qpms/lepaper/infinite-old.lyx

497 lines
10 KiB
Plaintext
Raw Blame History

This file contains ambiguous Unicode characters

This file contains Unicode characters that might be confused with other characters. If you think that this is intentional, you can safely ignore this warning. Use the Escape button to reveal them.

#LyX 2.4 created this file. For more info see https://www.lyx.org/
\lyxformat 583
\begin_document
\begin_header
\save_transient_properties true
\origin unavailable
\textclass article
\use_default_options false
\maintain_unincluded_children false
\language english
\language_package none
\inputencoding utf8
\fontencoding default
\font_roman "default" "default"
\font_sans "default" "default"
\font_typewriter "default" "default"
\font_math "auto" "auto"
\font_default_family default
\use_non_tex_fonts false
\font_sc false
\font_roman_osf false
\font_sans_osf false
\font_typewriter_osf false
\font_sf_scale 100 100
\font_tt_scale 100 100
\use_microtype false
\use_dash_ligatures true
\graphics default
\default_output_format default
\output_sync 0
\bibtex_command default
\index_command default
\float_placement class
\float_alignment class
\paperfontsize default
\spacing single
\use_hyperref false
\papersize default
\use_geometry false
\use_package amsmath 1
\use_package amssymb 0
\use_package cancel 0
\use_package esint 1
\use_package mathdots 0
\use_package mathtools 0
\use_package mhchem 0
\use_package stackrel 0
\use_package stmaryrd 0
\use_package undertilde 0
\cite_engine basic
\cite_engine_type default
\biblio_style plain
\use_bibtopic false
\use_indices false
\paperorientation portrait
\suppress_date false
\justification true
\use_refstyle 0
\use_minted 0
\use_lineno 0
\index Index
\shortcut idx
\color #008000
\end_index
\secnumdepth 3
\tocdepth 3
\paragraph_separation indent
\paragraph_indentation default
\is_math_indent 0
\math_numbering_side default
\quotes_style english
\dynamic_quotes 0
\papercolumns 1
\papersides 1
\paperpagestyle default
\tablestyle default
\tracking_changes false
\output_changes false
\html_math_output 0
\html_css_as_file 0
\html_be_strict false
\end_header
\begin_body
\begin_layout Subsection
Periodic systems and mode analysis
\begin_inset CommandInset label
LatexCommand label
name "subsec:Periodic-systems"
\end_inset
\end_layout
\begin_layout Standard
In an infinite periodic array of nanoparticles, the excitations of the nanoparti
cles take the quasiperiodic Bloch-wave form
\begin_inset Formula
\[
\coeffs_{i\nu}=e^{i\vect k\cdot\vect R_{i}}\coeffs_{\nu}
\]
\end_inset
(assuming the incident external field has the same periodicity,
\begin_inset Formula $\coeffr_{\mathrm{ext}(i\nu)}=e^{i\vect k\cdot\vect R_{i}}p_{\mathrm{ext}\left(\nu\right)}$
\end_inset
) where
\begin_inset Formula $\nu$
\end_inset
is the index of a particle inside one unit cell and
\begin_inset Formula $\vect R_{i},\vect R_{i'}\in\Lambda$
\end_inset
are the lattice vectors corresponding to the sites (labeled by multiindices
\begin_inset Formula $i,i'$
\end_inset
) of a Bravais lattice
\begin_inset Formula $\Lambda$
\end_inset
.
