Dynamics of Charge Transfer and Multiple Exciton Generation in the

Subscriber access provided by Kaohsiung Medical University

Surfaces, Interfaces, and Catalysis; Physical Properties of Nanomaterials and Materials

Dynamics of Charge Transfer and Multiple Exciton Generation in the Doped Silicon Quantum Dot -- Carbon Nanotube System: Density Functional Theory Based Computation Andrei Kryjevski, Deyan Mihaylov, and Dmitri S. Kilin J. Phys. Chem. Lett., Just Accepted Manuscript • DOI: 10.1021/acs.jpclett.8b02288 • Publication Date (Web): 10 Sep 2018 Downloaded from http://pubs.acs.org on September 11, 2018

Just Accepted “Just Accepted” manuscripts have been peer-reviewed and accepted for publication. They are posted online prior to technical editing, formatting for publication and author proofing. The American Chemical Society provides “Just Accepted” as a service to the research community to expedite the dissemination of scientific material as soon as possible after acceptance. “Just Accepted” manuscripts appear in full in PDF format accompanied by an HTML abstract. “Just Accepted” manuscripts have been fully peer reviewed, but should not be considered the official version of record. They are citable by the Digital Object Identifier (DOI®). “Just Accepted” is an optional service offered to authors. Therefore, the “Just Accepted” Web site may not include all articles that will be published in the journal. After a manuscript is technically edited and formatted, it will be removed from the “Just Accepted” Web site and published as an ASAP article. Note that technical editing may introduce minor changes to the manuscript text and/or graphics which could affect content, and all legal disclaimers and ethical guidelines that apply to the journal pertain. ACS cannot be held responsible for errors or consequences arising from the use of information contained in these “Just Accepted” manuscripts.

is published by the American Chemical Society. 1155 Sixteenth Street N.W., Washington, DC 20036 Published by American Chemical Society. Copyright © American Chemical Society. However, no copyright claim is made to original U.S. Government works, or works produced by employees of any Commonwealth realm Crown government in the course of their duties.

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

The Journal of Physical Chemistry Letters

Dynamics of Charge Transfer and Multiple Exciton Generation in the Doped Silicon Quantum Dot – Carbon Nanotube System: Density Functional Theory Based Computation Andrei Kryjevski,1 Deyan Mihaylov,1 and Dmitri Kilin2 1)

Department of Physics, North Dakota State University, Fargo, ND 58108, USA a) Department of Chemistry and Biochemistry, North Dakota State University, Fargo, ND 58108, USA

2)

(Dated: 5 September 2018)

We use Boltzmann transport equation (BE) to study time evolution of a photo-excited state, including phonon-mediated exciton relaxation, multiple exciton generation (MEG) and energy transfer processes. BE collision integrals are derived using Kadanoff-Baym-Keldysh many-body perturbation theory (MBPT) based on density functional theory (DFT) simulations, including exciton effects. We apply the method to a nanostructured p − n junction composed of a 1 nm hydrogen-terminated Si quantum dot (QD) doped with two phosphorus atoms (Si36 P2 H42 ) adjacent to the (6,2) single-wall carbon nanotube (CNT) with two chlorine atoms per two unit cells adsorbed to the surface. We find that an initial excitation localized on either the QD or CNT evolves into a transient charge transfer (CT) state where either electron or hole transfer has taken place. The CT state lifetime is about 40 f s. Also, we study MEG in this system by computing internal quantum efficiency (QE), which is the number of excitons generated from an absorbed photon during relaxation. We predict efficient MEG starting at 3 × Eg ≃ 1.5 eV and with QE reaching QE = 1.65 at about 5 × Eg , where Eg ≃ 0.5 eV is the lowest exciton energy, i.e., the gap. However, we find that including energy transfer and MEG effects suppresses CT state generation. I.

II.

