WO2023144512A1

WO2023144512A1 - Method and apparatus for rapid optoelectronic material screening

Info

Publication number: WO2023144512A1
Application number: PCT/GB2023/050090
Authority: WO
Inventors: Alan R. BOWMAN; Samuel D. STRANKS
Original assignee: Cambridge Enterprise Limited
Priority date: 2022-01-26
Filing date: 2023-01-19
Publication date: 2023-08-03
Also published as: GB202201015D0

Abstract

Broadly speaking, embodiments of the present techniques provide a method and apparatus to quickly and cost effectively screen (5100) a large number of semiconductor materials that may be used for optoelectronic devices. Advantageously, the present techniques use photoluminescence efficiency, PLQE, measurements (S102) and time- resolved photoluminescence, TRPL, measurements (S102) of a sample of a semiconductor material to extract (S104, S106) the parameters needed to screen or evaluate the material. This is advantageous because the measurements can be made using techniques and equipment that is readily accessible to many laboratories and therefore, does not require specialised expertise or equipment that is not readily available in research institutions or industry.

Description

Method and Apparatus for Rapid Optoelectronic Material Screening

Field

The present techniques generally relate to a method and apparatus for screening samples of optoelectronic materials. In particular, the present techniques provide a method and apparatus for rapidly screening and quantitative evaluation of optoelectronic materials to determine their suitability for optoelectronic devices.

Background

With global power consumption on the rise, there is a need for more efficient optoelectronic devices. Optoelectronic devices are electronic devices which operate on both light and electrical currents. Broadly speaking, optoelectronic devices are electrical-to-optical transducers or optical-to-electrical transducers. Optoelectronic devices include electrically-driven light sources such as laser diodes and light-emitting diodes, components for detecting light/electromagnetic radiation such as photodetectors, components for converting light into an electrical current such as solar and photovoltaic cells, and devices which control light/electromagnetic radiation. Optoelectronic devices are generally based on the quantum mechanical effects of light on electronic materials, such as semiconductors. New optoelectronic semiconductors, including halide perovskites, antimony selenide and bismuth-based materials, are being rapidly developed as the next generation of solar cells, light emitting diodes and X-ray scintillators. For example, halide perovskite solar cell laboratory efficiencies now rival those of silicon. But finding materials within these and other families with the right properties to create more efficient devices is a challenge.

An indicator of how well a semiconductor could perform as an optoelectronic device is the recombination rates associated with the semiconductor. When excited charge carriers (electrons and holes, excitons, polarons, etc.) in a semiconductor recombine, this may or may not result in a photon being released. Recombination involving a photon that is emitted to the surroundings is referred to as external radiative recombination, while other recombination is referred to as non-radiative recombination. Quantifying recombination rates, both radiative and non-radiative, can help predict the efficiency and performance of a new semiconductor material as an optoelectronic device.

Recombination rates in optoelectronic semiconductors are typically measured using time-intensive and expensive measurements.

The applicant has therefore identified the need for improved techniques for quickly evaluating and screening samples of optoelectronic material.

Summary

In a first approach of the present techniques, there is provided a computer- implemented method for rapid optoelectronic material screening, the method comprising: receiving, for a sample of a optoelectronic material to be screened, at least one screening criterion; receiving a plurality of inputs, comprising: a plurality of photoluminescence efficiency, PLQE, measurements of the sample of the optoelectronic material, and a plurality of time-resolved photoluminescence, TRPL, measurements of the sample of the optoelectronic material; calculating, using the received plurality of inputs, at least two parameters; and determining, using the at least two calculated parameters, whether the sample of the optoelectronic material meets the at least one screening criterion.

The at least two parameters comprise a radiative decay rate and a first order recombination rate. The radiative decay rate may be a first order radiative rate or a second order radiative rate.

The optoelectronic material may be used to fabricate, for example, photodiodes, solar cells, photomultipliers, phototransistors, charge-coupled imaging devices, light-emitting diodes, colour converters for display devices, photodetectors, and radiation detectors. It will be understood that this is a nonlimiting and non-exhaustive list of example optoelectronic devices, and that the optoelectronic material may be used for any optoelectronic device.

Receiving a plurality of inputs comprises receiving any one or more of: a sample thickness; a beam size of a beam of light used to optically interrogate the sample; a beam shape of a beam of light used to optically interrogate the sample; and a power of a beam of light used to optically interrogate the sample. The light source used to generate the beam of light may be a laser or white light source.

The at least two calculated parameters may comprise any two or more of: a radiative decay rate and a first order recombination rate; at least one first recombination rate associated with first order loss processes; at least one second recombination rate associated with second order loss processes; at least one third recombination rate associated with third order loss processes; at least one trap density; at least one background doping density; at least one diffusion length; at least one material recombination rate; at least one surface velocity; at least one diffusion rate; at least one photon absorption coefficient; and at least one recombination velocity. In some cases, there may be, for example, only a single first combination rate, a single second combination rate, etc, but in other cases, there may be multiple such rates. The number of each type of parameter depends on which model is being used to calculate the parameter from the PLQE and TR.PL measurements, and which material or type of material is being screened.

The at least one screening criterion enables quantitative evaluation of characteristics of the sample of the optoelectronic material, and may comprise any one or more of: a radiative decay rate; a first order recombination rate; a first recombination rate associated with first order loss processes; a second recombination rate associated with second order loss processes; a third recombination rate associated with third order loss processes; a trap density; a background doping density; a diffusion length; a material recombination rate; a surface velocity; a diffusion rate; a photon absorption coefficient; and a recombination velocity.

Calculating the at least two parameters may comprise: calculating, using the received plurality of PLQE measurements, at least one ratio of recombination rates, wherein each ratio of recombination rates has a unit of the form cm^3x s^x, where x is any number; and converting, using the received plurality of TR.PL measurements, the at least one ratio of recombination rates into at least one recombination rate. The at least one recombination rate may include a first recombination rate associated with first order loss processes and a second recombination rate associated with second order loss processes. In some cases, the PLQE measurements may be used to calculate a plurality of ratios of recombination rates.

Generally, calculating the at least one ratio of recombination rates comprises fitting the plurality of PLQE measurements using a minimal number of constants, and using the fit to calculate the at least one ratio of recombination rates. Converting, using the received plurality of TR.PL measurements, the at least one ratio of recombination rate into at least one recombination rate may comprise fitting the plurality of TR.PL measurements, using the fit to extract one or more parameters (such as any of the parameters mentioned above), and using the extracted one or more parameters to convert the at least one ratio of recombination rates into at least one recombination rate. The plurality of PLQE measurements and the plurality of TR.PL measurements may be fit independently. Alternatively, the plurality of PLQE measurements and the plurality of TR.PL measurements may be co-fit.

In a more specific example, calculating, using the received plurality of PLQE measurements, at least one ratio of recombination rates may comprise calculating the following plurality of recombination rates:

where p₀ is a background electron or hole concentration, p_esc is a probability an emitted photon escapes the material, b_r is a radiative recombination rate, a is a first order recombination rate, and b is a second order recombination rate.

In this specific example, calculating the plurality of ratios of recombination rates may comprise: fitting the received plurality of PLQE measurements to a PLQE curve using a curve fitting algorithm; extracting an approximate value of Po^ escbr using a portion of the PLQE curve corresponding to low laser power data; extracting an approximate value of . ^a using a portion of the PLQE curve j esc^r corresponding to intermediate laser power data; and extracting an approximate value of — — from a value close to a maximum value of the PLQE curve. Here, Vesc^r the term "low laser power" means an excitation regime where the first order terms dominate recombination, and the term "high laser power" means a regime where the highest order recombination terms dominate recombination. It will be understood that the term "intermediate laser power" therefore means a power between the low and high laser powers.

In this specific example, converting, using the received plurality of TR.PL measurements, the plurality of ratios of recombination rates into recombination rates may comprise: plotting the received plurality of TR.PL measurements to extract p₀, the background electron or hole concentration and a, the first order recombination rate; and using a to determine the second order recombination rate and radiative combination rate.

In another example, the at least two calculated parameters may comprise a first recombination rate associated with first order loss processes and a second recombination rate associated with second order loss processes. In this case, the method may further comprise: receiving, for the sample of the optoelectronic material, a plurality of absorption measurements for a set of wavelengths; calculating, using the absorption measurements, an intrinsic absorption value for the sample of the optoelectronic material; and calculating, using the first and second recombination rates and the intrinsic absorption value, a maximum potential optoelectronic efficiency for the sample optoelectronic material. This may be useful if the optoelectronic material is being used to fabricate a solar cell, for example.

In a second approach of the present techniques, there is provided an apparatus for rapid optoelectronic material screening, the apparatus comprising: at least one light source for optically interrogating a sample of a optoelectronic material to be screened; a mechanism for varying incident light source power on the sample; at least one detector for detecting light emitted by the sample after interrogation by the at least one light source; and at least one processor coupled to memory arranged to: receive, for the sample of the optoelectronic material, at least one screening criterion; receive a plurality of inputs, comprising: a plurality of photoluminescence efficiency, PLQE, measurements of the sample of the optoelectronic material from the at least one detector, wherein the PLQE measurements are made by varying an incident power of the at least one light source, and a plurality of time-resolved photoluminescence, TR.PL, measurements of the sample of the optoelectronic material from the at least one detector; calculate, using the plurality of inputs, at least two parameters; and determine, using the at least two calculated parameters, whether the sample of the optoelectronic material meets the at least one screening criterion.

The at least one light source may comprise one or more of: a white light source; a flash lamp; a continuous wave laser; and a pulsed laser.

