Structural and orientation effects on electronic energy transfer between silicon quantum dots with dopants and with silver adsorbates N. Vinson, H. Freitag, and D. A. Micha Citation: The Journal of Chemical Physics 140, 244709 (2014); doi: 10.1063/1.4884350 View online: http://dx.doi.org/10.1063/1.4884350 View Table of Contents: http://scitation.aip.org/content/aip/journal/jcp/140/24?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Quantum dynamical study of femtosecond photodesorption of CO from TiO2(110) J. Chem. Phys. 141, 084715 (2014); 10.1063/1.4893528 Cu adhesion on tantalum and ruthenium surface: Density functional theory study J. Appl. Phys. 107, 103534 (2010); 10.1063/1.3369443 Optimal surface functionalization of silicon quantum dots J. Chem. Phys. 128, 244714 (2008); 10.1063/1.2940735 Effect of displacement and distortion of potential energy surfaces and overlapping resonances of electronic transitions on surface-enhanced Raman scattering: Models and ab initio theoretical calculation J. Chem. Phys. 122, 094719 (2005); 10.1063/1.1859283 A combined molecular dynamics+quantum mechanics method for investigation of dynamic effects on local surface structures J. Chem. Phys. 120, 4939 (2004); 10.1063/1.1635802
This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 69.166.47.134 On: Sun, 07 Dec 2014 03:01:28
THE JOURNAL OF CHEMICAL PHYSICS 140, 244709 (2014)
Structural and orientation effects on electronic energy transfer between silicon quantum dots with dopants and with silver adsorbates N. Vinson, H. Freitag, and D. A. Michaa) Quantum Theory Project, Departments of Chemistry and Physics, University of Florida, Gainesville, Florida 32611, USA
(Received 3 April 2014; accepted 5 June 2014; published online 30 June 2014) Starting from the atomic structure of silicon quantum dots (QDs), and utilizing ab initio electronic structure calculations within the Förster resonance energy transfer (FRET) treatment, a model has been developed to characterize electronic excitation energy transfer between QDs. Electronic energy transfer rates, KEET , between selected identical pairs of crystalline silicon quantum dots systems, either bare, doped with Al or P, or adsorbed with Ag and Ag3 , have been calculated and analyzed to extend previous work on light absorption by QDs. The effects of their size and relative orientation on energy transfer rates for each system have also been considered. Using time-dependent density functional theory and the hybrid functional HSE06, the FRET treatment was employed to model electronic energy transfer rates within the dipole-dipole interaction approximation. Calculations with adsorbed Ag show that: (a) addition of Ag increases rates up to 100 times, (b) addition of Ag3 increases rates up to 1000 times, (c) collinear alignment of permanent dipoles increases transfer rates by an order of magnitude compared to parallel orientation, and (d) smaller QD-size increases transfer due to greater electronic orbitals overlap. Calculations with dopants show that: (a) p-type and n-type dopants enhance energy transfer up to two orders of magnitude, (b) surface-doping with P and centerdoping with Al show the greatest rates, and (c) KEET is largest for collinear permanent dipoles when the dopant is on the outer surface and for parallel permanent dipoles when the dopant is inside the QD. © 2014 AIP Publishing LLC. [http://dx.doi.org/10.1063/1.4884350] I. INTRODUCTION
Assemblies of quantum dots (QDs) show a large variety of optical and electronic properties relevant to photovoltaics and to biosensors, among many other applications.1–9 When a QD is electronically excited by light absorption, its energy can be transferred to other QDs at a rate that determines the efficiency of the energy transfer process. This rate depends on the atomic composition and structure of the involved QDs, their relative distance from one another as well as their relative orientation. Clusters of atoms making nanometer-sized QDs exhibit unique physical properties due to their three-dimensional quantum confinement of electrons. Arrays of interacting QDs are of interest in the preparation of novel photoconductive materials where electronic energy transfer between QDs in dielectric matrices plays a role. Of particular interest are the tunable optoelectronic properties of QD systems for multilayer solar cell developments. QDs are especially useful in this case since their broad absorption spectrum provides flexibility in choosing excitation light wavelengths. Silicon QDs are strong candidates for the active components in photovoltaic materials due to their convenient band gap structure,10–12 environmental compatibility, durability, and they provide alternatives to the current state of research focused on photoactive systems with heavy metal nanoparticle compounds such as CdSe QDs. In what follows a) Author to whom correspondence should be addressed. Electronic mail:
[email protected]
0021-9606/2014/140(24)/244709/10/$30.00
we concentrate on Si QDs with dopants and adsorbates, and their interaction through electronic energy transfer (EET). In our recent work,13–15 our research group has investigated the photodynamic properties of select Si QDs that were either doped with group III and V elements or contained adsorbed Ag clusters. Light absorption spectra were reported and compared with available theoretical and experimental results. Results indicate that the optical absorption properties of Si QDs are highly sensitive to system size as well as the addition of n-type and p-type dopants and the adsorption of clusters of Ag atoms. We have also calculated optical properties of Si slabs with dopants and adsorbed Ag clusters, finding similar trends.16–18 Other related studies have added to our understanding of doped Si QDs.19–21 The focus of this paper is on the potential of these structures to enhance EET between pairs of Si QDs. The characterization of the optoelectronic properties of nanostructures involving pairs of interacting Si QDs is essential to the understanding of the photodynamic properties of assemblies of these QDs and to the eventual implementation of related materials in photovoltaic devices. In addition, the following treatment combining Förster resonance energy transfer (FRET) and ab initio electronic structure from time dependent density functional theory (TDDFT) appears promising also to characterize the interaction between chromophores and quantum dots used to identify protein conformations. By serving as passive fluorescent probes, QDs can measure protein interactions, and in this connection the non-toxicity of Si QDs seems advantageous.
140, 244709-1
© 2014 AIP Publishing LLC
This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 69.166.47.134 On: Sun, 07 Dec 2014 03:01:28
244709-2
Vinson, Freitag, and Micha
The FRET treatment22 has been successfully used in many studies involving not only QDs but also their interaction with molecules and surfaces.23–25 Electronic energy transfer occurs when an excited QD emits light which is captured by a neighbor QD. Light emission and absorption involve electronic density fluctuations of light donor and acceptor and the interaction of their corresponding electric transition dipoles. Derivation of the rate of transfer using perturbation theory displays their related emission and absorption spectra and shows that transfer is likely where the spectra overlap, indication resonance energy transfer. The original treatment has been extended to account for the multipolar components of electronic densities in Coulomb interactions,26, 27 and temperature and many-electron effects.28, 29 The present study is an extension of our previous work, using here the FRET combined with ab initio electronic structure calculations of donor and acceptor spectra to obtain energy transfer rates between Si QDs. Crystalline nearly spherical structures are amenable to treatment of energy transfer within the dipole-dipole approximation and this is used in the present paper. Because our previous work showed that doping with either Al or P, or alternatively adsorbing Ag or Ag3 clusters on the Si QD surface, resulted in stronger light absorption compared with pure Si QDs, those structures have been chosen in the present study. Distances between QDs have been kept larger than QD diameters to avoid charge overlaps, and QD electric dipole orientations, parallel or collinear, have been chosen to learn about relative orientation effects. In what follows we present the theoretical and computational treatment applicable to oriented QDs, give results for rates of EET separately for QDs doped with P and Al inside and outside the dots, or with adsorbed Ag atoms, and discuss their relative values based on ab initio calculated electronic orbitals and overlaps of spectral densities. The Conclusions focus on the large effects of changing structures and orientations on rates of EET, and further comment on the relevance of the present work in other applications of FRET with oriented QDs.
J. Chem. Phys. 140, 244709 (2014)
II. THEORETICAL AND COMPUTATIONAL TREATMENT A. Quantum dot structures
The structures studied are those from our previous work.13–15 They were optimized to begin with using the PW91 density functional and a basis set of plane waves with the Vienna Ab initio Simulation Package (VASP),30 and were further allowed to relax to their final structures using the PW91, and PBE GGA functionals in the GAUSSIAN 09 package31 with the LANL2DZ basis set, and also similarly using the hybrid HSE functional,15 which has given excellent results for the electronic density of states of semiconductor compounds.32 Electronic energy transfer between structurally identical pairs of crystalline Si29 H36 and Si35 H36 systems was investigated. Studies were made for QDs doped with Al in the center of the QD (indicated by –i) as well as doped on the outside (–o), and also for Si29 H36 P with the dopant inside and outside. We also considered similar QDs with adsorbed Ag and Ag3 instead of dopants, to complement our characterization of the interaction between these nanoparticles. They are shown in Figs. 1 and 2, with Al in pink and Ag in pale blue. B. Förster treatment for oriented structures
Electronic energy transfer rates were calculated using the dipole approximation of the Förster treatment. As confirmed in Refs. 23 and 26 for sufficiently distant QDs and nearly spherical QDs, the dipole-dipole potential energy interaction provides an accurate characterization for transition probabilities. The energy transfer was calculated using both an isotropic average orientation and two specific orientations of the permanent electric dipoles of the QDs: Orientation I has the permanent dipole moments of the QDs pointed parallel to one another and Orientation II has the permanent dipole moments aligned end to end. They are shown in Fig. 3. Light absorption and energy transfer between QDs involve transition dipoles as shown in Fig. 4, where labels indicate state-to-state electronic transitions in acceptor and donor QDs.
