Quantifying Surface Recombination—Improvements in Determination and Simulation of the Surface Recombination Parameter J0s

The recombination parameter <italic>J</italic><sub>0</sub><italic><sub>s</sub></italic> provides an important metric to characterize surface recombination. For its calculation, numerous methods and models have to be applied. Since the models for the Auger and radiative recombination in crystalline silicon were recently revised, it is important to investigate the influence of these changes on <italic>J</italic><sub>0</sub><italic><sub>s</sub></italic>. The origin and possible ways of obtaining <italic>J</italic><sub>0</sub><italic><sub>s</sub></italic> from effective lifetime measurements as well as simulations are described in detail, including the potential to fit the full lifetime curve and a new approach that is based upon the reparameterization of the excess charge carrier density Δ<italic>n</italic>. Using the effective lifetime measurements, we find that <italic>J</italic><sub>0</sub><italic><sub>s</sub></italic> values determined with the older parameterization by Richter et al. will result in erroneous values up to 5 fA/cm<sup>2</sup>, depending on the chosen conditions. By simulating the recombination parameter <italic>J</italic><sub>0</sub><italic><sub>s</sub></italic> in near surface, highly doped structures, such as emitters, it is shown that these errors can even go up to 50%. If used in a simulation, we highlight the importance of having the parameterizations of surface recombination being determined with the corresponding parameterization of intrinsic recombination. Therefore, an update for the recombination at oxide-passivated and phosphorous doped surfaces is given that can be used with the new intrinsic recombination models. Finally, we give some best-practice examples on how recent improvements in effective lifetime measurements affect <italic>J</italic><sub>0</sub><italic><sub>s</sub></italic> values as well as possible pitfalls.


I. INTRODUCTION
T HE development of high-quality surface passivation technologies is one of the driving forces behind high-efficiency silicon solar cells based on the concepts, such as tunnel oxide passivating contacts (TOPCon) [1], [2] or heterojunctions [3], [4]. This led not only to high-efficiency solar cells but also to effective lifetime records of up to 500 ms in silicon wafers passivated with TOPCon [5]. Together with improvements in evaluation and measurement techniques [6], this allowed for an improved assessment of Auger recombination in silicon by Niewelt et al. [7] that was accompanied by an updated description of the radiative recombination, including photon recycling by Fell et al. [8].
As discussed in [7], the intrinsic recombination affects, for example, the single junction maximum efficiency, which is now almost invariant to the doping type. It was also briefly mentioned in [7] that another important quantity-the surface recombination parameter J 0s (or the analogously defined J 0E [9])-is strongly influenced by the improved description of intrinsic recombination. This parameter is used to describe the recombination occurring at the surface of a silicon wafer or in highly doped near-surface regions [9], [10]. It is often used for numeric cell simulation and to assess the impact of different process parameters, process steps or even different technologies.
This work intends to clarify the concept behind the surface recombination parameter and discuss different ways to determine it from experimental data. The fundamentals of J 0s are described in Section II, along with a review of concepts on how it can be calculated from effective lifetime measurements, as well as its determination in near-surface regions based upon electrical simulations. In Section III, the implications on J 0s determination due to the new updated Auger parameterization by Niewelt et al. [7] are discussed, as well as issues arising from using incompatible parameterizations, especially in simulations.

II. FUNDAMENTALS
The term J 0 in a basic form is known from the one-diodemodel describing the simple JV characteristic of a solar cell. The model in (1) can be seen as the difference between the current generated by photon absorption (J ph ) and the recombination current, which is mostly dominated by J 0 and the voltage V The recombination current J rec can also be described in terms of the normalized product of the electron and hole density, n and p with n i,eff = √ n 0 p 0 being the effective intrinsic charge carrier concentration, n and p the electron and hole densities (n 0 and p 0 the thermal equilibrium values under dark conditions), q the elementary charge, and U the net recombination rate. Note that (2) is more general with (1) being valid only in the nondegenerate case. As Cuevas pointed out [11], J 0 can be seen as a general recombination parameter with which the individual recombination mechanisms Auger-, radiative, Shockley-Read-Hall (SRH), or surface recombination can be described. One way of quantifying recombination is by measuring the effective lifetime τ eff , which relates to the total net recombination rate as U tot = Δn/τ eff at a certain excess charge carrier density Δn (often also termed injection density). τ eff can be written as the sum of the lifetime of the individual recombination processes The recombination at the surface is mostly dominated by SRH recombination at nonsaturated silicon (dangling) bonds [10]. In analogy to the concept of lifetime, the surface recombination velocity (SRV) S can be used to quantify how well a surface is passivated, with low SRV equaling low-surface recombination. However, near-surface band bending (and hence an electric field) is often present due to fixed charges in passivation layers or a change in net doping concentration (compare Fig. 1). Under these conditions, the commonly used approximation for the excess charge carrier concentrations of electrons and holes Δn s = Δp s is violated, which is why no analytical expression for S exists. As a solution, an effective SRV S eff is defined at a distance d close to the surface where no electric field is present anymore.
