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The final section of this chapter forms the background forChapter 6. Starting with the general laws of diffusion, the special conditions related to exciton diffusion within NCs will be derived. Based thereon, two concepts for the measurement of the exciton diffusion length will be introduced.

2.5.1 General Laws of Diffusion

Generally, diffusion denotes the movement of particles from a region of higher concentration towards a region of lower concentration.147This process could for example involve atoms or molecules within a closed system. The basic equations for diffusion date back to the nineteenth century and were established by Adolf Fick.148For the three-dimensional case, his first law states

J(r) = โˆ’๐ทโˆ‡๐‘(r) , (2.57)

whereJ(r)denotes the diffusion flux vector that defines the direction and magnitude of the correspond-ing particle flow at positionr.Equation 2.57shows thatJ(r)is directly proportional to the gradient of the concentration distributionโˆ‡๐‘(r)with a factor๐ทwhich is known as the diffusion coefficient. This means that a given initial gradient in concentration is the driving force for diffusion leading to the movement of the respective particles. No external force is involved in this process, it is rather referred to as a โ€œrandom walkโ€.149The temporal evolution of the concentration is considered in Fickโ€™s second law which states

๐œ•๐‘(r, ๐‘ก)

๐œ•๐‘ก = ๐ทโˆ‡2๐‘(r, ๐‘ก) . (2.58)

Here and in the following, the diffusion coefficient๐ทis assumed to be time-independent.

2.5.2 Exciton Diffusion in Nanocrystals

An important question for this work was, which role diffusion plays for excitons within individual LHP NCs or their assemblies. In contrast to other particles, excitons first have to be generated and also have a limited lifetime๐‘‡1until their decay. Equation 2.58has to be adapted accordingly. The corresponding following equation restricts itself to the representation of diffusion along one dimension, the z-direction in this case.

๐œ•๐‘›ex(๐‘ง, ๐‘ก)

๐œ•๐‘ก = ๐‘›โŸโŸโŸโŸโŸโŸโŸโŸโŸโŸโŸโŸโŸโŸโŸ0exp(โˆ’๐›ผ๐‘ง)


+ ๐ท๐œ•๐‘›ex(๐‘ง, ๐‘ก)




โˆ’ ๐‘›ex(๐‘ง, ๐‘ก) ๐‘‡1




Obviously, the equivalent to the concentration๐‘inEquation 2.58is the exciton density๐‘›ex. Additionally, the adapted formula includes a term attributed to exciton generation which is assumed to occur instantaneously and is based on the Beer-Lambert law considering an incident light wave in z-direction.150,151Finally, the last term accounts for the extinction of excitons, i.e., the decay with lifetime ๐‘‡1.

48 2.5 Diffusion

x,y x,y

z z




nex(z) (i)


(i) (ii)

t = 0 t = t1

Figure 2.21: Exciton Diffusion in Nanocrystals.A laser pulse (green) hits the surface of a LHP NC film (i) at normal incidence. Excitons (red) are generated close to the surface within this film due to absorption of the photons. Therefore, the initial distribution of excitons๐‘›ex(๐‘ก = 0)is basically determined by the Gaussian profile of the excitation beam in the xy-plane and by the exponential decay of the Beer-Lambert law along the z-axis. Through diffusion the distribution of excitons spreads out such that some excitons potentially reach the adjacent layer of material (ii) at a time๐‘ก1> 0before their decay.

Generally, one can distinguish between two types of diffusion in the NC systems investigated through-out the course of this work. The first case usually applies to NCs which are relatively large in at least one dimension, i.e., the length in this dimension should be much greater than the exciton Bohr radius ๐‘Žex. Within such a NC, an exciton is basically free to move in the non-confined directions as long as it is not bound to a defect. The second case plays a role for densely packed small NCs, for example a film of nanocubes. In such cases an exciton is usually confined to one NC, but has the possibility to transfer to an adjacent NC via Fรถrster resonance energy transfer (FRET),152also referred to as exciton hopping in this specific case of exciton transport.153,154

Figure 2.21illustrates the diffusion occurring in a NC ensemble right after the initial excitation with a laser pulse. Here, the pulse propagates towards the NC film at normal incidence. Hence, the Gaussian shaped beam profile of the excitation laser determines the initial density of excitons in the lateral directions๐‘›ex(๐‘ฅ, ๐‘ฆ, ๐‘ก = 0). Moreover, as mentioned above,๐‘›ex(๐‘ง, ๐‘ก = 0)is given by the Beer-Lambert law. Starting from this situation โ€” depicted in the left panel ofFigure 2.21โ€” the spreading of excitons throughout the NC film may be described byEquation 2.59and the respective versions of this equation for the x and y direction. The right panel of the figure shows the exciton distribution after a certain time๐‘ก1> 0. In agreement with the laws of diffusion, the exciton density is strongly reduced in the initial excitation spot, while excitons are distributed over a larger volume.

