Probing Fabry-Perot Interference in Self-Assembled Excitonic

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C: Plasmonics; Optical, Magnetic, and Hybrid Materials

Probing Fabry-Perot Interference in Self-Assembled Excitonic Microcrystals with Subgap Light Emission Adedayo M. Sanni, Shofikur Shuhag, and Aaron S. Rury J. Phys. Chem. C, Just Accepted Manuscript • DOI: 10.1021/acs.jpcc.9b05558 • Publication Date (Web): 20 Aug 2019 Downloaded from pubs.acs.org on August 28, 2019

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The Journal of Physical Chemistry

Probing Fabry-Perot Interference in Self-Assembled Excitonic Microcrystals with Subgap Light Emission Adedayo M. Sanni, Shofikur Shuhag, and Aaron S. Rury∗ Department of Chemistry, Wayne State University, Detroit, MI, USA 48202 E-mail: [email protected]

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Abstract Establishing a fundamental understanding of the range of optical effects possible in selfassembled excitonic materials remains crucial to the utility of these materials in future optoelectronic and photonics technologies. In this study we report the observation and subsequent analysis of Fabry-Perot interference intrinsic to two types of self-assembled excitonic crystalline materials: the hybrid organic-inorganic perovskite-like quantum well superlattice structure of hexyl ammonium lead iodide and the charge transfer co-crystal of anthracene and pyromellitic dianhydride. The observation of Fabry-Perot interference stems from strong suboptical gap photoluminescence (PL) from both materials in a spectral region of very low material absorption. We characterize this sub-gap PL in each material to propose permanent defects trap excitons, stabilize their energy, and cause sub-gap light emission in both materials. We use a model developed to explain interference in the photoluminescence of vacuum deposited thin films to estimate the thickness, mid-bandgap index of refraction, and surface roughness of our samples. These results indicate self-assembled excitonic materials may have use in applications such as laser amplifiers, frequency discriminators, and single photon emitters.

Introduction The formation of excitons in response to the confined dielectric environment of certain solids provides an avenue to opto-electronic technologies necessitating intense optical response. 1–7 The ability to observe and characterize excitonic resonances in self-assembled materials processed in solution makes such materials intriguing platforms for future opto-electronic technologies such as light emitting diodes, lasers, and integrated frequency discriminators. Despite this persistent interest, several open questions related to the surface morphologies of and subgap light emission from self-assembled excitonic materials remain unanswered. The development of functional devices for opto-electronic applications based on self-assembled excitonic materials necessitates a more complete understanding of the fundamental physical drivers of the wide range of optical phenomena observed in this class of materials. One important physical phenomenon often neglected in the 2


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context of visible light emission from self-assembled excitonic materials is optical interference. When the light emission spectrum of a material overlaps with a spectral region of minimal light absorption the optical losses associated with light propagation through the material are low. This fact means multiple spatial paths exist for the emitted photons to exit the material. When the appropriate phase between these optical paths can be maintained, then interference will occur and modulate the spectral intensity of the light emission, i.e Fabry-Perot interference fringes. Such fringes have been observed in the photoluminescence spectra of semiconductor thin films formed by vapor and electrochemical deposition in vacuum. 8–11 Despite this simple picture and its observation in vacuum formed materials, optical interference is often unobserved in solution-processed, self-assembled materials due to high surface roughness and a lack of appropriate mid-bandgap light emitters. Without clear signatures of Fabry-Perot interference fringes it is not clear whether self-assembled excitonic materials can act as active and passive components in solution-processed, integrated opto-electronic technologies such as laser gain sections, references for frequency stabilization, or broadband light emitting diodes. In this study we examine the characteristics and formation mechanism of Fabry-Perot interference fringes in the sub-optical gap light emission spectra of the perovskite-like layered material hexylammonium lead iodide (HA2 PbI4 ) and 1:1 co-crystals of anthracene and pyromellitic dianhydride (A-PMDA). The ability to process semiconducting hybrid organic-inorganic perovskite in solution gives them the potential to transform methods to fabricate opto-electronic technologies including photo-voltaic cells, 2,4,12,13 light emitting diodes, 1,14,15 and lasers. 1,16,17 A-PMDA has long been used as a model system to understand the properties of charge transfer excitons including fine structure and their coupling to material vibrations. 18–22 The ability to reliably control light emitted by this material may open interesting avenues to interrogate intermolecular interactions in the solid state. Features consistent with distinct single exciton transitions appear in the emission spectra of HA2 PbI4 and A-PMDA and centered near 532 nm and 545 nm, respectively. In addition to these single excitonic features, the photoluminescence spectra of both materials show broad features

