Manifestation of excitonic resonance in diffuse reflectance spectra of halide perovskites

Abstract

The application of variable-temperature UV-vis diffuse reflectance (DR) spectroscopy within the range of 100–300 K has facilitated a detailed investigation of the resonance-type feature observed near the absorption edge of several halide perovskites. This study focused on two 3D perovskites, CsPbBr₃ and MAPbBr₃ (where MA represents methylammonium), as well as two 1D perovskites, piperidinium lead iodide (PipPbI₃) and pyridinium lead iodide (PyPbI₃). The resemblance in shape, spectral positions, and temperature-dependent behavior of the resonance-type features in the DR spectra to those found in the well-documented reflection (specular) spectra of single crystals of the same compounds enabled us to attribute these features in the DR spectra to excitonic characteristics in all the studied perovskites. This finding suggests that specular reflection predominates in the DR spectra within the spectral region of strong excitonic absorption. Therefore, the application of the Kubelka–Munk (KM) transformation to the DR spectra—yielding pseudo-absorption spectra—should be restricted to regions where sub-excitonic absorption is relatively weak, as the KM approach only accounts for isotropically scattered light. Furthermore, our analysis revealed that the DR spectra, with excitonic resonances transforming into pseudo-absorption spectra characterized by two regularly spaced band structures, also encompass a lower energy “excess” absorption band. This structure is typical of the pseudo-absorption spectra observed in various low-dimensional halide perovskites at room temperature and contributes to the discrepancies observed in spectral interpretations.

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1. Introduction

UV-vis-NIR diffuse reflectance (DR) spectroscopy has emerged as a powerful tool for delving into the intricate absorption spectra of powdered halide perovskites. This technique is pivotal in estimating their bandgap energy (E_bg) and observing the dynamic shifts in the optical properties that arise from various factors including anion and cation modifications, phase transitions, fluctuations in synthetic methods, and post-synthesis treatments. Unlike the more rigid forms of single crystals and thin films, halide perovskite powders or microcrystals can be prepared through a myriad of synthesis techniques, offering a versatile playground for materials scientists. One of the most compelling advantages of DR spectroscopy lies in its ability to analyze samples without the need for elaborate preparations, which are often cumbersome and essential for obtaining transmittance spectra from single crystal samples. This ease of use allows for a more straightforward exploration of the rich optical landscape presented by these interesting materials.

At present, the prevailing technique for extracting the absorption spectrum from a diffuse reflectance (DR) spectrum is the Kubelka–Munk (KM) transformation.^1–3 This method employs the KM function, articulated as KMF = K/S = [(1 − R)²/2R], which connects the absorption (K) and scattering (S) coefficients to the measured diffuse reflectance (R) of a sample layer with a thickness of 1 millimeter or greater. In this context, K represents twice the true physical absorption coefficient of the medium exhibiting diffuse reflectance.¹ To distinguish these interesting absorption spectra from those derived through alternative methodologies, we shall designate them as pseudo-absorption spectra.^3–5

The optical characteristics of low-dimensional (2D and 1D) organic–inorganic halide perovskites have captured considerable interest, as these materials behave as quantum-confined systems. In these unique compounds, excitons—quasi-particles resulting from the strong interaction between electrons and holes—are effectively trapped within the inorganic layers. This occurs due to energetic barriers formed by the limited dielectric screening provided by the surrounding organic spacers, with the interlayer organic moiety acting as a barrier.⁶

The result of such confinement is a vivid and prominent excitonic signal observable in the pseudo-absorption spectra of many low-dimensional halide perovskites, even under room temperature conditions. At the absorption threshold, the pseudo-absorption spectra unveil a dual-band structure characterized by two distinguishable, regularly spaced bands, separated by approximately 0.2 eV. This intriguing duality is denoted as the P1 band (representing the lower-energy state) and the P2 band (indicating the higher-energy state), a phenomenon that will be visually illustrated in the forthcoming Fig. 1c–4c and 5a.

Fig. 1 Temperature dependences of the optical spectra of 3D perovskite CsPbBr₃. (a) The DR spectra; the insert presents the DR spectra within the energy range of 3.4 eV, averaged over two temperature intervals: 115–180 K (curve 1) and 240–300 K (curve 2). (b) DR spectra highlighting the resonance-like feature; the spectrum measured at 210 K is emphasized in olive-green color, while the insert illustrates the temperature dependence of the spectral position of the low-energy maximum. (c) Demonstration of the pseudo-absorption spectra using the Kubelka–Munk function (KMF).