Using the concept of recombination currents from (2), we can describe the recombination current J rec,d flowing toward a single (virtual) surface at x = d with where U d is the recombination rate and Δn d the excess charge carrier density at the (virtual) surface. However, McIntosh and Black [9] showed that, especially in undiffused samples, S eff as a metric to quantify and compare surface passivation is not recommended. This is due to S eff being The implied open-circuit voltage iV oc is another quantity that is sensitive to the passivation quality and is sometimes used to assess the quality of surface passivation, but as the iV oc is influenced by many nonsurface-related parameters, such as the bulk doping, thickness, or reflectivity [12], it is not well suited for this task, too.
A better quantity is the surface-related recombination parameter J 0s [9], as it is independent of N s and Δn [9]. The relation between J 0s and S eff can be calculated by combining (2) and (4) at a distance d from the surface, and assuming an either p-or n-doped sample with a doping concentration While the right side of (5) is dependent on the measured quantities (and therefore, the goodness of the measurement), the left side is influenced by the used n i,eff value. As especially n 2 i,eff can vary strongly with the chosen parameterization and temperature, it is important to always state both when reporting J 0s values. In principle, the important quantity for surface recombination is J 0s /n 2 i,eff , and reporting it would be beneficial, as it reduces uncertainties from the n i,eff parameterization and is less temperature dependent than the individual J 0s and n i,eff [9]. Due to the popularity of J 0s , we stick with it for the descriptions in this work. For n i,eff , the parameterization by Altermatt et al. is used [13], which incorporates the bandgap-narrowing (BGN) model by Schenk [14]. The calculations are done at T = 25°C, which is the temperature suggested as a standard test condition for silicon solar cells.
Note that, in case of emitters (p + layer on an n-type substrate and vice-versa), the term J 0E was introduced by Kane and Swanson in 1985 [15] to describe the recombination current flowing toward the higher doped region. Due to the higher doping, the recombination rate in these cases is strongly impacted by Auger recombination. However, the same methodology and equations can be applied, with the definition of a surface-related or emitter-related J 0s /J 0E being interchangeable.
In Section II-A, different ways to obtain J 0s from carrier lifetime measurements are discussed, and Section II-B focuses on the determination of J 0s with electrical simulations of highly doped near-surface regions (e.g., for emitters).

A. Determination of J 0s Using Carrier Lifetime Measurements
One common way of obtaining information on the surface recombination is from effective lifetime measurements, with the surface lifetime τ s being the important part of (3). τ s can be obtained by different approaches, which will be discussed in more detail in the following text. Note that the typical measurements of τ eff are based on photoconductance or photoluminescence methods that measure a thickness averaged Δn avg . In these cases, Δn d can be well approximated by Δn avg . For simplicity, we denote the measured excess carrier density in the following as Δn, independent on whether it was determined close to the surface (Δn = Δn d ) or as a (thickness-) averaged value (Δn = Δn avg ).
Having obtained τ s , the calculation of J 0s is usually based on a first, sometimes tacitly calculation of S eff , which is then translated into J 0s via (5). The relation between S eff and τ s can, in the most general way, be described by a sum of decaying exponential terms where the dominating first mode can be written as follows [16], [17]: where W is the sample's thickness and D the ambipolar-diffusion coefficient. This equation holds true for the transient case and samples with symmetric surface passivation, as denoted by the factor "2." A common approximation is the one given by Sproul [18] which deviates from (6) by less than 5% across all possible combinations of S eff ·W/D. This equation can further be approximated to the commonly used equation which is still accurate within 4% for low S eff values that fulfill S eff W/D < 0.25 [18]. Choosing the appropriate approximation can be challenging if the order of magnitude of the term S eff W/D is unknown. As a visual guidance, we have calculated S eff W/D values for a 1 Ω·cm n-and p-type sample in terms of J 0s , using (5), as depicted in Fig. 2 (7) and the simple approximation (8) are given in percent to the right of the color bar, according to Sproul [18]. S eff was calculated as a function of J 0s according to (5). The ambipolar-diffusion coefficient D was calculated using Klaassen's mobility model [19], [20] as it is for example depicted in [21]. The bright squares highlight the range of typical modern samples.
<4% deviation. The differences between both approximations vanish with lower surface recombination and thinner samples. The thickly hatched region with S eff W/D < 0.1 depicts the case where the difference between both approximations become marginal. Typically, silicon solar cells have a thickness significantly less than 200 μm, with current thicknesses ranging between 150 and 170 μm [22] while also featuring good surface passivation quality with J 0s values less than 100 fA/cm 2 (depicted by the bright squares in Fig. 2). In these cases, the simple approximation from (8) is very close to the accurate description, but to minimize errors, we recommend to always use the more accurate approximation by Sproul [18, eq. (7)] to determine J 0s values. While its advantage over the simple approximation is clear, its advantage over the general (6) comes mainly from the easy rearrangement if one wants to calculate a τ s based upon given J 0s or S eff values.
The S eff W/D value generally decreases with Δn and N dop , therefore increasing the area where the simple approximation is valid.

1) Sample Thickness Variation:
The accurate way to experimentally characterize surface recombination for a given type of surface treatment (e.g., dielectric or wet-chemical passivation) is its extraction from a set of samples with varying sample thickness. Assuming bulk recombination in the samples does not depend on their thickness, and the effective lifetime changes only due to the different relative impacts of the surface recombination. In terms of (3), this is reflected in changes of τ eff caused only by the changing τ s and within the approximation introduced in (8) Authorized licensed use limited to the terms of the applicable license agreement with IEEE. Restrictions apply.
