Role of Lead Vacancies for Optoelectronic Properties of Lead-Halide

Article Cite This: J. Phys. Chem. C 2018, 122, 5216−5226

pubs.acs.org/JPCC

Role of Lead Vacancies for Optoelectronic Properties of Lead-Halide Perovskites Dayton J. Vogel,† Talgat M. Inerbaev,‡,§ and Dmitri S. Kilin*,†,∥ †

Department of Chemistry, University of South Dakota, Vermillion, South Dakota 57069, United States L.N. Gumilyov Eurasian National University, Astana 010008, Kazakhstan § National University of Science and Technology MISIS, Moscow 119049, Russian Federation ∥ Department of Chemistry and Biochemistry, North Dakota State University, Fargo, North Dakota 58108, United States ‡

S Supporting Information *

ABSTRACT: Methylammonium lead iodide perovskite materials have been shown to be efficient in photovoltaic devices. The current fabrication process has not been perfected, leaving defects such as site vacancies, which can trap charge and have a detrimental effect on photogenerated charge carriers. Here focus is placed on the effect a Pb site vacancy has on the charge-carrier dynamics following photoexcitation. The electronic structure of materials with vacancies is often found in open-shell configurations with unpaired electrons in the conduction/valence bands To accurately describe unpaired electrons, spin-polarized and noncollinear spin calculations are performed on both neutral and charged vacancy systems. This work presents spin-polarized and noncollinear spin ground-state electronic structures and nonradiative rates of charge-carrier relaxation and introduces an extension to a novel procedure to compute photoluminescence spectra for open-shell models. This study describes nonadiabatic dynamics of MAPbI3 models within a Redfield formalism focusing on the role of a Pb vacancy defect on electronic relaxation processes. Results show the vacancy of the Pb ion introducing a new energy state within the unblemished material band gap region. This additional unoccupied state is expected to increase the nonradiative relaxation lifetime of the excited electron, allowing for a longer lifetime of the charge carrier and increased opportunity for secondary relaxation mechanisms or collection to take place.

1. INTRODUCTION The introduction of methylammonium lead halide perovskite materials into photovoltaic devices in 2009 has resulted in increased interest in the material’s wide range of applications.1 A large factor increasing interest in the material is that PV device efficiency has now been increased to 22.1%.2 Photoinduced processes within MAPbI3 are fundamentally important to optoelectronic applications and have been experimentally observed through a range of optical and spectral monitoring techniques.3 Within the available optical characterization data, photoluminescence signatures can provide keen insight into specific photoexcited processes and the corresponding material composition giving rise to unique findings. As the synthesis of MAPbI3 materials has been developed as a low-cost solutionbased process, it is not uncommon to find material defects and impurities.4,5 These defects can range from surface defects, interstitial positioning, ion migration, antisite populations, and vacancies.6−9 Each type of material defect has an impact on the electronic structure of the MAPbI3, with vacancies and surface defects providing possible locations for charge-carrier recombination and trapping states.10 Depending on the target application of the material and its desired properties, these defects can be unwanted or intentionally manufactured by choice of precursor ratios.11,12 © 2017 American Chemical Society

A removal of a positive atomic species from a hybrid organic−inorganic perovskite material induces a localized charge due to electrons left at the defect site. The excess charges need to be accounted for as the number of charges will modify the HOMO/LUMO energy levels and the resulting band gap energy. In the situation where there is excess charge, it should be noted that open-shell electronic configurations are possible and must be taken into consideration. Open-shell systems require spin-polarized calculations that have been readily applied to systems with heavy elements and high spin multiplicities. The “spin-polarized” density functional theory (DFT) is a typical work-horse implemented into electronic structure software packages, allowing qualitative analysis of electronic properties. The explicit description of spin components has been shown to be of importance within MAPbI3 materials.13,14 While spin-polarized approach provides a basis for qualitative analysis, the noncollinear spin approach helps us to build complete picture. To assess and minimize the suppressive influence of defects onto optoelectronic properties of MAPbI3, nonadiabatic Received: June 1, 2017 Revised: December 5, 2017 Published: December 18, 2017 5216

DOI: 10.1021/acs.jpcc.7b05375 J. Phys. Chem. C 2018, 122, 5216−5226

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The Journal of Physical Chemistry C

state electronic structure calculations. With each type of defect there can be changes to both the morphology and electronic relaxation channels within the material. Studying a single isolated point defect and the resulting optical properties can be a challenging task for an experiment. This work focuses on accurately studying the effect of a VPb and the resulting electronic structure, charge carrier dynamics, and emission mechanisms in three bulk cubic MAPbI3 systems. The systems provide a pure MAPbI3 material, VPb(σ), V+Pb(σ), and 2+ VPb(σ) . The charge system provides an opportunity for computational model conditions representing open-shell electronic systems due to excess charge. The use of approximations within this computational study allows for strong qualitative trends to be found regarding spin-polarized calculations. Computed data, even for simplistic atomistic models, can provide qualitative insight for others researching MAPbI3 materials. An important issue facing theoretical modeling of MAPbI3 materials is the necessity to include SOC due to the heavy Pb and I elements. The largest constraint for results accounting for SOC is the computational time required. In this work, basic data in the presence of SOC are provided. However, the meticulous development and testing of incorporating SOC into nonadiabatic dynamics has been applied to lanthanide-doped systems.32

