# Size Dependence of Plasmonic Response in Phosphorus-Doped Si Nanocrystals: Implication for Infrared Plasmonics

Clavel Berclis Kengne, Xavier Devaux, Mathieu Stoffel, Michel Vergnat, Hervé Rinnert*

Université de Lorraine, CNRS, Institut Jean Lamour, F-54000 Nancy, France

## ABSTRACT

Highly doped semiconductor nanocrystals are of great interest for applications in nanophotonics and appear as an exciting alternative for infrared plasmonics with detection and identification of molecules, covering applications in biology, medicine, air quality control, sanitary control, and safety issues. Among the main parameters that influence optical properties and plasmonic response, the nanocrystals size plays a major role. In this work, we report on the influence of the silicon nanocrystals size on the localized surface plasmon resonance obtained in n-type Si nanocrystals embedded in a silicon dioxide matrix. The size control of phosphorus-doped Si nanocrystals was achieved by using a (SiO/SiO₂) multilayer architecture. In this study, the nanocrystal diameter is varied from 7 to 16 nm while the P content is kept constant at 0.9 atom %. Here, we demonstrate that the mid-infrared plasmonic absorption exhibits a redshift as well as a broadening as the nanocrystal diameter decreases from 16 to 7 nm. The plasmonic response was successfully modelled in the framework of the Mie theory, considering the Drude model and a size-dependent scattering of free carriers. The redshift of the plasmon is explained not only by a size-dependent scattering but also by a size-dependent doping efficiency. Both charge carrier mobilities and free carrier densities are found to vary between 17 and 28 cm²V⁻¹s⁻¹ and between 1.89×10²⁰ cm⁻³ and 2.6×10²⁰ cm⁻³, respectively. In this work, we shed light on the key role of the Si/SiO₂ interface that needs to be optimized to reach plasmonic properties in very small nanocrystals that could support quantum plasmonics.

Keywords: nanocrystals, plasmon, silicon, doping, infrared
INTRODUCTION

Localized surface plasmon resonances (LSPR) induced by collective oscillations of free carriers have been investigated for decades because of their high potential for applications in sensing,¹ cancer therapy,² or photovoltaics.³ In metal nanoparticles such as Au or Ag, the high charge carrier concentration leads to resonance frequencies for LSPR in the visible spectrum. As the resonance frequency strongly depends on the dielectric constant of the environment, targeted molecules can be sensed by measuring the spectral shift of the plasmon resonance. Another sensing approach consists of coupling the molecular vibrational mode, characterized by a low cross section, with the near field around plasmonic nanoparticles, which leads to Fano resonances.² A high vibrational signal enhancement is expected if the overlap between LSPR and molecular vibrations is strong. With metals, micrometer sized particles or even more complex structures are usually required to reach an optical resonance in the near-infrared or mid-infrared range.⁴,⁵ To overcome the limitation of plasmonic metal nanoparticles, highly doped semiconductors (SC) nanocrystals (NCs) have emerged as alternative plasmonic particles. Within the last fifteen years, LSPR were obtained in SC NCs either intrinsically doped with vacancies,⁶,⁷ or extrinsically doped with impurities.⁸,⁹ Due to their low cost, their compatibility with the CMOS technology, their ease of fabrication and their non-toxicity, Si nanocrystals (Si-NCs) are particularly promising. Although some theoretical works suggested that insertion of dopants at the nanoscale is not favoured by thermodynamics,¹⁰ highly phosphorus (P) or boron (B) doped Si-NCs showing plasmonic responses in the near-infrared were nevertheless obtained by using a nonthermal plasma method.¹¹,¹²,¹³,¹⁴ Other approaches were successfully used to efficiently insert dopants into Si-NCs such as annealing of Si/SiOₓ particles in gaseous phosphorus,¹⁵ low energy ion implantation of Si and P atoms in a silica matrix followed by rapid thermal annealing,¹⁶,¹⁷ or thermal evaporation of P-doped (SiO/SiO₂) multilayers in ultrahigh vacuum.¹⁸ Plasmonic P-hyperdoped Si nanodisks were also obtained on a silicon-on-insulator substrate by using a combination of e-beam lithography and reactive ion etching.¹⁹ The optical properties of Si NCs are found to be size-dependent. Indeed, size effects have been strongly studied in Si-NCs particularly motivated by quantum size effects that led to a strong increase of the photoluminescence properties.²⁰,²¹ In P-doped Si-NCs, Pi et al. demonstrated that the LSPR energy of a doped Si-NC varies with the carrier concentration according to the Drude model when the NC size is larger than 4 nm. For smaller Si NCs, the LSPR energy also depends on the NC size due to size dependent screening.²² For LSPR, two main effects are expected upon size-reduction. The first is the discretization of the electronic states near the conduction band minimum and valence band maximum. With size reduction, the
charge carrier confinement could lead to a transition from continuous LSPR absorption to discrete intraband absorption,²³ or to a more complex situation where both intraband and plasmon excitations could coexist.²⁴ The second case appears when the NC size is lower than the electron mean free path, leading to surface scattering of free electrons. This effect is generally well described by adding to the overall LSPR scattering a size-dependent scattering contribution that can be expressed as a function of the Fermi velocity of electrons, the NC diameter, and a dimensionless empirical parameter whose value depends on the details of the electron-surface interaction.²⁵ This size-dependent scattering leads to a redshift of the LSPR peak and to a broadening of the LSPR response. Both effects have been observed in metal and semiconductor LSPR.²⁵,²⁶,²⁷ However, only few papers have discussed the influence of the SC-NCs size on the LSPR. Moreover, there is no experimental report concerning the influence of size on LSPR in highly doped Si-NCs.