The plurality of PLQE measurements may be obtained by varying an incident power of a fixed wavelength continuous wave laser. By varying the incident power, and therefore the intensity, of the laser being used to interrogate the sample, it is possible to measure how PLQE varies with intensity. The at least one detector may be any one of: a photodetector, a photodiode, a camera, and a charge coupled device.

The plurality of TR.PL measurements may be obtained by varying an incident power and/or a repetition rate of a pulsed laser or flash lamp. The at least one detector may be a silicon single photon avalanche diode. By varying the incident power (intensity) and/or repetition rate of the light source being used to interrogate the sample, it is possible to measure how TR.PL varies with intensity and/or repetition rate.

The at least two calculated parameters may comprise any two or more of: a radiative decay rate and a first order recombination rate; at least one first recombination rate associated with first order loss processes; at least one second recombination rate associated with second order loss processes; at least one third recombination rate associated with third order loss processes; at least one trap density; at least one background doping density; at least one diffusion length; at least one material recombination rate; at least one surface velocity; at least one diffusion rate; at least one photon absorption coefficient; and at least one recombination velocity.

Calculating, using the at least one processor, at least two parameters may comprise: calculating, using the received plurality of PLQE measurements, at least one ratio of recombination rates; and converting, using the received plurality of TR.PL measurements, the at least one ratio of recombination rates into at least one recombination rate. In some cases, the at least one recombination rate includes a first recombination rate associated with first order loss processes and a second recombination rate associated with second order loss processes. In some cases, the PLQE measurements may be used to calculate a plurality of ratios of recombination rates.

Generally, calculating the at least one ratio of recombination rates may comprise fitting the plurality of PLQE measurements using a minimal number of constants, and using the fit to calculate the at least one ratio of recombination rates. Converting, using the received plurality of TR.PL measurements, the at least one ratio of recombination rates into at least one recombination rate may comprise fitting the plurality of TR.PL measurements, using the fit to extract one or more parameters (such as any of the parameters mentioned above), and using the extracted one or more parameters to convert the at least one ratio of recombination rates into at least one recombination rate. The plurality of PLQE measurements and the plurality of TR.PL measurements may be fit independently. Alternatively, the plurality of PLQE measurements and the plurality of TR.PL measurements may be co-fit.

In a specific example, calculating, using the received plurality of PLQE measurements, at least one ratio of recombination rates may comprise calculating the following plurality of ratios of recombination rates:

In this specific example, the calculating of the plurality of ratios of recombination rates may comprise: fitting the received plurality of PLQE measurements to a PLQE curve using a curve fitting algorithm; extracting an approximate value of p₀^p_escb_r using a portion of the PLQE curve corresponding to low laser power data; extracting an approximate value of “ — using a portion of j esc^r the PLQE curve corresponding to intermediate laser power data; and extracting an approximate value of — — from a value close to a maximum value of the PLQE esc^r curve. Here, the term "low laser power" means an excitation regime where the first order terms dominate recombination, and the term "high laser power" means a regime where the highest order recombination terms dominate recombination. It will be understood that the term "intermediate laser power" therefore means a power between the low and high laser powers.

The apparatus may further comprise at least one white light source.

When the at least two calculated parameters comprise a first recombination rate associated with first order loss processes and a second recombination rate associated with second order loss processes, and the at least one processor may be arranged to: receive, for the sample of the optoelectronic material, a plurality of absorption measurements for a set of wavelengths made using the at least one white light source and the at least one detector; calculate, using the absorption measurements, an intrinsic absorption value for the sample of the optoelectronic material; and calculate, using the first and second recombination rates and the intrinsic absorption value, a maximum potential optoelectronic efficiency for the sample optoelectronic material. This may be useful if the optoelectronic material is being used to fabricate a solar cell, for example.

The plurality of absorption measurements may be UV-Vis absorption measurements.

In a related approach of the present techniques, there is provided a non- transitory data carrier carrying processor control code to implement any of the methods, processes and techniques described herein.

As will be appreciated by one skilled in the art, the present techniques may be embodied as a system, method or computer program product. Accordingly, present techniques may take the form of an entirely hardware embodiment, an entirely software embodiment, or an embodiment combining software and hardware aspects.

Furthermore, the present techniques may take the form of a computer program product embodied in a computer readable medium having computer readable program code embodied thereon. The computer readable medium may be a computer readable signal medium or a computer readable storage medium. A computer readable medium may be, for example, but is not limited to, an electronic, magnetic, optical, electromagnetic, infrared, or semiconductor system, apparatus, or device, or any suitable combination of the foregoing.

Computer program code for carrying out operations of the present techniques may be written in any combination of one or more programming languages, including object oriented programming languages and conventional procedural programming languages. Code components may be embodied as procedures, methods or the like, and may comprise sub-components which may take the form of instructions or sequences of instructions at any of the levels of abstraction, from the direct machine instructions of a native instruction set to high-level compiled or interpreted language constructs.

Embodiments of the present techniques also provide a non-transitory data carrier carrying code which, when implemented on a processor, causes the processor to carry out any of the methods described herein.

The techniques further provide processor control code to implement the above-described methods, for example on a general purpose computer system or on a digital signal processor (DSP). The techniques also provide a carrier carrying processor control code to, when running, implement any of the above methods, in particular on a non-transitory data carrier. The code may be provided on a carrier such as a disk, a microprocessor, CD- or DVD-ROM, programmed memory such as non-volatile memory (e.g. Flash) or read-only memory (firmware), or on a data carrier such as an optical or electrical signal carrier. Code (and/or data) to implement embodiments of the techniques described herein may comprise source, object or executable code in a conventional programming language (interpreted or compiled) such as C, or assembly code, code for setting up or controlling an ASIC (Application Specific Integrated Circuit) or FPGA (Field Programmable Gate Array), or code for a hardware description language such as Verilog (RTM) or VHDL (Very high speed integrated circuit Hardware Description Language). As the skilled person will appreciate, such code and/or data may be distributed between a plurality of coupled components in communication with one another. The techniques may comprise a controller which includes a microprocessor, working memory and program memory coupled to one or more of the components of the system. It will also be clear to one of skill in the art that all or part of a logical method according to embodiments of the present techniques may suitably be embodied in a logic apparatus comprising logic elements to perform the steps of the above-described methods, and that such logic elements may comprise components such as logic gates in, for example a programmable logic array or application-specific integrated circuit. Such a logic arrangement may further be embodied in enabling elements for temporarily or permanently establishing logic structures in such an array or circuit using, for example, a virtual hardware descriptor language, which may be stored and transmitted using fixed or transmittable carrier media.

In an embodiment, the present techniques may be implemented using multiple processors or control circuits. The present techniques may be adapted to run on, or integrated into, the operating system of an apparatus.

In an embodiment, the present techniques may be realised in the form of a data carrier having functional data thereon, said functional data comprising functional computer data structures to, when loaded into a computer system or network and operated upon thereby, enable said computer system to perform all the steps of the above-described method.

Brief description of the drawings

Implementations of the present techniques will now be described, by way of example only, with reference to the accompanying drawings, in which:

Figure 1 is a flowchart of example steps to perform rapid optoelectronic material screening;

Figures 2A to 2E show example approaches for fitting PLQE and TR.PL measurement data;

Figures 3A shows TAS measurement data and 3B shows PLQE, TR.PL and TAS measurement data;

Figures 4A to 4D show micro-PLQE, TR.PL (including calculations) and TAS measurement data;

Figure 5 shows optoelectronic current-voltage curves for an exemplary material;

Figure 6 shows an example experimental setup for taking PLQE measurements; Figure 7 shows an example experimental setup for taking time-correlated single photon counting (TCSPC) measurements, which are forms of TR.PL;

Figure 8 shows an example experimental setup for taking measurements using an intensified charge coupled device (ICCD, which are forms of TR.PL);

Figure 9 shows an example experimental setup for taking UV-Vis measurements; and

Figure 10 shows an example experimental setup for obtaining all the measurements required to perform the rapid optoelectronic material screening technique described herein.

Detailed description of the drawings

Broadly speaking, embodiments of the present techniques provide a method and apparatus to quickly and cost effectively screen a large number of semiconductor materials that may be used for optoelectronic devices. Advantageously, the present techniques use photoluminescence efficiency, PLQE, measurements and time-resolved photoluminescence, TR.PL, measurements of a sample of a semiconductor material to extract the parameters needed to screen or evaluate the material. This is advantageous because the measurements can be made using techniques and equipment that are readily accessible to many laboratories and therefore, does not require specialised expertise or equipment that is not readily available in research institutions or industry.

In order to characterise different semiconductor materials, identify remaining power loss pathways, and enable further advances, recombination mechanisms need to be fully quantified. Of particular importance is non-radiative recombination, which needs to be minimised in any developed device to maximize performance. Several methods have been used to understand excited charge carrier processes in films, including transient absorption spectroscopy (TAS), time- resolved photoluminescence (TR.PL), terahertz spectroscopy, impedance spectroscopy and time-resolved voltage response. Although powerful techniques, the majority are time-intensive, inaccessible to many academic and industry laboratories, and difficult to run successfully without specialised expertise, meaning recombination rates have only been fully quantified for a small subset of developing optoelectronic semiconductors, and material screening is not viable. The present techniques include a method to rapidly quantify ratios between recombination rates in luminescent semiconductor thin film absorbers using just photoluminescence quantum efficiency (PLQE) measurements. By combining the PLQE measurements with complementary TR.PL measurements - screenable techniques readily accessible to many laboratories - all recombination coefficients can be extracted. The method may streamline materials screening and optimisation for device performance using accessible techniques, and motivate further studies to explain unexpected recombination behaviour in semiconductors.

