Manipulating the Phonon Bottleneck in Graphene Quantum Dots

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Manipulating the Phonon Bottleneck in Graphene Quantum Dots: Phonon-Induced Carrier Relaxation within Linear Response Theory Jonathan Paul Trinastic, Iek-Heng Chu, and Hai-Ping Cheng J. Phys. Chem. C, Just Accepted Manuscript • DOI: 10.1021/acs.jpcc.5b06885 • Publication Date (Web): 08 Sep 2015 Downloaded from http://pubs.acs.org on September 15, 2015

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The Journal of Physical Chemistry

Manipulating the Phonon Bottleneck in Graphene Quantum Dots: Phonon-Induced Carrier Relaxation within Linear Response Theory Jonathan P. Trinastic, Iek-Heng Chu, and Hai-Ping Cheng∗ Department of Physics and Quantum Theory Project, University of Florida, 2001 Museum Road, Gainesville, FL, 32611 USA E-mail: [email protected] Phone: (1)352 3926256

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Abstract Slow phonon-induced relaxation of excited carriers in quantum dots is crucial for next-generation photovoltaics, however the phenomenon has been difficult to realize experimentally due to defects and ligands that introduce strongly coupled states. Graphene quantum dots are an intriguing system to avoid these problems due to chemical synthesis methods that minimize defects. Modeling electron-phonon interactions using a state-of-the-art method combining the reduced density matrix formalism with linear response theory, we find that 100 picosecond lifetimes are possible for electronhole pairs in graphene quantum dots due to large transition energies and weak coupling to excited states near the band edge. By calculating carrier relaxation rates in dots with and without ligands, with armchair or zigzag edges, and of increasing size, we show how excited state lifetimes are sensitive to structural changes. In contrast to other types of quantum dots in which ligands can increase phonon-induced relaxation, carbon ligands on graphene quantum dots extend lifetimes by nonadiabatically decoupling excited states. Changing carbon edge termination type between armchair and zigzag patterns increases and decreases the electron-phonon coupling, respectively, and geometrically symmetric dots significantly increase nonadiabatic electronic coupling. These results provide guidance for experimental routes to control excited state lifetimes.

1. Introduction In optoelectronics, a photoexcited electron or hole (carrier) initially transitions from highto low-energy excited states due to energy loss to lattice phonons, known as phonon-induced relaxation. This relaxation occurs quickly in semiconductors due to efficient, single-phonon processes that relax carriers to the band edge within a picosecond, contributing to the efficiency limit of photovoltaics (PVs). 1 Slowing phonon-induced relaxation and increasing carrier lifetimes in light-absorbing materials will improve open-circuit voltage in PVs, 2 charge separation in organic PVs, 3 and charge transfer in photodetection devices. 4

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Semiconductor quantum dots (QDs) provide a route to extend excited carrier lifetimes by taking advantage of discrete transition energies between excited states arising from electronic confinement. Energy must be conserved during phonon-induced relaxation such that the total energy of coupled phonons matches the transition energies in QDs, around 0.1 eV. 5 This transition energy often mismatches available phonon energies, typically tens of meV, 6 so that multiple phonons must facilitate the transition. This mismatch leads to inefficient carrier relaxation that should create long excited state lifetimes in QDs, known as the phonon bottleneck. Electronic coupling between involved states must also be weak, which depends on the wave functions and the momentum of phonon modes inducing the transition. 7 Understanding how to engineer phonon bottlenecks will improve optoelectronic devices, 2,3 however the promise of long excited carrier lifetimes beyond tens of picoseconds (ps) has not been realized experimentally in CdSe or PbS QDs due to Auger processes, 8 structural defects, and surface ligands that introduce additional phonon modes. 9–11 Defects introduce additional states that reduce transition energies and increase electron-phonon coupling, leading to shortened carrier lifetimes in the picosecond range. 9 Experiments suggest that a bottleneck can be achieved by eliminating these factors, 5 however consistent observation of long lifetimes in QDs is still elusive. Thus, theoretical modeling of phonon-induced relaxation in QDs is a priority to identify materials with a bottleneck and determine how structural changes control its timescale to improve optoelectronic performance. Graphene quantum dots (GQDs) are a promising material to achieve long carrier lifetimes because they can be made using solution-based methods that allow for nanoscale structural control to reduce defects and minimize impurity states. 12 Such structural precision makes GQDs a favorable material in which to observe relaxation bottlenecks and to understand how various geometries affect electron-phonon coupling. Insight into phononinduced relaxation in GQDs will be beneficial for their many optoelectronic applications in PVs, 13,14 photocatalysis, 15,16 light-emitting diodes, 17 and sensing. 18 Although previous work has shown that electronic relaxation in periodic graphene layers occur via disorder-assisted

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supercollisions at higher temperatures, 19,20 standard phonon-induced processes should be the dominant relaxation mechanism in GQDs with large band gaps. The two-dimensional nature of graphene and its unique electronic dispersion lead to increased carrier interactions and a dielectric constant of one when free-standing. 21 This limited screening makes it imperative to rigorously describe excited states that take into account interactions between electron-hole pairs in GQDs. The reduced density matrix (RDM) 22–24 and fewest switches surface hopping (FSSH) 25,26 methods have been applied extensively to study phonon-assisted relaxation between Kohn-Sham states within density functional theory (DFT), however this independent-particle picture does not accurately describe excited state properties. A FSSH method has been developed within linear-response time-dependent density functional theory, 27 which accounts for electron-hole pair interactions, however it is computationally unfeasible beyond tens of atoms. Thus, it is an open problem to study phonon-induced relaxation with rigorously treated excited states in large systems. In this paper, we introduce an ab initio method combining the RDM formalism with linear response theory to calculate phonon-induced carrier relaxation in systems containing hundreds of atoms. We apply this method to study relaxation dynamics in GQDs of varying shape and size and find that it is necessary to go beyond an independent-particle picture and use linear response theory to correctly describe excited states. Our results attribute a long, 100 picosecond (ps) decay seen in transient absorption (TA) measurements 28 to phononinduced carrier relaxation times, indicating the potential for a phonon bottleneck in these systems. To understand how geometrical changes influence this bottleneck, we investigate how relaxation times change with the addition of carbon-chain ligands or by changing the termination pattern of the GQD edges between armchair and zigzag types. To the best of our knowledge, these are the first ab initio findings that indicate the possibility of long excited state lifetimes in GQDs and illustrate the possible methods to tune them for various applications. The paper is organized as follows. Section 2 describes the graphene quantum dot struc-

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tures studied (2.1) as well as the theoretical and computational methods used to describe electronic structure (2.2), absorption spectra (2.3), and relaxation dynamics (2.4). Electronic excitations for various GQD geometries and sizes are discussed in Section 3.1 and calculations of phonon-induced relaxation dynamics are reported in Section 3.2. General conclusion are given in Section 4.

