Simulating Energy Transfer Between Nanocrystals and Organic Semiconductors by Nadav Geva B.S.E., University of Michigan (2013) Submitted to the Department of Materials Science and Engineering in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Materials Science and Engineering at the MASSACHUSETTS INSTITUTE OF TECHNOLOGY February 2018 @ Massachusetts Institute of Technology 2018. All rights reserved.

Author ...... Signature redacted ...... Department of Materials Science and Engineering Signature redacted January 3, 2017 Certified by / Troy Van Voorhis Haslam and Dewey Professor of Chemistry Signature redacted Thesis Supervisor Certified by ...... Jeffrey C. Grossman Professor of Materials ien l Engineering 7"/3 4~hesis Reader Accepted by...... Signature redacted ARCHIVES Donald R. S1dway John F. Elliot Professor of Materials Chemistry MASSACHUSETTS INSTITUTE OFTECHNOLOGY Chair, Department Committee on Graduate Theses MAR 01 Z018

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The images contained in this document are of the best quality available. 2 Simulating Energy Transfer Between Nanocrystals and Organic Semiconductors by Nadav Geva

Submitted to the Department of Materials Science and Engineering on December 22, 2017, in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Materials Science and Engineering

Abstract

Recent trends in renewable energy made silicon based photovoltaics the undisputed leader. Therefore, technologies that enhance, instead of compete with, silicon based solar cells are desirable. One such technology is the use of organic semiconductors and noncrystalline semiconductors for photon up- and down-conversion. However, the understanding of energy transfer in these hybrid systems required to effectively engineer devices is missing. In this thesis, I explore and explain the mechanism of energy transfer between noncrystalline semiconductors and organic semiconductors. Using a combination of density functional calculations, molecular dynamics, and ki- netic theory, I have explored the geometry, morphology, electronic structure, and coarse grained kinetics of these system. The result is improved understanding of the transfer mechanism, rate, and the device structure needed for efficient devices. I have also looked at machine learning inspired algorithm for acceleration of density func- tional theory methods. By training machine learning models on DFT data, a much improved initial guess can be made, greatly accelerating DFT optimizations. Gen- erating and examining this data set also revealed a remarkable degree of structure, that perhaps can be further exploited in the future.

Thesis Supervisor: Troy Van Voorhis Title: Haslam and Dewey Professor of Chemistry

3 4 Acknowledgments

What a ride. 9.5 years ago, I came to the US for undergrad. Since then, I finished my undergrad, got married, had 2 kids, bought a house, and now I'm finishing my PhD.

I would like to start by thanking my advisor, Prof. Troy Van Voorhis, for taking a chance on a Materials grad student with no computational background. You were my model for a scientist in my time at MIT. Next is my office mate Alex Kohn. Getting a PhD is mentally challenging, and having someone who is going through the same ordeal to share the misery and the triumph was indispensable. Thank you Alex, for being that person.

Another such person was James Shepherd. Our long conversions, on anything from science to politics, shaped me as much as all my previous experiences combined.

The life of a grad student can be very lonely, and I am grateful I had you to break from that loneliness.

While lonely, I was not alone. I would like to thank the present and past zoo community, for creating an inspiring, productive, and social work environment. I would like to thank Matt Welborn, Mike Mavros, Tianyu Zhu, Eric Hontz, Takashi

Tsuchimochi, Churn Chang, Dave McMahon, Thomas Avila, Hungzho Ye, Zhou Lin, Piotr deSilva, Lexie Mclsaac, Nathan Ricke, Kaitlyn Dwelle, Amr Dodin and all other visitors to the zoo. It's been an honor being your volleyball captain and colleague.

Finally, I would like to thank my family. My wife, Ceit, who's support and love kept me going through the worst of it, I couldn't have done it without you. And to my two kids, Adam and Ella. You made this twice as hard, and twice as worth it.

This thesis is dedicated to you.

5 "Every attempt to employ mathematical methods in the study of chemical questions must be considered profoundly irrationaland contrary to the spirit of chemistry...

If mathematical analysis should ever hold a prominent place in chemistry - an aberration which is happily almost impossible - it would occasion a rapid and widespread degeneration of that science."

-Auguste Comte, Cours de philosophie positive, 1830

6 Contents

1 Introduction 19

1.1 M otivation ...... 20 1.2 M ethods ...... 22 1.2.1 Molecular Dynamics (MD) ...... 22

1.2.2 Constrained Density Functional Theory - Configuration Inter- action (CDFT-CI) ...... 24

2 Morphology of passivating organic ligands around a nanocrystal 29

2.1 Introduction ...... 29

2.2 Simulations ...... 31

2.3 Experiment ...... 40

2.4 Theory ...... 43

2.5 Conclusion ...... 45

3 Reaching the Speed Limit for Triplet Transfer in Solid-State Lead

Sulfide Nanocrystal Sensitized Photon Upconversion 49

3.1 Introduction ...... 49

3.2 Results and Discussion ...... 51

3.3 C onclusion ...... 62

3.4 Materials and Methods ...... 63

4 Mean Field Treatment of Heterogeneous Steady State Kinetics 69

4.1 Introduction ...... 69

7 4.2 Working example ...... 71 4.3 The self consistent mean field approach .. .. . 74 4.4 R esults ...... 77 4.5 Conclusion ...... 81

5 A disordered kinetics model for triplet transfer 83 5.1 Introduction ...... 83 5.2 Methods ...... 85 5.3 Results and Discussion ...... 88 5.4 Conclusions ...... 90

6 WDA with exact condition: An acceleration scheme 93 6.1 Introduction ...... 93 6.2 Theoretical Background 95 6.3 Method ...... 96 6.4 Computational Details 97 6.5 Results and discussion 98 6.6 Conclusion ...... 102

7 Optimal Tuning in Machine Learning 103

7.1 Introduction ...... 103 7.2 Methods ...... 104 7.3 Results and Discussion . . 106 7.4 Conclusions ...... 107

8 Conclusion 109

8 List of Figures

2-1 Steps of the simulation are ordered as follows: a) the nanocrystal as

carved from bulk; b) which is then decorated with ligands in a 'spiky

ball' conformation; c) the geometry after energy minimization; d) the

geometry after molecular dynamics ...... 32

2-2 Thickness of the ligand shell as a function of stretched ligand length

for three dot sizes. Two linear eye-guides are also provided: the first

(- -) has a slope of one, shifted by the Cd-N bond length: x + bCd-N-

The second (-.-.) has a slope one half: x/2 + bCd-N. Representative

error bars are shown for the smallest dot size, where we averaged over

six independent runs of Fig. 2-1. These show that the shell thickness

does not vary significantly with diameter over the range shown here. 34

2-3 Calculation of the order parameter: For each ligand, we calculate

two vectors: the first is the vector from the geometric center of the

nanocrystal to the ligand head group, the second is the vector from

the head group to the tail group (shown in (a)). The order parameter

is then the dot product of the normalized vectors. This distribution

varies according to ligand length (shown in (b)), with ligand repeat

units shown in the key as n in CH3 (CH 2)nNH 2 . The dot size is 3.56nm. 35

9 2-4 Illustrative snapshots from simulations of a 3.56nm nanocrystal. From

left to right, these correspond to ligand lengths of n=3 to 17 in in-

crements of two. The surface ligands are colored by their orientation

relative to the surface. Red-colored molecules are sticking straight

out, as quantified by an order parameter greater than 0.6, whereas

turquoise-colored molecules lay flat...... 36

2-5 Mode of the order parameter distribution as a function of ligand size, for three nanocrystal sizes...... 37

2-6 Anisotropy of the relative ligand shell thickness is shown for a pair of

dots. The thickness (indicated by color) is plotted against two angular

parameters which describe a location on a dot's spherical surface (for

details, see Eq. (2.1)). Here, we have used the height of the location of

the end group (C in CH 3 , red points) as a way of mapping thickness. Color is used to indicate height as a heat map, where blue locations

are increased ligand height, white are decreased. At the start of the

simulation (left) the height distribution is uniform within projection

error but over the course of the simulation it becomes increasingly

anisotropic (right). Since we are measuring the height only at the

head group location, white areas are not necessarily bare. The dot size is 3.56 nm ...... 37

2-7 Transmission electron microscopy (TEM) was performed to make com-

parison with computational results. (a) An example of a TEM micro-

graph of a layer of PbS nanocrystals with carboxylic acid ligands. (b)

TEM data compares well with simulation data (here, taken from the

1.00 nm dot size). The thickness in this plot is corrected for the bond

length that joins the ligand to the surface i.e. Cd-N or Pb-O. The red,

dashed line shows the fit to Eq. (2.2) for the whole data set. The black, dashed line is a line of slope one...... 40

10 2-8 Two explanations for how dots are separated by a distance of approx-

imately one ligand length. Right panel shows interdigitation of the

ligands, which is the prevailing literature viewpoint; left panel shows

ligands crumpled over as in our simulations...... 42

2-9 In (a), relative frequency of gauche conformations along the chain is

plotted against chain location. Dihedrals are measured along the length

of the chain, with chain location being referenced with the N-C-C-C

dihedral as zero. For longer ligands, represented here by n = 17, gauche

conformations are more common at the beginning of the chain. In (b), relaxed ligand end-to-end distance does not increase linearly with the

stretched chain length of the ligand. Three dot sizes are shown. The red

and green points overlay each other due to the similarity between the

3.56 nm and 5.00 nm dot. The black, dashed lined labeled Monte Carlo

(MC) provides a link between the dihedral angle and the end-to-end

distance. In (c), thickness of an imagined ligand shell is measured for

the Monte Carlo simulation of a single ligand to show the contribution

that is made to the thickness by the dihedral angles. A curve similar

to the MD simulation is seen, despite the simplicity of the single-ligand

picture which does not include volume-filling effects. The dashed line

is a slope one linear scaling: x. The dashed-dot line is a slope one half linear scaling: x/2...... 44

3-1 Solid-state device structure (not to scale). In upconversion the PbS

NCs are directly excited at 785 nm, the photoexcitations undergo Dex-

ter energy transfer to sensitize the triplet state of rubrene. A singlet is

then formed by triplet-triplet annihilation (TTA), which then under-

goes FRET to the dopant dye DBP, where it is readily emitted at 610

nm. For clarity, the QD ligands have been omitted...... 52

11 3-2 Nonlinear modulation of the spacing between the NCs and rubrene by changing the ligand length. a, Atomistic molecular dynamics simula-

tions of the length dependent ligand morphology of aliphatic amines

on a model CdSe NC in the rock salt crystal structure. VMD is used

for image rendering. b, Relationship between the ligand shell thickness

and the number of carbon atoms in the ligand (black). The teal circles

show the experimental distances obtained for close packed PbS NCs by

TEM33 in close agreement with the theoretical model. The dashed and

dotted grey lines function as guides to the eye and show the extended

and half of the extended length LC-C, respectively. Inset: TEM image

illustrating the experimental spacing between close-packed PbS NCs for a dodecanoic acid (12C) ligand...... 54

3-3 Time resolved photoluminescence decay of the infrared emission of a

neat PbS NC film (black curve) and the quenched TRPL in the pres-

ence of rubrene: 0.5% DBP (red curve) to extract the TT dynamics for

different ligand lengths (blue curves). a, TT dynamics for stearic acid

(18C), showing a mono-exponential decay lifetime Ttransfer,,18C= 651

6 ns (dark blue). This results in an estimated characteristic trans-

fer time of Ttransfer,18C= 850 ns when accounting for the competition

with the intrinsic decay channels (black curve, TrPbS AEL 2.8 Ats). b,

Transfer dynamics for hexanoic acid (6C), with Ttransfer,6C = 116 3 ns (light blue). This corresponds to a characteristic transfer time of

Ttransfer,6C = 120 ns. Insets: cartoons to highlight the enhanced trans- fer rate (reduced transfer time) when replacing long ligands (dlA13)

with short ligands due to the reduced spacing d2. c, The deviation of

the TT time from the expected strictly exponential relationship (green)

is apparent at ligand shell thicknesses below 10 AE and the asymptote

at To AEL 100 ns (pink dotted) is highlighted. The unsaturated native

oleic acid ligand is highlighted in purple. The error bars depict the

standard deviation of multiple samples...... 56

12 3-4 DFT calculations of the electron density distribution in the excitonic

energy levels involved in the TT process. Kohn-Sham wavefunctions

of the initial state: the unconstrained exciton on the PbS NC (top), and the final state: the triplet exciton on rubrene (bottom). The

coupling V between the wavefunctions is computed using constrained

density functional theory- configuration interaction, and the rate of TT

(ktheory=0.13ns-1) is estimated from Marcus theory. VMD is used for im age rendering. [791 ...... 58

3-5 Correction of the extracted transfer rate by the calculated dielectric

constant. a, Calculated dielectric constant for a PbS monolayer with

different ligands based on the Bruggeman model. Inset: cartoon high-

lighting the change in the volume fraction of PbS vs. organic lig-

and when shortening the ligand (not to scale). b, Saturation of the

extracted rates (gray line) and deviation from the exponential trend

(green dotted line). c, Normalization of characteristic TT rates (grey

diamonds) by the dielectric constant: recovers the expected purely ex-

ponential trend (green). Error bars are based on an error propagation

with an estimated error for the dielectric constant AE= 0.2, and the

standard deviation of the extracted rate...... 61

4-1 Illustration of the mean field steady state idea. Beginning with a set of

sites with different rates (A) one simulates steady state for each site in

the presence of a set of averaged neighbors (B). The average of these

steady states (C) then self-consistently defines the average neighbor

used to obtain steady state...... 71

13 4-2 Steady state # (blue), H (green) and H2 (red) populations calculated with KMC (dots) and MFSS (solid lines) as a function of the standard

deviation of the log normal distribution of the catalytic rates k 1. At

ambient temperature, the upper limit - = 20 kBT - 0.5 eV sets a

realistic range of expected disorder in real systems. Also shown are the

MFSS populations for a null standard deviation but a mean adjusted

to ln(k) + -/2 (dotted lines). The homogeneous rates were: k, = 0.1, kd= 0.1, ka O 0.1 ...... 78

4-3 Steady state # (blue), H (green) and H2 (red) populations calculated with KMC (dots) and MFSS (solid lines) for a range of the mean log

of the catalytic rates ln(k). The homogeneous rates were: k' = 0.01, kd= 0.16, ka= 0.14 ...... 79

5-1 An illustration of the system and the various rates considered in the m odel ...... 85

5-2 A visualization of the shape of the transfer matrix. Each pixel repre-

sents a matrix element location. Light yellow pixels are zero entries, dark yellow pixel are the elements that model the 1st NC layer, Orange

are the 2nd NC layer. The red pixels of the off-diagonal connect the

two layers, and the brown pixel along the border are the connection

between the NC and OSC ...... 87

5-3 PL traces for 1 NC monolayers and the simulation, showing the "bendi-

ness" of a disordered PL trace ...... 89

5-4 PL traces for 1 and 2 NC monolayers, with and without rubrene (dots), vs the kinetic model (lines) ...... 90

5-5 Steady state PL traces for 1 and 2 NC monolayers. 2 ML gives slightly

more up-conversion output, but far from the expected doubling. . . . 91

14 5-6 The flow of energy in the system. On the left, 1 ML of NCs can

readily transfer to rubrene, while accepting little back. On the left,

the bottom ML cannot transfer to rubrene well, but can still accept a

similar amount back. The end result is an increase in loss for little gain. 91

6-1 The initial Fermi wave-vector from the LDA vs the final calculated

wave-vector from the WDA. Points are colored by the nearest heavy

element, C,N,O, or F. Two points are circled that have very similar

density, yet different solved kF. The corresponding gradient is marked

in Fig. 6-3, allowing the model to distinguish between the two...... 98

6-2 A zoomed-in version of Fig. 6-1 showing the hydrogen feature. .... 99

6-3 The reduced density gradient vs the final calculated wave-vector from

the WDA. Points are colored by the nearest heavy element, C,N,O,

or F. Two points are circled that have very similar density, yet dif-

ferent gradients, allowing the model to distinguish between the two.

By matching both the density and the gradient , the solution becomes

nearly one-to-one ...... 100

7-1 Plot comparing the predicted w values with the exact ones. We re-

produce the true optimal values with a mean absolute error of 0.015 bohr 1...... 105

7-2 The distribution of true omega values for the two databases employed

in this study. We find that they have little overlap, demonstrating how

each one only samples a limited region of w space...... 106

7-3 The distribution of true omega values for the two databases employed

in this study, plotted versus the molecule's characteristic radius. . . 107

15 16 List of Tables

6.1 Error of the functionals relative to exact HF ...... 99

17 18 1

Introduction

The role of theory is to take the measurements and observations of experiments, and synthesize them into a model. It is this model that gives us the understanding required to turn art into science, and it is this understanding that give rise to progress. In this thesis, I construct such a model for a very specific problem: transferring energy from one material to another. However, no scientific effort should be devoid of its context, we are not transferring energy for our amusement, but for a goal; enabling novel optoelectronic devices that improve solar cells and sensors. The structure of the rest of the thesis is as follows:

" In this intro chapter I outline the motivation and the methods used in con-

structing the model.

* Chapter 2 focuses on the geometrical considerations of the transfer using molec-

ular dynamics simulations.

" Chapter 3 deals with the electronic structure of the NC-OSC system, by directly

calculating the electronic coupling between the two transfer states.

