Accurate Extraction of Minority Carrier Lifetimes—Part II: Combined I–V C–V Methods

Many optoelectronic applications require precise knowledge of minority carrier lifetimes and diffusion lengths to ensure well-defined device characteristics under operation. To extract the minority carrier lifetimes, various methods have been proposed. For instance, current–voltage (<inline-formula> <tex-math notation="LaTeX">${I}$ </tex-math></inline-formula>–<inline-formula> <tex-math notation="LaTeX">${V}$ </tex-math></inline-formula>) and capacitance–voltage (<inline-formula> <tex-math notation="LaTeX">${C}$ </tex-math></inline-formula>–<inline-formula> <tex-math notation="LaTeX">${V}$ </tex-math></inline-formula>) measurements have been used in combination (cIVCV) to extract carrier lifetimes. However, some of the proposed methods rely on approximations and simplifications that do not necessarily hold for every real device. In this work, the basic theory for the <inline-formula> <tex-math notation="LaTeX">${I}$ </tex-math></inline-formula>–<inline-formula> <tex-math notation="LaTeX">${V}$ </tex-math></inline-formula> and <inline-formula> <tex-math notation="LaTeX">${C}$ </tex-math></inline-formula>–<inline-formula> <tex-math notation="LaTeX">${V}$ </tex-math></inline-formula> relations in p-n junctions is reviewed, and accurate extraction procedures for the generation and recombination lifetime out of the reverse and forward <inline-formula> <tex-math notation="LaTeX">${I}$ </tex-math></inline-formula>–<inline-formula> <tex-math notation="LaTeX">${V}$ </tex-math></inline-formula> and <inline-formula> <tex-math notation="LaTeX">${C}$ </tex-math></inline-formula>–<inline-formula> <tex-math notation="LaTeX">${V}$ </tex-math></inline-formula> characteristics of p-n junctions are discussed in detail. Our results demonstrate that the minority carrier recombination lifetimes extracted by the cIVCV method are significantly lower than the results from the transient methods presented in Part I. By means of computer simulations, we can show that this can be related to the interface between the epitaxial layer and the substrate, which is modeled using an effective surface recombination velocity. Furthermore, our results outline the main difference between the open-circuit voltage decay (OCVD) and reverse recovery (RR) method presented in Part I where carrier recombination at the back surface plays an important role on one side and the cIVCV method where the main signal originates from the p-n junction depletion zone and is, therefore, less affected by the back surface on the other side.


I. INTRODUCTION
V ARIOUS methods for the extraction of minority carrier lifetimes have been proposed in the literature. The majority of these methods is based on transient measurements, such as reverse recovery (RR) [1], [2], [3], open-circuit voltage decay (OCVD) [4], [5], or the pulsed MOS capacitor technique [6], [7], [8]. In addition, optical measurement techniques are frequently used to extract carrier lifetimes [9] which are often based on measured current or voltage transients after optical excitation, such as the photo-conductance decay method [10]. These methods are known to give reliable results but require additional optical tools which increase the experimental efforts. However, it is well-known that minority carrier lifetimes can also be extracted using combined current-voltage (I-V) and capacitance-voltage (C-V) (cIVCV) measurements, which do not require transient measurements or a special setup and can therefore be routinely used. The epitaxial carrier lifetime extracted by the cIVCV method is a parameter of great interest since typically epitaxial lifetimes are measured and specified on raw material, which is unprocessed wafers, and therefore, the results by the cIVCV method can be used to evaluate and monitor the lifetime degradation after a full semiconductor process. The basic equation describing p-n junctions' current-voltage characteristic was derived by Shockley [11]. Shockley's diode model (often referred to as diffusion model) was later complemented by Sah et al. [12] to account for the generation and recombination of carriers from the generation-recombination centers in the space charge region of a p-n junction (Sah Noyce Shockley (SNS) model). Different approximations for the generation/recombination current can be found in literature, as well as lifetime extraction methods, relying on those approximations.
In this work, we first review the main theory for p-n junction I-V and C-V relations. Second, we show how to accurately extract generation and recombination lifetimes from cIVCV measurements and benchmark our results against computer simulations based on the drift-diffusion model and measurement results. The article presented, i.e., Part II, is self-contained, however, for more details about the test structure and simulation models as well as a theoretical introduction to the lifetime concept, the reader is strongly encouraged to read Part I [13].

