Perovskite solar cells have recently seen rapid improvements in performance with certified efficiencies of above 23%. Fullerene compounds are a very popular electron-transfer material in these devices. In a previous report, it has been shown that while an ultrathin fullerene layer of just 1 nm is sufficient to achieve good device performance, removal of this layer causes a drastic decrease in performance. We provide an explanation to these observed effects by use of a numerical device model. This work provides theoretical support to the experimental understanding of the dominant role of fullerenes in perovskite solar cells.
I. INTRODUCTION
Thin-film photovoltaics based on hybrid halide perovskites have attracted great interest in recent years1–5 due to lower material and production costs than single-crystal devices as well as rapid improvements in perovskite device efficiencies to above 23%.6 A typical planar heterojunction perovskite solar cell is composed of a perovskite absorber layer sandwiched between an electron transport layer (ETL) and a hole transport layer (HTL). The perovskite layer absorbs light which results in the photogeneration of free electrons and holes. The charge carriers will diffuse from higher to lower concentration and also drift due to the built-in electric field in the device. Electrons will on average move towards the ETL and holes towards the HTL. The main role of the ETL and HTL are to transport the respective carriers to the electrodes while blocking the opposite carrier type. For example, holes are unlikely to pass into the ETL and electrons are unlikely to pass into the HTL due to energy barriers. Both radiative and non-radiative recombination of carriers results in decrease in device performance. Radiative recombination is the annihilation of an electron from the conduction band with a hole in the valence band which results in the emission of a photon. Non-radiative recombination, also known as trap-assisted recombination, occurs through a defect or trap which is an energy level in the band-gap. The trap can capture either an electron or hole (depending on the trap’s energy level) which can recombine with a hole in the valence band or an electron in the conduction band respectively.
There is increasing interest in the application of fullerene compounds to perovskite photovoltaics.7,8 Recently, Liu et. al.9 observed that an ultrathin fullerene (C60) layer can have a drastic impact on the performance of planar perovskite solar cells. Through a combination of fluorescence microscopy and impedance spectroscopy they showed that the main role of the C60 is to efficiently extract electrons from the perovskite film. They demonstrated that even an ultrathin vapor deposited 1 nm C60 layer works efficiently in the devices, while devices without C60 exhibit poor performance.
The device architecture is as follows. The cells use poly(3,4-ethylenedioxythiophene):poly(styre-nesulfonic acid) (PEDOT:PSS) for the hole transport layer and bathocuproine (BCP) for the electron transport layer. The devices have a 1 nm, 20 nm, or no C60 layer between the perovskite and BCP layers. All devices have a 80 nm Ag cathode and 100 nm indium-tin oxide (ITO) anode. Fig. 1 shows the interior layer thicknesses and electronic energy levels of the devices, with the rectangles representing the bandgap of the materials. The energy alignment allows electrons to flow from the perovskite to the cathode and holes to flow from perovskite to the anode. It has been reported that electrons travel through BCP via gap-states which are thought to be caused by diffusion of the Ag into the BCP layer.10–12 The energy levels of the gap-states as predicted by our numerical model are shown within the BCP rectangle.
Energy levels and thicknesses of the device layers with the rectangles representing the bandgap of the material. The levels were found from a combination of literature values and fitting of the numerical model (Table I). The levels within the BCP bandgap are gap-states due to diffusion of Ag into BCP.
Energy levels and thicknesses of the device layers with the rectangles representing the bandgap of the material. The levels were found from a combination of literature values and fitting of the numerical model (Table I). The levels within the BCP bandgap are gap-states due to diffusion of Ag into BCP.
In this work, we provide an explanation to the observed effects of ultrathin C60 on device performance by use of a numerical device model. The differences in electron extraction with C60 layer thickness are modeled by a reduced effective carrier mobility in the C60 layer or at the perovskite/BCP interface (for the devices without C60). Our numerical results support the conclusions of Liu et. al regarding the role of C60 as essential for efficient electron extraction. Based on fitting of the model to the experimental results, the reason for the differences in performance between the devices with and without C60 is mostly due to a much lower electron mobility at a perovskite/BCP interface than at a perovskite/C60 interface. These results help clarify the role of the C60 layer as part of the ETL in a perovskite solar cell.
