Understanding and controlling the energy transfer between silicon nanocrystals is of significant importance for the design of efficient optoelectronic devices. However, previous studies on silicon nanocrystal energy transfer were limited because of the strict requirements to precisely control the inter-dot distance and to perform all measurements in air-free environments to preclude the effect of ambient oxygen. Here, we systematically investigate the distance-dependent resonance energy transfer in alkyl-terminated silicon nanocrystals for the first time. Silicon nanocrystal solids with inter-dot distances varying from 3 to 5 nm are fabricated by varying the length and surface coverage of alkyl ligands in solution-phase and gas-phase functionalized silicon nanocrystals. The inter-dot energy transfer rates are extracted from steady-state and time-resolved photoluminescence measurements, enabling a direct comparison to theoretical predictions. Our results reveal that the distance-dependent energy transfer rates in Si NCs decay faster than predicted by the Förster mechanism, suggesting higher-order multipole interactions.

## I. INTRODUCTION

Understanding the charge and energy transfer (ET) processes in quantum dot or nanocrystal (NC) solids is essential for the design of high-performance optoelectronic devices.^{1–4} The control of exciton diffusion lengths and rates is crucial for achieving optimal device efficiency. For instance, effective exciton diffusion to the charge separating interface is desired in the design of solar cells.^{5–7} On the contrary, in light-emitting devices, exciton diffusion is unfavorable as the goal is for excitons to recombine radiatively to emit photons. Therefore, long-chain interparticle linkers are employed to prevent exciton migration to “dark” dots and to minimize photoluminescence (PL) quenching.^{8,9}

ET in ligand-functionalized quantum dot solids is mainly described by the Förster mechanism^{1,10–13} since there is no direct overlap of the molecular orbitals between donors and acceptors. In the literature, experimental studies in NC systems have been conducted in detail to validate the 1/*R*^{6} distance dependence of the Förster mechanism, with *R* being the donor–acceptor distance, where inter-dot distances are varied by altering ligand lengths^{14,15} or by constructing bilayer structures.^{16}

For silicon nanocrystals, however, only a few experimental studies have been reported,^{17–19} although ET studies on porous silicon were already performed decades ago.^{20–22} The lack of studies on silicon nanocrystals is partially due to the difficulty of synthesizing closely packed silicon quantum dots with sufficiently narrow size distribution and controlled inter-dot distances as well as maintaining an air-free environment during PL measurements. Due to the nature of silicon’s indirect bandgap, inter-dot ET is not efficient especially when surface ligands create excessive NC separations, and thus, the inter-dot separation needs to be carefully limited to a narrow range, generally between 1 and 3 nm. Electron transfer to atmospheric oxygen, a process hard to distinguish from inter-dot ET, can also contribute to the PL quenching of Si NCs.^{23–25} Another challenge is that the PL quenching experiments for the study of ET traditionally require the use of monodisperse NCs with two different sizes to construct donor–acceptor pairs.^{1,2,4,26,27}

In this work, we generalized the Förster relationship to consider the case of a donor embedded in an ensemble of NCs with known size distributions, enabling us to study ET of samples with relatively broad size distributions. Furthermore, we systematically investigate the distance-dependent resonance ET in alkyl-functionalized Si NCs and compare it with predictions from the Förster mechanism. By tuning surface alkyl coverages and ligand lengths using both solution-phase and gas-phase functionalization methods, we obtain a series of samples with controlled inter-dot distances and surface ligand coverages. An air-free environment is maintained throughout the synthesis to the PL lifetime measurements, and therefore, the observed trend in PL is solely affected by the resonance ET between Si NCs. Apparent differences of PL spectra and PL lifetime in Si NC films with varying ligand lengths and ligand coverages are observed, which provide evidence for interparticle ET. Previous theoretical and computational work has shown that the Förster mechanism only applies when NCs are separated at relatively large distances.^{28,29} The dipolar transitions in Si nanocrystals are relatively weak due to the indirect gap, and multipole interactions may play an important role when particles are in closer contact. Here, we find that in Si NC films, the ET rates as a function of donor–acceptor distances exhibit a faster decay than predicted by the Förster mechanism, providing experimental evidence that higher-order multipole interactions dominate the ET.

