Landau–Khalatnikov subcircuit based equivalent circuit model for hybrid perovskite solar cells

Hybrid perovskite solar cell technology has a distinct advantage over the conventional solar cell technologies due to its high predicted efficiency and low manufacturing cost. However, its commercialization is hindered by the unpredictability existing in its J-V characteristics leading to ambiguous efficiency estimation. Modeling the hysteresis in the J-V characteristics is a means of curtailing this ambiguity. It is established in literature that hysteresis models can be derived from the non-linear behavior of ferroelectric materials. Perovskite, which forms the light absorbing region of the solar cell is a ferroelectric material. In this paper, an equivalent circuit model for the hybrid perovskite solar cell is proposed in which the reasons for origin of hysteresis is characterized as varying capacitance to model hysteresis. A Landau–Khalatnikov subcircuit which portrays this variation is the principal addition to the conventional model to include hysteresis effect. The model parameters of the subcircuit are estimated from the inherent properties of perovskites. Hence, the proposed equivalent circuit model is completely physics based and it links the material property of perovskite to its equivalent circuit model parameters.


Introduction
Hybrid perovskite solar cells (HPSCs) are an expeditiously developing photovoltaic technology having a true potential to succeed the commercially popular silicon solar cells. It shows a large promise in efficiency reaching upto 25.8% and also has cheaper fabrication costs due to low cost solution processing (Kojima et al. 2009;Jeong et al. 2021). In spite of such advantages, this technology is still held back by numerous challenges impeding its performance in the commercial market. These limitations include device instability due to hysteresis in its current density-voltage (J-V) characteristics, toxicity due to the presence of lead and shorter lifetime.
The basic structure of a HPSC consists of a light absorbing intrinsic perovskite layer inserted between an n-type electron-transport layer (ETL) and a p-type hole transport layer (HTL) forming a p-i-n type structure . When light falls on the intrinsic region, excitons and unbound electron-hole pairs are generated. By the property of charge selectivity offered by ETL and HTL layers, the photogenerated pairs are dissociated to produce current. Figure 1 describes various layers of a typical HPSC.
The most significant limitation of HPSCs is the anomalous hysteresis observed in its J-V characteristics . Hysteresis refers to the variation occurring in its J-V graph, while the voltage is swept in the forward and reverse directions. A graph depicting hysteresis effect is shown in Fig. 2. This hysteresis phenomena can lead to ambiguous estimation of device efficiency, thereby challenging its reliability during actual performance. The origin of hysteresis have been attributed to four main reasons -capacitive effects, ferroelectric polarization, ion migration and charge trapping (Kalyanasundaram et al. 2016). Among these, the first two effects originate from the inherent properties of the intrinsic perovskite layer. The last two effects are caused due to the interfacial defects formed by ETL and HTL with the perovskite layer (Chen et al. 2016). Accordingly, HPSCs behave like conventional solar cells only during the forward scan. However, various physical effects mentioned above will cause the maximum power point to shift towards higher values during the reverse scan.
