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Doped Silicon Nanocrystal Plasmonics Hui Zhang, Runmin Zhang, Katelyn S. Schramke, Nicholas M. Bedford, Katharine Hunter, Uwe R. Kortshagen, and Peter Nordlander ACS Photonics, Just Accepted Manuscript • DOI: 10.1021/acsphotonics.7b00026 • Publication Date (Web): 28 Mar 2017 Downloaded from http://pubs.acs.org on March 28, 2017

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Doped Silicon Nanocrystal Plasmonics Hui Zhang,1§ Runmin Zhang,2§ Katelyn S. Schramke,3§ Nicholas M. Bedford,4 Katharine Hunter,3 Uwe R. Kortshagen,3 and Peter Nordlander1,2,* 1

Department of Electrical and Computer Engineering, Laboratory for Nanophotonics. Rice University, Houston, Texas 77005, United States.

2

Department of Physics and Astronomy, Rice University, Houston, Texas 77005, United States.

3

Department of Mechanical Engineering, University of Minnesota, Minneapolis, Minnesota 55455, United States 4 Applied Chemical and Materials Division, National Institute of Standards and Technology, Boulder, Colorado 80305, United States §

Equal contribution. *Correspondence should be addressed to P.N. (email: [email protected])

Doped semiconductor nanocrystals represent an exciting new type of plasmonic material with optical resonances in the infrared. Unlike noble metal nanoparticles, the plasmon resonance can be tuned by altering the doping density. Recently, it has been shown that silicon nanocrystals can be doped using phosphorous and boron resulting in highly tunable infrared plasmon resonances. Due to the band structure of silicon, doping with phosphorus contributes light (transverse) and heavy (longitudinal) electrons while boron contributes light and heavy holes and one would expect two distinct plasmon branches. Here we develop a classical hybridization theory and a full quantum mechanical TDLDA approach for twocomponent carrier plasmas and show that the interaction between the two plasmon branches results in strongly hybridized plasmon modes. The antibonding mode where the two components move in phase is bright and depends sensitively on the doping densities. The low energy bonding mode with opposite charge alignment can only be observed in the quantum regime when strong Coulomb screening is present. The theoretical results are in good agreement with the experimental data.

Keywords: Silicon Nanocrystals; quantum dots; plasmonics; plasmon hybridization

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Localized surface plasmon resonances (LSPR), the collective and coherent oscillations of carriers in a nanocrystal,1 have attracted much recent research interest. These excitations provide a mechanism for enhancing and controlling many important processes such as energy transfer,2,3 light-harvesting,4,5 and hot carrier generation.6,7 Plasmons also offer a path for manipulation of the optical response since they can be confined on a subwavelength scale. Plasmons couple strongly with incident light and generate large field enhancements as well as a significant absorption, which has been exploited in various applications including photocatalysis,8,9 photodetection,10,11 and increased efficiency of light-to-fuel devices.12

The tunability of plasmon resonances is a long-standing research goal in plasmonics. However, for noble metallic nanoparticles, only passive tuning can be achieved through modification of their size, shape, composition, or by embedding them in different dielectric media. Active tunability of plasmon resonances over a large spectral region is of significant interest in many relevant plasmonic applications. In order to achieve such a goal, different approaches have been followed, including mechanical stretching of nanoparticle arrays on elastomeric substrates,13 dynamic switching between the capacitive and conductive regimes through electrochemical oxidation-reduction processes14, and applied voltage controlled charge transfer plasmons.15 Recently, there has been increased interest in doped nanocrystals (NCs)16–18, since their free carrier densities can be changed dramatically through doping which in principle could be achieved in an active manner using a gate bias such as in a field effect transistor. For sufficiently large doping densities the NCs can support LSPRs in the infrared and visible parts of the spectrum. So far, the research effort has been focused mostly on II-VI nanocrystals,18–22 such as Cu2-xS, Cu2-xSe, Cu2-xTe, and other materials such as ZnO,23 transition-metal oxides,24 and indium tin oxide.25 Because the carrier density can be easily controlled, doped semiconductor NCs represent a particularly interesting approach for active control of plasmon resonances.

Recently it has been shown that silicon NCs can be doped to high densities using phosphorous (P) or boron (B), yielding highly tunable infrared plasmon resonances.26–31 Previous theoretical studies of the plasmonic behavior of doped semiconductors for single carrier doping have been published.32 However, due to the indirect band structure of silicon, both doping processes result in two carrier types, heavy and light electrons (holes) for phosphorus (boron), with different 2 ACS Paragon Plus Environment

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effective masses. The presence of two distinct but interacting charge carriers complicates the theoretical description of the optical response and to our knowledge, has not previously been investigated.

