Field

The present disclosure relates to methods and devices capable of providing controllable impedance. In particular, but not exclusively, the disclosure relates to the control of reactive impedance to modify capacitive and/or inductive qualities of an electrochemical structure. Furthermore, the disclosure provides a device and method of controlling the same which can offer high capacitance and/ or inductance in comparison with conventional approaches. Background

Improvements in electronics are often tied to improvements in two particular characteristics: speed and size. Amongst the passive elements required in electronic circuits are inductors and capacitors. The size of these components has been limited by the physics upon which they are based. While, for example, high dielectric materials may improve the capacitance of a given capacitor, there remains a limit on what can be achieved through conventional techniques.

There is an ongoing desire, therefore, to find new ways of providing capacitance and inductance to electrical circuits. This is particular challenging where large capacitance and/ or inductance values are required. Conventional techniques would suggest that circuit elements having these properties are themselves large. This creates further challenges for the miniaturisation process.

It is known that particular materials may display capacitive and/or inductive behaviour under some conditions. For example, perovskite has been proposed as potentially of use as an active region within the context of solar cells. The behaviour of this material has been shown to diverge from more conventional (e.g. silicon-based) solar cells in some circumstances. Attempts to model such behaviour have led to the development of equivalent circuit models including very large capacitors and/ or inductors. However, the origins of these virtual circuit elements are not well understood, lacking physically meaningful interpretations. As a consequence, the utility of existing models of perovskite behaviour and their application is limited. Summary

According to a first aspect, there is provided a device for providing controllable electrical impedance, comprising: an electrochemical structure comprising first and second contact elements and an active region comprising first and second charge carrier types, the active region having a first interface with the first contact element and second interface with the second contact element; wherein charge carriers of the first type are suitable to participate in faradaic charge transfer at the first and second interfaces, wherein charge carriers of the second type are not suitable to participate in faradaic charge transfer at the first and second interfaces, and wherein the device further comprises a controller configured to influence a DC current across the active region in order to affect an effective impedance of the electrochemical structure.

A method of providing a controlled impedance, comprising: providing an electrochemical structure comprising an electrochemical structure comprising first and second contact elements and an active region comprising first and second charge carrier types, the active region having a first interface with the first contact element and second interface with the second contact element; wherein charge carriers of the first type are suitable to participate in faradaic charge transfer at the first and second interfaces, wherein charge carriers of the second type are not suitable to participate in faradaic charge transfer at the first and second interfaces, and the method further comprising influencing a DC current across the active region in order to affect an effective impedance of the electrochemical structure.

The first and second aspects can provide a controllable impedance. In particular, it has been found that the interaction of two charge carrier types, one of which may participate in a faradaic current (e.g. through faradaic charge transfer) across an electrochemical structure and one of which may not, can cause a change to the impedance (for example, a change to the reactance) of the electrochemical structure which will be responsive to changes in DC current. In this way, a powered impedance structure maybe provided, using DC control to affect the effective capacitive or inductive impedance. The DC current maybe a function of an applied DC voltage and/or a

concentration of charge carriers within the active region. Where the active region is formed of a photoresponsive material (i.e. one in which charge carriers are photogenerated) the concentration of charge carriers may be controlled by controlling incident light on the active region. Thus, the controller, may act to control one or both of the light incident on the active region and the DC voltage across the electrochemical structure. For example, the controller may comprise a light source, and may adjust the amount of light incident upon the active region from the light source. The controller may additionally or alternatively include a DC potential generator, and may control the DC potential applied to the electrochemical structure.

The controller may control the DC current such that the effective impedance matches a desired impedance. In other examples, the DC current may be controlled such that the electrochemical structure displays a desired capacitance and/or inductance.

Optionally, one or both of the contact elements are semiconductors. For example, in some embodiments the first contact layer is an n-doped

semiconductor and the second contact later is a p-doped semiconductor. In other examples, one or both of the contact elements may be metallic. For example, at least one contact may form a rectifying contact with the active region. In general, an energetic barrier may be provided to faradaic current transfer of charge carriers of the first type at the first and second interfaces. An appropriate material for the first and second contact elements may be chosen to satisfy this criteria. Fluxes of the faradaic carriers (charge carriers of the first type) across the interfaces can be modulated for both direction due to

movement of non faradaic carriers (charge carriers of the second type).

The active region may be formed of a perovskite material such as a metal halide or lead halide perovskite material. This material has been shown to operate effectively under the conditions described in this disclosure. Alternatively, other materials in which more than one charge carrier type may be supported can be utilised.

In some embodiments, the first charge carrier type is electronic and/or the second charge carrier type is ionic. The behaviour of electronic charge carriers (i.e. electrons and/or holes) in recombination and/or injection processes has been studied extensively and so offers a convenient approach for faradaic charge transfer at the interfaces. Ionic charge carriers are likely to display substantially different mobility within a given material to electronic charge carriers; this difference in mobility can enhance the effects described herein.

As such, in some preferred embodiments the mobility and/or conductivity within the active region of the first and second charge carrier types is different. For example, the effective masses of the first and second charge carrier types may be different. In some preferred embodiments, the mobility and/ or conductivity of the first charge carrier type is greater than the mobility and/or conductivity of the second charge carrier type.

The first and second charge carrier types may have different concentrations within the active region. For example, the concentration of the second carrier type may be greater than the concentration of the first carrier type. In preferred embodiments, the mobility and/or conductivity of the first carrier type is greater than that of the second carrier type in the active region while the concentration the second carrier type is greater than that of the first carrier type in the active region. It has been found that this combination can provide an enhanced effect. Optionally, the electrochemical structure may be a thin film device. For example, each of the first and second contact elements and the active region may be layers formed by thin film techniques, such as thin film deposition techniques. In some examples, each layer may have a thickness of 1 micrometre or less. Indeed, the thickness of each layer may he significantly less than this. For example, the thickness of the electrochemical structure may be less than 50 nanometres. Brief Description of the Drawings

A preferred embodiment of the disclosure is now described with reference to the accompanying figures, in which:

Figure l illustrates a device providing tunable impedance within an electrical circuit;

Figures 2a to 2d show measured and simulated impedance of a device incorporating an electrochemical structure;

Figure 3a to 3e shows simulated impedance measurement of a device stack containing mobile ionic charges;

Figures 4a to 4d show simplified energy level diagrams and equivalent circuit models;

Figures 5a to sd show measured and modelled impedance for a device with inductive behaviour;

Figure 6 illustrates current- voltage characteristics of a spiro-OMeTAD/mixed- perovskite/Ti0 ₂ device;

Figure 7 shows performance and apparent efficiency of the spiro- OMeTAD/mixed-perovskite/Ti0 ₂ device;

Figures 8a and 8b show the effect of device stabilisation on impedance measurements;

Figure 9 shows complete impedance spectra of the devices shown in Figure 2;

Figure 10 shows measured and simulated impedance spectroscopy

measurements for the device shown in Figure 2 at short circuit;

Figure 11 shows similar spectroscopy values varying with applied potential; Figure 12

Figure 13 shows simulated impedance with applied voltage in the dark;

Figure 14 shows examples of equivalent circuit model Nyquist plots and impedance spectra for a recombination limited circuit; and

Figure 15 shows simplified energy level diagrams and circuit models using transistors to describe the ionic gating of electron processes at different interfaces.