Fig. 2 Illustration of the temperature dependence of the optical spectra of 3D perovskite MAPbBr₃. Panel (a) presents the DR spectra, with the insert showing the temperature dependence of the bandgap, E_bg (curve 1), alongside the spectral position of the low-energy maximum of the resonance-like feature in panel (b), E_ex (curve 2, shifted down by 0.04 eV). Panel (b) focuses on the DR spectra within the range of the resonance-like feature, while the insert displays the DR spectra in the vicinity of 3.34 eV. Panel (c) shows the pseudo-absorption spectra, based on the Kubelka–Munk function (KMF). For clarity, only the spectra at 100 K and 300 K, as well as those just before (151 K) the phase transition and after (162 K) the phase transition, are presented.

Fig. 3 Temperature dependence of the optical spectra for one-dimensional perovskite piperidinium lead iodide (PipPbI₃). (a) The diffuse reflectance (DR) spectra. (b) The DR spectra focus on the range of the resonance-like feature with an insert illustrating how the spectral position of the low-energy maximum varies with temperature. (c) The pseudo-absorption spectra.

Fig. 4 Temperature dependence of the optical spectra of 1D perovskite pyridinium lead iodide (PyPbI₃). (a) The diffuse reflectance (DR) spectra. (b) The DR spectra focus on the range of the resonance-like feature, with an insert illustrating the temperature dependence of the spectral position of the low-energy maximum of the resonance-like feature shown in (b). (c) The pseudo-absorption spectra.

Fig. 5 (a) Display of the DR spectra, R(hν), along with the pseudo-absorption spectra, KMF(hν), for powdered CsPbBr₃, and (b) presents the same for Bi-doped CsPbBr₃ at temperatures of 100 K and 200 K. Arrows highlight the relationship between the features observed in the DR and pseudo-absorption spectra at 100 K. The curves labelled R in panel (b) were increased and downward shifted in the range of hν > 2.2 eV to 0.67.

The structure of these two bands has sparked a variety of interpretations among researchers. Several studies have pinpointed the low-energy side of peak P2 as a defining marker for the onset of fundamental absorption.^7–13 In contrast, peak P1 has been closely linked to excitonic absorption phenomena. Within this framework, the bandgap value, E_bg, is determined by projecting the absorption edge (P2) along the energy axis. This method, however, is not without complications, as the presence of the excitonic peak (P1) hinders precise identification of the absorption edge. Consequently, the bandgap value exceeds the energy of the excitonic absorption peak by 0.06 to 0.09 eV.⁷

In addition, the fundamental absorption edge was observed prominently on the lower-energy side of peak P1, while a distinct secondary peak, designated P2, rose above this threshold. This additional absorption feature was linked to a long-lived excitonic state, which becomes trapped within the potent electric field created by localized positive organic ions.^14–17 Within this framework, the energy bandgap (E_bg) was determined to be less than the excitonic energy, measuring approximately 0.065 eV.¹⁵ In certain instances, the unmistakable presence of the secondary peak P2 left analysts grappling for a definitive explanation.¹⁸

Moreover, two distinct spectral bands emerged prominently at the pseudo-absorption edge in the spectra of the 2D (BA)₂PbI₄¹⁹ and (BA)₂Pb(Br_xI_1−x)₄²⁰ single crystals (BA: butyl-ammonium). These spaced bands, striking in their regularity, hint at the existence of dual optical bandgaps. This phenomenon likely arises from contrasting regions within the material itself: one band emanating from the intricate surface or sub-surface area, while the other reflects the deeper, bulk interior. This layering of optical characteristics underscores the complexity of the crystal structure and its unique light interactions.

The well-documented characteristics observed at the absorption edge of pseudo-absorption spectra for various low-dimensional halide perovskites have been attributed to contradictory explanations. To clarify these contradictions, this study employed variable temperature UV-vis diffuse reflectance (DR) spectroscopy, analyzing the diffuse reflectance spectra of several perovskites across a temperature spectrum ranging from 100 to 300 K. We focused on compounds for which comprehensive measurements of reflectance (specular) spectra from single crystals at various temperatures have already been reported in previous research. Our exploration encompassed two three-dimensional (3D) perovskites, cesium lead bromide (CsPbBr₃) and methylammonium lead bromide (MAPbBr₃), alongside two one-dimensional (1D) perovskites: pyridinium lead iodide (PyPbI₃) and piperidinium lead iodide (PipPbI₃).