dynamics may be used to highlight fundamentally important mechanisms that pertain to (a) the relative change in electronic structure across models representing multiple possible defect environments, (b) their respective excited-state relaxation mechanisms and (c) a protocol to convert general quantumstate dynamics into observables. Such modeling can be completed through a range of methods and approaches, with each unique defect condition necessitating the level of methodology needed. (a) Initial comparisons of ground-state electronic structure are calculated using DFT15,16 and used to highlight changes due to a specific defect and to show the necessity of spin-polarized calculations17,18,18 or noncollinear spin calculations. Electron dynamic calculations begin with molecular dynamic calculations, providing modeling of atomic positions under specific conditions. Interesting insights into the thermally induced restructurization- and photoexcitation-induced processes can be obtained by nonadiabatic molecular dynamics,19 excited-state molecular dynamics,20 or time-dependent excited-state molecular dynamics.21 (b) The excited-state dynamics is critically needed to assess the ability of a material with a defect to emit light. Photoemission can be computationally modeled by a synergy of radiative and nonradiative processes following a photoexcitation. The nonradiative dissipative dynamics of excited states primarily originates from interaction with thermalized nuclear degrees of freedom and can be modeled by a range of approaches. Most of atomistic approaches to nonradiative dynamics rest on the time-dependent electronic structure monitored along an adiabatic MD trajectory, allowing for computation of “on-the-fly” nonadiabatic couplings.22,23 The time propagation of the excited state can be pursued by multiple spawning methods,24 a family of methods based on surface hopping approach.25,26 Another option for modeling time evolution of the excited state stems from the density matrix approach,27,28 which has been recently implemented in a form appropriate for atomistic modeling of closed- and openshell configurations.29,30 Nonradiative relaxation rate calculations in any of these methods are facilitated by atomic motion. (c) Observables indicating excited-state phenomena, such as an excited state or a pair of hot carriers (electron and hole), can be calculated to follow charge-carrier migration through a material and the multiple processes that may occur. Photoluminescence is dependent on the spatial and energetic location of the charge carriers, which is calculated through electron dynamics.31 Along the dynamic trajectory, all possible emission events may be explored following a specific initial excitation. This work also reports an adaptation of spin-polarized nonadiabatic dynamics codes for computing photoluminescence of open-shell species. This can provide insight into spinpolarized systems and unique photoluminesence (PL) spectral features previously unattainable through closed-shell theoretical methods. This methodology is critically needed to computationally assess optical trends in open-shell electronic configurations, radicals, and dilute magnetic semiconductors. In this work, a single Pb point defect (VPb) is created within bulk cubic MAPbI3 to monitor the resulting changes in electronic structure. To study a single VPb this study is conducted computationally using ab initio DFT and nonadiabatic formalism to study electronic structure, molecular dynamics, and nonadiabatic transitions relating to the resulting optical properties. The importance of material defects in MAPbI3 has been extensively covered through various ground-

2. METHODS 2.a. Theoretical Approaches. Computational characterization of material’s properties is achieved in five directions: (i) ground state electronic structure, (ii) linear optical absorption, (iii) thermal motion of nuclei, (iv) non-radiative transitions, and (v) photoluminescence. 2.a.I. Electronic structure with spin, spinors, and momentum dispersion. Electronic structure with spin, spinors, and momentum dispersion. Accurate analysis of the open-shell system requires the use of spin-polarized DFT or noncollinear spin DFT calculations. Noncollinear spin DFT as implemented in VASP has been used to calculate the effects of spin orbit coupling upon the ground-state electronic structure of the modeled systems.33−35 This methodology has been detailed in previous work.32 The relativistic Hamiltonian, includes scalar relativistic and spin−orbit terms. These corrections are added to self-consistent Kohn−Sham equation on a noncollinear spin eff NCS φiσ′( r ⃗), with basis ∑σ = α , β ( −δσσ′∇2 + vσσ ′( r ⃗))φiσ ( r ⃗) = εi eff vσσ ′ =

δE δρσσ

. The noncollinear spin DFT procedure provides ′

components of spinor orbitals, which represent two-component ⎧ φ ( r ⃗)⎫ + v e c t o r s , ψiKS( r ⃗) = ⎨ φiα( r )⎬, ∫ dr ⃗ 3 (|φiα|2 + |φiβ |2 ) = 1. ⃗ ⎩ iβ ⎭

Note that for the noncollinear spin approach, veff αβ ≠ 0, while for the spin-polarized approach, vαβ = 0, and only diagonal densities take nonzero values: ραα(r)⃗ → ρα(r)⃗ , ρββ(r)⃗ → ρβ(r)⃗ .36 Solutions of the spin-polarized ground-state electronic structure are calculated according to the single-electron Kohn− Sham equation, with the solutions provided on the basis of KS Kohn−Sham orbitals, φKS iσ , and energies, εiσ , where σ = α, β specifies spin-up or spin-down. The spin-resolved densities N N ρα ( r ⃗) = ∑i =α1 |φiKS |2 and ρβ ( r ⃗) = ∑i =β1 |φiKS |2 are parame,α ,β trized by total number of electrons Nα, Nβ with α, β (+1/2 and −1/2) values of spin projections. A constraint to the sum, N = Nα + Nβ, and difference, ΔN = Nα − Nβ, of these parameters allows us to describe a broad variety of charged 5217


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The Journal of Physical Chemistry C species and nonsinglet electronic configurations.37 For example, in one of the created open-shell systems, it is found that there is one more alpha electron than beta, ΔN = Nα − Nβ = 1, yielding ΔN a doublet spin S = 2 and multiplicity 2S + 1 = 2. The periodic atomistic models are most efficiently treated by KS equation on the basis of plane waves, where each jth KS orbital is defined for a range of independent values of net m o m e n t u m k⃗ , t h u s r e p r e s e n t i n g t h e j t h b a n d ⃗

