Electron Bands in 2D Semiconductor

[StochasticHarmonic]

Homework Statement

The given energy dispersion is:
\begin{equation}
E(p) =
\begin{cases}
-p^2/2m_v & \text{if } E \leq 0 \\
p^2/2m_c+\Delta & \text{if } E > 0
\end{cases}
\end{equation}
Where $m_c$ and $m_v$ are the effective masses of the conduction and valence band electrons.

Part A begins with derivation of the energy dependent DOS, which is fairly simple to find using the 2D density of states relation:
\begin{equation}
\frac{2}{\pi^2 \hbar} \frac{d^2 p}{dE} = D(E)
\end{equation}
to be:
\begin{equation}
D(E)=\{\begin{array}{cc}\frac{m_v}{\pi \hbar^2 },&\mbox{ if } E\leq 0\\ \frac{m_c}{\pi \hbar^2 }, & \mbox{ if } E>\Delta\end{array}
\end{equation}
Then for part b we use the fact that the total number of electrons are fixed:

\begin{equation}
\int_{E_F}^{\infty} f(E)D(E) \, dE = \int^{E_F}_{-\infty} (1-f(E))D(E) \, dE
\end{equation}
To derive the equation:
\begin{equation}
-\beta m_c E_D+m_c ln[1+e^{-\beta(\mu-\Delta)}]=m_vln[1+e^{- \beta \mu}]
\end{equation}
Which gives us a relation for our chemical potential.

My Question comes from the following qualitative question: Assume the semiconductor is intrinsic, where $E_D=0$, or the valence band is filled and conduction band empty at T=0, How does the chemical potential change with increasing temperature for $m_c>m_v$?

Relevant Equations

When $E_D=0$, equation 5 becomes:

\begin{equation}
m_c ln[1+e^{-\beta(\mu-\Delta)}]=m_vln[1+e^{- \beta \mu}]
\end{equation}
Question is stated below:

In the questions solution, they conceptually discuss how the DOS for the conduction band becomes larger when $m_c$ is larger than $m_v$. This then implies that there is "more phase space for electrons than holes", which confuses me. How can you make a statement about the phase space of electrons based only on the conduction band DOS being larger than the valence band DOS?

They go on to make another few statements which confuse me; "at finite temperature the holes must spread wider in energy compared to electrons. This means that with increasing temperature, the half-filled state must shift down in energy so the Boltzmann Tail of the distribution |E-mu|>>T has larger overlap with the valence band."

If anyone could provide some clarity on this conceptually, that would be extremely beneficial. I'm confused on how the conduction band DOS is proportional to the phase space of the electrons, and I'm also confused on what the half filled state is, and also why the holes energy spreads out with finite temperature.

Thank you.