Absorption and Emission Processes in Organometal Trihalide Perovskites

UNIVERSITÀ DEGLI STUDI DI CAGLIARI

FACOLTÀ DI SCIENZE MATEMATICHE, FISICHE E NATURALI

SCUOLA DI DOTTORATO IN FISICA XXVIII CICLO

Supervisor: PhD Candidate: Prof. Michele Saba Michele Cadelano

ANNO ACCADEMICO 2014/2015

Ad Anna e alla mia famiglia

CONTENTS

1 Introduction 1

2 Optical spectroscopy 5 2.1 Introduction ...... 5 2.2 Light propagation in an optical medium ...... 5 2.2.1 Light absorption ...... 7 2.2.2 Rayleigh scattering ...... 8 2.3 Electronic bands in crystals ...... 9 2.4 Interband transitions in semiconductors ...... 11 2.4.1 Absorption in direct gap semiconductors ...... 12 2.5 Excitons ...... 14 2.6 Recombination in direct gap semiconductors ...... 17 2.6.1 Photoluminescence ...... 19 2.6.2 Ampliﬁed spontaneous emission ...... 20

3 Perovskites 23 3.1 Introduction ...... 23 3.2 Crystal structure ...... 23 3.3 Methylammonium lead trihalide perovskites ...... 24 3.3.1 Applications ...... 26 3.3.2 Sample fabrication ...... 28

4 Experimental techniques 31 4.1 Introduction ...... 31 4.2 Linear Absorption Spectroscopy ...... 31 4.2.1 UV-Vis absorption setup ...... 33 4.3 Photoluminescence Spectroscopy ...... 33

i 4.3.1 Ultrafast PL Spectroscopy setup ...... 35 4.3.2 PL Spectroscopy setup under cw pumping ...... 38 4.3.3 PL Spectroscopy setup under quasi-cw pumping . . . 38

5 Excitons vs free carriers 41 5.1 Introduction ...... 41 5.2 Radiative recombination processes ...... 42 5.3 Saha equation ...... 44 5.4 Quantum yield ...... 47 5.5 Analysis of the recombination processes ...... 49 5.6 Steady-state photoluminescence ...... 51

6 Exciton binding energy 55 6.1 Introduction ...... 55 6.2 Elliot’s theory of Wannier excitons ...... 56 6.2.1 Comparison with literature ...... 59 6.3 f-sum rule ...... 62

7 Optical ampliﬁcation 67 7.1 Introduction ...... 67 7.2 ASE under fs excitation ...... 68 7.3 ASE under ns excitation ...... 69 7.4 Warming processes ...... 71 7.4.1 Rate equation model ...... 74 7.5 Comparison with nitride semiconductors ...... 79

8 Conclusion and outlook 81

A Radiation-matter Interaction 85 A.1 Einstein coefﬁcients ...... 85 A.1.1 Absorption ...... 85 A.1.2 Stimulated emission ...... 86 A.1.3 Spontaneous emission ...... 86 A.2 Light dispersion ...... 86 A.2.1 Diffraction grating ...... 86 A.2.2 Condition for maxima ...... 87

ii A.2.3 Angular dispersion and resolving power ...... 88 A.2.4 Blazed grating ...... 88 A.2.5 Monochromator ...... 89

B Laser devices 91 B.1 Laser emission ...... 91 B.2 Pulsed lasers ...... 92 B.3 Nd:YAG and Nd:YLF lasers ...... 93

C Light detection 95 C.1 Quantum efﬁciency ...... 95 C.2 Responsivity ...... 95 C.3 p-n junction ...... 96 C.3.1 Photodetectors ...... 97 C.4 Microchannel plate ...... 97 C.5 Detectors ...... 98 C.5.1 Charge-Coupled Device (CCD) ...... 98 C.5.2 Streak camera ...... 99 C.5.3 Gated intensiﬁed CCD camera ...... 101

D Samples 103 D.1 Sample fabrication ...... 103 D.1.1 Spin coating ...... 103 D.1.2 Samples for absorption measurements ...... 103 D.1.3 Samples for optical ampliﬁcation measurements . . . . 105 D.2 Sample characterization ...... 106 D.2.1 X-ray diffraction ...... 106 D.2.2 Scanning probe microscopy ...... 107

Acknowledgements 111

References 115

iii iv 1|I NTRODUCTION

Among the various solution processed semiconductors, organometal halide perovskites represent a noteworthy class of materials thanks to their unique combination of optoelectronic properties: efficient charge transport, favor- able emission properties, strong light absorption and optical gap tunability stand for the key features of these novel semiconductors and make them appealing for the realization of a new generation of low-cost solar cells and optical emitters [1-19]. Recently, scientists have devoted large efforts in studying perovskites as absorbers in solar cells, reaching energy conversion efficiencies even higher than 20%. However, in spite of such an intense research, fundamental aspects of the photophysics remain still elu- sive, generating lively debates among researchers from all over the world. [20-29]. One of the more interesting debates concerns the nature of the photoexcited species. Being organometal perovskites hybrid materials, theoretically it is not clear if the excited states are dominated by bound or unbound electron-hole states. First publications showed perovskites exhibiting excitonic properties, feature typical of organic materials. On the contrary, recent optical spectroscopy reports converge to state that the majority of band-edge optical excitations at room temperature are free carriers, like happens in inorganic semiconductors [6,22,30]. If photoexcitations result in bound or unbound electron-hole pairs is not a problem of minor importance, because the design of optoelectronic devices strongly depends on it. As an example, if light absorption gives rise to bound electron-hole states, a heterojunction is necessary to split charges and produce a current flow in a solar cell, while such architecture is not necessary if electron- hole states are unbound, since only an electric field is needed to separate

1 2 Introduction charges. Another debate concerns the physical reasons that support the prevalence of free carriers over bound electron-hole pairs. A small value of the exciton binding energy could justify such finding, as thermal excitation at room temperature could ionize excitons. Consequently, a precise and reliable determination of the exciton binding energy results to be of primary importance. However, exciton binding energies have been reported from less than 5 meV to over 50 meV [22,30-36] and it has even been suggested that ionic screening could decrease the exciton binding energy with temperature [33,34,37]. At the present, a consensus concerning the exciton binding energy is lacking. In addition to the potential use in photovoltaics, organometal perovskites could be employed also as active media in light emission devices. Different works reported amplified spontaneous emission (ASE) at injected carrier densities comparable to ones typical of organic semiconductors [17-19], making perovskites promising for the realization of a new generation of lasers. However, such reports demonstrated ASE only under impulsive excitation, a regime far from the continuous operation of real lasers [17- 19,38-40], where different parasitic and warming processes are involved [19]. Hence, further investigations are needed to state if perovskites can be actually used in future laser devices. Herein, we will discuss in details experimental results obtained by optical spectroscopy measurements, providing both a deep understanding of the photophysical properties of organometal perovskites and the possible solution to the unsolved problems we mentioned before. Chapter 2 concerns the description of theoretical concepts useful to the reading, while Chapter 3 deals with a general description of perovskites and their potential applications. Both the techniques and experimental setups with which we carried out optical spectroscopy experiments are described in Chap- ter 4. After, we will start discussing the nature of the photoexcitations of organometal trihalide perovskites in Chapter 5, where we will show that an electron-hole plasma exists in a wide excitation range, ranging from light intensities much smaller than typical of solar illumination to those typical to obtain light amplification. The issue concerning the determination of an accurate exciton binding energy is addressed in Chapter 6, where we will 3 use a f-sum rule and the Elliot’s theory of Wannier excitons to study the absorption spectra at the band-edge. Finally, Chapter 7 concerns the study of optical amplification, where optical thermometry and rate equations will be used to estimate the magnitude of the warming processes establishing under cw regime. 4 Introduction 2|O PTICALSPECTROSCOPY

2.1 Introduction

The aim of this chapter is to introduce the basic concepts of optical spectroscopy and discuss the processes involved in the interaction between light and materials. In details, we will describe the mechanisms concerning light absorption and emission, the electronic structure in crystals and the features that arise in absorption and emission spectra in direct-gap inorganic semiconductors.

2.2 Light propagation in an optical medium

Considering a light beam of initial intensity I0 crossing an optical medium, some of the light will be reflected, part will propagate inside the material and part will be transmitted, as shown in Figure 2.1. Reflection is a process quantified by the reflectance R, which is defined as the ratio between the beam reflected intensity IR and the beam intensity I0 at the material surface. Transmission is quantified by the transmittance T, defined as the ratio between the beam transmitted intensity IT and I0. Fi- nally, the fraction of light that will be absorbed by the material is quantified by the absorbance A, defined by the ratio between IA and I0, where IA is the light intensity absorbed by the optical medium. Reflectance, transmittance and absorbance are connected by the following equation:

R + T + A = 1 (2.1)

In details, as the light beam propagates trough the material, the most common processes which can occur are refraction, absorption, luminescence

5 6 Optical spectroscopy

Figure 2.1 | Light propagation in an optical medium. Refraction gives rise to a variation of the speed of light with a consequent bending of the progagating beam with respect to the initial direction, with no attenuation of intensity. Absorption of light result in a decrease of the beam intensity during propagation and can be accompanied by photoluminescence, which is light emission in all directions. Scattering gives rise to a redirection of light that results in a decrease of the beam intensity with propagation, as happens for absorption.

and scattering. Refraction causes a reduction of the light velocity inside the material with respect to free space. The resulting beam will be bended at the interface between free space and the material, according to the Snell’s law, which deﬁnes the bending magnitude through the refraction index. Ab- sorption is a process where photons resonant with electronic transitions of the material are absorbed, resulting in an attenuation of the initial beam intensity I0. Such process is responsible for optical materials colouration, as the amount of light that is not absorbed determines the colour of the optical medium. Energy absorbed by the material will be returned to external environment trough different relaxation processes. Part can be dissipated through non-radiative processes that generate warming, and part can be re-emitted at lower frequencies in all directions trough photoluminescence (PL), that is light emission from the material. When both relaxation processes occur, the energy conservation dictates that emitted light will have a smaller frequency than the incident beam, an effect known as Stokes shift (Figure 2.2). Finally, scattering occurs when light changes direction without being absorbed during interaction with the atoms inside the medium. If the photon 2.2 Light propagation in an optical medium 7

Figure 2.2 | Excitation and relaxation processes in a three level system. An electron initially in the ground state of energy E0 can be excited to an excited state of energy E2 by the absorption of a photon of energy h¯ ω = E2 − E0. After excitation, the electron can relax towards the ground state ﬁrst occupying the excited state E1, emitting energy by non-radiative processes (as an example collisions with other electrons or atoms), and later by the emission of a photon of energy h¯ ω = E1 − E0. The photon emitted results to have energy smaller than the absorbed one, giving rise to the eﬀect known as Stokes shift.

frequency of the scattered beam is the same of the one incoming, scattering is deﬁned elastic. On the contrary, inelastic scattering occurs when interaction gives rise to a beam with different photon frequency with respect to the initial.

2.2.1 Light absorption

When a light beam of low intensity I0,E and energy E = h¯ ω crosses a sample of thickness l, the outgoing beam will have an intensity IE(l) dictated by the Beer-Lambert law (see Figure 2.3):

−α(E)l IE(l) = I0,Ee (2.2) where α(E) is the absorption coefficient of the sample at the energyh ¯ ω. As we have discussed in Section 2.2, when the external photon is resonant with an electronic transition of the material, it can be absorbed. If the external field has a spectral distribution ρ(h¯ ω), some components will be absorbed and others will pass through the sample or will be reflected. The difference between the spectral distributions of the incident and outgoing beams yields the absorption spectrum of the sample. 8 Optical spectroscopy

Figure 2.3 | Beer-Lambert law. A beam of initial intensity I0 that crosses an optical medium −αl of thickness l will be exponentially attenuated according to the Beer-Lambert law I(l) = I0e , where α is the absorption coeﬃcient of the optical medium.

Another quantity related to absorption and usually adopted in optical spectroscopy is the optical density (OD), which corresponds to the sample absorbance. The optical density can be derived from Equation 2.2 and is deﬁned as:

IE(l) α(E)l OD(E) = −log10 = = 0.434α(E)l (2.3) I0,E ln10 The spectral shape of both OD(E) and α(E) is the same, but the former includes the sample thickness, while the latter it is an intrinsic property of the material.

2.2.2 Rayleigh scattering

Inhomogeneities of the material, impurities and defects can heavily affect light propagation as a consequence of refractive index variations that occur in length scales smaller than the light wavelenght. The resulting beam will be attenuated similarly to what happens during absorption and the scattered intensity in the direction x decreases esponentially as the following equation:

−Nσsx I(x) = I0e (2.4) where N and σs represent the volume density of scattering centres and the scattering cross-section of each centre, respectively. Raylaigh scattering occurs when the scattering centre size is much smaller than the light 2.3 Electronic bands in crystals 9 wavelength, resulting in a cross section that increase as E4, where E is the photon energy.

2.3 Electronic bands in crystals

Until now we have discussed general processes that occur when light interacts with an optical medium. To have a deeper understanding of these processes, it is necessary a more detailed description of the internal structure of solid-state materials. In particular, we will focus our attention on crystals, which are characterized by long-range translational order, differently from amorphous solids. In an ideal one dimensional crystal, atoms are placed at fixed positions to form a lattice, separated each other by a distance R, which is the spatial lattice period, called lattice constant. Hence, they interact each other in order to form an ordered structure that repeats itself into the space indefinitely. In three dimensional crystals, each direction i can be characterized by its speficic lattice constant Ri, where i = x, y, z, the three spatial directions. One can immagine the lattice formed by a infinite sequence of a element that repeats itselft in the space. Such element is called primitive cell or unit cell and included the minimum set of atoms that make the lattice. Each atom have a number of electrons and each electron occupies an energetic state defined by a wavefunction. In a crystal, the wavefunction ψ associated to an electronic state in the space position r is expressed by the Bloch’s theorem:

ik·r ψk(r) = uk(r)e (2.5) where uk(r) is a function having the spatial periodicity of the crystal lattice. λ is the wavelength associated to an electron k is its associated wavevector |k| = 2π/λ in the momentum space, that is the dual of the real space. The unit cell has also its dual in the momentum space, called Brilloiuin zone, which have dimensions ai = 2π/Ri, where Ri are the lattice constants in the real space and i = x, y, z. The overlap between the electronic wavefunctions corresponding to the outer orbitals of different atoms results in a strong interaction, which gives rise to the broadening of the discrete energy levels of each single atom and to the formation of energy bands, as shown in Figure 2.4. Bands are 10 Optical spectroscopy

Figure 2.4 | Energy bands vs discrete levels. Separated atoms (i.e. a gas of a certain element) are characterized by discrete energy levels, for which photon absorption can occur only if the photon energy is resonant with the atomic transitions. The resulting absorption and emission spectra are thus formed by separated lines. When the interatomic separation decreases, that is when atoms begin to interact toghether to form a solid, the overlap between their outer orbitals results in generation of energy bands (where many energy levels are close each other to form a continuum) separated by energy gaps (where no energy levels are present).

separated each other by energy intervals where no electronic states are present. Hence, an optical transition between two different electronic bands, that is an interband transition, can occur over photon energies included between the energy gap and the energy difference between the upper energy of the highest band and the lower of the less energetic band, yielding a continuous absorption spectrum in this range. Such feature is typical of solid-state materials, as a gas of molecules or atoms yields a discrete absorption spectrum consisting in sharp lines. The number of states included in the range between energy E and E + dE, where dE is a inﬁnitesimal energy variation, is expressed by the density of states g(E), which is related to the density of states g(k) in momentum space as follows:

dk g(E) = g(k) (2.6) dE

The density of states clearly depends on the energy-momentum band dispersion and is fundamental for the calculation of both absorption and emission spectra. 2.4 Interband transitions in semiconductors 11

Figure 2.5 | Absorption in semiconductors. When a photon of energy higher than the energy gap Eg of a semiconductor is absorbed, an electron initially occupying an energy state Ei in valence band (VB) can undergo a transition to occupy a state of energy Ef in the conduction band (CB). The resulting vacancy in VB is called hole and can be considered as a positive charge with its own mass that can move in the semiconductor.

2.4 Interband transitions in semiconductors

Figure 2.5 shows the typical energy band structure of a semiconductor. Electrons ﬁll available band electronic states according to the Pauli exclu- sion principle in order of increasing energy. In semiconductors, state oc- cupation is such that, below the Fermi energy of the material, bands are completely occupied and above they are completely empty. The highest occupied band is called valence band (VB) and the lowest un- occupied is the conduction band (CB). The energy separation between the highest energy level in VB and the lowest in CB is called energy gap or band- gap energy Eg. An electron occupying a VB state of energy Ei can jump to a CB state of energy Ef absorbing a photon of energy E = h¯ ω = Ef − Ei, according to both selection rules and the Pauli principle. Interband transitions can occur over an energy range dictated by the band limits. It is clear that the lowest energy transition is the one corresponding to the minimum energy separation of the bands, that is Eg, while photons with energiesh ¯ ω lower than the energy gap will not be absorbed. Every transition from VB to CB promotes an electron in conduction 12 Optical spectroscopy band and leaves an empty state in valence band. Such vacancy is called hole and can be considered as a positive charge. As a result, the transition results in a generation of an electron-hole pair (e-h).

2.4.1 Absorption in direct gap semiconductors

Depending on the energy-momentum dispersion of both valence and conduction band, the band-gap of a semiconductor can be direct or indirect. Figure 2.6 is useful to understand such difference. Considering the energy- momentum band dispersion in the Brillouin zone, if the less energetic transition takes place at k = 0, the band-gap is direct. As a contrary, if the transition resonant with Eg occurs between two VB and CB states at different k, the band-gap is indirect. According to the momentum conservation, k-variation is supplied by absorption or emission of a phonon, that is a lattice vibration. In indirect gap semiconductors, the resonant Eg transition can occur only if both photon and phonon are absorbed (or emitted) simultaneously. As a consequence, light absorption and emission are more complicated in indirect gap semiconductors than in those with direct gap. For the purposes of this thesis, we will consider only absorption in direct gap semiconductors, among which GaAs and GaP represent the most fa- mous examples.