A thorough comparative analysis of the DR and reflectance spectra of identical compounds, particularly as they approach the critical absorption edge, has unveiled insights into the temperature-dependent behavior of their spectral features. Notably, at cryogenic temperatures, the excitonic resonance – a phenomenon marked by a low-energy peak positioned next to a high-energy trough – emerges vividly in the raw DR spectra for all compounds examined. This observation yields two pivotal conclusions:

(1) Specular reflection emerges as the dominant factor within the DR spectra amidst the region of intense excitonic absorption.

(2) The relevance of the Kubelka–Munk (KM) theory, which primarily concerns isotropically scattered light, should be limited to areas where sub-excitonic absorption remains relatively weak.

Additionally, applying the KM transformation to the DR spectra, which exhibit excitonic resonances, yields a pseudo-absorption spectrum characterized by a distinctive two-band structure. This structure features a low-energy “excess” absorption band, further complicating the interpretation of the spectra and adding nuance to our understanding.

2. Experimental section

2.1. Materials and synthesis

The synthesis and characterization of perovskite materials, specifically CsPbBr₃²¹ and MAPbBr₃,²² as well as PyPbI₃ and PipPbI₃,²³ have been previously documented through X-ray diffraction (XRD) analysis. These studies provide valuable insights into the structural properties of these compounds.

The diffuse reflectance (DR) spectra were measured using a state-of-the-art Cary-5000 spectrophotometer, equipped with an integrating sphere and a cryostat accessory with BaSO₄ as the relative reflectance reference.^24,25 To create the samples, a fine layer of perovskite—approximately 0.7 mm thick—was carefully applied to a small aluminum dish (foil, a mere 0.07 mm thick and 15 mm in diameter). This dish was first coated with a film of tetrafluoroethylene, measuring 0.1 mm, thereby providing an additional layer of protection for the powder beneath.

To safeguard the compressed layer during experimentation, a robust, 1 mm thick quartz window, was positioned securely over the sample, while the edges of the dish were rolled inward to ensure complete encasement. The assembled powder was then firmly affixed to the wall of the cryostat's cold finger, thereby enabling precise temperature measurements.

The temperature dependencies of the DR spectra were systematically measured using a temperature-programmed heating method, ramping at a steady rate of 5–10 K min⁻¹, spanning 100–300 K. Under these conditions, a compact cryogenic pump established and maintained a vacuum of 1 × 10⁻⁴ Torr within the cryostat.

3. Results and discussion

3.1. Manifestation of the excitonic resonance in diffuse reflectance spectra

Fig. 1a presents the diffuse reflectance (DR) spectra of powder CsPbBr₃ measured at various temperatures ranging from 100 to 300 K. Fig. 1b illustrates the temperature-dependent evolution of the resonance-like feature in the DR spectra at 2.35 eV (at 100 K) on a broad scale. This feature exhibits a similar shape to the excitonic resonance observed in the specular reflectance spectra of CsPbBr₃ thin films (100 nm) at 4 K²⁶ and in single crystals at cryogenic temperatures.^27–31 At 100 K, the low-energy maximum of the excitonic resonance in the reflectance spectra peaked at 2.35 eV.³¹ In contrast, the resonance inflection point (the point between the maximum and minimum) was approximately 2.375 eV.²⁹ Beginning at around 180 K, the broadened excitonic resonance in the reflectance spectra took the form of a single reflectance maximum.^29,31 This same shape is evident in the DR features shown in Fig. 1b for temperatures at or above 210 K. Fig. 1 also demonstrates a monotonic blueshift of the resonance-like feature in the DR spectra with temperature increase. This temperature behavior of the excitonic resonance has also been reported for CsPbBr₃ crystals.^29,31

The correspondence between the shape at different temperatures, the spectral position, and the temperature dependence of the resonance-like feature in the DR spectra of powder CsPbBr₃, when compared to the excitonic signal characteristics of CsPbBr₃ single crystals, leads us to conclude that the resonance-like feature observed in the DR spectra of this material is indeed excitonic.

