells[13].

Standard single-junction perovskite solar cells are presented in Figure 1.1 (b-c):

perovskite is the active layer, meaning the layer that absorbs the light and therefore produces photoexcited charge carriers, electrons and holes. It is sandwiched between the electron- and hole transport layers (ETL and HTL), which determine the direction of the current generated by the solar cell: by choosing materials with matching conduction and valence band energy levels with the perovskite, the ETL accepts photoexcited electrons from the perovskite but does not allow their transfer back. The

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(a) (b)

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Figure 1.1 (a) Perovskite crystal structure ABX₃, (b) prototype perovskite solar cells prepared on a glass substrate (photo credits: Arto Hiltunen), (c) a scanning electron microscope image of perovskite solar cell cross-section.

photoexcited electrons leave behind so-called positively charged holes, which are in turn accepted by the HTL. In other words, the HTL donates electrons to perovskite in order to compensate for the positive charges. These three layers are then covered by the electrodes, the front one having to be transparent. The entire thickness of the solar cell is just few micrometers, which is only a small fraction of the width of an average human hair.

Due to the variety of perovskite structures and compositions, their research requires many measurement techniques for characterization of their photophysical and other properties. Transient absorption spectroscopy (TA, also known as time- resolved absorption spectroscopy) is one such technique, which has been traditional employed to study the excited state lifetime, charge transfer, band structure and various other phenomena of photoactive materials. TA functions as follows: When a material absorbs light, in other words photons, the photon energy is transferred to an electron in the material, which jumps to a higher energy level known as the excited state. This excited electron absorbs light differently than the non-excited

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(a) (b)

WAVELENGTHNM

2AND4

2EFLECTANCE 4RANSMITTANCE

Figure 1.2 (a) Glasses are often coated with anti-reflective coatings that produce destructive thin-film interference, minimizing the reflectance. (b) The steady-state reflectance and transmittance spectra of a 500 nm thick perovskite film. The wavy pattern above 700 nm wavelength is caused by thin-film interference.

electrons that are in the so-called ground state. TA measures this difference between the ground state absorption and the excited state absorption (ΔA) by ﬁrst exciting the sample with a laser pulse known as the pump pulse, which is then followed by another probe light pulse for measuring the transmittance of the sample. By compar- ing the transmittance before and after the excitation, TA determines the absorption spectrum of the excited state. The timing between the pump and the probe pulses can be adjusted, allowing us to monitor how the excited state evolves over time from femtoseconds to nanoseconds.

However, TA had been in the past used primarily with solution samples, materials with low refractive index, or inhomogeneous thin films (such as mesoporous TiO₂), where reflectance effects were of little significance. These were for example dye molecules or semiconducting polymer films. With samples like perovskites and compact titanium dioxide (TiO₂) films the situation changed: higher refractive index and thin-film samples meant that the reflection played a far bigger role in the transmittance of the samples than it had previously (Figure 1.2 (b)). Absorption cannot be measured directly: we can only measure transmittance and reflectance of any sample, and then calculate the absorption based on theoretical models. TA studies are therefore transmittance measurements where the photoinduced change in absorbance (ΔA) was assumed to be the only relevant factor, but that was no longer the case with many perovskites and other thin-film samples. Proper analysis of thin-film TA spectra now requires a complementary transient reflectance (TR) study, which is essentially the same as a TA measurement expect that it is done re-

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flectance mode instead of transmittance mode. So far our research group has found significant reflectance effects in many thin-film samples such as perovskites, gallium arsenide (GaAs), TiO₂, and doped graphene. As a rule of thumb, any transparent to semitransparent thin-film sample should be measured in both reflectance and transmittance modes in order to correct the TA spectra for reflectance changes.

