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The Low Density of Conduction and Valence Band States Contribute to the High Open Voltage in Perovskite Solar Cells Yecheng Zhou, and Guankui Long J. Phys. Chem. C, Just Accepted Manuscript • DOI: 10.1021/acs.jpcc.6b10914 • Publication Date (Web): 04 Jan 2017 Downloaded from http://pubs.acs.org on January 4, 2017

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The Journal of Physical Chemistry

The Low Density of Conduction and Valence Band States Contribute to the High Open Voltage in Perovskite Solar Cells Yecheng Zhou† and Guankui Long∗,‡ †School of Chemistry, The University of Melbourne, Parkville, VIC 3010, Australia ‡School of Materials Science and Engineering, Nankai University, Tianjin 300071, China E-mail: [email protected]

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Abstract Hybrid perovskites are widely used for high performance solar cells. Large diffusion lengths and long charge carrier lifetimes are considered as two main factors for their high performance. Here, we argue that not only large diffusion lengths and long carrier lifetimes, but also the low densities of the conduction and valence band states (Nc , Nv ) contribute to highperformance perovskite solar cells. We estimated Nc and Nv of silicon, CdTe, and typical perovskites with two different methods. It was found that the Nc and Nv of perovskites and CdTe are much lower than that of silicon. Using numerical models, we found that the solar cell of a material with same characteristics but lower Nc and Nv can realize a higher open-circuit voltage (Voc ) and higher power conversion efficiency (PCE). We put forward and proved that the low Nc and Nv in hybrid perovskite is one of the factors for its high performance. This provides a new guideline for finding and developing new photovoltaic candidate materials.

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Introduction Open-circuit voltage (Voc ) of a solar cell is always lower than the absorber band gap. The difference between them is called open voltage deficit. A large Voc deficit means large potential energy lost and leads to a lower performance. The effort to reduce the open voltage deficit never stop. Most of this kind of work is focusing on fabrication technics and methods, 1–3 few of them focus on the influence from the intrinsic properties of perovskites. Here, we discuss the open voltage deficit from the aspect of material itself and propose that we can reduce the open voltage deficit by using materials with low DOS and small effective mass. Perovskite solar cells have realized Power Conversion Efficiencies (PCEs) up to 22 %. 4–9 Excellent electron and hole diffusion ability 10,11 and a very wide absorption wavelength range with high IPCE 12–14 are reckoned as two main factors for perovskite solar cell’s high performance. Both theoreticians 15–19 and experimentalists 10,11,20,21 are trying to find more fundamental reasons. They attribute the high performance of perovskite solar cell to its proper band gap and large static dielectric constant. 16,19,22,23 Others believe that polarization of perovskite benefits to its high PCEs through grain boundary 24–27 or screening effect. 23 If we examine it carefully, it will be found that the reason of excellent electronic properties found by both experimental and theoretical works only benefits to the high output current. Whereas the maximum Jsc (21 mA/cm2 ) of perovskite solar cells 8 is much lower than the Jsc (30 mA/cm2 ) of CdTe solar cells, 28 which has similar band gaps. Therefore, there must be other reasons. The other most possible reason is the high Voc , which hasn’t been fully discussed for perovskite solar cells. Before our investigation, we collected experimental band gaps and Voc related data of different type solar cells. For amorphous silicon, its band gap ranges from 1.55 to 2.10 eV, 29 whereas its highest Voc is only about 0.98V. 30 Its voltage deficit is 570 mV. For multi-crystalline silicon solar cells, the highest Voc is 660 mV, which is 460 mV lower than its band gap at 1.12 eV. 31 The maximum Voc of CdTe solar cells is 1.10 V, 32–34 which is 400 mV lower than its band gap of 1.5 eV at 300K. 35 However, the high Voc in CdTe 3