The insert in Fig. 1a highlights another spectral feature in the DR spectra at 3.41 eV. The position of the DR maximum is independent of temperature. A strong peak observed at approximately 3.44 eV at both 40 K²⁷ and 77 K²⁸ represents the second significant feature in the reflection spectrum of the CsPbBr₃ crystal. This finding confirms the consistency of the shape and spectral position of features in both the DR and specular reflection spectra of this perovskite.

The resonance-like feature in the DR spectra of polycrystalline MAPbBr₃ is not prominently observed at any temperature (Fig. 2a) and warrants examination on a larger scale (Fig. 2b). Nevertheless, the shape of this feature mirrors the excitonic resonance observed in the specular reflectance spectra of MAPbBr₃ single crystals.^22,32 A phase transition occurs in both the MAPbBr₃ single crystal³² and our sample around 150 K (see the insert in Fig. 2a), which emphasizes the low-energy maximum of the resonance. At 100 K, the spectral position of this maximum is recorded at 2.285 eV in Fig. 2b, which aligns closely with approximately 2.29 eV in the reflectance spectra of the single crystal at 100 K,³² as well as around 2.25 eV at 4 K²² and 2.26 eV at 5 K.³³ Both the resonance-like feature in the DR spectra (Fig. 2a, insert) and the excitonic resonance in the reflectance spectra³² exhibit a blue-shift with increasing temperature in both low-temperature and high-temperature phases. These observations corroborate earlier conclusions that the excitonic resonance appears in the DR spectra as a resonance-like feature, particularly at cryogenic temperatures.

The insert in Fig. 2b highlights the second spectral feature in the DR spectra at 3.34 eV. The optical absorption spectrum of MAPbBr₃ single crystals, measured at 5 K, also revealed an excitonic line at 2.26 eV alongside an absorption band at 3.33 eV.³³ DR spectra indicate that this band is temperature-independent and observable at temperatures below 151 K, specifically within the low-temperature phase (see Fig. 2b, insert). Therefore, the measured DR spectra of the two 3D perovskites, CsPbBr₃ and MAPbBr₃, align well with the reflectance spectra of the corresponding single crystals across a wide spectral range, encompassing both excitonic and above-excitonic absorption.

Fig. 3 and 4 illustrate the temperature-dependent DR spectra of two polycrystalline one-dimensional lead iodide perovskites, PipPbI₃ and PyPbI₃, respectively. The absorption spectra A = 1 − R of these materials were analyzed by us earlier²³ to establish the nature of low-energy sub-bandgap absorption bands typical of Pb-based 1D perovskites. Although these materials differ in their organic components, with piperidinium and pyridinium cations, they share a common framework of inorganic lead iodide chains. Both perovskites exhibit a resonance-like feature in their DR spectra, with low-energy maxima observed at 3.28 eV for PipPbI₃ and 3.20 eV for PyPbI₃ at 100 K. Additionally, a strong excitonic resonance has been recorded at 3.3 eV in the reflectance spectra of PipPbI₃ single crystals³⁴ and at 3.11 eV for methylviologen lead iodide (MVPb₂I₆) single crystals.³⁵ The pronounced polarization of the excitonic signal in both one-dimensional iodides suggests that exciton excitation occurs along the [PbI] chains. For these one-dimensional perovskites, the observed shape and spectral position of the resonance-like feature in the DR spectra are consistent with those of the excitonic resonance observed in single crystals. Consequently, we deduce (again) that the resonance-like feature in the DR spectra of the studied one-dimensional perovskites also possesses an excitonic character.

Very little is known about the temperature behavior of excitons in the studied 1D perovskites. The only relevant observations include reflectance spectra of single crystals measured at 37 K for PipPbI₃³⁴ and 300 K for MVPb₂I₆.³⁵ In both cases, the exciton was attributed to the same structural units, specifically the [PbI] chains. Therefore, it can be speculated that the exciton in the [PbI] chains experiences a redshift with increasing temperature, shifting from 3.3 eV at low temperatures to 3.11 eV at higher temperatures. This qualitatively supports the temperature dependence of the excitonic features observed in the DR spectra of PipPbI₃ and PyPbI₃ powders, as displayed in Fig. 3b and 4b.