Detailed Description Figure l shows a device 10 comprising an electrochemical structure 30 and a controller 20. The electrochemical structure comprises a first contact element 34 and a second contact element 36 together with an active region 32. The active region may be formed of perovskite or another suitable material capable of carrying first and second charge carrier types.

The active region 32 has a first interface with the first contact element 34 and second interface with the second contact element 36. The charge carriers of the first type are suitable to participate in faradaic charge transfer at the first and second interfaces while charge carriers of the second type are not suitable to participate in faradaic current transfer at the first and second interfaces. For example, the charge carriers of the first type may be electronic (e.g. electrons or holes) and the charge carriers of the second type may be ionic. One or both of the contact elements 34, 36 may be semiconductors and in the embodiment shown the first contact layer 34 is an n-doped semiconductor and the second contact later is a p-doped semiconductor.

The controller 20 is configured to influence a DC current across the structure 30. As will be demonstrated below, by doing so the controller can affect an effective impedance of the electrochemical structure 30. In particular, the controller may modify the reactance of the structure 30. To influence the DC current, the controller may modify a DC bias voltage across the structure 30 and/or may act to affect the number of charge carriers within the structure (for example, by modifying the energy applied to the structure). In particular, the controller 20 may modify the light incident upon the structure 30.

The device 10 may be electrically coupled to circuit components 40 such as power supply or other elements. In this manner, a variable and tunable impedance may be provided for implementation within an electrical or electronic circuit. Thus, devices and methods are provided in which the impedance of an electrochemical structure can be controlled by modifying a DC current across the structure. The DC current may be modulated by, for example, modifying the available charge carriers (e.g. by applying

electromagnetic radiation) and/or modifying an applied DC potential.

To understand the manner in which such a tunable impedance may be provided, the example of solar cells may be considered. Solar cells are typically

represented as diodes in equivalent circuit models. The current flowing through the diode varies exponentially with the height of its energy barrier which is controlled by the voltage between its contactsfj). However, solar cells based on the rapidly developing technology of hybrid perovskite semiconductors (2, 3 ) do not generally display pure diode-like behaviour. This is thought to be due to the presence of mobile ionic defects in the perovskite semiconductor phase which underlie the hysteresis often seen in their current-voltage characteristics (4-7) . An example of how mobile ionic charge can control electronic current is seen in electrochemical transistors where the density of electronic charges in a semiconductor film is determined by the gate potential induced by inert ions in a surrounding electrolyte(S); the effect can be used to amplify the lateral current through the semiconductor. Electrochemical doping by ionic charge has also been used to control electronic barrier heights in polymer diodes. (9) However, in hybrid perovskite solar cells, the full implications of mobile ions for the device physics are still being unravelled.

Impedance spectroscopy is routinely used to characterise many electrochemical systems including solar cells (10-14). Fitting an equivalent circuit model to an impedance spectrum gives insight by relating the circuit’s elements to the physical properties of the material or devicef/fy). This process has been problematic for hybrid perovskite devices because the models introduced to explain impedance measurements under illumination, or with a d.c. bias voltage, require very large capacitors (> 10^ F cm ² ) and inductors(> 1 H cm ² ) which lack physically meaningful interpretations!/?, 25-29). Ferroelectric effects, a photoinduced giant dielectric constantfyo), and accumulation of ionic charge(7, 22 ) have all been discounted as explanations (2 -24) . Bisquert et al. have proposed that giant capacitive and inductive bchaviour(2 -26) could be caused by phase-lagged accumulation or release of electronic charge from within a degenerate layer induced by fluctuations in the surface polarisation due to ionic charge. However, the energetics of these materials suggest the required interfacial degeneracy is unlikely to exist under normal operating

conditions. (27, 28) More promisingly, Pockett et al. have highlighted the link between rate of recombination and varying ion distribution as an explanation for the low frequency behaviour of perovsldte impedance spectra(iT). A physically convincing, quantitative, interpretation is needed to unlock this powerful tool for use with hybrid perovskite technologies. Here we show that the interfaces in perovskite solar cells behave like bipolar transistors(29) in which the electronic current through the cell is amplified by the voltage difference due to the ionic current in the perovskite phase. The accumulation of ionic charge at the interfaces in response to oscillating voltages naturally introduces an out of phase, capacitive charging/discharging ionic current. Importantly, the associated changes in electrostatic potential across the perovskite also modulate the rates of electronic recombination and injection across the interfaces. This results in an out of phase electronic current amplified with either a positive sign (for recombination) or a negative sign (for injection) in proportion to the ionic current, resulting in capacitor-like or inductor-like behaviour. Consequently, the device’s apparent capacitance or inductance can contain large contributions from the trans-carrier amplification of

recombination and injection. This effect explains giant capacitive and inductive behaviour without electronic charge accumulation. Modelling the amplification effect using bipolar transistor elements incorporated in a simple equivalent circuit allows us to explain and physically interpret the many peculiar features observed in perovskite impedance spectra.

Measured impedance spectra characteristics

Impedance spectroscopy involves the application of small periodic voltage perturbation, v, superimposed on a background voltage, V, and measurement of the amplitude and phase shift of the induced current, j, superimposed on a background current J. The complex impedance (Z = Z + iZ”) is given by Z =

1 / _/ ! exp(i¾ where Q = arctanCZ”/!?) is the phase shift of j relative to v. This is evaluated at different angular frequencies (w) resulting in a spectrum Z(w). We made impedance measurements on perovskite solar cells fabricated from a thin film of (~ 550 nm) perovskite semiconductor sandwiched between a p-type hole transporting material (HTM) and n-type electron transporting material (ETM) making a p-i-n device (see fabrication details in supporting information). The device was then allowed to stabilise prior to impedance measurement. This avoids inductive loops in the resulting Nyquist plot of (-Z” vs Z’) which are purely an artefact due to the incomplete stabilisation or degradation of the device during data collection (see Figure 8). Figure 2a and b show impedance data collected from an ultra-stable perovskite solar cell equilibrated at open circuit for different light intensities (Bode plots are shown in Figure 9).

