Analytical Model for Light Modulating Impedance Spectroscopy (LIMIS) in All-Solid-State p-n Junction Solar Cells at Open-Circuit

Non-circuit theory drift-diffusion numerical simulation of standard potentiostatic impedance spectroscopy (IS) is a well-known strategy for characterization of materials and electronic devices. It implies the time-dependent solutions from the continuity and Poisson's equations under small perturbation of the bias boundary condition at the electrodes. But in the case of photo-sensitive devices a small light perturbation can be also taken modulating the generation rate along the absorber bulk. In that focus, this work approaches a set of analytical solutions for the signals of IS and intensity modulated photocurrent and photovoltage spectroscopies, IMPS and IMVS respectively, from one-sided p-n junction solar cells at open-circuit. Subsequently, a photo-impedance signal named light intensity modulated impedance spectroscopy (LIMIS equals IMVS over IMPS) is analytically simulated and its difference with respect to IS suggests a correlation with the surface charge carrier recombination velocity. This is an illustrative result and starting point for future more realistic numerical simulations.


Introduction
The concept of impedance as a transfer function in a form of a ratio between two complex magnitudes has been widely tackled, since its introduction by Heaviside. 1 The most typical application is for the study of electrical current response to small voltage perturbation, as in the standard potentiostatic impedance spectroscopic (IS) where the impedance itself has units of Ohms. 2 The IS is a well-known and stablished characterization technique for the evaluation of the resistive, capacitive and inductive features of materials and electronic devices. Particularly, on photovoltaic solar cells IS typically informs on the recombination modes, 3 the doping densities, [4][5] deep defect levels 6 and the density of states. 7 One of the most common practices is to measure potentiostatic IS at open-circuit (OC) condition under a series of steady-state illumination intensities. In this way the recombination resistance , chemical capacitance and characteristic lifetimes can be evaluated, among other parameters.
Alternatively, in photo-sensitive samples the current or voltage responses to small light intensity perturbations can be also studied, which are the cases of the intensity modulated photocurrent spectroscopy (IMPS) [8][9][10][11][12][13][14][15][16][17][18][19] and the intensity modulated photovoltage spectroscopy (IMVS), [14][15][20][21][22] respectively. IMVS and IMPS individually characterize the current and voltage responsivities Ψ and Ψ , respectively. Hence, similarly to IS, we can take the ratio IMVS/IMPS to define a light intensity modulated impedance spectroscopy (LIMIS). This relation was first introduced by Song & Macdonald 23 who measured the spectra on n-Si in KOH solution, validating the transfer function by Kramers-Kronig transformation. Later Halme 24 applied the concept on dye sensitized solar cells, concluding the approximate equivalence between IS and LIMIS. Furthermore, simultaneously to this work, we have measured LIMIS in all-solid-state silicon, organic and perovskite solar cells showing qualitative and quantitative differences between IS and LIMIS. 25 In this article we further analyze this concept at OC and solve the drift-diffusion equations in an analytical approximation for the one-sided p-n junction solar cells that suggest a correlation between the difference among IS and LIMIS and the surface recombination velocity.
Let's consider a sinusoidal ̃( ) small perturbation applied to a generic sample at  15-16, 18, 22, 27 Now, as discussed before, 25 it can be advantageous to combine IMPS and IMVS instead or complementary to their individual analyses. Therefore we obtain the "light intensity modulated impedance spectroscopy" (LIMIS) as These concepts are also introduced in Section S1.1.