The absorption coefﬁcient is dictated by the transition rate Wi→ f between the initial state i in VB and the ﬁnal state f in CB, and it can be demonstrated that:

2π 2 W → = |M| g(h¯ ω) (2.7) i f h¯ where M = hi|H0| f i is an element of the transition matrix and g(h¯ ω) the joint density of states at the photon energy E = h¯ ω. H0 is the perturbation associated to the photon incoming to the lattice. The joint density of states takes into account all the possible direct transitions between valence and conduction band. For parabolic bands (E ∝ k2) it can be demonstrated that:

∗ 3/2 1 2m 1/2 g(h¯ ω) = (h¯ ω − Eg) (2.8) 2π2 h¯ 2 where m∗ is the electron-hole effective mass, obtained by the following equa- 2.4 Interband transitions in semiconductors 13

Figure 2.6 | Direct and indirect band gap. (a) When the minimum separation between two states in VB and CB, that is Eg, occurs at k=0 in the Brillouin zone, the band gap is defined direct. Absorption can occur without mediation of phonons. (b) If the minimum energy separation occurs between VB and CB states having different k, the band gap is defined indirect and absorption can occur only if mediated by a phonon having momentum q, equal to the momentum separation between the two initial and final states.

tion: 1 1 1 ∗ = ∗ + ∗ (2.9) m me mh ∗ ∗ me and mh are the effective masses of electrons in CB and holes in VB, which differ from the free electron mass m0 and can be calculated from the energy-momentum dispersion of both conduction and valence bands. Since near k = 0 both VB and CB can be approximated by parabolic bands, Equation 2.8 is valid for energies close to the energy gap in direct gap 1/2 semiconductors, thus α(h¯ ω) ∝ g(h¯ ω) ∝ (h¯ ω − Eg) for photon energies h¯ ω resonant or higher than Eg and α(h¯ ω) = 0 forh ¯ ω < Eg. As a consequence of the large number of atoms in a crystal, the density of state is large as well, giving rise to absorption coefﬁcients in the range 104 − 106cm−1. Since lattice constants depend on vibrations due to thermal energy, they can change with temperature, giving rise to band structure, energy gap, density of states and absorption variations. Figure 2.7 shows α2 as a function of energy for the direct III-V semiconductor InAs at room temperature.

For energies well above Eg, parabolic band approximation is not valid anymore, due to the particular shape of the energy-momentum band dispersion. Close to Eg, absorption can result higher than what estimated by 14 Optical spectroscopy

Figure 2.7 | Energy-gap determination in direct gap semiconductors. The value of Eg can be extracted by fitting the square of the optical absorption with a linear function. Eg is the value for which the fitting function crosses the energy axis. The figure shows the room temperature absorption relative to the direct-gap InAs semiconductor.2

this model, as a consequence of electron-hole Coulomb interactions that here are not taken into account.

2.5 Excitons

When Coulomb interaction is sufﬁciently intense, electrons and holes can form bound states called excitons. An exciton can be considered as a hy- drogenic system composed by an electron and a hole that orbit around each other. There are two kinds of excitons, which differ depending on the e-h interaction intensity: the Wannier-Mott and the Frenkel excitons. The former can move freely throughout the crystal and have an orbit ra- dius much larger than the lattice constant, including hundreds of atoms and resulting in a weak e-h interaction. The latter are tightly bound to speciﬁc atoms, resulting in radii comparable to the crystal unit cell and strong e-h interaction. Exciton formation is possible when the e-h attrac- tive potential, that is the binding energy Eb, is higher than thermal energy kBT, where kB is the Boltzmann constant and T the temperature. At room temperature, kBT ∼ 25meV, excitons having binding energies smaller tend to dissociate into an electron-hole plasma (a gas of unbound electrons and holes), while those with higher energies remain bound. In terms of bind- 2.5 Excitons 15

ing energy, Wannier-Mott excitons have typically Eb smaller than 10 meV, while Frenkel excitons binding energies rang from 100 meV to over 1 eV. Since in Wannier excitons e-h distance is much large than lattice constant, particles can be considered as moving throughout medium made by the lattice atoms having relative dielectric constant er. After applying the Bohr model to the exciton with reduced mass m∗ (see Equation 2.9), it can be demonstrated that bound states have quantized energies according to the following equation:

m∗ 1 R R ( ) = H = − X Eb n 2 2 2 (2.10) m0 er n n

where n = 1, 2, 3, .. and RH is the Rydberg constant of the hydrogen atom, ∗ 2 which is 13.6 eV. RX = RHm /(m0er ) represents the exciton Rydberg constant and corresponds to the exciton binding energy Eb for n = 1. We can observe that the more the dielectric constant er is large, the more Eb is small. The total energy E(n) of an excitonic state is obtained by summing the energy required to form an e-h pair, that is the energy gap Eg, and the potential energy Eb(n), thus:

R E(n) = E + E (n) = E − X (2.11) g b g n2 yielding an absorption spectrum similar to the one shown in Figure 2.8, which is composed by an excitonic contribution below the energy gap and by the interband absorption above. The intensity of each excitonic transition is proportional to the probability for an electron to occupy the n-state, thus intensity is maximum for the ground state n = 1 and decrease with n. Each transition is affected by thermal line broadening, thus lines assume a bell shape of a certain width rather than a vertical line and overlap each other at high temperature, becoming indistinguishable. Figure 2.9 shows the excitonic absorption in the direct gap GaAs semiconductor at different temperatures. 16 Optical spectroscopy

Figure 2.8 | Excitonic levels. Interband absorption in direct gap semiconductors can be accompanied by excitonic eﬀects close to the energy gap, which manifest themselve through a serie of atom-like transitions. The blue line accounts for both band-to-band and excitonic contributions, while the red dashed line represents the absorption in absence of excitonic features.

Figure 2.9 | Excitonic absorption in GaAs. (a) Excitonic absorption in GaAs between 21 and 294 K. The dashed line represents the absorption coeﬃcient at 294 if excitons would not be present, considering an energy gap of 1.425 eV.2 (b) Excitonic absorption in ultra pure GaAs at 1.2 K.2 2.6 Recombination in direct gap semiconductors 17

Figure 2.10 | Interband luminescence in direct-gap semiconductors. Electrons in both valence and conduction bands are represented by the shaded regions, while holes stay in the white regions. After being excited, electrons and holes can recombine radiatively with emission 2 of photons of the same energy of Eg.

2.6 Recombination in direct gap semiconductors

Until now we have described what happens when a photon of energyh ¯ ω is absorbed by a semiconductor. The processes that occur after absorption are the subject of this section. Figure 2.10 schematically shows the recombination processes occurring in semiconductors. After being excited from VB to CB, electrons relax towards the bottom of the conduction band emitting phonons, a process which takes place on time scales as short as 100 fs. The same happens for holes in valence band, which are equal in number with respect to electrons, as absorption of a photon creates an e-h pair. Once reached the lowest energy available states, electrons can drop down to the valence band recombin- ing with holes. Recombination can be radiative and non-radiative. The ﬁrst occurs when electrons and holes recombine emitting a photon, while the second occurs when energy is loss by emission of photons or transferred to lattice defects called traps. The spontaneous emission rate due to radiative 18 Optical spectroscopy e-h recombination between two states can be calculated by the following rate equation:

dN = −AN (2.12) dt radiative where N is the excited state population at time t and A is the Einstein coef- ﬁcient of the transition (a more detailed description of Einstein coefﬁcients is subject of Appendix ??). Solving Equation 2.12 gives:

N(t) = N(0)e−At = N(0)e−t/τR (2.13)

−1 where τR = A is the radiative lifetime of the transition. The bigger is A, the most efficient will be radiative recombination and the smaller will be τR. It can be demonstrated that the spontaneous emission Einstein coefficient A is directly proportional to the absorption Einstein coefficient B. Hence, materials exhibiting strong absortion can potentially exhibit strong light emission. In most of direct semiconductors, τR is typically includes in the range between 1 and 100 ns. Since radiative processes compete with non-radiative recombination, the rate equation accounting for both processes reads:

dN N N 1 1 = − − = −N − + (2.14) dt radiative τR τNR τR τNR where τNR is the non-radiative lifetime. The parameter which quantify the radiative efﬁciency of a material the quantum yield QY, deﬁned as the ratio between the radiative emission rate and the total relaxation rate:

AN 1 QY = = τ (2.15) 1 1 1 + R N + τNR τR τNR

If τR << τNR, electrons and holes recombine radiatively before being involved in some non-radiative relaxation, thus QY approaches values close to 1. On the contrary, materials exhibiting strong absorption but having high trap density with extremely short τNR are bad light emitters as QY tends to zero. 2.6 Recombination in direct gap semiconductors 19

2.6.1 Photoluminescence

Considering a direct gap semiconductor, the absorption of photons with energies higher than Eg initially promotes electrons in high energy CB states and simultaneously creates holes in VB. After excitation, ultrafast carrier thermalization by phonon emission results in non-radiative transition from higher to lower energy states in each band, and the states population is governed by the Fermi statistic. Each carrier population will have its own Fermi energy EF, thus

1 fe,h(E) = (2.16) E−Ee,h − F e kBT + 1 c h where EF and EF are the Fermi level for electrons and holes, respectively. At low carrier density, Fermi distribution can be approximated by a Boltz- mann function, and Equation 2.16 can be modiﬁed as follows:

E−Ee,h F − k T fe,h(E) ∼ e B (2.17) The luminescence intensity I(h¯ ω), resulting from e-h radiative transitions of energyh ¯ ω, will follow the relation:

I(h¯ ω) ∝ |M|2g(h¯ ω) f (h¯ ω − Eg) (2.18) where |M| is the transition matrix element between the initial and ﬁnal state, g(h¯ ω) the joint density of states and f (h¯ ω − Eg) the occupancy factor accounting for both electrons and holes distributions. Since g(h¯ ω) ∼ (h¯ ω − 1/2 (h¯ ω−Eg)/k T Eg) and f (h¯ ω − Eg) ∼ e B , Equation 2.18 reads:

1/2 (h¯ ω−Eg)/kBT I(h¯ ω) ∝ (h¯ ω − Eg) e (2.19) Hence, the typical emission in direct-gap semiconductors a PL spectrum peaked close to Eg, with a shape similar to the one shown in Figure 2.11, obtained from bulk GaAs. 1/2 The rapid increase close to Eg is due to the (h¯ ω − Eg) factor, while the high energy tail follows a Boltzmann exponential decrease, which can be used to estimate the e-h plasma temperature T. The photoluminescence full width at half maximum (FWHM) is of the order of kBT. 20 Optical spectroscopy

Figure 2.11 | Emission spectrum in GaAs at 100 K. Photoluminescence was excited by the 632.8 nm in wavelength He-Ne laser beam. Inset: semilogarithmic plot of the same data. The e-h plasma temperature can be extracted by ﬁtting the high energy tail of the photoluminescence with a Ae−E/kBT Boltzmann function.2

At high carrier densities, classic limit is no more valid and the band ﬁlling results in band saturation, which gives rise to I(h¯ ω) saturation.

2.6.2 Ampliﬁed spontaneous emission

Considering a semiconductor under photoexcitation, the photoluminescence due to some e-h pairs can induce other electrons in the excited states to recombine with holes, thus giving rise to stimulated emission (see Ap- pendix A.1.2). This process is the more efﬁcient the more intense is the light beam, as strong photoexcitation results in a large number of e-h pairs that can recombine radiatively. High intensity photoexcitation can lead to a situation where almost all electrons and holes occupy the excited states in CB and VB, respectively, thus reversing the initial situation where VB states are fully occupied by electrons and those of CB are empty: such condition is called population inversion. Hence, during propagation inside the semiconductor, the initial photoluminescence intensity IPL0 , due to spontaneous radiative emission, can be ampliﬁed by stimulated emission in a path x, following the equation: 2.6 Recombination in direct gap semiconductors 21

Figure 2.12 | ASE in semiconducting polymers. (a) Emission intensity as a function of the laser pumping. The emission slope increases once reached the ASE threshold. (b) Normalized emission spectra at diﬀerent excitation intensity. The dotted, dashed and solid lines correspond to pump intensities of 0.34, 0.61 and 5.2 kW/cm2, respectively. ASE gives rise to a narrowing of the emission peak.3

(g−α)x IPL(x) = IPL0 e (2.20) where g and α are the optical gain (due to stimulated emission) and the optical loss (due to light absorption and scattering) of the material, respectively. Due to the fact that the optical gain increases with excitation intensity, the photoluminescence amplification occurs only when g overcomes the optical loss. In addition, since the exponential of Equation 2.20 depends on the difference g − α, such amplification occurs in the PL peak but within a tiny spectral region where α is small, that is close to the energy-gap. Hence, when g > α, a peak large few nanometers emerges from the photoluminescence spectrum, giving rise to the process called amplified spontaneous emission (ASE). Figure 2.12a shows the emission intensity of a gain material, calculated in the ASE spectral region, as a function of the pump intensity P. A drastic slope increase of the intensity occurs at the ASE threshold PASE. Spectrally, the FWHM of emission is broad for values of pump intensity below the threshold and narrows above PASE, as shown in Figure 2.12b. The ASE threshold can be lowered increasing the optical feedback of the material, for example adopting waveguide architectures (1D or 2D struc- 22 Optical spectroscopy tures) that confine the light better and decreasing the material optical losses (low defect density and low surface roughness). 3|P EROVSKITES

3.1 Introduction

Perovskite identiﬁes a crystal structure that was ﬁrst discovered in the Ural Mountains by Gustav Rose in 1839 studying calcium titanate minerals, whose chemical formula is CaTiO3. The origin of the name perovskite is 1 due to Lev Peroski, a Russian nobleman and mineralogist. CaTiO3 represents only one of the hundreds of chemical compounds existing in nature that exhibit the same crystal structure. Hence, all the crystals with the same structure of CaTiO3 are included in the family of perovskites. Among them, hybrid perovskites are solution-processed semiconductors that have recently attracted great attention among scientists thanks to their unique combination of features that mixes the advantages of low-cost processabil- ity to highly tunable properties by chemical synthesis. This chapter will review the perovskite crystal structure, the features that make hybrid perovskites such appealing for optoelectronics and their potential applications.

3.2 Crystal structure

The perovskite chemical formula can be generally expressed as ABX3, where A and B are cations and X is an anion. Figure 3.1 shows the ideal perovskite structure, which is cubic in absence of distorsions. The larger cations A occupy the eight angles of the cube. The center of each face is occupied by an anion X, which forms an octahedron. The center of this latter is occupied by the smaller cation B, which is therefore octahedrally coordinated with X anions. Even if a compound can be theoretically classiﬁed as a perovskite by the

23 24 Perovskites

Figure 3.1 | Perovskite cubic structure. Larger cations A form a simple cubic structure, where the center of each face is occupied by an anion X, forming an octahedron. The centre of the cube is occupied by the smaller cation B. RA, RB and RX are the radii of the A, B and X ions, respectively. Perovskite can be formed only if the tolerance factor t, expressed by Equation 3.1, is included between 0.8 and 1.1.

ABX3 formula, it is not sufficient to state if the resulting structure could be effectively a perovskite. In order to have a perovskite, ions size must be confined by the tolerance factor, defined by the following equation:

RB + RB t = p (3.1) (RA + RX) where RA, RB and RX represents the ionic radii of A, B and X, respectively. In order to have a stable perovskite, t must be included between 0.8 and 1.1. A perfect cubic structure is obtained for t = 1, while deviations from this value are responsible for structural distortions that appear due to vi- brational motions and chemical bondings (ionic, covalent and hydrogen bonding). Such distortions modify the archetypical cubic structure into other simmetries, but manteining the same coordination numbers.

3.3 Methylammonium lead trihalide perovskites

Due to the large variety of chemical compounds that belong to perovskite family, such materials exhibit different physical properties depending on the atoms included in the structure, such as ferroelectricity, piezoelectricity, 3.3 Methylammonium lead trihalide perovskites 25 superconductivity and properties of optoelectronic interest. An novel class of these materials is represented by organometal halide perovskites, which are solution-processed semiconductors considered very promising for the future optoelectronic applications. They are composed by organic cations occupying sites A, a metal cation in sites B and a mixture of halide anions in sites X. Acting on chemical composition, organometal halide perovskites exhibit a wide band-gap tunability from the ultra-violet (UV) to the infra- red (IR) spectral regions. Such wide tunability makes these novel semiconductors particularly appealing for both light harvesting and emission, which can be exploited in solar cells, LEDs and lasers. Chemical substitution concerns the use of different organic molecules, metals and halides. Precisely, halide substitution is responsible for band-gap tunability, while the metal governs the kind of valence and conduction bands. The valence band is composed of hybridized Pb 6s and halide p orbitals, while the conduction band is primarily Pb 6p in character with minor contributions from the halide s orbitals [63 – 67]. Small modulations of the band-gap can be obtained also changing the organic cation, as it modifies the lead-halide bond distance [31,37]. Among variuos kind of hybrid perovskites, recently organometal halide perovskites have attracted great attention among scientists due to their noteworthy optoelectronic features. In fact, they show absorption coefficients of the order of 104 − 105cm−1 across the visible spectrum, thus their application in photovoltaics makes necessary only a thickness of few hundreds nanometers of material to harvest light efficiently. Many works report also rimarkable carrier diffusion lengths from some hundreds nm to ∼ 1µm, demonstrating that organometal perovskites can support efficient charge collection at the electrodes of a photovoltaic device. In this thesis we will discuss the optoelectronic properties of methylammonium lead trihalide perovskites, whose chemical formula is CH3NH3PbX3 + (where the methylammonium cation CH3NH3 occupies the site A, cation Pb2+ occupies the site B and the halide anion X− the site X). For notation simplicity, from now we will refer to this class of materials with MAPbX3 instead of CH3NH3PbX3. Such semiconductors are particularly interesting due to their large optical absorption coefficients and gap-tunability across the visible spectum due to halide substitution, as shown in Figure 3.2.

Our investigation is focused on MAPbI3, MAPbI3−xClx and MAPbBr3 26 Perovskites

Figure 3.2 | Band-gap tunability with halide substitution. Absorption in MAPbX3 perovskites can be modiﬁed changing the halide X, where X=I,Cl,Br. Absorption spectra relative to MAPbI3 and MAPbBr3 have been obtained obtained by our experiments, while the MAPbCl3 perovskite absorption spectrum has been extracted from the publication of Comin et al.6 through the software CurveSnap, freely available online.

perovskites. Both pure MAPbI3 and mixed MAPbI3−xClx exist at room temperature in the tetragonal phase, while room temperature MAPbBr3 is arranged in the cubic structure. The structural re-arrangement due to each phase transition is combined with signiﬁcant modiﬁcations in both absorption and emission properties of the material.