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significantly red-shifted from single exciton peaks. Analysis of the power and polarization dependences of these subgap light emission spectra suggests these red-shifted features result from exciton trapping at material defects. In addition to subgap light emission spectra arising from structural defects, we observe the appearance of significant modulations of these subgap emission spectra. We apply the methods developed by Holm et al. to show these spectral modulations arising from Fabry-Perot interference between subgap emitted light waves whose paths differ, but maintain a stable relative phase. We explain this origin of the measured spectra by modeling the spectral modulations using a fundamental physical treatment. Surprisingly, our theoretical results suggest self-assembled excitonic materials can attain surface smoothness on the order of λ/100. Our results indicate self-assembled excitonic materials could play an important role in a wider array of potential opto-electronic applications than previously anticipated.

Methods Microcrystals of hexyl ammonium lead iodide (HA2 PbI4 ) were formed by mixing HI (57% by weight in H2 O) solution of PbI2 , and completely neutralized hexyl ammonium in 1:2 molar ratios in a round bottom flask. The subsequent solution was then heated to 403 K under an inert, dry N2 atmosphere until all the initial materials were dissolved. We then allowed the solution to cool to room temperature at the rate of the silicon oil heat bath, ∼100 minutes. The cooled flask was then immersed in an ice bath to further cool the solution and cause rapid crystallization. The resulting crystals were extracted via vacuum filtration, washed with diethyl ether three times, and then dried in a vacuum oven at 338 K for 24 hours. Single co-crystals of anthracene and pyromellitic dianhydride (A-PMDA) were formed through slowly cooling a supersaturated solution of 1:1 molar ratio of the molecular constituents in 2butanone from 333 K to 270 K, as described previously. 22 Crystals form after being stored for several days at 270 K. The crystals were then extracted from the mother liquor by vacuum filtration

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in the dark, allowed to dry, and stored at room temperature in the dark until the spectroscopic measurements were undertaken. Typical crystals measured 10 µm by 50 µm by 100’s of µm in thickness, width, and length, respectively. To observe the excitonic peaks in the absorption spectra of each of our materials, films of APMDA and HA2 PbI4 were spun on glass substrates using a Laurell spin processor. For A-PMDA the substrate was spun at 1500 rpm for 20 s while for HA2 PbI4 the substrate was spun at 1000 rpm for 30 s. The HA2 PbI4 film was then annealed at 373 K for 20 minutes. Photoluminescence (PL) spectra were collected in a laser-based microspectrometer, the Horiba XploRA-PLUS, in a back-scattering geometry. All reported PL spectra were collected with a 600 gr/mm grating for varying incident laser powers. Laser powers were controlled with external neutral density filters, which attenuated both the incident and emitted light fields. Spectra at each incident power were corrected for the attenuation of the light emission according to the optical density of each filter. The sample temperature was controlled with a Linkham THMS-600 stage. Powder X-ray diffraction (PXRD) measurements were collected with a Bruker D8 diffractome˚ Diffractograms for ter operating at 40 kV and 40 mA, employing Cu Kα radiation (λ = 1.5418 A). structural analysis were collected in different ranges for each material. For HA2 PbI4 we placed the powdered sample in ambient conditions and used a 5-65◦ 2θ range with a step size of 0.0122◦ and an integration time of 1.5 s for each angular step. For the case of A-PMDA, powder samples were placed in an evacuated sample holder and cooled to 150 K. Using this holder minimized any degradation of the sample due to photo-oxidation and produced conditions similar to those considered in previous analysis of the material’s structure. 23 Details on the methods and results of the analysis used to characterize each material’s structure are found in the Supporting Information (SI).

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Results Crystal Structures Rietveld analysis of PXRD patterns from HA2 PbI4 shows our method of rapidly cooling the solution results in formation of a monoclinic phase similar to that reported by Lemmerer and Billing. 24 In addition, we do not observe peaks consistent with any other crystalline phase besides this monoclinic structure, as shown by Figure S1 of the SI. Le Bail analysis of the PXRD patterns of APMDA indicate we form the same triclinic phase reported by Boeyens and Herbstein, 23 as shown by Figure S3. Furthermore, we see no evidence of additional crystalline phases besides this triclinic structure.