The optical properties of layered perovskite compounds, specifically (BA)₂(MA)_n−1Pb_nI_3n+1 {where n = 1–4, BA = CH₃(CH₂)₂NH₃⁺, and MA = CH₃NH₃⁺}, are particularly intriguing due to the variety of methods used to obtain their absorption and reflectance spectra. These include pseudo-absorption spectra from both powders^7,14 and single crystals,¹⁹ as well as raw diffuse reflectance (DR) spectra (see Fig S1 in the Supporting Information of ref. 7). Most notably, polarized specular reflectance spectra of individual microcrystals have been measured within the temperature range of 80–300 K.³⁶ A comparison of these spectra reveals notable similarities in shape and spectral position, particularly between the resonance-like features observed in the raw DR spectra, which fall within the ∼520–650 nm range (2.38–1.91 eV) for n = 1–4,⁷ and the excitonic resonance evident in the polarized specular reflectance spectra.³⁶

3.2. Specular and diffuse reflectance in DR spectra

The presence of a similar excitonic resonance in the reflectance spectra of single crystals (specular reflectance) and in the diffuse reflectance (DR) spectra of polycrystalline materials indicates that the specular reflectance significantly influences the DR spectra in the spectral region where excitons are observed. A similar conclusion was reached by Eijkelenkamp³⁷ after considering the reflection spectra of powders and single crystals of several lead halides, including PbCl₂ and PbBr₂.

The specular and diffuse reflected light are essential components of the experimental diffuse reflectance (DR) spectra measured using an integrating sphere.^1,2 Specular reflection adheres to the Fresnel equation, which relates the two fundamental parameters of materials: the refractive index and the absorption coefficient.^1,2 According to the Fresnel equation, an increase in a sample's absorptivity results in a corresponding increase in specular reflectance.

Near the absorption edge of semiconductors, the absorption coefficient increases significantly, spanning several orders of magnitude. For instance, calculations of the absorption coefficient, α, derived from combined transmittance/reflectance and ellipsometric measurements on a MAPbBr₃ single crystal, indicated that within the absorption edge range of 2.1–2.3 eV (540–590 nm), α increased to as high as 10⁵ cm⁻¹.³⁸ In the case of the excitonic absorption maximum of the aforementioned layered perovskite single crystal (BA)₂(MA)_n−1Pb_nI_3n+1, the absorption coefficient ranged from (3.9 to 2.03) × 10⁵ cm⁻¹ for n = 1–4,³⁶ while in the one-dimensional PipPbBr₃ crystal, α reached approximately 1.3 × 10⁶ cm⁻¹.³⁹ The absorption coefficients of various known excitonic semiconductor materials fall between 7 × 10⁴ and 8 × 10⁵ cm⁻¹.⁴⁰ Thus, it can be concluded that in the region of the reflectance/absorption edge, incident light is diffusely reflected in the long-wavelength segment of the edge – characterized by weak absorption and a significant light pathway within the particle – whereas in the short-wavelength segment of the edge – where strong absorption occurs and reflection is limited to the particle surface – the incident light is reflected specularly. This widely accepted understanding, though theoretically sound, had little practical significance until the investigation of low-dimensional halide perovskites, which exhibit pronounced excitonic resonance even at room temperature, gained prominence (see the Introduction).

Direct evidence of the superposition of diffuse and specular reflected light in the reflectance spectra was obtained using a setup designed to measure the angular dependence of the reflectance spectra of a MAPbI₃ single crystal.⁴¹ In this configuration, the light beam was incident nearly normal to the crystal's surface, while the reflected beam was focused at an adjustable angle (θ) onto the detector. It was found that the normalized reflectance spectrum took on the characteristic shape of a conventional diffuse reflectance (DR) spectrum of a semiconductor, with the absorption edge or onset appearing at approximately 1.48 eV in the spectra recorded at θ = 60° to 70° (refer to Fig. S7 in ref. 41). In contrast, the reflectance spectra recorded at θ = 10° and 0° did not exhibit this onset. The authors of this study⁴¹ reasonably concluded that the specular reflection component primarily influenced the high-energy portion of the reflection spectrum, while the diffuse reflection dominated the low-energy region.

