Role of Spin-Orbit Coupling - American Chemical Society

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C: Physical Processes in Nanomaterials and Nanostructures 2+

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Spinor Dynamics in Pristine and Mn Doped CsPbBr NC: Role of Spin-Orbit Coupling in Ground and Excited State Dynamics Aaron Forde, Talgat M Inerbaev, and Dmitri S. Kilin J. Phys. Chem. C, Just Accepted Manuscript • DOI: 10.1021/acs.jpcc.8b05392 • Publication Date (Web): 10 Oct 2018 Downloaded from http://pubs.acs.org on October 10, 2018

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The Journal of Physical Chemistry

1

Spinor Dynamics in Pristine and Mn2+ Doped CsPbBr3 NC: Role of Spin-Orbit Coupling in Ground and Excited State Dynamics Aaron Forde1, Talgat Inerbaev2,3, Dmitri Kilin4* 1

Department of Materials Science and Nanotechnology, North Dakota State University, Fargo, North Dakota 58102, United States 2

L. N. Gumilyov Eurasian National University, Astana, Kazakhstan

3

National University of Science and Technology MISIS, 4 Leninskiy pr., Moscow 119049, Russian Federation 4

Department of Chemistry and Biochemistry, North Dakota State University, Fargo, North Dakota 58102, United States *Corresponding Author: [email protected] ABSTRACT

Fully inorganic lead halide perovskite nanocrystals (NCs) are of interest for optoelectronic and light emitting devices due to their photoluminescence (PL) emission properties which can be tuned/optimized by (I) surface passivation and (II) doping. (I) Surface passivation of the NC affects PL capabilities, as an under‐passivated surface can introduce trap states which reduces PL quantum yields (QY). (II) Doping NCs and quantum dots (QDs) with transition metal ions provides stable optical transitions. Doping perovskite NCs with Mn2+ ions provides high intensity 4T1  6A1 optical transitions in addition to the bright, intrinsic NC emission. Here we use noncollinear Density Functional Theory (DFT) to investigate the roles of surface passivation and doping on PL emission stability of perovskite NCs. Two models are investigated: (i) a pristine NC and (ii) a NC doped with Mn2+ ion. The noncollinear DFT includes spin‐orbit coupling (SOC) between different spin‐states and produces spin‐adiabatic molecular orbitals. These orbitals are used to calculate the transition dipoles between electronic states, oscillator strengths, radiative transition rates, and emission spectra. It was found that non‐collinear spin basis with spin‐orbit coupling slows down hole relaxation in the doped NC by two orders of magnitude compared to spin‐polarized basis. This is attributed to ‘spin‐flip’ transition from the perovskite NC to the Mn2+ dopant and low‐probability non‐radiative d‐d transition.

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2 I. INTRODUCTION All inorganic CsPbX3 (X=Cl, Br, I) lead halide perovskite nanocrystals (NCs) have gained considerable attention within many research communities after the first report of a facile colloidal synthesis in 20151, which showed 50% to 90% photo‐luminescence quantum yields (PLQY) with the ability to tune the PL emission across the visible spectrum by halide substitution. These materials have shown a wide range of properties which are of interest for commercial devices, such as solid state LEDs2‐ , as components in photovoltaics to improve efficiencies5‐6, and lasing7. These applications all depend

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on stable PL from the perovskite NC’s, which has shown to be the main challenge that currently limits the commercial implementation. PL is the result of a multi‐step excited‐state process. Under low fluence intensities, there are two main pathways for photo‐excited electrons and photo‐induced holes to recombine: radiative and non‐radiative. Surface trap states act as a ‘heat sink’ were energy that could be used to generate PL is instead transferred to the dark trap state and is converted to a phonon. This means any non‐radiative recombination channels present will reduce PLQY=

. Under high fluence, there are multi‐

particle interactions, such as Auger recombination and multi‐exciton generation, which will compete with the previously mentioned recombination pathways as well. From a structure to property perspective, PL instability in perovskite NCs can arise from two sources: incomplete passivation of the perovskite NC creating surface trap states 8 9 and intrinsic phase instability of the lattice turning the bright cubic phase into a dark orthorhombic 1 10. In this manuscript


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3 we focus on the issue of surface trap states and ways to mitigate their effect. Within the literature there are a variety of experimental synthetic methods used to improve surface passivation of CsPbX3 NCs, which include core‐shelling, polymer and silica coatings, post‐synthetic thiol treatment 11, zwitterions 12. An alternative method used to improve PL stability of perovskite NCs is through doping perovskite NCs with transition metal ions, such as Mn2+. Doping with transition metal ions has been a long‐held strategy to modulate physical and electronic properties of semiconductor QDs and NCs 13 14. Motivation for this strategy lies in the fact that PL from the dopant would primarily depend on the ligand/crystal field surrounding it and with PLQY being independent of shallow surface trap states. For Mn2+ doped CsPb1‐x MnxCl3 and CsPb1‐x MnxBr3 NCs it has been reported they exhibit dual emission from the intrinsic NC bandgap and the Mn2+ 4T1  6A1 transition (~600 nm) with high quantum yields while CsPb1‐x MnxI3 only shows NC emission 15 16. Trends observed in dual emission Mn2+ doped NCs has been attributed to an interplay between kinetics of excited‐state non‐radiative relaxation and relative band alignment of dopant Mn2+ 3d and perovskite NC states15, 17‐19. ab Initio computations can provide an atomistic description of trends observed in in pristine and doped perovskite QDs. Ground state electronic structure calculations for pristine NCs can give insight into how passivation ligands interact with the NC surface and its influence on optical properties. For doped NCs, geometry optimization can provide knowledge on the configuration of halides surrounding the Mn2+ dopant which will greatly influence 3d orbital splitting 20. Quantum dynamics simulations can provide insight into non‐radiative and radiative relaxation pathways that charges follow in these materials. Quantum dynamics simulations has shown to be a powerful tool to understand the role excited‐state processes in perovskite materials exceptional properties 21 22 23 24 25.



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4 Computationally modeling ground‐state electronic structure and optical properties of pristine and doped QDs/NCs has shown to be a challenging task. (1) typical colloidal perovskite NCs are on the scale of 10 nm and have simultaneous cationic and anionic surface passivation ligands, which would be computationally demanding to explicitly model. (2) for all APbX3 perovskites it is known that the LUMO is composed of Pb2+ 6p orbitals which requires non‐collinear spin basis26 with spin‐orbit coupling (SOC) for accurate electronic structure (3) any dopant within a QD/NC that experiences weak ligand field splitting will require spin‐flip transition to emit, which is dipole‐forbidden and requires SOC to facilitate the optical transition. (4) within the quantum confinement size regime, e‐ ‐ e‐ correlation needs explicit treatment for accurate bandgaps. Accounting for all of these constraints on a single model, at the moment, is computationally unfeasible. Thus, approximations have to be made for each case study to model a specific observable. Examples of this include the following: Size dependent ground‐state electronic structure for un‐ passivated CsPbBr3 NCs has been investigated with the incorporation of SOC and hybrid functionals to determine origin of size‐dependent Stokes shift 27. Mn2+ doped CdSe has been investigated using DFT, TDDFT, and CIS methods to show band alignment of the NC/QD relative to the dopant influences PL properties 28. Role of passivation ligands on equilibrium geometry and optical properties of magic sized CdSe clusters29. Modeling optical transitions in Mn2+ doped ZnO QDs 30. To computationally model excited‐state processes, such as non‐radiative relaxation and PL, it is required to use methodology that goes beyond Born‐Oppenheimer approximation (BOA). BOA separates nuclear and electron degrees of freedom and does not allow for energy‐flow between them. However, broad range of processes in excited states are related to energy being transferred from electronic degrees of


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5 freedom into nuclear degrees of freedom, giving raise to nonradiative relaxation, cooling, and recombination of charge carriers. These aspects of the excited‐state dynamics are often described in two practical steps: (a) assessing of electron‐to‐nuclear non‐adiabatic couplings31 (NACs) computed “on‐the‐fly” or by displacements along normal modes, (b) phonon‐assisted dynamics of electronic states, described with NACs, by surface hopping 32 or multi‐level density matrix methods in Redfield formalism 33‐34. Currently, non‐collinear spin calculations are on the way to be implemented into commercial software for excited state dynamics, yet in progress 35. Thus, to compute NACs with SOC a homesuite of codes have to been developed 36. The non‐collinear spin approach refers to use basis of spinor orbitals. The transitions between adiabatic states are determined by two factors: vibronic couplings between states and energy subgaps between corresponding states. SOC creates additional non‐radiative relaxation pathways by mixing spin states which would otherwise be orthogonal. This should allow for quicker non‐radiative relaxation as it effectively increases the DOS. For APbX3 perovskites in general, there is a giant SOC effect in conduction band edge which splits the J=1/2 and J=3/2 angular momentum states of Pb‐6p with the resulting energy splitting being greater than thermal energy 𝑘𝑇 ~ 26 𝑚𝑒𝑉 57. It would be expected that with the larger energy splitting the rates of non‐ radiative relaxation near the band edge would be slower than if SOC was not accounted for. New additions to the code enable the discussion of whether the additional relaxation pathways and energy sub‐gaps in conduction band has an important role in non‐radiative relaxation. In this manuscript we explore ground state electronic structure, molecular dynamics, and excited‐state electron dynamics of a fully passivated pristine CsPbBr3 QD and Mn2+ doped CsPb1‐x MnxBr3 QD using spin‐restricted (SR), spin‐polarized (SP), and SOC enabled NCS DFT with GGA functional. We use suite codes to compute NACs and oscillator strengths using each methodology, allowing us to



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6 quantify the impact of SOC on non‐radiative and radiative relaxation in both systems. Specifically, NACs computed with SOC NCS DFT allows for radiative and non‐radiative spin‐flip transitions. Spin‐flip transitions can be characterized by tracking the magnetization vector magnitude during the excited‐ state relaxation. The details of our manuscript are as follows: (i) ground state electronic structure and optical properties of a pristine CsPbBr3 QD and a Mn2+ doped CsPbBr3 QD, with both having their surface terminated by ethylammonium (EA) and anionic acetate (AC) ligands (ii) molecular dynamics simulation exploring the surface ligand dynamics of the intrinsic QD and ligand field dynamics in the doped QD. (iii) excited‐state dynamics in density matrix formalism to characterize non‐radiative relaxation of excited‐ states in both pristine and doped QD (iv) time‐resolved and time‐integrated emission for both the pristine and doped QD. II. METHODS This section starts with basic definitions for consistency of notations. For theory, we will describe relevant equations and observables in SP KSO basis first then followed by NCS SKSO basis. We placed description of spin‐restricted methodology into SI and provide references to previous work. Rational for how we constructed atomistic models is also provided. The rest of the section is outlined as follows: (1.a) ground state electronic structure calculations using SP DFT and SOC enabled NCS DFT (1.b) SP and SOC NCS observables (1.c) ab initio molecular dynamics (1.d) non‐radiative relaxation of excited‐states 1.e) radiative relaxation of excited‐states. (2.a) Atomistic construction of pristine and doped QD 2.b) computational details.


