Statistical analysis of recombination properties of the boron-oxygen defect in p-type Czochralski silicon

Abstract

This paper presents the application of lifetime spectroscopy to the study of carrier-induced degradation ascribed to the boron-oxygen (BO) defect. Specifically, a large data set of p-type silicon samples is used to investigate two important aspects of carrier lifetime analysis: ① the methods used to extract the recombination lifetime associated with the defect and ② the underlying assumption that carrier injection does not affect lifetime components unrelated to the defect. The results demonstrate that the capture cross section ratio associated with the donor level of the BO defect (k₁) vary widely depending on the specific method used to extract the defect-specific recombination lifetime. For the data set studied here, it was also found that illumination used to form the defect caused minor, but statistically significant changes in the surface passivation used. This violation of the fundamental assumption could be accounted for by applying appropriate curve fitting methods, resulting in an improved estimate of k₁ (11.90±0.45) for the fully formed BO defect when modeled using the donor level alone. Illumination also appeared to cause a minor, apparently injection-independent change in lifetime that could not be attributed to the donor level of the BO defect alone and is likely related to the acceptor level of the BO defect. While specific to the BO defect, this study has implications for the use of lifetime spectroscopy to study other carrier induced defects. Finally, we demonstrate the use of a unit-less regression goodness-of-fit metric for lifetime data that is easy to interpret and accounts for repeatability error.

Keywords

Czochralski silicon / boron-oxygen defect / injection dependent lifetime spectroscopy / goodness-of-fit / repeatability error

Introduction

The boron-oxygen (BO) defect is an important lifetime-limiting defect in p-type silicon, potentially reducing the efficiency of commercial cells by 1%–2% absolute [ 1]. Despite its importance however, there exist only a limited number of studies on the recombination characteristics of BO defects, none which employ large, statistically robust data sets. This study explores some important issues relating to the BO defect’s recombination properties and their implications for carrier lifetime analysis.

The recombination behavior of BO defects has been typically described by Shockley-Read-Hall (SRH) trap-assisted recombination [ 2], and has been studied using a range of different methods, namely deep level transient spectroscopy (DLTS) [ 3], injection-dependent lifetime spectroscopy (IDLS) [ 4– 14] and temperature-dependent lifetime spectroscopy (TDLS) [ 6, 15]. There is broad agreement in the literature about some aspects of the defect; however, the various studies differ somewhat in their conclusions regarding the SRH recombination characteristics of the defect.

What is known with confidence is that there are two distinct defect levels associated with the fully formed BO defect—a deep donor level (E_trap,1) situated 0.41 eV below the conduction band (E_C) [ 3– 6] and an acceptor level (E_trap,2) situated above the valence band energy level (E_V) [ 3, 8, 9]. The deep donor level acts as a minority carrier trap in p-type silicon and is largely responsible for lifetime degradation in p-type silicon as it dominates at low to medium injection conditions. The acceptor level, on the other hand, is a majority carrier trap in p-type silicon and has a smaller impact on carrier lifetime in p-type silicon and then too, is thought to affect lifetime only at high injection [ 4, 6]. In fact, a number of studies on the fully formed BO defect in p-type silicon have reported good fits to IDLS data using a single defect model with a single trap level (i.e. using E_trap,1 alone) [ 6, 10, 13]. In contrast to p-type silicon, carrier lifetime in compensated n-type material appears to be strongly affected by both the donor level (majority carrier trap in n-type silicon) and the acceptor level (minority carrier trap in n-type silicon) [ 5, 8, 9, 11, 12], and therefore requires a two-level defect model (i.e. using E_trap,1 and E_trap,2) to adequately describe the injection dependence of carrier lifetime.

The second well known property of the BO defect is that the CID effect occurs in two stages––a fast degradation stage and a slow degradation stage. This has been well-characterized in p-type silicon [ 7, 14] but is also suspected to occur in compensated n-type silicon [ 9, 12], albeit at a much slower rate than in p-type silicon. Further, the characteristic injection-dependence of carrier lifetime associated with the fast stage appears to require a significantly higher capture cross section ratio to adequately fit the measured data compared to the slow stage [ 7, 9, 12]. This has important implications for the study of the fully formed defect (i.e. after complete degradation) in compensated n-type material, in which the fast stage is more dominant and therefore requires a correction factor [ 9] to be applied when studying the slow stage in such material [ 12]. In contrast, lifetime in p-type silicon after complete degradation is known to be largely dominated by the slow degradation stage [ 7]. Considering also that E_trap,1 appears to be more dominant in p-type material, as previously noted, characterization of the fully formed defect in p-type material using a single defect model with a single trap level has therefore provided important insights into the property of the E_trap,1 level and is the primary focus of this work.

However, even when a single defect level in p-type silicon has been used to describe the fully formed BO defect, there have been three major inconsistencies in the literature regarding the determination of recombination properties of the BO defect based on IDLS data. First, the electron-to-hole carrier capture cross section ratio associated with E_trap,1 (k₁ = s_n1/s_p1) was proposed to be in the range of 9–10 based on IDLS and TDLS data assuming a single defect model with a single trap level [ 4– 6]; however, a small number of studies have determined k₁ values in p-type silicon to be higher—between 10 and 12 [ 8]. A similar value was obtained in another study on samples with the BO defects deactivated [ 13], although values as high as 16–23 have also been reported for the fully active BO defect [ 14]. The reason why such a broad range in observed k₁ values exists is currently unknown. Given that the few studies on the fully formed defect in which the p-type base doping levels were varied (1–30 W·cm in Ref. [ 4], 0.5–3.5 W·cm in Ref. [ 6]) do not explicitly report variations in k₁ with base doping—and indeed report a single k₁ value—it appears likely that the variation in reported k₁ is due to other unknown reasons. Attempts to explore this have also been limited by a lack of knowledge about the expected statistical variability in k₁ for a sample set of similar wafers.

Secondly, the primary assumption made in most studies is that processes used to manipulate the state of the BO defect only affects the SRH lifetime associated with the BO defect, t_SRH,BO. Since any changes in non-BO related lifetime components such as the surface/emitter lifetime (t_surf) and non-BO related bulk lifetime could potentially affect the estimation of t_SRH,BO and therefore the value of k₁, it is important that any such changes are accounted for. Although minor, evidence of such changes are visible in IDLS data presented in some early studies [ 4– 6]; however, there have been no explicit attempts to quantify such changes. The magnitude of such changes, and their impact on the determination of k₁ therefore remains unknown.

Thirdly, methods used to determine k₁ from IDLS data vary slightly between studies. In most studies, the inverse lifetime after complete deactivation of the BO defect via dark annealing (t_DA⁻¹) is subtracted from the inverse lifetime after complete activation of the defect after light soaking (t_LS⁻¹), with the difference (t_LS⁻¹–t_DA⁻¹) then fitted to determine k₁ [ 4– 6, 8, 11]. However, in other studies, k₁ is determined by attempting to fit t_LS rather than (t_LS⁻¹–t_DA⁻¹) based on certain assumptions about t_DA [ 10, 13, 14]. The impact of the various fitting methods, the assumptions made about t_DA and the ability of these methods to account for change in non-BO related lifetime on the fitted values of k₁ has not been quantified, making it difficult to determine which method is more accurate.

In this work, we present statistical analysis of IDLS data from a large data set of commercial p-type wafers to address these unanswered questions. In particular, multiple methods for analyzing IDLS data from BO-afflicted samples are evaluated to validate the underlying assumption regarding changes in non-BO related lifetime under exposure to illumination. We discuss the implications of these assumptions on IDLS and TDLS characterization and present revised values of k₁ for the BO defect assuming a single defect model with a single trap level.

Experimental method

In this study, commercial grade 156 mm × 156 mm pseudo-square boron-doped Cz wafers (1.6 Ω·cm) were used to create symmetric lifetime test structures. A saw damage etch was performed on the as-received wafers, followed by alkaline texturing and acidic neutralisation. Wafers were then subjected to an RCA clean and HF dip, then diffused at 795°C for 25 min in a POCl₃ tube furnace followed by a drive-in diffusion at 885°C for 30 min to achieve an emitter sheet resistance of 70 Ω/sq —a process known to be effective at gettering [ 16].

The phosphosilicate glass (PSG) and emitter thus formed were removed by etching ~2 mm from both surfaces, resulting in a wafer thickness of approximately 180 µm. This was done to ensure minimal concentrations of impurities other than BO within the wafer bulk. The wafers were then RCA cleaned and followed by a second, light POCl₃ diffusion at 780°C for 30 min with a 20 min drive-in step to achieve a sheet resistance of ~200 Ω/sq. This was done to aid in surface passivation and minimize the influence of changes in surface recombination on carrier lifetime during further processing. The PSG resulting from this second diffusion was etched off in dilute HF, followed by the deposition of SiN_x:H dielectric layers on both sides of the wafers using a Roth & Rau MAiA plasma-enhanced chemical vapor deposition (PECVD) system. All SiN_x:H layers deposited were 80 nm thick and had a refractive index of 2.08 at 633 nm.