3.3.1 Applications

Wide band-gap tunability across the visible spectrum, direct energy gap and high absorption coefﬁcient close to the band-edge make organometal halide perovskites particularly appealing for light harvesting (Figure 3.3). Perovskite-based solar cells have even reached rimarkable energy conversion efﬁciencies exceeding 20%. However, such excellent results can be further improved in the future, acting on device architecture and design parameters. The state-of-art perovskite solar cells are composed from different layers, each one has a precise impact on the device performance and can heavily affect its working if not proper optimized. Perovskite layer is typically sandwiched between a hole-transport layer and electron-transport layer. Both of them are connected to the external circuit through metal or transparent contacts. Fabrication of a solar cell implies a proper band align- 3.3 Methylammonium lead trihalide perovskites 27

Figure 3.3 | Perovskite-based solar cell. Solar light crosses some layers before being absorbed by perovskite. Electron-hole pairs created by photon absorption are kept away each other by the electron and hole transport layers, which are connected to electric contacts.7

ment that allows the ﬂow of one carrier and blocks the other, layer charge mobility comparable with that of perovskite, good interfaces and trans- parency of one contact to absorb light. Chemical stability represents at the moment the main hurdle for the realization of perovskite-based solar cells on a large scale, as atmosphere exposition increases perovskite decomposi- tion due to air humidity. Beyond photovoltaics, organometal halide perovskites are of great interest for optoelectronic applications thanks to the observation of intense photoluminescence under laser excitation both in pure and mixed-composition. Perovskite-based Light Emitting Diodes (LEDs) and lasers may represent the new frontier for optoelectronics. Due to their full gap-tunability across the visible spectrum, perovskites could be used as white-light emitters, if arranged in tandem stacked light-emitting diodes or in microscopically phase-segregated mixture (Figure 3.4). Despite these extremely intrigu- ing perspectives, there is still a very long way to go before obtaining a real working perovskite-emitting device. In fact, light emission from perovskites has been obtained only by laser excitation, while LEDs and lasers need direct electric current injection. Hence, as in the case of photovoltaics, the challenge is to ﬁnd proper contacts and transport layers that can make possible such direct injection. An additional complication is due to the fact that current densities in both LEDs and lasers are orders of magnitude 28 Perovskites

Figure 3.4 | Perovskite-based light emitters. Due to the band-gap tunability, also the light emission can be modiﬁed with halide substitution in MAPbX3 perovskites. The high photoluminescence eﬃciencies reported for such materials make promising the realization of perovskite-based light emitters in the near future.7

larger than those typical of photovoltaic devices. Overcoming such issues is therefore the ﬁrst step towards the a new generation of optoelectronic devices.

3.3.2 Sample fabrication

MAPbX3 provskites synthesis (X=I,Cl,Br) is easier than other traditional methods commonly adopted to fabricate solution-processed semiconductors, due to relatively simple experimental equipment and environmental conditions needed. The synthesis can be in general divided into two parts: the synthesis of the precursors salts (organic MAX and inorganic lead halides PbX2) and the perovskite crystallization. Generally, lead halides are commonly purchasable by companies, while organic salts must be synthe- sized in laboratory. MAI and MABr organic salts are obtained by reacting a methylamine-ethanol solution with halogen acids HI and HBr, respectively. Perovskite crystals are obtained by reacting the organic salts with lead halides with different methods. Herein, we adopted the single step and two-step method. The single step method consists in obtaining a solution by dissolving together MAX salt and PbX2, in a 3:1 molar ratio, in N-Dimethylformamide (DMF). A drop of such solution is then deposited on a glass substrate, and a thin film is obtained by rotating it at high speed by spin coating. After, 3.3 Methylammonium lead trihalide perovskites 29 thermal annealing allows perovskite crystallization via self-organization. The two-step method consists in first obtaining a thin lead halide film by spin coating a drop of lead halide solution deposited on a substrate; secondly, a solution of the MAX salt is deposited on the lead halide layer by evaporation in nitrogen atmosphere or by spin coating. A thermal annealing completes the crystallization. Organic salts residues that have not reacted are removed by rinsing the film in isopropanol a few times during spin coating. The two-step method is more suitable to fabricate samples for optical amplification than the single step, as the former produces thinner films that confine light more strictly than thick samples. Conversely, the single step method makes samples more suitable for absorption experiments. A more detailed description about the fabrication of our samples is reported in Appendix D. 30 Perovskites 4|E XPERIMENTAL TECHNIQUES

4.1 Introduction

In order to extract infromation about the optoelectronic properties of MAPbX3 perovskites, we carried out different optical spectroscopy experiments. Lin- ear absorption spectroscopy is useful to measure the energy gap and the absorption spectrum of a material, representing the ﬁrst and fundamental step of the investigation. Further information comes from photoluminescence spectroscopy experiments, where a laser source is used to excite the material and the emitted light is analyzed both spectrally and temporally in order to extract information such as the nature of the photoexcited species, the carrier dynamics, the relaxation rates, and the radiative efﬁciency.

4.2 Linear Absorption Spectroscopy

The optical density or the absorption coefficient of an optical medium can be obtained from linear absorption spectroscopy experiments, with which one measures the light transmitted from a thin sample. To carry out reliable measurements, the sample thickness must be sufficiently large in order to detect sensitive spectral OD variations and sufficiently small to avoid absorption saturation. If sample thickness is too large, almost all of the light will be absorbed from the energy gap upwards, thus only the scattered and not absorbed fraction pass through the sample. The resulting absorption spectrum appears distorted, consisting of a rapid increase of absorption in few meV close to the energy gap, followed by a plateau. To avoid saturation, since typically crystal absorption coefficients are included between 104 and 106 cm−1, sample with thicknesses of the order of 100 nm are used,

31 32 Experimental techniques

Figure 4.1 | Schematic diagram of a typical experimental arrangement adopted for linear absorption spectroscopy measurements. The light beam provider by a continuous wave source is focused to the sample to analyze trhough collimation lenses. The transmitted light is then collected, collimated and focuses on the entrance slit of a spectrometer by collection lenses. Then, light is dispersed by a grating installed in the spectrometer and ﬁnally collected by a detector??.

yielding OD ∼ 0.1 − 1. If such optical densities are obtained with larger crystals, absorption spectra are likely to be affected by saturation. On the contrary, too small thicknesses are such that almost all the light passes the sample, thus spectral absorbance variations could not be detected by instruments or drowned inside noise. Figure 4.1 shows the typical experimental conﬁguration adopted for light transmission measurements. A beam of intensity I0, provided by a continuous wave (cw) white lamp, is directed towards the sample to analyze through a system of collimation lenses. After crossing the sample, the beam intensity will be attenuated according to the Beer-Lambert law. Col- lection lenses harvest the transmitted light and focus it inside the entrance slit of a spectrometer coupled to a detector, which records the transmitted spectrum measuring the intensity of each spectral component. Sample OD is obtained by calculating the base 10 logarithm of the ratio between the light intensity measured without and with the sample, respectively, according to Equation 2.3. If the sample thickness l is known, absorption coefﬁcient α can be easily obtained from OD as follows: OD α = (4.1) 0.434l Temperature-dependent linear absorption measurements can be carried out by placing samples in cryostats. The most used coolant are liquid nitrogen and liquid helium, which allow measurements from room temperature 4.3 Photoluminescence Spectroscopy 33 down to 77 K and 2 K, respectively. Typically, linear absorption measurements covering the whole visible spectral range, from the ultra-violet (UV) to the near infra-red (NIR), are called UV-Vis measurements.

4.2.1 UV-Vis absorption setup

?? Figure 4.2 shows a scheme of the experimental setup that we adopted to carry out linear absorption measurements. All the used lenses are achromatic doublets that minimize effects due to chromatic aberration. Optical absorption is measured from samples kept under vacuum in a continuous-flow cold-finger cryostat (Janis ST-500), which uses liquid nitrogen as coolant to vary the temperature in the 80 – 300 K range. Mea- surements in the visible range are carried out using the collimated light beam provided by a source (Oriel 67005) equipped with a Xe lamp. The beam is first reflected at an angle of 90° by the removable mirror RM, and after focused on the sample surface by the lens L1. The transmitted light is collected and collimated by L2 and focused on the entrance slit of a spectrometer (ARC-SpectraPro 2300i, 600 g/mm grating, 500 nm blaze), coupled to a CCD camera (Andor Newton), by the lens L3. Measurements in the near infra-red spectral region are carried out removing the mirror RM and using the collimated light beam delivered by an incandescent tungsten plate. Sample imaging, useful to select a precise region of the sample before starting data acquisition and to quickly check the optics alignment, is obtained by placing the removable mirror RIM between the lenses L2 and L3. RIM deflects the transmitted light beam at an angle of 90° towards the lens

L4, which focuses the light on the sensor of a CCTV camera connected to a LCD monitor.

4.3 Photoluminescence Spectroscopy

Photoluminescence spectroscopy is an useful tool to establish if a material have the potentialities to be employed as an optical emitter in an optoelectronic device. Light emitting diodes (LEDs) emit light upon direct electric injection, while lasers deliver extremely intense and collimated beams, 34 Experimental techniques

Figure 4.2 | Linear absorption spectroscopy experimental setup. Vis source: Oriel 67005 equipped with a xenon lamp. NIR source: incandescent tungsten plate. Cryostat: Janis ST-500. Spectrometer: ARC-SpectraPro 2300i, 600 g/mm grating, 500 nm blaze. CCD camera: Andor Newton. Lenses are achromatic doublets.

within a extremely short wavelenth range, as a consequence of intense photoexcitation or electric injection of an active medium. Figure 4.3 shows the typical experimental arrangement used to perform PL measurements. The sample to analyze is placed in a sample holder. PL mesarurements at different temperatures can be carried out by placing samples in a cryostat. A monochromatic excitation source, i.e. a laser device, delivers a beam with photons having energy larger than the energy gap of the sample material (it is therefore necessary to know the absorption coefﬁcient of the sample beforhand). The laser beam excites the material and the resulting photoluminescence is emitted in all directions. A fraction of the PL is collected by an optical system and dispersed into a spectrometer coupled to a detector. Depending on the possibility of the detector to measure a time-resolved PL signal or not, the kind of PL spectroscopy is deﬁned Time-Resolved Photolu- minescence (TRPL) or Time-Integrated Photoluminescence (TIPL), respectively.

If the laser pulse duration τlaser is extremely short compared to the PL lifetime τPL, as an example τlaser ∼ 100 fs and τPL ∼ 1 ns, the excitation regime is deﬁned impulsive, since emission occurs orders of magnitude later 4.3 Photoluminescence Spectroscopy 35

Figure 4.3 | Schematic diagram of a typical experimental arrangement adopted for photoluminescence spectroscopy measurements. Luminescence is excited focusing on the sample surface a laser beam with photons of energy larger than the energy gap of the material. A part of the emitted light is collected, collimated and focused on the entrance slit of a spectrometer. Finally, the photoluminescence is dispersed by a grating installed in the spectrometer and a detector measures the intensity of each spectral component.2

than excitation and there is no interplay between the two processes. With such experiments, the PL temporal evolution of a sample yields direct information about radiative recobination processes, rates and lifetimes. The PL time dependence can be measured with temporal resolutions down to 1 ps and time ranges of some nanoseconds by using a streak camera. Lower resolutions combined with longer ranges are obtained using photomultipli- ers or photodiodes.

When τlaser is comparable or longer than τPL, the regime is deﬁned quasi-steady state, because there is interplay between excitation and relaxation. Pure steady state regime is obtained using a cw laser. This kind of experiment is an useful tool to study the material behaviour under continuous excitation, thus simulating what happens under the continuous operation of a real device and giving rise to processes and effects that do not appear under impulsive regime.

4.3.1 Ultrafast PL Spectroscopy setup

Figure 4.4 shows a scheme of our ultrafast PL spectroscopy setup. Sam- ples are kept under vacuum in a continuous-flow cold-finger cryostat (Ja- nis ST-500), fed with liquid nitrogen to vary the temperature in the 80 – 300 K range. Excitation is obtained by using separately two ultrafast laser sources, both delivering 130-fs-long pulses at a repetition rate of 1 36 Experimental techniques kHz. Source 1 is obtained by passing the intense near infra-red beam (780 nm in wavelength), delivered by a Ti:sapphire regenerative amplifier (Quantronix Integra, 1.5 mJ output energy), trough a BBO (beta barium bo- rate, βBaB2O4) optical non-linear crystal (placed between the mirrors LM1 and RLM2), which halves the incoming photon wavelength to generate the second harmonic at 390 nm. Source 2 is a wavelength-tunable optical parametric amplifier (Light Conversion Topas), pumped by the Ti:sapphire regenerative amplifier. Source 1 is used for both TRPL and TIPL, while source

2 only for TIPL. Mounting or removing the removable mirrors RLM1 and RLM2 makes possible the use of whether the source 1 or the source 2, respectively. Once selected the laser source, the excitation beam passes through a variable neutral density (ND) filter wheel, which attenuates the beam intensity at the desidered value. After being reflected off at an angle of 90° by the LM3 mirror, the beam is focused on the sample surface by the lens L1. A fraction of the photoluminescence is collected and collimated by the lens L2 and a colored glass filter stops the laser beam. The PL beam can be directed to the TRPL or TIPL lines by removing or placing the mirror RSM, respectively. Placing the RIM mirror between the glass filter and RSM, the PL beam can be deflected to an imaging system to quickly select a precise sample region and monitor the beam alignment. RIM is removed during data acquisition.

For time-integrated measurements, the optical emission is focused by the lens L5 on the entrance slit of the spectrometer 2 (ARC-SpectraPro 2500i, 150 g/mm grating, 500 nm blaze), coupled to a liquid-nitrogen cooled CCD camera (Princeton Instruments).

For time-resolved measurements, the PL beam is focused by the lens L3 on the entrance slit of the spectrometer 1 (ARC-SpectraPro 2300i, 50 g/mm grating, 600 nm blaze) coupled to a streak camera (Hamamatsu C5640 - more details about streak cameras are subject of Appendix C.5.2). This latter acquires the PL emission sinchronous with the laser excitation thanks to a trigger signal generated by the photodiode PD, which detects the weak laser pulses deﬂected by the semi-transparent mirror TM, placed between the mirrors RLM1 and LM1. A delay unit optimizes the delay between the beginning of the acquisition and the PL arrival. 4.3 Photoluminescence Spectroscopy 37

Figure 4.4 | Ultrafast photoluminescence spectroscopy experimental setup. Source 1: Ti:sapphire regenerative ampliﬁer Quantronix Integra, 1.5 mJ output energy. Source 2: optical parametric ampliﬁer Light Conversion Topas. Cryostat: Janis ST-500. Spectrometer 1: ARC- SpectraPro 2300i, 50 g/mm grating, 600 nm blaze. Spectrometer 2: ARC-SpectraPro 2500i, 150 g/mm grating, 500 nm blaze. Streak camera: Hamamatsu C5640. CCD camera: Princeton Instruments. Collection lenses are achromatic doublets, while to focus laser beams quarz lenses are used. 38 Experimental techniques

Figure 4.5 | Steady-state photoluminescence spectroscopy experimental setup. Laser source: Spectra-Physics Millennia VIs. Cryostat: Janis ST-500. Spectrometer: ARC-SpectraPro 2500i, 150 g/mm grating, 500 nm blaze. CCD camera: Princeton Instruments. Collection lenses are achromatic doublets, while to focus laser beams quarz lenses are used.

4.3.2 PL Spectroscopy setup under cw pumping

Figure 4.5 shows the experimental configuration adopted to carry out steady- state TIPL spectroscopy measurements. The source consists of a Nd:YAG laser (Spectra-Physics Millennia VIs) that delivers a cw monochromatic beam of 532 nm in wavelength. Excitation and imaging are obtained adopting a configuration similar to the one used for ultrafast PL spectroscopy measurements. Samples are kept under vacuum in a cold-finger cryostat (Janis ST-500) and the temperature is varied in the 80 – 300 K range using liquid nitrogen as coolant. Light emission is focused on the entrance slit of a spectrometer (ARC-SpectraPro 2500i, 150 g/mm grating, 500 nm blaze) and detected by a liquid-nitrogen cooled CCD camera (Princeton Instruments).

4.3.3 PL Spectroscopy setup under quasi-cw pumping

Figures 4.6 shows a schematic diagram of the setup used for quasi steady- state TRPL and TIPL spectroscopy experiments, which makes use of two separate excitation sources that deliver ns-long pulses. Even if the longest 4.3 Photoluminescence Spectroscopy 39

PL lifetime of our samples does not exceed 80 ns, both kinds of excitation are considered as a quasi steady-state regime due to their finite pulse duration. Source 1 is a Q-switched Nd:YLF laser delivering 527 nm in wavelength and 300-ns-long pulses at different repetition rate, which is tunable from 100 to 1000 Hz. For our purposes, the beam repetition rate was chosen in order to have a final frequency of 6 Hz, obtained by passing it through the blade of a phase-locked optical chopper synchronous with the laser source. Source 2 provides emission of 4-ns long pulses, at different wavelength and at 10 Hz in repetition rate. It consists of an optical parametric oscillator with a frequency doubler device (Spectra-Physics MOPO-730), pumped by the third harmonic of a pulsed Nd:YAG laser (Spectra-Physics Quanta Ray PRO-270). Placing and removing the mirror RLM makes possible to use source 1 or source 2, respectively. Samples are kept under vacuum in a liquid nitrogen cold-finger cryostat (Janis ST-500). Excitation and imaging are obtained adopting a configuration similar to the one used for ultrafast PL spectroscopy measurements. After being focused on the entrance slit of a spectrometer (ARC-SpectraPro 2300i, 600 g/mm grating, 500 nm blaze), light emission from samples is dispersed and detected by an image-intensified, time-gateable CCD camera (ICCD PI MAX Princeton Instruments) with 2 ns minimum gate (more details about ICCD cameras are subject of Appendix C.5.3). Such camera is triggered by a delay unit that provides signals sinchronous with the laser source. 40 Experimental techniques

Figure 4.6 | Quasi steady-state photoluminescence spectroscopy experimental setup. Source 1: Q-switched Nd:YLF laser BMI Industries. Source 2: optical parametric oscillator Spectra-Physics MOPO-730. Cryostat: Janis ST-500. Spectrometer: ARC-SpectraPro 2300i, 600 g/mm grating, 500 nm blaze. ICCD camera: PI MAX Princeton Instruments. Collection lenses are achromatic doublets, while to focus laser beams quarz lenses are used. 5|E XCITONSVSFREECARRIERS

5.1 Introduction

Establishing which ones between excitons and free carriers are the dominant photoexcitations in organometal perovskites is of primary importance, since the design of optoelectronic devices changes signiﬁcantly depending on the kind of photogenerated carriers. Initially, various publications explained the optoelectronic properties of perovskites in terms of excitons, independently from the particular kind of halide used to syn- thesize the crystal. In fact, MAPbX3 perovskites, where X represents the halide (I−, Cl−, Br−andmixed), show strong absorption close to the energy- gap and an evident excitonic feature at the band-edge. At the same time, emission occurs very close to the excitonic resonance. These two features are common to most organic materials, where optoelectronic properties are notoriously mediated by excitons. Since also perovskites contain an organic molecule inside their structure, there is no reason to doubt about such excitonic features. However, several recent papers have radically over- turned the initial interpretations, showing that photoexcitations at room temperature largely generate unbound electron-hole pairs instead of excitons, which represent only a small fraction of the photoexcited carriers. This chapter will address the issue concerning the prevalence of unbound over bound electron-hole pairs by studying the emission processes. In particular, relaxation rates and excited state dynamics are investigated by transient photoluminescence spectroscopy measurements.