" Chapter 4 introduces a kinetics theory we call Mean Field Steady State (MFSS)

that describes steady state kinetics in a disordered system.

" Chapter 5 uses a transfer matrix model to describe the time-dependent behavior

of a disordered film.

19 The last two chapters discuss a different topic: applications of machine learning

(ML) models to density functional problems. While not directly used for energy transfer modeling, the acceleration these ML models bring might one day allow for systems as large as those in the previous chapters to be studied at a higher level of theory:

" Chapter 6 discuss the structure of the non-linear system of equations arising in

the symmetrizing weighted density approximation, and how a k-nearest neigh-

bor interpolator can be used to accelerate the solution.

" Chapter 7 trains a neural network on a large dataset of molecules to find the

optimal range separation parameter that appear in modern density functionals

with the need for direct optimization.

Finally, the thesis concludes with remarks on the model, future work, and possible extensions.

Motivation

The goal of the center for exictonics, which funded most of this thesis, is "to develop the science and technology of excitons, to reveal the fundamental characteristics of these crucial quasi-particles, and enable new solar cells and lighting technologies."

Why is there a research center devoted to these quasi-particles? An exciton, a bound pair of an electron with its positive counter-quasi-particle a hole, is one of the basic components in light-matter interactions. When a photon hits a material and excites an electron, an exciton will be formed. In inorganic semiconductors like silicon, the binding energy between the electron and the hole is very small, on the order of kT thermal energy, and the exciton quickly separates. In organic materials, the binding energy is much larger, and therefore the exciton has a much longer lifetime. This long lifetime allows us to construct materials the directly manipulate the exciton. The key here is that an exciton can do things that an unbound electron or hole cannot do.

One of these tricks is exciton conversion. A high energy exciton can split into two

20 lower energy excitons in a process called singlet fission, and two low energy excitons can combine to form a high energy exciton in the reverse process called triplet fusion.

These processes are fast and highly efficient, and allow for improved solar-cells, lights, and sensors.

But there is a catch. So far, devices purely based on excitons are not over-all efficient. That is why most solar cells are silicon based, and organic light emitting diodes (oLED) appear in Tvs and cell phones, but not in light bulbs. Furthermore, excitons suffer from the curse of spin statistics. When an exciton forms electrically, it can be either as a bright singlet state or a dark triplet state. But as their name suggests, there are 3 triplets and only 1 singlet, and so only 25% of excitons formed will be bright and emmiting. Similarly, when a singlet undergoes fission into two triplets, these triplet are dark due to spin selection rules. Lastly, organic molecules do not absorb well in the IR. Therefore, using triplet fusion as a method of harvesting or detecting IR photons is challenging.

Luckily, there is a class of materials that unable us to mitigate most of these problems. Nanocrystaline semiconductors, better known as quantum dots, are nano- sized crystals of a semi-conducting material. Their small size, on the same order as th size of an exciton, allow for quantum effects to emerge. Through quantum confinement, the energy of a quantum dot-exciton can be controlled. Using this control, QD can be made to absorb well in a large range of wavelength, including in the IR. Furthermore, since they contain heavy elements with a large spin-orbit coupling, they can mitigate the spin curse by flipping triplets into singlets. Lastly, QD enable transfer of energy to highly efficient solid-state solar cells, like silicon and

CdTe, which dominate the market. Combined, a hybrid organic molecule-quantum dot device allows us to get the best of all worlds and take advantage of the progress made in many field. But to effectively engineer such devices, we must first understand the basic science of their interaction. This thesis aims to elucidated this interaction, with a focus on energy transfer between QD and organic molecules.

21 Methods

Most of the work in this thesis have used the bread and butter of atomistic modeling: Molecular dynamics (MD) and Density Functional Theory (DFT). In this section, I briefly outline the main equations and principals describing these methods.

Molecular Dynamics (MD)

Molecular dynamics is a method that relays on the fact that despite the inherent quantum mechanical behavior of nature, atoms are still big enough to be treated classically. The classical nuclear Hamiltonian partitions into a kinetic and potential energy piece: H = T + V where

M

T(PP, P2,. PM) = 2 P

and M M

V (R, R 2 ,..Rm) Z ZIZJ+Ee(RR 2,.... RI) (1.2) I=1 J>I i While molecules usually exist in a single electronic state (most often the electronic ground state), finite temperature causes the nuclei to move. Thus we when doing MD we are interested in the dynamics of this Hamiltonian. These dynamics are given by Hamilton's equations of motion:

dP1 OH dR1 _ OH dt OR1 dt P1)

Which we integrate numerically using the Velocity-Verlet algorithm. In a real system, temperature is maintained through energy transfer between the system and the rest of the world. In an MD simulation, the part that is responsible for simulating this effect is called a thermostat. In this thesis, we use the Andersen thermostat[4]. The Andersen thermostat randomly rescales the velocity of an atom such that the velocity distribution in the

22 system matches a Maxwell-Boltzmann distribution[142]:

f (v) ( )347rv 2 eT (1.4)

Integrating the classical equations of motion is computationally straightforward.

The expensive part of an MD calculation is the evaluation of the force on each nucleus:

dP1 OH _ VOV N M ZOZJ Ee (R1 , R 2 , ... , RM) -~~~ = -rTJj + R .(1.5) dt OR, OR, OR, I=1 J>I j OR(

Specifically, the expensive term is the derivative of the electronic energy with respect to the nuclear coordinates because it requires an electronic structure cal- culation. When MD is performed in this manner, it is called AAIJab-initioAAi or

AAIJBorn-OppenheimeraAi MD.

In order to massively speed up this calculation, E, (RI, R2 ,.. RN) is often ap- proximated by an analytical expression. The resulting approximate potential energy is called a force field.

Most common forcefield follow this form[84:

M M M M M VFF EL [Vcoul (jJ, I, J) + VNB (rIJ, I, J) + Vbnd (7IJ, I, J)]+L EI>3 Vangie (OIJK, I, J, K). I=1 J>I I=1 J>I K>J (1.6)

VO,, is the Coulomb interaction between two atoms:

Ve l (r, A, B) = QAQB (1.7) 0 r where Qx is the charge of atom X and is a parameter for the force field.

The non-bonded interaction, VNB, usually describes both hard-core short-range interactions as well as long range dispersion. One popular form of this interaction is the Lennard-Jones potential

VLJ (r, A, B) = 4EAB [(AB) 12 (AB) 6(18)

23 where -AB and EAB are empirical parameters. Vbcynd describes stretching of chemical bonds, and is most simply approximated by a harmonic potential:

1 Vbond (r, A, B) = kAB (r - bAB) 2 (1.9) 2

where kAB and bAB are parameters if atoms A and B are covalently bonded, and zero

otherwise. Finally, Vangie represents the energy of angular bond vibration and might be approximated by:

1 Vangle (0, A, B, C) = kABC [Cos (0) - cos (qABc)] 2 (1.10) 2

with kABC and OABc as parameters (again zero if B isn't bonded to both A and C). Looking at these equations we see that even a simple force field has many param- eters. Because these parameters are the only thing that changes for different atoms and molecules, the MD description of a chemical system is very sensitive to our choice of force field parameters.

Constrained Density Functional Theory - Configuration Inter- action (CDFT-CI)

The wavefunction is a costly description of a chemical system. Density Functional Theory (DFT) replaces the highly-dimensional wavefunction with the 3D density[76, 931:

P (r, r 2 ,... , rN) * (r, r 2, ... , TN) ' (rl, r2, ,rN) (1-11)

p (r) = (r, r2, .. ., rN) T (r, r2, - I rN) dr2 , -- -, drN (1.12)

Hohenberg and Kohn showed that there exists a universal functional of the density F, which does not depend on the details of the system, and gives its energy:

E [p(r)] = J vn(r)p(r)dr + F [p(r)] (1.13)

24 where v, is the external Coulomb potential created by the nuclei. Remarkably, this energy functional is minimized by the one-electron density of the ground state wave- function. Because the density can be expanded in a basis (in the same manner as the wavefunction), the density provides a complete description of an N electron system with only 0 (N) parameters.

Unfortunately, the exact functional F is not known and must be approximated130].

The primary difficulty in approximation of F comes from approximation of the kinetic energy functional. Thus, the vast majority of approximations for this functional are framed in the Kohn-Sham (KS) framework, due to its highly accurate treatment of the kinetic energy. In this formalism, a fictitious system of non-interacting electrons is created whose ground state density is the same as that of the physical system.

Because these fictitious electrons do not interact, their wavefunction is completely specified by a Slater determinant of orthonormal orbitals {/S} and the density is given by: N P(r) = KS* (r)KS(r) (1.14)

The kinetic energy of this wavefunction is readily evaluated, and turns out to be a very good approximation to the kinetic energy of the physical system[130. These orbitals are found by solving the Kohn-Sham equations:

v (r) 2dr ciOKS(ri). 1 + J 2 + vc(r)1 S _r) (1.15) _2 + nM+I rl 1 r2 +Vc|)

Note that this equation is essentially the Hartree Fock equation, differing only in the exchange-correlation potential, vxc(r), which becomes the only part the needs to be approximated. The price of this formalism is the re-introduction of orbitals which increase the cost of the calculation from O(N) to O(N3 ). However, KS-DFT can achieve much higher accuracy than Hartree Fock.

DFT however, is not a silver bullet. Sometimes, a state other than the ground state is desired. For example, in the energy transfer discussed in this thesis, we require

25 a localized triplet excitation (an exciton). Such a property can be described entirely through the one electron density by requiring that the spin density in one region of

space integrate to a certain number of electrons.

Constrained Density Functional Theory (CDFT) provides just such a description.

In CDFT, energy is minimized subject to constraints on the density or spin density.

This variation is performed by finding stationary points of the CDFT functional:

W [p, A] = E [p] + A (Jw(r)p(r)dr - V) (1.16)

where E is the energy functional from equation, A is a Lagrange multiplier, w is a

weight function that defines the desired constraint, and V is the desired value of the

constraint. For example, if we wished to constrain the number of electrons in a region

of space to a certain value Ne , w would be a binary function defining this region and

V would be Ne. For the purposes of our charge transfer reaction, w is a density-based

charge partitioning function (e.g. the LXfiwdin population ) and V represents an

excess electron on part of our system.

We can now compute the energy of both the ground state (from DFT) and the

triplet exciton excited state (from CDFT) of an energy transfer reaction.

Densities generated via CDFT represent diabatic states[177. These states do not

diagonalize the Hamiltonian, and the Hamiltonian matrix element between two such 4 states, 1i) and I T2), is called the AAIJcouplingAI

V (T11 Ht I'I2). This coupling is important, e.g., for determining the rate of condensed phase electron transfer reactions in Marcus theory[197:

1 27r(1.17) ktheory IV12 wkT -- 4A h 41rBTh

In the framework of CDFT, these couplings are given by:

1 1 V = (E 1 + E2) (Ti1 'P2) - I1 A(Zi + A2 tb2 |'P2) (1.18) 2 2

where the E% are the functional energies associated with the two constrained densities

26 and the wi and Ai are the weight functions and Lagrange multipliers used to specify and apply the constraints to these densities. However, DFT does not define a wave- function for the physical system, so we do not know IT,) and iT2) . That is to say,

CDFT computes pi and P2 which are the densities associated with IT1) and ["2)

P1 = I'I') (Til P2 = I2) (T21 (1.19)

However, we do not have access to the exact wavefunction thus, an approximation must be made.

In CDFT-CI, we approximate the exact wavefunctions by the Kohn-Sham Slater determinant. The final coupling is then calculated as[194]:

11 KS A,,& + A IS)KIS V -(E 1 + F 2 ) ( KS ITKS) 2 2 2 2 h

It is this approximation that we will employ throughout this thesis.

27 28 2

Morphology of passivating organic ligands around a nanocrystal

The starting point of atomistic modeling is to decide on where the atoms are. In this chapter, we use molecular dynamics to investigate the position of the atoms of a quantum dot, with the goal of gaining understanding of the distances energy transfer is going through. This chapter includes work by our experimental collaborators, Lea

Nienhaus from Prof. Bawendi's group and Mengfei Wu from Prof. Baldo's group as well as James Shepherd, a post-doc in our group. This chapter was published as an article in [541.

Introduction

Semiconducting quantum dots have attracted substantial attention due to their tun- able structure-property relationships [86, 139, 206, 133, 25, 1241. The ability to simul- taneously engineer their electronic and optical properties within a single device has made them a prime candidate material in a variety of applications. In solar cells and

LEDs, quantum dot size is used to tune band gaps, and this is commonly exploited to produce varied spectral properties [98, 131, 21, 157, 33].

These optoelectronic devices function through electronic processes that are often strongly dependent on distance [29, 144, 159, 66, 25]. For example, conductivity

29 in a quantum dot array is mediated by Marcus-type charge transfer events between dots[114, 200, 144, 102, 21, 107, 186, 1, 871. As the dot-to-dot distance increases, the charge transfer rate decays exponentially, making the conductivity extremely sensitive to the dot-to-dot distance 1114, 23, 201, 200]. Excitonic energy transfer, relevant in solar cells and light emitters, usually occurs through Forster resonant energy transfer (FRET)[51, 138, 125, 1, 104, 57, 85, 86] or Dexter processes [36, 121, 169]; these are also dependent on distance. Since the organic ligand shell is usually composed of insulating alkane chains, they behave as a spacer layer that can determine that closest approach distance [144, 1, 186, 77, 1071. Ligand exchange reactions [5, 39, 202, 171 give us in situ synthetic access to the ligand shell, and using this design space it is possible to achieve fine control of the aforementioned electronic processes.

Recently, the ability to control the energy gap and energy transfer has been ex- ploited for novel optoelectronic devices [131, 21, 157], allowing for down-conversion of a high energy UV photon into two lower energy photons [169, 138, 121], and up- conversion of two low energy IR photons into one higher energy photon [192, 113, 165]. The ability to up-, and down-convert photon energy can allow solar cells to capture more of the solar spectrum, thereby circumventing the Shockley-Queisser limit [160, 43, 108]. However, these conversion processes rely on the aforementioned energy transfer mechanisms and, therefore, are very sensitive to the structure and thickness of the ligand shell [74, 146].

Due to the importance of ligands numerous theoretical studies have been under- taken concerning the structure of the ligand sphere around nanocrystals. A commonly studied system is gold nanocrystals (NC) [152, 105, 106, 189, 56, 15], which.tend to have a dense ligand shell, typically around 4.5 per nm 2 [190]. This causes a strong ligand-to-ligand interaction, giving rise to concerted motion and ordering of the lig- ands in the shell [105, 106]. Typical semiconductor nanocrystals have a much lower density ligand shell - around 3.0 ligands per nm 2 [26] - because the ligands only pas- sivate one of the two atoms that compose the NC (e.g. the ligands bind to Cd but not

Se in CdSe). This situation will lead itself to much weaker ligand interactions, and comparatively little is known about the impact this will have on ligand shell struc-

30 ture. In electronic structure calculations [206, 89, 143], abstracted models tend to be used, where the ligands are replaced by passivating atoms for computational ease.

Even when ligands are included, they are typically coarse-grained so larger dots can be studied [105, 106, 189]. A major focus of these past studies has been a generation of a potential of mean force in a NC array [88, 189]. In contrast, we focus here on the detailed morphology of a single NC, and from that morphology make reasonable inferences about the influence the ligand sphere will have on both the inter-particle spacing in an NC array and the distance of closest approach for a molecule near a

NC. We present a study where we examine the structure of the ligand shell using a combination of computational simulations, electron microscopy, and theoretical anal- ysis. In particular, we undertake a molecular dynamics study on the amine ligand shell surrounding a CdSe nanocrystal, where both the core and the ligand shell are treated atomistically. In so doing, we found that the ligands show a tendency to lay flat against the surface, leading to an effective shell thickness reduction over the range of ligand lengths studied (0.3nm to 2.5nm). Quantum dots of nanocrystalline

PbS were then synthesized with carboxylic acid ligands and the dot-to-dot distance was measured by transmission electron microscopy. We were able to use these data to verify our simulations for ligand shell thickness, whilst also demonstrating the transferability of our theoretical findings. Theoretical analysis was then used to un- derstand the physical chemistry of the ligands in terms of the microscopic properties of the ligands, through the dihedral angle. Our results imply that energy and electron transfer processes involving dots may be significantly enhanced in practice due to the unique morphology of the ligand shell.

Simulations

We simulate the morphology of the ligand shell by performing classical molecular dynamics with atomistic detail. We choose CdSe quantum dots with amine ligands due to the quality of the experimental data [127, 5, 42, 174], for the ease of simulation

(i.e. amines are not charged) and because force fields are readily available[84, 151].

31 (a) (b)

(c) (d)

Figure 2-1: Steps of the simulation are ordered as follows: a) the nanocrystal as carved from bulk; b) which is then decorated with ligands in a 'spiky ball' conformation; c) the geometry after energy minimization; d) the geometry after molecular dynamics

The amine head group can only have one attachment site, as opposed to the two

attachment sites in carboxylic acid ligands, which facilitates ligand placement. The

amines we study have a simple alkane chain backbone of between three and 19 carbon

atoms (CH3 (CH2 )nNH 2)-

Our focus in this paper is on nanocrystals as they appear in a thin-film superlat-

tice, where all solvent from their synthesis is evaporated, leaving behind an ordered superlattice. Recent experimental studies [185, 186, 261 suggest superlattices are equi-

librium structures since they undergo reversible and reproducible phase transitions

between different equilibrium structures that would be very hard to rationalize if these structures were only kinetically trapped. While the solvent is clearly important in determining the kinetics of the formation of the dot films, the solvent cannot de- termine the thermodynamics of the effectively solvent-free outcome; the equilibrium structure is therefore the same independent of how the structure is made. As such, solvent is not included in our simulation.