2. Theoretical and Computational Details 2.1. Graphene Quantum Dot Structures Examples of the GQD structures investigated are shown in Figure 1. We have calculated excitation energies and excited state relaxation dynamics for i) the 132-carbon-atom GQD with carbon-chain ligands (C132-L) that has been studied experimentally, 28 ii) the same GQD without carbon chains (C132) to determine the ligands’ effect on relaxation rates (Figure 1(a)), iii) GQDs with armchair (AC) or zigzag termination (ZZ) edges (Figure 1(b)), and iv) GQDs of varying size from 42-222 carbon atoms (Figure 1(c)). This systematic examination will allow us to compare directly to experiment as well as make predictions about how specific geometric properties affect phonon-induced relaxation dynamics in these systems. Throughout the rest of the paper, GQDs will be labeled by their edge type (AC or ZZ) and number of carbon atoms (e.g., ZZ216).

2.2. Electronic Structure Electronic excitations in GQDs were obtained by solving for the twenty lowest eigenvalues of the Casida equation 29 within linear-response, time-dependent density functional theory (LR-TDDFT). 30 All calculations were completed using the Gaussian09 package, 31 employing the B3LYP hybrid exchange-correlation functional within the adiabatic approximation and the 6-31G(d) Gaussian basis set, which have been shown to provide accurate excitations for conjugated molecules 32 and graphene-based systems. 33 Ground and excited state 5



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Figure 1: a) 132-carbon-atom GQD with (C132-L) and without (C132) carbon-chain ligands used in experiment to prevent dot aggregation. b) GQDs with carbon atom edge termination following an armchair (AC) or zigzag (ZZ) pattern. c) GQDs of increasing size with diameters ranging from 1 to 3 nanometers. Medium-sized gray spheres are graphene C atoms, mediumsized blue spheres are ligand C atoms, and small pink spheres are H atoms. configurations were relaxed such that all atomic forces are less than 0.01 eV/Å. The exact exchange contribution included in the hybrid functional gives the correct asymptotic behavior of the exchange potential that is necessary to produce long-range attraction between electron-hole pairs and reproduce the experimental absorption spectrum. We have compared excitation energies computed with the B3LYP functional with those using the range-separated CAM-B3LYP functional, which may provide more accurate results when excitations demonstrate significant charge transfer. However, we have found that the B3LYP functional accurately describes the major features of the experimental absorption spectra 28 due to the lack of charge transfer character in GQD excitations (see Section 3.1).

2.3. Absorption Spectra The zero-Kelvin absorption spectrum, A0K (ω), for each GQD has been calculated by weightexc ing a delta function at each LR-TDDFT excitation energy, E0α =h ¯ ω0α , by its corresponding

oscillator strength, f0α : A0K (ω) =

X

f0α δ(¯hω − h ¯ ω0α ),

(1)

α

where f0α =

2me ω0α |hΨ0 |r|Ψα i|2 , 3¯ h

|Ψ0 i and |Ψα i are the ground and excited state TDDFT 6


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wave functions, me is the electron mass, and hΨ0 |r|Ψα i is the transition dipole moment. The delta function has been modeled using a Gaussian function with 80 meV broadening. Forbidden transitions with near-zero oscillator strengths at 0K , known as dark states, can be activated by thermal motion at room temperature. To better compare to experimental absorption spectra, we have modeled these finite temperature effects for the C132-L GQD by calculating ω0α and f0α over multiple time steps of a 300K ab initio molecular dynamics (AIMD) simulation. The thermally averaged absorption spectra (AT (ω)) becomes N 1 XX n n A (ω) = f0α δ(¯hω − h ¯ ω0α ), N n=1 α T

(2)

n n where N is the total number of time steps in the AIMD trajectory, and ω0α and f0α represent

excitations and oscillator strengths at the nth time step. The thermally averaged absorption spectrum has been used for all results in Section 3. Finally, Equation (1) does not account for vibrational state overlap that could impact the absorption spectra. To do so, we have first included the nuclear wave functions in the calculation of transition dipole matrix elements:

AF C (ω) =

X 2me ω0α X α

3¯h

fi |hφi |hΨ0 |r|Ψα i|φj i|2 δ(¯hω − h ¯ ω0α − ωij ),

(3)

ij

where |φi i and |φj i are the nuclear wave functions of the initial and final states, respectively, fi is the initial nuclear thermal distribution, and h ¯ ωij is the phonon energy. Invoking the Condon approximation, which states that optical absorption occurs on a time scale much faster than nuclear motion, the dipole matrix element does not depend on the nuclear wave functions and the absorption spectrum becomes

AF C (ω) =

X α

f0α

X

fi |hφi |φj i|2 δ(¯hω − h ¯ ω0α − ωij ) =

X

f0α FABS (ω),

(4)

α

ij

where FABS (ω) is known as the Franck-Condon-weighted density of states (FC-DOS). The 7



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inclusion of the vibrational overlap leads to smaller peaks in the absorption spectra at energies higher than the zero-phonon line (ZPL) located at each excitation energy. Known as phonon sidebands (SB), these peaks represent light absorption that excites the system to higher vibrational states in a given electronic state.

2.4. Phonon-Induced Relaxation Dynamics 2.4.1. Reduced density matrix formalism To calculate phonon-induced transition rates between electronic excitations, we have employed the RDM formalism that leads to the famous Redfield equations. 22,34 The formalism provides a computationally efficient method for calculating dissipative dynamics of a system embedded in a macroscopic environment. Here, we have considered electrons to be the system of interest coupled to phonons from the surrounding lattice acting as the environment or reservoir. Using the RDM formalism allows for our method to study both population transfer between electronic eigenstates as well as their decoherence due to electron-phonon interactions, although the current study only focuses on the former to study phonon-induced carrier relaxation. If the system and the reservoir make up a closed system, then we can write the total ˆ S , reservoir component, H ˆ R , and the Hamiltonian as a sum of the system component, H ˆ tot = H ˆS + H ˆ R + Vˆ . The system and bath system-reservoir coupling component, Vˆ : H ˆ S |Ψα i = Eα |Ψα i and can each be described by their respective eigenvalue equations: H ˆ R |φi i = Ei |φi i, where |Ψα i and Eα correspond to the system eigenstates and eigenvalues, H and |φi i and Ei correspond to the reservoir eigenstates and eigenvalues. ˆ tot can be expressed as ρˆ(t) = P wn |Φn ihΦn |, The density operator corresponding to H n

where |Φn i is a pure state of the closed, combined system and wn is the corresponding probability. However, we are only concerned with the system degrees of freedom, so we define a reduced density operator (RDO), σ ˆ (t), by taking the trace over the reservoir space: P ρ(t)|φi (t)i. σ ˆ (t) = T rR [ˆ ρ(t)] = i hφi (t)|ˆ 8


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The RDO only depends on the system’s degrees of freedom and contains information about its dynamics due to coupling to the reservoir. Assuming both a Markovian bath and that V is small enough such that the reservoir does not change over time, the Redfield equations describe system dynamics as a function of time (written in the interaction representation): I dσαβ (t) X I = σµν (t)Rαβ,µν ei(ωαβ −ωµν )t , dt µν