II. NUMERICAL MODEL
To model these devices, we use a one dimensional (1D) drift-diffusion approach based on the model reported in Sherkar et al.13 to self-consistently solve the Poisson, continuity, and drift-diffusion equations using Gummel iterations14 together with Scharfetter-Gummel discretization.15 The Poisson equation relates the electron and hole densities (n and p respectively) to the electric potential ψ
where q is the elementary charge and ϵ is the dielectric constant. The current density for electrons and holes (Jn and Jp respectively) is related to the net free carrier generation rate U(x) by the continuity equations
The charge generation profile is shown in Fig. 2 and was determined from a transfer-matrix optical model.16
Carrier photogeneration rate profile for the device without C60 as determined from the transfer-matrix optical model using the sum over wavelengths 350-750 nm. Position is displayed relative to the beginning of the HTL (PEDOT:PSS) layer.
Carrier photogeneration rate profile for the device without C60 as determined from the transfer-matrix optical model using the sum over wavelengths 350-750 nm. Position is displayed relative to the beginning of the HTL (PEDOT:PSS) layer.
The current densities are also related to the electric potential and charge carrier densities through the drift-diffusion equations
where Vt = kT/q is the thermal voltage.
The device is discretized into a one dimensional mesh with grid size of 0.25 nm with the anode and cathode located at x = 0 and x = L respectively. The boundary conditions on the electric potential are
where Vext is the applied bias and Vbi is the built-in field which is defined by the difference between anode and cathode work functions. The boundary conditions for the carriers are
where Nc is the density of states of both the conduction and valence band, Egap is the perovskite bandgap, and ϕa is the injection barrier at the anode. Similarly, for the cathode
where ϕc is the injection barrier at the cathode.
We include both radiative and non-radiative (trap-assisted) carrier recombination. Radiative recombination is modeled by Langevin bimolecular recombination with the recombination rate given by
where kb is the bimolecular recombination constant and ni is the intrinsic carrier density. Bimolecular recombination is assumed to occur only in the perovskite layer. It is not considered in the transport layers because of the very low density of minority carriers in those layers which makes the recombination rate many orders of magnitude less than in the perovskite. Non-radiative trap assisted recombination is assumed to occur only in 5 nm thick regions inside the perovskite layer near the HTL and ETL interfaces and is described by the Shockley-Read-Hall equation
where Cn and Cp are electron and hole capture coefficients and
and
The trap energies Et are assumed to be located in the middle of the bandgap since this energy gives most effective recombination.17 As in Sherkar et. al., the traps are assumed to have neutral charge. Ions are not considered since there is no experimental evidence that ions are a key factor affecting the physics of these devices. For example, there is no hysteresis in the JV curves of devices with C60, therefore the hysteresis observed in the 0nm C60 devices are likely due to space charge effects rather than ions.9
Table I provides the complete set of parameters used for the model with references for parameters taken from literature listed.9–13,18–23 Less efficient electron transport at the perovskite/BCP interface is modeled by a decreased electron mobility value in a 0.5 nm interface region. For the 1 nm C60, less efficient electron transport due to incomplete coverage of C60 is modeled by a reduced effective mobility for C60.
Complete set of parameters used for the model. References for parameters obtained from literature are listed.