The impedance measurements indicate there is a small but significant out of phase component of the current, which at low frequencies, results in a large apparent device capacitance, as defined by orTmCZ ^' - ¹ ). This increases linearly with light intensity and thus exponentially with the bias voltage (Figure 2(b)). This observation is consistent with the previous literature and has been the subject of considerable discussionQy, 18, 25). Qualitatively similar behaviour is seen from the same device measured at short circuit and a range of light levels (Figure 10) or with different applied biases in the dark (Figure 11). The absence of a light intensity dependent shift in the time constant of the low frequency feature indicates that the observed impedance behaviour cannot easily be attributed to photoinduced changes in ionic conductivity^, 31 ) (see Figure 11). Figure 2(a) shows A Nyquist plot of the real (Z ) vs imaginary ( Z’) impedance components, and Figure 2(b) the apparent capacitance, 6 ¹ Im(Z ¹ ) vs frequency of a spiro - OM eTAD / mixed-p er o vskite/TiO ₂ solar cell measured around the open circuit voltage, illuminated at different constant light intensities. The devices were stabilised to avoid loops in the Nyquist plot arising as artefacts due to incomplete stabilisation of the device during data collection. The inset of

Figure 2(b) shows the ratio of the out of phase component of the recombination current to the ionic current plotted vs the recombination transconductance. Figures 2(c) and 2(d) show corresponding simulated impedance measurements (filled squares) determined from a drift-diffusion model of a p-type/intrinsic/n- type (p-i-n) device structure containing mobile ionic charge. The dashed lines indicate the simulated contribution to the capacitance from mobile ionic charge. Figure 2€ shows a mixed conductor diode circuit model containing an ionically gated transistor used for the simultaneous 5 parameter global fit (continuous lines) to the experimental data (filled squares in Figures 2(a) and (b)) and to the drift-diffusion simulated data (filled squares in Figures 2(c) and 2(d)). Figure 2 (f) illustrates an alternative representation of the equivalent circuit model shown in Figure 2(e). he elements in the ionic circuit branches are related by apparent capacitance and recombination resistance elements in the electronic circuit branch, c _rec (c _<) ) and G _Gbo (w), have a frequency dependence controlled by the ionic circuit branch as derived from the transistor model. Numerical simulation of impedance spectroscopy

To elucidate the potential role of mobile ions within the intrinsic phase on the impedance of the interfaces in a p-i-n device we simulated impedance

spectroscopy measurements by adapting our open-source Driftfusion model (32) described previously (33). This one-dimensional, time-dependent, drift-diffusion simulation solves for the evolution of free electron, hole, and mobile ionic defect concentration profiles, as well as the electric potential. Figure 3a shows an example of the simulated electrostatic profiles of the conduction and valence band energies with corresponding quasi-Fermi energies under steady state conditions with an applied d.c. voltage equal to the open circuit potential at 1 sun equivalent illumination. There is no electric field in the bulk of the perovskite layer since the mobile ionic charge has migrated to accumulate at the interfaces screening the built-in potential (Figure 3a) consistent with previous observations and simulations explaining hysteresis. (33-36) By superimposing a small, periodically varying, voltage u(w) on the boundary conditions of the system we evaluated the impedance at a wide range of frequencies by measuring the amplitude and relative phase, Q, of the resulting current density ( ) oscillations . When a high frequency oscillation u(w) is applied to the boundaries, the resulting changes in the conduction band (and valence band energy) gradients show that the potential variations drop across perovskite layer (Figure 3b). The ionic resistance is too great for the movement of ions to compensate the potential oscillations on this time scale, and the small ionic current Ji _on is in phase with n(w). However, with low frequency n(w) oscillations, the ionic charge can move with sufficient speed to compensate the changes in potential which reduces the amplitude of oscillations across the perovskite (Figure 3c). At low frequency, the ionic current lags the phase of u(w ) by p/ 2 rad and the voltage oscillations at the interfaces now have a component in phase with this ionic current but out of phase with n(w) (Figure ). The simulations show that the rate of charge transfer across each interface (recombination and charge

collection/injection) is modulated by the potential across the interfaces which have a component varying out of phase with the applied voltage (Figure 3d).

The impedance, Z(w), evaluated from these simulations (Figures 2c and 2d) shows remarkably similar behaviour to the measurements. In the dark with no bias voltage or light, the capacitance, urTm ( ¹ ), of the device at low frequencies is dominated by contributions from ionic movement (dotted lines). However, the exponential increase of crTmCZ ¹¹ ) when the steady state voltage across the device was increased by light (or applied voltage in the dark, Figure 12) does not arise directly from the ions, and is also not due to an accumulation of electronic charge. Instead it arises from the out of phase modulation of recombination across the interfaces discussed above.

Figure 3 shows simulated impedance measurement of a device stack containing mobile ionic charges. In particular Figure 3(a) shows the steady state

electrostatic energy profile of the p-i-n simulated in Figure 2 at open circuit under 1 sun equivalent illumination, the insets show the concentration of ionic charges accumulated at the perovskite/HTM (left-hand side) and

perovskite/ETM (right-hand side) interfaces to screen the internal potentials. The effect of a high frequency (Figure 3(b)) and low frequency (Figure 3(c)) periodic voltage perturbation on the conduction band energy profile (limits indicated by the black and grey lines) is also illustrated. The perturbation is superimposed on the Vbc value. Figure 3(d) shows low frequency applied voltage oscillation (v) vs time and the resulting ionic current (Ji _on ) in the device. The out of phase part of the recombination current (;„ _E ) is in phase with Ji _on , while the out of phase part of the injection current ( _/ ,„,) is p rad out of phase with i _on - Figure 3(e) provides a diagram indicating the processes of ionic transport, electron recombination and injection during a small perturbation measurement. Ionic charge is represented by squares, free electrons by circles with a symbol and holes by circles with a“+” symbol.

Perovskite interfaces as ionically gated transistors

We now develop simple expressions to describe the impedance of the interfaces with a mixed conductor by considering how the current across each interface will vary with the potential difference across it in the presence of inert mobile ions. The interfacial electronic current can be related to the injection or collection of charge carriers across the charge collection interface, or the recombination of charge carriers which will occur predominantly at the opposite interface in non-degraded devices( 7, 38). Under most circumstances one or the other of these processes will dominate the impedance of the device, either for the free electron or free hole species. We assume transport of free electrons and holes is fast relative to injection and recombination processes, consistent with the long diffusion lengths observed in these materialsC^p-^i).