A linear fit to a plot of 1/τ eff (W) versus 1/W yields τ bulk from the y-axis intercept and S eff from the slope, as suggested by Yablonovitch et al. [23]. The approach can be used across the range of Δn covered by measurements to extract S eff at various levels of injection density, which in turn can be used to calculate J 0s according to (5). Depending on the range of investigated sample thicknesses, impacts of diffusion limitation may occur for thick samples. This can be accounted for by using (6) or (7). Another aspect that should be taken into account for samples with high bulk lifetime is that sample geometry may impact the lifetime due to radiative recombination. While the absolute rate of radiative recombination R rad is defined by intrinsic material properties, recent investigations by Kerr et al. [24] and Niewelt et al. [7] argued that reabsorption of the emitted photons within the sample will artificially reduce U rad . This effect is termed photon recycling and strongly depends on the optical properties of the sample-most prominently its surface texture-and is also affected by sample thickness. The amount of photon recycling can be quantified by numerical simulations and/or fitting of optical measurement data (e.g., reflection and transmission data), for example with the formalism presented recently by Fell et al. [8]. Once the impact of photon recycling is assessed for all samples in the set, it can easily be taken into account by subtracting the respective U rad (W) for each sample before further analysis.
The approach of determining S eff from thickness variation requires a high degree of control for sample processing to ensure comparable bulk quality and S eff as well as precise measurement of effective charge carrier lifetimes but is, otherwise, considered a very robust method of surface characterization.
2) Kane-Swanson Method and Its Refinements: It is also possible to determine the surface recombination parameter J 0s from a single τ eff (Δn) measurement. The approach is based on a method originally presented by Kane and Swanson in 1985 [15]. It has been refined over the years (e.g., by Mäckel and Varner [25], and Kimmerle et al. [21]) and is one of the most widely used. Taking the derivative of (5) with respect to Δn yields J 0s according to Kane and Swanson's original approach used the approximation from (8), leading to J 0s = qW n 2 i d dΔn (2τ s ) −1 (assuming a constant n i ). Instead of numerically differentiating the equation, the authors suggested a linear fit to the reciprocal effective lifetime at high excess carrier densities Δn > 10 · N dop . When using lowly doped wafers, this approach should reduce the influence of SRH recombination as τ SRH becomes independent of Δn. In addition, intrinsic recombination (namely Auger recombination) should not yet play a significant role. In their case, the measured τ eff (Δn) data are used instead of calculating τ s .
Over time, refinements of this method have been developed addressing several inaccuracies due to the simplifications made for obtaining (10). One important part is to take the injection dependence of the intrinsic carrier density resulting from BGN into account, e.g., using Altermatt's parametrization of n i,eff (Δn) [13] that needs to be accounted for in the derivative. In the same manner, using S eff equations that are less prone to errors [(6) or (7)] is recommended, especially in cases where doping or wafer thickness result in nonnegligible errors (c.f., Fig. 2). Finally, the determination of τ s can be especially important since the The refinements cumulated in the approach by Kimmerle et al., which we want to briefly discuss [21].
The first step toward obtaining τ s is to subtract the intrinsic lifetime from the measured effective lifetime [compare (3)]. Since the chosen parameterization has a strong impact on the resulting J 0s values, we strongly recommend to state which parameterization of τ Auger and τ rad was used when reporting J 0s . We recommend using the recent Auger recombination parameters by Niewelt et al. [7] and taking radiative recombination into account as discussed by Fell et al. [8]. The impact of using the older parameterization by Richter et al. [26] is discussed more thoroughly in Section III.
The bulk defect (SRH) recombination term should also be corrected for. In practice, this is not straightforward since precise information regarding τ SRH in a sample is rarely available. To get an estimation for τ SRH , Kimmerle et al. suggested an iterative procedure [21].
1) Calculate τ s from τ eff by subtracting intrinsic recombination and assuming τ SRH = Ý. 2) Using Sproul's approximation (7), one can use the calculated S eff to determine J 0s (10). The suggested approach is to use a linear fit to extract an initial J 0s from the slope. 3) Having a first value for J 0s , the next step is to recalculate τ SRH , which is done by using the obtained J 0s in a combination of (5) and (7) to obtain τ s . With τ s , one can use (3) to calculate τ SRH for every excess carrier density Δn within the fit range. Finally, Kimmerle et al. suggested to calculate a mean τ SRH . We would like to recommend at this point to use a harmonic mean (i.e., an arithmetic mean of the recombination rate). 4) Iterate the procedure (multiple times) by using the obtained mean τ SRH value in the calculation of τ s (step 1). One of the main challenges in this approach is to choose a sensible fit range. This is especially important for samples with resistivities in the range of 1 Ω·cm and below. Measuring these samples with common setups, such as Sinton Instruments' lifetime tester [27], makes it often impossible to follow Kane and Swanson's advice [15] to fit at high excess carrier densities Δn > 10 · N dop since barely any datapoints will be available. However, with increasing silicon bulk quality, recombination is often dominated by surface recombination, which lessens the need for evaluating at such high excess carrier densities. Although it is important to consider each case individually, in the context of this report, we have achieved good results using τ eff data with Δn > 0.5 · N dop .
