Steady-state microwave conductivity reveals mobility-lifetime product in methylammonium lead iodide

The recent emergence of solar cells based on hybrid halide perovskites, and related compounds, has been one of the most disruptive events in the field of solar energy for many years.^1–3 Despite being processable from solution at low temperatures (≤100 °C),^4,5 solar cells based on hybrid organic metal-halide, Pb-based compounds with the general perovskite formula ABX₃, exhibit certified power conversion efficiencies (PCEs) in-excess of those of polycrystalline silicon.⁶ These compounds possess optoelectronic properties that are, by many metrics, extraordinary.^3,7–18 They are strong absorbers in the visible and near-infrared portion of the solar spectrum,^11,19 have high charge-carrier mobilities,^15,20 exhibit carrier-diffusion lengths on the order of microns,^9,10 display an unusually low concentration^11,21 of energetically shallow¹⁵ electronic traps states, and demonstrate evidence of photon recycling.¹⁸ 

Understanding the optoelectronic properties of this class of semiconductors is an intense area of research^22,23 and is fundamental to determining their potential performance.²⁴ The power conversion efficiency of photovoltaic devices is intimately related to charge carrier mobilities and lifetimes of carriers in the semiconducting layer.²⁵ The mobility-lifetime product (μτ) of charge carriers helps to determine the fill factor of solar cells because of the importance for charge extraction after generation.^25–28 The charge carrier mobility of semiconducting materials can be evaluated using Hall effect measurements,²⁹ space charge measurements,³⁰ and measurements of thin-film transistors (TFTs).^31,32 Carrying out such measurements requires the formation of continuous thin films or growth of single crystals, and often involves inducing charge-carrier densities many orders of magnitude higher than those present in solar cells under typical operating conditions.³³ 

Charge carrier lifetimes are evaluated using a variety of spectroscopic techniques, such as time-resolved photoluminescence spectroscopy,^9,16 time-resolved microwave conductivity,^15,34–36 transient absorption spectroscopy,^37,38 and THz spectroscopy.^20,39,40 The unusually long lifetimes observed in hybrid organic metal halide compounds,^{9,17,20,22,41–46} for example, have led to the investigation of many physical phenomena such as photon-recycling,¹⁸ the Rashba effect,^47–50 and electron-phonon coupling.⁵¹ Time-resolved spectroscopic techniques are, therefore, highly valuable methods to investigate fundamental physical and material properties²² that inform future material and device design efforts.² However the drawback with many transient spectroscopic measurements is that they employ a high-fluence optical source, and photo-generated charge-carrier densities are typically many orders of magnitude higher than under normal solar cell operating conditions.⁵² Charge carrier dynamics in disordered semiconductors are known to be strongly dependent on carrier density,³³ and under high charge-densities, carrier dynamics will not necessarily be representative of those relevant for solar cell operation.²⁸ Charge-carrier mobilities in disordered semiconductors are also known to be carrier-density dependent,⁵³ and at high optical-fluence higher-order recombination processes (bimolecular and Auger) can dominate transient behavior.⁵⁴ In the interest of gaining greater insight into the physical processes relevant for solar cells, the incentive to study charge carrier dynamics under typical solar cell operating conditions is therefore considerable.

An alternative approach is to study carrier dynamics using steady-state illumination^28,43,55 rather than with high-fluence pulsed optical sources. In this report, we describe a simple and versatile contactless technique to study the photoconductivity of a semiconductor as a function of incident optical power-density at optical power densities of ≤1 sun. By employing thin solution-processed films of the highly-studied, hybrid-halide perovskite methylammonium lead iodide (MAPbI₃),^2,3,7,8 we evaluate a parameter $ϕΣμτ$⁠, a proxy for mobility-lifetime product, with a strong dependence on incident optical power densities even below 1 mW/cm².