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7 II.1.a Ground State Electronic Structure: Spin‐Polarized and NCS DFT SP calculations are needed due to the presence of Mn2+ within an octahedral Br‐ ligand field of the perovskite QD. The halide ligand field provides an electrostatic potential that determines the most favorable spin configuration of the valent Mn2+ 3d electrons. Also, SP calculations are far more cost‐ efficient for coarse‐grained geometry‐optimization than NCS calculations. SOC enabled NCS calculations are needed to complement SP for two reasons: 1) it is well known in the literature that the conduction band edge of pristine APbX3 perovskites are composted of Pb2+ 6p orbitals which experience strong energy splitting of degenerate angular momentum and spin states37. 2) In the literature it is proposed that the ~600nm emission in doped CsPb1‐x MnxBr3 QDs is due to a spin‐flip 4T1  6A1 optical transition15, 17

. In a SP basis, spin is a good quantum number and is conserved for dipole allowed optical transitions.

In contrast, when working in the SKSO basis total angular momentum 𝐽⃗

𝑆⃗ is the conserved

𝐿⃗

quantity and allows for changes in spin during excited‐state dynamics, such as the dipole‐forbidden triplet‐singlet optical transition or non‐radiative spin‐flip relaxation. Here we introduce the basic equations and quantities of SP and NCS DFT calculations needed to define relevant observables and keep consistent notations. ∑

,

,

𝛻

𝛿

𝑣

For equations (1)‐(2), 𝛿 𝑣

𝑟⃗ 𝜑

𝑣

𝑟⃗ =

𝑟⃗

𝜀,

𝑣

𝑟⃗

𝑣

𝑟⃗

𝑣

𝑟⃗

𝑣

𝑟⃗

𝜑

𝑟⃗

is the Kroenecker‐delta function, 𝜎 and 𝜎 define spin projections 𝛼 or 𝛽,

r⃗ is the spin‐dependent external potential, and 𝜑

𝑟⃗ is a spin‐dependent KSO. For SP DFT,


(1)

(2)


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8 𝜎′ i.e. v

equations (1)‐(2) have non‐zero components when 𝜎

r⃗

r⃗

v

0 26. Self‐consistent

DFT electronic structure calculations with spin‐dependent external potentials

𝑟⃗ =

𝑣

provide SP KSOs 𝜑 , 𝑟⃑ with energy 𝜀 , with 𝜎

r⃗

v

0

0

v

r⃗

(3)

𝛼 or 𝛽 defining orthogonal spin‐projections. SP KSOs

obey the orthogonality relation and normalization, equation 4 𝑑𝑟⃗𝜑 ∗ 𝑟⃑ 𝜑 𝑁

Spin polarization parameter Δ𝑁↑↓ of the magnetization vector |M|⃗

𝑟⃑

𝛿 𝛿

(4)

𝑁 defines the spin multiplicity S = Δ𝑁↑↓ /2 and the magnitude

Δ𝑁↑↓ . It is noted that in a SP basis the magnetization vector

magnitude is constant during excited‐state dynamics i.e. spin is conserved. For NCS DFT, equations (1)‐ (2) are not constrained by spin‐orthogonality allowing for non‐zero off‐diagonal effective potentials. These off‐diagonal elements produce a set of equations that couple the spin states 𝛼 and 𝛽. The solution to the set of NCS DFT equations is represented as a superposition of spin‐ 𝛼 and spin‐ 𝛽 states |𝛼⟩

1 and |𝛽⟩ 0

0 with spatially dependent coefficents 𝜑 1

𝑟⃗ and 𝜑

𝑟⃗ . These solutions are

commonly referred to as spinor KSOs (SKSOs) shown in equation (5).

𝜑

𝜑 𝜑

𝑟⃗

𝑟⃗ =𝜑 𝑟⃗

𝑟⃗ |𝛼⟩

𝜑

𝑟⃗ |𝛽⟩

(5)

SKSOs obey the normalization, equation (6), and orthogonality, equation (7) ∗ 𝑑𝑟⃗ 𝜑

𝜑∗

𝜑 𝜑

= 𝑑𝑟⃗ 𝜑 ∗ 𝑟⃑ 𝜑

𝑑𝑟⃗ 𝜑 ∗ 𝑟⃑ 𝜑

𝑟⃑

𝑟⃑

𝜑 ∗ 𝑟⃑ 𝜑


𝜑 ∗ 𝑟⃑ 𝜑

𝑟⃑

𝑟⃑

1

𝛿

(6)

(7)

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9 Within the NCS DFT framework, relativistic effects can be incorporated using 2nd order scalar relativistic corrections. 𝐻

𝐻

𝐻

(8)

The relativistic Hamiltonian from equation (8) is composed of two terms: 𝐻 is the scalar relativistic term and 𝐻 𝐻

is the SOC term. The 𝐻 term describes relativistic kinetic energy corrections and

describes energy shifts of spin occupations. Up to second‐order 𝐻 ℏ

𝐻

can be represented as

𝐿⃗∙𝑆⃗

(9)

where 𝐿⃗ is the angular momentum operator and 𝑆⃗ is composed of Pauli spin matrices σ , σ , σ . Magnetization vector M⃗ ρ ρ ρ . f

,

r⃗ r⃗

ρ ρ

(r⃑

∑

M , M , M in NCS DFT is constructed from SKSO density matrix ρ

(r⃑ =

r⃗ . Components within the density matrix are found from r⃗

,

f

φ

r⃑ φ

r⃑

is the occupation number of the ith SKSO and takes values between 0 and 1. Magnetization density

projections m r⃗ are found from expectation values of corresponding Pauli matrices σ , i=x,y,z, shown in equations (10a)‐(10e)38. ρ

r⃗

m r⃗

m r⃗

m r⃗

Tr ρ μ Tr ρ

r⃑

=ρ

r⃑

ρ

r⃑

r⃑ σ =μ

ρ

r⃑

ρ

iμ Tr ρ

μ Tr ρ

r⃑ σ

r⃑ σ

iμ

μ

ρ

ρ

r⃑


(10a)

r⃑

(10b)

r⃑

(10c)

(10d)

r⃑

ρ

ρ

r⃑


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10 M

dr m r⃗ , M

dr m r⃗ , M

dr m r⃗

(10e)

(11a)

II.1.b Ground State Observables: SP and NCS DFT Density of states (DOS) is computed for both 𝜎

∑, δ ε

ε

DOS

𝛼 or 𝛽 spin projections. ε ,

The Dirac delta function is approximated as a Gaussian distribution to provide thermal broadening. For visualization purposes β‐dependent DOS are multiplied by ‐1. The same procedure is used to compute density of states for SKSOs ∑ 𝛿 𝜀

𝐷𝑂𝑆

𝜀

SP KSOs found from DFT can be used to find transition dipole matrix elements 𝐷⃗ 𝑒 𝜑,

𝑟⃑ | 𝑟⃗ |𝜑

,

(11b)

,

𝑟⃑ where 𝑒 is the charge of an electron and 𝑟⃗ is the position operator. We work

under the independent orbital approximation (IOA) were excited‐states are described as a pair of orbitals, as opposed to a superposition of orbitals commonly used in TDDFT or Bethe‐Saltpeter approaches. IOA is applicable for systems with weak confinement and negligible e‐ ‐ e‐ correlation. With the transition dipole matrix elements we can find oscillator strengths for transitions between KS orbital i and j with spin σ, equation (12a), and compute optical spectra, equation (12b) 𝑓 𝛼 𝜔

𝐷⃗

,

∑

,

𝑓

,

,

ℏ

,

𝛿 𝜀

Δ𝜀

,


(12a)

(12b)

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11 SKSO transition dipole matrix elements are found in a similar manner as SP elements, but with SKSO elements being composed as a superposition of spin‐ 𝛼 and spin‐ 𝛽 transitions, equation (13a). From transition dipole can compute SKSO oscillator strengths (13b) and spectra (13c). ∗ 𝑒 𝑑𝑟⃗ 𝜑

〈𝐷⃗ 〉

𝜑 𝜑 ∗ 𝑟⃗ 𝜑

𝑒 𝑑𝑟⃗ 𝜑 ∗ 𝑟⃑ 𝑟⃗𝜑

𝑓

𝐷⃗

𝛼

𝜀

ℏ

∑ 𝑓 𝛿 𝜀

𝜑 ∗ 𝑟⃑ 𝑟⃗ 𝜑

𝑟⃑

𝛥𝜀

𝑟⃑

(13a)

(13b)

(13c)

II.1.b. ab initio Adiabatic Molecular Dynamics: Spin‐Restricted and Spin‐Polarized Molecular dynamics is used for two purposes: to generate NACs (covered in 1.c) and to observe how ionic fluctuations affects electronic structure. Specifically, trap state formation in the pristine QD and their influence on oscillator strengths. For adiabatic molecular dynamics, the total kinetic energy of the system is set equal to a thermostat, equation (14a), for an initial condition and is propagated by using Newtons equations of motion, equation (14b). The energy of KSOs are recorded for each timestep to observe fluctuations in time, equation (14c). ⃗

∑

𝑅⃗

𝜀, 𝑡

|

𝑁

𝑘 𝑇

(14a)

𝐹⃗ /𝑀

(14b)

𝜀,

𝑅⃗ 𝑡


(14c)


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12 To identify correlations between bond distances and SKSO computed oscillator strengths we consider average ligand‐surface bond distances in the pristine QD. We compute time‐resolved radial distribution function (RDF), equation (15a), for both models considering Pb‐O bonds for the pristine QD and Mn‐Br bonds for the doped QD. |𝑅⃗ 𝑡

𝑅⃗ 𝑡 | corresponds to time dependent bond distance between ion I

and J. ∑ 𝛿 𝑟

𝑅𝐷𝐹 𝑡, 𝑟

𝑅⃗ 𝑡

𝑅⃗ 𝑡

(15a)

SKSO oscillator strengths are computed after each timestep as a function of ionic position 𝑅⃗ 𝑡 , equation (15b).