One other noteworthy refinement was described by Bonilla et al. [17]. Their approach is quite similar to the one from Kimmerle et al., with two distinct differences: First, they suggested to use the general S eff (6). Second, they followed Mäckel and Varner's suggestion [25] to calculate a J 0s (Δn) by differentiating (10). According to Mäckel and Varner [25], the most accurate J 0s is determined by averaging the J 0s (Δn) ±10% around its flattest part. The flattest part can be determined by differentiating J 0s (Δn) and finding the minimum.
This approach is in theory the most accurate. In practice, we did not find significant differences in the determined J 0s values compared with the approach by Kimmerle et al. We attribute this mostly to the small difference introduced by Sproul's approximation (6) and (7). The main advantage of this method is the implementation of Mäckel and Varner's suggestion, making the determination of the best evaluation region straightforward.
3) Using the Linear Formulation of SRH Statistics in Case of Strong Bulk Recombination: As mentioned above, in the presence of significant bulk recombination at SRH defects, the determination of J 0s can be difficult due to the superposition of several recombination channels. As an alternative to Kimmerle's iterative approach (cf. A.2), linearization of the carrier lifetime as introduced by Murphy et al. [28] for SRH recombination has proven to be useful in such a case.
The basic idea is to describe lifetime as a function of the ratio between minority and majority carrier concentrations, e.g., n/p = Δn/(p 0 + Δn) ≡ X for p-type Si, instead of the excess charge carrier concentration Δn. For example, the electron lifetime τ n in p-type Si then reads is the product of electron (hole) capture cross section and thermal velocity, Q = α n /α p , N t is the defect concentration, and n 1 and p 1 are the SRH electron and hole concentrations, respectively. A similar expression can be obtained for n-type silicon.
As a result, a single dominant defect level leads to a straight line when plotting τ eff versus X; multiple defects are described by the superposition of an identical number of linear functions. Several key parameters of the defects' SRH properties are easily accessible in this representation. For a detailed discussion, the reader is referred to the article presented in [28].
This approach can be extended to the surface lifetime τ s by combining (5) and (7), and using the carrier ratio X Hence, the injection-dependent surface-related lifetime also becomes a straight line with a constant offset accounting for the carrier diffusivity. For low S eff values (i.e., when the simplified expression (8) is applicable, see Fig. 2), the last term in (12) can be neglected. Then, 1) the linear fit of τ s intercepts the abscissa at X = 1, as shown in Fig. 3, and 2) its slope equals the negative intercept of the ordinate. This is characteristic for surface recombination terms since Q is >0 by definition. Hence, conditions 1) and 2) can be used to discriminate between injection-dependent SRH bulk lifetime and surface-related lifetime in many cases. This allows a separation of τ s and τ SRH contributions via simply fitting the superposition of two straights to 1/(1/τ eff -1/τ intr ) as a function of X, as depicted in Fig. 3. Fig. 3. Effective lifetime corrected for intrinsic recombination versus the linearized X = n/p with the fit as described in the text. The SRH fit uses (11) and the J 0s fit is done through (12). The full fit is depicted in red and is the inverse sum of the two recombination processes. Fig. 4. Exemplary evaluation using the linear formulation for a sample during a degradation treatment. The upper graph shows τ eff evaluated at Δn = 10 15 cm −3 . The middle graph depicts J 0s using two different evaluation approaches and the lower graph defect-specific k-factor. The full fit of the first data point is shown in Fig. 3.
As the bulk lifetime is explicitly taken into account, the injection range for measurements and fitting is not as restricted as it is for the aforementioned Kane and Swanson method and its refinements. It suffices to cover the ranges where SRH limitation merges into surface limitation; high-injection measurements are not necessary. In our experience, good fits can be obtained when the injection range covers one to two orders of magnitude, including the intensity, where the excess carrier density equals the doping density. Fig. 4 shows an example of the advantages and disadvantages of this method: Here, τ eff is analyzed during a degradation investigation where the sample suffers from light-and elevatedtemperature-induced degradation (LeTID) (for details on LeTID compare, e.g., [29]), resulting in the formation and subsequent annihilation of the LeTID defect. As the linearization in [28] is often used to analyze defects, it can be used to determine the defect-specific k-factor with k = Q · v th,p /v th,n , as depicted in the lower graph of Fig. 4. It is often important to separate bulk and surface-related effects during such degradation treatments, as a surface-related degradation is often observed that follows the LeTID regeneration (compare, e.g., [30]).
However, using the method suggested by Kimmerle et al. [21] can lead to deceptive trends as depicted in the middle graph of Fig. 4 (blue dots). This is caused by the increasing importance of SRH recombination, effectively changing the shape of τ eff such that the algorithm results in a negative J 0s . The J 0s evaluation based on the linear formulation is much more robust against strong SRH defects and changes in SRH recombination, with the extracted J 0s staying constant throughout the degradation. The small increase in the last data point can be attributed to an onset of actual surface passivation degradation, as reported by Sperber et al. [30].
