Extraction technique of trap states based on transient photo-voltage measurement.

This article puts forward a technique for extracting the density of trap states (DOST) distribution based on the transient photo-voltage (TPV) measurement result. We prove that when the TPV result is linear, the DOST distribution is exponential type and vice versa. Compared to the approach based on the space charge limited current measurement, the method given in this paper has the advantage of requiring less calculation. The results obtained by our method provides a guidance for preparing less trap states solar cells.

Introduction

In order to improve the photo-voltaic performance of perovskite solar cells (PSCs), we need to further explore the mechanism such as carrier mobility[1-3], ion migration[4-8], density of trap states (DOST) distribution[9-13], carrier recombination[14-18] and so on. The DOST distribution is a crucial factor that determines the photovoltaic performance of PSCs[19-33]. According to the literatures in recent years[19-23], current density–voltage (J–V) hysteresis is caused by the ion migration and trap assisted carrier recombination. The DOST distribution affects the carrier recombination[24-27], influences the open circuit voltage[28-30], and hinders the enhancement of power conversion efficiency (PCE)[31-34]. The DOST distribution cannot be obtained by experimental measurement directly. There are only few methods for the extraction of DOST distribution. The space charge limited current (SCLC) method uses the deconvolution to extract the DOST distribution[35-40], based on the measured J–V data at different temperatures. Walter et al.[41-45] put forward the impedance spectroscopy (IS) method to extract the DOST distribution. They extract the DOST distribution based on the plot of equivalent chemical capacitance versus frequency given by IS measurement[41-45]. They regard the capture and de-capture of carrier by the trap states in the PSCs as charging and discharging of the equivalent chemical capacitance[41]. They put forward the formula DOST (E) = (Vbi/eW)(dC/dω)(ω/kB) to extract the DOST distribution[41-45]. Here, C is the equivalent chemical capacitance. ω is the angular frequency of the ac signal. Vbi is the built-in electric voltage. W is the depletion width. kB is the Boltzmann constant. e is the elementary charge. The corresponding energy level is calculated by the formula E = kBTln(ω0/ω), where T is the ambient temperature and ω0 is the attempt-to-escape frequency[41-45]. Wang et al.[46] proposed a transient photo-voltage (TPV) method for DOST distribution extraction. Based on the hypothesis of exponential type DOST distribution[46-49], multiple-trapping model[46,50-52], and the zero-temperature approximation[46,52], they find that when the DOST distribution is exponential type, the logarithm of carrier lifetime and the photo-voltage satisfy linear relation. They use this relation to extract the DOST distribution based on the TPV result[46-48]. Because of the hypothesis of exponential type distribution[46-49], their method is effective only when the TPV result is linear. The DOST distribution extracted by their method is an exponential type distribution[46-48].

However, according to the TPV experiments reported in recent years[46-48], the majority of TPV results are non-linear. In these cases, the method of Wang et al. is not effective to extract the DOST distribution. In this article, we put forward a new technique for extraction of DOST distribution based on the TPV measurement result. We give up the hypothesis of exponential type DOST distribution[46-49] and zero-temperature approximation[46,52]. The method given in our work is based on the single hypothesis of multiple-trapping model[46,50-52]. Our method is effective for arbitrary TPV results and can be used to extract general type DOST distribution.

Results and discussion

Establishment of theory and method

In this section, we establish the equation for the extraction of DOST distribution based on the TPV result.

Because of the trap states in the perovskite absorber layer, the behavior of trapping and de-trapping of carrier by the trap states determines the recombination rate, which can be described by the multiple-trapping model[46,50-52]. According to the multiple-trapping model, the relation of carrier lifetime τn and the free carrier lifetime τf satisfies[46,50-52]

Here n = nt + nc denotes the sum of electron density in trap states nt and electron density in conduct band nc.

Therefore, we have

which can be rewritten as

The density of electron in trap states satisfies[46,52]

Here denotes the DOST distribution. denotes the Fermi–Dirac distribution[53].

Therefore, we have

According to the Fermi–Dirac distribution[53], we have

We rewrite Eq. (6) as

The carrier density in conductor band satisfies[46,52,53]

Here Nc is the density of effective states in conduction band. Ec is the conduction band energy level position[53].