〈𝐷⃗ 〉 𝑅⃗ 𝑡

𝑒 𝑑𝑟⃗ 𝜑 ∗ 𝑅⃗ 𝑡

𝜑 ∗ 𝑅⃗ 𝑡

𝑟⃗

𝜑 𝜑

𝑅⃗ 𝑡 𝑅⃗ 𝑡

(15b)

II.1.c. Non‐Radiative Relaxation: Time evolution of electronic degrees of freedom is found from solving the Liouville‐von Neumann equation of motion 𝜌

ℏ

∑ 𝐹 𝜌

𝜌 𝐹

(16)

Where 𝐹 is the Fock matrix and 𝜌 is the density matrix. The first term corresponds to unitary time evolution while the second term corresponds to dissipative transitions due to coupling of electronic and nuclear degrees of freedom. Electronic and nuclear degrees of freedom are connected through NACs allowing for energy dissipation from hot charge carriers to phonons. NACs can be computed using the “on‐the‐fly” procedure along a nuclear trajectory


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13 𝑉

𝑡

ℏ ∆

𝑑𝑟⃗ 𝜑 ∗ 𝑟⃗, 𝑅⃗ 𝑡

𝜑 ∗ 𝑟⃗, 𝑅⃗ 𝑡

∙

𝑟⃗, 𝑅⃗ 𝑡 𝑟⃗, 𝑅⃗ 𝑡

𝜑 𝜑

Δ𝑡

ℎ. 𝑐. (17)

Δ𝑡

Fourier transform of the autocorrelation function, equation (18a)‐(18c), provides components for the Redfield tensor, equation (18d). 𝜏

𝑀

𝑡

𝑉

𝜏 𝑉

𝑡 𝑑𝑡

(18a)

𝛤

𝑀

𝜏 𝑒

𝑑𝜏

(18b)

𝛤

𝑀

𝜏 𝑒

𝑑𝜏

(18c)

, (

(18d)

𝛤

𝑅

𝛿 ∑ 𝛤

𝛤

𝛿 ∑ 𝛤

Redfield tensor controls dissipative dynamics of the density matrix, equation (19). ∑

𝑅

𝜌

(19)

Along the trajectory we can compute time‐resolved observables, such as changes in charge carrier occupation, equations (20a)‐(20b), and average charge carrier energy, equation (20c) 𝑛 ∆𝑛

,

,

〈∆𝜀 〉 𝑡

,

∑ 𝜌

𝜀, 𝑡 𝜀, 𝑡

𝑛 ∑

,

𝑡 𝛿 𝜀 𝜀, 𝑡

𝜀 𝑛

𝜌 𝑡 𝜀 𝑡

(20a)

𝜀, 𝑡

(20b)

(20c)

To get rates of charge carrier relaxation to band edges we convert energy expectation value from equation (20c) into a dimensionless energy, equation (21a). Assuming a single exponential decay, we fit equation (21a) to an exponential decay and solve for rate constant 𝑘 , equation (21b). 〈𝐸 〉 𝑡

〈∆

〉

〈∆

〉

〈∆

〉

〈∆

〉


(21a)


Page 14 of 55

14 𝑘

〈𝐸 〉 𝑡 𝑑𝑡

𝜏

;𝑡

0, 𝑡

∞

(21b)

In the case of long intermediate lifetimes during relaxation to band edges, such a spin‐flip transition, we compute rates as sums of single exponential decays, equation (22). ∑ 𝜏

𝑘

(22)

II.1.d. Excited State Emission: Along the excited‐state trajectory time‐resolved PL can be computed between states i and j by using oscillator strength 𝑓 and is based on concept of inverse population, equation (23a). Summing equation (23a) provides time‐integrated emission, equation (23b) 39‐40. ∑

𝐸 ℏ𝜔, 𝑡 𝐸

ℏ𝜔

𝑓 𝛿 ℏ𝜔

ℏ𝜔

𝜌

𝐸 ℏ𝜔, 𝑡 𝑑𝑡

𝑡

𝜌 𝑡

(23a)

(23b)

For accurate comparison between time‐integrated PL spectra we run the simulation based on the lifetime of the HO‐LU transition found from the inverse of the Einstein coefficient for spontaneous emission, equations (24a)‐(24b).

A



A

f

(24a)

(24b)

𝑣 is the transion frequency, f is the oscillator strength of the transition, g is the degeneracy of the ith state, and the rest are the familiar fundamental constants. PLQY can be computed for both models from their respective radiative and non‐radiative HO‐LU transition lifetimes



PLQY = 

=




(25)

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15 II.2.a. Atomistic Construction of Pristine and Doped QD The major constraints on constructing a fully‐passivated CsPbX3 perovskite NC are: (a) surface termination (b) ligand placement, (c) charge neutrality. (a) CsPbBr3 perovskites can have two distinct surface terminations in the cubic: X‐Cs or Pb‐X. We construct a ‘Pb rich’ Pb‐X surface termination from 2x2x2 unit cell configuration with a stoichiometry Cs8Pb27Br54. This leaves 8 corner Pb atoms 3 bonds under‐coordinated, 12 edge Pb atoms 2 bond under‐ coordinated, and 6 face Pb atoms which are 1 bond under‐coordinated giving 54 dangling Pb bonds. This gives an NC 1.5 nm in edge length. (b) Experimentally, oleic acid and oleylamine are used to passivate NC surfaces 1. Here we use short‐ chain cationic ethylammonium (EA) and anionic acetate (AC) as passivation ligands as an approximation. Results for short‐chain ligands will provide insight into interaction potential but will neglect entropic contribution that long‐chain ligands provide for binding energies. Anionic acetate (AC) ligands are used to passivate dangling Pb2+ bonds. The role of EA is three‐fold: (i) provide counter‐balance charge for AC ligands (ii) provide lattice strain in placement of Cs+ (iii) stabilize surface Br‐ atoms. (c) Ideally, charge neutrality should be maintained over the entire NC. Our Pb‐rich surface without passivation ligands has a charge of +8. With addition of 54 AC ligands to passivate surface Pb2+ gives a charge of ‐46 to be balanced. Under constraints mentioned previously along with steric hindrance, we introduce 48 EA ligands symmetrically over the NC surface to give an overall charge of 2+. For the doped NC, the Mn2+ ion substitutionally replaces the central most Pb2+ atom giving an atomistic configuration of Cs8Pb26 Mn1Br54 with the same surface passivation configuration as the pristine NC with



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16 overall charge 2+. Figure 1(a),(c) illustrates the pristine and doped NCs, respectively, with Figure 1(b) showing the atomistic constituents for each model. II.2.b. Computational Details: Electronic structure was found using DFT using the generalized gradient approximation (GGA) Perdew‐ Burke‐Ernzerhof (PBE) functional41 in a plane‐wave basis set along with projector augmented‐wave (PAW) pseudopotentials 42‐43 in VASP44 software. All calculations were performed at the gamma point. Both pristine and doped QDs have a simulation cell size of 31x31x31 Å with 7 Å of vacuum in each direction. Each model is composed of 995 atoms with the pristine and doped QD having 2806 and 2809 electrons, respectively. Both NC models have charged simulation cells. The spurious electrostatic interactions between replicas of charged species discussed by Neugebauer and Scheffler45 are avoided with the use of a background charge concept, as suggested by Makov and Payne46 and used for modeling of charged metal clusters in PBC.47 We have recently tested that this approach provides reasonable results for atomistic modeling of catalytic47 and fluorescent48 nanoparticles. Recently, this approach has been successfully applied to model perovskite nanoclusters49 Electron‐phonon coupling, which we refer to as NACs, can be computed using two different procedures50: ‘on‐the‐fly’ along a nuclear trajectory were 𝑉 along spatial coordinates51 52 were 𝑉 , ~ 𝜑 along normal mode 𝜉 and 𝑀

𝜑 𝑀

𝑡 ~ 𝜑

𝜑 and normal mode analysis

were 𝑑𝜉 symbolizes infinitesimal elongation

represents momentum of mode 𝜉 at a given temperature. For systems that

are investigated here (~1000 atoms, ~2800 electrons) it is more convenient to generate NACs using ‘on‐the‐ fly’ procedure along a molecular dynamics trajectory. The processes which we attempt to model are illustrated in Figure 2.


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17 III. RESULTS This section provides results on computational characterization of doped and intrinsic perovskite quantum dots organized in four subsections. Subsection III.1 reports ground state properties such as bandgap, ligand binding, high versus low spin configurations, magnetization, and linear absorption. Influence of thermal motion on properties of quantum dots is presented in subsection III.2. The nonradiative pathway of photoinduced excited states are shown in subsection III.3. Subsection III.4 presents result on the radiative pathways of photoexcitation in perovskite quantum dots. III.1. Ground State Properties.