TOC GRAPHIC

INTRODUCTION

Increasing efficiency of the photon-to-electron energy conversion in a nanosystem has received a lot of attention. For instance, it is envisioned that efficiency of the nanomaterial-based devices, such as solar cells, photooxidizing catalysts for water-splitting reaction, etc., can be increased due to carrier multiplication, or multiple exciton generation (MEG) process, where absorption of a single energetic photon generates several excitons1–3 . In this scenario the excess photon energy is diverted into creating additional charge carriers instead of exciting atomic vibrations3 . In fact, phonon-mediated relaxation is a major effect competing with the MEG. Any conclusion about MEG efficiency can only be made by simultaneously including MEG, phonon-mediated carrier

a) Electronic

mail: [email protected].

relaxation, and other processes, such as charge and energy transfer4,5 . In contrast to the bulk semiconductor materials where MEG in the solar photon energy range is inefficient 6–8 , in nanomaterials MEG is enhanced by spatial confinement, which increases electrostatic electron interactions responsible for this effect2,3,9–11 . Internal quantum efficiency (QE), which is the average number of excitons generated from an absorbed photon, is a useful characteristic of MEG efficiency. QE exceeding 100% has been measured in recent experiments on, e.g., silicon and germanium nanocrystals, silicon nanorods and in nanoparticle-based solar cells12–18 . Also, MEG has been observed in the single-wall carbon nanotubes (CNTs) using transient absorption spectroscopy13 and the photocurrent spectroscopy19; QE = 1.1 at the photon energy ~ω = 2.5Egopt , QE = 1.3 at ~ω = 3Egopt , where Egopt is the optical gap, was found in the (6,5) CNT. Work to create CNT-based solar cells is currently under way20 . MEG is dominated by the impact ionization process21,22 . Therefore, one has to calculate the excitonto-biexciton decay rate (R1→2 ) and the biexciton-toexciton recombination rate (R2→1 ), the direct Auger process. Description of CNTs, which are quasi onedimensional systems, requires inclusion of the exciton effects23 . Previously, we have developed an approach based on Boltzmann transport equation (BE) to study MEG, which includes exciton multiplication and recombination, i.e., the Auger processes, and the phononmediated exciton relaxation24. The Kadanoff-BaymKeldysh, or non-equilibrium Green’s function (NEGF), formalism allows one to use many-body perturbation theory (MBPT) to calculate collision integrals in the

ACS Paragon Plus Environment

The Journal of Physical Chemistry Letters

Page 2 of 7 2

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

transport equation, using Density Functional Theory (DFT) simulations25–29 . Also, the R1→2 , R2→1 rates and phonon couplings were computed previously using DFT combined with the many-body perturbation theory (MBPT) techniques30,31 . While BE coefficients are computed to a given order in many-body perturbation theory (MBPT), BE itself is non-perturbative. We have applied this technique to the (6,2) and (6,5) CNTs and found efficient low-energy MEG in these systems. For (6,5) we predicted QE(~ω = 2.5Egopt ) = 1.2 and QE(~ω = 3Egopt ) = 1.6. In this work we further develop the approach by including energy transfer into the DFT-based excited state relaxation description. For this we add exciton transfer terms into the BE collision integrals. The resulting technique is applied to a doped Si QD-CNT system, which is an example of a nano-structured p − n junction. Here we use a 1 nm hydrogen-terminated Si QD and (6,2) CNT. Dependence of CT state dynamics on the size and type of the QD, chirality of CNT, type of ligands, etc. is left to future studies. This work aims to provide basic insight into dynamics of photoexcited functionalized CNTs and its dependence on surface functionalization.

photoexcited nanoparticle24 . The collision integrals can be obtained using the Kadanoff-Baym-Keldysh formalism25–27 . All this results in a system of equations for the (slowly) varying spin-zero exciton state occupation numbers nα

+− n˙ α = iΣ−+ α (n; ωα ) (1 + nα ) − iΣα (n; ωα ) nα ,

(1)

+− where Σ−+ are the leading Keldysh exciton selfα , Σα energies, which depend parametrically on nα . Also, they depend on the spin-zero exciton energies E α = ~ωα and wavefunctions Ψα eh , electron-phonon couplings, etc.

III. BE INCLUDING MEG, PHONON RELAXATION AND EXCITON TRANSFER TERMS

Let us state a major simplifying approximations used in this work, which is necessary to simulate a nanosystem as sizable as a functionalized chiral CNT. We use KohnSham (KS) energies produced by the HSE06 functional instead of computing GW corrections to the KS energies ǫi 24,30–32 . For the spin non-polarized system studied here for the KS energies and eigenfunctions we have ǫi↑ = ǫi↓ ≡ ǫi and φi↑ = φi↓ ≡ φi .