41 42 Excitons vs free carriers

Figure 5.1 | Transient photoluminescence spectra from methylammonium lead halide perovskite films. PL spectra are collected from samples at 300 K, integrated in 60 ps (temporal resolution of our streak camera), compared with the absorption spectrum at the same temperature (gray shadow). Photoluminescence was excited by 130-fs-long laser pulses with a repetition rate of 1 kHz and 3.18 eV photon energy. The legend reports the injected carrier density, calculated from laser pulse fluence and film absorbance38.4

5.2 Radiative recombination processes

Studying the recombination processes is useful to extract information about the photoexcited carriers and to estimate the value of fundamental parameters like the bimolecular and the Auger recombination constants. Ultrafast transient photoluminescence spectroscopy measurements, which use extremely short pulses to excite light emission, represents a powerful tool to investigate these processes. In agreement with results previously published in literature by other groups, our experiments show that the photoluminescence emission in methylammonium lead perovskites occurs close to the excitonic resonance, with the the high energy side of the PL spectrum overlapping with the continuum of band-to-band transitions. Such overlap is observable in a wide excitation range, as shown in Figure 5.1. Further information comes from the analysis of the Time-Resolved PL. 5.2 Radiative recombination processes 43

Figure 5.2 | PL lifetimes from methylammonium lead halide perovskite films. (a,b) Transient PL signal as a function of the injected electron–hole pair density at the film surface in MAPbI3 (a) and MAPbI3−xClx (b) films. The initial decays of the signal are fitted by an exponential function (black dotted lines). (c) Time resolved PL signal from mixed MAPbI3−xClx 17 (blue line) and pure MAPbI3 (red line) films, relative to an excitation density of at 1.7 × 10 cm−3. The different mean PL lifetime is attributed to different trap densities in the samples38.4

The rise of the photoluminescence signal after the laser pulse arrival occurs within the temporal resolution of our experimental setup (about 60 ps) in both samples and independently from the excitation intensity. If some exciton formation is involved following excitation, such process is extremely fast and it is not observable, as it occurs within the temporal resolution of the streak camera. Pure iodide MAPbI3 perovskites have smaller mean lifetimes at low laser intensity than those of mixed MAPbI3−xClx, because the presence of chlorine in these latter improve the crystallization during the synthesis, reducing the formation of defects with a consequent longer excited state lifetime [49,50] (Figure 5.2). Different mean PL lifetimes at low laser intensity are therefore consistent with different trap concentrations between the samples. At higher laser ﬂuences, samples appear to be quite similar, since their PL lifetime decreases with increasing laser pumping, as a consequence of the activation of density-dependent recombination mechanisms. Such similarity results to be more evident analyzing the recombination rate kPL calculated just after the laser excitation (at t = 0), which reads:

dPL kPL = (5.1) dt t=0 kPL accounts only for radiative recombination processes, as other slow re- 44 Excitons vs free carriers combination mechanisms depopulate excited states at later times. Figure

5.3 shows the PL decay rate as a function of n0, which is the injected carrier density at the sample surface. n0 is obtained by the product between the laser fluence Φ and absorption coefficient of the film α(E), calculated at the excitation energy. As can be observed, kPL grows in n0 in both samples as a consequence of the activation of density-dependent recombination mechanisms. The fact that the initial PL decay rates are quite similar in the samples means that kPL is governed by intrinsic nonlinear processes [30], differently from what happens at low laser intensities, where extrin- sic mechanisms are involved (i.e. carrier trapping). The rapid increase of the decay rate at high excitation density is a consequence of the prevailing 3 non-radiative Auger recombination, whose rate grows as n0. Fundamental information about the nature of the photoexcited carriers can be obtained studying the intensity of the PL signal just after the laser excitation (PL0 = PLatt = 0) as a function of n0. The dependence of

PL0 on the injected carrier density reveals which kind of carrier prevails after the laser excitation. Our experimental results show that PL0 grows 16 −3 2 18 −3 quadratically in n0 from less than 10 cm (∼ 0.1µJ/cm ) to 10 cm 2 (∼ 10µJ/cm ) in both MAPbI3 and MAPbI3−xClx samples. For densities 18 −3 above 2.5 × 10 cm , the band ﬁlling gives rise to some saturation of PL0. The quadratic dependence below 1018cm−3 is consistent with the radiative recombination from a gas of unbound electron-hole pairs, otherwise PL0 would increase linearly in n0 if excitons were the main resulting photoexcitations [51]. This ﬁnding does not imply that excitons cannot be formed, but their concentration is negligible with respect to free carriers. The next section will explain in details this aspect by thermal equilibrium consider- ations.

5.3 Saha equation

In a gas of electrons and holes at a certain temperature T, charges can interact to form bound excitons or unbound pairs, depending on the thermal equilibrium between the two species. The following action-mass law is useful to estimate the fraction of bound (and unbound) electron-hole pairs 5.3 Saha equation 45

Figure 5.3 | Decay rates from methylammonium lead halide perovskite films as a function of the injected carrier density (bottom axis) and the laser pulse fluence (top axis) The rates, defined as kPL = [dPL/dt · 1/PL]t=0, are extracted from the data in Figure 5.2. Such rates represent the initial decay and should not be mistaken for the average photoluminescence decay rate obtained by fitting the entire decay with an exponential function. The error bars represent the standard deviation from a least square fit to an exponential decay and are reported only when they exceed the size of the marker. The rates are very similar for the two samples (although the average exponential decay rates are significantly different) and grow together for growing injected carrier densities. Such rates measure the intrinsic density-dependent bimolecular and Auger recombination processes. The dotted lines represent the results of the rate equation used to model the experimental data38.4 46 Excitons vs free carriers with respect to the total pairs:

nenh = nXneq (5.2) where ne, nh, nX and neq represent the free electron density, the free hole density, the exciton density and the equilibrium population density, respectively. This latter can be expressed as a function of the exciton binding energy Eb and the gas temperature T, through the Saha equation:

3/2 Eb µXkBT − k T neq = e B (5.3) 2πh¯ 2 where µX is the exciton reduced mass (∼ 0.15m0 in MAPbI3−xClx perovskites), and kB is the Boltzmann constant [c26,29,30nc]. Photoexcitation creates electron and holes in equal proportions, therefore we label the number of unbound electron-hole pairs per unit volume nc = ne = nh. Hence, under optical excitation, the injected electron-hole pair density n0 is connected to both the exciton density nX and the unbound electron-hole pair density nc through the following equation:

n0 = nX + nc (5.4)

Therefore, PL can be divided into two contributions, one coming from the bimolecular recombination between unbound electron-hole pairs, and one coming from the monomolecular excitonic recombination, as reported in the next equation:

2 PL = kbnenh + kXnX = kbnc + kXnX (5.5) where kb and kX are the bimolecular radiative recombination rate from unbound electron-hole pairs and the monomolecular radiative recombination rate from excitons, respectively. Setting Eb = 23meV, as obtained from the ﬁtting of the absorption spectrum, and T= 300 K, we found 17 −3 neq = 6.4 × 10 cm . For injected carrier densities much lower than the equilibrium population density, that is for n0 neq, the majority of the photoexcited electron-hole pairs are unbound, as can be calculated from

Equations 5.2 and 5.4. Under these conditions, nc ≈ n0, and also the exciton density scales quadratically in n0, according to Equation 5.2. Such 5.4 Quantum yield 47 condition results in a quadratic dependence of the PL signal in the injected carrier density from both the contributions of bound and unbound electron-hole pairs. Such scaling is in agreement with our experimental results.38

5.4 Quantum yield

Until now, only radiative recombination processes have been investigated. To have a wider insight about the recombination mechanisms involved, it is useful to include also non-radiative recombination processes. The quantum yield (QY), which represents the number of emitted photons with respect to the total absorbed photons, takes into account all the recombination mechanisms and quantifies the radiative efficiency of the material. Low values of the QY are due to non-radiative processes that dominate over radiative recombination, while a QY ∼ 1 is obtained when radiative processes pre- vail over non-radiative ones. One way to extract the quantum yield from experimental data is dividing the Time-Integrated Photoluminescence signal (TIPL) by the laser pulse fluence Φ. Investigating the QY dependence on n0 provides fundamental information about the interplay between radiative and non-radiative recombination processes. Figure 5.4 shows the QY in both pure iodide and mixed methylammonium lead perovskites as a function of the injected carrier density and the laser pulse fluence. Low values of the QY are obtained at low and high injected carrier den- 17 −3 sities. Carrier trapping is particulary efficient below n0 ∼ 10 cm , being the fastest recombination process in this regime. An increase of the laser intensity results in an increase of the bimolecular radiative recombination rate, with a consequent raise of the number of electron-hole pairs that recombine radiatively. Hence, also the QY grows and reaches the maximum at the remarkable high values of 30% in MAPbI3 and 70% in MAPbI3−xClx 18 −3 perovskites. Such values are obtained for n0 ∼ 10 cm , close to the amplified spontaneous emission threshold reported in literature for these 38 19 −3 materials CITARE XING. The decrease of the QY above n0 ∼ 10 cm is consistent with the predominance of Auger recombination over bimolecular radiative recombination at high intensity [30]. 48 Excitons vs free carriers

Figure 5.4 | Photoluminescence quantum yield from methylammoium lead halide perovskites. The emission quantum yields are calculated as TIPL/Φ, where Φ is the laser pulse fluence. The injected carrier density is calculated multiplying the laser pulse photon fluence by the absorption coefficient of the films. As a reference, the laser pulse fluence directly measured in the experiments is also reported on the top axis. Initially the QY grows with fluence for both films, as the bimolecular recombination becomes faster and a growing fraction of the injected excitations recombine radiatively, instead of being trapped. At excitations 4 × 1019 cm−3, non-radiative Auger processes dominate. The absolute QY is scaled to match theoretical predictions. The maximum QY values are about 30% for MAPbI3 and 70% for MAPbI3−xClx. The dotted lines represent predictions from a rate equation model accounting for the main relaxation channels for electrons and holes. Simulations take into account the exponential spatial profile of the electron–hole density created by laser pulses. The very good agreement between model and data indicates that the main photophysical processes are accounted for in the model38.4 5.5 Analysis of the recombination processes 49

5.5 Analysis of the recombination processes

This section provides an analysis of the recombination processes in pure

MAPbI3 and mixed MAPbI3−xClx perovskites. As a result of the calculation, we will be able to extract some photophysical parameters, such as the bimolecular and the Auger recombination coefﬁcients, which represent two constants characterizing the material and can be considered as ﬁgures of merit for different optoelectronic applications. We started estimating the emission properties of MAPbI3−xClx perovskites solving the Kubo-Martin- Schwinger relation for a gas of free carriers, since the experiments highlighted that the large majority of the photoexcited electron-hole pairs are unbound [c42,43nc]. The spontaneous radiative photon emission rate per unit volume, R(n), can be calculated as follows:

Z ∞ ωnr R(n) ≈ fe(ω) fh(ω) dω (5.6) ωg πc where ωg is the photon frequency corresponding to the energy gap of the material, nr the refraction index, c the speed of light, α0 the absorption coefﬁcient of the continuum states, fe and fh the Fermi distributions of electrons and holes, respectively. R(n) is proportional to the value of PL0 obtained experimentally. To calculate the spontaneous radiative recombination rate per electron-hole pair is sufﬁcient to divide R(n) by the carrier density n, thus obtaining:

R(n) k (n) = (5.7) rad n Starting from this equation, and making use of the absorption spectrum, it is possible to calculate the bimolecular recombination constant kb, a parameter that should be the same among samples of the same material, as it depends only on intrinsic processes. Evaluating the bimolecular constant from absorption data results to be more reliable than what obtained extracting it from the PL decay curves using multi-power decays. In fact, the former method makes use of the simmetry between absorption and emission and is completely independent on non-radiative recombination processes, while the second does not provide reliable estimates when applied to PL decays with several competing processes relevant at the same time, as 50 Excitons vs free carriers happens in perovskites (i.e. monomolecular carrier trapping, bimolecular radiative recombination and trimolecular non-radiative Auger recombination). Hence, from Equation 5.7 we obtained a value of 2.6 × 10−10cm3s−1 kb for MAPbI3 perovskites, a value included between the two extremes reported in literature for MAPbI3 and MAPbI3−xClx perovskites [20,30,52], that is 10−11 and 10−9cm3s−1, respectively. Such a wide range of values can be attributed to the different methods used to calculate it. Equation 5.7 is extremely powerful to study the recombination dynamics as a function of the carrier density n. At low excitation intensity, namely 17 −3 for n 10 cm , the Fermi distributions fe(ω) and fh(ω) are much smaller than 1, a situation where the regime can be considered dilute: the probability for an electron to meet a hole is proportional nh. Hence, krad can be calculated as the product between kb and nh. Since ne = nh = n, we can observe from Equation 5.7 that krad(n) grows linearly in n, and PL0 ∼ R(n) grows quadratically according to the experiment. High values of the injected carrier density result in a deviation from the quadratic de- 19 −3 pendence of R(n), as fe ∼ fh ≈ 1. Therefore, above 10 cm , the band

ﬁlling causes the saturation of krad. As a consequence, R(n) grows linearly in the injected carrier density, again in agreement with what observed in the experiment.38 In order to ﬁt TIPL and QY and evaluate the Auger recombination constant, we solved the following rate equation, which accounts for all the relevant recombination processes involved in perovskites:

dPL = −γ n − k n2 − γ n3 (5.8) dt t b A

γA represents the Auger constant and γt the carrier trapping. Auger recombination can be neglected below 1018cm−3, thus Equation 5.8 can be simpliﬁed as follows:

dPL = −γ n − k n2 (5.9) dt t b

The carrier trapping constant γt can be easily extracted from this equation, being the only free parameter, sincekb has been already extracted from absorption. Once set γt in order to ﬁt the experimental results for small n, we used Equation 5.8 at high densities, this time taking into 5.6 Steady-state photoluminescence 51

account also the Auger recombination and leaving γA as free parameter. The best fitting procedure is obtained for values of the Auger constant of −28 6 −1 −28 6 −1 2 × 10 cm s and 4 × 10 cm s , for MAPbI3−xClx and MAPbI3 perovskites, respectively [c19nc,saba]. Differently from the bimolecular constant, γA can change among samples remarkably, as it depends on the crystal defect density of a crystal. This latter is strongly sensitive to the chemical synthesis and also the surface morphology can contribute to increase −28 6 −1 the value of γA [55,56]. In fact, we obtained γA ≈ 2 × 10 cm s for −28 6 −1 mixed MAPbI3−xClx and 4 × 10 cm s for pure iodide MAPbI3. Auger recombination is more significant in pure iodide perovskites due to the less crystallinity with respect to films obtained including chlorine. Other publications report Auger recombination constants from 10−29 to 10−28cm6s−1 for pure and mixed methylammonium lead iodide perovskites, which are close to our results. [20,30].

5.6 Steady-state photoluminescence

Impulsive regime excitation is very useful to obtain fundamental information about basic properties of materials, but to have a wider insight into the oproelectronic properties of the samples it is necessary to go beyond the limits of this investigation. Experimental results obtained when laser pulse duration is much shorter than the excited-state lifetime are not suf- ﬁcient to show if a material is really promising for some optoelectronic application, and further analysis is required. As a matter of fact, real world optoelectronic devices work under continuous operation, which is a regime that can be simulated using pulses longer than the excited-state lifetime, as interplay between excitation and relaxation processes occur at the same time. Therefore, a reliable instrument of investigation of the steady-state properties is represented by Steady-state Photoluminescence experiments, which are carried out exciting samples with a continuous wave laser (cw pumping) or with pulses much longer in time than the PL lifetime of the excited states (quasi-cw pumping). We then excited samples under cw and quasi-cw regime over eight order of magnitude in laser intensity, spanning from 10−4 to 104W/cm2. Experimental results, reported in Figure 5.5a, 52 Excitons vs free carriers show that below 100W/cm2, the PL signal follows a 3/2 power law in the laser intensity for ﬁve decades in excitation intensity. Under cw excitation, if radiative recombination would be the dominant process that governs the electron-hole dynamics one would expect a linear dependence of the PL in the laser intensity. Hence, we attributed such 3/2 power dependence to the interplay between excitation, radiative relaxation between electron and holes and non-radiative carrier trapping from intra-gap states. In particular, such traps act only on one kind of carrier, meaning traps only for electrons or only for holes, but not traps of both species.38 The model is reported in Figure 5.5b and can be explained by the following system of rate equations:

dne = −R(n) − γ n − γ n3 + I = 0 (5.10) dt t e A e

dnh = −R(n) − γ n n − γ n3 + I = 0 (5.11) dt b t h A h

dnt = γ n − γ n n = 0 (5.12) dt t e b t h

R(n) = kbnenh (5.13)

The steady-state condition is achieved setting all the equations equal to zero [c48nc]. Equation 5.10 governs the electron population, Equation 5.11 the hole population, Equation 5.12 the electron trapping and the non- radiative bimolecular recombination between trapped electrons and holes, while Equation 5.13 describes the radiative bimolecular recombination between free electrons in conduction band and free holes in valence band. At low excitation intensity, the Auger recombination is negligible, thus the cubic terms in Equations 5.10 and 5.11 can be neglected. At the same time, considering the electron trapping the dominant process at low intensities, we can neglect also the term R(n) in both Equations 5.10 and 5.11. Upon these simpliﬁcations, we obtain that ne ∝ I from Equation 5.10 and that ntnh ∝ I from Equations 5.11 and 5.12. Assuming that the populations nt 1/2 and nh are similar, that is each one scales as I , and showing the dependence on I of ne and nh in the Equation 5.13, we obtain: 5.6 Steady-state photoluminescence 53

Figure 5.5 | Steady-state photoluminescence. (a) Photoluminescence signal as a function of the intensity of the exciting laser. The empty markers represent the measurements obtained with a cw Nd:Yag 532 nm laser; the ﬁlled markers are instead measured in quasi-cw conditions, exciting the samples with 300-ns-long laser pulses from a Q-switched 527 nm Nd:Ylf laser; the pulse duration is much longer than all the relevant relaxation rates, so that steady-state condition are expected to be achieved during laser excitation. The PL signal grows as I3/2 for a wide range of excitations. Investigations were extended from laser intensities much lower than the solar one, to intensities large enough to generate population inversion and optical gain. The unusual 3/2 power law is attributed to intra-gap trap states only electrons or only holes, but not both of them. The dashed black line shows the PL dependence in laser intensity as a result of the trap model, under the assumption that electrons are the trapped species. (b) shows a sketch of the relaxation of optical excitations under steady-state conditions (VB is the valence band and CB the conduction band)38.4 54 Excitons vs free carriers

1/2 3/2 R(n) = kbnenh ∝ I · I = I (5.14) Such result is in agreement with the PL dependence in light intensity found from the experimental results. At high excitation intensities, radiative recombination becomes more important than electron trapping, resulting in a deviation from the 3/2 power law to a linear dependence. Such deviation is observed in the experiment for intensities above 102W/cm2 [c.saba] The model extimates a trap density of the order of 1018 − 1017cm−3, a value in agreement with what reported in literature by other groups [6]. Such a large trap density does not prejudice electronic properties, as PL lifetimes exceed several nanoseconds from intensities smaller than solar illumination to those typical to obtain light amplification. Therefore, the resulting carrier mobility is sufficiently high to ensure efficient transport and charge collection in perovskite-based devices. 6|E XCITONBINDINGENERGY

6.1 Introduction

One of the main features that makes perovskites particularly appealing for photovoltaics is the efficient light harvesting due to the large absorption coefficient at the band-edge and across the visible spectrum. Absorp- tion in these semiconductors is characterized by an excitonic resonance at the band-edge, followed by the continuum of the interband transitions. In Chapter 5 we discussed the optoelectronic properties of perovskites in terms of free carriers, meaning that the majority of photoexcitations results in unbound electron-hole states even at intensities comparable to solar illumination, while only a fraction of them seems to contribute to form bound excitons. Electron-hole correlation strongly contributes to enhance the absorption coefficient close to the energy-gap in these materials and the exciton binding energy is a tool useful to quantify the magnitude of such correlation. Until now, several estimates of EB have been reported in literature, ranging from few meV to over 50 meV at room temperature. Establishing a reliable and precise value for the exciton binding energy in perovskites is then an open question of primary importance. In this chapter, we will analyze the absorption coefficient of MAPbI3 and MAPbI3−xClx perovskites, discussing the determination of EB by applying the Elliot’s theory of Wan- nier excitons, which has been successfully used to model the band-edge absorption in direct-gap inorganic semiconductors. After, we will compare our and other published results in order to find if there is some dependence of the exciton binding energy on the sample realization. Finally, we will develop a method based on a f-sum rule to establish univocally the exciton binding energy in MAPbX3 perovskites, avoiding the ambiguities

55 56 Exciton binding energy

Figure 6.1 | Absorption spectra from a methylammonium lead iodide perovskite ﬁlm. Experimental data have been collected from a ﬁlm of 800 nm in thickness, at 170 (blue line) and 300 K (red line) with the experimental setup described in Section ??38.4

arising from the least squared ﬁtting of experimental data.