2.5.3 Determination of the Exciton Diffusion Length

To compare exciton diffusion within several systems, it is best to assign a value to each of these systems that determines the average distance an exciton travels during its lifetime: the exciton diffusion length, described by

๐ฟD= โˆš2 ๐ท ๐‘‡1=

โˆš2 ๐œ‡๐‘˜B๐‘‡

๐‘’ ๐‘‡1, (2.60)

exhibiting a square-root dependence on the product of the diffusion coefficient๐ทand the average lifetime of an exciton๐‘‡1 within the pure material of the examined film.155,156In some parts of the literature, the factor2is dropped, but here the mathematically rigorous convention is used, such that

๐ฟDdenotes the minimum net displacement achieved in one dimension by1/๐‘’of the starting exciton population.157Moreover,Equation 2.60reveals the proportionality of the diffusion coefficient๐ทto the exciton mobility๐œ‡and the temperature๐‘‡. Large electron-hole diffusion lengths up to around 1 ยตm โ€” mainly due to long lifetimes โ€” have been reported for bulk films of organometal trihalide perovskites at room temperature and are also one of the main reasons for the success of this material type in the recent development of solar cells.158In this work NC assemblies were investigated to find out how the exciton diffusion properties differ compared to a bulk film. To this end, two approaches were applied to study the exciton diffusion length in such LHP NC systems, as explained briefly in the following.

Photoluminescence Mapping

The concept of photoluminescence mapping is based on diffusion along the lateral direction parallel to the film surface. In the scheme ofFigure 2.21this corresponds to diffusion within the xy-plane.

For this approach to be accurate, the film thickness should be as uniform as possible. In particular, it should not contain any empty spots void of NCs. Moreover, the supporting layer underneath the NC film โ€” i.e., layer (ii) inFigure 2.21โ€” should be an exciton blocking material in this scenario.

The Gaussian profile of the exciton distribution๐‘›ex(๐‘ฅ, ๐‘ฆ, ๐‘ก)can be studied by observing the in- and outgoing light waves from this film on a substrate. First, the ingoing wave, i.e., the excitation laser beam, determines the exciton distribution๐‘›ex(๐‘ฅ, ๐‘ฆ, ๐‘ก = 0). Therefore, the laser beam and๐‘›ex(๐‘ฅ, ๐‘ฆ, ๐‘ก = 0) exhibit the same Gaussian profile such that the FWHM of this distribution (๐น ๐‘Š ๐ป ๐‘€Laser) may be obtained by mapping the surface of the film while only measuring the reflected excitation beam. Then, the emitted wave shall be observed, i.e., the photoluminescence (PL) of the NC film which occurs at a longer wavelength as compared to the excitation. Not all exciton recombination processes exhibit a radiative nature, however, the magnitude of PL originating from a certain point is directly proportional to the local exciton density๐‘›ex(๐‘ฅ, ๐‘ฆ). It does not even matter if the excitation light source is a pulsed or continuous wave laser when mapping a time-integrated PL distribution along the surface. The averaged PL map will represent the Gaussian profile corresponding to the average exciton lifetime๐‘‡1 such that๐น ๐‘Š ๐ป ๐‘€PL= ๐น ๐‘Š ๐ป ๐‘€PL(๐‘‡1)may be extracted from this distribution.159

Obtaining ๐น ๐‘Š ๐ป ๐‘€Laser and ๐น ๐‘Š ๐ป ๐‘€PL enables the calculation of the standard deviation for both Gaussian profiles according to

๐œŽ = ๐น ๐‘Š ๐ป ๐‘€

2 โˆš2 ln(2) (2.61)

such that๐œŽLaserand๐œŽPLmay be obtained. Finally then, the exciton diffusion length

๐ฟD= โˆš๐œŽPL2 โˆ’ ๐œŽLaser2 (2.62)

may be calculated using the aforementioned standard deviation values. In the above description, identical diffusion along the x and y axes has been assumed. In NCs or NC assemblies that are not fully symmetric along the xy-plane, the behavior will likely deviate. InChapter 6this will be shown and discussed in greater detail.