(5)

where h ¯ ωαβ = Eα − Eβ is the transition energy between eigenstates. Rαβ,µν is a tetradic matrix that describes the rates of change of the elements of σ I (t). Invoking the secular approximation reduces the number of elements of Rαβ,µν we have calculated to investigate relaxation dynamics between excited states. 22 The righthand side of Equation (5) depends on an exponential term that oscillates rapidly. Over a long time average, the righthand side will vanish unless ωαβ − ωµν = 0. Imposing this constraint, the secular approximation reduces Equation (5) to two cases: 1) α = β, µ = ν, corresponding to population transfer between eigenstates, and 2) α 6= β, α = µ, β = ν, corresponding to coherence dephasing that describes phase differences between eigenstates. We can now write the Redfield equation for each of the two cases: 1) Population relaxation: I X dσαα (t) I =− Rαα,µµ σµµ (t) dt µ

Rαα,µµ = δαµ

X

kακ − kµα

(6)

(7)

κ

2) Coherence dephasing: I dσαβ (t) I = −Rαβ,αβ σαβ (t) dt X1 Rαβ,αβ = (kαµ + kβµ ) + γ0 , 2 µ

9


(8) (9)


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where γ0 is a pure dephasing constant. 35 The rates kαβ can be shown to be 22,36

kαβ =

E 2π D X |hφi Ψα |V |φj Ψβ i|2 δ(¯hωαβ − h ¯ ωij ) , h ¯ j

(10)

D E P P where ... = i fi = e−βEi /( i e−βEi ), the thermal average over initial conditions. The

sum over final reservoir states preserves conservation of energy. The principle of detailed balance ensures proper behavior as time goes to infinity and provides the following relationship between forward and backward transition rates kαβ = e¯hωαβ /kB T kβα . In the present study, we have only solved Equation (6) to calculate population transfer between LR-TDDFT eigenstates. 2.4.2. Nonadiabatic system-reservoir coupling Phonon-induced transitions between adiabatic eigenstates can be described using Equations (6) and (7) by modeling V as a nonadiabatic coupling peturbation defined by the nuclear P h2 ¯ ∇2k |φj Ψβ i, where Mk is kinetic energy operator: hφi Ψα |V |φj Ψβ i = Vαβ,ij = hφi Ψα | k − 2M k 22 the nuclear mass and k runs over the K nuclear degrees of freedom, |φi i = |φ1i i|φ2i i...|φK i i.

If we only include terms to first order in the nuclear gradient operator, previous work has shown that the system-reservoir coupling can be rewritten to good approximation as 7

Vαβ,ij ≈

X k

−

Y i¯h hΨα |∇k |Ψβ ihˆ pi hφki |φkj i, Mk k

(11)

where hˆ pi = −i¯hhφi |∇k |φi i is the expectation value of the nuclear momentum operator and follows classical equations of motion according to Ehrenfest’s theorem. Using this expression to represent the matrix elements of the system-reservoir coupling, the relaxation rate from Equation (10) becomes

kαβ =

E XY 2π D X i¯h |hφki |φkj i|2 δ(¯hωαβ − h ¯ ωij ) , | − hΨα |∇k |Ψβ ihˆ pi|2 h ¯ Mk j k k 10


(12)

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where the electronic nonadiabatic coupling, Vαβ =

P

k

− Mi¯hk hΨα |∇k |Ψβ ihˆ pi, describes how

atomic motion induces transitions between excited states. Due to the order of magnitude difference in electron and nuclear relaxation times, we apply a decorrelation assumption to separate the thermal averaging of the electronic and nuclear components: 7,37

kαβ =

EX Y 2π D fi |Vαβ |2 |hφki |φkj i|2 δ(¯hωαβ − h ¯ ωij ), h ¯ i,j k

(13)

To examine the excited population as a function of time, we consider a system bathed in light with frequency ω at t < 0. After the light is switched off at t = 0, a nonequilibrium carrier distribution will relax due to interactions with phonons as described by Equations P (6) and (13). The population change in σαβ is then defined as ∆η(E, t) = α σαα (t)δ(Eα −

E) − ηeq , where Eα is the energy of the αth excitation and ηeq is the equilibrium population. Thus, calculation of the transition rates between excitations reduces to the calculation of two major quantities: 1) the thermally averaged square of the electronic nonadiabatic E D 2 coupling, |Vαβ | , and 2) the Franck-Condon-weighted phonon density of states (FC-DOS), Q P ¯ ωji ). F (ωαβ ) = i,j fi k |hφki |φkj i|2 δ(¯hωβα − h 2.4.3. Electronic nonadiabatic coupling within TDDFT We have calculated Vαβ using the auxiliary many-body wave function originally proposed

by Casida 38 and Tavernelli 39 to describe excited state wave functions within LR-TDDFT. In this case, wave functions are expressed as linear combinations of singly excited Slater determinants of Kohn-Sham orbitals, determined from the solution of the Casida equation:

|Ψα i =

X

(Xαma + Yαma )ˆ a†a a ˆm |Ψ0 i =

ma

X

(Xαma + Yαma )|Ψm−>a i,

(14)

ma

where m and a index occupied and unoccupied Kohn-Sham orbitals, respectively, a ˆ†a and a ˆm are creation and annihilation operators, Xαma and Yαma are the normalized coefficients corresponding to an excitation from the mth to ath orbital and de-excitation from the ath 11



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to mth orbital, respectively, |Ψ0 i is the ground state Slater determinant, and |Ψm−>a i is a singly excited Slater determinant. This auxiliary wave function has been shown to give the exact first-order nonadiabatic coupling between the ground state and any excited state 39 and is an accurate approximation to the first-order coupling between excited states (exact within the Tamm-Dancoff approximation). 40 We have calculated Vαβ matrix elements by converting the nuclear gradient operator to a time derivative and using the finite difference method between adjacent timesteps in an AIMD simulation:

Vαβ = −i¯hhΨα |

∂ i¯h |Ψβ i ≈ − [hΨα (t)|Ψβ (t + ∆t)i − hΨα (t + ∆t)|Ψβ (t)i], ∂t 2∆t

(15)

where ∆t is the time step of the simulation. At each time step, we decompose the manybody wave function overlap from Equation (15) as a sum over singly excited Slater deterP P minant overlaps: hΨα (t)|Ψβ (t + ∆t)i = ma nb (Xαma (t) + Yαma (t))(Xβnb (t + ∆t) + Yβnb (t +

