Simon E. J.
O'Kane§
a,
Giles
Richardson§
b,
Adam
Pockett
c,
Ralf G.
Niemann
c,
James M.
Cave
*a,
Nobuya
Sakai
d,
Giles E.
Eperon
d,
Henry J.
Snaith
d,
Jamie M.
Foster
e,
Petra J.
Cameron
c and
Alison B.
Walker
a
aDepartment of Physics, University of Bath, UK. E-mail: j.m.cave@bath.ac.uk
bMathematical Sciences, University of Southampton, UK
cDepartment of Chemistry, University of Bath, UK
dDepartment of Physics, University of Oxford, UK
eDepartment of Mathematics, University of Portsmouth, UK
First published on 15th December 2016
The current decay in response to a sudden change of applied bias up to 1 V has been measured on a methylammonium lead triiodide perovskite solar cell with titania and spiro-OMeTAD transport layers, for temperatures between 258 and 308 K. These measurements are highly reproducible, in contrast to most other techniques used to investigate perovskite cells. A drift–diffusion model that accounts for slow moving ions as well as electrons and holes acting as charge carriers was used to predict the current transients. The close fit of the model predictions to the measurements shows that mobile ions in the perovskite layer influence transient behaviour on timescales of up to 50 s. An activation energy of 0.55 eV is inferred from fitting simulations to measurements made at room temperature.
Historically, the credibility of reported PCE values for perovskite solar cells has been undermined by widespread use of a voltage scanning protocol that takes advantage of hysteresis in the current density–voltage (J–V) characteristics of some cells to obtain high initial PCE values that decrease as the cell is held at the maximum power point (MPP) voltage. Zhang et al.7 measured how the efficiency fell over time for different cell architectures: in the most extreme case the PCE fell from 11.5% to zero. Hysteresis in the current–voltage characteristics of perovskite solar cells was first reported in 20148 and confirmed by subsequent studies,9–12 although hysteresis loops in conductance measurements of bulk perovskite had been reported earlier.13
In most perovskite solar cells, the organometal halide perovskite absorber is sandwiched between an electron transporting (hole blocking) layer and a hole transporting (electron blocking) layer.14 One of the most commonly used,15 and indeed longest used,1 perovskites is methylammonium lead iodide (CH3NH3PbI3), or MAPI. The methylammonium (CH3NH3+) cation has been successfully replaced with formamidinium16 (NH2CHNH2+) and/or caesium17 (Cs+); a mixture of these cations can also be used.18,19 There are efforts to replace lead (Pb2+) with tin (Sn2+), though this usually comes at the expense of performance and stability.20 Iodide can be replaced with other halides, either entirely or in a mixture.3,21,22 The highest efficiency perovskite cells employ mixed-cation lead mixed-halides.18,23–25
The hole transporting layer (HTL) usually consists of spiro-OMeTAD (2,2′,7,7′-tetrakis-(N,N-di-p-methoxyphenylamine)-9,9′-spirobifluorene) that forms a planar heterojunction with the perovskite. The most commonly used electron transporting layer (ETL) is titania (TiO2), which can form either a mesostructured1 or planar26 heterojunction with the perovskite.
Recently, there have been a number of well-documented ‘hysteresis-free’ cells27–31 in which the titania ETL is replaced with a different material, popularly the fullerene derivative PCBM (phenyl-C61-butyric acid methyl ester). Often the spiro HTL is replaced as well, such as with the polymer PEDOT:PSS (poly(3,4-ethylenedioxythiophene)polystyrene sulfonate). It has been argued that the removal of hysteresis by this change in the blocking materials demonstrates that ion motion in the perovskite is not the source of hysteresis.27,32 However it is notable that Bryant et al.33 demonstrate that apparently hysteresis-free cells employing PEDOT:PSS, MAPI and PCBM still exhibit significant hysteresis when cooled to 175 K and that hysteresis under normal operating conditions depends on the choice of transport layer.34
In addition to hysteresis in both conductance and current–voltage measurements, unusual features have been observed in impedance spectroscopy,8,35,36 intensity-modulated photocurrent spectroscopy (IMPS)35,36 and open-circuit voltage decay.36–38 These anomalies occur on the same timescale as the hysteresis, suggesting that they may have a common origin.
