Fundamental Efficiency Limit of Lead Iodide Perovskite Solar Cells

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Perspective Cite This: J. Phys. Chem. Lett. 2018, 9, 1703−1711

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Fundamental Efficiency Limit of Lead Iodide Perovskite Solar Cells Luis M. Pazos-Outón,* T. Patrick Xiao, and Eli Yablonovitch Department of Electrical Engineering and Computer Sciences, University of California, Berkeley, Berkeley, California 94720, United States S Supporting Information *

ABSTRACT: Lead halide materials have seen a recent surge of interest from the photovoltaics community following the observation of surprisingly high photovoltaic performance, with optoelectronic properties similar to GaAs. This begs the question: What is the limit for the efficiency of these materials? It has been known that under 1-sun illumination the efficiency limit of crystalline silicon is ∼29%, despite the Shockley−Queisser (SQ) limit for its bandgap being ∼33%: the discrepancy is due to strong Auger recombination. In this article, we show that methyl ammonium lead iodide (MAPbI3) likewise has a larger than expected Auger coefficient. Auger nonradiative recombination decreases the theoretical external luminescence efficiency to ∼95% at open-circuit conditions. The Auger penalty is much reduced at the operating point where the carrier density is less, producing an oddly high fill factor of ∼90.4%. This compensates the Auger penalty and leads to a power conversion efficiency of 30.5%, close to ideal for the MAPbI3 bandgap.

O

radiative limit assumes that all recombination processes are radiative and that every luminescent photon escapes the solar cell. We distinguish it from the Shockley-Queisser limit, which additionally assumes that the material is a hypothetical stepfunction absorber at the band edge. It is important to note that, unlike the SQ limit, the ideal radiative voltage, Voc,rad, is not a constant of the material, as it also varies with the optical design. When accounting for the properties of real materials, which can have weak but nonzero absorption below the band edge, the optical design of the device can change the effective bandgap.6,8 As an example, we can consider two distinct effects that texturing a solar cell will have on its performance: On one hand, it increases the absorptivity at long wavelengths, which produces a red-shift of the effective bandgap. This will produce an enhancement of the short-circuit current and a reduction of the voltage, Voc,rad. In effect, the device will operate as a solar cell with a slightly smaller bandgap. On the other hand, texturing favors light extraction, increasing the external luminescence efficiency. This gain in external luminescence efficiency increases the opencircuit voltage, Voc, bringing it closer to the ideal voltage, Voc,rad. Therefore, any enhancement in the external luminescence efficiency is beneficial for device performance. The reasoning behind eq 1 is the following: by the law of mass action for semiconductors, the internal quasi-Fermi level separation in a solar cell is proportional to the logarithm of the carrier density product, np. The steady-state carrier density is determined by the balance between carrier injection and carrier decay. Under open-circuit conditions, where no charge carriers

ver the past few years, the performance of lead halide perovskite solar cells has rapidly surged. Since the first demonstration of good performance with these materials,1,2 the power conversion efficiency increased from 12% to 22.1%3 in a span of just three years. Moreover, lead halides have demonstrated optoelectronic properties, including the presence of photon recycling,4 which approach those of the most efficient photovoltaic materials, like gallium arsenide. In this work, we analyze the internal carrier and photon dynamics that dominate the behavior of MAPbI3, and of similar perovskitebased compounds, to reveal the intrinsic photovoltaic limits of this class of materials. For any technology to approach the theoretical limit laid out by Shockley and Queisser,5 the device must be designed not only for the efficient collection of photogenerated carriers, but also for the efficient extraction of internal luminescence.6 The first barrier of entry for a high performance photovoltaic technology is efficient carrier extraction. Once this is achieved, light management becomes the limiting factor for device efficiency. Good light management manifests itself in an improved short-circuit current by enhancing the material’s absorptivity to incident sunlight. A yet more important consequence of good light management in a solar cell is an increase in its open-circuit voltage (and hence the operating voltage). The degree to which a solar cell’s voltage approaches ideal is directly related to its ability to extract its own internal luminescence, as derived in 1967 by Ross:7 Voc = Voc,rad +

kT ln(ηext) q

(1)

where Voc is the expected open-circuit voltage, Voc,rad is the radiative (ideal) limit of the open-circuit voltage, and ηext is the external photoluminescence quantum efficiency (PLQE). The © 2018 American Chemical Society

