Optical Properties of Doped Silicon Quantum Dots with Crystalline and

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Optical Properties of Doped Silicon Quantum Dots with Crystalline and Amorphous Structures Michael G. Mavros,† David A. Micha,*,† and Dmitri S. Kilin‡ †

Quantum Theory Project, Departments of Chemistry and of Physics, University of Florida, P.O. Box 119435, Gainesville, Florida 32611-8435, United States ABSTRACT: The interaction of silicon quantum dots with light is remarkable, as electronic transitions are influenced by the interplay of their atomic structure and by electronic quantum confinement in three dimensions. In this study, the optical properties of 4 undoped and 16 doped silicon quantum dots were calculated using time-dependent density functional theory. The HOMOLUMO gap, maximum absorption wavelength, and oscillator strength at that wavelength were calculated for two crystalline structures, c-Si29H36 and c-Si35H36, and two amorphous structures, a-Si29H36 and a-Si35H36; in addition, optical properties were calculated for each of the structures doped with either phosphorus or aluminum in one of two different positions: in the center of the cluster or at the surface of the cluster. The calculated optical properties reveal that the absorbance spectrum of the amorphous structures is red shifted compared to that of the crystalline structures, and doping causes the spectrum to shift even further toward the red. Additionally, absorption of light at the maximum wavelength in doped structures caused charge density to transfer from the center of the quantum dot to the surface. The combination of strong absorptions in the visible region of the electromagnetic spectrum and the observed charge transfer make doped silicon quantum dots promising candidates as materials for solar energy applications.

1. INTRODUCTION Quantum dots are nanometer-sized particles whose electronic structure is dictated by quantum confinement.13 Because they possess the property that their electronic structure is highly tunable, quantum dots have been used in a variety of technological applications, including bioimaging,46 light-emitting devices,7 photodetectors, lasers,8 and solar cells.912 Nanocrystalline solar cells are of particular interest because of the potential to raise the maximum theoretical efficiency of a solar cell from about 30%13 to about 60%,14 primarily due to the ability of quantum dots to undergo a process known as multiple exciton generation.15 Quantum dots can be made out of a variety of materials, and due to its unique properties, silicon shows promise for solar energy applications. Silicon is environmentally friendly and stable to prolonged exposure to sunlight.16 Additionally, crystalline silicon has a well-studied band structure: The band gap of bulk silicon is indirect, meaning that optical transitions are phonon mediated. However, the bulk selection rules are lifted or modified in confined structures.17 The implication is that exciton recombination occurs extremely slowly,18 an ideal quality for photovoltaic devices.1921 Previously, we calculated many of the optical properties of bulk crystalline and amorphous silicon slabs, both pure and functionalized with silver clusters and dopants;2225 we now extend our work to include doped silicon quantum dots, an equally important class of materials. A very recent publication26 provides results for small doped quantum dots using a highly accurate electronic structure method. By comparison, we present here results with more approximate methods which can be used r 2011 American Chemical Society

for larger quantum dots and across a larger range of excitation energies. Additionally, here we present and compare results for both crystalline and amorphous doped quantum dots. Phosphorus and aluminum have been specifically chosen as dopants to examine the effects of p-doping and n-doping silicon quantum dots with atoms very similar in size and electronic structure to silicon in order to minimize perturbation of structures for comparison purposes. Both species have also frequently been involved in experimental and theoretical work on doped silicon. In our recent work,25 the effects of systematically doping Si slabs with those and additional atoms were explored. The electronic structure of crystalline silicon quantum dots has been studied previously.2729 As expected, silicon quantum dots have very similar electronic properties to bulk silicon with one additional feature: The exact nature of their electronic properties hinges upon their size. Electrons localized in a quantum dot with a certain radius can be thought of as electrons confined in a spherical well with the same radius; the larger the quantum dot, the lower the energy levels of the system. Amorphous silicon quantum dots have also been studied previously.3033 Experimentally, amorphous structures are much easier to produce; thus, for photovoltaic applications, the optical properties of amorphous quantum dots are of great interest. Bulk amorphous silicon absorbs electromagnetic radiation in a similar range as its crystalline counterpart; however, the efficiency of Received: June 14, 2011 Revised: August 26, 2011 Published: August 29, 2011 19529

