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Commit d9efa4f5 authored by Nicolas Eßing's avatar Nicolas Eßing
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Changes proposed by Fnolte (thanks dude), unbreakable spaces for references, 

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tex/experimentaldata.tex
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......@@ -30,7 +30,7 @@ experiments as well.
The most direct method to determine the growth velocity is to measure the size
of a grain as a function of time.
It can be calculated as the slope of a linear fit on the radius $r(t)$.
An example is shown in figure \ref{fig:expfit_v}
An example is shown in figure~\ref{fig:expfit_v}.
The only thing important to consider is the domain on which the fit is done.
In an early stage, the velocity might be influenced by the small curvature of
......@@ -93,13 +93,13 @@ experiments as well.
To test this method, a few simulations were done with different nucleation
rates and different random number generation seeds.
The method was then used to recreate the used rate.
One comparison is shown in figure \ref{fig:expfit_J_test}.
One comparison is shown in figure~\ref{fig:expfit_J_test}.
In most of these tests, the result of the analysis was approximately
one sigma away from the value used in the simulation.
For the available data on \sbte{} however,
the assumption of a constant nucleation rate does not seem to hold.
This can be seen in the data shown in figure \ref{fig:expfit_J} and will also
This can be seen in the data shown in figure~\ref{fig:expfit_J} and will also
be discussed in the comparison with simulated data.
As a workaround, several rates were fitted over different time intervals.
Inhomogenous nucleation can be an explanation for decreasing rates,
......@@ -125,7 +125,7 @@ experiments as well.
\end{align}
and should be a linear function of time if $J$ is constant.
Different rates can be seen as different slopes.
For one measurement, this is shown in figure \ref{fig:expfit_J}
For one measurement, this is shown in figure~\ref{fig:expfit_J}
% TODO show results?
% Fitting a single nucleation rate to the data yields the following results:
......@@ -155,7 +155,7 @@ experiments as well.
($\SI[per-mode=reciprocal]{2.256\pm0.092}{\per\s}$ on the left and
$\SI[per-mode=reciprocal]{2.324\pm0.095}{\per\s}$ on the right).
Both images use the same parameters, but a different seed for random
number generation}
number generation.}
\label{fig:expfit_J_test}
\end{figure}
......@@ -192,7 +192,7 @@ experiments as well.
The values for $k$ resulting from both methods were compared,
with varying consistency.
Two examples with different consistencies are shown
in figure \ref{fig:expfit_jmak_comp}.
in figure~\ref{fig:expfit_jmak_comp}.
In these test however, a constant nucleation rate was assumed, which is not
in good agreement with the rates obtained from the counted nuclei.
More complex equations could be constructed taking different rates into
......@@ -228,7 +228,7 @@ experiments as well.
With only four data points available for low temperatures,
an Arrhenius was used.
It can sufficiently describe the behaviour, as can be seen in
figure \ref{fig:expfit_v_arrhenius}. % chisq/ndf of 3.0
figure~\ref{fig:expfit_v_arrhenius}. % chisq/ndf of 3.0
Fitting an Arrhenius behaviour to the mobility gives similar results.
For the use of a more complex model, more data is necessary.
......@@ -262,14 +262,14 @@ experiments as well.
To get an approximation providing values for the nucleation rate for other
temperatures, the mean rates of each measurement can be used.
The classical nucleation theory model (equatoin \ref{eq:cnt_J})
The classical nucleation theory model (equatoin~\ref{eq:cnt_J})
was fitted to this values.
For $\dGv$, the introduced model was used
together with values from other experiments \citep{expdata,sb2teStructure}
together with values from other experiments~\citep{expdata,sb2teStructure}
that are listed in the next chapter. %TODO ref
For $\eta$, an Arrhenius behaviour was assumed with the same activation energy
as obtained from the fit onto the velocities,
which is motivated by the discussion in section \ref{sec:arrheniusviscosity}.
which is motivated by the discussion in section~\ref{sec:arrheniusviscosity}.