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solar cell is obtained by scarifying its PCE. For CdTe solar cells with Voc larger than 1.0 V, the maximum PCE is 13.6%. 34 For the best performance CdTe solar cell the voltage deficit is 630 meV. 31 For a perovskite solar cell, its Voc reaches 1.14 V corresponding to the band gap of 1.57 V. 8 The voltage deficit is 430 mV. It is obvious that the PCE is higher when its voltage deficit is lower. Therefore, we are going to discuss some factors related to Voc and check whether there is a rule governing solar cell’s Voc and performance. Open-circuit voltage deficits were discussed in organic solar cells, 36–39 quantum dot solar cells, 2 a-Si:H/c-Si heterojunction solar cells. 40,41 In quantum dot solar cells, it was found that sub-bandgap states are the most likely origin of the high Voc deficit. 2 For bulk heterojunction solar cells, theoretical works showed that the Voc depends on (1) the donor-acceptor energy gap; (2) charge-carrier recombination rates; (3) illumination intensity; (4) the contact work functions (if not in the pinning regime); and (5) the amount of energetic disorder. 40,41 In 2013, Edri et al. showed that perovskite solar cells can realize high Voc up to 1.3 V. While, this high Voc is based on its large band gap of 2.3 eV. Its open-circuit voltage deficit is as large as 1.0 V. Shao’s experiment showed that reducing the energy disorder in the fullerene electron transport layer through a simple solvent annealing process increases the Voc from 1.04 V to 1.13 V. 42 It was also found that the tuning of the work function can improve Voc in the devices. 43 Chen et al. demonstrated that the limiting of Voc of CH3 NH3 PbBr3 perovskite solar cells is due to the interface loss induced by the charge extraction layer rather than by bulk dominated recombination losses. 44 In theory, Yang et al. illustrated that the reduced bimolecular recombination is the origin of high Voc . 45 With numerical simulations, we also showed the inert interface defects, long charge carrier lifetime and strong incident light intensity increase Voc . 46 Kirchartz et al. showed that photon recycling enhances the Voc in the range of 10-100 mV. 47 The upper limit of Voc was estimated to be 1.24-1.32 V based on different recombination mechanisms. 48 For organic solar cells, Garcia-Belmonte et al. found that small tails of DOS distribution will lead to high Voc . 36 Sulas et al. also showed that the increased density of charge-transfer

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states causes Voc loss. 37 The same conclusion was observed by Collins et al.. 39 So the question is whether the DOSs in perovskite contribute to its solar cell high Voc ? Especially, there are some works reported that perovskites exhibit low DOSs at the edges of the valence and conduction band. 49,50 In this paper, we are going to show that besides the defects, charge carrier lifetime and incident light intensity, there are other parameters that affects Voc . They are Nc and Nv . With two different methods based on ab initio calculations, we calculated the effective conduction band and valence band DOSs (Nc and Nv ) of silicon, CdTe and typical hybrid perovskites, such as CH3 NH3 PbI3 51,52 and HC(NH2 )2 PbI3 , 53–57 which have been widely used in solar cells. It was found that Nc and Nv of hybrid perovskites and CdTe are several times lower than that of silicon. The low Nc and Nv are in good agreement with the low DOS observed by Endres et al. 49,50 By implementing these Nc and Nv into the numerical model we developed for perovskite solar cell, 46 we found that the low Nc and Nv in perovskite materials could contribute 100 mV higher Voc compared to the ‘perovskite solar cell’ with silicon’s Nc and Nv .

Methods DFT calculation details In this paper, we calculated Nc and Nv of typical perovskites, silicon and CdTe in order to find out DOS effect on their power conversion performance. Perovskite structures came from Ref. 58. All of the calculations are performed by VASP. 59,60 Silicon band structure based on GGA-PS functional, 61 GW method 62,63 and HSE functional 64,65 are carried out and compared in order to find the optimum method. It is found that the GW method and HSE functional do give correct band gaps of 1.10 eV and 1.14 eV comparing to 1.12 eV of experiments. Whereas, their effective masses are very similar to the masses from GGA-PS functional. Effective mass calculations in MAPbI3 have been carried out with HSE functional 5



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and GW methods. 18,66 The harmonic mean effective electron and hole masses of HSE+SOC calculations are 0.18 m0 and 0.22 m0 respectively. 66 Their average effective masses are 0.19 m0 and 0.25 m0 for the method of SOC+GW. 18 Both of these effective masses close to 0.1920.199 m0 23 and 0.21 m0 (Shown in Table S6) in our GGA-PS calculations. The effective mass error between our GGA-PS method and these two highly accurate methods is smaller than the error between these two accurate methods themselves. This confirms that GW method and HSE functional do give correct band gaps while their effective masses are the same with GGA-PS functional. For the sake of computational time limitation and little accuracy sacrifice, we use GGA-PS functional in this work. The effective masses are the quadratic coefficients of the fitting quadratic function with five points near the high-symmetry k-point. Fittings are performed by R code. 67 Detailed processes and methods are described in SI.

