Applicability of the thin-film approximation in terahertz photoconductivity measurements

The sun is the largest energy source available to humankind and research towards harvesting this energy is crucially important. One feasible option is found in water-splitting dye sensitized photoelectrochemical cells (WS-DSPECs) that use solar energy to renewably produce chemical fuels.^1–6 A critical component of WS-DSPECs is the photoanode material, which consists of a wide bandgap metal oxide material that has a photoabsorbing dye attached to its surface. In the dark, the metal oxide material is an insulator, yet when the dye molecule is photoexcited, it can transfer its excited state electron into the conduction band of the metal oxide material.^4,7 The metal oxide allows the transport of the photogenerated charges, which travel through the porous nanoparticulate metal oxide film until being transported to a dark cathode to facilitate a proton reduction reaction to generate H₂. Understanding this transport mechanism, and in particular, carrier mobility and lifetime, is crucial for the optimization of WS-DSPECs.

Terahertz (THz) radiation has proven to be an excellent tool for studying photoexcited electron dynamics in photoanode materials.^7–13 THz radiation provides a noncontact electrical probe that monitors the concentration of free carriers in the system and their associated charge mobility. Since the THz electric field is measured coherently, both amplitude and phase information in the frequency-domain is obtained. As a result, the complex-valued, frequency-dependent refractive index of the photoexcited sample, n_p, at a particular time delay, t_pump, after photoexcitation can be determined. Previous publications have used the thin-film approximation.^7–11 This approximation was originally designed for microwave spectroscopy on thin superconductive layers.¹⁴ Application of this formula to THz studies of photoexcited layers is common in the field. In this letter, we compare this approximation with non-approximated, computationally expensive calculations of n_p.

For this study, two different photosensitizers were anchored on mesoporous SnO₂, 5-4(methoxycarbonylphenyl)-10,15,20-tris(2,4,6-trimethypheyl)-porphyrin (referred to as porphyrin) and (4,4′-diphosphonato-2,2′-bipyridine)bis(2,2′-bipyridine)ruthenium(II) bromide (which is referred to as RuP). Their structures are shown in the supplementary material. These samples were prepared following previously published techniques,^15,16 and a screen-printed paste of SnO₂ was prepared as described previously.^3,4,10 The deposition of the layer and sensitization was repeated one, three, and five times, resulting in a total layer thickness of 3.3 µm ± 0.3 µm, 8.2 µm ± 0.3 µm, and 12.6 µm ± 1.0 µm, respectively.

The absorption spectra are shown in Fig. 1. The two sensitizers are markedly different, with porphyrin exhibiting a strong Soret band absorption on the order of 10⁵ M⁻¹cm⁻¹, while the molar extinction coefficient for the metal to ligand charge transfer band in RuP is on the order of 10⁴ M⁻¹cm⁻¹.^17,18 At the wavelength of our pump laser (400 nm), this difference in absorption results in a pump beam penetration depth of ∼1 µm for porphyrin and of ∼15 µm for RuP, which is larger than 3.3–12.6 µm of thick SnO₂ films.

The use of these two photosensitizers allows us to access two distinct physical scenarios: The RuP-sensitized SnO₂ films generate charge carriers throughout the full thickness of the films, while the porphyrin-sensitized SnO₂ films exhibit a very thin photoexcited layer on top of a thicker layer of a nonphotoexcited porphyrin/SnO₂ film (see Fig. 2).

The THz spectrometer has been described in detail in previous publications.^10,19 In short, the output of a 1 kHz 4 mJ amplifier at 800 nm is split into a pump and a THz generation beam. The generation beam is frequency doubled to 400 nm for photoexcitation; the other beam is used to generate THz radiation via two-color air plasma. Based on previous studies, we chose a pump delay of 1350 ps.¹¹ This delay time is significantly larger than the injection time of ≈100 ps, and hence we can assume steady-state behavior of the mobile charges in the conduction band. The measured THz pulses at a particular t_pump are Fourier transformed, resulting in the frequency-dependent THz field E_pump(ω, t_pump). Additionally a THz spectrum without photoexcitation is measured and used as a reference, E_ref(ω), to calculate the spectral transfer function $Tpump(ω,tpump)=Epump(ω,tpump)Eref(ω)$⁠.

