3.8 Quantifying Hysteresis with EIS
3.8.3 Temperature dependence
Since hysteresis apparently stems from ion migration, a thermally activated process, it is natural to investigate the effect of temperature on susceptance measurements. This was done on mixed cation cells as part of this investigation with the results shown in Fig. 3.16. It is observed that the characteristic (peak) frequency shifts to higher frequencies at elevated temperatures, as expected. However, it is also clear that some other change occurs simultaneously because the curves fail to coincide on bringing the temperature back down to the starting point. This could either be indicative of an irreversible form of degradation (perhaps involving the production of excess ions, explaining the faster response) or of a very slow process (such as the migration of a very slow species) over the timescale of measurement. We therefore did not attempt to extract an activation energy from these measurements. The increase in peak height indicates more extensive recombination loss at higher temperatures, consistent with the detrimental effect of heat on cell performance.
102 103 104 f (Hz) 1 1.5 2.0 2.5 3.0 3.5 4.0 B (S cm -2 ) 25 32 39 46 53 60 25 finish start
Figure 3.16: Effect of temperature on the susceptance peak of a mixed cation cell measured at max power point under illumination. This cell failed to return to its initial condition on lowering the temperature again (note that as previously the cell was stabilized for at least 600 s at each temperature prior to the EIS measurement).
3.9
Summary and Conclusions
In this chapter we have extended the theory and models developed in Chapter 2 for addressing I-V measurements to cover EIS measurements as well. Indeed, the simulations referred to here are the same drift-diffusion COMSOL MultiphysicsR models applied in sec. 2.5, modified only
to simulate EIS instead of I-V measurements. We have used these models to interpret the most common features seen in EIS measurements of standard perovskite solar cells, and found that they are successful in providing sound explanations for these with little modification.
A natural consequence of ion-induced hysteresis is that delayed recombination contributes significantly to capacitance measurements, which ultimately measure the delayed electrical re- sponse and therefore do not discriminate between charging currents and other sources of delay. We have described in detail how the AC capacitance obtained from measurements of a solar cell can be considered as a sum of two components, one from conventional charging currents and the other from phase-delayed recombination. In MAPbI3 delayed recombination occurs due to
ion migration, but many physical mechanisms could have a similar effect, including dielectric relaxation, slow carrier trapping and carrier accumulation. Indeed, we presented simulated ev- idence that a small but noticeable light-induced high-frequency capacitance can result purely
from carrier dynamics, with a mechanism that is similar to the ones responsible for producing negative capacitance in conventional diodes [39]. At low frequencies, we have argued that an- tiphase recombination is almost certainly responsible for observations of giant photo-induced capacitance. Inductive loop features and measurements of low-frequency negative capacitance can also be explained simply with recombination currents in mind, but further studies will be required for experimental confirmation.
One of the most common applications for EIS in cell characterization is the determination of doping densities via capacitance-voltage profiling. On this topic we have presented experimental C-V measurements, interpretively backed by simulations, which indicate significant depletion in the titania layer below forward bias voltages of ≈ 0.6V. This depletion implies that Mott- Schottky analysis should be undertaken with great care to avoid confusing the titania doping with that of the perovskite layer, which appears essentially un-doped in the cells fabricated for this study.
Despite uncertainties concerning the mechanism by which a large ionic charge is accommo- dated at the contact layer interfaces, it is still reasonable to expect that many important features in EIS measurements are properly identified by ionic drift-diffusion models. This is because the mechanisms described in this work result from generic processes, such as the qualitative effect of ion migration on internal electric fields, and ionic relaxation, that are not expected to depend sensitively on the proper description of adsorption processes.
The success of the models presented here in explaining several major features of interest, such as giant photo-induced capacitance, inductive loop features and capacitance-voltage curves, builds on their earlier achievements in the domain of I-V hysteresis [8, 9, 11, 90, 92]. The inter- pretations that follow from these models therefore fit into a unified understanding of transient behaviour, which should enable more integrated usage of EIS as a tool to supplement stan- dard I-V measurements. On this topic we have shown that measurements of the imaginary admittance provide a straightforward and relatively unambiguous quantifier of hysteresis, with several advantages over the conventional hysteresis indices. The adoption of EIS as a method to quantify hysteresis would resolve the ambiguity in commonly encountered statements such as that certain cells are “hysteresis-free”, and better distinguish between genuinely suppressed hysteresis and mere changes in its characteristic timescale.