Nc and Nv calculation from DOS(E) When the charge concentration is low, the interaction between charges is weak and can be neglected. We assume there is no interaction between charge carriers, the charge carrier density follows the Maxwell-Boltzmann distribution. It is expressed by: f (E) = f0 exp(−

E−Ef ), kT

where, E is the energy level of charges, Ef is the Fermi energy level, and f0 is the charge density at Fermi level. Thus, the electron density in the conduction band is given by

n= =

Z

∞

c ZE∞

DOS(E) × f (E)dE = DOS(E − Ec )e

E−Ec −kT

Z

∞

DOS(E) × e

E−Ef n −kT

dE/Vcell

Ec

dE/Vcell × e

Ec −Ef n −kT

= Nc × e

Ec −Ef n −kT

(1)

Ec

where, Ec and Ef n are the lowest energy of the conduction band and quasi Fermi level, Vcell is the volume of the unit cell, T is the temperature and k is the Boltzmann constant, DOS(E)

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is shown in Figure 1. Nc is calculated by:

Nc =

Z

∞ Ec

DOS(E − Ec ) E−E c e −kT dE Vcell

(2)

The integration of DOS(E) is very sensitive to calculation methods, in this work we used tetrahedron method with Blochl corrections (ISMEAR=-5) as recommended by VASP for accurate electronic DOS calculation.

Nc and Nv calculation from effective mass Theoretically, the density-of-states function is given by 1 3 DOS(E) = 2 2π

Z

dSk , |∇k E|

(3)

where, 2 is for the spin, dSk = 4πk 2 for the integral of the energy surface. For free electrons, its energy is calculated by E =

¯ 2 k2 h . 2m0

Thus ∇k E =

¯2k h . m0

Substituting them into Equation 3,

we obtain

DOS(E) =

4π 1 4πk 2 (2m0 )3/2 E 1/2 . = 2 3 4π 3 h h ¯ k/m0

(4)

Equation 4 is based on free electrons with parabolic energy band. This equation can also be used for semiconductors with parabolic energy bands just by replacing the free electron mass with the effective mass of electron or hole. 68 Choosing the conduction band edge as the reference level, Equation 4 can be expressed by

DOS(E − Ec ) =

4π (2m∗ )3/2 (E − Ec )1/2 . h3

7


(5)


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Finally, the electron density can be calculated by 68,69 ∞ 1 f (E)DOS(E − Ec )dE = DOS(E − Ec )dE (Ef −E)/kB T ] Ec [1 + e Ec Z ∞ (E − Ec )1/2 dE 4π ∗ 3/2 = 3 (2m kB T ) (Ef −E)/kB T ] h Ec [1 + e

n=

Z

∞

Z

(6)

= Nc F1/2 (η), where T m∗c 3/2 ) 300m0 R∞ is the effective density of the conduction band states; F1/2 (η) = √2π Ec Nc = 2(2πm∗ kB T /h2 )3/2 = 2.5409 × 1019 (

(7) (E−Ec )1/2 dE [1+e

(Ef −E)/kB T

]

is

the Fermi integral.

For the non-spherical constant-energy surface, the effective mass is replaced by the reduced effective mass for DOS, which is given by

m∗ =

p 3 g 2 ml mt mt

(8)

where g is the number of equivalent band minima, ml and mt are the longitudinal and transverse effective masses respectively. Equation 7 is calculated for the DOS calculation of materials with parabolic bands. The Nc and Nv are dominated by the bottom of the conduction band or the top of the valence band, the bottom and the top can be fitted by parabolic curves without bringing about significant errors. For hybrid perovskites, the conduction band bottom is strongly affected by MA+ orientations. 23 Under certain cases, 70 hybrid perovskites can show two valleys. We think the Nc and Nv with and without disordered MA+ should be similar. On the other hand, in this paper, all of the conduction bands show one bottom. All of the fittings are reasonable without significant variation, as shown in the SI. For a material has several conduction bands, the total Nc is the summation of these conduction band Nc if their energy levels are the same. If their energy levels have Ei −E0 P small difference, the total DOS is calculated by N = ni=0 Ni × e −kT , where Ni and Ei are 8


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the effective DOS and the energy level of the i th band.

