This part talks about the electrodeless time-resolved microwave conductivity (TRMC) method, explains a kinetic model for quantitative data, and shows TRMC readings on MAPbI3 and MAPbBr3 and how they compare to other values that have been recorded. It also shows TRMC readings on metal halide perovskites (MHPs) with various CSTLs, including Spiro-OMeTAD as a hole transport layer and C60 as an electron transport layer. This chapter talks about what these mobilities and trap densities mean for the behaviour and collection of charges in a cell.
Time-Resolved Microwave Conductivity
The TRMC process is a way to look into how photo-induced charge carriers move in semiconductors that don’t carry electricity very well in general. It works by looking at how the electric field of microwaves (GHz range) interacts with mobile carriers. This lets us find out if a semiconductor is photoconducting without having to touch it with wires.
𝼏 = e^∑i+ini+𝼇i shows how the electrical conductivity, 𝼏, changes with the number of free charge carriers, n, and how quickly they move, 𝼇. To find out the photoconductance, TRMC is used. The photoconductance scales with the concentration n and movement 𝼇 of free electrons and holes over time. The undulating line shows how strong the microwave electric field is as it moves through the excited MHP.
Photo-excitation by laser bursts of 3–5 ns FWHM with a frequency that can be changed and a repeat rate of 10 Hz is used in the TRMC setup. Microwaves with a frequency in the 10 GHz range are made with a voltage-controlled generator. The important sample is put in a microwave cell that ends in a metal grating. This grating mostly lets the laser power through but completely blocks the microwaves. On top of the grate is glued a crystal window that keeps air out of the cell. The sensitive factor, K, is much higher when there is an iris than when there is no iris. The reaction time goes from 1 ns to 18 ns, though, because of this sensitive rise.
If you know the sensitivity factor K, you can use Eq. (9.3) to get a number value for ΔG from the measured ΔP(t)/P. The last two unknowns that can be used to describe the TRMC signal are the mobility, 𝼇, and the charge carrier generation yield, 𝼑. If each photon that is received makes one positive charge carrier and one negative charge carrier, then Eq. (9.1) is as simple as: 𝼏 = en(𝼇e + 𝼇h), where n is the concentration, 𝼇e is the electron mobility, and 𝼇h is the hole mobility. The yield can be written as 𝼑 = Ln FAI0, where I0 is the laser’s strength in photons per pulse per unit area and FA is the amount of light that is absorbed at the trigger wavelength.
By writing the TRMC signal as 𝼑(𝼇e + 𝼇h) products, it is possible to directly compare the TRMC readings of various samples. During a TRMC measurement, charges do not move around in the photo-active layer like they do in DC methods. This means that moving across grain borders doesn’t slow things down unless the grain size is less than 100 nm. People usually describe mobilities that are at least a factor of ten lower when extra carriers have to cross grain borders.
Most of the time, time-resolved photoluminescence (TRPL) measurements only find radiative recombination events. However, TRMC measurements give more information than TRPL measurements because they show the recombination of all free mobile charge carriers, whether they are radiative or not. So, TRMC is a great way to study how recombination works in MHPs that don’t let much light out and absorb a lot of the photons that are released. Examples of these are large MHP crystals or MHPs that have a transport material added to them, where most of the recombination is not radiation-based.
Global Modeling of TRMC Data
The TRMC method is used to get numerical information from TRMC readings, which take into account the different ways that photo-excited electrons and holes can recombine. The model assumes that charges are generated and lost in a uniform way. This can be shown in the lab by using an excitation wavelength close to the absorption start. Because most MHPs have a relatively low exciton binding energy compared to temperature energy, pairs of electrons and holes break apart into loose conduction band (CB) electrons and valence band (VB) holes.
The generation term, GC, talks about how charge carriers are made and takes into account the laser pulse’s total light strength and its shape over time. In most perovskite semiconductors, they are filled by accident because of flaws and fault states in the crystal structure. This makes more CB electrons (n-type) or VB holes (p-type) that were there before the photoexcitation. The amount of dark carriers, denoted by p0, doesn’t change the photoconductance because the TRMC method is an AC method, but it does change the rates of recombination.
The steps in Figure 9.3 are carried out by a set of linked differential equations that explain ne, nh, and nt as VB, CB, nh, ne, po, Gc, K2, KD, KT, and NT. k2, kT, and kD are the rate constants for electron–hole recombination from one band to another, trap filling, and trap emptying.
To get correct numbers for all the kinetic factors that describe a certain MHP, you should record a set of TRMC tracks that are caused by laser levels that change a lot. The only thing that changes about these TRMC lines is the laser strength, which is part of the generation term, GC.