Origin of apparent capacitive behaviour

Initially we will consider the impedance related to the recombination (and thermal generation) of electrons at the interface with p-type HTM assuming a negligible change in the barrier to collection or injection of charge carriers at the ETM. Close to the interface, electrons in the perovskite phase participating in recombination (with concentration n _rec ), may be considered a minority species relative to the holes in the neighbouring HTM. For simplicity we assume the electron recombination current density from the perovskite to HTM can be approximated by the first order process /,t _EC = where q is the electronic charge and s is the recombination velocity across the interface. We expect that T _hee will be controlled by the electron quasi-Fermi level in the perovskite at the interface with potential ¾ relative to the Fermi level in the dark at equilibrium (see Figure 4a). The system must obey the principle of detailed balance, so there will be a thermal generation current of electrons, from the HTM to the perovskite which will scale exponentially with changes in potential between the

Fermi levels in the HTM and perovskite, v ^~ _c . The net recombination current is given by the sum of these currents:

1 where / _rECD = QSH, e^[- ₀ /(ft _B r)], m is the intrinsic electron concentration in the perovskite, / _<¾ is Boltzmann’s constant, T is temperature and _D is the barrier potential of the HTM/perovskite interface under dark equilibrium conditions (see Figure 4(a)). Without mobile ions in the system, a potential, V, applied across the cell would be fully experienced by the perovskite electrons at the perovskite interface (¾ = V) with no change in the barrier to thermal generation (K. ^~ = 0) so that equation 1 would become the standard diode equation. However, in hybrid perovskites mobile ions redistribute to screen V so, at steady-state, the change in potential for electrons at the perovskite interface will be approximately half the ion-free case (¾. = v/2) assuming the potential drops evenly within each contact (Figure 4b). Consequently there will be a smaller change in recombination current than if the ions were not present. The current due to thermal generation would also drop as the barrier height increases to ¾r _c = -F/2.

Figure 4 shows simplified energy level diagrams and equivalent circuit models. The conduction and valence bands of the perovskite layer are sandwiched by the hole transporting material (HTM, left-hand side) and the electron transporting material (ETM, right-hand side), the vertical axis represents electrochemical potential energy which points down. The ionic accumulation layers are assumed negligibly thin. The height of the energy barrier for electron injection and recombination in the dark is given by and ionic charge is represented by the light grey squares. The electron and hole quasi Fermi- energies are indicated by the dotted blue and red lines, the other symbols are defined in the text. The equivalent circuit diagrams are colour coded blue, red and grey to indicate the paths for electrons, holes and ions. Figure 4a shows the energy levels of the conduction and valence bands in the dark before and after ionic equilibration. The ideal Schottky-Mott limit electronic energy barriers are indicated, these change with applied potential and ionic redistribution. Energy levels after application of a voltage (V) shown instantaneously (w ¥) and at steady state (w o) and corresponding circuit models for devices in the: Figure 4b recombination limited regime where Jreco < < Jinjo; Figure 4c the injection limited regime where Jreco > > Jinjo; and Figure 4d the mixed limit regime. Example model Nyquist plots are also shown for each regime, the mixed limit plot corresponds to a special case where Rion is comparable to the real parts of Zrec and Zinj.

On application of a voltage V, ions will redistribute with a time constant approximated by ( ionGon/ 2) where Rio _n is the specific resistance (W cm ² ) to ionic motion and Gon is the specific capacitance for accumulation of ions at each the interface (F cm- ² ). If the concentration of mobile ionic defects is large relative to the concentration of free charge carriers in the active layer then the changing concentration of ions at each interface (Df _0h ) will result in a change in potential across each interface of V - V _t = V ₂ = 2&Q, _on /C _iDa where V-V is the change in screening potential due to the ions across the HTM/perovskite interface and V ₂ is the corresponding change in screening potential across the

perovskite/ETM interface (Figure 4). The change in ¾. due to the non equilibrium distribution of ions is thus given by: u ¾ ⁺ C = V ^r .l— V

2 where V _n is the potential of the electrons in the perovskite (V _n = o if the barrier to electron injection at the HTM interface is negligible, as indicated in the Figure 4b example). The difference in the barrier height for thermal generation at the interface is given by (Figure 4): v T ^~ EE ft - V

3

If V varies with angular frequency w, the complex impedance of this ionic motion to or from the interfaces (described by i¾ _on in series with Ci _on at each interface, Figure 3) gives the changes in screening potential at each interface:

5

Substituting equations 2, 3 and 4 into equation 1 gives a general expression for the net recombination current across the interface in terms of V. We note that this expression is analogous to the expression used to describe a bipolar NPN transistor where the voltage of the base is driven by the change in potential Vi due to the accumulation of ions at the interface. Under dark forward bias conditions there is net flux of electrons from the perovskite (which acts as the emitter) to the HTM (which acts as the collector). The change in voltage of the base-emitter (interface-perovskite) and base- collector (interface-HTM) junctions are equivalent to ¾, and ¾ respectively. Note that we have modified the conventional bipolar transistor symbol used to represent the interfaces in Figure 4 to emphasise that the net electronic current through the transistor may be in either direction according to the electrical and light bias conditions. If ¾ then the assignment of the terms‘emitter’ and‘collector’ to the two sides of the interface would be reversed. If there is no reaction between ionic and electronic charge at the interface and no ionic penetration into the HTM, then the ionic-to-electronic current gain of the transistor (/¾o _n -eiect _ron ) is infinite.

Under forward bias (V > o) conditions in the dark / _r ⁺ _EC > / _r ^_ _EC so the second term of equation l can be neglected. Differentiating with respect to the applied voltage V gives an expression for the inverse of the recombination impedance, which in the small voltage perturbation (u) limit can be written: where the background recombination current across the interface with a potential difference V at steady state (w=o) is _re c(U,6)=o) = Jrecoexp[q V/ (2/CBT)] . Inverting the real part of equation 6 gives the small perturbation resistance of the interface:

7

Since J _rec (V,c =o) varies exponentially with V (when V _n = o) we see that r _rec is proportional to exp [-qV/ ( sT)] analogous to the resistance expected from a diode but also has a frequency dependence. When w o equation 7 reduces to T _ree = 2½r/[qJ _rec (V,co=o)] but when w ¥ it becomes r _rec = feT/ [qJreJV, w = o)] , such that the resistance of the interface is twice as small relative to the steady state condition. This is because at high frequencies the magnitude of the potential difference across the interface is greater because the mobile ions do not have time to screen the changing potential, this results in larger

perturbations in the recombination current relative to the perturbations resulting from low frequency changes in voltage. Under many operating conditions this change in r _Tec with applied frequency gives rise to two semi- circles in Nyquist plots (see Figure 4b) in agreement with the observations of Pockett et al.(i^). The real part of equation 6 divided by ίw gives an expression corresponding to the apparent small perturbation capacitance of the interface (c _rec ) due to the recombination component out of phase with v:

8

Several features of equation 8 are noteworthy. First, the interface appears to behave as a capacitor despite no accumulation of electronic charge being

i1o0 involved. This is due to the effect of the mobile ionic charge screening the

potential change across the interface with a component out of phase with changes in v when w > o. We still expect a real capacitance due to the ionic charge (Ci _on /2) in parallel with the apparent recombination capacitance of the device (c _rec ) as seen in the equivalent circuits in Figure 4. Second, the apparent

15 capacitance will be greatest at low angular frequencies since the denominator of the expression contains an w ² term, conversely when w ¥, c _rec = o. Third, the magnitude of c _{re c} is proportional to _< / _rec (V/ =o) which increases exponentially with the d.c. value of V as discussed above. Consequently, as the voltage across the interface increases, the apparent capacitance will be amplified

0 exponentially.