## II. METHODS

### A. Experimental

Si NCs were synthesized in a nonthermal plasma reactor previously described in Ref. 30. Alkyl-terminated Si NCs with ligand lengths varying from 8C to 18C and with controlled surface ligand coverages were prepared using thermal hydrosilylation.^{31} Separate mixtures of 5 mg Si NCs with either 1-octene, 1-decene, 1-dodecene, or 1-octadecene and 1 ml mesitylene were heated to 160 °C for 96 h under nitrogen. The ligand to the surface Si atom ratio was fixed at 1:2.5 to ensure surface ligand coverage of 40% for each ligand. After the hydrosilylation reaction, mesitylene was evaporated, and the functionalized Si NCs were redispersed in toluene. 20 *μ*l of the Si NC/toluene solution (10 mg/ml) were then dropcast onto glass substrates to form 1–2 *μ*m thick solid NC films.

Gas-phase functionalization was performed by introducing a vapor of the ligand molecules into the afterglow of the synthesis plasma, as previously described in Refs. 30 and 32. Here, 100 SCCM of hydrogen as the carrier gas was flown through the bubbler containing 1-octene, 1-decene, 1-dodecene, and 1-octadecene. Prior to film deposition, the gas-phase functionalized Si NCs were refluxed in mesitylene for 4 h to ensure that their photoluminescence quantum yields (PLQYs) are similar to the solution-phase functionalized samples. All solvents were degassed by bubbling nitrogen and including 4 Å molecular sieves prior to use.

The surface ligand coverages of both solution phase and gas phase funtionalized samples were determined by measuring the weight loss of Si NCs using carbon, nitrogen, and hydrogen (CNH) elemental analysis. Here, we estimate that Si NCs are 3.3 nm diameter spheres with 260 surface atoms. Assuming that the weight loss during the CNH measurement is solely due to loss of ligands, the ligand coverage is calculated by^{33,34}

Transmission electron microscopy (TEM) measurements were performed with a FEI Titan aberration-corrected scanning transmission electron microscope with a high-angle annular dark-field imaging (HAADF) detector. Fourier-transform infrared spectroscopy (FTIR) measurements were performed with a Bruker ALPHA spectrometer with the attenuated total refection (ATR) module. Si NCs were dissolved in toluene and dropcast onto the ATR crystal and then allowed to dry before each measurement.

Time-resolved photoluminescence (TRPL) measurements were taken with a Horiba DeltaFlex TCSPC lifetime fluorometer. The excitation source was a 407 nm laser with a delay time of 13 *µ*s. Each Si NC film deposited on a glass slide was placed in a nitrogen-filled borosilicate vial with screw caps and measured for 5 times at different spots. For each sample, TRPL measurements were taken at 700, 750, 800, and 900 nm with a 6 nm bandpass.

### B. Computational methodology

To estimate the surface distance between Si NCs in the presence of ligands, we simulated the interactions of two half Si spheres (to reduce the computational cost) over which ligands were deposited. Each half sphere was generated starting from a 3.2 nm diameter sphere with a 60% hydrogen surface coverage and consisted of 523 silicon atoms and 152 surface H atoms. Ligands were individually deposited with the iterative procedure outlined in Fig. S1. Specifically, 1-radicals of octane, decane, dodecane, or octadecane were placed adjacent to the Si surface at a random position (deposition step) and a brief (10 ps for gas phase and 1 ps for liquid phase) microcanonical simulation was performed (equilibration step). The resulting structure was then either accepted or rejected depending on whether or not the deposition was successful (whether the closest distance between the ligand atoms and the Si half sphere was smaller than 0.5 nm). These two steps were repeated until the desired ligand density was achieved, namely, 2 ligands/nm^{2} for gas-phase functionalization (11 ligands, equivalent to 20% coverage) or 3.2 ligands/nm^{2} for solution-phase functionalization (17 ligands, equivalent to 40% coverage). The final structures were further equilibrated at 300 K for 0.5 ns using a Berendsen algorithm^{35} with a damping factor of 10 ps. Finally, to estimate the surface distance between Si NCs, two ligand-coated Si half-spheres were placed facing each other (surface distance 0.5 nm) pointed toward the center of the other fragment. The system was simulated for 0.5 ns while keeping the average temperature at 300 K with a Berendsen algorithm, and the average surface distance was obtained by collecting the data over the last 200 ps. All the simulations were performed using the Large-scale Atomic/Molecular Massively Parallel Simulator (LAMMPS) software^{36} (version March 2020) using the Reax force field^{37} in combination with the QeQ charge equilibration method. A timestep of 0.1 fs was used in all the simulations.