Optical Spectroscopy Hexyl Ammonium Lead Iodide The optical spectra of hexyl ammonium lead iodide (HA2 PbI4 ) have been studied by several groups, especially over the last few years. 25,26 Figure 1 shows the absorption spectrum of HA2 PbI4 a thin film of spun onto a glass substrate at 298 K. Upon inspection, one observes a prominent peak centered at 514 nm due to the exciton confined to the PbI4 octahedral layers, as established previously. 27–29 One also can see the rise of the interband absorption feature at higher energies (lower wavelengths). The inset of Figure 1 shows a typical microcrystal of HA2 PbI4 formed through the synthetic route described above. To further characterize our microcrystalline samples, we undertook photoluminescence measurements exciting HA2 PbI4 at 532, on the red-edge of the excitonic absorption shown in Figure 1. As shown in Figure S5, excitation at this energy causes light emission in two separate spectral regions, the higher of which stems from free exciton recombination. Booker et al. report the appearance of broad, structure-less light emission over 100 meV below the excitonic gap of HA2 PbI4 at temperatures below 100 K. 25 The spectral position and width of the subgap PL spectra emitted

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Absorbance [O. D.]

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Wavelength [nm] Figure 1: Absorption spectrum of a thin film of hexyl ammonium lead iodide spun on a glass substrate showing the excitonic peak resonant with the 532 nm laser used in photoluminescence measurements. Inset: micrograph of a prototypical hexyl ammonium lead iodide microcrystal. The lateral dimension of the image is 300 µm. by HA2 PbI4 resemble those we observed in the case of butyl ammonium lead iodide. 30 In contrast to Booker et al., however, we begin to see significant subgap PL at temperatures above 150 K.

T = 173 K

Intensity [a. u.]

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Wavelength [nm] Figure 2: Linear scale photoluminescence spectrum of hexyl ammonium lead iodide excited at 532 nm at 173 K, which shows clear intensity modulations. We also observe spectral modulations on top of the PL spectra, as clearly seen in Figure 2. 7



Kagan and co-workers reported the appearance of similar structure in the PL spectra of thin films of hybrid lead iodide perovskite-like quantum wells cooled to 20 K. 31,32 However, these workers examined the region of the PL spectrum thought to arise from free exciton recombination. They proposed spectral modulations appear from strong electron-phonon coupling and hot exciton light emission. 31,32 To assess the origin of the spectral modulations on the subgap PL spectra of Figure 2, we undertook an analysis of the subgap PL to determine the fundamental physical drivers of its appearance in our measurements. Specifically, we measured the dependence of the integrated subgap PL intensity on the amount of incident laser power with which we excited the sample and the sample temperature. We cooled the sample to 173 K in order to balance the deposition of residual heat from the non-unity quantum yield of PL and laser excitation. The top panel of Figure 3 shows the results of the power dependent PL spectral measurements. Inspection of these results shows the basic structure of the subgap PL spectrum remains the same as a function incident power, but its intensity changes substantially. While we observe an initial rise in the peak intensity upon increasing the incident laser power from 100 µW to 500 µW, above 500 µW the peak intensity of the subgap PL reduces. At the highest measured power, 1 mW, we find the peak PL intensity reduces to less than a quarter of that measured at 500 µW. We fit the measured PL spectrum at each incident laser power to a Gaussian lineshape to extract the changes to the integrated intensity as a function of the incident laser power. The bottom panel of Figure 3 shows the results of this analysis. We find the integrated PL intensity tracks the peak intensity, with the most light emission at 500 µW incident power of 532 nm light. Given the 5 µm laser spot radius, this power converts into an incident light intensity of over 600 W/cm2 , as shown in the bottom panel of Figure 3. The behavior of the integrated PL intensity with incident light intensity suggests the subgap PL arises from the presence of structural defects in our samples. Previous studies correlate similar PL saturation behavior with structural defects in microcrystalline and thin film forms of butyl ammonium lead iodide 30 and (ethylenedioxyl)bis(ethylammonium) lead iodide, 33 respectively. Furthermore, we

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HA2PbI 4

12 10 8 6 4 2 0

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Incident Intensity [W/cm2] Figure 3: Top panel: comparison of the power dependent subgap photoluminescence (PL) spectra of hexyl ammonium lead iodide following excitation at 532 nm. Bottom panel: comparison of integrated intensities extracted from a Gaussian model of the spectra in the top panel indicating a reduction of the PL process at the highest incident laser intensities. know the subgap light emission cannot arise from quasi-particles such as exciton polarons. Thouin et al show such an emission should appear at energies significantly closer to the exciton optical gap than the spectra shown in Figures 2 and 3. 34 In the case of light emission stemming from permanent defects, the results of Figure 3 would be explained by a physical picture in which a finite density of structural defects capable of trapping an optically excited exciton will be filled at sufficiently high incident laser intensities. Any 9


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Integrated Int. [a. u.]