∆t))hΨm−>a (t)|Ψn−>b (t+∆t)i. The singly excited Slater determinant overlaps can be decomposed using Slater-Condon rules and calculated using the overlaps between single-particle Kohn-Sham orbitals making up each determinant. These overlaps are nonzero when either the determinants are the same or differ by only one single-particle orbital. Thus, the nonadiabatic coupling matrix elements (Vαβ ) are found by weighting the Kohn-Sham orbital overlaps

by the corresponding LR-TDDFT excitation coefficients (Xαma and Yαma ) and summing over all single-particle transitions that make up each many-body excitation. The thermally averaged electronic nonadiabatic coupling that enters into Equation (13) has been calculated by averaging the value of Vαβ calculated between adjacent time steps E D P 2 across a 300K AIMD trajectory consisting of N time steps: |Vαβ |2 = N1 N i=1 |Vαβ | . The

AIMD dynamics were performed using the PBE functional to produce input structures for calculating the nonadiabatic coupling matrix elements (Vαβ ) and room-temperature absorption spectra. Structures were initially relaxed at 0K and brought to 300K slowly through

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velocity rescaling. After equilibrating at 300K using the canonical NVT ensemble, a 1 ps simulation was performed within the NVE ensemble with a 0.5 femtosecond (fs) time step. Structures from the AIMD simulation were then used as input to calculate LR-TDDFT excitations and nonadiabatic coupling using the B3LYP functional. The PBE functional allows for the computationally efficient calculation of AIMD trajectories while accurately capturing the phonon modes of the systems. Figures 2(a)-(b) plot the temperature as a function of time during the AIMD trajectory for the C132 and C132-L GQDs as well as the thermally averaged, squared nonadiabatic coupling for two sample transitions in the C132-L GQD. Temperatures typically oscillate within 15-20% of room temperature and the nonadiabatic coupling converges within 150 fs, exemplifying the computational efficiency of the current method. 2.4.4. Franck-Condon-weighted density of states The final component to calculate phonon-induced transitions rates between LR-TDDFT eigenstates is the Franck-Condon-weighted density of states (F (ωαβ )). Assuming a quantum harmonic oscillator phonon bath and that the ground state normal modes are an accurate approximation for all excited state modes, F (ωαβ ) describes the density of phonon states coupled to a given nonadiabatic transition: 22 1 F (ωαβ ) = 2π¯h where G(t) =

P

k

Z

∞

dteiωαβ t eG(t) ,

(16)

−∞

k k Sαβ [(e−iωk t −1)(n(ωk )+1)+(eiωk t −1)n(ωk )]. Sαβ is the Huang-Rhys factor,

a dimensionless quantity representing the coupling strength of the kth normal mode to the transition, and n(ωk ) is the Bose-Einstein distribution for a phonon with energy h ¯ ωk at 300K. The function F (ω) peaks at energies corresponding to modes with nonzero Sαβ . F (ωαβ ) can be expressed using the nuclear wave function overlap and is connected to decoherence and the quantum Zeno effect. 8

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Figure 2: a) Temperature as a function of time for the ab initio molecular dynamics simulation for the C132-L and C132 GQDs. b) Thermal average of the squared nonadiabatic coupling (|Vαβ |2 ) for two sample transitions as a function of time for the C132-L GQD. For example, S1 -S0 corresponds to the transition between the first excited state and ground state. k To calculate Sαβ , we relaxed the atomic positions for all excited states and projected the

change in atomic configuration between two states α and β onto the normal modes of the k system. The Huang-Rhys factor is then calculated as Sαβ = (µk ωk d2kαβ )/¯h, where µk is the

modal reduced mass and dkαβ is the configurational difference between excitations α and k β projected onto the kth mode eigenvector. These values of Sαβ enter Equation 16 along

with phonon modes (ωk ) calculated within the harmonic approximation using the B3LYP functional. In summary, the calculation of phonon-assisted relaxation between excitations requires three ingredients: 1) the reduced density matrix constructed from the eigenstates of the

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Casida equation within LR-TDDFT; 2) thermally averaged nonadiabatic couplings Vαβ calculated along an AIMD trajectory within the LR-TDDFT framework; and 3) the FranckCondon-weighted density of states (FC-DOS) calculated within the harmonic approximation and requiring the ground state normal modes and excited state configurations of each system.

3. Results and Discussion 3.1. Electronic excitations and absorption spectra We first systematically describe the excitations in C132-L, C132, AC, and ZZ GQDs, paying particular attention to how edge type affects the excitation spacing that will be important when interpreting relaxation dynamics. When the transition energy between two excitations is greater than the characteristic phonon mode energy coupled to that transition, more than one phonon is required for the carrier to nonradiatively relax to the lower state, which can increase relaxation times. Thus, understanding how edge type, size, and ligands affect energy spacing is a key ingredient to determining and tailoring relaxation dynamics. 3.1.1. C132-L and C132 GQDs Table 1 lists the ten lowest excitation energies and corresponding oscillator strengths at 0K for the C132 and C132-L GQDs and also compares results employing the B3LYP and CAMB3LYP functionals. Using the B3LYP functional, the C132-L GQD demonstrates a lowest exc exc excited state at E0,1 = 1.918 eV (λexc 0,1 = 646 nm) and strong absorption peak at E0,3 =

2.274 eV (λexc 0,3 = 545 nm), both of which match well with the experimental fluorescence peak near 660-670 nm and absorption peak arond 540-560 nm. 28 A large, 0.30 eV transition energy exists between S3 and S2 that opens the possibility for a phonon bottleneck if the electronic coupling (Vαβ ) between these states is small, which we explore in Section 3.2.1. A continuum of excited states exists above 2.7 eV, at which point transition energies between adjacent states are all less than 0.05 eV. 15



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The C132 and C132-L GQDs have nearly identical excitation energies, indicating that the ligands used experimentally do not introduce defect states that could decrease carrier lifetimes, an important requirement for observing a relaxation bottleneck. The similarity in excitation energies is likely due to the fact that the ligands do not introduce any dangling bonds nor do they lead to substantial geometric changes in the minimum-energy configuration of the dot. exc Table 1: Comparison of the lowest ten vertical excitation energies (E0α ) and oscillator strengths (f0α ) at 0 K for the C132 and C132-L GQDs using either the B3LYP or range-separated CAM-B3LYP functional (C132-L only). Experimental excitation energies for the absorption onset and peak are included in parentheses. 28

Excitation S0 → S1 S0 → S2 S0 → S3 S0 → S4 S0 → S5 S0 → S6 S0 → S7 S0 → S8 S0 → S9 S0 → S10

C132 (B3LYP) exc E0α (eV) 1.920 (1.879) 1.981 2.283 (2.210) 2.340 2.344 2.460 2.515 2.567 2.655 2.657

f0α 0.000 0.033 2.487 0.049 0.850 0.005 0.001 0.004 0.001 0.001

C132-L (B3LYP) exc E0α (eV) 1.918 (1.879) 1.978 2.274 (2.210) 2.340 2.344 2.455 2.512 2.560 2.654 2.655

f0α 0.000 0.043 2.622 0.186 0.930 0.009 0.001 0.006 0.001 0.001

C132-L (CAM-B3LYP) exc E0α (eV) 2.307 (1.879) 2.420 2.817 2.832 (2.210) 2.975 2.981 3.243 3.291 3.329 3.367

f0α 0.000 0.046 0.000 4.213 0.186 2.305 0.002 0.001 0.002 0.007

In contrast to the B3LYP functional, employing the CAM-B3LYP functional overestimates the experimental absorption peak and lowest excited state by 0.6 and 0.4 eV, respectively. The B3LYP functional typically provides accurate results for excitations without significant charge transfer character, and we confirm that this is the case for these GQDs by plotting the charge difference isosurfaces (CDI) for several sample excitations in the left column of Figure 3. Both electron (yellow) and hole (blue) charge densities for each excitation spread equally across the dot without localized regions of electron or hole density alone that would increase the charge transfer character. This spatial quality of the excited state charge densities supports the fact that the B3LYP functional should describe excitation energies well. The first three excitations all exhibit charge density localized near the center of the 16