FIG. 1. (Left) Si29 H36 Al-i and (Right) Si29 H36 Al-o structures. Al is shown in pink. Similarly, Si29 H36 P-i as well as Si29 H36 P-o were investigated. For every Si29 structure considered, a similar Si35 structure was also tested.
This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 69.166.47.134 On: Sun, 07 Dec 2014 03:01:28
244709-3
Vinson, Freitag, and Micha
J. Chem. Phys. 140, 244709 (2014)
FIG. 2. (Left) Si29 H36 Ag and (Right) Si29 H36 Ag3 nuclear structures. For each Si29 structure considered, a similar Si35 structure was also tested. Ag is shown in pale blue.
The detailed expression for the EET depends on the orientation of transition dipoles. Indicating the state-to-state transitions of the energy donor and acceptor with δ = (I → I ) and α = (J → J ), respectively, the corresponding matrix element of the dipole-dipole interaction potenA,D = καδ dα(A) dδ(D) /(4π ε0 R 3 ) in terms of tial energy is Vαδ the transition dipole magnitudes and the orientation facnδ − 3 nα . nR ( nR . nδ ) from unit vectors along tor καδ = nα . the transition dipole vectors and the inter-particle position which is taken as the relative position between vectorR, the centers of each QD. The rate of transfer follows from the standard perturbation expression,33 KEET order second (AD) 2 = (2π/¯) δ ρδ α |Vαδ | , where the initial density per unit energy of excited donor states is included. A simpler form arises when averaging over all possible orientations of the QDs, done by taking into account only the transition dipole moments that follow the selection rules for circularly polarized light. The result is23 κ 1 2 ∞ 2π KEET = Ddip (ε) Adip (ε) dε, ¯ 4π ε0 R 3 0 where Ddip (ε) = ND
ρδ |dδ |2 δ (ε − εδ + 2λ)
δ
Adip (ε) = NA
α
|dα |2 δ (ε − εα ) .
Here, Ddip (ε) and Adip (ε) are the normalized donor emission and acceptor absorption spectra, respectively, R is the distance between QD centers, and κ is a relative orientation factor. For an isotropically averaged orientation, κ 2 is typically set equal to 2/3 as calculated in Ref. 23. The Stokes energy shift, 2λ, describes the average difference between light absorption and emission energy changes due to atomic rearrangement and vibrations, and is calculated from the displacement of potential energy minima and curvature of ground and excited electronic states. The normalization constants NA and ND have been defined by the equations
∞
1/NA = 0
0
δ (ε − εα ) dε,
α ∞
1/ND =
δ (ε − εδ + 2λ) dε,
δ
where α δ(ε − εα ) and δ δ(ε − εδ + 2λ) are the density of states of the acceptor and donor, respectively. The normalized density of states of the acceptor and donor are in turn used to calculate the spectral overlap J as defined in the literature as ∞ J = NA ND δ (ε − εα ) δ (ε − εδ + 2λ) dε. 0
α
δ
FIG. 3. Si29 QDs with Al on the surface, for coplanar parallel and collinear orientations of their permanent electric dipoles.
This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 69.166.47.134 On: Sun, 07 Dec 2014 03:01:28
244709-4
Vinson, Freitag, and Micha
J. Chem. Phys. 140, 244709 (2014)
of atomic vibrational motions are instead introduced by using the Stokes shift, 2λ, which was calculated by another group34 for Si29 and Si35 nanoparticles using potential energies from LDA density functionals. For the Stokes shift calculations, the 2λ values calculated in the literature and used here for the bare Si29 and Si35 systems were 1.65 eV and 1.43 eV, respectively. The influence of dopants and adsorbates on determining the Stokes shift has been assumed negligible, as the bare systems’ total potential energies were relatively unperturbed by the addition or substitution of one or three atoms.
III. RESULTS AND DISCUSSION A. P and Al doped structures
FIG. 4. Transition dipole moments and angles used for KEET calculations. Orientations were selected such that transition dipole vectors and R were coplanar.
C. Spectral densities
The required spectral densities of donor and acceptor have been obtained within the TDDFT framework, using the GAUSSIAN 09 software with LANL2DZ as the basis set. In our recent publication,15 it was demonstrated that the semi-local hybrid functional, HSE/PBE, provides a significant advantage in terms of quantitative accuracy over the PW91/PW91, which explains the present choice in utilizing the HSE/PBE hybrid functional for calculations of excited states of the investigated structures. The absorption spectra were calculated using the normalized density of states and included the first 50 excited states of the acceptor. The emission spectra used the normalized density of states of only one excited state of the donor corresponding to the transition with the largest oscillator strength. The quantity ρ δ is the normalized Boltzmann weighting factor for the initially excited donor electronic states. In this work, vibronic structure of the emission band is not explicitly calculated, and therefore ρ δ is set equal to 1 for the excited electronic state and 0 for the ground electronic state. The effects
Table I shows the parameters that were used to calculate the Förster excitation rates KEET for each system. The donor transition energies and dipole moments squared were derived directly from the GAUSSIAN output, and correspond to the energy transition with the highest oscillator strength. The spectral overlap J as well as the values for D dip (ε) A dip (ε) dε are also given as computed by the equations in Sec. II. As observed in the table, the spectral overlap tends to be higher in Si35 H36 compared to Si29 H36 , and is expected to increase as the diameter of the QD becomes larger. Because the Stokes shift tends to decrease with increasing QD size, the distance between the peaks of the emission and absorption spectra also become smaller as the diameter of the QD increases, allowing greater spectral overlap between emission and donor. Compared to the spectral overlaps of CdSe quantum dots observed in recent studies,35 the spectral overlaps of Si QDs in this work are approximately two orders of magnitude smaller for bare Si and one order of magnitude smaller for doped silicon. An explanation for this result is the much smaller size of the QDs used in this work, with 4 nm-diameter CdSe quantum dots experimentally observed compared to the 1 nm-diameter of Si QDs used in this work. Also shown in Table I are larger transition dipoles for the smaller doped QDs due to larger MO overlaps in their more confined environments, and a weak positive correlation between the magnitude of the spectral overlap and the strength of the donor transition dipole moment squared. Both donor
TABLE I. Physical values for Förster energy transfer calculations on doped Si quantum dots.
System Si29 H36 Si35 H36 Si29 H36 -Al-i Si29 H36 -Al-o Si35 H36 -Al-i Si35 H36 -Al-o Si29 H36 -P-i Si29 H36 -P-o Si35 H36 -P-i Si35 H36 -P-o
Donor transition energy (eV)
Donor transition dipole moment squared (AU2 )
J spectral overlap (×10−7 cm)
5.45 5.29 3.20 2.94 3.06 2.90 3.27 2.59 4.07 2.59
0.264 0.281 0.837 0.529 0.794 0.377 0.203 0.357 0.172 0.282
15.5 23.9 370 50.2 284 143 131 36.9 247 177
D dip (ε) A dip (ε) dε (×10−120 C4 m4 eV−1 ) 0.421 0.947 104 5.69 17.1 7.35 5.47 17.6 6.82 40.7
This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 69.166.47.134 On: Sun, 07 Dec 2014 03:01:28
244709-5
Vinson, Freitag, and Micha
and acceptor transition dipole values are used in the calculation of the energy transfer rates. The trends between QDs in Table II show that energy transfer rates tend to increase with the addition of dopants. This is especially noticeable for Al added to the center and also evident for P added to the surface. This trend can be partially explained by the larger transition dipole moments of doped QDs and the greater overlap between emission and absorption spectra. Consistent with the data in Table I, bare Si quantum dots are about 4–6 orders of magnitude lower than the ones for the CdSe QDs in Ref. 35. This can again be explained by the larger size of the CdSe QDs (4 nm vs 1 nm in present diameters). Figure 5 shows the donor QD emission spectral densities (blue lines) and the acceptor QD absorption spectral densities (green lines) for doped QDs with the highest KEET rates. The QDs with the highest KEET rates tended to be those with Al doped in the center or P doped on the surface. The addition of these dopants considerably increased the amount of overlap between the emission and acceptor spectral densities compared to bare silicon.
J. Chem. Phys. 140, 244709 (2014) TABLE II. Förster energy transfer rates for doped Si quantum dots. KEET (kHz) isotropic average orientation R (Å)
Si29 H36
Si29 H36 Al-i
Si29 H36 Al-o
Si29 H36 P-i
20 40 60 80
132 2.07 0.182 0.0324
32500 508 44.6 7.93
1770 27.8 2.44 0.434
1710 26.7 2.35 0.418
Si29 H36 P-o 5490 85.8 7.53 1.34
KEET (kHz) isotropic average orientation R (Å) 20 40 60 80
Si35 H36 296 4.63 0.406 0.0723
Si35 H36 Al-i 5340 83.5 7.33 1.30
Si35 H36 Al-o 2300 35.9 3.15 0.561
Si35 H36 P-i 2130 33.3 2.92 .520
Si35 H36 P-o 12 700 198 17.4 3.10
For the aluminum-doped systems, the greatest rates were observed for systems with collinearly oriented permanent electric dipole moments. An additional increase in electronic energy transfer rate was observed when Al was positioned in the center of these systems. This may be attributed the p-type
FIG. 5. Spectral densities of crystalline (a) Si29 H36 , (b) Si29 H36 Al-i, (c) Si29 H36 P-o, (d) Si35 H36 , (e) Si35 H36 Al-i, (f) Si35 H36 P-o. Donor QD emission and acceptor QD absorption spectral densities are shown by blue and green lines, respectively.