Received: November 17, 2017 Accepted: March 14, 2018 Published: March 14, 2018 1703

DOI: 10.1021/acs.jpclett.7b03054 J. Phys. Chem. Lett. 2018, 9, 1703−1711

The Journal of Physical Chemistry Letters

Perspective

Bext = Bint × Pesc

can be extracted by the electrodes, carriers are removed from the semiconductor through (1) radiative recombination, and subsequent escape out of the device as photons, (2) internal nonradiative recombination, or (3) parasitic absorption, after being converted to internal photons. The first is a necessary phenomenon since, as dictated by detailed balance,5 the absorptivity and the emissivity are directly linked. Because light must enter the solar cell from the front surface, the extraction of internally generated photons through the same surface cannot be blocked. This process determines the minimum possible rate of carrier loss. Therefore, the only way to maximize carrier density is by minimizing nonradiative recombination and internal photon losses. The result of minimizing these losses is the maximization of the external luminescence efficiency, which translates into an enhanced open-circuit voltage as given by eq 1. In order to determine the fundamental efficiency limit of lead halide perovskites, we need the intrinsic recombination rates, similar to prior fundamental efficiency analysis on GaAs6 and crystalline silicon.9 The only intrinsic recombination processes are radiative recombination and Auger recombination. We will call the efficiency in the presence of these two processes the “Auger limit”, in contrast to the “perfect radiator” radiative limit, which includes only radiative recombination. Shockley− Read−Hall (SRH) recombination is a nonfundamental decay channel mediated by defects, and is excluded from our intrinsic efficiency limit calculation. This is justified since the usefulness of the lead halide perovskites can be attributed to the surprisingly small SRH recombination. Disregarding all forms of nonradiative decay, the radiative limit can be derived from the detailed-balance approach used by Shockley and Queisser,5 which invokes the equality of absorptivity and emissivity at all energies E: a(E) = ε(E). Rather than a step function absorber, as assumed in the original SQ work, we use the real absorptivity spectrum of perovskite materials. When constrained by the actual absorption spectrum, the radiative limit10 provides a reference versus the real performance limit, which includes both radiative recombination and Auger nonradiative recombination. The competition between radiative and Auger decay determines the maximum achievable external luminescence efficiency, and therefore the performance limit of the material.

(2)

In the case of an angle independent absorptivity, as in textured devices, the escape probability into vacuum can be directly calculated using11 ∞

Pesc =

∫0 πaΦBBdE ∞

∫0 4πnr2αdΦBBdE

(3)

where nr is the refractive index of the active material, a is the absorptivity, d is the thickness, and α is the band-to-band absorption coefficient. For perovskites, we use the refractive index measurements from Löper et al.,12 which has a value of nr ≈ 2.6 near the band edge. The integrand in the numerator is the external luminescence flux spectrum, and the integrand in the denominator is the spectrum of the internal radiative flux. The numerator is multiplied by π, which is the solid angle of external emission out of the front surface with a Lambertian distribution, and the denominator is multiplied by 4π, due to internal isotropic emission. The n2r term accounts for the greater density of optical modes inside the material. ΦBB is the spectral density of blackbody radiation, which can be expressed under the Boltzmann approximation as ΦBB(E) =