A total of 25 wafers were so processed. All samples were then dark annealed at 200°C for 15 min (to ensure complete temporary deactivation of the BO defect) followed by a light soak under halogen lights at an intensity equivalent to 0.78 suns at 35°C for 48 h to ensure complete activation of the BO defect. The effective carrier lifetime was measured on nine sampling points for all wafers after complete deactivation and after complete activation of the BO defect using the WCT-120 (Sinton Instruments) photoconductance decay (PCD) lifetime tester [ 17]. Ambient light exposure prior to measurement was minimized by maintaining all samples in the dark between the dark anneal step and the subsequent lifetime measurement. Further, the measurement tool was maintained in a darkened chamber during measurement to minimize any unintentional biasing of the sample during measurement. A flash lamp with an added infrared filter was used as the variable light source for measurements. The measured IDLS data was analyzed using the generalized method [ 18] and corrected for intrinsic recombination (i.e. Auger and radiative recombination) using the Richter model [ 19]. The resulting injection dependent data obtained after accounting for intrinsic recombination is hereby referred to as t_DA (measured after dark annealing) and t_LS (measured after light soaking). Unless otherwise specified, all lifetime values reported in this work refer to measured values at an injection level corresponding to Dn = 0.1 × N_A after accounting for intrinsic recombination.

Analytical approach

A description of the lifetime components used, the approach used to determine each component and the metrics relevant to describing the BO defect properties are discussed in this section.

Carrier lifetime model

The inverse lifetime of a sample containing BO defects can be modeled as the sum of a series of inverse lifetime components to account for recombination related to BO defects as well as from other sources of recombination. In this study, t_DA was assumed to be described by three lifetime components representing recombination in the surface region (t_surf), an injection independent bulk recombination component (t_bulk,non-BO) and an injection dependent SRH recombination component (t_SRH,non-BO). t_LS was then assumed to contain an additional lifetime component representing SRH recombination due to the BO defect (t_SRH,BO). Having already accounted for intrinsic recombination, as noted earlier, t_DA⁻¹ and t_LS⁻¹ were thus modeled as follows:

(1) {\displaystyle \frac{\mathrm{1}}{{\tau }_{\mathrm{DA}}}}={\displaystyle \frac{\mathrm{1}}{{\tau }_{\mathrm{surf}}}}+{\displaystyle \frac{\mathrm{1}}{{\tau }_{\mathrm{bulk},\mathrm{fixed}}}}+{\displaystyle \frac{\mathrm{1}}{{\tau }_{\mathrm{SRH},\mathrm{non}-\mathrm{BO}}}},

(2) {\displaystyle \frac{\mathrm{1}}{{\tau }_{\mathrm{LS}}}}={\displaystyle \frac{\mathrm{1}}{{\tau }_{\mathrm{surf}}}}+{\displaystyle \frac{\mathrm{1}}{{\tau }_{\mathrm{bulk},\mathrm{fixed}}}}+{\displaystyle \frac{\mathrm{1}}{{\tau }_{\mathrm{SRH},\mathrm{non}-\mathrm{BO}}}}+{\displaystyle \frac{\mathrm{1}}{{\tau }_{\mathrm{SRH},\mathrm{BO}}}}.

Note that the lifetime components specified in Eq. (1) were chosen in order to achieve acceptable fits to t_DA across a broad range of injection levels. This is explained further below and in Section 4. For now, we discuss details about the individual lifetime components. t_surf is assumed to be dependent on the emitter saturation current (J_0e) through the following relation which applies for samples with symmetric surfaces/emitters [ 20]:

(3) {\displaystyle \frac{\mathrm{1}}{{\tau }_{\mathrm{surf}}}}=\mathrm{2}{J}_{\mathrm{0}\mathrm{e}}\times {\displaystyle \frac{\left(N_{\mathrm{A}}+\Delta n\right)}{qW{n}_{\mathrm{i}}^{\mathrm{2}}}}.

N_A, Dn, n_i², W and q in Eq. (3) refer to the bulk doping concentration, excess minority carrier concentration, intrinsic carrier concentration, sample thickness and fundamental charge constant respectively. N_A for the samples in this study was determined to be 9.14 × 10¹⁵ cm⁻³ via 4-point probe measurements performed prior to the emitter diffusion. All carrier concentrations in the model (n, p, and n_i²) were calculated at steady-state as a function of Dn using Fermi-Dirac statistics taking into account band-gap narrowing [ 21, 22] and potential degeneracy effects [ 23]. The SRH recombination terms (t_SRH,BO and t_SRH,non-BO) are assumed to be described by a single defect with a single trap level as follows:

(4) {\displaystyle \frac{\mathrm{1}}{{\tau }_{\mathrm{SRH}}}}={\displaystyle \frac{\left(np-n_{\mathrm{i}}^{\mathrm{2}}\right)}{\Delta n\left[{\tau }_{\mathrm{n}\mathrm{0}}\left(n+n_{\mathrm{1}}\right)+{\tau }_{\mathrm{p}\mathrm{0}}\left(p+p_{\mathrm{1}}\right)\right]}}.

Here, n and p are the total electron and hole concentrations, t_n0 and t_p0 are the SRH-related electron and hole recombination lifetimes, and n₁ and p₁ represent the SRH-related electron and hole populations characteristic of the trap level within the bandgap. t_n0 and t_p0 are further related to the defect capture cross section ratio, k, through the following relation:

(5) k={\displaystyle \frac{{\sigma }_{\mathrm{n}}}{{\sigma }_{\mathrm{p}}}}={\displaystyle \frac{{\tau }_{\mathrm{p}\mathrm{0}}}{{\tau }_{\mathrm{n}\mathrm{0}}}}\times {\displaystyle \frac{v_{\mathrm{h}}}{v_{\mathrm{e}}}}.

Here, v_e and v_h are the electron and hole thermal velocities respectively [ 24]. Note that a number of studies in the literature have ignored the contribution to k of (v_e/v_h), which has a value of 1.208 at 300 K. This has resulted in an overestimation of k in such studies [ 5, 10, 13]. For the sake of consistency, the impact of (v_e/v_h) has been taken into account where values of k are quoted from literature.

t_SRH,non-BO was found to be best described by a mid-gap trap level with a fixed k value of 100. Although this is similar to the value expected for interstitial iron, samples were confirmed to have negligible concentrations of iron [ 25] and it was ruled out that t_SRH,non-BO represents any carrier-induced defects. Therefore, t_SRH,non-BO represents other bulk defects that are not light-activated. Together with t_bulk,fixed, t_SRH,non-BO so defined was found to adequately describe any injection-dependence in bulk lifetime in t_DA, thus allowing injection dependent effects in t_LS specific to the BO defect to be fully described by t_SRH,BO.

t_SRH,BO was also assumed to be described by a single trap level, E_trap,1 at E_C−0.41 eV [ 3, 6], with the capture cross section ratio, k₁, allowed to vary during fitting. Note that the contribution of the acceptor level of the BO defect (E_trap,2) has not been included in t_SRH,BO. As will be discussed later, this approach was justified for the p-type samples used in this work. Nevertheless, any potential changes in lifetime related to E_trap,2 that display strong injection-dependence in high injection [ 4, 6] or are approximately injection-independent in p-type silicon [ 11, 14] are expected to be accounted for by changes in t_surf and t_bulk,fixed respectively. Separate from this, any injection-dependent changes in the bulk unrelated to the BO defect are expected to be accounted for by changes in t_SRH,non-BO. A detailed discussion about the influence of E_trap,2 is presented later in Sections 5.3 and 5.5.

IDLS modeling approach

Considered together, t_DA and t_LS can be thought of as containing information about the four components of the lifetime model noted in Eqs. (1) and (2). The parameters describing these components can be deduced in a number of ways depending on which IDLS curve is used to extract the parameters and how the parameters are constrained during regression fitting. While t_surf, t_bulk,fixed, and t_SRH,non-BO can be extracted from t_DA, from t_LS or from a combination of t_LS and t_DA, t_SRH,BO can only be extracted from t_LS or from the difference in inverse lifetime (t_LS⁻¹–t_DA⁻¹). Further, in the case where all lifetime components from t_DA are assumed to apply to t_LS, the constraints on possible values of the fitting parameters can be different depending on whether t_DA and t_LS are fitted separately or simultaneously. Thus, a number of fitting methods may be used, with the overall goal being to determine methods of extracting the lifetime components that not only result in the best fit to the experimental data but are also physically meaningful.

To guide the choices of methods, we attempted to test two hypotheses in this study. First, we test the hypothesis that light soaking causes only t_SRH,BO to change between t_DA and t_LS. To do this, the measured IDLS data was modeled in 2 ways: by constraining all of the three components comprising t_DA during t_LS fitting (referred to as constrained methods), or by constraining some but not all of the three components (partially unconstrained). Comparing results from the constrained and partially unconstrained methods is therefore expected to reveal not only the presence of changes in lifetime between t_LS and t_DA not attributable to t_SRH,BO, but also which of the three remaining lifetime components best describes the changes.