This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 69.166.47.134 On: Sun, 07 Dec 2014 03:01:28
244709-6
Vinson, Freitag, and Micha
J. Chem. Phys. 140, 244709 (2014)
character of Al which induces the delocalization of electron density imposed by impurity within the system, causing the overall electronic structure to polarize charge more efficiently and thus produce an electric dipole. This spatial configuration provides a stronger electric dipole moment in the system and contributes to the overlap necessary for electronic energy transfer. To develop a physically intuitive argument as previously done for the doped systems in Sec. III A, the spatial orbitals of the adsorbed system that were involved with the highest electronic energy transfer rates are presented in Figures 6(a)–6(c). The orbitals with the highest probability of transition (associated with the maximum oscillator strength for the system) of the studied systems are displayed. Because degenerate transitions did exist, there were systems with multiple possible transitions for energy transfer, as illustrated in Figure 6(c). It is important to notice that the excitations are not necessarily from HOMO to LUMO, and it is common for non-valence electronic excitation to occur for such systems. As evidenced in Figure 6(c), for the Si29 H33 Ag3 system there were several degeneracies that were observed in the first 50 excited states; in fact, the HOMO to LUMO (MO 103 to 104) transition in this case was the 13th in probability for transition. The following discussion is based on results from the isotropic average orientation for the permanent electric dipole moments. For the phosphorous-doped systems, it was observed that the systems were most conducive to energy transfer when the P atom was placed on the surface of the structure. A similar coupling effect was observed with these systems with the parallel arrangement of the systems’ permanent electric dipole moments. The n-type character of the P atom contributes a localization of electron density on the surface of the QD (as opposed to the delocalization of electron density imposed by Al) that establishes a permanent electric dipole moment that encourages energy transfer within the studied systems. Overall, the doped systems responded well to electronic excitations and exhibited a particular preference for the collinear permanent dipole orientation. Between the optimal aluminum-doped and phosphorous-doped systems, the Si29 H36 Al-i system produced rates double those of the Si35 H36 P-o system. In order to understand this difference in rate, it is necessary to consider the underlying physical influences affecting the process. There is a quantum confinement effect compounding with a dopant effect. Although the phosphorous-doped system is a priori electronically
FIG. 6. (a). Si29 H36 Al-i orbitals involved with the excited electronic state transition of the greatest probability. The yellow arrow indicates the energy of excitation transition. For reference the HOMO is number 75 and the LUMO is 76. E = 3.20 eV is the transition energy. (b). Si35 H36 P-o orbitals involved with the excited electronic state transition of the greatest probability. The yellow arrow indicates the energy of excitation transition. For reference the HOMO is number 75 and the LUMO is 76. E = 2.59 eV transition. (c). Si29 H33 Ag3 orbitals involved with the excited state transitions of the greatest probability of occurrence. In this system, there were seven equal energy transitions, and only two transitions are shown for simplicity. Each yellow arrow indicates a specific transition. For reference the HOMO = MO 103 and LUMO = MO 104. E = 3.52 eV transition.
TABLE III. Parameters in Förster calculations for Ag adsorbed on Si quantum dots.
System Si29 H36 Si29 H35 Ag Si29 H33 Ag3 Si35 H36 Si35 H35 Ag Si35 H36 Ag3
Donor transition energy (eV)
Donor transition dipole moment squared (AU2 )
J spectral overlap (×10−7 cm)
5.45 3.38 3.52 5.29 3.47 3.50
0.264 5.46 3.62 0.281 6.66 3.43
15.4 4.50 86.3 23.8 6.23 49.9
D dip (ε) A dip (ε) dε (×10−120 C4 m4 eV−1 ) 0.421 40.7 7200 0.942 81.3 2860
This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 69.166.47.134 On: Sun, 07 Dec 2014 03:01:28
244709-7
Vinson, Freitag, and Micha
J. Chem. Phys. 140, 244709 (2014)
TABLE IV. Förster transfer rates for Ag adsorbed on Si quantum dots (isotropic average orientation). KEET (kHz) R (Å)
Si29 H36
Si29 H35 Ag
Si29 H33 Ag3
20 40 60 80
131 2.06 0.180 0.0322
12 700 199 17.5 3.12
225 000 3520 309 55.0
KEET (kHz) R (Å) 20 40 60 80
Si35 H36 295 4.61 0.404 0.0720
Si35 H35 Ag 25 400 398 34.9 6.21
Si35 H36 Ag3 89 700 1400 123 21.9
better suited for energy transfer with an induced dipole, the electronic energy transfer rates appear to benefit to a larger extent from the quantum confinement effect in the aluminum-doped system. B. Structures with adsorbed Ag and Ag3
Energy transfer rates and spectral function overlaps were calculated for each pair of isotropically oriented adsorbed systems and are listed in Tables III and IV. Structures containing adsorbed Ag were as generated in Ref. 14. From the calculated results in Table IV, the adsorption of Ag appears to significantly impact the electronic energy transfer rates. Silver presence displays a non-linear effect on the rate as can be shown by a comparison of the magnitude of the rates of singly and triply substituted systems. The single Ag atom replacement of a passive hydrogen atom on the system results in an increase of electronic energy transfer rates by two orders of magnitude. However, the effect of the Ag
presence appears to decrease as the triply substituted system only undergoes in addition an approximate 20-fold increase in transfer rates for Si29 and a four-fold increase in transfer rates for Si35 . As was observed with the doped systems, the quantum confinement effect couples with the adsorbate presence to determine the electronic energy transfer rates. The singly substituted Si29 system shows rates that are half of the equivalent Si35 system to demonstrate the size effect; however, the triply substituted Si29 system shows rates that are twice the equivalent Si35 system to demonstrate the quantum confinement effect for Ag3 . Overall, the net effect from confinement and the adsorbate is confirming that the former maintains a larger role in determining electronic energy transfer rates than the adsorbate effect alone on these systems as similarly described with the doped systems in Sec. II A. Figure 7 contains the spectral densities for donor and acceptor, respectively, in blue and red. It shows that going from a single Ag atom to three has a large effect on the acceptor spectral density, and it gives a larger overlap with the donor density. In the spectra above, adsorbed QDs had greater spectral density overlap than the corresponding pure systems. A relevant observation is that the overlap for the doped systems in Figure 5 was greater than for the adsorbed systems in Figure 7. However, the electronic energy transfer rates were superior in the adsorbed systems; this difference is explained by the transition electric dipole moment strength that is larger for the adsorbed systems than for the doped systems.
C. Effects of orientation
Table V shows rates of energy transfer for QDs oriented coplanar as seen in Figure 4, for parallel and collinear orientations of permanent dipoles. The orientation factors are calculated for each transition dipole vector (usually different from
FIG. 7. Spectral densities of emitting (blue lines) and absorbing (red lines) bare silicon dots and dots with adsorbed silver.
This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 69.166.47.134 On: Sun, 07 Dec 2014 03:01:28
244709-8
Vinson, Freitag, and Micha
J. Chem. Phys. 140, 244709 (2014)
TABLE V. Effects of orientations I and II on energy transfer rates of doped Si and Ag adsorbed on Si QD.
Orientation
QD pair
Transition energy (eV)
(κ αδ )2
KEET at R = 30 Å (kHz)
KEET at R = 30 Å (kHz) [isotropic average]
Si29 H36 Si29 H36 -Al-i Si29 H36 -Al-o Si29 H36 -P-i Si29 H36 -P-o Si29 H35 Ag Si29 H33 Ag3 Si35 H36 Si35 H36 -Al-i Si35 H36 -Al-o Si35 H36 -P-i Si35 H36 -P-o Si35 H35 Ag Si35 H33 Ag3
5.45 3.20 2.94 3.27 3.76 3.38 3.52 5.29 3.06 3.79 4.07 2.59 3.47 3.50
0.993 0.192 0.937 0.768 1.11 0.999 0.999 0.800 0.970 0.0168 0.0293 0.999 0.994 0.999
17.2 824 219 173 1020 1680 29 600 31.1 682 9.09 8.22 1670 3330 11 800
11.5 2850 156 150 612 1120 19 700 25.9 468 360 186 1115 2230 7880
Si29 H36 Si29 H36 -Al-i Si29 H36 -Al-o Si29 H36 -P-i Si29 H36 -P-o Si29 H35 Ag Si29 H33 Ag3 Si35 H36 Si35 H36 -Al-i Si35 H36 -Al-o Si35 H36 -P-i Si35 H36 -P-o Si35 H35 Ag Si35 H33 Ag3
5.45 3.20 2.94 3.27 3.76 3.38 3.52 5.29 3.06 3.79 4.07 2.59 3.47 3.50
0.923 1.37 0.990 0.507 0.791 3.95 3.99 0.143 0.691 0.991 0.364 3.98 3.86 3.99
16.0 5890 231 114 727 6640 118 000 5.57 486 536 102 6660 12 900 47 200
11.5 2850 156 150 612 1120 19 700 25.9 468 360 186 1115 2230 7880
D A
dD dA ↑
↑
D A
dD dA → →
FIG. 8. Electronic energy transfer rates plotted against radial distance between QDs. The systems with the highest and lowest rates in the doped series and the adsorbed series were chosen to compare with the pure systems. For isotropic average orientation, κ 2 is set equal to 2/3. [cNX-z means crystalline SiN Hsaturation Xinner or SiN Hsaturation X-outer]
This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 69.166.47.134 On: Sun, 07 Dec 2014 03:01:28
244709-9
Vinson, Freitag, and Micha
permanent dipole orientations) and the value shown in Table V is for the acceptor absorption with the largest oscillator strength. The last column shows results for isotropic averages for comparison, and is the same for both arrangements. As shown in Table V, both orientations resulted in an increased κ αδ 2 factor compared to the average orientation and therefore an increased energy transfer rate. From Figure 8, it is evident that the introduction of adsorbates enhances the electronic energy transfer for all systems in an isotropic permanent dipole arrangement. This is also shown for all oriented permanent dipole arrangements except for one case in parallel configuration, where the parallel arrangement of the dipoles hinders energy transfer in the Si35 H36 P- i system. It is clear graphically in Figure 8 that the ideal orientation for both doped and adsorbed systems is the collinear configuration. This can be attributed to the appearance of various dipoles created by the adsorbates, that encourages delocalization of electron density and contributes to increased energy transfer rates.