Therefore, we have

According to Eqs. (3), (7), and (9), we have

We rewrite Eq. (10) as

Substituting the Fermi–Dirac distribution into Eq. (11), we have

We rewrite Eq. (12) as

We define a derivation factor

and rewrite Eq. (13) as

According to the TPV result, the carrier lifetime is a function of photo-voltage[46-49]. Therefore, we rewrite Eq. (15) as

Equation (16) is the fundamental equation of our method. The right-hand side of Eq. (16) can be obtained from experimental measurement. τn(Vph) can be measured from TPV experiment. nc can be obtained from differential charging method[5] or from SCLC under different intensity of illumination[39]. We can use absorbance spectrum, Kelvin probe (KP), ultraviolet photoemission spectroscopy (UPS), and X-ray photoelectron spectroscopy (XPS) to get conduction band energy level position Ec, valence band energy level position Ev, Fermi energy level position EF0 and band gap Eg[35], respectively. After getting the conductor band electron density in dark n0 and in different intensity of illumination nc, according to the relation EFn = EF0 + kBTln(nc/n0)[53], we obtain the electron quasi Fermi energy level EFn corresponding to the photo-voltage Vph in the different intensity of illumination. The free carrier lifetime τf can be measured or can be calculated according to the equationτf = 1/CnNt[54]. Here, Nt is the electron trap concentration. Cn denotes the capture coefficients for electrons. Since the left-hand side of Eq. (16) is the convolution integral of the DOST distribution and the derivative factor, we can get the DOST distribution by deconvolution. We use the numerical deconvolution function ([q,r] = deconv(u,v)) of MATLAB to solve the equation for DOST distribution.

For intrinsic perovskite, the electron density in conductor band satisfies[53]

Here is the electron density in conductor band at dark state[53].

Substituting Eq. (17) into Eq. (16), we have

Equation (18) is the fundamental equation for DOST distribution extraction of intrinsic perovskite. Similarly, we obtain the DOST distribution using numerical deconvolution function in MATLAB. The parameters used for calculation are list in Table 1[27,35].

Table 1

Parameters used for DOST distribution calculation[27,35].

Sym	Description	Value
e	Elementary charge	1.60 × 10–19 C
kB	Boltzmann constant	1.38 × 10–23 JK−1
Ec	Conduction band minimum	− 4.18 eV
Eg	Band gap	1.52 eV
EF0	Fermi energy level	− 4.3 eV
T	Ambient temperature	300 K
Nc	Conduction band DoS	1 × 1024 m−3
Cn	Electron trap coefficients	1 × 10–15 m3 s−1
Nt	Electron trap concentration	1 × 1019 m−3

Exponential type DOST distribution

In this section, we explore the exponential type DOST distribution. Wang et al.[46] find that for the exponential type DOST distribution, the logarithm of carrier lifetime is proportional to the photo-voltage (TPV result). The proof of this relation is shown as follows.

The exponential type DOST distribution satisfies[46-49]

Here, EB is the characteristic energy and NT is the total density of the trapped state. Substituting formula (19) into formula (4), and taking zero-temperature approximation[46,52], we have

Making the approximation of n ≈ nT[50], we rewrite the multiple-trapping model[46,50-52] as

which is equivalent to

The electron density in conductor band satisfies[53]

According to Eqs. (20), (22), and (23) and from the relation of EFn = EFp + eVph, we have

Here .

We write Eq. (24) as linear mathematical form lnτn = aVph + b.

Here, a = e/EB − e/kBT, b = lnA.

Therefore, we finish the proof of this relation. Note that for the intrinsic perovskite, Eq. (24) can be written as

Here . We write Eq. (25) as linear form lnτn = aVph + b. Here, a = e/2 EB− e/2kBT, b = lnA.

Similarly, we can also prove that when TPV result is linear, the extracted DOST distribution is exponential type. Details of the proof are given in the supporting information. We can use this relation to extract the DOST distribution when the TPV result is linear. Below, this method is called analytic method.