This thesis is split into two parts: The first part addresses the modelling of transient reflectance signals (Sections 2-4), and the second part presents the new discoveries revealed by the TR measurements (Sections 5 and 6). Modelling the TR requires factoring in the thin-film interference (TFI), and despite it being a very old and well- known concept in physics, it had been ignored in TR studies to simplify the data analysis. In short, TFI occurs when multiple reflections from different interfaces, such as the air-film and film-substrate, interfere with each other either constructively or destructively. It is a very useful phenomenon for instance in anti-reflective coatings (Figure 1.2 (a)), as we can control the reflectivity of a surface by adding a thin (nanoscale) transparent film on it. As mentioned previously, TR is an uncommon measurement technique because it was rarely required in the past, and the inadequate analysis models may have also led some scientists to abandon their TR results as too difficult to analyse. My initial work focused on programming the TFI based TR models which were absent from the literature, as the previous non-TFI models were far too simplified to explain the TR signals we were seeing.

The few TR studies reported in the literature had focused on three main topics: 1) charge carrier mobility[14], 2) carrier surface recombination or surface traps[15], and 3) extracting the real photoinducedΔAspectra from the measuredΔRandΔT spectra[16]. Our practical ﬁndings presented in this thesis and the publications we base it on impact all three of these main applications, and thanks to the more sophis- ticated data analysis we were able to better evaluate phenomena such as the charge trapping at the sample surface, charge carrier diffusion in single crystals and across grain boundaries, defects such as shallow traps, unreacted precursors on the sample surface, and so on. Many of these ﬁndings have practical implications for photovoltaic devices, and they can be utilized in characterization and quality assessment of perovskites and various other materials. The background and prior publications of each of these subjects will be addressed in greater detail in their respective sections:

Section 2 for the transient reﬂectance methodology, and Section 5 for the perovskite photophysics.

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1.1 Hypothesis

1. Accounting for the TFI will achieve consistent interpretation of the pump- probe data of both reﬂectance and transmittance modes, and this will in turn 2. allow us to better extract new quantitative information on the key carrier dynamics that would otherwise be inaccessible, including charge carrier diffusion perpendicular to the sample surface, hot-carrier dynamics, PbI₂impurities, and so on.

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2 TRANSIENT REFLECTANCE LITERATURE BACKGROUND

The debate around TR and its importance began with attempts to correct the bandgap TA spectra of perovskites in order to better analyse the hot-carrier cooling and other phenomena [15, 16, 17, 18, 19]. However, many of the proposed approaches are either very difficult to implement experimentally or have led to wildly different in- terpretations of the importance and scale of the reflectance effects on the TA spectra [19]. For instance, Ruhman et al[19]. compared the TA spectra of a perovskite thin- film to its nanoparticle counterpart in solvent to reveal any potential effects from reflectance. Obviously, this method only works if one can produce both versions of the material without alterations to the energy states or other properties, which is very unlikely. Meanwhile, Tamming et al. [16]modified a standard TA setup with a white light pulse interferometer to probe the ultrafast refractive index changes but severely limiting the observable timescale. Liu et al.[20]reduced the impact of reflectance on the TA spectra by applying an oil droplet on the sample, which reduces the reflectance from the film by reducing the gap in the refractive indexes.

Herein are two new methods presented for correcting the TA spectra, by which I mean extracting the real photoinducedΔAbased on the photoinducedΔRandΔT: 1. An efﬁcient linear approximation for calculating theΔnandΔkbased on the

measuredΔRandΔT, and a TFI model.

2. A simple ΔA approximation that only requires the steady-state reﬂectance spectra in addition to the measuredΔRand ΔT, without any need for TFI modelling.

If the main focus of the study is on the carrier lifetimes or only the changes in absorption are of interest, where information onΔnis not needed, the latter method provides a quick and easy approximation of theΔAfrom theΔRandΔT. I will

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also provide an analysis on the accuracy of this approximation, and I believe the methods developed by us are more accurate and universally applicable than the ones previously established in the literature.

