Light-trapping in perovskite solar cells

We numerically demonstrate enhanced light harvesting efficiency in both CH3NH3PbI3 and CH(NH2)2PbI3-based perovskite solar cells using inverted vertical-cone photonic-crystal nanostructures. For CH3NH3PbI3 perovskite solar cells, the maximum achievable photocurrent density (MAPD) reaches 25.1 mA/cm2, corresponding to 92% of the total available photocurrent in the absorption range of 300 nm to 800 nm. Our cell shows 6% absorption enhancement compared to the Lambertian limit (23.7 mA/cm2) and has a projected power conversion efficiency of 12.9%. Excellent solar absorption is numerically demonstrated over a broad angular range from 0 to 60 degree for both S- and P- polarizations. For the corresponding CH(NH2)2PbI3 based perovskite solar cell, with absorption range of 300 nm to 850 nm, we find a MAPD of 29.1 mA/cm2, corresponding to 95.4% of the total available photocurrent. The projected power conversion efficiency of the CH(NH2)2PbI3 based photonic crystal solar cell is 23.4%, well above the current world record efficiency of 20.1%.


I. INTRODUCTION
Metal halide perovskites have been intensively studied in solar cell applications since the first perovskite solar cell was reported. 1The excellent optical and electrical properties of perovskites make them among the most promising next generation solar cells.][4][5][6][7][8][9][10][11][12] There are currently two distinct perovskite family materials, CH 3 NH 3 PbI 3 and CH(NH 2 ) 2 PbI 3 , used in solar cell research.The absorption spectrum of CH 3 NH 3 PbI 3 is from 300 nm to 800 nm while CH(NH 2 ) 2 PbI 3 has a broader spectrum from 300 nm to 850 nm.Consequently, CH(NH 2 ) 2 PbI 3 enables higher power conversion efficiency than CH 3 NH 3 PbI 3 based solar cells.Most recently, a 20.1% PCE solar cell has been certified in a CH(NH 2 ) 2 PbI 3 planar device architecture. 13While previous attention has been focused on improving electronic quality of the perovskite layer, there remain opportunities to optimize optical absorption through light-trapping.For planar perovskite solar cells, the optical properties have been well studied 14 and attempts to improve the solar absorption in the perovskite active layer have focused on tandem solar cell structures. 15,168][19][20][21][22] Light absorption enhancement and PCE improvement have been demonstrated both experimentally and theoretically.However, light-trapping enhancements have not yet been fully exploited in perovskite solar cells.In this paper, we numerically simulate solar absorption in nanostructured perovskite solar cells.By direct solution of Maxwell's equations, we identify optimized light-trapping architectures consisting of inverted-vertical-cone arrays.The active layer thickness is limited to about 300 nm of perovskite material to ensure charge carrier collection.This corresponds to an "equivalent bulk thickness" of 180 nm.Absorption and reflection are calculated using the finite-difference time-domain (FDTD) method for the absorption wavelength ranges of both perovskites using a 5 nm step size. 23The lattice constant of the photonic crystal and the base radius of the inverted cone are optimized to provide the best absorption, expressed as a maximum achievable photocurrent density (MAPD).
For CH 3 NH 3 PbI 3 perovskite, the total available photocurrent from sunlight from 300 nm to 800 nm is 27.2 mA/cm 2 .We identify an inverted-vertical-cone photonic crystal with MAPD of 25.1 mA/cm 2 .This is 92% of the total available photocurrent and exceeds the Lambertian limit (23.7 mA/cm 2 ) 19 for an equivalent thickness of 180 nm.For the CH(NH 2 ) 2 PbI 3 perovskite, the total available photocurrent from sunlight from 300 nm to 850 nm is 30.5 mA/cm 2 .In this case, we identify a photonic crystal architecture that yields 29.1 mA/cm 2 , corresponding to 95.4% of the total available sunlight.