It is often assumed that the superposition of diffuse and specular reflected light poses a significant challenge in DR spectroscopy. However, in the context of perovskites, it is evident that only the specular reflection carries informative value, though it requires careful interpretation of the features within the DR spectra. Moreover, it is important to note that the KM theory, employed to analyze data from DR spectroscopy, does not account for the specular reflection component, as it focuses exclusively on isotropically (diffusely) scattered light.^1,2 Nonetheless, this theory continues to be the sole theoretical framework utilized in practice.

3.3. Manifestation of the excitonic resonance in the pseudo-absorption spectra of perovskites

To confirm the presence of exciton-like features in the diffuse reflectance (DR) spectra of various low-dimensional halide perovskites at room temperature (as discussed in the Introduction) it is essential to analyze the results of the Kubelka–Munk (KM) transformation applied to these DR spectra. This analysis will provide valuable insights into the optical properties of the materials under investigation.

Fig. 5a presents two diffuse reflectance spectra of CsPbBr₃ measured at 100 K and 200 K (represented by the black curves), which are also illustrated in Fig. 1a and b. The wine-colored curves depict the pseudo-absorption spectra derived from the DR spectra through the Kubelka–Munk (KM) transformation, also shown in Fig. 1c. Arrows indicate the transformation of minima and maxima in the DR curve into maxima and minima in the KM function, highlighting the emergence of a two-band structure. The two maxima (bands) of the pseudo-absorption spectra, denoted as P1 and P2 in Fig. 5a, are present in the pseudo-absorption spectra of our 3D perovskites only at cryogenic temperatures as shown in Fig. 1c and 2c. In contrast, these bands are relatively well-resolved even at room temperature in 1D compounds, illustrated in Fig. 3c and 4c. This dual structure is characteristic of the pseudo-absorption spectra of other low-dimensional perovskites discussed in the Introduction. The reasons for the appearance of the two-band structure at the edge of the pseudo-absorption spectra become particularly clear when the DR spectra and pseudo-absorption spectra are depicted together, as seen in Fig. 5a. However, raw DR spectra are seldom discussed in the literature, with very few exceptions.^7,10

Additionally, Fig. 5a clearly illustrates that the KM transformation substitutes the excitonic resonance in the DR spectra with a structure comprising two pseudo-absorption bands (or one P1 band at higher temperatures; dotted curves), which invites interpretation (see Introduction). Due to the nonlinearity of the KM function in the range R < 0.3 (see Fig. S1 in SI), a weak minimum of R at the excitonic resonance at 2.37 eV (100 K) in the DR spectra transforms into a pronounced maximum, labelled P2 at 2.37 eV, in the pseudo-absorption spectra (wine-colored curves) (left black arrow in Fig. 5a). At the same time, the minimum R at 2.33 eV (100 K), situated between the reflection/absorption edge and the maximum of the excitonic resonance, also transforms into a pronounced maximum, labelled P1 at 2.33 eV, in the pseudo-absorption spectra (red arrow in Fig. 5a). So, the P1 band, which is prominent in the pseudo-absorption spectra at all temperatures, corresponds to a region of relatively low reflectance (minimum) in the DR spectra which has no physical significance. Transformation of the maximum of the excitonic resonance at 2.35 eV (100 K) into the minimum of the pseudo-absorption spectra at this energy (right black arrow in Fig. 5a) makes the excitonic resonance unrecognizable.

Fig. 5b strongly supports the statement that the P1 band is an “artifact” of the KM transformation. Doping CsPbBr₃ with Bi shifts the apparent absorption edge from 2.31 eV to 2.03 eV (at 100 K) and results in a noticeable decrease in the amplitude of the excitonic resonance. It is important to note that the spectral position of the exciton remains unchanged at 2.35 eV at 100 K. Consequently, the diffuse reflectance spectrum between the reflection edge and the excitonic resonance (2.17–2.27 eV) shows no minimum but rather a plateau, and the band P1 observed in the pseudo-absorption spectra is completely absent. Therefore, the low-energy quasi-absorption band P1 lacks any physical significance and is merely an “excess” band. Nonetheless, band P1 has often been mischaracterized in the literature as an excitonic feature.

The dual structure of the pseudo-absorption spectra obtained from the application of the KM function leads to conflicting interpretations of the absorption features. It is important to note also that, in the case of CsPbBr₃, the band labelled P1 in the pseudo-absorption spectra appears to be temperature-independent (see Fig. 1c and 5a). The maximum shift of this band with temperature is minimal, not exceeding 0.01 eV. This finding stands in stark contrast to the well-known blue shift of the excitonic resonance in CsPbBr₃,^29,31 which is distinctly observable in the raw DR spectra (Fig. 1a and b). Consequently, the transformation of DR spectra into pseudo-absorption spectra not only complicates the interpretation of features at the perovskites absorption edge but may also significantly alter the understanding of their temperature-dependent behavior.

