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@@ -228,6 +228,8 @@ distributions in silicon nanoparticle around a magnetic resonance.}
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\label{fgr:concept}
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\end{figure}
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+Recently, highly localized plasma inside the nanoparticles, irradiated by femtosecond laser, has been directly observed using plasma explosion imaging \cite{Hickstein2014}. Additionally, inhomogeneous resonant scattering patterns inside single silicon nanoparticles have been experimentally revealed \cite{Valuckas2017}.
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+
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In this Letter, we show that electron-hole plasma (EHP) generation in
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a spherical dielectric (e.g., silicon) nanoparticle leads to strongly
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nonhomogeneous EHP distribution. To reveal and study this effect, we
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@@ -280,7 +282,7 @@ excitation~\cite{leuthold2010nonlinear}. Furthermore, silicon
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nanoantennas demonstrate a sufficiently high damage threshold due to
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the large melting temperature ($\approx$1690~K), whereas its nonlinear
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optical properties have been extensively studied during last
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-decays~\cite{Van1987, Sokolowski2000, leuthold2010nonlinear}.
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+decays~\cite{Van1987, Sokolowski2000, leuthold2010nonlinear}. High melting point for silicon preserves up to EHP densities of order $n_{cr} \approx 5\cdot{10}^{21}$ cm$^{-3}$ \cite{Korfiatis2007}, for which silicon acquires metallic properties ($Re(\epsilon) < 0$) and contributes to the EHP reconfiguration during ultrashort laser irradiation.
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Since the 3D modeling of EHP photo-generation in a resonant silicon
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nanoparticle has not been done before in time-domain, we develop a
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@@ -293,119 +295,53 @@ order to simplify our model, we neglect diffusion of EHP, because the
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aim of our work is to study EHP dynamics \textit{during} laser
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interaction with the nanoparticle.
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-\subsection{Light propagation} The propagation of light inside the
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-silicon nanoparticle is modeled by solving the system of Maxwell's
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-equations, written in the following way
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+\subsection{Light propagation}
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+
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+The propagation of light inside the silicon nanoparticle is modeled by solving the system of Maxwell's equations, written in the following way
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\begin{align} \begin{cases} \label{Maxwell}$$
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-\displaystyle{\frac{\partial{\vec{E}}}{\partial{t}}=\frac{\nabla\times\vec{H}}{\epsilon_0\epsilon}-\frac{1}{\epsilon_0\epsilon}\vec{J}}
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-\\
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-\displaystyle{\frac{\partial{\vec{H}}}{\partial{t}}=-\frac{\nabla\times\vec{E}}{\mu_0}},
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+ \displaystyle{\frac{\partial{\vec{E}}}{\partial{t}}=\frac{\nabla\times\vec{H}}{\epsilon_0\epsilon}-\frac{1}{\epsilon_0\epsilon}(\vec{J}_p+\vec{J}_{Kerr})} \\
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+ \displaystyle{\frac{\partial{\vec{H}}}{\partial{t}}=-\frac{\nabla\times\vec{E}}{\mu_0}},
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$$ \end{cases} \end{align}
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-where $\vec{E}$ is the electric field, $\vec{H}$ is the magnetizing
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-field, $\epsilon_0$ is the free space permittivity, $\mu_0$ is the
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-permeability of free space, $\epsilon = n^2 = 3.681^2$ is the
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-permittivity of non-excited silicon at $800$ nm wavelength [green1995]
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-\cite{Green1995}, and $\vec{J}$ is the nonlinear current, which
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-includes the contribution due to heating of the conduction band,
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-described by the differential equation derived from the Drude model
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-\begin{equation} \label{Drude}
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- \displaystyle{\frac{\partial{\vec{J}}}{\partial{t}} = - \nu_e\vec{J}
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- + \frac{e^2n_e(t)}{m_e^*}\vec{E}}, \end{equation} where $e$ is the
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-elementary charge, $m_e^* = 0.18m_e$ is the reduced electron-hole mass
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-[sokolowski2000]\cite{Sokolowski2000}, $n_e(t)$ is the time-dependent