Three types of constrained methods were used in this study. The first of these is the subtraction method (method A), in which t_DA⁻¹ was subtracted from t_LS⁻¹ and the resulting curve was modeled using t_SRH,BO alone. t_DA was then separately modeled to determine the other three lifetime components (t_surf, t_bulk,fixed, and t_SRH,non-BO), leaving t_LS to be described by the sum of the contributions of all four components. Note that explicit fits to t_LS were not attempted in this method. The second constrained method was the ‘separate’ method (method B) in which t_DA was first fitted to determine the parameters describing t_surf, t_bulk,fixed, and t_SRH,non-BO. These parameters were then fixed in the subsequent fitting of the t_LS curve, which included the t_SRH,BO component. Thus, t_SRH,BO was determined from the fit to t_LS in this method rather than the fit to (t_LS⁻¹ – t_DA⁻¹) as in the subtraction method. Lastly, in the ‘simultaneous’ method (method C), parameters for all lifetime components are solved for simultaneously rather than separately. This is done by simultaneous regression fitting of t_LS and t_DA, where t_SRH,BO was assumed to contribute only to t_LS. Thus, in summary, information about t_surf, t_bulk,fixed, and t_SRH,non-BO is effectively determined from t_DA in methods A and B and from a combination of t_LS and t_DA in method C, whereas information regarding t_SRH,BO is determined from (t_LS⁻¹ – t_DA⁻¹) in method A, and from t_LS alone in methods B and C.

The partially unconstrained methods followed a similar procedure to the ‘separate’ method, where t_DA was fit first to determine t_surf, t_bulk,fixed, and t_SRH,non-BO. However, in contrast to the constrained methods, the fitting parameters were not all unconstrained during t_LS fitting. In methods D, E and F, only t_surf, t_bulk,fixed, and t_SRH,non-BO respectively were unconstrained. Thus, method D corresponds to the case where only a change in surface lifetime was assumed to occur between t_DA and t_LS measurement (i.e. during light soaking or between measurements). Methods E and F, in contrast, correspond to cases where only the bulk-related lifetime components are assumed to change. Whereas method E allows for an injection-independent change in bulk lifetime via t_bulk,fixed, method F allows for changes in bulk lifetime that are injection-dependent in the low injection regime (Dn≤0.01 × N_A) via t_SRH,non-BO. We also allowed for the possibility that changes in surface lifetime and injection-independent bulk lifetime occur simultaneously by allowing both t_surf and t_bulk,fixed to be free (method G), as well as the possibility that surface lifetime and injection-dependent bulk lifetime in low injection occur simultaneously (method H) by allowing t_surf and t_SRH,non-BO to vary between t_DA and t_LS fitting.

The second hypothesis tested in this study is whether it is possible to determine t_SRH,non-BO in the absence of t_DA measurements, as was done in some previous studies [ 14]. To do this, we perform fitting only to t_LS using all four lifetime components (method I). t_DA is then assumed to be described by removing the contribution of t_SRH,non-BO. Lastly, to determine the impact of other SRH defects in the bulk on the determination of t_SRH,BO, we attempted to fit t_LS without t_SRH,non-BO in method J (i.e. using only three of the four components: t_surf, t_bulk,fixed and t_SRH,BO). Comparing fits to t_DA and t_LS from methods I and B is therefore expected to reveal if accurate parameters for all components can be determined from t_LS alone, whereas comparing methods I and J is expected to reveal the impact of ignoring non-BO related SRH defects on the determination of k₁.

Note that in all partially and fully unconstrained methods, information regarding t_SRH,BO was deduced from fits to t_LS only (as in method B) rather than from simultaneous fitting of t_DA and t_LS (method C) or from (t_LS⁻¹ – t_DA⁻¹) (as in method A).

A summary of the methods described above and the IDLS curves from which the different lifetime components are extracted are presented in Table 1. Where relevant, references to previous IDLS studies utilizing the methods are also noted.

Regression and statistical analysis

Fitting of IDLS curves was performed by minimizing the sum of the squared residual terms (SSQ) using a quasi-Newton nonlinear regression algorithm. In addition to the SSQ between the measured inverse lifetime data and model output (SSQ_data), an extra SSQ term related to the difference in slopes of the inverse lifetime for the injection range 10¹⁴<Dn<10¹⁶ cm⁻³ (SSQ_slope) was calculated and given a 10% weight. The sum of the weighted terms was then minimized. The use of both terms was found to result in faster and more reliable convergence as the regression algorithm tended to converge at local minima in the parameter space when SSQ_slope was not weighted.

Goodness-of-fit metric

The goodness-of-fit (GOF) between the IDLS data and the model output was determined using a reduced Chi-square metric (Q), which is calculated as follows:

(6) Q={\displaystyle \frac{\sum _{i=\mathrm{1}}^N{\left[\left(x_i-{\mu }_i\right)/{\sigma }_i\right]}^{\mathrm{2}}}{\mathrm{DOF}}}.

In the above equations, x_i and m_i refer to the inverse lifetime from the model and measured data respectively after correction for intrinsic recombination. s_i refers to the variance in m_i. DOF is the degrees of freedom remaining after model fitting and is calculated as the difference between the total number of fitted data points (N) and the number of variable fitting parameters used in the model (r).

Since the overall accuracy of a model being tested depends on the GOF associated with both t_LS and t_DA, it is useful to define a global GOF for the model (Q_global) in addition to the Q value associated with t_LS (Q_LS). Q_global is calculated similarly to Eq. (7) by including x_i and m_i terms from both t_LS and t_DA, and adding the corresponding N and r terms for DOF calculation. Note that fit parameters to describe t_DA are obtained from the t_LS model fits for methods I and J to reflect the source of the fitted components. In contrast, for the partially unconstrained methods (methods D–H), parameters for t_DA are sourced from method B to reflect the assumption underlying these methods that light soaking causes changes in fit parameters, as described in Section 3.2. Finally, the constrained methods (methods A, B and C) use the t_DA parameters determined from the respective methods.

The reduced Chi-square metric defined above is essentially a unit-less ratio of the residual error associated with the model relative to the expected point-to-point variability in the data. Hence, a Q value of 1 signifies a good fit to the data within the expected measurement uncertainty; whereas Q values significantly greater than 1 indicate poor fits wherein the residual error is greater than what might be expected due to variance in the data. Q values may also be less than 1, with values significantly lower than 1 suggesting that the model is ‘over-fitting’ the data. This can occur when r is a large fraction of N, or when s_i is overestimated. This simplicity of interpretation is a significant advantage of Q over GOF metrics used in other IDLS studies such as mean squared error [ 26] and non-normalized Chi-square [ 6], which are not unit-less and do not account for repeatability error. This makes them impractical for comparing GOF measures between studies. Also, while GOF metrics for nonlinear models that are more robust than Q do exist, they are computationally intensive [ 27] and were therefore not considered for this analysis.

However, the ability to draw meaningful conclusions from Q depends on accurate estimation of DOF and s_i. We note that DOF calculated as above is likely an underestimation as IDLS curves are nonlinear with respect to Dn and, strictly speaking, DOF can vary between 0 to (N – r) depending on the degree of nonlinearity of the model [ 27]. Further, since the inclusion of t_SRH,BO causes significant changes in the degree of nonlinearity of the IDLS curves, and increases r between t_DA fitting (r≤4) and t_LS fitting (r≤5), the Q_LS for a particular sample analyzed using one method may not be directly comparable to Q_DA or Q_global from another sample fit using a different method. However, considering that the maximum number of free parameters (r≤5) is at least an order of magnitude less than the number of data points within an IDLS curve (80<N<179 for all samples), Q values for a given sample are still comparable. Further, the large number of IDLS curves (225) allows aggregate behavior across samples to be compared, making Q an acceptable metric for comparative analysis of fitting methods.

We now consider the implications of s_i. There are three contributors to total error for any measurement depending on the source of the error: repeatability error, reproducibility error, and measurement artifacts. While repeatability error quantifies differences in measured values with time using the same instrument, reproducibility error quantifies differences in measured value between instruments [ 28, 29]. In contrast, measurement artifacts are deviations from the true value that stem from the specific measurement method or technique. Note that for practical purposes, repeatability and/or reproducibility error may be quantified and used as is in Eq. (6), whereas measurement artifacts either need to be corrected, prevented or removed by screening the measured data.

The approach used in this study to quantifys_i was to (a) minimize the tool-specific sources of error where feasible such that the difference between the reproducibility and repeatability errors are minimal, (b) minimize the impact of artifacts, and (c) quantify the remaining error via measurements of the repeatability error. Tool-specific errors were minimized via careful and regular calibration of the tool between measurements. While the component of reproducibility error specific to the tool could not be feasibly quantified and may be nonzero, we make the assumption here that reproducibility and repeatability error are now equal. We now discuss the issue of minimizing measurement artifacts.