Boltzmann transport equation (BE) is a potent tool for a comprehensive study of time evolution of a

Next, for the spin-zero E α , Ψα eh one solves the BetheSalpeter equation (BSE), which is33,34

([ǫe − ǫh ] − E α ) Ψα eh +

X 4πe2 (2KCoul − Kdir )e,h;e′ ,h′ Ψα = 0, e′ ,h′ V ′ ′

(2)

e h

where X X ρeh (q)ρ∗′ ′ (q) e h , ρji (p) = φ∗j (k − p)φi (k), 2 Vq k q6=0 X  2 −1 ∗ = ρee′ (k) k δk,p − Π(0, −k, p) ρhh′ (p), (3)

KCoul = Kdir

k,p6=0

ijkln

4π X 2 R = 2| Wjlnk (ωα )θ−l θn Ψβnl θ−i θj θk (Ψγji )∗ Ψα ki | , ~ ijkln X X X X θ−i = , θi = , h

where Π(0, k, p) is the static random phase approximation (RPA) polarization insertion (see, e.g., 35 ). BE equation including phonon emission and absorption, exciton-to-biexciton decay and biexciton-to-exciton recombination, and exciton transfer terms is n˙ γ = Cphonon [n] +

In the above (cf. Eq. 21 in24 )   γ ˜p + R ˜ h δ(ωγ − ωα − ωβ ), Rαβ = Rp + Rh + R 4π X ∗ 2 Wjlnk (ωα )θl θ−n (Ψβln )∗ θi θ−j θ−k Ψγij (Ψα Rp = 2 | ik ) | , ~

X

i

i>HO

Wjlnk (ω) =

CMEG [n] + Ctransf er [n] ,

X 4πe2 ρ∗kj (q)ρln (q) V (q 2 − Π(ω, −q, q))

(6)

q6=0

X dnα dnβ + nβ =− CMEG , dt dt γ

γ CMEG = Rαβ (nα nβ − nγ (nα + nβ + 1)) .

(5)

i≤HO

where

α,β

nα

i

(4)

is the (approximate) dynamically screened Coulomb matrix element; HO is the highest occupied orbital index. ˜ h and R ˜ p are the same as the ones The expressions for R h p for R , R with Wjlnk (ωα ) replaced by Wjlkn (ωγ − ωni )

ACS Paragon Plus Environment

Page 3 of 7

The Journal of Physical Chemistry Letters 3

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

−

+ +

+

+

−

−

+

−

−

−

+

+

β

α

α

+

+

−

+

+ −

β

α

−

α

+

=

−

+

α

β

−

+

−

β

α +

+ ···

−

+

+

−

+

+ α

+ − +

−

+

β

α

+

+ α

−

− α

FIG. 1. Feynman diagram for the Coulomb-driven exciton transfer process. Thick lines are the exciton propagators; α, β are the exciton state labels. Thin solid lines with arrows are the KS fermion propagators. Zigzag line stands for the Coulomb interaction. The -+ contribution obtains when + and - are interchanged.

˜ h and by Wjlkn (ωγ − ωin ) for R ˜ p , where ~ωij = for R ǫi − ǫj , and divided by 2. Phonon-exciton collision integrals Cphonon [n] , which include one- and two-phonon emission/absorption processes, are identical to those used in 24 . (See Eqs. 1619 of 24 .) We refer the reader to 24 for the description of the electron-phonon coupling and all the accompanying approximations. A widely-used method for the phonon-mediated relaxation is the surface-hopping technique36,37 . However, currently it does not include, e.g., the exciton effects and, so, is not directly suitable for our purposes. Exciton transfer terms Ctransf er will be discussed in the next subsection.

A.

Exciton transfer contribution to BE

Collision integral contribution that describes exciton transfer due to residual Coulomb interactions between excitons is depicted in the last diagram in Fig. 1. This is a result of re-summation of various Coulomb corrections that contribute to the exciton transfer process shown in the top part of Fig. 1. However, they are already included by the the BSE, which results in the last diagram in Fig. 1. The corresponding addition to the r.h.s. of BE Eq. 4 is Ctransf er =

X

Tαβ (nα − nβ ) ,

α

Tαβ = 2π|

X

  β ∗ Ψα (ωβ − ωeh )|2 δ (ωβ − ωα ) ,(7) eh Ψeh

eh

where Tαβ is the transition rate between exciton states |αi and |βi. This approach may be seen as an extension of the Forster method38 .