Figure 2 presents a schematic diagram of the samples. The reference sample is not photoexcited, in the RuP-sensitized sample, the entire film is photoexcited, and in the porphyrin-sensitized sample, only a fraction of the film is photoexcited. n_a, n_q, n_n, and n_p are the complex-valued refractive indices of air n_a (ν = 1 THz) = 1, fused quartz n_q (ν = 1 THz) = 1.95, and non-photoexcited SnO₂ n_n (ν = 1 THz) = 2.2 (see supplementary material), and photoexcited SnO₂, respectively. For the presented calculations, frequency resolved measured refractive indices are used. d_n is the thickness of the SnO₂ layer and d_p is the thickness of the excited layer, which is equal to d_n in the case of RuP. In the case of porphyrin sensitization, $dn′$ is the thickness of the non-photoexcited material and $dp′$ is the thickness of the photoexcited material. It is seen that $dn′+dp′=dn$ (and also d_p). The $FPijk$ terms account for the Fabry-Perot reflections within a layer of material “j” sandwiched between materials of “i” and “k.” Furthermore, we define the Fresnel reflection coefficient as $rji=ni−njni+nj$ and t_ij = 1 + r_ij and the propagation function as $Pdini=e−idinik0$⁠, with the wavevector of vacuum k₀.

The quartz substrate is thick enough such that no internal reflections from it are collected. These considerations allow us to define the complex transfer functions of a known input field E₀ for photoexcited or non-photoexcited SnO₂. The simplest case is non-photoexcited SnO₂ to give the transmission T

The equation for the RuP-sensitized sample is identical except that the quantities for the photoexcited material are used

Finally, the most general case of porphyrin-sensitized SnO₂ that has both photoexcited and non-photoexcited layers is given by

All measurable Fabry-Perot reflections are captured within our 6.5 ps THz transient; therefore, the infinite series of reflections reduces to the form shown in the second line of Eq. (4). This infinite sum converges for non-gain natural materials. To calculate the refractive index n_p, Eq. (3) is normalized to a reference measurement through non-photoexcited SnO₂

with a similar equation for the RuP-sensitized case where the entire SnO₂ layer is photoexcited. Inserting the Fresnel coefficients and propagation functions yields

Alternatively, if the penetration-depth for the pump beam is larger than the layer thickness (as is the case with RuP/SnO₂), the complete SnO₂ film is photoexcited and the transfer function is described by

Equations (6) and (8) are numerically solved to extract n_p. This equation is frequency-dependent and complex-valued. The refractive indices are obtained from the generalized permittivity of the photoexcited sample, the lattice permittivity of the non-photoexcited sample, and the photoconductivity of the photoexcited sample. The lattice permittivity, $ϵlattice≈nn2$⁠, is assumed to be unchanged upon photoexcitation, and it is also assumed that the sample is non-magnetic

This expression relates the change in the sample's refractive index to a change in the conductivity of the film, σ(ω, t_pump). Numerically solving these equations is computationally expensive. Therefore, several previous publications have expanded upon the expression derived by Glover and Tinkham¹⁴ in 1957 for a thin superconducting material, and this is referred to as thin-film approximation.^{10,11,20–22} If the photoexcited layer fulfills n_pk₀d_p ≪ 1 and k_pk₀d_p ≪ 1, the conductivity is given by^21,23

with Z₀ = 376.7 Ω being the impedance of free space and T(ω, t_pump) the spectral transmission at a given pump delay t_pump.

While this equation was originally developed for microwave studies of thin-layer superconductors on insulating substrates, it has been widely used for terahertz spectroscopy of photoexcited materials.^{10,11,20–22}

The primary question addressed in this letter is: what value is considered to be a value much less than 1? Is it 0.5? Or is it 0.1? Or is it 0.01? Therefore, we compare the measured photoconductivity of several samples, ranging from n_pk₀d_p = 0.04 to 0.5, using the thin-film formula [Eq. (10)] and the non-approximated Eqs. (6) and (8).

From Fig. 1(a), it is apparent that porphyrin-sensitized SnO₂ has a very high extinction coefficient at 400 nm. The optical penetration length was determined using an integration sphere to be d_p ∼ 1 µm, which results in the SnO₂ film not being fully photoexcited, as seen in Fig. 2. Therefore, an additional 2.3 µm (1 layer) to 11.6 µm (5 layers) of nonphotoexcited SnO₂ remains with a refractive index of n_n.