32 Figure 2-1 shows the simulation protocol we followed. The nanocrystal core struc- ture is carved from the bulk Wurtzite crystal structure. We remove surface atoms with only a single bond while maintaining the stoichiometry. The nanocrystal is then decorated in a configuration in which every ligand points directly away from the center of the nanocrystal, with the head of the ligand placed a Cd-N bond distance away from a surface Cd. This initial structure owes its inspiration to the conventional picture for the ligand structure around a nanocrystal, the so-called 'spiky ball' [1861.

We then perform a minimization step to reduce stress from unfavorable surface con- ditions and nearby ligands. This configuration then undergoes a molecular dynamics simulation to get the structure of the ligands at room temperature. Finally, we re- move any ligand whose head group is more than 0.3 nm away from the surface, which we deemed to be detached from the surface during the simulation.

Following this procedure we calculated the coverage of available surface sites to be 95 - 100% for the small dots and 85 - 95% for the larger dots corresponding to an absolute surface density of 2.8 - 3.1 nm- 2 ; this is comparable with experiments [5, 26].

For completeness, we re-ran some of our simulations with much stronger Cd-N bond strengths, which result in 100% coverage in all cases. The results were qualitatively similar. For simplicity, we discuss only the simulations with labile ligands in what follows, as they are closer to the experimental situation. However, it should be noted that, in terms of the ligand morphology, the experimentally accessible ligand densities are essentially complete coverage.

We simulated the same nanocrystal with ligands of varying lengths, from 0.30 nm to 2.5 nm, since a key engineering aspect of organic shell ligands is the ability to change the length of the ligands through ligand exchange reactions [5]. We also chose three initial nanocrystal cores of sizes 2.00 nm, 3.56 nm, and 5.00 nm in diameter to study the influence of dot size on the conclusions. One core structure per size was carved from bulk CdSe to yield equal stoichiometry and used for all simulations for this dot size. In this way, we focus on the effect of the ligands themselves without being concerned with variability in the core structure.

A simple measurement to take for our nanocrystals following molecular dynamics

33 E 3.5 %C . 3.0 - e * * 1.00 nm ai)e 1.78 nm., 2.5 S50 5-. 2.50 nm ~2.0 P 1.5 0 -

0 0 0.5 1.0 1.5 2.0 2.5 Stretched end-to-end (nm)

Figure 2-2: Thickness of the ligand shell as a function of stretched ligand length for three dot sizes. Two linear eye-guides are also provided: the first (- - -) has a slope of one, shifted by the Cd-N bond length: x + bCd-N. The second (-.-.) has a slope one half: x/2 + bCd-N. Representative error bars are shown for the smallest dot size, where we averaged over six independent runs of Fig. 2-1. These show that the shell thickness does not vary significantly with diameter over the range shown here.

34 rc-N rN-C

Ic-N N-C

(a)

3.0 . 2.5 .- n= 3 -n =5 c 2.0 S1.5 -n=7 -n=9 .0 --n= 11 0 0.5 x -~ -n=13 0n .- ~~------~ 0.0 0.2 0.4 0.6 0.8 1 .0 Order Parameter (b)

Figure 2-3: Calculation of the order parameter: For each ligand, we calculate two vectors: the first is the vector from the geometric center of the nanocrystal to the ligand head group, the second is the vector from the head group to the tail group (shown in (a)). The order parameter is then the dot product of the normalized vectors. This distribution varies according to ligand length (shown in (b)), with ligand repeat units shown in the key as n in CH3 (CH2),NH 2 . The dot size is 3.56 nm.

35

. 3 5 7 9 11 13 15 17

Figure 2-4: Illustrative snapshots from simulations of a 3.56nm nanocrystal. From left to right, these correspond to ligand lengths of n=3 to 17 in increments of two. The surface ligands are colored by their orientation relative to the surface. Red-colored molecules are sticking straight out, as quantified by an order parameter greater than 0.6, whereas turquoise-colored molecules lay flat. is the thickness of the ligand shell. The thickness distribution for one dot is measured as the distance between the tail group (C in CH3) and the closest Cd or Se atom. The effective thickness is then calculated as the average of these measurements. We find that shell thickness increases with ligand length as expected, but there is clearly a sub- linear component at large ligand lengths. When plotted against ligand length, Fig. 2-2 shows that the thickness is approximately piecewise linear. The slope changes from approximately unity to approximately one half between 0.5 nm to 10 nm. Previous simulations [88] may have also reflected this behavior, but a transition like this has never been explicitly highlighted in any previous study.

One can imagine several possibilities to explain this transition: perhaps the ligands are doubled over on themselves, or lay flat against the surface, or form tight coils.

By inspection (e.g. in Fig. 2-1(d)), the ligands tend to lay flat. To quantitatively analyze this behavior, we constructed an order parameter for the ligand arrangement.

For each ligand, we calculate two vectors: the first is the vector from the geometric center of the nanocrystal to the ligand head group, the second is the vector from the head group to the tail group. The order parameter is then the dot product of the normalized vectors. Figure 2-3(a) illustrates the calculation of the order parameter.

This construction approximates the cosine of the angle between the ligand and the normal to the surface of the nanocrystal and was chosen to be simple and easy to calculate, while still able to capture the current state of the system. When a ligand sticks straight out of and is normal to the surface, its order parameter is 1, and when the ligand lies completely flat and it is tangential to the surface its order parameter

36 .- 1.0 I I I I I I I a) E 00- 5.00 nm 0.8 - 0 3.56 nm *00 2.00 nm atL.. 0.6 0.4 0 0.2 - 0 I I I I I I I 2 4 6 8 10 12 14 16 18 Number of Repeat Units

Figure 2-5: Mode of the order parameter distribution as a function of ligand size, for three nanocrystal sizes.

rn

-o -

0-Tn n T 0 TE + R + 3n - TI 3n n 0 + n + T + 3n 6 (azimuth angle) 6 (azimuth angle)

Figure 2-6: Anisotropy of the relative ligand shell thickness is shown for a pair of dots. The thickness (indicated by color) is plotted against two angular parameters which describe a location on a dot's spherical surface (for details, see Eq. (2.1)). Here, we have used the height of the location of the end group (C in CH3 , red points) as a way of mapping thickness. Color is used to indicate height as a heat map, where blue locations are increased ligand height, white are decreased. At the start of the simulation (left) the height distribution is uniform within projection error but over the course of the simulation it becomes increasingly anisotropic (right). Since we are measuring the height only at the head group location, white areas are not necessarily bare. The dot size is 3.56 nm.

37 is 0.

Figure 2-3(b) shows the distribution of this order parameter for a representative

sample of ligand lengths in this study. At short ligand lengths (3-7 repeat units) the

ligands have a high order parameter of around 0.9. As the ligands grow longer (9-13

repeat units) a transition occurs, and most ligands have a low order parameter of 0.3

because they lie flat.

The order parameter forms a useful tool for the visual inspection of the structures

that we found. Rendered images of the dots are shown in Fig. 2-4, where we have

colored the ligands on the surface of the dot by their order parameter. Figure 2-3(b)

allows us to choose a sensible dividing value for the change in color; red ligands have

an order parameter greater than 0.6, while turquoise ligands have an order parameter less than 0.6.

We can therefore relate the observed conformation change to the surface thickness

relationship found in Fig. 2-2. At a ligand length of around 1 nm, corresponding to

chains with n = 7 to n = 9, the slope starts to transition in Fig. 2-2, and this is

then due to a change in ligand shell conformations shown in Fig. 2-4. The leftmost

nanocrystal, with the smallest ligand length, looks like a 'spiky ball' with the ligands

projecting upward from the surface of the dot. The rightmost nanocrystal, with the

longest ligand length, looks like a 'hair ball', with the ligands bunched together and

lying flat on the surface. As we move from left to right, we can see a tendency for

the ligands to be flatter against the surface (change from red to turquoise). When

seen through the lens of this conformation change, the simple heuristic that the shell

thickness is equal to the half the ligand length can be seen to be serendipitous and

part of a larger transition. When the ligands are much longer than the radius of

the CdSe core, we would expect volume-filling effects to dominate and the linear

relationship to break down asymptotically. But for physically realistic dot sizes and

ligand lengths, the intermediate transition regime dominates.

When the dot size is changed, the same qualitative observations described above

persist. However, there is a quantitative difference in order parameter distribution.

Figure 2-5 shows the effects of dot size on the transition between the spiky ball

38 to the wet hair. As the dot grows bigger, we observe that for intermediate ligand lengths (n = 5 - 12) there is a pronounced increase in the percentage of ligands either sticking straight out (n = 3 - 7) or a more significant mix of straight out and lying flat (n = 9 - 11) as compared to the smallest dot size. This fits our intuition well, as, in the limit of a flat interface, we expect the ligands to stand completely upright, as they do in self assembled mono-layers [67].

Next, we investigate the anisotropy of the height distribution on the surface of the nanocrystal. Anisotropy in thickness can affect the way in which nanocrystals assemble, and will determine the distance of closest approach of a molecule. For example, if the dots were amorphous and roughly spherical, they would adopt a body centered cubic structure. Figure 2-6 shows a height/heat map of the ligands at the start and end of the simulation. This distribution is made continuous using the von Mises-Fisher function [48]. Beginning with the coordinate of each tail group

(corresponding to the red points in Fig. 2-6, and the C in the CH 3 group), fi, a spherical gaussian of width r, is centered at that point. The continuous distribution is then the sum over the height-weighted functions as follows:

I Npoin s fp (r; f, r,) = Npois hi exp (r, jri -r) (2.1) where hi are the heights, and r are points on the sphere. Finally, the function on the surface of a sphere is projected onto a two-dimensional heat map shown in Fig. 2-6 using a Mercator projection.

From Fig. 2-6 we can make the observation that the spatial distribution of the ligands around the dots is not isotropic. Initially, the ligands are distributed al- most uniformly around the nanocrystal. After the simulation, there is a clustering of the ligands, leaving parts of the dots more exposed, while other parts become more crowded. This suggests that both the length of the ligand and the average distance to the surfaces are not fully adequate measures of the thickness of the ligand shell, since even in a well-covered nanocrystal with a thick shell there exist patches of sub- stantially lower thickness. This can be thought of in terms of anisotropy classes, with

39 (a)

1.4 I TEM 1.2 . Simulation E1.0- 0.8 0.6 A 0.4 I- 0 ' I I I 0 0.5 1.0 1.5 2.0 2.5 Length, L (nm)

(b)

Figure 2-7: Transmission electron microscopy (TEM) was performed to make com- parison with computational results. (a) An example of a TEM micrograph of a layer of PbS nanocrystals with carboxylic acid ligands. (b) TEM data compares well with simulation data (here, taken from the 1.00 nm dot size). The thickness in this plot is corrected for the bond length that joins the ligand to the surface i.e. Cd-N or Pb-0. The red, dashed line shows the fit to Eq. (2.2) for the whole data set. The black, dashed line is a line of slope one. our nanocrystals making a transition in surface coverage (corresponding to class A anisotropy, as described by Glotzer et. al.[59J).

Experiment

If the simulations are correct, the structure of the ligand sphere around a dot should have fairly clear implications in terms of how close a molecule or surface can get to a dot and how close two dots can get to each other. In this section, we test these implications by carefully examining the dot-to-dot spacing in a quantum dot array to determine if there is evidence for the spiky ball-to-hairball transition.

40 We synthesized lead sulfide quantum dots using a modified hot-injection method. [71].

In order to modify the ligands covering the surface, ligand exchange was performed ex-situ in toluene. We obtained the quantum dot arrays by drop-casting onto a TEM grid. We analyzed TEM micrographs (e.g. Fig. 2-7(a)) by sampling the image inten- sity using a Fourier transform technique. The dot diameter (d=2.67 +/- 0.35 nm) was subtracted from the peak-to-peak distance to yield the spacing between the edges of two dots, which amounts to double the ligand shell thickness. To make compar- ison between the PbS dots synthesized and the CdSe dots simulated, the relevant ligand-surface bond distance was subtracted from the measurements - Pb-O (0.23 nm)[63] and Cd-N (0.215 nm, measured from the simulation) respectively. The TEM micrographs and raw data can be found in the supplementary information.

The carboxylic head-group in the experiment is quite different to the amine head- group in the simulation. It has a strong bridge bond to the surface, which is expected to cause the ligand angle to the surface to be more normal, and increase the surface density. Yet we note that the surface coverage is already close to 100%, and the angle in the amine simulation is also close to normal. Therefore, we expect the different head-groups to have a similar effect on the morphology.

We found the measurements of ligand shell thickness (Fig. 2-7(b)) compared well with the simulations described in the previous section. In particular, this applies both not only in the region between 1.2 nm and 2.0 nm, where the shell thickness is propor- tional to one-half the ligand length, but also at shorter lengths where transitionary behavior is seen. We can thus conclude that it is very likely that the experimental quantum dots are undergoing the transition seen in our molecular dynamics simula- tions.

In previous experimental studies of nanocrystal-to-nanocrystal distance, the lig- ands are viewed to be interdigitated 'spiky balls'. This model was invoked in the explanation of X-ray experiments, which suggested that nanocrystals form superstruc- tures with dot-to-dot distances comparable to a single stretched ligand length [186].

However, the interdigitated 'spiky ball' would not explain the data in Fig. 2-7(b).

This is because interdigitation predicts a slope of 1/2 at the origin, when there is

41 Figure 2-8: Two explanations for how dots are separated by a distance of approxi- mately one ligand length. Right panel shows interdigitation of the ligands, which is the prevailing literature viewpoint; left panel shows ligands crumpled over as in our simulations. perfect interdigitation, transitioning to a slope of 1 at long distances, as there is less and less interdigitation. Our data show the opposite of this. In contrast, our 'hairball' picture shows a different path to achieve this dot-to-dot distance, as two halves of a ligand plus two Cd-N bond lengths (see Fig. 2-8). We conclude that the hairball picture is the correct explanation for the observed behavior.

Since we have a design space in the choice of ligand length, but the design objective is engineering distance-mediated transfer processes, it is useful to provide here an approximate equation linking the ligand length to distances in the system:

h A ((3L/k + 1)' - 1)

d = Ddot + 2 bdot-ligand + 2h) (2.2)

A ~ 1.2nm.

In this equation h is the thickness of the ligand shell; d is the dot-to-dot distance;

Ddot is the diameter of the dot; bdot-ligand is the bridging bond between the dot and the ligand. The final term represents twice the height of the ligand shell with ligands of length L, with A a fitting parameter. This functional form was chosen to reproduce limits of h = L in the small L limit and h oc L 1/3 in the large L limit, the latter corresponding to volume filling. The parameter A was found from fitting the data shown in Fig. 2-7(b), and, overall, affords a more accurate way to calculate approximate dot-to-dot distances. While A does depend on the size of the dot in principle, within the common size ranges, it seems reasonable to assume it is roughly constant.

42 As this fit agrees well with both the system modeled and the system measured, we believe it can by used in general for any system of NC surrounded by alkane chain ligands of similar density. We wish to emphasize that there is no direct physical meaning to this fit parameter.

Theory

Given that simulations and experiments both predict similar behavior for the ligand sphere thickness, there might be some more fundamental theoretical explanation for what is occurring. Here, we theoretically analyze the ligands from a microscopic perspective. We seek to isolate the role of the dihedral angle, since the degree of flexibility of a polymeric chain is usually attributed to the freedom in the dihedral angle [491. There are minima in the potential at dihedral angles corresponding to

60 degrees (gauche) and 180 degrees (trans), with the gauche conformation being somewhat higher in energy. Interatomic interactions mean that it can be favorable to form the gauche conformation, and this can cause substantial directional changes in the chain.

Figure 2-9(a) shows the distribution of gauche conformations along the chain for two representative dots. The distribution of gauche conformations in the spiky ball is relatively uniform. In contrast, the 'wet hair' structure has slight stabilization of early gauche conformations - that is to say, the chains twist preferentially near their base rather than in the middle or the end of the chain. A similar result has been experimentally seen in a gold NC system[7]. Steric crowding is unlikely to be an explanation for this trend, since it would suppress - rather than enhance - gauche defects near the surface, where crowding is most severe. A better explanation is based on interchain attraction: when one ligand lays flat, nearby ligands also have a tendency to lay flat to benefit from energetically favorable interchain interactions.