In this paper, we report on the influence of the NC size on the LSPR by considering an ensemble of uniformly P-doped Si-NCs formed in P-(SiO/SiO₂) multilayers. The multilayer approach has been successfully applied to achieve highly P-doped Si-NCs showing a plasmonic response.¹⁸ This approach offers several advantages: firstly, thanks the SiO₂ barrier layers, the size of the Si-NCs can be controlled by adjusting the SiO layer thickness, and secondly, several layers of heavily doped Si-NCs can contribute to the LSPR thereby improving the signal. The structure of the doped multilayers, which is obtained by combining advanced scanning transmission electron microscopy (STEM) and electron energy loss spectroscopy (EELS) mapping at the atomic scale, is presented. The size-control of highly doped Si-NCs is demonstrated for sufficiently thick SiO₂ barrier layers. The optical response measured by Fourier transform infrared spectroscopy (FTIR) is then presented and modelled in the framework of the Mie theory. The dielectric function of the material is modelled by the Drude theory. It is shown that a successful understanding of the experimental results requests the consideration of a size-dependent scattering effect of free carriers at the Si/SiO₂ interface. The modelling of the experimental results allows us to determine both concentration and mobility of free charge carriers as a function of the Si-NC size.

## EXPERIMENTAL SECTIONS

P-doped (SiO/SiO₂) multilayers were elaborated by successively evaporating silicon oxide (SiO) and silica (SiO₂) using electron guns. SiO and SiO₂ were deposited at a constant rate of 0.1 nm.s⁻¹ which is controlled by a quartz microbalance. (SiO/SiO₂) multilayers were grown on
rotating Si(100) substrates held at room temperature. The SiO₂ layer thickness was kept at 12.5 nm and that of SiO was varied from 7 to 16 nm. The total number of bilayers is 20. Phosphorus atoms were incorporated in both SiO and SiO₂ layers by a heated GaP decomposition source held at 800 °C during the whole deposition for all samples. The fixed GaP cell temperature allows us to keep a constant phosphorus content in the multilayers as well as in the Si-NCs regardless of their size. More details on P-content, NCs size, and GaP cell temperature dependence are given in reference.²⁸ As-deposited multilayers were amorphous. Samples were then post-growth-annealed in a rapid thermal annealing furnace at 1100 °C for 15 minutes under a constant nitrogen flow (200 sccm). As a result, the SiO layers will be destabilized leading to the formation of Si-NCs. The thickness of the SiO₂ layer is chosen to act as an efficient barrier and to prevent alteration of the multilayer architecture.²⁹ The multilayer architecture hence enables accurate tuning of the Si-NCs size as shown in Figure 1. The microstructural properties of P-(SiO/SiO₂) multilayers were investigated by energy-filtered transmission electron microscopy (EFTEM) and STEM energy-dispersive spectroscopy (STEM-EDS). Spatially resolved chemical maps were obtained by STEM electron energy loss spectroscopy (STEM-EELS). Electron microscopy and the associated spectroscopies were carried out with a probe-corrected JEOL ARM 200 F equipped with a cold-field emission gun. EELS spectrum images were recorded with a GATAN GIF quantum ER filter. More detailed information regarding the microstructural characterization can be found in our previous publications.¹⁸,²⁸,²⁹ The infrared (IR) absorption spectra were obtained using a Fourier transform infrared (FTIR) absorption spectrometer. The experiments were carried out using a Nicolet 6700 ThermoFisher spectrometer, under nitrogen flow in the sample cavity to avoid the absorption bands of the atmosphere. The spectra of the thin films were obtained after subtraction of a reference spectrum measured on a Si substrate.

