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.


extraction technique of trap states based on transient photo-voltage measurement
Zedong Lin this article puts forward a technique for extracting the density of trap states (DoS t ) distribution based on the transient photo-voltage (tpV) measurement result. We prove that when the tpV result is linear, the DoS t 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.
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][5][6][7][8] , density of trap states (DOS T ) distribution [9][10][11][12][13] , carrier recombination [14][15][16][17][18] and so on. The DOS T distribution is a crucial factor that determines the photovoltaic performance of PSCs [19][20][21][22][23][24][25][26][27][28][29][30][31][32][33] . According to the literatures in recent years [19][20][21][22][23] , current density-voltage (J-V) hysteresis is caused by the ion migration and trap assisted carrier recombination. The DOS T distribution affects the carrier recombination [24][25][26][27] , influences the open circuit voltage [28][29][30] , and hinders the enhancement of power conversion efficiency (PCE) [31][32][33][34] . The DOS T distribution cannot be obtained by experimental measurement directly. There are only few methods for the extraction of DOS T distribution. The space charge limited current (SCLC) method uses the deconvolution to extract the DOS T distribution [35][36][37][38][39][40] , based on the measured J-V data at different temperatures. Walter et al. [41][42][43][44][45] put forward the impedance spectroscopy (IS) method to extract the DOS T distribution. They extract the DOS T distribution based on the plot of equivalent chemical capacitance versus frequency given by IS measurement [41][42][43][44][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 DOS T (E ω ) = (V bi /eW)(dC/dω)(ω/k B ) to extract the DOS T distribution [41][42][43][44][45] . Here, C is the equivalent chemical capacitance. ω is the angular frequency of the ac signal. V bi is the built-in electric voltage. W is the depletion width. k B is the Boltzmann constant. e is the elementary charge. The corresponding energy level is calculated by the formula E ω = k B Tln(ω 0 /ω), where T is the ambient temperature and ω 0 is the attempt-to-escape frequency [41][42][43][44][45] . Wang et al. 46 proposed a transient photo-voltage (TPV) method for DOS T distribution extraction. Based on the hypothesis of exponential type DOS T distribution [46][47][48][49] , multiple-trapping model 46,[50][51][52] , and the zero-temperature approximation 46,52 , they find that when the DOS T distribution is exponential type, the logarithm of carrier lifetime and the photo-voltage satisfy linear relation. They use this relation to extract the DOS T distribution based on the TPV result [46][47][48] . Because of the hypothesis of exponential type distribution [46][47][48][49] , their method is effective only when the TPV result is linear. The DOS T distribution extracted by their method is an exponential type distribution 46-48 . However, according to the TPV experiments reported in recent years [46][47][48] , the majority of TPV results are nonlinear. In these cases, the method of Wang et al. is not effective to extract the DOS T distribution. In this article, we put forward a new technique for extraction of DOS T distribution based on the TPV measurement result. We give up the hypothesis of exponential type DOS T distribution [46][47][48][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][51][52] . Our method is effective for arbitrary TPV results and can be used to extract general type DOS T distribution.
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][51][52] . According to the multiple-trapping model, the relation of carrier lifetime τ n and the free carrier lifetime τ f satisfies 46,[50][51][52] Here n = n t + n c denotes the sum of electron density in trap states n t and electron density in conduct band n c . Therefore, we have which can be rewritten as The density of electron in trap states satisfies 46,52 Here 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 N c is the density of effective states in conduction band. E c is the conduction band energy level position 53  We define a derivation factor (1) τ n = (∂n/∂n c )τ f . According to the TPV result, the carrier lifetime is a function of photo-voltage [46][47][48][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. τ n (V ph ) can be measured from TPV experiment. n c 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 E c , valence band energy level position E v , Fermi energy level position E F0 and band gap E g 35 , respectively. After getting the conductor band electron density in dark n 0 and in different intensity of illumination n c , according to the relation E Fn = E F0 + k B Tln(n c /n 0 ) 53 , we obtain the electron quasi Fermi energy level E Fn corresponding to the photo-voltage V ph in the different intensity of illumination. The free carrier lifetime τ f can be measured or can be calculated according to the equationτ f = 1/C n N t 54 . Here, N t is the electron trap concentration. C n denotes the capture coefficients for electrons. Since the left-hand side of Eq. (16) is the convolution integral of the DOS T distribution and the derivative factor, we can get the DOS T distribution by deconvolution. We use the numerical deconvolution function ([q,r] = deconv(u,v)) of MATLAB to solve the equation for DOS T distribution.