II. THEORY
In this work, Si n + in p-doped epitaxial layer p-n junctions are considered, which implies that electrons (holes) are considered as minority (majority) carriers. However, similar conclusions can be drawn with the opposite configuration.

A. p-n Junction Current Components
The total p-n junction current density j consists of a diffusion current density j d and a generation/recombination current density j U . Neglecting the device's series resistance, the current density j can be calculated via where j d0 , q, k B , and T are the diffusion saturation current, the elementary charge, Boltzmann's constant, and the device temperature, respectively. Note that the separation of current and capacitance in area and perimeter component [14] is not required because of the large device geometry; see device description in Part I [13]. 1) Diffusion Current: Within the depletion approximation and assuming 1-D constant doping profiles, j d0 is given by [11] where n i , N A(D) , D n( p) , and τ n( p) are the intrinsic carrier concentration, acceptor (donor) doping concentration, electron (hole) diffusivity, and electron (hole) minority carrier lifetime. F is a correction factor accounting for finite device lengths and epitaxial layer thicknesses [15].
2) Recombination Current: The space charge region generation/recombination current j U can be calculated by integrating the net recombination rate R along the depletion zone For single defect-dominated devices, R can be calculated according to the Shockley-Read-Hall (SRH) theory presented in Part I [13]. In Section III, approximations of this integral in the forward and reverse bias regimes are discussed.

B. p-n Junction Capacitance and Depletion Width
According to the simple model of a parallel-plate capacitor with area A, the voltage (V )-dependent junction capacitance C is related to the depletion width W via Therefore, a measurement of C(V ) allows for simple experimental extraction of the voltage-dependent depletion width W (V ). It should be noted that (4) neglects the diffusion capacitance, which is a valid assumption in the reverse and small forward bias regimes [16].

III. CIVCV LIFETIME EXTRACTION
In this section, selected cIVCV lifetime extraction methods are discussed on the basis of the experimental results and TCAD simulations. The test samples consist of epitaxial layer p-n junctions, where a detailed sample description and description of the simulation setup are given in Part I [13] of this work. Commonly, the C-V measurement of the p-n junction is used to extract the voltage-dependent width of the depletion region W (V ). This information can be used in conjunction with the I-V measurements to extract lifetime parameters. We briefly discuss the approximations to derive the lifetime extraction equations in the reverse and forward bias regimes. Fig. 1 shows measurement and simulation data of the I-V and C-V sweeps. As can be seen, the simulated and measured C-V curves do not vary a lot; however, a significant dependence of the I-V characteristics on the epitaxial lifetime can be observed. The impact of carrier lifetime on the reverse and forward bias regimes is further discussed in Sections III-A and III-B where explicit formulas for the current in the corresponding regimes are stated.

A. Reverse Bias Extraction
The reverse bias extraction method (RM) [17], [18] relies on a widely used approximation of the SRH recombination rate in the reverse bias regime where n 1 and p 1 are the auxiliary variables introduced in Part I [13], and τ 0 is the minority carrier recombination lifetime. In (5), the generation lifetime τ g was introduced as follows: The minus sign in (5) indicates net generation in the depletion zone. Since (5) is independent of the integration variable x in (3), j U can be straightforwardly calculated by multiplying with the depletion width W and the total current density for a larger reverse bias can then be written as follows: Consequently, the reverse current density − j is expected to rise linearly with W , and a linear fit of − j versus W allows to extract τ g and j d0 .
As can be seen in Fig. 2, (5) is indeed an accurate approximation for R; however, R is not constant along the full depletion width W but rather along some effective  width W eff . For an abrupt junction and N D > >N A , W eff can be approximated via [19], [20] We, therefore, propose to fit the current density j in (8) against W eff instead of W to correct for the otherwise present overestimation of the effective generation width. Such fits, for measurement and simulation data, are shown in Fig. 3. It should be noted that W eff and W differ by a constant offset meaning the correction will only affect the extracted value of j d0 not τ g . However, as outlined in [21], a correction according to (9) might still not be sufficient, and in [21] also a more advanced formula for W eff is presented. For the purpose of this work, however, the method is used to extract τ g only, where good extraction accuracy is achieved even without any correction. Still the correction via (9) improves the accuracy of the extracted j d0 . For a more reliable extraction of j d0 , we use a direct fit in the forward bias regime according to the two-diode model (TDM) presented in Section III-B.