This would promote early distortion of the carbon backbone, and is our most plausible explanation for the distribution seen. The ligand lies flat, rather than curling up, due to the presence of other neighbors with favorable intermolecular interactions. Overall,

43 0.20 2.5 M 2.00 nm n=9 --- MC 2.00 nmn .5n=17 2.0 .- MC 5.00 nm SE 0.15 - "0s 2.00 nm O c S1 0", 5.003.56nm nm 0 0.10 -- ' 1.0 S0.05

0 5 10 15 0 0.5 1.0 1.5 2.0 2.5 3.0 Location on the chain Stretched end-to-end (nm)

(a) (b)

E 3.0 2.5 - Monte Carlo 2.0 - -

1.5-- 4~.00

~0.5

F 0 0.5 1.0 1.5 2.0 2.5 Stretched end-to-end (nm)

(c)

Figure 2-9: In (a), relative frequency of gauche conformations along the chain is plotted against chain location. Dihedrals are measured along the length of the chain, with chain location being referenced with the N-C-C-C dihedral as zero. For longer ligands, represented here by n = 17, gauche conformations are more common at the beginning of the chain. In (b), relaxed ligand end-to-end distance does not increase linearly with the stretched chain length of the ligand. Three dot sizes are shown. The red and green points overlay each other due to the similarity between the 3.56 nm and 5.00 nm dot. The black, dashed lined labeled Monte Carlo (MC) provides a link between the dihedral angle and the end-to-end distance. In (c), thickness of an imagined ligand shell is measured for the Monte Carlo simulation of a single ligand to show the contribution that is made to the thickness by the dihedral angles. A curve similar to the MD simulation is seen, despite the simplicity of the single-ligand picture which does not include volume-filling effects. The dashed line is a slope one linear scaling: x. The dashed-dot line is a slope one half linear scaling: x/2.

44 then, this looks like 'wet hair'.

If the dihedral angle distribution is indeed responsible for the surface thickness trend with increasing ligand length, then we should be able to devise a model which reproduces simulation observables with dihedral angles alone. Towards this end, we make a Monte Carlo simulation of an isolated ligand based on rigid C-C bonds with constant bond angles, randomly sampling the dihedral angles in Fig. 2-9(a). Data shown in Fig. 2-9(b) demonstrates that the end-to-end distance of the ligands modeled in this way agrees with corresponding lengths collected from the molecular dynamics simulations. When we use this dihedral angle distribution to measure the surface thickness due to the ligands (Fig. 2-9(c)), we find that the thickness values are con- sistent with the trend seen elsewhere in the manuscript. While this model involves only simulating the dihedral angles from a single ligand, they account for the aver- age effect of interactions with other ligands through the dihedral angle distributions

(which were drawn from a model in which the ligands interact with one another).

To test whether the non-uniformity of the dihedral angles seen in Fig. 2-9(a) is important for reducing surface thicknesses, dihedral angles are now sampled over both a uniform and a non-uniform distribution with the same average probability as shown in Fig. 2-9(a). For the 5.00 nm nanocrystal with ligand length n = 17 the average gauche conformation frequency is 12%. Both the uniform and non-uniform distributions reproduce the results shown in Fig. 2-9(b). Therefore, while the average kink probability from the simulation is needed to reproduce the trend, a non-uniform distribution is not required. This is unexpected - it suggests that perhaps weakly interacting ligands (or ligands that interact strongly with solvent) might also behave as wet hair, and this hypothesis is worth further testing.

Conclusion

In conclusion, we investigated the physical chemistry of the surface of a quantum dot with an emphasis on the ligands. We performed molecular dynamics simulations on

CdSe dots, treating the ligands and the core atomistically. We find that as the ligand

45 length increases, the average thickness of the shell transitions from approximately one full ligand length to something that is clearly less than one full ligand length. This was verified by transmission electron microscopy of PbS nanoarrays that determined a dot-to-dot separation consistent with the transition predicted by the simulations. The ligands are generally in an intermediate regime caused by flexibility in the dihedral angle and adoption of the gauche conformation. This causes the ligands to fall over on the surface of the dot lending a'wet hair' appearance to the surface of the nanocrystal.

The average distance between a nanocrystal and a nearby object (e.g. a nearby

2D material surface, organic semiconductor, etc.) will grow as the ligand length for short ligands and then grow sub-linearly for longer ligands. By coincidence, for commonly used ligand lengths, these distances are typically close to half the ligand length. However, the observed behavior is achieved not by interdigitation, but by a flopping over the surface of the ligands. These scaling relations suggest that transfer rates between dots and nearby objects will be enhanced due to the shorter distance, consistent with previous findings [1691. We should also emphasize that because of the non-uniformity of the thickness of the ligand sphere, the distance of closest approach could be even smaller than the average distance of approach we have focused on here.

Such effects would likely have a modest effect on dot-to-dot energy or charge transfer

(where bald spots would always be compensated for by thicker spots elsewhere) but could have a significant effect on molecule-to-dot transfer, where a single molecule on a bald spot could have greatly accelerated transfer. Agreement in this paper between simulated CdSe/amine dots and experimental PbS/carboxylic acid dots shows that our findings are broadly applicable, and may well be general for ligands with alkyl groups.

The availability of an accurate sense of where atoms are on the surface of a nanocrystal has advantages that go beyond the scope of this paper, and provide direction for further work. The role of the ligand shell in energy transfer to organic molecules and, ultimately, up-conversion is poorly understood; the atomic config- urations we have developed through molecular dynamics will be of use in providing realistic interfaces for electronic structure calculations of couplings and transfer rates.

46 The anisotropy that we see in the dots will also play an important role in how the dots pack and adopt their superstructure, which is important when considering which ligands to use in synthesis and the resultant quality of the arrays [1861.

There are some obvious future directions suggested by the present study. The simulations contained only a single QD, and therefore do not include dot-to-dot in- teraction [189, 881. There was also no solvent in the simulation, so the results for

QDs in solution will likely be very different. In particular, it would be interesting to see how solvent polarity influences the transition from a spiky ball to more of a wet hair configuration. Another important aspect is the role of surface coverage.

While not studied explicitly here, characterization of the role of surface coverage in the observed behavior, both theoretically and experimentally, remains an outstanding challenge. Finally, on the experimental side, with a quantitative understanding of the ligand sphere in hand it will be extremely interesting to return to the questions of fission and up-conversion and quantitatively assess the underlying rates as a function of ligand shell thickness. Such a study could give insight into the incorporation of molecules into the ligand shell structure.

47 48 3

Reaching the Speed Limit for Triplet Transfer in Solid-State Lead Sulfide Nanocrystal Sensitized Photon Upconversion

Armed with new understading of distance, we now turn to electronic strcutre modeling to provide rate prediction. In this chapter, we calculate the expected fastest possible transfer rate using constraint functional density theory - configuration interaction. This chapter was published in [1291. All experimental work was done by Lea Nienhaus from Prof. Bawendi's group and Mengfei Wu from Prof. Baldo's group, while all computional work was done by me.

Introduction

Recent observations of efficient exchange-mediated energy transfer between organic molecules and inorganic colloidal nanocrystals demand a detailed understanding of this archetypal organic/inorganic interface.178, 137, 121, 1921 In particular, it is im- portant to refine our understanding of the dipole-less process by which 'dark' spin- triplet excitons transfer, and identify which hurdles - if any - must be overcome

49 for the development of efficient devices. To address this question, we study the ki-

netics and distance dependence of triplet energy transfer across the hetero-interface in solid-state films for excitonic photon upconversion. These devices employ exci-

tons as intermediates to enable two absorbed low-energy photons to be converted

into one emitted higher-energy photon at low incident intensities.[122, 112, 1231 this technology has the potential to overcome the Shockley-Queisser limit[160 in single- junction photovoltaics by sensitizing silicon to sub-bandgap light,[171] and also has

other possible applications in biological imaging, cost-effective night-vision cameras

and photocatalysis.[207] The basic process in excitonic upconversion is triplet-triplet

annihilation (TTA) in organic semiconductors, where two spin-triplet excitons on

neighboring molecules, termed as annihilators, interact to generate one higher-energy

singlet excited state[162, 153, 1561 A fundamental advantage over direct nonlinear frequency conversion is that the incident energy is stored in long-lived triplet exci-

ton states, which reduces the required light intensity for efficient upconversion.[1621

However, these triplets are generally dark (not directly optically accessible) due

to spin-selection rules. Therefore, a sensitizer is required to absorb the incident

light and transfer this energy to the triplet state of the annihilator prior to TTA.

While this has conventionally been accomplished with phosphorescent metal-organic

complexes,[122, 162, 3, 1721 recent studies have shown that lead chalcogenide (PbS, PbSe) nanocrystals (NCs) are efficient sensitizers for the triplet state of the annihila-

tor rubrene. The simple synthetic bandgap tunability and cost-efficient scalability of

these nanomaterials make them prime candidates as infrared sensitizers. The sensi-

tization of the triplet state in rubrene occurs via exchange-mediated energy transfer

(Dexter transfer).[192, 112] Such transfer is possible because of the electronic fine-

structure of the PbS NCs: the exchange interaction between singlet and triplet states

at the band-edge is on the order of 1-25 meV .[155, 90] Hence, the exciton will have

both singlet and triplet character due to rapid spin mixing at room temperature. As

a result, the initial directly optically excited exciton on the NC is able to undergo

energy transfer into a dark, spin-forbidden triplet state in rubrene. This occurs while

conserving the spin state, and thus, making the transfer a spin-allowed process.[199]

50 This exchange-mediated energy transfer mechanism is known to be based on a direct wavefunction overlap between the donor and acceptor states,[36 which results in an electronic coupling between the two states. Due to the requirement of a wavefunction overlap, the magnitude of this coupling is expected to be exponentially dependent on the distance between the donor and the acceptor. [199, 36] However, there are additional parameters, which have the potential to influence the magnitude of the exchange interaction: (i) The exact nature of the electronic state on the PbS NC par- ticipating in the exchange mediated energy transfer is unknown. Due to the electronic fine-structure at the band-edge[155, 90] there are many states that can potentially electronically couple to the rubrene wavefunction. Additionally, defects,[11] mid-gap states[128 and surface states[149] have been shown to participate in charge transport and energy transfer processes. (ii) Large donor - acceptor separations, exceeding the

10E maximum length scale,[154] have been observed in efficient exchange-mediated energy transfer involving NCs.[170, 1661 This has been attributed to a large wave- function leakage in the lead chalcogenide NC,[167 or an incomplete surface ligand coverage resulting in a reduced spacing between the donor and acceptor.[166] (iii)

Lead chalcogenides have a very large dielectric constant, the dielectric screening of the exciton prior to triplet transfer is expected to have a large impact on the elec- tronic coupling between the donor and acceptor wavefunctions.[68 To elucidate the triplet energy transfer mechanism for the solid-state devices, we investigate the effect of the ligand length, which functions as the spacer or tunneling barrier between the

NC and rubrene, on the triplet transfer (TT) rate.

Results and Discussion

For the upconversion devices reported here, the PbS NCs are chosen to have the first excitonic absorption feature at A = 790 nm (1.57 eV) and the emission peak at A

= 970 nm (1.28 eV) in solution . As a result, TT to the first excited triplet state in rubrene (1.14 eV) is expected to be exothermic. The device structure is depicted in Figure 3-1. Devices are fabricated by spin coating a monolayer film of PbS NCs

51 PbS NCs

Figure 3-1: Solid-state device structure (not to scale). In upconversion the PbS NCs are directly excited at 785 nm, the photoexcitations undergo Dexter energy transfer to sensitize the triplet state of rubrene. A singlet is then formed by triplet-triplet annihilation (TTA), which then undergoes FRET to the dopant dye DBP, where it is readily emitted at 610 nm. For clarity, the QD ligands have been omitted.

with aliphatic ligands ranging from stearic acid (18 carbon atoms) to butyric acid

(4 carbon atoms) on a glass substrate. (See Methods for details of nanocrystal syn- thesis and ligand exchange). Then, an 80-nm-thick film of rubrene doped with 0.5 vol.% dibenzotetraphenylperiflanthene (DBP)[35] is thermally evaporated to form a host-guest/annihilator-emitter layer, an approach which was previously adopted in organic light-emitting diodes (OLEDs).[94] The DBP enhances the quantum yield of the rubrene film, which is otherwise low due to singlet fission.[136, 1101 Instead, the singlets formed in rubrene via TTA are rapidly transferred to DBP by FAfister reso- nance energy transfer (FRET), a process competitive with singlet fission (-r AEL 110 ps) in rubrene.[198] As discussed previously, typical exchange-mediated TT decays exponentially with distance from donor to acceptor. Modulating the NC ligand length in C-atoms will allow us to test whether transfer from PbS NCs follows the same ex- ponential trend as in the reverse (downconversion) process.[169] In a recent study we showed that the ligand shell thickness follows a non-linear trend with aliphatic chain length. Using both atomistic molecular dynamics (MD) and Transmission Elec- tron Spectroscopy (TEM), we determined the NC spacing of close-packed films to be approximately one extended ligand length.[54] Previous studies have attributed this reduced spacing to interdigitating ligands.[187] However, the atomistic simulation suggests a slumping of the ligands as the underlying cause. While this recent work was based on zinc blende CdSe NCs with amine ligands, here we experimentally in-

52 vestigate PbS NCs in the rock-salt structure. To model our NCs, we modify the CdSe crystal structure to the rock salt lattice and obtain a very similar trend in the ligand behavior as the previous study (Figure 3-2). We continue to attach amine ligands due to the ease of simulation based on the single, uncharged binding group in comparison to the charged carboxylate binding group in the experiment. Despite the differences in the simulation and experiment - CdSe and amine ligands vs. PbS and carboxylic acid ligands - the generality of the force fields allows us to bridge the gap to the lead-based NCs used here experimentally. The separation between NCs found exper- imentally by TEM of close-packed PbS arrays (inset Figure 3-2) is in good agreement with the simulated distances. Since we find that the results are transferable, we use the equation found for the relationship between the ligand shell thickness LC and the extended ligand length LC-C to translate the ligand length in carbon atoms into a real-space distance (Figure 3-2): [54]

Lc = dpb-coo- + 12 * (3Lcc + I)A - 1) (3.1) 12

Here, LC-C is the length of the extended ligand, dPb-coo- is the bond length between the carboxylate and the surface lead atom (estimated at 2.3 AE).[631 To reveal the transfer kinetics based on the ligand length, we measure the time-resolved photoluminescence (PL) dynamics of the hybrid films, seen in Figure 3-3. To enable direct comparison, each sample consists of an area of neat PbS NCs and an area containing the bilayer of the PbS NCs and DBP-doped rubrene. The PL dynamics of the close-packed NCs (black curve) are consistent with previous reports,[92, 1951 a faster multi-exponential decay gives way to a slow mono-exponential decay with

TPbS 2.4 - 2.9 Iijs, a lifetime similar to the PL decay of isolated NCs in solution .

The initial fast quenching has been attributed to interactions between adjacent NCs, in the form of energy transfer to non-emissive NCs[32, 188] or rapid non-radiative recombination.[195, 183J The addition of rubrene in the bilayer region adds a new exciton decay channel and the PL dynamics are clearly accelerated for NCs with all ligand lengths (red curve). As in our previous study,[192 we find that the resulting

53 a nc=3 nc=9 n=17

D + 2LC b 20

15

C theory TEM -C 1

0 0 2 4 6 8 10 12 14 16 18 20 nc (number of C-atoms)

Figure 3-2: Nonlinear modulation of the spacing between the NCs and rubrene by changing the ligand length. a, Atomistic molecular dynamics simulations of the length dependent ligand morphology of aliphatic amines on a model CdSe NC in the rock salt crystal structure. VMD is used for image rendering. b, Relationship between the ligand shell thickness and the number of carbon atoms in the ligand (black). The teal circles show the experimental distances obtained for close packed PbS NCs by TEM33 in close agreement with the theoretical model. The dashed and dotted grey lines function as guides to the eye and show the extended and half of the extended length LC-C, respectively. Inset: TEM image illustrating the experimental spacing between close-packed PbS NCs for a dodecanoic acid (12C) ligand.