Carrier lifetime measurements using the PCD technique are known to introduce measurement artifacts– particularly in low injection—due to trapping effects [ 30] and depletion region modulation (DRM) [ 31]. Further, any ambient light present during measurement could lead to unintentional biasing, leading to additional error in low injection conditions. Measurement error could also be introduced at high injection if the reference cell used to measure the illumination intensity of the flash lamp is saturated. Considering the cumbersome nature of quantifying the magnitude of error introduced by each of these effects and the large data set involved, it was decided to minimize and screen for artifacts, rather than apply corrections, which may themselves be a source of error. As noted in Section 2, the impact of unintentional bias at low injection from stray light was first minimized by performing all measurements in the dark. Secondly, only lifetime data within a specific range of Dn was used for regression. The upper limit of Dn varied between samples and was ensured to be below the injection level that caused saturation of the reference cell (roughly Dn = 3.3 × 10¹⁶ cm⁻³), whereas the lower limit of acceptable Dn was set to a single value for all samples (Dn = 6 × 10¹³ cm⁻³ ≈ 0.006 × N_A), which was chosen to be greater than the onset injection level below which DRM and trapping effects were found to dominate lifetime for a wide range of samples. Using these methods, measurement artifacts were largely eliminated from the data set used for regression.

Since all measurements in this study were performed on similarly processed wafers on the same instrument, with lifetime extracted using the same method (generalized), s_i can now be estimated by the repeatability error alone.

Estimation of uncertainty

The reproducibility error associated with PCD-based IDLS measurements analyzed using the generalized method is known to have a significant injection dependence and is estimated to be about 10% for Dn = 10¹⁵ cm⁻³ [ 25]. Since attempting to set s_i = 10% of x_i resulted in Q values significantly less than 1, it was concluded that the repeatability error was lower than 10%. We therefore performed direct measurements of the repeatability error as a function of Dn. This was done by removing and re-placing the same sample at approximately the same wafer position on the measurement area between 10 measurements. The standard deviation of the measured lifetime was then calculated at each injection level to obtain the repeatability error in inverse lifetime. This procedure was repeated for all 9 measurement points on the wafer, resulting in a total of 90 IDLS curves for s_i estimation. The standard deviations were then averaged across all measurement points and fitted empirically. The resulting average s_i as a function of Dn is plotted below in Fig. 1.

As Fig. 1 shows, s_i was almost constant (about 0.80%) for Dn>3.45 × 10¹⁴ cm⁻³ and increased almost linearly on a log-log scale for lower Dn. s_i (Dn) could be modeled using an empirical function given in Eq. (7). Note that although s_i is parameterized specifically for inverse lifetime (e.g. t_DA⁻¹), the repeatability error in lifetime (e.g. t_DA) is also approximately the same.

(7) {\sigma }_i(\Delta n)=\{\begin{array}{ll}\mathrm{13.62}\mathrm{\%},\hfill & \Delta n\le \mathrm{2}\times {\mathrm{10}}^{\mathrm{13}}{\mathrm{cm}}^{-\mathrm{3}}\hfill \\ {\mathrm{e}}^{[-\mathrm{0.9737}\times \ln (\Delta n)+\ln (\mathrm{1.14}\times {\mathrm{10}}^{\mathrm{12}}\left)\right]},\hfill & \mathrm{2}\times {\mathrm{10}}^{\mathrm{13}}\le \Delta n\le \mathrm{3.45}\times {\mathrm{10}}^{\mathrm{14}}{\mathrm{cm}}^{-\mathrm{3}}\hfill \\ \mathrm{1.413}\mathrm{\%},\hfill & \Delta n\ge \mathrm{3.45}\times {\mathrm{10}}^{\mathrm{14}}{\mathrm{cm}}^{-\mathrm{3}}\hfill \end{array}.

While the underlying causes of error are beyond the scope of this study, we note that the dependence of s_i on Dn is comparable to that of the reproducibility error reported in Ref. [ 25] and suggests that the same uncertainties that lead to high reproducibility error may also be responsible for increased repeatability error in low injection conditions. Further, measurement points on the wafer closer or further from the edges may be differently affected by edge recombination and trapping effects [ 30]. Likewise, doping non-uniformity and thickness variations may cause depletion region modulation as well as trapping effects to be triggered at slightly different Dn [ 31]. Such variations can explain the increased s_i in low injection.

Finally, we note that while the repeatability error measured above was performed for Dn>2 × 10¹³ cm⁻³, only data for Dn>6 × 10¹³ cm⁻³ was actually used for regression fitting in order to minimize the influence of measurement artifacts. The maximum value for s_i (Dn) was therefore less than 4% over the entire injection range used for regression fitting, and was less than 1% for Dn>10¹⁵ cm⁻³.

Results

A total of 225 IDLS measurements were performed in this study both after dark annealing and after light soaking. Given the large number of IDLS measurements, it was not feasible to present an in-depth analysis of individual IDLS curves; however, we summarize key lifetime metrics and plot a few typical IDLS curves here for the benefit of the reader. After dark annealing, the lifetime of the samples (t_DA) was found to range between 100 and 160 ms, with a majority of the samples displaying t_DA values of 140±12 ms. t_LS was found to be 73±4 ms, with the full range being 61–82 ms. This represented a decrease in lifetime due to light soaking of about 48% (±3%). J_0e values between 88±7 fA/cm² were obtained when fitted using method B, indicating a passivation quality typical of commercial PECVD SiN_x:H dielectrics applied on shallow emitters.

Figure 2 shows a typical set of lifetime curves fitted using one constrained method (method B) and one partially unconstrained method (method D) using data set from the same sample.

A number of important observations can be made from Fig. 2. First, clear evidence of BO-related CID can be observed, with t_LS (red points) displaying a strong injection-dependence in lifetime compared to t_DA (black points). Secondly, both t_DA and t_LS appear to be well-described by the lifetime components presented in Eqs. (1) and (2) respectively, evident from the good visual fit between the data (points) and model (lines), and relatively close-to-unity values for the GOF metrics (Q_DA, Q_LS and Q_global) for both methods.

The modeled curves (solid and dotted lines) in Fig. 2 also highlight the limiting recombination mechanisms affecting t_DA and t_LS. In the case of t_DA, it can be seen that t_surf (green dotted line) is the dominant recombination mechanism, with t_SRH,non-BO (purple dotted line) and t_bulk,fixed (yellow dotted line) being less limiting to overall lifetime. In contrast, t_LS is primarily limited by t_SRH,BO in low injection (Dn<3 × 10¹⁵ cm⁻³) and by t_surf in high injection (Dn>3 × 10¹⁵ cm⁻³). This has two main implications. First, this confirms that t_LS is sensitive to recombination from the BO defect over a sufficiently broad injection range, which is an important requirement for the determination of k₁ for the BO defect. Secondly, although t_surf dominates lifetime in high injection for t_LS, the high sensitivity of t_DA to t_surf ensures the accuracy of t_surf when determined from t_DA as in method B (dotted green line) or from t_LS as in method D (solid green line).

In fact, for this particular sample, the J_0e obtained from fitting t_DA (84.7 fA/cm², method B) appears to be about 4.5% higher than that obtained from fitting t_LS (80.8 fA/cm², method D). The implications of an apparent change in J_0e after light soaking of the sample are discussed later, but we emphasize two points here. First, this demonstrates the utility of comparing the fits from constrained and unconstrained fitting methods to detect potential light-induced changes in the sample. Secondly, it is worth noting that the failure to account for this change causes an anomalously low value for (t_LS⁻¹ – t_DA⁻¹) in high injection (Dn>2 × 10¹⁶ cm⁻³).

Figure 2 also demonstrates the power of regression and appropriate GOF metrics to determine suitability of the model and methods. A cursory visual inspection would suggest that the fits to t_LS for method B and D are both acceptable based on the visual overlap between red data points and lines. However, a comparison of the corresponding GOF metric, Q_LS, shows that method D provides a significantly better fit to t_LS (Q_LS = 1.02) compared to method B (Q_LS = 2.93), the latter being noticeably higher than unity. In fact, a closer look in the high injection regime dominated by t_surf (Dn>2 × 10¹⁶ cm⁻³) shows that the modeled t_LS curve from method B (solid red line) deviates from the observed values (red squares). Although this difference may be hard to discern visually from t_LS, the clear deviation of (t_LS⁻¹ – t_DA⁻¹) from t_SRH,BO from method B confirms that failure to account for changes in t_surf has an impact on the fitted value of k₁. This demonstrates that Q metrics with values closer to 1 are good indicators of fit quality and validates the approach presented in Section 3.3.1 to calculate Q.

Finally, it can be seen in Fig. 2 that the optimal value for k₁ that leads to the least Q_LS varies between methods even when the same data set is used. In the case of the presented set of measurements, k₁ varies by ~10% between methods (k₁ = 12.98 for method B and k₁ = 11.74 for method D). As will be shown later, this represents an extreme case within the data set. Nevertheless, it highlights the sensitivity of fitted k₁ to potential changes in non-BO related lifetime components and to the fitting method.