50 2.5 Diffusion Photoluminescence Quenching

In contrast to photoluminescence mapping, the second approach to determine the exciton diffusion length๐ฟDis based on an exciton quenching layer adjacent to the NC film. Such a quenching layer could for example consist of a fullerene such as C60155or the fullerene derivative phenylโ€C61โ€butyric acid methyl ester (PCBM)22โ€” materials that are generally known to be strong electron acceptors.160 If an exciton reaches the interface to the adjacent layer via diffusion, the exciton will be dissociated and the resulting free electron will transfer to layer (ii) whereas the free hole stays in layer (i) (see Figure 2.21). Since this e-h pair is no longer available for radiative recombination, this method is referred to as photoluminescence quenching. This possibility of surface quenching represents an additional way for an exciton to decay.161,162In the previous case without a quenching layer, the only way for an exciton to decay was recombination (radiatively or nonradiatively), occuring at a rate ๐‘˜re= 1/๐‘‡1as described in the previous sections. The additional path for exciton decay is referred to as charge transfer and consists of diffusion towards the interface and subsequent exciton dissociation.

Therefore, the rate for the charge transfer process is๐‘˜ct= ๐‘˜diff+๐‘˜diss. Consequently, the overall exciton decay within such a system of two layers may then be described by

๐‘˜total = ๐‘˜re+ ๐‘˜ct= 1 ๐‘‡1 + 1

๐‘‡ct = 1

๐‘‡total . (2.63)

Moreover, two boundary conditions may be defined forEquation 2.59.157All coordinates with๐‘ง = 0 represent the surface of the NC film and the ones with๐‘ง = ๐ฟrepresent the interface between layer (i) and (ii) (cf. Figure 2.21). That is, the NC film thickness is given by๐ฟ. There is no exciton movement out of the surface of the film into the air at๐‘ง = 0, therefore the first boundary condition is

๐œ•๐‘›ex(๐‘ง = 0, ๐‘ก)

๐œ•๐‘ง = 0 . (2.64)

Moreover, assuming perfect quenching, the exciton population decays completely at the interface, i.e.,

๐‘›ex(๐‘ง = ๐ฟ, ๐‘ก) = 0 . (2.65)

Considering these boundary conditions in combination withEquation 2.59andEquation 2.63, it can be shown that the overall exciton lifetime in the presence of a quenching layer๐‘‡totalmay be estimated as

๐‘‡total โ‰ˆ ๐‘‡1(1 + ๐œ‹2 8 (๐ฟD

๐ฟ )




. (2.66)

This estimation is reasonably accurate as long as๐›ผ๐ฟ < 1, where๐›ผis the absorption coefficient of the NC film.157 Finally, solvingEquation 2.66for the exciton diffusion length leads to

๐ฟD โ‰ˆ 2 ๐ฟ

๐œ‹ โˆš2 ( ๐‘‡1

๐‘‡total โˆ’ 1) . (2.67)

Hence,๐ฟDmay be calculated from the exciton lifetimes with and without an adjacent quenching layer (๐‘‡totaland๐‘‡1, respectively) if the NC layer thickness๐ฟis known.

Materials and Methods 3

Having laid the ground with fundamental explanations concerning the experiments conducted in the course of this thesis, the following chapter serves the purpose of introducing the respective employed materials and methods. Firstly, a typical perovskite NC synthesis is described, followed by an introduction to the different NC compositions and morphologies that may be achieved. Next, the three lasers used most commonly in the experiments for this thesis shall be explored with special emphasis on the optical nonlinear processes occurring within these systems. It goes without saying that none of the presented work would have been possible without a well-functioning, stable and reliable light source โ€” in this case, pulsed lasers whose maintenance and optimization represented a key part of my efforts. In the final section of this chapter, various spectroscopic experiments that were used to study the light-matter interaction in perovskite NCs will be explained in detail.


52 3.1 Perovskite Nanocrystal Synthesis and Variety