2 2nr,env E2

h3c 2

⎛ −E ⎞ ⎟ exp⎜ ⎝ kT ⎠

(4)

where nr,env = 1 is the vacuum refractive index, h is Planck’s constant, and c is the speed of light. We extract the above bandgap absorption coefficient from the absorptivity measurements of Soufiani et al.13 An incipient exciton can be observed at the band-edge, illustrating the high quality of the sample. Below band-edge measurements were obtained from the work of Sadhanala et al.14 where photothermal deflection spectroscopy was used to accurately measure the weak, long-wavelength absorptivity. The result is shown in Figure 1. The shape of the below band edge absorption is well fit by an Urbach tail, exp(E/E0), with an Urbach energy E0 = 15 meV, as reported elsewhere.14,15 Recent results suggest that lead iodide perovskites may have both a direct and indirect bandgap,16,17 a consequence of the Rashba effect.18,19 This could influence the shape of the band-

The competition between radiative and Auger decay determines the maximum achievable external luminescence efficiency, and therefore the performance limit of the material. In semiconductors, luminescence is produced when electrons recombine with holes, so that the external photon flux is proportional to the np product with a proportionality constant Bext, which is called the external bimolecular radiative coefficient. Although the external photoluminescence can be accessed experimentally, the more intrinsic rate constant is the internal radiative coefficient Bint, which relates the carrier densities to the internal rate of radiative recombination. Bint is a material constant, and is independent of the optical design of the solar cell. The two quantities are related by Pesc, the probability of escape of an internally emitted photon:

Figure 1. Band-to-band absorption coefficient of MAPbI3. The blue circles show the above-bandgap values obtained from measurements by Soufiani et al. Red circles represent the below-bandgap values extracted from Sadhanala et al. Below-bandgap data was fitted to an Urbach tail with an Urbach energy of 15 meV, as reported elsewhere,14,15 and shifted 38 meV to match the data sets. 1704



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Figure 2. External and internal luminescence efficiencies in the Auger limit, both in (a) open-circuit and (b) at the operating point under 1-sun illumination. In open-circuit conditions, the external luminescence is limited to ∼95% for submicron thicknesses, while at the operating point this rises to ∼99%. This is due to the decrease in carrier concentration in the latter case. In both cases, an increase in thickness leads to a carrier concentration drop, reducing the effect of Auger, and increasing the internal luminescence efficiency. Despite this, the external luminescence efficiency decreases with thickness, as internal photons have a smaller chance of escape from a thicker device. The external luminescence efficiency difference between the “perfect radiator” radiative limit and the Auger limit can be as small as 5% at open-circuit and less than 1% at the operating point. The calculations assume surface texturing and a perfect rear mirror.

generation rate due to photon absorption is balanced by the total recombination rate of carriers, including radiative and Auger recombination. The internal luminescence efficiency is the probability that a given recombination event is radiative. This is described with the equation

edge absorption. Our approach relies on the empirical band-toband absorption spectrum as shown in Figure 1, so that the Rashba effect, to the extent that it is present, is already included in our efficiency analysis. We use the external radiative coefficient (Bext = 0.9 × 10−11 cm3 s−1) measured by Richter et al.20 The external escape probability is calculated using a modified version of eq 3 to account for outcoupling through the glass substrate (Pesc = 6.7%)21 in Richter’s experiment. Note that this value was corrected from the original publication. See the SI section 2 for details. With these values, we estimate an internal bimolecular recombination coefficient of Bint = 1.34 × 10−10 cm3 s−1. Knowing the internal radiative coefficient, we can estimate the intrinsic carrier density via the van Roosbroeck−Shockley relationship.22 This relationship establishes the internal quasiequilibrium between charge carriers and photons in a material, such that emission of photons must exactly match the optical generation of carriers per unit volume: Bint ni2

=

∫0

∞

4πnr2αΦBB

dE

ηint =

Bint n2 2

Bint n + CAugern3

(6)

where n is the excess carrier density and CAuger (= 1.1 × 10−28 cm6 s−1) is the Auger coefficient, obtained from the work of Richter et al.20 The external luminescence efficiency accounts for the probability of escape of the internally emitted photons, and can be written as ηext =

