497 lines
10 KiB
Plaintext
497 lines
10 KiB
Plaintext
#LyX 2.4 created this file. For more info see https://www.lyx.org/
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\lyxformat 583
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\begin_document
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\begin_header
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\shortcut idx
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\end_header
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\begin_body
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\begin_layout Subsection
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Periodic systems and mode analysis
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\begin_inset CommandInset label
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LatexCommand label
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name "subsec:Periodic-systems"
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\end_inset
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\end_layout
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\begin_layout Standard
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In an infinite periodic array of nanoparticles, the excitations of the nanoparti
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cles take the quasiperiodic Bloch-wave form
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\begin_inset Formula
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\[
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\coeffs_{i\nu}=e^{i\vect k\cdot\vect R_{i}}\coeffs_{\nu}
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\]
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\end_inset
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(assuming the incident external field has the same periodicity,
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\begin_inset Formula $\coeffr_{\mathrm{ext}(i\nu)}=e^{i\vect k\cdot\vect R_{i}}p_{\mathrm{ext}\left(\nu\right)}$
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\end_inset
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) where
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\begin_inset Formula $\nu$
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\end_inset
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is the index of a particle inside one unit cell and
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\begin_inset Formula $\vect R_{i},\vect R_{i'}\in\Lambda$
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\end_inset
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are the lattice vectors corresponding to the sites (labeled by multiindices
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\begin_inset Formula $i,i'$
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\end_inset
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) of a Bravais lattice
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\begin_inset Formula $\Lambda$
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\end_inset
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.
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The multiple-scattering problem (
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\begin_inset CommandInset ref
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LatexCommand ref
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reference "eq:multiple scattering per particle a"
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\end_inset
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) then takes the form
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\end_layout
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\begin_layout Standard
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\begin_inset Formula
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\[
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\coeffs_{i\nu}-T_{\nu}\sum_{(i',\nu')\ne\left(i,\nu\right)}S_{i\nu,i'\nu'}e^{i\vect k\cdot\left(\vect R_{i'}-\vect R_{i}\right)}\coeffs_{i\nu'}=T_{\nu}\coeffr_{\mathrm{ext}(i\nu)}
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\]
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\end_inset
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or, labeling
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\begin_inset Formula $W_{\nu\nu'}=\sum_{i';(i',\nu')\ne\left(i,\nu\right)}S_{i\nu,i'\nu'}e^{i\vect k\cdot\left(\vect R_{i'}-\vect R_{i}\right)}=\sum_{i';(i',\nu')\ne\left(0,\nu\right)}S_{0\nu,i'\nu'}e^{i\vect k\cdot\vect R_{i'}}$
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\end_inset
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and using the quasiperiodicity,
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\begin_inset Formula
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\begin{equation}
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\sum_{\nu'}\left(\delta_{\nu\nu'}\mathbb{I}-T_{\nu}W_{\nu\nu'}\right)\coeffs_{\nu'}=T_{\nu}\coeffr_{\mathrm{ext}(\nu)},\label{eq:multiple scattering per particle a periodic}
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\end{equation}
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\end_inset
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which reduces the linear problem (
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\begin_inset CommandInset ref
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LatexCommand ref
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reference "eq:multiple scattering per particle a"
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\end_inset
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) to interactions between particles inside single unit cell.
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A problematic part is the evaluation of the translation operator lattice
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sums
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\begin_inset Formula $W_{\nu\nu'}$
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\end_inset
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; this is performed using exponentially convergent Ewald-type representations
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\begin_inset CommandInset citation
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LatexCommand cite
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key "linton_lattice_2010"
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literal "true"
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\end_inset
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.
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\end_layout
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\begin_layout Standard
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In an infinite periodic system, a nonlossy mode supports itself without
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external driving, i.e.
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such mode is described by excitation coefficients
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\begin_inset Formula $a_{\nu}$
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\end_inset
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that satisfy eq.
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(
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\begin_inset CommandInset ref
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LatexCommand ref
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reference "eq:multiple scattering per particle a periodic"
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\end_inset
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) with zero right-hand side.
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That can happen if the block matrix
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\begin_inset Formula
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\begin{equation}
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M\left(\omega,\vect k\right)=\left\{ \delta_{\nu\nu'}\mathbb{I}-T_{\nu}\left(\omega\right)W_{\nu\nu'}\left(\omega,\vect k\right)\right\} _{\nu\nu'}\label{eq:M matrix definition}
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\end{equation}
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\end_inset
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from the left hand side of (
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\begin_inset CommandInset ref
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LatexCommand ref
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reference "eq:multiple scattering per particle a periodic"
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\end_inset
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) is singular (here we explicitly note the
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\begin_inset Formula $\omega,\vect k$
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\end_inset
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depence).
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\end_layout
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\begin_layout Standard
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For lossy nanoparticles, however, perfect propagating modes will not exist
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and
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\begin_inset Formula $M\left(\omega,\vect k\right)$
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\end_inset
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will never be perfectly singular.
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Therefore in practice, we get the bands by scanning over
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\begin_inset Formula $\omega,\vect k$
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\end_inset
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to search for
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\begin_inset Formula $M\left(\omega,\vect k\right)$
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\end_inset
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which have an
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\begin_inset Quotes erd
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\end_inset
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almost zero
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\begin_inset Quotes erd
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\end_inset
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singular value.
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\end_layout
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\begin_layout Section
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\begin_inset ERT
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status collapsed
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\begin_layout Plain Layout
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{
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\end_layout
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\end_inset
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Symmetries
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\begin_inset ERT
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status collapsed
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\begin_layout Plain Layout
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}
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\end_layout
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\end_inset
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\end_layout
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\begin_layout Standard
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\begin_inset CommandInset label
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LatexCommand label
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name "sm:symmetries"
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\end_inset
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\end_layout
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\begin_layout Standard
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A general overview of utilizing group theory to find lattice modes at high-symme
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try points of the Brillouin zone can be found e.g.