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18 The geometry optimized pristine QD, SR DOS PBE calculations, see Figure 3(a), showed a band gap of 3.12 eV with the valence band edge being composed of Br 4p and Pb 6s KSOs and the conduction band edge being Pb 6p KSOs (SI Table S1). For the SOC NCL PBE DOS calculation, see Figure 3(b), the band gap reduced to 2.38 eV with the band edges having the same composition as the pristine QD (SI Table S1). It is also observed that the ground state magnetization is zero, which is expected as stoichiometric, defect free lead halide perovskites are closed‐shell systems. We note that for each energy level there is two‐ fold degeneracy, SI Table S2. pDOS calculations, see Figure 4(a), show contributions of electron density from the surface ligands are deeper in the valence and conduction band for both models, as indicated by the red line in Figures 4(a)‐(b). This suggests that the model is well passivated and does not contain any surface trap states 29. The average Pb‐O bond distance between a surface Pb2+ atom and acetate anion is 2.61 A, as computed from RDF (SI Figure S1). We computed binding energies of the anionic AC and EA ligands to the crystalline faces of the pristine NC surface, shown in Table 1. Three cases are considered: (1) remove one AC anion ligand which is sequentially protonated after removal from surface 𝑁𝐶

𝐻 → 𝑁𝐶

𝐴𝐶

𝐻 → 𝑁𝐶

𝐴𝐶

(2) remove one EA cation ligand which is sequentially deprotonated 𝑁𝐶

→ 𝑁𝐶

𝐸𝐴

→ 𝑁𝐶

𝐸𝐴

𝐻

(3) remove one AC anion ligand and an adjacent EA cation ligand which protonate/deprotonate each other


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19 𝑁𝐶

→ 𝑁𝐶

𝐸𝐴

𝐴𝐶

→ 𝑁𝐶

𝐸𝐴

𝐴𝐶

Comparing Case (1)‐(2) with case (3) would show if it is more favorable for the ligands to desorb in a correlated manner or if they act independent of one another with the assistance from free protons that would be available from solution. It is observed that the binding energy of case (3) is less than the sum of binding energies for (1) and (2). Further analysis of binding energy is left for discussion. For the doped NC, the Mn2+ ion takes the place of a Pb2+ ion and is placed in an octahedral ligand field of Br‐ ions. From ligand field theory, Br‐ ions provide weak field energy splitting of Mn2+ 3d orbitals and predicts a high spin configuration S=5/2. SI Table S3 shows geometry optimization calculations for S=1/2, 3/2, 5/2. S=5/2 configuration provides the lowest total energy for the doped NC. During geometry optimization for the S=5/2 spin configuration, the Br‐ ligand field surrounding the Mn2+ dopant adapted to a square pyramidal geometry (SI Figure S2). For the spin α DOS, see Figure 3(c), it is seen that there are two states that appear within the bandgap, above the valence band edge. pDOS analysis, see Figure 4(c), shows that these states are dopant Mn2+ hybridized with atoms of the perovskite QD. Further analysis shows these states are Mn2+ 3dz2 and 3dx2‐y2 states hybridizing with Br 4p (SI Table S4). For the spin β component, there is a manifold of unoccupied Mn2+ 3d states that appear 0.19 eV below the conduction band edge, see Figure 3(c)‐4(c). The bandgap for the spin α and spin β components are 2.37 eV and 3.14 eV, respectively. The Mn2+ 3d orbital ordering, its relation to the square‐pyramidal geometry, and influence on non‐radiative relaxation is explored further in the discussion section.



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20 The NCL SOC PBE DOS calculation, see Figure 3(d), shows a bandgap of 1.77 eV. pDOS, see Figure 4(d), shows that the valence band edge is composed primarily of spin α components. In the conduction band, it is seen that the band edge orbitals were composed of perovskite states, which screen the dopant states and places them deep within the conduction band. Further analysis shows the band edge is composed of Pb 6p SKSOs (SI Table S4). In the discussion section we explore how our computed band alignment with SP and NCS methods correspond to experiment. Comparing the SKSO electronic structure of the doped NC to the intrinsic NC, we do not observe any degenerate energy states. We attribute this to spin‐spin interactions of the Mn2+ dopant perturbing the perovskite states. This is reflected in the ground state magnetization of doped NC SKSOs not being strictly anti‐parallel, as observed in the intrinsic NC magnetization (SI Table S2). Optical properties were computed using equations 12(a)‐13(c) for both intrinsic and doped QD under different basis and are shown in Figures 5(a)‐(b). Table 2 shows transitions that are labeled in Figures 5(a)‐(b). Labels a‐c indicate spin‐conserving transitions and d‐f indicate spin‐flip transitions with alphabetic progression of labels guiding from UV to IR ranges. Table 2 shows the greatest contributing transition for each of the labeled peaks in Figures 5(a)‐(b). SI Table S5 shows most probable transitions for each system and corresponding basis. For the intrinsic QD, SR and NCS computed spectra are shown in Figure 5(a). The spinor spectra show lower transition energies than the SR spectra. This agrees with DOS in Figures 3(a)‐(b) were SOC reduces the bandgap. For both SR and spinor spectra, the lowest energy transitions had the highest oscillator strengths which favors radiative recombination, see SI Table S5. For the spinor spectra, the lowest transition energy has 4 contributions which are features c in Figure 5(a) (from highest to lowest oscillator strength): HO‐1LU+1, HO‐1LU, HOLU+1, HOLU. This is


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21 due the lowest (highest) energy conduction (valence) band being twice degenerate. It would be expected that optical transitions which minimize the change in magnetization would show the largest oscillator strengths, but that is not what is observed. The HO‐1LU+1 and HOLU transition show Δ|𝑚⃑| = 0.77 Bohr and have the highest and lowest oscillator strengths for the lowest energy transition, respectively. For the doped NC, spectra for SP and NCS calculations are shown in Figure 5(b). The inset of Figure 5(b) shows the lowest energy transitions for the spin α and spinor spectra, respectively. The spinor spectra show the lowest transition energy, which is an indication of low probability d‐d transitions. From the pDOS Figure 4(d) it is known that the lowest energy transition, feature f in Figure 5(b), is a hybridized Mn2+ 3d to Pb2+ 6p transition. Transitions from 1.76 eV to 2.20 eV are of this same character. From 2.20 eV to 2.37 eV there are a manifold of d‐d transitions which have oscillator strength values shown in SI Table S6. Higher energy transitions are characteristic of pristine NC transitions with c,b, and a in Figure 5(b) being Br 4p and Pb 6s to Pb 6p. For spin α and spin β their lowest energy transitions, features cα and cβ in Figure 5(b), are low in intensity. The cα feature is a Mn 3d to Pb 6p transition and cβ corresponds to a Br 4p to Mn 3d ligand‐to‐metal transition (SI Table S4). Higher energy transitions, such as features bα and bβ, are characteristic of pristine NC transitions from Br 4p and Pb 6s to Pb 6p (SI Table S$). So, it appears that the effect of the Mn2+ dopant on optical spectra is to locally suppress the pristine HO‐LU transition while retaining absorption characteristics for higher energy transitions. III.2. Molecular Dynamics.



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22 Here we explore how ionic motion influences electronic structure and oscillator strengths of the pristine and doped QD along an adiabatic molecular dynamics trajectory. It is expected that bond vibrations of surface passivation ligands will produce irregular fluctuations in HO‐LU oscillator strength while d‐d transitions in the doped QD show periodic, low‐intensity fluctuations. For both models we track the energy of KSOs from HO‐x to HO (lower half of plot) and LU to LU+y (upper half of plot) during the trajectory, as shown in Figure 6. Each solid colored line corresponds to a distinct KSO. The KSOs experience fluctuations in energy due to static coupling of KSOs to thermal motion of ions (ie diagonalizing BO Hamiltonian at each timestep). For the intrinsic QD, we computed 1000 1‐fs timestep MD trajectory with an average temperature of 300K using SR method. KS energy fluctuations demonstrate response of electronic energies to thermalization of the model. As seen in Figure 6(a), there are states that appear within the bandgap which is characteristic of surface state formation. Figure 6(b) shows a DOS calculation corresponding to the specific timestep that is ‘boxed’ in Figure 6(a). The DOS shows a state within the bandgap that forms around ‐1 eV. To track the average Pb‐O bond distance along the MD‐trajectory we compute a time‐ resolved RDF and track the value of the first maximum (SI Figure S3). Along with tracking KSO energies, we compute oscillator strengths in SKSO basis along 50‐steps of the MD trajectory and track HO‐LU transition (SI Figure 4). It is observed that the oscillator strength fluctuates from its maximum value to going completely dark, similar to a ‘blinking’ process. It should be mentioned that during the trajectory we do not see any ligands desorb from the surface. We explore this topic further in discussion. For the doped QD we tracked KS orbital energy fluctuations for both spin α and spin β components. For spin α, the two highest states in the valence band edge correspond to the Br 4p


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23 hybridized 3dz2 and 3dx2‐y2 orbitals described in Figure 4(c). For spin β, the lowest five states in the conduction band correspond to Mn2+ 3d orbitals described in Figure 4(d). Along the SP trajectory, as seen in Figure 6(c)‐(d), it is observed that trap states do not form within the gap. Looking at the SKSO computed d‐d oscillator strength transitions along the trajectory, they remain relatively constant but show some fluctuations (SI Figure S5). The large fluctuations are likely due to the bands being composed of Pb 6p orbitals as well, which shown previously will have ‘blinking’ like features along the trajectory. III.3. Non‐Radiative Relaxation. Non‐radiative relaxation dynamics of photoexcited states were found from computing non‐adiabatic couplings between KSOs i and j using equations 18‐19. We use shorter trajectory for couplings, assume average couplings to stay intact forever, and reformulate problem as system of ODE with constant coefficients. Such SODE is solved by diagonalization method, which is valid for arbitrary instant of time. This allows to assess ns lifetime by ps trajectory. Within Redfield formalism working under the secular approximation, the Redfield tensor modulates rates of transitions between populations ii and jj by Riijj. In simplified explanation, the magnitude of each Redfield tensor element is proportional to 𝑉 accordance to Fermi Golden Rule. The 𝑉

, in

matrix elements are averaged over the trajectory as described by

Eq 18a‐18d to generate time independent rates. These rates are then used to construct the Redfield tensor which modulates the non‐radiative relaxation. The values of Redfield tensor which depend on NACs are visualized in Figure 7(a)‐(e) and have values in the range of 10

𝑝𝑠

to 50 𝑝𝑠

.Redfield tensors for both

pristine and doped NCs are shown in Figure 7(a)‐(b) and Figure 7(c)‐(e), respectively. KSO indicies i and j



Page 24 of 55

24 are plotted on x‐axis and y‐axis with matrix elements Riijj plotted on the z‐axis. SI Figure S6 shows an enhanced image of typical HO‐LU Riijj elements indicating low probability of non‐radiative relaxation across the bandgap. For the pristine QD, Figure 7(a) corresponds to Redfield tensor elements computed using SR KSO basis and Figure 7(b) corresponds to SKSO basis. SR KSO and SKSO tensor elements show drastically different trends. For the SR SKO basis, tensor elements follow a trend of higher Riijj values away from the band edges and decreases continuously closer to the band edge R

,

𝑅 ∙ |𝜀 𝑅 ∙ |𝜀

𝜀 |𝛿 , 𝜀 |𝛿 ,

,𝑖 ,𝑖

𝐿𝑈 , 𝐻𝑂

in accordance with the gap law53. In the SKSO basis, Riijj elements show a discontinuous, step‐wise change R

,

,

𝑖

𝑅 𝛿,

1, 𝑖 𝑖𝑠 𝑒𝑣𝑒𝑛 . Reason for such treads are put forth in the 0, 𝑖 𝑖𝑠 𝑜𝑑𝑑

discussion section. It is also observed that there is a wider spread of off‐diagonal Redfield tensor elements in the SKSO basis compared to SR basis. This can be attributed to the mixing of spin states due to SOC. For the doped QD, Figure 7(c)‐(d) correspond to SP KSO basis with Figure 7(c) for spin α and Figure 7(d) spin β components, respectively. Figure 7(e) is Redfield tensor computed in SKSO basis. In the SP KSO basis, both spin α and spin β components show high magnitude Riijj elements that decrease near the band edge. In the SKSO basis, the Riijj elements are roughly half in magnitude compared those computed in SP KSO basis.