We can use the analytic method to verify the validity of our numerical method. We use both our numerical method and the analytic method to extract the DOST distribution and make a comparison. In Fig. 1a–c, we set the b = − 5 and the slope parameter a as − 1.5, − 2 and − 2.5, respectively. Using the analytic method, we derive EB and NT (see Table 2). Figure 1d–f shows the extracted DOST distributions from Fig. 1a–c using our numerical method. As expected by the analytic method, the DOST distribution is exponential type. In order to compare with the DOST distribution extracted by analytic method, we use the exponential fitting (f(E) = cexp(dE)) to calculate the EB and NT (see Table 2). As shown in Table 2, the EB and NT calculated by our numerical method are consistent with the EB and NT calculated by the analytic method, indicating that the numerical algorithm to do the deconvolution in our calculation is reliable. We explain the slight deviation of EB and NT extracted by the two methods as follows. The analytic method is established based on the three hypothesis of multiple-trapping model[46,50-52], zero-temperature approximation[46,52], and exponential type DOST distribution[46-49]. Our numerical method is established based on the unique hypothesis of multiple-trapping model[46,50-52]. Therefore, the slight deviation of EB and NT extracted by the two methods is attributed to the zero-temperature approximation and the fitting error.

Figure 1

(a)–(c) Plots of carrier lifetime versus photo-voltage. We set b =  − 5 and a as − 1.5, − 2 and − 2.5, respectively. (d)–(f) DOST distributions extracted from (a)–(c), respectively. The blue dots represent the extracted DOST distributions, the red lines represent the exponential fitting of extracted DOST distributions.

Table 2

Exponential fitting coefficients, distribution coefficients calculated from Fig. 2.

Coefficients	Example 1	Example 2	Example 3
Chosen parameters
a	− 1.5	− 2.0	− 2.5
b	− 5	− 5	− 5
Exponential fitting
c (exponential fitting)	1.525 × 1034	7.536 × 1033	3.666 × 1033
d (exponential fitting)	− 2.542 × 1020	− 2.466 × 1020	− 2.388 × 1020
R-square	1	1	1
Calculation result
EB(analytic method)	28.0 meV	28.8 meV	29.7 meV
EB(our method)	24.6 meV	25.3 meV	26.1 meV
NT(analytic method)	7.3050 × 1013 cm−3	7.5158 × 1013 cm−3	7.7392 × 1013 cm−3
NT(our method)	5.9992 × 1013 cm−3	3.0560 × 1013 cm−3	1.5352 × 1013 cm−3

Figure 2

Plots of carrier lifetime versus photo-voltage given by TPV measurement[47]. The black lines represent the result of TPV measurement. The red lines represent linear fitting in the subintervals. (a) shows the linear fittings of TPV in two subintervals, respectively. (b) shows the linear fittings of TPV in three subintervals, respectively.

Non-exponential type DOST distribution

In this section, we investigate the non-exponential type DOST distribution. We take the TPV data in Ref.[47] as an example. As shown in Fig. 2a,b, the TPV result is non-linear. Hence, we cannot use the analytic method to extract the DOST distribution.

To overcome this difficulty, Wang et al.[47] made a linear fitting in the subintervals of 0.05–0.6 V and 0.6–0.91 V (see Fig. 2a). They used formula (24) to get two values of EB in these two subintervals, respectively[47]. They explained these two EB as two types of exponential type DOST distribution (They called them as deep trap type and shallow trap type)[47]. However, there are some difficulties in their method. (1) Equation (24) is derived from one exponential type DOST distribution, not the two types of DOST distribution (deep trap state type DOST distribution and shallow trap state type DOST distribution). We cannot derive Eq. (24) based on these two types of distribution. (2) There is no clear boundary for deep and shallow trap. So the definition of deep and shallow trap is ambiguous. (3) It is more accurate to take linear fittings in three subintervals, respectively (see Fig. 2b). According to the differential theory, we can divide the photo-voltage interval into infinite differential subintervals and use Eq. (24) to calculate the EB in these differential subintervals, respectively. These EB cannot be explained by the concept of deep and shallow trap.

Our method is effective for arbitrary TPV results, which can be used to extract general type DOST distribution. Using our method, we extract the DOST distribution (as illustrated in Fig. 3). It can be seen that the extracted DOST distribution is not an exponential type distribution. Compared to the method given by Wang et al., our method is more accurate and it can give the fine structure of DOST distribution.

Figure 3

DOST distribution calculated by our method given in this article. The blue dots represent the extracted DOST distribution. The red, yellow and purple lines represent the exponential, double exponential and Gauss fittings for the extracted DOST distribution, respectively.