Similar fitted curves with varying fit quality and fitted parameters were obtained when other methods were used to analyze the same data set. While the subtleties relating to each of the fitting methods cannot be discussed here at length, we focus on overall trends relating to the fitting methods drawn from the large data set analyzed in this work. In particular, we present key statistical metrics and discuss results from the various fitting methods.

Goodness-of-fit for various fitting methods

As discussed in Section 3.3.1, GOF metrics provide important clues regarding the accuracy of the methods and of the assumptions behind the various methods. Since the goal here is to compare the validity of different methods, and their accuracy in modeling t_SRH,BO, we focus on two key GOF metrics: Q_LS and Q_global, which refer to the GOF associated with t_LS fitting, and the overall GOF for t_LS and t_DA fitting respectively. These are plotted in Fig. 3 and Fig. 4 respectively. We note that because parameters for fitting t_DA were the same for all methods except methods C, I and J, the GOF metric for t_DA (Q_DA) was equal for methods A, B and D-H. For the sake of brevity, Q_DA is therefore not plotted; however, qualitative information about the GOF for t_DA can be deduced from the difference between Q_global and Q_LS, and is commented upon below where appropriate. Note that the centrality and variability metrics reported in this work are the median and median absolute deviation (MAD) rather than the mean and standard deviation since the latter metrics are more sensitive to outliers and skews.

It can be seen from Fig. 3 and Fig. 4 that not all methods result in good overall fits to t_LS and t_DA. Constrained methods (methods A, B and C), for example, display Q_LS and Q_global values noticeably greater than 1 (Q_LS = 1.82–2.05, Q_global = 1.62–2.63). The same can also be concluded for method F, which also displays values for Q_LS (1.69) and Q_global (1.53) that are similar to the constrained methods.

These results have two implications for interpreting the Q values for other methods. First, the low GOF of methods A, B and C indicates that attempts to fit t_LS using parameters obtained from t_DA fitting (t_surf, t_bulk,fixed and t_SRH,non-BO) result in poor fits. Secondly, the poor fitting to method F, indicated by high Q_LS and high Q_global demonstrates that t_SRH,non-BO is not affected by light soaking. Thus, other methods that rely on these assumptions are also unlikely to be accurate. Such is the case with method H (free t_surf and t_SRH,non-BO), which displays lower Q_LS (1.27) and Q_global (1.33) but relies on the inaccurate assumption that t_SRH,non-BO is impacted by light soaking.

Similarly, methods I and J display excellent fits to t_LS (Q_LS ~ 0.66). However, interpreting this as a simultaneous change in all non-BO lifetime components due to light soaking would be inaccurate. Instead, the low Q_LS values, which are noticeably less than 1, are an indication that these methods ‘over-fit’ t_LS data owing to the large degrees of freedom (free parameters) available for fitting t_LS data. Moreover, when t_{SRH, BO} is subtracted from t_LS to obtain t_DA for methods I and J, this results in extremely poor fits to t_DA as demonstrated by the high Q_global values of ~20. While not plotted, the poor fits using methods I and J could be confirmed visually. Thus, methods I and J can also be ruled out as being inaccurate. This also rules out the possibility that fitting to t_LS alone can accurately determine the lifetime components defining t_DA.

Methods D (free t_surf) and E (free t_bulk,fixed), on the other hand, display Q values close to 1 (Q_LS = 1.31–1.35, Q_global = 1.20–1.28). These are noticeably lower than the constrained methods and also lower than method F. This suggests that either t_surf or t_bulk,fixed could be affected by light soaking. It may also be possible that both of these components are affected, as indicated by the low Q_LS (0.81) and Q_global (1.12) for method G. However, since the Q_LS value for method G is less than 1, the issue of over fitting cannot be ruled out. Thus, of the various methods tested, methods D, E and G— in that order —are most likely to be accurate.

Impact of fitting methods on k₁

While GOF metrics help to identify inaccurate methods, methods with comparable Q_LS or Q_global that are close to unity may not necessarily all be accurate. This could be caused, for example, by the regression algorithm attempting to adjust k₁ in order to achieve a better GOF (lower Q values) —but at the expense of accuracy. Hence, it is instructive to consider the fits to k₁ resulting from the various methods, which are plotted below in Fig. 5.

We first discuss the variability of fitted k₁ (MAD) between methods as this is particularly important for evaluating the accuracy of the methods. In particular, methods which cause k₁ to be adjusted to account for inaccuracies in other lifetime components can be expected to display higher MAD in k₁ compared to more accurate methods.

This is apparent when comparing Fig. 4 and Fig. 5. It can be seen that unconstrained methods which display high MAD in Q_global (methods F, I and J) also display a correspondingly high MAD in k₁. Once the higher median Q_global for these methods is also considered, it is clear that k₁ values for these methods were indeed influenced by attempted changes in other fitting parameters to achieve the best possible GOF. Similarly, constrained methods (methods A, B and C) also display moderately high MAD values for both Q_global as well as k₁.These results confirm the inaccuracy of the assumptions underlying the constrained methods (A, B and C) as well as some of the unconstrained methods (F, H, I and J).

Comparing the MAD in k₁ also helps distinguish between unconstrained methods with low Q_global (methods D, E, and G). It can be seen that despite comparable Q_global values (1.12–1.35), these methods display different MAD in k₁, with methods E (0.98) and G (1.01) displaying almost twice as much MAD in k₁ as method D (0.55). Note that the MAD in k₁ of method D is also the least of all methods tested. This strongly suggests that the assumption underlying method D (changing t_surf) more accurately reflects the changes due to light soaking than that underlying method E (changing t_bulk,fixed). However, since method G results in lower Q_LS and Q_global than both methods D and E, a combined change in t_surf and t_bulk.fixed cannot be ruled out.

The comparison of GOF and MAD in k₁ therefore suggests than method D is the most accurate modeling method, with method G possibly also being accurate. These results not only confirm that light soaking causes changes in lifetime components other than t_SRH,BO, but also confirm that these changes can be successfully modeled by allowing t_surf (and possibly also t_bulk,fixed) to change.

Comparison of the fitted k₁ between methods also leads to a few other interesting observations. First, it can be seen that the median k₁ ranges from 11.89 to 14.72 between methods. This clearly shows that different methods lead to noticeable different k₁ values.

Secondly, k₁ for individual samples for a given method are significantly different. For example, even for the method with the least variability in k₁ (method D), k₁ ranges from 11 to 13 (excluding extreme outliers). This represents a maximum deviation of 8% from the median. For the method with the largest variance (method F), k₁ deviates 35% from the median. This demonstrates the importance of using large data sets to determine k₁, particularly for IDLS data.

Further, when comparing median k₁ values between methods are compared, it can be seen that inaccurate methods either over- or under-estimate k₁. Compared to the most accurate method (method D, k₁ = 11.96), other unconstrained methods display either significantly higher k₁ (methods I and J, k₁ = 14.55–14.72) or moderately higher k₁ (methods E, F and H, k₁ = 12.12–12.38). In contrast, the constrained methods display either lower or comparable median k₁ values (11.89–11.96).

As previously mentioned, this is likely the result of the regression algorithm attempting to change k₁ in order to compensate for inaccurate parameter values for other lifetime components. Higher k₁ values are indicative of an overestimation in the change in t_bulk,fixed (methods E, F, I and J), or an overestimation in t_SRH,non-BO (methods F, H, I and J). Similarly, an overestimation of t_surf in the case of the constrained methods (methods A, B and C) could lead to an underestimation in k₁.

Note that although it is possible that changes in components other than t_surf do occur, or that no changes other than t_SRH,BO occur due to light soaking, the poor overall GOF for the inaccurate methods suggests that these changes are either restricted to individual samples or are negligible.

Finally, based on a reduced data set consisting of 208 measurements with outliers rejected, the final median and MAD of k₁ using method D was determined to be 11.90±0.45 (11.91±0.57 using the average and standard deviation).

Changes in non-BO related lifetime components

A careful analysis of the GOF in various injection ranges (not plotted) confirmed an injection-dependent change in lifetime, particularly in high injection, that can be modeled well by either allowing for changes in t_surf alone (as in method D) or in both t_surf and t_bulk,fixed (as in method G). We first consider changes in t_surf. Since method B does not involve a changing t_surf component, the relative difference in fitted J_0e from methods B and D represents the percentage change in J_0e due to light soaking. If changes in J_0e were due to the random errors in fitting, they would be expected to display a roughly normal distribution centered at 0%; however, our analysis showed that this relative change in fitted J_0e due to light soaking ranged from –10% to+ 5%, with ~75% of the samples undergoing a decrease in J_0e due to light soaking. Similarly, when method G was used, the relative change in fitted J_0e ranged from –8% to+1%, with ~90% of the samples displaying a decrease in J_0e due to light soaking. In both cases, the apparent light-induced change in J_0e displayed a roughly normal distribution but was centered at roughly –1.5% rather than 0%. This represents an average light-induced decrease in fitted J_0e of roughly 2 fA/cm². Although this change is less than the average deviation in J_0e between samples, this effect was statistically significant when compared to the uncertainty in J_0e estimation. In the case of the data set presented in Fig. 2, the difference in fitted J_0e between methods B and D is ~4 fA/cm², which is higher than the median decrease in J_0e, but well within the range observed across the data set.