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25 Non‐radiative excited‐state dynamics for the pristine and doped QD are shown in Figure 8(a)‐(e). Briefly we describe how to interpret the figures. For each figure, the green color represents ground‐ state charge density, yellow represents gain in charge density Δn>0 = e‐ and blue represents loss of charge density Δn 1. Even with lower absolute rates of relaxation for the

SKSO basis, compared to the SR basis, non‐radiative relaxation to band edges is slightly quicker in the SKSO basis. This can be attributed to the SKSO basis mixing with a larger number of states and providing a greater number of non‐radiative relaxation pathways ie the SKSO Redfield tensor has more off‐ diagonal elements compared to SR tensor. In general, a single exponential relaxation can be assumed when there is only one pathway available. For typical cases of spin‐conserved NAC transitions in semiconductors at room temperature, this is a good approximation as the NACs only couple electronic states within the energy window of kT  25 meV. Here we introduced SOC which provides additional pathways within this energy window. SI Figure 9 compares the computed dimensionless energy decay to single exponential decay for LU+10 to LU, SI Figure 10(a), and HO‐9 to HO, SI Figure 10(b), respectively. The error due to assuming single exponential fit is quantified using root‐mean‐square method

〈𝐸 〉 𝑡

〈𝐸 〉

∑ ∆〈𝐸 〉 𝑡

were ∆〈𝐸 〉 𝑡

𝑡 and t is the number of timesteps. The RMS for single exponential fits of LU+10


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33 to LU and HO‐9 to HO are 1.97x10‐3 and 3.4x10‐3, respectively. It is interesting to note that electron and hole relaxation show opposite trends in carrier cooling. For the photo‐excited electron its carrier cooling slows down near the band edge. This can be attributed to the low density of states within 0.50 eV of band edge due to strong SOC splitting (see Figure 3b). Indirect consequence of SOC coupled with NAC results in slower cooling of carriers in regions of large energy splitting. For the doped NC it was observed that the hole relaxation rate is 102 faster in the spin α basis than the SKSO basis. This is interesting, as the spin α and SKSO valence bands are basically identical in atomic orbital composition and in energy differences between states, compare Figure 4(c) and (d). Non‐ adiabatic transitions are generally inversely proportional to the energy difference between states and 𝝋𝑲𝑺 𝒊 proportional to the mixing of states 𝑉

𝒅𝑯 𝒅𝑹

𝝋𝑲𝑺 𝒋

𝑬𝒊 𝑬𝒋

, per the Hellman‐Feynman theorem. Since the

energetics are the same between the two basis the difference must be a result of a difference in ‘spin mixing’. The difference between the spin α and SKSO relaxation is in the HO‐2 HO‐1 and HO‐1  HO transitions. A population ‘bottleneck’ appears in the SKSO basis compared to spin α (SI Figure S11). We attribute the ‘bottleneck’ to two factors: (1) The HO‐2 to HO‐1, or HO, is a low probability spin‐flip transition, see SI Table S2, which is not considered in the spin α basis. (2) the HO‐1 and HO transition is a transition between Br‐ 4p hybridized Mn2+ 3d states. Contributions from mixing between Mn2+ 3dx2‐y2 and 3dz2 will be a negligible contribution to non‐radiative relaxation, so the relaxation between HO‐1 and HO is driven by the hybridized Br‐ 4p.

Note that in NCL approach the direction of spin magnetization may deviate from the

conventionally used z‐axis. However, SKSOs which are contributed by mostly α or β component are expected to have expectation values of magnetization oriented in opposite direction, antiparallel to



Page 34 of 55

34 each other. For the pristine NC it is seen that there are certain characteristic non‐radiative transitions: (1) ‘fast’ resonant transitions between degenerate bands of anti‐parallel magnetization vectors, which is of spin‐flip character (2) transitions to non‐degenerate bands that have magnetization vectors more in parallel, which is of spin‐conserved character (3) transitions to non‐degenerate bands with more anti‐ parallel magnetization, which is of spin‐flip character. To illustrate these transitions, we project the Redfield tensor on to transitions within window of HO‐5 to HO with inset table of vector magnetization in SI Figure S12. For the doped NC, there will be non‐radiative transitions of type (2) and (3) since the high spin configuration of Mn2+ dopant breaks 2‐fold degeneracy of electronic states.

IV.4. Excited State Emission. The radiative HO‐LU lifetimes are only an order of magnitude different with the pristine and doped NC being 2.35 and 52.77 ns, respectively. The ns ‐ scale lifetime of the intrinsic NC in SKSO basis agrees well with what is seen in experiment for weakly confined CsPbBr3 NCs at room temperature 54 65 and is slightly overestimated for lifetimes measured at 5 K 66. Even though the HO‐LU transition in the doped NC is not a d‐d transition due to the distorted band alignment, it’s oscillator strength is only an order of magnitude greater than the d‐d transitions. The lowest energy d‐d transition lifetimes are on the scale of sub‐s to s, based on their computed transition energies and oscillator strengths. This is 103 faster than what is seen in experiment where d‐d lifetimes are generally on the order of ms16, 18. One source of error is likely due to over‐delocalization of SKSOs computed using a GGA functional. Since transition dipoles are based on spatial overlap of SKSOs the oscillator strengths will be overestimated. Another factor is


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35 that the transition energy between Mn2+ 3d states is also overestimated which will overestimate radiative lifetimes as well. A common metric used to quantify the competing radiative and nonradiative charge recombination is PLQY. From the excited state calculations, one generates these rates and compares the computed PLQY to experimental values. SKSO PBE computed PLQYs for pristine and doped NCs are 33% and 0.20%, respectively. In experiment, it is observed that pristine CsPbBr3 NCs have high PLQYs even when defects are present on the surface creating non‐radiative relaxation pathways. Within our model our NC is well passivated and shows a low PLQY of 33%. One possible reason for this is due to neglect of Rashba effect which is suspected to be responsible for suppressed non‐radiative carrier recombination across the bandgap 67 68. To include Rashba effect into non‐radiative relaxation dynamics would require computing NACs between momentum dependent KSO/SKSOs, which to date has not been implemented. V. CONCLUSIONS



Page 36 of 55

36 We incorporate spin‐orbit coupling into excited‐state non‐adiabatic dynamics simulations using density matrix formalism to describe radiative and non‐radiative relaxation processes. NACs between nuclear and electronic degrees of freedom were computed in the basis of SKSOs found from NCS DFT with PBE functional. We apply this method to a pristine CsPbBr3 and a Mn2+ doped CsPb1‐x MnxBr3 perovskite NC to reveal how spin‐orbit interaction influences ground state electronic/optical properties and excited‐state electron dynamics. We found interesting features in (1) equilibrium geometry static electronic structure (2) role of dual ligand motif on binding energies of passivation ligands to NC surface (3) features of thermal MD nuclear degrees of freedom altering electronic and optical properties. (4) Excited state dynamics of electronic degrees of freedom. (1) Equilibrium geometry of the pristine NC showed expected trends of perovskite bandgaps, with SR GGA calculations giving bandgaps that resemble experiment while SR with SOC reduced the gap by 0.74 eV. For the doped NC the SP electronic structure shows two Br‐ 4p hybridized Mn2+ 3d KSOs above the valence band edge and a manifold of unoccupied Mn2+ 3d KSOs near the conduction band edge. With SOC, the Pb2+ 6p KSOs reduce the bandgap and screen dopant states, giving a band structure alignment that does not allow for d‐d transition PL. Calculations with DFT+U or hybrid functionals give better agreement with experiment. (2) Comparing binding energies of correlated verses independent desorption pathways for passivating ligands shows that it is more energetically favorable for the ethylammonium and acetate anion to desorb as a pair. (3) Along the molecular dynamics trajectory, the oscillator strength of the pristine NC displayed a ‘flickering’ behavior. In the doped NC d‐d oscillator strengths remained relatively constant during the trajectory. (4) For the pristine NC, NACs computed with SKSOs provided different trends in NRR relaxation rates. SKSOs showed an alternating pattern with rates being highest near the band‐gap while SR KSOs


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37 showed maximum rates away from the bandgap and continuously decreasing closer to the bandgap. This is due to degeneracies in the SKSO electronic structure that displays fast transitions between degenerate states and slower transitions to other states. HO‐LU radiative rate for the pristine NC agrees well with experiment while non‐radiative recombination is over‐estimated likely due to neglect of Rashba effect giving a low PLQY of 33%. For the doped NC, SKSO computed dynamics show much slower relaxation near the band‐gap compared to SP SKO NACs. Specifically, hole relaxation from perovskite valence band edge states to the Br‐ 4p hybridized Mn2+ 3d states inside the gap are 2 orders of magnitude slower with SKSO basis than SP basis. This is attributed to a low‐ probability spin‐flip transition from HO‐2 to HO‐1 and low‐probability d‐d transition from HO‐1 to HO. Two most needed continued areas of research would include determining optimal functional to model radiative and non‐radiative relaxation pathways in open shell systems containing transition metals. Also, to include momentum dependent NACs in radiative and non‐radiative relaxation. VI. ACKNOWLEDGEMENTS