<figure><img src="image_1.png" /></figure>
Figure 1. P-doped (SiO/SiO₂) multilayer structure grown on a Si(100) substrate before and after annealing at 1100°C for 15 minutes under N₂ flow (200 sccm). The annealing destabilizes the SiO layer thus leading to the formation of Si nanocrystals embedded in a SiO₂ matrix.

RESULTS

To control the size of the P-doped Si-NCs, a first set of multilayers was prepared with a fixed
SiO layer thickness equal to 10 nm and a SiO₂ barrier layer thickness that is varied between 6
and 20 nm. The dopant concentration was measured by STEM-EDS and was kept constant at

<figure><img src="image_2.png" /><figcaption>**Figure 2.** STEM-EELS maps of P-doped (SiO/SiO~2~) multilayers and their corresponding Si-NCs size distribution histograms. (a) Dependence of the SiO~2~ layers thickness with the SiO layer thickness being fixed at 10 nm (b) Dependence of the SiO layers thickness with the SiO~2~ layer thickness being fixed to 12.5 nm.</figcaption></figure>
0.9 at % for all samples studied in this work. Figure 2a shows the chemical maps relative to Si⁰ for post-growth annealed P-doped (SiO/SiO₂) multilayers and the corresponding histograms of the NCs size distributions. The maps are obtained after processing of EELS spectrum images.<sup>18,28</sup> The Si-NCs as well as the Si substrate (left side of the maps) appear in red while the silica matrix appears dark. As expected, the incorporation of P atoms in silicon oxide layers eases the diffusion of atoms during the annealing process and results in the loss of the multilayer structure for samples with the thinnest silica barrier (i.e. 6 nm). A similar behaviour, which is due to the P-induced softening of silicon oxide layers, was already reported.<sup>28,30</sup> The alteration of the multilayer structure by the P-induced softening leads to an increased NCs size. For thicker SiO₂ layers of 12.5 and 20 nm and for this dopant concentration, the softening still occurs but is not sufficient to alter the multilayer architecture. The multilayer structure therefore remains well preserved thanks to the large thickness of the SiO₂ layer, which acts as an efficient barrier for the NCs growth thus resulting in rather narrow NCs size distributions. Moreover, the chosen dopant concentration allows to obtain nanoparticles whose mean size is roughly equal to the thickness of the evaporated SiO active layer. This preliminary study allows us to conclude that a SiO₂ barrier thickness equal to 12.5 nm is well adapted for controlling the P-doped Si-NCs size. All multilayers were then grown with a barrier thickness equal to 12.5 nm and the SiO layer thickness was varied from 7 to 16 nm, while keeping the P concentration equal to 0.9 at%, the GaP cell temperature being fixed at 800 °C. Figure 2b shows the chemical maps relative to Si⁰ for P-doped (SiO/SiO₂) multilayers grown with a fixed SiO₂ barrier thickness of 12.5 nm and for SiO layer thicknesses varying between 8 and 12 nm. Whatever the thickness of the SiO layers, the multilayer architecture is well preserved. The histograms of the Si-NC sizes reveal narrow size distributions whose mean values are very close to the thickness of the deposited SiO layers.

