A Silicon-Singlet Fission Parallel Tandem Solar Cell Exceeding 100 % External Quantum Efficiency

Silicon solar cells dominate the solar cell market with record lab efficiencies reaching almost 26%. However, after 60 years of research, this efficiency saturated close to the theoretical limit for silicon, and radically new approaches are needed to further improve the efficiency. Here we present parallel-connected tandem solar cells based on down-conversion via singlet fission. This design allows raising the theoretical power conversion efficiency limit to 45% with far superior stability under changing sunlight conditions in comparison to traditional series tandems. We experimentally demonstrate a silicon/pentacene parallel tandem solar cell that exceeds 100% external quantum efficiency at the main absorption peak of pentacene, showing efficient photocurrent addition and proving this design as a realistic prospect for real-world applications.

Conventional single-junction solar cells are limited in efficiency to about 34%, mainly due to non-absorbed below-bandgap photons and the loss of energy via thermalization of high-energy electron-hole pairs. This limit is called the Shockley-Queisser limit. 1 Singlet fission is a downconversion process in organic semiconductors that spontaneously converts one high-energy spin-singlet electron-hole pair (exciton) into two spin-triplet excitons. 2 Each triplet exciton carries half the energy of the initial singlet exciton. Utilized in solar cells, this process could lift the theoretical limit of a single junction 3,4 when combined with a lower-bandgap semiconductor.
In previous work, we and others have shown successful examples which incorporated pentacene as the singlet fission sensitizer for lead chalcogenide quantum dots [5][6][7] or amorphous silicon. 8 Here we use a novel architecture, combining a conventional monocrystalline silicon solar cell with a pentacene cell connected electrically in parallel. In such a parallel-tandem architecture the efficiency of silicon photovoltaics can be enhanced with singlet fission by potentially doubling the current obtained from high-energy photons. Tandem solar cells already overcome 9 the single-junction Shockley-Queisser limit by stacking two or more solar cells with a different bandgap in series such that light passes the high-bandgap material before it reaches the lower-bandgap sub-cell(s) (see Figure 1 (A)). In this configuration, steady-state is reached when the voltages of the sub-cells add, and the currents match. A mismatch between the current generated by each sub-cell forces a shift on their corresponding operation voltages from their optimal points. For this reason a mismatch in current leads to a drop in efficiency.
The design and manufacturing for obtaining current-matching is challenging and very costly, and this match cannot be maintained as the solar spectrum changes, particularly under diffuse illumination. As a result, tandem cells are only viable in locations with direct illumination, and are restricted to niche markets, for example concentrator solar cells.
In contrast, when two solar cells are electrically connected in parallel, they operate at the same voltage and the currents add. Voltage scales only logarithmically with light intensity rather than linearly, 10 hence, as we show here, voltage matching is far easier to achieve for changing sunlight conditions as compared to current-matching, and more robust against fabrication constraints and materials mismatch. For conventional solar cells the voltage is mostly determined by the bandgap, hence a two-bandgap parallel tandem configuration could not achieve voltage-matching without complex contacting schemes combining different numbers of sub-cells. However, when the high-bandgap sub-cell is a singlet fission solar cell, voltage matching is possible in a single, two-terminal solar cell.
The singlet fission down-conversion from one high-energy exciton to two lower-energy excitons allows the current from the solar cell to double while the voltage is reduced by about a factor of two compared to a conventional cell of the same bandgap. On its own, such a singlet fission cell allows no higher conversion efficiency than a conventional cell, since the extracted power (current × voltage) remains constant for this single-bandgap cell. However, connecting a highbandgap singlet fission solar cell in parallel with a conventional solar cell of lower bandgap can form a voltage-matched two-bandgap system (see Figure 1 (B)). When the two sub-cells are optically connected in series, the light can pass both layers successively resulting in a solar cell with theoretical efficiencies greater than those for any single-junction solar cell. To find the limiting efficiency we use a detailed balance model following Shockley and Queisser 1 with modifications for singlet fission similar to Hanna and Nozik. 