The technique we employ is a steady-state analog of the time-resolved microwave conductivity technique (TRMC)^{15,34,56–59} and based upon a system we have described previously.^54,60,61 A circuit diagram of our apparatus is shown in Fig. 1(a), and a description of its operation is provided in the supplementary material Sec. S1.3. This technique, which we henceforth refer to as steady-state microwave conductivity (SSMC), employs a high-power green [ $λ m a x$ = 525 nm, see Fig. 1(b) for spectrum] light-emitting diode (LED) to illuminate a semiconducting sample in a microwave cavity. Because the method is contactless we do not need to form a device, deposit electrical contacts, or even have a continuous film. As with TRMC, this approach allows one to significantly reduce the number of optimization-parameters, and hence extract the properties of a semiconductor with little ambiguity with respect to contact-/device-based techniques. As with TRMC this technique is a probe of local transport in a direction parallel to the substrate based on the configuration of the sample in the cavity.^34,60,61

The normalized emission spectrum of this LED is shown in Fig. 1(b) along with an absorption spectrum of a thin film of MAPbI₃. By passing a collimated beam of incident light through an optical chopper (with modulation frequency $f m o d$ ∼1 kHz), the power density of the incident light is varied between 0 and 36 mW/cm² [Fig. 1(c)]. Using the microwave conductivity system, we are able to detect the photo-induced change in the conductance of the sample as a function of time [Fig. 1(d)]. The change in photoconductance as a function of time (⁠ $ΔG$⁠) shown in Fig. 1(d) is taken from 2000 accumulated averages. The fact that $ΔG$ is not a square wave is attributed to instability of the chopper wheel and oscilloscope triggering mechanism over many averages. The modulation frequency of the incident light is sufficiently low (<10 kHz) to ensure that one half-cycle is enough time for the carrier density due to recombination, generation and diffusion to be in a steady state. This assumption is validated in the supplementary material Sec. S3, where it is determined for our film of MAPbI₃, illumination timescales of <1 μs are required for $ΔG$ to be affected by deviations from steady-state conditions. The time-constant of our cavity is approximately 60 ns,^54,60,61 which is much shorter than illumination timescales under consideration for this study. The band gap of MAPbI₃ is known to be approximately 1.5–1.6 eV,²¹ hence the energy of the incident photons (∼2.36 eV) is sufficient to ensure that carriers are generated by band-to-band transitions, rather than de-trapping, for example. While many thin-film organic metal halide semiconductors possess exciton binding energies on the order of 100 meV,⁶² MAPbI₃ is known to have an exciton binding energy in the range of 5–60 meV and band-gap excitation generates carriers at room temperature.²² 

The limit of detection of the directly detected data shown in Fig. 1(d) is not adequate to study the conductivity as a function of optical power density over a wide range. The background in the experiment could be reduced through better electrical isolation, further optimization of the cavity dimensions, or numerical filtering techniques. Instead we improved the sensitivity of the measurement by using a lock-in amplifier to measure the peak-to-peak $ΔG$ as a function of optical power density, at modulation frequencies that still represent the steady state. The range of accessible generation rates could also be increased using an optical laser, instead of an LED, as the source.

Using lock-in detection, we examined the photoconductivity as a function of incident light intensity near the range of 1 sun illumination. With knowledge of the sample dimensions, the conductivity $Δσ$ was calculated from the conductance and plotted as a function of absorbed photons/cm³ (⁠ $F$⁠) in Fig. 2. 1 sun is roughly equivalent to 10²¹–10²² photons/cm³ in MAPbI₃. Photoconductivity is known to follow a power law, with the exponent (⁠ $γ$⁠) revealing information on the dominant recombination mechanism(s), where F is assumed to be equal to the generation rate because of the high quantum efficiency of MAPbI₃²⁸ 

Levine et al.²⁸ recently reported a value of $γ$ = 0.40 for MAPbI₃ measured using surface photovoltage (SPV) measurements, and Chen et al.⁴³ reported $γ$ = 0.5 via electrical photoconductivity for thin films and single crystals of MAPbI₃, at similar fluxes used in our study. We have fitted Eq. (1) to our data in Fig. 2, and observe an exponent of $γ$ = 0.47, in good agreement with previous reports.^28,43 Differences in steady-state photocurrent-dependence are observed in the literature for different processing conditions^43,63 and it is likely that trap states^28,35,64 in the MAPbI₃ play a strong role in the recombination dynamics at low carrier densities. The location of mid-gap traps in MAPbI₃ have been suggested to depend on the substrate properties,⁶⁴ and surface trapping is known to play a significant role in carrier recombination.⁶⁵ While larger grains have been shown to lead to higher carrier mobilities⁶⁶ and longer lifetimes¹⁶ in MAPbI₃, the nature of trap states remains under debate and an area for future investigation.^28,35,64