Bint Pescn2 Bint Pescn2 + CAugern3

(7)

The relationship between voltage and carrier concentration is directly given by the mass action law:

(5)

n2 = ni2e qV / kT

Using the radiative recombination rate obtained above, we calculate an intrinsic carrier density of ni = 2.74 × 105 cm−3, similar to the value found by Staub et al.21 We note that while the effective mass is often used to estimate the intrinsic carrier density from band structure calculations, we infer it from optical measurements directly. The work of Richter et al.20 and that of Staub et al.23 highlight the potential presence of a nonradiative bimolecular mechanism. In this work, we disregarded this mechanism, as there are no intrinsic bimolecular nonradiative mechanisms known to this date. Furthermore, there are reports showing >99% external luminescence efficiency in a single crystal of MAPbI3.24 Based on this, we assume that the measured bimolecular nonradiative recombination is a specific property of polycrystalline films. Since we assume that Auger recombination is the sole nonradiative loss mechanism, it is optimal to leave the material undoped, in which case the electron and hole carrier densities are equal, n = p, and use n as the sole variable. In steady state, the volumetric rate of generation of carriers must be balanced by the rate of carrier recombination. At open-circuit, the carrier

(8)

For photons that interact weakly with the material, a randomly textured device can greatly enhance the absorption relative to a planar device. Incident photons that would otherwise have only a single pass through the absorber are contained inside the high-index absorber by elastic scattering at the roughened surface, leading to a longer optical path length for absorption. The resulting expression for the absorptivity follows the form:25

a( E ) =

4nr2αd 4nr2αd + 1

(9)

However, it is worth noting that this approximation is only valid for weakly absorbed photons, αd ≪ 1, and results in an underestimate of the absorption at the more strongly absorbed energies above the bandgap. As previously mentioned, the principle of detailed balance establishes that, for any device, the absorptivity and the emissivity must match for all energies. The rate of external emission is determined by the overlap between the absorptivity (emissivity) spectrum of the solar cell and the spectrum of 1705



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monotonically for thicker devices as the effective bandgap becomes better matched to the solar spectrum. In a real device, the efficiency at thicknesses beyond 1 μm will likely by limited by SRH recombination. As noted in Figure 3, there is a slightly higher fill factor achieved by the solar cell when Auger recombination is present (Figure 3c). This can be explained by the fact that in decreasing the voltage from open-circuit to the maximum-power operating point, the rate of Auger recombination falls faster than that of radiative recombination due to its n3 dependence on carrier density. Therefore, the operating point voltage can be closer to the open-circuit voltage than expected in a cell in which only radiative bimolecular recombination (∼n2) is present. The operating voltage of the cell under Auger recombination is almost equal to the operating voltage in the radiative limit at all thicknesses. This is one reason why the efficiency of the cell under 1-sun illumination is largely unaffected by Auger recombination. Due to the possibility of experimental uncertainty in the measurement of the Auger coefficient, we recalculate the parameters shown in Figures S2 and S3 in the Supporting Information with an Auger coefficient that is 1 order of magnitude larger and 1 order of magnitude smaller than the value used in this work. We find that if the Auger coefficient were 10 times larger, the external luminescence efficiency at open-circuit would drop from ∼95% to ∼70% in a 500 nm thick textured device under 1-sun illumination. Nonetheless, due to the smaller role of Auger recombination at the operating point, the power conversion efficiency is affected very little by the larger Auger coefficient, only dropping from 30.45% to 30.40%. Similarly, if the Auger coefficient were 10 times smaller, the external luminescence efficiency of the device described above could be as high as 99.5% at open-circuit, with a negligible effect on the power conversion efficiency. Figure 4 illustrates the internal operation under solar illumination of a 500 nm thick device with a textured surface and a perfect mirror. At open-circuit (Figure 4a), the intensity of internal luminescence is about 10 times larger than that of the incoming sunlight. Due to the refractive index difference between the absorber and the surrounding air, most of this radiation is absorbed and re-emitted, leading to high external luminescence efficiency. At the operating voltage, Vop (Figure 4b), the carrier concentration is reduced, and therefore the relative contribution of Auger decreases. It has been discussed in several publications that, for a solar cell to reach optimal performance, an intense photon gas must build up inside the active material.27,28 Inside a medium with a refractive index of nr > 1, the density of optical modes is enhanced by a factor of n2r relative to vacuum.15 Therefore, a higher index material supports a more intense internal photon gas, seemingly putting perovskites (nr ≈ 2.6) at a disadvantage compared to semiconductors like GaAs (nr ≈ 3.5) used in conventional photovoltaics. However, this intuition can be misleading because the voltage is determined by the carrier density, rather than the internal photon concentration. The voltage measured across a photovoltaic device characterizes the ratio between the densities of photoexcited carriers and intrinsic carriers. Consider first the radiative limit of the device. At open-circuit, photoexcited carriers build up until a balance between solar photon absorption and external emission is reached. The probability of escape of an internally emitted photon is proportional to 1/n2r (as given by eq 3), since the photons see a narrower cone of escape out of a medium with