⃗

expressed via partial contributions from each value of the momentum

⃗

∑ δ(εi,σ − ε)

2

N PI⃗ 2mI

d2 ⃗ RI = FI⃗ [ρα , ρβ ]/MI dt 2

(2a)

For noncollinear spin DFT, the transition dipole moment can be found using the spinor form of KS orbitals ⎧φ ⎫ ⟨Dij⃗ ⟩ = ∫ {φi*α φi*β } r ⃗⎨ φjα ⎬ d r ⃗ . Note that, in ⎩ jβ ⎭

RDF(t , r ) =

j

noncollinear spin basis is explained in ref 43. An autocorrelation of NACs, Mijkl,σσ′(τ) = ⟨Vσ,ij(t)Vσ′,kl(t + τ)⟩, followed by a Fourier transform44 provides components of the terms of Redfield tensor, Rσ,jklm. The Redfield tensor provides parameters of the equation of motion of electronic degrees of freedom within eq 7

3ℏe

∑ fσ , ij (t )δ(ε − Δεσ , ij(t ));

a(ε) = aα(ε) + aβ (ε)

σ , ij

(2b)

ρσ̇ , ij = −

where explicit time dependence can allow monitoring of structural rearrangements and resulting spectral signatures.39 In the case that a single k-point is insufficient in correctly reproducing experimental spectra, a method of partial contributions from multiple k-points may be used. For multiple k-points the matrix elements for the transition dipole moment are computed under the assumption of momentum conservation, D⃗ ik,jk ⃗ ’⃗ = D⃗ ij,kδ ⃗ k,k ⃗ ’⃗ , and can be used to build the 4πmevij , k ⃗ corresponding oscillator strengths, f ⃗ = |Dij⃗ , k ⃗ |2 2 . The ij , k

⎛ dρσ , jk ⎞ ⎟⎟ ⎝ dt ⎠diss

∑ (Fσ ,ikρσ ,kj − ρσ , jk Fσ ,ki) + ⎜⎜ σ ,k

(6)

where the change in density matrix due to nuclear motion is defined as

dρσ , jk

( ) dt

diss

= ∑σ , lm 9σ , jklmρσ , lm. Diagonalization of

the Liouville and Redfield superoperators acting on the density matrix

3ℏe

(3σ + 9σ)ρσ(ξ) = Ω(σξ)ρσ(ξ)

(7)

provides a superposition of eigenvectors, Ω(ξ) σ , and eigenvalues, ρ(ξ) σ . The superposition of eigenvectors is used to compose the density matrix at each instant of time, allowing for solution of expansion coefficients for each transition at time zero according to

∑ fij ,k ⃗ δ(ℏω − ℏωij ,k ⃗) ij

i ℏ

= (3σ + 9σ)ρσ

partial contributions to absorption from fixed values of momentum read ak ⃗(ℏω) =

(5)

IJ

d

the absorption, aσ, according to aσ (ε , t ) =

∑ δ(r − |RI⃗ (t ) − RJ⃗ (t )|)

Vσ , ij(t ) = −iℏ φσKS | |φ KS 41,42 An analogous procedure for , i dt σ , j .

i

The transition dipole moment is then used in calculating the oscillator strength, fσ,ij, for the corresponding transition using 4πmevσ , ij f = |Dσ⃗ , ij |2 2 . The oscillator strength is used to weigh σ , ij

1 4πr 2

with R⃗ I(t) being positions of the Ith ion. 2.a.iv. Non-radiative transitoions. To connect the equilibrium thermal nuclear motion and effect on electronic structure, on-the-fly nonadiabatic coupling (NAC) in spinpolarized basis, V σ , i j , is calculated according to

= ∫ d r ⃗ (φi*α ( r ⃗) r φ ⃗ jβ ( r ⃗)) ⃗ jα( r ⃗) + φi*β ( r ⃗) r φ plane-wave basis set, eqs 2a and 2b are cast as iℏe ⟨i|D̂ |j⟩ = ⟨i|p ̂|j⟩ m (ε − ε ) with details provided in the SI. e

(4)

The atomic coordinates are collected at each time step of the MD trajectory, {R⃗ I(t)}, allowing for single-point electronic structure analysis. In some cases, thermalization may induce structural rearrangements for the model and facilitate adsorption, desorption, or bond formation.40 Calculation of the radial distribution function (RDF) for specific interatomic distances is found along the molecular dynamic trajectory according to

*

∫ φσKS,i r φ⃗ σKS,j d r ⃗

3

= 2 NI kBT . Upon reaching thermal

stability, the atomic momenta are used as input to calculate an adiabatic ground-state MD trajectory by

with an appropriate modification of eqs 2a and 2b for noncollinear spin and momentum resolved data. 2.a.ii. Linear absorption with spin, spinors, and momentum dispersion. The transition dipole moment in spin-polarized calculations, D⃗ σ,ij, is calculated within the independent orbital approximation (IOA),38 and under neglect of spin-flip transitions between each set of two specified orbital pairs, i and j, with spin conservation, according to Dσ⃗ , ij = e

(3b)

k⃗

at each time step ∑I I

(1)

i

∑ ak ⃗(ℏω)

with normalization N. 2.a.iii. Thermal motion. Modeling atomic motion is accomplished by heating and adiabatic molecular dynamic (MD) simulations. The atomistic system is interacted with a thermostat of a chosen temperature with rescaled momenta, P⃗ I,