IV.

COMPUTATIONAL DETAILS

The optimized QD-CNT geometries, KS orbitals and energy eigenvalues have been obtained using the ab initio total energy and molecular dynamics program VASP (Vienna ab initio simulation package) with the hybrid Heyd-Scuseria-Ernzerhof (HSE06) exchange correlation functional39,40 using the projector augmented-wave (PAW) pseudopotentials41,42 . Using conjugated gradient method for atomic position relaxation the structure was relaxed until residual forces on the ions were no greater than 0.05 eV /˚ A. Momentum cutoff defined by ~2 k 2 /(2m) ≤ Emax ,

(8)

m is the electron mass, was chosen at Emax = 400 eV. The energy cutoffs determined by the number of KS orbitals included in the simulations were chosen so that ǫimax − ǫHO ≃ ǫLU − ǫimin ≥ 3 eV, where imax , imin are the highest and the lowest KS labels included in simulations and LU = HO + 1. Normal frequencies and mode decompositions have been calculated using VASP DFT software with the Perdew-Burke-Ernzerhof (PBE) xc-functional43,44 . Atomistic model was placed in a finite volume simulation box with periodic boundary conditions. In the axial direction the box size was chosen to accommodate two unit cells of CNT (6,2). In the other two directions about 1 nm of vacuum between surfaces was included in order to exclude spurious interactions between periodic images of the model. Previously, we have found reasonably small (about 20%) variation of ǫi (K), where K is lattice momentum, over the Brillouin zone when two unit cells were included in the DFT simulations, and a smaller (about 10%) variation for three unit cells30 . However, due to high computational cost, we have only been able to include two unit cells of CNT (6,2). The simulations have been done at the Γ point. In our calculations lattice momenta of the KS states suppressed

ACS Paragon Plus Environment

The Journal of Physical Chemistry Letters

Page 4 of 7 4

FIG. 2. Atomistic model of two unit cells of (6,2) CNT doped with two Cl atoms (p-type doping) adjacent to a hydrogen-passivated 1 nm Si QD doped with two P atoms (Si36 P2 H42 )), which is n-type doping. Also shown are the electron densities of the LU and HO KS states localized on the QD and CNT, respectively, which are the dominant states in the CT exciton state.

4 absorption, a.u.

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

3 2 1 0

0.5

1.0

1.5

2.0 2.5 ℏω, eV

3.0

3.5

4.0

FIG. 3. Low-energy absorption spectrum of the doped QDCNT system shown in Fig. 2 is the solid (red) line. Dashed (blue) line shows exciton density of states (DOS). Marked with a black arrow is the CT state energy.

by the reduced Brillouin zone size are neglected. The rationale for including multiple unit cells instead of the Brillouin zone sampling in the DFT simulations is that surface of the CNT is functionalized30 . Inclusion of several unit cells keeps the concentration of surface dopants reasonably low. Atomistic model of functionalized CNT (6,2) with two chlorine atoms adsorbed to the surface has been provided by the Kilina’s group (NDSU). The optimized atomistic model is shown in Fig. 2. The DFT simulations have been done in a vacuum which approximates properties of this QD-CNT system in a non-polar solvent.

V.

RESULTS AND DISCUSSION

First, let us describe some method improvements aimed at increasing accuracy of the BE MEG technique. As already noted in Section III several simplifying approximations have to be employed in order to ensure

