Multicolumns with equation format

Question

I'm currently writing a summary for my semiconducter devices course and I'm trying to do it with Latex (just started today). I have the following problem:

I have two chapters with two formulas each. But they writing is too close together. Here my first attempt:

\section{Quantum Theory of solids}
\begin{multicols}{2}
\subsection{Density of States} 
$D_c(E)=\displaystyle\frac{4\pi (2m^*_n)^{3/2}}{h^3}\displaystyle\sqrt{E-E_c}$\\
$D_v(E)=\displaystyle\frac{4\pi (2m^*_p)^{3/2}}{h^3}\displaystyle\sqrt{E_v-E}$

\columnbreak

\subsection{Fermi-Dirac Distribution}
$f_e(E)=\displaystyle\frac{1}{1-exp(\frac{E-E_F}{k_BT})}$\\
$f_h(E)=1-f_e(E)$

\end{multicols}

Which looks like this:

then i tried to add the command \flushright like this:

\columnbreak

\begin{flushright}
\subsection{Fermi-Dirac Distribution}
$f_e(E)=\displaystyle\frac{1}{1-exp(\frac{E-E_F}{k_BT})}$\\
$f_h(E)=1-f_e(E)$
\end{flushright}

The result:

I would be really grateful if you were to help a noob like me to make this formation better.

PS: here is the full text

\documentclass[11pt]{article}
\usepackage{graphicx}
\usepackage{wrapfig}
\usepackage{color} 
\usepackage{comment}
\usepackage{multicol}

\begin{document}
\title{Semiconducter Devices}
\author{Jean-Marc Sujata}
\date{\today}
\maketitle
\newpage

\begin{wrapfigure}{r}{0.25\textwidth} %this figure will be at the right
    \centering
    \includegraphics[width=0.25\textwidth]{DC.png}
\end{wrapfigure}
\section{Unit Cells}
$package\ density = \displaystyle{\frac{N_{Atoms} \cdot V_{Atoms}}{V_{unit cell}}}$
\\
\begin{tabular}{|c|c|c|c|}
\hline
$Type$ & Atoms & r-a ratio & Volume\\ \hline
$simple\ cubic$ & $N=1$ & $2r=a$ & $V_a=\frac{\pi}{6}r^3$\\ \hline
$body\ centered$ & $N=2$ & $2r=\frac{\sqrt{3}}{2} a$ & $V_a=\frac{8\pi}{3}r^3$\\ \hline
$face\ centered$ & $N=4$ & $2r=\frac{\sqrt{2}}{2}\cdot a$ & $V_a=\frac{16\pi}{3}r^3$\\ \hline
$simple\ cubic$ & $N=8$ & $2r=\frac{\sqrt{3}}{4}a$ & $V_a=\frac{32\pi}{3}r^3$\\ \hline
\end{tabular}

\includegraphics[width=8cm, height=2cm]{Faces.png}
\section{Quantum Theory of solids}
  \begin{multicols}{2}
    \subsection{Density of States}
    $D_c(E)=\displaystyle\frac{4\pi (2m^*_n)^{3/2}}{h^3}\displaystyle\sqrt{E-E_c}$

    \noindent $D_v(E)=\displaystyle\frac{4\pi (2m^*_p)^{3/2}}{h^3}\displaystyle\sqrt{E_v-E}$

    \columnbreak

    \subsection{Fermi-Dirac Distribution}
    $f_e(E)=\displaystyle\frac{1}{1-exp(\frac{E-E_F}{k_BT})}$

    \noindent $f_h(E)=1-f_e(E)$

  \end{multicols}


\end{document}
\

If it helps: I'm using Texmaker

Answer 1

Note: This is an answer to the original question. It does not answer the current question.

If I complete your code in a standard way, I don't see any problem. However, I have replaced \\ by paragraph breaks and added \noindent just to avoid the problems which \\ in regular text mode can cause.

