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<li class="toctree-l2"><a class="reference internal" href="#methods-for-calculating-the-curvature-effective-mass">Methods for calculating the curvature effective mass</a></li>
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<div class="section" id="implementation">
<h1>Implementation<a class="headerlink" href="#implementation" title="Permalink to this headline">¶</a></h1>
<div class="section" id="methods-for-calculating-the-curvature-effective-mass">
<h2>Methods for calculating the curvature effective mass<a class="headerlink" href="#methods-for-calculating-the-curvature-effective-mass" title="Permalink to this headline">¶</a></h2>
<p>Three methods are used to calculate the curvature effective mass (Eqn.1 in the main text).</p>
<ul>
<li><p><strong>Finite difference</strong></p>
<p>We use a three point forward finite difference equation to calculate the curvature at point i:</p>
<div class="math notranslate nohighlight">
\[\frac{\partial^2E}{\partial k^2} = \frac{E_{i+2} - 2E_{i+1} + E_{i}}{\left|k_{i+1} - k_i\right|},\]</div>
<p>where <span class="math notranslate nohighlight">\(E_i\)</span> is the energy eigenvalue at position <span class="math notranslate nohighlight">\(k_i\)</span> in reciprocal space.</p>
</li>
<li><p><strong>Unweighted least-squares fit</strong></p>
<p>To obtain estimates for the coefficient of a parabolic dispersion</p>
<div class="math notranslate nohighlight">
\[E = ck^2,\]</div>
<p>we use the least-squares method as implemented in the NumPy Python library to minimise the summed square of residuals</p>
<div class="math notranslate nohighlight">
\[\sum^{5}_{i=1}(ck_i^2 - E_i)^2.\]</div>
<p>We fit to five points; three points from the DFT-calculated dispersion plus two from the symmetry of the dispersion (<span class="math notranslate nohighlight">\(E(k)=E(-k)\)</span>).</p>
</li>
<li><p><strong>Weighted least-squares fit</strong></p>
<p>To obtain estimates for the coefficients of the dispersion</p>
<div class="math notranslate nohighlight">
\[E = {c}k^2,\]</div>
<p>we use the least-squares method as implemented in the NumPy Python library to minimise the summed square of residuals</p>
<div class="math notranslate nohighlight">
\[\sum^{n}_{i=1}W_i(ck_i^2 - E_i)^2.\]</div>
<p>The summation is over all points up to an energy of <span class="math notranslate nohighlight">\(0.25\,eV\)</span>, including points generated from the symmetry of the dispersion, <span class="math notranslate nohighlight">\(E(k)=E(-k)\)</span>.
<span class="math notranslate nohighlight">\(W_i\)</span> is given by</p>
<div class="math notranslate nohighlight">
\[W_i(E_i,T) = \frac{1}{\exp\left(\frac{E_i-E_f}{k_BT}\right)+1}.\]</div>
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