applicability of our methods to sizable systems, such as functionalized chiral CNTs24,30–32 . 1. One such′ approximation for the RPA polarization is Π(0, x, x ) ≃ ′ Π(0, x − x ), i.e., treating the system as a uniform medium24,30–32 . In the momentum space this corresponds to Π(0, −k, p) ≃ δk,p Π(0, −k, k), which greatly simplifies evaluation of the screened Coulomb interaction in, e.g., direct part of the BSE kernel in Eq. (3). In this work we have solved BSE for the QD-CNT system ′ using both Π(0, x − x ) approximation and keeping full ′ Π(0, x, x ), for which we had to judiciously reduce spatial resolution, i.e., the number of the space grid points. For the nanostructure under consideration we found that the exciton energies in the two cases differ from each other by 10%, at the most, which suggests that the uniform medium screening in this case is adequate for a semiquantitative description. Based on this, for the rest of this work we employ uniform medium screening approx′ ′ imation Π(ω, x, x ) ≃ Π(ω, x − x ). 2. We have noted before that in the impact ionization process typical energy exchange is greater than the gap and the dynamical screening effects can be significant24,30,31 . Therefore, accurate MEG calculations are likely to require dynamical screening treatment as included in Eqs. (5,6) above. However, full implementation of this approach would require evaluation of Coulomb matrix elements Wjlnk (ω) for multiple values of ω, which is very computationally expensive. Therefore, as a first step, we have approximated Wjlnk (ω) ≃ Wjlnk (¯ ω ), where ω ¯ ≥ Eg is a typical energy flowing through the Coulomb interaction during an MEG decay. In order to check the effect of this change we have re-computed QE in CNT (6,5) using this dynamical screening approximation. We found QE(~ω = 2.5Egopt ) = 1.1 and QE(~ω = 3Egopt ) = 1.4, which agrees better with the experimental values of 1.1 and 1.3 for the same two energies than our earlier static screening results of 1.2 and 1.624 . Therefore, this is the dynamical screening approximation used in this work. Next, let us describe properties of the photoexcited QD-CNT system, which is an example of a nanostructured p − n junction. Exciton density of states is shown in Fig. 3, dashed line. In the energy spectrum of our QD-CNT system there exists a charge transfer (CT) exciton, i.e., a bound electron-hole state where the dominant KS orbital for the hole is localized on the QD and the dominant particle KS state is localized on the CNT. The corresponding KS orbital densities are shown in Fig. 2. The CT exciton energy ECT ≃ 0.88 eV is marked with an arrow in Fig. 3. Low-energy absorption spectrum is shown in Fig. 3, solid line. The lowest peak is associated with the Cl CNT surface defect and is, actually, at the exciton gap energy E1 = 0.49 eV. The next 2 peaks are the standard E11 and E22 peaks of the CNT (6,2). Peaks for energy above 3 eV are associated with the QD localized states. Solving BE for various initial excitations where the initial exciton state is localized on either QD or the CNT we have always found a transient CT state generated in

ACS Paragon Plus Environment

Page 5 of 7

The Journal of Physical Chemistry Letters a)

n1-5, n7, localized on CNT

0.2 0.1 0.0

0

50

100 t, fs

150

200

0.15 QE

n6, CT state

0.3

CT exciton occ.

0.4

0.10 0.05 0.00

0

50

5

c)

0.20 exciton occ. n1-7

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

b)

0.5

100 t, fs

150

200

1.7 1.6 1.5 1.4 1.3 1.2 1.1 1.0 0.9

2

3

4 Eexciton/Eg

5

6

FIG. 4. a) Typical low-energy exciton occupation numbers; b) CT exciton occupation with phonon terms only (thin dashed red line), with phonons and exciton transfer terms (thick dashed green line) and with phonon, exciton transfer and MEG terms included (solid blue line); c) QE in the doped QD-CNT system as a function of excitation energy.

the course of relaxation. A typical prediction for the few low-energy exciton occupation numbers n1 (t), ..., n7 (t) is shown in Fig. 4, a). Note that states are numbered in the order of increasing energy with E1 = 0.49 eV being the exciton gap. The analysis of the orbital electron density shows that exciton states 1 through 5 and 7 are all localized on the CNT. However, state number 6 is of the CT type and has a lifetime of order of 40 f s. Next, we study dependence of the CT state dynamics on the strength of different relaxation channels. The results are shown in Fig. 4, b), which are the three n6 (t) curves due to excitation at E ≃ 1.5 eV. The thin (red) dashed line is the CT exciton occupation number when only phonon-mediated relaxation is included. The thick (green) dashed line shows n6 (t) when both phononmediated and exciton transfer terms are included. Finally, the solid (blue) line is for the case when phononmediated, exciton transfer terms and MEG terms are taken into account. We see that inclusion of the exciton transfer and then of the 1 → 2 and 2 → 1 MEG decays decreases the CT state occupation number magnitude. This is not surprising since CT, exciton transfer and MEG decays are competing effects. This supports earlier observation that photoexcited dynamics is often governed by an intricate interplay between different relaxation channels24 . Next, we computed QE in the QD-CNT system using the approach developed in 24 . Initial excitation with energy E is evolved by solving BE including a. phonon emission and absorption, b. exciton-to-bi-exciton decay and recombination, c. exciton transfer terms. One adds up the occupation numbers of excitons generated after the occupancies have plateaued as t → ∞. (Recombination occurring on a much longer time scale is not included here.) After the initial state averaging P nα (t → ∞) QE(E) = α , (9) nin where nin is the initial state occupation number. As discussed above, we have approximated Wjlnk (ω) ≃ Wjlnk (¯ ω ), where ω ¯ = 0.675 eV. The result is shown in Fig. 4, c). We predict efficient low-energy MEG in this system: QE reaches ∼ 1.3 at about 3 × Eg ≃ 1.5 eV and further grows to QE = 1.65 at about 5×Eg . As discussed