4 The main difference for parallel tandem solar cells compared to conventional series-tandem cells is that the generation and recombination current of both sub-cells adds for the complete tandem cell (see Supplementary Information S2 for details). In a series tandem the current of both sub-cells equilibrates and the voltages are added. Figure 1   Contrary to the series tandem cell, the parallel tandem configuration does not require current matching of the two sub-cells. The voltage needs to be matched as closely as possible, but since changes in bandgap lead to smaller changes of voltage than current (see Supplementary Information S3), high efficiencies in a parallel tandem solar cell are easier to achieve for a broader range of materials with different bandgaps, without compromising the efficiency by incomplete absorption. This difference is illustrated in Figure 1 (D) where we plot the combination of bandgaps exceeding the single-junction limit. Crucially, the efficiency limit of the parallel tandem cell is less affected by changing spectral conditions. The spectral shape can change due to the angle between the cell and the sun, atmospheric conditions, time of the day, cloud coverage etc.; such changes alter the relation between direct and diffuse sunlight. To illustrate the stability of the tandem solar cell efficiency against changes in the spectrum we calculate the limiting efficiency for a series-and a parallel tandem solar cell as well as a single-junction cell, all three optimized for standard AM1.5G illumination, when the spectral shape changes. Figure 2 (top) shows the average limiting efficiency for spectra measured near Rotterdam (Netherlands) for each day of 2014. Even though the series tandem cell has a much higher efficiency limit under AM1.5G conditions when compared to the single-junction, its average efficiency limit under the real spectra in a cloudy country clusters around the same efficiency limit as the single-junction cell. This arises because the ratio of diffuse and direct light changes during the day, and these two components of sunlight have a very different spectral shape (see inset Figure 2). As a result one of the subcells in the tandem stack receives less light than the other, resulting in a mismatch in current.
The series tandem solar cell is always limited by the lowest of the currents, making its efficiency limit vulnerable to the change in the ratio of diffuse vs. direct light (see Supporting Information S4). In contrast, parallel tandem solar cells are far more stable against those changes, and the efficiency advantage over the single-junction solar cells remains almost independent of the incoming spectrum. The power output of an ideal series tandem, parallel tandem, and singlejunction solar cell under the same spectra is shown in the bottom part of Figure 2. The difference between the series tandem and parallel tandem cells is smaller, because under clearsky conditions, when the efficiency difference is marginal, the irradiated power is largest.
However, there is still a very significant difference in the possible power output, with a parallel tandem cell potentially providing 33% more power over the year than a single-junction cell, compared to only 18% more power in case of the series tandem cell.
We note that the analysis presented here holds also for parallel connection of any voltage matched architecture, being that achieved via down-conversion, up-conversion or combinations of series-connected sub-cells arranged to achieve voltage matching from cells with different V OC 11 ; this may be an effective approach for two-terminal lead halide perovskite cells in parallel tandem with silicon. We also note that in realistic architectures, the fill factor is sub-ideal, which makes the efficiency slightly less sensitive to changes in illumination spectrum.  Table 1 shows that the limiting efficiency for the pentacene/silicon tandem (43.9%) is relatively close to the maximum efficiency and about a third above the limit for a single-junction solar cell. Other singlet fission sensitizers which have an even better suited triplet exciton energy to match the silicon bandgap and voltage, such as terrylenes, 17 and perylenediimides, 18 have not been applied in solar cells yet, and singlet fission solar cells made from tetracene 19 and its derivatives, 20 have lower theoretical efficiency in a parallel tandem cell. We build the singlet fission solar cell following work by Congreve et al. 13 The pentacene device To illustrate the potential of our technology we measure the tandem cells in a slightly modified configuration where the singlet fission sub-cell features a reflective silver back-contact. It is placed at a small off-normal angle from the incoming light, such that light passes through the pentacene layer twice, before and after being reflected at the back-contact, and then reaches the silicon solar cell (see inset Figure 4 (B)). Glass-air and ITO-air interfaces as well as parasitic absorption account for approximately 20 % of light losses in the singlet fission device.
The EQE for this configuration is shown in Figure 4  Above unity EQE would not be possible without the singlet fission carrier multiplication process, and is something that has not been achieved with a two-bandgap solar cell to date. Magnetic field dependent measurements: The devices were placed between the poles of an electromagnet (GMW Model 3470). A cw-diode laser (Thorlabs CPS635S) at 637 nm with an intensity below 10 mW/cm 2 , chopped at 467 Hz, was used as the pump. The photocurrent was measured with a lock-in amplifier (Stanford SR830). For each magnetic field B, we averaged the B-field response over many cycles. In each cycle, we ramped up the B-field linearly from 0 T over 15 s, then waited 10 s, measured the photocurrent under B-field by averaging over 10 s with 0.5 s between each data point and ramped the B-field linearly down to 0 T over 15 s. Then we waited 10 s and measured the photocurrent with no magnetic field as above.
The photocurrent response was calculated by averaging over 10 cycles. We chose the excitation intensity low to avoid degradation artifacts and made sure that the photocurrent changes by less than 1% during the full measurement.