Note that there exists a significant difference in the initially determined J 0s values between the two investigated methods. This shows exemplarily the previously mentioned impact of using nonideal parameterizations. In the approach by Kimmerle et al., we have used Sproul's approximation (7) as well as an injection-dependent n i,eff parameterization, while the linearization was done using the simple approximation (8) as well as a constant n i,eff . This impact can also be seen in Fig. 3, where the full fit slightly deviates from the lifetime data at high X (or Δn).
This approach delineates recombination channels with distinct differences. Shallow defects have an SRH lifetime τ SRH with an injection dependence very similar to τ s . Hence, if their contribution to the effective lifetime is significant, the delineation may fail and only the sum of the surface-related and the shallow-defect recombination currents can be quantified. However, this limitation would affect all presented approaches, except the first, as τ s and τ SRH differ in their thickness dependence. As is the case for the other approaches, for high accuracy, the injection dependence of n i,eff needs to be taken into account.

4) Modeling of the Full Lifetime Data:
An alternative approach is to model the effective lifetime data over the full or a limited measured Δn-range. For the accurate representation of the measured data, all relevant recombination paths must be well known. In addition to intrinsic bulk and surface recombination, this can also include independently determined defect-specific recombination for one or more defects as well as effects, such as edge recombination. The advantage over the aforementioned Kane and Swanson method is that it is not restricted to certain ranges of the lifetime curve, but instead the whole lifetime data are regarded. This can be of interest, e.g., if defect lifetimes are modeled or in case of nonideal surface recombination, such as SiO 2 (see e.g., [31]).
Another advantage is the easy assessment of the model result, as it can be directly compared with the used effective lifetime data. An example of such a determination is shown in Fig. 5, with the fit result (in red) and its individual contributions of intrinsic (Auger and radiative), bulk (here: SRH recombination at Fe i ), and surface recombination.
The model is based upon (3), with the surface lifetime being described by Sproul's approximation [(5) and (7)] and a single Fig. 5. Exemplary fit to the full lifetime data of a 1 Ω·cm, p-type sample passivated with Al 2 O 3 /SiN x¸m easured with a WCT120 from Sinton instruments. For the fit, the data from Δn = 5×10 13 cm −3 onward are used, with the top five data points being ignored. The resulting fit is depicted in a red line. The surface lifetime is depicted in blue with a fitted J 0s = 4.6 fA/cm 2 . As SRH defect, Fe i is assumed, using the defect parameters from the article presented in [34]. The SRH lifetime is shown in green, using the fitted defect concentration N t = 2×10 9 cm −3 . The intrinsic recombination (Auger and radiative) is depicted in orange.
SRH defect. We applied a nonlinear curve fitting toolbox in Python based on the Levenberg-Marquardt method [32]. A useful addition is the calculation of the propagation of systematic uncertainties as it is possible with, e.g., the uncertainties package [33]. This allows further assessment of the systematic uncertainty of J 0s with respect to individual uncertainty sources, such as the measured data, the intrinsic recombination, and the temperature of the measurement. The lower the J 0s value, the more important these systematic uncertainties become, especially as τ eff and τ intr get closer. In the example, as depicted in Fig. 5, the J 0s is determined as 4.66 fA/cm 2 . The systematic uncertainties are calculated as ±1.75 fA/cm 2 and are much larger than the uncertainty resulting from the model optimization with ±0.02 fA/cm 2 . We want to highlight the importance of such systematic uncertainties, especially for low J 0s values, and suggest that they should be included when reporting J 0s values.
Having such a model allows for further fine-tuning, such as selectively excluding measurement artefacts (light gray data in Fig. 5) or specialized weighing of more/less important regions. It can also be adjusted to be comparable to the approach by Kimmerle et al. [21] by limiting the fit region to high Δn and assuming a constant τ SRH .

B. Determination of J 0s Via Electrical Simulation of Highly Doped Near-Surface Regions
J 0s can also be determined via a drift-diffusion simulation of highly doped near-surface regions when its electrical properties are known. Those properties comprise the following: 1) doping profile introduced, e.g., by thermal diffusion or ion implantation; 2) SRH defect profile, e.g., from inactive doping [35]; often assumed negligible; 3) electric field at the surface from fixed surface charges or from an electrical barrier resulting from a work function difference to the adjoining material layer; this electric field can well be neglected when high-surface doping concentration exists; 4) surface defects, often modeled via an effective midgap SRH defect using the SRVs for electrons and holes S n and S p ; this property is hard to determine independently and is, thus, usually determined indirectly by fitting near-surface J 0s simulations to J 0s values determined via lifetime measurements (compare Section II-A). The simulated J 0s also depends on fundamental semiconductor properties, namely the bandgap, density of states, BGN, and intrinsic recombination (radiative and Auger). A change of such fundamental properties, e.g., by a more precise determination leads consequently to a change in simulated J 0s values. The implications are highlighted in more detail for the case of the recently updated intrinsic recombination models for silicon, see Section III-B.
With the exception of the software PC3S [36], the near-surface simulations are almost exclusively performed in one dimension only, even if the actual surface investigated experimentally is textured. The texture leads to a surface enlargement, which is usually accounted for by a so-called "texture multiplier" applied to the recombination rates. Due to the three-dimensional transport effects, as well as the not perfectly conformal nature of the near-surface properties, the texture multiplier is commonly lower than the geometrical surface area enlargement (∼1.7 for pyramidal texture) and hard to determine [37].