## III. RESULTS AND DISCUSSION

### A. Spectral and structural characterization of Si NCs

Alkyl-terminated Si NCs with ligand lengths varying from 8C to 18C and with controlled surface ligand coverages were prepared using thermal hydrosilylation and dropcast into films as described above and schematically shown in Fig. 1(a). The Si NCs functionalized with different-length ligands exhibit the same PLQY (∼40%) and steady-state PL spectra when measured in toluene (Fig. S2). However, the PLQY and PL spectra of Si NC films show a drastically different trend, as shown in Fig. 1(b). Films with short ligands (8C–Si) show lower PLQYs than films with longer ligands (18C–Si), consistent with the expected trend due to ET to “dark” NCs. Specifically, for the longest ligand of 1-octadecene, the maximum PLQY is about 44%, equaling that of the solution phase measurements.

Time-resolved PL spectra of Si NCs in the form of solution and films also exhibit different characteristics. For Si NCs dispersed in toluene where ET is negligible, the emission exhibits an intrinsic redshift as delay time increases [Fig. 1(c)]. The intrinsic redshift is presumably due to the NC size distribution within the ensemble. As larger NCs with smaller emission energies exhibit slower radiative decays, the detected PL at longer wavelengths is enhanced at longer delay times, leading to the dynamic redshift in the PL spectra even when the NCs are non-interacting in solution. In NC films, the time-resolved PL spectra exhibit a narrower linewidth compared to the solution-phase counterparts [Fig. 1(d)]. The redshift becomes more pronounced as the delay time increases with a rapid reduction in the time-resolved PL intensity. The redshift is an indication of exciton migration to larger NCs.^{2} By plotting the peak energies against delay time, we obtain the temporal evolution of PL peak energy, where the film exhibits a much stronger redshift than that found from NCs in solution [Fig. 1(e)]. As shown in Fig. 1(e), the 8C–Si film data exhibit a notably faster redshift, while the 18C–Si film differentiates little compared to the solution data (Fig. S3). All these characteristics of time-resolved PL confirm that the ET becomes more pronounced when the ligand length decreases. The gas-phase functionalized Si NCs were also dropcast into films and characterized with the same procedure as the solution-phase functionalized samples (Fig. S4) and exhibit a similar behavior as the liquid-phase functionalized Si NC films.

To verify that the differences in the Si NC film PL originate from the ligand length variation, the alkyl-functionalized Si NCs were characterized with TEM, FTIR, carbon hydrogen and nitrogen (CHN) elemental analysis, and dynamic light scattering (DLS). Figures 2(a) and 2(b) show the dark field TEM image of 8C–Si NCs and the size distribution obtained from TEM. The average diameter obtained from TEM images is 3.3 nm with a standard deviation of 0.5 nm. The size distribution shown in Fig. 2(b) reflects the Si NC core size since the surface ligands dissociate drastically under electron beam irradiation. The size distribution also agrees with the size from XRD measurements and previous work by our group.^{38} FTIR measurements confirm that the CH_{x}/SiH_{x} area ratios increase as ligand lengths increase [Fig. 2(c)].

