Untangling free carrier and exciton dynamics in layered hybrid perovskites using ultrafast optical and terahertz spectroscopy

Thin films of (PEA)₂PbI₄ were fabricated by spin-coating onto both glass and crystalline quartz substrates for terahertz and transient absorption spectroscopy measurements, respectively.

2.1.1. Precursor preparation

2.1.2. Film deposition

The main time-resolved techniques used in this study are terahertz spectroscopy and transient absorption spectroscopy. As such their experimental details are discussed below. Supporting characterisation techniques were also utilised including scanning electron microscopy (SEM), x-ray diffraction (XRD), steady-state UV–vis absorption, photoluminescence (PL), and time-resolved PL. More specific information about these techniques can be found in sections S1-S4 in the supporting Information (SI).

2.2.1. Terahertz spectroscopy

2.2.2. Transient absorption spectroscopy (TAS)

Thin films of (PEA)₂PbI₄ on quartz and glass were prepared and optimised for OPTP and TAS measurements, respectively (See Experimental - section 2.1). Both samples were then analysed using a variety of techniques to evaluate their composition, purity, and comparability. The thickness of the samples was evaluated with scanning electron microscopy (SEM), as shown in figure S1 in the supporting Information. We found that the thickness of the sample deposited on quartz was 1.8 μm (±0.36 μm) (Shown in figure S1(a)). This value is much thicker than that for the sample deposited on glass (shown in figure S1(b)), which had a thickness of 125 nm (±25 nm). The thicker film was used to optimize optical absorption and thus signal for OPTP measurements, and the thinner film was selected to avoid saturation for TAS measurements.

Past studies on LPKs have shown that their layers can often preferentially align either parallel or perpendicular to the surface of the substrate on which they are grown [1, 8, 58]. To determine the preferential orientation within our films, x-ray diffraction (XRD) measurements were performed on the (PEA)₂PbI₄ films on quartz and glass. Figures S2(a) and (b) show the XRD patterns of the LPK films on quartz and glass, respectively. Figure S2(c) compares the XRD pattern of (PEA)₂PbI₄ on quartz with the theoretical pattern modeled using single crystal data. The overlap in the XRD patterns confirms that the crystals are predominantly in the (001) orientation such that the lead-iodide octahedra layers are parallel to the substrates [10].

The composition of the LPK samples was further confirmed by analysis of their optical absorbance (UV–vis) and steady-state photoluminescence (PL) spectra as seen in figure 2 for the film on quartz. The absorption spectrum displays a strong and sharp excitonic absorption with a peak at 514 nm and a continuum absorption feature which is clearly distinguished from the exciton peak starting around 490 nm. The absorbance of the LPK sample on glass displays similar features (figure S3).

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The PL spectrum of the sample on quartz displays a sharp emission line at 530 nm, which corresponds to the excitonic emission [10, 59, 60]. For the sample on glass (figure S4), this peak is shifted to 522 nm and has a tail towards longer wavelengths. This shift is attributed to the thicker LPK film exhibiting increased photon reabsorption [61].

To determine if there are any differences in trap state population between the two films, we performed time-resolved PL measurements as described in detail in section S4 in the SI. The results reveal that the PL decays for both films have similar average lifetimes and thus indicate that the trap state populations are similar and that the films should therefore be comparible in other time-resolved measurements.

To investigate free charge-carriers in (PEA)₂PbI₄, we performed optical pump terahertz probe (OPTP) measurements. The photoinduced differential transmission of THz radiation (∆T/T) is proportional to the photoconductivity (σ) of the film, which is in turn linearly dependent on the charge-carrier density (n) and the effective charge-carrier mobility (μ) [10, 62].