III. IMPLICATIONS OF THE REPARAMETERIZATION OF THE AUGER RECOMBINATION
Any calculation of J 0s is closely linked to the chosen parameterization for the intrinsic recombination. A change in the assumed intrinsic recombination will impact the evaluated J 0s values and, therefore, the used models for Auger and radiative recombination must be considered when comparing J 0s values. To illustrate this effect, we compare J 0s values evaluated with both the recent Niewelt parameterization [7] and the still widely adapted Richter parameterization [26]. First, the impact on J 0s values determined from effective lifetime data is investigated (cf. Section II-A). Then, its impact on simulating J 0s from the near-surface properties (cf. Section II-B) is discussed. We demonstrate the effect further by re-evaluating the latest SRV parametrization for oxide-passivated phosphorous-doped surfaces by Wolf et al. [43].

A. Determination of J 0s From τ eff Data
To estimate the impact of the updated Auger parametrization, we calculate synthetic effective lifetime data τ eff by considering the intrinsic recombination, as suggested by Niewelt et al. [7], for different J 0s values between 0.1 and 100 fA/cm 2 , with a thickness of 200 μm at 25°C. The J 0s is then recalculated from the resulting τ eff curves by assuming the widely used previous Richter parameterization [26]. This is done for every data point using (3).
The difference J 0s (Niewelt) − J 0s (Richter) is plotted in Fig. 6. In the upper two panels, this difference is shown for very low (1×10 12 cm −3 ) and high (5×10 16 cm −3 ) Δn and for a variation of the base resistivity and doping type. The lower two panels display the difference as a function of Δn for 1 and 10 Ω·cm base resistivity. The starting J 0s (Niewelt) is given in the color scale ranging from extremely low (0.1 fA/cm 2 ) up to relatively high (100 fA/cm 2 ) values. In case of very low Δn (upper left panel), it can be observed that there is an almost constant offset for p-type material. The main reason for this offset is that Niewelt et al. took photon recycling into account, which was not considered by Richter et al. Since this effect reduces the effective radiative recombination rate and radiative recombination shares the proportionality to np, an increase in J 0s would occur even if the same Auger recombination rate was assumed. In addition, the low-injection Auger recombination rate at low base resistivity was overestimated by the Richter model (especially on n-type but to lesser extent also on p-type). Thus, the difference in J 0s increases between n-and p-type toward lower base resistivity.
For the case of high Δn, as shown in the upper right panel, the situation is completely different since the difference in J 0s is negative for both n-and p-type. The reason for this discrepancy in J 0s is a result of the choices which the available literature data for the Auger lifetime of highly doped n-and p-type material should be included. Here, Niewelt et al. [7] favored a slightly different dataset, as suggested by Black and Macdonald [44]. Due to the adoption of Black and Macdonald's C eeh and C ehh values, the Niewelt model describes the average of the literature data for highly doped material, whereas the Richter model tends to envelop the data points [44].
In addition, the injection density Δn is incorporated into the equations of the Auger lifetime in both models in different ways. While Richter et al. introduced additional terms for Δn with separate exponents, Niewelt et al. used terms that represent the sums (n 0 +Δn) and (p 0 +Δn), meaning that the charge carriers are treated in the same manner independent of their provencance. Due to this equivalency, the difference in J 0s is negative on lowly doped material as well.
The bottom two graphs in Fig. 6 depict the transition from low to high injection for normally doped 1 Ω·cm (left panel) and lowly doped 10 Ω·cm (right panel) base material. This indicates that determining J 0s with the updated model will lead to higher or lower values, dependent on the chosen Δn range. In the exemplary case of 1 Ω·cm n-type samples, an evaluation with the Richter parameterization of well passivated samples with J 0s = 1 fA/cm 2 would even lead to negative J 0s values for Δn < 10 16 cm −3 .

B. Simulation of J 0s in Heavily Doped Near-Surface Regions
When simulating J 0s from near-surface properties, the results depend on the choice of the intrinsic recombination model. In particular, for the case of highly doped near surfaces, Auger is a dominant recombination loss influencing J 0s , and thus, substantial changes are expected when switching from the previous Richter [26] to the latest Niewelt [7] Auger model in the simulations.
To quantify the extent of the change, we performed J 0s simulations with Quokka3 [39] for various doping profiles. As the first case, we choose a phosphorus doping with a surface concentration of N s = 1.7×10 20 cm −3 and a hole SRV of S p = 3×10 4 cm/s, which roughly represents a phosphorus emitter in industrial solar cells. In the second case, we define a relatively low surface concentration of N s = 5×10 19 cm −3 and zero surface recombination in order to maximize the impact of Auger recombination and, thereby, identify the maximum extent of the change. In the third case, we switch the dopant from phosphorus to boron, keeping the low surface concentration and zero surface recombination. For each case, the depth of an erfc profile is varied so that the resulting sheet resistances cover a typical range of interest, i.e., 50 Ω/ to 300 Ω/ .