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109 108 107 106 105

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-3

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-1

1/T [10 K ] Figure 4: Temperature dependence of the spectrally integrated, subgap photoluminescence emitted by a microcrystalline hexyl ammonium lead iodide. Blue squares correspond to the measured intensities while the solid line is a guide. subsequent increase in the laser power would not increase the PL intensity due to the lack of available traps from which emission could occur. In contrast, these subsequent increases in laser power could increase non-radiative relaxation processes that compete with exciton trapping, such as Auger recombination. 35 In addition, to this non-radiative recombination process, the deposition of residual heat in the sample from the laser excitation could affect the intensity of subgap PL. To test this hypothesis, we undertook temperature dependent measurements of the subgap PL spectrum of HA2 PbI4 and processed our results in the same ways as the power dependent spectra. Figure 4 shows the results of this analysis on a logarithmic intensity scale plotted against 1/T. The trend observed in Figure 4 suggests the presence of an energy barrier between the free exciton of HA2 PbI4 and the trap state giving rise to the subgap light emission, as observed previously for the case of self-trapped excitons in layered, hybrid lead perovskite films. 36 Assessing these results more closely one finds a linear increase in the log of integrated PL intensity between room temperature (0.003356 K−1 ) and 173 K (0.00578 K−1 ) possessing a slope of -1.73 x 10−2 log counts/K. Using this slope we find our sample temperature would need to increase by 45 K to completely explain the decrease in integrated PL intensity observed in the bottom panel of Figure 3 excited at 1200 W/cm2 . Given such a dramatic temperature increase over such a small change in 10


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the incident light intensity, we conclude a change in the sample temperature cannot account for the total decrease in the integrated PL intensity at the highest incident powers. A smaller temperature change must also be working in concert with other non-radiative recombination processes such as Auger recombination. Further power dependent, time-resolved studies will more clearly establish this delineation of relevant processes, but are beyond the scope of the current study. By tentatively assigning structural defects as the cause for the appearance in our temperature and power dependent PL measurements, we seek an explanation for the appearance of modulations

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on the spectra of Figure 2 that contrasts with those observed by Kagan and co-workers.

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Wavelength [nm] Figure 5: Comparison of the sub-bandgap photoluminescence spectrum of a single hexyl ammonium lead iodide microcrystal measured at 173 K (blue) to the modeled spectrum (red) constructed by convolving the Gaussian contribution to the 500 µW -excited spectrum in Figure 3 (green) with an interference transfer function (black) calculated by the methodology described in the text. As stated above, optical interference has been observed in the PL spectra of several thin film material systems prepared under high and ultrahigh vacuum conditions. To test for the contribution of optical interference to the spectra of Figure 2 we applied a model developed by Holm et al. 8 and expanded by Larsen et al. 11 to our data. This model essentially treats the PL arising from a point source, but taking multiple paths through the material to the detector. Interference between the optical paths taken to the detector depends on the thickness of the material, the angle of detec11



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tion, and refractive indices of the sample and its surrounding media. In addition, we modeled the dispersion of the index of refraction using Cauchy’s formula, n(λ) = B + C/λ2 . For a sample of thickness d, the emitted PL power can then be written, 11

P

(s)

2 (s) 1 n2 (s) 2 1 + r23 exp(ıδ) (θ1 , λ) = t 2 , (s) (s) 8π n1 21 1 − r21 r23 exp(ıφ)

(1a)

h

i (p) 2 (p) 1 + r + 2Re{r exp(ıδ)}cos(2θ ) 2 23 23 1 n2 (p) 2 , P (p) (θ1 , λ) = t21 (p) (p) 1 − r21 r23 exp(ıφ) 2 8π n1

(1b)

where rij and tij are the Fresnel reflection and transmission amplitude coefficients at the interface between materials with complex indices of refraction ni and nj , θ2 is the incident angle of the PL light out of the sample with respect to the surface normal, and φ = δ =

