Generalized Method to Extract Carrier Diffusion Length from Photoconductivity Transients: Cases of ${\mathrm{BiVO}}_{4}$, Halide Perovskites, and Amorphous and Crystalline Silicon

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Generalized Method to Extract Carrier Diffusion Length from Photoconductivity Transients: Cases of ${BiVO}_{4}$ , Halide Perovskites, and Amorphous and Crystalline Silicon

Markus Schleuning, Moritz Kölbach, Fatwa F. Abdi, Klaus Schwarzburg, Martin Stolterfoht, Rainer Eichberger, Roel van de Krol, Dennis Friedrich, and Hannes Hempel

PRX Energy 1, 023008 – Published 29 September 2022

Abstract

Long diffusion lengths of photoexcited charge carriers are crucial for high power conversion efficiencies of photoelectrochemical and photovoltaic devices. Time-resolved photoconductance measurements are often used to determine diffusion lengths in conventional semiconductors. However, effects such as polaron formation or multiple trapping can lead to time-varying mobilities and lifetimes that are not accounted for in the conventional calculation of the diffusion length. Here, a generalized analysis is presented that is valid for time-dependent mobilities and time-dependent lifetimes. The diffusion length is determined directly from the integral of a photoconductivity transient and can be applied regardless of the nature of carrier relaxation. To demonstrate our approach, photoconductivity transients are measured from 100 fs to 1 µs by the combination of time-resolved terahertz and microwave spectroscopy for ${BiVO}_{4}$ , one of the most studied metal oxide photoanodes for photoelectrochemical water splitting. The temporal evolution of charge carrier displacement is monitored and converges after about 100 ns to a diffusion length of about 15 nm, which rationalizes the photocurrent loss in the corresponding photoelectrochemical device. The presented method is further validated on $a − Si : H$ , $c − Si$ , and halide perovskite, which underlines its potential to determine the diffusion length in a wide range of semiconductors, including disordered materials.

Received 14 April 2022
Revised 2 August 2022
Accepted 12 August 2022

DOI:https://doi.org/10.1103/PRXEnergy.1.023008

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

Published by the American Physical Society

Physics Subject Headings (PhySH)

Carrier generation & recombination Diffusion Electrical conductivity Optical conductivity Photoconductivity Photocurrent

Amorphous materials Amorphous semiconductors Complex materials Energy applications Energy materials Thin films

Microwave techniques Terahertz spectroscopy

Condensed Matter, Materials & Applied Physics

Authors & Affiliations

Markus Schleuning ^1,2, Moritz Kölbach ¹, Fatwa F. Abdi ¹, Klaus Schwarzburg¹, Martin Stolterfoht³, Rainer Eichberger¹, Roel van de Krol ^1,2, Dennis Friedrich ^1,*, and Hannes Hempel ^4,†

¹Institute for Solar Fuels, Helmholtz-Zentrum Berlin für Materialien und Energie GmbH, Hahn-Meitner-Platz 1, 14109 Berlin, Germany
²Institut für Chemie, Technische Universität Berlin, Straße des 17. Juni 124, 10623 Berlin, Germany
³Institute of Physics and Astronomy, Universität Potsdam, 14476 Potsdam-Golm, Germany
⁴Department Structure and Dynamics of Energy Materials, Helmholtz-Zentrum Berlin für Materialien und Energie GmbH, Hahn-Meitner-Platz 1, 14109 Berlin, Germany

^*friedrich@helmholtz-berlin.de
^†hannes.hempel@helmholtz-berlin.de

Popular Summary

Solar cells, photodetectors, and photoelectrochemical cells rely on the generation of charge carriers by photoexcitation and the subsequent transport to the contacts or to the reaction site. The average distance over which such carriers can transport by diffusion before they are lost is the diffusion length. Hence, the diffusion length is a widely used predictor for the development of photoabsorbers. However, the conventional calculation of the diffusion length assumes constant mobility and a constant lifetime, which is not valid for many materials due to the complexity of the transport and loss processes. In this work, the authors present a generalized analysis that yields the diffusion length directly by integrating the measured transient photoconductivity and that can be applied independent of the type of carrier relaxation. This analysis is validated by numerical simulations and demonstrated on ${BiVO}_{4}$ , a material widely used as a photoanode for solar water splitting, as well as on the photovoltaic materials $a − Si : H$ , crystalline $Si$ , and lead halide perovskite. To this end, the authors measure photoconductivity transients by contactless terahertz and microwave spectroscopy. The techniques cover different time windows and in combination are able to cover a range from 100 fs up to several µs. The analysis allows the authors to monitor the diffusion of the charge carriers over time, obtain the final diffusion length, and predict the photocurrent loss in the corresponding devices. The presented fundamental relationship facilitates future determinations of carrier diffusion lengths, as it is simpler to apply and more general than previous approaches. It can guide a highly predictive characterization of emerging semiconductors and identify losses in state-of-the-art devices.

