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Methylammonium Lead-Iodide Perovskite: Recombination and Photoconversion of an Inorganic Semiconductor Within a Hybrid Body Alessio Filippetti, Pietro Delugas, and Alessandro Mattoni J. Phys. Chem. C, Just Accepted Manuscript • DOI: 10.1021/jp507430x • Publication Date (Web): 03 Oct 2014 Downloaded from http://pubs.acs.org on October 5, 2014

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The Journal of Physical Chemistry

Radiative Recombination and Photoconversion of Methylammonium lead-iodide Perovskite by FirstPrinciples: the Properties of an Inorganic Semiconductor within a Hybrid Body Alessio Filippetti1,*, Pietro Delugas1,2, and Alessandro Mattoni1,* 1

CNR-IOM, UOS Cagliari, c/o Dipartimento di Fisica, Cittadella Universitaria, Monserrato

09042-I (CA), Italy. 2

Istituto Italiano di Tecnologia - IIT, Via Morego 30, 16163 - Genova, Italy.

E-mail: [email protected]; [email protected] KEYWORDS: PHOTOVOLTAICS, DENSITY-FUNCTIONAL THEORY, RADIATIVE RECOMBINATION

ACS Paragon Plus Environment

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ABSTRACT

The excellent photoconversion properties of lead-iodide hybrid perovskites, used as absorber in solar cell devices with power conversion efficiencies exceeding 15%, are explained on the basis of ab-initio calculated radiative recombination rates and minority carrier lifetimes. We obtain Brad ∼ 0.5-1.5 x 10-9 s-1cm3 and minority lifetime ∼ 103 ns for a doping concentration n ∼ 1015 cm3

at room temperature. These values, comparable to that of typical optoelectronic semiconductors

(e.g. GaAs) reflect the very nature of the perovskite: fully solution-processable owing to its hybrid nature, and yet a truly inorganic semiconductor for what concerns photoconversion properties. Recombination rates are also used to quantitatively describe the maximum limit of power conversion efficiency potentially achievable by these systems, e.g. 21% for a 200 nmthick perovskite film, and 23% for a 300 nm thickness.

INTRODUCTION The search for the highest possible efficiency in solar cells based on methylammonium leadiodide (MPbI3)

1,2,3,4,5

has been following various strategies: device and material processing

optimization6,7,8,9,10,11,12, molecule substitution13, cation substitution14, and the selection of the best possible electron transporter15,16,17 and hole transporter

18,19,20,2122

. Simultaneously, a

significant amount of activity23,24,25,26,27,28,29,30,31 is carried out to understand the excellent optical and photoconversion properties of this absorbing material at the basis of the high-performance solar cells. This is important not only for conceptual reasons, but also to draw useful guidelines for further device improvements. Recent photoluminescence (PL) experiments27,28,30,31 have


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shown

long lifetimes and diffusion lengths for the perovskite. Particularly striking is the

observation27 that the inclusion of Cl-doping in solution can increase the carriers lifetime (and in turn the diffusion length) by about an order of magnitude, a result of difficult interpretation considering that the amount of Cl actually embodied in the sample is probably limited to a few percent fraction. The basic properties of hybrid perovskites were studied in a number of theoretical works 32,33,34,35,36,37,38,39,40,41

. All of them substantially agree in drawing a well established picture for the

electronic properties of this material: the energy region around the band gap (EG) is dominated by Pb(6s,6p) and I(5p) states (see the calculated density of states (DOS) in Fig.1); the electronic energies of the ionized methylammonium molecule (M1+) are placed far away from this region: the highest occupied molecular orbitals (HOMO) is ∼4.5 eV lower than the valence band top (VBT) which derives from I(5p)-Pb(6s) hybridized states, whereas the LUMO is ∼2.0 eV higher than the conduction band bottom (CBB) which is mainly Pb(6p) in orbital character. It follows that the molecule and the inorganic part are electronically decoupled, and only the latter contributes to the electronic spectrum in the range important for the photoconversion. For this reason the hybrid perovskites, for what concerns photo absorption, is, in fact, an inorganic semiconductor. Consistently with this analysis, there is a large consensus based on various optical measurements

27,30

on the fact that photoabsorption and photoemission are largely

dominated by band-to-band recombination involving extended electronic states. The role of the excitons in hybrid perovskites, typically dominant in organic absorbers, is minor or negligible (at least at room temperature). In a previous work

41

we put in evidence that the optical absorption of this material in the

important region of solar irradiance (400-600 nm) is similar to that of other popular inorganic


