Letter pubs.acs.org/JPCL
Small Photocarrier Effective Masses Featuring Ambipolar Transport in Methylammonium Lead Iodide Perovskite: A Density Functional Analysis Giacomo Giorgi,*,†,‡ Jun-Ichi Fujisawa,†,§ Hiroshi Segawa,† and Koichi Yamashita*,‡ †
Research Center for Advanced Science and Technology (RCAST), The University of Tokyo, 4-6-1 Komaba, Meguro-ku, 153-8904 Tokyo, Japan ‡ Department of Chemical System Engineering, School of Engineering, The University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo, Japan § Japan Science and Technology Agency (JST), Precursory Research for Embryonic Science and Technology (PRESTO), 4-1-8 Honcho, Kawaguchi, Saitama 332-0012, Japan ABSTRACT: Methylammonium lead iodide perovskite (CH3NH3PbI3) plays an important role in light absorption and carrier transport in efficient organic−inorganic perovskite solar cells. In this Letter, we report the first theoretical estimation of effective masses of photocarriers in CH3NH3PbI3. Effective masses of photogenerated electrons and holes were estimated to be me* = 0.23m0 and mh* = 0.29m0, respectively, including spin−orbit coupling effects. This result is consistent with the long-range ambipolar transport property and with the larger diffusion constant for electrons compared with that for holes in the perovskite, which enable efficient photovoltaic conversion. SECTION: Energy Conversion and Storage; Energy and Charge Transport reported by Even et al.14 In this Letter, we report the first theoretical estimation of effective mass of photocarriers in CH3NH3PbI3, a characteristic related to the pronounced carrier transport property. We show that the estimated effective mass of the photocarriers is comparable to the values15,16 of silicon used in commercially available solar cells, suggesting the high potentiality in photovoltaic cells. Spin-polarized DFT calculations have been performed with the generalized gradient approximation as implemented in the VASP code17,18 by means of the electron exchange−correlation functional proposed by Perdew−Burke−Ernzerhof (PBE). The electron−ion interaction is described by the projector augmented wave method (PAW).19−21 Also, still concerning the PAW potential, a 5d106s26p2 valence electron potential was used for the Pb atom.22,23 A plane wave basis set energy cutoff of 500 eV was considered in the calculations, while the force convergence threshold was set to 2.0 e−3 eV/Å. A 6 × 6 × 6 (corresponding to 216 points) centered k-point sampling of the Brillouin zone was used in the structural optimizations, while for the study of electronic properties, denser meshes have been employed (700 k-points).
M
ethylammonium metal halide perovskites (CH3NH3PbX3, M = Sn, Pb, X = Cl, Br, I) are solution-processable organic−inorganic hybrid semiconductors that consist of a three-dimensional (3D) metal halide semiconducting framework and an intervening organic molecule.1,2 Recently, it has been reported that the lead iodide perovskite (CH3NH3PbI3) plays an important two-fold role, as a light absorber and as a carrier transporter, in organic− inorganic hybrid solar cells.3−8 Such devices exhibit high power conversion efficiencies exceeding 15%.7,8 This efficient conversion is ascribed to the ambipolar properties of the perovskite material. In addition, the long-range electron−hole diffusion lengths of at least 100 nm have been demonstrated for CH3NH3PbI3 crystals.9,10 Stoumpos et al. reported relatively high mobility (∼66 cm2/V sec) of carriers in the ground state, even using pressed pellet samples of annealed polycrystalline CH3NH3PbI3.11 In order to clarify the attractive electronic properties of CH3NH3PbI3, theoretical investigations of the electronic structure are mandatory. The band structure of CH3NH3PbI3 was calculated without and with intervening methylammonium ions by Umebayashi et al. and Mosconi et al., respectively, by density functional theory (DFT).12,13 Their calculations indicate that the conduction and valence bands are comprised of the lead iodide framework, with little effects of the organic molecule. In addition, quite recently, the significant effect of spin−orbit coupling on the band structure was © XXXX American Chemical Society
Received: November 6, 2013 Accepted: November 25, 2013
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Figure 1. Optimized structure for (pseudo)-cubic CH3NH3PbI3. (Left) The unit cell. (Right) The 4 × 4 × 4 supercell. Large dark gray: lead; purple: iodine; brown: carbon; small light gray: nitrogen; white: hydrogen atoms.