The term "optoelectronic material" is used interchangeably herein with the terms "material", "semiconductor" and "semiconductor material".

The method is now described with reference to the Figures.

Figure 1 is a flowchart of example steps to perform rapid optoelectronic material screening. The method shown in Figure 1 may be used to perform two broad types of screening. For example, the method may be used to screen or evaluate a sample of a specific semiconductor material. The screening may be determine whether the potential of the material is ever going to be high enough to be used in a optoelectronic device. This screening method only needs to be performed once per material. Additionally or alternatively, the method may be used to screen or evaluate different material treatments or passivation, to determine which treatment/passivation is best for overall optoelectronic optimisation. In this case, the same material may be treated in different ways, and the screening method is used to determine which treated material is best to take to a subsequent round of optimisation.

The method comprises receiving, for a sample of a optoelectronic material to be screened, at least one screening criterion (step S100). The at least one screening criterion may depend on which type of screening is being performed. For example, if the screening is being performed to determine whether the material is suitable for a solar cell device, then the at least one screening criterion may be selected accordingly. It will be understood that the at least one screening criterion may depend on what the optoelectronic material is to be used for. As noted above, the optoelectronic material may be used to fabricate, for example, photodiodes, solar cells, photomultipliers, phototransistors, charge-coupled imaging devices, light-emitting diodes, colour converters for display devices, photodetectors, and radiation detectors, and each of these device types may be associated with different screening criteria.

The at least one screening criterion enables quantitative evaluation of characteristics of the sample of the optoelectronic material, and may comprise any one or more of: a radiative decay rate; a first order recombination rate; at least one first recombination rate associated with first order loss processes; at least one second recombination rate associated with second order loss processes; at least one third recombination rate associated with third order loss processes; at least one trap density; at least one background doping density; at least one diffusion length; a material recombination rate; a surface velocity; a diffusion rate; a photon absorption coefficient; and at least one recombination velocity. In an example, the first and second recombination rates may be useful for determining whether a material is suitable for being used in a solar cell device.

At step S102, the method comprises receiving a plurality of inputs. The plurality of inputs comprise a plurality of photoluminescence efficiency, PLQE, measurements of the sample of the optoelectronic material, and a plurality of time-resolved photoluminescence, TR.PL, measurements of the sample of the optoelectronic material.

The plurality of inputs may further comprise one or more of: a sample thickness (i.e. thickness of the sample of the optoelectronic material); a beam size of the beam of light used to optically interrogate the sample; a beam shape of the beam of light used to optically interrogate the sample; and a power of the beam of light used to optically interrogate the sample. These inputs may help to define a generation rate, which is the number of excited states generated by photons within the material (per unit time, per unit volume). The relevance of the generation rate is explained below.

The plurality of PLQE measurements may be obtained by varying an incident power of a fixed wavelength continuous wave laser that is incident on the sample of the optoelectronic material. By varying the incident power, and therefore the intensity, of the laser being used to interrogate the sample, it is possible to measure how PLQE varies with intensity.

The plurality of TR.PL measurements may be obtained by varying an incident power and/or a repetition rate of a pulsed laser or flash lamp that is incident on the sample of the optoelectronic material. By varying the incident power (intensity) and/or repetition rate of the light source being used to interrogate the sample, it is possible to measure how TR.PL varies with intensity and/or repetition rate.

At step S104, the method comprises calculating, using the received plurality of inputs, at least two parameters. The at least two calculated parameters may comprise any two or more of: a radiative decay rate and a first order recombination rate; at least one first recombination rate associated with first order loss processes; at least one second recombination rate associated with second order loss processes; at least one third recombination rate associated with third order loss processes; at least one trap density; at least one background doping density; at least one diffusion length; at least one material recombination rate; at least one surface velocity; at least one diffusion rate; at least one photon absorption coefficient; and at least one recombination velocity. In some cases, there may be, for example, only a single first combination rate, a single second combination rate, etc, but in other cases, there may be multiple such rates. The number of each type of parameter depends on which model is being used to calculate the parameter from the PLQE and TR.PL measurements, and which material or type of material is being screened. A general model is described below, and details about a specific model (used for specific semiconductor materials) are also provided to illustrate how the present techniques work.

At step S106, the method comprises determining, using the at least two calculated parameters, whether the sample of the optoelectronic material meets the at least one screening criterion.

In one specific, non-limiting example, the at least one screening criterion may be a maximum potential efficiency for the sample. The samples may be screened to determine their suitability to form photovoltaic devices such as solar cells. In this case, it may be useful to determine the maximum potential efficiency for the sample, because those samples with a efficiency over a threshold value may be determined to be suitable for photovoltaic devices. The maximum potential efficiency may require needing to know a first recombination rate associated with first order loss processes and a second recombination rate associated with second order loss processes. Thus, at step S104, these parameters may be calculated and step S106 may further comprise: receiving, for the sample of the optoelectronic material, a plurality of absorption measurements for a set of wavelengths made using at least one white light source and at least one detector. The method may comprise calculating, using the received absorption measurements, an intrinsic absorption value for the sample of the optoelectronic material, and calculating, using the first and second recombination rates and the intrinsic absorption value, a maximum potential optoelectronic efficiency for the sample optoelectronic material. The plurality of absorption measurements may be UV-Vis absorption measurements.

Step S104 is now described in more detail with respect to calculating particular parameters. By combining the PLQE measurements with complementary TR.PL measurements - screenable techniques readily accessible to many laboratories - all recombination coefficients can be extracted. The technique is demonstrated on halide perovskite thin films and excellent agreement when benchmarking this approach against TAS measurements is found.

This new technique is used to reveal that a non-radiative second-order recombination component is present, both in bulk and on the microscale, in a range of semiconductor or intrinsic semiconductor materials used for optoelectronic devices. For example, the present techniques may be used to screen halide perovskite thin film compositions being utilised for optoelectronics, such as in methylammonium lead iodide (MAPbh), mixed-cation mixed-halide formamidinium (FA)-containing mixtures (FAo.79MAo.i6Cso.o5)Pb(Io.83lo.i?)3 and low bandgap FAPbo.5Sno.5I3 samples. It will be understood that these are non-limiting example materials. The present techniques may be used to screen families of intrinsic semiconductor, excitonic semiconductor or doped semiconductor material. More generally, the present techniques may be used to screen semiconductor materials which exhibit second order non-radiative recombination.

Evidence suggesting parasitic absorption and photon recycling alone cannot explain second order non-radiative recombination is presented. The maximum efficiency of a optoelectronic fabricated from a measured MAPbh film is calculated and it is demonstrated that second order non-radiative recombination will play an important role in device efficiency when time-resolved photoluminescence lifetimes (at low excitation fluence) are longer than 1 ps. Therefore, second order non-radiative recombination is an important phenomenon to understand for ongoing optoelectronic device development. The method and apparatus presented herein, will streamline materials screening and optimisation for device performance using accessible techniques, and motivate further studies to explain unexpected recombination behaviour in semiconductors.

General model

As mentioned above, the calculated parameters, which are used to determine whether an optoelectronic material meets the at least one screening criterion, depend on the model used to calculate the parameters from the plurality of inputs and in particular, from the PLQE and TR.PL measurements. A general model is now described.

Light incident on a semiconductor populates excited states. Depending on the material, the excited states may be excited electrons and holes, increased trap state populations, excitons, polarons, polaritons etc. In general, the set of excited states is referred to as {nj(x,y,z)}, where i refers to excited state i, and x,y and z are spatial positions within the material. For example, in halide perovskites i = 1,2,3 may be said to correspond to electrons, electron traps and holes. (One distinction that can be made is that when i spans from 1 to n, the states may involve excited electrons (including excitons), while when i spans from n + 1 to m, the states purely consist of holes (i.e. no electrons are involved, so no excitonictype states, noting m > n).)

During continuous wave excitation, the generation rate is defined as Gj(x,y,z). This is the number of excited states i generated by photons at position (x,y,z) within the material (per unit time, per unit volume). For example, in halide perovskites G_r = G₃ everywhere (as the generation of electrons and holes is equal) while G₂ = 0 (as traps are not directly filled by incident photons). Gt can be directly related to the incident laser beam for any material, and it is noted that E”=iJ f f Gt(x,y, z)dxdydz is equal to the total number of photons absorbed in the material.

The local recombination rate per unit time per unit volume can be defined as ffj(x,y,z). The form of the recombination is fully material dependant, but in general can be a function of N free parameters, fixed parameters, and the set of excited states. The N free parameters may comprise any one or more of: material recombination rates, surface velocities, diffusion rates, photon absorption rates, and trap densities. The fixed parameters may comprise sample thickness. The set of excited states is {nj(x,y,z)}, as noted above. For any material, a model can be defined (or at least proposed) that relates {n_i x,y, z')} to Gj(%,y,z), allowing for the local excitation density of all excited states to be calculated.

In the steady state, total generation is equal to total recombination within the material for any form of excited state, that is f f f Gt(x,y, z)dxdydz = f f f Ri(x,y, z)dxdydz.

It is also possible to define the total light emitted from the material, as given f f esc ^x>y> ^z)Lout,t(.^x>y> ^z)dxdydz, where L_{out i} is the light emitted by state i (per unit volume, per unit time) and T _esc is the probability that an emitted photon escapes the material. The sum is only over the states which involve an electron, as to emit light an electron must recombine with a hole.