The second main application of TR is the determination of charge carrier diffusion and mobility. The TR signal, in part, depends on the distribution of charge carriers inside the film. First of all, the reflectance of light at an interface depends on the difference in refractive index between the two media. The strength of the pho- toinducedΔndepends on the number of excited charge carriers, so the more carriers there are at the surface the stronger the TR signal should be. This lead to what I call the surface-carrier concentration (SCC) model[13, 14, 15, 21], which assumes that in the absence of TFI (when the film absorption is high enough to prevent interference from the film bottom interface), the TR signal strength only depends on the carrier concentration at the surface. Another assumption that was often made was that theΔncaused by a single carrier is constant across time, in other words, there is no change in the state of the carrier apart from recombination. Under these conditions, one could easily model the decay in TR signal as a function of recombination and carrier diffusion away from the surface.

However, what was not taken into account was the gradient in the refractive index inside the film: when theΔnis not uniform due to inhomogeneous carrier distribution, the refractive index cannot be uniform either. This gradual change in refractive index generates reflections and interference from inside the film even when the absorption and film thickness are enough to prevent interference from the film bottom interface. The new TFI based model developed by us and presented in Section 4 is the first to account for this effect, which proved to be very significant for modelling carrier diffusion from TR measurements. I will compare the two models, the SCC and TFI models, to validate under which conditions the earlier assumptions made in the literature were valid.

Nonetheless, TR is a powerful method for charge carrier diffusion and mobility studies. Most applications rely on carrier mobility perpendicular to the ﬁlm surface, which is something the other contact-free methods cannot measure. These other methods, such as combining time-resolved photoluminescence (TRPL)[22] or transient absorption[23, 24]with optical microscopy can only measure the diffusion parallel to the ﬁlm surface. In polycrystalline materials such as perovskites, these methods typically end up measuring diffusion across multiple grain bound-

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aries rather than the real diffusion in the devices. Contact-free methods also include time-resolved terahertz spectroscopy (TRTS) and time-resolved microwave conductivity (TRMC), which measure the sum mobility of the carriers across very short dis- tances inside single grains[25, 26, 27]. Even combining these different techniques leaves open major questions: if one technique gives the minimum mobility across grain boundaries and the other the maximum mobility inside single grains, and the two are orders of magnitude apart[28, 29], what is the real carrier mobility inside the devices? PL decay has been employed to study the perpendicular diffusion by adding a charge extraction layer[30], but those results suffer from poor reliability because they require the ETL/HTL to accept the charge carriers as fast as they dif- fuse to the interface. PL decay can therefore measure either the charge transfer rate or the diffusion, but only the slower factor of the two. The contact-based methods for measuring perpendicular diffusion also have their own problems because they require using high voltage across the sample, which may result in ionic movement in addition to the electrode-film interface complications, in addition to other interfacial problems[31, 32, 33, 34]. Despite its own difficulties, TR remains the only contact-free method for probing the carrier diffusion and mobility perpendicular to the film surface, and it is also one of the few with ultrafast time resolution.

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3 STEADY-STATE REFLECTANCE MODELLING

Before we look at modelling the transient reflectance signals, we need to establish how the reflectance is modelled in general. This section goes through these equations and models from the literature that are relevant for the basic modelling of steady-state and therefore transient reflectance signals.

3.1 Complex refractive index

The complex refractive index ñ combines the refractive index n, which gives the speed of light in the material, with the imaginary partkcalled extinction coefﬁcient, which is linked to the absorption of the material via

α=4πk/λ, (3.1)

where α is the attenuation coefﬁcient andλ is the wavelength [35]. The relative absorption of a layer is then given bye^−αLwhereLis the thickness of the absorbing medium.

In order to model steady-state reflectance, one only needs to know the sample structure (complex refractive index and the thickness of each material or layer), as well as the incident angle and polarization of the probe light. Once we know how to model the steady-state spectra, the transient reflectance models are simply about modelling the reflectance first in the ground-state with some complex refractive in- dexñ_{g s} =n_{g s}+i k_{g s}, and then modelling the excited state withñ_{e s}=ñ_{g s}+Δñ= n_{g s}+Δn+i(k_{g s}+Δk), and taking the difference in reflectance between these two cases.

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Figure 3.1 Reﬂection of light at an interface:I is the original intensity,T is the transmitted light andR is the reﬂected light. Reprinted under CC BY-SA 3.0 licence from Wikipedia [36].