The values of surface ligand coverage for solution-phase and gas-phase functionalized samples are listed in Table I. The surface ligand coverages for solution-phase functionalized samples are all around 40%, which agrees with the 1:2.5 ligand-to-surface site ratio used for functionalization. In contrast, the gas-phase ligand grafting method leads to surface ligand coverages of around 25% for each ligand. The solvodynamic diameters, which are between 7 and 12 nm, indicate good dispersibility of both solution and gas-phase functionalized Si NCs before film deposition; thus, the NC films are expected to be composed of uniformly dispersed NCs rather than agglomerates. It is noteworthy that the solvodynamic diameters of solution-phase functionalized Si NCs are slightly larger than the gas-phase functionalized Si NCs with a difference of 2–3 nm. The observed differences in solvodynamic diameters between NCs with different ligand coverages have previously been observed and were associated with different ligand configurations in terms of ligands extending out toward the solvent or remaining more confined.^{39}

. | Solution-phase functionalization . | Gas-phase functionalization . | ||
---|---|---|---|---|

Surface . | Solvodynamic . | Surface . | Solvodynamic . | |

. | coverage (%) . | diameter (nm) . | coverage (%) . | diameter (nm) . |

8C | 37 | 9.6 | 25 | 7.1 |

10C | 38 | 10.5 | 23 | 8.1 |

12C | 37 | 9.5 | 26 | 8.0 |

18C | 44 | 11.9 | 28 | 8.9 |

. | Solution-phase functionalization . | Gas-phase functionalization . | ||
---|---|---|---|---|

Surface . | Solvodynamic . | Surface . | Solvodynamic . | |

. | coverage (%) . | diameter (nm) . | coverage (%) . | diameter (nm) . |

8C | 37 | 9.6 | 25 | 7.1 |

10C | 38 | 10.5 | 23 | 8.1 |

12C | 37 | 9.5 | 26 | 8.0 |

18C | 44 | 11.9 | 28 | 8.9 |

### B. Impact of NC size distribution on the Förster ET rate

In previous work, Förster ET has been mainly studied in highly monodisperse NCs. Si NCs in this study have a comparatively broad size distribution. Hence, to appropriately analyze our data, it is important to understand how the size distribution will affect ET dynamics if the Förster mechanism would apply. Therefore, we generalize the traditional Förster formulation to obtain the ET rate as a function of the average donor–acceptor distance given an NC size distribution. As we show below, the exponent of the power law is only slightly modified to 5–7 from the traditional exponent of 6 for highly monodisperse NCs.

To model the scattered size distribution of donor–acceptor distances within the Si NC ensemble, the donor–acceptor distance is estimated using a linear relationship $R=x+d\u0304/2$, where *x* is the sum of the donor radius (1.6 nm for 800 nm emitting Si NCs) plus the surface spacing obtained by MD simulation and is dependent on the ligand length [Fig. 3(a)]. The expression for the Förster radius *R*_{0} is given by

where *η*_{PL} is the quantum efficiency of the donor, *κ* is the dipole orientation factor between the donor and the acceptor and typically assumed to be 2/3 for randomly oriented dipoles, *n* is the refractive index of the sample, *F*_{D}(*λ*) is the normalized donor PL spectrum, and *σ*_{A}(*λ*) is the acceptor absorption coefficient.

Our main assumptions are as follows:

For each donor, the number of nearest acceptors

*N*is a fixed number regardless of the ligand length.The donor–acceptor distance

*R*_{i}for a donor with each adjacent acceptor is described by*R*_{i}=*x*+*d*_{i}/2, where*x*is the sum of the donor radius and the NC surface distance and is, thus, a function of the ligand length, and*d*is the diameter of the acceptor.The diameter

*d*is described by a distribution*f*(*d*) with a mean value $d\u0304$.The total ET rate measured for the donor

*k*_{ET}is the sum of ET rates of each possible acceptors*k*_{ET,i}. According to the traditional Förster formulation, each*k*_{ET,i}has the same expression $kET,i=(R0/Ri)6/\tau D$, where*R*_{0}is the Förster radius and*τ*_{D}is the donor lifetime. Because of the expected 1/*d*^{6}scaling, only the nearest neighbors have substantial contributions, and thus, $kET=\u2211i=1NkET,i$.