At the same time, there are no compelling reasons to exclude the application of the KM function in the sub-excitonic spectral region. Specifically, the bandgap energy (E_bg) of CsPbBr₃, as determined from the first derivative low-energy maximum of dR/dλ^25,42 (see Fig. 6a) and the d(KMF)/dλ (Fig. 6b) curves, was found to be very similar (as indicated in the insert of Fig. 6b). The E_bg derived from the KM function is higher than that obtained from the DR spectra by 0.007 eV at 100 K and by 0.023 eV at 300 K. These discrepancies are significantly smaller than the underestimation of E_bg that occurs when analyzing the apparent onset of the DR spectra that we discuss next.

Fig. 6 (a) First derivative with respect to wavelength of the DR spectra, dR/dλ as a function of photon energy, and (b) first derivative with respect to wavelength of the pseudo-absorption spectra, dKMF/dλ of CsPbBr₃; the insert in (b) shows the temperature dependence of E_bg determined over the maximum of dR/dλ, E_bg/R, and that of dKMF/dλ, E_bg/KMF.

3.4. On the occurrence of the excitonic feature at energies exceeding the absorption edge in the DR spectra

Attributing resonance-like features in the diffuse reflectance spectra to excitonic effects necessitates an explanation for the observation of these excitonic features at energies that surpass both the perovskite absorption edge and the bandgap determined from the DR spectra.

In the present study, the bandgap energy (E_bg) of the perovskites was estimated using the spectral position of the first derivative of the diffuse reflectance spectrum with respect to wavelength.^25,42 For CsPbBr₃, the excitonic low-energy maximum (E_ex) is observed at 2.35 eV at 100 K, while E_bg is estimated to be 2.31 eV at this temperature (see Fig. 1a). In the case of MAPbBr₃ at 100 K, E_ex is measured at 2.285 eV (Fig. 2a) with an E_bg of 2.24 eV. For PipPbI₃ at 100 K, E_ex is 3.28 eV and E_bg is 3.14 eV (Fig. 3a). Similarly, for PyPbI₃ at the same temperature, E_ex is 3.20 eV (Fig. 4a) with an E_bg of 3.05 eV. Thus, for the perovskites studied, the excitonic energy exceeds the bandgap energy derived from the DR spectra by 0.04 to 0.15 eV.

Detailed investigations of the optical properties of MAPbBr₃ single crystals were conducted by Wenger and coworkers³⁸ using transmittance and reflectance spectroscopies, along with ellipsometry. These studies revealed that the exciton-like resonance feature observed in the specular reflectance spectra occurs approximately 0.14 eV above the transmittance edge. This finding aligns with the bandgap determined from a combination of transmittance and ellipsometry data. These authors³⁸ concluded that due to the significant optical density of the crystal (thickness, 2.3 mm), the absorbance was saturated at energies lower than the bandgap, leading to an underestimation of the true bandgap energy by about 0.1 eV for the examined crystal.

A blue shift of approximately 0.04 eV was observed in the transmittance spectral edge of GaP crystals as the thickness decreased from 4 mm to 0.5 mm.⁴³ In the case of ZnO single crystals, the absorption edge measured via conventional transmittance spectroscopy was redshifted compared to the values obtained using spectroscopic ellipsometry in the reflection mode.⁴⁴ These two techniques provided differing bandgap values for ZnO: 3.15 eV and 3.30 eV, respectively. Conversely, the bandgap values derived from the transmittance spectra of single crystals generally align well with those obtained from the diffuse reflectance spectra of powdered perovskites.⁴⁵ This suggests that the absorption edge observed in DR spectra tends to underestimate the actual spectral position of the absorption edge and the corresponding bandgap value. Therefore, it can be concluded that the systematic underestimation of E_bg extracted from the DR spectra may reach approximately 0.15 eV.