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in
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\begin_inset CommandInset citation
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LatexCommand cite
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after "chapters 10–11"
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key "dresselhaus_group_2008"
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literal "true"
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\end_inset
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; here we use the same notation.
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\end_layout
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\begin_layout Standard
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We analyse the symmetries of the system in the same VSWF representation
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as used in the
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\begin_inset Formula $T$
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\end_inset
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-matrix formalism introduced above.
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We are interested in the modes at the
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\begin_inset Formula $\Kp$
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\end_inset
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-point of the hexagonal lattice, which has the
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\begin_inset Formula $D_{3h}$
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\end_inset
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point symmetry.
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The six irreducible representations (irreps) of the
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\begin_inset Formula $D_{3h}$
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\end_inset
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group are known and are available in the literature in their explicit forms.
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In order to find and classify the modes, we need to find a decomposition
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of the lattice mode representation
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\begin_inset Formula $\Gamma_{\mathrm{lat.mod.}}=\Gamma^{\mathrm{equiv.}}\otimes\Gamma_{\mathrm{vec.}}$
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\end_inset
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into the irreps of
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\begin_inset Formula $D_{3h}$
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\end_inset
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.
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The equivalence representation
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\begin_inset Formula $\Gamma^{\mathrm{equiv.}}$
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\end_inset
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is the
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\begin_inset Formula $E'$
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\end_inset
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representation as can be deduced from
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\begin_inset CommandInset citation
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LatexCommand cite
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after "eq. (11.19)"
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key "dresselhaus_group_2008"
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literal "true"
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\end_inset
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, eq.
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(11.19) and the character table for
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\begin_inset Formula $D_{3h}$
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\end_inset
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.
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\begin_inset Formula $\Gamma_{\mathrm{vec.}}$
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\end_inset
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operates on a space spanned by the VSWFs around each nanoparticle in the
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unit cell (the effects of point group operations on VSWFs are described
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in
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\begin_inset CommandInset citation
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LatexCommand cite
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key "schulz_point-group_1999"
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literal "true"
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\end_inset
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).
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This space can be then decomposed into invariant subspaces of the
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\begin_inset Formula $D_{3h}$
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\end_inset
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using the projectors
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\begin_inset Formula $\hat{P}_{ab}^{\left(\Gamma\right)}$
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\end_inset
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defined by
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\begin_inset CommandInset citation
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LatexCommand cite
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after "eq. (4.28)"
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key "dresselhaus_group_2008"
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literal "true"
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\end_inset
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.
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This way, we obtain a symmetry adapted basis
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\begin_inset Formula $\left\{ \vect b_{\Gamma,r,i}^{\mathrm{s.a.b.}}\right\} $
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\end_inset
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as linear combinations of VSWFs
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\begin_inset Formula $\vswfs lm{p,t}$
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\end_inset
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around the constituting nanoparticles (labeled
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\begin_inset Formula $p$
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\end_inset
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),
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\begin_inset Formula
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\[
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\vect b_{\Gamma,r,i}^{\mathrm{s.a.b.}}=\sum_{l,m,p,t}U_{\Gamma,r,i}^{p,t,l,m}\vswfs lm{p,t},
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\]
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\end_inset
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where
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\begin_inset Formula $\Gamma$
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\end_inset
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stands for one of the six different irreps of
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\begin_inset Formula $D_{3h}$
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\end_inset
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,
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\begin_inset Formula $r$
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\end_inset
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labels the different realisations of the same irrep, and the last index
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\begin_inset Formula $i$
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\end_inset
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going from 1 to
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\begin_inset Formula $d_{\Gamma}$
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\end_inset
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(the dimensionality of
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\begin_inset Formula $\Gamma$
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\end_inset
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) labels the different partners of the same given irrep.
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The number of how many times is each irrep contained in
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\begin_inset Formula $\Gamma_{\mathrm{lat.mod.}}$
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\end_inset
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(i.e.
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the range of index
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\begin_inset Formula $r$
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\end_inset
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for given
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\begin_inset Formula $\Gamma$
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\end_inset
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) depends on the multipole degree cutoff
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\begin_inset Formula $l_{\mathrm{max}}$
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\end_inset
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.
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\end_layout
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\begin_layout Standard
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Each mode at the
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\begin_inset Formula $\Kp$
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\end_inset
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-point shall lie in the irreducible spaces of only one of the six possible
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irreps and it can be shown via
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\begin_inset CommandInset citation
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LatexCommand cite
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after "eq. (2.51)"
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key "dresselhaus_group_2008"
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literal "true"
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\end_inset
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that, at the
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\begin_inset Formula $\Kp$
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\end_inset
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-point, the matrix
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\begin_inset Formula $M\left(\omega,\vect k\right)$
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\end_inset
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defined above takes a block-diagonal form in the symmetry-adapted basis,
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\begin_inset Formula
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\[
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M\left(\omega,\vect K\right)_{\Gamma,r,i;\Gamma',r',j}^{\mathrm{s.a.b.}}=\frac{\delta_{\Gamma\Gamma'}\delta_{ij}}{d_{\Gamma}}\sum_{q}M\left(\omega,\vect K\right)_{\Gamma,r,q;\Gamma',r',q}^{\mathrm{s.a.b.}}.
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\]
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\end_inset
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This enables us to decompose the matrix according to the irreps and to solve
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the singular value problem in each irrep separately, as done in Fig.
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\begin_inset CommandInset ref
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LatexCommand ref
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reference "smfig:dispersions"
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\end_inset
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(a).
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\end_layout
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\end_body
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\end_document
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