Parameter Layer Value(s) Figures Description
NA−ND(cm−3) ABS 0, 4·1017,0 3.5, 3.6, 3.8-3.9 Net doping
r ABS 62[25] All Perimittivity
Eg(eV) ABS 1.6 All Band gap
χ(eV) ABS 3.9 All Electron affinity
Nc, Nv(cm−3) ABS 8·1018,6·1018 All Density of states t(nm) ABS 300, 500, 500, 350 3.5, 3.6, 3.8, 3.9 Mobility
µn(cm2V−1s) ABS 1 All Mobility
µp(cm2V−1s) ABS 1 All Mobility
σn(cm−2) ABS 1·10−15 All SRH Cross-section
σp(cm−2) ABS 1·10−17 All SRH Cross-section
Ntd(cm−3) ABS 1·1014,1·1014,1·1014to 1·1016, 1·1015 3.5, 3.6, 3.8, 3.9 Defect Density (Donor type)
Et ABS mid-gap All SRH Energy
Nion(cm−3) ABS 1·1019,2·1018,1·1018,1·1019 3.5, 3.6, 3.8, 3.9 SRH Energy D1(cm2s−1) ABS 1·10−10,1·10−9,to 1·10−11,1·10−11 3.5, 3.6, 3.8, 3.9 Ionic Diff. Const. (fast)
D2(cm2s−1) ABS 1·10−13,0, 0, 0 3.5, 3.6, 3.8, 3.9 Ionic Diff. Const. (slow)
ND(cm−3) ETL 1·1017,1·1017to 6·1018,1·1017 3.5-3.6, 3.8, 3.9 Net doping
r ETL 55,[110] 24[111] 3.5-3.8, 3.9 Perimittivity
Eg(eV) ETL 3.39 All Band gap
χ(eV) ETL 4.1 All Electron affinity
Nc, Nv(cm−3) ETL 8·1018,6·1018 All Density of states
t(nm) ETL 120 All Thickness
µn(cm2V−1s) ETL 0.1 All Mobility
µp(cm2V−1s) ETL 0.1 All Mobility
NA(cm−3) HTL 1·1019 All Net doping
r HTL 3 All Perimittivity
Eg(eV) HTL 3.0 All Band gap
χ(eV) HTL 2.1 All Electron affinity
Nc, Nv(cm−3) HTL 8·1018,6·1018 All Density of states
t(nm) HTL 200 All Thickness
µn(cm2V−1s) HTL 0.1 All Mobility
µp(cm2V−1s) HTL 0.1 All Mobility
vn(cm s−1) ABS-ETL 1·104,10, 1·10−3to 1·102, 1·104 3.5, 3.6, 3.8, 3.9 Recombination velocity vp(cm s−1) ABS-ETL 1·104,10, 1·10−3to 1·102, 1·104 3.5, 3.6, 3.8, 3.9 Recombination velocity
Et ABS-ETL mid-gap All SRH energy
Dit(cm−2eV−1) ABS-ETL 3·1014,0 to 3·1014, 0 3.5, 3.6, 3.8-3.9 Trap density vn(cm s−1) HTL-ABS 1·104,10, 1·10−3to 1, 1·104 3.5, 3.6, 3.8, 3.9 Recombination velocity vp(cm s−1) HTL-ABS 1·104,10, 1·10−3to 1, 1·104 3.5, 3.6, 3.8, 3.9 Recombination velocity
Et HTL-ABS mid-gap All SRH energy
Dit(cm−2eV−1) HTL-ABS 3·1014,0 3.5, 3.6-3.9 Trap density
φm(eV) metal(ETL) 4.4 All Work function
φm(eV) metal(HTL) 5 All Work function
Table 3.1: Device parameters used in our EIS simulations, chosen to emulate a “normal- structure” TiO2/MAPbI3/Spiro-OMeTAD cell. Values without references are estimates, with
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