Similarly, when illumination-induced changes in t_bulk,fixed were analyzed by comparing fits from methods G, a majority of the samples (89%) appeared to display a decrease in fitted t_bulk,fixed due to light soaking rather than an increase, with a distribution in apparent light-induced change in t_bulk,fixed ranging from –60% to+20% with a peak at roughly –35%. This corresponds to an average apparent light-induced decrease in t_bulk,fixed of ~200 ms. Despite the seemingly large value, the small contribution of t_bulk,fixed to overall recombination (as seen in Fig. 2) suggests that the inherent uncertainty in parameter fitting is quite large for t_bulk,fixed. This explains why the GOF for method G is only marginally better than method D in spite of allowing for large changes in t_bulk,fixed (±20%). Nevertheless, the fact that the decrease in t_bulk,fixed is statistically significant suggests this may be a real effect.

Finally, in contrast to t_surf and t_bulk,fixed, attempting to model potential changes in t_SRH,non-BO (from methods F and H) did not display any clear statistically meaningful trends.

Discussion

The results above highlight four important points. First, it is clear that assuming t_SRH,BO to be the only variable parameter between t_DA and t_LS fitting—in other words, assuming light soaking only causes a change in t_SRH,BO—is most likely an inaccurate assumption for the samples used in this work. Further, light soaking appears to cause noticeable changes in lifetime at high injection (attributable to changing t_surf) and may also be causing minor and apparently injection-independent changes (modeled in this work by changing t_bulk,fixed).

Secondly, the results show that allowing the wrong fitting parameters to vary does not necessarily improve the variability of fitted k₁. Thus, not all methods to determine k₁ are accurate. Thirdly, considering the variability in k₁ for a given method, it is clear that the use of smaller IDLS data sets to determine k₁ can be error-prone.

Finally, median values of k₁ for all fitting methods are significantly different from the oft-quoted value of 9.3 obtained from a combination of method A and TDLS analysis [ 6]. In fact, only outliers in this study produced fits to k₁≤10 for any of the methods used in this study. We now discuss the implications of these results in detail in the following sections.

Influence of modeling methods on k₁

The median k₁ of 11.9 for method A in this study is significantly different from a majority of the reported values in the literature using the same methods for the fully formed BO defect. Statistical tests confirm that k₁ values from method A in this study are greater compared to other reported values of 8.3 [ 5], 9.3 [ 6] and 9.9 [ 13] with 95% confidence. Finally, only one reported k₁ value of 12.1 [ 10] (using method B) appears to correlate well with the results in this study.

There are multiple possible reasons why values in this work differ significantly from previous work; however, we focus here on the influence of modeling methods. An important point to note is that none of the other studies thus far have used large data sets of similar samples and therefore values reported in those studies have an inherent degree of uncertainty that may be larger than the statistical variability reported in this work. This could be caused by the use of samples with different concentrations of the BO defect (for example due to varying N_A [ 4] or degree of compensation [ 8, 11]) or different overall lifetime, both of which may cause differences in the regression uncertainty for k₁. Moreover, GOF metrics were not quantified in some studies [ 5, 8, 13], making it difficult to assess the models used and therefore the accuracy of fitted k₁ values. Thus, it is possible that the lower fitted values of k₁ in previous studies are due to limited sample sizes.

Secondly, a number of studies employing method A have ignored IDLS data in high injection where t_DA appears to be greater than t_LS, which results in negative values for (t_LS⁻¹– t_DA⁻¹) [ 4, 6]. Since such data cannot be accurately fitted without accounting for potential light-induced changes in t_surf, it may have been more correct to use method D in such cases. Further, the lack of a diffused emitter in prior work may have resulted in larger changes in t_surf than seen in this work. Accounting for such changes using method D would have most likely resulted in larger k₁ values and therefore closer to the values reported in this work.

The impact of a diffused emitter on k₁ fitting is also highlighted in the case of Ref. [ 10], in which k₁ determined from an emitter-diffused p-type wafer was found to be 12.1 using method B. This is not only closer to the value of 11.9 determined in this work but is also significantly different from k₁ values of 10 or less reported for p-type wafers without an emitter diffusion [ 4– 6, 13]. The presence of an emitter is also expected to increase J_0e, and therefore it is possible that fitted k₁is related to high J_0e values rather than the presence of an emitter. This possibility is further discussed in Section 5.2 below.

Finally, when methods I or J is used, fitted k₁ in this work appears to be significantly lower (k₁ = 14.5 to 14.7) compared to a reported value of 19.3 [ 14] for those methods. However, we note that explicit modeling of E_trap,2 in the latter work could have led to the higher k₁ value reported. We comment further on this in Section 5.5. Nevertheless, the latter result does confirm that the use of methods I and J can lead to larger k₁ values than methods A and B.

Influence of sample type and measurement methods on k₁

Based on the comparison to reported k₁ values from the literature, it is evident that modeling methods could influence the determination of k₁ and can explain some of the inconsistencies in reported k₁. However, the noticeably larger values of k₁ from this work and in Refs. [ 10] and [ 14] suggests other underlying causes. One possible explanation is that k₁ values determined in this work are specific to the wafers used in this study. For example, the commercial p-type wafers used in this study contained other SRH defects beside BO. Although these defects were confirmed not to cause CID, they necessitated the use of the t_SRH,non-BO component to achieve good fits to t_DA. Attempting to solve for two SRH defect components (t_SRH,BO and t_SRH,non-BO) simultaneously during t_LS fitting, as in methods I and J, resulted in low-Q, but inaccurate fits. This issue may be less relevant in wafers that do not contain defects other than BO, such as the p-type float-zone silicon wafers fitted with method J in Ref. [ 14].

Further, in the single study where k₁ was found to be similar to that determined in this work, the wafers used included an emitter diffusion step [ 10]. This is similar to the wafers in this study and unlike the wafers used in other studies with comparable N_A values [ 4, 6]. The use of a shallow emitter for lifetime characterization, such as that used in this study, may have either have a neutral or negative influence on the accuracy of k₁ estimation. On the one hand, wafers with emitter diffusion are known to display DRM [ 28], which artificially increases the measured lifetime in low injection. Hence, this is expected to lead to a reduction in the fitted k₁. While this was the case with some samples in this study, the effect was found to be small as excluding these samples did not appear to appreciably change the median k₁. Moreover, DRM effects do not explain why k₁ values in this work are higher, rather than lower, compared to other work on samples without emitters.

Another potential drawback of using emitter-diffused samples is that an emitter with a high J_0e (low t_surf) may reduce the injection range in which t_SRH,BO (and therefore k₁) is sensitive to model fitting. It is likely that a more lightly diffused emitter or better surface passivation would have increased the injection range in which t_SRH,BO is the limiting component of lifetime in t_LS (see Fig. 2). Considering that k₁ appears to be highly sensitive to light-induced changes in J_0e in the samples used in this work, the question arises whether lower overall J_0e in the samples could lower the fitted k₁. However, a detailed analysis on our data set revealed that even the best passivated samples in this study (J_0e ≈ 75 fA/cm²) displayed a similar range of fitted k₁ values as the worst passivated samples (J_0e ≈ 100 fA/cm²). Moreover, no discernible correlation could be found between k₁ and the absolute value of the fitted J_0e despite a strong correlation between k₁ and light-induced changes in J_0e. While this does not rule out the influence of absolute J_0e on the fitted k₁ for samples with J_0e lower than that seen in this study, we point out that this is unlikely to be the case. Nevertheless, the possibility of emitter diffusion skewing the fitting of k₁ is a topic worthy of further investigation.

On the other hand, the presence of an emitter could also be beneficial since an emitter is expected to reduce the impact on lifetime of light-induced changes near the silicon-dielectric interface. This is because t_surf is a lumped parameter that accounts for recombination in the emitter as well as from the surface. Thus, the absence of an emitter may increase t_surf (i.e. reduce J_0e), but would also make t_surf more sensitive to potential light-induced changes at the silicon-dielectric interface and to any non-homogeneities at the surface (e.g. surface damage). We also note that the emitter diffusion step is a necessary process in solar cell fabrication. Hence, even in the unlikely case that diffused emitters play a role in k₁ fitting, the k₁ values determined in this work remain highly relevant to the characterization of BO defects in commercial p-type solar cells and precursors. The large data set used in this work provides further statistical confidence in the fitted k₁ values in this work.