The multiple-scattering problem (
\begin_inset CommandInset ref
LatexCommand ref
reference "eq:multiple scattering per particle a"
\end_inset
) then takes the form
\end_layout
\begin_layout Standard
\begin_inset Formula
\[
\coeffs_{i\nu}-T_{\nu}\sum_{(i',\nu')\ne\left(i,\nu\right)}S_{i\nu,i'\nu'}e^{i\vect k\cdot\left(\vect R_{i'}-\vect R_{i}\right)}\coeffs_{i\nu'}=T_{\nu}\coeffr_{\mathrm{ext}(i\nu)}
\]
\end_inset
or, labeling
\begin_inset Formula $W_{\nu\nu'}=\sum_{i';(i',\nu')\ne\left(i,\nu\right)}S_{i\nu,i'\nu'}e^{i\vect k\cdot\left(\vect R_{i'}-\vect R_{i}\right)}=\sum_{i';(i',\nu')\ne\left(0,\nu\right)}S_{0\nu,i'\nu'}e^{i\vect k\cdot\vect R_{i'}}$
\end_inset
and using the quasiperiodicity,
\begin_inset Formula
\begin{equation}
\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}
\end{equation}
\end_inset
which reduces the linear problem (
\begin_inset CommandInset ref
LatexCommand ref
reference "eq:multiple scattering per particle a"
\end_inset
) to interactions between particles inside single unit cell.
A problematic part is the evaluation of the translation operator lattice
sums
\begin_inset Formula $W_{\nu\nu'}$
\end_inset
; this is performed using exponentially convergent Ewald-type representations
\begin_inset CommandInset citation
LatexCommand cite
key "linton_lattice_2010"
literal "true"
\end_inset
.
\end_layout
\begin_layout Standard
In an infinite periodic system, a nonlossy mode supports itself without
external driving, i.e.
such mode is described by excitation coefficients
\begin_inset Formula $a_{\nu}$
\end_inset
that satisfy eq.
(
\begin_inset CommandInset ref
LatexCommand ref
reference "eq:multiple scattering per particle a periodic"
\end_inset
) with zero right-hand side.
That can happen if the block matrix
\begin_inset Formula
\begin{equation}
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}
\end{equation}
\end_inset
from the left hand side of (
\begin_inset CommandInset ref
LatexCommand ref
reference "eq:multiple scattering per particle a periodic"
\end_inset
) is singular (here we explicitly note the
\begin_inset Formula $\omega,\vect k$
\end_inset
depence).
\end_layout
\begin_layout Standard
For lossy nanoparticles, however, perfect propagating modes will not exist
and
\begin_inset Formula $M\left(\omega,\vect k\right)$
\end_inset
will never be perfectly singular.
Therefore in practice, we get the bands by scanning over
\begin_inset Formula $\omega,\vect k$
\end_inset
to search for
\begin_inset Formula $M\left(\omega,\vect k\right)$
\end_inset
which have an
\begin_inset Quotes erd
\end_inset
almost zero
\begin_inset Quotes erd
\end_inset
singular value.
\end_layout
\begin_layout Section
\begin_inset ERT
status collapsed
\begin_layout Plain Layout
{
\end_layout
\end_inset
Symmetries
\begin_inset ERT
status collapsed
\begin_layout Plain Layout
}
\end_layout
\end_inset
\end_layout
\begin_layout Standard
\begin_inset CommandInset label
LatexCommand label
name "sm:symmetries"
\end_inset
\end_layout
\begin_layout Standard
A general overview of utilizing group theory to find lattice modes at high-symme
try points of the Brillouin zone can be found e.g.
in
\begin_inset CommandInset citation
LatexCommand cite
after "chapters 1011"
key "dresselhaus_group_2008"
literal "true"
\end_inset
; here we use the same notation.
\end_layout
\begin_layout Standard
We analyse the symmetries of the system in the same VSWF representation
as used in the
\begin_inset Formula $T$
\end_inset
-matrix formalism introduced above.
We are interested in the modes at the
\begin_inset Formula $\Kp$
\end_inset
-point of the hexagonal lattice, which has the
\begin_inset Formula $D_{3h}$
\end_inset
point symmetry.
The six irreducible representations (irreps) of the
\begin_inset Formula $D_{3h}$
\end_inset
group are known and are available in the literature in their explicit forms.
In order to find and classify the modes, we need to find a decomposition
of the lattice mode representation
\begin_inset Formula $\Gamma_{\mathrm{lat.mod.}}=\Gamma^{\mathrm{equiv.}}\otimes\Gamma_{\mathrm{vec.}}$
\end_inset
into the irreps of
\begin_inset Formula $D_{3h}$
\end_inset
.