2πn2 cosθ2 (2d λ

2πn2 cosθ2 2d. λ

In addition,

− 2z) is a phase that accounts for the difference between the thickness of the

sample and the position of the emitter embedded in the sample, z. The spatial distribution of emitters in the sample is dictated by the attenuation of the electric field of the incident laser, El (z) = exp[−αl z/2cos(θ2 )], where αl is the material absorption coefficient at the laser frequency. By integrating the convolution of the emitted PL power for each polarization with the spatial excitation profile over the thickness of our samples, we can model their interference transfer functions. To examine our ability to explain the spectra in Figure 2 with the theoretical model proposed above, we multiplied the fit to the 500 µW spectrum in Figure 3 by interference transfer functions for different values of the real part of the index of refraction of HA2 PbI4 and sample thicknesses. To determine the imaginary part, we extracted the tail of the spectrum shown in Figure 1, fit it to a polynomial in wavelength, and constructed a model function using the same wavelength scale as that of the fits to subgap PL spectra. Figure 5 shows the results of this analysis. Upon inspection one sees that by convolving the Gaussian contribution to the subgap PL spectrum with an appropriately constructed interference transfer function we are able to recover the most prominent features of the measured spectra possessing Fabry-Perot fringes. The modeled spectra shown in Figure 4 presumed Cauchy coefficients of B = 1.6105 and C = 83,500 nm2 with a thickness of 4.9975 µm 12


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stemming from illumination by an electric field impinging on the sample at normal incidence. The values of the Cauchy coefficients conform with the index of refraction reported in a recent study of dielectric functions of BA2 PbI4 and its mixed cation counterparts. 37 Anthracene-Pyromellitic Dianhydride Co-crystals Figure 6 shows the absorption spectrum of a polycrystalline film formed by spin coating a 1:1 molar mixture of anthracene and pyromellitic dianhydride in 2-butanone onto a glass substrate. A discernible peak centered at 540 nm appears on top of the rising background. Based on previous studies, we assign this peak as the CT exciton resonance. 18,21 It is denoted explicitly in Figure 6. 0.1

Absorbance [O. D.]

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Wavelength [nm] Figure 6: Absorption spectrum of a spin-cast film of the 1:1 co-crystal of anthracene and pyromellitic dyanhride (A-PMDA) showing the charge transfer resonance peaked near 540 nm. Inset: micrograph of a typical A-PMDA co-crystal studied with photoluminescence spectroscopy. The lateral dimension of the image is 300 µm. In addition to the exciton absorption feature, we resolve indications of light emission through exciton recombination in photoluminescence (PL) spectra excited at 532 nm. Figure 7 shows PL spectra at 230 K and 330 K. Three main facets of the spectra stand out at both temperatures. First, there is a relatively weak band of emission centered near 560 nm spanning nearly 40 nm at its half maximum. Second, there is a more intense, significantly broader feature centered near 675 nm that asymmetrically tails towards longer wavelengths. The peak intensities of both of these 13



features increase as we cooled the sample from 330 K to 230 K. Third, modulations ride on top of the broad, lower energy feature of both spectra. In fact, the inset of Figure 7 shows that the phase of these modulations shifts by nearly π upon cooling the sample.

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230 K 330 K

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Wavelength [nm] Figure 7: Comparison of the temperature dependent photoluminescence spectra of 1:1 co-crystals of anthracene and pyromellitic dianhydride (A-PMDA) following excitation at 532 nm. Inset: expanded view of the top panel more clearly showing the π phase shift of spectral fringes upon cooling the sample from 330 K to 230 K. To determine the origin of the PL spectra in Figure 7, we made measurements on crystals whose emission spectra were not modulated under varying laser intensities. The top panel of Figure 8 shows the results of these intensity dependent measurements. Upon inspection one notices the relative intensity of the higher and lower energy features of the spectra change as a function of incident laser power. To better quantify these trends, we fit these features to Gaussian and gamma line shapes. That is, we modeled the spectra as,

2

2

I(λ) = AG exp −[λ − λ0 ] /2∆λ

Ag + λΓ(α)

λ λβ

α exp (−λ/λβ ) ,

(2)

where AG , λ0 , and ∆λ are the amplitude, center wavelength, and width of the higher energy feature. The terms Ag , λβ , and α are the amplitude and pertinent parameters of the gamma distribution we use to model the lower energy feature. We aim to develop a model of the origin of the PL fea14


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tures by examining their integrated intensities. The bottom panel of the Figure 8 shows the results of this analysis. 104

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Emission Wavelength [nm] Integrated PL Intensity [a. u.]