blackbody radiation. Strong band-edge optical absorption therefore implies more intense external luminescence. Texturing increases the absorptivity of the solar cell at low energies, accelerating external emission relative to nonradiative loss mechanisms, such as Auger or SRH recombination. The enhanced absorption at low energies also slightly reduces the effective bandgap of the device. It is observed that the external luminescence efficiency of a lead halide perovskite film can increase from 20% to 57% by texturing the top surface,20 as expected for any light-emitting device as discussed above. Flat devices are therefore far from ideal, and so we treat only the case of a textured solar cell with a perfectly reflecting rear mirror (See Figure 2). A detailed treatment of flat solar cells can be found in the Supporting Information. Our results show that like crystalline silicon,9 the performance of lead halides is affected by Auger recombination. However, owing to the strong optical absorption at the bandedge, this effect is much smaller in MAPbI3. Therefore, in the absence of SRH decay channels, we will show that, in spite of Auger recombination, the photovoltaic performance of lead halide solar cells comes very close to the radiative limit. Figure 2 shows the internal and external luminescence efficiency as a function of thickness, evaluated at open-circuit and at the operating point, under 1-sun illumination using the AM 1.5G solar spectrum.26 We note that this model may not capture the full physics for devices thinner than ∼100 nm.

The principle of detailed balance establishes that, for any device, the absorptivity and the emissivity must match for all energies. The internal luminescence efficiency increases with thickness, as the carrier density drops and the effect of Auger is diluted. The external luminescence efficiency, however, declines slightly with increasing thickness despite the gain in internal luminescence efficiency, owing to the reduced probability of escape of internally emitted photons. A thicker device is more likely to reabsorb the emitted photon, which can then potentially be lost by nonradiative Auger recombination. Voc also decreases with thickness, as a result of both the reduced luminescence efficiency and a decrease in the effective bandgap of the absorber. At the operating voltage, where the carrier concentration is lower, Auger recombination has a negligible penalty. In a textured submicron device, the deviation of the external luminescence efficiency from the radiative limit can be smaller than 5% at open-circuit Voc, and below 1% at the operating point Vop. When accounting also for the increase in short-circuit current (Figure 3a), the solar conversion efficiency increases with thickness (Figure 3d). Below ∼500 nm, the current increases with thickness because of improved above-bandgap absorption, while for thicker devices there is an effective decrease in bandgap that enhances the absorption of long wavelength photons. Figure 3 shows the performance parameters of the resulting photovoltaic device. Since the open-circuit voltage depends logarithmically on the external luminescence efficiency, which is limited to ∼95% in open-circuit, Voc (Figure 3f) is penalized by less than 2 mV from the radiative limit by Auger recombination. The intrinsic efficiency limit of MAPbI3 is nearly identical to the radiative limit, which is 30.5% at 500 nm and increases 1706