The results in Fig. 7 show that, indeed, the impact of the change of Auger models is substantial. For the Auger-limited profiles, the J 0s is almost 50% higher with the new Niewelt Auger model, and it is still quite relevant also for the industrial emitters despite those being more limited by surface recombination. It should be kept in mind that a substantial change of simulation results can be expected when using existing solar cell simulation setups and updating to the newest Auger model in the solar cell simulation tool.
The Niewelt model resulting in higher surface recombination might seem counterintuitive to the claim that the new model accounts for higher Auger lifetime measurements. The reason is that the higher lifetimes are in the lowly doped regime only, but for the highly doped regime, the new parameterization results, in fact, in lower lifetimes [7], as explained in Section III-A. Although the new parameterization did not include new measurements in this highly doped regime, we still believe that the new model and, thus, the new J 0s values are closer to reality being based on a thorough re-evaluation of old lifetime measurements [44] as well as a more physically sound parameterization formula compared with Richter's formula [7].

C. Reparameterization of SRV for Oxide-Passivated Phosphorus-Doped Surfaces
As surface recombination (SRV) at a highly doped near surface is always only indirectly determined by matching the simulated J 0s with the J 0s measurement, see Section II-B, a consequence of the new Auger model is also that the SRV values determined with the old Auger model are not valid anymore. On the one hand, using the new Auger model will lead to better accuracy of derived SRV values. On the other, and more important, hand, it is inconsistent and erroneous to use SRV values determined with the old Auger model within current simulations using the new Auger model. This is, in particular, cumbersome for prominent published SRV parameterizations, like e.g., [35], [43], [45], and [46], all being derived using the Richter Auger model.
To overcome this issue and estimate the possible error associated with it, we re-evaluate the latest SRV parameterization for oxide-passivated phosphorus-doped surfaces by Wolf et al. [43]. In a first step, all lifetime measurements for the various doping profiles are reanalyzed with the Niewelt Auger model to increase the accuracy of the J 0s measurements, as explained in Sections II-A. and III-A. We find, however, that this had only minimal impact (∼1%), as those samples are dominated by J 0s surface recombination and, thus, Auger recombination in the bulk plays a minor role for J 0s determination. . (13) Notably, for the prominent SRV parameterization of recombination at Al 2 O 3 passivated surfaces by Black et al. [46], we find that a reparameterization is not required. The SRV parameters were in fact determined directly by capacitance-voltage measurements. A comparison of simulated J 0s and values determined via lifetime measurement was only performed for validation purposes. The reason why this validation was successful despite using the inaccurate Richter Auger model might be that the, at this time most recent, BGN model by Yan and Cuevas [47] was used. The differences to the Schenk BGN model in the J 0s calculations broadly compensate the Auger differences. We, therefore, suggest that the existing SRV parameterization by Black et al. can be applied as-is within simulations using the Niewelt Auger model.

IV. BEST-PRACTICE EXAMPLES FOR MEASURING AND EVALUATING J 0s VALUES
The improved Auger parametrization was accompanied and, in fact, has only been made possible by refinements in measuring and evaluating τ eff data. In this section, we want to highlight the impact of these refinements on the calculated J 0s and give The impact is shown exemplarily for a FZ-Si, p-type wafer with a bulk resistivity of 3 Ω·cm, a thickness of 170 μm, and passivated with n-doped amorphous Si. The sample was measured with a Sinton Instrument Lifetime Tester WCT-120, with an 80 μm thin paper between the chuck and the sample.
The first important aspect is the dependence of the WCT-120s coil sensitivity on the sample thickness and lift-off, as described by Black and Macdonald [6]. The usual calibration procedure consists of 525 μm thick wafers with different resistivities. Without accounting for the thickness dependence, measuring wafers with thicknesses different from 525 μm will lead to wrong conductance values and, thus, wrong τ eff values. As shown by Black and Macdonald, this effect is Δn-dependent, and especially strong for high Δn-values. Thus, ignoring this effect will not only influence J 0s values that are determined using a thickness-dependent approach (compare Section II-A1) but also approaches using the Δn-dependent behavior of τ eff (such as Section II-A2-A4).
Using the evaluation method of Kimmerle et al. (compare Section II-A2), the impact is demonstrated by plotting the q·S eff ·n ieff 2 values after the last iteration of τ SRH correction and the resulting linear fit to calculate J 0s in Fig. 9.
As shown in Fig. 9, ignoring the coil attenuation will lead to lower J 0s values, in this case, by around 20%. Correctly accounting for the thickness dependence, but still ignoring the lift-off introduced (in this case) by a thin paper will increase the determined J 0s .
Additionally shown in the top graph of Fig. 9 is the case when the thickness dependence and the lift-off are taken into account correctly, but the measurement temperature is not accounted for. In the case of the WCT-120, the operating temperature is generally given as around 30°C. The resistivities of the calibration wafers, therefore, need to be known at 30°C. To show the impact of using, e.g., room-temperature measurement, without correcting for the resistivity's temperature dependence, the resistivity of the calibration wafers at 22°C is used as input for the calibration. As can be seen, this leads to the largest discrepancies observed here, reducing J 0s by almost half. All of the above-mentioned factors are directly related to the goodness of the coil calibration. Therefore, we strongly recommend an optimized coil calibration for reporting accurate J 0s values.