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-free carrier density and $\nu_e = 10^{15} s^{-1}$ is the electron
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-collision frequency [sokolowski2000]\cite{Sokolowski2000}. Silicon
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-nanoparticle is surrounded by air, where the light propagation is
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-calculated by Maxwell's equations with $\vec{J} = 0$ and
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-$\epsilon = 1$. The system of Maxwell's equations coupled with
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-electron density equations is solved by the finite-difference
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-numerical method [rudenko2016]\cite{Rudenko2016} , based on the
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-finite-difference time-domain (FDTD) method [yee1966] \cite{Yee1966}
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-and auxiliary-differential method for disperse media
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-[taflove1995]\cite{Taflove1995}. At the edges of the grid, we apply
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-absorbing boundary conditions related to convolutional perfect matched
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-layers (CPML) to avoid nonphysical reflections
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-[roden2000]\cite{Roden2000} . Initial electric field is introduced as
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-a Gaussian focused beam source as follows
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+where $\vec{E}$ is the electric field, $\vec{H}$ is the magnetizing field, $\epsilon_0$ is the free space permittivity, $\mu_0$ is the permeability of free space, $\epsilon = n^2 = 3.681^2$ is the permittivity of non-excited silicon at $800$ nm wavelength \cite{Green1995}, $\vec{J}_p$ and $\vec{J}_{Kerr}$ are the nonlinear currents, which include the contribution due to Kerr effect $\vec{J}_{Kerr} = \epsilon_0\epsilon_\infty\chi_3\frac{\partial\left(\left|\vec{E}\right|^2\vec{E}\right)}{\partial{t}}$, where $\chi_3 =4\cdot{10}^{-20}$ m$^2$/V$^2$ for laser wavelength $\lambda = 800$nm \cite{Bristow2007}, and heating of the conduction band, described by the differential equation derived from the Drude model
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+\begin{equation} \label{Drude} \displaystyle{\frac{\partial{\vec{J_p}}}{\partial{t}} = - \nu_e\vec{J_p} + \frac{e^2n_e(t)}{m_e^*}\vec{E}}, \end{equation}
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+where $e$ is the elementary charge, $m_e^* = 0.18m_e$ is the reduced electron-hole mass \cite{Sokolowski2000}, $n_e(t)$ is the time-dependent free carrier density and $\nu_e = 10^{15}$ s$^{-1}$ is the electron collision frequency \cite{Sokolowski2000}. Silicon nanoparticle is surrounded by vacuum, where the light propagation is calculated by Maxwell's equations with $\vec{J} = 0$ and $\epsilon = 1$. The system of Maxwell's equations coupled with electron density equation is solved by the finite-difference numerical method \cite{Rudenko2016}, based on the finite-difference time-domain (FDTD) \cite{Yee1966} and auxiliary-differential methods for dispersive media \cite{Taflove1995}. At the edges of the grid, we apply absorbing boundary conditions related to convolutional perfectly matched layers (CPML) to avoid nonphysical reflections \cite{Roden2000}. Initial electric field is introduced as a Gaussian focused beam source as follows
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\begin{align}
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\begin{aligned}
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-\label{Gaussian} {E_x}(t, r, z) =
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-\frac{w_0}{w(z)}{exp}\left(i\omega{t} - \frac{r^2}{{w(z)}^2} - ikz -
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-ik\frac{r^2}{2R(z)} + i\varsigma(z)\right)\\
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+\label{Gaussian}
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+{E_x}(t, r, z) = \frac{w_0}{w(z)}{exp}\left(i\omega{t} - \frac{r^2}{{w(z)}^2} - ikz - ik\frac{r^2}{2R(z)} + i\varsigma(z)\right)\\
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\times\;{exp}\left(-\frac{(t-t_0)^2}{\theta^2}\right),
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\end{aligned}
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-\end{align} where $\theta$ is the pulse width at half maximum (FWHM),
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-$t_0$ is a time delay, $w_0 = 3{\mu}m$ is the waist beam, $w(z) =
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-{w_0}\sqrt{1+(\frac{z}{z_R})^2}$ is the Gaussian's beam spot size,
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-$\omega = 2{\pi}c/{\lambda}$ is the angular frequency, $\lambda = 800
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-nm$ is the laser wavelength in air, $c$ is the speed of light, ${z_R}
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-= \frac{\pi{{w_0}^2}n_0}{\lambda}$ is the Rayleigh length, $r =
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-\sqrt{x^2 + y^2}$ is the radial distance from the beam's waist, $R_z =
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-z\left(1+(\frac{z_R}{z})^2\right)$ is the radius of curvature of the
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-wavelength comprising the beam, and $\varsigma(z) =
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-{\arctan}(\frac{z}{z_R}) $ is the Gouy phase shift.