As excitons are an uncharged, bound species, they only strongly absorb THz radiation under resonance conditions [62, 63]. For (PEA)₂PbI₄, the exciton binding energy range is between 193–270 meV [14, 18–20, 64–67], and the corresponding resonant feature would be situated between 46.6–65.3 THz. With a bandwidth of 2–16 THz, Burgos-Caminal et al were able to observe the low frequency tail of a Lorentzian line-shape towards the higher frequency end of their measurement window which they associated with this exciton resonance [19, 55]. As we are probing with a narrower and much lower bandwidth at a relatively low frequency (0.5–3.5 THz), we expect that the excitonic feature would be far enough away from this frequency range to not observe any resonant excitonic contributions.

To confirm that we are only observing free-charge carriers and not excitons, photoconductivity spectra between 0.5–3.5 THz were obtained at excitation wavelengths of 410 and 514 nm (see SI, figure S6). In both cases, the real and imaginary photoconductivity components have a free charge-carrier (Drude-like) response that has been observed in the THz photoconductivity spectra of perovskites including MAPbI₃ and LPKs [23, 55, 68]. There is also no evidence of a resonance feature, confirming that we are likely probing only a free-carrier response.

Having confirmed that our OPTP measurements are probing free carriers, we proceed by estimating the effective mobilities and analysing the charge-carrier dynamics after photoexcitation into both continuum (λ_exc = 410 nm) and excitonic states (λ_exc = 514 nm). Figure 3 shows the decay dynamics when (PEA)₂PbI₄ is excited at 410 nm and 514 nm at fluences ranging from 36–182 μJcm⁻². Additional representations of these dynamics as well as dynamics obtained following photoexcitation at 450 nm are included in section S6 in the SI.

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From the differential absorption amplitude (-∆T/T) immediately after photoexcitation (i.e., before charge-carrier recombination has occured), the effective charge-carrier mobility φμ can be calculated as described in section S7 in the SI. For excitation at 410 nm, it can be assumed that φ ≈ 1 at early times since the excitation energy is well above the excitonic absorption feature and an equilibrium between excitions and free carriers has not yet been established [20]. For (PEA)₂PbI₄ excited at 410 nm, we find the effective charge-carrier mobility to be 5.8 ± 0.9 cm² V⁻¹s⁻¹ , consistent with other reports in literature [10, 20, 23]. Similarly, photoexciation at 450 nm yeilds a charge-carrier mobility value of 5.4 ± 0.2 cm² V⁻¹s⁻¹

An effective charge-carrier mobility of 1.9 ± 0.5 cm² V⁻¹s⁻¹ is estimated from the decay traces following resonant exciation to the excitonic states at 514 nm. This value is somewhat suprising considering that the free-carrier concentration should be much lower here and closer to its equilibrium value, so we would expect the charge-carrier mobility to be significantly lower. This discrepency between the expected mobility and the what is predicted by the Saha equation was also mentioned in the work by Motti et al and was attributed to a sustained, long-lived free charge-carrier population being present [23].

One possibility to explain the presistence of free-carriers is that the photoexcitation density is above the Mott density where excitons dissociate due the presence of a reduced exciton binding energy caused by screening effects [19]. The Mott criteria (n_c ) is expressed as follows [69]:

$$\begin{eqnarray}&&{n}_{c}={\left(\displaystyle \frac{0.26}{{a}_{B}}\right)}^{3}\end{eqnarray} \tag{ 1 }$$

where a_B is the exciton Bohr radius. Using a_B = 1.35 nm [70]. We estimate the Mott transition to be 1.4 × 10¹⁹ cm⁻³ which is higher than the highest carrier density of 2.24 × 10¹⁸ cm⁻³ used in this study (see section S8 in SI). We can therefore confirm that we are exciting below the Mott transition.

Another possible explaination for the high mobility value obtained at 514 nm excitation can be understood, however, by considering the overlap between the broadband spectrum of the ultrafast laser pulse and the contributions to the steady-state absorption spectrum from the continuum and excitonic states as determined by a fit to the Elliot model (section S9 and S10 in the SI), which is typically used in literature to model the aborption spectrum in semiconductor materials including halide perovskites (17 {Singh, 2016 #121, 20, [71–73].