A few approaches were suggested in literature for modeling the hysteresis of HPSCs. Some of the models are numerical (Nandal and Nair 2018;Kumar 2021;Nandal and Nair 2017;Agarwal et al. 2014), while the others are analytical (Pedesseau et al. 2016;van Reenen et al. 2015;Ravishankar et al. 2017). A few equivalent circuit models (ECMs) are also proposed for the same (Nemnes et al. 2017a, b;Anghel et al. 2019;Seki 2016;Liao et al. 2018;Cojocaru et al. 2015Cojocaru et al. , 2017Todinova et al. 2017;Guerrero et al. 2021;Bisquert et al. 2021;Bisquert and Guerrero 2022). The equivalent circuit based approach for modeling the J-V characteristics of HPSCs is derived from the ECM of a conventional solar cell. The most notable factor in the equivalent circuit-based approach was the introduction of a capacitor or an inductor in the circuit to explain the phenomenon of hysteresis. G. A. Nemnes et al., proposed a dynamic electrical model (DEM) to explain the hysteresis of HPSCs (Nemnes et al. 2017a). Here, a capacitor was introduced to represent both the linear and non-linear capacitive effects occurring while performing the forward and reverse scans of the cell. The aforementioned model was extended to include inverted hysteresis phenomena along with normal hysteresis (Nemnes et al. 2017b). An extension of the DEM was suggested by D. V. Anghel et al., where the capacitance was replaced by a subcircuit model which accounts for the non-linearity occurring in the J-V graph due to hysteresis (Anghel et al. 2019). In another model, K. Seki proposed an equivalent circuit representation based on interface charge accumulation (Seki 2016). P. Liao et al., proposed another equivalent circuit model in which double heterojunctions were used to simulate the J-V curves of planar perovskite solar cells (Liao et al. 2018). A universal double diode model was suggested for fitting the J-V graph of the heterojunction perovskite solar cell. In a model proposed by L. Cojocaru et al., the origin of J-V hysteresis has been related to the capacitance created at the interfaces (Cojocaru et al. 2015). Another model uses an equivalent circuit with an inductance to simulate the hysteresis effect (Cojocaru et al. 2017). Few ECMs also stemmed from the analysis of impedance spectra (IS) of perovskite solar cells. IS techniques are widely used to characterize HPSCs by employing simple elements like resistors and capacitors. The circuital model so developed is used to fit the measured IS spectra (Todinova et al. 2017;Guerrero et al. 2021;Bisquert et al. 2021;Bisquert and Guerrero 2022).
In this paper, each physical effect which contributes to hysteresis is analyzed individually. This leads to the development of a physics based equivalent circuit model which addresses all the reasons behind the origin of hysteresis. In Sect. 2, an investigation on the reasons behind the origin of hysteresis and the method of development of the proposed ECM is presented. The scheme of extraction and estimation of the parameters of   ECM is given in Sect. 3. In Sect. 4, the developed model is validated against experimental data from literature and Sect. 5 concludes the work.