II. ELECTROMAGNETIC SIMULATION
A single unit cell of our perovskite photonic crystal solar cell is shown in Fig. 1.From bottom to top, the device consists of (i) perfect electronic conductor (PEC) lower contact, (ii) PCBM hole blocker, (iii) perovskite active region (CH 3 NH 3 PbI 3 or CH(NH 2 ) 2 PbI 3 ) and (iv) PEDOT electron blocker.The ITO top contact is a 60 nm thick conformal coating on the PEDOT layer and the remainder of the conical pore is filled with glass (SiO 2 ) for encapsulation.The lattice constant, a, of the periodic structure and the cone base radius, R, are parameters that are optimized for maximum solar absorption.For simplicity, we neglect the imaginary parts (k) of the refractive indices of PEDOT, PCBM and ITO.The real parts of the refractive indices of PEDOT and PCBM are chosen as 1.6 and 2.1 respectively. 11The real part of the refractive index of ITO is chosen as 1.5, 24 the same as glass.These numbers are approximate, but illustrate the solar absorption enhancement in perovskite photonic crystals due to light-trapping.
A full three-dimensional FDTD method is employed for all numerical calculations.Periodic boundary conditions are used in the x-, y-directions and perfectly matched layers (PMLs) are used in the z-direction.The simulation region in the x-and y-directions is chosen to be one unit cell.Light is incident on the cell from the top of the structure in Fig. 1.During the numerical simulation, the amplitudes of the reflected and transmitted waves are recorded, transformed to the frequency domain and normalized by the incident spectrum.In this way, we obtain the transmission T(ω) and reflection R(ω) coefficients.The absorption coefficient A(ω) is then determined by Knowing the fraction of incident light absorbed by the active layer and assuming all generated carriers are collected, the MAPD is given by: Here, e is the electron charge, λ is the vacuum wavelength, h is Plank's constant and c is the speed of light in vacuum.For CH 3 NH 3 PbI 3 , λ max = 800 nm, whereas, for CH(NH 2 ) 2 PbI 3 , λ max = 850 nm.I(λ) is the solar intensity of the Air Mass 1.5 (AM 1.5) solar spectrum, 25 and A(λ) is the absorption coefficient in the active layer obtained from Eqn. (1).

III. CH 3 NH 3 PbI 3 BASED SOLAR CELLS
The refractive index of CH 3 NH 3 PbI 3 26 is shown in Fig. 2. In our numerical simulation of sunlight absorption, the thicknesses of PCBM and PEDOT are both fixed at 10 nm.The total height of the active perovskite region is fixed at H = 300 nm (see Fig. 1(a)) to ensure reasonable carrier collection efficiency. 27The optimization process is performed by varying the lattice constant a from 200 nm to 1000 nm and the base radius R of the inverted cone from a/4 to 3a/4.The MAPD map is shown in Fig. 3.When a = 600 nm and R = 380 nm, the maximum MAPD of 24.0 mA/cm 2 is realized.The total equivalent bulk thickness of perovskite material in this case is 180 nm of a corresponding planar device.
The absorption and reflection spectra of the optimized device structure are shown in Fig. 4. The spectrum of a corresponding planar device is also shown for reference.For the planar reference cell, the thickness of PCBM and PEDOT are both fix at 10 nm.The thickness of ITO is likewise 60 nm.Clearly, absorption in the optimized photonic crystal is better than the planar one throughout wavelength range from 300 nm to 800 nm.We observe a large absorption enhancement in the wavelength range from 750 nm to 800 nm with an absorption peak at 781 nm, where the absorption is very low for the planar device.
Fig. 5 shows the field intensity distribution and the Poynting vector at 781 nm (absorption peak in Fig. 4) in the x-z plane bisecting the cone.The high field intensity spot near the bottom of the conical pore has a peak enhancement of 50 as indicated by the color bar.This localized resonance mode extends into the nearby perovskite with intensity enhancement in the range of 10-15.The  Poynting vector exhibits a circulation pattern around the localized hot spot, consistent with a long optical dwell time and strong absorption in the perovskite film.
In order to further reduce the reflection from the top surface, a glass dome is placed on top of each unit cell as shown in the insert of Fig. 6 (2D side view of the device structure).The base radius of the glass dome is fixed at 380 nm (the same as the base radius of the inverted vertical cone) and antireflection behavior is optimized by varying the height (indicated as h in the insert of fig. 6) of the dome from 100 nm to 500 nm.The optimized height of the glass dome is h = 300 nm.The absorption spectra of the optimized inverted cone device with and without the dome are shown in Fig. 6.With this anti-reflection coating, the absorption is enhanced for wavelengths longer than 500 nm, leading to an improved MAPD of 24.3 mA/cm 2 .For an equivalent bulk thickness of L = 180 nm of CH 3 NH 3 PbI 3 , the absorption spectrum in the "Lambertian limit" 19,28 is given by: Here α = 4πk λ is the absorption coefficient, n and k are the real and imaginary parts of the refractive index of CH 3 NH 3 PbI 3 .The Lambertian absorption benchmark is shown in Fig. 6, as a dashed black line and yields the MAPD of 23.7 mA/cm 2 .
The ability to absorb sunlight at off-normal incidence over a wide angular range for both polarizations is important for solar cell designs without an expensive solar tracking system.In Fig. 7, we present numerical results on the dependence of MAPD on incident angle for our optimized CH 3 NH 3 PbI 3 perovskite, inverted cone, solar cell (with glass dome on top) for both S-(no electric field in z direction) and P-(no magnetic field in z direction) polarizations.The MAPD dependence on incident angles of both S-and P-polarizations for the planar cell are listed as a reference.For both polarizations, the MAPD of our optimized, inverted cone, solar cell has no substantial degradation for incident angles up to 60 degrees.This is in contrast to the planar perovskite solar cell where both S-and P-polarizations show lower and gradually decreasing MAPD for incident angles from 5 to 70 degrees.
Finally, we consider the insertion of a thin layer of bulk CH 3 NH 3 PbI 3 between the inverted cone layer and the bottom PCBM layer.The thickness, l, of the extra bulk CH 3 NH 3 PbI 3 is optimized in FIG. 6. Simulated absorption spectra of the optimized CH 3 NH 3 PbI 3 based inverted cone perovskite solar cell with (red curve) and without (blue curve) antireflection glass dome (h = 300 nm) compared with the "Lambertian limit" (dashed line), for an equivalent bulk CH 3 NH 3 PbI 3 thickness of 180 nm. the range from 20 nm to 100 nm as shown in Table .I. The optimum thickness of bulk CH 3 NH 3 PbI 3 is 40 nm, providing a MAPD of 25.1 mA/cm 2 , corresponding to absorption of 92% of sunlight.
The PCE of a real-world planar cell with a 180 nm CH 3 NH 3 PbI 3 perovskite is 11.2%. 11We obtain a projected efficiency for our photonic crystal solar cell by assuming the same open circuit voltage and fill factor as in the planar cell.Our numerical simulation yields a MAPD of the planar cell of 21.8 mA/cm 2 compared to 25.1 mA/cm 2 for the optimized device structure with a = 600 nm, R = 380 nm, h = 300 nm and l = 40 nm.Accordingly, we project an efficiency of 12.9% for our CH 3 NH 3 PbI 3 based, perovskite photonic crystal solar cell.