With 410 nm excitation, the OPTP decay traces exhibit a fast relaxation component in the first 1–2 ps followed by slower decay on a 0.4 ns (400 ps) timescale (figure 3(a)). Previous studies on materials with high exciton binding energies such as metal dichalcogenides, quantum dots and LPKs have attributed similar fast decays to carrier cooling and exciton formation [19, 74–80]. Comparing the results of exciting into the continuum at 410 nm and at the excitonic resonance at 514 nm leads us to a similar assignment for PEA₂PbI₄. As shown in figure 3(b), fast dynamics are not observed when exciting at 514 nm, similar to the results seen by Burgos-Caminal et al obtained using a higher frequency THz probe [19, 54]. At 514 nm, carriers are resonantly excited into the excitonic sub-band and the minority of free carriers is generated near the band edge. The slower, longer-lived features of the 514 nm and 410 nm kinectics exhibit similar timescales, suggesting a similar recombination process for a minority population of free charge-carriers. To highlight these longer timescale dynamics, normalised decay traces for which the first picosecond has been removed are shown in figures 3(c) and (d). These dynamics also exhibit a fluence dependence, where the total recombination is faster at higher fluences, which indicates the presence of higher order processes such as bimolecular and Auger recombination [10].

To further understand the charge-carrier recombination mechanism in (PEA)₂PbI₄, we fitted the decay traces after the first 2.5 ps to the following charge-carrier recombination equation (see section S11 in SI):

$$\begin{eqnarray}&&\displaystyle \frac{d{n}_{FC}}{dt}=-{\varphi }^{2}{k}_{3}{{n}_{FC}}^{3}-\varphi {k}_{2}{{n}_{FC}}^{2}-{k}_{1}{n}_{FC}\end{eqnarray} \tag{ 2 }$$

where n_FC is the free charge-carrier density, k₁ is the monomolecular recombination rate constant, k₂ is the bimolecular recombination rate constant, k₃ is the Auger recombination rate constant and, φ photon-to-charge-branching ratio. We then extrapolate the fits to t = 0 ps and fit the residuals to a single exponential to obtain the exciton formation rate constant k_FC-Ex . The results of the global fits are given in figures S10 (a) and (b).

The kinetics of the OPTP traces at 410 nm and 514 nm yielded effective bimolecular recomination rate constants (φk₂) of 9.38 × 10⁻⁹ cm³ s⁻¹ and 8.86 × 10⁻⁹ cm³ s⁻¹. This value agrees well with literature as bimolecular reocombination rates constants for (PEA)₂PbI₄ have been shown to vary between 2 × 10⁻⁸ cm³ s⁻¹ and 9.0 × 10⁻⁹ cm³ s⁻¹ [10, 20]. The similarity in bimolecular rate constants further confirms the similar kinetics between the traces at 410 and 514 nm from 2.5 ps onwards as seen in the figure 3(b). Additionally, a k_FC-Ex value of 3.16 × 10¹² s⁻¹ (lifetime of 316 fs) was attained. Similar values of around 350 fs have been reported for excitonic materials such as metal dichalcogenides [74–79].

We complement our OPTP measurements with transient absorption spectroscopy (TAS) experiments. As with OPTP, the sample was photoexcited at 410 and 514 nm (exciting into the continuum and excitonic bands, respectively), with fluences ranging from 36–182 μJ/cm² (figure 4). The change in absorption of the white light probe (in mOD) was measured as a function of delay time between the optical pump and white light probe, enabling the resolution of spectral features in the 450–570 nm region.

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Representive transient absorption spectra for both continuum (410 nm) and excitonic (514 nm) excitations are shown in figure 4(a) (see figures S11 and S12 for the rest of the kinetic transients). In both cases, the transient spectra are composed of two positive features on either side and a stronger negative feature centered at 519 nm. The peak centroid of the negative feature (approminately at 519 nm) is in good agreement with the position of the excitonic feature of the same sample in the steady-state UV–vis (approximately 514 nm), strongly suggesting it originates from bleaching of the the excitonic absorption, in agreement with previous reports [12, 30, 37, 38, 44]. It is thus assigned to a ground state bleach (GSB). The two positive excited-state absorption peaks—which are assigned to excited state absorption (ESA) features and labeled ESA1 and ESA2— exhibit peaks at at roughly 490 nm and 525 nm, respectively.