Results
In our simultaneous work 28  formalisms already implemented and exploited in the literature. [29][30][31][32][33] In the case of LIMIS, or individually IMPS and IMVS, the complete development of the time dependent numeric solutions is still not reported in all-solid-state devices, to the knowledge of the authors. Such task is beyond the scope of this paper.
Nevertheless, we here introduce and discuss the main equations and boundary conditions, as well as an analytical solution for a particular case of one-sided abrupt p-n junction thin film solar cell.
In Section S1.2. we present the continuity equation (S9) including the drift and diffusion terms, the Poisson's equation, the drift-diffusion currents and the boundary conditions (S10) for the electrostatic potential and the current in the assumption of ohmic contacts. In this formalism we highlight that IS and LIMIS are different regarding "where" the perturbation is included. In IS ̃ directly affects the boundary condition, which defines the electric field after the Poisson's equation. Later, its effect will be particularly related with recombination and its influence in the space charge region in the continuity equation.
On the other hand, the perturbed term in LIMIS is directly affecting the continuity equation via the generation rate . Assuming a light intensity independent incident light spectrum, can be expressed in dc and ac real terms as where here is nearly the absorber layer thickness and Ψ is the photo-current responsivity at short-circuit that depends on the incident light spectrum, the absorption coefficient and geometry of the absorbing materials. Note that ̃= |̃| and ̃= |̃| are the perturbation, and similarly to ̃, we will omit the modulus notation in the next.
Also note that only in thin film devices (5) can be approached to an space independent constant , otherwise the Beer-Lambert law should be considered. Subsequently, the inclusion of ̃ in the continuity equation defines the diffusion currents out of the space charge region, or in situations for low field effects. This can be particularly significant for the current boundary condition.
Keeping this in mind, in Sections S1.1-8 the analytical solution of the charge carrier concentrations around OC under ̃ and ̃ perturbations, for IS and LIMIS respectively, are presented. The main idea is to structure the solutions in the form where ̅ 0 = ̅ + 0 , 0 is the dark equilibrium concentration, ̅ the steady-state overequilibrium-concentration (under dc bias and/or illumination) and ̃ is the complex amplitude response to ̃ or ̃ which includes the phase shift , i.e., ̃= |̃|exp[ ].
The current boundary condition was taken as (S11) in the form of ohmic contact selectivity with negligible drift current, where minority carriers recombine with surface recombination velocity . No significant difference was considered between IS and LIMIS regarding the constrain. On the other hand, the potential boundary condition was chosen as the depletion approximation (S12) expressing how the different measurement ways ideally affect the charge carrier distributions and hence the energy diagram. This is summarized in Figure 1. For IS the ̃ small perturbation changes the depletion region width and creates small charge carrier gradients around the steady-state OC distribution. For the IMVS the ̃ small perturbation also changes , but without gradients, so the charge profile is flat all the time. Finally, for IMPS no change of is assumed and the opposite gradient direction takes place. Accordingly, as deducted in section S1.5. the impedance from IS at OC can be written as Where ̅ = √ / 0 is the diffusion length, the diffusion coefficient, 0 a characteristic recombination frequency as (S19.b) and ̃ is a surface recombination factor as (S24.c).
Furthermore, in sections S1.6,7 the ac voltage and current responsivities were deducted as with the complex diffusion length ̃ as (S21.b), the factor ̃ as (S24.c) and Interestingly, (8) suggest that at the low frequency limit Ψ ∝ ̅ −1 but Ψ should be nearly ̅ -independent. Regarding the -dependency, towards higher frequencies both are expected to behave with arc-shape-type spectra since Finally, the impedance from LIMIS at OC was deducted in section S1.8 as (7) and (9) we can see that the IS-LIMIS normalized impedance difference is meaning that it may inform on the surface recombination at the electrodes. More particularly, the model suggests that perfect contact selectivity, with only majority carriers current towards the electrodes may deliver no difference between LIMIS and IS. These results could be used to compare different samples or different states of the same sample regarding the interface recombination parameters.

Conclusions
In summary, a particular case on one-sided p-n junction was analytically solved suggesting that the difference between LIMIS and IS respective impedances, informs on the interface recombination velocity.

Conflicts of interest
There are no conflicts to declare.

S1.1. Modulated magnitudes
The impedance is defined as the ratio between the oscillating voltage ̃ and current ̃, In the potentiostatic IS case, a voltage small alternating current (ac) perturbation ̃ is applied in addition to the direct current (dc) bias ̅ , so the applied voltage has the form Note that here ̃= |̃| is a real amplitude (the one is set by the instrument), differently to ̃ and the rest of the perturbed magnitudes. For simplicity we will use ̃ in the following. Under the bias perturbation the current may similarly evolve as a modulated ac current ̃ added to the dc current ̅ as In this case, similarly to ̃, ̃ is a complex amplitude which carriers the phase shift information introduced by ̃. Its relations can be approached by drift diffusion equations in the infinite mobility approximation and discarding second order small terms, so it results as [1] Here is the elementary charge, is approximately the distance between electrodes but in practice the integral should be within the space charge region and diffusion lenghts, is the non-radiative surface/Shockley-Read-Hall (SRH) recombination lifetime, is the radiative recombination coefficient and is the generation rate, which is the perturbation for the LIMIS with ̅ and ̃ as dc and ac real amplitudes, respectively, in the form Where Ψ is the photo-current responsivity at short-circuit that depends on the incident light spectrum, the absorption coefficient and geometry of the absorbing materials, and ̅ and ̃ are the dc and ac real amplitudes of the incident light power, respectively. here is nearly the absorber layer thickness. Note that ̃= |̃| and ̃= |̃| are the perturbation, and similarly to ̃, we will omit the modulus notation in the next.

Under a perturbation like Equation (S6) a photo-voltage can be measured with ̅
and ̃ as the dc and ac amplitudes, respectively, in the form Here ̃ is a complex amplitude carrying the information of the phase shift, which can be deducted from standard equation and with the use of McLaurin series as [2] ̅ ≅ + 2 ln [ where is the absorber energy band-gap for the device, is the Boltzmann constant, is absolute temperature, and = √ is the square root of the average density of states at the conduction and valence bands, respectively, or the corresponding one in case of a one-side abrupt junction.