Ionic-to-electronic current amplification

Current amplification is a key property shown by bipolar transistorsfsq), an alternating electronic current flowing to the device’s gating terminal (base) 5 amplifies the flux of electrons or holes between the collector and emitter

terminals. Here, at sufficiently low frequencies (w o) the ionic current will be given by / _jMl = iaC _im v/2, from equation 8 we can infer that the out of phase component of the electronic current across the interface will be given by

/ _{re c} = w jf _{lcn CjBn i q} / _rec <y, w = o) / (4 ¾ T) . Since both quantities vary with the same frequency, taking the ratio gives the amplification of the out of phase

recombination current by the ionic current:

9 where g _Tec is the recombination transconductance, analogous to the classic amplification result for a bipolar transistor. The magnitude of the out of phase component of the electronic current recombining across the interface is proportional to the resistance to ionic motion in the perovskite, independent of the value of Cion and will also increase exponentially with background bias voltage, V. Interestingly, this result implies that the ionic resistance (and thus conductivity) can be inferred from measurements of the apparent capacitance since can easily be inferred from the measurements of apparent capacitance at short circuit dark conditions and with a background voltage V (see Figure 2). The result also suggests that the ionic amplification phenomenon could be used as sensor for ionic motion, for example to read out ionic conduction in biological systems requiring neural interfacing in a manner related to electrochemical

transistors. (8)

Circuit model

The expressions we have derived (equations 6, 7 and 8) can explain the majority of unusual features observed in the impedance spectroscopy measurements of hybrid perovskite solar cells. Global fits to both measurements and drift- diffusion simulations are shown in Figure 2 using the expression for Z _Tec given by equation 6 incorporated within the equivalent circuit model shown in Figure 4b which also contains an element in parallel corresponding to the geometric capacitance of the device This allows parameters quantifying the interface to be derived. Agreement is seen between the values of C _on and i¾ _on determined from the equivalent circuit fit compared to the values derived from the inputs to the Driftfusion model, helping to confirm our interpretation of the system.

However, impedance spectra of perovskite devices can be more complex than those shown in Figure 2 {18, 26 ); an example displaying inductive behaviour is shown in Figure 5. Similar arguments can be used to derive expressions for the impedance to recombination of holes at the perovskite/ETM interface also yielding capacitive behaviour. Thus the functional form of Z _rec (either for electrons or holes) is unable to adequately explain the response of perovskite devices displaying inductive behaviour either in the small perturbation regime or in response to large perturbations of voltage or light intensity where the current slowly evolves towards a new steady stated).

Origin of apparent inductive behaviour

We now consider the role of charge injection across the perovskite / collection layer interface. Charge injection of a carrier (free electron or hole) will occur in series with the corresponding recombination process described above, described by the difference in the current densities in each direction (¾ and J¾) across the interface. Confining our analysis to injection of electrons across the ETM interface, the net injection current density is given by:

^ ini ¹¹¹ mi

/mj /inj ^~ J in _] /in] 0 ^{s —} /injD ^e

10 where J _mjo is the saturation current density of the interface at equilibrium in the dark and the changes in barrier potentials ¾¾ and t¾ are given by:

¾ = ¾

11 and

12

If the change in potential driving recombination of electrons is negligible (¾ ^¾ t _ree ) then v _n = v (Figure 4c). Substituting equations 11, 12 and 5 into equation 10 and differentiating with respect to V allows gives the inverse of the injection impedance in the small perturbation limit:

13

From equations 4 and 5 it is apparent that V ₂ varies p rad out of phase with V so ion motion will modulate ¾ n rad out of phase with Z _{re c} , consequently the out of phase component of Zmj will behave like an inductor despite no release of accumulated electronic charge. The impedance behaviour of this interface is discussed below where the amplification of the out of phase injection current by the ionic current at low frequencies is shown to be negative:

-R _ism qf _iai (V. = 0} /2 k _E T. Under operating conditions where impedance of the two interfaces is similar (Figure 4d) the value of V _n (which will no longer be o or V) must be established in order to quantify Z _rec and Zi _nj using the steady state values of / _r ⁺ _{K(V w} = o],

/¾(y. w = o ) and ¾cy w = O) required in equations 6 and 13. The steady state value of U _n (or V _p for holes) can be found by substituting in the above expressions into the current continuity relation for steady state (w=o) conditions:

/rec ^" /ph ^— inj

14 and numerically solving for V _n given that the applied potential V and

photogeneration current density, J _ph , are measureable inputs. The complex component of Vn in the small perturbation regime can then be found

analytically. The inclusion Z _rec (V,J _ph ,&>) (capacitor-like) and Z _mj ( V, _Ph ,<y) (inductor like) elements within an equivalent circuit model can reproduce peculiar features observed in Nyquist plots of perovskites devices, an example is shown in Figure 4d along with an equivalent circuit considering these processes for both electrons and holes.

Modulation of interfacial recombination by ions can also lead to apparently inductive behaviour if additional ionic redistribution processes are involved. For example the phase of/ _rec can lag v if ionic charge penetrates, or reversibly reacts at, a dominant recombination interface. Fits from an equivalent circuit allowing ion penetration into an interface are shown in Figure 5 such that electronic recombination shows both capacitive and inductive behaviour. The model can be generalised to consider the fraction of interfacial screening potential dropped in the perovskite rather than the contacts, asymmetric interfacial ionic capacitances, non-ideal recombination and injection, recombination in the perovskite bulk (for cases where interfacial recombination is sufficiently low) and the effect of interface screening by electronic charge.

Figure 5 shows measured and modelled impedance for a device with inductive behaviour. Figure 5(a) shows a Nyquist plot of the real (Z’) vs imaginary (Z”) impedance components, and Figure 5(b) the apparent capacitance, Re[i/ (icuZ)] vs frequency of a spiro-OMeTAD/mixed-perovskite/Sn0 ₂ solar cell measured around the open circuit voltage, illuminated at different constant light intensities. Corresponding Nyquist plot (Figure 5(c)) and apparent capacitance (Figure 5(d)) derived from an equivalent circuit model impedance in which ions may penetrate the Sn0 ₂ contact. In addition to facilitating the interpretation of complex impedance spectra of perovskite devices for identification of the key bottlenecks in device

performance, this elegant description of the interfaces as ionically gated transistors has a number of interesting broader implications. As disclosed herein, the amplification of the out of phase electronic current by ionic current facilitates the design of two terminal thin film devices (such as that illustrated in Figure l) which may display a huge, tuneable, effective capacitances or inductances without the volume required for similar physical capacitances or inductances. Such devices have the possibility to be directly powered by photogenerated charge carriers. Furthermore, the model will be generally applicable to other mixed conducting systems in addition to defect laden semiconductors. For example, charge transfer across interfaces in

electrochemical redox systems supported by a high concentration of low mobility inert ions should show similar behaviour.