It was originally suspected13,39 that hysteresis was caused by the creation of ferroelectric domains via dipoles in the methylammonium (CH3NH3+) ions, but this hypothesis has now been ruled out by electrical measurements.40 Two other possible origins of hysteresis have been proposed: large-scale trapping of electrons27,41,42 and mobile ions in the perovskite film.40,43,44 Yang et al.45 identified that iodide (I−) ions were moving, either via interstitials or vacancies. Walsh et al.46 calculated that the activation energy required to form Schottky defects (vacancy pairs) was much lower than that required to form Frenkel defects (an interstitial and a vacancy). Several computational studies47–50 have been conducted in order to calculate the activation energies for the migration of each ionic species in MAPI and other perovskites; all agree that I− is the most mobile ion species, although the calculated values of activation energy for I− ions vary between 0.08 eV and 0.58 eV.
While activation energies allow calculation of the rate at which ions move, the only way to evaluate the effect of ion motion on current–voltage characteristics is with electrical transport modelling. Similar to ions in an electrolyte, ionic charge accumulates in Debye layers (also termed double layers) at the edge of the perovskite under the influence of the built-in field.51 Li et al.52 used X-ray photoemission spectroscopy to measure the iodine to lead (I/Pb) ratio at various points in a MAPI film. They found that application of an external field increased the I/Pb ratio at the positive electrode, indicating a build-up of I− ions, and decreased the ratio at the negative electrode, indicating a depletion of I− ions. Electron and hole densities are negligible in comparison to ion vacancy densities at all applied voltages up to and including the built-in voltage. The result of this build-up of net charge is band bending, as postulated by Li et al.52 and others.12,47,51,53 The band structure of the film as seen by charge carriers is modified electrostatically by the Debye layers, which act to screen the built-in field.
The Debye length is a measure of the characteristic width of these layers and is over two orders of magnitude smaller then the perovskite layer width (∼1.5 nm compared to hundreds of nm) at the high ion vacancy densities encountered in perovskites. These narrow layers create a major computational challenge when computing the solution over the entire width of the perovskite layer as a very fine mesh spacing is required to represent the structure of the Debye layers. Richardson et al.51 argued that electron and hole densities are negligible in comparison to ion vacancy densities at all applied voltages up to and including the built-in voltage. They could then use the method of matched asymptotic expansions to avoid this pitfall and model the Debye layers accurately, providing quantitative agreement with experimental current–voltage curves in a model assuming mobile ion vacancies and trap-assisted electron–hole recombination. Van Reenen et al.54 also found that both ion motion and electron trapping were required to explain the current–voltage hysteresis of the type observed in the literature but were unable to reproduce results in the physically relevant regime, nor to achieve quantitative comparison with experiment. We suggest a contributory factor is that the resolution of the finite difference method employed by these authors was too coarse (4 nm) to accurately resolve the narrow Debye layers.
In this article we explore the effects of ion motion in perovskite solar cells by measuring their dark current decay transients. These measurements are made in the dark by causing the applied potential difference to jump from equilibrium, at 0 V, to Vjump and measuring the resulting current flow through the cell as a function of time. Prior to the measurement the cell is held in the dark at open circuit for sufficient time to allow it to equilibrate (up to an hour at low temperatures). No more than 1 mV of residual voltage could remain after each measurement for reproducibility on this level. This procedure ensures reproducibility of the observed current trace by eliminating preconditioning effects; indeed we have shown that repeating the experiment on the same device by allowing it to equilibrate again after a measurement gives a result that exactly overlays the previous one. In contrast, most reported measured current–voltage characteristics suffer from irreproducibility – each time the measurement is made the results are different, even for the same device, as a result of residual preconditioning from previous measurements, slight changes in illumination etc.
Obtaining this repeatability allows us to eliminate extraneous effects such as cell degradation and material changes arising from exposure of the perovskite to light, the latter was discussed in Gottesman et al.55 The measured current through a cell depends on the ability of charge carriers to traverse the energy landscape of the materials. As we later discuss in detail, ion motion has a significant effect on the electrostatic potential within the perovskite layer. By probing the physics in this relatively simple setup, we gain crucial insight into the operation of perovskite cells during standard lab tests such as J–V curve sweeps and stabilised power conversion efficiency measurements, and how, and to what extent, ion motion may affect the results. We therefore improve our understanding of how perovskite-based cells behave under normal generating conditions, and what material advantages may be exploited and what barriers need to be overcome.