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-\subsection{Material ionization} To account for material ionization,
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-we couple Maxwell's equations with the kinetic equation for EHP
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-generation and relaxation inside silicon nanoparticle.
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-\begin{figure*}[ht!] \centering
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+\end{align}
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+where $\theta$ is the pulse width at half maximum (FWHM), $t_0$ is a time delay, $w_0 = 3{\mu}m$ is the waist beam, $w(z) = {w_0}\sqrt{1+(\frac{z}{z_R})^2}$ is the Gaussian's beam spot size, $\omega = 2{\pi}c/{\lambda}$ is the angular frequency, $\lambda = 800 nm$ is the laser wavelength in air, $c$ is the speed of light, ${z_R} = \frac{\pi{{w_0}^2}n_0}{\lambda}$ is the Rayleigh length, $r = \sqrt{x^2 + y^2}$ is the radial distance from the beam's waist, $R_z = z\left(1+(\frac{z_R}{z})^2\right)$ is the radius of curvature of the wavelength comprising the beam, and $\varsigma(z) = {\arctan}(\frac{z}{z_R}) $ is the Gouy phase shift.
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+
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+\subsection{Material ionization}
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+
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+To account for material ionization, we couple Maxwell's equations with the kinetic equation for EHP generation and relaxation inside silicon nanoparticle.
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+\begin{figure*}[ht!]
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+\centering
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\includegraphics[width=120mm]{fig2.png}
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-\caption{\label{fig2} Free carrier density snapshots of electron
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-plasma evolution inside Si nanoparticle taken a) $30$ fs b) $10$ fs
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-before the pulse peak, c) at the pulse peak, d) $10$ fs e) $30$ fs
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-after the pulse peak. Pulse duration $50$ fs (FWHM). Wavelength $800$
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-nm in air. Radius of the nanoparticle $R \approx 105$ nm,
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-corresponding to the resonance condition. Graph shows the dependence
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-of the asymmetric parameter of electron plasma density on the average
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-electron density in the front half of the nanoparticle. $n_{cr} =
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-5\cdot{10}^{21} cm^{-3}$ is the critical plasma resonance electron
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-density for silicon.}
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+\caption{\label{fig2} Free carrier density snapshots of electron plasma evolution inside Si nanoparticle taken a) $30$ fs b) $10$ fs before the pulse peak, c) at the pulse peak, d) $10$ fs e) $30$ fs after the pulse peak. Pulse duration $50$ fs (FWHM). Wavelength $800$ nm in air. Radius of the nanoparticle $R \approx 105$ nm, corresponding to the resonance condition. Graph shows the dependence of the asymmetric parameter of electron plasma density on the average electron density in the front half of the nanoparticle. $n_{cr} = 5\cdot{10}^{21}$ cm$^{-3}$ is the critical plasma resonance electron density for silicon.}
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\end{figure*}
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-The time-dependent conduction-band carrier density evolution is
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-described with a rate equation, firstly proposed by van Driel
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-[van1987] \cite{Van1987}, taking into account photoionization,
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-avalanche ionization and Auger recombination as
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-\begin{equation} \label{Dens}
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-\displaystyle{\frac{\partial{n_e}}{\partial t} =
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-\frac{n_a-n_e}{n_a}\left(\frac{\sigma_1I}{\hbar\omega} +
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-\frac{\sigma_2I^2}{2\hbar\omega}\right) + \alpha{I}n_e -
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-\frac{C\cdot{n_e}^3}{C\tau_{rec}n_e^2+1},} \end{equation} where
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-$I=\frac{n}{2}\sqrt{\frac{\epsilon_0}{\mu_0}}\left|\vec{E}\right|^2$
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-is the intensity, $\sigma_1 = 1.021\cdot{10}^3 cm^{-1}$ and $\sigma_2
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-= 0.1\cdot{10}^{-7} cm/W$ are the one-photon and two-photon interband
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-cross-sections [choi2002, bristow2007, derrien2013] \cite{Choi2002,
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-Bristow2007, Derrien2013}, $n_a = 5\cdot{10}^{22} cm^{-3}$ is the
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-saturation particle density [derrien2013] \cite{Derrien2013}, $C =
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-3.8\cdot{10}^{-31} cm^6/s$ is the Auger recombination rate
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-[van1987]\cite{Van1987}, $\tau_{rec} = 6\cdot{10}^{-12}$s is the
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-minimum Auger recombination time [yoffa1980]\cite{Yoffa1980}, and
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-$\alpha = 21.2 cm^2/J$ is the avalanche ionization coefficient
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-[pronko1998] \cite{Pronko1998} at the wavelength $800$ nm in air. Free
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-carrier diffusion can be neglected during and shortly after the
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-excitation [van1987, sokolowski2000]\cite{Van1987, Sokolowski2000}.