CH3NH3PbI3 crystals have cubic and tetragonal lattice systems at room temperature.24−26 Here, using the experimental cubic coordinates as an initial guess,24 we calculated a (pseudo)-cubic structure (Pm3m, Z = 1) for the CH3NH3PbI3 crystal whose optimized dimensions of the unit cell are a = c = 6.545 Å and b = 6.488 Å. The three Pb−I bond lengths were estimated to be 3.292, 3.150, and 3.340 Å. As a result of our attempts from several initial structures, we found that the C N bond of the methylammonium cation oriented along the (111) direction is the most stable. The optimized structure is shown in Figure 1. The band structure calculated for this optimized (pseudo)-cubic structure is shown in Figure 2a. The top of the valence band at the R point is composed of the antibonding linear combination of Pb 6s and I 5p orbitals, that is, 21.0% 6s(Pb), 14.9% 5px(I1), 14.6% 5py(I2), and 14.9% 5pz(I3). On the other hand, the bottom of the conduction band at the R point is predominantly comprised of Pb 6p orbitals with minor antibonding contributions of I 5s orbitals, that is, 12.3% 6px(Pb), 10.7% 6py(Pb), 12.6% 6pz(Pb), 1.3% 5s(I1), 1.5% 5s(I2), and 1.6% 5s(I3). These orbital characteristics of the valence and conduction bands are consistent with the previously reported theoretical results.12−14 From the band structure, the direct band gap is estimated to be 1.64 eV at the R point, which is in good agreement with the experimental data (1.5 eV).3−8 However, as reported by Even et al., this agreement is accidental coincidence, as will be described below. Indeed, owing to the large spin−orbit coupling constant of lead atoms, effects of spin−orbit coupling on the band structure of CH3NH3PbI3 are considered to be significant. The band structure of (pseudo)-cubic CH3NH3PbI3 calculated with spin−orbit coupling is shown in Figure 2b. As reported by Even et al., the conduction band structure is drastically changed by the inclusion of the spin−orbit coupling effects. In particular, the energy of the bottom of the conduction band decreases significantly. This reduction results from the splitting of the
Figure 2. Band structures of (pseudo)-cubic CH3NH3PbI3 without (a) and with (b) spin−orbit coupling. R = (0.5, 0.5, 0.5), Γ = (0, 0, 0), M = (0.5, 0.5, 0), and X = (0.5, 0, 0).
conduction band into two-fold degenerate states |1/2, ±1/2⟩ corresponding to light electrons and four-fold degenerate states |3/2, ±3/2⟩, |3/2, ±1/2⟩ corresponding to heavy electrons at the R point. As a result, the band gap energy is reduced from 1.64 to 0.52 eV, similarly to the result reported by Even et al. Figure 3 shows the charge density of the bottom ((a) and (b)) of the conduction band and the top ((c) and (d)) of the valence band. From these figures, one can see that photogenerated electrons around the bottom of the conduction band and holes around the top of the valence band exist separately, results related to the ambipolar transport nature of the material. 4214
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Figure 3. Charge density (yellow isosurfaces) of the two-fold degenerate states ((a) and (b)) of the bottom of the conduction band and those ((c) and (d)) of the top of the valence band at the R point calculated with spin−orbit coupling for (pseudo)-cubic CH3NH3PbI3. (Large dark gray: lead; purple: iodine; brown: carbon; small light gray: nitrogen; white: hydrogen atoms.)