dx.doi.org/10.1021/jp2055798 | J. Phys. Chem. C 2011, 115, 19529–19537

The Journal of Physical Chemistry C

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Figure 1. Models of crystalline silicon quantum dots; in the following labels, c- indicates crystalline and a- indicates amorphous whereas -i- and -oindicate the position of the dopant inside at the center of the dot and outside on the surface of the dot, respectively. The dopant is colored and marked with an arrow; X = P, Al. (a) Undoped c-Si29H36; (b) c-Si28H36-i-X; (c) c-Si28H36-o-X; (d) undoped c-Si35H36; (e) c-Si34H36-i-X; (f) c-Si34H36-o-X; (g) undoped a-Si29H36; (h) a-Si28H36-i-X; (i) a-Si28H36-o-X; (j) undoped a-Si35H36; (k) a-Si34H36-i-X; (l) a-Si34H36-o-X. The structures shown are those optimized for phosphorus; structures were also optimized for aluminum.

absorption is significantly lower, often around 6%.34,35 With their ease of production and size-tunable density of electronic states, both crystalline and amorphous quantum dots may optimize the light absorbance embodied in bulk crystalline and amorphous slabs.

The primary interest in the study of silicon quantum dots arises from their potential for chemical modification. Like all semiconductors, silicon quantum dots can be doped; however, because of their small size, doping often has a much more drastic effect on the electronic structure of quantum dots than of bulk 19530

dx.doi.org/10.1021/jp2055798 |J. Phys. Chem. C 2011, 115, 19529–19537

The Journal of Physical Chemistry C silicon.36 However, while doping silicon quantum dots is expected to have a large effect on the optical properties of the structures, the precise nature of the changes is largely unknown.37 In this paper, we present the electronic structure and optical properties of 20 atomic models of doped and undoped crystalline and amorphous silicon quantum dots, calculated using timedependent density functional theory. Trends in the excitation gap, wavelength of maximum absorption, oscillator strength at that wavelength, and exciton lifetime are highlighted. Additionally, ground-state and excited-state molecular orbitals of interest are plotted to showcase trends in charge redistribution upon the absorption of a photon.

2. METHODS 2.1. Modeling and Computational Details. In order to calculate ground- and excited-state properties of our large systems, approximate methods are required. For the systems that we studied, we found density functional theory (DFT) and time-dependent density functional theory (TD-DFT) to be expedient while still maintaining sufficient accuracy: our aim is not to provide results to high numerical accuracy but to compare the optical properties of a series of related compounds. Normal usage of TD-DFT involves finding solutions to the time-dependent KohnSham equations38 when the time-dependent part of the potential is “small” (i.e., in the pertubative regime), such as the absorption of light by a molecule, and linear response theory can be applied to obtain reasonable approximations,39,40 as was done in the present work. Four primary structures were generated in this work: crystalline Si29H36, crystalline Si35H36, amorphous Si29H36, and amorphous Si35H36. Small structures were used to save computational resources; however, the trends reported here are readily generalizable to larger structures.41 The crystalline structures were generated by initially using an A4 crystal structure of bulk silicon reported in the literature;42 then, the quantum dots were prepared by selecting a fragment of the bulk crystalline structure that includes the minimal number of atoms needed to mimic an approximately spherical shape. Specifically, we first selected a central atom and then identified its nearest neighbors and then the neighbors of the neighbors; the number of such iterations determined the diameter of the nanocrystal. The coordinates of the selected atoms were saved; then, any unsaturated bonds were hydrogenated and the ground electronic-state geometry of the resulting model was optimized with DFT using the Vienna Abinitio Simulation Package (VASP).43 The final crystalline structures generated were highly symmetric, belonging to the Td point group. Amorphous quantum dots were modeled using simulated melting and quenching44 in the HyperChem software package as follows.45 The crystalline structures were imported into HyperChem; then, the structures were unbonded and the hydrogen atoms removed to leave only unbonded silicon atoms. The entire structure was enclosed in a periodic box, and a molecular mechanics simulation was performed. The “silicon gas” was heated to 3000 K over 0.1 ps, allowed to diffuse for 10 ps, and then cooled back to 0 K over 0.1 ps. The silicon atoms were rebonded such that all silicon atoms were saturated and that the same number of hydrogen atoms passivated the surface of the quantum dot in the amorphous structure as in the crystalline structure. As before, the ground electronic state was optimized using DFT. Additionally, the pair correlation distribution