| Parameter | Value | Source |
|---|---|---|
| Anode (ITO) work function | -4.8 eV | Ref. 18 |
| HTL (PEDOT) HOMO Level | -5.1 eV | Ref. 19 |
| HTL (PEDOT) LUMO Level | -3.5 eV | Ref. 19 |
| Perovskite conduction band minimum | -3.9 eV | Ref. 13, 18, and 19 |
| Perovskite valence band maximum | -5.4 eV | Ref. 13, 18, and 19 |
| ETL (C60) HOMO level | -5.9 eV | Fit |
| ETL (C60) LUMO level | -4.2 eV | Fit |
| BCP gap-state levels | -3.93 eV, -4.32 eV | Fit/Ref. 10–12 |
| BCP HOMO | -7.0 eV | Ref. 20 |
| Cathode (Ag) work function | -4.32 eV | Ref. 20/Fit |
| Density of states (DOS) | 8.1 × 1024 m−3 | Ref. 13 |
| Perovskite relative permitivity | 24.1 | Ref. 13 |
| PEDOT:PSS relative permitivity | 3.0 | Ref. 13 |
| C60 relative permitivity | 4.25 | Ref. 21 |
| BCP relative permitivity | 4.25 | Fit |
| Carrier mobility in perovskite | 10−4 m2V−1s−1 | Ref. 22/Fit |
| Carrier mobility in PEDOT:PSS | 4.5 × 10−6 m2V−1s−1 | Ref23 |
| Carrier mobility in 20 nm C60 | 1.6 × 10−4 m2V−1s−1 | Ref. 19 |
| Hole mobility in 1 nm C60 | 1.6 × 10−4 m2V−1s−1 | Ref. 19 |
| Electron mobility in 1 nm C60 | 1.6 × 10−5 m2V−1s−1 | Fit |
| Carrier mobility in BCP bulk | 2 × 10−9 m2V−1s−1 | Fit |
| Electron mobility at perovskite/BCP interface | 5 × 10−13 m2V−1s−1 | Fit |
| Maximum charge generation rate | 1.97 × 1028 m−3s−1 | Fit |
| Electron and hole capture coefficients | 10−13 m3 s−1 | Ref. 13 |
| Bimolecular recombination constant | 6 × 10−17 m3 s−1 | Ref. 13 |
| HTL/perovskite interface trap density | 5 × 1021 m−3 | Ref. 13 |
| Perovskite/ETL interface trap density | 6 × 1020 m−3 | Ref. 13/Fit |
| PEDOT:PSS thickness | 1.5 nm | Ref. 9 |
| Perovskite thickness | 320 nm | Ref. 9 |
| BCP thickness | 7.5 nm | Ref. 9 |
| Grid spacing | 0.25 nm |
| Parameter | Value | Source |
|---|---|---|
| Anode (ITO) work function | -4.8 eV | Ref. 18 |
| HTL (PEDOT) HOMO Level | -5.1 eV | Ref. 19 |
| HTL (PEDOT) LUMO Level | -3.5 eV | Ref. 19 |
| Perovskite conduction band minimum | -3.9 eV | Ref. 13, 18, and 19 |
| Perovskite valence band maximum | -5.4 eV | Ref. 13, 18, and 19 |
| ETL (C60) HOMO level | -5.9 eV | Fit |
| ETL (C60) LUMO level | -4.2 eV | Fit |
| BCP gap-state levels | -3.93 eV, -4.32 eV | Fit/Ref. 10–12 |
| BCP HOMO | -7.0 eV | Ref. 20 |
| Cathode (Ag) work function | -4.32 eV | Ref. 20/Fit |
| Density of states (DOS) | 8.1 × 1024 m−3 | Ref. 13 |
| Perovskite relative permitivity | 24.1 | Ref. 13 |
| PEDOT:PSS relative permitivity | 3.0 | Ref. 13 |
| C60 relative permitivity | 4.25 | Ref. 21 |
| BCP relative permitivity | 4.25 | Fit |
| Carrier mobility in perovskite | 10−4 m2V−1s−1 | Ref. 22/Fit |
| Carrier mobility in PEDOT:PSS | 4.5 × 10−6 m2V−1s−1 | Ref23 |
| Carrier mobility in 20 nm C60 | 1.6 × 10−4 m2V−1s−1 | Ref. 19 |
| Hole mobility in 1 nm C60 | 1.6 × 10−4 m2V−1s−1 | Ref. 19 |
| Electron mobility in 1 nm C60 | 1.6 × 10−5 m2V−1s−1 | Fit |
| Carrier mobility in BCP bulk | 2 × 10−9 m2V−1s−1 | Fit |
| Electron mobility at perovskite/BCP interface | 5 × 10−13 m2V−1s−1 | Fit |
| Maximum charge generation rate | 1.97 × 1028 m−3s−1 | Fit |
| Electron and hole capture coefficients | 10−13 m3 s−1 | Ref. 13 |
| Bimolecular recombination constant | 6 × 10−17 m3 s−1 | Ref. 13 |
| HTL/perovskite interface trap density | 5 × 1021 m−3 | Ref. 13 |
| Perovskite/ETL interface trap density | 6 × 1020 m−3 | Ref. 13/Fit |
| PEDOT:PSS thickness | 1.5 nm | Ref. 9 |
| Perovskite thickness | 320 nm | Ref. 9 |
| BCP thickness | 7.5 nm | Ref. 9 |
| Grid spacing | 0.25 nm |
III. RESULTS AND DISCUSSION
We find a good agreement between the experimentally measured and numerically predicted current-voltage (JV) curves (Fig. 3). We see that the C60 layer is essential for good device performance and that a 1 nm C60 layer is sufficient for good performance, with only a small improvement achieved when the layer thickness is increased to 20 nm. The fit to experiment suggests that the electron mobility at a perovskite/C60 interface is significantly higher than at a perovskite/BCP interface (case of no C60). This higher mobility models a more efficient extraction of electrons from the perovskite to the ETL.