Under steady-state conditions, we can equate the rate of carrier generation (⁠ $n ̇ g$ for holes, $p ̇ g$ for electrons), to the rate of carrier recombination (⁠ $n ̇ r$ for holes, $p ̇ r$ for electrons)

Assuming band-to-band excitation dominates, the rate of carrier density generation is equal for holes and electrons. For a film of thickness d with a fractional absorption F_A ∈ [0,1] and carrier generation efficiency ϕ ∈ [0,1], illuminated with light of optical power density P and wavelength of λ, the generation rate is expressed as

where h and c are the Planck constant and speed of light in a vacuum, respectively. Since we are concerned with low carrier densities, and hence predominantly trap-mediated recombination, we describe recombination via a single effective lifetime for each carrier type: $τ i$ (⁠ $i=e$ for electrons, $i=h$ for holes) rather than specific rate constants for trap-assisted, bimolecular and Auger recombination, respectively. Under these circumstances the recombination rate at a specific carrier density (⁠ $n$ for electrons, $p$ for holes) can be approximated as

The photo-induced change in conductivity (⁠ $Δσ$⁠) is described in terms of hole and electron density, and hole and electron mobility through

where $e$ is the magnitude of the fundamental unit of charge, and $μ e$ and $μ h$ are the electron and hole mobilites in the film, respectively. From Eqs. (2) to (4), we know that all generation and recombination rates are equal: $n ̇ g = p ̇ g = n ̇ r = p ̇ r$⁠. Substituting Eqs. (4)–(6) into Eq. (7), we can therefore define a new parameter $ϕΣμτ$ (where $Σμτ= τ e μ e + τ h μ h$⁠) given by Eq. (8)

Because microwave conductivity experiments do not (conventionally) allow one to separate the charge dynamics of holes and electrons, parameters are typically extracted involving the sum of two parameters.^15,34 The TRMC figure of merit $ϕΣμ$ for example contains the sum of electron and hole mobilities: (⁠ $Σμ= μ e + μ h$⁠). The parameter $ϕΣμτ$ in Eq. (8) has the same dimensions as the mobility-lifetime product: $μτ$⁠, but carrier-type-specific information remains obscured, in an analogous manner to $ϕΣμ$ in TRMC experiments. The influence of $μτ$ in solar cell performance is well known,^25–28 and the ability to rapidly evaluate $ϕΣμτ$ at low carrier densities, without the need to form an electronic device is clearly valuable. Despite losing access to carrier-specific information, SSMC is a contactless technique and eliminates the static electric fields, which are a potential source of ionic diffusion processes in materials like MAPbI₃,⁶⁷ and can use low optical flux minimizing potential damage to the semiconductor.

$ϕΣμτ$ is plotted as a function of optical power density for our thin film of MAPbI₃ in Fig. 3. For systems with low exciton binding energy (such as MAPbI₃)²² we know that ϕ is very close to unity, and can hence interpret $ϕΣμτ$ in a similar way as to one would interpret $μτ$⁠. Carrier lifetimes in the range of roughly 200 ns–5 μs,^{9,20,22,41,64} and mobilities in the range of roughly 0.5–70 cm²/V s,^{9,10,39,64,66,68–70} have previously been reported for thin-films of MAPbI₃. The values shown in Fig. 3 are therefore comparable with previous reports, but we draw attention to our observed intensity-dependence of $ϕΣμτ$⁠. By assuming that the electron mobility $μ e$⁠, hole mobility $μ h$⁠, and carrier-generation-efficiency ϕ, are approximately constant as function of carrier density, we conclude that the electron and/or hole lifetime is strongly dependent upon optical power density. In their report, Levine et al. observe ambipolar diffusion-lengths (⁠ $L D$⁠) in MAPbI₃ which are independent of excitation intensity.²⁸ Because our technique does not isolate carrier-specific dynamics, and contains the carrier-generation efficiency ϕ, we cannot evaluate a value of $L D$ for direct comparison to their study. Yamada et al. have observed a carrier lifetime that was independent of excitation intensity at low intensities and was dependent on intensity at high intensities.⁷¹ Our observed carrier-density-dependence of lifetime can be interpreted in terms of trap states,^28,64 with the proximity of the Fermi level (moving under photoexcitation) to this trap state affecting the rate at which carriers are trapped or de-trapped. It is therefore not unreasonable to suggest that microstructure plays a significant role in the carrier-density-dependence of lifetime, as it does for absolute values of the lifetime.¹⁶ 