In this article, we theoretically and experimentally investigate the plasmonic properties of phosphorus- and boron-doped silicon NCs. We first generalize the quantum mechanical timedependent local density approximation (TDLDA) approach to include two different charge carriers. The NCs are described as jellium spheres and the interaction between the two components are described within the random phase approximation (RPA).22,33,34 The doped silicon nanocrystals are synthesized using a non-thermal plasma process. The optical properties of the nanocrystals are measured as-produced, then again after post-synthesis treatments including low-temperature annealing and oxidation at atmospheric conditions. The calculated absorption spectra are compared with experimental results and show good agreement for different doping levels. The theory predicts one dominant plasmon resonance in most cases, as is also observed in the majority of the experiments. This is surprising since the particles contain two different carriers with very different effective masses, and therefore different plasma frequencies, leading one to expect two distinct plasmon branches. To provide a physical understanding of the absorption spectra, and in particular why only one plasmon mode is visible, we generalize the classical plasmon hybridization approach to the two-component situation.35,36 In this model the degrees of freedom are the incompressible deformations of the two electron fluids, which interact with each other through the Coulomb interaction between their surface charges. In the classical limit, all surface charges are delta-functions, i.e. localized at the boundary of the particle. The hybridization picture reveals that two hybridized plasmon modes are formed: an antibonding high energy mode where the two components move in phase and a low-energy bonding mode where the two components move with opposite phase. The energy of the antibonding mode depends on the doping densities of both carriers. However, due to the strong interaction between the two components, the bonding mode is redshifted to zero energy and absent from the absorption spectrum but could in principle be observed in the limit of few carriers where quantum effects are important. 37–40


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Results

Figure 1: (a) Schematic of the experimental nanoparticle synthesis setup. (b) Illustration of our model: two kinds of carriers are present in an 8 nm-diameter Si nanospheredepicting the charge distribution of anti-bonding mode as an example. (c and d) Carrier distribution for surface boron- (c) and bulk P-doping (d) with 2.5, 5, 10, 20 × 1020 𝑐𝑐−3doping densities. (e and f) The corresponding absorption spectra calculated with a damping parameter δ=0.1 𝑒𝑒.

The experimental approach for synthesizing the doped Si NCs is illustrated in Fig. 1a and our

model in Fig. 1b. Theoretical simulations show that for Si NCs with a radius larger than 2.5 nm, the band structure is bulk-like.41 The Si NCs are therefore modeled as jellium spheres with a background polarizability of 11.97 as appropriate for bulk Si.42 Boron (B) doping is hole-type doping and results in holes at the top of the two valence bands at the Γ-point of the Brillouin zone. The effective masses are mH= 0.537𝑚𝑒 for the heavy holes and mL= 0.153𝑚𝑒 for the light

holes where 𝑚𝑒 is the mass of a free electron. Phosphorous (P) doping is electron-type doping and results in electrons in the 6 degenerate pockets near the X-points. The anisotropy of these pockets results in two types of carriers with effective masses being 0.916𝑚𝑒 for the heavy

(longitudinal) and 0.19𝑚𝑒 for the light (transverse) electrons.42 When two carriers of different 4


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mass are present, their relative doping densities will be different. For the present B- and Pdoping, the relative doping densities are related to the total doping density (𝑛 𝑇𝑇𝑇 ) as 𝑛𝐻 =

3

𝑚 2 � 𝑚𝐻 � 𝐿

3 𝑛 𝑇𝑇𝑇 , 𝑚𝐻 2

1 + �𝑚 � 𝐿

𝑛𝐿 =

1

3 𝑛 𝑇𝑇𝑇 , 𝑚𝐻 2

1 + �𝑚 � 𝐿

1 2 𝑛𝐿𝐿𝐿𝐿 = 𝑛 𝑇𝑇𝑇 , 𝑛 𝑇𝑇𝑇𝑇𝑇 = 𝑛 𝑇𝑇𝑇 3 3

(1)

For both P- and B-doping, we will refer to the carriers as heavy (subscript ‘H’ and ‘Long’) or light carriers (subscript ‘L’ and ‘Trans’). It makes no difference in the theoretical modeling whether the carriers are electrons or holes or whether the origin of the mass differences is due to band dispersion or anisotropy.