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function was calculated to show that the amorphous quantum dot structures that we generated are representative of all amorphous quantum dots. From each of the four parent structures, four doped structures were prepared for a grand total of 20 structures studied (4 undoped, 16 doped). In one structure, the parent was doped with aluminum in the center, in another with aluminum on the surface, in another with phosphorus in the center, and in the last with phosphorus on the surface. The location of the surface dopants was chosen such that the dopant would bond to four silicon atoms, thus ensuring that hydrogen atoms passivating the surface of the structures would not play a role in determining the optical properties of the quantum dots. Representatives of the 20 structures generated are shown in Figure 1. In this work, the electronic structure of the systems studied was calculated using DFT in Gaussian0346 and their optical properties using TD-DFT. All calculations were performed with the LANL2DZ basis set47 using the ab initio PW91/PW91 exchange-correlation functionals.48 The PW91 functionals were chosen because they do not contain any empirical parameters; additionally, they have given reliable physical results on HOMO LUMO gaps in previous work done in our group.2225 While TD-DFT is a very useful tool for excited-state calculations, it has known issues predicting the transition energies and oscillator strengths with numerical accuracy for open-shell systems:49,50 All of the doped quantum dots reported on here are open-shell systems because they contain an odd number of electrons. To investigate these issues on the results obtained for the doped structures, we calculated excitation energies with TDDFT for related ions and found that the trends were the same as what is reported here for the open-shell systems; thus, the trends among the results presented here for the open-shell quantum dots appear to be reliable. In this work, different energy eigenvalues were obtained for spin-up (α) and spin-down (β) electrons; the eigenvalues which resulted in a smaller HOMO LUMO gap are reported here. 2.2. Calculation of Optical Properties. Several properties were extracted from each Gaussian03 calculation, including the binding energy of the dopant, the HOMOLUMO gap, the electronic density of states, the exciton lifetime, and the spectral density for absorption. The binding energy of a dopant was calculated as Ebinding = (Eundoped + EP) (Edoped + ESi), where Eundoped and Edoped refer to the energy of the undoped and doped compound, respectively, and ESi and EP refer to the ground-state energies of the Si and P atoms in vacuum, respectively.41 Similarly, the energy of the HOMOLUMO gap was calculated as Egap= εHOMO εLUMO, where εHOMO and εLUMO refer to the energies of the highest occupied and lowest unoccupied molecular orbitals, respectively. The density of states was calculated using DðEÞ ¼

∑i δðE Ei Þ

where εi is the energy of a given orbital (calculated using DFT) and the index i runs over all orbitals calculated. The function δ(ε εi) is the Dirac delta function, modeled using a Lorentzian δðxÞ ¼

1 σ 2 π πσ þ x2

Here, σ is a parameter with the same dimensions as the argument of the delta function which gives the width of the distribution. The value σ = 0.05 eV was used to simulate spectral line 19531



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Table 1. Electronic Structure Dataa excitation gap binding energy of dopant (eV)

system crystalline

Si29H36 undoped

0

4.39

i-Al

4.35

3.95

o-Al

4.09

4.11

i-P

2.50

3.69

o-P

2.47

4.37

0 4.37

4.35 3.91

o-Al

4.22

3.96

i-P

2.51

4.20

o-P

2.50

4.17

0

2.27

i-Al

3.51

1.58

o-Al

3.30

1.11

i-P o-P

1.57 1.68

1.57 0.53

Si35H36 undoped i-Al

amorphous

(LUMO HOMO) (eV)

Si29H36 undoped

Si35H36 undoped

0

2.70

i-Al

3.26

1.68

o-Al

3.04

2.09

i-P

1.29

1.52

o-P

1.24

1.87

a

The prefixes i- and o- indicate the position of the dopant inside at the center of the dot and outside on the surface of the dot, respectively.

broadening in an experimental measure of the density of states. At room temperature (300 K), kBT = 0.027 eV, with additional effects such as inhomogenous broadening raising the uncertainty to the order of 0.05 eV. One of the parameters included in the output of a TD-DFT calculation is the oscillator strength defined as fij ¼

4πme ωij jDij j2 3he2

where ωij is the angular frequency required to excite an electron from state i to state j, Dij is the transition dipole from state i to state j, and me, h, and e are the fundamental constants. The oscillator strength can be thought of as the probability that a quantum dot which absorbs a photon of frequency ωij will undergo an electronic transition; it is thus related to the rate of absorption. The Einstein coefficient for absorption is given by B = (4π2|Dij|2)/(6ε0h2), where ε0 is the fundamental constant. Similarly, the Einstein coefficient for spontaneous emission is given by A = (hω3ijB)/(2π3c3). Thus, the lifetime τij of an excited state, defined as the inverse of the rate of spontaneous emission (A), is related to the oscillator strength by τij ¼