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absorbers (GaAs, Ge, InP). In this work we push further the comparison between MPbI3 and GaAs by including the analysis of radiative band-to-band recombination rate and minority lifetime as a function of doping and temperature, and we provide evidence of a remarkable similarity between the two materials. The determination of recombination rate is also used to quantify the current-voltage characteristic, and power conversion efficiency in perovskite-based cells, in the ideal situation of lack of any current and/or potential loss due to shunt or parasitic resistances or band disalignment between absorber and transporting materials. The calculated efficiency as a function of the perovskite film thickness fixes the upper limit of attainable power conversion efficiency for this material. The fact that molecular levels and bands are decoupled should not lead to the erroneous conclusion that the molecule is inessential to the photoconversion properties of the perovskite. In fact, the molecule orientation can affect the structural properties, and in turn the band structure. It is well known that for inorganic perovskites the electronic band structure is crucially influenced by the detail of the atomic structure (octahedral tilting, Pb-I distances, structural distortions), which, in turn, changes with the characteristics (primarily its ionic radius) of the cation located at the A-site. Furthermore, having a molecule in place of an atom increases the degrees of freedom and remarkably complicates the picture. In this work we provide evidence that different but energetically competitive molecular orientations may give rise to EG variations as large as ∼0.1 eV; such a large gap fluctuation can reflect in sizable differences in the recombination rate and photoconversion efficiency.

2.THEORY AND TECHNICALITIES


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Local Density Functional Theory (LDA), using a plane-waves and ultrasoft pseudopotential approach as implemented in the PWSIC code42, was used to calculate the structural and electronic properties of the perovskites. As well known from previous reports, LDA describes with good accuracy the band gap of the perovskite, although this agreement is fortuitous and related to the cancelation of correlation and relativistic (spin-orbit) effects, both large but opposite in sign38,43. However, the correct description of the band gap is a key aspect for the determination of radiative band-to-band recombination, thus we considered the adoption of the LDA functional as the best-suited option for our aims. Atomic structures are calculated by forces minimization below a threshold of 10-3 Ry/Å on each atom. The electronic structure is selfconsistently determined using a 6x6x6 special k-point grid and 30 Ry cut-off energy. The Density of states is calculated on extremely dense 20x20x20 k-point grid (essential for the precise determination of the intrinsic charge doping concentration).

The optical functions

(dielectric function, absorption, and refraction index) are calculated in random-phase approximation on an ultra-dense mesh of more than 5000 k-points and 150 bands. We also calculated the optical properties of GaAs (for which we have abundance of experimental data) as a test case to validate the accuracy of our technicalities, and as scientifically sound term of comparison for the perovskite. However, in order to treat both materials at the same level of accuracy, for GaAs (whose band gap is notoriously underestimated in LDA) we used a model pseudo-self-interaction density functional (VPSIC44) so that the calculated band gap could be fixed at the experimental value. In this way a very good description of the optical properties of GaAs could be obtained.


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Following Van Roosbroeck and Shockley45, from the ab-initio calculated absorption function () and refraction index (), the radiative recombination rate (number of events of electronhole recombination per unit volume and time) can be calculated as:

( ) = (, )() () (1)

where (ε)=c/ () is the photon velocity in the matter, and: (, ) =

8 1 ' (2) ( ()ℎ) ! "/#$ % − 1

is the equilibrium photon distribution. Another important quantity is the intrinsic carrier concentration: ) ( ) =

-$$

()

1+

1

! ("+,)/#$

'= %

.$/

+

()

1+

1

! ("+,)/#$ %

' (3)

where () is the electronic density of states (DOS) and µ=EF(T) the chemical potential. From ni we can calculate the so called B-factor (Brad = Rrad/ni2) which is more commonly used in literature than Rrad. The calculated radiative recombination rate can be also exploited to make quantitative estimates of the power conversion efficiency for perovskite-based solar cells. According to the detailed balance principle first introduced by Shockley and Queisser46, the total current flowing through the absorber in a solar cell circuit under a bias potential Vb is: 1232 = 145 − 1 # (67 ) (4) where Jsc is the short-circuit current flowing through the circuit at zero bias, given by two terms:


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145 = 19 − 1 (0)

!2 !2 @9 () = ; = "/# % ? @9 () − ; = "/# % ? (5) < ℎ ℎ ! $ >−1 ! $ − 1 ()

JL is the illumination current due to the solar flux (AM0 black-body radiation) incident on the surface of the absorber, while the second term is the current lost due to radiative recombination at zero bias. The geometric factor

f = sin 2 (0.2670 ) takes into account the solar cone incident

on the surface; Ts= 5759 K the sun temperature, and the absorption probability @9 : @9 () = 1 − ! +B(")9 (6)

where L is the absorber thickness along the incident light direction. In practice, JL is largely dominant over the recombination term, which can thus be safely discarded. If zero-bias recombination

is

ignored, PL (ε ) corresponds to the so-called "external quantum

efficiency" or "incident photon-to-current conversion efficiency" (IPCE, percentage of incident photons transformed to current) in the hypothesis that the "internal quantum efficiency" (IQE, percentage of electron-hole couples transformed to current) is set to 100% (IPCE =IQE⋅PL) The bias-induced "dark" current is given by: 1 # (67 ) = −

!2 @9 ()

= ? ℎ ! "/#$ % − 1 () +

!2 @9 ()

= ? (7) ℎ ! ("+DEF )/#$ % − 1 ()

where the first term compensates the Vb=0 contribution of the second (1 # (0) = 0). The

delivered power @ is calculated from the current as P = J⋅Vb and the power conversion efficiency

(PCE) as the ratio between the maximum power Pmax and the incident power Pin given by: 2; @)H = = ("/# % ? = 135 mWcm+ (8) $ > −1 < ℎ I !