Depending on the chemical properties of the dopants and the synthesis process, the dopant distribution in Si NCs can either be primarily bulk (in which substitutional dopants in the lattice are essential), or primarily surface doping (in which dopants situated at the surface of the Si NCs surface are dominant donors).28,43 As shown in Figs. 1c and 1d, the resulting carrier distribution can be very different and depend on the spatial distribution of impurity potential. Previous studies concluded that P-doping is surface-like while B-doping would be bulk-like.30,31 However, subsequent studies where the dopant distribution was probed with X-ray photoelectron spectroscopy (XPS), showed that for B-doping, the relevant doping profile is surface doping while for P-doping the profile is bulk-like.28 In Figs. 1c and 1d, we show the carrier distributions for B- and P-doped silicon NCs modeled as 8 nm-diameter nanospheres for different doping densities. The holes tend to be located near the surface giving the particles a non-uniform carrier profile, rather than the nearly uniform carrier density shown in Fig. 1d for P-doping. The optical properties of the NCs depend on the carrier distributions. Figs. 1e and 1f show the calculated optical absorption spectra (to be described below) for the carrier distributions shown in Figs. 1c and 1d.

To model the optical properties we extend the TDLDA approach previously developed for single component carriers.22,33,44 First, we use Kohn-Sham (KS) equations45,46 to compute the ground state:


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�−

ℏ2

2𝑚𝑖

(𝑖)

(𝑖) (𝑖)

(𝑖)

∇2𝑟 + 𝑉𝑒𝑒𝑒,𝑙 (𝑟)� 𝑢𝑙𝑙 (𝑟) = 𝐸𝑙𝑙 𝑢𝑙𝑙 (𝑟) (𝑖 = 1, 2) (𝑖)

(𝑖)

(2) (𝑖)

where 𝑚𝑖 is the effective mass of the i:th component and 𝑢𝑙𝑙 (𝑟)=𝑟 ∙ ℛ𝑙𝑙 (𝑟) with ℛ𝑙𝑙 (𝑟) being (𝑖)

the radial wave functions. The effective potential 𝑉𝑒𝑒𝑒,𝑙 (𝑟) depends on the angular momentum l

and is given by,

(𝑖) (𝑖) 𝑉𝑒𝑒𝑒,𝑙 (𝑟) = 𝑉𝑃𝑃 (𝑟) + 𝑉𝑥𝑥 [𝑛(𝑟)] + 𝑉ℎ [𝑛(𝑟)] + 𝑉𝑖𝑖𝑖 �𝜌𝑖𝑖𝑖 ; 𝑟� + (𝑖)

ℏ2 𝑙(𝑙 + 1) 2𝑚𝑖 𝑟 2

(𝑖 = 1, 2), (3) (𝑖)

where 𝑉𝑃𝑃 is a confining spherical well potential. Due to the different effective mass, 𝑉𝑃𝑃 is

adjusted self-consistently to preserve the nominal relative doping concentrations of the different components in Eq. (1). Also, in this way, the work function 4.5eV is preserved for both two (𝑖)

components. In the calculations, the values of 𝑉𝑃𝑃 ranges between 4.60 eV and 4.64 eV. 𝑉𝑥𝑥 is

the exchange-correlation potential,22 𝑛(𝑟) is the carrier density at position 𝑟, 𝑉ℎ is the carrier-

carrier Hartree potential due to Coulomb interactions, while 𝑉𝑖𝑖𝑖 denotes the dopant impurity

potential (Supporting Information) which depends on the spatial distribution of impurities 𝜌𝑖𝑖𝑖 . The impurities are assumed to be uniformly distributed in the Si NCs for bulk doping. For

surface doping, the impurities are assumed to be on the surface of Si NCs. Although we cannot obtain the exact impurity distributions in Si NCs, our model captures the main properties of different doping distribution profiles. Equations (2) and (3) are solved self-consistently to ensure that the Fermi energies of the different components are equal, 𝐸𝐹,1 = 𝐸𝐹,2 . From the ground

state electronic structure, we use the Random Phase Approximation (RPA)22,33,44 to calculate the absorption spectra (Supporting Information). In the calculations we use a phenomenological damping parameter δ which is chosen to provide the best fit of the experimentally observed linewidths. In the infrared region of the spectrum, intra-band transitions will broaden the absorption peaks, however as shown in Fig. S6 in the SI, there are no significant shifts of the peak positions. In the experiments, the NCs are produced using a non-thermal, low-pressure plasma process with all gas-phase precursors.26,28As illustrated in Fig. 1a, Silane (SiH4) is the silicon precursor, while


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diborane (B2H6) diluted to 10% in hydrogen and phosphine (PH3) diluted to 15% in hydrogen are flown as boron and phosphorus precursors, respectively (more details in Supporting Information). All doping percentages indicated for experimental data are fractional precursor flows defined as Χ𝐵 = �2𝑄𝐵2𝐻6 /�𝑄𝑆𝑆𝑆3 + 𝑄𝐵2𝐻6 �� × 100% and Χ𝑃 = �𝑄𝑃𝑃3 /�𝑄𝑆𝑆𝑆3 + 𝑄𝑃𝑃3 �� × 100% where Q is the volumetric flow rate of the indicated precursor gasses.