The photoluminescence efficiency, PLQE, of the material can be defined as

It is noted that E”=i f f f Rt(x,y, z)dxdydz is equal to the total number of photons absorbed in the material, per unit time.

To fit the PLQE, a transform is used to convert N free parameters into ratios between N-l free parameters, which are ratios between the original parameters. Generally speaking, the ratio of recombination rates has a unit of the form cm^3x s^x, where x is any number. This corresponds to reducing to the number of independent variables that can be fitted from the PLQE. This is advantageous because the complexity of the fit is reduced until the data fits well with the fewest arbitrary constants.

This enables at least one parameter to be calculated using the PLQE measurement, i.e. at least one ratio of recombination rates. As noted above, the present techniques also use time-resolved photoluminescence, TR.PL, measurements to calculate the at least one parameter. For example, to extract all decay rates one parameter must be fit absolutely from a time resolved decay method. In this method it can be said that d

— nt (x, y, z, t) = Rt (x, y, z, t) dt where t is time (as we are no longer in steady state). All equations can be solved (either analytically or numerically) to establish nt as a function of time (for all j). It is assumed that the laser pulse is much shorter than decay times in the material so that boundary conditions can be defined as follows: the total excited charges immediately following a laser beam hitting the sample are the sum of excited charges from photons absorbed from the laser beam and excited charges left over from the previous pulse (i.e. defined by the repetition rate of the laser beam, there is a built up of excited charges).

The light emitted at time t is given by f f f rj_esc(x, y, z)L_{0Ut i}(x, y, z, t)dxdydz, which is proportional to the TRPL signal at that time.

There are two options for fitting the PLQE and TRPL measurements. Firstly, the PLQE and TRPL measurements may be fit independently. To explore the parameter space, a stochastic fitting approach may be used for the PLQE measurements. The decay rate ratios obtained using this fitting approach may then be used to calculate recombination rates by separately fitting the TRPL decay measurements. Secondly, the PLQE and TRPL measurements may be co-fit. This may comprise finding the minimum in the difference between fitted and actual results from both sets of measurements simultaneously. Again, a stochastic fitting approach may be used.

In both approaches the number of free parameters (N-l in the first fitting approach 1, or N in the second fitting approach) is reduced to the fewest number of free parameters which can fit all the data well.

The general model may be used for any semiconductor materials, such as doped semiconductors, intrinsic semiconductors, and excitonic semiconductors.

Example Specific Version of the General Model

A more specific version of the general model is now described, to exemplify the general model. As model may depend on the material or type of materials being screened. This example specific model has been generated with respect to particular semiconductor materials, and a specific experimental set-up. It will be understood that this is just one way the general model may be adapted for particular materials and set-ups. The example specific model described below is based on halide perovskites (an intrinsic semiconductor) and similar materials.

PLQE fitting model Recombination rates of electrons and holes in optoelectronic semiconductors can be described by first, second and third order loss processes with respective rates a, b and c. These rates are typically interpreted to be charge trapping (assuming most traps are rapidly filled), second order processes (which typically describes radiative recombination of electrons and holes) and Auger recombination, respectively. (It will be understood that these processes are based on electrons and holes combining, and that the processes may vary for other types of excited charge carriers. For example, in the case of excitons, there is a first order component (radiative and non-radiative), and a second order component (Auger recombination). The following description of the specific model is based on the electron and hole combination, but it will be understood that this model is specific to this type of recombination). An accessible technique to measure optoelectronic semiconductors, requiring only a laser, integrating sphere and detector, is the measurement of photoluminescence quantum efficiency - the ratio of emitted to absorbed photons.

An intrinsic optoelectronic semiconductor is defined to be one in which the number of excited electrons and holes (i.e. electrons and holes above background doping densities) are approximately equal, as generation rates produce excitation densities well in excess of trap densities. This corresponds to a semiconductor with a reasonably high (>0.1 %) external PLQE, as long as traps are not luminescent and thus do not contribute to this value. For example, trap densities are on the order of 10¹⁵ cm^-3 in halide perovskites and 10¹³ cm^-3 in copper indium gallium selenide thin films, so these assumptions are valid when excitation densities are larger than this, as in typical illumination conditions for optoelectronics and the measurements presented here (see below). With these assumptions, when a region of the optoelectronic semiconductor is excited by a steady-state laser beam, the generation of charges is balanced by recombination. That is

G(x, y) = an(x,y) + bn x,y)² + cn(x,y)³. (1)

Here G(x,y) is the number of charges generated per unit time per unit volume in the film by an external laser beam at location x,y (with coordinates denoting the two in-plane directions of the film) and n(x,y) is the number of excited charges per unit volume. It is assumed that charges are distributed uniformly from the front to the back of the thin film. This condition can be readily checked in other materials by testing whether TR.PL measurements at different excitation fluences overlay when offset in time. If this does not hold diffusion lengths and/or surface recombination velocities should be included as an additional parameter, which is beyond the scope of this work (though this is explored further specifically for TR.PL measurements below). It is noted that in all experiments carried out the incident laser beam was sufficiently large (>0.1 mm diameter) that lateral carrier diffusion effects can be ignored.

The number of photons emitted to the surroundings at position x,y (per unit volume, per unit time) is escbr (p_on x, y) + n(x, y)²) (2) where p_esc is the probability an emitted photon escapes the material, b_r the radiative recombination rate and p₀ a background electron or hole concentration which contributes to luminescence upon recombination with an excited charge.

The PLQE can now be defined as the light emitted over the sample volume divided by the total (external) generation over the volume:

Here D is the sample thickness. This is a general form which can be applied to any laser beam distribution. It reduces to forms previously discussed for a uniform excitation density, while the focus of here is on a Gaussian shaped laser beam.

Information can be extracted from PLQE measurements when varying the incident laser power. Importantly, ratios between recombination rates can be extracted from PLQE alone. This is explained by considering a case when excitation density is halved, i.e., n(x,y) ->

. The same PLQE value can be obtained if a -> 2a, b -> 4b and so on, such that the products an(x,y), bn x,y)²... are unchanged. However, specific ratios between recombination coefficients are unaffected by a change in the value of n(x,y). These ratios are

A fitting method has been developed to rapidly explore parameter space and extract these ratios, alongside an estimation of the error in each value, from PLQE measurements at different excitation powers. In addition to ratios between different order trapping rates, the ratio of total to radiative second order recombination, — — , is extracted from PLQE fitting alone, which is explored further esc^r below.

A fitting approach for

Further information is now provided on the fitting approaching. This approach is based on a Gaussian laser beam. As noted above, the beam size, shape and power/intensity may impact or help to define a generation rate. Thus, the following description is based on the assumption that the beam used to optically interrogate the sample is Gaussian. It will be understood that the fitting approach may vary for other shapes of laser beam.

Begin by considering the laser generation rate. The intensity of a macroscopic laser beam, I(x,y), can be described by a two-dimensional Gaussian distribution. Defining the coordinate system such that the laser is most powerful at the origin, is incident perpendicular to the sample surface, and its ellipse axes align with the coordinates axes, it can be said that

Here I_o is the laser beam intensity at the origin and its diameter in the x and y directions is 2V2%₀ and 2V2y₀ respectively (noting that the diameter of a laser beam is typically defined as where the laser intensity falls to

of its maximum value).

It can be shown that I_o = where P is the power as measured on a n-XoVo

(macroscopic) power meter. The local external generation rate can be defined as

Here A is the fraction of laser light absorbed by the sample (which is measured in a PLQE measurement), E_ph the energy of a photon at the laser wavelength, D the sample thickness and P_min the lowest power used in any measurement.

For a Gaussian beam, equation 3 above can be simplified to one integral by using the transformations:

and

This gives a computational method to calculate the PLQE with four free parameters:

1. Calculate n(r) from

2. Calculate the PLQE using

The four free parameters are

all of which fall between 0 and 1000 for typical measurements, allowing for easier computation. These can be readily converted to the terms shown above in (4) by multiplication or division of ^G_0:min. Roughly, a and p determine the intercept of PLQE and the slope of the curve at low laser powers, y is a value close to the maximum PLQE value and 6 gives the decrease of PLQE following the maximum.

To fit PLQE data a statistical approach is used. Each dataset is fitted a number of times with random initial parameters. The fitting algorithm is re-run until a converged output is achieved. Following each of these initial fittings, a further three fits are done with random starting parameters within the regions bounded by each initial fit. Again, the fitting algorithm is re-run until a converged output is achieved.

Experimental methods

To test the example specific model, and screening technique, described above, a number of experiments were performed. These are now described. As the example specific model is based on a halide perovskite (specifically, an intrinsic semiconductor perovskite) and similar materials, the experimental methods are performed using the same materials. It will be understood therefore that the experimental methods described below are specific to these materials (3D halide perovskites), and other materials which can be screened using the general model described above may be prepared in different ways. Generally speaking, the materials to be screened may be thin films on substrates.

Sample fabrication

For all samples glass substrates and cover slips were cleaned via sonication in acetone and then isopropanol, each for 15 minutes, in an ultrasonic bath. Substrates were then cleaned by UV-Ozone treatment. After substrate cleaning all fabrication steps were in a N2 filled glovebox.

