Distribution of Relaxation Times Based on Lasso Regression: A Tool for High-Resolution Analysis of IMPS Data in Photoelectrochemical Systems

Intensity-modulated photocurrent spectroscopy (IMPS) has been largely employed in semiconductor characterization for solar energy conversion devices to probe the operando behavior with widely available facilities. However, the implementation of IMPS data analysis to complex structures, whether based on the physical rate constant model (RCM) or the assumption-free distribution of relaxation times (DRT), is generally limited to a semi-quantitative description of the charge carrier kinetics of the system. In this study, a new algorithm for the analysis of IMPS data is developed, providing unprecedented time resolution to the investigation of μs to s charge carrier dynamics in semiconductor-based systems used in photoelectrochemistry and photovoltaics. The algorithm, based on the previously developed DRT analysis, is herein modified with a Lasso regression method and available to the reader free of charge. A validation of this new algorithm is performed on a α-Fe2O3 photoanode for photoelectrochemical water splitting, identified as a standard platform in the field, highlighting multiple potential-dependent charge transfer paths, otherwise hidden in the conventional IMPS data analysis.


DRT-Lasso algorithm description
The DRT analysis models the admittance ( ) of a system with the integral equation where is the frequency at which the admittance ( ) is measured, ( ) represents the unknown real-valued distribution over the times that we want to find, and = √−1 is the imaginary unit. From the physics of the system under investigation, it is reasonable to assume that the set of such that ( ) ≠ 0 is discrete, i.e. there are very few characteristic times that fully describe the admittance of the system. The goal is to find this set of characteristic times with great precision. Once a characteristic time is found, the corresponding value ( ) can be considered as its weight in describing the system. A positive (negative) weight means that the process related to that characteristic time increases (decreases) the photocurrent. Moreover, the weight can either be fully assigned to a single characteristic time using a Dirac distribution ( ) ( − ), or be distributed around using a Gaussian function with a physically reasonable standard deviation (see paragraph "Results and Discussion"). This problem will be tackled numerically with a discretization of the integral. The time range is subdivided in > evenly logarithmically spaced intervals [ , +1 ], so that Eq. S1 becomes for = 1, . . . , . Typically, = × and a reasonable value is = 10, meaning that the number of considered characteristic times is 10 times larger than the number the experimental frequencies. This results in a ( × ) complex linear system where ∈ ℂ × is the matrix ( , ) = 1/(1 + ) and ∈ ℂ is the vector with the data ( ( 1 ), . . . , ( )). An additional requirement for the vector ∈ ℝ N is to be sparse, i.e. with 'many' zero-entries.
The linear system is first split it in real and imaginary part: where the two matrices ′ and ′′ have entries and ′ and ′′ are, respectively, the real and imaginary parts of .
Then it is necessary to solve the following minimization problem where ∥ − ∥ denotes, as usual, the -norm. In addition to the least square problem, the regularization term || || 1 helps avoiding overfitting. Different choices can be made: in this work, we decide to introduce a Lasso regularization, as it is often used to achieve sparse solutions, penalizing solutions with a high number of non-zero entries 2 .
The parameter ∈ ℝ ≥0 weights how much regularization is introduced in the minimization problem. Some validation tests have been developed 2 for identifying the best regularization parameter but these value may depend also on the quality of the measured data 3 . In general, if is too small, the regularization term becomes negligible and the solution is overfitted; if it is too big ( > 1), the regularization becomes prevalent, resulting in a solution which is S3 identically zero. In the following, we will always use = 0.5, since this value represents a reasonable trade-off for the present data analysis.
The algorithm was implemented in Python (main code attached as supporting material) and it is available here 4 .