54 decay dynamics cannot be wholly accounted for by simple first-order kinetics, and assert that there is a sub-population of NCs not undergoing TT with unchanged PL dynamics. Thus we subtract the multi-exponential emission dynamics from these inactive NCs from the total PL decay to isolate the TT dynamics in active NCs (blue curves) .192] This results in largely mono-exponential decay dynamics, with an initial fast decay, which we attribute to parasitic IR emission from the organic film . We observe slow transfer for long ligands (e.g. stearic acid (18C), Ttransfer,1C all 850 ns)

(Figure 3-3, dark blue), and more rapid transfer with shorter ligands (e.g. hexanoic acid (6C), Ttransfer,60 AEL 120 ns) (Figure 3-3, light blue). In Figure 3-3, we plot the obtained characteristic TT times (transfer) as a function of the ligand shell thickness

(compare Figure S4 for additional aliphatic ligand lengths) on a semi-logarithmic scale. The unsaturated native oleic acid ligand (OA) is highlighted in purple, showing slightly faster transfer in comparison to the saturated 18C ligand (dark blue) due to the decreased ligand length resulting from the cis-double bond.[54, 1871 As established previously, TT occurs via the Dexter mechanism, a simultaneous exchange between the ground and excited states of donor and acceptor. [113] Therefore, we anticipate a direct exponential relationship between the characteristic TT rate and the spacing

LC between the rubrene and the NC:

2LC ktransjer A * exp LTT (3.2)

where LTT is the characteristic length of TT. While the lifetime follows the ex- pected exponential trend for longer ligands, the transfer time appears to saturate at To aEl 100 ns for ligands with an effective length shorter than ca. 10 AE. This extracted

TT rate is orders of magnitude slower than most previously reported rates for electron transfer36 or resonance energy transfer in systems involving similar NCs.[182 How- ever, there are also reports of TT on a 70 ns timescale by Piland et al. using CdSe

NCs.[137 To further investigate the unusually slow extracted characteristic transfer rates ktransfer, we use constrained density functional theory (DFT) to calculate the

55 101 a T PbS -236 OU2 p s

E NC only: 18 C 0 10^' NC1 ubrene rubrene C T=651t5ns T re 850nsSlow S~ 10.2 TseBBOS ~ ~ se SNC difference on

104 0 1 2 3 4 Time (ps) b '

ET =2.60 0.04ps 0

NC only: 6 C

NC-rubrene rubrene 2d3nlsl6transfer 102 1

1 1

C Time (ps) ~10- 1000 18OA 6 101 1 14

neatL E PbS NCfl bakcre n h unhdTP ntepeec frbee

6 8 10 12 14 Ligand shell thickness (A)

curves).,. TT. dyam.~10 fo ster ani (1C) shsn.. oexoeta Figure 3-3: Time resolved photoluminescenceea decay of the infrared emission of a neat PbS NC film (black curve) and the quenched TRPL in the presence of rubrene: 0.5% DBP (red curve) to extract the TT dynamics for different ligand lengths (blue curves), a, TT dynamics for stearic acid (18C), showing a mono-exponential decay lifetime Ttransfer,18C= 651 6 ns (dark blue). This results in an estimated character- istic transfer time of Ttransfer,18C = 850 ns when accounting for the competition with the intrinsic decay channels (black curve, TPbS AEL 2.8 Ais). b, Transfer dynamics for hexanoic acid (6C), with Ttransfer,6C =116 3 ns (light blue). This corresponds to a characteristic transfer time of transfer,6C =120 ns. Insets: cartoons to highlight the enhanced transfer rate (reduced transfer time) when replacing long ligands (dlArj) with short ligands due to the reduced spacing d2. c, The deviation of the TT time from the expected strictly exponential relgionship (green) is apparent at ligand shell

thicknesses below 10 AE and the asymptoYe at T0 Al 100 ns (pink dotted) is high- lighted. The unsaturated native oleic acid ligand is highlighted in purple. The error bars depict the standard deviation of multiple samples. maximum triplet transfer rate. For this we place the rubrene perpendicular to the NC surface (85% surface-passivated with propylamine (C3) ligands) at a donor-acceptor spacing of 3.5 AE, which is the closest possible spacing based on short-range repulsive

interactions. We compute the Kohn-Sham wavefunctions of an unconstrained exciton

in the PbS NC (Figure 3-4, top) and the triplet state in rubrene (Figure 3-4, bottom), evaluate the coupling between the exciton in the NC and the triplet state, and then

obtain the TT rate ktheory from a Marcus-type expression.[114, 115, 1971 We are

obliged to make several general assumptions for the terms in the Marcus equation

(see Methods), and so the resulting rates are useful as order-of-magnitude estimates. Our approach yields a computed coupling between the NC and rubrene of V = 5.4

AaL2 10-5 eV, which corresponds to a rate of ktheory = 0.13 ns-1 (-r AEL 8 ns) for

TT transfer. While this calculated rate is an order of magnitude faster than the

fastest experimental transfer rate, it is also the rate calculated for a shorter spacing

of 3.5 AE. However, the simulation does not explain the experimentally observed

asymptotic behavior for short ligands, a deviation from the Dexter mechanism.

Hence, we postulate that the saturation of the characteristic TT rate ktransfer

points to an additional parameter that could influence the TT rate other than the

direct donor-acceptor spacing determined by the ligands or a possible wavefunction

leakage. To investigate the plateauing of the transfer rate, we examine the ligand

dependent properties of single NCs and the macroscopic NC arrays. Incomplete

passivation during the ligand exchange process can result in dangling bonds which

can act as rapid non-radiative recombination centers. [195, 183] The TT must indeed

compete with the non-radiative and radiative decay pathways, however, the neat PbS

NC kinetics should already accounted for via subtraction to first order, and should

not play a role in the TT kinetics. As a result, the overall dynamics of the exciton in

a single NC undergoing TT are not expected to be greatly affected by a change solely

in the ligand length. Indeed, we do not observe a significant change in the PbS NC PL

lifetime in solution upon ligand exchange. Changing the ligand length, however, not

only changes the spacing between adjacent NCs and between the NCs and rubrene, but also changes the volume fraction of PbS in the NC monolayer, [1831 which has

57 initial state exciton on PbS NC

k Q013 ns L Dexter transfer final state triplet excdton on rubrene

Figure 3-4: DFT calculations of the electron density distribution in the excitonic energy levels involved in the TT process. Kohn-Sham wavefunctions of the initial state: the unconstrained exciton on the PbS NC (top), and the final state: the triplet exciton on rubrene (bottom). The coupling V between the wavefunctions is computed using constrained density functional theory- configuration interaction, and the rate of TT (ktheory=0.13ns-1) is estimated from Marcus theory. VMD is used for image rendering. [79]

58 a large effect on the dielectric screening of the exciton prior to TT. Studies have previously shown a solvochromic redshift in the optical properties of CdSe NCs[103 and PbSe NC arrays[191j due to a change in the dielectric environment. We observe a similar redshift in the emission of our PbS NCs when going from solution into solid state. (Figure S7a) This redshift is often attributed to energy transfer to larger

NCs,[28] and not to the dielectric effect. However, the nearly symmetrical line shape and lack of narrowing of the line width of our emission peak is inconsistent with energy transfer (Figure S7b), as the enhancement of the low energy emission and the quenching of the higher energy emission will yield narrower and asymmetric emission line shapes. On the other hand, the long (hijs) lifetime of PbS NCs has been attributed to the large dielectric constant of PbS,[82 and a change in the lifetime as a result of a change in the refractive index of the NC solvent has been observed.[28 This is in agreement with our observations when comparing the long component of the

PbS lifetimes in solution (T= 3.1 hijs) and in solid state: r = 2.6 Jijs for shorter ligands where the dielectric effects are expected to be increased (compare Figure

S2 and Figures 3b (black curve)). We estimate the dielectric constant of the PbS

NC monolayer using an effective medium approximation following the Bruggeman model:[27, 147]

6 0 (33) a IEPbS - Ligand + (1 - Ligand - ELigand EPbS 2 ELigand CLigand + 2 6Ligand

where itPbS represents the static dielectric constant of bulk PbS, it'LigandArj is the dielectric constant of the ligand, and itLigand is the calculated effective medium dielectric constant. Decreasing the ligand length from 18C to 6C (4C) results in a two-fold increase in the PbS volume fraction a from 8.5% to 17.3% (21.2%), based on a hexagonal close-packing model of the PbS NCs in a monolayer. The high static dielectric constant of PbS itPbS = 169 combined with the much lower dielectric constant of the ligand it'Ligand ranging from it'18C = 2.3 to it'6C = 2.6 (it'4C 3) results in a large change in the effective medium dielectric constant itLigand from it18C = 3 to It6C = 5.2 (it4C = 7.1) when exchanging the NC ligand from

59 18C to 6C (4C) (compare Figure 3-5). The coupling term or exchange integral V in exchange-mediated energy transfer is inversely proportional to the dielectric constant of the medium: , which in turn results in a lower coupling as the dielectric constant increases. The energy transfer rate is proportional to the square of the coupling term: k IV12 and therefore, inversely proportional to the square of the dielectric constant.

We multiply the determined characteristic TT rates ktransfer (Figure 3-5) by their respective dielectric constants normalized to the effective medium dielectric constant of the C18 NC film as a reference value:

kTT = ktransfer - fLigand/8C -4

As a result, we obtain the expected exponential relationship between the resulting

TT rate kTT and the ligand shell thickness (Figure 3-5). A fit to the extracted ligand shell thickness dependent transfer lifetimes yields LTT = 3.8 0.1 AE (21 = 0.52

AE-1) as the characteristic length of TT. This value for the characteristic length is similar to previously reported values (LTT AEL 4.7 AE, 21s = 0.43 AE-1) by Li et al. for upconversion using CdSe and tetracene.53 Extrapolating the normalized TT rate to a donor-acceptor spacing of 3.5 AE yields k3.5AE = 0.19 ns-1, which is now in good agreement with our theoretically predicted rate (ktheory = 0.13 ns-1), despite the considerable assumptions made in the simulation. To relate the increased transfer rate for short ligands to the expected device improvement, we estimate the transfer efficiency by assuming first-order kinetics for active nanocrystals and comparing the quenched lifetime of NCs in a bilayer ID to the inherent NC-only lifetime of the PbS monolayer TPbS:

7t = (1 - T/TPbs) - 100% (3.5)

This yields anticipated transfer efficiencies of hit AEL 76% in the active NCs for

long ligands (18C). For short ligands (4C, 6C), nearly 100% of excitons in active NCs

should transfer into the rubrene layer, resulting in a significant enhancement in the

expected device efficiency. We measure the intrinsic upconversion efficiency , defined

60 a ligand bf k k- exchange 4 47 V) 18C

.~ 54 6Co10., 6 &ACC 4 102 8 10 12 14 0 j5 3 10- 2 pure l8ligand 1 10 3

6 8 10 12 14 6 8 10 12 14 Ligand shell thickness (A) Ligand shell thickness (A)

Figure 3-5: Correction of the extracted transfer rate by the calculated dielectric constant. a, Calculated dielectric constant for a PbS monolayer with different ligands based on the Bruggeman model. Inset: cartoon highlighting the change in the volume fraction of PbS vs. organic ligand when shortening the ligand (not to scale). b, Saturation of the extracted rates (gray line) and deviation from the exponential trend (green dotted line). c, Normalization of characteristic TT rates (grey diamonds) by the dielectric constant: recovers the expected purely exponential trend (green). Error bars are based on an error propagation with an estimated error for the dielectric constant AE= 0.2, and the standard deviation of the extracted rate.

as the fraction of absorbed low-energy photons that lead to higher-energy emissive

states in the annihilator, for devices sensitized by A = 790 nm PbS NCs with short

6C ligands. This probes the combined efficiency of TT and TTA only, isolated from

the PL quantum yield of the organic layer, which varies with thickness and dopant

concentration.[193] The 6C ligand is chosen over the 4C ligand due to their essentially

identical TT rates, yet superior colloidal stability. We use a modified device structure

(see Methods) recently reported,[1931 where we enhance both the infrared absorption and visible emission by employing an optical spacer and a silver back reflector. The

stronger absorption in the monolayer NC and the higher upconverted PL improve

the signal-to-noise ratio of the measurement, allowing more accurate determination

of . When pumped at A = 808 nm, we obtain an efficiency of (7 1) % upconverted

photons per pair of absorbed infrared photons. Previously, we reported = (1.2 t

0.2) % for devices sensitized by A = 850 nm NCs capped with native oleic acid

ligands excited at A = 808 nm.4 The substantial improvement in can be attributed

61 to a combination of faster TT transfer with the use of shorter ligands, smaller-size (higher-energy) NCs, improved device structure and a more accurate measurement technique.

Conclusion

In conclusion, we demonstrate an order of magnitude enhanced triplet transfer rate between PbS NCs and rubrene upon reduction of the aliphatic ligand length from 18C to 4C, which based on theoretical results by Geva et al.33 corresponds to a de- crease in the ligand shell thickness from ca. 13 AE to 6 AE. However, the extracted characteristic TT rate ktransfer unexpectedly saturates for ligand shell thicknesses AId' 10 AE, which we can correlate to the rapid increase in the medium dielectric constant due to a higher volume percentage of PbS in the monolayer. The electronic coupling V is inversely proportional to the dielectric constant It; by normalizing the characteristic TT rates ktransfer by the square of the dielectric constant, we are able to recover the expected exponential trend between the kTT and the donor-acceptor spacing. We observe an increase in the upconversion efficiency from previously re- ported values of (1.2 0.2) % for long oleic acid ligands4 to a solid-state efficiency of (7 1) % measured here for shorter ligands (6C). This is attributed to the increased TT rate, in combination with the smaller-size (higher-energy) NCs, improved device structure and a more precise measurement technique. Unlike in many optoelectronic devices that include NCs, where the use of extremely short ligands to maximize carrier mobility is a significant concern, we show here that this strategy yields diminishing returns in term of the TT rate for ligands shorter than 10 AE - a trend which is accentuated when considering the transfer efficiency given the very slow competing decay channels. Accordingly, our results show that engineers are free to choose ligands to address other considerations, such as colloidal stability and reduction of parasitic non-radiative decay channels and ease of device manufacturing.

62 Materials and Methods

Nanocrystal Synthesis. Lead sulfide (PbS) nanocrystals (NCs) with a first excitonic

feature at 790 nm (emission A AEL 970nm) are synthesized following a modified ver-

sion of the hot-injection method by Hines and Scholes.[751 Lead oleate is prepared

ex-situ as reported elsewhere.[71 A three-neck roundbottom flask is charged with 2.0 g Pb(oleate)2 and 20 mL octadecene (ODE) (technical grade, 90% Sigma-Aldrich) and

is degassed under vacuum for 12 h at 12OkfC. The injection solution is prepared in a

nitrogen-filled glovebox and consists of 0.27 mL trimethylsilylthiane (Sigma-Aldrich)

in 4 mL ODE. The reaction temperature is lowered to 90AIC and the flask is backfilled

with nitrogen. The solution is quickly injected and the reaction immediately quenched

by an ice bath. The nanocrystals are purified by a standard solvent-nonsolvent pro-

cedure, redispersed in toluene (Sigma-Aldrich) and stored in the nitrogen glovebox.

Ligand Exchange. All carboxylic acids are purchased from Sigma-Aldrich. For ligand

exchange the stock solution is diluted by adding four times the volume of toluene. 0.1 mL of a 0.1 M ligand stock solution in toluene (4C, 6C, 16C and 18C) or 0.4 M ligand

stock solution (8C, 10C, 12C and 14C) is added to 0.5 mL of the diluted nanocrystal

solution and stirred air-free at room temperature for 20 min (4C, 6C, 16C and 18C)

or 3 h (8C, 10C, 12C and 14C). Excess ligand is removed by repeated washing with a

solvent-nonsolvent procedure. The purified NCs are redispersed in toluene and stored

air-free. TEM. Transmission electron microscopy (TEM) is carried out on a JEOL

2010 HR microscope at 300000x magnification. TEM samples are prepared by drop-

casting the diluted purified ligand-exchanged nanocrystal solution onto TEM grids

(UC-A on holey 400 mesh Cu, Ted Pella). Images are analyzed using ImageJ. De-

vice fabrication. Glass substrates are cleaned sequentially with 2% Micro-90 solution, deionized water, acetone, boiling isopropanol and oxygen plasma. All following steps

of fabrication are performed in a nitrogen-filled glovebox. A monolayer of PbS NCs is deposited on clean glass by spinning the NCs dissolved in toluene (ca. 3 mg/ml) at

1500 rpm for 45 s. An 80 nm-thick film of rubrene doped with 0.5 vol.% of dibenzote- traphenylperiflanthene (DBP) is then thermally evaporated through a shadow mask

63 at a base pressure below 3 * 10-6 Torr. The mask defines areas of the NC film with and without doped rubrene on the same device for the transient photoluminescence

(PL) measurement. For measurement of the upconversion efficiency, the device has two more layers in the thin-film stack. Following the doped rubrene layer, 20 nm of tris-(8-hydroxyquinoline)aluminum (AlQ3) and 100 nm of silver are evaporated sequentially. All devices are encapsulated using two-part epoxy (Devcon 5 Minute) and a second piece of glass, and allowed to cure for 30 minutes before being removed from the glovebox for testing. Rubrene was purchased from Luminescence Technology

Corp. and used as received. All solvents, DBP, AlQ3 and silver were purchased from

Sigma Aldrich and used as received. TCSPC lifetimes. To measure the PL quenching caused by triplet transfer (TT) to the organics we use time-correlated single-photon counting (TCSPC). The nanocrystals are selectively excited at A 785 nm with a pulsed laser (PicoQuant LDH-P-C-780), which is sent through a band-pass cleanup filter (ThorLabs FBH780-10). The sample is excited by a pulse train at 100 kHz, and the pump power is adjusted to obtain a AEd' 5 % count rate in each measurement to avoid pile-up artifacts in the detector. The SWIR nanocrystal emission is collected by parabolic mirrors and focused onto InGaAs/InP single-photon counting avalanche photodiode (Micro Photon Devices $IR-DH-025-C). Excess laser scatter is removed by a long-pass filter (Chroma Technology Corp. ET900LP). Photon arrival times are recorded by a PicoQuant PicoHarp 300. To extract the dynamics of the triplet transfer (blue), we subtract a scaled copy of the NC-only dynamics (black) from the quenched dynamics in the bilayer region (red).4 The extracted dynamics are fit to a mono-exponential function at longer times (t dD 200 ns). We assume a simple kinetic model in which the extracted decay is composed of a competition between the extracted TT time ID and the intrinsic PbS decay TPbS, resulting in the characteristic transfer time Ttransfer given by:

Ttransfer = (1/T - 1/Tps) (3.6)

Molecular dynamics simulation of the ligands. To determine the structure of

64 ligands on the surface of the NC, we preform molecular dynamics calculations. We run an NVT simulation at 300 K for 2 ns with a timestep of 2 ps using the Velocity

Verlet integrator. The temperature is controlled using the Anderson thermostat with a coupling time of 100 fs. We employ the OPLS force field for the amine ligands,58 and following Schapotschnikow et al.,59 the Lennard-Jones-Coulombic potential for the NC. The NC is carved from a bulk PbS crystal in the rock salt structure with a

Wulff parameter of 0.8 between the 100 and the 111 planes. Comparable to the NC used here experimentally, the NC simulated is about 2.5 nm in diameter and contains

100 atoms. We decorate the NC with 70 amine ligands. Simulations of PbS-rubrene coupling. We calculate the theoretical rate of TT kthary from the NC and rubrene following the protocol by Yost et al.44 starting from a Marcus-like expression.