We grew single crystals of MAPbI₃, using a previously reported inverse-crystallization technique.⁴⁴ The crystal dimensions are roughly 2 mm × 2 mm × 2 mm, and the non-radiative and radiative recombination rates have been reported⁴⁶ to be significantly lower than in thin films of MAPbI₃: $k n r$ ∼1 × 10⁵ s⁻¹ and $k r a d$ ∼5 × 10⁻¹⁰cm³/s, respectively. For these reasons, we expected the timescales for the system to reach equilibration between generation and recombination to be significantly longer in a crystal than in a thin film. The absolute conductance of a thin film and of a single crystal of MAPbI₃ are shown as a function of modulation (chopper) frequency at an optical power density 46 mW/cm² in Fig. 4. For a large (mm-scale) single crystal of MAPbI₃, there is a clear dependence of microwave detected absolute photoconductance on modulation frequency; the value is generally lower than that of the thin film due to the difference in sample size. This difference in frequency dependence is attributed to the much-larger sample depth (∼2 mm for crystal vs. ∼100 nm for film) where carriers in the single crystal will not have diffused sufficiently to reach an equilibrium depth over the time-scales involved in our measurement. The much longer lifetimes reported⁴⁶ in single crystals will require longer times for recombination and generation to equilibrate, even in the absence of an inhomogeneous carrier distribution. For these reasons, and the difficulty in determining the effective depth of the sample due to diffusion (the depth in which carriers are present), it is impossible to evaluate the parameter $ϕΣμτ$ for a single crystal using this technique, without a priori detailed knowledge of the carrier mobilities. Nonetheless, this data illustrates that carrier dynamics in single crystals of MAPbI₃ occur over very long timescales (>100 μs), even in the absence of ionic diffusion effects,^72,73 which should not be present in the GHz-frequency field employed in our measurement system, but could be present under constant-bias based techniques.¹⁷ 

In conclusion, we have demonstrated a contactless technique to characterize the conductivity of carriers in semiconductors using a simple LED light source at low illumination near effective 1 Sun conditions. This technique allows one to extract the parameter $ϕΣμτ$⁠, a proxy for the mobility-lifetime product, as a function of optical power density, at very low charge densities, without the need to construct a device/apply contacts, and in the absence of any field driven ionic motion. The ability to probe dynamics under typical operating conditions is relatively rare^28,55 in the field of contactless optical spectroscopy, and we expect the method to provide information about regimes of carrier density that were previously inaccessible. As an example, we find that the observed $ϕΣμτ$ of thin films of MAPbI₃ to be highly dependent on incident optical power density, even under sub-AM1.5 illumination conditions revealing the role of trap states in recombination.

See supplementary material for experimental methods, scanning electron microscope images, X-ray diffraction spectra, a theoretical verification of steady-state conditions in methylammonium lead iodide thin films, chopper-frequency-dependent conductance data, and details of the baseline correction employed.

Research on microwave conductivity and device properties was supported by the Defense Threat Reduction Agency (HDTRA1-15-1-0023). Research on crystal growth supported by DOE under Grant Award No. DE-SC-0012541. J.G.L. gratefully acknowledges Virgil Elings and Betty Elings Wells for financial support through the Elings Fellowship Awards. The authors thanks one of the anonymous reviewers for comments on the details of the experimental method.