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-\begin{figure*}[ht!] \centering
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+The time-dependent conduction-band carrier density evolution is described with a rate equation, firstly proposed by van Driel \cite{Van1987}, taking into account photoionization, avalanche ionization and Auger recombination as
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+\begin{equation} \label{Dens} \displaystyle{\frac{\partial{n_e}}{\partial t} = \frac{n_a-n_e}{n_a}\left(\frac{\sigma_1I}{\hbar\omega} + \frac{\sigma_2I^2}{2\hbar\omega}\right) + \alpha{I}n_e - \frac{C\cdot{n_e}^3}{C\tau_{rec}n_e^2+1},} \end{equation}
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+where $I=\frac{n}{2}\sqrt{\frac{\epsilon_0}{\mu_0}}\left|\vec{E}\right|^2$ is the intensity, $\sigma_1 = 1.021\cdot{10}^3 cm^{-1}$ and $\sigma_2 = 0.1\cdot{10}^{-7}$ cm/W are the one-photon and two-photon interband cross-sections \cite{Choi2002, Bristow2007, Derrien2013}, $n_a = 5\cdot{10}^{22} cm^{-3}$ is the saturation particle density \cite{Derrien2013}, $C = 3.8\cdot{10}^{-31}$ cm$^6$/s is the Auger recombination rate \cite{Van1987}, $\tau_{rec} = 6\cdot{10}^{-12}$s is the minimum Auger recombination time \cite{Yoffa1980}, and $\alpha = 21.2$ cm$^2$/J is the avalanche ionization coefficient \cite{Pronko1998} at the wavelength $800$ nm in air. Free carrier diffusion can be neglected during and shortly after the excitation \cite{Van1987, Sokolowski2000}.
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+\begin{figure*}[ht!]
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+\centering
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\includegraphics[width=0.9\textwidth]{Ne_105nm_800}
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-\caption{\label{plasma-105nm} Split figure \ref{fig2} into two, this
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-is first part.}
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+\caption{\label{plasma-105nm} Split figure \ref{fig2} into two, this is first part.}
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\end{figure*}
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-The changes of the real and imaginary parts of the permeability
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-associated with the time-dependent free carrier response
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-[sokolowski2000] \cite{Sokolowski2000} can be derived from equations
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-(\ref{Maxwell}, \ref{Drude}) and are written as follows
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-\begin{align} \begin{cases} \label{Index} $$
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-\displaystyle{Re(\epsilon) = \epsilon
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--\frac{{e^2}n_e}{\epsilon_0m_e^*(\omega^2+{\nu_e}^2)}} \\
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-\displaystyle{Im(\epsilon) =
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-\frac{{e^2}n_e\nu_e}{\epsilon_0m_e^*\omega(\omega^2+{\nu_e}^2)}.}
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+The changes of the real and imaginary parts of the permittivity associated with the time-dependent free carrier response \cite{Sokolowski2000} can be derived from equations (\ref{Maxwell}, \ref{Drude}) and are written as follows
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+\begin{align} \begin{cases} \label{Index} $$
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+ \displaystyle{Re(\epsilon) = \epsilon -\frac{{e^2}n_e}{\epsilon_0m_e^*(\omega^2+{\nu_e}^2)}} \\
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+ \displaystyle{Im(\epsilon) = \frac{{e^2}n_e\nu_e}{\epsilon_0m_e^*\omega(\omega^2+{\nu_e}^2)}.}
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$$ \end{cases} \end{align}
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\section{Results and discussion}
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\subsection{Effect of the irradiation intensity on EHP generation}
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-
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-
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Fig. \ref{fig2} demonstrates EHP temporal evolution inside silicon
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nanoparticle of $R \approx 105$ nm during irradiation by
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high-intensity ultrashort laser Gaussian pulse. Snapshots of electron
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@@ -428,8 +364,7 @@ nanoparticle, increasing the asymmetry factor $G$ in
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Fig. \ref{fig2}(b). Approximately at the pulse peak, the critical
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electron density $n_{cr} = 5\cdot{10}^{21} cm^{-3}$ for silicon, which
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corresponds to the transition to quasi-metallic state $Re(\epsilon)
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-\approx 0$ and to electron plasma resonance [sokolowksi2000]
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-\cite{Sokolowski2000}, is overcome. At the same time, $G$ factor
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+\approx 0$ and to electron plasma resonance \cite{Sokolowski2000}, is overcome. At the same time, $G$ factor
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reaches the maximum value close to $2.5$ in