Model development
In this section, a complete ECM for HPSC is developed from the ECM of conventional solar cell incorporating all the reasons for hysteresis. Various reasons for the origin of hysteresis, namely, capacitive effects, ferroelectric polarization, ion migration and charge trapping are individually analyzed to develop subcircuits corresponding to each effect. These subcircuits are combined together to obtain the complete model.
The ECM of a typical solar cell is shown in Fig. 3. The equation for the ECM of a solar cell including losses (Martil and Diaz 1992) is given by where, I 0 is the reverse saturation current, is the ideality factor of the diode, V T is the thermal equivalent voltage and I ph is the photogenerated current. Here, R s represents the internal resistance offered by the cell and R sh is a parameter to show the leakage current (Zhang et al. 2011).

Ferroelectric polarization and capacitive effect
The property of negative capacitance in ferroelectrics is used to develop a subcircuit to emulate the hysteresis in J-V characteristics. Naturally occurring perovskite is a calcium titanium oxide mineral with the chemical formula CaTiO 3 , and these perovskites are inherently ferroelectric by nature. The active layer in HPSCs is made of artificially synthesized organicinorganic hybrid halide perovskites, e.g., CH 3 NH 3 PbI 3 (MAPbI 3 ) and these materials are also found to exhibit ferroelectric polarization (Rakita et al. 2017).
A ferroelectric material is capable of sustaining a polarization even after removal of the applied electric field. Perovskites tend to exhibit spontaneous polarization due to the geometrical imbalance in its structure. The relationship between electric displacement, D, field, E and polarization, P is given by  (Martil and Diaz 1992) The polarization-hysteresis loop of a ferroelectric material is shown in Fig. 4. A ferroelectric material is characterized by two different polarizations. The linear polarization of the material in the presence of an electric field is P E = E , where represents the electric susceptibility of the medium. The spontaneous polarization, P S is caused due to the alignment of dipoles in the material (Dragan et al. 2005). The relation between E and P S is non-linear and the value of P S is non-zero even when the field is made zero. The non-zero polarization value on the hysteresis loop in the absence of a field is its remanent polarization, P r and the negative value of field at which the total polarization is made zero is the coercive field, E C of the ferroelectric.
All perovskites are ferroelectric by nature and exhibits negative capacitance. Hence, HPSCs behaves as ferroelectric capacitors. Since all ferroelectric materials exhibit remanent polarization, the capacitance of a ferroelectric capacitor varies while shifting from one stable polarization to the other (Yadav et al. 2019).
Landau model is the most widely used equation to fit experimental data of spontaneous polarization with electric field (Khan et al. 2015). Its corresponding circuit model is the Landau-Khalatnikov (L-K) circuit for negative capacitance and was found suitable to incorporate the theory of negative capacitance into the hysteresis model. Another model used by researchers for accounting negative capacitance is the Miller model (Saha et al. 2018). But, the Miller circuit for negative capacitance when placed as an additional parallel branch in the conventional ECM, is ineffective in reflecting the variation in capacitance due to hysteresis effects to a corresponding change in voltage. Hence, Landau model was chosen for the same.
The Landau-Khalatnikov (L-K) dynamical equation for a ferroelectric capacitor is given as where, Q F represents the charge in a ferroelectric capacitor, U gives the energy stored in the capacitor and is a material dependent parameter having the dimensions of resistance and accounts for ferroelectric switching. This gives rise to an expression for voltage across the ferroelectric capacitor given by where, C F (Q F )represents a capacitor whose capacitance variation with charge Q F is non-linear in nature and hence, referred to as a non-linear capacitor (Khan et al. 2015).
Polarization-hysteresis loop of a ferroelectric material (Dragan et al. 2005) Equation (4) gives the voltage contributed by the ferroelectric property of the material. Its corresponding equivalent circuit representation is given in Fig. 5a. For any given capacitor, voltage across it can be expressed in terms of its charge. The total charge in a capacitor is the sum of charges on the metal electrodes when placed in vacuum and the charge due to polarization in the ferroelectric medium between the electrodes. Maxwell's displacement field within a material with polarization is given by Eq.
(2). Hence, the total charge across the material is obtained as Therefore where and V F is obtained from Eq. (4).
Here, Q tot denotes the total charge in the capacitor, Q vac is the charge in the capacitor when the metal electrodes are placed in vacuum and Q F represents the charge in the ferroelectric capacitor caused due to polarization.
The L-K dynamical equation for total charge given by Eqs. (4), (6) and (7) can be represented by an equivalent circuit (Sivasubramanian et al. 2003) as shown in Fig. 5b. The capacitor on the upper branch of the circuit, C 0 is a linear steady-state capacitor.
The capacitor, C F on the lower branch is a non-linear capacitor, which stores the polarization energy of the ferroelectric material. The ohmic resistor, R F describes the dissipation due to variations in ferroelectric polarization.
Here, the equivalent circuit for the L-K dynamical equation is adopted to capture the ferroelectric nature of perovskite material. Hence, the proposed ECM which incorporates the ferroelectric polarization and capacitive effects is shown in Fig. 6. 5 a Circuit equivalent for a ferroelectric capacitor using a resistive element and a non-linear capacitor (Khan et al. 2015); b Equivalent circuit to the Landau-Khalatnikov dynamical equation (Sivasubramanian et al. 2003) Page 7 of 20 176