IV. CH(NH 2 ) 2 PbI 3 BASED SOLAR CELLS
Given the broader (300 nm -850 nm) absorption spectrum of CH(NH 2 ) 2 PbI 3 , a higher MAPD and PCE are realized for this type of perovskite.Since, the refractive index of CH(NH 2 ) 2 PbI 3 , to the best of our knowledge, is not fully tabulated in the wavelength range from 300 nm to 850 nm, we infer it from the following considerations.The imaginary part of the refractive index of CH(NH 2 ) 2 PbI 3 from 500 nm to 850 nm is tabulated. 29However, the absorption spectra for both CH(NH 2 ) 2 PbI 3 and CH 3 NH 3 PbI 3 are nearly the same in wavelength from 300 nm to 500 nm. 30ccordingly, we adopt the imaginary part of the refractive index of CH 3 NH 3 PbI 3 to represent that of CH(NH 2 ) 2 PbI 3 in wavelength from 300 nm to 500 nm.Small differences in the real part of the refractive index have little influence on the absorption.Accordingly, we adopt the real part of the refractive index of CH 3 NH 3 PbI 3 in wavelength from 300 nm to 800 nm and stretch it linearly from 300 nm to 850 nm.We use this stretched real part of the refractive index to represent that of CH(NH 2 ) 2 PbI 3 .The inferred real and imaginary parts of the refractive index of CH(NH 2 ) 2 PbI 3 are shown in Fig. 8.The additional range of weak absorption from 800-850 nm leads to a different set of optimum structural parameters for the two different materials.Our light-trapping strategy is based on parallel-to-interface refraction of incident sunlight into slow-light modes within the active range. 19,20The efficient capture of this additional sunlight requires a modified structure.As in the case of CH 3 NH 3 PbI 3 , we now optimize the MAPD of CH(NH 2 ) 2 PbI 3 using our inverted cone photonic crystal architecture.As a start, the cone height is fixed at H = 300 nm, the lattice constant a is varied from 300 nm to 1000 nm and the cone base radius is varied from a/4 to 3a/4.The calculated MAPD map is shown in Fig. 9(a).The optimum structure with lattice constant a = 700 nm and radius R = 435 nm provides a MAPD of 26.9 mA/cm 2 .
For a planar cell structure, high quality CH(NH 2 ) 2 PbI 3 films with thicknesses of 500 nm can be obtained. 13Accordingly, we consider the influence of increased cone height, H, on the MAPD.We consider H = 300 nm to 500 nm in steps of 50 nm.For each cone height, the lattice constant a and the base radius R are allowed to vary and the best MAPD for each cone height is shown in Fig. 10.Second peak MAPD for cone height H = 450 nm is provided as a reference (red dot in Fig. 10).The MAPD increases with H from 300 nm to 450 nm but reaches a maximum value of 28.3 mA/cm 2 for H = 450 nm, with a = 400 nm and R = 230 nm.MAPD then decreases for H > 450 nm.An interesting outcome of our simulation is the discovery of three competing structures (see Fig. 9(b)) for 400 nm < H < 500 nm.The optimized lattice constant a ≈ 700 nm and cone base radius R ≈ 450 nm remain roughly the same for cone heights of 300 nm, 350 nm and 400 nm.However when the cone height H = 450 nm, there is a sudden switching to a second optimum structure with a= 400 nm and R = 230 nm.When H = 500 nm, the optimum architecture switches to a large lattice constant a = 900 nm and radius R = 595 nm (third peak in Fig. 9(b)), but with a slightly lower MAPD.For CH(NH 2 ) 2 PbI 3 based conical pore perovskite solar cells, the optimum structure with H= 450 nm, a = 400 nm and R = 230 nm, yields the MAPD of 28.3 mA/cm 2 .We note that the MAPD of the first optimum at a = 800 nm and radius R = 510 nm (Red dot in Fig. 10) is 28.2 mA/cm 2 which is only slight lower than the MAPD of the optimum one (28.3mA/cm 2 ).
A slight improvement on this photocurrent density is obtained by adding a glass-dome antireflection layer on top of the solar cell compared to the solar cell without a glass-dome.The cap heights are optimized for both cases of a = 400 nm, R = 230 nm and a = 800 nm, R = 510 nm for H = 450 nm as shown in Fig. 11.For a = 400 nm, R = 230 nm, the dome base radius is fixed at 230 nm (the same as the base radius of the vertical cone).The optimum dome height is found to be h = 300 nm (dome spherical radius is 238 nm).This yields the best MAPD of 29.1 mA/cm 2 .For a = 800 nm, R = 510 nm, the dome base radius is fixed at 510 nm (the same as the base radius of the vertical cone).The optimum dome height is found to be h = 420 nm (dome spherical radius is 520 nm).The optimum MAPD is 28.8 mA/cm 2 , which is only slightly lower than the best MAPD obtained here.Finally, we consider inserting an extra bulk layer of CH(NH 2 ) 2 PbI 3 between the inverted vertical cone CH(NH 2 ) 2 PbI 3 layer and the PCBM layer.This strategy improved absorption in CH 3 NH 3 PbI 3 , which otherwise absorbed 89% (24.3 mA/cm 2 ) of the total available sunlight from 300-800 nm.However, no improvement in MAPD was found by inserting a bulk layer below our optimized conical pore structure of CH(NH 2 ) 2 PbI 3 with a = 400 nm, R = 230 nm and H = 450 nm.
Our best MAPD of 29.1 mA/cm 2 for the CH(NH 2 ) 2 PbI 3 based conical pore, glass-encapsulated solar cell corresponds to absorption of 95% of all available sunlight in the 300 nm -850 nm range.On the other hand, a world-record power-conversion efficiency of 20.1% has been reported 13 with a 500 nm planar film of CH(NH 2 ) 2 PbI 3 .In order to obtain a projected power-conversion efficiency for our photonic crystal design, we first calculated the MAPD of a planar CH 3 NH 3 PbI 3 device structure (glass/ITO 80 nm)/PEDOT:PSS (15 nm)/PCDTBT (5 nm)/ Perovskite (350 nm)/PC60BM (10 nm)/Ag (100 nm)) reported in Ref. 11.The short circuit in this structure is measured to be 20 mA/cm 2 . 11For this planar architecture, our calculation yields the MAPD of 21.3 mA/cm 2 , using the model refractive indices obtained in Fig. 2 and assuming no absorption loss in the materials surrounding the perovskite active layer.We use this difference of 1.3 mA/cm 2 (6% of the total MAPD) as a guide to amount of additional losses not accounted by our MAPD calculation.Given the similar device structures reported in Ref. 11 and Ref. 13, we also assume the same ratio of short circuit current to MAPD (as calculated by our model) in the device 13 with reported short circuit current of 23.5 mA/cm 2 and 20.1% power conversion efficiency.Under these assumptions, the estimated MAPD of the world-record-setting CH(NH 2 ) 2 PbI 3 perovskite solar cell 13 is 25.0 mA/cm 2 .The projected efficiency of our CH(NH 2 ) 2 PbI 3 -based photonic-crystal solar cell is then estimated to be 29.1 25 × 20.1% = 23.4%.