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Fig. \ref{fig2}(c). Further irradiation leads to a decrease of the
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asymmetry parameter down to $1$ for higher electron densities in
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@@ -459,17 +394,15 @@ average electron density $n_e^{front}$ in the front half of the
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nanoparticle for indicated radii (1-7).}
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\end{figure}
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-Previously, the EHP kinetics has been demonstrated only for a silicon
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-nanoparticle of a fixed radius $R \approx 105$ nm (TEI :REF
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-???). Here, we investigate the influence of the nanoparticle size on
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+We have discussed the EHP kinetics for a silicon nanoparticle of a fixed radius $R \approx 105$ nm. In what follows, we investigate the influence of the nanoparticle size on
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the EHP patterns and temporal evolution during ultrashort laser
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irradiation. A brief analysis of the initial intensity distribution
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inside the nanoparticle given by Mie theory for a spherical
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-homogenneous nanoparticle [mie1908] \cite{Mie1908} can be useful in
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+homogenneous nanoparticle \cite{Mie1908} can be useful in
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this case. Fig. \ref{fig3}(a, b) shows the scattering efficiency and
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the asymmetry parameter for forward/backward scattering for
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non-excited silicon nanoparticles of different radii calculated by Mie
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-theory [mie1908]\cite{Mie1908}. Scattering efficiency dependence gives
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+theory \cite{Mie1908}. Scattering efficiency dependence gives
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us the value of resonant sizes of nanoparticles, where the initial
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electric fields are significantly enhanced and, therefore, we can
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expect that the following conditions will result in a stronger
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@@ -510,7 +443,7 @@ along $Ox$ generates asymmetric EHP inside silicon nanoparticle,
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whereas the second pulse of lower pulse energy and polarization
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$Oz$interacts with EHP after the first pulse is gone. The minimum
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relaxation time of high electron density in silicon is $\tau_{rec} =
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-6\cdot{10}^{-12}$ s [yoffa1980] \cite{Yoffa1980}, therefore, the
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+6\cdot{10}^{-12}$ s \cite{Yoffa1980}, therefore, the
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electron density will not have time to decrease significantly for
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subpicosecond pulse separations. In our simulations, we use $\delta{t}
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= 200$ fs pulse separation. The intensity distributions near the
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@@ -520,12 +453,12 @@ shown in Fig. \ref{fig4}. The intensity distribution is strongly
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asymmetric in the case of EHP presence. One can note, that the excited
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nanoparticle is out of quasi-resonant condition and the intensity
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enhancements in Fig. \ref{fig4}(c) are weaker than in
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-Fig. \ref{fig4}(b).
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+Fig. \ref{fig4}(b). Therefore, the generated nanoplasma acts like a quasi-metallic nonconcentric nanoshell inside the nanoparticle, providing a symmetry reduction \cite{Wang2006}.
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\begin{figure}[ht] \centering
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\includegraphics[width=90mm]{fig4.png}
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\caption{\label{fig4} a) Electron plasma distribution inside Si
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-nanoparticle $R \approx 95$ nm $50$ fs after the pulse peak; Intensity
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+nanoparticle $R \approx 95$ nm $50$ fs after the pulse peak; (Kostya: Is it scattering field intensity snapshot XXX fs after the second pulse maxima passed the particle?) Intensity
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distributions around and inside the nanoparticle b) without plasma, c)
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with electron plasma inside.}
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\end{figure}
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rua14212