Here, we report measured dark current decay transients and use the asymptotic method of Richardson et al.51 along with the experimental results to perform a drift–diffusion simulation on the system. We assume the diffusion coefficient D+(T) of iodide ions shows Arrhenius behaviour as a function of temperature.47 By measuring and simulating the transients at a fixed temperature for a range of Vjump, we deduce a value for the activation energy for iodide ion movement for our cell. We then use the obtained D+(T) to simulate transients for a fixed Vjump at a variety of temperatures and compare to experiment. Experiment and model employ a FTO:TiO2:MAPI:spiro-OMeTAD:Au cell, as shown in Fig. 1. The cell was fabricated using the chloride precursor route as described in Section 2.4.
Fig. 1 Band diagram of the device. Band edges for the perovskite and transport layers are represented by solid lines, with energies relative to the vacuum level in eV labelled.56 Fermi levels for the contacts are depicted by dotted lines – values for these levels were not required by the model and are difficult to measure accurately, so are not given. |
By using the asymptotic model, most of the features observed in the experimental transients can be understood in terms of ion motion. The structure in our transient results (Fig. 5 and 8) provides the first direct evidence that the rate of charge injection into the device, i.e. the current, is dependent on the position of mobile ions within the cell. This result is a corroboration of our earlier work suggesting ion motion is responsible for the widely observed current–voltage hysteresis in perovskite devices.51 Analysis of the effect of temperature variation on the timescale of the transients confirms the thermally activated mechanism of ion migration assumed by Eames et al.47 and others.48–50
We implement this model in a two-step process. Firstly, the electric potential ϕ(x,t) – a function of position and time – is found using a time-dependent asymptotic model that considers solely the mobile I− vacancies and their effect on the potential, assuming that the contribution of electrons and holes to the potential is negligible. A schematic diagram of the Debye layer at the MAPI:spiro interface as modelled in this step is illustrated in Fig. 2. We denote the position of the interface of the perovskite with the titania as x = 0, and the interface of the perovskite with the spiro as x = b. As the Debye layers are very thin compared to the perovskite film width, we further denote the edge of the Debye layer on the titania side of the perovskite film as x = 0+ and the edge of the layer on the spiro side as x = b−. The distinction between b and b− is illustrated in Fig. 2. Secondly, a drift–diffusion calculation of electron and hole motion is performed using this potential profile which can then be used to determine the output current density J(t). An accurate calculation of the potential ϕ(x,t) and I− vacancy motion can be made by using an asymptotic method51 (based on earlier work57) that overcomes the numerical difficulties encountered when tackling this stiff problem and accounts for the vacancy charge build-up in the narrow Debye layers (of size comparable to the Debye length Ld) in terms of a nonlinear capacitance.
The model and its asymptotic solution have previously been described in detail51 and are briefly summarized here. For volume-averaged vacancy concentrations N0 > 1017 cm−3 it can be assumed that the main contributions to space charge in the perovskite are concentrated in the Debye layers adjacent to the TiO2 and spiro blocking layers. The bulk of the perovskite between these two layers is therefore free of net charge.
Assuming that only the positively charged I− vacancies can move and the negatively charged CH3NH3+ and Pb2+ vacancies are static, the charge per unit area Q in these layers is related to the voltage drop V across them by the asymmetric capacitance relation
(1) |
(2) |
By considering charge conservation and the charge neutrality of the system, it can be shown51 that the problem of determining the electric field can be reduced to determining the solution of the single ordinary differential equation
(3) |
The difference between the model here and that reported in Richardson et al.51 is that here we model recombination by a combination of a bimolecular scheme within the perovskite and so-called surface recombination between an electron in the TiO2 conduction band and a hole in the MAPI valence band across the interface. The bimolecular recombination rate Rb is given by