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For a more accurate comparison with the experiments, we compute Pin and JL (Eq.5) using, in place of the AM0 black-body radiation, the numerical AM1.5 solar illumination spectrum which deliver a Pin = 100 mW/cm2.

3.RESULTS AND DISCUSSION 3.1 Absorption, Recombination rate and Lifetime The determination of radiative electron-hole recombination requires the accurate ab-initio calculation of absorption and refraction index. In Fig.1 A) we display the calculated 3D-averaged absorption due to inter-band electron-hole excitations. The excitonic contribution to absorption is not included in the calculation. However at room-temperature, which is our main interest in this work, the excitonic peak is smoothed away and can be safely discarded (see e.g. the optical spectra measured in Ref.30). We considered four different MPbI3 structures: P0 and P1 with orthorhombic Pnma symmetry and I0 and I1 withinin tetragonal I4/mcm symmetry. These are stable at temperature lower than 161 K and from 161 K to 327 K, respectively47; we report for both symmetries the lowest energy ground-state configuration (P0 and I0) and the second-lowest configuration (P1 and I1) corresponding to a different molecular orientation pattern. The structural and electronic properties of these states will be described in Sections 3.3 and 3.4. First of all we analyze their different electronic gap EG (reported in Tab.1) and its effect on the recombination properties. Notice that the energy gap of P0 is very close to the measured value 1.57 eV, thus this structure is likely the closest reference to the actual experimental values. GaAs is also included in the analysis as a relevant term of comparison. For energies higher than EG our calculated absorption () compares well with a recent absorption measurement 26 which serves as a useful validation of our method. In agreement with


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the experiment, the calculated () at the onset is ∼104 cm-1 and grows linearly (on a logarithmic scale) with the energy. Nicely, the slope change visible near = 2.5 eV reproduces a similar

feature of the experimental spectrum. The MPbI3 and GaAs absorption curves are similar in a wide energy range but for the energy onset corresponding to the different EG of the materials. For energies lower than EG we model the absorption with a Urbach exponential tail48, following the finding of Ref. 26 which estimates a Urbach energy (15 meV) similar to that of GaAs 49,50.

Figure 1: Upper panel) DOS calculated for the most stable tetragonal structure (I0). The different orbital characters are highlighted by different colors: black for molecular states, cyan and red for I(5s) and I(5p), respectively, blue and light green for Pb(6s) and (6p). At the VBT the green states indicate Pb(6s)-I(5p) hybridization. Lower panel) calculated absorption (averaged


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over x, y, and z components) for the energy-lowest tetragonal (I0) and orthorhombic state (P0), as well as for the two most energy-competitive states (I1 and P1) relative to different molecular orientations. The calculated absorption for GaAs is also reported for comparison. Our results for the recombination properties at T=298 K are reported in Tab.1 for the four perovskite structures and for GaAs. In Fig.2 we report the same quantities as a function of T (this T-dependence is at a qualitative level, since it does not include the excitonic peak and the Tdependence of EG, which is fixed to the calculated values shown in Tab.1). Rrad is mainly controlled by EG and by the absorption coefficient at the onset of the interband transition; for the examined perovskites Rrad increases by more than one order of magnitude while going from the ground-states (I0, P0) to the excited states (I1, P1), as a consequence of the smaller EG of the latters. Since both Rrad and ni decrease exponentially with EG, these two large and opposite variations mostly compensate in Brad which is thus much less EG-dependent than Rrad or ni2 separately. However, sensible differences in Brad for the examined perovskites still remain (see Tab.1); typically, factor ni2 overcompensates the difference in Rrad, so that the larger Brad occurs for the system with larger EG. For what concerns the T-dependence, in Fig.2A we can notice that ln Rrad grows roughly linearly with T, as a consequence of the increase in photon population. The intrinsic carrier concentration (Fig.2B) on a logarithmic scale shows a typical -1/T dependence, due to the FermiDirac distribution; notice the change of slope around T=250 K due to a sudden decrease of the effective DOS; for the resulting ln Brad (Fig.2C) we can neatly distinguish two regions: a linear decrease up to T = 250 K, and nearly constant behavior above it in correspondence to the ni slope change. Finally, assuming quasi-equilibrium conditions (np∼ ni2), we can determine the radiative lifetime (also called minority lifetime) as τrad = (Brad n)-1. In Tab.1 we also compare the lifetimes


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for the perovskite with that of GaAs at a doping concentration n=2x1016 cm-3 for which an experimental reference51 is available for validation. For this doping level the radiative lifetime of the perovskites is in the range of few tents up to hundred ns, thus similar or larger than in GaAs.