φjKS ( r ) = ∑|G⃗| ≤ G CGj,⃗ k ei(G + k )· r ⃗ in terms of expansion coef,k⃗ ⃗ cut ⃗ iG⃗ ·r⃗ ficients Cj,k G⃗ for each plane wave e . Characterization of the ground-state electronic structure is visualized by a density of states computed as

nσ (ε) =

1 N

a tot(ω) =

(3a)

where the resonance condition follows the dispersion of bands i ⃗ The total absorption spectrum is and j ℏωij,k⃗ = εj(k)⃗⃗ − εi(k). 5218


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Figure 1. Model of 3 × 3 × 3 cubic MAPbI3 perovskite with central Pb-vacancy (Vac). Anionic configuration facilitates the formation of quasi-stable I2 molecules at the place of Pb vacancy. Formation of the I2 molecule can be quantitatively monitored via decrease in I−I bond length and increment of the Pb−I bond length (dashes). The model under study shows the location of the Pb vacancy as a 2D slice (A), 3D model (B), and an example of band structure (C), which shows negligible dispersion of defect state; see Figure S3 for details.

ρσ , ij (t ) =

2.b. Computational Details. The presented models are used as a simplistic structure for testing isolated defects and a basis for continuing work of calculating qualitative properties of MAPbI3. The models allow for the study of photocarrier dynamics in a range of approximated electronic environments that are possible within MAPbI3 materials. In this work, four systems with a combination of morphological and electronic parameters are specifically highlighted for comparison. The models used are a supercell of cubic MAPbI3 with dimensions of 3 × 3 × 3 using periodic boundary conditions to simulate a bulk material. The systems under consideration are (1) MAPbI3 bulk cubic material with no defects, (2) V+2 Pb, which is bulk cubic MAPbI3 with the central Pb atom removed, as seen in Figure 1, (3) V+Pb(σ), and (4) VPb,S=0 and VPb,S=1,σ, which are the V+2 Pb systems with two additional electrons added to the system in both a singlet, S = 0, and triplet, S = 1, electronic configuration. Note that the notation of V+2 Pb below should be interpreted in the following way: A 2+ lead cation is removed to form vacancy, while anionic environment is kept intact.

(ξ) σ t

∑ ⟨ρσ(a,ij, b)(0) ρσ(ξ)⟩ρσ(ξ)eΩ

(8)

ξ

ρ(a,b) σ,ij (0)

where is the initial condition. This solution is often visualized using energy distribution nσ (ε) =

∑ ρii ,σ (t )·δ(εi ,σ − ε)

(9)

i

Expectation values of charge carriers are calculated according to

∑ ρii (t )εi(t )

⟨Δεe⟩(t ) =

⟨Ee⟩ =

i

⟨Δεe⟩(t ) − ⟨Δεe⟩(∞) ⟨Δεe⟩(0) − ⟨Δεe⟩(∞)

Assuming a single exponential fit, the rate of charge-carrier relaxation is found through ⟨Ee⟩(t ) = e−ket ;

ke = {τ e}−1 =

{∫0

−1

∞

⟨Ee⟩(t ) dt

}

(10)

MA +n Pb+n 2 I−3n1 → MA +n Pb+n −2 1I−3n1 + Pb+2   

Rates of electrons and holes, for alpha and beta spin projections, are calculated in the same fashion, and thus equations for only the electron with alpha projection are provided. 2.a.v. Photoluminescence. Calculation of the charge carrier density at each time allows for the time dependent photoluminescence spectra to be calculated according to eq 11a Eσ (ω , t ) =

∑

neutral

fσ , ij δ(ℏω − ℏωσ , ij){ρσ , jj (t ) − ρσ , ii (t )} (11a)

depending on the population between two states at some time, as shown by ρσ,jj(t) − ρσ,ii(t). The transition is characterized by the energy, ℏω − ℏωσ,ij, and the weight of the possible transition, fσ,ij. Summation of the total emission is integrated over all time and possible transition energies defined by the time-integrated emission (TIE)

∫0

∞

dt Eσ (ω , t )

Eint(ω) = Eαint(ω) + Eβint(ω)

(11b)

Also available from radiative relaxation rates are calculation of quantum yields (QYs). The QYs are shown in the panel τ corresponding to each system where QY = (τ +r τ ) . r

⏟ cation

+ For all spin-polarized models, VPb(σ) and VPb,S=1,σ, the distinction of spin projection, σ = α,β, is specified throughout the paper. Models (3) and (4) have manually increased total electron counts of one and two, respectively, to account for the two valence electrons lost due to the Pb2+ vacancy in a stepwise fashion. Utilizing the Vienna ab initio Software Package (VASP), any artificial excess charge interactions between unit cells due to the small size of the super cell and PBCs have been addressed and eliminated by VASP authors through an intrinsic tool of homogeneous compensating charge, which makes each cell neutral.45−47 The stoichiometry of the four systems are as follows: (1) MAPBI3 = Pb27I81MA27, where MA = H3CNH3; + − (2) V+2 Pb = Pb26I81MA27; (3) VPb(σ) = Pb26I81MA27 + e ; and (4) − VPb,S=0 and VPb,S=1,σ = Pb26I81MA27 + 2e . The VPb model is shown in Figure 1B, with a 2D slice showing the vacancy in the center of the unit cell (Figure 1A). Ground-state electronic structure calculations were computed within VASP48,49 using the generalized gradient approximation (GGA) Perdew−Burke−Ernzerhof (PBE)50 functional with projector-augmented-wave (PAW)51,52 potentials in a plane-wave basis set.53,54 Initial analysis on the effect of a VPb, and subsequent added charge, onto the periodic cubic MAPbI3 system begins with ground-state electronic structure

