Markdown to LaTeX Converter

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\usepackage{hyperref}
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\lstset{basicstyle=\ttfamily\small,breaklines=true,frame=single,columns=fullflexible}

\title{Heat transfer in thin films}
\author{R. Okonkwo}
\date{September 2026}

\begin{document}
\maketitle

\section{Introduction}

Thin films conduct heat \emph{anisotropically}; the in-plane conductivity $k_\parallel$ can exceed the cross-plane value $k_\perp$ by \textbf{an order of magnitude} \footnote{See Cahill et al., *Nanoscale thermal transport*, 2003.}. The ratio depends on film thickness $d$ and the phonon mean free path $\Lambda$.

\subsection{Model}

The steady-state temperature field satisfies

\[\nabla \cdot (k \nabla T) = -q\]

where $q$ is the volumetric heat source. We solve it with a finite-difference scheme:

\begin{lstlisting}[language=Python]
def solve(k, q, dx, iterations=1000):
    T = np.zeros_like(q)
    for _ in range(iterations):
        T[1:-1] = 0.5 * (T[2:] + T[:-2] + dx**2 * q[1:-1] / k)
    return T
\end{lstlisting}

\subsection{Results}

\begin{table}[h]
\centering
\begin{tabular}{rrr}
\toprule
Thickness (nm) & $k_\parallel$ (W/mK) & $k_\perp$ (W/mK) \\
\midrule
50 & 148 & 21 \\
200 & 152 & 63 \\
800 & 155 & 140 \\
\bottomrule
\end{tabular}
\end{table}

\begin{itemize}
  \item Conductivity converges to the bulk value above 800 nm
  \item Below 100 nm boundary scattering dominates
  \begin{itemize}
    \item Roughness matters more than composition
  \end{itemize}
\end{itemize}

\begin{quote}
Measurements were taken at 300 K with a 3ω method; see \href{https://example.org/setup}{the setup}.
\end{quote}

\begin{figure}[h]
  \centering
  \includegraphics[width=0.8\linewidth]{figures/profile.png}
  \caption{Temperature profile}
\end{figure}

\end{document}