27r AGA2+1 ktheory = |V|2 e- (3.7) h 4TrkBTA

Here, V is the coupling, A is the reorganization energy, and AG is the difference in free energy between the triplet before and after transfer. To obtain an order of mag- nitude estimate for the ktheory from the coupling, we must make assumptions about the remaining terms in the Marcus theory expression. We take the reorganization energy A to be approximately half of that of rubrene-to-rubrene TT: 0.15 eV.44 Fur- ther, we assume AG EL A, and therefore the exponential in the Marcus expression equals one. A PbS NC is cut out from the bulk (rock salt) using a Wulff construction polygon.60 To model the passivation of the nanocrystal, we use amine ligands placed at a distance of 2 AE from surface lead atoms. Amines are used for computational convenience because monodentate ligands are more straight-forward to place on the surface than carboxylic acids which have a variety of potential coordinations. Due to the computational cost of the simulation, we wish to undertake a single-point cal- culation. In order to provide an estimate for the maximum rate possible, we make a judicious choice about where and how to place the rubrene molecule. As such, a rubrene molecule is placed at a distance of 3.5 AE (nearest hydrogen-to-sulfur dis- tance) from a bare sulfur facet, following structural optimization using a 6-31G/PBEO

65 DFT calculation. The placement is such that the plane of the polyacene backbone is perpendicular to the surface. Constrained density functional theory is used to compute Kohn-Sham wavefunction states for the triplet constrained to the rubrene or the NC. The coupling is then computed using constrained density functional the- ory - configuration interaction (CDFT-CI).[194] The dot triplet is unconstrained and spans the whole of the NC. All calculations use LANL2DZ basis set and effective core potentials, with PBEO as the functional. Upconversion Efficiency. To determine the upconversion efficiency as defined in the main text, we measure the absorption and

PL of the interference-enhanced thin-film devices with an integrating sphere (Lab- sphere RTC-060-SF). The laser beam is run through a chopper and is incident on the sample at the center of the sphere at an angle of ca. 35 degrees. The light out of the exit port of the sphere is focused onto a photodetector (Newport 818-SL) connected to a lock-in amplifier. Appropriate dielectric edge-pass filters are mounted on the photodetector to isolate either the laser or the PL signal. The signal spectral shape is checked with a spectrometer. Absorption (Abs) is obtained by measuring the output laser signal (L) with and without the sample inside the sphere:

Abs = (1 Lsamp e"" )100% (3.8) Lsample out

We notice that for a control sample comprised of glass/AlQ3/silver/glass, there is a finite decrease in the output laser signal when the sample is placed inside the sphere. This is due to parasitic absorption in silver and non-idealities of the sphere.

As a result, to get the actual absorption by the active material, we subtract the absorption of the control from the absorption measured of the active devices. The devices are excited and measured at two wavelengths: A = 808 nm and A = 450 nm.

The upconversion efficiency efficiency is determined experimentally by:

P(A = 808nm) (I(A = 450nm)A(A = 450nm)) rIUC -2-100% (3.9) (I(A = 808nm)A(A = 808nm)) (P(A = 450nm))

where P is the number of visible photons emitted, I is the number of photons in

66 the incident laser beam, and A is the absorption by the active material. Subscripts indicate the pump wavelength. A factor of 2 is included by convention, to make unity the maximum upconversion efficiency. Here, P is measured by the lock-in photodetec- tor with edge-pass filters on (ThorLabs FELH500 and FESH750); I is calculated from the power measured by a calibrated power meter (Thorlabs S130C and PM100A). To improve the accuracy for excitation at A = 808 nm, we repeat the measurement of

Lsample in, Lsample out and P on each device (including the control) 6 to 8 times, and compute the absorption and emission from the averaged data. For excitation at A = 450 nm, we measure an organic-only control - glass/rubrene:DPB/Alq3/silver/glass - to calibrate the emissivity of doped rubrene without any influence from the NCs.

67 68 4

Mean Field Treatment of . Heterogeneous Steady State Kinetics

We now take a short digression. In the previous chapters, our experimental collab- orators relied on kinetic fitting of monoexponentials to their data, combined with background subtraction to extract rates. However, in many cases the fit is poor. The reason is that a QD system is inherently disordered, exhibiting a continuum of rates.

In this and the following chapter, we develop new kinetic models that specifically target disordered systems. This chapter was published as [55J. Work on this chapter has been done with Valerie Vassier and James Shepherd, post-docs in our group.

Introduction

Recent single molecule experiments have revealed that beneath the ensemble aver- age common to macroscopic observations, heterogeneity plays an astonishing role in chemical kinetics. From light harvesting [721 to chemical catalysis [801 to signaling

[91 to enzyme function [441 there is a common theme: chemical function is not deter- mined by the typical or average member of the ensemble. Rather, the rate of a given process can be strongly influenced by outliers.

Stochastic approaches have proven indispensable in the theoretical understanding of how this kind of molecule-by-molecule heterogeneity governs the dynamics. Kinetic

69 Monte Carlo (KMC) provides an in principle exact method for dealing with both spatial and temporal fluctuations in rate constants.[501Meanwhile, stochastic master equations provide a powerful tool for simulating systems in which the rate constant is spatially uniform but fluctuates in time. This kind of purely dynamic disorder is, for example, thought to be widespread in cooperative or allosteric interactions.[161, 37, 109, 8] The challenge of using any of these techniques is that they are often frustratingly slow, either because of the range of timescales involved [180, 24, 145 or because of the influence of long-range forces on the rates. [100, 180, 22, 119] The time-consuming nature of these methods can sometimes prevent systematic study of the importance of individual rates on the dynamics, since such exploration requires running dozens or hundreds of already time consuming simulations.

In this article, we report the development of a new method, the Mean Field Steady

State (MFSS) approximation, that retains the speed and simplicity of ensemble ki- netics while also accounting in an averaged sense for the static disorder that is evident in so many chemical situations. MFSS makes a different use of the mean field ap- proximation than the Langmuir-Hinshelwood[99, 62, 83] or Bragg-Williams[148, 1111 theories employed in the modeling of catalytic surfaces. Indeed, these techniques as- sume a perfect species mixing by relying on a uniform distribution of catalytic sites.

As a result, they quickly fail for inherently heterogeneous phenomena as considered here. To circumvent this issue, other groups have introduced an inhomogeneity factor, effectively allowing for local variations in species occupancy probabilities within the mean field approximation.[34, 205, 73, 163, 164] However, this approach still requires homogeneous catalytic rates. In contrast, our MFSS finds self consistent steady state populations of species subject to inhomogeneous chemical activity.

The general MFSS scheme is illustrated in Figure 4-1. Envisioning each reaction as occurring at a fixed lattice position, static disorder induces changes in the underlying rates at individual sites. Focusing on one active site we can approximately replace the neighboring sites by their ensemble averages (the mean field approximation). We can then examine the influence of fluctuations in the rate constants at the active site in the presence of the averaged neighbors to approximately describe the influence

70 A) B) C)

Figure 4-1: Illustration of the mean field steady state idea. Beginning with a set of sites with different rates (A) one simulates steady state for each site in the presence of a set of averaged neighbors (B). The average of these steady states (C) then self- consistently defines the average neighbor used to obtain steady state.

of local heterogeneity on the overall kinetics. We finally make the entire simulation self consistent by forcing the average neighbor state to match the computed average over fluctuations at the active site. We should emphasize that here we use the word heterogeneity to refer to variations in chemical activity (disorder in chemical rates) as distinguished from heterogeneous catalysis, where heterogeneity refers to the phase of the catalyst as compared to the substrate. While self-consistent mean field methods like this are quite common in a variety of fields, for instance in electronic structure, [14, 69]and statistical mechanics [10, 53] the analogous formalism does not appear to be widely used in the study of rate equations. This work begins to remedy that situation.

Working example

It is perhaps easiest to explain how these ideas function by using an example. Toward that end, consider the following hypothetical mechanism for the catalytic hydrogen

71 evolution (2 H+ + 2 e- - H2 ) at a metal surface in aqueous solution:

H+(aq) + e- H

H+H esH2

H2 - H 2 (g)

where H and H2 refer to chemisorbed species at the surface. For a given chemical system, this may or may not be the appropriate mechanism, but for the purposes of illustration this example is sufficient. If these are the elementary steps in our reaction then at an ensemble level these steps imply a steady state rate expression in the following familiar way. Using ([$], [HI, [H2]) to refer to the fraction of surface sites that are vacant, occupied by an H atom and occupied by H2 , respectively, the rate expressions for each step are:

H+(aq) + e- H Rate1 - kA[H+(aq)][0] = kA[0]

H -4 H+(aq) + e- Rate 2 = kD[H]

H+H -% H2 Rate3 k1 [H][H]

H2 -1 H+ H Rate4 = k_[H2 ] [0]

H 2 H 2 (g) Rate5 =kR[H2] where, for simplicity, we assume that the reaction occurs at constant pH and electrical potential so that [H+] can be absorbed into the rate constant kA. We intentionally build a model as simple as possible, for example ignoring the diffusion of adsorbed

H2 , as is commonly done in method development studies of chemical kinetics.[34, 73, 164] The diffusion of adsorbed hydrogen is effectively accounted for by the sequence of desorption and adsorption steps (slow diffusion limit). At steady state, these macroscopic rate expressions imply that the fraction of H2 gas, [H2]ss, would be (see Supporting Information for details):

72 [H2 -2klkA(kA + 2kR) + (kA + kD)(k-lkD + (kA + kD)kr) - (kA + kD) / 2 [k_(kA + kD)(kD - 2kr) + kl(kA+ 2kr) 2] -=kk21 2(2klkA(2k-l + kA) + k-lkD(kA + kD))kr + (8klkA + (kA + kD)2 )k2 4.)

Now, the steady-state rate (Eq. 4.1) has the clear weakness that it assumes that the system is homogeneous: every potential absorption site possesses the same catalytic rate k, as every other. However, in reality each site might have its own rate as a result of point defects, doping, surface reconstruction, charging, grain boundaries, etc. A specific example in the context of heterogeneous catalysis would be bimetallic surfaces, where the random distribution in the nearest neighbor shell gives rise to non-uniform rates. Alternatively, disorder in catalytic rates can arise from weak adsorbate-adsorbate interactions that does not allow for a significant ordering of the adsorbed layer. Assuming that the activation energy of the bimolecular chemical step is normally distributed, we will vary the rate ki (proportional to the exponential of the activation energy) from site to site according to a log-normal distribution:[1801

1 - ln(k)) 2 P (ln(ki)) = exp ((ln(ki) 2 2 , (4.2) V/27rg22 where ln(k) and a are the mean and standard deviation respectively. Note then, that o- takes the units of kBT such that a value of o-=5 kBT corresponds to a standard deviation of 5 kBT in activation energy.

A log-normal distribution is also used in the popular Gaussian disorder model

(GDM). [176, 16, 951 In GDM, site energies are distributed in an uncorrelated fash- ion according to a normal distribution. In our case, we assume that the activation barrier fluctuates in this manner, which would seem a particularly interesting case for kinetics. This distribution might arise, for example, in the presence of randomly distributed point defects near the active site. While physically reasonable, this choice of probability distribution is arbitrary. The method presented here can, in princi- ple, deal with any distribution that is dictated by the underlying chemistry (e.g. a

73 binomial distribution arising from terrace and ledge sites or a logistic distribution representing the cumulative effect of subsurface defects).

Note that by detailed balance, the variations in k1 imply similar variations in k_ 1 because k, = Keqk_ where Keq is the equilibrium constant for the overall reaction at the given conditions. For the purpose of this paper, we focus on introducing disorder in the bimolecular rate. Therefore, we will assume that the rate constants for all other steps - kA, kD and kR - are homogeneous, as also seen elsewhere.[34, 73, 163] We should also note that the method described here would be perfectly capable of accounting for disorder in all variables. We have performed a single study of disorder

in all rates (Figure 4 in the SI), and note that disorder improved the H2 generation. For simplicity, in the rest of this letter we restrict our attention to disorder in a single variable, but further explorations of simultaneous disorder in multiple variables could be illuminating.

The self consistent mean field approach

In this situation, the typical solution is to employ KMC [501 to simulate the kinetics: one generates a large lattice of sites with rates k, drawn from Eq. 4.2 (as in Figure 4-

1A) and then monitors the dynamics of all species at all sites until steady state is

established. For example, the rate of change of H 2 at site i at any given instant would be:

d[H212 - -kR[H ]1 + 2 k [HH]|j - k'1 [H2 ]ij, (4.3) dt1

where k' and kU 1 are the catalytic rates at site i, [H2 ]i refers to the probability that site i is occupied by H2 and [AB]ij refers to the probability of A being at site i while B occupies the neighboring site j. Note that we assume that when H2 forms or dissociates on a given site, it is the value of k, or k_ 1 at that site that determines the reaction rate.

Clearly, the difficult things to obtain in Eq. 4.3 are the joint probabilities [HH]j

74 and [H2 #]ij We can simplify these terms by assuming the populations on neighbor sites are independent:

[HH] J [H]i[H [H2 0]ii ~ [H21i[l]j (4.4) in which case Eq. 4.3 becomes

d[ H2 -kR[H ]i+ 2 k'[H] [H]j - k 1 [H2 Z O]j

-kR [H2] + k'[H] d H - k' 1 [H2]i d # (4.5) where in the second line, we have made the mean field approximation, replacing the sum over d neighbor sites (Z,) with d times the equilibrium average over all sites.

Like most mean field approaches, we expect this approximation to be more accurate when there are many neighbors, [97, 961 because then the sum over neighbors will more closely reflect the ensemble average. However, the results below were obtained with a square lattice, so 4 nearest neighbors is already sufficiently many for the discussed case. We also note that this mean field approximation may fail if there is strong spatial correlation between the heterogeneities, such as those occurring at a step edge. In this case, the model would need to be extended to address such correlations.

Applying the same mean field approximation to the other two absorbed species gives:

d[H' - d (k'H + Hk )[H] dt = kA[#]i - kD[H]i 1

+d H2k_[1]i + d kL10[H2 ]i, dO]_ = -kA[]i+ kD[H] + d Hki[H] dt

-d H2 k_1 [] + kR[H2 ]. (4.6)

Here, in addition to H, H2 and 0 the joint averages Hk1 and H2 k_ 1 appear. These arise from reactions that can occur at a neighbor site but either produce or consume

75 a species at the active site. For example, Hi + Hj -+ H2 + 0 gives a contribution to d[H], of dt

d Z(kl + kj)[H]j[H] 3 = d ki[H]iZ[H]

+d [H] i kij.[H]j

d kz[H]iH + d [H]ik1[H]

where the first term corresponds to the formation of H2 at site i and the second term corresponds to the formation of H2 at a neighbor site. Note that the mean field approximation converts the non-linear ensemble rate expressions into a set of linear equations for [H2]i, [H]i and [O]j.

Setting Eqs 4.5 and 4.6 to zero to enforce steady state gives the populations

a[H2]i = d k H(d H2 k_1+kA)

a[H]i = (d H2 k + kA)(k' 15+kR) (4.7)

a#]i, = kR(d Hk1 + d khI + kD)+ k" 10(d Hk1 + kD)

where a is a constant. For simplicity we will choose a so that [H2 ] + [H]i + [c]i = 1.

Eqn. 4.7 predicts the steady state behavior given a set of known rate constants kt, kU1 , kA, kD and kR and a set of averages H2 , H, 0, HkI and H2 k_ 1 , which we don't know a priori. However, Eqn. 4.7 suggests a means of obtaining these averages. By solving for the mean field steady state of each potential active site in the lattice, one can compute the averages as

H 2 = Z[H2]i

Hk1 = NL k'[H], (4.8)

where N is the number of lattice sites and similarly for H, 0 and H2 kj. Note that these averages can equivalently (and perhaps more conveniently) be computed

76 as integrals over the probability distribution of k 1 . For example

H2 = [H2](k)P(k)dk (4.9)

where [H2](k) is given by Eq. 4.5 with k = ki. No matter how the averages are computed numerically, the key point is that the steady state equations (Eqs. 4.5 and 4.6) need to be solved self-consistently with the ensemble average relationship (Eq. 4.8). In practice we have found that a simple iterative approach of solving one set of equations and then feeding the results into the other converges quickly for this example. In more difficult MFSS equations, it might be necessary to use a convergence acceleration technique similar to the Direct

Inversion in the Iterative Subspace (DIIS). [52]

Taken together, Eqs. 4.5,4.6 and 4.8 provide a simple, approximate means of accounting for static fluctuations in the rate constant that might occur in the catalytic production of H2 at a disordered surface. For the key catalytic step, each site interacts with a set of neighbors that are representative of the steady state distribution averaged over the whole system. Within that average environment, the steady state at each site fluctuates according to its local catalytic rate. The resulting MFSS solution should

give at least a reasonable estimate of the effects of heterogeneity on the rate of H 2 production. We should note that similar approaches have been used occasionally in treating metal alloys [150, 1731 but overall this kind of approximation appears to be seldom used in chemical kinetics.

Results

To validate our approach we solved the MFSS equations (Eq. 4.6) self-consistently until convergence of the average populations. These were computed by quadrature using Eq. 4.9 with a log normal distribution of rates. Figure 4-2 shows the resulting

steady state 0, H and H2 populations for an increasing heterogeneity in the catalytic rates (increasing a).