S1.2. General equations for the numeric approach
The continuity equations for charge carrier concentrations of electrons ( ) and holes ( ) within the space between the electrodes [3] = − + ∇ + ∇ + ∇ 2 Here and are the generation and recombination rates, respectively, Ohmic. [4] In the latter case the surface recombination effects are the typical approach, which results Here and are the surface recombination velocities for electrons and holes, respectively. In (S10.b) and are not necessary the same among them and at each electrode. For instance, in the perfect selectivity approximation and taking 0 and at the interfaces with the electrons and holes transport layers, ETL and HTL respectively, we obtain ( ) = (0) = 0.

S1.3. Boundary conditions
The current boundary condition is set at the electrodes as in (S10.b), stating negligible drift current so the diffusion current at such interfaces is proportional to the surface recombination as ̅ │ − , ≈ ̅ (S11.a) where is the diffusion coefficient and is the effective surface recombination velocity. Note that when referring to minority holes the sign of the corresponding D should be negative. Also note that in practice there is an individual for each carrier type at each interface, only that here for simplicity we take the average or the relevant minority carrier one.
For the potential boundary condition, note that the connection between the distribution of charge across the junction and the electrostatic potential -mirrored by the intrinsic energy level -is usefully expressed in terms of the quasi-fermi levels of electrons and holes . In our deductions we take as an average minority carrier density when the space gradients can be neglected or one of the minority carriers at quasineutral region of the selective contacts in the case of one-sided abrupt p-n junction.

S1.4. Open circuit dc problem
The focus of our analysis is the OC regime, which makes a good situation for using (S15) Where is the generation rate, is the surface/SRH non-radiative recombination lifetime, is the radiative recombination coefficient and is the diffusion coefficient.
Note that here we consider the main "ingredients" for the simulation of a rectifying semiconductor device.
By substituting Equation (S2) into (S15) and considering ̃≪ ̅ 0 we can approach to where ̅ is the dc part of the generation rate.
Second, the modulated solution may be obtained from 22 −̃( + 1 + 2 ̅ 0 ) +̃= 0 (S18) where ̃ is the ac part of the generation rate when it is present. Here note that (S18) is a second order non-linear differential equation without solution in terms of elementary functions. Accordingly, the appropriate approximations will be made.
The steady-state problem in (S17) consist on a quadratic equation with exact solution and binomial series approximation as Where we use the characteristic generation-recombination frequency and McLaurin series approximation as Here note that by discarding the corresponding term directly in (S17), for predominant non-radiative recombination ( → 0) we obtain the same approximation. However, for predominant radiative recombination → ∞, we define ≈ 2√ ̅ , and thus More importantly, in all the cases there is an evident proportionality with the generation rate.

S1.5. IS: ac current problem
The problem of the modulated carrier density for potentiostatic IS requires ̃= 0 in (S18), and by considering 2̅ 0 ≫̃ it can be re-written as or in the case of predominant radiative recombination using (S19.c) it will result the same but changing 0 by in (S20). The general solution of (S20) is with the complex diffusion length as and the space constants ̃1 and ̃2 defined by the boundary conditions.
Applying our first constriction (S13) implies modulation of the depletion layer width and minority carrier concentrations at its border. By substituting (S3) in (S14) and taking the binomial series approximation for ̃≪ ( − ̅ ) the dc and ac components of the depletion layer can be approached as Note that (S22.b) loses validity as ̅ → and the minus sign expresses that ̃ shrinks when ̃ increases. Accordingly, in the one side abrupt junction case the condition (S13) may be evaluated for = ̅ . Hence, by substituting (S3) in (S13), considering the forward bias low injection range with ̅ > 4 / , and using the McLaurin series approximation, we can get an expression for the average charge carrier concentration with the same structure of (S2) where Here, first note that relation (S23.a) approaches (S19) when ̅ is that of the OC regime.
For our second boundary condition, by substituting (S21.a) in (S11.b) it results The solution of the system gives Subsequently, we can evaluate ̃ in (S5.b) and then substituting in (S1) it results

S1.6. IMVS: ac voltage problem
In the case of light modulation at OC, we take the steady-state dc solution of the average charge density ̅ 0 as Equation (S19), which is the best approximation for the IMVS. On the other hand, for the ac solution the problem should be split again, in solving first the transfer function for the IMVS and later with the IMPS.
In the case of finding the photo-voltage solution, the OC condition makes plausible to neglect the diffusion term in Equation (S18). Hence, we can discard the quadratic term (2̅ 0 ≫̃) and the solution and binomial series approximation are or half of that in case of predominant radiative recombination. Importantly, note that in the low frequency limit Ψ ∝ ̅ −1 , which is a very useful result in order to validate our calculus. Towards higher frequencies (1 + 0 ) −1 delivers a typical arc-shapetype spectrum.

S1.7. IMPS: ac current problem
Moreover, regarding the IMPS the ac solution comes by setting Equation (S20) equal to ̃, whose general solution is that of (S21) plus the term ̃/ 0 (1 + / 0 ). For the determination of the constants the first boundary condition fixes the carrier density at the border of the depletion region to the dc value: Importantly, note that in the low frequency limit Ψ is nearly independent on the dc illumination intensity. This is a clear difference with respect to the photovoltage responsivity and a useful argument to validate our theoretical approximations.