6.2 Elliot’s theory of Wannier excitons

We first analyzed the band-edge absorption in MAPbI3 and MAPbI3−xClx perovskites, limiting the investigation only to the tetragonal phase of the crystals, which extends from ∼160 K to about 340 K. Figure 6.1 shows the experimental results carried out on MAPbI3 films of 800 nm in thickness, at 170 and 300 K. Similar results were obtained with mixed MAPbI3−xClx samples. The absorption spectrum near the energy gap is characterized by a clear excitonic peak close to the absorption edge, followed by the contribution of band-to-band transitions. Usually, the absorption coefficient α(h¯ ω) near the band-edge for a direct bandgap semiconductor is described in the framework of the Elliot’s theory of Wannier excitons, which models the shape of the absorption spectrum by taking into account both the excitonic and continuum contributions, as shown in Figure 6.2 [c36nc]. Such theory is valid for bulk semiconductors having exciton binding energies much 6.2 Elliot’s theory of Wannier excitons 57

Figure 6.2 | Absorption spectrum computed according to Elliott’s formula. The band- edge absorption in semiconductors with small exciton binding energy can be modelled in the framework of the Elliot’s theory of Wannier excitons. The red and blue dotted line account for the excitonic peak and the continuum of the interband transitions, respectively. The solid black line takes into account both the two contributions. The horizontal axis is calculated with respect to the energy of the exciton peak384.5

smaller than the energy gap (Wannier excitons), and accurately describes the optical transitions near the band-gap in inorganic semiconductors such as GaAs and GaP [c38,39nc]. We can consider the following equation for α(h¯ ω) as starting point for our investigation:

2 µcv 2 α(h¯ ω) ∝ ∑ |φj(r = 0)| δ(h¯ ω − Ej) ∝ h¯ ω j q √ (6.1) 2 4π E3 µcv h b b 2π Ebθ(h¯ ω − Eg) i ∝ δ(h¯ ω − E ) + r ) ∑ 3 j E h¯ ω j j −2π b 1 − e h¯ ω−eg

Such equation takes into account the transition dipole moment µcv between conduction and valence bands (which quantifies the oscillator strength), while 58 Exciton binding energy h¯ ω is the photon energy involved in the transition. The absorption coefficient depends on the weighted density of electron-hole pair states, with the weight provided by the probability for an electron and a hole to be at the 2 same position, that is |φj(r = 0)| , while φj represents the wavefunctions of bound and unbound states. In the third expression of Equation 6.1, the first b term describes the transitions to bound exciton states Ej , while the second term takes into account the band-to-band transitions above the energy gap

Eg. δ(x) and θ(x) are the Dirac-delta and the Heaviside step function, respectively. To simulate the absorption coefﬁcient, we overcome the limits of approximation of the parabolic bands approximation, introducing the joint valence-band energy-momentum dispersion

h¯ 2k2 E (k) − E (k) ≈ − bk4 + E (6.2) c v 2µ g where Ec(k) and Ev(k) are the energy-momentum dispersion of the conduction and the valence bands, respectively. b is a parameter that takes into account the non-parabolicity of the bands and µ is the reduced electron- hole mass (0.15 m0, where m0 is the electron mass [c1-3SIsaba]). The joint density of states associated to this band dispersion is described by the following equation, obtained developing in series for small b at the ﬁrst order:

1 2µ3/2h 10µ2b iq ρ(E) ≈ 1 + (E − Eg) E − Eg (6.3) 2π2 h¯ 2 h¯ 4 Equation 6.3 recovers the usual density of states obtained for parabolic bands if b = 0. Hence, upon these improvements, we modiﬁed Equation 6.1 in order to obtain the following expression, which is to the one we used to ﬁt the experimental results:

b α h 2E h¯ ω − Ej ( ) = scal b + α h¯ ω ∑ 3 sech h¯ ω j j Γ b Z ∞ h¯ ω − Ej 1 h 10µb i i + sech r 1 + (E − Eg) dE +αc E 4 Eg Γ b −2π − h¯ 1 − e E Eg (6.4) We convoluted the excitonic transitions with a bell-shaped function, in this case a hyperbolic secant of width Γ, which accounts for thermal and 6.2 Elliot’s theory of Wannier excitons 59

inhomogeneous broadening. αscal is a constant that is adjusted in order to obtain the right scaling between the model and the experimental absorption spectrum. We also introduced the term αc, accounting for absorbance offsets often found in experimental curves due to unbalanced reference or zero spectra in spectrophotometers. Excitonic and free-carrier contributions weight differently in different spectral regions: while excitonic transitions are very important at the band edge, for photon energies well above the bandgap the excitonic contribution becomes negligible and the absorption coefficient follows the density of states expressed by Equation 6.3, as expected for band-to-band transitions. To calculate the exciton binding energy, we fitted the experimental absorption spectra with the least square method, leaving the exciton binding energy and the thermal line broadening as free parameters. As a matter of fact, line broadening grows with temperature and one expects to have a higher Γ value at 300 K than at 170 K. We then extended the fitting range as much as possible at low energy in order to limit the possible values of αc and at high energy to limit the possible values of αscal. Fitting was first calculated for the absorption data at 170 K, since the excitonic resonance is sharper at low than at room temperature, leading to a better sensitivity of the fit to Eb. The final result returns the best agreement between theory and experiment for an exciton binding energy of 25 meV38 (Figure 6.3). We than fitted the absorption spectrum at 300 K using the same parameters obtained by fitting the absorption at 170 K, except for the thermal line broadening and the energy gap, which were left free. Again, we obtained a satisfactory agreement between theory and experiment. Thermal broadening is higher at room than at low temperature, as we expected by theory, and the exciton binding energy does not change. 25 meV is an intermediate value between the binding energy typical of Frenkel excitons in organic semiconductors (>100 meV) and the one typical of Wannier excitons in inorganic ones (<10 meV) [c40,34nc].

6.2.1 Comparison with literature

To have a more general validity of our results and verify if there was some dependence of Eb on both sample morphology and growth conditions, we applied the Elliot’s theory to other published absorption spectra, extracting 60 Exciton binding energy

Figure 6.3 | Analysis of the band-edge absorption at 170 and 300 K for MAPbI3 perovskite ﬁlms. Panel (a) and (b) shows the analysis relative to absorption at 170 K and 300 K, respectively. In both panels, the red empty circles represent the theoretical ﬁts to the experimental data (continuous black lines). The contributions to the absorption due to both excitonic and band-to-band transitions are modelled in the framework of the Elliott’s theory of Wannier excitons. The dotted green lines are relative to excitonic transitions, while the continuous blue lines are relative to band-to-band contributions with the inclusion of Coulomb interactions between electrons and holes38.4 6.2 Elliot’s theory of Wannier excitons 61

Figure 6.4 | Absorption spectra in MAPbI3 perovskite ﬁlms. The red empty circles represent the theoretical ﬁts to the experimental data (continuous black lines). The contributions to the absorption due to both excitonic and band-to-band transitions are modelled in the framework of the Elliott’s theory of Wannier excitons. The dotted green lines are relative to excitonic transitions, while the continuous blue lines are relative to band-to-band contributions with the inclusion of Coulomb interactions between electrons and holes. (a,b) Absorption spectrum reported by Saba et al. at 170 and 300 K, respectively38.4 (c,d) Absorption spectra reported at room temperature by D’Innocenzo et al. at 160 K and 290 K, respectively30.4

from the pdf ﬁles using CurveSnap, a software freely available online.4 Figure 6.4 shows a comparison between the absorption spectra reported by D’Innocenzo et al. and our experimental data, recorded at cryogenic and room temperature38.30

After applying Equation 6.4, extracted Eb values are comparable between the two samples, both at low and room temperature. At low temperature they coincide within the uncertainty and at room temperature they do not differ more than 20% from the average value of 25 meV.

Hence, the overall picture that emerges from optical band-edge absorption analysis seems to converge towards a temperature-independent exciton binding energy of 25 meV. Such ﬁndings contrast with ones reported by some recent works based on magnetometry, which found that Eb depends on temperature, due to e-h interaction screening as a consequence of the organic cation rotation. In particular, they report exciton binding energy values to be about 15 meV at low temperature and a few meV at room temperature [33,34,37,47]. Currently, the origin of this discrepancy between what observed from optical spectroscopy and magnetometry is not clear, and further investigation is needed. 62 Exciton binding energy

6.3 f-sum rule

All physical quantities obtained from least squared fitting of experimental data strongly depend on both the fitting range and the number of procedure parameters. Satisfactory results can be in fact provided with different combinations of them. We have discussed that fitting of experimental absorption by the Elliot formula provides reliable estimates of the exciton binding energy only when the excitonic peak is well resolved with respect to the continuum contribution, that is when the ratio between binding energy and thermal line broadening is large. At high temperatures, the estimate of Eb results less reliable due to the fact that excitonic contribution is not well resolved with respect to the continuum, as a consequence of the thermal line broadening. It is therefore necessary to define a method that can be readily applied to experimental data and independent from the particular choice of fitting parameters. A possible solution can be found analyzing the variations of the absorption coefficient due to Eb and Γ. In fact, when exciton binding energy or thermal line broadening assume different values with temperature, absorption spectrum results modified following the particular kind of variation. Figure 6.5 shows the absorption spectrum modification in the two cases when Eb is variable but Γ is constant and when Eb is constant but Γ is variable.

When Eb changes and Γ stays constant, line transitions do not change and the absorption modiﬁcations at the band-edge result governed from the variations of the excitonic resonance: at high Eb values α(E) is dominated by the excitonic peak (tightly bound e-h pairs), while assumes the p typical E − Eg dependence when Eb is very small (unbound e-h pairs).

On the contrary, if Eb = constant and Γ varies, the oscillator strength does not change and the absorption modiﬁcations result only in broadening of the transitions. To discriminate if absorption modiﬁcation is due either to

Eb or to Γ variations, it is sufﬁcient to calculate the following normalized integral of absorption:

R En α(E)dE I = 0 (6.5) α(En) where α(E) is calculated from Equation 6.1 and En is an arbitrary energy 6.3 f-sum rule 63

Figure 6.5 | UV-Vis absorption spectra computed according to the Elliott formula. Spectra are modelled leaving whether the linewidth (a) or the exciton binding energy (b) as a variable parameter. Parameters similar to ones were adopted (EB=50 meV in (a), Γ=20 meV in (b), Eg=2.3 eV everywhere). Spectra are plotted versus the diﬀerence energy with respect to the n=1 exciton transition and normalized at their value for En = E1x + 0.3 eV. The inset in panel (b) reports the spectra before normalization. Panel (c) shows that the integral of normalized absorption spectra R En grows as a function of the exciton binding energy 1/α(En) 0 α(E)dE EB (red line), while it is constant when only the linewidth Γ is varied (blue line).5

value above Eg. I is a quantity independent from both the temperature variations of Γ and the oscillator strenght, as normalization for α(En) deletes their contributions. Once ﬁxed b, I results an univocal and universal function of the exciton binding energy, thus Equation 6.5 can be deﬁned a f-sum rule. If Eb varies, also I(Eb) varies accordingly. On the other hand, if I(Eb) stays constant with temperature, Eb does not change and absorption variations are due only to broadening of the transitions. The choice of En should be done in order to have comparable excitonic and continuum contributions to absorption within the energy range from 0 to En. Since Eg changes with temperature, also the excitonic peak at energy E1X shifts with energy, and so does En. A reasonable choice is setting En = E1X + δE, where δE is an arbitrary energy amount. This latter should be of the order of some Γ to balance the contributions of both the exciton peak and continuum of interband transitions, as for δE < Γ only the excitonic peak contributes to I, while for δE >> Γ the continuum weights much more than the exciton.

To avoid spurious contribution to the calculation of I at energies below Eg, it is better not to calculate the integral of Equation 6.5 from E = 0 but instead from E = E1X − e, where e should be sufﬁciently small not to ac- 64 Exciton binding energy

Figure 6.6 | Analysis of the experimental absorption spectra for MAPbI3 films. (a) UV-Vis absorption spectra for different temperatures, normalized at the energy En = E1x + 0.2 eV. The gray dotted line is the continuum contribution αc calculated according to Elliot formula with the b=0.1 eV−1 and Γ=0. Spectra are shown with 20 K temperature interval for clarity purposes, although measurements were taken with 10 K intervals. (b) Normalized absorbance integrated up to energy (defined as R En as a function of temperature. En I = 1/α(En) 0 α(E)dE (c) Shift of the excitonic peak in the absorption spectra as a function of temperature; when the exciton peak was not well-resolved in absorption, the energy of the photoluminescence peak was taken instead, taking advantage of the negligible Stokes shift. The vertical dashed lines mark the temperatures of the phase transitions.5

count for absorption due to scattering and sufﬁciently large to include the low-energy tail of the excitonic peak at high temperature. Once set b, δE and e, it is possible to monitor the I(Eb) variation as a function of temperature. Figures 6.6 and 6.7 show such analysis for the absorption spectra of both MAPbI3 and MAPbBr3 perovskites, respectively, collected from 80 K to room temperature at steps of 10 K, where the energy axis is calculated with respect to the energy of the ﬁrst exciton peak E1X.

Since the integral is a function of only the exciton binding energy, Eb can be calculated simply as the inverse function of I(Eb). We applied this method to absorption spectra of both MAPbI3 and MAPbBr3 perovskites, as shown in Figure 6.8. We found that I(eb) calculated for MAPbBr3 is temperature-independent in the investigated range, yielding an exciton binding energy of 60 ± 3 meV that stays constant in all the three crystal phases, whereas integrals obtained from MAPbI3 absorption show a discontinuity at the phase transition between ortorhombic and tetragonal 6.3 f-sum rule 65

Figure 6.7 | Analysis of the experimental absorption spectra for MAPbBr3 films. (a) UV-Vis absorption spectra for different temperatures, normalized at the energy En = E1x + 0.2 eV. The gray dotted line is the continuum contribution αc calculated according to Elliot formula with the b=0.11 eV−1 and Γ=0. Spectra are shown with 20 K temperature interval for clarity purposes, although measurements were taken with 10 K intervals. (b) Normalized absorbance integrated up to energy (defined as R En in the text) as a function En I = 1/α(En) 0 α(E)dE of temperature. (c) Shift of the excitonic peak in the absorption spectra as a function of temperature. The vertical dashed lines mark the temperatures of the phase transitions.5

phase, for which Eb = 34 ± 3 from 80 to 140 K (ortorhombic phase) and

Eb = 29 ± 3 from 170 to 300 K (tetragonal phase). Data between 140 and 170 K are not reported because in this range both ortorhombic and tetragonal phases coexist and the value of Eg is not deﬁned precisely during the transition. Differently from what reported by recent publications, our ﬁndings show that, except for the Eb jump at the low temperature phase transition in MAPbI3 perovskite, there is no evidence of some exciton binding energy temperature dependence in MAPbX3 perovskites. 66 Exciton binding energy

Figure 6.8 | Exciton binding energy vs temperature in lead halide perovskite ﬁlms.

The binding energy in MAPbI3 (filled red circles) and MAPbBr3 (filled blue circles) films, are extracted with the f-sum rule on the integrated absorption, as described in the text. The height of the error bars on the first two points for each materials represent the uncertainties in the absolute values. Empty circles are instead the linewidths as extracted from the absorption spectra with the formula R E1x .5 Γ = ( 0 α(E)dE)/α(E1x) 7|O PTICAL AMPLIFICATION

7.1 Introduction

We have seen that, due to their excellent light harvesting capability, organometal halide perovskites are particularly promising for the realization of a new generation of solar cells. Nevertheless, such materials exhibit excellent emission properties as well as absorption. Our measurements show high

QY values for bulk MAPbI3 perovskite samples, and recent publications have reported lasing and ASE from perovskite thin films, at injected carrier densities comparable to those of the best organic semiconductors [17- 19,38,44,57]. Despite these results are of great perspective, further investigation is required to verify if perovskites can effectively represent the beginning of a new era for laser devices. Many experiments investigated emission from thin films arranged in both cavity resonators and cavity-free configuration, exciting them under impulsive excitation, which represents a regime well away from the continuous operation of real lasers. A wider insight into the emission properties at high laser pumping is provided by PL experiments where the excitation source delivers ns-long pulses, in order to simulate what happens in a real device. This chapter will discuss emission from MAPbI3 and MAPbBr3 perovskite thin films in cavity-free configuration at high injected carrier densities, exciting samples with both impulsive and quasi-cw regime. We will indentify the main processes that allow and inhibit ASE in the different excitation regimes, suggesting the possible improvements that could be made in the outlook of the realization of a real perovskite-based laser.