This leaves the interfacial energy and a prefactor as free parameters,
where the latter one is the product of several quantities occouring in the
theoretical equation.
......@@ -297,4 +297,4 @@ experiments as well.
The interfacial energy is in the expected order of magnitude,
while the prefactor is several orders of magnitude below values for similar
materials.
The fit is shown in figure \ref{fig:expfit_cnt}.
The fit is shown in figure~\ref{fig:expfit_cnt}.
tex/introduction.tex
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......@@ -10,25 +10,25 @@
This includes several electrical and optical properties.
In chalcogenide phase change materials for example both the
real and the imaginary part of the dielectric function change
about two orders of magnitude \citep{Usecases}.
about two orders of magnitude~\citep{Usecases}.
Other changes include the refractive index
($\Delta n > 1$) \citep{PCMsInOptics}
and the electrical resistivity \citep{phaseChangeRam}.
($\Delta n > 1$)~\citep{PCMsInOptics}
and the electrical resistivity~\citep{phaseChangeRam}.
A transition from one phase to the other can be achieved by
electrical, thermal or optical stimuli \citep{PCMsInOptics}.
electrical, thermal or optical stimuli~\citep{PCMsInOptics}.
The timescale of the phase change can be as low as nanoseconds
or even femtoseconds \citep{Usecases}.
or even femtoseconds~\citep{Usecases}.
These characteristics make phase change materials usable in
a wide field of applications. \\
In optical memory, such as rewriteable DVDs and BlueRay discs,
phase change materials are in use
for over 20 years now \citep{phaseChangeMemory}. \\
for over 20 years now~\citep{phaseChangeMemory}. \\
%TODO optical devices
Phase change material random access memory (PCRAM) is already
in use and might become the next leading technology in computer
memory \citep{phaseChangeMemory},
memory~\citep{phaseChangeMemory},
offering non volatile data storage with high access speed and
long lifetime. \\
Another very interesting development is the coupeling of this
......@@ -36,7 +36,7 @@
These consist of structures on a length scale smaller than the
wavelength of the light they interact with.
With them, a wide range of optical and electrical properties can
be constructed \citep{Usecases},
be constructed~\citep{Usecases},
e.g. a negative refractive index or perfect absorption.
Combining this technology with phase change materials enables to alter
these properties post-production.
......@@ -55,18 +55,18 @@
% This material has a metal to insulator transition between two crystalline phases
% with different structure.
% Despite being volatile it might be useable in optical applications due to its
% large change in refractive index and electrical resistivity. \citep{PCMsInOptics}
% large change in refractive index and electrical resistivity.~\citep{PCMsInOptics}
\section{Goal of this Work}
As mentioned, one of the phase change materials that has been studied most
is \GSTxxx{2}{2}{5}.
It has also been in the focus of the group in which this work was created.
For this material, a simulation has been written by \citet{zhiyang} using
For this material, a simulation has been written by~\citet{zhiyang} using
the phase field method.
It can be coupled to thermo-electrical simulations
as done by \citet{sebastiansThesis}.
as done by~\citet{sebastiansThesis}.
There are however other materials being investigated,
for example \sbte{} by \citet{expdata},
for example \sbte{} by~\citet{expdata},
who kindly provided some of his data.
During the work, interest was expressed in comparing data from other groups at
this institute with simulations.
......
tex/simulationmodel.tex
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......@@ -7,7 +7,7 @@ and simple simulations to test the limits of the calculated behaviour.
\subsection{Laplace Operator}
In each time step, the Laplace operator of the phase field $\Phi$ has to be
calculated (cf. Eq. \ref{eq:pfderivative}),
calculated (cf.~eq.~\ref{eq:pfderivative}),
next to only a few elementwise multiplications.
It is a crucial part regarding performance.
The implementation of the numerical Laplace operator in the original code
......@@ -32,7 +32,7 @@ and simple simulations to test the limits of the calculated behaviour.
a runtime shorter by approximately $69\%$ compared to the original code.