For MAPbh samples Pb (2 mmol, 0.922 g) and MAI (2 mmol, 0.318 g) was dissolved in 1 mL of DMF: DMSO (4: 1) solvent at room temperature with continuous stirring for 30 minutes. This halide perovskite solution was used as a stock solution for making thin films with different thicknesses. Perovskite solution was then dynamically spin coated at 2000 rpm (10 sec) and 4500 rpm (20 sec). Spin coated film were then annealed at 100 °C for 30 minutes. In order to achieve various thicknesses, the stock solution was diluted in following ratios (stock solution = 2.0mmol):

• Sample 1 = 0.5 mmol (50 pL stock solution + 150 pL of DMF:DMSO);

• Sample 2 = 0.75 mmol (75 pL stock solution + 125 pL of DMF: DMSO);

• Sample 3 = 1.0 mmol (100 pL stock solution + 100 pL of DMF: DMSO);

• Sample 4 = 1.25 mmol (125 pL stock solution + 75 pL of DMF: DMSO).

This produced samples of 230 nm, 270 nm, 330 nm and 760 nm respectively (termed samples 1, 2, 3 and 4 in main text respectively). MABF4 solution was spin coated in isopropanol (lmg/mL, 4000 rpm, 20 sec) on the surface of annealed perovskite films.

For Cso.o5FAo.79MAo.i6Pb(Io.83Bro.i7)3 samples, formamidinium iodide (FAI) (1 M, Greatcell Solar), methylammonium bromide (MABr) (0.2 M, Greatcell Solar), Pb (1.1 M, TCI) and PbBrz (0.2 M, TCI) was dissolved in anhydrous dimethylformamide/dimethyl sulfoxide (DMF: DMSO 4: 1 (v:v), Sigma). 5% CsI (Sigma) dissolved in DMSO (1.5 M) was then added to the precursor solution. To fabricate the halide perovskite films, 50 pL of prepared solution was spread onto the substrate and spun in a two-step spinning process: 1000 rpm for 10 s and 6000 rpm for 20 s. During the second spinning, 100 pL of chlorobenzene was dropped in the middle of film 5 s before the end of the process. After spinning, the substrates were transferred to a hotplate and annealed at 100°C for 1 hour.

FASno.5Pbo.5I3 samples were prepared following the description in Bowman et al (A. R. Bowman, M. T. Klug, T. A. S. Doherty, M. D. Farrar, S. P. Senanayak, B. Wenger, G. Divitini, E. P. Booker, Z. Andaji-Garmaroudi, S. Macpherson, E. Ruggeri, H. Sirringhaus, H. J. Snaith and S. D. Stranks, ACS Energy Lett., 2019, 4, 2301-2307.) Unpassivated and passivated in this work corresponds to 0 % and 5 % content of Zn .

MAPbh and FASno.5Pbo.5I3 samples were encapsulated with transparent epoxy immediately following fabrication.

Photoluminescence quantum efficiency

Photoluminescence quantum efficiency (PLQE) measurements were recorded using an integrating sphere, following the three measurement approach of De Mello et al (J. C. De Mello, H. F. Wittmann and R. H. Friend, Adv. Mater., 1997, 9, 230-232). In both photoluminescence and PLQE measurements a continuous wave temperature controlled Thorlabs 520 nm laser was used to photo-excite samples and excitation fluence varied with an optical filter wheel. The emission was recorded using an Andor IDus DU420A Silicon detector. The integrating sphere was regularly cleaned and re-painted, and calibrations were recorded specifically for the measurements.

Time-resolved photoluminescence Time-resolved PL spectra were recorded using a gated intensified CCD camera (Andor iStar DH740 CCI-010) connected to a calibrated grating spectrometer (Andor SR303i). A Ti :sapphire optical amplifier (1 kHz repetition rate, 90 fs pulse width) was used to generate narrow bandwidth photoexcitation (20 nm full-width at half maximum) with a wavelength of 520 nm, via a custom- built noncollinear optical parametric amplifier (NOPA).

Transient absorption spectroscopy

For the pump, a TOPAS optical amplifier was pumped with the output from a Spectra Physics Solstice Ace Ti : Sapphire amplifier (1 kHz) to produce a beam at 520 nm. The probe beam was generated with a LEUKOS Disco 1 UV supercontinuum laser (STM-l-UV, 1 kHz). The probe was split into a reference and probe and both were focused onto the sample. A pair of line image sensors (Hamamatsu, G11608) mounted on a spectrograph (Andor Solis, Shamrock SR- 303i) were used to detect the signal, using a custom-built board from Stresing Entwickslungsburo to read out the signal.

Local PLQE maps

Hyperspectral mapping of perovskite films was performed using an IMATM Vis microscope (Photon etc.). The setup uses a volume Bragg grating that splits light onto a silicon CMOS camera (Hamamatsu) allowing both spatial and spectral resolution of light. Photoluminescence maps were performed using a 405 nm continuous wave laser using a dichroic beam splitter to direct the laser onto the sample and remove the laser from the detected light. In order to calibrate the system for absolute photon counts, first a 658 nm continuous wave laser was coupled into an optical fibre. The laser was reduced in power using several optical density filters into the nW power regime to avoid saturation of the camera. The power of the laser at the end of the fibre was measured using a power meter (Thorlabs). The fibre was then coupled into the objective lens used for the measurements, in this case a Nikon 20 x, 0.45 NA, chromatic aberration corrected objective. The laser spot was imaged and total counts calculated. This gave a conversion between absolute numbers of photons and counts at this wavelength. The objective lens was then coupled into an integrating sphere along with a calibrated white light source (Ocean Insight, HL-3P-INT-CAL). A hyperspectral image of the diffuse light from the integrating sphere was measured to give spatial and spectral sensitivity. Combined with the laser measurement, this gave an absolute calibration of the system.

To measure absolute local photoluminescence quantum efficiency, the total absorbed and emitted photons per second must be calculated at each point. The total incident photon intensity was calculated by measuring both the power and spot size of the 405 nm beam. The reflection spectrum of the sample at 405 nm was found using a white light lamp and calibration mirror with known reflectance. A hyperspectral image of the mirror was measured, dividing the measured spectra at each point of the mirror by the known reflectance spectrum, giving the full incident spectrum of the white light source at each point. The reflection spectrum of the sample was then measured and by dividing by the incident white light spectrum, local absolute reflectance spectra are obtained. While ideally the reflectance at 405 nm would be used to match the laser, the lamp lacked sufficient power and there was low efficiency of the grating in this region. Therefore, the average value about 450±10 nm is taken. It is assumed that at 405 nm for the sample measured no light is transmitted. Knowing the fraction of light absorbed at each point and the incident intensity, the absorbed photon irradiance can be calculated. After measuring a calibrated hyperspectral photoluminescence maps, and integrating the photoluminescence spectra at each point, the emitted photon irradiance can be found. Dividing the two values point by point gives the local absolute PLQE.

Beam size measurements

Spot size was recorded using a Thorlabs beam profiler, with the beam profile being calculated using the Thorlabs beam profile software.

Atomic force microscopy

Sample thickness was recorded using an Asylum Research MFP-3D atomic force microscope in non-contact AC mode. A scratch on the surface on an unencapsulated sample was made using metal tweezers and the average difference in height between the material surface and the glass below as recorded (after 0^th order flattening and 1^st order plane fit were applied). All measurements and data processing were carried out on Asylum Research AFM Software version 15.

Fitting experimental PLQE data

To test this fitting approach, and ascertain its reliability, measurements on spin-coated passivated MAPbh films of different thicknesses were performed (with samples increasing in thickness from sample 1 to sample 4. Unless otherwise stated, samples were excited with 520 nm sources.

The PLQE is calculated (using the de-Mello et al. three measurement approach) as

Here P_c is the emission from the sample when directly excited, P_b sample emission when it is indirectly excited, L_a the laser counts measured when no sample is present in the sphere and A the fraction of light directly absorbed by the sample (equal to 1 - — ). As A is calculated for every PLQE measurement, it is possible to estimate the error in this value, o_A, from the variation in values measured. Errors in P_b, P_c and L_a, termed a_Pb, a_Pc and a_La, can be estimated during the measurement process (an error of 5 % is estimated in all these values). By standard error analysis it can be shown that the error in a PLQE measurement is given by:

Figure 2A shows the PLQE of a MAPbh film versus both incident laser intensity and excitation density n (as estimated from TAS and TR.PL measurements). As the laser intensity is increased the PLQE rises from ~0.1 % to ~4 %. This PLQE curve was fitted with several models based on equation 3, with different constraints on the parameters. Specifically, the complexity of the fit was reduced until the data fit well with the fewest arbitrary constants. This allowed for extracting of all ratios except — - — 3 for all samples.

( lesc^r ²

In order to determine the best fit of the PLQE data, several fits to the data were made, as shown in Figure 2B. Specifically, 'Full fit' uses the model presented in equation 3, 'no Auger' models have c = 0, 'no doping' have p₀ = 0, 'only trapping' have b = 0, and 'only radiative bimolecular' have p_escb_r = b. Each function fits the data to different degrees of accuracy.

It was found that 'Full fit', 'No Auger' and 'Only radiative bimolecular' fit the data best at all generation rates, while for other models there are more significant discrepancies. There are discrepancies for all models at low generation rates. This is attributed to not all traps being filled in this region, which is not included in equation 3. Including this effect requires three extra constants, making it much harder to reliably fit data at higher generation rates. By observing the 'no doping' fit, it was found that p₀ determines the value of the fitted PLQE at low excitation densities. Therefore, it is anticipated that the value of p₀ is relatively unreliable in the measurements.

The model 'No Auger' fits the data as well as 'Full fit', so 'No Auger' should be used to fit the data (as it only requires three fitting constants). Both 'No Auger' and 'Only radiative bimolecular' fit the data equally well and require the same number of fitting constants. However, 'Only radiative bimolecular' relies on a large Auger coefficient, as can be seen in the fit peaking at a generation rate of ~2 x 10²⁴cm“³s^-1. For this fit, even assuming an optimistic first order trapping rate of 10⁵ s ¹ (i.e. on the order of the best ever observed for MAPbh) gives an Auger recombination rate of ~10’²⁴ cm⁶s ¹, four orders of magnitude higher than has been observed in the literature. Therefore, 'No Auger' is the model which best fits the data with the fewest constants required, while giving physically reasonable parameters.