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Figure 3. Characteristic photovoltaic parameters of methylammonium lead halide perovskite-based devices vs thickness are shown in panels a through e. The red dotted lines represent a “perfect radiator” with 100% luminescence efficiency, and the solid lines represent the material’s performance limit in the presence of Auger recombination. The Auger limit of performance of MAPbI3-based photovoltaics is nearly identical to its radiative limit, with an open-circuit voltage loss of less than 2 mV, as shown in panel f. This minor loss is compensated by an increase in fill factor, which increases the operating voltage relative to the open-circuit voltage, reducing the loss even further to 80%) of the internally generated radiation is self-absorbed, as depicted in Figure 5. Photons that are self-absorbed require radiative recombination to rejoin the photon gas and ultimately escape, highlighting the requirement for high internal luminescence efficiencies. The presence of parasitic optical losses, however, leads to a quenching of the concentration of internal photons that can otherwise build up in the material. To maximize performance, it is therefore paramount to suppress these losses, and to use surface texturing and photon recycling to accelerate the extraction of photons out of the device before they are

a(E ) =

4nr2αd 4nr2αd + 1 + nr2(1 − R )

(10)

This also reduces the emissivity, due to the principle of detailed balance, a(E) = ε(E). Note that this equation is only reliable in the ergodic limit,25 where photons are isotropically distributed inside the semiconductor, which requires reflectivities close to 100%. Owing to the presence of optical losses, photons can now leave the device both through the front surface and through the nonideal mirror. Furthermore, the flux of photons incident on the mirror is n2r times larger than the flux of photons escaping the material through the front surface, which is limited by the cone of escape into vacuum. The probability of parasitic absorption by the mirror can be written as a function of the probability of escape as Ppar = Pesc × nr2(1 − R )

(11)

This illustrates how the penalty for imperfect reflectivity, as well as for other parasitic optical losses, increases with n2r due to 1707



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Figure 6. Dependence of photovoltaic performance on the quality of the rear mirror under 1-sun illumination. In panel a, we illustrate that a textured MAPbI3 device with a thickness of 500 nm is less dependent on mirror quality than a similar, 2 μm thick, GaAs device of comparable optical depth. This is largely due to the smaller refractive index of MAPbI3, leading to a smaller trapped internal photon density, reducing parasitic photon absorption. In panel b, we show the effect of the mirror on photovoltaic performance.

Figure 4. Internal currents per centimeter square (a) at open-circuit, and (b) at the operating point of a 500 nm thick textured solar cell with a perfect mirror at 1-sun illumination. It is worth noting that the luminescence efficiency increases at lower voltages, as the effect of Auger is reduced for lower carrier concentrations. The internal currents shown in this figure are calculated using eqs 2 and 7.

We note that the luminescence efficiency of MAPbI3 is higher than that of GaAs, despite the higher Auger recombination, for reflectivities below 99% under 1-sun illumination. Furthermore, for devices operating under lower carrier concentrations where Auger plays a smaller role, MAPbI3 will overtake GaAs even for higher reflectivitites. The external luminescence efficiency in the presence of SRH recombination can be written as ηext =

the greater intensity of internal luminescence. The luminescence efficiency is then described by Bint Pescn2 Bint (Pesc + Ppar)n2 + CAugern3