4π2 C3 E0 me fij ω2ij e2

The spectral density of absorption was calculated analogously to the density of states αðωÞ ¼

∑ fI δðpω pωI Þ

Iij

Here, the double index I = ij has been introduced to indicate that the sum runs over each transition and not the individual energy

levels. Each delta function is weighted by the oscillator strength corresponding to the transition, so that more probable transitions are given more weight in the total absorbance density spectrum. Here, the parameter σ in the approximation for the delta function is set to 0.05 eV, which physically corresponds to spectral line broadening due to coupling of electronic excitations to atomic vibrations in the clusters. For each absorbance spectrum calculated in this work, the 50 lowest energy excited states were used in the calculation.

3. RESULTS AND DISCUSSION 3.1. Electronic Structure Properties. Table 1 summarizes the electronic structure properties of the 20 systems calculated, showing the binding energies and the HOMOLUMO gaps for each system. It can be seen that the binding energy of the aluminum structures are larger than those for the phosphorus structures. The calculated binding energies reveal a small preference for the surface-doped structures over the center-doped structures by (0.21 ( 0.04) eV for the aluminum-doped dots and (0.030 ( 0.015) eV for the phosphorus-doped dots. For small silicon systems like quantum dots, dopant location is hard to control experimentally; thus, this preference indicates that most doped quantum dots prepared by an experimentalist will be doped on the surface. The dopant binding energy is also relatively constant for each dopant type among all systems calculated, showing the mutual consistency of the calculations reported in this work. The HOMOLUMO gaps (also reported in Table 1) reveal that the gap is smaller for amorphous quantum dots than for crystalline ones by almost 2 eV; amorphous quantum dots, then, have a substantially reduced gap. This compression can be explained primarily by symmetry. Since molecular orbitals can only be made from combinations of atomic orbitals of compatible symmetry, it follows that for the amorphous structures every atomic orbital can combine with every other atomic orbital to form a wide variety of molecular orbitals over a broad energy range.51 The amorphous quantum dots prepared here belong to the C1 point group, meaning that the symmetry-adapted linear combinations of atomic orbitals which make up the molecular orbitals of the amorphous molecules are quite unrestricted by symmetry and need not be classified. Since the crystalline clusters have much higher symmetry (and thus are more restricted in formation of molecular orbitals), the excitation gap is significantly larger, since the energy range the molecular orbitals can form over is narrower. The tabulated HOMOLUMO gaps for the four parent undoped quantum dots agree with previously reported values, both experimental and theoretical. Other calculated values for the excitation gap of Si29H36 include 3.92 eV with BLYP-DFT27 and 5.2 eV using the B88/PW91 exchange-correlation functionals and the 6-31G(d,p) basis.28 Other calculated values for the energy gap of Si35H36 include 3.7 eV using the BP functional and TZVP basis29and 5.1 eV using the B88/PW91 functionals and the 6-31G(d,p) basis.28 The data also agree with high-level ab initio calculations done on the excitation gap of quantum dots.52 Experimental values obtained using UV excitation have placed the gap of crystalline quantum dots, around 1 nm in diameter (the approximate size of both dots in this experiment), between 3.5 and 4.5 eV.53 Amorphous quantum dots around 1 nm in diameter have also been made experimentally, and the gap was determined to be approximately 2.7 eV using 19532



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Figure 2. Densities of states D(ε) of the (a) c-Si29H36, (b) c-Si35H36, (c) a-Si29H36, and (d) a-Si35H36 series. All D(ε) plots were calculated using energy eigenvalues from Gaussian03. Black solid lines correspond to the undoped quantum dot, red dashed lines to the aluminum-doped quantum dots, and green dashed lines to the phosphorus-doped quantum dots. All plots shown are for center-doped compounds; the surface-doped compounds have very similar plots for D(ε).