To determine the reason for the experimentally observed performance loss in the 0 nm C60 devices, the influence of electron mobility and traps at the perovskite/BCP interface was computationally studied (Fig. 4). For the mobility trials, we set all parameters as in Table I and only vary the perovskite interface mobility. For testing the effects of the interface trap density, we first raise the perovskite interface mobility to the same value as in the device with 20 nm C60 (2 × 10−9 m2/Vs) to remove the effect of lower mobility, and then vary the trap density. We find that mobility mostly influences the short-circuit current, while trap density mostly influences the open-circuit voltage and fill factor. Fig. 4a shows how a decrease in perovskite interface mobility significantly affects the short-circuit current and fill factor of the device. For the devices without a C60 layer, fitting the model to the experimental results suggests an electron mobility of 5 × 10−13 m2/Vs. An increased trap density at the interface (Fig. 4b) does not result in the drastic short-circuit current loss which is experimentally observed; therefore the dominant factor is a low interface mobility.
JV curves showing the effects of perovskite/BCP interface carrier mobility (a) with ETL trap density set to 6 × 1020 m−3 and trap density (b) on the 0 nm device with perovskite/BCP interface mobility set to 2 × 10−9 m2/Vs (the same value as for the 20 nm C60 device). All other parameters are kept the same as in Table I.
JV curves showing the effects of perovskite/BCP interface carrier mobility (a) with ETL trap density set to 6 × 1020 m−3 and trap density (b) on the 0 nm device with perovskite/BCP interface mobility set to 2 × 10−9 m2/Vs (the same value as for the 20 nm C60 device). All other parameters are kept the same as in Table I.
These results support the fluorescence microscopy experiments of Liu et al. which showed that while adding thin C60 layers to perovskite cause a significant decrease in photoluminescence (PL), placing BCP alone onto perovskite does not significantly change the PL efficiency (neat perovskite films are as bright or brighter with BCP).9 A quenched PL intensity indicates a smaller accumulation of electrons. If the improvement in performance due to C60 was due to small molecules passivating defect states, we would expect the PL quenching with BCP to be similar to that of C60. Since this is not the case, it was concluded that C60 acts to greatly improve the efficiency of electron extraction from perovskite. Similar results were reported in the literature for perovskite devices using TiO2 as the ETL; addition of fullerene compounds to the ETL enhanced electron extraction at the perovskite/ETL interface.24,25
We gain more insight into the space-charge effects in these devices by considering the charge density profiles for devices with different C60 thicknesses at zero applied bias (Fig. 5). In the 0 nm C60 device, due to the low mobility at the perovskite/BCP interface the electron extraction is poor and results in a lower extraction of electrons than holes from the device as seen from the low electron density in the BCP layer at the right side of Fig. 5. This can also be understood as an imbalance in mobilities of the majority charge carrier through their corresponding transport layer or equivalently as the addition of an extraction barrier for electrons. A charge accumulation at the perovskite/BCP interface occurs. In contrast, the devices with C60 have no electron accumulation within the perovskite layer. This supports the fluorescence microscopy measurements done on these devices which show a much stronger photoluminescence, indicating a larger electron accumulation for the devices without C60 than with C60.9 In the devices with C60, due to the efficient electron extraction from the perovskite, there is a high electron density in the C60 together with a very low hole density and an injection barrier which prevents electrons from moving back into the perovskite. The high electron transfer efficiency also reduces the loss of free carriers due to carrier recombination within the perovskite layer. This means that more electrons can be extracted to the BCP than in the 0 nm C60 device, resulting in a higher current.