ASRHn + Bint (Pesc + Ppar)n2 + CAugern3

(13)

where ASRH is the SRH recombination constant. This is related to the SRH lifetime (assumed equal for electrons and holes) via τSRH = 1/ASRH. The presence of SRH recombination hinders both the internal and external luminescence efficiencies. In Figure 7 we show the dependence on the SRH lifetime of (a) the external luminescence efficiency, and other critical performance parameters such as the (b) open-circuit and operating voltages, (c) power conversion efficiency, and (d) fill factor. This was calculated for a 500 nm thick device with a perfect rear mirror under 1-sun illumination. In previous reports,30,31 it has been shown that a nearly ideal open-circuit voltage can be achieved for SRH lifetimes exceeding ∼10 μs, which agrees with our findings shown in Figure 7b. However, due to the extra penalty paid by the operating voltage, we show that only with lifetimes larger than ∼100 μs can a device expect to reach nearly perfect performance. The loss in open-circuit voltage is always linked to the external luminescence efficiency following eq 1. However, unless the material is doped, as is the case for optimized GaAs

Figure 5. Spectral dependence of the luminescence emission rate for a textured 500 nm thick device with a perfect rear reflector in opencircuit under 1-sun illumination. Most of the internally emitted photons are self-absorbed by the semiconductor generating electrons and holes, thus large internal quantum efficiencies are required to achieve effective photon extraction.

ηext =

Bint Pescn2

(12)

The penalty on the performance of a textured 500 nm thick photovoltaic device as a function of the rear reflectivity is shown in Figure 6. The drop in power conversion efficiency is caused by both a loss in Jsc, due to the lower absorptivity, and a loss in Voc, due to the lower luminescence efficiency. 1708



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Figure 7. Performance limit of the (a) external luminescence efficiency, (b) open-circuit and operating point voltages, (c) power conversion efficiency, and (d) fill factor vs SRH lifetime. This is calculated for a textured 500 nm thick device with a perfect rear mirror under 1-sun illumination. Beyond a lifetime of ∼100 μs, nearly perfect performance can be achieved.

solar cells, the loss in operating voltage is not directly linked to the open-circuit voltage by a simple expression.6 It is, however, strongly dependent on the underlying physical process causing the loss in luminescence efficiency. The effect on the operating voltage is related to the dependence of the nonradiative decay mechanism on carrier density. In Figure 2, we have shown that when Auger recombination (∼n3) dominates near the open-circuit point, we observe an increase in fill factor. Therefore, Auger-induced losses in luminescence efficiency will have a smaller toll on the operating voltage than on the open-circuit voltage. An imperfect mirror (∼n2) causes the open-circuit voltage and the operating voltage to drop by similar amounts, while in the presence of SRH recombination (∼n), the operating voltage falls more rapidly than the open-circuit voltage, as shown in Figure 7b. In Figure 8 we re-express the effect of SRH recombination on photovoltaic performance in terms of its effect on external luminescence efficiency. In this calculation, we again assume a textured 500 nm device, with a perfect mirror and a variable SRH lifetime. Auger recombination has been included. While eq 1 predicts that the open-circuit voltage always depends linearly on the logarithm of the external luminescence efficiency, the same does not necessarily hold for the operating voltage. For an equivalent decrease in the external luminescence efficiency, SRH recombination results in a greater degradation of the power conversion efficiency compared to mirror losses or Auger recombination. SRH recombination can be minimized with adequate trap passivation, or with improvements in light outcoupling. The former eliminates nonradiative recombination channels, while the latter enables internal photons to be extracted more rapidly, outcompeting nonradiative decay, and bringing the device closer to ideal. The fill factor rises rapidly once the external luminescence efficiency exceeds ∼30%. This emphasizes how

SRH recombination can be minimized with adequate trap passivation, or with improvements in light outcoupling. trap passivation and optical design are critical to approach the limits of power conversion efficiency. In optimized solar cells, the active material is doped to minimize SRH recombination. This, however, leads to an increase in Auger recombination. If the doping density is greater than the photogenerated carrier density, the external luminescence efficiency takes the form ηext =