77 1.0 1- MF Vac - - Adjusted Mean Vac . KMC Vac -- MF H - - Adjusted Mean H . KMC H 0.8 - MF H2 - Adjusted Mean H2 * KMC H2

aZ 0.6

0.2 - -

0.0 0 5 10 15 20 sigma

Figure 4-2: Steady state q (blue), H (green) and H2 (red) populations calculated with KMC (dots) and MFSS (solid lines) as a function of the standard deviation of the log normal distribution of the catalytic rates k1. At ambient temperature, the upper limit a = 20 kBT - 0.5 eV sets a realistic range of expected disorder in real systems. Also shown are the MFSS populations for a null standard deviation but a mean adjusted to ln(k) + o/2 (dotted lines). The homogeneous rates were: k, = 0.1, kd= 0.1, ka = 0.1

Comparing with the KMC populations (details of the KMC algorithm can be found in the Supporting Information), we observe that the two methods give essen- tially identical results. When scanning over the disorder in the bimolecular rate, as measured by the value of sigma, KMC and MF are within 10-' of each other in the range tested. Clearly the MFSS approach gives an accurate and yet highly efficient description of the heterogeneous kinetics of this system. The method easily recovers the results of homogenous steady state (4.1), as can be seen in Table 1 in the SI. Most notably, the surface fraction of H2 , which is proportional to the (rateprod Cx k,[H2])7 increases from 3 x 10- 3 k, (at a = 0 kBT) to 0.11k, (at a = 20 kBT), vividly illustrat- ing the simple truth that the average of the rates is not the same as the rate of the average site. It is somewhat shocking that something as simple as MFSS can capture this kind of significant change. As a point of comparison, it should be noted that because of the asymmetry of the log normal distribution, changing a as we have done leaves the mode of k fixed, but adjusts the mean to ln(k) + a/2. One might then suspect that using this adjusted mean in the ensemble SS expression (Eq. 4.1) might also give good agreement with KMC. As illustrated in Figure 4-2, this approach vastly overestimates the effect of heterogeneity, emphasizing that MFSS is doing something non-trivial here. To check that these results hold when the diffusion of adsorbed

78 hydrogen is fast, we ran the same calculations only adding an explicit diffusion step with independent rate constant (see Supporting Information). We observe that fast diffusion does not affect the results in the parameter space considered here.

As an alternative test, we show in Figure 4-3 the MFSS and KMC steady state populations when tuning the log average rate ln(k), at fixed disorder (a = 10 kBT in the Figure). Because ln(k) ~ one can change the log rate either by adjusting the mean activation energy or the overal temperature. Thus, this scheme corresponds to the other, more "conventional" way one could increase H 2 production: tuning the average activation energy. Here again we observe quantitative agreement between the two methods. When scanning the typical mean of the bimolecular rate, KMC and

MF are on average within 10-2 of one another, with the result being qualitatively different than predicted by the ensemble SS with the average rate.

1.0 J -L - MF Vac - - Adjusted Mean Vac . KMNC Vac -- MIF H - Adjusted Mean H * KMC H 0.8 - - F H2 - - Adjusted Mean H2 . KMIC H2

-0.6------

0.4 '

0.2 -

0.0 -10 -9 -8 -7 -6 -5 -4 mu

Figure 4-3: Steady state # (blue), H (green) and H 2 (red) populations calculated with KMC (dots) and MFSS (solid lines) for a range of the mean log of the catalytic rates ln(k). The homogeneous rates were: kr = 0.01, kd = 0.16, ka = 0.14

Having worked through one example, it is now fairly clear how MFSS could be applied to a general heterogeneous system:

Step 1 Write out the elementary steps of the reaction and determine (either by exper-

iment, computation or estimation) the rate constants, including the probability

distribution of any rate(s) that are heterogeneous.

Step 2 Guess approximate ensemble average populations for all species (X). For

example, these might be the values obtained from the average rate constants.

79 Step 3 Solve the pseudo-first order rate equations (e.g. Eqs. 4.5 and 4.6) to obtain an

expression for the steady state populations for each site ([X]j) or equivalently for

every rate constant ([X](k)). Because the equations will be linear, this should be

feasible for mechanisms involving as many as several thousand distinct reactions.

Step 4 Integrate over the probability distribution P(k) to obtain new steady state

values:

X = [X](k)P(k)dk

Equivalently, one could sum over all sites to obtain the averages (e.g. Eq. 4.8).

For simple mechanisms this might be done analytically, but other cases this

integral can be evaluated to high accuracy by quadrature as long as there are

not too many rate constants that vary independently.

Step 5 Compare the new X to the old X. If the two agree, the calculation is

complete. If they differ significantly, return to the second step with the new

averages replacing the old.

It seems likely that this scheme could be useful for describing the effect of disorder on the rates of numerous chemical systems. For example, catalytic processes like methanol oxidation [1841, oxygen evolution [201, and the hydrogenation of olefins

[204] could all potentially be described using a formulation similar to the one discussed here. In a somewhat different context, the generation and destruction of charges in OLEDs [178, 1261 and OPVs [91, 203] could also benefit from an analysis like this one.

At a basic level, in any system where the rate constants have some static disorder and where one of the steps is bimolecular (so that the active site interacts with sites around it) MFSS provides an attractive means to quickly approximate the ensemble averaged kinetics.

80 Conclusion

The resulting methodology thus seems like a worthwhile tool for chemical kinetics

quite generally. It is extremely easy to apply, involving only some algebraic manipu-

lations and a modest amount of numerical integration. To put it another way, MFSS

allows us to study heterogeneity without being limited by the speed of the computer on which we run the simulation.

Of course, as with any inexpensive method, MFSS is highly approximate. For

the simple example we have discussed here (H2 formation) it is rather difficult to find situations where it fails, but it is not hard to imagine what these would be. In

cases where there are not many near neighbors (e.g. in low dimensional systems) the assumption that the average over neighbors approximates the ensemble average

is likely to be poor. Alternatively, when the mechanism involves multiple bimolecular

steps, it seems likely that the populations of neighboring sites will actually be cor-

related: the presence of A on one site might appreciably reduce the probability that

B appears on a neighbor site. Since these are a set of coupled non-linear equations, many solutions might exist. This might require a global optimization scheme to find the most realistic solution. The severity of these limitations is not immediately clear, but they create an obvious area for future study.

There are a number of other clear directions to extend this work. Beyond the

above-mentioned applications, it would be interesting to try to build correlated fluc- tuations into MFSS theory in much the same way that correlation is built into other mean field theories. [701 In this case, the most obvious steps are to either include self-consistent fluctuations in the neighbor sites (as opposed to just a self-consistent mean) and/or simply including more than one active site in the MFSS kinetics. It would also be interesting to go beyond the steady state situation and apply these kinds of approximations to examine the average time evolution of heterogeneous re- action networks. Finally, in addition to being able to treat static disorder in this way, it would also be useful to have a simple mean field picture for dynamic disorder in the rate constant.

81 82 5

A disordered kinetics model for triplet transfer

In the previous chapter we saw the role disorder can play in steady-state kinetics.

In this chapter we apply a different method based on transfer matrix kinetics, to investigate time-dependent photoluminescence (PL) traces. Again, our experimental collaborators were Lea Nienhaus from Prof. Bawendi's group and Mengfei Wu from

Prof. Baldo's group. This chapter will be submitted to Nano Letters.

Introduction

Colloidal nanocrystals (NCs) - organic semiconductors (OSCs) hybrid interfaces rep- resent a unique opportunity for new avenues in optoelectronic research[78, 137, 1211.

By combining the tunable electronic structure of NCs with the unique features of

OSCs, such as singlet fission and triplet fusion, devices that overcome the Shockley-

Queisser limit in single-junction photovoltaics[156, 172], allow for improved biological imaging, and reduce the cost of IR camera are possible[207j.

Recent progress on such conversion devices has shown that energy transfer between

NCs and OSC can be made fast and efficient[78, 137, 121, 1921. However, so far the progress in reported efficiency has focused on increasing the internal efficiency; the percentage of absorbed light that was converted. The external efficiency, the

83 percentage of total incident light that is converted, remains low. The main barriers to improving the external efficiency are light absorption and energy transfer. To achieve a high device efficiency, a large fraction of incoming photons must be absorbed and subsequently be transferred into the up-converting layer. In previous studies[193,

192, 1291, the goal was to extract the underlying triplet exciton transfer mechanism, and to minimize effects caused by inter-dot energy transfer, the sensitizing layer was made extremely thin: one or sub-monolayer.

In this paper, we study the thickness dependence of the internal up-conversion efficiency. We show that the external up-conversion efficiency increases slightly going from one to two monolayers, however, the internal efficiency drastically decreases.

To investigate the underlining reason for this behavior we turn to transient photo- luminescence (PL). However, this hybrid NC-OSC system represent a challenge for

PL measurement and fitting due to the disordered nature of the NCs.

In crystalline and highly ordered systems, a PL measurement can be easily fitted to a mono-exponential function. A mono-exponential represents the decay of a sin- gle population, therefore any deviation from a mono-exponential behavior is usually attributed to the presence of multiple populations, with multiple decay rates. The dif- ferent rates can then be captured by a multi-exponential fit. However, as the number of exponentials in the fit increases, so does the number of degrees of freedom. With many degrees of freedom, any PL measurement can be fit, even if the underlining kinetics do not come from a multi-exponential solution.

Disordered systems can be thought of as being composed of many sub-populations, each with its own decay rate. Trying to fit a large number of exponentials to a disordered system results in an ill-conditioned fit, and the extracted decay rate cannot be trusted.

To solve this problem, we present here a disordered kinetics model based on the transfer matrix formalism[117 By using a random sample of exponentials out of a log-normal distribution, we reduced the number of fit parameters to only two: the mean and the variance of the decay rate. We then apply this model to investigate the thickness dependence of transfer efficiency.

84 ET PbS NCs /00 1100nm 785nm ,

612nm 1W

Figure 5-1: An illustration of the system and the various rates considered in the model

Methods

Figure 5-1 illustrates our system and the various rates that are present in the model.

We start by writing a single-population first order kinetics equation:

d[X ] = -kd[X] (5.1) dt

Where [X] is the concentration and kd is the decay rate. This can be extended to a group of populations: d[XJ (5.2) dt -K== [X]

Where now [X] is a vector of populations and K(j a matrix of transfer rates. If no transfer can occur between populations, Kd is a diagonal matrix, where the decay rate of each population is on the diagonal. To model a disordered system, we sample the decay rate from a log-normal distribution:

kdi = exp(-Af(E,, E,)) (5.3)

85 We follow here the energy Gaussian disorder model (EGDM), where .A denotes a Gaussian distribution, with E, the average barrier and E, the barrier standard vari- ation . It is important to clarify that we are generating a random set of numbers for each simulation. This results in run-to-run variation, therefore care should be taken to choose a large enough sample to ensure convergence. The resultant set of equation can then be propagated in time. We find that a simple forward-Euler[6] algorithm is sufficient:

[X = [X]t - Kd [X] At (5.4)

The next step is to couple different populations together. If the transfer is a first-order reaction, the coupling will appear on the off-diagonal of the matrix Kd. To model the energy transfer between NCs of different sizes, we use the Bell-Evans-Polanyi principle[13, 45]:

kNC-to-NC = koexp(-AE/kT) (5.5)

The means that energy will follow preferentially from small NCs (higher energy) to large NCs (lower energy). The distribution of NC energies is again taken to be the normal distribution.

Next, the energies are placed on a hexagonal close-packed grid, and Eq. 5.5 is used to calculate the coupling between the nearest neighbors (6 for a single layer of NCs, 9 for the bottom or top layer in a multi-layer film, 12 for a middle layer). Finally, the NCs can be coupled to a bulk OSC phase. For simplicity, we chose to model this as a simple unimolcular reaction, without any variance in the rate constants. Figure 5-2 shows the location of the non-zero elements of the final Kd matrix. As can be seen, this is a very sparse matrix, and therefore calculations using many populations are possible.

The last ingredient needed for the simulation is the initial condition. Here, we chose to uniformly populate all NCs at time t = 0. While an idealization of the pump process, we find that it works well.

86 U

Figure 5-2: A visualization of the shape of the transfer matrix. Each pixel represents a matrix element location. Light yellow pixels are zero entries, dark yellow pixel are the elements that model the 1st NC layer, Orange are the 2nd NC layer. The red pixels of the off-diagonal connect the two layers, and the brown pixel along the border are the connection between the NC and OSC

87 Results and Discussion

We used the above method to model several device structures. The first device modeled is a NC only monolayer (Figure 5-3). The main feature of note is the ability of the model to capture the "bendiness" (the degree to which the PL curve varies from linear) of a PL curve (the degree to which the PL curve varies from linear). This effect is usually accounted for using a multi-exponential fit. However, if there is no a-priori knowledge of multiple discrete populations present in the sample, it is hard to associate meaning with the fitting parameter. Furthermore, a tri-exponential fit is sometimes needed to fit very non-linear curves. With 5 degrees of freedom in a tri-exponential, a good-looking fit can be obtained that is simply an over-fit. In the present model, a very non-linear curve is fitted using only two parameter: the mean and variance of the decay constant.

Next, we look at an NC-OSC device with multiple layers. Figure 5-5 shows the steady state PL measurement as a function of NC layer thickness. As can be seen, up- conversion output increases slightly with the thickness, yet it is far from the expected increase. We conclude therefore, the the internal efficiency must have gone down. We noted earlier that this is a key barrier to improving the technology; without a thicker NC layer to absorb more light, the total external device efficiency remains low. We turn again to the model to explain the results. As can be seen in Fig. 5-4 the model fit the data very well. We note that the degree of disorder, defined as the variance in the rate distribution, is quite small energetically, on the order of 1kT - 2kT. As expected, both model and experiment show very little dependence of the PL on the number of monolayers. The interesting part comes in the measurement of the NC monolayer with a rubrene layer. The model is able to fit both curves with all fit parameters remaining constant. Fig. 5-6 summarize the flow of energy in the system. The added layer of NCs is too far from the rubrene interface to transfer, since transfer from NC to OSC is thought to occur via Dexter transfer[192. Yet the back-transfer occur through FRET, which has a longer transfer range and can transfer from rubrene back into the top as well as bottom layers. We arrive at the

88 1(2

- 10-'

1o-2 L 0 21 Time (Ps)

Figure 5-3: PL traces for 1 NC monolayers and the simulation, showing the "bendi- ness" of a disordered PL trace

conclusion that the presence of a second layer does more harm than good; while more photons are absorbed, the added loss from the increased back-transfer, together with the poor transfer from the bottom later to the rubrene, results in decreased performance. Using the model, we are also able to explain the non-parallelity in later time that we observed in our previous papers. By including the back-transfer from

the rubrene film to the NC film, a much slower process, we can easily fit the late time

deviation from parallelity. The model also seamlessly acounted for the increase in the

back-transfer (and the associated increase in non-parallelity) when going from 1 to

2 monolayers. Combining all of the above, we validated the rate extraction method

of our previous paper. By following the methodology described in [192, 129] on data

generated by the model, we can recover the NC-to-rubrene rate entered in the model to within %10.

89 101

100

10-1

10-2

10-3

10-1 0 1 2 3 4 5 Time (ps)

Figure 5-4: PL traces for 1 and 2 NC monolayers, with and without rubrene (dots), vs the kinetic model (lines)

Conclusions

In conclusion, we have presented here a kinetic model for disordered system. We than applied the model to a system of NC with OSC, and deduced the relevant rate constants. We have seen the the model predicts the decline in efficiency with increase of the NC layer thickness.

Possible avenues for improvement should therefore aim to increase the energy transfer from the NCs to the active OSC layer, while reducing the amount of back- transfer from the OSC to NC. One way to achieve this is by use of antennas; structures that readily accept and convey energy, such as dye aggregates or emissive bulk layers.

90 x10-4 5. 5L I ML] 12 ML 5

4.5F

4

3.5 1 0 U 3

2.51

2

1.5

0.5

flu 550 600 650 700 750 Wavelength (nm)

Figure 5-5: Steady state PL traces for 1 and 2 NC monolayers. 2 ML gives slightly more up-conversion output, but far from the expected doubling.

r! II

Figure 5-6: The flow of energy in the system. On the left, 1 ML of NCs can readily transfer to rubrene, while accepting little back. On the left, the bottom ML cannot transfer to rubrene well, but can still accept a similar amount back. The end result is an increase in loss for little gain.

91 92 6

WDA with exact condition: An acceleration scheme

While most of my thesis work has focused on energy transfer modling, a substitnaital part was devoted to improving density functional theory method. In this chapter, we look into methods of solving a set of non-linear equations arising from the sym- metrized weighted density approximation. The work in this chapter has been done in collaboration with Prof. Timo Thonhauser from Wake Forest University. This chapter will be submitted to Physics Review Letters.

Introduction

Kohn-Sham Density functional theory (KS-DFT)[93] has been remarkably success- ful at bringing electronic structure calculations to the masses.[18 By combining the scaling of mean-field methods with corrections from an approximate exchange- correlation functional, large-scale semi-quantitiave predictions have been made. To- day, DFT calculations are a cornerstone of many papers, both experimental and theoretical.[175, 120, 158]

Much of this success is due to the steady march of functional development. Fol- lowing the road map illustrated by the so-called AAIJJacobAA2s ladderAAi,[135 functionals have successively incorporated the density, then first derivatives of the

93 density[134, second derivatives[1681, exact Hartree-Fock exchange,{12] Moller-Plasset perturbation corrections[651, van der Waals corrections[181, 641, and more. The intro- duction of each piece further reduced the error in DFT, resulting in today's successful functionals.