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Gaussian Contribution Gamma Contribution

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Incident Laser Intensity [W/cm ] Figure 8: Top panel: comparison of the power dependent photoluminescence spectra of a 1:1 co-crystal of anthracene and pyromellitic dianhydride (A-PMDA) following excitation at 532 nm. Bottom panel: log scale comparison of integrated intensities extracted from a two mode model of the PL spectra of A-PMDA in the top panel where the shorter wavelength peak is fit to a Gaussian (blue circles) and the longer wavelength peak is fit to a Gamma distribution (red x’s). Two facets of the analysis in the bottom panel of Figure 8 stand out. First, there is a noticeable difference in the integrated intensity of each contribution to the measured PL at the lowest incident laser intensities. Specifically, while the intensity of the Gaussian contribution to the spectra does not change dramatically as we change the intensity from 50 W/cm2 to 100 W/cm2 , there is an

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increase by about a factor of 2 in the integrated intensity of the gamma contribution to the spectra. Second, despite this difference at the lowest incident intensities, both predominant features of the PL spectra of A-PMDA show the same general dependence on the incident laser intensity. The bottom panel of Figure 8 shows the integrated intensities of both the Gaussian and gamma contributions peak for an incident laser intensity of nearly 500 W/cm2 and then decrease very similarly with increasing laser intensity. This saturation of the total integrated PL emission as we increase the incident laser intensity qualitatively resembles the behavior observed in HA2 PbI4 in Figure 3. 4

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Polarized Depolarized

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Wavelength [nm] Figure 9: Comparison of the photoluminescence spectrum of a 1:1 co-crystal of anthracene and pyromellitic dianhydride (A-PMDA) following excitation at 532 nm for the case of parallel excitation and detection polarizations (blue) to the spectrum measured in a perpendicular configuration (red). To further determine the origin of the measured PL spectra of A-PMDA at the highest incident laser intensities we measured the PL spectra as a function of incident and detected light polarizations, as shown in Figure 8. Since the structure of A-PMDA is highly anisotropic due to the stacking of electron donor and acceptor sites, we would expect PL emission due to recombination of the CT exciton to be preferentially polarized along the crystal’s long axis as this is the intermolecular axis. 38 Any deviation from this expectation would suggest the participation of additional electronic states due to structural arrangements unlike that of the bulk crystal, i.e. structural

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defects. Our polarization-sensitive measurements show the higher energy Gaussian contribution is substantially more polarized along the long axis of the crystal than its Gamma distribution counterpart as would be expected from emission due to the recombination of the CT exciton. Taken with the saturation of the total PL intensity as we increase the incident laser power manifest in the analysis of Figure 8, the lack of discernible polarization in the lower energy peak of the PL suggests this part of the spectra arises from structural defect emission, as observed in some singlet fission materials. 39 Based on the significant depolarization of the lower energy peak, we conclude these structural defects must introduce mid-bandgap electronic states whose transition dipole moment points more along the short axis of the crystal than its long, intermolecular axis. This tentative conclusion suggests interchain coupling around structural defects such as linear dislocations or

280 K Data IF TF 500 W -Fit Model Spectrum

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vacancies could play a role in the photophysics of our A-PMDA samples.


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0 850

Wavelength [nm] Figure 10: Comparison of the sub-bandgap photoluminescence spectrum of a single 1:1 co-crystal of anthracene and pyromellitic dianhydride measured at 280 K (blue) to the modeled spectrum (red) constructed by convolving the Gaussian contribution to the 500 µW -excited spectrum in Figure 8 (green) with an interference transfer function (black) calculated by the methodology described in the text. We applied the same model developed in the previous section to the spectra in Figure 7 to understand the role of interference in the observed spectral modulations. Figure 10 shows the results 17



of this analysis, which indicate interference can explain the observe spectral modulations. The models used to make the plots shown in Figure 10 presumed a material with Cauchy’s coefficients B = 1.235 and C = 0 nm2 and a thickness of 10.75 µm stemming from illumination by an electric field impinging on the sample at normal incidence. To model the measured spectrum, we assume a complete smooth material surface, as was also the case for HA2 PbI4 . The measured and modeled spectra were compared by shifting the center of the Γ-fit function extracted from the 500 µWexcited measurement in Figure 8 downward by 12 nm to best overlap with the fringed spectrum measured at 280 K. In contrast to the case of HA2 PbI4 , the fact we were able to sufficiently model the spectral modulations in the subgap light emission from A-PMDA without dispersive corrections to the index of refraction suggests this optical property of the crystal must remain almost constant across the band of wavelengths for which we measure the PL spectrum.