3.2 Fresnel equations

Modelling the reﬂection from the interface between two different media, as in Fig- ure 3.1, is done via Fresnel equations[36]. The Fresnel coefﬁcients are

r_s= ñ₁c osθ_i−ñ₂c osθ_t

ñ₁c osθ_i+ñ₂c osθ_t (3.2) r_p=ñ₁c osθ_t−ñ₂c osθ_i

ñ₁c osθ_t+ñ₂c osθ_i (3.3) for s- and p-polarized light, respectively.ñ₁is the complex refractive index of medium 1,ñ₂of medium 2,θ_i is the angle of incidence before the interface andθ_t after the interface, which can be calculated from the Snell’s law:

n₁s i nθ_i=n₂s i nθ_t. (3.4) The reﬂectanceRis then simply the square of the coefﬁcient:

R=|r|² (3.5)

for both the s- and p-polarizations. We can similarly calculate the Fresnel coefﬁcients

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(a) (b)

Figure 3.2 (a) Constructive and (b) destructive interference. Reprinted under GNU Free Documentation License from Wikipedia [37].

for transmittance at the interface:

t_s= 2ñ₁c osθ_i

ñ₁c osθ_i+ñ₂c osθ_t (3.6) t_p= 2ñ₁c osθ_i

ñ₁c osθ_t+ñ₂c osθ_i. (3.7)

3.3 Transfer matrix and thin-ﬁlm interference

The total reflectance of thin films is more complicated than in the case above. Thin- film interference is a phenomenon where reflections from multiple interfaces, such as the top and bottom interfaces of a thin layer, interfere with each other either constructively or destructively (Figure 3.2)[38]. In other words, if the phases of the reflections match as they exit the system, they are combined for increased reflectance. Similarly, if the phases are opposite to one another, the destructive interference reduces the reflectance. The phase of the light as it exits the layer depends on the wavelength, distance it has travelled, and the refractive index. Thus, depend- ing on the layer thickness and incident angle, the reflectance is maximized at some wavelengths while minimized at others. TFI requires a very well-ordered material with minimal scattering and thickness variation, which is why it mainly appears in nano- to micrometer scale materials. Thick layers, such as one-millimeter thick glass plates, have too incoherent reflections to generate interference, and reflections from the two interfaces can be simply summed up. Reflectance, including transient reflectance, can be measured from non-uniform samples with incoherent reflections or high scattering, but they require an alternative analysis method.

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Figure 3.3 Light initially propagates through medium 0, presented by the matrixP_M0. Then some of it is reflected at the first interface (I_M₁) while the rest continues propagation in medium 1 (P_M1). The transfer matrix method presents these propagations and interfacial reflections as matrices and multiplies them together to acquire the total reflectance and transmittance of the system.

Transfer matrix is a method for modelling thin-film interference by representing each interface and propagation through a layer as a 2-by-2 matrix[38]. By multiply- ing these matrices, we acquire the reflectance coefficients and the total reflectance.

An interface matrix (I_M) is

I_M = 1 t

⎡

⎣1 r r 1

⎤

⎦ (3.8)

where r andt are the coefﬁcients from Fresnel relations, and a propagation matrix (P_M) is

P_M =

⎡

⎣eⁱ²^{πñLc osθ/λ} 0 0 e⁻ⁱ²^{πñLc osθ/λ}

⎤

⎦. (3.9)

The full transfer matrixM contains all the interface and propagation matrices as in Figure 3.3:

M=P_M₀I_M1P_M1I_M₂P_M₂...=

⎡

⎣M₁₁ M₁₁ M₂₁ M₂₂

⎤

⎦ (3.10)

and the coefﬁcients are

r_M= M₂₁

M₁₁ (3.11)

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Figure 3.4 Reflections from the substrate. If coated with a thin film, the respectiveRândT ^{need to be} replaced with the reflectance and transmittance of the film either from the air-film interface or substrate-film interface.

t_M = 1

M₁₁. (3.12)

The reﬂectance and transmittance are calculated similarly to Equation 3.5:

R=|r_M|² (3.13)

T = n_sc os(θ_s)

n₀c os(θ₀)|t_M|² (3.14) where n_s and θ_s are the refractive index and the angle of incidence of light in the substrate, andn₀andθ0the same for air, assuming a system of air-ﬁlm-substrate.