Results Known factors influence on Voc Open circuit voltages are always lower than their band gaps, which is due to charge carrier densities are lower than the effective DOSs of conduction band and valence band. In silicon solar cells, it was already found that Voc is related to the charge density. 71–73 We pointed out that the output voltage of a solar cell is determined by its charge densities at two electrodes and Nc and Nv : 46,48 p|x=d n|x=0 ) − (Ev − kT ln( )) Nc Nv n|x=0 p|x=d = Ebgap + kT ln( ) + kT ln( ) Nc Nv

eV = Ef n − Ef p = Ec + kT ln(

(9)

= Ebgap + kT ln(n|x=0 p|x=d ) − kT ln(Nc Nv ) According to Equation (9), there are three approaches to reduce the voltage deficit. The first is operating solar cells at low temperature. If the working temperature close to zero Kelvin, then kT ≈ 0, which leads to V ≈ Ebgap . But keeping solar cells work under low temperature may cost more. This conclusion agrees with experiment and theory done by Sachenko et al.. 74 The second is increasing the charge carrier densities at two boundaries (n|x=0 p|x=d ). To do so, both increasing the generation rate by using concentrated incident light and efficient charge transport are feasible. Using high-density incident light source to generate high density charge carriers is already applied in concentrator solar cells that higher concentrated solar cells have higher Voc . 75–77 We also verified that with numerical simulations. 46 The other method is to improve charge transfer efficiency by either reducing recombination or increasing carriers lifetime and mobility. Numerical simulations showed that longer lifetime is, higher Voc outputs. 46 We also demonstrated that the presence of 9



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interface recombination lowers Voc , 46 which was verified by recent publications that Voc is primarily limited by the interface losses. 44,45 Here, we found that besides the light intensity, charge carries lifetime and mobility and recombination, there is another parameter controls Voc .

Nc and Nv from DOS(E) The last approach to improve Voc is using materials with low Nc and Nv . Here, we calculated DOS(E) of typical perovskites, CdTe and silicon by VASP. DOS(E) are displayed in Figure 1. Nc and Nv were then calculated by integrating these DOS(E), as illustrated by Equation (2). Table 1 shows Nc and Nv of these materials. It was found that all the perovskites have similar Nc and Nv , with product around 6 × 1036 cm−6 , which is about one third of silicons. CdTe has a very low Nc of 3.0×1016 cm−3 as a result of the very sharp DOS(E) at bottom of conduction (Figure 1D). While, its Nv is the largest. According to Equation (9), only the product of Nc and Nv influences Voc as the output potential is the relative potential difference between quasi hole and electron Fermi levels. Among them, Si has the highest Nc and Nv product, which is up to 1037 cm−6 . The lower Nc and Nv in perovskite and CdTe are supposed to have higher Voc than silicon under same parameters and conditions. The Nc and Nv calculated from DOS(E) integration is very sensitive to calculation methods, although we used recommended settings for accurate electronic DOS(E), the integration of discrete DOS(E) brings about errors as well. The accuracy of Nc and Nv calculated from DOS(E) integral here is poor. Only the relative magnitude of the calculated Nc and Nv are reliable. Table 1: Nc and Nv calculated from integration of conduction and valence band DOS(E) of unit cells through Equation (2). Compound α-FAPbI3 MAPbCl3 MAPbI3 CdTe Si

V (˚ A3 ) 765.85 716.61 984.66 271.72 40.05

Nc -integral 4.55×10−3 8.56×10−3 1.03×10−2 8.19×10−6 4.56×10−4

Nv -integral 8.99×10−4 3.11×10−4 6.26×10−4 5.53×10−3 6.17×10−5

10

Nc (cm−3 ) 5.94×1018 1.19×1019 1.04×1019 3.00×1016 1.14×1019


Nc (cm−3 ) 1.17×1018 4.34×1017 6.36×1017 2.03×1019 1.54×1018

Nc Nv (cm−6 ) 6.97×1036 5.18×1036 6.62×1036 6.09×1035 1.75×1037

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A)

B)

D)

E)

C)

Figure 1: Density of states of A), α-FAPbI3 ; B), MAPbCl3 ; C), MAPbI3 ; D), CdTe and E), Si respectively.