The calculated photoconductivity of the one, three, and five layer porphyrin-sensitized SnO₂ samples using both Eqs. (6) and (10) is plotted in Fig. 3(a). The dashed lines depict the results of using the thin-film approximation [Eq. (10)], and the solid lines represent the non-approximated numerical solution [Eq. (6)]. The percent difference between the two methods is plotted in Fig. 3(b). The results agree within 1% for the real conductivity, and within 5% for the imaginary conductivity.

Given the small absorption depth for the porphyrin/SnO₂ samples, the total number of carriers is independent of the layer thickness, and is defined by the pump fluence (which remains constant for all of the measurements). As the overall layer thickness increases, the same number of photoexcited free electrons is generated, and it is seen that the photoconductivity for all three samples is nearly the same.

The conductivity is measured 1350 ps after photoexcitation. This allows the carriers to diffuse into the nonphotoexcited layer which reduces the carrier density in the thickest sample with respect to the thinnest sample. The lower electron density reduces the electron-electron scattering probability.¹¹ As a result, the carrier mobility and lifetime is higher in the thicker layer, and results in an increase in conductivity with increasing layer thickness. While there are small deviations between the two calculations on the order of 1%–5% in the imaginary conductivity, the thin-film approximation nicely reproduces the photoconductivity calculated from the non-approximated method.

The RuP-sensitized SnO₂ sample has an overall optical absorption length of ∼15 µm, which is larger than the thicknesses of the SnO₂ films (3.3–12.6 µm). As seen in Fig. 1(b), even the five layer film does not reach an absorptance higher than 65% when excited at 400 nm. Therefore, the SnO₂ layer will be fully photoexcited and described by the complex photoexcited index of refraction n_p. The photoconductivity calculations of the one, three, and five layer RuP/SnO₂ samples are plotted in Fig. 4(a). The dashed line depicts the results of using the thin-film approximation [Eq. (10)] and the solid line represents the non-approximated numerical solution [Eq. (8)].

A comparison of Figs. 3 and 4 shows that the photoconductivity in the short absorption length case (porphyrin/SnO₂) is about an order of magnitude higher than the RuP/SnO₂ sample. Furthermore, when the two and three layer cases are compared, we note a 15-fold higher conductivity for the porphyrin/SnO₂ sample, which is caused by the stronger total absorption. The RuP/SnO₂ sample is thinner than the absorption length so less than 1/e of the pump photons are absorbed. In the case of porphyrin/SnO₂, all photons are absorbed, resulting in an approximately 2–3 times higher number of carriers. These carriers are also confined in a significantly thinner layer (1 µm versus 12.6 µm), which causes a higher carrier density and therefore a higher conductivity.

There is a significant difference in the RuP/SnO₂ conductivities calculated using the non-approximated numerical solution versus the thin-film approximation. The percent difference between the two methods is plotted in Fig. 4(b). Similar to the case of porphyrin-sensitized SnO₂, the results for the real part of the photoconductivity agree within 5%, similar to previous results on GaAs.²⁴ However, a very significant change in the imaginary conductivity is noted. The difference between the thin-film approximation and the correct calculation for the thinnest sample with d_p = 3.3 µm ≈ λ/100 is −10%, for n_pk₀d_p ≈ 0.1. It increases with the total layer thickness to reach a maximum of −45% for the 12.6 µm sample, for n_pk₀d_p ≈ 0.5.

In conclusion, we have compared the widely used thin-film approximation with a numerical data-workup without any approximations. The difference between the two approaches is demonstrated for mesoporous SnO₂, sensitized with a dye that results in a short absorption length (porphyrin) or one that results in a long absorption length (RuP). The difference between the two methods is less than 5% if a strong photo absorber is used. In contrast, if a dye with a smaller extinction coefficient (RuP) is used, which thereby leads to a large absorption length, the discrepancy between the calculated conductivities is up to 45%. This large error in the approximation clearly demonstrates that the thin-film formula should be applied with care.

See supplementary material for the Matlab script used which can be freely downloaded from our homepage at https://THz.yale.edu, chemical structure of the used compounds, and the refractive index of non-excited mesoporous SnO₂.

This work was supported by the Office of Basic Energy Sciences, Division of Chemical Sciences, Geosciences, and Energy Biosciences, Department of Energy, under Contract No. DE-FG02-07ER15909 for the data collection and the National Science Foundation under Grant No. NSF CHE-CSDMA 1465085 for the data analysis, as well as by a generous donation from the TomKat Charitable Trust. We would like to thank Shin Hee Lee and Jianbing Jiang for providing us with the dyes for sensitization.