Ion migration
Ion migration was found to be one of the major contributors for the occurence of hysteresis in HPSCs (Azpiroz et al. 2015). When light falls on the active layer, which is an intrinsic region of the solar cell, bound charges are separated by the photogenerated potential (Eames et al. 2015). For example, when light falls on the perovskite, CH 3 NH 3 PbI 3 ,it is split into the following ions The ions or dipoles thus formed, migrates to the corresponding interfaces. The surface dipoles that accumulate at the two interfaces acts as opposing potentials to the internal barrier. This lowers the effective potential barrier and increases the current injection at the interfaces (Ebadi et al. 2019). This phenomenon also adds a negative capacitance effect in HPSCs. From the above, it is concluded that the capacitor, C dl that represents the effect of ion migration in the proposed model should be placed along with the ferroelectric capacitor, C F in the L-K branch. The expanded ECM which includes the effect of ion migration is shown in Fig. 7.
A step increase in voltage, ΔV leads to a capacitive ionic response with a single time constant, ion given by This current, flowing through R ion charges a capacitor by forming a Debye layer, couple of nanometers wide at the interface, between the perovskite layer and the charge-transport layer. Accumulated ionic charge influences both the electric field and the barrier potential, V bi . The piled up charge acts as a surface dipole, and hence changes the barrier potential by where, ΔU(t) represents the change of voltage drop over the depletion layer capacitance, C dl which is given by Hence, the variation in the internal barrier potential due to ion migration is reflected in the equivalent circuit by the depletion layer capacitance, C dl .

Charge trapping
The trap assisted recombination at the interfaces of perovskite and the charge transport layers was found to be one of the most dominant recombination mechanisms invoking hysteresis effect in perovskite solar cells (Yu et al. 2016). The cause of hysteresis is attributed to the presence of injection barrier and trap assisted recombination in the bulk and interfacial sites. Application of voltage at the electrodes of the HPSC creates an electric field within the device that forces the carriers to move with an average drift velocity. The change in the injection barrier leads to charge accumulation in the bulk region near the interfaces. This, in turn increases the trap-assisted recombination, resulting in a reduction of current density. The carrier density at the interfaces due to the presence of injection barriers (Shi et al. 2018) are given by At the cathode interface, At the anode interface, where, L is the thickness of the absorber region or the bulk, N c and N v are the effective density of states in the charge transport materials, n and p are the injection barriers at the cathode and anode sides respectively. The bulk recombination occurs in the absorber region and the amount of trap-assisted bulk recombination is largely affected by the change in barrier potential. This trap-assisted bulk recombination being accumulative in nature, cumulatively alters the series resistance of a solar cell (Seki 2016). Hence, the accumulative capacitance, C acc depicting the series leakage due to charge trapping is placed along with the series resistance, R s . The proposed ECM incorporating all the reasons behind hysteresis is shown in Fig. 8.

Parameter extraction and estimation
For validating the developed ECM with experimental data from literature, model parameters have to be extracted from its J-V characteristics or estimated from the device parameters. The four basic circuit parameters of the ECM are initially extracted from the corresponding forward scan J-V characteristics. The other circuit elements added to incorporate the hysteresis effects are calculated from the device parameters of the chosen perovskite.
Observing the hysteresis in the J-V characteristics of different HPSCs, it is found that the V oc obtained from the reverse scan is a little greater than the V oc from its forward scan. For a conventional solar cell ECM whose shunt losses are not negligible, i.e., R sh << ∞ , the ideal diode equation cannot be used to find the value of V oc . The V oc of practical solar cell is hence found from Eq. where, V oc is the open-circuit voltage, I ph and I 0 are the photogenerated current and reverse saturation current of the diode respectively, is the diode ideality factor, V T is the thermal voltage and W −1 is the negative branch of the Lambert-W function. The values of V oc extracted from the reverse and forward scan characteristics are represented by V ocr and V ocf respectively. This variation in V oc is accounted by a shift in the value of ideality factor of the diode. It is found from literature that HPSCs exhibit a uniqueness in ideality factor due to two different current regimes occuring during the forward and reverse scans (Agarwal et al. 2014). It exhibits an ideality factor of 1.5 or more depending on the permittivity of the perovskite material used. Hence, a shift in occurs from 1 to 1.5 when the scanning direction of HPSCs change from forward to reverse. Parameter extraction and estimation is performed using the experimental data set of two devices from literature (Snaith et al. 2014;Kim et al. 2015). Table 1 gives the materials of the various regions of corresponding HPSCs used. The structure of both the devices is as shown in Fig. 1. Device parameters used for model parameter extraction and estimation is given in table 2.