photoluminescence.30 Therefore, our calculated results are consistent with available experimental results. Another trend which can be seen in Table 1 is that doping a quantum dot decreases the HOMOLUMO gap. This effect occurs because doping adds states into excitation gap—electron states for phosphorus and hole states for aluminum—which causes the HOMOLUMO gap to decrease.43 As originally reported in the literature54,55 and confirmed here, doping a quantum dot in different locations adds states into the gap between the HOMO and the LUMO at different energies. The effect of doping can be seen visually by comparing the densities of states of the doped quantum dots with that of the undoped quantum dots, shown in Figure 2. The densities of states also confirm the “compression” of orbitals reported on earlier and agree qualitatively with the results presented previously.56 The density of states for crystalline silicon has sharp individual peaks, similar to small structures; in contrast, the density of states of amorphous silicon tends to have a smoother shape, as a continuous line, showing that the distribution of orbital energies has higher homogeneity. Figure 2 also shows that dopants add additional states into the excitation gap, as they do to the band gap in larger silicon slabs.25 The n-doped quantum dots exhibit additional states at 3 eV, near the “conduction band” minimum of the undoped model; by contrast, the p-doped quantum dots exhibit two peaks at 5.5 and 5.8 eV, near the “valence band” maximum of the undoped model. Careful study of the trends reported in Table 1 for the amorphous structures reveals some apparent inconsistencies; for example, the HOMOLUMO gap for the p-doped amorphous quantum dots shrinks when Al is moved from center to surface for Si29H36, yet it grows for the same change in the larger Si35H36. In addition, the gap for o-P in Si29H36 is appreciably smaller than the others. These discrepancies may be artifacts of

the generation of amorphous silicon quantum dots. A single model of an amorphous dot contains only one of many possible atomic conformations; an average over multiple amorphous conformations, not attempted here, might show more consistent trends. Analysis of Figure 2, 2c, and 2d reveals that the band gap in the amorphous quantum dots is determined by dopant atoms placed in the irregular amorphous structure; the shape and energy of orbitals of these dopants are significantly affected by the location of silicon atoms in the entire amorphous model. Details of these DOS might change in other amorphous structures, but the explanation for decreased gaps would remain the same. 3.2. Optical Properties. The highest intensity electronic transitions occurring upon photon absorption, calculated using TD-DFT, are shown in Table 2. The highest intensity transition was chosen as the transition with the highest oscillator strength. We chose to tabulate the results using the highest intensity transition in order to be unbiased of different quantum dots sizes and structural differences. We believe that the most intense transition occurs between orbitals with the same symmetry. Among different models, the absolute value of the transition energy may change but the intensity, which is related to the orbital symmetry, is a clear indication of the relevant optical properties of the quantum dots that is independent of model. Interestingly, the maximum value of the oscillator strength for undoped structures is in most cases lower than those for doped structures. Because doping inserts new states into the density of states distribution, doped quantum dots undergo lower energy electronic transitions than undoped quantum dots. Additionally, a remarkable trend was observed among the 16 doped structures studied: any perturbation from crystalline quantum dots doped in the center caused light absorption to be shifted toward the red end of the electromagnetic spectrum. Specifically, amorphous 19533



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Table 2. Calculated Optical Propertiesa system crystalline

Si29H36

Si35H36

amorphous

Si29H36

Si35H36

a

wavelength of max absorption (nm)

oscillator strength at max absorption

lifetime of state (ms)

undoped

256.36

0.0096

0.65

i-Al

420.64

0.0524

0.32

o-Al

507.29

0.0337

0.72

i-P

420.19

0.013

1.28

o-P

546.12

0.0172

1.63

undoped

266.40

0.016

0.42

i-Al

464.13

0.0236

0.86

o-Al i-P

507.58 567.53

0.017 0.0084

1.43 3.62

o-P

531.99

0.0141

1.89

undoped

318.14

0.0192

0.50

i-Al

636.11

0.0244

1.56

o-Al

754.73

0.0126

4.26

i-P

682.65

0.0207

2.12

o-P

815.26

0.0112

5.60

undoped i-Al

368.48 609.19

0.0105 0.0045

1.22 7.78

o-Al

780.43

0.0082

7.00

i-P

570.36

0.0056

5.48

o-P

582.78

0.0074

4.33

The prefixes i- and o- indicate the position of the dopant inside at the center of the dot and outside on the surface of the dot, respectively.