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GQD (the second excitation is nearly identical to the first and is not shown). These first three excitations are all composed of transitions between the first two occupied (HOMO,HOMO-1) and unoccupied (LUMO, LUMO+1) Kohn-Sham orbitals. For excited states S4 and higher, Kohn-Sham states beyond these four significantly contribute and the charge density spreads considerably to the edges of the dot. All excitations demonstrate charge densities above and below the GQD, consistent with the π orbitals of graphene. Due to its accuracy in describing the electronic structure, we have implemented the B3LYP functional for all calculations discussed below.

Figure 3: Charge difference isosurfaces (CDI) for low-lying excitations in GQDs for C132 (left column), AC114 (middle column), and ZZ96 (right column). Yellow and blue isosurfaces indicate regions of electron accumulation and depletion, respectively, for each excitation compared to the ground state density. In all cases, the CDI for S2 is very similar to S1 and is not shown. The C132-L GQD has nearly identical CDI to C132 and is not shown for clarity. Medium gray spheres correspond to carbon atoms and small pink spheres are hydrogen atoms.

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As shown in Figure 4(a), the C132-L absorption spectrum at 300K (ATF C (ω), dotted blue line) reveals similar trends to the zero-temperature results. Excitations are redshifted by about 0.04 eV at 300K compared to their energies at 0K (compare the positions of the zero phonon lines (ZPL) to the energies in Table 1), however the relative energy spacing between them does not change. The first two excitations (S1 and S2 ), which are dark states at 0K, activate due to thermal motion at room temperature, increasing their oscillator strength enough that they reproduce the slight rise in absorption near 1.90 eV seen experimentally (dotted blue line in Figure 4(a)). The large, 0.30 eV transition energy between S3 and S2 persists at room temperature, which will strongly influence the relaxation dynamics explored in Section 3.2.1. Although ATF C (ω) captures the critical elements of the experimental spectrum, one unexplained feature is the small absorption peak occurring near 2.05 eV in the Exp A spectrum (dotted red line, Figure 4(a)). Examining ATF C (ω) without broadening (solid blue line) provides a possible explanation. Although the smearing better reflects the broad experimental spectra, removing the smearing reveals a phonon sideband (SB) near 2.1 eV that does not correspond to an excited state but rather represents absorption to a singly excited, S2 vibrational state. This small peak is blueshifted by 0.16 eV compared to the S2 ZPL at 1.94 eV, an energy difference that corresponds to a 1328 cm−1 disorder mode associated with edge states or defects in graphene quantum dots and nanoribbons. 41,42 In the calculated results, the S2 oscillator strength at 300K is still not strong enough to allow the SB to appear in the broadened absorption spectrum (dotted blue line), however the SB position corresponds closely to the small peak at 2.05 eV in the Exp A spectrum. This small peak does not appear in the Exp B spectrum, indicating it is sensitive to the chemical environment and likely not a dominant feature of the absorption spectrum. Figure 4(b) compares the absorption spectrum with and without the FC-DOS (ATF C (ω) and AT (ω), respectively). ATF C (ω) leads to a slightly broadened spectrum compared to AT (ω) as expected due to absorption into vibrationally excited states that create phonon sidebands.

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However, this effect is quite small and does not change any of the qualitative aspects of the absorption spectrum. These results suggest a weak electron-phonon coupling in the C132-L GQD, since phonon sidebands are not large enough to significantly alter the spectrum, which we discuss in more detail in the context of Huang-Rhys factors and relaxation dynamics in Section 3.2. In general, the LR-TDDFT spectrum accurately models key spectral features, including the absorption rise and peak, and indicates a large 0.3 eV gap between S2 and S3 that will play an important role in carrier relaxation dynamics that we expore in Section 3.2.1..

Figure 4: a) Thermally averaged, normalized absorption spectra of C132-L GQD including the Franck-Condon-weighted density of states (FC-DOS) (ATF C (ω)) with (blue dotted line) and without (blue solid line) 80 meV smearing compared to experiment (Exp A 28 and B 13 ). ZPL indicates zero phonon lines and SB labels a phonon sideband at 2.1 eV. b) Absorption spectra of C132-L GQD without FC-DOS (AT (ω), green dotted line) and with FC-DOS (ATF C (ω), blue dotted line).

3.1.2. AC and ZZ GQDs Tables 2 and 3 list the six lowest-lying excitations for AC and ZZ GQDs, respectively. In all cases, a single excitation has a large oscillator strength and dominates the absorption profile. The oscillator strength at the peak increases with dot size, indicating that larger GQDs will likely manifest more peaked absorption spectra at these energies. All excitations beyond S6 have an oscillator strength less than 0.1. 19



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exc Table 2: Vertical excitation energies (E0α ) and oscillator strengths (f0α ) for lowlying states at 0K for armchair (AC) GQDs.

AC42 exc Excitation E0α (eV) S0 → S1 2.894 S0 → S2 3.028 S0 → S3 3.291 S0 → S4 3.476 S0 → S5 3.624 S0 → S6 3.834

f0α 0.000 0.000 0.000 0.722 0.000 0.000

AC114 exc E0α (eV) 1.900 1.970 2.306 2.309 2.462 2.739

f0α 0.000 0.000 1.328 0.000 0.000 0.000

AC222 exc E0α (eV) 1.418 1.462 1.738 1.783 1.873 2.139

f0α 0.000 0.000 2.029 0.000 0.000 0.000

For AC GQDs, as dot size increases, the energy spacing between S1 and S2 decreases from 0.134 eV for AC42 to 0.045 eV for AC222. Both of these excitations are composed of Kohn-Sham transitions between HOMO-1 or HOMO and LUMO or LUMO+1 and lead to a similar charge density localized near the center of the dot similar to C132 (Figure 3, middle column). Higher energy excitations exhibit densities spread near the dot edges similar to S3 and S4 . Similar to the C132 GQD, a large transition energy exists between S2 and S3 in AC GQDs that stays relatively constant across AC dot sizes, from 0.263 eV for AC42 to 0.276 eV for AC222. This is somewhat counterintuitive because transition energies typically decrease with QD size. We investigate this result by plotting the excitation levels in Figure 5(a) and examining the Kohn-Sham transitions for particular excitations. Excitation energies for each AC GQD are plotted as blue horizontal lines, and the red horizontal line indicates a particular stsate of interest composed of Kohn-Sham transitions from HOMO-2 to LUMO and from HOMO to LUMO+2. In the AC42 GQD, this state is below the absorption peak at S4 , however its excitation energy shifts to higher values with dot size such that the absorption peak occurs at S3 for AC114 and AC222. Thus, the S3 -S2 transition energy stays relatively constant across AC GQD size due to this state’s increase in excitation energy with dot size. Beyond the absorption peak, transition energies range between 0.15 to 0.20 eV. Around 1.4 to 1.6 E/EG , the excitations become more dense and the continuum region is reached 20


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(solid blue regions in Figure 5), in which most states are less than 0.05 eV apart and rapid nonradiative relaxation is expected.