S2 DETAILED BALANCE CALCULATION
The calculation for the maximum theoretical efficiency of our parallel tandem solar cells follows Shockley et al., 1 with modification for singlet fission similar to Hanna et al. 2 The generation current of a solar cell is determined by the overlap of the solar spectrum and the quantum yield ( ) of charge generation from the solar cell as where Γ( ) is the photon flux, the bandgap of the absorbing semiconductor, and the maximum photon energy in the solar spectrum. The maximum quantum yield of a conventional semiconductor is unity above its bandgap ( ) and zero below, while singlet fission allows for a quantum yield of two from the point where the singlet fission sensitizer absorbs ( ). So the generated current in the parallel tandem solar cell from a singlet fission and a conventional semiconductor is In the ideal situation all recombination current is radiative such that the recombination current can be written as Where is the elementary charge, is the speed of light, ℎ is Plank's constant, is Boltzmann's constant, = 300 the temperature of the device and the applied voltage. The recombination current for both cells adds in the parallel tandem architecture, leaving Then the conversion efficiency from the solar cell becomes where is the power extracted from the solar cell and the incoming power from the sun. Maximizing for voltage allows finding the maximum power converted.
For the series tandem solar cells, generation and recombination current are calculated in the same fashion, but the current matching means that the efficiency is calculated as 1 ( 1 ) = 2 ( 2 ) 1 − 1 ( 1 ) = 2 − 2 ( 2 ) = 1 + 2 Where 1 and 1 refer to the current and voltage of cell 1 respectively. The two equations are then simultaneously optimized for the product of current and voltage. 2 Table 1 of the main text shows a few examples for various possible bandgap configurations for singlejunction and Table 1 marked.
The spectral data for Figure 2 of the main text was generated from data taken at the Cabauw weather station, which is operated by the Koninklijk Nederlands Meterologisch Instituut (KNMI). The data is deposited on via the World Radiation Monitoring Center Baseline Surface Radiation Network on the Pangaea website. The data used here for the year 2014 can be found in ref. [3][4][5][6][7][8][9][10][11][12][13][14] . It contains the power received for the direct and diffuse component of the solar radiation. The direct and diffuse part of the AM1.5G spectrum were then scaled to that power and used for the detailed balance calculations as described above.
For Figure 2 (b) of the main text, the efficiency limit was calculated for all spectra of the month of June 2014 and overlaid in the figure. For Figure 2 (c) of the main text, the efficiency limit was first calculated for every spectrum measured at a particular day, and then the efficiency limit was averaged over that day. Each datapoint in Figure 2 (c) represents such an average.

S3 V OC AND J SC LIMITS COMPARISON
The J SC and V OC for a single-junction solar cell were calculated as in Shockley & Queisser. 1 Figure 2 shows both values normalized to their lowest value in the data range. The J SC decreases by a factor of 9 between 0.8 eV and 2.5 eV, while the V OC changes only by a factor of less than 4.

S4 EFFICIENCY LIMIT UNDER DIRECT AND DIFFUSE LIGHT
For the efficiency limit calculation under diffuse and direct light the direct and diffuse component of the AM1.5G spectrum as shown in Figure 2(a) of the main text were used to calculate the efficiency limit. The ratio of direct vs. diffuse light was varied such that = × + (1 − ) × . Figure 3 shows the efficiency limit as a function of that ratio.