IV. CONCLUSIONS
A combination of the Förster perturbation theory of electronic energy transfer and ab initio electronic structure calculations with hybrid time-dependent density functionals has provided a versatile tool for modeling a series of Si-based quantum dots. In review, it has been observed that size and composition are effective independent variables for controlling the electronic energy transfer rates between the Si quantum dots. Magnitudes of energy transfer rates have been shown to depend strongly on both quantum confinement and atomic composition, leading to changes of several orders of magnitude. Doping with either Al or P dramatically increases energy transfer rates by up to two orders of magnitude as demonstrated for the collinearly oriented Si35 H36 P-o system compared to the equivalent isotropic pure Si QD system. However, the highest energy transfer rates were exhibited by the systems with adsorbed Ag, with a peak rate observed for the collinearly oriented Si29 H33 Ag3 at four orders of magnitude over the equivalent isotropic pure Si QD system. From the results and discussion discussed in Sec. III, it appears that these nanoparticles could play a potentially large role in the design of novel photovoltaic designs. To summarize, the present calculations with adsorbed Ag show that: (a) addition of Ag increases rates up to 100 times; (b) addition of Ag3 increases rates up to 100 times; (c) collinear alignment of permanent dipoles increases transfer rates by an order of magnitude compared to parallel orientation; and (d) smaller QD-size increases transfer due to greater electronic orbitals overlap. Calculations with dopants show that: (a) p-type and n-type dopants enhance energy transfer up to two orders of magnitude; (b) surface-doping with P and center-doping with Al show the greatest rates; and (c) KEET is largest for collinear permanent dipoles when the dopant is on the outer surface and for parallel permanent dipoles when the dopant is inside the QD. The spectral overlap between our QDs are comparable to the overlap for other reported QDs: of the order of 10−5 cm
J. Chem. Phys. 140, 244709 (2014)
in our work compared to 10−4 cm for CdSe QDs.35 Related energy transfer rates are also smaller for the present QDs, and this can be attributed to the much smaller size of our QDs. As obtained in Ref. 22 Stokes shift values are typically calculated utilizing the HOMO-LUMO transitions, which were subsequently used in this investigation, as it was assumed that the shifts did not vary significantly across system transitions. This is a limitation which could be validated by doing more extensive calculations of potential energy changes with bond distances for each electronic transition, but it is not likely to change our conclusions involving rate changes of several orders of magnitude. Regarding the implications of this work for future developments of photovoltaic materials, issues of solar cell efficiencies and of toxicity of QDs are of interest. Since theoretical constraints were established on maximum solar energy harvesting by the Shockley-Queisser limit,36 the scientific community has strived to develop photovoltaic materials that can approach the theoretical maximum efficiency for light conversion into electricity. The efficiency limit was derived for a single p-n junction, and has since inspired the development of tandem solar cells1–5 as a means to approach higher energy conversion efficiency by introducing multi-junction cells. In brief, the interest in tandem solar cells is motivated by their spread of the photonic load across optimized layers of optically active materials; this spread allows for controlled absorption of incident photons at layers with chosen energy band gaps. Doped and adsorbed Si quantum dots have been both theoretically9 as well as experimentally validated37 as worthy candidates for implementation in such matrices. The combination of Förster resonance energy transfer and ab initio electronic structure from TDDFT appears promising also to characterize the interaction between chromophores and quantum dots used to identify protein conformations, as described in the biomedical literature.38,24 By serving as passive fluorescent probes,39 QDs can measure protein interactions by experiencing nonradiative electronic energy excitations that do not themselves influence the desired observed phenomenon, and in this connection the nontoxicity of Si QDs seems advantageous.
ACKNOWLEDGMENTS
This material is based upon work supported financially by the National Science Foundation grant (Grant No. NSF CHE-1011967) and by the Dreyfus Foundation. Computations were performed at the University of Florida High Performance Computing facility. 1 A.
P. Alivisatos, Science 271, 933 (1996). J. Nozik, Ann. Rev. Phys. Chem. 52, 193 (2001). 3 N. S. Lewis, Science 315, 798 (2007). 4 P. V. Kamat, J. Phys. Chem. C 112, 18737 (2008). 5 G. Conibeer, M. Green, E-C. Cho, D. Koenig et al., Thin Solid Films 516, 6748 (2008). 6 A. J. Nozik, Nano Lett. 10, 2735 (2010). 7 Y. Kanai, J. B. Neaton, and J. C. Grossman, Comput. Sci. Eng. 12, 18 (2010). 8 W. A. Tisdale and Y. X. Zhu, Proc. Natl. Acad. Sci. USA 108, 965–970 (2011). 9 J-W. Luo, P. Stradins, and A. Zunger, Energy Environ. Sci. 4, 2546 (2011). 2 A.
This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 69.166.47.134 On: Sun, 07 Dec 2014 03:01:28
244709-10 10 T.
Vinson, Freitag, and Micha
Takagahara and K. Takeda, Phys. Rev. B 46, 15578 (1992).
11 I. Vasiliev, S. Ogut, and J. R. Chelikowsky, Phys. Rev. B 65, 115416 (2002). 12 A.
Puzder, A. J. Williamson, J. C. Grossman, and G. Galli, J. Am. Chem. Soc. 125, 2786 (2003). 13 M. G. Mavros, D. A. Micha, and D. S. Kilin, J. Phys. Chem. C 115, 19529 (2011). 14 D. M. Stewart, M. G. Mavros, and D. A. Micha, J. Phys. Chem. C 116, 23107 (2012). 15 H. Freitag, M. G. Mavros, and D. A. Micha, J. Chem. Phys. 137, 144301 (2012). 16 D. S. Kilin and D. A. Micha, J. Phys. Chem. C 115, 770 (2011). 17 J. J. Ramirez, D. S. Kilin, and D. A. Micha, Int. J. Quantum Chem. 112, 300 (2012). 18 T. Vazhappilly, D. S. Kilin, and D. A. Micha, J. Phys. Chem. C 116, 25525 (2012). 19 Z-Y. Zhou, M. L. Steigerwald, R. A. Friesner, L. Brus, and M. S. Hybertsen, Phys. Rev. B 71, 245308 (2005). 20 S. A. Fischer and O. V. Prezhdo, J. Phys. Chem. C 115, 10006 (2011). 21 J. Chen, A. Schmitz, and D. S. Kilin, Int. J. Quantum Chem. 112, 3879 (2012). 22 Th. Foerster, in Modern Quantum Chemistry, Part III, edited by O. Sinanoglu (Academic Press, New York, 1965), p. 93. 23 G. D. Scholes, Annu. Rev. Phys. Chem. 54, 57 (2003). 24 A. R. Clapp, I. L. Medintz, and H. Mattoussi, ChemPhysChem 7, 47 (2006).
J. Chem. Phys. 140, 244709 (2014) 25 C.
F. Kaminski, E. Sackmann, and K. Schulten, ChemPhysChem 12, 423 (2011); Special issue on Th. Foerster’s science. 26 C. Curutchet, A. Franceschetti, A. Zunger, and G. D. Scholes, J. Phys. Chem. C 112, 13336–13341 (2008). 27 R. Baer and E. Rabani, J. Chem. Phys. 128, 184710 (2008). 28 J. Megow, B. Roeder, A. Kulesza, V. Bonacic-Koutecky, and V. May, ChemPhysChem 12, 645 (2011). 29 Z. Lin, H. Li, A. Franceschetti, and M. T. Lusk, ACS Nano 6, 4029 (2012). 30 J. Hafner, J. Comput. Chem. 29, 2044 (2008). 31 M. J. Frisch, G. W. Trucks, H. B. Schlegel et al., Gaussian 09, Revision D.01, Gaussian, Inc., Wallingford CT, 2009. 32 A. V. Krukau, O. A. Vydrov, A. F. Izmaylov, and G. E. Scuseria, J. Chem. Phys. 125, 224106 (2006). 33 P. W. Atkins and R. S. Friedman, Molecular Quantum Mechanics, 3rd ed. (Oxford University Press, Oxford, England, 1997), Chap. 6. 34 M. L. del Puerto, M. Jain, and J. R. Chelikowsky, Phys. Rev. B 81, 035309 (2010). 35 G. D. Scholes and D. L. Andrews, Phys. Rev. B 72, 125331 (2005). 36 W. Shockley and H. J. Queisser, J. Appl. Phys. 32, 510 (1961). 37 E. Cho, S. W. Park, X. Hao, D. Song, G. Conibeer, S. C. Park, and M. A. Green, Nano 19, 245201 (2008). 38 A. P. Alivisatos and W. Gu, Annu. Rev. Biomed. Eng. 7, 55 (2005). 39 R. Q. Zhang, A. DeSarkar, T. A. Niehaus, and T. Frauenheim, Phys. Status Solidi B 249, 401 (2012).