Finally, we also need to take into account the substrate in case the film is not fully opaque. In addition to the reflections and absorption of the film, the substrate also reflects light at the substrate-air interface. This light then bounces back and forth between the film-substrate and substrate-air interfaces (Figure 3.4), and these can be added to the reflectance of the film without interference modelling because the substrate typically only produces incoherent reflections.

3.4 Kramers-Kronig relations

When a complex function is analytic in the upper half-plane, the real part of a function relates to its imaginary part via Kramers-Kronig (KK) relations, which allows

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us to calculate one from the other[39]. This applies to the complex dielectric func- tionεˆ(ω), whereω is the angular frequency. While the KK relations do not apply directly to the complex refractive index, theñcan nonetheless be calculated from the ε

ˆ(ω). Therefore, by using an approximation, we can estimate the refractive index of a material from its absorption spectrum similarly to the KK relations:

n(ω)≈1+ c πP

_∞

0

α(Ω)

Ω²−ω²dΩ, (3.15)

wherec is the speed of light,P is the Cauchy principal value, andΩis the angular frequency we integrate over. While the KK relations are often employed for determining the steady-state refractive index from the absorption of the material, they can also be utilized in TR spectroscopy. For example, let us assume that in the excited state the sample has reduced absorption at the bandgap, as in Figure 3.5. KK-relations suggest that the photoinduced change in refractive index has to be positive at lower wavelengths than the band gap and negative at higher wavelengths. The exactΔn can be difficult to determine with this method due to some practical challenges, such as the inability to probe the entire frequency range with transient absorption setups, the difficulty of determining the exact value ofP, and the fact that we cannot measure theΔα(orΔk) directly, but have to fit it simultaneously with theΔn. However, it does provide us with an approximation, which is useful for ensuring that theΔn acquired from the fitting process is correct based on theΔk. This fitting process itself is described in detail in the next section.

Figure 3.5 Relation betweenΔn^andΔkaccording to KK relations: a photoinduced change in absorption also changes the refractive index.

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4 TRANSIENT REFLECTANCE AND TRANSMITTANCE MODELLING

4.1 Basics of pump-probing and data presentation

The pump-probe setup in transmittance mode is described in Figure 4.1. The re- ﬂectance mode is otherwise similar except that the reﬂected light is collected via a mirror placed in front of the sample. This requires tilting the sample slightly, but as long as the angle of incidence is small (<12^◦) it does not make a practical difference.

With larger angles, the two polarization types will also differ in their reﬂectance.

In the literature, the transmittance mode measurements are commonly called transient absorption measurements. However, because in our case this so-called "transient absorption" does not necessarily correspond to the real photoinduced change in absorption, we instead refer to the transmittance mode measurements as transient transmittance (TT).

Pump-probe measurement devices typically present the data as a change of optical density

ΔOD=−log₁₀ I

I₀ =−log₁₀

1+ΔI

I₀ (4.1)

whereI₀is the probe light intensity after the sample before excitation,I is the intensity after excitation andΔI =I −I₀. The spectra are measured as function of both time and wavelength:ΔOD=ΔOD(t,λ),ΔI =ΔI(t,λ), andI₀=I₀(λ).