scatterings in (pseudo)-cubic CH3NH3PbI3 crystals. Similarly, taking advantage of the relationship between photocarrier effective masses and their mobility
On the basis of the parabolic approximation, we estimate the effective mass (m*) of carriers existing around the bottom of the conduction band or the top of the valence band as result of the fitting of the dispersion relation ⎡ ∂ 2ε(k) ⎤−1 m* = ℏ ⎢ ⎥ ⎣ ∂k 2 ⎦
m=
2
(1)
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Table 1. Effective Masses of Photogenerated Electrons and Holes Estimated from the Calculated Band Structures along the R−Γ Direction without spin−orbit coupling with spin−orbit coupling
mh*/m0
0.32 0.23
0.36 0.29
(2)
where q is the elementary charge and τ ̅ the average scattering time, we can predict, considering a constant τ ̅ value for both carriers, a slightly larger mobility for electrons compared with that for holes, in fair agreement with the experimental results of Stranks et al.10 We report the first theoretical estimation of effective masses of photocarriers in (pseudo)-cubic CH3NH3PbI3 taking into account spin−orbit coupling effects. The calculated effective masses of photogenerated electrons and holes are comparable to those for silicon used in inorganic solar cells. This result is consistent with the experimentally observed result of longrange photocarrier transport in CH3NH3PbI3.
where ε(k) are the band edge eigenvalues and k is the wavevector. Because photogenerated electrons and holes in CH3NH3PbI3 thermally relax to the bottom of the conduction band or the top of the valence band, respectively, their effective masses (me* and mh*) in CH3NH3PbI3 are calculated from the band structure in Figure 2a to be 0.32m0 and 0.36m0, respectively according to eq 1. Including the spin−orbit coupling effects, me* and mh* values are reduced to 0.23m0 and 0.29m0, respectively, as shown in Table 1. Comparing such values with those of other
me*/m0
⎛ q ⎞ ⎜ ⎟τ ⎝ m* ⎠ ̅
AUTHOR INFORMATION
Corresponding Authors
*E-mail:
[email protected] (G.G.). *E-mail:
[email protected] (K.Y.). Notes
The authors declare no competing financial interest.
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semiconductors, me* in silicon is estimated to be 0.19m0 at the Δ point, while mh* for heavy and light holes is 0.53m0 and 0.16m0, respectively.15,16 Thus, our calculated me* and mh* values of (pseudo)-cubic CH3NH3PbI3 are comparable to those in silicon used in commercially available inorganic solar cells. In the actual crystal, the effective masses would increase by several elastic scatterings with phonons and structural defects and impurities. Thus, the estimated values correspond to the maximum effective masses attainable with minimizing of these
ACKNOWLEDGMENTS This research is supported by the Japan Society for the Promotion of Science (JSPS) through its “Fundamental Program for World-Leading Innovative R&D on Science and Technology (FIRST Program)”. G.G. wants to thank Prof. Maurizia Palummo of the University of Rome “Tor Vergata” for the always very useful suggestions and fruitful scientific discussions. 4215
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(22) Perdew, J. P.; Burke, K.; Ernzerhof, M. Generalized Gradient Approximation Made Simple. Phys. Rev. Lett. 1996, 77, 3865−3868. (23) Perdew, J. P.; Burke, K.; Ernzerhof, M. Errata: Generalized Gradient Approximation Made Simple. Phys. Rev. Lett. 1997, 78, 1396. (24) Poglitsch, A.; Weber, D. Dynamic Disorder in Methylammoniumtrihalogenoplumbates (II) Observed by Millimeter-Wave Spectroscopy. J. Chem. Phys. 1987, 87, 6373−6378. (25) Kawamura, Y.; Mashiyama, H.; Hasebe, K. Structural Study on Cubic−Tetragonal Transition of CH3NH3PbI3. J. Phys. Soc. Jpn. 2002, 71, 1694−1697. (26) Baikie, T.; Fang, Y.; Kadro, J. M.; Schreyer, M.; Wei, F.; Mhaisalkar, S. G.; Graetzel, M.; White, T. J. Synthesis and Crystal Chemistry of the Hybrid Perovskite (CH3NH3)PbI3 for Solid-State Sensitised Solar Cell Applications. J. Mater. Chem. A 2013, 1, 5628− 5641.
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