Extraction of conventional circuit elements
The conventional circuit elements can be extracted from its corresponding forward scan data using the Eqs. (15−18). The Lambert-W method is used here to extract the conventional circuit elements. R s , R sh , I 0 and I ph are extracted from the procedure available in literature (Cubas et al. 2014).
To extract R s : where, W −1 is the negative branch of the Lambert-W function. V oc depicts the open-circuit voltage, I sc is the short circuit current, V mp and I mp represent the voltage and current intercepts respectively at the maximum power point and gives the ideality factor of the diode. To extract R sh : To extract I 0 : To extract I ph : Conventional circuit elements for are extracted from the experimental forward scan J-V characteristics (Snaith et al. 2014). For device I, the following values from the forward scan characteristics, namely, J SC = 22.9 mA/cm -2 , V OCf = 0.97 V, J mp = 19.65 mA/cm -2 , and V mp = 0.67 V are used for the same. The extracted circuit parameter values for the conventional ECM of device I are given in Table 3. The same procedure is also followed for the parameter extraction of device II (Kim et al. 2015). The validation of conventional ECM with extracted parameters is shown in Fig. 9.
The open circuit voltage values of forward and reverse scan vary slightly as can be seen from the J-V characteristics. The open circuit voltage from reverse scan, V OCr is required Fig. 9 Comparison of simulated forward scan J-V characteristics of conventional ECM with experimental forward scan J-V characteristics (Snaith et al. 2014) for the estimation of additional circuit parameters. The value of V OCr = 1.07V , is hence, taken from the reverse scan characteristics of device I. The same is again adopted for device II. The estimation of additional subcircuit elements is explained in-detail in the next sub-section.

Specific to ferroelectric polarization and capacitive effect
The property of variation of capacitance in ferroelectric material is used to estimate the sub-circuit parameters of the L-K model. For estimating the sub-circuit capacitance parameters, C 0 and C F , the Landau free energy equation is used (Íñiguez et al. 2019). The Landau free energy equation, F for a ferroelectric capacitor is given by where, and are the anisotropy constants, P is the electric polarization and E is the externally applied field. The equilibrium equation is obtained by equating the free energy, F to zero. Therefore, the corresponding equilibrium equation becomes For a parallel-plate capacitor with area A, distance d between the plates carrying a charge Q, with a vacuum permittivity 0 and an applied voltage V across the plates, the electric polarization P is given by Therefore, the equilibrium condition can be re-written as Page 13 of 20 176 where, C 0 is the capacitance associated with an empty capacitor given by and capacitance of the ferroelectric material at zero applied voltage, C F is given by The derivation for C 0 and C F from Eq. (21) is given in Appendix. The values of C 0 and C F in the sub-circuit can be estimated using the Eqs. (23) and (24).
To obtain the value of , the Clausius-Mossotti equation is used (Kittel et al. 1996).
where r is the relative permittivity of the material, 0 is the vacuum permittivity, is the molecular polarizability, and N is the number of dipoles per unit volume.
For any given ferroelectric sample, the value of N and 0 are constant. Therefore This relation can be used to map the variation occurring in the ferroelectric material due to polarization at the molecular level to a capacitance value in the L-K model sub-circuit. The sub-circuit element, R F can be estimated from the loss tangent of the polarizationhysteresis loop (Sheikholeslami and Gulak 1997). The loss tangent, tan is given by where is the angular frequency used in the polarization loop measurement, C F is the capacitance in the parallel branch and R F is the ac parallel resistance. The parameter is obtained from the polarization-hysteresis loop (Wang et al. 2017) which yields where E C is the coercive electric field, P r and P s are the remnant polarization and the saturation polarization respectively, obtained from its corresponding polarization-hysteresis loop.