Rb(x,t) = B(np − ni2), | (4) |
Rs(x = 0+,t) = vsp(x = 0+,t), | (5) |
(6) |
(7) |
(8) |
(9) |
Symbol | Description | Value | Source |
---|---|---|---|
b | Perovskite layer width | 450 nm | Measured |
D ∞ | High T vacancy diffusion coefficient | 1.35 × 10−3 cm2 s−1 | Eames et al.47 |
E C | Perovskite conduction band minimum | −3.7 eV | Schulz et al.56 |
E Fs | Spiro-OMeTAD Fermi level | −5.0 eV | Schulz et al.56 |
E Ft | TiO2 Fermi level | −3.95 eV | Schulz et al.56 |
E g | Perovskite bandgap | 1.7 eV | =EC − EV |
E V | Perovskite valence band maximum | −5.4 eV | Schulz et al.56 |
ε p | Static dielectric constant | 24.1ε0 | Brivio et al.59 |
ĝ c | Conduction band density of states | 8.1 × 1018 cm−3 | Brivio et al.59 |
ĝ v | Valence band density of states | 5.8 × 1018 cm−3 | Brivio et al.59 |
L d | Debye length | 1.5 nm | =(εpVT/(qN0))1/2 |
N 0 | I− volume-averaged vacancy density | 1.6 × 1019 cm−3 | Walsh et al.46 |
n(x = 0) | Density of electrons in perovskite at TiO2 boundary | 4.8 × 1014 cm−3 | Eqn (6) |
n i | Intrinsic carrier density | 3.0 × 104 cm−3 | Eqn (8) |
p(x = b) | Density of holes in perovskite at spiro boundary | 1.0 × 1012 cm−3 | Eqn (7) |
V T | Thermal voltage | 0.0257 V | Eqn (9) |
In practice, since the simplified asymptotic model is solved only in the charge-free bulk region between the thin Debye layers, we require boundary conditions for the carrier densities at the edge of the Debye layers: n(x = 0+) for electrons and p(x = b−) for holes. These conditions are obtained by assuming that within the Debye layers both electron and hole distributions are close to quasi-equilibrium and so are approximately Boltzmann distributed such that
(10) |
(11) |
(12) |
(13) |
Parameters found by fitting to dark current decay transient measurements are shown in Table 2. The room-temperature value of the I− vacancy diffusion coefficient D+ was tuned such that the time for the simulated dark current transient, as shown in Fig. 5, to reach a stable value matched the experiment. The electron and hole diffusion coefficients n = p were adjusted to achieve the best possible agreement regarding the current transient behaviour observed prior to stabilisation at T = 298 K, and were taken to be temperature independent. Savenije et al.60 find that the carrier diffusion coefficients are likely to vary as T−0.6 – this changes the values of D+ and D− only very slightly over our temperature range, and negligibly compared to other parameters that vary exponentially with T. The need to fit D+, n and p to our cell's experimental data arises from sensitivity of the diffusion coefficients to grain morphology, which varies from cell to cell.
Symbol | Description | Value |
---|---|---|
B | Bimolecular recombination factor | 4.1 × 10−5 cm3 s−1 |
D + | I− vacancy diffusion coefficient | 7.5 × 10−13 cm2 s−1 |
n | Electron diffusion coefficient | 0.21 cm2 s−1 |
p | Hole diffusion coefficient | 0.21 cm2 s−1 |
E A | I− vacancy activation energy | 0.55 eV |
V bi | Built-in voltage | 1.05 V |
v s | Surface recombination velocity | 0.058 cm s−1 |
According to Eames et al.,47 the temperature dependence of D+ is given by
(14) |
Current decay measurements were performed in the dark using a Solartron Modulab XM electrochemical system. Care was taken to ensure that all measurements were started from the same equilibrium condition by leaving the cell in the dark at open-circuit between measurements until the residual voltage dropped below 1 mV. This would take up to one hour at low temperatures. Temperature controlled measurements were performed using a Peltier element and DC temperature controller (Meerstetter Engineering).
We note that some MAPI cells fabricated using chloride precursors have been reported to contain residual chloride ions, and it is still an open question as to the effect of these ions (if any) on cell behaviour. Sakai et al.16 discuss the crystal growth in films produced using PbCl2. XRD shows that initially grains of MAPbCl3 form readily over MAPI grains. During the annealing phase, the XRD peak corresponding to MAPbCl3 disappears as the peak for MAPI grows. They note that, during crystal growth, the MAPbCl3 acts as a template for MAPI phases. During annealing, I− ions replace Cl− ions, forming apparently pure MAPI.