Infrared absorption measurements were carried out to probe the optical properties of the P-
doped (SiO/SiO₂) multilayers. The spectrum shown as a solid black line in Figure 3a represents
the absorbance of the P-doped (SiO/SiO₂) multilayer grown with 16nm thick SiO layers.
Several contributions can be identified. Narrow and well-defined peaks in the low wavenumber
range are attributed to vibrational modes of Si-O-Si bonds of the silica matrix.<sup>31</sup> Peaks at 461,
816 and 1085 cm<sup>-1</sup> correspond to the bending, symmetric elongation and asymmetric elongation
mode of Si-O-Si bonds, respectively. Moreover, the baseline is characterized by a periodic
<figure><img src="image_3.png" /></figure>

<figure><img src="image_4.png" /><figcaption>**Figure 3.** (a) FTIR spectrum of a multilayer with a SiO thickness of 16 nm (black solid line) and simulated baseline using an interference-based model (solid red line). (b) Evolution of the infrared absorption measured on P-doped (SiO/SiO₂) multilayers for different SiO layer thicknesses (stacked for clarity).</figcaption></figure>

oscillation, which originates from interferences that are due to multiple reflections at the multilayer/substrate and air/multilayer interfaces. The red solid line on the same plot represents
the interferences simulated using a three media model taking into account a semi-infinite Si substrate, a thin layer modelled by its thickness and its optical index, and the air.32 This approach allows to extract a baseline that can be subtracted from the experimental absorbance. Finally, a rather broad absorption band located at around 2000 cm⁻¹ is observed. This contribution, which does not correspond to a vibrational mode, can be attributed to the plasmonic signature of free electrons generated in Si-NCs through doping, in good agreement with recently published results.17,18

Figure 3b shows FTIR spectra measured on P-doped (SiO/SiO₂) multilayers after subtraction of the contribution related to interferences, for SiO layer thickness varying between 7 nm and 16 nm. Spectra were normalized with respect to the overall SiO thickness. For all Si-NCs sizes considered here, a plasmonic contribution can be identified. As the Si-NCs mean diameter decreases, the intensity of the plasmonic contribution decreases significantly. Meanwhile, the peak shows a redshift and concomitantly, a broadening of the plasmonic response can be observed. According to Drude's model, the redshift could be due to a decrease of the free carrier concentration. This is, however, not expected since the P concentration is kept constant for all samples. The redshift could also arise from the interaction of the plasmon with the Si-NC/SiO₂ interface. To determine the free carrier density as well as the scattering value in P-doped Si-NCs, the absorption cross section was calculated assuming the theory of Mie for a spherical particle within the quasi-static approximation:

$$ \sigma_{abs} = \frac{8\pi^2 r^3 \omega \sqrt{\varepsilon_m}}{c} \mathrm{Im} \left[ \frac{\varepsilon(\omega) - \varepsilon_m}{\varepsilon(\omega) + 2\varepsilon_m} \right] \quad (1) $$

where ω is the frequency, c is the speed of light, r is the radius of the NC, εₘ is the dielectric constant of the surrounding medium (assumed to be frequency-independent over the spectral range of interest), and ε(ω) is the complex dielectric function of doped Si-NCs. ε(ω) was derived from the Drude model:

$$ \varepsilon(\omega) = \varepsilon_{\infty} - \frac{\omega_p^2}{\omega^2 + i\omega\gamma_{\text{bulk}}} \quad (2) $$
<figure><img src="image_5.png" /><figcaption>**Figure 4.** Infrared absorption spectra measured on P-doped (SiO/SiO₂) multilayers with SiO thicknesses of 14 nm (top) and 8.5 nm (bottom). Experimental (resp. simulated) curves are represented in solid black lines (resp. red dotted lines).</figcaption></figure>

Here, $\epsilon_{\infty}$ is the high frequency dielectric constant of the material, $\gamma_{bulk}$ is the damping constant of the bulk material that represent the free carrier scattering and $\omega_p$ is the bulk plasma frequency, that can be written as:

$$ \omega_p = \sqrt{\frac{ne^2}{m^*\epsilon_0}} \quad (3) $$
Here $e$ is the electron charge, $n$ is the free carrier density and $m^*$ is the free carrier effective mass. $\gamma_{bulk}$ is inversely proportional to the bulk carrier mobility $\mu_{bulk}$, and is written as,