Based on the above assumptions, *k*_{ET} is a function of *d* with a known distribution *f*(*d*), with its mean value $k\u0304ET$ calculated by

with

The integral in Eq. (3) does not have an analytical solution. To verify the deviation from the traditional Förster expression due to the influence of NC diameter distribution *f*(*d*), we plot the numerically calculated values of $lnk\u0304ET$ vs $ln(2x+d\u0304)$. Assuming constants *R*_{0} and *τ*_{D}, we test the influence of the size distribution for a normal distribution of *d* $(f(d)\u223cN(\mu ,\sigma 2))$. The calculated values are plotted in Fig. 3(b) with *μ* = 2 and *σ* = 0.1, 0.75, and 5. For *σ* = 0.1 representing a narrow distribution of *d*, the slope of $lnk\u0304ET$ vs $ln(2x+d\u0304)$ approaches −6, which is the limit of a uniform size sample described by the Förster model. As *σ* increases, corresponding to the broader size distribution, the slope gradually deviates from −6. It is noteworthy that for typical size distributions, represented by the *σ* = 0.75 case, the deviation in the slope shown in Fig. 3(b) from −6 is not significant. In other words, the relationship between average ET rate and average NC distance can be roughly described with $k\u0304ET\u221d1/(2x+d\u0304)6$, even in a non-size-purified NC sample if the ET process follows the Förster model.

The above analysis is based on the simplifying assumption of a constant Förster radius *R*_{0}. From Eq. (1), *R*_{0} is, in fact, proportional to the overlap integral of donor emission and acceptor absorption spectra. As the acceptor absorption varies with the acceptor diameter *d*, the actual *R*_{0} is also a function of *d*. Here, we use the emission and absorption models from Ref. 40 to quantify the effect of varying *R*_{0}. The PL spectrum and absorption coefficient are predicted by

and

where *E*_{g}(eV) = 1.081(eV) + 3.73/*d*(nm)^{1.39} is the bandgap using a linear combination of atomic orbital (LCAO) model,^{41} and Δ*E* is the spectral width with a typical value Δ*E* = 70 meV at room temperature.

The predicted PL spectrum for 3.2 nm donor NC and absorption spectra for 3–5 nm NCs are plotted in Fig. 3(c). As the acceptor size increases, their absorption edges redshift, and the overlap between donor-emission and acceptor-absorption increases. We also describe the size distribution of our Si NCs by a more realistic lognormal size distribution as observed in our plasma synthesis,

with *μ* = 1.2 and *σ* = 0.2 derived by fitting our experimental data in Fig. 2(b). Substituting *f*(*d*) into Eq. (3) with *R*_{0} either a constant or size-dependent following Eq. (1), slight differences of slopes can be observed. With the constant *R*_{0} assumption, the average ET rate closely follows the inverse sixth-power donor–acceptor distance dependence. After taking into consideration the varying *R*_{0}, the exponent of the power-law relationship is modified to 5.4. The above analysis indicates that in the modified Förster expression, where the influence of the NC size distribution and the changing Förster radius is taken into consideration, the exponent of the power-law may be slightly modified to ∼5–7.

### C. Extracting ET rates

The decay curves of Si NCs exhibit a stretched-exponential behavior due to the dispersion of decay rates.^{21} Measured decay rate dispersions are the result of the coupling of (a) the distribution of the NCs’ radiative and non-radiative decay rates and (b) the distribution of ET rates. In order to analyze the distance-dependence of ET rates, it is necessary to deconvolute these two effects. To explicitly obtain the distribution of decay rates, we adapt the method by van Driel *et al.*^{42} and fit the decay curve assuming a continuous distribution *f*(*k*_{tot}) of total decay rates *k*_{tot}. The decay curve *I*(*t*), thus, has the following form:

with the distribution of *k*_{tot} following a lognormal distribution:

where *σ* and *μ* are the parameters to be determined. Figures 4(b) and 4(e) plot the distribution of *k*_{tot} obtained from fitting the experimental PL decay curves. The average total decay rate $k\u0304tot$, or the arithmetic mean of *k*_{tot}, can thus be deduced as