Finally, we note that fitting of k₁ may also be influenced by the measurement technique used to obtain IDLS data. The use of photoluminescence (PL) lifetime spectroscopy to determine k₁ [ 10] could, for example, eliminate the influence of DRM, if present. Therefore, PL spectroscopy may be better suited for characterizing BO defects in p-type samples with heavily doped emitters. Nevertheless, both PCD and PL techniques are sensitive to edge recombination and carrier trapping effects, which may either cause an increase or decrease in fitted k₁ depending on the sample type and the range of injection levels considered for analysis.

Apparent changes in non-BO related lifetime components

The statistically significant changes in lifetime components not modeled by t_SRH,BO in this study point to either underlying physical reasons for light-induced changes in t_surf, and to a smaller extent in t_bulk,fixed, or indicate deficiencies in the model describing t_SRH,BO that led to apparent changes in t_surf and t_bulk,fixed. We first consider possible physical mechanisms for such changes.

Illumination-induced changes in t_surf have been observed in other studies [ 32– 35] and are thought to be related to charge trapping in the dielectric layer [ 33, 35]. Such effects could lead to an improvement in surface passivation when SiN_x:H layers are deposited on a diffused n-type surface (as is the case in this study). This could explain the light-induced reduction in J_0e seen in a majority of the samples in this study, while a varying magnitude of the charge trapping effect could help explain the distribution of differences in fitted J_0e between methods B and D. Likewise, the small increase in J_0e observed in some samples could either be due to small uncertainties in the fitted parameters, or due to minor surface damage induced during wafer handling. However, the presence of an emitter is expected to make t_surf less sensitive to such changes reduce the impact of such damage on the observed J_0e.

Although samples in this work largely display a decrease in J_0e due to light soaking, it is important to consider that a statistically significant increase in J_0e due to light-soaking could occur in other sample types. For example, the charge trapping effect in SiN_x:H that improves t_surf on emitter-diffused p-type wafers may have a negative or neutral effect on t_surf on p-type wafers without an emitter, or on wafers passivated using a different dielectric. In addition, samples without an emitter may be more susceptible to changes in t_surf due to surface damage. In fact, there may be some evidence of this in early studies [ 4– 6] wherein an additional shallow trap level at roughly E_V + 0.15 eV was found to be required to fully describe the high-injection behavior of p-type wafers without an emitter. Interestingly, not all wafers were found to require this trap level. This is particularly relevant since our attempts to model a small increase in J_0e using an SRH defect revealed that a minor increase in J_0e of 4 fA/cm² could be effectively modeled by a shallow defect located in a narrow range between E_V + 0.05 eV and E_V + 0.2 eV. Thus, when IDL data are fitted using method B, a sample displaying an increase in J_0e would appear to require the shallow defect level, whereas samples with no change (or a decrease) in J_0e would appear not to require this level. Further, the lack of an emitter diffusion would be expected to make the samples in the other studies even more sensitive to changes in t_surf than samples used in this study. As discussed in Section 5.2, any potential increase in J_0e between t_DA and t_LS in such studies could also affect the fitted value of k₁. While further studies are required to understand this effect, the results in this work suggest that accounting for potential changes in t_surf due to light soaking (methods D or G) is a more accurate approach to IDLS modeling.

On the other hand, physical reasons for the apparent decrease in t_bulk,fixed between t_DA and t_LS are harder to justify. The samples in this study were confirmed to have negligible quantities of iron and no other known CID causing defects with the exception of the BO defect. In any case, most known CID-causing defects such as iron and copper are known to display a significant injection dependence and therefore would not be expected to cause changes in, which is assumed to be injection independent. The minor changes in t_bulk,fixed observed from the analysis of method G therefore point to a deficiency in the model used to describe t_SRH,BO. As previously noted, Eq. (3) used to describe t_SRH,BO does not include the recombination due to E_trap,2, the acceptor level of the BO defect [ 3, 11]. The additional recombination introduced by E_trap,2, if largely injection-independent, could manifest as a decrease in t_bulk,fixed and therefore explain the improved fit obtained from method G.

To check if E_trap,2 could explain the apparent decrease in t_bulk,fixed, an additional SRH lifetime component described by Eq. (4) was used to represent E_trap,2. This model was then used to analyze some of the curves in the data set in lieu of a variable t_bulk,fixed component in method G. However, since allowing k₂ and k₁ to vary would allow any value of E_trap,2 in the lower half of the bandgap to replicate the roughly injection-independent effect modeled by t_bulk,fixed, two constraints were applied. First, k₁ was not allowed to vary and was fixed to the value determined using method G and applying the single defect model (Eq. (4)). Secondly, only values of E_trap,2 that required less than unity values of k₂ were considered. Preliminary results indicate that only for E_trap,2>E_V + 0.2 eV was the latter condition satisfied. Further, upper limits of k₂ for reported acceptor levels from the literature could be determined. The upper limit of k₂ was found to be 10⁻¹ to 10⁻² for E_V + 0.26 eV [ 9, 11] and 10⁻³ to 10⁻⁴ for E_V + 0.35 eV [ 3]. These limits largely agree with other IDLS studies incorporating the E_trap,2 level at E_V + 0.26 eV, which report k₂ to be roughly 2 × 10⁻³ [ 11, 14]. The higher k₁ values resulting from method E and G relative to method D also suggest a minor influence of the acceptor trap level, since a decrease in such apparent injection-independent lifetime would require larger k₁ values to provide good fits to the data [ 14]. Thus, it would appear that the apparent changes in t_bulk,fixed could be explained by the recombination activity of E_trap,2.

However, similar efforts to model changes in t_surf from method D by incorporating E_trap,2 via an additional SRH defect and keeping k₁ fixed were not successful. This is because as long as other lifetime components are fixed, introducing an additional SRH component in the model only allows for a net decrease in lifetime to be modeled. However, a large majority of the samples in this study actually displayed a decrease in fitted J_0e and therefore an apparent net increase in surface/emitter lifetime. In fact, for some samples, t_LS was actually found to display a higher lifetime compared to t_DA, leading to negative values for (t_LS⁻¹ – t_DA⁻¹), particularly in high injection (Dn>10¹⁶ cm⁻³). Similar negative values for (t_LS⁻¹ – t_DA⁻¹) are also alluded to in other studies on the fully formed BO defect [ 4– 6]; however, these have largely been rejected for the fitting of k₁ in such studies. Nevertheless, it must be acknowledged that this does not rule out the possibility of explaining some, but perhaps not all, of the decrease in lifetime associated with a changing J_0e component using E_trap,2.

While further work is required to clarify the impact of E_trap,2, we conclude for now that there is indirect evidence of a real, but minor light-induced change in J_0e for the samples in this study. In contrast, the minor apparent change in seen in most samples is most likely related to E_trap,2 rather than due to an underlying change in the background lifetime. Attempts to model the data set in this study using a more accurate, two-level description of t_SRH,BO [ 11, 14, 36] are currently underway to clarify these effects.

Influence of fast and slow degradation effects on k₁

It must be recognized that the conclusions regarding the validity of the modeling methods presented in this work are not only specific to the single defect model used (Eq. (4)), but also to the analysis of the fully formed BO defect in p-type silicon. As mentioned earlier, BO-related CID occurs in two stages, which appear to require very different capture cross section ratios to accurately model their impact on carrier lifetime. In the case of p-type silicon, carrier lifetime after complete formation of the BO defect in p-type silicon is largely limited by the slow forming BO defect, which can be fitted by a mid-gap defect with a capture cross section ratio similar to that determined in this study (k = 9 to 19) [ 4– 6]. This property has been used in this work and in a majority of other studies on p-type silicon to determine k₁ and E_trap,1 assuming a single defect model with a single trap level. However, a sufficiently large contribution to the lifetime after complete degradation from the fast-forming defect (k = 65 to 100) [ 7, 37] could result in a higher fitted value for k₁ [ 14].

Due to the lack of time-resolved data during the light soaking process, it was not possible to verify if the higher k₁ value of ~12 reported in this work and in Ref. [ 10] could be due to greater contribution to lifetime from the fast-forming defect compared to other studies reporting a lower k₁ value. However, we note that it is unlikely that the samples in this study or in Ref. [ 10] are significantly more influenced by the fast forming defect compared to other studies. First, the Cz samples used in this study and in Ref. [ 10] appear to be of similar quality and have N_A values that are well within the range studied in other works that report lower k₁ values [ 4– 6]. Thus, differences in the net doping-normalized concentration of BO defects are likely to be minimal between the various studies. Moreover, any potential differences in the fraction of BO defects undergoing fast and slow degradation [ 9, 14] are also expected to be minimal between this work and other studies on non-fired samples. We also note that dark annealing was performed at the same temperature (200°C) in this and other work. Unless there are other unknown factors that could increase the contribution of the fast defect center to the lifetime after complete formation of the defect, the improved estimate of k₁ presented in this work is still valid for single-defect modeling of the fully formed BO defect in commercial p-type silicon.