quantum dots absorb lower energy photons than crystalline quantum dots, and quantum dots doped on the surface absorb lower energy photons than ones doped in the center. The observed trends can be confirmed visually through the plotted spectral density for absorbance shown in Figure 3. Though a variety of modifications can potentially lower the energy of a silicon quantum dot, the physical mechanism is different for each. A previous study confirms that amorphous silicon absorbs lower energy photons than crystalline silicon.57 Because the orbitals in amorphous quantum dots are energetically more localized, the HOMOLUMO gap is smaller, causing red-shifted excitations. Additionally, previous studies confirm that doped silicon absorbs lower energy photons than undoped silicon,23,5860 an effect which occurs because new states inserted into the energy gap allow low-energy electronic transitions from the “valence band” to the hole state or from the electron state to the “conduction band”. Finally, a previous study54 shows that the closer a phosphorus dopant is located to the surface of a quantum dot, the closer in energy the added electron state is to the “conduction band”, explaining the red-shifted excitations observed when a dopant was moved from the center to the surface of a quantum dot in this study. It seems as if moving an aluminum dopant closer to the surface analogously moves the added hole state closer to the “valence band”. Besides knowing trends in wavelengths of absorption, knowing the probability of undergoing a transition after absorption of electromagnetic radiation is critical in cataloging the optical properties of a class of molecules. The oscillator strengths of the maximum intensity absorptions are also reported in Table 2. The oscillator strength is related both to the probability that an absorption will occur and to the lifetime of the excited state created by that absorption. The oscillator strength is found to scale inversely with quantum dot size, meaning electrons remain in excited states for longer in smaller dots. This effect may be

related to quantum confinement: smaller quantum dots have orbitals which are more localized spatially and thus overlap with each other more, causing higher probability transitions. In agreement with the present results, previous studies56,6163 report exciton lifetimes of the order 0.110 ms in quantum dots, of the same order of magnitude as those calculated in this study. However, a more complete study would analyze vibronic couplings to relate the observed trend to a vibrational bottleneck effect: An electronic transition of higher energy has a lower chance to undergo a nonradiative relaxation event, as high-energy vibrations are less readily available to accept the energy given off by nonradiative relaxation than lower energy vibrations. Figure 4 shows the exciton recombination rate (1/τij, the lifetime) plotted versus the wavelength of absorption. The figure shows a negative correlation between the radiative recombination rate and wavelength, in agreement with previous results;64 this shows that quantum dots which absorb light primarily in the red region of the electromagnetic spectrum form excitons with longer radiative lifetimes. Since both doped quantum dots and amorphous quantum dots absorb lower energy photons than undoped and crystalline nanocrystals, it is possible to tune the lifetime of excitons formed upon light absorption through controlling the quantum dot structure. Following our previous work65 on photoinduced surface photovoltage in silicon slabs, we expect that photon absorption in doped silicon quantum dots will cause an analogous transfer of charge between the location of the dopant and regions away from it. In our structures, n-doped quantum dots experience charge transfer away from the dopant whereas p-doped quantum dots undergo charge transfer toward the dopant. Figure 5 displays relevant molecular orbitals of the representative phosphorus-doped c-Si29H36 systems, showing the realization of this charge transfer in the systems studied 19534



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Figure 5. Molecular orbitals showing the highest oscillator strength electronic transition for (a) c-Si29H35-i-P and (b) c-Si29H35-o-P. Orbital numbers are labeled in boldface for the final state; in both cases, the transition began from orbital 77, the HOMO. Multiple orbitals may be excited in the final many-electron state of the transition, appearing in degenerate states; the true final state in a is a combination of degenerate states, with shown linear combination coefficients below the orbital numbers. In each case, the final state of the transition is a very diffuse orbital with a significant fraction of charge spread out over the surface of the quantum dot.

Figure 3. Spectral density for absorption α(ε) for the (a) c-Si29H36-Al, (b) c-Si35H36-Al, (c) a-Si29H36-Al, (d) a-Si35H36-Al, (e) c-Si29H36-P, (f) c-Si35H36-P, (g) a-Si29H36-P, and (h) a-Si35H36-P series. In each panel, the black line corresponds to the undoped structure, thick dashed lines correspond to dopants located inside the structure at the center, and thin dotted lines correspond to dopants located outside the structure on the surface. In addition to the peaks shown here, doped structures are expected to have additional absorbance peaks similar to the ones in the parent structures; however, these peaks were not generated in this work due to computational limitations on the number of excited states generated.