67 68 Optical ampliﬁcation

Figure 7.1 | ASE in methylammonium lead halide perovskite films at room temperature under femtosecond excitation. (a,b) Time integrated photoluminescence spectra of MAPbI3 (a) and MAPbBr3 (b) perovskites at 300 K. Emission was excited by 130-fs-long laser pulses with a repetition rate of 1 kHz and 3.18 eV photon energy. Spontaneous and amplified spontaneous emission were detected by a CCD camera. From top to bottom: laser fluence = 90 µJ/cm2, 70 µJ/cm2, 40 µJ/cm2, 30 µJ/cm2, 20 µJ/cm2, 10 µJ/cm227.4 (c,d) Emission intensity in

MAPbBr3 (c) and MAPbI3 (d) ﬁlms as a function of the laser ﬂuence. In both panels the red circles mark the spectrally integrated photoluminescence spectrum, excluding the ASE peak, while the blue circles represent the amplitude of the ASE peak27.4

7.2 ASE under fs excitation

We investigated ASE in thin films of MAPbI3 and MAPbBr3 perovskites, deposited by spin coating on a glass substrate. Samples were studied under femtosecond UV excitation. Figure 7.1 reports room temperature MAPbI3 and MAPbBr3 emission spectra obtained at different laser fluences. Amplified spontaneous emission appears through a sharp peak in the low energy side of the emission spectrum of both films, at injected carrier densities of the order of 1018cm−3, which corresponds to a laser fluence 2 of ∼ 10µJ/cm . The threshold density nthr is calculated by the following equation:

nthr = α(Eexc) · Φ (7.1) 7.3 ASE under ns excitation 69

where Φ is the laser fluence and α(Eexc) the film absorption coefficient at the excitation energy. Even though samples have been fabricated following the same procedure, they show slightly different nthr values between 18 −3 18 −3 2 × 10 cm and 5 × 10 cm [19]. nthr dispersion can be explained in terms of different optical losses due to the film surface morphology, which can depend on the sample deposition. Our ASE thresholds are included into the range of values reported in literature, from the low value of 1.5 × 1016cm−3 (obtained by Zhu et al. with high crystalline perovskite nanowires?) to fluences of 1mJ/cm2 (reported by Deschler et al. for thin films in free-cavity configuration?). Such wide range can be attributed to different sample architectures (nanowires, thin films, etc.), high cristallinity (synthesis method) and optical quality (density of traps). Despite such results make perovskites appealing in the outlook of the realization of a new generation of optical emitters, impulsive excitation experiments do not provide sufficient information to state if the material can preserve the same features under continuous operation, as happens in any electronic device. Under short-pulse excitation, emission occurs when pumping has already stopped. Differently, other processes are involved during continuous carrier injection, due to the interplay between excited states population and the relaxation processes that occur in the meantime. Therefore, to verify if the material can effectively act as an optical gain medium, it is necessary to observe if optical amplification is supported until the population inversion condition is kept. In fact, warming due to non-radiative processes established under cw excitation can considerably affect light emission at high injected carrier densities, significantly reducing the radiative efficiency [19,30].

7.3 ASE under ns excitation

In order to have a more complete insight into the emission processes, we extended our experiments to regimes where laser pulse duration is comparable or longer than the PL lifetime at nthr. To achieve this condition, we used ns-long pulsed sources, as samples PL lifetime ∼1 ns at ASE threshold. As we have already discussed, under these condition the excited electron-hole 70 Optical ampliﬁcation

Figure 7.2 | Quasi-steady-state ASE in trihalide perovskite ﬁlms. Emission was excited at diﬀerent temperatures by 300-ns-long laser pulses with a repetition rate of 6 Hz and 2.35 eV photon energy. (a-d) Time-integrated photoluminescence spectra of MAPbI3 (red lines) and MAPbBr3 (blue lines) perovskites detected by a CCD camera; ASE occurs at cryogenic temperatures (180 K), but disappears around 220 K27.4

plasma density is no more dictated only by the absorbed laser photons, but by the interplay between laser excitation rates and relaxation processes. This is why, in this regime, we will consider the instantaneous light intensity instead of the laser fluence. We used the experimental setup described in Section 4.3.3, exciting samples with 4-ns long excitation pulses. ASE is observable at room temperature in both MAPbI3 and MAPbBr3 films at threshold densities of the order of 10 kW/cm2. A regime even more similar to cw pumping was obtained exciting samples with 300-ns-long pulses. This time, no ASE peak was observed at room temperature. We attribute the absence of ASE to the intense warming provided by such a long pulses. In order to decrease warming, samples were cooled down to cryogenic temperatures. A very small hint of amplification appears at 220 K and becomes more and more sharper with decreasing temperature, as can be observed from Figure 7.2. These results are consistent with some warming process that inhibits ASE under cw excitation. Three primary effects can activate warming under cw-pumping: Auger recombination, excess energy and the use of long pulse excitation. As we have shown in Section 5.4, Auger recombination is off course responsible for the radiative efficiency loss above 1018 − 1019cm−3. Excess energy is due to non-resonant pumping, for which the laser energy Eexc is higher than the energy gap Egap, and the resulting difference is dissipated non-radiatively. Finally, longer pulses deposit more 7.4 Warming processes 71 energy than what can be obtained under impulsive excitation, resulting in more warming. We investigated all these mechanisms resorting an accurate analysis based on optical thermometry, which is discussed in the next section.

7.4 Warming processes

First of all, we veriﬁed how much excess energy is relevant in terms of plasma warming, keeping samples at room temperature and exciting them at different laser intensities under both impulsive and quasi-cw regime. As reported in the inset of Figure 7.3, plasma temperature can be extracted by ﬁtting the high energy tail of the PL spectrum with a Ae−E/kBT Boltzmann function, where A is an arbitrary multiplication constant, E the photon energy, kB the Boltzmann constant and TP the plasma temperature [58].

Figure 7.3 reports experimental TP extracted from MAPbI3 PL measurements as a function of the laser intensity for both impulsive and quasi- cw regimes (4-ns-long pulses). As we expected, plasma temperatures obtained under 4-ns-long excitation are higher than those obtained under impulsive regime in the whole excitation range, as a consequence of the higher energy amount deposited on the sample. Experimental curves corresponding to different laser energy excitation show that larger excess energy causes larger warming, as for the same excitation intensity TP grows with Eexc − Egap. Warming due to Auger recombination is evident by the rapid increase of the plasma temperature with the excitation intensity in each curve. Since ASE appears under 300-ns-long excitation only at cryogenic temperatures, we verified if there is some dependence of nthr on sample temperature. Hence, samples were cooled down to 160 K, kept in a cold-finger cryostat under vacuum, and excited with the UV femtosecond laser. Sam- ple temperature was not decreased below 160 K, as MAPbI3 perovskite undergoes a phase transition, which causes a modification in absorption, emission, PL lifetime and ASE threshold fluence [c22aom]. We found that the ASE threshold fluence follows a quadratic dependence in temperature in both MAPbI3 and MAPbBr3 perovskites (see Figure 7.4), and since ab- 72 Optical amplification

Figure 7.3 | Plasma temperatures of a MAPbI3 film as a function of the laser fluence. The empty markers represent the measurements obtained exciting the sample at a lattice temperature of 300 K with 130-fs-long laser pulses at repetition rate of 1 kHz and different photon energies; the filled markers represent the opposite regime, with 5-ns-long laser pulses at a repetition rate of 10 Hz and different photon energies. The arrows represent the ASE thresholds for the two excitation regimes. The corresponding average excitation power density during the ns laser pulses is also reported on the top axis as a reference; such axis does not apply to measurements under fs excitation. The excess energy was calculated as the difference between the excitation photon energy and the energy gap (1.55 eV); the wavelength indicated in brackets is the actual central wavelength of the excitation laser. Inset: plasma temperatures are extracted fitting the high energy tail of photoluminescence spectra with a Ae−E/kBTP exponential function, representing a Boltzmann thermal distribution, E being the photon energy, TP the temperature, kB the Boltzmann constant and A an arbitrary multiplication factor. The blue lines represent the fitting functions27.4 7.4 Warming processes 73

Figure 7.4 | ASE threshold vs temperature. ASE threshold ﬂuence is shown as a function of the lattice temperature in MAPbI3 and MAPbBr3 perovskites, under short pulse excitation. The dashed lines are ﬁts of measured values to a quadratic dependence of the ASE threshold on temperature27.4

sorption does not change signiﬁcantly from room temperature to 160 K, such dependence can not be ascribed to a variation of the injected electron- hole pair density with temperature. We attribute such variation to some thermal inﬂuence on both bimolecular recombination and loss rates, as it is known that the radiative rate of an electron-hole plasma is inversely proportional to the plasma temperature [c34aom]. The ASE threshold lowering with decreasing temperature, makes non-radiative Auger recombination less important, as it begins to be dominant over 1018 − 1019cm−3.

We then studied warming effects in MAPbI3 films under 300-ns-long pulse excitation. A deeper insight about warming processes is provided by both time-resolved PL spectroscopy and time-resolved optical thermometry at different lattice temperatures TL and, for each TL, at different excitation intensities. Acquiring emission with the experimental setup described in Section 4.3.3, which uses a gated-intensified CCD camera, it is possible to study both the spectral and temporal PL evolution during excitation. The camera, triggered by the excitation source, acquires PL spectra at different delay times with respect to the trigger signal, providing then the temporal 74 Optical amplification evolution of spectral emission. Hence, experimental data can be organized into a spectrogram, which is a matrix where one dimension represents the spectral emission and the other represents the time-resolved signal. Ex- tracting TP from spectrograms is useful to investigate plasma temperature variations during laser pumping. Figure 7.5 reports both PL intensity and the plasma temperature as a function of time for different lattice temperatures. The time-resolved signal follows instantaneously the laser pulse at low intensity, while a sensitive PL temporal reshape is observed for intensities close and higher than ASE threshold. Temporal reshape is present even when ASE does not appear in PL spectra, and it is accompanied by a TP increase. Such finding suggests that, under cw pumping, plasma warming is responsible for both the decrease of the radiative efficiency with increasing laser intensity and the inhibition of ASE at higher lattice temperature [19].

As a confirmation, every time ASE is observed, it stops when TP is higher than 370 K, independently from the initial lattice temperature. To verify if some reshape is present also under impulsive excitation, we turned again to ultrafast PL transient spectroscopy, acquiring PL spectrograms with the streak camera from room temperature MAPbI3 films. Exciting them at laser fluences much lower and close to ASE threshold, no plasma cooling was observed during relaxation (see Figure 7.6), meaning that plasma and lattice achieve thermal equilibrium within the temporal resolution of our streak camera. Such finding is typical of ultrafast plasma thermal relaxation in inorganic semiconductors [c35,36aom]. Hence, since plasma and lattice ex- change thermal energy instantaneously each other, the origin of warming could be due to both the lattice and the substrate dissipation efficiency.

7.4.1 Rate equation model

The physics underlying the processes we have previously described can be explained solving a system of rate equations which govern the temporal evolution of the electron-hole population n, the modal emitted photon density nphot and the plasma temperature T. Such equations are listed as follows: 7.4 Warming processes 75

Figure 7.5 | Analysis of the warming processes. The photoluminescence was excited at different temperatures by 300-ns-long laser pulses with a repetition rate of 6 Hz and 2.35 eV photon energy. (a-d) Time-resolved photoluminescence spectrograms of a MAPbI3 film, acquired with a gated intensified camera at different temperatures. (e-h) Corresponding photoluminescence temporal profiles, demonstrating a narrowing of the emission profile with respect to the exciting laser pulse duration; we attributed the effect to amplified spontaneous emission and Auger recombination. (i-l) Plasma temperature extracted from MAPbI3 photoluminescence spectrograms in panels e-h, at different lattice temperatures, with the fitting procedure reported in inset of Figure 7.3. Significant warming occurred during the excitation pulse and limited the duration of the amplified spontaneous emission. The gray circles in each panel mark the temperatures and times for which amplified spontaneous emission started and stopped. The gray shadows represent the laser profile and highlight the fact that the temperature increase was delayed with respect to the laser pulse27.4 76 Optical amplification

Figure 7.6 | Ultrafast relaxation in a MAPbI3. Temperature extracted as a function of time from the streak camera spectrograms applying serially the ﬁtting procedure illustrated in inset of Figure 7.3.?

dn = −k n − k n − B (n − n )n − γn3 + P (7.2) dt t rad stim thr phot

dnphot = −k n + B (n − n )n + βk n (7.3) dt loss phot stim thr phot rad

dT 3 Egap h¯ ωlaser − Egap = ηw γn + P −kth(T − T0) (7.4) dt kB kB

T 2 nthr = nthr,0 (7.5) T0 To simplify our reasoning, we did not take into account any spatial in- homogeneity of the plasma density due to both the population depletion and the experimental conditions. Another key simplification comes from the assumption of the thermal equilibrium between the plasma and the lattice, which was already probed during the ultrafast PL spectroscopy measurements. Equation 7.2 describes how the electron-hole density n evolves in time T under excitation. kt is a non-radiative recombination rate due to traps, while krad represents the electron-hole radiative recombination rate. This latter depends on both temperature and the saturation density nsat, accordingly to the equation krad = ksat/(1 + nsat/n)T0/T. In particular, ksat is the radiative recombination rate at the saturation density, while T0 is the initial lattice temperature (and therefore the plasma temperature) before the excitation. Bstim is the stimulated emission coefficient, γ the non-radiative Auger recombination rate and P the rate of injected 7.4 Warming processes 77 electron-hole pairs per unit volume from the laser excitation. Equation 7.3 models the evolution of the modal photon density per unit volume. It takes into account the modal loss rate per unit volume kloss, while β is a dimensionless parameter which represents the fraction of photons acting as seed for amplified stimulated emission. Equation 7.4 regulates the dynamics of the lattice warming, and thus of the electron-hole plasma, since we assumed they are in thermal equilibrium. As we reported in the previous section, heating is mainly provided by Auger recombination and ex- 3 cess energy. Such warming sources are described by the terms γn Egap/kB and (h¯ ωlaser − Egap)/kBP in the right hand of the equation, respectively. The warming rate of the lattice is represented by the constant ηw, while kth takes into account the thermalization rate at which the electron-hole plasma returns to the initial value T0. Equation 7.5 governs the variation of the ASE threshold density nthr with temperature, starting from the initial value n(thr,0) at temperature T0. The threshold density increases as the second power of temperature, according to the experimental results reported in section 6. The system of equations 7.2.7.5 was solved numerically, under assumptions that take into account our experimental conditions. Particu- larly, we modelled the pump temporal profile with a Gaussian function that simulates a pulse of 200 ns. In addition to this, all the fundamental rates 7 −1 were set in order to match the experimental values, such as kt (10 s ), ksat (1.2 × 109s−1) and γ (4 × 1028cm−6s−1). The remaining parameters, that is kloss, Bstim, ηw and kth, were freely adjusted to reproduce the experimental results as well. The model provides results in satisfactory agreement with what observed experimentally, especially concerning the electron-hole density and the plasma temperature. Such results are reported in Figure 7.7. As we can observe, the model accurately reproduces both the observed sudden rise in temperature and the corresponding increase in the ASE threshold density during the laser pumping. Under long pulse excitation, the photo generated electron-hole pairs density grows during the laser pulse, giving rise to an enhancement of both temperature and then of the ASE threshold density. For a certain initial temperature T0, once the electron-hole density reaches the value of nthr, ASE can occur. Such situation can be reached only if there is an overlap between the predicted temporal variations of both the electron-hole and ASE threshold densities. 78 Optical amplification

Figure 7.7 | Simulation of quasi-steady state stimulated emission with a rate equation model. (a-d) Panels show the calculated excited state density (continuous lines) as a function of time for an e-h plasma, at diﬀerent temperatures, according to the rate equation model outlined in the text. The dotted lines represent the lasing threshold density calculated by the model, while the dashed line simulates the excitation pulse. (e-h) Calculated plasma temperature as a function of time27.4

Therefore, ASE is observable only within the temporal window where n overlaps with nthr. Once the plasma temperature reaches a value for which such overlap cannot occur, ASE is inhibited. These ﬁndings explain why ampliﬁed stimulated emission is sustained longer at low temperatures and disappears at high temperatures. In addition, they highlight the strong dependence of ASE on plasma temperature (and therefore on the lattice temperature). The temperature equilibration rate C and the constant ηw are the parameters that mainly control the magnitude of warming. The former determines the temporal delay between the peak intensities of both laser pumping and plasma temperature. The latter is inversely proportional to the thermal capacity of the lattice, thus it sets the highest temperature achieved from the plasma for a given laser excitation intensity. Setting the values in order to match the experiment, the simulation provides the 7 −1 3 −1 reliable estimate of 10 s for nthr, while ηw = 330Kns eV . 7.5 Comparison with nitride semiconductors 79

−3 −6 −1 −1 2 nthr (cm ) τthr (ns) γA (cm s ) Rth (K kW cm ) Perovskites 1016 − 1018 1-3 4 × 10−28 2-4 Nitrides 1018 − 1019 2-5 5 × 10−30 0.5

Table 7.1 | Comparison between perovskites and nitrides. ASE threshold density nthr, lifetime at ASE threshold τthr, Auger recombination constant γA and thermal resistance Rth are relative to room temperature27.4

7.5 Comparison with nitride semiconductors

As a conclusion of this analysis, we will compare perovskites with nitride semiconductors. It is interesting to observe that warming processes that occur in perovskites are very similar to those known for nitrides at room temperature. In perspective of the realization of a perovskite-based cw laser, warming issues in perovskites could be solved following the same improvements that characterized the advancement of nitride-based lasers. Table 7.1 reports the main parameters responsible for warming for both materials.