We fit the solution and film PL decay curves to obtain $k\u0304tot$ for both solution and film samples, $k\u0304tot,s$ and $k\u0304tot,f$, respectively. The fitted curves and the distribution of *k*_{tot} are plotted in Figs. 4(a), 4(b), 4(d), and 4(e). The trend that ET becomes more significant as the ligand length decreases is confirmed in both solution-phase and gas-phase functionalized samples [Figs. 4(a) and 4(d)]. The films exhibit faster decays than the solution samples with a decrease in the lifetime as the ligand length becomes shorter. The PL decay curves exhibit similar trends as the solution-phase functionalized samples with the shorter ligand film exhibiting faster decay. Moreover, for each ligand length, the decay lifetimes in gas-phase functionalized samples are shorter than their solution-phase functionalized counterparts, indicating more efficient ET due to lower surface ligand coverages. As shown in Fig. 4(b), $k\u0304tot$ for solution-phase samples is 0.0051 *µ*s^{−1}, corresponding to a radiative lifetime around 200 *µ*s. This timescale agrees with the average lifetime of 238 *µ*s obtained by applying a stretched-exponential fitting. For both solution-phase and gas-phase functionalized Si NCs, it is clear that the average $k\u0304tot,f$ increases as the ligand length decreases.

The ET rates for each film sample can be calculated by comparing the ET rates of solution and film samples. The *k*_{ET} values of Si NCs in the solution form are expected to reflect the recombination rate without ET events, since the DLS measurements indicate little to no agglomeration and thus minimum NC interaction. Assuming the measured PL is dominated by “bright” NCs showing radiative recombination in both the solutions and films, the total recombination rate measured in solution *k*_{tot,s} can be written as

and the total recombination rate in film *k*_{tot,f} will be

The *k*_{tot,f} and *k*_{tot,s} are correlated based on the effect of the local dielectric environment on the emission time. Different theoretical models have been developed to predict the radiative recombination rate for emitters embedded in an environment with a refractive index *n*. The most prominent models include the virtual cavity (VC) model, empty cavity (EC) model, and fully microscopic (FM) model.^{43–45} To establish a relationship between the radiative recombination rate of Si NCs and the local refractive index, we measured the PL lifetime of Si NCs dispersed in solvents spanning a range of refractive indices. All the solvents with Si NCs form clear and uniform solutions and have refractive indices ranging from 1.375 (hexane) to 1.516 (styrene). The measured PLQY of Si NCs in these solvents are all around 35% (Fig. S5a). As the PL emission of Si NCs features a broad spectrum, the PL lifetime is measured across the PL spectrum at detector wavelengths of 650, 700, 750, 800, and 850 nm. As shown in Fig. S5b, the VC model well describes the experimental data.

The refractive indices of the NC films are experimentally measured by ellipsometry. For ellipsometry measurements, Si NCs dissolved in toluene at ∼20 mg/ml are spin-cast onto silicon substrates at 600 rpm. As shown in Fig. S5c, modeling the 8C–Si film using a Cauchy layer provides good fits with *n* = 2.08 and the film thickness of 39 nm. Figure S5d shows the measured refractive indices of the solution-phase and gas-phase functionalized nanoparticle films. A decrease in the measured refractive index with an increase in the ligand length is observed in both series of samples.

In Figs. 4(c) and 4(f), the calculated average ET rates, $k\u0304ET$, are plotted as a function of the alkyl chain length for solution-phase and gas-phase functionalized samples, respectively. At 800 nm detector wavelength, the ET rates for solution-phase functionalized Si NCs are on the order of 0.01 *µ*s^{−1} corresponding to an ET timescale of 100 *µ*s and a Förster radius of 3.6 nm, while the ET rates for gas-phase functionalized Si NCs are on the order of 0.1 *µ*s^{−1} with an ET timescale of 10 *µ*s and a Förster radius of 5.2 nm. For each sample series, $k\u0304ET$ is measured for several detector wavelengths, corresponding to the ET rates of donors emitting at these wavelengths. For each ligand length, the measured ET rates increase with a decrease in wavelengths as an exciton in smaller NCs has a higher probability of transferring energy to nearby larger NCs with smaller bandgaps.^{2} At each wavelength, the ET rates decrease as the ligand lengths increase due to the larger inter-dot separation by the alkyl chains.