The left panels of Figs. 2a and 2b present the measurements of absorption spectra for the oxidation process of silicon NCs under B- and P-doping cases, respectively. The sharp features in the spectra are related to absorptions from C-Hx, Si-Hx, or Si-O-Si surface species.

Figure 2: Measured (left) and calculated (right) absorption spectra. (a) Oxidized B-doped Si NCs ( Χ𝐵 = 10%) for as-produced, 2, 18, and 98 hour oxidization. The fitted doping concentrations are 1.5, 1.9, 2.0, 2.3 × 1020 𝑐𝑐−3 . (b) Oxidized P-doped Si NCs (Χ𝑃 = 30%) for as-produced, 20-minute, 1-hour, and 18-hour oxidization. The fitted doping densities are 2.1, 2.0, 1.6, 1.1 × 1020 𝑐𝑐−3 . (c) Annealed B-doped Si NCs (Χ𝐵 = 10%). The fitted doping densities are 1.5, 2.5, 3.2, 3.3 × 1020 𝑐𝑐−3. (d) As-produced P-doped Si NCs (Χ𝑃 = 10, 20, 30, 40, 50%). The fitted doping densities are 1.2, 1.3, 1.4, 1.7, 2.0 × 1020 𝑐𝑐−3 . The damping used is 𝛿 = 0.12 eV (a,c), 0.04 eV (b,d).

For the as-produced B-doped case, there is no obvious plasmonic absorption resonance. With

oxidation, a plasmon resonance appears and becomes stronger and blueshifts from 1000 𝑐𝑚−1 7


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to~1800 𝑐𝑚−1 . The phosphorus-doped NCs show the opposite behavior with oxidation. The plasmon resonance is initially at ~1600 𝑐𝑚−1 and redshifts and weakens. Fig. 2c shows the

effect of annealing the B-doped Si NCs. As the temperature is increased from 100℃ to 300℃,

the plasmon strengthens and blueshifts. The annealing process can free some trapped carriers to the bulk by passivating dangling bond defects, 26,47 resulting in an increase of free carrier density inside NCs and a blueshift of the plasmon resonance.48 Fig. 2d shows the effect of

fractional precursor flow 𝑋𝑃 on the spectra. As the doping density is increased, the plasmon

resonance in the as-produced P-doped Si NCs strengthens and blueshifts. To model the measured spectra we use the doping density and profile as fitting parameters. Figs. 1d and 1f show that the absorption spectrum blueshifts monotonously with increasing doping density. The Cabrera-Mott mechanism49 predicts that oxidation will result in a decreased carrier density for P-doping. The

doped electrons will recombine with the oxygen ions at the surface of the Si NCs and decrease the effective carrier concentration. For B-doping, the carrier generation mechanism is still vague. It may be related to surface dopants which accept electrons from Si NC’s valence bands and generate free holes.43,50 Annealing results in an increase in carrier densities for both B- and Pdoping as charges trapped at the surface become free with increasing temperature.26,47 The variations of the fitted doping densities in Figs. 2(a-d) are all consistent with these physical doping mechanisms and match the experimental data well.

Figure 3. Comparisons of carrier concentrations of as-produced Si NCs. For each fractional phosphine flow rate (10%to 50%), the height of the blue or red bars shows the measured atomic dopant concentration or fitted doping densities. The fitted carrier concentrations are extracted from calculations of Fig. 2d, where the diameter of the NCs is 8 nm, and the effective masses are 0.916𝑚𝑒 and 0.19𝑚𝑒 .