σ ,j>i

Eσint(ω) =

anionic

nr

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Figure 2. Density of states for closed-shell systems: perfect MAPbI3 (A), VPb+2 (B), and VPb singlet (C). Occupied and unoccupied states are shown as filled and unfilled, respectively.

of spin projections. In such systems φiα ≠ φiβ, εiα ≠ εiβ, and f iα ≠ f iβ . 57 In general, one can define spinor orbitals

calculations. Ground-state electronic structure calculations are performed at both the gamma point and for a 2 × 2 × 2 k-point mesh using the Monkhorst packing scheme.55

where () ∫ d3r (⃗ |φiα( r ⃗)|2 + |φiβ ( r )⃗ |2 ) = 1. However, in spin-polarized

|ψi = φiα( r ⃗)|α + φiβ r ⃗ |β ,

3. RESULTS To identify the effect of a Pb-vacancy point defect on the + optimized ground-state electronic structure, the V+2 Pb, VPb(σ), VPb,S=0, and VPb,S=1,σ systems are compared using their respective density of states (DOS). In Figure 2, the DOS, introduced in eqs 2a and 2b, is presented for each of the three closed-shell systems. Comparison of the DOS for the MAPbI3, defect V+2 Pb, and VPb,S=0 systems highlights two things. With the creation of Pb-vacancy defect, the relative shape and distribution of the available electronic states broadens at the band edges. Partial charge density distributions of the HOMO and LUMO orbitals are available (Figure S1). Of greater importance is the addition of a new unoccupied state located within the original band gap. Pb vacancies have been shown to be a p-type dopant in MAPbI3, providing an additional state very near the top of the valence band.6 The addition of a new unoccupied defect state is present in the absorption spectra (eq 4), which show new red-shifted peaks, shown in Figure S2. The new state not only reduces the band gap energy and the absorption onset but also provides new possible pathways for photoexcited charge-carrier relaxation processes. The energy of defect states, contributed due to the presence of the VPb, is expected to remain independent of the k-vector value (Figure S3) because at low concentrations the tunneling of electrons between defects states vanish. An increased number of unit cells for the considered simulation cell also indicates that the number of electronic bands will grow. Doubling the number of unit cells per simulation cell, one increases the number of bands within a system, observing that the k-points sampled will fold back on themselves as the Brillouin zone is reduced, presenting a direct gap at the gamma point.56 In the limit of the infinite size of the simulation cell, one expects there only to be a gamma point, indicating nonradiative transitions at the gamma point to be of significant value. It has also been found that the number of unit cells within the super cell will bend the direct band gap back to the gamma point for N × N × N super cells, where N = even integer. The calculated band structure for a 2 × 2 × 2 super cell is shown in Figure S4. Effects of the super cell size can also be seen in the absorption onset in Figure S5. The additional electrons placed into the V+Pb(σ) and VPb,S=1,σ systems create an open-shell electronic structure. The nomenclature of identifying the two spin components, α and β, allows for consideration of the fact that there are two values

approach, φiα(r) and φiβ(r) form an independent set of orbitals. Singlet configurations are defined by equal occupations f iα = f iβ. The DOS of the intrinsic and VPb systems are mirror images showing the expected symmetry of equal α and β electron counts. This mirrored nature of α and β DOS validates the use of closed-shell dynamics moving forward for the perfect and VPb systems. However, it will be necessary to treat the V+Pb(σ) and VPb,S=1,σ systems with spin-polarized methodology throughout the calculations. One benefit of utilizing benchmarked spinpolarized DFT is that unique spin-multiplicity mechanisms are available for study. Within lead halide perovskite materials triplet emission states are known to be bright emitting states. This radiative relaxation is a competing mechanism to the nonradiative pathway and is only accessible through methods that allow spin polarization. Previously published work highlights the need to understand spin polarization for specific defects within lead halide materials.58 To characterize the electronic structure of spin-polarized ground state structures, spin-polarized DOS may be used (Figure 3). The spin-polarized DOS exhibits nonsymmetric density located at the valence band edge between α- and βlabeled electrons. However, it is this difference in electronic state density distribution that allows for possible spindependent pathways of absorption and relaxation. In the V+Pb(σ) and VPb,S=1,σ systems the unoccupied β state near the VB edge is comparable to the unoccupied state in the VPb system. Because each spin system is unique, it is expected that electronic transitions will also be dependent on the spin. The energy values and the relative absorption energies of the spin system can be seen in Figure S6. The panels in Figure 3 provide the spin-polarized DOS for + the V+Pb(σ) and VPb,S=1,σ systems, respectively. The VPb(σ) and VPb,S=1,σ systems require spin-polarized calculations due to the uneven number of electrons, producing a doublet and triplet spin configuration in the ground-state electronic structures. For congruency, the DOS of the closed shell systems are calculated with spin-polarized methodology (Figure S7). For heavy elements and high spin multiplicities, spin-polarized DFT is a typical work-horse that allows for qualitative analysis. The DOS of the perfect system (Figure 2A) shows the band gap energy to be close to 3 eV. This is larger than the experimentally found 1.6 eV Eg,59,60 but in this case the interest