### D. Distance-dependent ET rates and comparison with the Förster mechanism

To establish a relationship between the average ET rate $k\u0304ET$ and the donor–acceptor distance, measuring inter-NC distances at each alkyl chain length is necessary. However, it is difficult to directly measure the inter-NC distance in TEM due to the dissociation of ligands under electron beam irradiation. Therefore, we performed molecular dynamics (MD) simulation to simulate the ligand packing structure and to extract the particle surface distances for each ligand length. MD simulations have been widely employed to reveal the detailed morphology of the organic ligand shell around NCs, providing remarkably good agreement with experimental measurements.^{46} This agreement is expected as the NC-NC distance is dictated by the attractive and repulsive interactions of the species in the ligand shell, a problem closely associated with estimating the species density, which MD has been repeatedly shown to be able to perform accurately. The average error in estimating the volumetric density of species with different force fields (including the one used here) is typically less than 2%, which includes species with complicated solvent structures and H-bonding (linear hydrocarbons have a much smaller error). Therefore, we can estimate the error in the predicted NC-NC distance reasonably below 1%.^{47,48}

As shown in Fig. 5(a), the surface distances in the MD simulation first drastically decrease due to the opposite movement and then oscillate around an average value. The average distances for 8C, 12C, and 18C ligands with 25% or 40% ligand coverages are plotted in Fig. 5(b). For each surface ligand coverage, the average surface distances roughly follow a linear relationship as a function of the ligand length. The ligand tilting angles calculated from Fig. 5(b) are 22° and 29° for 25% or 40% ligand coverage, respectively. These tilting angles appear to be very small due to the interlacing between ligands when two ligand-terminated surfaces interact with each other.

In Fig. 5(c), we also compare the MD simulation results with two simpler models: the collapsed shell model^{49} and the distance assuming that the ligands stick out perpendicular to the Si NC surface.

According to the Förster expression, $kET=1/\tau D\u22c5(R0/R)6$, where *τ*_{D} is the donor lifetime and *R* is the donor–acceptor separation, the product of *k*_{ET} and *n*^{4} should be proportional to *R*^{−6} for a given sample. However, applying a fit to

where

and is a constant for a given sample, yields a slope of about −14, i.e., a distance scaling exponent of 14, indicating deviation from the Förster expression [Fig. 5(d)]. Replacing the inter-dot distances obtained by MD simulation with other simpler models leads to distance scaling exponents between 10 and 14 (Table II).

. | . | Collapsed . | Ligands . |
---|---|---|---|

. | MD results . | shell model . | sticking out^{a}
. |

Solution-phase samples | 14.3 | 13.6 | 9.4 |

Gas-phase samples | 14.3 | 9.5 | 6.5 |

. | . | Collapsed . | Ligands . |
---|---|---|---|

. | MD results . | shell model . | sticking out^{a}
. |

Solution-phase samples | 14.3 | 13.6 | 9.4 |

Gas-phase samples | 14.3 | 9.5 | 6.5 |

^{a}

For solution-phase functionalized samples, this provide a lower limit, while for gas-phase functionalized samples, it clearly overestimates the inter-NC distance and underestimates the slope.

As shown in Sec. III B, the scaling exponent in Förster’s formulation is expected to be within 5–7 that for an NC ensemble with our size distribution. Therefore, this range is not consistent with the exponent between 10 and 14 obtained from the experimental data. Such strong deviations from Förster’s theory have been predicted for Si NCs in very close (<4 nm) contact,^{29} and to our knowledge, this is the first known experimental observation of this behavior in close-contact Si NCs. A 1/*R*^{14} dependence was predicted to dominate at very short distances due to the octopole–octopole interaction with the quadrupole–quadrupole interactions vanishing due to the T_{d} symmetry of the NCs.^{29} This work indicates a power law exponent of ∼14 based on MD simulations consistent with theory.