8


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Fig. 3 summarizes the measured atomic dopant concentration in experiments and the free carrier densities that were extracted from our simulations. The measured doping concentrations were determined using energy dispersive X-ray spectroscopy (EDS). The free carrier density is always smaller than the actual atomic concentrations and the ratio of precursors as shown in Fig. 3. This mismatch is due to a substantial amount of inactive dopant atoms at the surface and inside the nanocrystals. This could be caused by imperfect coordination between the dopants and the Si atoms, by internal defects inside nanocrystals created during synthesis, or by carrier localization effects similar to the situation in copper-deficient chalcogenides.20

The absorption spectra in Fig. 2 exhibit a single resonance. Due to the presence of two different charge carriers, associated with two different plasma frequencies, for both for P- and B-doping, two distinct plasmon resonances would be expected, rather than just one. One explanation for this behavior may be that the relative doping densities of the heavy and light carriers are related by the SCF doping constraint in Eq. (1). However, through the calculations of the optical absorption spectra for arbitrary and independent doping concentrations for the different carriers, we found that the absorption spectra are still characterized by a single plasmon resonance in most cases. To gain a deeper understanding of the optical resonance for a two-component system and specifically to understand why only one plasmon resonance appears in the absorption spectrum, we develop a classical hybridization model for the optical properties of a two-component system. Specifically, we extend the incompressible fluid model35,36 to a spherical two-component electron fluid. In order to keep the formalism analytical and transparent, we neglect the dielectric background screening of the SiNC in the description and all equations below. The effect of background dielectrics can straightforwardly be included using formalism previously reported.35,36 This polarizability results in a redshift of the plasmon energies and is fully accounted for in all of the calculation of the mode energies and absorption spectra presented in this paper. The Lagrangian for incompressible multipolar deformation of the conduction fluids is:


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𝐿=

𝑛1 𝑚1 𝑛 𝑚 ̇ (𝑡) + 2 2 � 𝑆2,𝑙𝑙 ̇ � 𝑆1,𝑙𝑙 (𝑡) 2 2 𝑙𝑙

2,𝑙𝑙

2𝜋𝑛12 𝑒 2 𝑙 2 2𝜋𝑛12 𝑒 2 𝑙 2 − �� 𝑆1,𝑙𝑙 (𝑡) + 𝑆 (𝑡) 2𝑙 + 1 2𝑙 + 1 2,𝑙𝑙 𝑙𝑙

+ (1 − 𝛽 )

4𝜋𝑛1 𝑛2 𝑒 2 𝑙 𝑆1,𝑙𝑙 (𝑡)𝑆2,𝑙𝑙 (𝑡)� 2𝑙 + 1

(4)

where the 1st and 2nd terms are the kinetic energy of the two components, respectively and the 3rd and 4th terms correspond to the repulsive potential energy from the Coulomb interactions of their surface charges. The last term represents the interaction between the two components. The doping density of the i:th component with effective mass 𝑚𝑖 is 𝑛𝑖 , and 𝑆𝑖,𝑙𝑙 is the amplitude of the deformation field of the i:th component.51 The multipolar order of the deformations is 𝑙 and 𝑚 and e is the electron charge. For the small nanostructures in the present study, retardation effects can be neglected and only the dipolar plasmons (l=1) are relevant.

The hybridization model is schematically illustrated in 4a. The plasmon resonances associated with the two components 𝜔1 and 𝜔2 interact and results in two hybridized modes: a redshifted

bonding mode 𝜔− with opposite alignment of the surface charges and a blueshifted antibonding mode 𝜔+ where the surface charges of the two components are aligned resulting in a repulsive

interaction. In a classical model, all induced charges are two-dimensional surface charges.

However, in a quantum mechanical model, these surface charges exhibit nonlocal effects, i.e. their distributions are shifted, broadened, and display an oscillatory spatial behavior (dynamical Friedel oscillations).52 The spatial characteristics of the broadened surface charge distribution depend on the doping densities and effective masses of the specific components, i.e. will be different for the two components. The Coulomb interactions associated with each carrier would be relatively independent of nonlocal effects since the dominant repulsive Coulomb term is the selfinteraction of their surface charge. However, interaction between the surface charges induced by the interference of different components could occur due to their different spatial distributions, i.e. their offset with respect to each other and their oscillatory pattern. 10 ACS Paragon Plus Environment

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To qualitatively account for this effect, we introduce an ad hoc screening parameter 𝛽 in the

interaction between the two components. A value 𝛽 =0 corresponds to classical theory without

screening. A finite 𝛽 indicates a reduced interaction between the two components. For 𝛽 = 1, the

interaction is zero and the motion of the two components occurs independently with their respective plasmon frequencies.