Nevertheless, accounting for the influence of the fast and slow degradation centers in the fully formed defect could well lead to more accurate estimates of k₁ than that determined in this study. The exact approach to account for the influence of the fast and slow degradation centers is currently a matter of debate [ 9, 12, 14]; however, eliminating the influence of either the slow or fast forming defect has been shown to simplify analysis [ 5]. Time-resolved IDLS measurements leading to complete degradation such as that performed in Refs. [ 11, 14, 37] are expected to lead to more accurate estimates of k₁.

Approaches to modeling E_trap,2 in p-type silicon

Ignoring E_trap,2 has been justified in this and prior work on fully formed BO defects in p-type silicon considering the minimal additional injection-dependence in carrier lifetime introduced by the acceptor level as noted in Section 5.3 and in Ref. [ 7]. However, E_trap,2 is known to be dominant and non-negligible in compensated n-type material [ 8, 11] and may have implications for the determination of k₁ in p-type silicon as well [ 14]. There also appears to be some evidence that E_trap,1 and E_trap,2 are related to the slow and fast degradation centers [ 14, 38], further complicating the analysis of E_trap,1 and E_trap,2 in fully degraded p-type silicon. We briefly discuss these implications here and outline the challenges in characterizing E_trap,2 in p-type silicon.

As noted earlier, the exact location of E_trap,2 within the bandgap is not known with certainty. A recent DLTS study on p-type silicon has suggested an acceptor trap level at E_V + 0.35 eV; however, this was based on preliminary results and there appears to be some ambiguity in the DLTS data regarding E_trap,2 that requires further analysis [ 3]. In contrast, IDLS characterization of compensated n-type material has suggested a trap level at E_V + 0.26 eV [ 9, 11], although E_V + 0.15 eV has also been shown to result in acceptable visual fits in such material [ 8]. While it is suspected that the estimates of E_trap,2 from compensated n-type silicon may not apply to p-type silicon due to differences in the charge state of the defect between the two material types [ 9], a good fit to t_LS in p-type silicon has nevertheless been achieved [ 14]. Even so, the absence of a comprehensive analysis via TDLS or DLTS makes it difficult to establish E_trap,2 unambiguously in p- and n-type material.

Compounding the situation further is the fact that the capture cross section ratio associated with E_trap,2 (k₂) also appears to vary widely between studies. k₂ has been variously reported in p-type silicon as 1.5 × 10⁻² [ 9], 2.1 × 10⁻³ [ 14] and 2.55 × 10⁻³ [ 11] for E_trap,2 = E_V + 0.26 eV, and simply as k₂<<1 for E_trap,2 = E_V + 0.15 eV [ 4]. In contrast, k₂ in compensated n-type silicon is reported to be 6 × 10⁻² for E_trap,2 = E_V + 0.15 eV [ 8] and 2.55 × 10⁻³ [ 11] for E_trap,2 = E_V + 0.26 eV. k₂ has not been estimated to date in p- or n-type silicon for E_trap,2 = E_V + 0.35 eV (the trap level estimated from the DLTS study [ 3]).While estimates of k₂ all indicate that k₂ is<<1, the uncertainty in k₂ is considerable. A large part of this uncertainty stems from the fact that the injection dependence attributable to E_trap,2 is not very sensitive to k₂ [ 11]. The varying choice of E_trap,2 further compounds this effect since there could exist multiple combinations of E_trap and k to describe the injection-dependent behavior of any given defect. Moreover, if E_trap,2 is specific to either the fast or slow recombination centers, then k₂ may also be a function of the charge state of the defect at E_trap,2, and therefore the substrate doping type [ 9]. Finally, the uncertainty in k₂ could be related to the assumed value of k₁. We note that most attempts to determine k₂ within a two-level defect model have assumed k₁ to be in the range of 9–10 [ 8, 11, 12]; however, since this value of k₁ was itself determined using a single defect model with a single trap level (Eq. (4)) such as that used in this study, the resulting estimates of k₂ may be inaccurate. All these reasons make it particularly difficult to study E_trap,2 using data from the fully formed BO defect in p-type silicon, in which the slow degradation center appears to dominate.

Thus, in order to investigate E_trap,2 in p-type material using a two-level defect model, it may be necessary to not only distinguish between the fast and slow degradation effects, but also to solve for all unknowns describing t_SRH,BO (k₁, k₂, E_trap,2, t_n0 andt_p0) simultaneously. While the issues regarding fast and slow degradation can be solved via time-resolved measurements of t_LS [ 11, 14, 37], the disambiguation of parameters defining t_SRH,BO within a two-level defect model cannot be easily resolved. This is primarily because the effective DOF associated with fitting t_SRH,BO using data from a single sample type is lower than the number of variables to be solved. However, analyzing samples containing variations in N_A [ 9, 11], in sample temperature (via DLTS or TDLS) or potentially isolating the recombination activity of E_trap,2 or E_trap,1 (as suggested in Ref. [ 14]) are all approaches that could simplify parameterisation or reduce the number of unknown parameters.

Attempts to perform regression on the current data set using a two-level defect model are currently underway to check if statistically variation could help establish a likely range of fit parameters.

Implications for lifetime characterization of CID defects

The potential invalidity of the assumption that light soaking causes changes only in t_SRH,BO, has significant implications for the determination of other SRH recombination characteristics of the BO defect. In particular, changes in t_surf are expected to result in an over- or under-estimation of normalized defect density [ 5] and introduce additional uncertainty in the determination of the BO defect state transition kinetics [ 39], both of which rely on IDLS measurements.

These effects are expected to be even more important for TDLS analysis, which relies on determining the temperature dependence of t_SRH,BO. All TDLS studies done on BO defects so far have determined recombination properties based on the temperature dependence of t_LS [ 6] or t_DA [ 15] rather than t_SRH,BO alone. If the temperature dependence of t_surf [ 40] compensates that of t_SRH,BO, and changes in t_surf in those studies were non-negligible, this could have resulted in an under- or over-estimation of k₁ and E_trap,1. However, since the analysis was restricted to low injection conditions in those studies, the degree of uncertainty introduced due to changing t_surfis expected to be small. Nevertheless, this effect may be non-negligible for TDLS analysis at higher injection levels or for the analysis of shallower trap levels such as the acceptor level of the BO defect (E_trap,2).

Although BO defects are the primary focus in this work, these effects are expected to be equally relevant for the characterization of other similar SRH defects such as the CID defect in multi-crystalline silicon [ 41], which may be particularly vulnerable to changes in t_surf due to the long annealing and illumination exposure times required to cause defect state transitions.

Finally, the fitting methods used in this study could be useful for modeling of lifetime data involving SRH defects in general, as well as other types of data sets that require nonlinear regression. In particular, the robust, unit-less measure for goodness of fit used in this work offers an improvement over visual fitting or mean-squared-error based metrics with arbirary units used in previous studies. As demonstrated in this work, a reduced Chi-square metric accounting for repeatability error not only allows for easy interpretation of the fit quality but could also be used to compare fitting results from data sets measured using the same measurement tool. Additionally, accounting for the reproducibility error could easily enable comparison of results between different tools.

Conclusions

In this work, we performed modeling and regression fitting of IDLS data from a large number of commercial p-type wafers with lightly diffused emitters that contain BO defects. Analysis of the IDLS data set containing N = 225 measurements performed after complete deactivation and complete activation of the BO defect revealed that exposure to illumination causes minor but statistically significant changes in lifetime that cannot be attributed to the BO defect alone. This violates a common assumption made during IDLS and TDLS characterization of BO and other CID defects. We also introduced a reduced Chi-square metric that allows for easier interpretation of goodness-of-fit and allows for comparison of goodness-of-fit between samples, analysis methods and potentially, between tools. Based on an analysis of goodness-of-fit and the statistical variability of the capture cross section ratio (k₁) determined from regression fitting for the donor level of BO defect using a number of different methods, it was shown that apart from the BO CID effect, illumination exposure also causes a statistically significant, injection-dependent change in lifetime and a less significant, apparently injection-independent change in lifetime. The former was found to be well modeled by changes in surface passivation while the latter, if present, is hypothesized to be the effect of the acceptor level of the BO defect. While further investigation is required to determine the underlying cause of these effects, ignoring them is expected to introduce additional uncertainties in the determination of k₁ from IDLS and TDLS measurements, in the calculation of normalized defect density and in the determination of defect transition kinetics for BO defects as well as other CID defects.

This study also suggests that the substrate material quality, the presence of diffused emitters, the measurement techniques, and methods used to determine the parameters describing the various components of carrier lifetime can all potentially influence the determination of k₁. A combination of these and other effects explains why a wide range of k₁ values for the BO defect have been previously reported in the literature. Overall, modeling methods that account for changes in surface recombination are expected to be the most accurate for determining k₁. Based on the large data set used in this work, k₁ is determined with a high degree of confidence to be 11.90±0.45 as applied to the fully formed BO defect in low-resistivity, commercial p-type silicon with lightly diffused emitters when modeled using a single-level donor defect. It is predicted that modeling of the acceptor level of the BO defect, performed only in a limited way in this study, is likely to result in a larger and more accurate value of k₁.

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