The source of controversy here is whether all of the chloride ions are actually removed from the film. In any case, there is a consensus that iodide ions are present in far greater numbers than chloride ions. Dar et al.62 used scanning transmission electron microscopy-energy dispersive spectroscopy (STEM-EDS) to find that films they produced using the chloride precursor route were ‘chloride-free’. Yu et al.63 used energy dispersive X-ray spectrometry (EDX) and surface-sensitive X-ray photoelectron spectroscopy (XPS) and found what they called a ‘negligible’ amount of Cl in a fully annealed film. However, Yu et al.63 do note a weak XPS signal corresponding to 1% Cl compared to I in the top 20 nm of their film. In addition, chloride and iodide ions attach to the same sites on the perovskite lattice. As halide migration in perovskites is vacancy mediated, this means that it is only necessary to account for one species of vacancy (see Fig. 3). Unless chloride ions are orders of magnitude more mobile than iodides, the values of vacancy diffusion coefficient and density will be determined solely by the values for iodide ions alone. Even in the case of extremely mobile chlorides the effect on the model is only to increase the effective vacancy mobility by a small amount. As MAPI cells that are not produced using chloride precursors exhibit the same hysteresis as ours, we are confident that it is iodide migration playing the major role in our cell and any chloride contamination does not significantly impact the results. We chose the chloride precursor method for this study as cells produced in this manner have been found to exhibit better photovoltaic performance.61
The evolution of the conduction band profile for the first 8 s after jumping from the equilibrium condition to Vjump = 1 V is depicted in the lower panels of Fig. 4. When the voltage is first switched from 0 V to 1 V, Q+ is still equal to Qeq, causing Ebulk to be large and negative; specifically, Ebulk = −Vjump/b at t = 0. As time progresses, the positively charged iodide vacancies are driven by the negative Ebulk to flow away from the spiro and back towards the TiO2, reducing Q+, |Q−| and |Ebulk|. After 8 s, |Ebulk| has decayed to less than 5% of its initial value. The decay is non-exponential.
While these results are specifically for TiO2:MAPI:spiro, we suggest the principles found here transfer over to other transport layer configurations as we can see no mechanism whereby the choice of transport layer could completely stop all ion motion. Other materials would likely alter the size of the layers or shift the timescale of ionic behaviour. The stages in different fabrication techniques for different layers and chemical effects between the layers and perovskite during and after fabrication may also play a role.
Assuming a diffusion coefficient D+ = 7.5 × 10−13 cm2 s−1 for the I− vacancies, the agreement between simulation (Fig. 5a) and experiment (Fig. 5b) is good, both in terms of the magnitudes of the current densities and the timescale of the decay. Crucially, the current density maximum observed at 3 s for Vjump = 0.95 V in the measurement is predicted in the simulation – its origin is described below. However, the simulations fail to predict the minima observed for all Vjump at around 1 s, labelled minimum 1 on Fig. 5b, as well as the minima seen at around 20 s for Vjump ≥ 0.9 V, labelled minimum 2.
We believe the ‘spike’ at t = 0 in the experiment is due to electron/hole displacement currents; these are beyond the scope of this work and provide no insight into ion motion. Minimum 1 occurs over short time scales on the order of 1 s and could have its origins in competing recombination pathways or motion of relatively fast Li+ ions in the spiro hole transporting layer. Minimum 2 occurs at much longer timescales, on which the electric field Ejump has been mostly compensated by the mobile ions. Possible causes of this second minimum include the very slow motion of methylammonium vacancies within the Debye layers, the creation of additional vacancies at grain boundaries in the perovskite film, or ions moving between the perovskite and electron/hole transporting layers (as alluded to by Richardson et al.51). Further development of the model will be necessary to explain these discrepancies.
Fig. 6 shows the evolution of the bulk electric field Ebulk, the areal charge density Q+ in the Debye layer adjacent to the spiro and the voltage drop across the same Debye layer. The bulk electric field Ebulk is equal to −Vjump/b at t = 0 (when the jump from 0 V to Vjump takes place) but returns to zero within 30 s. The Debye charge Q+ decays from Qeq to a smaller stable value Qs over the same timescale. This decay occurs because vacancies move out of the layer on the spiro side of the perovskite into the bulk and from the bulk into the layer on the side with titania due to the reduced potential difference across the perovskite.
Fig. 6 Simulated (a) electric field Ebulk between the Debye layers; (b) charge per unit area in the Debye layer next to the spiro; and (c) voltage drop across the same Debye layer. |
For values of Vjump less than the built-in voltage, 0 < Qs < Qeq, there is still a net potential difference across the device, though it is reduced to Vbi − Vjump. As there is a smaller potential difference, charge flows out of the Debye layers into the bulk, reducing Q+, |Q−| and the widths of the layers to the extent needed to screen this smaller voltage. Lower values of Vjump thus correspond to larger Qs, and , each in a nonlinear manner.
If Vjump equals Vbi exactly, Qs = 0. In this case there is no potential difference across the perovskite, and so nothing to keep the vacancies in the Debye layers. The Debye layers fully discharge such that the vacancy density is uniform across the entirety of the perovskite.