Table 1: Calculated energy gap (EG), radiative recombination rate (Rrad), intrinsic carrier density (ni), B-factor (Brad) and minority carrier radiative lifetime τrad for MPbI3 in 4 different structures (see text) and GaAs at T=298 K. The radiative lifetime is calculated for a reference doping n=2x1016 cm-3, to be compared with an experimental value for GaAs at the same doping given in literature51. Material

EG (eV) Rrad (s-1cm-3) ni (cm-3)

Brad (s-1cm3)

τrad (ns)

MPbI3 P0

1.584

4.9

0.78x105

0.80x10-9

64

MPbI3 P1

1.508

80.5

4.14x105

0.47x10-9

106

MPbI3 I0

1.614

2.0

0.36x105

1.57x10-9

31

MPbI3 I1

1.510

70.6

2.33x105

1.30x10-9

38

GaAs

1.424

4446

1.87x106

1.3x10-9

39

GaAs(expt 51 )

1.424

4500

1.8x106

1.4x10-9

36

In Fig.2 D) we show the lifetime of perovskite P0, in comparison with GaAs, for various reference doping concentrations. Like Brad, τrad is nearly constant with T in the T=250-300 K range, and decreases linearly with doping, spanning a wide range of values, from τrad ∼ 1 ns at n ∼ 1018 cm-3 up to τrad ∼103 ns at n ∼ 1015 cm-3. The range τrad ∼102-103 ns is especially interesting


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since it matches the lifetime measured by PL. In Ref.27 the reported lifetime is 10 ns for pure MPbI3 and 280 ns for MPbI3-xClx. This huge difference may appear surprising since the amount of Cl actually embodied in the perovskite is believed to be only a few percent, thus a fraction which can hardly change the intrinsic properties of the perovskite. To push the comparison with PL measurements up to a quantitative level, in Fig.2 E) we plotted the calculated τrad as a function of doping at room-T, together with the measured lifetimes, indicated by dashed and dotted horizontal lines, respectively. The MPbI3 lifetime intercepts the calculated τrad for a doping interval between n ∼ 7x1016 cm-3 and n ∼2x 1017 cm-3 (delimited by dashed vertical lines), while the Cl-doped lifetime matches τrad in the range between n ∼ 2x1015 cm-3 and n ∼8x1015 cm-3 (delimited by dotted vertical lines). Thus, a possible understanding of the PL measurements is to think of them as corresponding to two different doping levels, the lowest of which corresponding to the Cl-doped perovskite. The scenario implied in this hypothesis is that inclusion of Cl in solution somehow helps to clean up the processing, allowing the production of perovskites with a much reduced (one or two order of magnitude smaller) amount of native defects. Clearly, a limit of our lifetime estimate is that non-radiative contributions are not included. However, there are indications in literature that lead-iodide perovskites are not easily affected by deep electronic traps, since typical defects such as molecule or cation vacancies produce shallow acceptors and donors, respectively36 whereas isovalent Cl-I substitutions are obviously ineffective for what concerns the introduction of electronic traps34. Thus, the assumption of electron-hole recombination dominated (or primarily determined) by radiative processes should be seen as a sound possibility.


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Figure 2: A) Radiative recombination rate Rrad as a function of T for the four different perovskite structures (see text) indicated in the legend and for GaAs; B) the intrinsic carrier concentration ni for the same systems; C) the B-factor Brad =Rrad / ni2; D) the quasi-equilibrium radiative lifetime τrad = (Brad⋅n)-1 for perovskite P0 and GaAs and various doping levels n; E): τrad for all the perovskites at varying doping n and fixed T=300 K; dashed and dotted horizontal lines indicate the PL measured values27, and the vertical lines the doping concentration regions where these measurements intercept the calculated τrad (see text). 3.2 Current-voltage characteristics and power conversion efficiency The calculation of current-voltage characteristics is reported in Fig.3 for the P0 perovskite. A crucial parameter is the thickness (L) of the absorber. The perovskite film thickness reported in literature varies in the range of few hundreds (100-400) nm, in order not to overcome the diffusion length. Thus, we performed the calculations for a set of different thicknesses in this range.