In particular, where charge injection dominates the impedance of the circuit, at low frequencies, the out of phase injection current is negatively amplified by the ionic current (hypothetical examples are shown in Fig. S4C and d, ESI). The trans-carrier amplification factor is—R _ion /2 [q/ _inJ (V, w = 0)/ k _B T] resulting in inductive behaviour. The effect opens the possibility to design thin film devices with huge tuneable effective inductances per unit volume (> 104 H cnr3) without relying on the elements coupling to a magnetic flux. Moreover, given the influence of the ionic circuit on the electronic impedance described here, more complex interactions of ionic charge with electronic charge or contact materials would also modulate interfacial electronic processes. For example, the phase of ec can lag v if ionic charge penetrates, or undergoes a reversible chemical reaction, at a dominant recombination interface. Under these circumstances the transistor interface model facilitates control of the ionic gating of the electronic recombination process in order to obtain both apparent capacitive and inductive behaviour.

Further details of device fabrication and characterisation of device in Figure 2.

Device Fabrication: spiro-OMeTAD/mixed-perovskite/Ti0

Chemicals: Lead (II) Iodide (Pbl2, 99.99% TCI UK Ltd.), Lead Bromide (PbBr2, TCI UK Ltd.), Formamidinium Iodide (FAI), Methylammonium Bromide (MABr), FK209 Co(III) TFSI and 30NTD T1O2 paste were purchased from Dyesol Ltd. Dimethylformamide (DMF anhydrous), Dimethyl sulfoxide (DMSO, anhydrous), Chlorobenzene (anhydrous), Acetonitrile (anhydrous). Titanium di- isopropoxide bis-acetylacetonate (TiPAcAc, 75 wt% in IPA), Butyl Alcohol (anhydrous), Bis(trifluoromethane)sulfonimide lithium salt (Li-TFSI), 4-tert- butyl pyridine (96%), Cesium Iodide (99.9%) were purchased from Sigma Aldrich. Spiro-MeOTAD (Sublime grade 98%) and Fluorine doped Tin Oxide (FTO, 8f!/ _□ ) substrates were purchased from Ossila Ltd. UK. All chemicals were used without further purification.

FTO substrates were patterned to desired geometry using chemical etching with Zinc metal powder and Hydrochloric Acid (4M, Sigma Aldrich). Substrates were cleaned by sequential ultra-sonication in diluted Hellmanex (Sigma Aldrich), De-ionized water and Isopropyl-Alcohol. Compact-Ti0 ₂ layer (-30 nm) was deposited on patterned FTOs using spray pyrolysis of TiPAcAc (0.5 M in butyl alcohol) at 450 °C and post-heated at 450 °C for 30 min. Mesoporous Ti0 ₂ layer (-150 nm) was then deposited by spin coating 30NRD solution (1:6 wt:wt in butyl alcohol) at 5000 rpm for 30 sec and heated at 150 °C for 10 min. Substrates were then heat-treated at 480 °C for 30 min to remove organic contents in the 30-NRD paste.

Triple cation (Cso. ₀₅ FAo.8 ₁ MAo. ₁₄ PbI _2.55 Bro. ₄₅ ) perovskite solution was prepared using a reported protocol (42). Typically Csl, FAI, MABr, Pbl ₂ and PbBr ₂ were mixed in appropriate ratio in mixed solvents DMF:DMSO (4:1 v:v) to get 1.2 M concentration of Pb ²⁺ ions. This solution was filtered using 0.4 pm PTFE syringe filter before use. Perovskite films were deposited by anti-solvent quenching method in which 70 pL solution was spin coated initially at 2000 rpm for 10 s (ramped 200 rpm S ^'1 ) and then at 6000 rpm for 20 s (ramp 2000 rpm s ^-1 ) with 100 pL chlorobenzene dripped at 10 s before the end of second spin cycle. Spin coated perovskite films were crystalized by heating at 100 °C for 30 min. After cooling, hole-transport layer (HTL) of spiro-OMeTAD was spin coated at 4000 rpm for 30 s. HTL solution was prepared by dissolving 86 mg mL ¹ spiro-OMeTAD (Ossila Ltd. sublime grade) in 1 mL chlorobenzene, Li-TFSI (20 pL from 500 mg mL· ¹ stock solution in Acetonitrile), FK209 Co-TFSI (ll pL from 300 mg mL ¹ stock solution in acetonitrile) and tert-butyl pyridine (34 pL). HTL coated perovskite cells were aged in dry air (RH < 20%) for 12 hrs before depositing Au (80 nm) top electrodes using thermal evaporation. Fabricated devices were then encapsulated first using 250 nm Al ₂ 0 ₃ deposited by e-heam process and then using UV-Vis curable epoxy (Ossila Ltd.) with glass cover-slip. The thickness of the perovskite layer was 550 ±20 nm. The active area of the device was 0.12 cm ² .

Device Fabrication: spiro-OMeTAD/mixed-perovskite/SnO _x

For the fabrication of perovskite solar cells on SnO _x compact layers, patterned and cleaned FTO-glass (7il/sq, Hartfordglass Inc.) was covered with a 10 nm SnO _x layer using an atomic layer deposition (ALD) process.

Tetrakis(rH ethyl amino)tin (TV) (TDMSn, Strem, 99.99%) was used as a tin precursor and held at 75 °C during depositions. The deposition was conducted at 118 °C with a base pressure of 5 mbar in a Picosun R-200 Advanced ALD reactor. Ozone gas was produced by an ozone generator (INUSA AC2025).

Nitrogen (99.999%, Air Liquide) was used as the carrier and purge gas with a flow rate of 50 seem per precursor line. The growth rate was 0.69 A per cycle. Double cation (FA _o .ssMA _o . sPbL _j ) perovskite solution was prepared by dissolving FAT (182.7 mg, 06 mmol), MAI (29.8 mg, 0.19 mmol) and Pbl ₂ (576.2 mg, 1.25 mmol) in a mixture of 800 pL DMF and 200 pL DMSO. The solution was filtered using a 0.45 pm PTFE syringe filter before use. FA _o .ssMAo. sPbls perovskite films were prepared on the compact SnO _x layer by spin-coating 75 pL solution at first 1000 rpm, then 5000 rpm for 10 s and 30 s, respectively. 500 pL chlorobenzene was dripped as an anti-solvent 15 s before the end of the second spin cycle. Spin-coated perovskite films were annealed at 100 °C for 10 min. For the hole transporter layer, l mL of a solution of spiro-OMeTAD (Borun Chemicals, 99.8%) in anhydrous chlorobenzene (75 mg mL ¹ ) was doped with 10 pL 4-terf-butylpyridine and 30 pL of a Li-TFSI solution in acetonitrile (170 mg mL-^and deposited by spin-coating at 1500 rpm for 40 s and then 2000 rpm for 5 s. After storing the samples overnight in air at 25% relative humidity, 40 nm Au was deposited through a patterned shadow mask by thermal evaporation.