Figure 4. (a) Hybridization picture for a two-component system. (b) Plasmon energies 𝜔+ and 𝜔− as a function of the screening parameter 𝛽 for 𝑛𝐻 = 𝑛𝐿 = 1.5 × 1021 𝑐𝑐−3. (c) The corresponding evolution of the total absorption spectrum, where the screening parameter 𝛽 increases along the arrow. (d) Total and partial absorption spectra from each component for different doping densities. From top to bottom: 𝑛𝐻 = 𝑛𝐿 = 1.5 × 1021 𝑐𝑐−3; 𝑛𝐻 = 0.4, 𝑛𝐿 = 1.5 × 1021 𝑐𝑐−3; 𝑛𝐻 = 1.5, 𝑛𝐿 = 0.4 × 1021 𝑐𝑐−3. In all calculations, we assume a background permittivity (𝜀∞ = 11.97) for silicon.

We first investigate the classical limit 𝛽 = 0. The dipolar plasmon modes obtained from equations of motion of Eq. (4) are:

2 2 𝜔2+ = 𝜔𝑠,1 +𝜔𝑠,2 ,

𝜔−2 = 0

(5)


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2 where 𝜔𝑠,𝑖 =

4𝜋𝑛𝑖 𝑒 2 3𝑚𝑖

is the dipolar surface plasmon frequency of the 𝑖𝑖ℎ component as also

predicted from Mie theory. Thus the low-energy bonding mode is shifted to zero energy and disappears. This finding is interesting and physically reasonable because the surface charges from the two components are located at the same spot resulting in a very strong attractive

interaction. In the classical limit, the absorption spectrum is thus characterized by a single peak corresponding to the antibonding plasmon resonance with an energy determined by both carrier densities and effective masses, 𝜔+ = �

4𝜋𝑒 2 𝑛1 𝑛2 (𝑚1 +𝑚2 ). 3

Figure 3b shows the dispersion of the

hybridized plasmon resonances as a function of 𝛽. As the screening parameter 𝛽 is increased, the

interaction between the two modes decreases resulting in a blueshift of the bonding mode and a redshift of the antibonding mode. When the interaction is completely screened, 𝛽=1, the two

modes are Mie resonances of the individual conduction liquids. 4c shows the evolution of the absorption spectrum with increasing 𝛽. The lower energy bonding modes start to appear around

𝛽 = 0.5. Figure 3d shows the partial absorption of the two conduction components as well as the total absorption spectrum for several different carrier concentrations and 𝛽 = 0. The energy of the absorption resonance decreases monotonically with decreasing carrier densities.

To further explore whether the dopant distribution may play a role in the absorption spectra, we performed high-energy X-ray diffraction (HE-XRD) coupled to atomic pair distribution function (PDF) analysis53,54 and reverse Monte Carlo (RMC) structural simulations55,56 to experimentally elucidate atomic-scale dopant distribution in the P-doped Si NCs (Subsection E in the Supporting Information). In this instance, however, the calculated absorption spectra (Fig. S5 in the Supporting Information) showed no significant dependence on the spatial location of dopant distribution and are all consistent with a bulk dopant profile for as-produced P-doped Si NCs.


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Figure 5. (a) The absorption for different doping densities in boron doped Si NCs. The corresponding number of electrons inside NCs, from bottom to top, are 67, 93, 107, 120, and 147. The NC diameter is fixed to be 8 nm. The bulk-doping case is employed here, and the damping parameter is 0.06eV. Other unmentioned parameters are identical to Fig. 2a. (b) The absorption with varying size of NCs under a constant total doping density of 4.0× 1020 𝑐𝑐−3. The diameters of NCs are labeled in the legend, with the corresponding number of electrons inside NCs, from bottom to top, being 45, 71, 107, and 152. Other unmentioned parameters are identical to (a).

To further understand the origin of the double-peak feature revealed in 4(c) and why we can only observe one peaks in most systems, the TDLDA approach is used to show the crossover from the quantum plasmon regime to the classical regime for the two-component model. As discussed previously, quantum mechanical effects are essential for the double peak absorption feature to be observed. Quantum effects become important when the number of carriers is small. Although no definite number has been established, a study of the optical properties of silver nanorods clearly demonstrates the presence of a strong quantum effect in structures containing less than ~200 electrons.52 In Figs. 5a and b we show the evolution of the absorption spectra for increasing doping density and nanoparticle size respectively in a theoretical particle. The calculations are carried out for a hypothetical system: B-doped Si NCs with a bulk doping profile, and with the same effective masses as in Fig.2a (i.e., 0.537𝑚𝑒 and 0.153𝑚𝑒 ). As the number of electrons

increases from 67 to 147, we can clearly see a crossover from the quantum plasmon regime (i.e., smaller size and lower doping concentration cases), where two modes are apparent, to the classical regime where only the single peak resonance is present. More specifically, with 13 ACS Paragon Plus Environment