Standard pump-probe measurements are done in transmittance mode, and the monitoring intensity changes as I = I_{i n}(T₀+ΔT) = I₀+I_{i n}ΔT = I₀(1+ ^ΔT_T

0 ), whereI_{i n}is the probe intensity before the sample. Thus, theΔT in OD units is

ΔT_OD=−log₁₀

1+ΔT

T₀ (4.2)

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Figure 4.1 Pump-probe schematics: The sapphire laser produces ultrashort (about 100 fs) pulses with a wavelength of 800 nm. 1000 of these pulses are generated each second, but the optical modulator reduces this by half. The Topas-C unit converts the original 800 nm pump beam to the desired pump wavelength. The probe beam is directed to a tunable delay setup which adjusts the travel distance of the probe pulses: it adjusts the delay time, in other words the time difference between the pump and the probe pulses hitting the sample. Before hitting the sample, the probe pulse passes through a medium that turns it into white light: a cuvette ﬁlled with heavy water for producing visible light or a sapphire crystal for producing stronger NIR light. The probe beam is split twice to acquire one reference measurement before the white light conversion and another after the conversion. The pump spot is between 1-2 mm², while the probe spot is much smaller, a fraction of a square millimeter.

and similarly for reﬂectance mode measurements:

ΔR_OD=−log₁₀

1+ΔR

R₀ . (4.3)

In both cases the relative change of either transmittance or reﬂectance can be acquired from the measured data and the steady-state (ground state) reﬂectanceR₀and transmittanceT₀:

ΔR(t) =R₀

10^−ΔR^OD^(t)−1 ΔT(t) =T₀

10^−ΔT^OD^(t)−1

. (4.4)

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4.2 Correcting the transient absorption spectra: Δ n ^and Δ k ^ﬁtting

The basic fitting ofΔn andΔk is simple. We first model the reflectance and transmittance of the ground state using the transfer matrix and the knownnandkof the material. We then model the excited stateRand T by introducing a small change in thenandk. By using standard error minimization algorithms, we can match the modelledΔRandΔT with the measuredΔRandΔT, which gives us theΔnand Δk. However, rather than using these slow error minimization algorithms, we devel- oped a faster method for fitting theΔnandΔk. Because the photoinduced changes are typically small in comparison to the ground state, we can use linear approximation to directly calculate theΔn andΔk from theΔRandΔT. Computationally this can be orders of magnitude faster than the brute-force algorithms, which is very important when other factors such as the sample thickness also need to be adjusted for accurate results.

When the photoinduced changes inRandT are small, we can assume that they linearly depend on theΔnandΔk:

R(t)R₀+R_nΔn(t) +R_kΔk(t) (4.5)

T(t)T₀+T_nΔn(t) +T_kΔk(t) (4.6) whereR_n,R_k,T_nandT_kare derivatives ofRandT overnandk, respectively, and R₀andT₀are transmittance and reﬂectance in the ground state (steady-state). From these equations,ΔnandΔkcan be solved as

Δn= T_kΔR−R_kΔT

R_nT_k−T_nR_k (4.7)

Δk= R_nΔT −T_nΔR

R_nT_k−T_nR_k . (4.8)

where ΔR = R(t)−R₀ and ΔT = T(t)−T₀. The 4 derivatives required by the above equations are acquired by modelling the transmittance and reﬂectance with the transfer matrix or an equivalent method. By applying a smallΔnto the model,

Discovering Perovskite Photophysics with Transient Reflectance Spectroscopy

Discovering Perovskite Photophysics with 7UDQVLHQW5HÀHFWDQFH Spectroscopy

HANNU PASANEN

HANNU PASANEN

Discovering Perovskite Photophysics with Transient Reflectance Spectroscopy

PREFACE

ABSTRACT

TIIVISTELMÄ

CONTENTS

ABBREVIATIONS

SYMBOLS

ORIGINAL PUBLICATIONS

1 INTRODUCTION

1.1 Hypothesis

2 TRANSIENT REFLECTANCE LITERATURE BACKGROUND

3 STEADY-STATE REFLECTANCE MODELLING

3.1 Complex refractive index

3.2 Fresnel equations

3.3 Transfer matrix and thin-ﬁlm interference

3.4 Kramers-Kronig relations

4 TRANSIENT REFLECTANCE AND TRANSMITTANCE MODELLING

4.1 Basics of pump-probing and data presentation

4.2 Correcting the transient absorption spectra: Δ n and Δ k ﬁtting

4.2 Correcting the transient absorption spectra: Δ n ^and Δ k ^ﬁtting