This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 69.166.47.134 On: Sun, 07 Dec 2014 03:01:28
This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 69.166.47.134 On: Sun, 07 Dec 2014 03:01:28
THE JOURNAL OF CHEMICAL PHYSICS 140, 244709 (2014)
Structural and orientation effects on electronic energy transfer between silicon quantum dots with dopants and with silver adsorbates N. Vinson, H. Freitag, and D. A. Michaa) Quantum Theory Project, Departments of Chemistry and Physics, University of Florida, Gainesville, Florida 32611, USA
(Received 3 April 2014; accepted 5 June 2014; published online 30 June 2014) Starting from the atomic structure of silicon quantum dots (QDs), and utilizing ab initio electronic structure calculations within the Förster resonance energy transfer (FRET) treatment, a model has been developed to characterize electronic excitation energy transfer between QDs. Electronic energy transfer rates, KEET , between selected identical pairs of crystalline silicon quantum dots systems, either bare, doped with Al or P, or adsorbed with Ag and Ag3 , have been calculated and analyzed to extend previous work on light absorption by QDs. The effects of their size and relative orientation on energy transfer rates for each system have also been considered. Using time-dependent density functional theory and the hybrid functional HSE06, the FRET treatment was employed to model electronic energy transfer rates within the dipole-dipole interaction approximation. Calculations with adsorbed Ag show that: (a) addition of Ag increases rates up to 100 times, (b) addition of Ag3 increases rates up to 1000 times, (c) collinear alignment of permanent dipoles increases transfer rates by an order of magnitude compared to parallel orientation, and (d) smaller QD-size increases transfer due to greater electronic orbitals overlap. Calculations with dopants show that: (a) p-type and n-type dopants enhance energy transfer up to two orders of magnitude, (b) surface-doping with P and centerdoping with Al show the greatest rates, and (c) KEET is largest for collinear permanent dipoles when the dopant is on the outer surface and for parallel permanent dipoles when the dopant is inside the QD. © 2014 AIP Publishing LLC. [http://dx.doi.org/10.1063/1.4884350] I. INTRODUCTION
Assemblies of quantum dots (QDs) show a large variety of optical and electronic properties relevant to photovoltaics and to biosensors, among many other applications.1–9 When a QD is electronically excited by light absorption, its energy can be transferred to other QDs at a rate that determines the efficiency of the energy transfer process. This rate depends on the atomic composition and structure of the involved QDs, their relative distance from one another as well as their relative orientation. Clusters of atoms making nanometer-sized QDs exhibit unique physical properties due to their three-dimensional quantum confinement of electrons. Arrays of interacting QDs are of interest in the preparation of novel photoconductive materials where electronic energy transfer between QDs in dielectric matrices plays a role. Of particular interest are the tunable optoelectronic properties of QD systems for multilayer solar cell developments. QDs are especially useful in this case since their broad absorption spectrum provides flexibility in choosing excitation light wavelengths. Silicon QDs are strong candidates for the active components in photovoltaic materials due to their convenient band gap structure,10–12 environmental compatibility, durability, and they provide alternatives to the current state of research focused on photoactive systems with heavy metal nanoparticle compounds such as CdSe QDs. In what follows a) Author to whom correspondence should be addressed. Electronic mail:
[email protected]
0021-9606/2014/140(24)/244709/10/$30.00
we concentrate on Si QDs with dopants and adsorbates, and their interaction through electronic energy transfer (EET). In our recent work,13–15 our research group has investigated the photodynamic properties of select Si QDs that were either doped with group III and V elements or contained adsorbed Ag clusters. Light absorption spectra were reported and compared with available theoretical and experimental results. Results indicate that the optical absorption properties of Si QDs are highly sensitive to system size as well as the addition of n-type and p-type dopants and the adsorption of clusters of Ag atoms. We have also calculated optical properties of Si slabs with dopants and adsorbed Ag clusters, finding similar trends.16–18 Other related studies have added to our understanding of doped Si QDs.19–21 The focus of this paper is on the potential of these structures to enhance EET between pairs of Si QDs. The characterization of the optoelectronic properties of nanostructures involving pairs of interacting Si QDs is essential to the understanding of the photodynamic properties of assemblies of these QDs and to the eventual implementation of related materials in photovoltaic devices. In addition, the following treatment combining Förster resonance energy transfer (FRET) and ab initio electronic structure from time dependent density functional theory (TDDFT) appears promising also to characterize the interaction between chromophores and quantum dots used to identify protein conformations. By serving as passive fluorescent probes, QDs can measure protein interactions, and in this connection the non-toxicity of Si QDs seems advantageous.
140, 244709-1
© 2014 AIP Publishing LLC
This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 69.166.47.134 On: Sun, 07 Dec 2014 03:01:28
244709-2
Vinson, Freitag, and Micha
The FRET treatment22 has been successfully used in many studies involving not only QDs but also their interaction with molecules and surfaces.23–25 Electronic energy transfer occurs when an excited QD emits light which is captured by a neighbor QD. Light emission and absorption involve electronic density fluctuations of light donor and acceptor and the interaction of their corresponding electric transition dipoles. Derivation of the rate of transfer using perturbation theory displays their related emission and absorption spectra and shows that transfer is likely where the spectra overlap, indication resonance energy transfer. The original treatment has been extended to account for the multipolar components of electronic densities in Coulomb interactions,26, 27 and temperature and many-electron effects.28, 29 The present study is an extension of our previous work, using here the FRET combined with ab initio electronic structure calculations of donor and acceptor spectra to obtain energy transfer rates between Si QDs. Crystalline nearly spherical structures are amenable to treatment of energy transfer within the dipole-dipole approximation and this is used in the present paper. Because our previous work showed that doping with either Al or P, or alternatively adsorbing Ag or Ag3 clusters on the Si QD surface, resulted in stronger light absorption compared with pure Si QDs, those structures have been chosen in the present study. Distances between QDs have been kept larger than QD diameters to avoid charge overlaps, and QD electric dipole orientations, parallel or collinear, have been chosen to learn about relative orientation effects. In what follows we present the theoretical and computational treatment applicable to oriented QDs, give results for rates of EET separately for QDs doped with P and Al inside and outside the dots, or with adsorbed Ag atoms, and discuss their relative values based on ab initio calculated electronic orbitals and overlaps of spectral densities. The Conclusions focus on the large effects of changing structures and orientations on rates of EET, and further comment on the relevance of the present work in other applications of FRET with oriented QDs.
J. Chem. Phys. 140, 244709 (2014)
II. THEORETICAL AND COMPUTATIONAL TREATMENT A. Quantum dot structures
The structures studied are those from our previous work.13–15 They were optimized to begin with using the PW91 density functional and a basis set of plane waves with the Vienna Ab initio Simulation Package (VASP),30 and were further allowed to relax to their final structures using the PW91, and PBE GGA functionals in the GAUSSIAN 09 package31 with the LANL2DZ basis set, and also similarly using the hybrid HSE functional,15 which has given excellent results for the electronic density of states of semiconductor compounds.32 Electronic energy transfer between structurally identical pairs of crystalline Si29 H36 and Si35 H36 systems was investigated. Studies were made for QDs doped with Al in the center of the QD (indicated by –i) as well as doped on the outside (–o), and also for Si29 H36 P with the dopant inside and outside. We also considered similar QDs with adsorbed Ag and Ag3 instead of dopants, to complement our characterization of the interaction between these nanoparticles. They are shown in Figs. 1 and 2, with Al in pink and Ag in pale blue. B. Förster treatment for oriented structures
Electronic energy transfer rates were calculated using the dipole approximation of the Förster treatment. As confirmed in Refs. 23 and 26 for sufficiently distant QDs and nearly spherical QDs, the dipole-dipole potential energy interaction provides an accurate characterization for transition probabilities. The energy transfer was calculated using both an isotropic average orientation and two specific orientations of the permanent electric dipoles of the QDs: Orientation I has the permanent dipole moments of the QDs pointed parallel to one another and Orientation II has the permanent dipole moments aligned end to end. They are shown in Fig. 3. Light absorption and energy transfer between QDs involve transition dipoles as shown in Fig. 4, where labels indicate state-to-state electronic transitions in acceptor and donor QDs.
FIG. 1. (Left) Si29 H36 Al-i and (Right) Si29 H36 Al-o structures. Al is shown in pink. Similarly, Si29 H36 P-i as well as Si29 H36 P-o were investigated. For every Si29 structure considered, a similar Si35 structure was also tested.
This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 69.166.47.134 On: Sun, 07 Dec 2014 03:01:28
244709-3
Vinson, Freitag, and Micha
J. Chem. Phys. 140, 244709 (2014)
FIG. 2. (Left) Si29 H36 Ag and (Right) Si29 H36 Ag3 nuclear structures. For each Si29 structure considered, a similar Si35 structure was also tested. Ag is shown in pale blue.