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Figure 3: Current-voltage characteristic (Eqs.4-7) of the solar cell with P0 perovskite as absorber. Each panel shows short-circuit (Jsc) and total (Jtot) current densities (current per unit surface), as well as and output power density (Pout), (power per unit surface) for a given perovskite layer thickness (L). For each calculation the considered power input is Pin = 100 mW/cm2, that is the power under maximum AM1.5 solar illumination spectrum; since PCE = Pout/Pin, Pout also represents the percent efficiency. We can see that the calculated Jsc substantially matches the values reported in literature, ranging from 15 mA/cm2 up to about 20-21 mA/cm2

52,17

. As an example, Jsc calculated for

L=300 and L=400 is in satisfactory agreement with the very high 21.5 mA/cm2 obtained in Ref.17 for a 330 nm-thick perovskite layer, vapour-deposited within a planar heterojunction solar cell52 (it is found that vapour deposition increases the homogeneity of the perovskite film in comparison to solution deposition). According to our results, going above these Jsc values would require excessively large thicknesses. As a useful indication, in Fig.3 we inserted the calculation for L=800 nm, for which Jsc becomes close to the maximum (100% absorption) limit of 25.7


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mA/cm2. The good agreement with the experimental Jsc is an indication that radiative recombination probably captures (most of) the dominant mechanism of electron-hole recombination in the perovskite. The second fundamental parameter for the solar cell performance is the open-circuit potential (Voc), i.e. the value for which short-circuit and bias currents cancel. We obtain Voc ∼1.28 eV, against typical measured values around 1.0 eV (among the highest, we can mention Voc =1.05 eV in solar cells with organic transport layers

11 12

,

and

1.07 eV in vapour-deposited perovskite films52). This overestimation is rather obvious, considering the ideality of our calculation which do not include current leakage (shunt resistance), series resistances, and most importantly, the voltage loss due to band offsets between perovskite and electron/hole transporters25. A direct consequence of larger Voc is that the calculated PCE are considerably larger than the presently measured efficiencies (to the best of our knowledge the record-high value reported in literature is 15.9% for the meso-superstructured cell9). Looking at the calculated Pout in Fig.3 (since Pin = 100 mW/cm2, Pout is also the PCE in percent) we obtain PCE = 16% for L=100 nm, 21% for L=200 nm, and more than 23% for L=300 nm. We remark that our PCE values represent the "intrinsic" limit to the PCE, i.e. the efficiency of the material itself and solely due to the fundamental absorption and recombination properties of the absorber. While probably not reachable in practice (since shunt and series resistances could never made completely negligible) they nevertheless represent a useful reference point for the device performance, and indicate that a large PCE improvement should still be expected for the perovskite-based solar cell, in terms of better morphological characteristics, device optimization, and optimal band alignment with the carrier transporting materials.


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Figure 4: J-V characteristic curve for the solar cell with perovskite P0 (solid lines) and P1 (dashed lines) as absorber. Each panel is relative to a specific film thickness L indicated in the figure. The difference between the two perovskites stems from their band gap difference (see text). Another fundamental aspect for what concerns the device functionality is the energy gap. To explore its impact on the perovskite efficiencies, in Fig.4 we compared currents and powers for P0 and P1 perovskites for three different thicknesses. The gap of P1 is about 0.08 eV lower than that of P0 (see Tab.1), a difference large enough to produce remarkable differences in the current-voltage characteristics. Fig.4 clearly shows that the current Jsc increases remarkably with the gap closure and this effect grows with L (i.e. ∆Jsc ∼1 mA/cm2 for L=100 nm, ∆Jsc ∼2 mA/cm2 for L=400 nm). This is easily understood, since the maximum current gain for decreasing EG occurs for 100% absorption probability. On the other hand, the recombination rate, and in turn the bias current, grows with decreasing EG. Indeed, we can see that for P1 the total current has a faster decay than for P0, and the corresponding Voc decreases to ∼ 1.25 eV for L=100 nm, and 1.20 eV for L=400 nm. For the L=100 nm sample, the combined increase of Jsc and decrease of Voc nearly compensate in the resulting power, which thus remain substantially constant for the two perovskites. Only for L=400 nm the gain in Jsc overcomes the decrease of Voc, so that output power and efficiency of P1 becomes larger than in P0 by about 1%. In words, a decreases of EG in


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the order of about 0.1 eV does not bring significant efficiency improvements, at least for thicknesses in the range of interest (100-300 nm). This is consistent with Ref.13 where the methylammonium molecule is replaced by formamidinium. The molecule substitution induces a smaller band gap (1.48 eV) and, in turn, a very high Jsc = 23 mA/cm2, but the measured efficiency of 14.2% is not sizably higher than in MPbI3. 3.3 Atomic Structures Present theoretical description of recombination rates and current-voltage characteristics is directly linked to the basic properties of the perovskites since it takes explicitly into account the actual (i.e. ab-initio simulated) material structures. In what follows we describe the perovskites included in our previous analysis, starting from their atomic structure obtained by atomic force minimization.

