A quantitative model for charge carrier transport, trapping and recombination in nanocrystal-based solar cells

Improving devices incorporating solution-processed nanocrystal-based semiconductors requires a better understanding of charge transport in these complex, inorganic–organic materials. Here we perform a systematic study on PbS nanocrystal-based diodes using temperature-dependent current–voltage characterization and thermal admittance spectroscopy to develop a model for charge transport that is applicable to different nanocrystal-solids and device architectures. Our analysis confirms that charge transport occurs in states that derive from the quantum-confined electronic levels of the individual nanocrystals and is governed by diffusion-controlled trap-assisted recombination. The current is limited not by the Schottky effect, but by Fermi-level pinning because of trap states that is independent of the electrode–nanocrystal interface. Our model successfully explains the non-trivial trends in charge transport as a function of nanocrystal size and the origins of the trade-offs facing the optimization of nanocrystal-based solar cells. We use the insights from our charge transport model to formulate design guidelines for engineering higher-performance nanocrystal-based devices.

As these parameters vary, the spatial spread of the recombination rate remains the constant, indicating that the width of the recombination region is independent of β, α1, and α 2 .   We evaluated the goodness of fit of the current-voltage model (Eq. 1-4 in the main text) for different temperatures and voltage ranges by the coefficient of determination (R 2 ), which is defined as , where the total data variance (SStot) and the residual sum of squared (SSres) are given by and and .
The fitting was performed on the logarithm of the diode current, i.e. yi = log(JD,i).
In Supplementary Figure 7a, we show R 2 for the device shown in Figure 1 of the main text by taking into account only data points between 0 V and a maximum voltage Vmax. For higher voltages, we find that the model starts to deviate more from the data (i.e. SSres increases), but also the total data variance increases (SStot). The later effect dominates the R 2 value, so that R 2 improves for large voltage ranges up to 2V.
In Supplementary Figure 7b, we show R 2 for different temperatures down to 150K. While for 200K -300K the R 2 value is 0.9946±0.0050, it starts to significantly decrease below 200K. We conclude that our model gives a very good description of the data above 200K. In this section, we discuss the approximations used to simplify Eq. 5 in the main text: to Eq. 6 in the main text: Specifically, we assumed that (1) recombination occurs homogeneously in a region of width WR, To validate these assumptions, we (1)  First we must write the recombination rate as a function of device thickness such that the spatial profile of recombination can be assessed and can be numerically integrated to evaluate the recombination current. Following the notation of Ref. 2, the recombination rate in the Shockley-Read-Hall-model is given by: where the parameters are: and find: Fn Fp The total recombination current in the device is now given by the integral over the device thickness: We solve this equation numerically using the following assumptions and parameters: If not otherwise mentioned, we use the parameter values shown in Supplementary Table 1.

The width of the recombination region
Sah et al. 1 find that the largest fraction of this recombination current occurs in a narrow region with the approximate width of: where E is the electric field. Assuming a homogeneous electric field over the film thickness in our devices, we estimate the electric field from the built-in voltage (Vbi): E=(Vbi-VD)/d.
for the relevant voltage and temperature range (VD between 0.1 V and 0.4 V in steps of 0.05 V and T between 150 K to 300 K in steps of 30 K).
In Supplementary Figure 5a, we plot a histogram over all calculated values of JRG/JRG,est and find that it is in most cases close to 1. We conclude that the approximation is good for the purposes of the discussion in the main text.
In our analysis of the size dependence of J00, we assume that the effective density of states of the In Ref. 39, it is shown that ∆ scales roughly like r -3 , so that both effects can be expected to approximately compensate for each other.