Page 38 of 55

38 Authors thank DOE BES NERSC facility for computational resources, allocation award #91202, “Computational Modeling of Photo-catalysis and Photo-induced Charge Transfer Dynamics on Surfaces” supported by the Office of Science of the DOE under contract no. DE-AC02-05CH11231. The authors would like to thank Douglas Jennewein for support and maintaining the High-Performance Computing system at the University of South Dakota. 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. 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). DSK thanks Center for Integrated Nanotechnology and Sergei Tretiak for discussions and hospitality during manuscript preparation. VII. Supporting Information Supporting information includes: pDOS of models investigated, ligand and ligand field geometry/bonding, magnetization projections and magnitudes, optical transition oscillator strengths/transition energies, oscillator strength along MD trajectory, projected Redfield tensors, population dynamics/PL emission along excited state trajectory, dimensionless carrier cooling, spinor KSO fluctuations VIII. REFERENCES


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39 1. Protesescu, L.; Yakunin, S.; Bodnarchuk, M. I.; Krieg, F.; Caputo, R.; Hendon, C. H.; Yang, R. X.; Walsh, A.; Kovalenko, M. V., Nanocrystals of cesium lead halide perovskites (cspbx(3), x = cl, br, and i): Novel optoelectronic materials showing bright emission with wide color gamut. Nano Lett 2015, 15 (6), 3692‐6. 2. Song, J.; Li, J.; Li, X.; Xu, L.; Dong, Y.; Zeng, H., Quantum dot light‐emitting diodes based on inorganic perovskite cesium lead halides (cspbx3 ). Adv. Mater. 2015, 27 (44), 7162‐7. 3. Kim, Y.; Yassitepe, E.; Voznyy, O.; Comin, R.; Walters, G.; Gong, X.; Kanjanaboos, P.; Nogueira, A. F.; Sargent, E. H., Efficient luminescence from perovskite quantum dot solids. ACS Appl Mater Interfaces 2015, 7 (45), 25007‐13. 4. Zhang, X.; Lin, H.; Huang, H.; Reckmeier, C.; Zhang, Y.; Choy, W. C.; Rogach, A. L., Enhancing the brightness of cesium lead halide perovskite nanocrystal based green light‐emitting devices through the interface engineering with perfluorinated ionomer. Nano Lett. 2016, 16 (2), 1415‐20. 5. Morozov, Y. V.; Zhang, S.; Brennan, M. C.; Janko, B.; Kuno, M., Photoluminescence up‐ conversion in cspbbr3 nanocrystals. ACS Energy Letters 2017, 2 (10), 2514‐2515. 6. Manzi, A.; Tong, Y.; Feucht, J.; Yao, E. P.; Polavarapu, L.; Urban, A. S.; Feldmann, J., Resonantly enhanced multiple exciton generation through below‐band‐gap multi‐photon absorption in perovskite nanocrystals. Nat Commun 2018, 9 (1), 1518. 7. Yakunin, S.; Protesescu, L.; Krieg, F.; Bodnarchuk, M. I.; Nedelcu, G.; Humer, M.; De Luca, G.; Fiebig, M.; Heiss, W.; Kovalenko, M. V., Low‐threshold amplified spontaneous emission and lasing from colloidal nanocrystals of caesium lead halide perovskites. Nat Commun 2015, 6, 8056. 8. Huang, H.; Bodnarchuk, M. I.; Kershaw, S. V.; Kovalenko, M. V.; Rogach, A. L., Lead halide perovskite nanocrystals in the research spotlight: Stability and defect tolerance. ACS Energy Letters 2017, 2071‐2083. 9. Giansante, C.; Infante, I., Surface traps in colloidal quantum dots: A combined experimental and theoretical perspective. J Phys Chem Lett 2017, 8 (20), 5209‐5215. 10. Protesescu, L.; Yakunin, S.; Kumar, S.; Bar, J.; Bertolotti, F.; Masciocchi, N.; Guagliardi, A.; Grotevent, M.; Shorubalko, I.; Bodnarchuk, M. I.; Shih, C. J.; Kovalenko, M. V., Dismantling the "red wall" of colloidal perovskites: Highly luminescent formamidinium and formamidinium‐cesium lead iodide nanocrystals. ACS Nano 2017, 11 (3), 3119‐3134. 11. Koscher, B. A.; Swabeck, J. K.; Bronstein, N. D.; Alivisatos, A. P., Essentially trap‐free cspbbr3 colloidal nanocrystals by postsynthetic thiocyanate surface treatment. J. Am. Chem. Soc. 2017, 139 (19), 6566‐6569. 12. Krieg, F.; Ochsenbein, S. T.; Yakunin, S.; ten Brinck, S.; Aellen, P.; Süess, A.; Clerc, B.; Guggisberg, D.; Nazarenko, O.; Shynkarenko, Y.; Kumar, S.; Shih, C.‐J.; Infante, I.; Kovalenko, M. V., Colloidal cspbx3 (x = cl, br, i) nanocrystals 2.0: Zwitterionic capping ligands for improved durability and stability. ACS Energy Letters 2018, 641‐646. 13. Norris, D. J.; Efros, A. L.; Erwin, S. C., Doped nanocrystals. Science 2008, 319 (5871), 1776‐1779. 14. Fainblat, R.; Barrows, C. J.; Gamelin, D. R., Single magnetic impurities in colloidal quantum dots and magic‐size clusters. Chemistry of Materials 2017. 15. Liu, W.; Lin, Q.; Li, H.; Wu, K.; Robel, I.; Pietryga, J. M.; Klimov, V. I., Mn2+‐doped lead halide perovskite nanocrystals with dual‐color emission controlled by halide content. J Am Chem Soc 2016, 138 (45), 14954‐14961. 16. Akkerman, Q. A.; Meggiolaro, D.; Dang, Z.; De Angelis, F.; Manna, L., Fluorescent alloy cspbxmn1–xi3 perovskite nanocrystals with high structural and optical stability. ACS Energy Letters 2017, 2 (9), 2183‐2186. 17. Yuan, X.; Ji, S.; De Siena, M. C.; Fei, L.; Zhao, Z.; Wang, Y.; Li, H.; Zhao, J.; Gamelin, D. R., Photoluminescence temperature dependence, dynamics, and quantum efficiencies in mn2+‐doped



Page 40 of 55

40 cspbcl3 perovskite nanocrystals with varied dopant concentration. Chemistry of Materials 2017, 29 (18), 8003‐8011. 18. Rossi, D.; Parobek, D.; Dong, Y.; Son, D. H., Dynamics of exciton–mn energy transfer in mn‐ doped cspbcl3 perovskite nanocrystals. The Journal of Physical Chemistry C 2017, 121 (32), 17143‐ 17149. 19. Parobek, D.; Roman, B. J.; Dong, Y.; Jin, H.; Lee, E.; Sheldon, M.; Son, D. H., Exciton‐to‐dopant energy transfer in mn‐doped cesium lead halide perovskite nanocrystals. Nano Lett 2016, 16 (12), 7376‐ 7380. 20. Yao, G.; Huang, S.; May, P. S.; Berry, M. T.; D., K., Non‐collinear spin dft for lanthanide ions in doped hexagonal nayf4 Molecular Physics 2014, 112 (3‐4), 546‐556. 21. Vogel, D. J.; Kryjevski, A.; Inerbaev, T.; Kilin, D. S., Photoinduced single‐ and multiple‐electron dynamics processes enhanced by quantum confinement in lead halide perovskite quantum dots. The Journal of Physical Chemistry Letters 2017, 8 (13), 3032‐3039. 22. Jankowska, J.; Long, R.; Prezhdo, O. V., Quantum dynamics of photogenerated charge carriers in hybrid perovskites: Dopants, grain boundaries, electric order, and other realistic aspects. ACS Energy Letters 2017, 2 (7), 1588‐1597. 23. Nguyen, T. S.; Parkhill, J., Nonradiative relaxation in real‐time electronic dynamics oscf2: Organolead triiodide perovskite. J Phys Chem A 2016, 120 (34), 6880‐7. 24. Forde, A.; Kilin, D., Hole transfer in dye‐sensitized cesium lead halide perovskite photovoltaics: Effect of interfacial bonding. The Journal of Physical Chemistry C 2017, 121 (37), 20113‐20125. 25. Vogel, D. J.; Inerbaev, T. M.; Kilin, D. S., Role of lead vacancies for optoelectronic properties of lead‐halide perovskites. The Journal of Physical Chemistry C 2018, 122 (10), 5216‐5226. 26. Barth, U. v.; Hedin, L., A local exchange‐correlation potential for the spin polarized case. I. Journal of Physics C: Solid State Physics 1972, 5 (13), 1629‐1642. 27. Brennan, M. C.; Herr, J. E.; Nguyen‐Beck, T. S.; Zinna, J.; Draguta, S.; Rouvimov, S.; Parkhill, J.; Kuno, M., Origin of the size‐dependent stokes shift in cspbbr3 perovskite nanocrystals. J. Am. Chem. Soc. 2017, 139 (35), 12201‐12208. 28. Proshchenko, V.; Dahnovsky, Y., Tunable luminescence in cdse quantum dots doped by mn impurities. The Journal of Physical Chemistry C 2014, 118 (48), 28314‐28321. 29. Kilina, S.; Ivanov, S.; Tretiak, S., Effect of surface ligands on optical and electronic spectra of semiconductor nanoclusters. Journal of the American Chemical Society 2009, 131 (22), 7717‐7726. 30. Badaeva, E.; Isborn, C. M.; Feng, Y.; Ochsenbein, S. T.; Gamelin, D. R.; Li, X., Theoretical characterization of electronic transitions in co2+‐ and mn2+‐doped zno nanocrystals. The Journal of Physical Chemistry C 2009, 113 (20), 8710‐8717. 31. Hammes‐Schiffer, S.; Tully, J. C., Proton transfer in solution ‐ molecular dynamics with quantum transitions. Journal of Chemical Physics 1994, 101 (6), 4657‐4667. 32. Tully, J. C., Molecular dynamics with electronic transitions. The Journal of Chemical Physics 1990, 93 (2), 1061‐1071. 33. Redfield, A. G., On the theory of relaxation processes. Ibm Journal of Research and Development 1957, 1 (1), 19‐31. 34. Jean, J. M.; Friesner, R. A.; Fleming, G. R., Application of a multilevel redfield theory to electron transfer in condensed phases. Journal of Chemical Physics 1992, 96 (8), 5827‐5842. 35. Egidi, F.; Sun, S.; Goings, J. J.; Scalmani, G.; Frisch, M. J.; Li, X., Two‐component noncollinear time‐dependent spin density functional theory for excited state calculations. Journal of Chemical Theory and Computation 2017, 13 (6), 2591‐2603. 36. Yao, G.; Meng, Q.; Berry, M. T.; May, P. S.; S. Kilin, D., Molecular dynamics in finding nonadiabatic coupling for β‐nayf4: Ce3+ nanocrystals. Molecular Physics 2015, 113 (3‐4), 385‐391.