Figure 5: Top-view of the 4 structures analyzed in the text, resulting from our ab-initio total energy minimization. Those are √2x√2x2 supercells with 4 formula units and 48 atoms per cell; the (a,b) plane is 45o-rotated with respect to the (a0,b0) plane of the cubic perovskite cell, while c


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and c0 axis are parallel and orthogonal to the page. P0 and P1 are the orthorhombic Pnma structures with the lowest (ground-state) and second-lowest (first excited state) energies: I0 and I1 are the corresponding states for the tetragonal I4/mcm symmetry. The two (001) layers of the supercell are shown separately, to better distinguish the molecular orientation in the two layers. Top panels: z=0; bottom panels: z=c/2. Atomic color code: C (yellow), N (gray, small balls), H (cyan), I (red), Pb (gray, large balls). We started our calculations from lattice parameters and symmetries reported by the experiments 47,53. At low temperature the structure is orthorhombic Pnma, and characterized by a pattern of Glazer octahedral rotations (tilting) of the kind a-a-c+. At room temperature, MPbI3 is tetragonal I4/mcm with octahedral tilting pattern a0a0c-. Both symmetries are described by a √2×√2×2 4-formula unit supercell, whose (a, b) plane is rotated by 45o with respect to the (a0,b0) plane of the cubic cell (see Fig.5). A peculiar difficulty when dealing with this hybrid structure is the determination of the molecule atomic positions: in absence of a well defined high-symmetry configuration, the molecule can be seen as a C-N dimer whose center is located nearby the A-site of the perovskite; however, the dimer orientation remains substantially undetermined to X-ray diffraction. However, the most likely orientations are those with C-N axis parallel to high symmetry directions [110], [100], and [111], due to their large configurational entropy. Furthermore, not only the absolute orientation, but also the relative orientation of the molecules with respect to each other should be considered. In our study, we include all the possible molecular patterns compatible with tetragonal and orthorhombic supercells. Having four molecules per unit cells, this greatly expands the number of nonequivalent molecular configurations included in the analysis. Since each configuration corresponds to a structural local minimum, the determination of the absolute ground state requires the structural optimization of a


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large number of configurations assumed as a possible starting point. After the full atomic relaxation, the ground state is selected as the structure with lowest energy. At the end of this lengthy process, we obtained the structures displayed in Fig.5. Our results draw a complex scenario formed by a large number of possible molecular sublattice configurations in subtle competition with each other. The most stable molecule orientations are those with C-N dimers nearly parallel to the [110] axis of the supercell; in comparison, configurations with molecules parallel to [100] and [111] directions are highly unfavorable, and can be discarded. Within the [110] molecular orientation there are still a large variety of possible structures. We can distinguish two main typologies: parallel ("chain-wise") and orthogonal ("checkboard-wise") molecule orientation. In the former the dimers align in parallel forming [110]-oriented parallel chains. P0, the orthorhombic ground-state, belong to this class (see Fig.5). Notice also that the chains in two consecutive (001) monolayers are flipped by 90o with respect to each other. The first excited state P1 belongs instead to the class of orthogonal orientation: now any two adjacent dimers are tilted by 90o along [110], giving rise to a checkboard molecular ordering in the (a, b) plane. Again, the molecules in two adjacent planes are tilted by 90o with respect to each other. For the tetragonal symmetry the situation is reversed: the lowest-energy structure I0 displays a checkboard molecular sublattice, while in the first excited state I1 the dimers are oriented in [110]-parallel chains. At variance with the other cases, for I1 the molecules in two consecutive (001) monolayers of the cell are also parallel to each other.

Table 2: Atomic structure for the MPbI3 perovskite in four different molecular configurations obtained by total energy minimization. Cell parameters a, b, c, are fixed to the experimental


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values of the √2×√2×2 supercell measured at T=100 K47 for orthorhombic phase, and T=220 K for the tetragonal phases53; apc, cpc are lattice parameters of the equivalent-volume pseudocubic supercell (see text). θa is the cell-averaged Pb-I-Pb angle related to octahedral rotations around axis a0 and b0, θc the cell-averaged Pb-I-Pb angle corresponding to octahedral rotations around axis c; da is the average lead-iodine distance in the plane, dc the average lead-iodine distance along c; Etot is the total energy per formula units, relative to the energy-lowest P0. P0

P1

I0

I1

a (Å)

8.836

8.836

8.80

8.80

b (Å)

8.555

8.555

...

...

c (Å)

12.58

12.58

12.68

12.68

a/apc

1.008

1.008

0.994

0.994

b/apc

0.976

0.976

...

...

c/cpc

1.015

1.015

1.010

1.010

θa (deg)

164.3

175.5

180

180

θc (deg)

150.6

151.5

151.3

151.9

da (Å)

3.182

3.174

3.211

3.211

dc (Å)

3.175

3.156

3.167

3.167

Etot (meV)

0

4

9

25

The most important structural parameters are summarized in Tab.2. We can see that the difference in energy per formula unit is rather small, in the order of a few tents meV; thus it is reasonable to assume that the actual equilibrium structure could be best represented as a thermodynamic average over fluctuating molecular orientations. Consider first the orthorhombic Pnma structure. It is useful to introduce the pseudocubic lattice parameters a pc = 2 (Ω / 4)1 / 3 =