B. Forward Bias Extraction
The forward bias method (FM) allows extracting τ 0 and τ g (6) with a single fit [14]. However, we propose a correction scheme to get improved and more accurate results.

1) Derivation of Lifetime Extraction Equation:
The starting point is the integral for the recombination current, (3), with the full expression for the SRH recombination rate (10) where n( p) is the electron (hole) concentration, E t is the trap's energetic position, E i is the intrinsic energy, and we used the simplification τ n0 = τ p0 = τ 0 . The exact integral in (10) cannot be solved analytically; however, the integrand shows a maximum R max [22] R max = 1 2n i τ 0 Via (11), an upper limit j U,max = W R max can be defined, whereas the exact value of I is given by the following equation: with R avg < R max and f R > 1. In (12), we introduced f R as the ratio between j U and j U,max (or equivalently R avg and R max ). Equation (10) is then given by the following equation: Defining the saturation bulk recombination current density j rb0 as follows: the following relation can be derived: Fits for measurement and simulation data according to (15) without correction, i.e., f R = 1 are shown in Fig. 4. It should be noted that the correction factor f R was not considered in [14] which might lead to an enormous overestimation of the extracted lifetime (see Table I). Therefore, the results without correction might serve as upper bounds for the extracted lifetime parameters but can deviate from the real values. However, in the next section, we outline an analytical correction scheme. Fig. 5 shows a plot of recombination rate R versus position x along a cut perpendicular to the n + -p EPI interface for the cIVCV simulation with τ EPI = 10 µs. One can see that j U,max considerably overestimates j U .
2) Analytical Correction: In the forward bias regime, the recombination R is sharply peaked at the plane x = x 0 , where p = n and the electric field is at its maximal value F 0 [23] Authorized licensed use limited to the terms of the applicable license agreement with IEEE. Restrictions apply.   In the vicinity of this plane, the carrier concentrations can be calculated via a first-order expansion in the electrostatic potential [23], [24] where the positive sign in the exponent refers to electrons, and the negative sign refers to holes. Substituting (17) into (10) yields (18) where β was introduced as follows: Since the integrand in (18) is sharply peaked and decreases toward 0 rapidly, the integration limits W 1 and W 2 can be extended to ±∞, allowing the integral, (18), to be solved analytically [25] where V t is the thermal voltage V t = k B T /q, and the function f b (β) is stated in the Appendix. Considering (12)- (20), the correction factor f R can be calculated Consistent with the results shown in Table I the correction factor does not depend on the lifetime τ 0 ; however, it depends on E t via (19). Therefore, we propose an iterative correction scheme as shown in Fig. 6. First, the linear fit, (15), without correction, i.e., f R = 1, is made to extract τ init 0 and E init t . Then the correction factor (21) is computed and the fit is . This procedure is repeated until convergence is reached, where a convergence criterion such as < δ can be defined, where E i t is the extracted trap energy of the ith iteration and δ is a small energy difference, e.g., δ = 1 meV.

C. Extraction of Diffusion Current Density
In (14), j d0 needs to be subtracted. We outline a possible extraction of this parameter in the following.
For a forward bias larger than ≈0.25 V and considering the diode area A and series resistance R s , the forward current of a p-n junction can be well described by a two-diode model (TDM) [26]  consisting of a diffusion-dominated diode with ideality factor 1 and a recombination-dominated diode with ideality factor 2. Equation (22) is implicit in j, still, it can be easily fitted to measurement/simulation data with numerical routines. Therefore, we propose to extract j d0 via minimizing residuals of measurement/simulation data to the model (22) on a log scale. This procedure gave the most reliable results compared with other methods such as the reverse method presented in Section III-A or the forward bias separation method [26].