Figure 4. Calculated exciton decay rate (defined as 1/τij, inverse of the excited-state lifetime) plotted as a function of the wavelength of the highest intensity absorption. The longest lifetime transition for each atomic model is represented by a symbol in this plot. The values are negatively correlated: quantum dots having intense absorption of longer wavelength electromagnetic radiation form excitons which recombine more slowly, implying a longer exciton lifetime.

here. Electrons were observed to be promoted from an orbital centered on the dopant to a more diffuse orbital with larger amplitudes at the surface of the quantum dot, qualitatively representing a delocalization of charge from the dopant and some transfer to the surface.

Consistent with the qualitative results reported, the maximum oscillator strength of phosphorus-doped quantum dots was observed to be larger for compounds doped on the surface. This trend can be explained largely by the breaking of symmetry when the phosphorus dopant is moved from the center to the surface of the quantum dot. Moving the location of the dopant changes the symmetry of the entire quantum dot, which, in turn, breaks the symmetries of its molecular orbitals. Overlapping molecular orbitals that once destructively interfered now constructively interfere. In at least one case an occupied orbital has a larger overlap with an unoccupied orbital, allowing for more probable excitation to an excited state when the dopant is located on the surface. This trend can also be explained by considering that it represents more probable transitions from the dopant-localized HOMO to an orbital diffuse about the surface of the quantum dot when the dopant was itself located on the surface of the structure. Since phosphorus is an n-type dopant, with an extra electron compared to silicon, the transitions are stronger because the orbital at which this extra electron is located overlaps more with orbitals diffuse about the surface, and this gives larger oscillator strengths. Conversely, aluminum dopants undergo more probable transitions when the dopant is located in the center. This is likely because the HOMO of an aluminum-doped quantum dot has several nodal planes which intersect on the aluminum dopant—a consequence of aluminum being a p-type dopant. This means that a quantum dot doped in the center with aluminum has much of its ground-state electron density away from the center of the molecule, so that transitions to the surface are more probable, whereas a quantum dot doped on the surface with aluminum has much of its ground-state electron density localized toward the center, decreasing the probability of transitions to the surface. The result is an increase in probability of transitions from the delocalized “valence band” orbitals of silicon to more localized dopant levels. 19535



4. CONCLUSIONS In summary, the TD-DFT-calculated optical properties of doped crystalline and amorphous silicon quantum dots are reported here for a series of structures that provide helpful comparisons. Amorphous silicon quantum dots absorb electromagnetic radiation at longer wavelengths than crystalline silicon quantum dots, though with lower probability. Additionally, dopants positioned in the center of quantum dots cause the structures to absorb light at longer wavelengths, and dopants located at the surface absorb at even longer wavelengths. Depending on the type of impurity, different dopant positions (center or surface) cause the structures to undergo electronic transitions with different probability. It is difficult for an experimentalist to control the location of an impurity in a small compound; thus, knowing how dopant position affects optical properties is extremely important. While recently some progress has been made on site-specific doping,66,67 doping is generally carried out by controlling the dopant concentration in the solution or colloidal mixture in which crystalline silicon quantum dots are synthesized or the silane gas in which amorphous silicon quantum dots are synthesized. Additionally, evidence suggests that phosphorus dopants tend to get ejected to the surface of small silicon nanocrystals,68 which the binding energy calculation presented here supports; thus, the optical properties of quantum dots doped on the surface may be even more relevant than those doped in the center. The optical properties of doped silicon quantum dots suggest the use of quantum dots in a wide variety of technological applications. The quantum dots studied in this paper are especially suitable for solar energy applications, where absorption of sunlight (which is peaked in the near-IR part of the electromagnetic spectrum) is desired to form excitons with a long lifetime. The quantum dots most suited for this application are the amorphous ones, which, according to our observed trends, absorbs low-energy photons to induce electronic transitions with the slowest exciton decay rate. For a photovoltaic application, a relevant investigation would involve finding a method to extract charge from the surface of a quantum dot after an optically induced electronic transition, effectively turning light into an electrical current; the issue has been experimentally studied in a recent paper.69 Since the studied transitions describe excitons which place negative charge on the surface of a quantum dot for time scales on the order of 1 ms, extracting charge should be relatively simple. The LUMOs of the n-doped quantum dots studied in this experiment are delocalized about the surface of the dots, so the LUMO of a second quantum dot could be used as an electron acceptor should the two quantum dots be spatially close. Experimentally, this may be realized by preparing the dots in a SiO2 matrix,70 where the quantum dots are spaced regularly at known distances apart. Gaining insight into the charge extraction process would require considerable knowledge of how the quantum dot interacts with a medium or a solvent and how it interacts with neighboring quantum dots, something that could be attained using methods similar to the ones presented here. In addition, it would be necessary to explore different exchange/correlation functionals besides the PW91 functionals explored here to include effects of electronic self-interaction and long-range electron transfer. Work on comparisons with other functionals is ongoing. The rich optical properties of silicon nanocrystals and quantum dots result from two features: crystalline silicon’s indirect band gap and three-dimensional confinement in quantum dots,