TRMC Measurements on MAPbI3 and MAPbBr3
The research is mostly about how microparticles (MHPs) move charge carriers and how that changes photoconductance, band-to-band recombination rate constants, and the density of the trap and background. The results show that the signal’s rise at t = 0 comes from the creation of mobile extra charges. The highest signal level is equal to the product of the charge carrier generation yield and mobility. Higher levels make the breakdown dynamics of MAPbI3 faster, which is what you’d expect to see in a situation where higher-order recombination is the main process. When the excitation density is very low, the number of extra carriers created is less than the trap density. This means that the band-to-band second-order recombination process is no longer the main decay process because the minority carriers are quickly trapped.
For MAPbBr3, the decline rates are very different. There is a very quick, sharp drop followed by a long-lasting signal that stays almost the same. When the intensity goes up, the normalised TRMC gets smaller, just like with MAPbI3. A single set of kinetic factors is used to fit the kinetic model to the TRMC tracks for both MHPs. The results are shown in Table 9.1.
There are some big differences between the two MHPs, such as differences in mobilities, band-to-band recombination rate factors, and the numbers for the trap and background densities. The mobilities for MAPbI3 layers are about 50 to 100 cm2 (Vs)−1. These are similar to values found before using TRMC or optically pump terahertz probe measurements (OPTP). Between MAPbI3 and MAPbBr3, Table 9.1 shows a clear difference in how they move. The basic Drude model says that mobility is equal to 𝼇 = em∗ (9.11), where 𝼏 is the scattering time and m* is the effective mass.
It is more than two orders of magnitude bigger between MAPbI3 and MAPbBr3 in terms of the number of traps they have. As you can see from the TRMC decay rates shown in Figure 9.4a,b, the number of dark carriers has a big impact on how charges move. We can figure out how long a carrier will last in the conduction or valence band by looking at the factors. We can get an idea of the charge carrier diffusion length, R, by knowing the half life, 𝼏1/2, and the mobilities.
R = − Dₜ^1−2 (9.12). You can find the diffusion coefficient, D, by fitting the TRMC lines and using the Einstein–Smoluchowski equation, which says that D = kT₼/e.
As you can see in Figure 9.4c, R for the minority carriers in MAPbI3 is not nearly as big as R for the majority carriers. When the activation level is high enough to produce a charge carrier concentration just below NT, the values of R for electrons and holes are the same and highest. Because band-to-band recombination is stronger at higher numbers, R gets smaller for both types of carriers. So, the lengths of the electron and hole diffusion paths are most balanced when the charge density is just above the trap density. This is what you want for a working solar cell to collect electrons and holes as efficiently as possible.
TRMC Measurements on MAPbI3 and MAPbBr3
with Charge Selective Contacts
The study’s goal is to figure out what kinds of carriers and traps are responsible for the charge carrier behaviour seen in perovskite solar cells. Time-Resolved Microwave (TRMC) is a method that doesn’t tell the difference between holes and electrons. Both types of carriers add to photo conductivity. A Charge Selective Contact (CSTL) was put on top of the MHP to find out what kind of carrier is the minority carrier for each layer. We used CSTL as a layer that takes in electrons and Spiro-OMeTAD as a layer that moves holes around.
The signal for all bilayers drops quickly below the signal for the bare perovskite layer. This shows that the TRMC signal is mostly caused by countercharges still being in the perovskite. It was found that the charge carrier lifetimes for MAPbI3-based bilayers are very long, lasting many microseconds. This means that it takes a long time for the CSTL to recombine the removed carrier with the leftover counter charge. It is possible to figure out how mobile the counter charge is if either electrons or holes are fully extracted from a CSTL. However, extraction is probably not finished yet because charge binding and band-to-band recombination happen at the same time.
For MAPbI3, both charge carriers are easily taken out, and their speeds are about the same.
The signal quickly drops to almost nothing for bilayers containing MAPbBr3. However, there aren’t many changes between the bare layer and the MAPbBr3/spiroOMeTAD layer. When we put this information together with the results of the fits from Section 3, which showed that there are a lot of deep trap states, we can say that these states trap holes and that most of the mobile carriers are electrons. This may help explain why solar cells made from MAPbBr3 usually don’t work very well.
The time-resolved microwave approach (TRMC) is a direct method that doesn’t need long-term exposure to electric fields, which could cause ions to move around in perovskites. In the 1-sun equivalent state, photosynthesis can happen at a lot of different rates. The method was used on MAPbI3 and MAPbBr3 with and without CSTLs in Sections 9.3 and 9.4. The kinetic parameters that describe how the charge carriers move are found by looking at the fits on the bare layers that were taken at different light levels. You can get specific details about the electron and hole mobility as well as the sign of the carriers from the bilayers.
Finally, the TRMC method can show how charge carriers change over time on timescales important for optical devices because it is flexible and doesn’t require any electrodes.
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