X-Git-Url: https://hackdaworld.org/gitweb/?p=lectures%2Flatex.git;a=blobdiff_plain;f=posic%2Fthesis%2Fbasics.tex;h=243d262f8e4b93d670100675d34fbb6a86fd92ea;hp=cba8c5626362e9512c93717412d7a88c16e6a85e;hb=cbed3c8589a70f445c4f2c7ae6532261afb65351;hpb=e3a0b64c27ef578fffd44e26b43953917b858e86

diff --git a/posic/thesis/basics.tex b/posic/thesis/basics.tex
index cba8c56..243d262 100644
--- a/posic/thesis/basics.tex
+++ b/posic/thesis/basics.tex
@@ -421,7 +421,7 @@ for the exchange-correlation energy, where $\epsilon_{\text{xc}}(\vec{r};[n(\til
 Expressing $n(\tilde{\vec{r}})$ in a Taylor series, $\epsilon_{\text{xc}}$ can be thought of as a function of coefficients, which correspond to the respective terms of the expansion.
 Neglecting all terms of order $\mathcal{O}(\nabla n(\vec{r}))$ results in the functional equal to LDA, which requires the function of variable $n$.
 Including the next element of the Taylor series introduces the gradient correction to the functional, which requires the function of variables $n$ and $|\nabla n|$.
-This is called the generalized gradient approximation (GGA), which expresses the exchange-correlation energy density as a function of the local density and the local gradient of the density
+This is called the generalized-gradient approximation (GGA), which expresses the exchange-correlation energy density as a function of the local density and the local gradient of the density
 \begin{equation}
 E^{\text{GGA}}_{\text{xc}}[n(\vec{r})]=\int\epsilon_{\text{xc}}(n(\vec{r}),|\nabla n(\vec{r})|)n(\vec{r}) d\vec{r}
 \text{ .}
@@ -532,6 +532,7 @@ Using PPs the rapid oscillations of the wave functions near the core of the atom
 More importantly, less accuracy is required compared to all-electron calculations to determine energy differences among ionic configurations, which almost totally appear in the energy of the valence electrons that are typically a factor $10^3$ smaller than the energy of the core electrons.
 
 \subsection{Brillouin zone sampling}
+\label{subsection:basics:bzs}
 
 Following Bloch's theorem only a finite number of electronic wave functions need to be calculated for a periodic system.
 However, to calculate quantities like the total energy or charge density, these have to be evaluated in a sum over an infinite number of $\vec{k}$ points.
@@ -599,6 +600,22 @@ E_{\text{f}}=\left(E_{\text{coh}}^{\text{defect}}
 where $N$ and $E_{\text{coh}}^{\text{defect}}$ are the number of atoms and the cohesive energy per atom in the defect configuration and $E_{\text{coh}}^{\text{defect-free}}$ is the cohesive energy per atom of the defect-free structure.
 Clearly, for a single atom species equation \eqref{eq:basics:ef2} is equivalent to equation \eqref{eq:basics:ef1} since $NE_{\text{coh}}^{\text{defect}}$ is equal to the total energy of the defect structure and $NE_{\text{coh}}^{\text{defect-free}}$ corresponds to $N\mu$, provided the structure is fully relaxed at zero temperature.
 
+However, there is hardly ever only one defect in a crystal, not even only one kind of defect.
+Again, energetic considerations can be used to investigate the existing interaction of two defects.
+The binding energy $E_{\text{b}}$ of a defect pair is given by the difference of the formation energy of the defect combination $E_{\text{f}}^{\text{comb}} $ and the sum of the two separated defect configurations $E_{\text{f}}^{1^{\text{st}}}$ and $E_{\text{f}}^{2^{\text{nd}}}$.
+This can be expressed by
+\begin{equation}
+E_{\text{b}}=
+E_{\text{f}}^{\text{comb}}-
+E_{\text{f}}^{1^{\text{st}}}-
+E_{\text{f}}^{2^{\text{nd}}}
+\label{eq:basics:e_bind}
+\end{equation}
+where the formation energies $E_{\text{f}}^{\text{comb}}$, $E_{\text{f}}^{1^{\text{st}}}$ and $E_{\text{f}}^{2^{\text{nd}}}$ are determined as discussed above.
+Accordingly, energetically favorable configurations result in binding energies below zero while unfavorable configurations show positive values for the binding energy.
+The interaction strength, i.e. the absolute value of the binding energy, approaches zero for increasingly non-interacting isolated defects.
+Thus, $E_{\text{b}}$ indeed can be best thought of a binding energy, which is required to bring the defects to infinite separation.
+
 The methods presented in the last two chapters can be used to investigate defect structures and energetics.
 Therefore, a supercell containing the perfect crystal is generated in an initial process.
 If not by construction, the system should be fully relaxed.
@@ -631,10 +648,8 @@ The path exhibiting the minimal energy difference determines the diffusion path
 
 \begin{figure}[t]
 \begin{center}
-\subfigure[]{\label{fig:basics:crto}
-\includegraphics[width=0.5\textwidth]{crt_orig.eps}}
-\subfigure[]{\label{fig:basics:crtm}
-\includegraphics[width=0.5\textwidth]{crt_mod.eps}}
+\subfigure[]{\label{fig:basics:crto}\includegraphics[width=0.45\textwidth]{crt_orig.eps}}
+\subfigure[]{\label{fig:basics:crtm}\includegraphics[width=0.45\textwidth]{crt_mod.eps}}
 \end{center}
 \caption{Schematic of the constrained relaxation technique (a) and of a modified version (b) used to obtain migration pathways and corresponding configurational energies.}
 \label{fig:basics:crt}
@@ -655,10 +670,9 @@ In the modified version respective energies could be higher than the real ones d
 
 Structures of maximum configurational energy do not necessarily constitute saddle point configurations, i .e. the method does not guarantee to find the true minimum energy path.
 Whether a saddle point configuration and, thus, the minimum energy path is obtained by the CRT method, needs to be verified by caculating the respective vibrational modes.
-
 Modifications used to add the CRT feature to the VASP code and a short instruction on how to use it can be found in appendix \ref{app:patch_vasp}.
-Due to these constraints obtained activation energies can effectively be higher.
 
 % todo
 % advantages of pw basis with respect to hellmann feynman forces / pulay forces
+% crt sketch needs increased text