V. CONCLUSIONS
We have designed vertical, inverted cone, photonic-crystal light-trapping architectures for the two different perovskite solar cells involving either CH 3 NH 3 PbI 3 or CH(NH 2 ) 2 PbI 3 .In our calculations, we have limited ourselves to perovskite film thicknesses in which the carrier collection probability (internal quantum efficiency) is close to unity.In the case of the CH 3 NH 3 PbI 3 -perovskite, the efficacy of our light-trapping architecture was shown by its ability to absorb sunlight beyond the Lambertian limit in the wavelength range of 300 nm -800 nm.For this material, our photonic crystal design enabled a MAPD of 25.1 mA/cm 2 and a projected power-conversion efficiency of 12.9%.This corresponds to absorption of 92% of all available sunlight in the relevant wavelength range.Excellent solar absorption was retained over a broad incident angular range from 0 to 60 degrees.A similar light-trapping enhancement of solar absorption in the wavelength range of 300 nm -850 nm was found in the case of the CH(NH 2 ) 2 PbI 3 -perovskite.For this material, our inverted cone photonic crystal design yields a MAPD of 29.1 mA/cm 2 using a film height of 450 nm.Several competing photonic crystal architectures appear near this film height.Interestingly, no further improvement in overall solar absorption was found by increasing the film thickness beyond 450 nm.By comparison of our calculated results with published experimental results on planar perovskite solar cells, we estimate a power-conversion efficiency of 23.4% in a corresponding, real-world, photonic crystal solar cell based on CH(NH 2 ) 2 PbI 3 .For both perovskites, we estimate a 15-17% improvement in power conversion efficiency relative to those of the best planar solar cell geometries.It is of interest to explore nano-imprinting methods for the fabrication of thin-film, inverted-cone photonic crystals.It is also of interest to study the possibility of deposition of perovskites onto thin-film Silicon photonic crystals 31 to realized very high-efficiency tandem solar cells.