Figure 5: Excitation levels for a) armchair (AC) and b) zigzag (ZZ) GQDs. Excitation energy is given as a function of E/EG , where EG is the lowest LR-TDDFT excitation energy. Blue horizontal lines indicate excitation energies. The red excitation indicates a state of interest that changes its position relative to other states as a function of size in AC GQDs (see text for details). C stands fo the continuum level, at which point transition energies between states are only several meV. A similar analysis of ZZ GQDs as a function of size indicates a different pattern of excitation energies (Table 3 and Figure 5(b)). The transition energies between the lowest two excitations are roughly 0.14-0.16 eV across sizes. As in the C132 and AC GQDs, these states are comprised of Kohn-Sham transitions from HOMO-1 or HOMO to LUMO or LUMO+1. However, in contrast to the other GQDs, the electronic charge densities for these two states are spread across the center and edges of the dot (Figure 3, right column), which will impact electronic coupling and which phonon modes couple to transitions to these states. A large transition energy between S2 and S3 decreases with dot size from 0.51 eV for ZZ54 to 0.19 eV for ZZ216. Such a large transition energy is a likely candidate for a phonon bottleneck, which we explore in Section 3.2.2. Across all sizes, excitations become very dense between 1.4 and 1.6 E/EG , however the spacing widens again at higher energies in the ZZ96, ZZ150, and ZZ216 GQDs before reaching the continuum. This will be an intriguing region to examine carrier lifetimes given the wider energy spacing. 21



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exc Table 3: Vertical excitation energies (E0α ) and oscillator strengths (f0α ) for lowlying states at 0K for zigzag (ZZ) GQDs.

ZZ54 exc Excitation E0α (eV) S0 → S1 2.239 S0 → S2 2.391 S0 → S3 2.905 S0 → S4 3.122 S0 → S5 3.140 S0 → S6 3.205

f0α 0.000 0.000 0.929 0.000 0.000 0.000

ZZ96 exc E0α (eV) 1.667 1.806 2.179 2.367 2.386 2.429

f0α 0.000 0.000 1.143 0.000 0.000 0.000

ZZ150 exc E0α (eV) 1.280 1.422 1.688 1.818 1.865 1.905

f0α 0.000 0.000 1.279 0.000 0.000 0.000

ZZ216 exc E0α (eV) 0.992 1.139 1.324 1.379 1.474 1.497

f0α 0.000 0.000 1.311 0.000 0.000 0.000

3.2. Phonon-induced relaxation dynamics 3.2.1. C132-L and C132 GQDs We first discuss phonon-induced relaxation dynamics in the 132-carbon-atom GQD with carbon ligands (C132-L) studied experimentally. 28 Figure 6(a) shows the general excitation and phonon-induced relaxation process predicted by our model results in Figure 6(b). To match transient absorption (TA) measurements, 28 we initially excite an electron-hole pair with a 3 eV photoexcitation (I), creating a "hot" electron-hole pair weakly bound by their Coulomb attraction and delocalized across the edges of the GQD. Over time, the pair loses energy to phonons (II), rapidly transitioning downward through states close in energy via single-phonon processes over the first 0.1-1 ps. Within 10 ps, the electron-hole pair transitions to the state near 2.25 eV (S3 in Figure 6(a)-(c)) that corresponds to the absorption peak of the molecule near the fundamental band gap (Figure 6(c)). The S3 state, however, does not correspond to the lowest excited state in GQDs, since two lower-energy states (S1 and S2 in Figure 6(c)) define the band edge 0.30 eV below the absorption peak, matching the experimental fluorescence peak and rise in absorption near 1.85 eV (red curves in Figure 6(c)). The large transition energy to these lower-energy states necessitates a few-phonon process that drastically slows carrier relaxation, giving S3 a long 104 ps lifetime before transitioning to S1 /S2 , localized near the center of the dot. This 100 ps timescale is 1-2 orders of magnitude longer than relaxation in bulk graphene 43 and an order

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Figure 6: a) Diagram illustrating physical processes relevant to our model. b) Isocontour of ∆η(E, t) after 3 eV photoexcitation in the C132-L GQD. Colors from blue to red indicate population changes from 0 to 1, where 1 represents the entire population in one state. Experimental lifetimes taken from transient absorption measurements 28 are included in parentheses. c) Normalized absorption spectra of C132-L GQD comparing experimental (Exp A 28 and B 13 ) to calculated results (AT (ω)). Vertical lines indicate calculated excitation energies of the system (heights do not indicate a physical quantity). The black line indicates the HOMO-LUMO electronic band gap. Yellow and blue isosurfaces on the GQD structures indicate electron and hole charge densities of each excitation. d) Relaxation dynamics for GQD without ligands (C132) similar to (b). of magnitude longer than lifetimes in other QDs. 9,44 The electron-hole stays in S1 /S2 for two nanoseconds before nonradiatively relaxing to the ground state through a many-phonon process. Both the S3 (104 ps) and S1 (2.2 ns) lifetimes match extremely well with 76-185 ps and 1.5 ns lifetimes 28 observed at similar energies in TA measurements, providing evidence that experiment was probing phonon-induced processes. The 100 ps lifetime provides the potential to extract excited carriers from both the higher-energy excited states and those near the band edge (Figure 6(a), green region) in photovoltaic or other applications, as