The detailed expression for the EET depends on the orientation of transition dipoles. Indicating the state-to-state transitions of the energy donor and acceptor with δ = (I → I ) and α = (J → J ), respectively, the corresponding matrix element of the dipole-dipole interaction potenA,D = καδ dα(A) dδ(D) /(4π ε0 R 3 ) in terms of tial energy is Vαδ the transition dipole magnitudes and the orientation facnδ − 3 nα . nR ( nR . nδ ) from unit vectors along tor καδ = nα . the transition dipole vectors and the inter-particle position which is taken as the relative position between vectorR, the centers of each QD. The rate of transfer follows from the standard perturbation expression,33 KEET order second (AD) 2 = (2π/¯) δ ρδ α |Vαδ | , where the initial density per unit energy of excited donor states is included. A simpler form arises when averaging over all possible orientations of the QDs, done by taking into account only the transition dipole moments that follow the selection rules for circularly polarized light. The result is23 κ 1 2 ∞ 2π KEET = Ddip (ε) Adip (ε) dε, ¯ 4π ε0 R 3 0 where Ddip (ε) = ND
ρδ |dδ |2 δ (ε − εδ + 2λ)
δ
Adip (ε) = NA
α
|dα |2 δ (ε − εα ) .
Here, Ddip (ε) and Adip (ε) are the normalized donor emission and acceptor absorption spectra, respectively, R is the distance between QD centers, and κ is a relative orientation factor. For an isotropically averaged orientation, κ 2 is typically set equal to 2/3 as calculated in Ref. 23. The Stokes energy shift, 2λ, describes the average difference between light absorption and emission energy changes due to atomic rearrangement and vibrations, and is calculated from the displacement of potential energy minima and curvature of ground and excited electronic states. The normalization constants NA and ND have been defined by the equations
∞
1/NA = 0
0
δ (ε − εα ) dε,
α ∞
1/ND =
δ (ε − εδ + 2λ) dε,
δ
where α δ(ε − εα ) and δ δ(ε − εδ + 2λ) are the density of states of the acceptor and donor, respectively. The normalized density of states of the acceptor and donor are in turn used to calculate the spectral overlap J as defined in the literature as ∞ J = NA ND δ (ε − εα ) δ (ε − εδ + 2λ) dε. 0
α
δ
FIG. 3. Si29 QDs with Al on the surface, for coplanar parallel and collinear orientations of their permanent electric dipoles.
This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 69.166.47.134 On: Sun, 07 Dec 2014 03:01:28
244709-4
Vinson, Freitag, and Micha
J. Chem. Phys. 140, 244709 (2014)
of atomic vibrational motions are instead introduced by using the Stokes shift, 2λ, which was calculated by another group34 for Si29 and Si35 nanoparticles using potential energies from LDA density functionals. For the Stokes shift calculations, the 2λ values calculated in the literature and used here for the bare Si29 and Si35 systems were 1.65 eV and 1.43 eV, respectively. The influence of dopants and adsorbates on determining the Stokes shift has been assumed negligible, as the bare systems’ total potential energies were relatively unperturbed by the addition or substitution of one or three atoms.
III. RESULTS AND DISCUSSION A. P and Al doped structures
FIG. 4. Transition dipole moments and angles used for KEET calculations. Orientations were selected such that transition dipole vectors and R were coplanar.
C. Spectral densities
The required spectral densities of donor and acceptor have been obtained within the TDDFT framework, using the GAUSSIAN 09 software with LANL2DZ as the basis set. In our recent publication,15 it was demonstrated that the semi-local hybrid functional, HSE/PBE, provides a significant advantage in terms of quantitative accuracy over the PW91/PW91, which explains the present choice in utilizing the HSE/PBE hybrid functional for calculations of excited states of the investigated structures. The absorption spectra were calculated using the normalized density of states and included the first 50 excited states of the acceptor. The emission spectra used the normalized density of states of only one excited state of the donor corresponding to the transition with the largest oscillator strength. The quantity ρ δ is the normalized Boltzmann weighting factor for the initially excited donor electronic states. In this work, vibronic structure of the emission band is not explicitly calculated, and therefore ρ δ is set equal to 1 for the excited electronic state and 0 for the ground electronic state. The effects
Table I shows the parameters that were used to calculate the Förster excitation rates KEET for each system. The donor transition energies and dipole moments squared were derived directly from the GAUSSIAN output, and correspond to the energy transition with the highest oscillator strength. The spectral overlap J as well as the values for D dip (ε) A dip (ε) dε are also given as computed by the equations in Sec. II. As observed in the table, the spectral overlap tends to be higher in Si35 H36 compared to Si29 H36 , and is expected to increase as the diameter of the QD becomes larger. Because the Stokes shift tends to decrease with increasing QD size, the distance between the peaks of the emission and absorption spectra also become smaller as the diameter of the QD increases, allowing greater spectral overlap between emission and donor. Compared to the spectral overlaps of CdSe quantum dots observed in recent studies,35 the spectral overlaps of Si QDs in this work are approximately two orders of magnitude smaller for bare Si and one order of magnitude smaller for doped silicon. An explanation for this result is the much smaller size of the QDs used in this work, with 4 nm-diameter CdSe quantum dots experimentally observed compared to the 1 nm-diameter of Si QDs used in this work. Also shown in Table I are larger transition dipoles for the smaller doped QDs due to larger MO overlaps in their more confined environments, and a weak positive correlation between the magnitude of the spectral overlap and the strength of the donor transition dipole moment squared. Both donor
TABLE I. Physical values for Förster energy transfer calculations on doped Si quantum dots.
System Si29 H36 Si35 H36 Si29 H36 -Al-i Si29 H36 -Al-o Si35 H36 -Al-i Si35 H36 -Al-o Si29 H36 -P-i Si29 H36 -P-o Si35 H36 -P-i Si35 H36 -P-o
Donor transition energy (eV)
Donor transition dipole moment squared (AU2 )
J spectral overlap (×10−7 cm)
5.45 5.29 3.20 2.94 3.06 2.90 3.27 2.59 4.07 2.59
0.264 0.281 0.837 0.529 0.794 0.377 0.203 0.357 0.172 0.282
15.5 23.9 370 50.2 284 143 131 36.9 247 177
D dip (ε) A dip (ε) dε (×10−120 C4 m4 eV−1 ) 0.421 0.947 104 5.69 17.1 7.35 5.47 17.6 6.82 40.7
This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 69.166.47.134 On: Sun, 07 Dec 2014 03:01:28
244709-5
Vinson, Freitag, and Micha
and acceptor transition dipole values are used in the calculation of the energy transfer rates. The trends between QDs in Table II show that energy transfer rates tend to increase with the addition of dopants. This is especially noticeable for Al added to the center and also evident for P added to the surface. This trend can be partially explained by the larger transition dipole moments of doped QDs and the greater overlap between emission and absorption spectra. Consistent with the data in Table I, bare Si quantum dots are about 4–6 orders of magnitude lower than the ones for the CdSe QDs in Ref. 35. This can again be explained by the larger size of the CdSe QDs (4 nm vs 1 nm in present diameters). Figure 5 shows the donor QD emission spectral densities (blue lines) and the acceptor QD absorption spectral densities (green lines) for doped QDs with the highest KEET rates. The QDs with the highest KEET rates tended to be those with Al doped in the center or P doped on the surface. The addition of these dopants considerably increased the amount of overlap between the emission and acceptor spectral densities compared to bare silicon.
J. Chem. Phys. 140, 244709 (2014) TABLE II. Förster energy transfer rates for doped Si quantum dots. KEET (kHz) isotropic average orientation R (Å)
Si29 H36
Si29 H36 Al-i
Si29 H36 Al-o
Si29 H36 P-i
20 40 60 80
132 2.07 0.182 0.0324
32500 508 44.6 7.93
1770 27.8 2.44 0.434
1710 26.7 2.35 0.418
Si29 H36 P-o 5490 85.8 7.53 1.34
KEET (kHz) isotropic average orientation R (Å) 20 40 60 80
Si35 H36 296 4.63 0.406 0.0723
Si35 H36 Al-i 5340 83.5 7.33 1.30
Si35 H36 Al-o 2300 35.9 3.15 0.561
Si35 H36 P-i 2130 33.3 2.92 .520
Si35 H36 P-o 12 700 198 17.4 3.10
For the aluminum-doped systems, the greatest rates were observed for systems with collinearly oriented permanent electric dipole moments. An additional increase in electronic energy transfer rate was observed when Al was positioned in the center of these systems. This may be attributed the p-type
FIG. 5. Spectral densities of crystalline (a) Si29 H36 , (b) Si29 H36 Al-i, (c) Si29 H36 P-o, (d) Si35 H36 , (e) Si35 H36 Al-i, (f) Si35 H36 P-o. Donor QD emission and acceptor QD absorption spectral densities are shown by blue and green lines, respectively.