Charge carrier densities for the simulated devices with 0 nm, 1 nm, and 20 nm C60 under illumination at zero applied bias. The perovskite/C60 and C60/BCP interfaces are marked by dotted black lines. The parameters used are listed in Table I.
Charge carrier densities for the simulated devices with 0 nm, 1 nm, and 20 nm C60 under illumination at zero applied bias. The perovskite/C60 and C60/BCP interfaces are marked by dotted black lines. The parameters used are listed in Table I.
A possible reason for the reduction of mobility at the perovskite/BCP interface could be the formation of a back-to-back Schottky barrier.26 The charge transport through this barrier could be a tunneling process which is exponentially related to the height of the barrier. However, using an energetic barrier at the perovskite/BCP interface instead of a low interface effective mobility does not reproduce the experimental results (see supplementary material S1). A more likely reason for the less efficient transport could be related to a difference between the two interfaces at the molecular level causing the perovskite/C60 interface to have a better electron transport pathway.
Next we offer an explanation for the origin of the S-shaped JV curve observed in the 0 nm C60 device (Fig. 3). Several groups have proposed explanations for the appearance of S-shaped JV curves in solar cells. They have been attributed to various factors including interface dipoles,27 charge accumulation,20,28 injection or extraction barriers,29 imbalanced mobilities,30 and poor extraction of electrons near the cathode due to a mobility drop-off.31 In the device studied here the origin is likely a combination of several of the above, namely an injection barrier, mobility drop-off (equivalently imbalanced mobilities), charge accumulation, and interfacial dipole.
Fig. 6 shows the 0 nm C60 device under 0.7 V and 0.9 V applied bias which correspond respectively to points before and after the onset of forward current in the device. At 0.7 V, we see that there is no longer a dip in electron density within the BCP which indicates that electrons are being injected from the cathode into BCP. However, the electrons are not being injected into the perovskite layer as seen by the steep difference in electron density on opposite sides of the perovskite/BCP interface. The electron accumulation in BCP is caused by a combination of the 0.42 eV injection barrier for electrons (see Fig. 1) and the low electron mobility at the perovskite/BCP interface which prevents efficient injection into the perovskite layer. The injection barrier and low interface electron mobility prevent an injection current from flowing. As we further increase the applied voltage from 0.7 V, there is an even larger accumulation of electrons within the BCP near the perovskite/BCP interface and now also an accumulation of holes on the other side of the interface, forming a dipole. The dipole effectively decreases the injection barrier for electrons, however as long as there still remains an injection barrier, no injection current will flow. Therefore, the net current in the device is near zero until the applied voltage is sufficient to overcome the injection barrier (i.e. about 0.9 V). Notice that the width of the flatter region in the JV curve increases with decreasing interface mobility (Fig. 4a) since a lower interface mobility requires a stronger net field (and thus a higher applied voltage) to establish an injection current across the interface.
Charge carrier densities for device with 0 nm C60 under illumination at 0.7 V (a) and 0.9 V (b) applied bias. The parameters used are listed in Table I.
Charge carrier densities for device with 0 nm C60 under illumination at 0.7 V (a) and 0.9 V (b) applied bias. The parameters used are listed in Table I.
IV. CONCLUSION
A 1D drift-diffusion model for planar perovskite solar cells has been used to describe recent experimental results for inverted planar structure perovskite devices which use fullerene layers for electron transport. By systematically studying the effects of interface mobility and trap density on the JV curves, we find that poor electron extraction at the perovskite interface modeled by a decreased carrier mobility is the main cause of the severe decrease in device performance when there is no C60 layer. We demonstrate that the decrease in performance is not due to an increased trap density at the interface. We have also analyzed and provided an explanation for the S-shaped JV curve of the 0 nm C60 device. The main origin of the S-shape is likely due to the non-equal mobilities in the HTL and ETL which results in charge accumulation and the formation of an interface dipole. Our results support the conclusion that very little C60 is needed to enhance carrier extraction. Ultimately, this work helps clarify the role of fullerenes in perovskite solar cells.
SUPPLEMENTARY MATERIAL
See supplementary material for results showing use of an energetic barrier at the perovskite/BCP interface and the effects of varying the HOMO/LUMO levels of the HTL/ETL in the 0 nm C60 device.