Page 41 of 55 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


41 37. Filip, M. R.; Giustino, F., Gw quasiparticle band gap of the hybrid organic‐inorganic perovskite ch3nh3pbi3: Effect of spin‐orbit interaction, semicore electrons, and self‐consistency. Physical Review B 2014, 90 (24). 38. Bulik, I. W.; Scalmani, G.; Frisch, M. J., Noncollinear density functional theory having proper invariance and local torque properties. Physical Review B 2013, 87 (3), 035117. 39. Vogel, D. J.; Kilin, D. S., First‐principles treatment of photoluminescence in semiconductors. Journal of Physical Chemistry C 2015, 119 (50), 27954‐27964. 40. Han, Y.; Vogel, D. J.; Inerbaev, T. M.; May, P. S.; Berry, M. T.; Kilin, D. S., Photoinduced dynamics to photoluminescence in ln3+ (ln = ce, pr) doped β‐nayf4 nanocrystals computed in basis of non‐ collinear spin dft with spin‐orbit coupling. Molecular Physics 2017, 1‐11. 41. Perdew, J. P.; Burke, K.; Ernzerhof, M., Generalized gradient approximation made simple [phys. Rev. Lett. 77, 3865 (1996)]. Phys. Rev. Lett. 1997, 78 (7), 1396‐1396. 42. Blöchl, P. E., Projector augmented‐wave method. Physical Review B 1994, 50 (24), 17953‐17979. 43. Kresse, G.; Joubert, D., From ultrasoft pseudopotentials to the projector augmented‐wave method. Physical Review B 1999, 59 (3), 1758‐1775. 44. Kresse, G.; Furthmuller, J., Efficiency of ab‐initio total energy calculations for metals and semiconductors using a plane‐wave basis set. Computational Materials Science 1996, 6 (1), 15‐50. 45. Neugebauer, J.; Scheffler, M., Adsorbate‐substrate and adsorbate‐adsorbate interactions of na and k adlayers on al(111). Physical Review B 1992, 46 (24), 16067‐16080. 46. Makov, G.; Payne, M. C., Periodic boundary conditions in ab initio calculations. Phys. Rev. B 1995, 51 (7), 4014‐4022. 47. Meng, Q. G.; Chen, J. C.; Kilin, D., Proton reduction at surface of transition metal nanocatalysts. Molec. Simulation 2015, 41 (1‐3), 134‐145. 48. Brown, S. L.; Hobbie, E. K.; Tretiak, S.; Kilin, D. S., First‐principles study of fluorescence in silver nanoclusters. The Journal of Physical Chemistry C 2017, 121 (43), 23875‐23885. 49. Forde, A.; Inerbaev, T.; Kilin, D., Role of cation‐anion organic ligands for optical properties of fully inorganic perovskite quantum dots. MRS Advances, 1‐7. 50. Giustino, F., Electron‐phonon interactions from first principles. Reviews of Modern Physics 2017, 89 (1), 015003. 51. Han, P.; Bester, G., First‐principles calculation of the electron‐phonon interaction in semiconductor nanoclusters. Physical Review B 2012, 85 (23). 52. Kryjevski, A.; Mihaylov, D.; Kilina, S.; Kilin, D., Multiple exciton generation in chiral carbon nanotubes: Density functional theory based computation. The Journal of Chemical Physics 2017, 147 (15), 154106. 53. Englman, R.; Jortner, J., Energy gap law for radiationless transitions in large molecules. Molecular Physics 1970, 18 (2), 145‐164. 54. Koscher, B. A.; Swabeck, J. K.; Bronstein, N. D.; Alivisatos, A. P., Essentially trap‐free cspbbr3 colloidal nanocrystals by postsynthetic thiocyanate surface treatment. J Am Chem Soc 2017, 139 (19), 6566‐6569. 55. Krieg, F.; Ochsenbein, S. T.; Yakunin, S.; ten Brinck, S.; Aellen, P.; Süess, A.; Clerc, B.; Guggisberg, D.; Nazarenko, O.; Shynkarenko, Y.; Kumar, S.; Shih, C.‐J.; Infante, I.; Kovalenko, M. V., Colloidal cspbx3 (x = cl, br, i) nanocrystals 2.0: Zwitterionic capping ligands for improved durability and stability. ACS Energy Letters 2018, 3 (3), 641‐646. 56. Liu, F.; Zhang, Y.; Ding, C.; Kobayashi, S.; Izuishi, T.; Nakazawa, N.; Toyoda, T.; Ohta, T.; Hayase, S.; Minemoto, T.; Yoshino, K.; Dai, S.; Shen, Q., Highly luminescent phase‐stable cspbi3 perovskite quantum dots achieving near 100% absolute photoluminescence quantum yield. ACS Nano 2017, 11 (10), 10373‐10383.



Page 42 of 55

42 57. Even, J.; Pedesseau, L.; Jancu, J.‐M.; Katan, C., Importance of spin–orbit coupling in hybrid organic/inorganic perovskites for photovoltaic applications. The Journal of Physical Chemistry Letters 2013, 4 (17), 2999‐3005. 58. De Roo, J.; Ibanez, M.; Geiregat, P.; Nedelcu, G.; Walravens, W.; Maes, J.; Martins, J. C.; Van Driessche, I.; Kovalenko, M. V.; Hens, Z., Highly dynamic ligand binding and light absorption coefficient of cesium lead bromide perovskite nanocrystals. ACS Nano 2016, 10 (2), 2071‐81. 59. Heyd, J.; Scuseria, G. E.; Ernzerhof, M., Hybrid functionals based on a screened coulomb potential. Journal of Chemical Physics 2003, 118 (18), 8207‐8215. 60. Dudarev, S. L.; Botton, G. A.; Savrasov, S. Y.; Humphreys, C. J.; Sutton, A. P., Electron‐energy‐loss spectra and the structural stability of nickel oxide: An lsda+u study. Physical Review B 1998, 57 (3), 1505‐ 1509. 61. Ekaterina, B.; Yong, F.; Daniel, R. G.; Xiaosong, L., Investigation of pure and co 2+ ‐doped zno quantum dot electronic structures using the density functional theory: Choosing the right functional. New Journal of Physics 2008, 10 (5), 055013. 62. Klimov, V. I.; Mikhailovsky, A. A.; Xu, S.; Malko, A.; Hollingsworth, J. A.; Leatherdale, C. A.; Eisler, H. J.; Bawendi, M. G., Optical gain and stimulated emission in nanocrystal quantum dots. Science 2000, 290 (5490), 314‐317. 63. Frantsuzov, P.; Kuno, M.; Janko, B.; Marcus, R. A., Universal emission intermittency in quantum dots, nanorods and nanowires. Nature Physics 2008, 4 (7), 519‐522. 64. Yarita, N.; Tahara, H.; Saruyama, M.; Kawawaki, T.; Sato, R.; Teranishi, T.; Kanemitsu, Y., Impact of postsynthetic surface modification on photoluminescence intermittency in formamidinium lead bromide perovskite nanocrystals. The Journal of Physical Chemistry Letters 2017, 8 (24), 6041‐6047. 65. Makarov, N. S.; Guo, S.; Isaienko, O.; Liu, W.; Robel, I.; Klimov, V. I., Spectral and dynamical properties of single excitons, biexcitons, and trions in cesium‐lead‐halide perovskite quantum dots. Nano Lett 2016, 16 (4), 2349‐62. 66. Becker, M. A.; Vaxenburg, R.; Nedelcu, G.; Sercel, P. C.; Shabaev, A.; Mehl, M. J.; Michopoulos, J. G.; Lambrakos, S. G.; Bernstein, N.; Lyons, J. L.; Stöferle, T.; Mahrt, R. F.; Kovalenko, M. V.; Norris, D. J.; Rainò, G.; Efros, A. L., Bright triplet excitons in caesium lead halide perovskites. Nature 2018, 553, 189. 67. Etienne, T.; Mosconi, E.; De Angelis, F., Dynamical origin of the rashba effect in organohalide lead perovskites: A key to suppressed carrier recombination in perovskite solar cells? The Journal of Physical Chemistry Letters 2016, 7 (9), 1638‐1645. 68. Isarov, M.; Tan, L. Z.; Bodnarchuk, M. I.; Kovalenko, M. V.; Rappe, A. M.; Lifshitz, E., Rashba effect in a single colloidal cspbbr3 perovskite nanocrystal detected by magneto‐optical measurements. Nano Letters 2017, 17 (8), 5020‐5026.


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43 VIII. FIGURES AND TABLES

Figure 1: (a) shows the geometry optimized pristine CsPbBr3 QD constructed from a 2x2x2 unit cells and passivated by short chain carboxylic acid and amine ligands. The pristine QD has an atomistic composition of Cs8Pb27Br54 + 48 ethylammonium cations+ 54 acetate anions. Binding energies of each ligand are summarized in Table 1. (b) shows the elements/molecules used to construct the perovskite QD models. (c) shows the Mn2+ doped CsPb1‐x MnxBr3 QD which has the same morphology as the intrinsic QD, but with the central most Pb2+ atom substitutionally replaced with Mn2+.