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8.76 Å and c pc = 2(Ω / 4)1 / 3 = 12.39 Å, i.e. the lattice parameters of a cubic cell with equivalent volume Ω. Since the (a, b) average is 8.70 Å and c =12.58 Å, it follows that the Pnma structure is squeezed in the (a,b) plane, and stretched along c with respect to the corresponding pseudocubic structure. As a result of the planar shrinking, the Pb-I-Pb angles rotate orthogonally to the c (or c0) axis (θc), sizably decreasing from the ideal 180o value. Rotations around a0 and b0 axis (θa) are much smaller, consistently with the c axis elongation. On top of the tetragonal distortion there is also a large planar distortion (a/b=1.03, √2a/c=0.993, √2b/c=0.962), which lowers the symmetry to orthorhombic. The difference between P0 and P1 is especially evident in the value of θa, which for the latter is close to the ideal 180o value. For what concerns the tetragonal structures, only the octahedral tilting around c axis are present, whereas θa=180o. The pseudocubic parameters (apc = 8.85 Å, cpc = 12.52 Å) are slightly expanded with respect to the orthorhombic values. This expansion plays against the tetragonal distortion, which still persists, but is slightly reduced (√2a/c = 0.98). To summarize, for both symmetries the octahedra are strongly tilted around the c=c0 axis, and weakly tilted (for P states) or not tilted at all (for I states) around a0 and b0 axes. It follows that the Pb-I-Pb bonds are nearly straight segments along c0, and strongly bent in the (a0, b0) plane. This Pb-I-Pb bending crucially affects the overlap of Pb(6s,6p) and I(5p) states, and in turn, electron hopping and bandwidth in the plane: of course the larger the tilting, the smaller the Pb-I electron hopping and bandwidth. From this analysis we should expect the Pb-derived bands to be remarkably anisotropic, i.e. broader and with lighter masses along kz than in the (kx, ky) plane, as shown in the next Section. Finally, the driving force of the octahedral tilting and its relation with molecular patterns leads back to the concept of tolerance factor, for which a detailed discussion can be found in Ref.41.


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3.4 Band structures The band energies for the four perovskites are reported in Fig.6. For clarity, only a small energy interval surrounding the band gap is shown. Let us analyze first the conduction bands that are dominated by Pb(6p) states in the 2 eV-wide region above the CBB. Due to the tetragonal distortion, at the Γ-point they split into a singlet and a doublet; being anti-bonding, the stretching along the c axis favors the stabilization of 6pz with respect to (6px, 6py) states. Consider first the P0 band structure: along Γ-Z (i.e. [001]) we can observe the splitting between a dispersed, highly parabolic band derived from the ligand 6pz orbital, of energy 1.58 eV at Γ, and two flat bands located 163 meV above, and derived from 6px, 6py states. The doublet actually splits (by ∼50 meV) due to the orthorhombic distortion. Moving to higher energy, another doublet is visible just above 2 eV, which is nothing but the downfolding of the same bands due to the cell doubling along the c axis. Finally, the other dispersed band of 6pz orbital character (there are two Pb per plane in the supercell) is present at 2.3 eV. Looking now at the Γ-L direction (the [110] direction of the rotated supercell, corresponding to [100] of the cubic lattice), we have that the lowest 6pz band becomes heavy (mL* = 0.89 me ) and the doublet splits into a light 6px (m*L = 0.11 me ) and a heavy 6py band (mL* = 1.59 me ) . Due to the large octahedral tilting around c, the 6pz mass along z is slightly lighter than the 6px mass along x (or the 6py mass along y). For what concerns the valence bands, at the very top there is a singlet with rather light and nearly isotropic effective masses, (mZ* = 0.21 me , m*L = 0.33 me ) mainly derived by Pb(6s) states of bandwidth 0.246 eV along Γ-Z, and 0.318 eV along Γ-L. The qualitative features described above for P0 are found also for the P1 structure though small but crucial quantitative differences occur; for example, the band gap is about 0.08 eV smaller for


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P1. The decrease of the band gap comes from very subtle band broadening effects: in conduction, the band bottom splitting between the broad 6pz and the flat (6px, 6py) bands is 0.152 eV for P1, against 0.163 eV for P0. This increase of p-state splitting is consequence of a slightly stronger tetragonal distortion for the latter, due to its parallel-chain molecular pattern. On the other hand, in the valence, the 6s band dispersion is 0.370 eV for P1, against 0.246 eV for P0. The addition of these two opposite changes produces a total bandwidth increment of 0.11 eV for P1 which substantially accounts for the reduced EG with respect to P0: in words, the gap decrease in P1 comes from the valence s-state broadening, partially compensated by the smaller p-state splitting (smaller tetragonal distortion) of P1.

Figure 6: Ab-initio calculated band energies for MPbI3 in four different structures (described in the text). K-points are Z=[0,0,π/c], L=[π/a,π/a,0] in the (a, b, c) cartesian reference of the


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√2x√2x2 supercell; Z=[0,0,π/(2a0)] and L=[π/(2a0),0,0] in the (1x1x1) reference system. Energy zero is fixed at the VBT.