This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 69.166.47.134 On: Sun, 07 Dec 2014 03:01:28
244709-6
Vinson, Freitag, and Micha
J. Chem. Phys. 140, 244709 (2014)
character of Al which induces the delocalization of electron density imposed by impurity within the system, causing the overall electronic structure to polarize charge more efficiently and thus produce an electric dipole. This spatial configuration provides a stronger electric dipole moment in the system and contributes to the overlap necessary for electronic energy transfer. To develop a physically intuitive argument as previously done for the doped systems in Sec. III A, the spatial orbitals of the adsorbed system that were involved with the highest electronic energy transfer rates are presented in Figures 6(a)–6(c). The orbitals with the highest probability of transition (associated with the maximum oscillator strength for the system) of the studied systems are displayed. Because degenerate transitions did exist, there were systems with multiple possible transitions for energy transfer, as illustrated in Figure 6(c). It is important to notice that the excitations are not necessarily from HOMO to LUMO, and it is common for non-valence electronic excitation to occur for such systems. As evidenced in Figure 6(c), for the Si29 H33 Ag3 system there were several degeneracies that were observed in the first 50 excited states; in fact, the HOMO to LUMO (MO 103 to 104) transition in this case was the 13th in probability for transition. The following discussion is based on results from the isotropic average orientation for the permanent electric dipole moments. For the phosphorous-doped systems, it was observed that the systems were most conducive to energy transfer when the P atom was placed on the surface of the structure. A similar coupling effect was observed with these systems with the parallel arrangement of the systems’ permanent electric dipole moments. The n-type character of the P atom contributes a localization of electron density on the surface of the QD (as opposed to the delocalization of electron density imposed by Al) that establishes a permanent electric dipole moment that encourages energy transfer within the studied systems. Overall, the doped systems responded well to electronic excitations and exhibited a particular preference for the collinear permanent dipole orientation. Between the optimal aluminum-doped and phosphorous-doped systems, the Si29 H36 Al-i system produced rates double those of the Si35 H36 P-o system. In order to understand this difference in rate, it is necessary to consider the underlying physical influences affecting the process. There is a quantum confinement effect compounding with a dopant effect. Although the phosphorous-doped system is a priori electronically
FIG. 6. (a). Si29 H36 Al-i orbitals involved with the excited electronic state transition of the greatest probability. The yellow arrow indicates the energy of excitation transition. For reference the HOMO is number 75 and the LUMO is 76. E = 3.20 eV is the transition energy. (b). Si35 H36 P-o orbitals involved with the excited electronic state transition of the greatest probability. The yellow arrow indicates the energy of excitation transition. For reference the HOMO is number 75 and the LUMO is 76. E = 2.59 eV transition. (c). Si29 H33 Ag3 orbitals involved with the excited state transitions of the greatest probability of occurrence. In this system, there were seven equal energy transitions, and only two transitions are shown for simplicity. Each yellow arrow indicates a specific transition. For reference the HOMO = MO 103 and LUMO = MO 104. E = 3.52 eV transition.
TABLE III. Parameters in Förster calculations for Ag adsorbed on Si quantum dots.
System Si29 H36 Si29 H35 Ag Si29 H33 Ag3 Si35 H36 Si35 H35 Ag Si35 H36 Ag3
Donor transition energy (eV)
Donor transition dipole moment squared (AU2 )
J spectral overlap (×10−7 cm)
5.45 3.38 3.52 5.29 3.47 3.50
0.264 5.46 3.62 0.281 6.66 3.43
15.4 4.50 86.3 23.8 6.23 49.9
D dip (ε) A dip (ε) dε (×10−120 C4 m4 eV−1 ) 0.421 40.7 7200 0.942 81.3 2860
This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 69.166.47.134 On: Sun, 07 Dec 2014 03:01:28
244709-7
Vinson, Freitag, and Micha
J. Chem. Phys. 140, 244709 (2014)
TABLE IV. Förster transfer rates for Ag adsorbed on Si quantum dots (isotropic average orientation). KEET (kHz) R (Å)
Si29 H36
Si29 H35 Ag
Si29 H33 Ag3
20 40 60 80
131 2.06 0.180 0.0322
12 700 199 17.5 3.12
225 000 3520 309 55.0
KEET (kHz) R (Å) 20 40 60 80
Si35 H36 295 4.61 0.404 0.0720
Si35 H35 Ag 25 400 398 34.9 6.21
Si35 H36 Ag3 89 700 1400 123 21.9
better suited for energy transfer with an induced dipole, the electronic energy transfer rates appear to benefit to a larger extent from the quantum confinement effect in the aluminum-doped system. B. Structures with adsorbed Ag and Ag3
Energy transfer rates and spectral function overlaps were calculated for each pair of isotropically oriented adsorbed systems and are listed in Tables III and IV. Structures containing adsorbed Ag were as generated in Ref. 14. From the calculated results in Table IV, the adsorption of Ag appears to significantly impact the electronic energy transfer rates. Silver presence displays a non-linear effect on the rate as can be shown by a comparison of the magnitude of the rates of singly and triply substituted systems. The single Ag atom replacement of a passive hydrogen atom on the system results in an increase of electronic energy transfer rates by two orders of magnitude. However, the effect of the Ag
presence appears to decrease as the triply substituted system only undergoes in addition an approximate 20-fold increase in transfer rates for Si29 and a four-fold increase in transfer rates for Si35 . As was observed with the doped systems, the quantum confinement effect couples with the adsorbate presence to determine the electronic energy transfer rates. The singly substituted Si29 system shows rates that are half of the equivalent Si35 system to demonstrate the size effect; however, the triply substituted Si29 system shows rates that are twice the equivalent Si35 system to demonstrate the quantum confinement effect for Ag3 . Overall, the net effect from confinement and the adsorbate is confirming that the former maintains a larger role in determining electronic energy transfer rates than the adsorbate effect alone on these systems as similarly described with the doped systems in Sec. II A. Figure 7 contains the spectral densities for donor and acceptor, respectively, in blue and red. It shows that going from a single Ag atom to three has a large effect on the acceptor spectral density, and it gives a larger overlap with the donor density. In the spectra above, adsorbed QDs had greater spectral density overlap than the corresponding pure systems. A relevant observation is that the overlap for the doped systems in Figure 5 was greater than for the adsorbed systems in Figure 7. However, the electronic energy transfer rates were superior in the adsorbed systems; this difference is explained by the transition electric dipole moment strength that is larger for the adsorbed systems than for the doped systems.
C. Effects of orientation
Table V shows rates of energy transfer for QDs oriented coplanar as seen in Figure 4, for parallel and collinear orientations of permanent dipoles. The orientation factors are calculated for each transition dipole vector (usually different from
FIG. 7. Spectral densities of emitting (blue lines) and absorbing (red lines) bare silicon dots and dots with adsorbed silver.
This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 69.166.47.134 On: Sun, 07 Dec 2014 03:01:28
244709-8
Vinson, Freitag, and Micha
J. Chem. Phys. 140, 244709 (2014)
TABLE V. Effects of orientations I and II on energy transfer rates of doped Si and Ag adsorbed on Si QD.
Orientation
QD pair
Transition energy (eV)
(κ αδ )2
KEET at R = 30 Å (kHz)
KEET at R = 30 Å (kHz) [isotropic average]
Si29 H36 Si29 H36 -Al-i Si29 H36 -Al-o Si29 H36 -P-i Si29 H36 -P-o Si29 H35 Ag Si29 H33 Ag3 Si35 H36 Si35 H36 -Al-i Si35 H36 -Al-o Si35 H36 -P-i Si35 H36 -P-o Si35 H35 Ag Si35 H33 Ag3
5.45 3.20 2.94 3.27 3.76 3.38 3.52 5.29 3.06 3.79 4.07 2.59 3.47 3.50
0.993 0.192 0.937 0.768 1.11 0.999 0.999 0.800 0.970 0.0168 0.0293 0.999 0.994 0.999
17.2 824 219 173 1020 1680 29 600 31.1 682 9.09 8.22 1670 3330 11 800
11.5 2850 156 150 612 1120 19 700 25.9 468 360 186 1115 2230 7880
Si29 H36 Si29 H36 -Al-i Si29 H36 -Al-o Si29 H36 -P-i Si29 H36 -P-o Si29 H35 Ag Si29 H33 Ag3 Si35 H36 Si35 H36 -Al-i Si35 H36 -Al-o Si35 H36 -P-i Si35 H36 -P-o Si35 H35 Ag Si35 H33 Ag3
5.45 3.20 2.94 3.27 3.76 3.38 3.52 5.29 3.06 3.79 4.07 2.59 3.47 3.50
0.923 1.37 0.990 0.507 0.791 3.95 3.99 0.143 0.691 0.991 0.364 3.98 3.86 3.99
16.0 5890 231 114 727 6640 118 000 5.57 486 536 102 6660 12 900 47 200
11.5 2850 156 150 612 1120 19 700 25.9 468 360 186 1115 2230 7880
D A
dD dA ↑
↑
D A
dD dA → →
FIG. 8. Electronic energy transfer rates plotted against radial distance between QDs. The systems with the highest and lowest rates in the doped series and the adsorbed series were chosen to compare with the pure systems. For isotropic average orientation, κ 2 is set equal to 2/3. [cNX-z means crystalline SiN Hsaturation Xinner or SiN Hsaturation X-outer]
This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 69.166.47.134 On: Sun, 07 Dec 2014 03:01:28
244709-9
Vinson, Freitag, and Micha
permanent dipole orientations) and the value shown in Table V is for the acceptor absorption with the largest oscillator strength. The last column shows results for isotropic averages for comparison, and is the same for both arrangements. As shown in Table V, both orientations resulted in an increased κ αδ 2 factor compared to the average orientation and therefore an increased energy transfer rate. From Figure 8, it is evident that the introduction of adsorbates enhances the electronic energy transfer for all systems in an isotropic permanent dipole arrangement. This is also shown for all oriented permanent dipole arrangements except for one case in parallel configuration, where the parallel arrangement of the dipoles hinders energy transfer in the Si35 H36 P- i system. It is clear graphically in Figure 8 that the ideal orientation for both doped and adsorbed systems is the collinear configuration. This can be attributed to the appearance of various dipoles created by the adsorbates, that encourages delocalization of electron density and contributes to increased energy transfer rates.