Specific to ion migration
The variation in ions migrating to the interfaces is reflected in the change of internal barrier potential. The depletion layer capacitance, C dl that relates ion migration to the change in barrier potential (Bisquert 2003;Lopez-Varo et al. 2018) is given by where, q is the elementary charge and w is the space charge width. Here, 0 , r and N eff are the absolute dielectric constant, relative dielectric constant and the effective concentration of fixed ionized species respectively of the layer where the depletion zone is placed. V bi is the built-in voltage or barrier potential and V is the applied voltage. Here, the applied voltage, V is zero and V bi is equal to the open circuit voltage of the reverse scan, V ocr of the solar cell.

Specific to charge trapping
The accumulative capacitance, C acc due to trap-assisted bulk recombination (Lopez-Varo et al. 2018;Zarazua et al. 2016) is given by where, p b is the bulk carrier density corresponding to native defects, q is the elementary charge, 0 and r are the absolute dielectric constant and relative dielectric constant respectively. k B is the Boltzmann constant, T is the temperature, V bi is the built-in voltage or barrier potential, V is the applied voltage and V T is the thermal voltage. Here, the applied voltage, V is zero and V bi is equal to the open circuit voltage of the reverse scan, V ocr of the solar cell. Extracted and estimated parameters using the proposed procedure for devices I and II are listed in Tables 3 and 4 respectively.

Model validation
The experimental data set taken from literature is used for model validation (Snaith et al. 2014;Kim et al. 2015). The perovskite, methyl ammonium lead iodide or CH 3 NH 3 PbI 3 is used as the absorber material in the structure and is of thickness 300 nm. An active layer thickness of 300 nm was found to be optimal for attaining maximum cell efficiency (Pavu et al. 2022). The device parameters given in table 2 is used for extraction and estimation of model parameters for validating the proposed model (Frost et al. 2014).
The circuit simulations are done in LTspice [52]. The J-V characteristics obtained using the proposed model is validated against the experimental data reported in literature (Snaith et al. 2014;Kim et al. 2015) and are shown in Figs. 10 and 11 respectively. The standard deviation (SD) of the forward scan J-V characteristics of the proposed model with that of experimental forward scan data was found to be 0.5723 and similarly, SD for the corresponding reverse scan J-V characteristics was 0.5529. From these values of SD, it can be concluded that the J-V characteristics of the proposed model has a very good fit with that in literature. Also, since the SD of reverse scan is slightly less than that of the forward scan data, it can be concluded that the estimated parameters using the properties of the perovskite leads to a fairly accurate model for modeling the hysteresis in HPSCs.
The experimental forward scan J-V characteristics serves as the base for extracting the conventional circuit parameters of the proposed model. The proposed ECM includes both extracted and estimated circuit parameters. The hysteresis effect in the reverse scan characteristics is, therefore, realized by introducing additional circuit parameters estimated from the physical properties of the perovskite material used as the active layer. Hence, it is a completely physics based model with good accuracy.

Conclusion
This work reported an L-K subcircuit based equivalent circuit model for modeling hysteresis in hybrid perovskite solar cells. The main reasons for the occurence of hysteresis in HPSCs namely, capacitance effect, ferroelectric polarization, ion migration and charge trapping are analyzed individually to develop the proposed ECM. The proposed ECM was developed using the ECM of a conventional solar cell as the basic building block. Simple circuit elements such as resistances and capacitances are utilized in this model to emulate the reasons for hysteresis. The parametric values of the newly introduced subcircuit model elements are estimated from the inherent ferroelectric and semiconducting properties of perovskite. The proposed equivalent circuit model is completely physics based with commendable accuracy and can be used to model various solar cell structures including complex tandem perovskite solar cells.
The corresponding equilibrium Eq. (20) is given by Substituting for P, it is re-written as where A is the area, d is the distance and V is the voltage between the two electrodes of the capacitor.
Re-arranging terms, By re-substituting C 0 or From this, the equation for C F is obtained as where, is the molecular polarizability of the material under study. Data availability Data sharing not applicable to this article as no datasets were generated or analysed during the current study.

Conflict of Interest
The authors declare that they have no conflict of interest. The authors have no financial or proprietary interests in any material discussed in this article