Parameter estimation
Finally, we assess how well the charge transport parameters J00, E µ , and nid can be estimated by fitting Eq. 1-2 (Ref. 3) to measurement data. We use the numerically calculated JRG with randomly chosen device parameters (Eg, Ebi, ET, α1, α2) and fit Eq. 1-2 from the main text (for Rs = R -1 p = 0) to the JRG data by adjusting the parameters J00, E µ , nid.
Since J00 does not enter the simulation directly we calculate the theoretical value for J00 from the simulation parameters. Using the equations developed in the main text, , , and R,est where k is the Boltzmann constant and the temperature dependence of T 2 cancels.
data with the theoretical value based on the input parameters for the simulation (Supplementary Equation 6) and find good agreement. In other words, the parameter J00 can be estimated well from the JRG data for a broad range of parameters. The spread along the unity line (gray) results from the random variation of the built-in voltage.
In Supplementary Figure 6b, we show the ratio of the E µ estimated by fitting from the simulated data and the E µ used in the simulation of the JRG data, as a function of the ideality factor (nid). For ideality factors close to 2, we find a ratio of ~1 indicating that E µ is estimated well from the simulated data. For lower ideality factors (nid < 2) the mobility band gap (E µ ) will be underestimated from its real value, but our analysis shows that the error in the estimation of E µ is in all simulated cases limited to less than 25% of the correct value of E µ .
The trend of the free carrier mobility with NC size, showing a strong increase in mobility with decreasing NC size, that is predicted theoretically and found to be in quantitative agreement with our measurements of J00 and R -1 s00, may at first seem contradictory to previous experimental work, which reported a decreasing or non-monotonic trend with increasing band gap (decreasing NC size). [5][6][7] Here we explain the reasons behind this seeming inconsistency.
In most studies, carrier mobility is measured in field-effect transistors (FET) structures, where mobility is determined by dividing the source-drain-current by the total charge carrier density (often written as the oxide-capacitance) and accounting for the geometry. Since there is no differentiation of the fractions of charge carriers that are free and trapped, the obtained mobility is an effective mobility. 8 The fraction of free versus trapped charge carriers is exponentially dependent on the energy difference between the trap-states and the conduction band. Because this energy difference increases with larger NC band gap (as seen from our measurements of Ers and ET), a significantly smaller fraction of charge carriers are free in films composed of larger band gap NCs. In this situation, the effective mobility is measured to decrease even if the free charge carrier mobility increases for larger band gap NC-solids. Said differently, the increase in mobility is compensated by a larger activation energy. This leads to non-monotonic trends in the effective mobility as reported by Liu et al. 7 and is analogous to our measurement of the series resistance Rs0, which includes both Rs00 and Ers, shown in Fig. 4b.
It should further be noted that in other regimes of charge transport, different effects are important.
In particular, the semi-metallic regime with charge carrier densities of 1 electron per NC (~10 19 cm -3 ), which is about 2-3 orders of magnitude higher than in our solar cell devices, it was shown that charge carrier mobility is not anymore limited by NC-NC coupling, but by the charging energy of a NC. 5,6 We demonstrate the generality of our findings in the main text by applying our analysis methods and model to a heterojunction device with 1,2-ethanedithiol as a cross-linking ligand and a MSM device using 1,4-benzenedithiol as a cross-linking ligand.
The heterojunction device has the following structure: shows temperature dependent current-voltage (IVT) and thermal admittance spectroscopy (TAS) data. All charge transport parameters extracted from these measurements are summarized in Supplementary Table 2. In agreement with our findings for the ethanedithiol MSM diode in the main text, we find that trapassisted recombination can explain the diode current (nid = 1.62) and that the mobility band gap is close to the optical band gap (E µ = 0.83 Eg). Further, the series resistance follows the barrier lowering shape of Eq. 3 in the main text. Both the activation energy (Ers = 0.26 eV) and prefactor (R -1 s00 = 894) are in agreement with the data in Figure 2 in the main text. The trap energy (ET1 = 0.285eV) determined by TAS matches the series resistance (Ers = 0.26 eV).
In contrast to the MSM devices, we observe a second activation energy of the series resistance at lower temperatures (Ers,2 = ~0.6 eV). The occurrence of two activation energies can arise from the fact that the Fermi-level is pinned only in one portion of the device, which is supported by the finding that the effective distance (d) of the Poole-Frankel conduction region is shorter (60nm) than the total film (160nm). The value for J00 = 334 is also high compared to the values obtained for the MSM diode. We attribute this in part to a lower electric field, which leads to a wider recombination zone (See Supplementary Equation 6). In summary, the data from the heterojunction device can be satisfactorily explained with the model developed for MSM diodes.
For the device using the 1,4-Benzenedithiol ligand (BDT), we use the same procedure described in the main text to fabricate a MSM-device. The crosslinking solution in this case consists of 10mM BDT in anhydrous acetonitrile.
In Supplementary Figure 10, we show the basic characterization of the NCs and the current-voltage characteristics of the device under AM1.5G illumination. In Supplementary Figure 11 we show temperature dependent current-voltage (IVT) and thermal admittance spectroscopy (TAS) data.
All charge transport parameters extracted from these measurements are summarized in Supplementary Table 3.
In agreement with the main text we find that trap-assisted recombination can explain the diode current (nid = 2.00) and that the mobility band gap is close to the optical band gap (E µ = 0.76 Eg).
The prefactor J00 = 0.78, is in good agreement with the data in Figure 1f. Further, we find the series resistance to follow the barrier lowering shape of Eq. 3 in the main text. Both activation energy (Ers = 0.38 eV) and prefactor (R -1 s00 = 168) are in agreement with the data in Figure 2 in the main text.
In contrast to the MSM devices, we find that the trap energy determined by TAS (ET1 = 0.27eV) is slightly smaller than the activation energy of the series resistance (Ers = 0.38 eV). Also the ratio of the mobility band gap to the optical band gap is found to be slightly lower than for the EDT devices (E µ /Eg = 0.76).
The data from the BDT-device is in line with the main points developed in the main text.
For the convenience of the reader, we summarize all equations used for data fitting and interpretation of temperature-dependent current-voltage data. An overview of all parameters is given in Supplementary Table 4