The performances for different numbers of runs are compared in
figure \ref{fig:laplacian_performance}.
figure~\ref{fig:laplacian_performance}.
\begin{figure}[htbp]
\centering
......@@ -86,7 +86,7 @@ and simple simulations to test the limits of the calculated behaviour.
performance for basic operations like additions or comparisons. \\
A stencil algorithm was written that implements the specifications above
comparing a point with one neighbour at a time.
A ``branch to composition'' technique \citep{computerOrganizations} was used
A ``branch to composition'' technique~\citep{computerOrganizations} was used
to avoid expensive branching operations.
Overall, the new implementation needs about $9\%$ the time of the old one
......@@ -121,7 +121,7 @@ and simple simulations to test the limits of the calculated behaviour.
\end{figure}
\subsection{Nucleation}
The existing nucleation algorithm (described in ref. \citep{zhiyang}) worked
The existing nucleation algorithm (described in ref.~\citep{zhiyang}) worked
only for a limited range of parameters.
Problems include the convergence of the growth process:
For $r_c > h$, a single point was set to and kept at $\Phi = 1$
......@@ -203,7 +203,7 @@ and simple simulations to test the limits of the calculated behaviour.
nucleation, where initial nuclei are inserted that may differ from the
actual critical shape.
The predicted behaviour was numerically tested for different settings,
which is shown in figure \ref{fig:tau_demo}.
which is shown in figure~\ref{fig:tau_demo}.
With a dimensionless grid spacing of $1$, a shape with a thickness $\zeta = 3$
can be sufficiently represented and fit on a sample of size $100$.
......@@ -246,7 +246,7 @@ and simple simulations to test the limits of the calculated behaviour.
To test this limitation, the ratio of $\zeta$ to $h$ was analysed in
analytically solveable, one dimensional setups.
An example is shown in figure \ref{fig:num_v_DIT}.
An example is shown in figure~\ref{fig:num_v_DIT}.
The remaining free parameter $r_c$ was varied over a few orders of magnitude
compared to $\zeta$.
This showed that it has no strong influence in this setting apart from
......@@ -282,7 +282,7 @@ and simple simulations to test the limits of the calculated behaviour.
The directions to these are treated different from other directions.
In this model, neighbouring points effect each other only through the
differential. The rest of the time evolution for a point
(equation \ref{eq:pfderivative}) does only depend on the phase at the point
(equation~\ref{eq:pfderivative}) does only depend on the phase at the point
itself, which is done as an elementwise calculation in the discretized
implementation.
This means that a nucleated point leads to nucleation in the neighbouring
......@@ -300,7 +300,7 @@ and simple simulations to test the limits of the calculated behaviour.
should grow in spherical shape.
This idea was tested in several simulations and could be confirmed.
Examples are shown in figure \ref{fig:numlim_rc}.
Examples are shown in figure~\ref{fig:numlim_rc}.
Starting from an diffuse with only slight initial anisotropies, these
effects can still be seen for $h \geq 0.1 r_c$,
but vanishes for $h \leq r_c$.
......@@ -318,7 +318,7 @@ and simple simulations to test the limits of the calculated behaviour.
One additional effect has to be considered that could also be the origin of
the observed phenomena.
Normally, the time step $dt$ is choosen according to the numerical stability
criterion given in equation \ref{eq:numstab}.
criterion given in equation~\ref{eq:numstab}.
For smaller values of $r_c$, this leads to larger time steps and thus to
less precision.
To exclude this, simulations were done using the smallest necessary time
......@@ -388,7 +388,7 @@ and simple simulations to test the limits of the calculated behaviour.
Apart from these points, no restrictions were found in the tested cases.
Togehter with $dt$, the ratio of $v dt$ to the length scales if of interest.
The numerical stability criterion eq. \ref{eq:numstab} yields that the ratio
The numerical stability criterion eq.~\ref{eq:numstab} yields that the ratio
of $v dt$ to $h$ is about a third of the one of $h$ to $r_c$,
for which a restriction has been discussed.