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extraction times under 100 ps are possible to acceptor materials such as titanium oxide. 45,46 It is important to note that the HOMO-LUMO gap calculated by density functional theory (DFT) using the B3LYP functional predicts a lowest excitation energy of 2.25 eV, corresponding to the absorption peak (black vertical line in Figure 6(c)), but does not reproduce the lower energy excitations between 1.8-2.0 eV seen experimentally and captured using LR-TDDFT. Therefore, the large transition energy between S3 and S2 , as well as the corresponding 100 ps phonon bottleneck that matches experimental results, depends on the use of linear response theory to describe excited states. In particular, previous work studying relaxation dynamics in GQDs using DFT to describe excited states found a shorter phononinduced relaxation timescale, 47 emphasizing the importance of the current methodology to rigorously describe excited states. Looking at the effect of ligands, Figures 6(b) and 6(d) plot relaxation dynamics for C132-L and C132, respectively. Lifetimes at high-energy states are similar, however the S3 lifetime (τS3 ) is shortened by half in C132 (69 ps) compared to C132-L (104 ps). More dramatically, the S1 lifetime (τS1 ) decreases by an order of magnitude from 2.2 ns to 332 ps in C132, indicating that ligands delay nonradiative recombination to the ground state and could increase fluorescence yields. Previous work on other types of QDs, such as PbS or CdSe, has indicated that some types of ligands can expedite relaxation by introducing defects or hybridized states, 48 however our findings indicate that ligands on GQDs could increase lifetimes through different physical mechanisms we explore below. Further experimental work in this area will be important to investigate this finding. The above results suggest that the phonon bottleneck and nonradiative relaxation to the ground state are sensitive to the existence of the carbon chain ligands. To explore the physical mechanisms behind these results, we consider the factors influencing transition rates in our model (see Equation 13). Examining Vαβ and Sαβ as a function of phonon frequency (which determines F (ωαβ )) will provide insight into the physical mechanisms behind the carrier lifetimes discussed above.

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Figures 7(a)-(f) plot Vαβ for the grond state and lowest 10 excited states for the GQDs shown in Figure 1. We first consider the ligands’ effect on the above factors. Figures 7(a)-(b) plot Vαβ matrix elements for C132 and C132-L, respectively. Vαβ is smaller for C132-L across most matrix elements, indicating that the ligands suppress electronic coupling. Ligands weaken coupling to S1 /S2 (V32C132 = 2.3 meV, V32C132−L = 2.0 meV) and to the ground state (V10C132 = 2.5 meV, V10C132−L = 0.8 meV), thus reducing kαβ and rationalizing the longer S3 and S1 lifetimes in C132-L compared to C132 (Figures 6(b) and 6(d)). The reduced Vαβ likely occurs because ligands stabilize the dot from motion perpendicular to the graphene plane as shown by the AIMD snapshots in Figure 8(a). During AIMD, edge C atoms in C132 (Figure 8(b)) oscillate by several Angstroms whereas ligands in C132-L limit this motion. Such substantial nuclear motion in C132 increases electronic coupling between states that decreases carrier lifetimes. We next examine the Huang-Rhys factors (Sαβ ) that describe which phonon modes couple strongly to each transition. Figures 9(a)-(c) plot Sαβ as a function of phonon frequency for the C132-L/C132, AC, and ZZ GQDs, respectively. Both C132 and C132-L couple to lowfrequency breathing modes as well as 1300 cm−1 (0.16 eV) and 1600 cm−1 (0.20 eV) defect and optical G modes typical in graphene, 42 as shown in Figure 9(a) by the dominant HuangRhys factors that occur at these energies. These modes match the transition energies for higher-energy states between 2.3 and 3.0 eV, allowing for single-phonon processes that lead to rapid relaxation on a picosecond timescale. In contrast, no available phonon energies match the 0.3 eV (2400 cm−1 ) transition energy from S3 to S1 /S2 , creating the phonon bottleneck and an inefficient few-phonon relaxation process that results in the 100 ps carrier S3 lifetimes and phonon bottleneck. In general, the S3 -S2 transition only weakly couples to the defect and optical modes (S32 < 0.01). This is also true of most other transitions, indicating the weak electron-phonon coupling intrinsic to these systems already indicated by the similarity in absorption profiles with and without including the FC-DOS (Figure 4(b)). The ligands in C132-L introduce additional phonon modes at 1000 cm−1 and 3000 cm−1 ,

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Figure 7: Electronic nonadiabatic coupling (Vαβ ) for GQDs with different structural modifications. (a)-(f) plot Vαβ matrix elements for the ground state (state index 0) and first ten excited states (state indices 1-10) for the C132, C132-L, AC114, ZZ54, ZZ150, and ZZ216 GQDs, respectively. The dashed blue line highlights matrix elements corresponding to coupling between states at the absorption peak, S3 , and band edge, S2 (V32 ) and S1 (V31 ), relevant to the phonon bottleneck. however they only weakly couple to low-energy transitions and therefore do not significantly impact lifetimes (Figure 9(a)). These results, combined with the reduced electronic coupling, provide a rationale for the 100 ps lifetimes seen experimentally. 28 3.2.2. AC and ZZ GQDs Finding ways to modify state lifetimes would provide a path to tailor relaxation dynamics for various optoelectronic devices. Thus, we next show how nanoscale structural changes to carbon edge type (Figure 1(b)) and size (Figure 1(c)) reveal a dramatic ability to tune the phonon bottleneck to the lowest excited states. Comparing GQDs similar in size, we plot the excited carrier energy as a function of time for AC114 and ZZ150 in Figures 10(a)-(b). In

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Figure 8: Configuration of a) C132-L and b) C132 GQD at representative timestep in the 300K molecular dynamics trajectory. Medium gray spheres are carbon atoms in the body of the GQD, medium blue spheres are carbon atoms in the ligands, and small pink spheres are hydrogen atoms. both cases, the excited carrier relaxes to the absorption peak (S3 ) in less than a picosecond. However, the S3 lifetime for AC114 (τSAC114 ) dramatically reduces to 0.4 ps, breaking the 3 phonon bottleneck and providing a fast relaxation channel to S1 and S2 . Across all AC sizes (Figure 10(c)), S3 lifetimes are never more than 2 ps and higher-energy state lifetimes are within 10-50 fs (E/EG > 1.5, where EG is the first excitation energy). In contrast, the S3 lifetime is two orders of magnitude larger for ZZ150 (τSZZ150 = 19.9 ps) and further lengthened 3 by decreasing ZZ GQD size (τSZZ54 = 39.1 ps (Figure 10(d))). Increasing ZZ GQD size leads 3 to a sharp decrease in S3 lifetime to 3.1 ps for ZZ216. An intriguing effect of size on lifetimes occurs for high-energy states in ZZ GQDs (Figure 10(d)). As discussed in Section 3.1.2, ZZ GQDs exhibit wider energy spacing at E/EG > 1.7, which leads to increased lifetimes over 100 fs for these states as a function of size for ZZ150 and ZZ216. Multiexciton generation (MEG) can occur in states with E/EG > 2 if phonon-induced relaxation is slow enough to allow high-energy carriers to Coulombically interact with another electron-hole pair and decay into multiple low-energy carriers. This 27