FIG. 1 .
FIG. 1.(a) Three-dimensional perspective of the inverted vertical cone perovskite unit cell, (b) Two-dimensional side view of the device unit cell.

FIG. 2 .
FIG. 2. Refractive index (real part n and imaginary part k) of CH 3 NH 3 PbI 3 as reported in Ref. 26.
FIG. 3. MAPD optimization map for CH 3 NH 3 PbI 3 based perovskite photonic crystal with lattice constant a from 200 nm to 1000 nm and cone base radius R from a/4 to 3a/4.

FIG. 4 .
FIG. 4. Simulated absorption and reflection spectra of the optimized (a = 600 nm, R = 380 nm) inverted cone photonic crystal and planar CH 3 NH 3 PbI 3 solar cells.The CH 3 NH 3 PbI 3 thickness of the planar cell is 180 nm, corresponding to the "equivalent bulk thickness" of CH 3 NH 3 PbI 3 in the inverted-cone photonic-crystal device.

FIG. 7 .
FIG.7.MAPD at oblique incidence from 5 to 70 degrees for both s and p polarizations for our optimized CH 3 NH 3 PbI 3 conical pore photonic crystal (with a glass dome anti-reflection) with a = 600 nm, R = 380 nm and dome height h = 300 nm.Also shown (dashed curves) are MAPD at oblique incidence from 5 to 70 degrees for both s and p polarizations for the planar counterpart with 180 nm equivalent bulk thickness of CH 3 NH 3 PbI 3 .

FIG. 9 .
FIG. 9. MAPD optimization of CH(NH 2 ) 2 PbI 3 based perovskite photonic crystal for fixed cone height (a) H = 300 nm, (b) H = 450 nm.The lattice constant a is varied from 300 nm to 1000 nm and cone base radius R is varied from a/4 to 3a/4.

065002- 9 FIG. 10 .
FIG. 10.Optimized MAPD versus conical pore height for CH(NH 2 ) 2 PbI 3 based photonic crystal solar cell.The optimized lattice constant a and cone base radius R for each cone height are given.The second best MAPD for cone height H = 450 nm is provided for reference (red dot).

TABLE I .
MAPD of optimized device with an extra thickness (l) of bulk CH 3 NH 3 PbI 3 layer inserted between inverted cone layer and PCBM layer.FIG.8.Estimated Refractive index (real part, n and imaginary part, k) of CH(NH 2 ) 2 PbI 3 .