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carrier photophysics. The study of multiple environments may highlight unique relaxation mechanisms previously unseen. The ground-state electronic structures previously discussed provide adequate information to understand charge density distributions and possible optical transitions within the systems. However, they lack the ability to describe the molecular dynamics and nonadiabatic transitions within the materials. To simulate molecular motion at ambient temperatures, heating and molecular dynamic (MD) simulations are completed and used to evaluate trends in nonradiative cooling of hot carriers. The adiabatic MD trajectory propagated in the ground state provides new detailed atomic coordinates at each 1 fs time step. At each time step, the energies of all Kohn−Sham orbitals (KSO), εi(t), are calculated for the VPb system and can be seen in Figure S8. The fluctuations indicate that the individual KSO energies change due to the atomic motion within the system. At t = 0 a KSO energy in the V+2 Pb MD trajectory is seen in the fluctuation plot (Figure S8), corresponding to the additional vacancy state. However, along the dynamic trajectory, the KSO energy rises and settles at the bottom of the conduction band. This unique feature of a rising KSO energy has a corresponding molecular motion of an I2 molecule forming. The I2 molecule formation is found to occur at the VPb site, as seen in the 2D slice of Figure 1A. A time-dependent radial distribution function of I−I bond distances (see SI animated movie), shows a peak transition, shortly after 200 fs, toward an I2 bond length of 2.7 Å.63 Previous studies regarding I2 formation in relation to MAPbI3 defects have been established.64,65 However, in previous work, I2 formation is discussed as an interstitial I defect rather than the VPb presented here. In the current model, we show the formation of an I2 molecule at the defect site without the introduction of additional iodine atoms (Figure S9). It is the breaking of two Pb−I bonds and formation of one I2 molecule. It can also be postulated that I3− is a likely defect byproduct candidate given I3− is readily formed in DSSC devices that utilize I−/I3− mechanisms to replace oxidized dye molecules.66 The rate of I3− formation is relatively fast when it is within organic solvents, as implemented in DSSCs.67 This is dependent on the concentration and availability of I− within the solution. For MAPbI3 materials I− is bound to Pb2+, making the removal of an I− atom more difficult than in organic solvent. The PBE total energies for each of the systems can be found in Table S1. In comparing the systems, the trend in energies is found to be

Figure 3. Spin-polarized DOS for the open-shell V+Pb(σ) (a) and VPb,S=1,σ (b) systems. The two spin components, α and β, are visually presented as positive (red) and negative (blue) density, respectively.

lies in the trends due to the VPb. The incorrect energy of the band gap is known due to the band structure at the chosen kpoint, Γ. For the cubic morphology of MAPbI3, calculated band structures show a direct band gap at the R-point.61,62 Consideration of multiple k-points away from the gamma point allows for the direct band gap to be sampled, lowering the calculated band gap value close to the known experimental value, as seen in Figure S3. To verify a correct electronic structure for the models currently under study electronic structure calculations using a kpoint mesh were performed. A comparison of the DOS with and without a k-point mesh for the three closed-shell systems is presented in Figure 4. The DOS calculated utilizing a k-point mesh highlight the unique electronic structure of the material. With the inclusion of multiple k-points, the band gap values are corrected to experimental values. Although quantitative values of band edges shift with inclusion of multiple k-points, the qualitative ordering and location of defect states is maintained. Calculation of charge-carrier relaxation within multiple environments provides an entry level understanding of mechanisms contributing to experimentally measured phenomena. Given the large range of factors influencing electronic structure calculations of perovskite materials, the proposed models allow for method applicability and qualitative descriptions of charge-

Figure 4. Comparison of closed-shell systems density of states calculated at the gamma point (red) and with a 2 × 2 × 2 k-point mesh (blue). The systems shown are perfect MAPbI3 (A), VPb+2 (B), and VPb singlet (C). 5221


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extended electronic lifetime, τe. The new unoccupied state creates a subgap within the conduction band, requiring that a larger amount of energy be dissipated before relaxing to the final LUMO state. The nonradiative transitions across the subgap described in Figure 5 must be facilitated by either a single or multiple phonon modes. The increased energy dissipated accounts for the lengthening of the nonradiative lifetime. The relative lifetimes for each of the systems and their respective trends are shown in Figures S10 and S11, and numerical values are provided in Table S4, computed by eqs 11a and 11b. The extended lifetime of the electron also provided longer charge-carrier separation lifetimes within the system. Extended lifetimes are of high desire in areas such as photocatalysis and also provide environments to increase photovoltage within a device if the hot electron can be collected before cooling. Extended nonradiative lifetimes also play an important role in dictating the radiative yields a material may have. The nonradiative relaxation mechanisms have been proposed to occur on two time scales: a fast, subpicosecond time scale and a secondary time scale on the order of 50 ps. The shorter time scale is contributed to fast thermalization to the band edges following photoexcitation.69 The long-lived time scale has been discussed as a thermalization of charge carriers from high-value momentum points to the gamma point.72−75 However, the longer time scale is shown to occur when the excitation energy is >2.6 eV. The time scales seen in the neutral and spin-down method, which contain the midgap state, provide nonradiative relaxation lifetimes between 50 and 100 ps. Nonradiative relaxation rates have been recomputed utilizing a 2 × 2 × 2 kpoint mesh during the MD simulation. The computed rates for electrons and holes show a decrease in relaxation time for specific environments relative to gamma point MD trajectories. The rates for all systems under study are presented numerically, Table S5, and visually, Figures S12 and S13, in the SI. Multiple relaxation processes compete following photoexcitation and must be taken into account when calculating relaxation rates. The creation of a VPb and the subsequent charging of the system play a significant role in the radiative properties of the MAPbI3 material. Emission spectra are calculated according to possible transitions as a function of time-dependent state occupation. Following photoexcitation, the charge carriers migrated between states, as shown in the nonradiative rate calculations. At each time step, radiative relaxation may be possible provided there is sufficient spatial overlap of the orbitals of the electron and hole states. The timeintegrated emissions (TIEs), computed via eqs 11a and 11b, for the vacancy, charged, and charge-corrected systems, are shown in Figure 6, respectively. To explore for ability of the system to emit light, initial electron and hole occupations were chosen that correspond to a high-energy excitation. This initial electronic configuration provides the highest number of states to be accessed during relaxation of the charge carrier and therefore provides the largest number of possible radiative transitions. To compare the effect of the Pb vacancy defect, the radiative and nonradiative rates are used to calculate quantum yield. From the QY values obtained for each system, it is apparent that the vacancy decreases the QY. Because the QY is the ratio of the radiative and total radiative plus nonradiative rates, either a decrease in radiative rates or an increase in nonradiative rates is responsible for the reduced QY.