Physical approximations in the generalized RCM
In order to extract the height of the peaks that appears in the GL-DRT curves, it is necessary to rearrange Eq. 5 in a form of an admittance similar to Eq. S2, which allows the ( ) factor for every addendum to be identified, as follows: where = + is the fraction of accumulated minority carriers that recombine with majority carriers coming from the bulk. Apart from the normalization factor ℎ / , which is constant and does not depend on , Eq. S6 differs from Eq. S2 because of the factor (1 + ) −1 in front of the summation; however, this factor becomes appreciably different from 1 only at high frequencies, say ≳ 0.1. In this range, the summation vanishes because ≫ , leading to ≫ 1 and (1 + ) −1 ≈ 0. This situation reflects the fact that the summation represents the low-frequency limit of the admittance, which is dictated by the long (compared to ) characteristic times . It is therefore legitimate to approximate the second, low-frequency term of Eq. S6 by setting (1 + ) −1 ≈ 1: In Eq. S7 it is useful to set ( ) = ℎ / and ( ) = ( ) . If ≫ then = 1, meaning that all the minority carriers accumulated at the -th sites recombines before going into the solution. On the other hand, if ≪ most of the minority carriers escape recombination and pass in the solution.
As expected, Eq. S8 suggests that the total steady-state photocurrent ℎ (0) is given by the sum of the photocurrent that comes from every accumulation site. Eq. S8 can be also written by making explicit the efficiencies of the main optoelectronic processes involved in PEC systems, i.e. light harvesting efficiency ( ), charge separation efficiency ( ) 5 , transfer efficiency and external quantum efficiency ( ). This results in: and therefore ℎ = ( ) = ×

Ti-doped hematite photoanode preparation
The photoanode fabrication and structural characterization was made by adapting and combining various previously reported synthetic approaches. 6,7 Ti(IV) doped nanostructured hematite photoanodes (Ti:Fe2O3) were prepared on a fluorine-doped SnO2 (FTO) conductive glass by an hydrothermal approach. Briefly, the FTO glass was cleaned through ultrasonication in isopropanol, and then rinsed with deionized water. The fabrication of hematite nano-rod electrodes involves the deposition of a Ti(IV) doped iron oxide seed layer by dip coating (0,625 mm/s) the cleaned 2 mm thick FTO/glass slides (10 mm (wide) X 25 mm (long) in a Fe(III) oleate precursor containing 15 mM Titanium (IV) isopropoxide in order to obtain a 10 mm X 10 mm coated area. The dip coating solution was prepared following the procedure described by D. K. Bora. 7 The Fe(III)oleate layer was converted into hematite following a 30 minute heat treatment at 500 °C. Solvothermal synthesis was carried out in a teflon-lined stainless steel autoclave by using an aqueous precursor containing 0,91 M sodium nitrate (NaNO3, Carlo Erba Reagents), at a pH value of 1.5 adjusted with 6 M HCl, 0.136 M of ferric chloride (FeCl3 · 6 H2O, Alfa Aesar), 2.5 mM Ti2CN (Sigma-Aldrich) and a 5 % (v/v) ethanol (Carlo Erba Reagents). The seed-layered electrodes were inserted into the autoclave, lying with an angle of ca. 45° with respect to the vertical liner walls. Heating at 95 °C was applied for 4 h. A uniform layer of yellowish colour film (FeOOH) was formed on the electrodes. The FeOOH-coated substrates were washed with deionized water to remove weakly interacting residues from the hydrothermal batch before sintering in air at 550 °C for 1 h, during which conversion of FeOOH to Fe2O3 occurred. Finally, the resulting hematite thin films were modified by chemical bath treatment in a 0.2 M TiCl4 solution heated at 50 °C for 1 hour, followed by a final thermal annealing at 760 °C for 10 minutes affording the Ti-Fe2O3 electrodes used for this study.

Photoelectrochemical characterization: Linear Sweep Voltammetry (LSV) and Photoelectrochemical Impedance Spectroscopy (PEIS)
LSV and PEIS were performed in order to have a solid basis as a starting point for our analysis. The illumination condition adopted for these measurements was the same used for IMPS measurements, namely 5 mW/cm 2 at 470 nm. LSVs were performed in dark, light and chopped illumination with a scan rate of 20 mV/s. Results are reported in Fig. S1. The equivalent circuit used to fit the PEIS data is reported in Fig. S2a and was taken from the work by Bisquert et al. 8 The presence of a maximum in the capacitance Ctrap (associated to trapping states on the surface) centered at around 0.5 -0.6 VAg/AgCl is a characteristic feature of hematite photoanodes and indicates the accumulation of charge carriers at surface states before the transfer to the electrolyte, that takes place at higher applied potentials. In order to validate the use of this circuit model, it is useful to compare the quantity dI/dV calculated from the LSV (Fig. S2b) in light with , the total resistance of the system, which in this case is given by where , and , were extracted directly from PEIS. These quantities are in good agreement, especially in the potential range where the photocurrent is different from zero, confirming the good choice of the circuit model.