above, the CT state is not the lowest in the exciton energy spectrum. So, the additional excitons generated via MEG tend to accumulate in the lowest energy states partially bypassing populating the CT exciton.

VI.

CONCLUSIONS

We have developed a comprehensive approach to a photo-excited nanosystem, which is based on BE with collision integrals computed from the KadanoffBaym-Keldysh MBPT using results of DFT simulations. Phonon-mediated exciton relaxation, multiple exciton generation (MEG) and exciton transfer processes have been included. The method has been applied to a nanostructured p − n junction made of a 1 nm Si quantum dot (QD) doped with two phosphorus atoms (Si36 P2 H42 ) adjacent to the (6,2) single-wall carbon nanotube (CNT) with chlorine atoms adsorbed to the surface. Solving BSE has revealed that this system possesses a CT-type low energy exciton. Solving BE for various initial excitations has shown that generation of a transient long-lived CT state is a robust feature of this system. We have also studied interplay between different relaxation channels and have found that CT state dynamics is determined by their interplay. We find that competition between phonon-mediated relaxation, exciton transfer and MEG inhibits somewhat the CT state generation. However, in a more realistic setting a charge extraction mechanism, such as metal leads, should be included. It is likely that CT state generation might be enhanced in this case since CT state occupation depleted via charge extraction would be replenished from the nearby low-energy exciton states. In our future work we will check this conjecture by studying a nano-structured p − n junction with metal leads included into the atomistic model, continuing earlier work on the subject45,46 .

ACKNOWLEDGMENTS

Authors acknowledge financial support from the NSF grant CHE-1413614. The authors acknowledge the use of computational resources at the Center for Computa-

ACS Paragon Plus Environment

The Journal of Physical Chemistry Letters

Page 6 of 7 6

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

tionally Assisted Science and Technology (CCAST) at North Dakota State University and the National Energy Research Scientific Computing Center (NERSC) allocation award 31857, supported by the Office of Science of the DOE under contract No. DE-AC02-05CH11231. 1 W.