IV. DISCUSSION
The extracted lifetimes of the forward and backward cIVCV method are shown in Table I. It summarizes data from simulations with varied epitaxial layer lifetimes and experimental results, where the test structure and the simulation setup are described in detail in Part I of this work [13]. Considering the details and precautions stated in Section III, the methods give reliable results. The RM allows for an extraction of the generation lifetime where excellent accuracy is achieved without any correction. This method is a reliable candidate for the extraction of generation lifetimes. However, the extraction of the diffusion current density j d0 with this method yields unreliable results even after a correction by (9). Just for the high lifetime case, it seems that with this correction j d0 might be estimated. In our work, we use the TDM to extract j d0 , which gave better results. It should be noted that there are many publications on the extraction of TDM parameters in the photovoltaic community [27], [28], [29], [30]; however, the extracted parameters may differ significantly between the different formulations of the model and the extraction routines [31], yielding results that might be useful for the purpose of modeling p-n junction currents but suffer from being physically unsound. Typically, the accuracy of the models is measured and compared in terms of residuals. In this work, however, we physically validate the extracted values of j d0 to the values extracted from TCAD simulations, where good agreement is achieved; see Table I. Comparing the results for the corrected and uncorrected FM in Table I, it becomes evident that a correction according to our proposed scheme or simulation results should be performed to avoid erroneous results. RM and FM measurements can be performed in a single sweep. A combination of these methods is beneficial because it allows us to compare and verify the extracted generation lifetimes τ g . Unlike the effective lifetime which is discussed in Part I of this work, the epitaxial layer lifetime discussed in this work and extracted by the FM method is not affected by the epitaxial layer thickness t EPI . To further investigate the (in-)dependency of the extracted epitaxial layer lifetimes on t EPI , simulations without a substrate layer are performed. In that case, the epitaxial layer width is enlarged to cover the full device and no epitaxial layer to substrate transition region is present. The results of these simulations are shown in the lower section in Table II. Unlike the effective lifetimes extracted by the OCVD and RR methods discussed in Part I, where a strong dependence on the epitaxial layer thickness is observed, the epitaxial layer lifetimes extracted by the cIVCV method are similar for both the structures (i.e., with and without substrate, as described above).

V. CONCLUSION
The experimental efforts to extract minority carrier lifetime by cIVCV measurements are low. However, the data analysis might be challenging. We reviewed the basic theory for the extraction of carrier lifetimes and gave lifetime extraction guidelines for the forward and reverse bias regimes. In Table II, the results for the extracted minority carrier recombination lifetimes are summarized, where also the results for structures without substrate are presented. The values for the lifetime extracted by the cIVCV method correspond to those after our proposed correction as described in Section III-B2. The table also includes the extracted lifetimes from the transient RR and OCVD methods from Part I [13]. As can be seen for the low lifetime case, i.e., when the diffusion length L EPI is smaller than the epitaxial layer (t EPI = 20 µm), all the methods give the same results corresponding to the simulation parameter. For larger lifetimes and thus diffusion lengths, the results for the transient measurements (OCVD and RR) differ significantly from the cIVCV results. As pointed out in Part I [13], OCVD and RR are methods to measure effective lifetimes, while the main signal for the cIVCV method originates from the p-n junction depletion zone and is, therefore, less affected by the epitaxial layer to substrate transition. Therefore, the forward cIVCV method is a candidate for the extraction of the epitaxial layer lifetime. This parameter is important for assessing the purity of devices or semiconductor processes. Our measurement results obtained using the OCVD, RR, and cIVCV methods exhibit similar extracted lifetimes. From this, we conclude that for our sample τ EPI is about 10 µs.

APPENDIX
The function f b (β) is the solution to the integral given in (18) with the integration limits extended to ±∞ [33]