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which partially lifts and modifies the bulk Si selection rule. Quantum dots are highly tunable, as their diameter can, in principle, be controlled by the conditions under which they are made. Crystalline and amorphous quantum dots have significantly different optical properties, and dopants additionally allow more intense absorption of light due to their introduction of new states into the excitation gap. Therefore, silicon quantum dots are highly adaptable and can provide optical properties as desired for a relevant application.

’ AUTHOR INFORMATION Corresponding Author

*E-mail: [email protected]fl.edu. Present Addresses ‡

Department of Chemistry, University of South Dakota, 414 East Clark Street, Vermillion, South Dakota 57069, United States.

’ ACKNOWLEDGMENT This work was supported financially by the National Science Foundation and the Dreyfus Foundation. Computational resources were provided by the University of Florida High Performance Computing Facility. Related aspects of this work were presented at the 2011 Sanibel Symposium and submitted to the University of Florida Department of Chemistry as part of an undergraduate thesis by Michael G. Mavros. ’ REFERENCES (1) Ekimov, A. I.; Onushchenko, A. A. JETP Lett. 1981, 34, 345–349. (2) Reimann, S. M.; Manninem, M. Rev. Mod. Phys. 2002, 74, 1283– 1342. (3) Tisdale, W. A.; Zhu, X. Y. Proc. Natl. Acad. Sci. 2011, 108 (3), 965–970. (4) Yong, K.-T.; Ding, H.; Roy, I.; Law, W.-C.; Bergey, E. J.; Maitra, A.; Prasad, P. N. ACS Nano 2009, 3 (3), 502–510. (5) Mauro, J. M.; Mattoussi, H.; Medintz, I. L.; Goldman, E. R.; Tran, P. T.; Anderson, G. P. In Defense Applications of Nanomaterials; ACS Symposium Series 891; American Chemical Society: Washington, DC, 2005; pp 1630. (6) Lin, C. A. J.; Liedl, T.; Sperling, R. A; Fernandez-Arg€uelles, M. T.; Costa-Fernandez, J. M. J. Mater. Chem. 2007, 17, 1343–1346. (7) Anikeeva, P. O.; Halpert, J. E.; Bawendi, M. G.; Bulovic, V. Nano Lett. 2009, 9 (7), 2532–2536. (8) Bukowski, T. J.; Simmons, J. H. Crit. Rev. Solid State Mater. Sci. 2002, 27, 119–142. (9) Nozik, A. J. Nano Lett. 2010, 10, 2735–2741. (10) Nozik, A. J. Phys. E 2002, 14, 115–120. (11) Cho, E. C.; Park, S.; Hao, X.; Song, D.; Conibeer, G.; Park, S. C.; Green, M. A. Nanotechnology 2008, 19, 245201. (12) Kongkanand, A.; Tvrdy, K.; Takechi, K.; Kuno, M.; Kamat, P. V. J. Am. Chem. Soc. 2008, 120 (12), 4007–4015. (13) Shockley, W.; Queisser, H. J. Detailed balance limit of efficiency of p-n junction solar cells. J. Appl. Phys. 1961, 32, 510–519. (14) Ellingson, R. J.; Beard, M. C.; Johnson, J. C.; Yu, P.; Micic, O. I.; Nozik, A. J.; Shabaev, A.; Efros, A. L. Nano Lett. 2005, 5 (5), 865–871. (15) Kamat, P. V. J. Phys. Chem. C. 2008, 112, 18737–18753. (16) Lewis, N. S. Science 2007, 315, 798–801. (17) Kovalev, D.; Heckler, H.; Ben-Chorin, M.; Polisski, G.; Schwartzkopff, M.; Koch, F. Phys. Rev. Lett. 1998, 81, 2803–2806. (18) Kittel, C. Introduction to Solid State Physics, 8th ed.; John Wiley & Sons, Inc.: Hoboken, NJ, 2005; pp 188192. (19) W€urfel, P. Physics of Solar Cells; Wiley-VCH: Weinheim, Germany, 2005. 19536


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