Nc and Nv from effective mass Table 2: Nc , Nv and simulated solar cell performance parameters for α-FAPbI3 , MAPbCl3 , MAPbI3 , CdTe and silicon. Compound α-FAPbI3 MAPbCl3 MAPbI3 CdTe Si Si 78

Nv (cm−3 ) 2.90×1018 3.64×1018 2.49×1018 4.54×1019 2.32×1019 2.85×1019

Nc (cm−3 ) 1.20×1019 1.36×1019 6.98×1018 4.25×1017 3.35×1019 1.85×1019

Nc N v 3.47×1037 4.94×1037 1.74×1037 1.93×1037 7.76×1038 5.27×1038

Voc (mV) 1132.02 1123.32 1149.14 1146.46 1055.70 1065.21

PCEs (%) 19.70 19.53 20.04 19.99 18.18 18.37

Nc and Nv calculated from effective mass are expected to be more accurate. The error comes from the evaluation of effective mass. Here, we calculated both Nc and Nv from effective electron mass and effective hole mass respectively. The effective mass is calculated according to its definition and estimated by fitting band curves near conduction band bottoms and valence band tops. Details about the calculation method can be found in Ref. 11



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79 and 23. Related band structures and fittings are available in SI. The calculated reduced effective mass of MAPbI3 for conductivity with the same method is 0.09 m0 , 23 similar to experimental values ranging from 0.09-0.15. (0.09-0.12 m0 , 80 0.12 m0 , 81 0.11 m0 , 82 0.10 m0 83 and 0.15 m0 . 84 ) This agreement confirms that our calculations are reliable. The estimated silicon hole effective masses are 0.12-0.76 m0 , and effective electron mass is 1.20 m0 . Both the calculated Nc and Nv are comparable to reference data. 78 Effective electron masses of the lowest conduction bands are 0.31-0.70 m0 and 0.17-0.53 m0 , for MAPbCl3 and MAPbI3 . Both of them are similar to the SOC-GW calculated 0.28-0.40 m0 and 0.18-0.30 m0 . 85 These comparisons suggest that our method is reliable. According to Equation (9), the product of Nc and Nv determines the voltage deficit from DOS. Therefore, Nc , Nv and their products are calculated in Table 2. The product sequence is Si > MAPbCl3 ≈ α-FAPbI3 ≈ MAPbI3 ≈ CdTe, which is in concert with the result calculated from integration. The absolute value difference between the Nc and Nv from effective mass and the Nc and Nv from the integral is due to the poor accuracy of Nc and Nv from DOS(E) integral. Although CdTe has a very small Nc , its Nv is large as it has three flat valence bands. The Nc and Nv products of CdTe and perovskites are in the same scale, which is much smaller than that of silicon. Considering Equation (9), the voltage deficits of perovskites should be about 100 mV smaller than silicon’s due to the products of hybrid perovskites are about 45 times lower than that of silicon.

Numerical simulations In order to figure out how the Nc and Nv influences solar cell performance and eliminate other parameters influences, such as band gap and carrier mobility influences, we assumed all of these materials have same parameters except the Nc and Nv . Several other parameters are correlated with Nc and Nv , such as effective mass and absorption. A low Nc or Nv is related to the small effective electron or hole mass, which also contributes to the high performance. As the small effective mass produces high charge carrier mobility and small 12


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Figure 2: J-V curves of various materials with same electronic parameters except for Nc and Nv . The thickness of simulated solar cell is 350 nm. Diffusion coefficients of the perovskite were assumed to be 0.017 cm2 s−1 and 0.011 cm2 s−1 . 11 The band gap is set as 1.55 eV. 8 The interface recombination width is 5 nm and its lifetime is 5 ns. The charge carrier lifetime is assumed to be 736 ns. 8 Cells operate at 300 K and 1 sun (1.5AM). More simulated details can be found in Ref. 46. exciton separation energy, 23 mobilities and IPCEs are assumed the same in order to avoid the influence from the small effective mass. The low Nc and Nv may lead to low absorption. The absorption influence was excluded as we use same absorption coefficients. Using the numerical model we have developed for perovskite solar cells, 46 we simulated the working J-V curves of these materials. All of the simulated solar cells are ‘perovskite’ solar cells. For the simulated silicon and CdTe solar cells, they are not real silicon solar cells or CdTe solar cells, they are perovskite solar cells just with the Nc and Nv of silicon and CdTe. Figure 2 shows performances of their solar cells with different Nc and Nv . For the solar cell with silicon Nc and Nv , its Nc × Nv is 7.76×1038 cm−3 , the corresponding simulated Voc is 1055.70 mV. For the MAPbI3 , its Voc is calculated to be 1149.14 mV, which is about 93 mV higher than that of the silicon solar cell. If we lower the silicon Nc and Nv to the Nc and Nv of MAPbI3 , its PCEs will be improved from 18.18 % to 20.04 %. The relative improvement is 13



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about 10 %.