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Vol. 1, Iss. 2 — September - November 2022

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Figure 1
Confirming the generalized diffusion length by numerical modeling. (a)–(c) Evolution of Gaussian carrier distributions modeled numerically with Eq. (1) for (a) mobility decaying with a decay time, $τ_{μ}$ , of 10 ns from an initial value of 1 cm² V⁻¹ s⁻¹; (b) constant mobility of 1 cm² V⁻¹ s⁻¹ and a lifetime, $τ$ , of 10 ns; and (c) mobility decaying with a decay time of 100 ps from an initial value of 1 cm² V⁻¹ s⁻¹ and a lifetime of 10 ns. Final distribution of the recombined carriers is estimated at t = 100 ns, $≫ τ$ , and is shown in red. (d)–(f) Normalized transients of the mobility (blue line), sheet carrier concentration (red line), and sheet conductivity (thick gray line). (g)–(i) Evolution of the mean absolute displacement. Precise numerical value (red line) is, in general, between the values of analytical Eq. (8) with the factor $\sqrt{1 / 2} < a < \sqrt{2 / π}$ . For decaying mobility and without recombination (case 1), $a = \sqrt{2 / π}$ . For constant mobilities and recombination (case 2), it converges to a diffusion length with a factor $a = \sqrt{1 / 2}$ .
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Figure 2
Diffusion and recombination of photogenerated charge carriers in ${BiVO}_{4}$ . Photoconductivity transients of a PLD-deposited ${BiVO}_{4}$ thin film on fused silica are measured by (a) OPTP, (b) ST- and OC-TRMC, and (c) TRMC. Mobilities, $μ_{Σ}$ , and decay times, $τ$ , of the individual techniques are obtained from the initial amplitudes and from modeling the initial decays with $\exp (- t / τ)$ , respectively. (d) Superimposed measured photoconductivity transients, $Δ σ_{Σ} (t)$ , and a parameterized joint transient as a guide for the eye. (e) Temporal evolution of the decay time, $τ (t)$ , obtained from the parameterized joint transient. (f) Temporal evolution of the mean absolute displacement, $⟨ x ⟩ (t)$ , of photogenerated charge carriers in ${BiVO}_{4}$ obtained from the OPTP- and TRMC-derived photoconductivity transients by Eq. (10). Value approaches a diffusion length of about 15 nm at sufficiently long times (>100 ns).
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Figure 3
Rough estimation of photocurrent collection by diffusion. Modeled profile of charge carrier generation, G, in a ${BiVO}_{4}$ thin film based on its absorption coefficient and the AM1.5G sun spectrum. Total absorbed sunlight generates a photocurrent $J_{\max} = q \int G d x$ . Photocurrent collected by diffusion, $J_{D}$ , is estimated by the carriers that are generated within the diffusion length L_D.
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Figure 4
Charge carrier diffusion and recombination in $a − Si$ . (a) Photoconductivity transients measured by OPTP, OC-TRMC, and ST-TRMC. (b) Temporal evolution of the mean displacement of photogenerated charge carriers obtained by Eq. (10) converges to a diffusion length L_D.
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Figure 5
Application and validation for simple well-known materials. (a) Photoconductivity transients of a single-crystalline silicon wafer, a lead halide perovskite thin film, as well as $a − Si$ and ${BiVO}_{4}$ measured by OPTP, ST- or OC-TRMC, and conventional TRMC. (b) Transient of $c − Si$ appears approximately monoexponential with a fitted lifetime of 322 µs, and the mobility is taken from the initial amplitude. Conventional analysis of the diffusion length yields 900 µm. (c) Temporal evolution of the mean displacements of photogenerated charge carriers obtained by Eq. (10). They converge to finite diffusion lengths that are in line with conventional analysis.
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PRX Energy

Generalized Method to Extract Carrier Diffusion Length from Photoconductivity Transients: Cases of BiVO4, Halide Perovskites, and Amorphous and Crystalline Silicon

Markus Schleuning, Moritz Kölbach, Fatwa F. Abdi, Klaus Schwarzburg, Martin Stolterfoht, Rainer Eichberger, Roel van de Krol, Dennis Friedrich, and Hannes Hempel

PRX Energy 1, 023008 – Published 29 September 2022

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Generalized Method to Extract Carrier Diffusion Length from Photoconductivity Transients: Cases of ${BiVO}_{4}$ , Halide Perovskites, and Amorphous and Crystalline Silicon