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increasing doping concentration (bottom to top in Fig. 5a), the high-energy plasmon resonance blueshifts22 and strengthens. For decreasing particle size (top to bottom in Fig. 5b) from 9.0nm to 6.0nm in diameter, the high-energy mode shows a tiny red shift first and then followed by a blue shift. The red shift of the high-energy mode can be attributed to the spill-out effect on electrons, while the blue shift is related to a confinement-induced change of permittivity.57 In SI we also give the absorption spectra calculated in a real B-doped Si NCs. In this case, we can only observe one single peak when the surface doping profile is employed, indicating a strong Coulomb screening effect, which is consistent with previous experimental work.58

Conclusion The optical properties of doped Si NCs are experimentally and theoretically investigated and compared. Due to the band structure of Si, both P- and B-doping results in two distinct carriers with different effective masses. The fully quantum mechanical TDLDA and classical plasmon hybridization approaches are extended to two component plasmon resonances. The interaction between the two different carrier components results in two hybridized modes: a lower energy bonding mode and a higher energy antibonding mode with an energy that depends on the sum of the doping densities of both carriers. In the classical limit, the interaction is very strong and the low energy bonding modes is redshifted to zero energy and only the higher energy antibonding mode is observed in the absorption spectra.


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Methods Quantum mechanical TDLDA description of a two-component system. For a two-component system, the two-fold Kohn-Sham (KS) equations are given in Eq.(2), along with the effective potential defined in Eq. (3). After obtaining the ground state of the structure, the optical resonance under external field can be calculated within the RPA approach. Specifically, the dipolar polarizability is:22,33,44 𝛼 (𝑟, 𝜔) = 𝛼0 (𝑟, 𝜔) + � 𝑑𝑟 ′ 𝜒(𝑟, 𝜔)𝛼 (𝑟′, 𝜔),

(𝑀1)

where 𝛼(𝑟, 𝜔) = 𝑟 2 ∙ 𝛿𝛿(𝑟, 𝜔) with 𝛿𝛿(𝑟, 𝜔) being the induced dipolar charge density at radial position 𝑟 under field frequency 𝜔. Both 𝛼0 (𝑟, 𝜔) and 𝜒(𝑟, 𝜔) are proportional to the dipolar response function for independent electrons Π (0) (𝑟, 𝑟 ′ , 𝜔),22which can be split into two terms, corresponding to the two component, respectively: (0)

(0) Π (0) (𝑟, 𝑟 ′ , 𝜔) = Π1 (𝑟, 𝑟 ′ , 𝜔) + Π2 (𝑟, 𝑟 ′ , 𝜔), (𝑀2)

where the subscript 1, 2 refers to the two components. The absorption for each component can be calculated straightforwardly using: 𝛼 (𝑟, 𝜔) 1 − 𝜒1 (𝑟, 𝜔) �=� � 1 𝛼2 (𝑟, 𝜔) −𝜒2 (𝑟, 𝜔)

The total polarizability is given by

−𝜒1 (𝑟, 𝜔) � 1 − 𝜒2 (𝑟, 𝜔)

−1

𝛼0,1 (𝑟, 𝜔) � �. 𝛼0,2 (𝑟, 𝜔)

(𝑀3)

𝛼 (𝑟, 𝜔) = 𝛼1 (𝑟, 𝜔) + 𝛼2 (𝑟, 𝜔) = [1 − 𝜒1 (𝑟, 𝜔) − 𝜒2 (𝑟, 𝜔)]−1 �𝛼0,1 (𝑟, 𝜔) + 𝛼0,2 (𝑟, 𝜔)� (𝑀4)

Finally, the absorption rate 𝜎𝑎𝑎𝑎 can be obtained through the relation 𝛼 (𝜔 ) =

4𝜋 𝜔 � 𝑑𝑟 ′ 𝑟 ′ 𝛼 (𝑟 ′ , 𝜔), 𝜎𝑎𝑎𝑎 (𝜔) = 𝑰𝑰[𝛼(𝜔 + 𝑖0+ )] 3 𝑐

where 𝑐 is the speed of light.