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44

Figure 2: Energy diagram which shows how dual emission occurs in doped semiconductor NC/QDs. A photoexcitation event occurs in the NC with the photo‐excited electron e‐ and photo‐induced hole h+ relaxing to their respective band edges (green rectangles). Once at the band edges e‐ and h+ can either radiatively combine to produce PL with energy hvperovskite or non‐radiatively transfer to either surface trap states (dashed line) or Mn2+ 3d states (solid black line) inside the perovskite NC bandgap. If both the e‐ and h+ non‐radiatively relax to Mn2+ 3d states there can be radiative emission with energy hvdopant typically associated with the 4T1  6A1 transition.


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45

Figure 3: (a) – (d) show DOS for each respective model computed under different methodologies. Shaded regions represent occupied states. (a) Spin‐restricted DOS for the pristine CsPbBr3 QD which shows a bandgap of 3.12 eV. (b) Spinor DOS for the pristine CsPbBr3 QD which shows a bandgap of 2.38 eV. The spinor DOS shows a reduced bandgap due to SOC splitting degenerate Pb2+ 6p spin states in the conduction band. (c) Spin‐polarized DOS for the doped CsPb1‐x MnxBr3 QD where the upper and lower curves represent spin α and spin β components, respectively. The Mn2+ dopant takes a high spin S=5/2 configuration with two Mn2+ d‐orbital states appearing inside the spin α bandgap. The spin α and spin β components show a bandgap of 2.35 eV and 3.13 eV, respectively. (d) Spinor DOS for the doped CsPb1‐x MnxBr3 QD which shows a bandgap of 1.76 eV.



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46

Figure 4: (a) – (d) show pDOS for each respective QD model computed under different methodologies. Solid red, dashed green, and dashed‐hyphen orange lines correspond to states located on ligand atoms (C,H,O,N), perovskite atoms (Cs, Pb, Br), and the dopant atom (Mn), respectively. (a) Spin‐restricted pDOS and (b) spinor pDOS for the pristine CsPbBr3 QD. The band edges for both (a) and (b) show perovskite states, indicating the QD is well passivated. (c) Spin‐polarized pDOS and (d) spinor pDOS for the doped CsPb1‐x MnxBr3 QD were for (c) upper and lower curves represent spin α and spin β components, respectively. The insets for (c) and (d) are 100x magnification of the states between ‐3 and ‐2 eV and 10x magnification for states between ‐0.2 and 0.5 eV. The valence band edge spin α states in (c) show hybridization between Mn2+ 3d‐orbitals and ligand field Br 4p KSOs while the conduction band edge spin β states show mostly pure Mn2+ 3d KSOs (SI Table 1). Note that in (c) an optical transition from HOMO‐LUMO would correspond to an explicit spin‐flip. In (d) the same features as described in (c) appear, but the conduction band Mn2+ 3d KSOs are covered up by Pb2+ 6p KSOs due to SOC reducing the perovskite QD bandgap.


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47 Figure 5: Optical spectra (in arbitrary units) for (a) pristine CsPbBr3 QD and (b) doped CsPb1‐x MnxBr3 QD. Labels show notable optical transitions are described in Table x. (a) The solid red line corresponds to SR spectra and the dashed green line corresponds to spinor spectra. The spinor spectrum shows lower transition energies due to SOC reducing the bandgap. Spinor spectra also shows enhanced on‐set absorption compared to spin‐restricted spectra (vertical black dashed lines). (b) Solid red, dashed green, and dashed‐hyphen orange lines correspond to spinor, spin α, and spin β spectra, respectively. The inset shows low‐intensity absorption onset for spinor and spin α spectra. The spinor spectrum shows earlier onset of absorption compared to SP spectra.



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48

Figure 6: (a), (c), and (d) show energy fluctuations of KSOs along a MD trajectory due to static coupling of KSOsto lattice vibrational modes. Lines in the lower and upper half of the figure correspond to energies near the valence band edge εi to εHO and conduction band edge εi to εLU, respectively. Each colored line corresponds to an energy of a distinct KSO. (a) KSO fluctuations for the pristine CsPbBr3 QD along MD trajectory. Along the trajectory sharp peaks form and disappear, suggestive of trap state formation. (b) DOS calculation corresponding to the specific time ‘boxed’ snapshot in (a) which illustrates trap state formation at that time‐step. The insets show partial charge density localized on surface ligands, confirming surface trap state formation. (c) KSO fluctuations for the spin α component of the doped CsPb1‐x MnxBr3 QD. It is observed that there are no sharp peaks that form above the pink curve in the valence band, which corresponds to the dopant state being ‘protected’ from surface states. (d) KSO fluctuations for the spin β component of the doped CsPb1‐x MnxBr3 QD.


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49

Figure 7: (a) – b) show Redfield tensors computed for the pristine CsPbBr3 QD and (c) – (e) show tensors computed for the doped CsPb1‐x MnxBr3 QD. Redfield tensors describe the average coupling of electronic states i (x‐axis) and j (y‐axis) along a MD trajectory due to thermal fluctuations. Each Riijj component (z‐ axis) is proportional to the rate of transition between states i and j. For the pristine QD, (a) shows greater coupling away from the band edge and lower coupling near the band edge, while (b) shows the opposite trend. (b) displays a step‐wise symmetry with Rii,i‐1,i‐1(i) fluctuating between high and low values 1, 𝑖 𝑖𝑠 𝑒𝑣𝑒𝑛 Rii,i‐1,i‐1(i) ~ . For the doped QD, (c) and (d) show 2x higher magnitude of Riijj 0, 𝑖 𝑖𝑠 𝑜𝑑𝑑 components compared to (e).



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50 Figure 8: (a) – (e) illustrate the non‐radiative relaxation of a hole and electron to the band edge after a photo‐excitation where (a)‐(b) are for the pristine QD and (c)‐(e) correspond to the doped QD. e and h represent time of relaxation from LU+y to LU and HO‐x to HO, respectively. Initial conditions HO‐x and LU+y are choosen based on oscillator strengths. Green background represents equilibrium distribution with blue and yellow regions specify energy intervals were charge density deviates from equilibrium. Blue represents loss of electron density Δn0 = e‐. Solid and dashed horizontal lines represent average electron energy 〈∆𝜀 〉 𝑡 and hole energy 〈∆𝜀 〉 𝑡 during relaxation, respectively. For the pristine QD, both (a) and (b) show sub‐picoseond electron and hole relaxation with (b) showing slightly faster relaxation. For the doped QD, (c) and (d) show sub‐ picosecond electron and hole relaxation while (e) shows picosecond electron relaxation and 101 picosecond hole relaxation, as summarized in Table 3.


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51

Figure 9: (a)‐(b) and (c)‐(d) correspond to time‐resolved and time‐integrated emission for the pristine NC and doped NC using SKSO basis, respectively. (a),(c) correspond to time‐resolved radiative transitions that compete with non‐radiative transitions along the excited‐state trajectory. The x‐axis is log scale normalized to picosecond and y‐axis shows emission energy in eV. Emission events occur during the trajectory when krad > knon‐rad . The intensity of the emission is indicated by the color of the band, with yellow indicating most intense transition and blue being less intense. The length of the trajectory is chosen based on radiative and non‐radiative lifetimes of HO‐LU transition. The pristine and doped HO‐LU transitions have radiative/non‐radiative lifetimes of 2.35 ns/1.18 ns and 52.7 ns/97.8 ps, respectively. (b),(d) sums the time‐resolved spectra to generate a PL spectrum, in arbitrary units. It is observed that the pristine NC has 105 more intense PL than the doped PL. PLQY is computed for both models with the pristine and doped NC having 33% and 0.20%, respectively.



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52

Ligand Removed

Binding Energy [eV]

(1) Acetate Anion

0.61

(2) Ethylammonium

3.61

(3) Acetate Anion + Ethylammonium 1.47 Table 1: Binding energies were computed using E(products) – E(reactants) for the reactions pathways explored. We explore whether it is more favorable for the acetate anion and ethylammonium ligands to desorb independently, case (1) and (2), or to desorb together in a correlated way, case (3). It is seen that case (3), where an adjacent ethylammonium protonates an acetate anion, is more energetically favorable than the sum of pathways (1) and (2) where an acetate anion desorbs and gained a proton from solution donated by a desorbed ethylammonium.


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53 Pristine QD ‐ Spin Restricted

i

j

eV

c' HO LU 3.11 b' HO‐3 LU+16 3.72 a' HO‐37 LU+2 4.01 Doped QD – Spin α α

c HO LU α b HO‐3 LU+1 aα HO‐20 LU+9

Pristine QD ‐ Spinor OS

6.73 C 4.52 B 2.61 A

j

eV

HO HO‐6 LU+29

LU LU+11 HO‐28

2.38 0.22 i 3.18 0.47 f HO 3.69 0.15 e HO

Doped QD ‐ Spin β 2.37 3.27 3.96

β

0.47 c HO 3.70 bβ HO‐7 7.12 aβ HO‐17

LU LU+8 LU+14

OS

Doped QD ‐ Spinor

i

d HO 3.13 0.08 c HO‐5 3.62 5.20 b HO‐7 3.98 1.76 a HO‐31

j LU LU+4

eV OS 1.76 0.02 1.91 0.01

LU+12 2.20 0.01 LU+1 2.63 0.46 LU+4 2.80 0.46 LU+29 3.62 0.08

Table 2: Optical transitions for the pristine and doped QDs using different basis. These transitions are shown in Figures 5(a)‐(b).



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Model : Method

ke

kh

[1/ps]

[1/ps]

Pristine : Spin‐Restricted 8(a)

3.34

2.27

Pristine: Spinor 8(b)

4.76

8.34

Doped: Spin α 8(c)

4.35

4.01

Doped: Spin β 8(d)

9.09

10.12

Doped: Spinor 8(e)

1.04

0.09

Table 3: Non‐radiative relaxation rates, in units of 1/ps, which correspond to Figures 8(a)‐(e). For the pristine NC, the spinor basis enhances relaxation rates while for the doped NC the spinor basis slows down relaxation. Reasons for these trends are put forth in discussion section.


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55 TOC Graphic


Role of Spin-Orbit Coupling - American Chemical Society

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