Additional evidence for the redshift of the absorption edge relative to the fundamental transition, as observed in DR spectra, includes data on the so-called above-bandgap photoluminescence in CsPbBr₃⁴⁶ in MAPbX₃ (where X = Cl, Br, I).⁴⁷ In both instances, the peak of the excitonic luminescence was found at energies higher than the absorption edge noted in the absorption spectra derived from DR measurements. For CsPbBr₃, the difference between the spectral position of the absorption edge and the excitonic peak was approximately 0.1 eV.⁴⁶ In the current study of CsPbBr₃, the excitonic energy (E_ex) exceeds the bandgap energy (E_bg) at room temperature by 0.13 eV. Previous studies^46,47 did not consider that DR spectroscopy values underestimate the bandgap, and the aforementioned spectral differences were attributed to the involvement of structural defects in photoluminescence.

The occurrence of exciton-like features above the absorption edge of the DR spectra is quite understandable. In this context, the increase in the E_bg energy of the PyPbI₃ sample upon dilution in a non-absorbing BaSO₄ matrix – from 3.05 eV to approximately 3.14 eV – supports our hypothesis. Additionally, it is worth noting that the blue shift of the DR spectral edge and the corresponding increase in E_bg by about 0.14 eV, resulting from dilution with BaSO₄ or MgO, have also been observed in powdered anatase TiO₂.⁴²

A redshift of the absorption edge in diffuse reflectance spectra, relative to the true spectral position, and the subsequent underestimation of the energy bandgap (E_bg) have become apparent in the DR spectra of perovskites exhibiting well-defined excitonic resonance. In cases where the excitonic signal is absent – due to the material's inherent properties or temperature-induced quenching – it becomes challenging to reveal this characteristic by DR spectroscopy. Currently, an understanding of the limitations of DR spectroscopy in determining the true spectral position of the absorption edge in semiconductors will enable more accurate interpretation of results. Nevertheless, DR spectroscopy remains a powerful and straightforward technique for the simultaneous study of both extrinsic factors (e.g., impurities and defects) and intrinsic properties (including fundamental and excitonic optical absorption) in solids.

4. Concluding remarks

The application of variable temperature UV-vis diffuse reflectance (DR) spectroscopy across the range of 100–300 K has enabled a detailed examination of the resonance-type feature present in the DR spectra of various halide perovskites. The similarity in shape, spectral position, and temperature dependence (for 3D perovskites) of this feature to that observed in the reflectance spectra of single crystals suggests that it is attributable to excitonic behavior. A direct implication of this attribution is that specular reflection plays a dominant role in the DR spectra within the region characterized by strong excitonic absorption. Consequently, the application of the Kubelka–Munk (KM) theory, which only accounts for isotropically scattered light, should be restricted to the spectral region of relatively weak sub-excitonic absorption.

Furthermore, an analysis of the transformation of the DR spectra relative to the excitonic resonance, utilizing the Kubelka–Munk (KM) function, revealed that the resulting pseudo-absorption spectra display a dual structure near the absorption edge, characterized by a low-energy “excess” absorption band. This feature is commonly observed in the pseudo-absorption spectra of various low-dimensional halide perovskites at room temperature and contributes to the discrepancies in spectra interpretation.

Author contributions

V. N. Kuznetsov: conceptualization, methodology, original draft, review and editing; Y. V. Chizhov: review and editing; N. I. Glazkova: data curation, methodology; R. V. Mikhaylov: data curation, methodology; V. K. Ryabchuk: review and editing; G. V. Kataeva: review and editing; A. V. Emeline: conceptualization; N. Serpone: original draft, data discussions, review and editing.

Conflicts of interest

The authors declare no conflict of interest.

Data availability

The data supporting this article are included in the supplementary information (SI). Supplementary information: the figure demonstrates the dependence of the KM function and its numerator, (1 − R)², on R at low values of R. See DOI: https://doi.org/10.1039/d5cp03576g.

Acknowledgements

This research has been supported by Saint-Petersburg State University (project 125022002745-6). The authors are grateful to the experts of the Resource Centers “Nanophotonics”, “Center for Optical and Laser Materials Research”, “X-ray Diffraction Studies”, “Nanotechnology”, and “Chemical Analysis and Materials Research Centre” of the Research Park of SPbU for their valuable technical and analytical support. One of us (NS) would like to further thank the staff of the PhotoGreen Laboratory in the Chemistry Department of the University of Pavia for their continued hospitality.

Notes and references

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