(𝑀5)

Classical theory for a two-component system under external excitations. As described in the main text, the Lagrangian with the screening parameter 𝛽 is introduced in Eq. (4). Using the equation of motions, the eigenmodes obtained in the presence of screening parameter are: 𝜔± 2 =

2

𝜔𝑠,1 2 + 𝜔𝑠,2 2 ± ��𝜔𝑠,1 2 − 𝜔𝑠,2 2 � + 4(1 − 𝛽)2 𝜔𝑠,1 2 𝜔𝑠,2 2 2

15


, (𝑀6)

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2 where 𝜔𝑠,𝑖 =

𝑙

2 𝑙+1

4𝜋𝑛𝑖 𝑒 2 𝑚𝑖

is the sphere surface plasmon frequency of the i:th component.

Assuming a multipolar external field35,36 𝜙𝑒𝑒𝑒 (𝑟, 𝑡) = 𝐸0 𝑒 𝑖𝑖𝑖 𝑟 𝑙 𝑌𝑙𝑙 (𝜃, 𝜑), with 𝐸0 being the field amplitude, 𝜔 is the angular frequency, 𝑌𝑙𝑙 is spherical harmonics with index 𝑙, 𝑚, then in the absence of dielectric background screening effects, the total Lagrangian becomes: 𝐿=

𝑛1 𝑚1 𝑛 𝑚 ̇ (𝑡) + 2 2 � 𝑆̇2,𝑙𝑙 (𝑡) � 𝑆1,𝑙𝑙 2 2 𝑙𝑙

2,𝑙𝑙

2𝜋𝑛12 𝑒 2 𝑙 2 2𝜋𝑛12 𝑒 2 𝑙 2 −�� 𝑆 (𝑡) + 𝑆 (𝑡) 2𝑙 + 1 1,𝑙𝑙 2𝑙 + 1 2,𝑙𝑙 𝑙𝑙

+ (1 − 𝛽 )

4𝜋𝑛1 𝑛2 𝑒 2 𝑙 𝑆1,𝑙𝑙 (𝑡)𝑆2,𝑙𝑙 (𝑡) 2𝑙 + 1

𝑙 + 𝐸0 (𝑡)𝑅𝑙+2 � 3 �𝑛1 𝑒𝑆1,𝑙𝑙 (𝑡) + 𝑛2 𝑒𝑆2,𝑙𝑙 (𝑡)� � 𝑅

(𝑀7)

Following the original plasmon hybridization approach,35,36 the absorption rate can be obtained as: 𝛼𝑖 (𝜔) = (2𝑙 + 1)𝑅2𝑙+1

𝜔2 𝜔𝑠,𝑖 2 − 𝛽𝜔𝑠,1 2 𝜔𝑠,2 2 𝜔 ( ) , 𝜎 𝜔 = 𝑰𝑰[𝛼𝑖 (𝜔 + 𝑖0+ )], (𝑀8) 𝑎𝑎𝑎,𝑖 2 2 2 2 𝑐 ( )� 4𝜋 𝜔 − 𝜔+ 𝜔 − 𝜔_ �

where 𝜔± are given in Eq. (M6). In the presence of dielectric background screening effect, the calculations of absorption are also straightforward.35,36 Associated Content Supporting Information

The supporting information is available free of charge on the ACS Publications website at DOI: Experimental setup, discussion of surface and bulk doping, absorption calculations using RPA, double peak feature, dopant distribution, Drude-Smith model and intraband transitions Conflict of Interest The authors declare no competing interest. Author contributions H. Z., R. Z., and P. N. worked on the theory, H. Z. and R. Z. performed all calculations based on the theory. K. S. S. and U. R. K. conducted the experiments, prepared samples and performed measurements. K. H. and K. S. S. prepared doped Si samples for N. M. B. who performed the 16 ACS Paragon Plus Environment

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PDF measurements and simulations to determine dopant distribution as shown in the SI. All authors contributed to the writing of the manuscript, and participated in discussions of the work. Acknowledgements We thank Alejandro Manjavacas and Alexander O. Govorov for helpful discussions on the work, and Karena W. Chapman for the help on collecting very high quality beamline data. This research was financially supported by the Army Office of Research under MURI Grant W911NF-12- 1-0407. The theoretical effort was also supported by the Robert A. Welch Foundation under grant C-1222. The experimental work was carried out in the College of Science and Engineering Characterization Facility, University of Minnesota, which has received capital equipment funding from the NSF through the UMN MRSEC program under Award Number DMR-1420013. Part of this work also used the College of Science and Engineering Minnesota Nanocenter, University of Minnesota, which receives partial support from NSF through the NNIN program. The use of beamline 11-ID-B of the Advanced Photon Source, a U.S. Department of Energy (DOE) Office of Science User Facility operated for the DOE Office of Science by Argonne National Laboratory was facilitated under Contract No. DE-AC0206CH11357.

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