To convert ratios extracted from PLQE into absolute decay rates, a second measurement is needed. Here, TR.PL was employed, an accessible technique in many research labs, where the change in sample photoluminescence following photoexcitation is monitored, allowing values for p₀ and a to be obtained. During proof of concept measurements a laser beam with a 1 kHz repetition rate was used, which is long enough for most charge traps to be depopulated in halide perovskites (on the order of 10-100 ps). This modelling can be readily extended to systems with faster repetition rates.

Extracting absolute decay rates At low excitation densities, it can be approximated that ^dw^’^y'⁾ = -an(x,y), where t is time. As the photoluminescence is due to recombination of excited electrons and holes (i.e, TRPL K n²), it can be shown that

]n TRPL(t)) = k — 2at (5) where k is an arbitrary constant. Figure 2C shows In (TRPL counts) plotted as a function of time for two different initial excitation densities stated in the legend (for sample 1). The solid lines show a fit to each data set. The value of a may be extracted from the plot. For all spots measured, two measurements with different initial excitation densities were carried out (as indicated on Figure 2C) to ensure that the values of a were robust. It can be seen in Figure 2C that at higher excitation density there is a faster initial drop in the signal. The faster decay component at higher excitation density is attributed to the effects of second-order recombination, but restricts the fits (equation 5) to the portion of the data which is linear, as shown on the plot.

It is assumed that the pulsed laser beam also has a Gaussian distribution (as was observed in all measurements). It can be said that immediately following photoexcitation the number of locally excited charges is

where E_ph is the energy of a photon at the laser wavelength and f the frequency of the pulsed laser beam. The light emitted in time dt following photoexcitation (assuming that time bins are sufficiently short that n(x,y) does not significantly change during this time) is

Using the distribution of n(x,y), this integral can be solved to find that

Therefore, fitting TCSPC₀ versus P with TRPL₀ = MP_pu + NP_pu allows one to obtain

AM n = - ■

2nx_oy_oDE_phfN

Hence, accounting for spatial variation in the laser beam it can be shown that, immediately following excitation, the initial TR.PL signal is given by

Here P_pu s the measured laser power, A the fraction of incident laser light absorbed by the sample (which is measured through a typical PLQE measurement¹³), x₀ and y₀ describe the spatial distribution of the incident laser beam, E_ph the energy of a photon at the excitation wavelength and f repetition rate of the incident laser beam. The second term gives the initial excitation density multiplied by P_pu. This equation again assumes that charges are uniformly distributed from the front to the back of the film, even at early times. This is approximately justified as charges diffuse relatively rapidly from the front to the back of the film (on the order of a few ns at most). However, for the model to be more generalizable, the case where charges have not redistributed was also explored.

Measuring the scaling of TRPL₀ against P_pu allows for the absolute value of p₀ to be estimated (or if it is below instrument resolution, an upper bound). This fitting approach is demonstrated in Figure 2D for a MAPbh sample. Figure 2D shows the scaling of initial time-resolved photoluminescence counts for sample 1 on a log scale, TRPL₀, for different excitation densities, used to estimate p₀. In all measurements only an upper bound of approximately 10¹⁴ cm^-3 could be placed on p₀, which is a reasonable value as MAPbh films are generally thought to have low doping levels. Four spots on two different samples were measured to quantify the variation in a, as presented in Figure 2E. Figure 2E shows first order decay rates extracted from fittings in b), for measurements at four different positions across the surface of sample 1 and 2. Decay rates were found to be relatively uniform across the sample surface (~±25 % at most) , in spite of notable morphological, compositional and thickness variations reported within samples resulting from lab-based solution processing of perovskite thin films. This justifies the approach in combining values extracted from TRPL with those from PLQE measurements, even though different regions on the sample surface are measured. p₀ values obtained from the combined PLQE/TRPL approach are compared with upper bounds from TRPL measurements alone and it is found that values from the combined approach are convoluted with trap densities.

Benchmarking the extracted rates

The PLQE/TRPL approach is benchmarked by comparing values of p_escb_r and b with those obtained from TAS. TAS is a pump-probe measurement that is more specialised than PLQE or TRPL and requires more experimental infrastructure. Importantly, n is directly measured as a function of time. In all TAS measurements it was ensured that the pump beam was large compared to the probe beam so that a region of uniform excitation density was measured. Within this region spatial variation in the pump and probe beams and any carrier diffusion effects and state can be neglected dn

— = —an — bn — cn . (7) dt

TAS signals are measured at excitation powers where Auger recombination, which was not ascertained in PLQE measurements, could be neglected. Here

where n_t is the excitation density when t = 0. Furthermore, if one only focuses on the region where second order recombination dominates then

1 1

- = — + bt. (9) n n_t

Therefore, to fit the TAS data, versus time was plotted and the data was fit with both equation 8 and equation 9 to extract values of b, depending in each case on whether a could be reasonably extracted from the data. Figures 3A and 3B show TAS measurement data. Specifically, Figure 3A shows an example of these fitting approaches. Figure 3A shows versus time, as measured for sample 1 in TAS. Two fits to this data are shown - a linear fit to the high power data (equation 9), and an exponential fit (equation 8), to all the data. The inset figure shows recorded TAS signal with time (with each fluence offset in time to overlay the decays). In both plots each colour represents a different incident fluence. Again, for two samples b was measured at three different positions across the sample surface. It was found that there was more significant variation in b values (~±33 %) than the variation in a from TRPL measurements, further validating the approach of combining PLQE measurements with TRPL (which have <25 % spot-to-spot variation) rather than with TAS values.

To ascertain whether the PLQE/TRPL approach agrees with TAS measurements, all second order rates are plotted (j _escb_r and b) in Figure 3B. Figure 3B shows all second order rates extracted from PLQE/TRPL and TAS measurements. For samples 1 and 2, TAS and TRPL error bars are obtained from the spread of data at different points on the surface, while for sample 3 and 4 20 % and 30 % error bars are assumed, based on quality of data taken. Thus, there is good agreement between values of b obtained from PLQE/TRPL and TAS, validating this new analysis and fitting. Importantly, the PLQE/TRPL analysis also reveals that 7]_escb_r deviates from b values extracted in TAS, giving additional information and demonstrating a proportion of second order recombination is radiative. Notably, it was not possible to fit PLQE data and obtain physically meaningful results with any model where b = rj_escb_r, confirming a real discrepancy between the radiative and total second-order recombination. Non-radiative second order recombination has previously been reported in halide perovskites using different techniques and thus its presence here is validated.

Extracting decay rate ratios on the microscale

The framework presented here can be extended to the microscale by local PLQE maps at different (405 nm) laser excitation densities. This allows one to understand recombination occurring on the microscale using only a continuous wave laser (while a large spot size allows us to neglect diffusion effects). A local PLQE map of a passivated halide perovskite sample is presented in Figure 4A. Figure 4A shows a Micro-PLQE map (for a 2 mWcnr² excitation density) of a halide perovskite MAPbh film (sample 3). The PLQEs with power were fitted for this map. Poor fits were removed from the data. a b

Results for . and — — are presented in Figure 4B. Overlaid on these Wlescbr T|escb_r plots are the maximum, minimum, median, upper and lower quartiles as box and whisker plots. Samples were excited by a 405 nm beam. Both first and second order decay ratios are relatively uniformly distributed about their central value (with upper and lower quartiles within ~ 20 % of median value), meaning bulk results are representative of a large region of the sample and not due to a few abnormal regions. PLQE values are dominated by “ — (so are proportional to j esc^r first order charge trapping), as is expected at the laser fluences used. Again, only a proportion of second order recombination is radiative. This approach opens up studies in which local recombination can be analysed using continuous wave microscopic setups.

Second order non-radiative recombination

Second order non-radiative recombination may be present in certain optoelectronic semiconductor materials, such as 3D halide perovskites. The following description only applies to intrinsic optoelectronic semiconductors that have a non-radiative second order component.

Figure 4C shows measured PLQEs alongside those predicted from TAS and TRPL for samples 1 and 2.

Figure 4D shows the ratio of ^b from PLQE measurements for the four ^lesc^r

MAPbh samples measured. Sample 4 was extremely thick (>800 nm) and had low PLQE, so there is larger error in some values obtained. All PLQE error bars represent a 95 % confidence interval for fits (for samples 1 and 2, these are the maximum intervals obtained from any measurement on the sample surface).

To explore non-radiative second-order recombination further, in Figure 4C measured (macroscopic) PLQE for two samples are presented, alongside predicted PLQEs from time-resolved measurements (i.e. TRPL and TAS), using equation 3 and assuming only radiative second order recombination. There is a large discrepancy between measured and calculated values in this case. This discrepancy is more noticeable at higher generation rates, which have been explored less by others. Furthermore, the variation in — — is significantly beyond esc^r what would be expected from outcoupling, as is shown by the ratio of - in esc^r

Figure 4D. To confirm observations of a fraction of the bimolecular component being non-radiative are not isolated to these passivated MAPbh samples, measured low-bandgap FAPbo.5Sno.5I3 and triple cation (FAo.79MAo.i6Cso.o5)Pb(Io.83lo.i7)3 halide perovskites were measured (where FA is formamidinium. There was also a discrepancy between b and 7]_escb_r on the same order of magnitude in these samples suggesting the presence of non-radiative second order recombination is a general phenomenon in the range of halide perovskite sample compositions relevant for photovoltaic materials.