We make the exponential, double exponential and Gauss fittings for the extracted DOST distribution, respectively (see Fig. 3). The R-square (Coefficient of determination) of these fittings are given in Table 3. It can be see that the R-square of double exponential fitting is larger than the exponential fitting and Gauss fitting, indicating that the DOST distribution is more consistent with the double exponential type distribution than with the exponential type distribution. Therefore, when the TPV result is non-linear, the extracted DOST distribution is non-exponential type.

Table 3

Different type fitting for extracted DOST distribution.

Fitting type	Exponential	Double exponential	Gauss
R-square	0.9679	0.9864	0.9331

Calculation example

In this section, we give some examples of DOST distribution extraction using our method. We take the TPV data in Ref.[47] as an example. Figure 4a–c shows the TPV results for meso-structured perovskite solar cells with large, middle and small size of perovskite grain, respectively. Using our method, we extract the DOST distributions from Fig. 4a–c, respectively, (as shown in Fig. 4d–f).

Figure 4

(a)–(c) TPV results for meso-structured perovskite solar cells with large, middle and small size of perovskite grain[47]. (d)–(f) DOST distributions extracted by our method from (a)–(c), respectively.

For comparing, we plot the extracted DOST distributions in Fig. 5. Using the formula , we calculate the total amount of trap states for three solar cells. We derived Ttotal = 5.4431 × 1020 cm−3 (large size), Ttotal = 5.9096 × 1020 cm−3 (middle size) and Ttotal = 4.5628 × 1020 cm−3 (small size). It can be seen that the solar cell with small size of perovskite grain has the least amount of trap states. This result could give a guidance for preparing perovskite solar cells with less trap states.

Figure 5

DOST distributions for meso-structured perovskite solar cells with large, middle and small size of perovskite grain extracted by our method given in this paper.

Comparison to SCLC method

SCLC measurement gives the relation of current density and voltage at different temperatures (j = j (U, T)). For SCLC method, we extract the DOST distribution adopting the equations listed as follows[35-40]

Here j denotes the current density. U denotes the voltage. Ea is the activation energy. m, B, C are the parameters for calculation. L is the thickness of perovskite absorber layer. e is the elementary charge. ε is the dielectric constant of perovskite absorber layer. Based on the SCLC measurement result, we need to calculate Ea, m, B, C and finally use Eq. (30) to extract the DOST by deconvolution[35-40]. The calculations of Ea, m, B, C are complicated. For our method, we only need to use formula (18) to extract the DOST distribution by deconvolution. Therefore, our method takes less computation.

Conclusion

In conclusion, this article presents a new technique for DOST distribution extraction based on the TPV measurement result. The approach given in this paper is effective for extraction of general type DOST distribution. We prove that when the TPV result is linear, the DOST distribution is exponential type and vice versa. Our method needs less computation than the SCLC method. The obtained results provide a guidance for preparing perovskite solar cells with less trap states.

Methods

The transient photo-voltage measurement is an effective technique for the study of carrier recombination[3,46-49]. Figure 6a shows the typical TPV measurement result. Figure 6b shows the mechanism of TPV experiment. Photo-voltaic device is held under open circuit. At the start of the TPV experiment, we use a steady-state bias light to illuminate the device until the equilibrium between generation and recombination is established. The steady-state bias light produces a bias photo-voltage Vph. As a result, the Fermi energy level EF0 changes to electron quasi Fermi energy level EFn and hole quasi Fermi energy level EFp. Thereafter, we apply an additional small light pulse to the device. With the perturbation of small light pulse, the photo-voltage increases to Vph + ΔV. The electron quasi Fermi energy level EFn shifts to , and the hole quasi Fermi energy level EFp to . After switching off the small light pulse source, due to carrier recombination, the photo-voltage decays exponentially until it reaches the bias value Vph. The electron quasi Fermi energy level comes back to EFn from and the hole quasi Fermi energy level comes back to EFp from .The carrier lifetime τn is defined as the time taken for the photo-voltage to decay from Vph + ΔV to Vph + ΔV/e. Here, e is the natural constant. By tuning the intensity of steady-state bias light I, we get the relation between the photo-voltage Vph and carrier lifetime τn, which is written as τn = τn(Vph) (TPV result)[46-49].

Figure 6

(a) Result of TPV measurement. (b) Mechanism of TPV experiment.

Supplementary Information.