While the assignment of the GSB to the excitonic absorption is well understood [51], the assignment of the ESA bands is more controversial with features generally being attributed solely to excitons and with no general consensus on the recombination mechanisms. For example Giovanni et al assigned the origins of the ESA2 band to be due to excitons and ultrafast (sub-ps) exciton spin relaxation [51]. Fu et al assigned the ESA components in (PEA)₂PbI₄ to excitonic contributions but instead proposed that its cause was exciton-induced linewdith broadening [81]. Furthermore, a similar conclusion was given by Ni et al in which the ESA features in BAPbI₄ (BA = butylammonium) and HAPbI₄ (HA = hexylammonium) were also described to have exciton origins, but from exciton–phonon coupling [49]. Deng et al, however, report that both free carriers and excitons contribute to the observed spectra. In this work, they ascribed the ESA features in BAPbI₄ to be due to both the excitonic (1st and 2nd order) and free carrier (Auger recombination) populations [52]. Collectively, these studies emphsize the lack of agreement in the assignment of bands and the need for further investigation into other species that may be contributing to the kinentics.

Although both the ESA1 and ESA2 bands are present under continuum and excitonic excitation (see figures S11 and 12), the ESA2 band is markedly smaller with excitonic excitation than when the continuum is excited. This difference suggests that the ESA2 band is dominated by free carrier effects but also highlights the difficulty with assigning these bands to any one species since a significant amplitude is obsereved when the free carrier concentration should be low.

Futhermore, the kinetics of the ESA2 band contain an oscillatory component that is overlaid upon the relatively slower decay profiles (figure S12 (b)). We find this oscillatory component to be significantly more pronounced when the 514 nm excitonic band of the perovskite is excited in comparison to the continumm excitation at 410 nm. Oscillations of this nature have been observed previously in both 3D perovskites such as MAPbI₃ and quasi-2D LPKs such as (PEA)₂PbI₄ and assigned to carrier phonon coupling [82]. In addition, n = 1 LPKs have recently been shown to be capable of increased collective lattice motion as compared to their 3D and higher n-valued counterparts [83].Through Fourier analysis of the oscillations, which is described in the SI section S13, we observed frequency components consistent with our measurements of phonon modes using THz-TDS and previously reported Raman modes obtained for single-crystal (PEA)₂PbI₄ [51]. Given the strong enhancement of the mode under excitonic stimulation, we believe this feature likely arises due to exciton–phonon coupling.

To further analyse the kinetics, care must be taken to ensure that the kinetics at any particular wavelengths do not contain contributions from multiple bands as the features are broad and overlapping. To facilitate kinetic modelling of the TAS data to study the underlying dynamics, it is therefore necessary to deconvolute these overlapping spectral features.

Previous studies have attempted to disentangle transient spectra by fitting the GSB and ESA features with a series of fixed Lorentzian functions [44]. Whilst this approach yields approximately correct solutions for the amplitudes of the various transient features, it does not account for changes in the lineshape or peak position of the bands, which can result from processes such as carrier thermalization or bandgap renormalization [84, 85]. To better track the true amplitude of the spectral components within the material, we also use a set of three Lorentzian functions as seen in figure 4(b), but allow the positions and linewidths of the GSB and ESA1 bands to shift over time (See figure S16 in the SI).

The kinetics extracted from the modelling are presented in figures 4(c) and (d). We find that the kinetics for the ESA2 band for both the continuum (λ_ex = 410 nm) and excitonic excitation (λ_ex = 514 nm) are both biphasic in shape with a fast relaxation component in the first 1–2 ps accompanied with a slower component on a 0.2 ns (200 ps) timescales. These dynamics are similar to the results of the OPTP kinetics presented previously and suggest a link to free carriers in which the fast decay is associated with charge-carrier cooling and exciton formation and the slower component was assigned to bimolecular recombination.