The devices were encapsulated using epoxy (Liqui Moly GmbH) and glass cover- slips. The active area was 0.158 cm ² for the impedance measurements. Photovoltaic characterisation

The current-voltage characteristics of the spiro-OMeTAD/mixed- perovskite/Ti0 ₂ device was measured with forward and backward scans between -o.i V to 1.2 V with scan rate of 400 mV s ^_1 under a Newport 92251A- 1000 AM 1.5 solar simulator calibrated against an NREL certified silicon reference cell. An aperture mask of 0.0261 cm ² was used to define the active area. The device was aged using an ATLAS Suntest CPS+ solar simulator with a 1500 W xenon lamp and internal reflector assembly to provide continuous illumination (~ioo mW cm- ² ) to the unmasked device for 200 hrs prior to the impedance measurements. Current-voltage measurements were made every 10 mins (reverse sweep 1.15 V to oV) in lifetime tester. These are illustrated in Figure 6

Performance and apparent ejficiency of the spiro-OMeTAD/mixed- perovskite/Ti0 ₂ device for a forward scan after aging for 200 hrs at open circuit with 1 sun equivalent illumination. Scan rate: 0.2 V sA

This is shown in Figure 7.

Impedance measurement

Impedance measurements were performed using an Ivium CompactStat potentiostat. The perovskite solar cell devices were masked using a mask slightly bigger than the total active area defined by the overlap between the FTO layer and the top metal contact. All impedance measurements were run by applying a 20 mV sinusoidal voltage perturbation to the cell superimposed to a DC voltage. The potentiostat measures the resulting current which is used to calculate the impedance spectrum as described in the main text. The frequency of the perturbation was varied between 1 MHz to 0.1 Hz. The measurement was performed after a stabilization time of at least loo seconds at the (light and voltage) bias condition used in the measurement, unless stated otherwise. When different stabilization protocols were used to investigate the effect of

preconditioning on the impedance measurements, these are specified in the main text and in this supporting document. Different light bias conditions were obtained using white LEDs and the sun equivalent light levels were calibrated against a silicon photodiode in turn calibrated by an AM1.5 solar simulator. Stabilization of the cell was performed as follows. Chronopotentiometry (for impedance measurements under light at open circuit) or chronoamperometry (for impedance measurements under light at short circuit or in the dark with an applied potential bias) measurements were run before the stabilization stage to monitor the cell behaviour while settling to the set measurement condition. For each measurement at open circuit under light, we ran a chronopotentiometry measurement and we used the open circuit voltage measured after at least 100 seconds as the DC voltage bias condition during the impedance measurement. This voltage was applied for an additional 100 seconds before the beginning of the impedance measurement. For measurements at short circuit under light or at an applied potential in the dark, a chronoamperometry measurement was run for 100 seconds to monitor the evolution of the current in the device at the applied voltage. The same voltage was then applied for additional 100 seconds before the start of the impedance measurement. In some cases we noticed that changes in cell potential or current still occurred after 100 second stabilization time. One could expect that these slow variations would not significantly vary the features probed for frequencies that range down to about 10 times the inverse of the stabilization time (in our case about 0.1 Hz). We find that this is not the case as discussed later in this document. In particular, some peculiar features (loops in the Nyquist plots) can disappear upon long enough

stabilization (see supporting Figures 8 and 11). While these features might still be representative of the state of the device at the time of the measurement, they represent a transient state rather than the equilibrated state. For quasi equilibrium measurements it is therefore recommended that different stabilization times are used to quantify the influence of this parameter on the impedance spectrum and identify the minimum time needed for the spectra to reach acceptable convergence.

Figure 8 shows the effect of device stabilisation on impedance measurements. Figure 8(a) shows VOC vs time for f = o.i sun illumination following

preconditioning at o V in the dark. Figure 8(b) shows Nyquist plot of the imaginary vs real parts of the impedance over a frequency range o.i Hz to l MHz, before (dotted line) and after (full line) stabilisation. The individual impedance measurements were collected in order of decreasing frequency (opposite direction to arrow).

Figure 9 shows complete impedance spectra of the devices shown in 2.

Figure 10 shows measured and simulated impedance spectroscopy

measurements for the device shown in Figure 2 at short circuit. Figure 11 shows similar spectroscopy values varying with applied potential. Figure 13 shows simulated impedance with applied voltage in the dark.

Figure 12 illustrates possible consequences of in photoinduced changes in ionic resistance for impedance spectra of a simplified hybrid perovskite solar cell calculated using an equivalent circuit model assuming C _m is constant. In this equivalent circuit model, the interfacial transistor element seen in Figure 4b has been replaced with a diode element representing a conventional recombination process. Three light intensities are shown corresponding to potentials V across the device of 0.1 V Qeft-hand side), 0.2 V (central), and 0.3 V (right-hand side) and respective ionic resistances of i¾ _on = 2 x 10 ⁶ , 4 x 104, 1 x ic>3 W cm ² . The other elements are Cio _n = 1 x io ^-8 F cm ^-2 , C _g = 1 x to ^-8 F cm ² and J _Tec0 = 1 x io ^-11 A cm ² . It is apparent that although capacitance of the device shows a shift in its frequency dependence, there is no change in the capacitance of the device at low frequencies. This is in contrast to observation where the apparent capacitance increases at low frequency but there is no shift in the frequency of this feature (Figure 2a and b). We note that if there were also photoinduced changes in Ci _on then it is possible that Ci _on and i¾ _on could co-vary such that the time constant of the ionic response remained unchanged. However, since C _on will be

predominantly controlled by the width of the interfacial space charge regions, which have contributions from both the accumulation/depletion of mobile ions in the perovskite as well as a contribution from depletion of electrons or holes in the contacts. Any change in Ci _on is likely to be dominated by changes in the electronic depletion layer which to a first approximation scales with the WX Thus perfect co-variance of i _on and J¾ _on is unlikely.

Details of the drift-diffusion impedance simulation

For simplicity we used electron and hole transporting contacts with the same band-gap, but work functions that differ from the intrinsic perovskite, to create a built-in potential in the simulated perovskite layer.