Discussion Analysis of the power dependent sub-bandgap photoluminescence spectra of of hexyl ammonium lead iodide (HA2 PbI4 ) and 1:1 co-crystals of anthracene and pyromellitic dianhydride (A-PMDA) suggests structural defects cause the appearance of the subgap light emission. In addition, the modeling results shown in Figures 5 and 10 suggest optical interference can account for the spectral modulations observed on the subgap photoluminescence spectra of smooth samples of HA2 PbI4 and A-PMDA. Despite the excellent agreement between the experimentally measured and theoretically modeled subgap photoluminescence spectra observed for both HA2 PbI4 and A-PMDA, an explanation of the spectral modulations different from Fabry-Perot interference does exist. In the presence of sufficiently long-lived cavity modes capable of capturing the majority of the emitted light, spontaneous photoluminescence can be significantly enhanced 40 or inhibited. 41 However, given the parameters of the subgap light emitted from our materials, our results cannot be explained by these so-called cavity quantum electrodynamic effects. A simple model shows why one should be skeptical of these explanations.

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Specifically, the control of spontaneous emission of light by the presence of a shaped photonic density of states caused by an electromagnetic cavity can be quantified by the Purcell factor, Fp , given by, 42,43

Fp =

λ 2 6 , Q π 2 n2 d

(3)

where λ is the wavelength of the emitted light, d is the diameter of the excitation beam, n is the material index of refraction, and Q is the quality factor of the cavity mode. Given the reasonable estimates of the index of refraction and thickness of HA2 PbI4 , we find the quality factor, Q, of the cavity modes formed by the two parallel material-air interfaces would be approximately 65. From this quality factor, the wavelength of the emitted light at the peak of the spectrum, and the diameter of the incident light beam, we can estimate Fp using the Eq. (3). Using the experimental estimates of the sample thickness, its index of refraction, and our 10 µm excitation beam diameter, we find a Purcell For light emission at 748 nm from HA2 PbI4 , on a maximum of the spectral modulations shown in Figure 4, would be approximately 0.05. For the case of A-PMDA we find a Q-factor nearly double of that of HA2 PbI4 at 138, but the Purcell factor still remains at very low values. Specifically, Eq. (3) allows to estimate a Purcell factor of 0.1 at the peaked emission wavelength of 692 nm. In contrast to previous studies on other organic crystals and inorganic quantum well structures formed within electromagnetic cavities, 42,43 these small values of Fp at the emission maxima for both materials are totally inconsistent with spontaneous light emission enhanced by cavity quantum electrodynamics, i.e. the Purcell effect. Surprisingly, our ability to model all of the important features of the experimental data did not necessitate the use of any estimate of surface roughness, as described in the method of Larsen et al. 11 Based on this fact we presume that the root mean square (RMS) surface roughness is much less than 50 nm, as established previously for vacuum deposited thin films. 11 To better establish this conclusion about the roughness of our sample surfaces, we modeled the subgap PL emission spectrum of HA2 PbI4 shown in Figure 2 using reflection and transmission coefficients for each

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= 0 nm

RMS

= 50 nm

RMS

= 100 nm

RMS

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0.4 173 K Data IF TF Model Spectrum

0.2 0 600

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0.2

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600

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Modeled Int. [norm.]