As long as this restriction is fulfilled, the expected growth in one time
......
tex/simulationresults.tex
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......@@ -15,7 +15,7 @@
Two simulations for \sbte{} were implemeted using both physical and
numerical values for $r_c$.
For the velocity, an Arrhenius model is used together with the parameters
obtained from the fit presented in section \ref{sec:fit_v}.
obtained from the fit presented in section~\ref{sec:fit_v}.
The nucleation rate is implemented as the fitted classical nucleation theory
model, which assumed an Arrhenius behaviour for the viscosity.
Thus, the model is only applicable for lower temperatures.
......@@ -43,7 +43,7 @@
$2$ to $\SI{3}{\nm}$, a purely numerical value of $r_c = \SI{2}{\um}$
was used.
The result of these simulation is compared to the experimental data in
figure \ref{fig:sbte_res_const_J}.
figure~\ref{fig:sbte_res_const_J}.
While for some experiments this gave acceptable results,
huge deviations occoured whenever unusual behaviour of the nucleation rate
was observed.
......@@ -73,7 +73,7 @@
The parameter $J$ in the simulation was manually altered after according
time intervals.
The results of this method are compared to the experiment in
figures \ref{fig:sbte_res_115} and \ref{fig:sbte_res_125} and show better
figures~\ref{fig:sbte_res_115} and~\ref{fig:sbte_res_125} and show better
consistence.
This is first of all a confirmation of the phase field model.
As it is however not possible to extrapolate the behaviour for different
......@@ -97,7 +97,8 @@
\begin{subfigure}{.49\columnwidth}
\includegraphics[width=\columnwidth]{../img/simres_sb2te_115_N.png}
\end{subfigure}
\caption{Comparison of experimental data and simulation results ($115\degC$).
\caption{Comparison of experimental data and simulation results
($\SI{115}{\celsius}$).
Different nucleation rates were used, which can be seen as kinks in the
number of grains. Apart from some time in the middle, where there are
considerably more nuclei in the simulation than in the experiment,
......@@ -115,7 +116,8 @@
\begin{subfigure}{.49\columnwidth}
\includegraphics[width=\columnwidth]{../img/simres_sb2te_125_N.png}
\end{subfigure}
\caption{Comparison of experimental data and simulation results ($125\degC$).
\caption{Comparison of experimental data and simulation results
($\SI{125}{\celsius}$).
The number of nuclei in the experiment shows unusual behaviour, which
is approximated by three different nucleation rates.
There are small but systematic differences visible in the crystallised
......@@ -126,16 +128,16 @@
\begin{figure}[tbp]
\begin{subfigure}{.49\columnwidth}
\includegraphics[width=\columnwidth]{../img/simres_sb2te_jmak_115.png}
\caption{$115\degC$}
\caption{$\SI{115}{\celsius}$}
\end{subfigure}
\hfill
\begin{subfigure}{.49\columnwidth}
\includegraphics[width=\columnwidth]{../img/simres_sb2te_jmak_125.png}
\caption{$125\degC$}
\caption{$$\SI{125}{\celsius}$$}
\end{subfigure}
\caption{Comparison of JMAK model and simulation to experimental data.
For $115\degC$, there are barly differences visible.
For $125\degC$ however, where some unusual behaviour in the grain count
For $\SI{115}{\celsius}$, there are barly differences visible.
For $\SI{125}{\celsius}$ however, where some unusual behaviour in the grain count
appears, the simulation describes the data significantly better.}
\end{figure}
......@@ -154,11 +156,11 @@
below $T_glass$ and the MYEGA model in the first form for higher
temperatures was used.
This glass transition temperature was estimated from the intersect of the
Arrhenius model for the velocity reported in ref. \citep{SbTeArrhenius}
with the one reported in ref. \citep{AISTpaper} (calculated from Frenkel
Arrhenius model for the velocity reported in ref.~\citep{SbTeArrhenius}
with the one reported in ref.~\citep{AISTpaper} (calculated from Frenkel
model with a MYEGA viscosity) to be $T_{glass} = \SI{480}{\K}$.