As discussed by Richardson et al.51 the quantities that have a major influence on recombination are Ebulk, and . Negative Ebulk hinders charge extraction as charges in the bulk are pushed to the opposite side of the perovskite bulk to their respective transport layers, causing carriers to build up. The increase in carrier density enhances recombination, as in (4) and (5). After the initial jump in the applied potential Ebulk jumps to a negative value, causing a sudden increase in recombination, before gradually relaxing to zero.
The effect of can be seen from the boundary condition (13). This equation demonstrates that a reduction in Q+ from its initial value Qeq reduces , and thereby increases p(x = b−) at the edge of the spiro-side Debye layer and thus also the resulting recombination. The same logic applies for by considering the relation (12) between the electron density at the MAPI:TiO2 interface n(x = 0) and at the edge of the Debye layer on the titania side of the perovskite n(x = 0+).
Fig. 7 shows the recombination pathways through the device. Due to the very deep valence band maximum of TiO2 and very shallow conduction band minimum of spiro, electrons must cross from the CB of the TiO2 to the VB of the spiro in order to form a complete circuit. This trajectory requires an electron in the conduction band to enter a hole in the valence band – in other words, recombination. A greater rate of recombination thus results in a more negative current as it is recombination which allows reverse current to traverse the energy landscape of a device. We identify three likely current pathways: (i) an electron in the titania CB may recombine across the interface with a hole in the MAPI VB, (ii) an electron in the MAPI CB may recombine with a hole in the MAPI VB and (iii) an electron in the MAPI CB may recombine across the interface with a hole in the spiro VB. Here, TiO2:MAPI surface recombination (i) is modelled using the boundary condition (5) and bulk MAPI recombination (ii) using the bimolecular scheme (4). MAPI:spiro surface recombination (iii) is not modelled, as it was not found to have as significant an effect as (i).
It follows that a more negative Ebulk results in a greater reverse current, while greater and result in less reverse current. As such, the decay of Ebulk from negative to zero in Fig. 6a results in a decrease in magnitude of reverse current. At the same time, the decay of from ∼0.1 V to a smaller value in Fig. 6c, and similar for , causes an increase in magnitude of reverse current.
The balance between these two effects explains the behaviour of the current density transient J(t) in Fig. 5. For Vjump ≤ 0.85 V the influence of Ebulk dominates causing J to increase then stabilise at a greater (less negative) value. For the case of Vjump ≥ 0.9 in the experimental results, Ebulk dominates for a few seconds but then the V dominate after, causing J to increase to the maxima observed at t of a few seconds, then decrease again before settling to a stable value. The simulated results show these maxima for Vjump ≥ 0.95 V, but not Vjump = 0.9 V. Notably, the maximum for Vjump = 1 V occurs at t = 1.5 s for both simulation and experiment.
Fig. 8 Simulated (panels a) and measured (panel b) evolution of current density over 50 s following a jump from 0 V to 1 V in the dark, for six different values of absolute temperature T. |
As with the results in Fig. 6, neither of the current density minima observed in the measurements are present in the simulation results, although the agreement is otherwise good. In particular, the simulated and measured current densities decay on similar timescales for all the temperatures considered. If vacancy motion is a thermally activated mechanism (14), as proposed by Eames et al.,47 this temperature dependence supports the hypothesis that the transients are a consequence of ion motion.
The large reduction in the saturation current density reached at large t as the temperature is reduced is also predicted by the simulations. Notably saturation current density does not depend on ion motion; at long times the ion distribution reaches an equilibrium that is independent of temperature. Increases in temperature lead to increases in carrier concentrations on the perovskite boundaries (through (6) and (7)), hence to increased carrier concentrations throughout the perovskite layer and increased recombination. In turn, this increased recombination causes negative saturation currents of greater magnitude at high temperature (where carrier recombination is high) in comparison to low temperature (where carrier recombination is low). The fact that the experimental and theoretical saturation current densities do not match up is attributable to small errors in the Fermi levels used to calculate the boundary conditions (6) and (7) on the carrier concentration.
Footnotes |
† The data used to produce Fig. 4–6 and 8 and ESI Fig. S1 are available at DOI: http://10.15125/BATH-00251. |
‡ Electronic supplementary information (ESI) available. See DOI: 10.1039/c6tc04964h |
§ Authors contributed equally to this work. |
This journal is © The Royal Society of Chemistry 2017 |