E(MAPbI3) < E(V +Pb2) < E(V +Pb(σ )) < E(VPb,S = 0) < E(VPb,S = 1, σ )

To understand the possible charge-carrier relaxation mechanisms and rates following photoexcitation, methods outside of standard software packages are utilized. The primary result of time-dependent charge-carrier dynamics is correlated to atomic motion through the generation of nonadiabatic coupling (NAC) values. Processing the NAC values provides components of the Redfield Tensor, indicating transition probabilities between sets of KSOs. Subsequent analysis of transition probabilities, coupled to chosen initial nonequilibrium electronic states, allows for calculation of charge-carrier excitation energies. The time-dependent energy distributions of both electron and hole charge carriers are displayed visually in Figure 5, as computed by eq 10.

Figure 5. Visualization of a probable excitation, chosen via oscillator strength, and subsequent time-dependent relaxation for the MAPbI3 model. The image represents nonequilibrium electron density as a function of energy and time following photoexcitation with charges represented by no change (green), gain - electron (red), and loss - hole (blue). Each image shows initial electronic states of chosen transition, excitation (vertical arrow), lifetime of electron (τe), and hole (τh) and final LU and HO states.

Initial nonequilibrium electronic conditions are chosen according to relative oscillator strengths of absorption. Shown in Figure 5 is an example of a high oscillator strength transition within the MAPbI3 bulk material. Within the images, energies of initial states are shown relative to the HOMO and LUMO levels and final states are always the HOMO and LUMO states. The rates of nonradiative relaxation (Table S3) for the pure system are the correct order of magnitude when compared with experimentally observed values with holes calculated on a subpicosecond time scale and electrons on a t < 10 ps time scale.68−71 The longer lifetime of the electron may be artificially lengthened through the use of the current method via averaging nonadiabatic coupling values along the MD trajectory. The relative changes in charge-carrier lifetimes between the pure system and vacancy system can provide insight into alternative relaxation pathways within the material. Comparing the images, there are similarities and differences between the different systems. Within all systems, it is shown that the lifetime of the electron, τe, is calculated to be longer than that of the hole, τh. It should also be noted that the hole lifetimes, τh, are on a similar time scale when generated in the energy region in all of the systems, indicating that the relative differences in the systems are primarily found within the conduction band. To distinguish the systems, the electronic lifetime of the spin-up, τeα, is found to be shorter than those in the vacancy, τvac, and spin-down system, τeβ. As noted in discussing the ground-state DOS, both the vacancy and spin-down systems had an addition unoccupied state generated close to the valence band edge. The effect of the new state is also seen in the 5222


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excitations is more important in defect systems. The presented PL spectra provide transient radiative emission that changes in time. The shown PL spectrum is specific for the interval of time prior to reaching the morphological equilibrium. The timeresolved PL spectra can exhibit characteristics of short-lived or rare events, which may help to monitor the formation of new structural rearrangements in the vicinity of the defect, including the generation of new atomic species. From the TIE spectra, it can be seen that for similar electronic structures there are similar PL spectra. For the V+Pb(β) and V+2 Pb TIE there are a larger number of possible emission energies. This is due in part to the additional unoccupied electronic state at the bottom of the conduction band. The additional state observed before thermally activated rearrangement provides a number of new transitions to compete with those found in the spin-up system. However, when comparing VPb,S=1,β with other systems that have a gap state, there is only an increase in low-energy PL peaks. The gain of low energy transition intensity may signify increased nonradiative pathways, resulting in a darkened material due to specific defect types. From the computed values of QY it is shown that the additional unoccupied state created due to a VPb, observed prior to thermally activated structure-rearrangement, greatly reduces the ability of the MAPbI3 for light-emitting applications. To determine the probability of increased or decreased emission due to spin-polarized vacancies, the charge-carrier lifetimes and quantum yield must be viewed together because photoemission is a synergy of radiative and nonradiative processes. If the spin alpha component is known to be more probable to excite than the spin beta component, one can conclude that emission is increased by defects. However, if the spin beta pathway is favored, then one expects emission to be quenched due to a VPb. This offers an additional option to tune emission intensity. It has also been discussed that trap states due to defects may act as carrier transport rather than exclusively recombination centers, providing further interest in possible isolated vacancy bands.78 To further explore spin-polarized photoluminescence

Figure 6. Time-integrated emission spectra for closed-shell systems provide probable radiative transition intensities with corresponding energies for the V+2 Pb (a) and VPb,S=1,σ (b) systems. High-intensity peaks at 1.15 eV (left) and 2.35 eV (right) correspond to band gap energies.