To understand whether this large deviation from the Förster exponent of 6 is due to errors in inter-QD distance estimation, we performed a sensitivity analysis of the scaling exponent as a function of the absolute error in the inter-NC distance (Fig. S6). Even with an inter-QD distance error of ±50%, much larger than that expected from MD the simulations, the scaling exponent ranges between 12 and 24, which is inconsistent with the Förster exponent of 6. Furthermore, also the other, simpler models in Table II, columns 2 and 3 indicate a power law exponent larger than 6 and closer to 10, which would be indicative of the quadrupole–quadrupole interactions.

While we cannot conclusively say which higher-order multipole interaction leads to our observed ET dynamics, our results do strongly suggest that it is higher-order multipole moments which govern the exciton transfer rates in Si NCs at short distances and not the Förster model. The correct nature of the multipole interaction may be resolved by future experimental studies that carefully measure the NC-NC separation directly, either by direct low-energy TEM measurements^{14} or observing changes in the packing density as a function of the ligand length with ellipsometry.^{50}

According to the expression of the Förster radius [Eq. (1)], the quantum efficiency of the donor *η*_{PL} proportionally influences the ET rate. When preparing the gas-phase functionalized Si NCs, we heated the samples at 160 °C, during which their PLQYs are improved from 2% to 35%–40%, to make sure that their PLQYs are similar to the solution-phase functionalized Si NCs. It is noteworthy that the ET rates of 3.2 nm Si NCs before and after heating, in fact, show no significant difference (Fig. S7). This constant ET rate supports our previous conclusion that heating increases PLQY by reducing the “dark” NC fraction within the ensemble without influencing the internal quantum efficiency (IQE) of NCs.^{30} The ET rates of the “bright” donor NCs contributing to the measured PL remain constant regardless of ensemble PLQY, and thus, the measured PL remains unchanged.

## IV. CONCLUSIONS

In this paper, we demonstrated tunable energy transfer in Si NC films by varying the alkyl ligand lengths used for Si NC functionalization. Steady-state and time-resolved PL measurements confirm that the energy transfer becomes more efficient when the ligand length becomes shorter. To obtain the NC spacing in the presence of surface alkyl ligands, we modeled the ligand packing structure using reactive molecular dynamics simulation. The combined analysis of experimental ET rate measurements and the MD simulation revealed that the distance-dependent ET rates in Si NCs decay faster than predicted by the Förster mechanism. The exponent of the power law is ∼14 rather than 6 as predicted by the Förster mechanism, possibly indicating octopole–octopole interactions, although we cannot rule out quadrupole–quadrupole interactions with an exponent of 10. When we consider the effect of NC size distribution, the exponent of the Förster mechanism power law changed only slightly to 5–7 for a moderately broad size distribution and can thus not explain the experimentally observed exponent of ∼14.

## SUPPLEMENTARY MATERIAL

See the supplementary material for flowchart of MD simulations, additional steady state and time-resolved photoluminescence data, data showing the influence of the refractive index on PL properties, ellipsometry data, sensitivity analysis of the inter-NC distance error, PL data for gas-phase functionalized Si NCs before and after heating, and the derivation of radiative decay rates.

## ACKNOWLEDGMENTS

This work was supported by the National Institutes of Health under Award No. R01DA045549. U.R.K., P.E., and A.V. acknowledge support by the Army Research Office under MURI Project under Grant No. W911NF-18-1-0240. Parts of this work were carried out in the Characterization Facility, University of Minnesota, which receives partial support from the National Science Foundation through the MRSEC Program (Grant No. DMR-1420013).

We thank Boris Shklovsii, Konstantin Reich, and Michael Sammon for inspiring this study and their generous assistance with initial theoretical considerations.

## AUTHOR DECLARATIONS

### Conflict of Interest

The authors have no conflicts to disclose.

## DATA AVAILABILITY

The data that support the findings of this study are available from the corresponding author upon reasonable request.

## REFERENCES

_{2}