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Wavelength [nm] Figure 11: Comparison of the sub-bandgap photoluminescence spectrum of a single hexyl ammonium iodide micro-crystal measured at 173 K (blue) to interference transfer functions (black) and modeled spectra (red) for values of root mean squared surface roughness of 0 nm (left-most panel), 50 nm (left middle panel), 100 nm (right middle panel), and 150 nm (right-most panel) calculated by the methodology described in the text. material-air interface that include corrections for roughened surfaces found from scalar scattering theory. 44 The results of this analysis are shown in the panels of Figure 11. These panels show decreasing agreement between the measured and model spectra as we treat the PL spectrum with increasing RMS surface roughness values of 50 nm, 100 nm, and 150 nm. Even at 50 nm RMS surface roughness the fringes of the modeled interference transfer function are significantly damped at shorter wavelengths while they become more prominent at longer wavelengths. These changes in the fringe contrast create very poor agreement between the measured spectra and its resulting model counterpart. For the largest value of 150 nm, the fringes are no longer visible and the interference transfer function significantly distorts the shape of the modeled spectrum. These modeled spectra suggest to us the surface roughness of our samples is significantly less than 50 nm. Despite the broad conclusions of spectral fringes arises from Fabry-Perot interference and very minimal surface roughness, two facets of our data remain unclear. First, there are small deviations between the measured and modeled spectra possessing spectral modulations. Second, our data cannot definitely assign the types of defects present in our samples. We will briefly discuss the importance of each of these issues and paths forward to better address them in future studies. The small deviations between the modeled and measured spectra likely stem from two different effects. First, the temperature dependent and power dependent subgap PL spectra were 20


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collected on completely different sample batches several months apart from one another. So while we are encouraged that our synthetic approach produces defects whose emission spectra are generally reproducible, there are small changes from crystal such as the exact center wavelength of the emission and the width of the emission band. These small differences between the samples could cause the deviations between our modeled and measured spectra seen in Figure 5. Second, deviations between the measured and model spectra could stem from our inability to accurately calculate how the index of refraction changes in the mid-bandgap region due to the presence of the electronic states caused by structural defects. As such, a more extensive study of the interference fringes could serve as a sensitive probe to small changes in the dielectric properties of defective semiconductors. However, such an extension is beyond the scope of the current study. The determination of the structural identity of crystalline defects remains a daunting challenge, especially for the case of self-assembled materials. While techniques such as hard X-ray diffraction, electron microscopy, and scanning tunneling microscope can probe local structure, the use of these methods necessitates materials capable of withstanding the through-put of high energy particles, which often destroy self-assembled materials. Optical spectroscopic techniques can also probe local structure if developed appropriately. Specifically, the positions of peaks in vibrational spectroscopic measurements can report the magnitude and direction of electric fields on length scales approaching 1 nm. 45–47 With the right probe one may be able to map electrostatic environments able to trap excitons and drive sub-optical gap photoluminescence like that observed in this study. The addition of coherent structural motion would further enhance these vibrational approaches to defect identification, as has been demonstrated by Rury and co-workers. 39

Conclusion In conclusion, we have used experimental measurements and theoretical modeling to demonstrate self-assembled excitonic materials can maintain Fabry-Perot modes capable of spectral interference. Power dependent photoluminescence measurements indicate this interference was manifest

21



in the light emission from structural defects in both hexyl ammonium lead iodide and 1:1 cocrystals of anthracene and pyromellitic dianhydride. The tentative assignment of emission from structural defects was established by observing emission saturation at increasing incident laser powers. In the case of A-PMDA it seems that non-radiative relaxation of optically induced charge transfer excitons limits the formation of defects at higher laser intensities. Future power-dependent, time-resolved spectroscopic studies can assess this assignment more clearly. The observation of Fabry-Perot fringes in self-assembled excitonic materials indicates these systems may have application in important opto-electronic technologies. For instance, the stabilization of light sources to particular frequencies often necessitates the passive or active locking of the source to a standard, such as a cavity resonance. If one could optimize the solution-based process of forming the materials examined in this study, then one could prepare integrable frequency standards into micro-photonic platforms with methods such as inkjet printing. In addition, the presence of electronic states capable of participating in optically allowed transitions suggests the possibility of forming active optical amplifiers for power conversion or lasing applications. As demonstrated by our results, the continued study of the optical properties of self-assembled excitonic materials will help enable new, more sustainable opto-electronic technologies.

Acknowledgement The authors gratefully acknowledge the assistance of Prof. Federico Rabuffetti with analysis of powder X-ray diffraction measurements, Aleksandr Avramenko with preparation of spin caste thin film samples, and financial support from Wayne State University. The authors declare no competing financial interests.

Supporting Information Available Supplemental Information including results from the analysis of powder X-ray diffraction measurements on each material and a photoluminescence spectrum measured over the complete band 22


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possible with the instrument described in Methods can be found on-line at:

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Graphical TOC Entry

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Probing Fabry-Perot Interference in Self-Assembled Excitonic

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