High-energy photoexcitation has shown to promote hot carrier radiative recombination on a time scale close to ∼100 ps.76 High-energy PL peaks shown in the TIE, Figures 5 and 6, agree with the ability of MAPbI3 materials to have fast radiative recombination of hot carriers. The TIE photoluminescence spectra provided in Figures 6 and 7 are produced by sampling all possible radiative relaxation channels following a specific initial excitation. The spectra are composed of any inter- and intraband transition possible in the material during thermalization to the band edges, according to eqs 11a and 11b. Computed low-energy (below gap) emission is contributed by intraband transitions that are not seen experimentally because they are in the IR region and not scanned experimentally, or the experimental IR spectra are contaminated to vibration− vibration transitions. The defects and vacancies may spoil the ability of MAPbI3 to “hold energy” or may be corrected due to a structure selfhealing mechanisms that compensate for trap states. In this work, undesired relaxation pathways are explored in detail to understand how to minimize energy loss. In defect-free models substantial Stoke’s shift is not seen or expected.77 This provides evidence that nuclear reorganization induced by photo-

Figure 7. Time-integrated spectra for open-shell systems V+Pb(σ)(left) and VPb,S=1,σ(right) for alpha (top) and beta (bottom). High-energy photoexcitation has shown to promote hot carrier radiative recombination on a time scale close to ∼100 ps.76 High-energy PL peaks shown in the TIE, Figure 4, agree with the ability of MAPbI3 materials to have fast radiative recombination of hot carriers. 5223



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and competing nonradiative relaxation, one must consider spin−orbit coupling when calculating electron densities and energies.79 The description on the basis of spinor orbitals can provide a much clearer picture of the competing relaxation pathways. Initial calculations of ground-state absorption spectra utilizing DFT+SOC are presented (Figure S14), with methodology presented in the SI. The noncollinear spin approach and relativistic/SOC effects are critically needed for understanding electronic structure of perovskites composed of heavy elements. However, the intuitive interpretation of spin−orbit enabled data is hard without a reference point. The spin-polarized DFT serves as an important reverence point for qualitative intuitive interpretation of main trends of electronic structure, especially for the open-shell configurations.

Article

ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.jpcc.7b05375. Absorption spectra of explored models, analysis of nonradiative relaxation rates, influence of momentum spacing on the gap, electronic structure rearrangement acquired during thermal dynamics, evidence of I 2 molecule formation, and images of frontier orbitals. (PDF) Radial distribution as a function of time, illustrating the formation of a specific peak at 2.6 Å as an additional evidence of I2 molecule formation. (MPG)

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4. CONCLUSIONS The impact of a VPb on the optical and electronic properties within a 3 × 3 × 3 supercell of bulk cubic MAPbI3 has been explored. An open-shell methodology was applied to correctly account for the additional charge required to balance a VPb. The creation of a VPb in the bulk material introduced an additional unoccupied electronic state within the band gap of the pure material. The additional VPb state can provide new relaxation mechanism pathways for photoexcited charge carriers. A substantial change in rates of charge-carrier relaxation and in mechanisms of relaxation has been identified. For the intrinsic, nondefect system, nonradiative hole and electron relaxation lifetimes were found on the order of subpicosecond and 1−7 ps, respectively. In the V+2 Pb system, the nonradiative hole and electron lifetimes were calculated to be on the order of 1−10 and 30−50 ps, respectively. The data for charged V+Pb(σ) in a doublet spin configuration highlight the difference in electronic dynamics for alpha and beta spin. The nonradiative spin alpha hole and electron lifetimes, τhα and τeα respectively, were calculated to be on the order of 1−30 ps for both charge carriers, dependent on the initial excitation criteria. However, the nonradiative spin beta hole and electron lifetimes, τhβ and τeβ, respectively, were calculated to be subpicosecond and ∼50 ps, respectively. Comparing the nonradiative and radiative lifetimes of the spin-polarized systems shows a decrease in quantum yield + within the V+2 Pb, VPb(β), and VPb,S=1,β systems. The decreased quantum yield can be contributed to the fast, nonradiative relaxation rates, promoted by the new unoccupied state. Following thermal treatment and atomic rearrangement the vacancy state energy migrated to the valence band edge. The optoelectronic properties are expected to be different before and after such a rearrangement and will be examined in future work. Given that the rates for nonradiative and radiative relaxation processes are qualitatively determined to guide in the application of MAPbI3 in photovoltaic and optoelectronic + processes, the increased electron lifetime in the V+2 Pb, VPb(β), and VPb,S=1,β configurations provides an excellent opportunity to increase photovoltage if the hot charge carriers can be extracted. At the same time, the additional unoccupied state created due to a VPb reduces the ability of the MAPbI3 for lightemitting applications. The reported approach of spin-polarized nonadiabatic dynamics for computing photoluminescence could be useful in assessing optical trends in systems with open-shell electronic configurations such as radicals and dilute magnetic semiconductors.

AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS This research was supported by NSF award CHE-1413614 and by DOE, BES Chemical Sciences, NERSC Contract No. DEAC02-05CH11231, allocation Awards 85213, 86185, 86898, 31857, ‘Computational Modeling of Photocatalysis and Photoinduced Charge Transfer Dynamics on Surfaces’. T.M.I. thanks the Center for Computational Materials Science, Institute for Materials Research, Tohoku University (Sendai, Japan) for their continuous support of the SR16000 M1 supercomputing system. The calculations were partially performed at supercomputer cluster “Cherry” provided by the Materials Modeling and Development Laboratory at NUST “MISIS” (supported via the Grant from the Ministry of Education and Science of the Russian Federation No. 14.Y26.31.0005). We thank Svetlana Kilina, Sergei Tretiak, Amanda Neukirch, Yulun Han, Aaron Forde, and Dane Hogoboom for inspiring discussions and suggestions. T.M.I. gratefully acknowledges financial support of the Ministry of Education and Science of the Russian Federation in the framework of the Increase Competitiveness Program of NUST MISIS (No. K3-2017-026) implemented by a governmental decree dated 16th of March 2013, N 211.

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