For intrinsic perovskite, the electron density in conductor band satisfies 53 is the electron density in conductor band at dark state 53  exponential type DoS t distribution. In this section, we explore the exponential type DOS T distribution. Wang et al. 46 find that for the exponential type DOS T 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 DOS T distribution satisfies [46][47][48][49]  According to Eqs. (20), (22), and (23) and from the relation of E Fn = E Fp + eV ph , we have Here, a = e/E B − e/k B T, b = lnA. Therefore, we finish the proof of this relation. Note that for the intrinsic perovskite, Eq. (24) can be written as Similarly, we can also prove that when TPV result is linear, the extracted DOS T distribution is exponential type. Details of the proof are given in the supporting information. We can use this relation to extract the DOS T 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 DOS T distribution and make a comparison. In Fig. 1a-c, we set (20) Table 2). Figure 1d-f shows the extracted DOS T distributions from Fig. 1a-c using our numerical method. As expected by the analytic method, the DOS T distribution is exponential type. In order to compare with the DOS T distribution extracted by analytic method, we use the exponential fitting (f(E) = cexp(dE)) to calculate the E B and N T (see Table 2). As shown in Table 2, the E B and N T calculated by our numerical method are consistent with the E B and N T 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 E B and N T 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 DOS T distribution [46][47][48][49] . Our numerical method is established based on the unique hypothesis of multiple-trapping model 46,[50][51][52] . Therefore, the slight deviation of E B and N T extracted by the two methods is attributed to the zero-temperature approximation and the fitting error.
non-exponential type DoS t distribution. In this section, we investigate the non-exponential type DOS T distribution. We take the TPV data in Ref. 47 as an example. As shown in Fig. 2a,b, the TPV result is nonlinear. Hence, we cannot use the analytic method to extract the DOS T 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 E B in these two subintervals, respectively 47 . They explained these two E B as two types of exponential type DOS T 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  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 E B in these differential subintervals, respectively. These E B 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 DOS T distribution. Using our method, we extract the DOS T distribution (as illustrated in Fig. 3). It can be seen that the extracted DOS T 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 DOS T distribution.
We make the exponential, double exponential and Gauss fittings for the extracted DOS T 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 DOS T 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 DOS T distribution is non-exponential type. calculation example. In this section, we give some examples of DOS T 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 DOS T distributions from Fig. 4a-c, respectively, (as shown in Fig. 4d-f).
For comparing, we plot the extracted DOS T distributions in Fig. 5. Using the formula T total = ρ t (E)dE , we calculate the total amount of trap states for three solar cells. We derived T total = 5.4431 × 10 20 cm −3 (large size), T total = 5.9096 × 10 20 cm −3 (middle size) and T total = 4.5628 × 10 20 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.

conclusion
In conclusion, this article presents a new technique for DOS T distribution extraction based on the TPV measurement result. The approach given in this paper is effective for extraction of general type DOS T distribution. We prove that when the TPV result is linear, the DOS T 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 V ph . As a result, the Fermi energy level E F0 changes to electron quasi Fermi energy level E Fn and hole quasi Fermi energy level E Fp . Thereafter, we apply an additional small light pulse to the device. With the perturbation of small light pulse, the photo-voltage increases to V ph + ΔV. The electron quasi Fermi energy level E Fn shifts to E * Fn , and the hole quasi Fermi energy level E Fp to E * Fp . After switching off the small light pulse source, due to carrier recombination, the photo-voltage decays exponentially until it reaches the bias value V ph . The electron quasi Fermi energy level comes back to E Fn from E * Fn and the hole quasi Fermi energy level comes back to E Fp from E * Fp .The carrier lifetime τ n is defined as the time taken for the photo-voltage to decay from V ph + ΔV to V ph + 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 V ph and carrier lifetime τ n , which is written as τ n = τ n (V ph ) (TPV result) [46][47][48][49] .