Furthermore, the oscillatory behaviour observed in the single-wavelength kinetics, shown in the SI (section S12) is very weak in the amplitude of the reconstructed ESA1 band, instead manifesting itself as an oscillation in the peak centroid (figure S16).

To further explore the link between the ESA2 band and the free carrier population we fit the kinetics of the ESA2 band using equation (1) and an extrapolation, which we have previously used to model free carrier behaviour in our OPTP data (see section S16 for details). The results of the fittings for k_FC-Ex and k₂ for the ESA2 band are presented in table 1. For k_FC-Ex, we find excellent agreement with the OPTP data as both rates are on the order of 10¹² s⁻¹ (i.e. < 400 fs). Additionally, we find the effective bimolecular rate constants (φk₂) of the ESA2 band TAS and OPTP at λ_exc = 514 nm and λ_exc = 410 nm to be statistcally close (between 1.9–9.4 × 10⁻⁹ cm³s⁻¹). The results for k₂ fitting further supports the conculsion that dynamics observed for the ESA2 band in TAS can be mostly attributed to free carriers rather than excitons.

Next, we move our attention to the ESA1 and the GSB bands. Figure S19 shows the TA colour map of (PEA)₂PbI₄ for the first 10 ps at 410 nm with a fluence of 146 μJ/cm². In this figure, we plot the peak position as a function of time and observe the ESA1 band redshifts on a sub-2ps timescale. This behaviour agrees with previous studies on other 2D material such as MoS₂ in which they assign this redshift to carrier thermalisation and bandgap renormalisation [39]. Using the fitting model discussed earlier in section 3.4, we are now able to model the peak dynamics of the of the ESA1 band. The results of this fitting are displayed in figure S14 (b).

When looking at extracted kinetics for the ESA1 and GSB in figures S20–24, there is no fluence dependence present, which is in contrast to obeservations for the ESA2 and OPTP data. This suggests that there is another species that is dictating the dynamics instead of free carriers and further supports the excitonic origins of the ESA1 and GSB bands. Due to this lack of fluence dependence, using the same fitting for the ESA2 band for the ESA1 and GSB bands would not be appropriate. We instead fit the dyanamics using a biexponential model as it provides the best fit without over parametising, and the results of the fit are shown in table 2 (for the full data set and explanation of the fitting see section S16 in the SI). We find that the time constant t₁ for the ESA1 (14.20 ps ± 0.85) and the GSB bands (14.66 ps ± 0.86) at λ_exc = 410 nm to be statistically close (table 2). We find this is also the case for the time constant t₂ at λ_exc = 410 nm (105.52 ps ± 9.71 versus 107.43 ps ± 9.67) for the ESA1 and GSB, respectively. Similar results are obtained at λ_exc = 514 nm (see section S16 in the SI). This similarity in the ESA band and the GSB has beeen observed previously in other layered perovskites such as (BA)₂PbI₄ (BA = butylammonium) [86].

Following from this, based on the lifetimes and behaviour of the GSB and ESA1 dynamics we tentatively assign t₁ exciton–exciton annihiliation, which has been reported to be ∼10 ps in other LPK material including (PEA)₂PbI₄ [51, 87]. The t₂ rate constant would be attributed to monomolecular (geminate) recombination which has a reported lifetime of ∼100 ps [51].

Overall, this kinetic analysis further supports that OPTP data is useful for disentangling excitonic and free carrier contributions in the TAS data. Figure 5 gives an overview of the various processes when the LPK is photoexcited at 410 nm. In the first ∼150–350 fs, free carriers are photoexcited into the continuum and then relax to band edge and form excitons. The excitons then relax further via exciton–exciton annihilation (t₁ ∼ 11 ps) and monomolecular (geminate) recombination (t₂ ∼ 83 ps) processes. Simultaneously, there are some free carriers that do not form excitons and recombine through bimolecular recombination at a rate on the order of ∼10⁹ cm³s¹.

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