Another effect that has a direct impact on the resulting J 0s value is the automatic smoothing of lifetime data that can be chosen in the WCT-120s software and is implemented using a Savitzky-Golay filter. Data smoothing used to be the only option until v.5.71 that added the option to disable automatic smoothing. While the smoothing might reduce noise in low injection density, it drastically distorts the effective lifetime at higher injection densities, as can be seen in the bottom graph of Fig. 9. The shape of q·S eff ·n ieff 2 is not linear anymore but seems to rather have a quadratic Δn-dependence; applying a linear fit is inappropriate.
We want to mention that the updated Auger parametrization by Niewelt et al. has been implemented since software version v.5.74, the use of which is strongly recommended. Note that if one wants to use advanced methods of determination J 0s , such as described in Section II-A2-A4, an independent evaluation is still inevitable.
Finally, we want to highlight the importance of an exact knowledge of the sample properties for correct J 0s determination. The two most important parameters are the doping concentration and the thickness of the sample. They not only influence the calculation of effective lifetime from the measured conductivity (changes) directly but they are also used for the calculation of J 0s . Erroneous sample parameters, therefore, influence the J 0s value twice. Exemplarily, these influences are shown for a sample with a resistivity of 3 Ω·cm and a thickness of 455 μm. The sample was passivated with a TOPCon structure (TOPCon, for process details compare [48]), resulting in a J 0s of 1.1 fA/cm 2 . As shown in Fig. 10, thickness and doping were individually changed to simulate wrong input parameters during lifetime measurements and during J 0s evaluation. As can be seen, wrong input parameters can have severe consequences for the resulting J 0s values. For the chosen variation, the deviations in J 0s range from physically unreasonable negative values up to values exceeding the true value by a factor 4, when falsely assuming a larger thickness.
The reason for this effect is that wrong input parameters affect the shape of τ eff (Δn), and with this, the relation of τ eff to its respective intrinsic lifetime τ intr . This relation is essential, with the calculation of their difference being the first step in Fig. 10. Calculated J 0s values for a p-type, 3 Ω·cm sample passivated with a TOPCon layer, resulting in a J 0s of 1.1 fA/cm 2 . The impact of a wrong thickness and resistivity on the evaluated J 0s is shown in blue and green, respectively. Lines serve as guide-to-the-eyes. the approach described in Section II-B, for example. As seen in Fig. 10, underestimating the thickness can even lead to τ eff values above the intrinsic limit because lower thickness shifts τ eff (Δn) to higher values. Although the doping shifts both the effective lifetime curve and the intrinsic limit in the same direction, both effects do not cancel each other and, thus, wrong resistivity values also lead to wrong J 0s values.

V. CONCLUSION
With large improvements in surface passivation quality due to passivation structures, such as TOPCon, leading to an update of the intrinsic recombination, it is necessary to take a closer look at the surface-related recombination parameter J 0s , which is best suited to describe recombination at the surface and in highly doped structures close to the surface. Different ways to calculate J 0s are shown. Based on the lifetime data, the accuracy of the calculated result depends mostly on the used assumptions. Kimmerle's improvement on Kane and Swanson's approach is an example of how more physically reasonable assumptions can lead to more accurate results. Additionally, a new method is presented that calculates J 0s using Murphy's parametrization, which has the distinct advantage that it takes injection-dependent SRH recombination into account with little extra effort. Finally, we explore the possibility to determine J 0s by modeling the full lifetime data.
The typical procedures to calculate J 0s from a known doping profile are shown, too, with an overview of necessary input parameters, influencing factors, and different ways to do the calculation.
The recent update of the models for intrinsic recombination in silicon by Niewelt et al. has substantial implications for the determination of J 0s . When using injection-dependent carrier lifetime measurements for this task, we show the error by assuming the older Richter parametrization for the usual injection ranges to amount up to 5 fA/cm 2 on lifetime samples without diffused surfaces, depending on doping, the doping type, and the injection range chosen for the evaluation.
For the high dopant densities met in typical doping profiles, the Auger recombination rates, as determined by Niewelt et al., are up to 50% higher compared with the previous Richter model. For doping profiles, where surface recombination does not dominate, this directly translates to up to 50% deviation in the determined J 0s/E values.
Therefore, care must be taken when updating to the Niewelt parameterization in the existing simulation environments. In particular, simulations that have been matched to the measured J 0s values, e.g., by fitting surface SRV values, need to be redone in addition to re-evaluating J 0s from the lifetime measurements. Consequently, the published parameterizations of SRV values need to be checked with regard to the employed Auger model before using them in combination with the new Niewelt parameterization. This is, e.g., the case for the SRV parameterization by Wolf et al. [43] for which we provide an updated parameterization in this work consistent with Niewelt et al. Notably, the popular SRV parameterization of an Al 2 O 3 coated Si surface by Black et al. [46] is compatible as-is with J 0s simulations irrespective of the used Auger model.
In addition to the impact of intrinsic recombination, we have looked at potential error sources due to the measurement of effective lifetime based on the most common tool, Sinton Instruments lifetime tester WCT-120. Exemplarily, the importance of the coil calibration, including the coil's attenuation and temperature dependence of the calibration wafers' resistivity, and the impact of smoothing on the calculated J 0s are shown. Using wrong values for sample properties, such as the thickness or bulk resistivity, can also lead to large errors in the J 0s determination.