Yet now, at the top of JacobA2s ladder, the path forward is no longer clear. Fur- thermore, modern functionals achieve their accuracy through a Frankenstein stitching together of the different parts. For example, the wB97X-V[116] functional calculates each part of the functional (the semi-local, range separated exact exchange, and non- local) separately, then combinse the result through a linear fit. The resulting func- tionals using this linear-fit method can give accurate energies, yet do so at the expense of good physics[118]. Clearly, a method relaying on exact conditions is required.

A potential avenue for improvement has been presented by Paul Ayers and co- workers based on the venerated Weighted Density Approximation (WDA).[31, 30 In it a single, non-local form for the XC-functional is used, seamlessly integrating the various pieces into one. This non-local form is exact for the true exchange-correlation hole. However, this form is unknown. Instead, this method enforces various exact conditions on an approximate form of the exchange-correlation hole. By enforcing these exact conditions, the results of the original WDA can be greatly improved.

The strength of the method lays in the direct enforcement of conditions based on the given density. However, enforcement comes at a cost: direct enforcement requires solving a large set of coupled non-liner equations. In practice, this severely limits the size of the molecules that can be calculated.

In this paper, we explore the structure of solutions to the coupled system of equations and find that they are remarkably predictable. We exploit this by building a Machine Learning (ML) k-nearest neighbors (kNN)[40 fit that replaces the costly solver. With the most expensive step removed, the method can be applied to a wider variety of molecules. Furthermore, other similar-in-form methods based on the WDA can use this acceleration scheme to speed-up calculations. Lastly, the structure of the solution shown here can lead to further development of novel functionals, not based on the WDA.

94 Theoretical Background

For the purpose of this paper, we will only discuss the exchange hole. However, extending the method to encompass correlation is fairly trivial and should follow the same outline. We will also assume closed-shell, spin-restricted (p, = p8) molecules to simplify the discussion. We refer the reader to the original work by Ayers and co-worker for a spin-unrestricted version of the equations[30.

The starting point is a formulation of the exact exchange functional in a nonlocal form:

Ex[p] = IIp(r) 2)hx(rir2) drdr2 (6.1) 2 ff fri - r2l

Where hx is the exchange hole (x-hole): the difference between the interacting and non-interacting densities. So far, this is a transfer of ignorance. We do not know the form of the x-hole, just like we donaAZt know the form of the exact functional.

However, following the traditional path of functional development, we can take the functional form of the x-hole from the homogeneous electron gas (HEG):

hx(ri, r 2 ) =

hx(kF ' r1 - r2f) (6.2)

-[jo(kF(r) ' ri - r2l) + j 2 (kF(r1) ' r, - r2 1)12

Where jo[z] and j 2 [z] are the zero-th and second spherical Bessel functions, and kF is the Fermi wave-vector: kF(r) = {/6-rp(r)4 (6.3)

The results from substituting this expression into Eq. 6.1 are surprisingly poor, even under-performing the similar Local Density Approximation (LDA). One reason for this poor performance is that the HEG x-hole, when applied to a non-uniform density, no longer obeys the sum rule for the x-hole:

p(rI)hx(ri,r2 )drI = -1 (6.4)

95 The idea behind the WDA is to use the Fermi wave vector that appears in the HEG x-hole as a variable, which we solve under the condition that the sum rule is obeyed.

In practice, this is achieved by solving a system of non-linear equations coming from a discrete integration grid:

Ngrid

wip(ri)f (kF(rj)lri - rj|) = -1 (6.5)

Now, kF is the solution to the system of equations. While the results are improved, they are still lacking. This is because the x-hole is formally symmetric under exchange of particles. In the form above, we have made a choice of direction, choosing kF at ri.

Where we symmetrize the Fermi wave-vector using a p-mean (we use p = 1, a simple average, in this work):

kF(rirj) kF(ri) + kF(rj)P)l/P (6.6) 2 the system of equations become coupled:

Ngrid

Swip(ri )h (kF (ri,rj)I -i- -1 (6.7) i=1

When solved, this symmetrized x-hole gives much better results - comparable to

GGAs, but without gradient information. The full details of the above outlined method can be seen in the work of Ayers and co-workers [31, 30]

Method

Eq. 6.7 enforces two conditions at once: symmetry and the sum rule. However, this couples all grid points together. Since the number of grid points is large, on the order of 2000 points per atom for the SG-1 grid, the system becomes challenging to solve, both from a computational effort perspective and a numerical stability perspective.

Furthermore, a non-linear solver needs a good initial guess. In our studies, we find that neither the LDA fermi wave-vector, nor the non-symmetrized WDA (Eq. 6.5) solution

96 provide an adequate guess. Therefore, to solve the coupled system of equations, first the uncoupled (Eq. 6.5) set is solved. We than employ a continuation scheme[2]:

Ngrid w p(r )(Ah,(kF(rj, r)fri - rj1) + (1 - A)h.,(kF(r9Iri - rj)) -1 V J (6-8)

When A equals 0, we get the uncoupled system. When A equals 1, we get the full, coupled system. We therefore solve at fixed A, and use the solution as an initial guess to the next, higher, A, until a A of 1 is reached. While greatly enhancing the numerical stability of the solver, the continuation scheme is quite slow, requiring about 5-10 lambda steps. However, with the acceleration scheme outlined below, this robust solver can be used to generate the data needed to explore the structure of the solution, and to train a machine-learning model to accelerate it.

Using the calculated wave-vectors, the density, and the gradient of the density, we train a k Nearest neighbor (kNN) using sk-learn{132. We split the data into a training set and a test set, roughly in half. Hyper-parameters of the model were optimized using a random search. The final model evaluates a point using the 50 nearest neighbors.

Computational Details

We use the graphical database (gDB)[47] of small molecules. This database contains all closed-shell molecules containing 11 or fewer heavy atoms, defined here as C, N, 0, or F. We then truncate this database at molecules with 3 or fewer heavy atoms.

For each molecule, a geometry optimization with PBE0/6-31g* was performed. The final density was then extracted, and the coupled system of equations was solved.

97 8

2 --

-20 00-

0 5 10 15 20 25 LDA kF

Figure 6-1: The initial Fermi wave-vector from the LDA vs the final calculated wave- vector from the WDA. Points are colored by the nearest heavy element, C,N,O, or F. Two points are circled that have very similar density, yet different solved kF- The corresponding gradient is marked in Fig. 6-3, allowing the model to distinguish between the two.

Results and discussion

Figure 6-1 shows the solved vs. initial Fermi-wave vector for several molecules. The atomic core densities can be clearly observed in this data, with a one-to-one mapping

(for a given element) of the solved wave-vector on the initial wave-vector. The bonding region, where the wave-vector (and therefore the density) is smaller, has a less clear correlation. Of note is a feature in the low density region arising from the hydrogen density.

Fig. 6-2 shows an enlarged version. Similarly, Figure 6-3 shows the solved wave-vector as a function of both density and gradient. Here, a one-to-one pattern emerge again. We also note that many features remain similar between different molecules. This makes the problem amenable to an interpolative fit. Figure 6.5 shows a predict-actual correlation plot of the kNN model. We note that in the core

98 4

3

-

1

0 2 34 LDA k,

Figure 6-2: A zoomed-in version of Fig. 6-1 showing the hydrogen feature. region (high Fermi wave-vector), the fit function performs very well, reproducing the solved wave vector almost exactly. Performance drops in the bonding region, but remains relatively high. Again, the hydrogen feature appears, which the model has some trouble capturing. At this point, there are two possible uses to the

Table 6.1: Error of the functionals relative to exact HF PBE WDA kNN MAE% 1.0% 5.6% 2.7% MSE% -0.1% 5.6% -0.03%

model. The first is as an accelerator. By providing a good initial guess, much of the numerical work is reduced. In practice, the kNN initial guess eliminated the need for a continuation scheme and reduced the number of iterations needed to solve the full coupled equations by an order of magnitude. Used in this way, novel forms for the xc- hole that would otherwise be prohibitively expensive to fully solve can be developed.

A second possibility is to skip the solver altogether and directly use the wave-vectors produced by the kNN. The results of using the model in this way are presented in

99 8 I

e* *- 6-6N 0

40 4-6 (9S * * -

29

-2 0-

3 102 10 104 Gradient

Figure 6-3: The reduced density gradient vs the final calculated wave-vector from the WDA. Points are colored by the nearest heavy element, C,N,O, or F. Two points are circled that have very similar density, yet different gradients, allowing the model to distinguish between the two. By matching both the density and the gradient , the solution becomes nearly one-to-one

100 7

6

5 * S

4

-o S. a, 3 S. U- -0 2

1 -6

0

-1 1 0 1 2 3 4 5 6 7 Solved kr

101 Table 6.1. While fortuitous cancellation of errors makes the KNN better than the solved WDA, and is even comparable to PBE, we caution that this is expected to be highly interpolative. Therefore, only molecules similar to the molecules in the training set should be employed. Nevertheless, for applications such as high-throughput design where even the accelerated solver is still too expensive, direct application of the kNN model may be beneficial.

Conclusion

We presented here a novel insight into the structure of the coupled system of non-linear equations that appear in the symmetrized WDA. Using this insight, we develop a kNN model that aims to either accelerate the solver for these equations or to completely replace it. The speed-up allowed by the model opens the door for a new avenue of DFT research based on the symmetrized WDA.

102 7

Optimal Tuning in Machine Learning

In the previous chapter, we explored the structure of the WDA solution, and applied machine learning techniques to accelerate the solver. In this chapter, we employ a neural network on a similar problem: accelerating the range seperation parameter tuning in optimal tuning functionals. The work in this chapter has been done with

Alex Kohn, my fellow group member. This work will be submitted to arxiv.

Introduction

In recent years, machine learning has risen in prominence as a useful tool for quantum chemistry problems. It has found applications in various domains: molecular rational design[61], construction of many-body force fields[1961, augmentation of electronic structure methods[140], and more[601. For rational design in particular, great ad- vances have been made in the past few years in transforming molecular graphs into a fixed-length vector that can then be readily used for high-throughput screening[58.

This has been demonstrated recently in accelerated explorations of molecular space for novel organic light-emitting diodes [61] and solar cells [1791.

For many molecular applications, excited state properties are critical for device performance. The most commonly-used excited state method is time-dependent den- sity functional theory (TDDFT) [191. However, TDDFT generally has accuracy of around 0.3 eV [811, in addition to critical failure modes involving charge-transfer

103 states [381. These can be partially remedied through the use of optimally-tuned range-separated functionals (OTRS) [1411. OTRS functionals are costly because they

require the performance of many successive density-functional calculations to com- pute the range-separation parameter. This multiplies the cost usually by around 10-30 times, making it unfeasible for high-throughput screening or dynamics calculations.

Herein, we present a method for computing an approximate range-separation pa- rameter using a neural-network fingerprint. Because optimal tuning is completely ab

initio method, we can compute these parameters for a large set of molecules, enabling

us to effectively train a neural network. We show that we can recover OTRS-quality

results at a fraction of the cost, allowing high-throughput rational design to use opti-

mal tuning. Additionally, we discuss the distributions of range-separation parameters for the datasets we use and suggest ways of exploiting and interpreting their statistics.

Methods

We employed the neural fingerprinting method described in Duvenaud et. al. [411

on a database of 4000 molecules taken from two sources: the 6-element members of

GDB-11 [46] and 2000 randomly sampled molecules from the Harvard Clean Energy

Project [I. We divide our data set into halves, with one half used as the training set

and the other half used as the test set. We optimized hyperparameters the same way

as Duvenaud et. al. [411, and use a single-layer network to convert the fingerprint

into a range-separation value.

We used RDKit to construct xyz coordinates from the SMILES strings [1011.

Following Refaely-Abramson et. al., we then computed the range-separation values

[1411. One important thing to note about this methodology is that it employs no

input from experiment and therefore can be extended to an arbitrarily large data set.

104 800

0 S 700

600

* .* 0 500 0 0

0 s 400

0 0 0. 300 *0

*, 0 0

200

100

0 0 0 100 200 300 400 500 600 700 800 predicted w (mbohr 1)

Figure 7-1: Plot comparing the predicted w values with the exact ones. We reproduce 1 the true optimal values with a mean absolute error of 0.015 bohr- .

105 0.030 - CEP dataset - GDB dataset 0.025 -

0.020 -

0.015

0.010 -

0.005 -

0.0001 0 100 200 300 400 500 600 700 800 w (mbohr 1)

Figure 7-2: The distribution of true omega values for the two databases employed in this study. We find that they have little overlap, demonstrating how each one only samples a limited region of w space.

Results and Discussion

We show how our model performs in Figure 7-1. We predict the optimal tuning parameters for species in our test set with a mean absolute error of 0.015 bohr'.

The distributions of the range-separation values of the two databases we use are given in Figure 7-2. We find that the distributions are very narrow and well- separated, likely due to the disjoint distributions of molecular sizes, as the optimal range-separation parameter is quite sensitive to the extent of the electronic system

[141]. This is a plausible explanation for the success of our model. However, this also suggests that a viable strategy for optimal tuning within similar high-thoughput screening databases is simply to use the average of a small sample of species, as the intra-database variation is quite small. One interesting critique that emerges here is that although these databases are supposed to span a large chemical space, from a range-separation perspective the databases are quite homogeneous.

To discuss the correlation between the molecular size and the optimal range-

106 9001 CEP dataset 800 - * eoGDB dataset-

700 -- * 0 600 * * *

500 00

400

300 .*

200 -

100 -

01 0.5 1.0 1.5 2.0 2.5 Molecular Radius (nm)

Figure 7-3: The distribution of true omega values for the two databases employed in this study, plotted versus the molecule's characteristic radius. separation value, we plot in Figure 7-3 the relationship between the characteristic radius of a molecule (the standard deviation of the distance between atomic positions and the mean atomic position) and its OTRS value. Refaely-Abramson and coworkers examined how the optimal w value is strongly determined by the molecule's charac- teristic radius, which led us to suspect that our neural net was only picking up on that aspect of the molecule. However, upon examination of our databases, we find little correlation between molecular radius and the optimal w value, despite our databases spanning a very large range of characteristic radii. This suggests that our neural network is picking up on something more subtle than the molecular radius, as that would not be a useful metric.

Conclusions

We have shown that the cost of optimal tuning can be greatly reduced through the use of a neural network to provide an approximately optimal range-separation value.

This can greatly accelerate high-throughput screening efforts that need optimally-

107 tuned quality results. We also suggest the use of other simple models to capture the variation in the optimal range-separation value without the full optimization procedure. Future work can focus on extending these simple models by looking at molecular features that are strongly correlated with the optimal range-separation parameters or on the usage of other optimal tuning schemes.

108 8

Conclusion

In this thesis we have developed a model for energy transfer between organic semi- conductors and noncrystalline semiconductors. At this point, it is worth reflecting on the generality of each part of the model.

For the morphology studies, all the action occurs at the organic ligands. Two main characteristics are important for the model presented in this thesis to be applicable to a material system: saturation and density. All the organic ligands studied here were fully saturated, allowing for the the statistics of the saturated carbon-carbon bond to dominate. The second is surface density. In a VI-II system, like the PbS and CdSe in the model, the ligands are quite spaced apart, since only the metal atoms tend to bond with the ligand head group. This reduced surface density compared to metallic nanocrystals, such as gold and silver, results in a weaker ligand-ligand interaction and therefore a reduction in the correlation of the ligand motion. Combined, we expect most systems the exhibit these properties to follow the trend seen.

Next, the electronic structure presented here is also likely to be common for VI-I systems. Since the main transfer mechanism we observe is Dexter transfer, with its characteristic exponential decay with distance, morphology dominates the coupling.

More recent work in our group on surface bound ligands have shown that the electronic coupling trends of CdSe and PbS are similar. Furthermore, simulation of the surface coverage dependence of the electronic structure have shown a remarkable robustness.

Even as the NC is stripped of its passivating ligands, the electronic structure remains

109 largely the same. While trap states do appear in the band gap, the do not menage to close is completely, even in the case of the small band gap of PbS.

Finally, the kinetics model for disorder system is expected to be widely applicable in any system the exhibits Gaussian Energy Disorder. As the assumptions that went into the making of the model were broad, the conclusion coming from the model also should be broad. While it lacks the atomistic detail of the other models presented in this work, it can nevertheless provide useful insight into macro-structure consid- erations, when making photon conversion devices. Since it is a general framework, other systems exhibiting similar transfer and disorder characteristics could be model through similar means.

While we have greatly expanded our understanding, there are still remaining ques- tions. Foremost, while we have explored the statistically average dot morphology, the question of distance remains: how close can an organic molecule get to a QD layer?

We saw that morphology dominates the transfer, therefore answering this question is the most obvious next step. Further more, in chapter 3 we saw that there appears to be a limit to the transfer rate. Can we overcome this limit? Is the dielectric modu- lation explanation the correct one? Finally, our collaborators would like to directly transfer the energy to a bulk semi-conductor like silicon, but are currently unsuccess- ful. What is it about QDs that enables the transfer to occur? Is it the nature of the defect states? All of these questions are currently being looked at in our groupr, and it is left to my successors to answer them.

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