\documentclass{article}
\usepackage{multicol}

\begin{document}
  \section{Quantum Theory of solids}
  \begin{multicols}{2}
    \subsection{Density of States}
    $D_c(E)=\displaystyle\frac{4\pi (2m^*_n)^{3/2}}{h^3}\displaystyle\sqrt{E-E_c}$

    \noindent $D_v(E)=\displaystyle\frac{4\pi (2m^*_p)^{3/2}}{h^3}\displaystyle\sqrt{E_v-E}$

    \columnbreak

    \subsection{Fermi-Dirac Distribution}
    $f_e(E)=\displaystyle\frac{1}{1-exp(\frac{E-E_F}{k_BT})}$

    \noindent $f_h(E)=1-f_e(E)$

  \end{multicols}
\end{document}

But you might be best served by one of the environments provided by amsmath. For example:

\documentclass{article}
\usepackage{multicol,mathtools}

\begin{document}
  \section{Quantum Theory of solids}
  \begin{multicols}{2}
    \subsection{Density of States}
    \begin{equation*}
      \begin{gathered}
        D_c(E)=\displaystyle\frac{4\pi (2m^*_n)^{3/2}}{h^3}\displaystyle\sqrt{E-E_c}\\
        D_v(E)=\displaystyle\frac{4\pi (2m^*_p)^{3/2}}{h^3}\displaystyle\sqrt{E_v-E}
      \end{gathered}
    \end{equation*}
    \columnbreak
    \subsection{Fermi-Dirac Distribution}
    \begin{equation*}
      \begin{gathered}
      f_e(E)=\displaystyle\frac{1}{1-exp(\frac{E-E_F}{k_BT})}\\
      f_h(E)=1-f_e(E)
      \end{gathered}
    \end{equation*}
  \end{multicols}
\end{document}

Answer 2

This is an answer to the revised question. It involves issues not evident in the original question.

You cannot use wrapfigure without ensuring that there is sufficient text in regular paragraphs to accommodate it. In particular, you can't use it too near things like lists and other special environments, such as multicols.

I would use minipage environments to handle this layout by hand. I also recommend booktabs for the table and the use of the amsmath environments, with mathtools enhancements.

However, I'm not sure what to do about the second subsection heading....

\documentclass[11pt]{article}
\usepackage[demo]{graphicx}
\usepackage{multicol,mathtools,booktabs}

\begin{document}

  \section{Unit Cells}
  \begin{minipage}{.75\textwidth}
    \centering
    \begin{equation*}
        package\ density = \frac{N_{Atoms} \cdot V_{Atoms}}{V_{unit cell}}
    \end{equation*}

    \begin{tabular}{*{4}{c}}
      \toprule
      $Type$ & Atoms & r-a ratio & Volume\\ \midrule
      $simple\ cubic$ & $N=1$ & $2r=a$ & $V_a=\frac{\pi}{6}r^3$\\
      $body\ centered$ & $N=2$ & $2r=\frac{\sqrt{3}}{2} a$ & $V_a=\frac{8\pi}{3}r^3$\\
      $face\ centered$ & $N=4$ & $2r=\frac{\sqrt{2}}{2}\cdot a$ & $V_a=\frac{16\pi}{3}r^3$\\
      $simple\ cubic$ & $N=8$ & $2r=\frac{\sqrt{3}}{4}a$ & $V_a=\frac{32\pi}{3}r^3$\\ \bottomrule
    \end{tabular}
    \medskip

    \includegraphics[width=8cm, height=2cm, keepaspectratio=true]{Faces}
  \end{minipage}
  \begin{minipage}{.25\textwidth}
    \centering
    \includegraphics[width=\linewidth]{DC.png}
  \end{minipage}

  \section{Quantum Theory of solids}
  \begin{multicols}{2}
    \subsection{Density of States}
    \begin{flalign*}
        D_c(E)&=\frac{4\pi (2m^*_n)^{3/2}}{h^3}\sqrt{E-E_c}\\
        D_v(E)&=\frac{4\pi (2m^*_p)^{3/2}}{h^3}\sqrt{E_v-E}
    \end{flalign*}
    \columnbreak
    \subsection{Fermi-Dirac Distribution}
    \begin{flalign*}
        f_e(E)&=\frac{1}{1-exp(\frac{E-E_F}{k_BT})}\\
        f_h(E)&=1-f_e(E)
    \end{flalign*}
  \end{multicols}
\end{document}