Carrier lifetime at ASE threshold density τthr is slightly longer in nitrides than in perovskites, due to the considerably lower value of the Auger recombination coefficient γA in nitride semiconductors with respect to perovskites. Auger recombination is the major responsible for the efficiency reduction in nitride-based LEDs at high power and the value of γA is higher in presence of disorder due to defects [55,56]. Reducing Auger recombination in perovskites is therefore of paramount importance. A possible solution could be decreasing the trap density acting in order to enhance both sample crystallinity and surface morphology. Also the semiconductor thermal resistance Rth and the substrate represent a crucial parameter in terms of thermal energy dissipation. Perovskites show a thermal resistance which is about 4-8 times higher than in nitrides and the substrate could se- riously contribute to enhance the value of the thermal resistance increasing the ASE threshold. 80 Optical amplification 8|C ONCLUSIONANDOUTLOOK

In this work we resorted different optical spectroscopy experiments to investigate the absorption and the emission properties of organolead trihalide perovskites. In details, we have studied the nature of the photogenerated species, measured the photoluminescence quantum yield, estimated both the carrier trap density and the exciton binding energy, studied the amplified spontaneous emission and suggested the possible solutions to overcome the limits that prevent perovskites to be employed as optical gain media in future laser devices. From ultrafast photoluminescence spectroscopy experiments we found that the photoluminescence is peaked close to the excitonic resonance, with the high energy side overlapping with the continuum of the interband transitions. The PL intensity grows quadratically in the injected carrier density at room temperature, from light intensities comparable to solar illumination to those typical to obtain light amplification. Such finding is consistent with the recombination from unbound electrons and holes, which therefore represent the prevailing photogenerated species, forming an electron-hole plasma. MAPbX3 perovskites result to be particularly efficient in terms of light emission, achieving remarkable quantum yields of 30% in MAPbI3 and 70% in mixed MAPbI3−xClx perovskites. Non-radiative Auger recombination is responsible for the efficiency droop from injected carrier densities of the order of 1019 cm−3 upwards. From steady-state PL measurements, we found a significant trap density of the order of 1016 − 1017 cm−3, which is a value typical of most solution processed semiconductors. Although carrier trapping has a negative impact in optoelectronic devices, its capture cross section results to be low, since the mean PL lifetimes in perovskites exceed tens of nanoseconds, jus-

81 82 Conclusion and outlook tifying the reported both remarkable values of carrier mobility higher than 10cm2V−1s−1 and the efficient charge collection for perovskite-based PV devices. Within the excitation range of our interest, excitonic population is negligible with respect to free carriers at room temperature, but the strong influence of excitonic states appears from linear absorption spectroscopy measurements. Electron-hole correlation is responsible for the reported perovskite large absorption coefficients at the band-edge, with a beneficial effect for light harvesting. To have a measure of such degree of correlation, we estimated the exciton binding energy in MAPbX3 perovskites by mod- elling the absorption spectrum near the band-edge in the framework of the Elliot’s theory of Wannier excitons. We started the analysis from the most studied MAPbI3 perovskites, measuring the absorption spectra between 160 K and room temperature, that is the tetragonal phase of the crystal. Our calculations, obtained from least squared fitting procedures, show that

EB = 25 ± 3 meV and such value does not change in the investigated range of temperature. Such exciton binding energy comparable to thermal energy at room temperature is consistent with the existence of the electron-hole plasma evidenced from ultrafast PL spectroscopy experiments. Despite the hybrid organic-inorganic nature of MAPbX3 perovskites, such findings, in addition to the PL emission peak close to the energy gap, make these novel materials very similar to inorganic semiconductors. To check the validity of our results about the determination of the exciton binding energy, we applied the Elliot’s theory to other MAPbI3 absorption spectra published in literature, extracting the experimental data from the pdf files with the software CurveSnap, and obtaining a satisfactory agreement with our experiments. However, such findings are in contrast with what evidenced from other recent works based on magnetometry measurements, which found temperature-dependent exciton binding energies in MAPbI3 perovskites, ranging from about 15 meV at cryogenic to few meV at room temperature. At the moment, the origin of such discrepancy between what observed by optical spectroscopy and magnetometry is not clear and could be subject of investigation for the future researches. In addition, such results are to be included within a framework where consensus about the determination of the exciton binding energy has not been 83

reached yet. As an example, a wide range of values for EB has been reported in literature for the most studied MAPbI3, from few to over 50 meV, due to the different methods used to estimate it. We then developed a method to estimate the exciton binding energy that goes beyond the limits of the standard procedures. Such method is independent from the parameters of typical adopted fitting procedures and is based only on the experimental absorption spectra. The as developed f-sum rule consists in calculating the normalized integral of absorption = R En ( ) ( ) I 0 α E dE /α En , where En is a value of energy chosen in order to balance the weigths owning to both excitonic and interband absorption contributions. The normalization makes the integral dependent only on the exciton binding energy, since the normalization cancels the dependence of both the thermal line broadening and the oscillator strength. We applied such f-sum rule to absorption spectra of MAPbI3 and MAPbBr3 perovskite films, acquired each 10 K from 80 to 300 K. The use of the f-sum rule evidenced that the exciton binding energy is temperature-independent in bromide perovskites in the whole investigated range, with EB = 60 ± 3 meV. Differently, measurements carried out on MAPbI3 perovskites showed that the exciton binding energy holds a value of 34 ± 3 meV in the orthorhom- bic phase (up to ∼140 K) and drops to 29 ± 3 meV at the transition to the tetragonal phase, staying constant up to room temperature. Such finding confirms our initial assumption arisen from the application of the Elliot’s theory, for which we found a temperature-independent exciton binding energy for MAPbI3 perovskites. In addition, also the numeric values are in agreement within the experimental uncertainty, confirming the validity of our initial results. Finally, the investigation of emission properties at high laser fluences highlighted the promising features of perovskites in terms of optical emitters. At room temperature, ASE under femtosecond excitation occurs in thin perovskite films at injected carrier densities similar to those of the best state-of-art organic semiconductors, that is 1018 cm−3. Our experiments showed that ASE thresholds can be heavily lowered due to their quadratic dependence on temperature. Under quasi-cw pumping, ASE is observable only at cryogenic temperatures, while warming effects due to excess energy, Auger recombination and long pulsed excitation, inhibit 84 Conclusion and outlook light amplification at room temperature. Such kind of processes are similar to those responsible for the efficiency droop in nitride semiconductors. In perspective of the realization of perovskite-based lasers, we think that light emission performances in perovskites can be enhanced following the same improvements that marked the success of nitrides, as there are strong similarities between these two semiconductors. A|R ADIATION-MATTER INTERAC- TION

A.1 Einstein coefﬁcients

A.1.1 Absorption

Let’s consider a two level energy system, placed inside a space region where there is a radiation ﬁeld. Initially, the system is in the ground state, with an electron occupying the state with energy E1 and the excited state of energy E2 is empty. If an external photon of energy E = E2 − E1 is absorbed by the electron, this latter undergoes a transition from the ground to the excited state of energy of the system. The probability per unit time dP12/dt for the system absorbing a photon is proportional to the number of photons with energy E = h¯ ω per unit volume, whereh ¯ and ω are the reduced Planck constant and the photon frequency, respectively. dP12/dt is expressed by the following equation:

dP 12 = B ρ(ω) (A.1) dt 12 where B12 is the Einstein coefﬁcient for absorption and ρ(ω) the radiation spectral energy density. B12 is a function of the electronic wavefunctions associated to the system levels.

85 86 Radiation-matter Interaction

A.1.2 Stimulated emission

If the two level system is in the excited state, the radiation ﬁeld can stim- ulate the electron to drop down from E2 to E1 level. Such transition is accompanied by emission of a new photon of energyh ¯ ω = E2 − E1. This process is called stimulated emission and the emitted photon has the same features of the one causing emission. The probability per unit time dP21/dt for the system emitting a photon by stimulated emission is expressed by the following equation:

dP 21 = B ρ(ω) (A.2) dt 21 where B21 is the Einstein coefﬁcient for stimulated emission. Coefﬁcients B12 and B21 are the same.

A.1.3 Spontaneous emission

The system in the initial state E2 can spontaneously undergoes a transition to the ground state, emitting a photon of energyh ¯ ω = E2 − E1, without be- spont ing perturbed by external ﬁelds. The probability per unit time dP21 /dt for the system emitting spontaneously a photon depends only on its electronic structure and is independent from any radiation spectral density. spont Hence, dP21 /dt reads:

dPspont 21 = A (A.3) dt 21 where A21 is the Einstein coefﬁcient for spontaneous emission.

A.2 Light dispersion

A.2.1 Diffraction grating

A diffraction grating is a device based on light diffraction, which is an interference process occurring when radiation interacts with objects having dimension similar to its wavelength. The diffraction grating consists of a planar or concave surface of length W, on which there is a large number of N parallel slits or grooves, separated each other by a short spacing d. A.2 Light dispersion 87

Figure A.1 | Light diﬀraction. An light beam incident on a grating at an angle i with respect to its normal gives rise to diﬀraction maxima at angles θ according to Equation A.5.8

When crossed by light, slits act as coherent sources and the light beams emitted from them interact toghether giving rise to constructive interference at certain angles and distructive at others. The resulting outcoming beam is composed by a repetition of interference maxima and minima. If the grating works in transmission, it is called transmission grating, otherwise reﬂection grating if it works reﬂecting light.

A.2.2 Condition for maxima

Figure A.1 shows a planar wave incident on a reﬂection grating at an angle i with respect to its surface normal. The path difference ∆ between the contribution due to two adjacent slits, diffracted at an angle θ, is:

∆ = d(sini + sinθ) (A.4)

The maximum condition for a wavelength λ is obtained when:

∆ = d(sini + sinθ) = nλ (A.5) 88 Radiation-matter Interaction where n is an integer and it is called order number. Zero order (n = 0) corresponds to specular reﬂection, thus |θ| = |i|. Different maxima, obtained for different values of n, correspond to different θ angles.

A.2.3 Angular dispersion and resolving power

Differentiating Equation A.5 one obtains the angular dispersion dθ/dλ as follows:

dθ n = cosθ (A.6) dλ d which represents the angle within two adjacent wavelength, differing each other of dλ, are separated. The resolving power R of a grating quantiﬁes how much two adjacent spectral wavelengths can be resolved. Considering the Rayleigh criterion, two adjacent spectral wavelengths are considered resolved when the maximum of one coincides with the minimum of the other. R is deﬁned by the following equation: R = nN (A.7) where n is the diffraction order and N the numer of illuminated slits.

A.2.4 Blazed grating

A blazed grating acts in order to disperse light in its spectral components reaching the maximum efficience within a certain wavelength range. The efficiency maximum is obtained at the blaze wavelength λB, while efficiency reaches values lower than 60-70% at wavelengths λ < 2/3λB and

λ > 3/2λB. A blazed grating is typically characterized by a glass plate on top of which it is deposited a thin aluminum film. This latter is engraved in order to obtain oblique grooves, parallel each other, giving rise to a saw- tooth profile. The groove inclination with respect to the grating plane is given by the blaze angle θB. Figure A.2 shows the typical configuration adopted in blazed gratings. FN is the groove surface normal, GN the grating plane normal, α the angle of incidence with respect to GN and β the angle of reflection with respect to GN. A.2 Light dispersion 89

Figure A.2 | Blazed grating. A light beam incident on a blazed grating groove at an angle

α with respect to the grating normal is diﬀracted to an angle β. θB is the blaze angle of the grating??.

A.2.5 Monochromator

A monochromator (or spectrometer) is an instrument used to disperse light into its spectral components. Generally, a monochromator is composed by an entrance slit, a set of blazed gratings installed on a rotating turret, a set of reflecting mirrors and an exit slit coupled to a detector. The light to analyze comes into the instrument through the entrance slit. A grating separates the spectral components of the light and the detector measures their intensity. If the detector has a small dimension sensor, for which it is possible to analyze wavelengths only one by one, the instrument is called monochromator. To change the exit wavelength it is sufficient to rotate the grating. Instead, if the sensor dimensions are such that it is possible to collect many spectral lines at the same time, the instrument is called spectrometer. The grating position defines the central wavelength of the spectrum. There exist different configurations with which a monochromator can be arranged, but the most common is the so called Czerny-Turner configuration, whose scheme is shown in Figure A.3. Such configuration adopts two concave mirrors M1 and M2. M1 colli- mates the beam coming from the entrance slit of the monochromator on the diffraction grating. The diffracted light is then focalized by M2 on the exit slit of the monochromator, where the detector is placed. 90 Radiation-matter Interaction

Figure A.3 | Czerny-Turner conﬁguration. A white light beam crosses the entrance slit of a spectrometer and is collimated on a blazed grating by the mirror M1. Light is dispersed and the mirror M2 focuses one spectral component on the exit slit of the spectrometer.8

Spectral bandwidth

The spectral bandwidth B is the smallest spectral range that the monochromator is able to isolate. Neglecting both the grating width and system aberrations, the bandwidth can be calculated as follows:

dλ B ≈ max(W , W ) (A.8) dx in out where Win and Wout are the entrance and the exit slit width, respectively. Experimentally, B is deﬁned as the full width at half maximum (FWHM) of the spectral proﬁle acquired from a monochromatic source. B|L ASERDEVICES

B.1 Laser emission

Laser is the acronym of Light Amplification by Stimulated Emission of Radia- tion and identifies the process where a system emits amplified and coherent light by stimulated emission. To have laser emission, it is necessary to achieve the pupulation inversion condition. At equilibrium and without any perturbation field, electrons are distributed in order to minimize their energy, thus the ground state is almost fully occupated, while the excited state population results to be negligible. To invert such situation occurs a proper pumping system. Figure B.1 shows schematically the steps occurring to generate a beam in a cw laser device. The optical active medium represents the device region where population inversion and stimulated emission occur, while the optical resonator is a cavity of a certain length where there are two mirrors at its extremes, one completely and one partially reflecting. The pumping system excites the active medium, giving rise to the population inversion. During optical pumping, a fraction of electrons relaxes towards the ground state emitting photons by spontaneous emission, thus emptying the excited state. Such photons propagate inside the active medium and go back and forth inside the cavity. Part of them will escape from the semi-reflecting mirror, originating the laser beam, and part will activate stimulated emission per- turbing electrons in the excited state, thus amplifying the emitted beam intensity. Once established equilibrium between excited state population due to optical pumping and depopulation due to both spontaneous and stimulated emission, the device works under steady-state regime, delivering a monochromatic, coherent, polarized and collimated beam of constant

91 92 Laser devices

Figure B.1 | Continuous wave laser action. The optical cavity consists of an active medium placed between a reflecting and a semi-reflecting mirror. Pumping provided by a flash lamp excites electrons in the active medium. During relaxation, electrons emit photons by spontaneous emission, which are reflected back and forth inside the cavity and cross the active medium several times before leaving the resonator. In the meantime, the intense pumping gives rise to the population inversion and the photons previously emitted from the active medium induce stimulated emission, giving rise to an intense, collimated, coherent, monochromatic and polarized laser beam.

intensity in time.

B.2 Pulsed lasers

Pulsed lasers emit light beams characterized by a repetition of pulses having specific duration and high power. There are several techniqes adopted to obtain pulsed beams, but here we will focus only on Q-switching. By Q-switching, the laser beam is turned on and off periodically, increasing and reducing the energy losses inside the optical resonator by moduling the quality factor Q, defined as the ratio between the stored and the lost energy per unit time inside the cavity. Q is modulated in time changing the cavity optical losses at the desidered laser repetition rate. Such losses are maximized and minimized by an electro-optic absorber plate, placed between the active medium and the semi-reflecting mirror, that acts as a shutter. The absorber is connected to an electric circuit that applies a periodic voltage of several hundreds volts at the frequency cor- B.3 Nd:YAG and Nd:YLF lasers 93 responding to the laser repetition rate. The absorber optical properties change at the same frequency of the voltage, alternating phases where optical losses are maximized and minimized. During the phase when optical loss is maximum, the active medium stores energy by population inversion. When population inversion reaches its highest value, a voltage variation makes optical loss minimum and the excited states relax almost instantaneously, releasing a large amount of energy by emission of a pulse having high power (∼GW) and short duration (∼ 1 − 100 ns). Typical repetition rates rang from several Hz to some kHz.

B.3 Nd:YAG and Nd:YLF lasers

Nd:YAG (neodymium-doped yttrium aluminium garnet, Nd : Y3Al5O12) is a crystal used as active medium in solid-state lasers. The active medium is usually pumped by a ﬂash lamp or by a 808 nm laser diode. This latter conﬁguration permits to obtain a more compact laser source. The outgoing beam consists of a 1064 nm near infra-red laser wavelength. By using proper nonlinear crystals that double or triple the laser photon frequency, it is possible to obtain the green 532 nm and the ultra-violet 355 nm wavelength, respectively.

Nd:YLF (neodymium-doped yttrium lithium ﬂuoride, Nd : LiYF4) is another lasing medium, pumped similarly to Nd:YAG. Differently from Nd:YAG laser, the Nd:YLF transition used to deliver laser beams occurs at 1053 nm. Halving such wavelength trough non-linear crystals, it is possible to obtain the green 527 nm laser wavelength, which is usually adopted to pump Ti:sapphire crystals, which are active media that emit in the near infra-red. 94 Laser devices C|L IGHTDETECTION

C.1 Quantum efﬁciency

The quantum efficiency η is defined as the propability for a photon absorbed by a detector creating an electron-hole pair. Such probability is obtained by the ratio between the carrier flux due to charge carriers and the flux due to incident photons. When no carriers are generated, η = 0. If all absorbed photons result in creation of e-h pairs, η = 1. The quantum efficiency is a function of λ, as it depends on the absorption coefficient of the detector active medium. Generally, the quantum efficiency is large within a spectral region and drops down at small and large wavelengths. If the active medium is a semiconductor, photons having energy smaller than the semiconductor energy gap will not be absorbed. The wavelength corresponding to energy gap is defined by the following equation: hc λg = (C.1) Eg where h is the Planck constant and c the speed of light. Hence, all photons having wavelengths larger than λg do not contribute to create e-h pairs, thus λg can be defined as cutoff wavelength. Another efficiency drop occurs at small wavelengths due to the photon absorption at the device surface. In this region, e-h recombination is faster than charge collection at the electrodes.

C.2 Responsivity

The responsivity ρ is the ratio between the electric current ip that ﬂows through the detector and the optical power P incident on it. Since ip = ηeΦ

95 96 Light detection

Figure C.1 | p-n junction. (a) Schematic view of a p-n junction. Electrons in n-side move towards the p-side and holes do the opposite. The final result consist in the generation of a depletion region accompanied by an internal electric field that inhibits further charge accumulation. (b) Reverse biased p-n junction. The positive and negative voltages applied on the n- and p-sides, respectively, enlarge the depletion region. (c) Forward biased p-n junction. The negative and positive voltages applied on the n- and p-sides, respectively, decrease the depletion region. (d) J-V characteristics of a p-n junction. When in reverse bias, a small negative current flows across the junction. When in forward bias, the current flows easily trough the p-n interface and it increases with the applied potential, as happens in diodes.

and P = h¯ ωΦ, where Φ il the photon ﬂux, the detector responsivity reads:

ip ηeΦ ηe ρ = = = (C.2) P h¯ ωΦ h¯ ω Hence, ρ depends only on the active medium quantum efﬁciency and the incindent photon frequency.