The equivalence representation
\begin_inset Formula $\Gamma^{\mathrm{equiv.}}$
\end_inset
is the
\begin_inset Formula $E'$
\end_inset
representation as can be deduced from
\begin_inset CommandInset citation
LatexCommand cite
after "eq. (11.19)"
key "dresselhaus_group_2008"
literal "true"
\end_inset
, eq.
(11.19) and the character table for
\begin_inset Formula $D_{3h}$
\end_inset
.
\begin_inset Formula $\Gamma_{\mathrm{vec.}}$
\end_inset
operates on a space spanned by the VSWFs around each nanoparticle in the
unit cell (the effects of point group operations on VSWFs are described
in
\begin_inset CommandInset citation
LatexCommand cite
key "schulz_point-group_1999"
literal "true"
\end_inset
).
This space can be then decomposed into invariant subspaces of the
\begin_inset Formula $D_{3h}$
\end_inset
using the projectors
\begin_inset Formula $\hat{P}_{ab}^{\left(\Gamma\right)}$
\end_inset
defined by
\begin_inset CommandInset citation
LatexCommand cite
after "eq. (4.28)"
key "dresselhaus_group_2008"
literal "true"
\end_inset
.
This way, we obtain a symmetry adapted basis
\begin_inset Formula $\left\{ \vect b_{\Gamma,r,i}^{\mathrm{s.a.b.}}\right\} $
\end_inset
as linear combinations of VSWFs
\begin_inset Formula $\vswfs lm{p,t}$
\end_inset
around the constituting nanoparticles (labeled
\begin_inset Formula $p$
\end_inset
),
\begin_inset Formula
\[
\vect b_{\Gamma,r,i}^{\mathrm{s.a.b.}}=\sum_{l,m,p,t}U_{\Gamma,r,i}^{p,t,l,m}\vswfs lm{p,t},
\]
\end_inset
where
\begin_inset Formula $\Gamma$
\end_inset
stands for one of the six different irreps of
\begin_inset Formula $D_{3h}$
\end_inset
,
\begin_inset Formula $r$
\end_inset
labels the different realisations of the same irrep, and the last index
\begin_inset Formula $i$
\end_inset
going from 1 to
\begin_inset Formula $d_{\Gamma}$
\end_inset
(the dimensionality of
\begin_inset Formula $\Gamma$
\end_inset
) labels the different partners of the same given irrep.
The number of how many times is each irrep contained in
\begin_inset Formula $\Gamma_{\mathrm{lat.mod.}}$
\end_inset
(i.e.
the range of index
\begin_inset Formula $r$
\end_inset
for given
\begin_inset Formula $\Gamma$
\end_inset
) depends on the multipole degree cutoff
\begin_inset Formula $l_{\mathrm{max}}$
\end_inset
.
\end_layout
\begin_layout Standard
Each mode at the
\begin_inset Formula $\Kp$
\end_inset
-point shall lie in the irreducible spaces of only one of the six possible
irreps and it can be shown via
\begin_inset CommandInset citation
LatexCommand cite
after "eq. (2.51)"
key "dresselhaus_group_2008"
literal "true"
\end_inset
that, at the
\begin_inset Formula $\Kp$
\end_inset
-point, the matrix
\begin_inset Formula $M\left(\omega,\vect k\right)$
\end_inset
defined above takes a block-diagonal form in the symmetry-adapted basis,
\begin_inset Formula
\[
M\left(\omega,\vect K\right)_{\Gamma,r,i;\Gamma',r',j}^{\mathrm{s.a.b.}}=\frac{\delta_{\Gamma\Gamma'}\delta_{ij}}{d_{\Gamma}}\sum_{q}M\left(\omega,\vect K\right)_{\Gamma,r,q;\Gamma',r',q}^{\mathrm{s.a.b.}}.
\]
\end_inset
This enables us to decompose the matrix according to the irreps and to solve
the singular value problem in each irrep separately, as done in Fig.
\begin_inset CommandInset ref
LatexCommand ref
reference "smfig:dispersions"
\end_inset
(a).
\end_layout
\end_body
\end_document