It is considered (quantitatively) whether the non-radiative second order rate can be explained by experimental error or parasitic absorption processes. Specifically, (i) a systematic error in PLQE measurements, (ii) an error in thickness or power measurements, and (iii) parasitic absorption within the halide perovskite thin film are considered. It has been demonstrated that none of these processes are able to fully explain non-radiative second order processes. Non-radiative second order rates are further explored by noting that when samples 1 and 2 measured above were first fabricated the samples had relatively low PLQEs (< 1 %). However, after two months storage in a nitrogen flushed box, where some passivation occurs due to low levels of oxygen present, their PLQEs had risen significantly. For sample 1, TAS measurements showed b changed from (5.8±1.5)xl0 ¹⁰ cnr³s ¹ to (8.5±2.0)xl0 ¹¹ cnr³s ¹ over this time. — — could also esc^r be measured reliably from PLQE before and after storage and comparable values of esc^br were obtained, of (1.4±0.7)xl0 ¹¹ cnr³s ¹ and (2.1±0.8)xl0 ¹¹ cnr³s ^-1, meaning the radiative rate is unchanged. Similar results were found for sample 2. Additionally, previously reported changes in b but constant 7]_escb_r when passivating FAPbo.5Sno.5I3 samples of similar thicknesses were found, as well as variation in b following passivation on MAPbh. Therefore, the results suggest that b can vary between samples (with passivation reducing b ) while 7]_escb_r remains constant.

There are several processes which could cause non-radiative second order recombination, including Auger recombination with a trap and processes related to ion motion. It is suggested that this is likely due to excited charges interacting with shallow traps following a Shockley-Reed-Hall type process (noting shallow traps have much higher densities, resulting in second order processes). Calculating device potential of measured films

The power of rapidly extracting decay rates by calculating the efficiency potential for a optoelectronic fabricated from a measured MAPbh sample is demonstrated. First, the sample's absorptance is measured using for example UV- Vis spectroscopy (another technique accessible to many laboratories), allowing calculation of the fraction of light absorbed and, for the purposes of modelling, an Urbach tail is fitted to the low energy edge of this absorptance spectrum.

To model a optoelectronic (or light emitting diode) a measure of the background doping density is required. To calculate the background doping density it is assumed that the sample has no Stokes shift, allowing for the use of the van- Roosbroeck-Shockley relation (for external recombination rates), that

Here n₀ is the background electron density, a_f and a_b the front and back absorptances of the encapsulated sample and (p_bb the black-body emission flux (per unit area, per unit solid angle, per unit energy). As background carrier density was below instrument resolution in experiments on halide perovskite films described here, it was assumed that n₀ = p₀ (i.e. an intrinsic semiconductor) and, additionally, it was assumed that the direct and diffuse absorptances are equal. This approach gives a background carrier density of 1.7xl0⁵ cnr³, in good agreement with other analyses. With this value calculated it is possible to calculate optoelectronic efficiencies using the same model as Pazos-Outon et al (L. M. Pazos- Outon, T. P. Xiao and E. Yablonovitch, J. Phys. Chem. Lett., 2018, 9, 1703-1711), but now non-radiative second order processes are also included in the modelling. Finally, the same Auger recombination rate as has been previously measured via TAS is assumed (though for reasonable values Auger recombination has only a small effect on resulting optoelectronic efficiencies).

Predicted current-voltage curves for a device comprised of one of the studied perovskite (MAPbh) absorbers are presented in Figure 5. Specifically, Figure 5 shows current-voltage curves for decay rates measured, with no first order loss (a = 0), no non-radiative (NR.) second order loss (— — = 1), and the (fully idealized esc^r or optimised optoelectronic) case of only intrinsic recombination (i.e. radiative and Auger recombination). The optoelectronic's absorptance is that measured for the encapsulated sample at its given thickness. The current voltage curve is almost

identical for the measured decay rates and — = 1 because first order non- Vesc^r radiative trapping is the dominant loss in these films. Therefore a should be the first process to focus on when optimising optoelectronics. Once a is reduced the maximum power point voltage can be further increased (by ~ 0.07 V) by reducing second order non-radiative processes.

Optoelectronic efficiency as a function of first and second order recombination rates was also plotted (not shown) for the same absorber layer but with optimal absorption i.e., the sample with a perfect back reflector and no incident light losses. The efficiency achievable with measured trapping rates is 22.7 %. Removing first order recombination raises this efficiency to 27.2 %, but to obtain the maximum efficiency of 28.6 % second order non-radiative recombination must be reduced. A key conclusion is that when a < 3 x 10⁵ s^-1, second order non-radiative recombination plays an important role in the optoelectronic. This corresponds to TRPL lifetimes at low excitation density of ~ 1 ps, relevant for many halide perovskite thin films. Second order non-radiative recombination plays an even more important role in light-emitting diodes. Here needs to be less than 10 to obtain external quantum efficiencies of greater

than 20 %.

Thus, a new method for rapidly extracting decay rate ratios from optoelectronic semiconductor thin films on both the bulk and microscale using photoluminescence quantum efficiency measurements is introduced. Combining the photoluminescence quantum efficiency measurements with time-resolved photoluminescence measurements allows for extraction of all absolute decay rates. It was demonstrated that the results are consistent with those obtained from the more infrastructure-intensive transient absorption spectroscopy. Importantly, the new measurement approach allows for a direct quantification of second order radiative and total recombination rates. Finally, it is shown how these extracted rates, alongside UV-Vis measurements, allow for the calculation of predicted optoelectronic efficiencies from the measured samples, and the effect of different decay rates on the limiting efficiency of a optoelectronic is quantified. For the measured sample reducing first order recombination rates should be focused on first, but when time-resolved photoluminescence lifetimes are longer than 1 ps, attention must be paid to second order non-radiative recombination processes. This work represents a fast method for linking simple spectroscopic measurements to optoelectronic performance criteria and revealing new recombination pathways, enabling fast screening of potential optoelectronic materials.

Figures 6 to 9 illustrate the various experimental setups that may be used to obtain the measurements mentioned above, while Figure 10 illustrates a single apparatus that may be used to obtain all the measurements required to screen optoelectronic materials is now described.

Figure 6 shows an example experimental setup for taking PLQE measurements. The setup comprises a laser 100, which may be a fixed wavelength continuous wave laser. The plurality of PLQE measurements may be obtained by varying an incident power of the laser 100 laser that is incident on the sample of the optoelectronic material. By varying the incident power, and therefore the intensity, of the laser being used to interrogate the sample, it is possible to measure how PLQE varies with intensity. The power of the laser may be at least lmW. More generally, the laser intensity on the sample being interrogated may be between 1-10,000 mWcm². This means that any suitable laser may be used in combination with suitable optics (e.g. focussing optical components), such that lower power lasers may be used for the PLQE measurements. The setup comprises filters and/or lenses 102. The setup comprises an integrating sphere 104, and a sample 106 to be analysed is placed within the integrating sphere 104, typically near the centre of the sphere. The setup comprises a spectrometer and camera system 108. In the example shown here, this may comprise a number of mirrors 110, a grating 112, and a detector 114. The detector 114 may be a silicon camera, such as a charge coupled device. A photodiode with a grating that can change the wavelength incident on the photodiode may be used instead, but this usually results in slow measurement speeds.

Figure 7 shows an example experimental setup for taking time-correlated single photon counting (TCSPC) measurements, which is a form of TR.PL. Here, a pulsed laser 200 is used to interrogate the sample 106, but a flash lamp may be used instead of the laser. Measurements may be made by varying an incident power and/or a repetition rate of the pulsed laser 200. By varying the incident power (intensity) and/or repetition rate of the light source being used to interrogate the sample, it is possible to measure how TR.PL varies with intensity and/or repetition rate. The pulsed laser 200 may be have a tunable repetition rate (e.g. 20MHz to 1kHz, which may require a pulse generator), and may be a fixed wavelength laser with a pulse length of less than a nanosecond. The setup may comprise a mirror 110 to direct the laser beam towards the sample 106. The setup comprises at least one detector 202, which may be a silicon single photon avalanche diode. The set-up may comprise filters and/or lenses 102. The filters may remove laser scatter, while the lenses may focus the signal on the detector 202. Signals from the detector 202 may be received by a computer.

Figure 8 shows an example experimental setup for taking measurements using an intensified charge coupled device (ICCD), which is a form of TR.PL. The setup comprises a laser 200, which may be a pulsed laser. The pulsed laser may have a relatively low frequency, e.g. around 1kHz. The laser 200 is used to interrogate the sample 106. The setup may comprise a filter 102 to remove laser from the signal before it is received by the spectrometer and camera system 108. In the example shown here, this may comprise a number of mirrors 110, a grating 112, and a detector 114. The detector 114 may be a gated silicon camera, which is an intensified charge coupled device.

Figure 9 shows an example experimental setup for taking UV-Vis measurements, which allows for the calculation of predicted optoelectronic efficiencies from the measured samples. The setup comprises a continuous wave white light source 300. Light from the white light source 300 passes through a grating 302 before reaching the sample 106. The sample 106 is provided within an integrating sphere 104. The sample 106 may be mounted in the centre of, at the front of, or at the back of the integrating sphere 104. Light emitted by the sample 106 passes through a grating 302 so that the same wavelength of light is observed/detected by detector 114. The detector 114 may be a silicon photodiode.