Discussion Since the effective mass is calculated from the second derivative of bands, and number of K point in the band structure calculation are limited, this second derivative is not accurate. The error in effective mass evaluation is approximately 10 % in our calculations. , and the error for the reduced mass (m∗ ) is estimated to be approximately 15 %. The error in Nc or Nv is therefore approximately 23 %. The error in these two methods is obvious, but the sequence of Nc and Nv magnitude calculated from these two methods are the same. This confirms the low Nc and Nv in perovskite. An important question is does low Nc and Nv lead to low charge density near electrodes (n|x=0 p|x=d )? We find there will be no enhancement if the changing of n|x=0 p|x=d is the same as the changing of Nc ×Nv . Charge carriers are generated and then migrate to the electrodes. Hence, charge carrier densities at the two ends should increase with generation rate and mobilities, and decrease with thickness and recombination rate. The DOS has no effect on the recombination rate as the recombination rate is mostly dependent on the charge carrier density itself, defect density and defect energy levels according to the Shockley-Read-Hall Model. 68,86–88 Low Nc and Nv are related with small effective masses according to Equation 7, and produce high mobilities. The mobility effect was excluded in our simulations by using same mobilities. If take this into consideration, Voc will be further improved. The generation rate and thickness greatly influence each other. Theoretically, the genRdRλ RdRλ eration rate is calculated from G = 0 0 0 G(λ, x)dλ = 0 0 0 ηg [1 − R(λ)] × I(λ) × α(λ) × Rλ e−α(λ)x dλdx = 0 0 I(λ) × IP CE(λ, d)dλ, where α is the absorption coefficient and I(λ) is

incident photon density; R(λ) is the reflect coefficient and ηg is the generation quantum effi-

ciency; IP CE(λ, d) is the quantum efficiency of a whole device, which is also called External Quantum Efficiency (EQE). 89 In some experiments, solar cells show IP CE(λ) of approxi-

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mately 80% in the visible light range. 6,12,90–93 For a small α material, a thicker thin film is required to reach the same IP CE. The low Nc and Nv in perovskite might lead to low absorption, which will lower solar cell performance. As shown in Ref. 94, saturable absorption happens at extreme incident light intensities. The intensity to exhibit saturable absorption for Bi2 Se3 is about 41-53 MW/cm2 , 95,96 which is about 108 times strong than 100mW/cm2 (the light intensity of AM1.5). There is no experimental saturable absorption data for perovskite materials, but the intensity should be of similar magnitude. Hence, the Nc and Nv ’s influence on absorption of sunlight is negligible for normal solar cells. Furthermore, the absorption of MAPbI3 is higher than silicon as shown in Ref. 97. As such, to reach the same IP CE, perovskite solar cell could be thinner. This thin film also leads to high n|x=0 p|x=d . In summary, if we consider the Nc and Nv related parameter influences, higher n|x=0 p|x=d are expected. The high Voc calculated from numerical models will be further improved.

Conclusion In conclusion, using two different methods based on ab initio results, we estimated Nc and Nv of silicon, CdTe and typical perovskites. It was found that the DOSs of perovskites and CdTe are much lower than that of silicon. The low Nc and Nv consist with the small effective mass, which directly contributes to high mobility. To avoid the small effective mass induced improvement, the mobility in these simulations is fixed. Analytic and numerical results show that the material with lower Nc and Nv can realize higher open circuit voltage and higher PCEs. With systematic discussions, we determined how the Nc and Nv of material works on its solar cell performance. The lower Nc and Nv of perovskites lead to about 100 mV higher Voc and 10 % (relative) higher PCEs. Considering low Nc and Nv consist with small effective masses, the performance of solar cells with low Nc and Nv will be further improved. In this paper, we show that besides inert interface defects, 44,46 long charge carrier lifetime, 46 and reduced bimolecular recombination, 45 that low Nc and Nv in perovskites contribute to the

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high Voc and high performance of perovskite solar cells. Therefore, the materials with low Nc and Nv should be good candidates for solar cells.

Acknowledgments Y. Zhou is grateful to the National Computational Infrastructure (NCI, Australia) for High Performance Computing resources. Y. Zhou thanks Dr. Angus Gray-Weale for discussion.

ASSOCIATED CONTENT Supporting Information The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.jpcc.00000. Method choosing, calculation details of effective masses and DOSs of silicon, FAPbI3 , MAPbCl3 , MAPbI3 , and CdTe.

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