The previous description can be identically repeated for the tetragonal structures as well. Consider that no differences in band degeneracy are visible with respect to P0 and P1 since, due to the molecular sublattice, tetragonal point symmetries are lost anyway. For what concerns the I0 vs. I1 band gap analysis, the p-state splitting in conduction gives 0.188 eV for I0 and 0.254 eV for I1; i.e. coherently with the case of orthorhombic structures, the I1 parallel-chain molecular pattern enhances the tetragonal distortion (and in consequence the 6p orbital splitting) with respect to the checkboard molecular pattern of I0. The s-state bandwidth in the valence is 0.266 eV for I0 and 0.327 eV for I1. The addition of these two band broadening effects (0.127 eV) accounts for the decrease of I1 gap with respect to I0. 3.5 Optical functions The optical properties of the hybrid perovoskite are reported in Fig.7 for the P0 case (for the other structures they are quite similar). The difference among the three cartesian components indicates a remarkable tetragonal anisotropy. Interestingly, the absorption along z is visibly higher than along the in-plane directions. This may seem in contrast to the results for the effective masses, lower along c and higher in the plane. However, it can be understood considering that, due to the tetragonal distortion (elongation along c axis), the dominant dipole transition at Γ-point involves Pb(6s) and Pb(6pz) orbitals as initial and final states. By symmetry considerations, it follows that only the z-component of the dipole does not vanish. Clearly, outside Γ-point, the symmetry cancelation does not apply.


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According to our results, the perovskite is characterized by high-frequency electronic dielectric constant ∼ 5, 5.2, and refraction index ∼2.2, 2.4.

Figure 7: Ab-initio calculated optical properties for MPbI3 (P0 structure).

4 CONCLUSIONS In conclusion, we studied from first-principles the radiative electron-hole recombination properties of the MPbI3 starting from realistic atomic structures determined by total energy minimization, and from the accurate determination of the optical functions. Electron-hole recombination is a key quantity to furnish a solid conceptual background to the interpretation of PL lifetime measurements, and an essential ingredient in the accurate determination of what we call 'intrinsic' power conversion efficiency, i.e. the efficiency purely derived by the perovskite used as absorber in the solar cell, thus unaffected by contributions of spurious series or shunt resistances, or by the band misalignment of the absorber with electron and/or hole transporters.


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Our analysis highlights the great qualities of the perovskite as absorber in solar cells: strong absorption coefficient, comparable with that of GaAs (used in this work as the reference prototype of semiconducting absorbers); small radiative recombination rate (with respect to GaAs) and long radiative recombination lifetimes (in the order of 103 ns for n=1015 cm-3 doping concentration); large short-circuit currents and (potentially) huge power conversion efficiencies, ranging between 16% for L=100 nm perovskite thickness, up to 25% for L=400 nm.

Ultimately, these excellent values derive from a peculiar aspect of these materials: while 'hybrids' in their capability of being processed in solutions, they are nevertheless fully 'inorganic' for what concerns optical and photoconversion properties, since the electronic states in the important energy region around the band gap are entirely represented by extended states (bands) derived by Pb and I orbitals. Pictorially, in this region the perovskite is like a GaAs with inverted band character (mainly s-states in the valence, p-states in conduction) and increased band gap. The increase of band gap largely reduces both the recombination rate and the intrinsic carrier concentration; these two changes mainly compensate in the radiative lifetime which is thus similar in the perovskite and in GaAs; most importantly, the gap increase is not detrimental for power conversion efficiency, since, for the thicknesses of interest, the decrease of short-circuit current due to the higher band gap and lower illumination is compensated by lower dark current and higher open-circuit voltage.


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SUPPORTING INFORMATION Atomic structure for the MPbI3 perovskite in the four different configurations (P0, P1, I0, I1) obtained by total energy minimization are available free of charge via the Internet at http://pubs.acs.org.

CORRESPONDING AUTHORS *to whom correspondence should be addressed: [email protected], phone: +39 070 675 4875 (A.F.) [email protected], phone: +39 070 675 4868 (A.M.) ACKNOWLEDGMENTS The Authors acknowledge financial support by the Italian Institute of Technology (IIT) under “Platform Computation”, by Regione Autonoma della Sardegna under L. R. 7/2007 (CRP24978 and CRP-18013), by Consiglio Nazionale delle Ricerche (Progetto Premialità RADIUS), by Fondazione Banco di Sardegna Projects n.5794 and n.7454. They also acknowledge computational support by CINECA (Casalecchio di Reno, Italy) and CRS4 (Loc. Piscina Manna, Pula, Italy).

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TOC: The excellent photoconversion properties of lead-iodide hybrid perovskites are explained on the basis of ab-initio calculated radiative recombination rates and minority carrier lifetimes comparable to that of typical optoelectronic semiconductors (e.g. GaAs) reflecting the very nature of the perovskite: fully solution-processable own to its hybrid nature, and yet truly inorganic semiconductor for what concerns photoconversion properties.


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