The volume per atom was estimated from the latent heat per volume and the
one per atom $L \cdot V_m = \SI{173 \pm 3}{\meV}$ \citep{AistExperiments}.
one per atom $L \cdot V_m = \SI{173 \pm 3}{\meV}$~\citep{AistExperiments}.
The nucleation rate was obtained from classical nucleation theory.
\begin{center}
......@@ -187,11 +189,11 @@
\section{GeTe}
For GeTe, the only values for the interfacial energy that could be found
are values from $0.2$ to $\SI{0.4}{\J\per\m\squared}$ obtained from
ab initio simulations. \citep{geteAbInitioSimulations}
ab initio simulations~\citep{geteAbInitioSimulations}.
This is an order of magnitude larger than the values for similar materials
and leads to vanishing nucleation in the classical theory.
Nucleation was thus implemented using an Arrhenius function with values
obtained from ref. \citep{geteNucleationAndGrowth}.
obtained from ref.~\citep{geteNucleationAndGrowth}.
These were obtained from experiments around $\SI{400}{\K}$ and do
most probably not describe the behaviour of this material for larger
temperatures.
......
tex/theory.tex
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tex/thesis.tex
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......@@ -22,10 +22,6 @@
\newcommand{\sbte}{$\text{Sb}_{\text{2}}\text{Te}$} % <- Sb2Te and GST
\newcommand{\GSTxxx}[3]{$\text{Ge}_{\text{#1}}\text{Sb}_{\text{#2}}\text{Te}_{\text{#3}}$}
\newcommand{\unit}[1]{\, \text{#1}} % units in math mode
\newcommand{\degC}{\ensuremath{{}^{\circ}\text{C}}} % degree celcius sign
\begin{document}
\frontmatter
......@@ -46,26 +42,39 @@
\input{simulationresults}
\chapter{Summary and Outlook}
%TODO
Models and parameters for different phase change materials have been reviewed
and implemented on the basis of an existing simulation for GST.
The simulation implementation has improved regarding useability, extendability
and performance, which was increased about $150\%$.
Numerical limits have been analysed.
In this thesis, simulations of the crystallization process of different phase
change materials have been implemented on the basis of an existing
simulation for \GSTxxx{2}{2}{5}.
For AIST, the model should be useable over the whole temperature range of
interest. For the other materials however, the models are only applicabe for
lower temperatures. To simulate these materials at higher temperatures,
additional parameters or experiental data would be necessary.
The implementation was improved regarding useability and extendability by
restructuring the code.
By several enhancements, the runtime was decreased to less than $40\%$
compared to the original code.
Numerical limits of the implementation were analysed.
It was found that $h \lesssim r_c$ and $h \lesssim \zeta$ have to be
fullfilled.
Thus, it was proposed and tested to use unphysical values for $r_c$ to
Motivated by this, it was proposed to use unphysical values for $r_c$ to
simulate larger samples.
This was tested with simulations on \sbte{} and compared the experiental
data, which confirmed this approach.
In the future, my code could be used to simulate arbitary
phase change materials.
It is written to be coupled to thermoelectrical simulations.
In the future, the implemented models should be useable to simulate
different phase change materials on a wide range of length scales.
It can be coupled to thermoelectrical simulations to consistently describe
the whole process of an experiment.
%TODO
\Citet{zhiyang} extendes the model to consider stresses,
which could also be analysed and included for other materials.
Nucleus shape is a weakness that could be fixed.
\Citet{zhiyang} extended the model for \GSTxxx{2}{2}{5} to consider stresses,
which could be analysed and included in the simulations for other materials.
Another weakness of the existing model is the initial shape of grains
added by the ad-hoc implementation of nucleation.
The current version is little more than a guess of an appropriate form and
not able to reproduce all details of the phase field model.
\chapter{Acknowledgements}
%TODO
......
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