Shockley and H. Queisser. Detailed balance limit of efficiency of p-n junction solar cells. J. Appl. Phys., 32(3):510–519, 1961. 2 R. Ellingson, M. Beard, J. Johnson, P. Yu, O. Micic, A. Nozik, A. Shabaev, and A. Efros. Highly efficient multiple exciton generation in colloidal pbse and pbs quantum dots. Nano Letters, 5(5):865–871, 2005. 3 A. J. Nozik. Quantum dot solar cells. Physica E: Lowdimensional Systems and Nanostructures, 14(1-2):115 – 120, 2002. 4 W. D. A. M. de Boer, E. M. L. D. de Jong, D. Timmerman, T. Gregorkiewicz, H. Zhang, W. J. Buma, A. N. Poddubny, A. A. Prokofiev, and I. N. Yassievich. Carrier dynamics in si nanocrystals in an sio2 matrix investigated by transient light absorption. Phys. Rev. B, 88:155304, 2013. 5 J. Stewart, L. Padilha, W. Bae, W. Koh, J. Pietryga, and V. Klimov. Carrier multiplication in quantum dots within the framework of two competing energy relaxation mechanisms. The Journal of Physical Chemistry Letters, 4(12):2061–2068, 2013. 6 J. Bude and K. Hess. Thresholds of impact ionization in semiconductors. J. Appl. Phys., 72(8):3554 –3561, 1992. 7 H. K. Jung, K. Taniguchi, and C. Hamaguchi. Impact ionization model for full band monte carlo simulation in gaas. J. Appl. Phys., 79(5):2473–2480, 1996. 8 D. Harrison, R. A. Abram, and S. Brand. Characteristics of impact ionization rates in direct and indirect gap semiconductors. J. Appl. Phys., 85(12):8186–8192, 1999. 9 A. Nozik. Spectroscopy and hot electron relaxation dynamics in semiconductor quantum wells and quantum dots. Annual Review of Physical Chemistry, 52(1):193–231, 2001. 10 J. McGuire, M. Sykora, J. Joo, J. Pietryga, and V. Klimov. Apparent versus true carrier multiplication yields in semiconductor nanocrystals. Nano Letters, 10(6):2049–2057, 2010. 11 Nathaniel M. Gabor. Impact excitation and electronhole multiplication in graphene and carbon nanotubes. Accounts of Chemical Research, 46(6):1348–1357, 2013. 12 O. Semonin, J. Luther, S. Choi, H.-Y. Chen, J. Gao, A. J. Nozik, and M. C. Beard. Peak external photocurrent quantum efficiency exceeding 100% via meg in a quantum dot solar cell. Science, 334:1530–1533, 2011. 13 S. Wang, M. Khafizov, X. Tu, M. Zheng, and T. Krauss. Multiple exciton generation in single-walled carbon nanotubes. Nano Letters, 10(7):2381–2386, 2010. 14 Marcus L. Bohm, Tom C. Jellicoe, Maxim Tabachnyk, Nathaniel J. L. K. Davis, Florencia Wisnivesky-Rocca-Rivarola, Caterina Ducati, Bruno Ehrler, Artem A. Bakulin, and Neil C. Greenham. Lead telluride quantum dot solar cells displaying external quantum efficiencies exceeding 120%. Nano Letters, 15(12):7987– 7993, 2015. 15 Marcus L. Bohm, Maxim Tabachnyk, Florencia WisniveskyRocca-Rivarola, Tom C. Jellicoe, Caterina Ducati, Bruno Ehrler, and Neil C. Greenham. Multiple-exciton generation in lead selenide nanorod solar cells with external quantum efficiencies exceeding 120%. Nature Communications, 6:8259, 2015. 16 S. Saeed, P. de Weerd, C. Stallinga, F. Spoor, A. Houtepen, L. Siebbeles, and T. Gregorkiewicz. Carrier multiplication in germanium nanocrystals. Light: Science and Applications, 4:672– 679, 2015. 17 M. T. Trinh, R. Limpens, W. de Boer, J. Schins, L. Siebbeles, and T. Gregorkiewicz. Direct generation of multiple excitons in adjacent silicon nanocrystals revealed by induced absorption. Nature Photonics, 6:316–321, 2012. 18 C. Stolle, X. Lu, Y. Yu, R. Schaller, and B. Korgel. Efficient carrier multiplication in colloidal silicon nanorods. Nano Letters, 17(9):5580–5586, 2017.