% Device Dimensions [cm]

p.tp = 200e~7; % p-type layer thickness

p.pp = 6o; % p-type layer points

p.ti = 500e- ; % Intrinsic layer thickness

p.mue_i = 20; % electron mobility

p.muh_i = 20; % hole mobility

p.mui = ie-10; % ion mobility

% Energy levels

p.EA = 0; % Conduction band energy

p.IP = -1.6; % Valence band energy

p.PhiC = -0.15; % Cathode workfunction

p.PhiA = -1.45; % Anode workfunction

p.Eg = p.EA-p.IP; % Band Gap

% Effective density of states and doping concentration and band bending p.No = ie20; % effective Density Of States (eDOS) %%%%%

MOBILE ION DEFECT DENSITY %%%%%

p.NI = iei9; % [cm-3]

% Radiative recombination, U = k(np - ni ^A 2) p.krad = ie-12; % [cm3 s-i] Bulk Radiative Recombination coefficient

[nominally ie-10]

% SRH recombination in the contact regions,

% U = (np-ni ^A 2)/(taun(p+pt) +taup(n+nt))

p.taun_etl = se-io; % [s] SRH time constant for electrons

p.taup_etl = 5e-io; % [s] SRH time constant for holes

p.taun_htl = ge-io; %%%% USE a high value of (e.g.) 1 to switch off p.taup_htl = 5e-io; %%%% NOT o- these variables are in the denominator p.taun_i = ie6;

p.taup_i = ie6;

p.sn = 0;%ie7; % [cm s-i] electron surface recombination velocity (rate constant for recombination at interface)

p.sp = o;%sn; % [cm s-i] hole surface recombination velocity (rate constant for recombination at interface)

The impedance characteristics of the circuit models and the interpretation of impedance spectra Recombination limited impedance spectra

Figure 14 shows examples of equivalent circuit model Nyquist plots and impedance spectra (magnitude \Z(w)\, phase Q, and apparent capacitance Re[i/ (ΐwZ)]) for a recombination limited circuit. The arbitrary parameters Ri _on = l x 109 W cm ² , Ci _on = 1 x lcr ⁸ F cm- ² , C _g = 1 x io ^~9 F cm- ² and J _rec0 = 1 x icr ¹¹ A cm ^-2 , evaluated at V= o V (yellow) and V = 0.8 V (blue). The corresponding circuit model cyclic voltamogram for _ph = 25 mA cm- ² (solid lines) and J _ph = o (dashed lines) with a scan rate of 0.1 V s ^-1 between o and 0.8 V forward (purple) and reverse (light blue). Applied voltage V, ionic interface potentials V and V ₂ and electron potential V _n vs time are also shown.

Impedance of the injection interface If there is significant forward bias (V > o) across the perovsldte/ETM interface and the light intensity is sufficiently small, then ¾ » J _mj so that the second term in equations 10 and 13 can be neglected. Inspection of the first term suggests that if charge injection processes limit the current through the device it will display behaviour similar to an inductor. In the small perturbation regime with v superimposed on a background voltage V across the interface, the real part of the injection impedance is given by:

15

The corresponding negative value of the imaginary part of equation 13 divided by the angular frequency (when /¾ » /,/,,) gives an expression which is analogous to an apparent inductance to injection Zi _nj of charge carriers across the interface:

16

This apparent inductance is independent of frequency, but varies with exp[- qV/kaT ] so will be more dominant at low applied voltages. This has the potential to lead to loops in Nyquist plots (Figure 3(c)). The corresponding amplification of the out of phase injection current by the ionic current at low frequencies is 180° out of phase, and given by:

J -L nn, ⁽ v =D 2

17 where -gi _n} is the charge injection transconductance. Calculation of the complex component ofV _n Under circumstances where V _n is not given by V or o, the steady state value of Vn must be established as described in the main text, this allows the steady state values evaluated. Assuming current continuity across both interfaces, the complex form of V _n in the small perturbation regime is then given by:

i8 Simplified energy level diagrams and circuit models using

transistors to describe the ionic gating of electron processes at different interfaces.

This is illustrated in Figure 15, in which the dark equilibrium barrier height is indicated by the unfilled rectangles. In non-equilibrium situations, a reduction in barrier height is indicated by the filled section. In Figure 15 (a) the energy levels of the conduction and valence bands in the dark after equilibration of ionic charge are illustrated. Due to detailed balance the interfacial currents are equal and opposite ( sen = Jrec =/ _Ei at interface 1 and at interface 2) at dark equilibrium. The corresponding energy level profiles after applying a voltage V, in the dark ( _n = o), for a device whose impedance is limited by electron recombination is illustrated in Figure 15(b) immediately after the voltage is applied (w ¥) and in Figure 15(c) after the redistribution of ionic charge has reached steady state (w o). The changes in barrier heights (¾en, hec, _{col an} d ¾) for the various interfacial electron transfer processes in response to an applied potential V and the electron quasi-Fermi potential (U _n ) are indicated. Figure 15(d) shows the corresponding change in the electrostatic potential profile (dashed line - instantaneous, solid line - steady state). The changes in electrostatic potential at interfaces l and 2 are indicated by V _\ and V ₂ . The relationship between these changes is given in Table 1. Figure 15(e) shows a general example for a device in the light (where the electron quasi Fermi level ¾ ¹ °). In this case the device impedance has contributions from both interfaces and the ions have not reached a steady state distribution. Figure 14(f) shows the equivalent circuit model for the impedance of the ionic circuit branch in response to high frequency voltage perturbation, Ή" °°) where perovskite ions are effectively frozen, and at lower frequencies, ⁿ ( ^w < ¥ ^') where perovskite ionic motion is described by Cion-i¾on-Cion series elements. Here we assume the dopant ions in the HTM and ETM (red and blue squares) are static. Figure 15(g) shows an equivalent circuit model for the device in which the impedance to electron transfer for both interfaces are modelled as bipolar transistors with impedance Z and Z ₂ , the base terminals are gated by the ionic potentials Vi and V ₂ . The curved arrows indicate the potential differences between the‘terminals’ on the transistor elements. Figure 15(h) shows a general circuit model considering both electrons (n) and holes (p) with a (negative) photogeneration current ( ph), where the potential of the electrons (Vn) and holes (V _p ) in the perovskite layer correspond to the electron and hole quasi Fermi levels.

With these definitions and the above description in mind, the skilled person will appreciate the following expressions for potentials driving electron transfer processes, and circuit branch impedances. The impedance for the electronic branch of the circuit is given for the specific case where impedance due to recombination of a single carrier dominates. The impedance of the electronic circuit branch, _rec , is given in terms of the apparent capacitance and resistance of the interface c _rec and r _rec which are represented in Fig. 2f.

Change in barrier potential for electron

transfer relative to equilibrium (V)

Electron generation F _gen = V _t — V

Electron

com _Ί bi .nat .i. V rec = V

on ' 1.— V

re n

Electron collection V _col = V ₂ - F _n

Electron injection F _in} = v ₂

Electrostatic potential from ionic circuit (V)

Impedance of ionic circuit branch (il cm ² )

Impedance of electronic circuit branch* (il

In these terms, equation 1 for the net electron recombination current across an interface given above can be expressed as f _L - J _sl g

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