IV. CONCLUSIONS
A combination of the Förster perturbation theory of electronic energy transfer and ab initio electronic structure calculations with hybrid time-dependent density functionals has provided a versatile tool for modeling a series of Si-based quantum dots. In review, it has been observed that size and composition are effective independent variables for controlling the electronic energy transfer rates between the Si quantum dots. Magnitudes of energy transfer rates have been shown to depend strongly on both quantum confinement and atomic composition, leading to changes of several orders of magnitude. Doping with either Al or P dramatically increases energy transfer rates by up to two orders of magnitude as demonstrated for the collinearly oriented Si35 H36 P-o system compared to the equivalent isotropic pure Si QD system. However, the highest energy transfer rates were exhibited by the systems with adsorbed Ag, with a peak rate observed for the collinearly oriented Si29 H33 Ag3 at four orders of magnitude over the equivalent isotropic pure Si QD system. From the results and discussion discussed in Sec. III, it appears that these nanoparticles could play a potentially large role in the design of novel photovoltaic designs. To summarize, the present calculations with adsorbed Ag show that: (a) addition of Ag increases rates up to 100 times; (b) addition of Ag3 increases rates up to 100 times; (c) collinear alignment of permanent dipoles increases transfer rates by an order of magnitude compared to parallel orientation; and (d) smaller QD-size increases transfer due to greater electronic orbitals overlap. Calculations with dopants show that: (a) p-type and n-type dopants enhance energy transfer up to two orders of magnitude; (b) surface-doping with P and center-doping with Al show the greatest rates; and (c) KEET is largest for collinear permanent dipoles when the dopant is on the outer surface and for parallel permanent dipoles when the dopant is inside the QD. The spectral overlap between our QDs are comparable to the overlap for other reported QDs: of the order of 10−5 cm
J. Chem. Phys. 140, 244709 (2014)
in our work compared to 10−4 cm for CdSe QDs.35 Related energy transfer rates are also smaller for the present QDs, and this can be attributed to the much smaller size of our QDs. As obtained in Ref. 22 Stokes shift values are typically calculated utilizing the HOMO-LUMO transitions, which were subsequently used in this investigation, as it was assumed that the shifts did not vary significantly across system transitions. This is a limitation which could be validated by doing more extensive calculations of potential energy changes with bond distances for each electronic transition, but it is not likely to change our conclusions involving rate changes of several orders of magnitude. Regarding the implications of this work for future developments of photovoltaic materials, issues of solar cell efficiencies and of toxicity of QDs are of interest. Since theoretical constraints were established on maximum solar energy harvesting by the Shockley-Queisser limit,36 the scientific community has strived to develop photovoltaic materials that can approach the theoretical maximum efficiency for light conversion into electricity. The efficiency limit was derived for a single p-n junction, and has since inspired the development of tandem solar cells1–5 as a means to approach higher energy conversion efficiency by introducing multi-junction cells. In brief, the interest in tandem solar cells is motivated by their spread of the photonic load across optimized layers of optically active materials; this spread allows for controlled absorption of incident photons at layers with chosen energy band gaps. Doped and adsorbed Si quantum dots have been both theoretically9 as well as experimentally validated37 as worthy candidates for implementation in such matrices. The combination of Förster resonance energy transfer and ab initio electronic structure from TDDFT appears promising also to characterize the interaction between chromophores and quantum dots used to identify protein conformations, as described in the biomedical literature.38,24 By serving as passive fluorescent probes,39 QDs can measure protein interactions by experiencing nonradiative electronic energy excitations that do not themselves influence the desired observed phenomenon, and in this connection the nontoxicity of Si QDs seems advantageous.
ACKNOWLEDGMENTS
This material is based upon work supported financially by the National Science Foundation grant (Grant No. NSF CHE-1011967) and by the Dreyfus Foundation. Computations were performed at the University of Florida High Performance Computing facility. 1 A.
P. Alivisatos, Science 271, 933 (1996). J. Nozik, Ann. Rev. Phys. Chem. 52, 193 (2001). 3 N. S. Lewis, Science 315, 798 (2007). 4 P. V. Kamat, J. Phys. Chem. C 112, 18737 (2008). 5 G. Conibeer, M. Green, E-C. Cho, D. Koenig et al., Thin Solid Films 516, 6748 (2008). 6 A. J. Nozik, Nano Lett. 10, 2735 (2010). 7 Y. Kanai, J. B. Neaton, and J. C. Grossman, Comput. Sci. Eng. 12, 18 (2010). 8 W. A. Tisdale and Y. X. Zhu, Proc. Natl. Acad. Sci. USA 108, 965–970 (2011). 9 J-W. Luo, P. Stradins, and A. Zunger, Energy Environ. Sci. 4, 2546 (2011). 2 A.
This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 69.166.47.134 On: Sun, 07 Dec 2014 03:01:28
244709-10 10 T.
Vinson, Freitag, and Micha
Takagahara and K. Takeda, Phys. Rev. B 46, 15578 (1992).
11 I. Vasiliev, S. Ogut, and J. R. Chelikowsky, Phys. Rev. B 65, 115416 (2002). 12 A.
Puzder, A. J. Williamson, J. C. Grossman, and G. Galli, J. Am. Chem. Soc. 125, 2786 (2003). 13 M. G. Mavros, D. A. Micha, and D. S. Kilin, J. Phys. Chem. C 115, 19529 (2011). 14 D. M. Stewart, M. G. Mavros, and D. A. Micha, J. Phys. Chem. C 116, 23107 (2012). 15 H. Freitag, M. G. Mavros, and D. A. Micha, J. Chem. Phys. 137, 144301 (2012). 16 D. S. Kilin and D. A. Micha, J. Phys. Chem. C 115, 770 (2011). 17 J. J. Ramirez, D. S. Kilin, and D. A. Micha, Int. J. Quantum Chem. 112, 300 (2012). 18 T. Vazhappilly, D. S. Kilin, and D. A. Micha, J. Phys. Chem. C 116, 25525 (2012). 19 Z-Y. Zhou, M. L. Steigerwald, R. A. Friesner, L. Brus, and M. S. Hybertsen, Phys. Rev. B 71, 245308 (2005). 20 S. A. Fischer and O. V. Prezhdo, J. Phys. Chem. C 115, 10006 (2011). 21 J. Chen, A. Schmitz, and D. S. Kilin, Int. J. Quantum Chem. 112, 3879 (2012). 22 Th. Foerster, in Modern Quantum Chemistry, Part III, edited by O. Sinanoglu (Academic Press, New York, 1965), p. 93. 23 G. D. Scholes, Annu. Rev. Phys. Chem. 54, 57 (2003). 24 A. R. Clapp, I. L. Medintz, and H. Mattoussi, ChemPhysChem 7, 47 (2006).
J. Chem. Phys. 140, 244709 (2014) 25 C.
F. Kaminski, E. Sackmann, and K. Schulten, ChemPhysChem 12, 423 (2011); Special issue on Th. Foerster’s science. 26 C. Curutchet, A. Franceschetti, A. Zunger, and G. D. Scholes, J. Phys. Chem. C 112, 13336–13341 (2008). 27 R. Baer and E. Rabani, J. Chem. Phys. 128, 184710 (2008). 28 J. Megow, B. Roeder, A. Kulesza, V. Bonacic-Koutecky, and V. May, ChemPhysChem 12, 645 (2011). 29 Z. Lin, H. Li, A. Franceschetti, and M. T. Lusk, ACS Nano 6, 4029 (2012). 30 J. Hafner, J. Comput. Chem. 29, 2044 (2008). 31 M. J. Frisch, G. W. Trucks, H. B. Schlegel et al., Gaussian 09, Revision D.01, Gaussian, Inc., Wallingford CT, 2009. 32 A. V. Krukau, O. A. Vydrov, A. F. Izmaylov, and G. E. Scuseria, J. Chem. Phys. 125, 224106 (2006). 33 P. W. Atkins and R. S. Friedman, Molecular Quantum Mechanics, 3rd ed. (Oxford University Press, Oxford, England, 1997), Chap. 6. 34 M. L. del Puerto, M. Jain, and J. R. Chelikowsky, Phys. Rev. B 81, 035309 (2010). 35 G. D. Scholes and D. L. Andrews, Phys. Rev. B 72, 125331 (2005). 36 W. Shockley and H. J. Queisser, J. Appl. Phys. 32, 510 (1961). 37 E. Cho, S. W. Park, X. Hao, D. Song, G. Conibeer, S. C. Park, and M. A. Green, Nano 19, 245201 (2008). 38 A. P. Alivisatos and W. Gu, Annu. Rev. Biomed. Eng. 7, 55 (2005). 39 R. Q. Zhang, A. DeSarkar, T. A. Niehaus, and T. Frauenheim, Phys. Status Solidi B 249, 401 (2012).
This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 69.166.47.134 On: Sun, 07 Dec 2014 03:01:28