19 N.

Gabor, Z. Zhong, K. Bosnick, J. Park, and P. McEuen. Extremely efficient multiple electron-hole pair generation in carbon nanotube photodiodes. Science, 325(5946):1367–1371, 2009. 20 J. Harris, R. Headrick, M. Semler, J. Fagan, M. Pasquali, and E. Hobbie. Impact of swcnt processing on nanotube-silicon heterojunctions. Nanoscale, 8:7969–7977, 2016. 21 K. Velizhanin and A. Piryatinski. Numerical study of carrier multiplication pathways in photoexcited nanocrystal and bulk forms of pbse. Phys. Rev. Lett., 106:207401, 2011. 22 Kirill A. Velizhanin and Andrei Piryatinski. Numerical analysis of carrier multiplication mechanisms in nanocrystalline and bulk forms of pbse and pbs. Phys. Rev. B, 86:165319, Oct 2012. 23 S. Kilina, D. Kilin, and S. Tretiak. Light-driven and phononassisted dynamics in organic and semiconductor nanostructures. Chemical Reviews, 115(12):5929–5978, 2015. 24 Andrei Kryjevski, Deyan Mihaylov, Svetlana Kilina, and Dmitri Kilin. Multiple exciton generation in chiral carbon nanotubes: Density functional theory based computation. The Journal of Chemical Physics, 147(15):154106, 2017. 25 L.P. Kadanoff and G. Baym. Quantum Statistical Mechanics. W.A. Benjamin Inc., New York, 1962. 26 L. V. Keldysh. Diagram technique for nonequilibrium processes. Zh. Eksp. Teor. Fiz., 47:1515–1527, 1964. [Sov. Phys. JETP20,1018(1965)]. 27 E. M. Lifshitz and L. P. Pitaevskii. Physical Kinetics. Pergamon Press, New York, 1st edition, 1981. 28 G. Kolesov and Y. Dahnovsky. Correlated electron dynamics in quantum-dot sensitized solar cell: Kadanoff-baym versus markovian approach. Phys. Rev. B, 85:241309, 2012. 29 G. Kolesov and Y. Dahnovsky. Chapter 5 - quantum correlation effects in electron dynamics in molecular wires and solar cells: The nonequilibrium green’s function approach. 30 A. Kryjevski, B. Gifford, S. Kilina, and D. Kilin. Theoretical predictions on efficiency of bi-exciton formation and dissociation in chiral carbon nanotubes. The Journal of Chemical Physics, 145(15):154112, 2016. 31 A. Kryjevski, D. Mihaylov, B. Gifford, and D. Kilin. Singlet fission in chiral carbon nanotubes: Density functional theory based computation. The Journal of Chemical Physics, 147(03):034106, 2017. 32 A. Kryjevski and D. Kilin. Enhanced multiple exciton generation in amorphous silicon nanowires and films. Molecular Physics, 114(3-4):365–379, 2016. 33 M. Rohlfing and S. Louie. Electron-hole excitations and optical spectra from first principles. Phys. Rev. B, 62:4927–4944, 2000. 34 G. Strinati. Effects of dynamical screening on resonances at inner-shell thresholds in semiconductors. Phys. Rev. B, 29(10):5718–5726, 1984. 35 A. L. Fetter and J.D. Walecka. Quantum Theory of ManyParticle Systems. McGraw-Hill, New York, 1971. 36 Walter R. Duncan and Oleg V. Prezhdo. Theoretical studies of photoinduced electron transfer in dye-sensitized tio2. Annual Review of Physical Chemistry, 58(1):143–184, 2007. 37 Colleen F. Craig, Walter R. Duncan, and Oleg V. Prezhdo. Trajectory surface hopping in the time-dependent kohn-sham approach for electron-nuclear dynamics. Phys. Rev. Lett., 95:163001, 2005. 38 G. Coker Ill B.W. van der Meer and S.-Y. Chen. Resonance Energy Transfer: Theory and Data. VCH Publishers (Now WileyVCH), New York, 1994. 39 O. Vydrov, J. Heyd, A. Krukau, and G. Scuseria. Importance of short-range versus long-range hartree-fock exchange for the performance of hybrid density functionals. The Journal of Chemical Physics, 125(7), 2006. 40 J. Heyd, G. Scuseria, and M. Ernzerhof. Erratum: “hybrid functionals based on a screened coulomb potential” [j. chem. phys. 118, 8207 (2003)]. The Journal of Chemical Physics, 124(21), 2006. 41 P. E. Bl¨ ochl. Projector augmented-wave method. Phys. Rev. B, 50:17953–17979, 1994.

ACS Paragon Plus Environment

Page 7 of 7

The Journal of Physical Chemistry Letters 7

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

42 G.

Kresse and D. Joubert. From ultrasoft pseudopotentials to the projector augmented-wave method. Phys. Rev. B, 59:1758–1775, 1999. 43 J. Perdew, K. Burke, and M. Ernzerhof. Generalized gradient approximation made simple. Phys. Rev. Lett., 77:3865–3868, Oct 1996.

44 J.

Perdew, K. Burke, and M. Ernzerhof. Generalized gradient approximation made simple [phys. rev. lett. 77, 3865 (1996)]. Phys. Rev. Lett., 78:1396–1396, Feb 1997. 45 Y. Han, D. Micha, and D. Kilin. Ab initio study of the photocurrent at the au/si metalsemiconductor nanointerface. Molecular Physics, 113(3-4):327–335, 2015. 46 Y. Han, S. Tretiak, and D. Kilin. Dynamics of charge transfer at au/si metal-semiconductor nano-interface. Molecular Physics, 112(3-4):474–484, 2014.

ACS Paragon Plus Environment