Bint PescN ASRH + Bint (Pesc + Ppar)N + CAugerN 2

(14)

where N is the doping density. Note that this equation does not depend on carrier concentration and is therefore voltage independent until high level injection occurs. This equation can be directly derived from eq 13 by applying the mass action law to a doped material. Optimization of the doping density, trading off SRH recombination with Auger recombination, is common in solar cell and LED design. The relatively large Auger coefficient in MAPbI3 reduces the value of the optimal doping density. Novel types of perovskites with mixed ions are being developed, with performance exceeding that of MAPbI3.31 The approach used in this manuscript can be used to calculate the efficiency limit of those materials. Critical to this limit is the Auger coefficient of those materials relative to the strength of their band-to-band absorption. We note that while some of these materials might have an Urbach energy E0 that is smaller than that of MAPbI3, this may not necessarily represent a great advantage. Losses due to a poor Urbach tail are meaningful 1709



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sun, inducing an open-circuit voltage loss of ∼1.5 mV from its radiative limit. The increase in fill factor compensates this loss, rendering the operating voltage unaffected. The intrinsic efficiency limit of lead halide perovskite is ∼30.5%, which is nearly identical to its radiative limit.

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ASSOCIATED CONTENT

* Supporting Information S

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.jpclett.7b03054. A MATLAB program to perform calculations on the efficiency limits of a given material can be found at https://gitlab.com/ luispazosouton/PV-Efficiency-Limits. Further details related to the calculations shown in this manuscript (PDF)

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. ORCID

Luis M. Pazos-Outón: 0000-0002-3157-5208 T. Patrick Xiao: 0000-0001-9066-2961 Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS This work was supported by the Light-Material Interactions in Energy Conversion, an Energy Frontier Research Center funded by the U.S. Department of Energy, Office of Science, Office of Basic Energy Sciences, under Contract DE-AC0205CH11231, part of the EFRC at Caltech under DESC0001293. LMPO was supported by the Kavli Energy NanoScience Institute Heising-Simons Junior Fellowship of the University of California, Berkeley, and the Winton Programme for the Physics of Sustainability, from the University of Cambridge. T.P.X. was supported by the National Science Foundation Graduate Research Fellowship Program under Grant No. DGE 1106400. The authors thank Aditya Sadhanala and Arman Soufiani for providing the absorption data, and Johannes Richter for useful discussions.

Figure 8. Performance as a function of material quality, parametrized by its effect on external luminescence efficiency. The x-axis represents the external luminescence efficiency, for a variable SRH lifetime, and CAuger = 1.1 × 10−28cm6s−1. The open-circuit voltage decreases linearly with the logarithm of the external luminescence efficiency, as predicted by eq 1. However, the large drop in fill factor from external luminescence ∼95% to ∼10% results in a greatly diminished power conversion efficiency. This highlights the need for near-perfect external luminescence efficiencies to reach optimal performance. This was modeled on a textured 500 nm thick device, with a perfect rear reflecting mirror under 1-sun illumination.

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only when E0 is comparable to kT,32 and the material considered in this work already attains E0 = 15 meV. Recently reported trap-passivating methods enable long SRH lifetimes, exceeding 8 μs33 in thin films, showing an external luminescence efficiency of 35% in a flat film on glass under 1sun illumination. With such lifetimes, a textured 500 nm thick device with a perfect rear mirror could reach external luminescence efficiencies around 60%, under 1-sun illumination. This would lead to potential solar power conversion efficiencies of ∼29%. We conclude that the main limitation of current perovskite photovoltaics comes from SRH recombination. We have shown that nearly ideal device performance can be achieved with SRH lifetimes longer than 100 μs. A reduction in SRH recombination not only leads to an enhancement in opencircuit voltage, but an even larger enhancement in the operating voltage. This produces a dramatic increase in power conversion efficiency, particularly when external luminescence efficiencies exceeding 30% are achieved. We have shown that Auger recombination limits the external luminescence efficiency of MAPbI3 devices to ∼95% under 1

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Fundamental Efficiency Limit of Lead Iodide Perovskite Solar Cells

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