$$ \gamma_{bulk} = \frac{e}{m^* \mu_{bulk}} \quad (4) $$

Experimental plasmon bands were simulated with the characteristic values for silicon and silica, using $m^* = 0.3m_0$, $\epsilon_\infty = 11.7$ and $\epsilon_m = 3.9$. $n$ and $\gamma$ are free parameters of the simulation which were extracted for each experimental curve. Figure 4 shows experimental infrared absorption spectra (black solid lines) with the corresponding simulated spectra (red dotted lines) for SiO layer thicknesses of 8.5 and 14 nm. Both measured spectra can be nicely fitted. The free electron density as well as the mobility are found to be greater than for the multilayer having larger Si-NCs. Figure 5a shows the dependence of both free electron density and mobility as a function of the Si-NCs diameter. Both values are increasing functions of the Si-NCs size. Considering the Si-NCs size range studied here, which is close to the electron mean free path in Si, an increase of the mobility (or decrease of the scattering) is expected. For smaller sized Si-NCs, the surface scattering is enhanced, thus leading to a reduction of the charge carrier mobility. When the Si-NCs mean diameter increases, the surface scattering effect becomes less significant which then results in an increased mobility. Following this reasoning, for very large crystals, surface scattering should be almost negligible thus giving rise to a bulk-like behaviour. The size effect does not explicitly appear in the equation defining $\epsilon(\omega)$. However, the influence
of the Si-NCs size can be phenomenologically introduced in the scattering value mentioned above by writing:

<figure><img src="image_6.png" /><figcaption>**Figure 5.** (a) Evolution of both densities and mobilities of free carriers extracted from the simulation of plasmons as a function of the Si-NCs diameter. (b) Inverse of the mobility as a function of the inverse of the NCs diameter.</figcaption></figure>

$$ \gamma = \gamma_{\text{bulk}} + \frac{A v_F}{D} \quad (5) $$
Here $\gamma$ is the total scattering of electrons which governs the broadening of the plasmon related peak, $\gamma_{bulk}$ is the electron scattering in bulk silicon, $A$ is a dimensionless parameter which is considered to be equal to unity, $v_F$ is the Fermi velocity and $D$ is the Si-NC diameter. By introducing the mobility defined in equation (4), equation (5) can be rewritten as:

$$ \frac{1}{\mu} = \frac{Am^*v_F}{eD} + \frac{1}{\mu_{bulk}} \quad (6) $$

Figure 5b shows the dependence of $1/\mu$ as a function of $1/D$. The obtained plot is linear thus demonstrating that the broadening of the plasmon related peak is indeed well described by the phenomenological modelling of the scattering using equation (5). Moreover, by considering the intercept of the linear fit shown in Figure 5b, one can determine the mobility in bulk Si which is equal to $59 \text{ cm}^2 \cdot \text{V}^{-1} \cdot \text{s}^{-1}$. This result is consistent with mobility values obtained from the literature for bulk Si containing dopant concentrations close to $10^{20} \text{ cm}^{-3}$.33 The slope of the dependence of $1/\mu$ as a function of $1/D$ allows to determine the Fermi velocity that is related to the free carrier density with the expression:

$$ n = \frac{1}{3\pi^2} \left( \frac{m^*v_F}{\hbar} \right)^3 \quad (7) $$

Here, $\hbar$ is the reduced Planck constant. The slope $Av_F m^*/e$ is equal to $1.32 \times 10^{-8} \text{ kg} \cdot \text{A}^{-1} \cdot \text{cm} \cdot \text{s}^{-2}$ and results in $v_F = 7.74 \times 10^3 \text{ cm} \cdot \text{s}^{-1}$ and $n = 2.7 \times 10^{20} \text{ cm}^{-3}$. The latter value is in good agreement with free carrier densities extracted from the simulation of the plasmon related peak. The results of the simulation hence suggest that the Drude model, associated with a size-dependence of the scattering, satisfactorily describes the plasmon response of P-doped Si-NCs. The increase of scattering induced by the decreasing Si-NCs size is accompanied by a redshift of the plasmon related peak (Figure 3b). This redshift may be explained by the enhanced scattering and by a decrease of the free carrier concentration from $2.6 \times 10^{20} \text{ cm}^{-3}$ for Si-NCs size close to 15 nm to $1.9 \times 10^{20} \text{ cm}^{-3}$ for the smallest Si-NCs studied here. Assuming that the P concentration inside the NC remains constant regardless of their size, the change in the carrier's density with the NC size only depends on the activation efficiency of these dopants. In a previous work, on similar samples, we demonstrated that the activation efficiency of dopants was close to 8% for Si-NC diameters equal to 10 nm.18 This activation efficiency is closely connected to the nature of the energy levels created by dopant atoms in the Si-NC band gap, which strongly depend on their localization within the Si-NC. Calculations show that the closer
a dopant atom is to the interface of the NC, the higher the activation energy.³⁴,³⁵ This location-dependence of the activation energy is attributed to the dielectric environment surrounding the dopant. At the nanocrystal/silica matrix interface, the dielectric function of the surrounding environment is no more only that of Si, but dopants undergo also screening from the SiO₂ matrix in which NCs are embedded. Dopant atoms located near the interface, will generate deep energy levels with large binding energies. In the case of small NCs with typical sizes of about 7 nm, most of the dopants will be located near the NC/SiO₂ interface. A rather large energy, significantly larger than the thermal energy at room temperature, is then required to ionize the dopants. As a result, they cannot supply free electrons in the Si-NCs at room temperature. As the NC size increases, most dopants are located far away from the interface. In this case, the environment of the dopants is close to that of bulk Si. As a result, the corresponding doping energy levels are shallow, thereby minimizing the energy necessary to activate such dopants and leading to a significant increase of the free electron density. This effect probably explains the size dependence of the free electron density shown in Figure 5a, which tends to reach a bulk-like behaviour for Si-NCs larger than 12 nm.