As mentioned above, the samples may be screened to determine their suitability to form photovoltaic devices such as solar cells. In this case, it may be useful to determine the maximum potential efficiency for the sample, because those samples with a efficiency over a threshold value may be determined to be suitable for photovoltaic devices. Thus, the method of the present techniques may further comprise: receiving, for the sample of the optoelectronic material, a plurality of absorption measurements for a set of wavelengths made using the at least one white light source 300 and the at least one detector 114. The method may comprise calculating, using the received absorption measurements, an intrinsic absorption value for the sample of the optoelectronic material, and calculating, using the first and second recombination rates and the intrinsic absorption value, a maximum potential optoelectronic efficiency for the sample optoelectronic material. The plurality of absorption measurements may be UV-Vis absorption measurements.

Figure 10 shows an example apparatus for obtaining all the measurements required to perform the rapid optoelectronic material screening technique described herein. It can be seen that Figure 10 incorporates the different setups shown in Figure 9. For this reason, the setup is not described in detail. Advantageously, this means a single apparatus may be used to perform the screening technique described herein. It will be understood that this is an example, non-limiting experimental arrangement.

Those skilled in the art will appreciate that while the foregoing has described what is considered to be the best mode and where appropriate other modes of performing present techniques, the present techniques should not be limited to the specific configurations and methods disclosed in this description of the preferred embodiment. Those skilled in the art will recognise that present techniques have a broad range of applications, and that the embodiments may take a wide range of modifications without departing from any inventive concept as defined in the appended claims.

Claims

1. A computer-implemented method for rapid optoelectronic material screening, the method comprising: receiving, for a sample of an optoelectronic material to be screened, at least one screening criterion; receiving a plurality of inputs, comprising: a plurality of photoluminescence efficiency, PLQE, measurements of the sample of the optoelectronic material, and a plurality of time-resolved photoluminescence, TR.PL, measurements of the sample of the optoelectronic material; calculating, using the received plurality of inputs, at least two parameters; and determining, using the at least two calculated parameters, whether the sample of the optoelectronic material meets the at least one screening criterion.

2. The method as claimed in claim 1 wherein receiving a plurality of inputs comprises receiving any one or more of: a sample thickness; a beam size of a beam of light used to optically interrogate the sample; a beam shape of a beam of light used to optically interrogate the sample; and a power of a beam of light used to optically interrogate the sample.

3. The method as claimed in claim 1 or 2 wherein the at least two calculated parameters comprises any two or more of: a radiative decay rate and a first order recombination rate; at least one first recombination rate associated with first order loss processes; at least one second recombination rate associated with second order loss processes; at least one third recombination rate associated with third order loss processes; at least one trap density; at least one background doping density; at least one diffusion length; at least one material recombination rate; at least one surface velocity; at least one diffusion rate; at least one photon absorption coefficient; and at least one recombination velocity.

4. The method as claimed in claim 1, 2 or 3 wherein the at least one screening criterion enables quantitative evaluation of characteristics of the sample of the optoelectronic material, and comprises any one or more of: a radiative decay rate; a first order recombination rate; a first recombination rate associated with first order loss processes; a second recombination rate associated with second order loss processes; a third recombination rate associated with third order loss processes; a trap density; a background doping density; a diffusion length; a material recombination rate; a surface velocity; a diffusion rate; a photon absorption coefficient; and a recombination velocity.

5. The method as claimed in any preceding claim wherein calculating at least two parameters comprises: calculating, using the received plurality of PLQE measurements, at least one ratio of recombination rates, wherein each ratio of recombination rates has a unit of the form cm^3x s^x, where x is any number; and converting, using the received plurality of TR.PL measurements, the at least one ratio of recombination rates into at least one recombination rate.

6. The method as claimed in claim 5 wherein calculating the at least one ratio of recombination rates comprises fitting the plurality of PLQE measurements using a minimal number of constants, and using the fit to calculate the at least one ratio of recombination rates.

7. The method as claimed in claim 6 wherein converting, using the received plurality of TR.PL measurements, the at least one ratio of recombination rates into at least one recombination rate comprises fitting the plurality of TR.PL measurements, using the fit to extract one or more parameters, and using the extracted one or more parameters to convert the at least one ratio of recombination rates into at least one recombination rate.

8. The method as claimed in claim 7 wherein the plurality of PLQE measurements and the plurality of TR.PL measurements are fit independently.

9. The method as claimed in claim 7 wherein the plurality of PLQE measurements and the plurality of TR.PL measurements are co-fit.

10. The method as claimed in any one of claims 5 to 9 wherein calculating, using the received plurality of PLQE measurements, a at least one ratio of recombination rates comprises calculating a plurality of ratios of recombination rates :

11. The method as claimed in claim 11 wherein the calculating of the plurality of ratios of recombination rates comprises: fitting the received plurality of PLQE measurements to a PLQE curve using a curve fitting algorithm; extracting an approximate value of p₀^p_escb_r using a portion of the PLQE curve corresponding to low laser power data; extracting an approximate value of . ^a using a portion of the PLQE curve j esc^r corresponding to intermediate laser power data; and extracting an approximate value of — — from a value close to a maximum esc^r value of the PLQE curve.

12. The method as claimed in claim 10 or 11 wherein converting, using the received plurality of TR.PL measurements, the plurality of ratios of recombination rates into recombination rates comprises: plotting the received plurality of TR.PL measurements to extract p₀, the background electron or hole concentration and a, the first order recombination rate; and using a to determine the second order recombination rate and radiative combination rate.

13. The method as claimed in any preceding claim wherein the at least two calculated parameters comprise a first recombination rate associated with first order loss processes and a second recombination rate associated with second order loss processes, and the method further comprises: receiving, for the sample of the optoelectronic material, a plurality of absorption measurements for a set of wavelengths; calculating, using the absorption measurements, an intrinsic absorption value for the sample of the optoelectronic material; and calculating, using the first and second recombination rates and the intrinsic absorption value, a maximum potential optoelectronic efficiency for the sample optoelectronic material.

14. A non-transitory data carrier carrying code which, when implemented on a processor, causes the processor to carry out the method of any of claims 1 to 13.

15. An apparatus for rapid optoelectronic material screening, the apparatus comprising: at least one light source for optically interrogating a sample of a optoelectronic material to be screened; a mechanism for varying incident light source power on the sample; at least one detector for detecting light emitted by the sample after interrogation by the at least one light source; and at least one processor coupled to memory arranged to: receive, for the sample of the optoelectronic material, at least one screening criterion; receive a plurality of inputs, comprising: a plurality of photoluminescence efficiency, PLQE, measurements of the sample of the optoelectronic material from the at least one detector, wherein the PLQE measurements are made by varying an incident power of the at least one light source, and a plurality of time-resolved photoluminescence, TR.PL, measurements of the sample of the optoelectronic material from the at least one detector; calculate, using the received plurality of inputs, at least two parameters; and determine, using the at least two calculated parameters, whether the sample of the optoelectronic material meets the at least one screening criterion.

16. The apparatus as claimed in claim 15 wherein the at least one light source comprises one or more of: a white light source; a flash lamp; a continuous wave laser; and a pulsed laser.

17. The apparatus as claimed in claim 15 or 16 wherein the plurality of PLQE measurements are obtained by varying an incident power of a fixed wavelength continuous wave laser.

18. The apparatus as claimed in claim 17 wherein the at least one detector is any one of: a photodetector, a photodiode, a camera, and a charge coupled device.

19. The apparatus as claimed in any of claims 15 to 18 wherein the plurality of TR.PL measurements are obtained by varying an incident power and/or a repetition rate of a pulsed light source.

20. The apparatus as claimed in any of claims 15 to 19 wherein the at least two calculated parameters comprise any two or more of: a radiative decay rate and a first order recombination rate; at least one first recombination rate associated with first order loss processes; at least one second recombination rate associated with second order loss processes; at least one third recombination rate associated with third order loss processes; at least one trap density; at least one background doping density; at least one diffusion length; at least one material recombination rate; at least one surface velocity; at least one diffusion rate; at least one photon absorption coefficient; and at least one recombination velocity.

21. The apparatus as claimed in any of claims 15 to 20 wherein the at least one screening criterion enables quantitative evaluation of characteristics of the sample of the optoelectronic material, and comprises any one or more of: a radiative decay rate; a first order recombination rate; a first recombination rate associated with first order loss processes; a second recombination rate associated with second order loss processes; a third recombination rate associated with third order loss processes; a trap density; a background doping density; a diffusion length; a material recombination rate; a surface velocity; a diffusion rate; a photon absorption coefficient; and a recombination velocity.

22. The apparatus as claimed in any of claims 15 to 21 wherein calculating, using the at least one processor, at least two parameters comprises: calculating, using the received plurality of PLQE measurements, at least one ratio of recombination rates; and converting, using the received plurality of TR.PL measurements, the at least one ratio of recombination rates into at least one recombination rate.

23. The apparatus as claimed in any of claims 15 to 22 further comprising at least one white light source.

24. The apparatus as claimed in claim 23 wherein the at least two calculated parameters comprise a first recombination rate associated with first order loss processes and a second recombination rate associated with second order loss processes, and the at least one processor is arranged to: receive, for the sample of the optoelectronic material, a plurality of absorption measurements for a set of wavelengths made using the at least one white light source and the at least one detector; calculate, using the absorption measurements, an intrinsic absorption value for the sample of the optoelectronic material; and calculate, using the first and second recombination rates and the intrinsic absorption value, a maximum potential optoelectronic efficiency for the sample optoelectronic material.

25. The apparatus as claimed in claim 24 wherein the plurality of absorption measurements are UV-Vis absorption measurements.