C.3 p-n junction

A p-n junction is an interface between two semiconductors, where one is p-type and the other n-type, as shown in Figure C.1. The two types can be obtained by introducing atomic species different from the intrinsic ones of the semiconductor. The p-type semiconductor has an excess of free holes, while the n-type an excess of free electrons. Contacting the two doped materials, electrons in n-type tend to diffuse towards the p-type material, filling holes, and so do holes diffusing towards the n side. The accumulation charges in the two sides results in the generation of a depletion region. This latter gives rise to an electric field that tends to inhibit further charge accumulation. At equilibrium, part of the C.4 Microchannel plate 97 n-type semiconductor will still have free electrons, part of the n-type free holes and the central region is neutral. The magnitude of the internal electric field can be modified when the p-n junction is connected to an external circuit. Reducing the internal field results in a smaller depletion region, which increases the probability of charge diffusion between the two sides, giving rise to a current flow. Conversely, an increment of the internal field enlarge the depletion region and hinders any charge flow. Such feature makes p-n junctions particularly useful as diodes. The applied voltage can be defined forward or reverse. If the electric current flows easily, the p-n junction is polarized in forward bias. On the contrary, if voltage application results in little or no current flow across the interface, the p-n junction is polarized in reverse bias.

C.3.1 Photodetectors

A reverse biased p-n junction can be used as a photodetector. As we have recently discussed, the application of a reverse bias results in an increment of the depletion region of the p-n junction. Such region can be exploited as the active region of a detector. The absorption of a photon in this region results in the creation of an e-h pair. The presence of the electric field rapidly separates the two charges (the electron moves towards the n side and the holes towards the p side). The more intense is the external photon flux, the larger will be the current flow generated by absorption. The p-n junction can be modified placing a layer of intrinsic semiconductor between the p and n sides, in order to enlarge as much as possible the depletion region and the active area of the detector.

C.4 Microchannel plate

The microchannel plate (MCP) is a planar device used in light detection as image intensiﬁer. Such component works similarly to an electron multi- plier, as the electric signal resulting from light absorption is intensiﬁed by electron multiplication due to secondary emission. Figure C.2 shows the schematic view of a typical microchannel plate. Parallel channels having micrometric diameter are realized drilling trough- 98 Light detection

Figure C.2 | Scheme of a microchannel plate. Impacts due to accelerated photoelectrons with the atoms of the microchannel plate give rise to secondary emission, intensifying the initial electron beam.

out a millimeter-thick and high-resistive disc. Two electrodes placed at the two disc sides apply a voltage, giving rise to an electric field across the channels. Hence, electrons coming into channels are accelerated by voltage and during their motion secondary emission occurs due to collisions with the material. The initial signal is then exponentially intensified at the output electrode. Typically, the multiplication factor is about 10k, but it can be modified acting on the gain voltage applied across the MCP.

C.5 Detectors

C.5.1 Charge-Coupled Device (CCD)

A CCD is a device based on a Metal-Oxide-Semiconductor (MOS) capacitor, which is used as sensor in digital imaging. The CCD structure, which is schematically shown in Figure C.3, is such that electric charge can be shifted between adjacent pixels simply acting on the voltage applied on electrodes. Typically, CCD are fabricated using silicon dioxide as oxide layer and C.5 Detectors 99

Figure C.3 | Scheme of a silicon-based CCD sensor. During the integration time, a positive voltage is applied between the gate electrode and the silicon substrate. Electrons due to photoabsorption are conﬁned in the potential well. The more the voltage is applied, the more electrons are stored, proportionally to the light intensity. Charge transfer to an adjacent pixel is possible applying the positive voltage to the next gate and setting to ground the previous one.

intrinsic or light p-doped silicon as semiconductor. A set of N metal gate electrodes is placed on the oxide layer. The application of a positive voltage to one electron (when all the other are kept at zero voltage) gives rise to a potential well that extends from the oxide-semiconductor interface to the silicon bulk region, thus holes are pushed towards the lower part of the device. The time within the voltage is applied is called integration time. During this phase, the absorption of external photons gives rise to e-h pairs creation and the applied electric ﬁeld immediately separates charges in order to push holes to the CCD lower part and electrons towards the oxide, conﬁning them in the potential well. Electron storage keeps on until voltage is applied, that is during the integration time. Typically, about 105 electrons can be stored in each pixel before saturation. The accumulated charge can be shifted to adjacent pixels properly applying positive voltages to the electrodes in order to shift the potential well from one pixel to the closest one in a precise direction. The stored charge is then converted into a voltage signal and then digitalized to create an image.

C.5.2 Streak camera

The streak camera is an instrument adopted in time-resolved spectroscopy that measures both the temporal and spectral intensity evolution of a light 100 Light detection

Figure C.4 | Schematic diagram of a streak camera. A transient light signal passes trough the horizontal entrance slit of the streak camera and after induces photoelectric effect crossing a photocathode. Photoelectrons are accelerated towards a MCP that intensifies the beam, but before they are vertically deflected by a sweep voltage synchronous with a trigger signal. The as intensified beam crosses a phosphor screen coupled to a CCD sensor that storages the time-resolved information.

signal. Such instrument typically allows measurements with picosecond temporal resolution and nanosecond time range. Figure C.4 shows schematically the working principle of a streak camera coupled to a spectrometer.

After being spectrally dispersed by the spectrometer, the light signal crosses an horizontal slit and then a photocathode, placed in an evacuated tube and coated in the inside surface, which emits electrons proportionally to the radiation intensity via photoelectric effect. A couple of vertical electrodes deflects the electron beam through a sweep voltage generated by an external trigger and syncronous with the light signal. Thanks to the sweep voltage, electrons coming first will have different deflection with respect to those coming later, as the voltage intensity change with time. After passing trough a gain-tunable MCP that intensifies the signal, the resulting beam crosses a phosphor screen, which emits photons proportionally to the electron beam intensity. The phosphor screen image is focused by an optical system on CCD array detector that storages the acquired signal and con- verts it into a voltage. A readout system reads and digitalizes the acquired information and returns an image where the horizontal axis represents its spectral information, while the vertical axis its temporal evolution. C.5 Detectors 101

Figure C.5 | Schematic view of an intensified gated-CCD camera. (a) Typical internal structure of a ICCD camera. (b) After passing through an input window, photons cross a photocathode giving rise to photoelectric effect. The so generated electron beam is intensified by a MCP and then directed towards a phosphor screen coupled to a fiber optics system. The applied voltage between photocathode and MCP governs the acquisition.

C.5.3 Gated intensiﬁed CCD camera

A gated-intensified CCD camera (ICCD) is an instrument used in time- resolved spectroscopy. Similar to the streak camera, it provides information about temporal and spectral evolution of a light signal, but with ns resolution and microsecond time-range. Figure C.5 shows the scheme of ICCD coupled to a spectrometer. The dispersed light coming from the exit slit of a spectrometer crosses a photocathode that emits electrons via photoelectric effect. Like in the streak camera, the photocathode is placed in a evacuated tube and is coated in the inside surface. Photoelectrons cross a gain-tunable MCP where an applied voltage intensifies the signal. The outcoming intensified electron beam crosses a phosphor screen and the generated photons are collected by lenses or a fiber optics system coupled to a CCD array, which storages the information. The acquired signal is then digitalized by a readout system and elaborated by a computer. The data acquisition process is different from that of the streak camera. While the streak camera continuously acquires photoelectons and deflects them through a sweep voltage, the ICCD acquires the signal only within 102 Light detection a definite time window triggered by an electronic circuitery, synchronous with the light source. Such time window is called gate time. In fact, a voltage applied between photocathode and MCP can accelerate photoelectrons far away or towards the MCP. When a negative voltage is applied, electrons go towards the microchannel plate and the camera is gated on. Conversely, a positive voltage pushes electrons far away from the MCP and the camera is gated off. The shortest time window within gate can be switched on and off is called minimum gate time, and depends on both the electronic circuitery and the photocathode material. Typically, the minimum gate time is of few nanoseconds. With this method it is possible to acquire spectra within a definite time window at different delay times with respect to the trigger signal, thus obtaining a time-resolved spectral evolution image. D|S AMPLES

D.1 Sample fabrication

D.1.1 Spin coating

One of the procedures used to obtain nanoscale and uniform films is the spin coating, for which a small amount of coating material, is deposited on the center of a flat substrate. This latter is coupled to a rotating plate of a spin coater (Figure D.1), initially fixed or spinning at low speed. The centrifugal force due to a high speed rotation of the plate spreads the material in order to obtain a film, whose thickness depends on rotation speed, solution viscosity and its concentration.

D.1.2 Samples for absorption measurements

MAPbI3 and MAPbI3−xClx perovskite solutions were prepared by single step method, under nitrogen atmosphere at room temperature, by dissolving together the methylamine iodide salt (MAI) and PbI2 or PbCl2 in dimethylformamide, in a 1:1 or 3:1 molar ratio, respectively. Similar procedure has been adopted for MAPbBr3 perovskites, where MABr and PbBr2 were dissolved in 3:1 molar ratio. The solutions were stirred for 12 h before ﬁlm deposition. MAI was prepared by reacting methylamine, 33 wt% in ethanol, with hydroiodic acid (HI), 57 wt% in water. In a typical synthesis, 12 ml of methylamine were added to 50 ml of absolute ethanol and stirred under nitrogen atmosphere at room temperature. Five millilitres of HI were then added by a syringe and left reacting for 2 h. MABr was prepared by reacting methylamine, 33 wt% in ethanol, with hydrobromic acid (HBr), 48 wt% in water. 4.3 ml of HBr were then added and left re-

103 104 Samples

Figure D.1 | Spin coating film deposition. Solution is deposited on the substrate when the spin coater plate is fixed. High-speed plate rotation spreads uniformely the solution on the substrate surface. The solvent evaporation results in a thickness decrease of the so formed film.

acting for 2 hrs at 0 °C. Both MABr and MAI salts were crystallized at 50 °C using a rotary evaporator until the powder reached a white and white- brown color, respectively. The salts were then washed and ﬁltered three times with diethyl ether and dried in vacuum overnight to ﬁnally obtain a white powder. All the reactants were purchased from Sigma-Aldrich.

Films were obtained by spinning the solution on top of soda lime glass substrates at 7,000 rpm for 30 s at room temperature inside a nitrogen- filled glove box. Before deposition the substrates were cleaned with water and soap, rinsed with acetone and ethanol and dried with dry air. The as-deposited films were thermally treated at 100 °C for 5 min on a hot plate inside the glove box to obtain the perovskite structure crystallization. Crystal formation was confirmed by color change of films during treat- ment, as MAPbBr3 turned from transparent to vivid yellow, while MAPbI3 and MAPbI3−xClx was pale yellow as a fresh film, then becoming brown by the end of the process. D.1 Sample fabrication 105

D.1.3 Samples for optical ampliﬁcation measurements

Perovskite samples were obtained by a two steps procedure involving the reaction of a lead halide (PbX2, where X = I, Br) film followed by its immer- sion in a solution of methylammonium halide (MAX). MAI was prepared by reacting methylamine, 33 wt% in ethanol, with hydroiodic acid (HI), 57 wt% in water. In a typical synthesis, 12 ml of methylamine were added to 50 ml of absolute ethanol and stirred under nitrogen atmosphere. Five milliliters of HI, 57% in water, were then added drop by drop and left reacting for 2 h at 0 °C. Analogous procedure was used for MABr synthesis. In particular 4,3 ml of HBr, 48% in water, were added drop by drop to a solution of 12 ml of methylamine solution, 33% in ethanol, and 50 ml of absolute ethanol under nitrogen atmosphere at 0 °C. The MAX salt was crystallized at 60 °C using a rotary evaporator until the complete evaporation of solvents. The salt was then washed and filtered three times with diethyl ether and dried in vacuum overnight to finally obtain a white powder. All the reactants were purchased from Sigma-Aldrich. Film deposition was carried out on soda lime glass substrates cleaned with water and soap, rinsed with acetone and ethanol and dried with dry air before film deposition. MAPbI3 films were obtained by spinning a 0.3M

PbI2 solution at 70 °C, on top of glass substrates at 7000 r.p.m. for 30 s at room temperature inside a nitrogen-filled glove box. The as-deposited films were thermally treated at 70 °C for 15 min on a hot plate inside the glove box and then left cooling down. A solution of MAI 10 mg/ml in 2-propanol was prepared and poured in a small container, then the glass with PbI2 film was immersed abruptly in the solution and left reacting for 20 s until it became light brown. The substrate was then spun and rinsed with 2-propanol during spinning and finally thermally treated at

100 °C for 1 hour. MAPbBr3 films were obtained by spinning a 0.5M PbBr2 solution at 70 °C, on top of glass substrates at 6500 r.p.m. for 30 s at room temperature inside a nitrogen-filled glove box. The as-deposited films were thermally treated at 70 °C for 5 min on a hot plate inside the glove box and then left cooling down. A solution of MABr 7 mg/ml in 2-propanol was prepared and poured in a small container, then the glass with PbBr2 film was immersed abruptly in the solution and left reacting for 20 s until 106 Samples it became light yellow. The substrate was then spun and rinsed with 2- propanol during spinning and finally thermally treated at 100 °C for 45 minutes.

D.2 Sample characterization

D.2.1 X-ray diffraction

X-Ray diffraction (XRD) is one of the fundamental techniques adopted to study crystals. Due to their long-range order, crystals can be considered as gratings, where atoms assume the role of slits when interacting with an external radiation. Diffraction occurs when light has wavelength comparable to the interatomic spacing, which is typically ∼ 0.1 nm. The spectral region where diffraction can be obtained from crystals is that of X-Rays, which extends from 10−3 to about 10 nm. Placing a crystal between a X-Ray beam and a screen, light beams scattered from atoms interact together giving rise to a diffraction pattern visible on the screen. It consists of a serie of maxima and minima of radiation intensity, due to the constructive and distructive interference between scattered beams, respectively. Such kind of process is called Bragg diffraction and follows the following equation:

2dsinθ = nλ (D.1) where d is the interatomic spacing, θ the angle between the X-Ray beam and the crystal plane normal, n the diffraction order and λ the radiation wavelength. Equation D.1 is called Bragg law and it is fundamental to measure the interatomic spacings of a crystal and the plane directions in the space, deﬁned each one by a precise triple of numbers (h, k, l). Depending on the values of h, k, l, the crystal can be associated to a precise group simmetry, thus obtaining its three-dimensional information. The XRD pattern can be acquired with a diffractometer, an instrument that consists of a X-Ray source, a X-Ray detector and a sample holder placed between them. Figure D.2 shows a diffractometer in Bragg-Brentano conﬁguration. The source is obtained using a X-Ray tube delivering a D.2 Sample characterization 107

Figure D.2 | Bragg-Brentano geometry. Materials to analyze are placed on a sample holder, which is installed in correspondence of the rotation center of both the X-Ray tube and the detector. For simmetric θ − θ scansions, X-ray tube and detector form each one an angle θ with respect to the sample plane.

monochromatic and collimated light beam, and the X-Ray wavelength depends on the metal used as anode in the tube. Copper is typically used for these purposes, thus a lot of X-Ray tubes emit the Cu Kα wavelength, which corresponds to 0.15 nm. Once generated by the tube, the beam is focused on the surface of the crystal to analyze, which is placed on the sample holder. Both source and detector are installed on independent mo- bile goniometers that can change simultaneously their orientation θ with respect to the crystal surface normal, thus mapping the diffraction pattern from 0 to 180°. Figure D.3 shows the XRD pattern acquired from our samples at room temperature with a Bruker D8-Discover diffractometer for thin ﬁlms with parallel beam geometry and Cu Kα wavelength. The three Bragg peaks in Figure D.3a for iodide-based- and the two peaks in Figure D.3b for bromide-based perovskites are consistent with a perovskite structure in the tetragonal and the cubic phase, respectively, according to what reported in literature.

D.2.2 Scanning probe microscopy

Scanning probe microscopy (SPM) is a branch of microscopy that yields the image of a sample surface morphology by scanning it trough a mechanical system. Differently from optical microscopy, SPM is not affected 108 Samples

Figure D.3 | XRD patterns from methylammonium lead halide perovskite films. Pat- terns are collected by using a Bruker D8-Discover diffractometer. (a) Symmetric θ − θ scans collected for films prepared by spin coating from different solutions of precursors: stoichiometric

1:1 mixture of MAI of PbCl2 in DMF to obtain the pure iodine phase (black pattern); non stoichiometric 3:1 mixture of MAI and PbCl2 to promote the formation of the I1−xClx mixed phase (green pattern). For both films, all Bragg peaks were indexed as (hh0) Bragg reflections √ on a tetragonal I4/mcm cell with a = b ≈ 2ac, c ≈ 2ac, indicating a highly oriented growth, with films having the face of the cubic perovskite cell (with ac lattice parameter) parallel to the 38 5 deposition plane. (b) XRD pattern for a MAPbBr3 film acquired with the same setup.

by the diffraction limit, for which the highest resolution achievable from a microscope is limited by both the numerical aperture of its objective and the observed light wavelength. Typically, optical microscopes using visible light can not resolve objects smaller than few hundrends nanometers. Higher resolutions can be obtained with electronic microscopes, which can resolve nanometric objects. However, these latter have big dimensions, are very expensive, and samples must be kept under vacuum due to the use of electron beams. SPM microscopes combine the advantages to be smaller and more cheap than electronic microscopes and allow sample measurements in atmosphere with nanometric resolution. SPM measurements can be carried out using an atomic force microscope (AFM), which is useful to measure the roughness of a sample surface and to evaluate the thickness of thin films deposited on substrates. Surface roughness is a parameter that can be fundamental for light amplification, as scattering increase the optical losses, while thickness determination is useful for calculating the absorption coefficient of a film, once its optical density is known. Typi- cally, an AFM provides a complete surface scan of a 104 nm2 square area D.2 Sample characterization 109

Figure D.4 | Atomic force microscope. The sample surface is scanned by a nanometric tip placed at the end of a ﬂexible cantilever. A laser beam is focused on the cantilever surface and is collected by a photodetector. The surface morphology variations are probed by the laser beam deviations induced from the cantilever deﬂections.

in few minutes. Figure D.4 shows the schematic view of an AFM. It consists of a cantilever with a nanometric sharp tip placed at its end. This latter is placed close to the sample surface and interacts with it, scanning the surface line by line. The interaction between tip and surface gives rise to a deflection of the cantilever proportional to the surface height variation. Such deflection is measured by an array of photodiodes, which detects the laser spot reflected from the cantilever upper surface.

Figure D.5 shows different AFM images obtained from our MAPbI3

MAPbBr3 thin ﬁlms analyzed for optical ampliﬁcation experiments. 110 Samples

Figure D.5 | AFM images of MAPbI3 and MAPbBr3 perovskite thin ﬁlms. Film thickness and surface morphology were obtained by atomic force microscopy (AFM) with a NT-MDT Solver P47H-Pro in semi contact mode by a high-resolution non-contact silicon tip.27 ACKNOWLEDGEMENTS

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