## CONCLUSION

In conclusion, by using a SiO₂ barrier thickness of 12.5 nm, size-controlled highly P-doped Si-NCs were successfully formed in (SiO/SiO₂) multilayers containing 0.9 at% of P, after annealing at 1100°C. The annealing process induced a phase separation in the SiO layers leading to the growth of P-doped Si-NCs whose size is limited by the thickness of the SiO layer. For all the Si-NCs sizes studied here, ranging from 7 nm to 16 nm, a plasmonic response was obtained in the mid-infrared, close to 5 µm. The influence of the Si-NCs size on the plasmonic response was modelled in the framework of the Mie theory, implementing the Drude model to describe the frequency-dependent dielectric function of Si. The modelling made it possible to determine both the density and the mobility of free carriers, which were found to increase as the Si-NCs size increases. The broadening of the plasmon related peak as the Si-NC size decreases, which is induced by the decrease of the mobility of carriers, is well modelled by a size-dependent scattering of free carriers at the Si/SiO₂ interface. The redshift of the plasmon peak as the size decreases is explained not only by the size-dependent scattering but also by a size-dependent doping efficiency. It is concluded that a multilayer structure is an efficient architecture to engineer plasmonic devices in the infrared based on highly doped and size-controlled Si-NCs. Because of the high scattering process at the NCs/matrix interface for small NCs, molecular sensing applications can be suitable provide the NCs size does not reach two
small values that could lead to a low coupling efficiency between plasmon and molecular vibration. The results also demonstrated the key role of the Si/SiO₂ interface that needs to be optimized to reach plasmonic properties with nanocrystals sizes below 10 nm.

## AUTHOR INFORMATION

### Corresponding Author

Hervé Rinnert – Institut Jean Lamour, Université de Lorraine, CNRS, F-54000 Nancy, France
; orcid.org/0000-0001-8777-1307

*E-mail: [herve.rinnert@univ-lorraine.fr](mailto:herve.rinnert@univ-lorraine.fr)

### Authors

Clavel Berclis Kengne – Institut Jean Lamour, Université de Lorraine, CNRS, F-54000 Nancy, France

Xavier Devaux – Institut Jean Lamour, Université de Lorraine, CNRS, F-54000 Nancy, France
; orcid.org/0000-0001-5694-464X

Mathieu Stoffel – Institut Jean Lamour, Université de Lorraine, CNRS, F-54000 Nancy, France
; orcid.org/0000-0003-3489-4519

Michel Vergnat – Institut Jean Lamour, Université de Lorraine, CNRS, F-54000 Nancy, France
; orcid.org/0000-0001-5237-9014

## ACKNOWLEDGMENTS

This work was supported by the French PIA project "Lorraine Université d'Excellence", reference ANR-15-IDEX-04-LUE and experiments were performed using equipments from the vibrational spectroscopy platform "Centre de Compétences Optique Lasers" at IJL funded by FEDER (EU), ANR, Région GrandEst and Metropole Grand Nancy.

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