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Figure 9: Huang-Rhys factors (Sαβ ) as a function of phonon frequency. a) C132-L and C132, b) AC, and c) ZZ GQDs of varying sizes. The left and right columns refer to Huang-Rhys factors for modes coupling to the S1 -S0 and S3 -S2 transitions, respectively. process leads to multiple carriers per absorbed photon and increased photovoltaic efficiency. MEG occurs on timescales of 100 fs to 100 ps in CdSe 49 and 250 fs in PbSe or PbS 50 QDs, competitive with the high-energy lifetimes seen in larger ZZ GQDs, encouraging future study in these systems. Examining both Vαβ and Sαβ also explains the different relaxation patterns in AC and ZZ GQDs. We focus on V32 and V31 , highlighted by the dashed blue line in Figures 7(a)-(f), as these couplings determine relaxation rates from the absorption peak (S3 ) to to the lowest excited states (S1 and S2 ) and are relevant to the existence or breaking of the bottleneck. Both AC and ZZ GQDs exhibit strong electronic coupling between S3 and S1 or S2 (Vαβ > 10 meV). Whereas Vαβ in AC GQDs does not significantly change with size (V32AC114 = 12.8 meV, V32AC222 = 11.0 meV), coupling to S1 or S2 slowly decreases with size in ZZ GQDs (V32ZZ = 9.7-21.0 meV). However, ZZ GQDs show extremely strong coupling around 10 meV between most excited states, with these values only slightly decreasing with size for the ZZ216 dot. These values are significantly larger compared to C132 and C132-L and explain the shorter S3 lifetimes, indicating that geometry and asymmetry play a crucial role in minimizing coupling 28


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Figure 10: Isocontours of ∆η(E, t) after a 1.6 E/EG excitation in a) AC114 and b) ZZ150 GQD, where EG is the first excitation energy. Colors from blue to red indicate population changes from 0 to 1 as in Figure 6. S3 corresponds to the absorption peak close to the electronic band gap and S1 is the lowest excited state. Relaxation time constants τ as a function of E/EG and size for c) AC and d) ZZ GQDs. and relaxation rates. Our future work will systematically explore why symmetry or the lack thereof influences nonadiabatic coupling in this way. The difference in relaxation rates between AC and ZZ GQDs arises due to large changes in electron-phonon coupling. Transitions in AC GQDs couple more strongly (Sαβ > 0.01) to a range of phonon modes, including optical G and defect modes as well as hydrogen modes over 3000 cm−1 (Figure 9(b)) due to significant nonplanar reorganization. Such large Huang-Rhys factors for the optical and defect modes in AC dots allow multiple phonon quanta to absorb the energy necessary to make the transition and thus break the phonon bottleneck. Strong electronic couplings between 5-10 meV also lead to the fast 10-100 fs lifetimes for E/EG > 1.5 across AC GQD sizes (indices 5-10 in Figure 7(c)). In contrast, the ZZ edge type tends to more planar geometries and extremely weak electron-phonon coupling, 29



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as shown in Figure 9(c). For the S3 -S2 transition, optical G and defect modes exhibit Sαβ < 0.01 even for the smallest dot. This weak coupling to phonons reduces F (ωαβ ) by orders of magnitude compared to the AC GQDs, leading to 20-40 ps lifetimes in ZZ GQDs. This weak electron-phonon coupling makes ZZ GQDs a promising material to generate long lifetimes if functionalization or geometries can be found to reduce the strong electronic coupling to the lower excited states. In contrast to AC GQDs, the S3 lifetime decreases with increasing ZZ GQD size due to the fact that the electron-phonon coupling is already so weak in the smallest ZZ GQD. As dot size increases, two competing effects determine kαβ . The transition energy decreases with size, requiring less phonons to induce the transition and increasing rates. However, electron-phonon coupling also decreases with size, which reduces Sαβ and F (ωαβ ). Whereas in AC GQDs the decrease of ω32 with size is balanced by reduced Sαβ , in ZZ GQDs the effect of a smaller transition energy dominates and leads to shorter carrier lifetimes. In addition, ω32 decreases to 0.19 eV in ZZ216, matching the 0.20 eV (1600 cm−1 ) optical G mode and creating a unique resonance condition in which a single-phonon relaxation process breaks the bottleneck and leads to the short 3.1 ps S3 lifetime. Further increasing ZZ GQD size should increase lifetimes because the transition energy would decrease and the resonance condition would no longer be satisfied, leading to a decrease in F (ωαβ ).

4. Conclusions In summary, we have deveoped ab initio methods that combine the reduced density matrix formalism with linear response theory to study phonon-induced relaxation of excited carriers in finite systems. The method provides a computationally efficient pathway to rigorously treat excited states and study relaxation dynamics in systems with hundreds of atoms. We have applied this methodology to reveal long excited carrier lifetimes in GQDs, indicating the possibility of a phonon bottleneck in these materials. Large transition energies exist in GQDs

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between higher-energy states near the absorption peak and those closest to the optical band edge. These transition energies do not match available phonon energies for most GQD sizes and open up the possibility of creating a phonon bottleneck. In a 132-carbon-atom dot with carbon chain ligands (C132-L), our method finds a 100 ps bottleneck that matches extremely well with experimental transient absorption measurements, supporting the accuracy of the current method. Surprisingly, we find that the carbon ligands, used in experiment to prevent GQDs from aggregating in solution, nonadiabatically decouple excited states by stabilizing motion perpendicular to the GQD plane. In addition, they do not introduce any additional defect states or strongly coupled phonon modes. This behavior prolongs carrier lifetimes in the first several excitations compared to the same GQD without ligands. This is an important example of how ligands, a property of QDs typically seen as disadvantageous to long carrier lifetimes, can be turned into something helpful depending on the type of QD studied. Future experimental work should investigate this property further to understand how ligands affect phonon-induced relaxation rates in GQDs. Extending to applications beyond the experimentally studied GQD, we have also found that the phonon bottleneck is sensitive to nanoscale structural changes that affect electronic and electron-phonon coupling, providing a playground to change carrier lifetimes by orders of magnitude. Terminating GQD edges with a zigzag pattern minimizes electron-phonon coupling, whereas GQDs with armchair edge types exhibit the largest coupling to phonons. However, symmetric GQDs with either armchair or zigzag edges show significantly increased electronic coupling between excited states, leading to shorter carrier lifetimes compared to the asymmetric C132-L GQD. Additional investigations should focus on understanding why the symmetric AC and ZZ GQDs show stronger electronic coupling between lower excited states compared to the asymmetric C132 GQD and how functionalization or other structures can minimize coupling to excited states and maximize carrier lifetimes. Results from the present work provide a range of possibilities to tune carrier lifetimes for specific applications. For

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example, the weak electronic coupling in asymmetric dots and minimized electronic coupling in ZZ GQDs should be of interest for hot carrier extraction in photovoltaics. Conversely, the rapid relaxation in AC GQDs could be beneficial in light-emitting applications. It is important to note that one possible limitation of the current study is the theory’s ability to correctly describe multiphonon relaxation processes because it assumes a weak interaction with the phonon bath. However, the function F (ωαβ ) does account for the possibility of multiple phonon quanta of the same mode or a combination of modes being absorbed to induce a particular transition. 22 Our present results match extremely well with available experimental transient absorption data, suggesting that this level of theory is adequately describing the phonon relaxation process.

Acknowledgement The authors gratefully acknolwedge support from the Department of Energy (Grant No. DOE/BES DE-FG02-02ER45995) and thank NERSC for computing resources.

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