= Solution
A <band effective mass> describes the acceleration of an electronic wavepacket in a periodic lattice through the local <band curvature>. It incorporates the periodic lattice potential into a useful transport parameter without changing the bare <Electron> <mass>. For a <Bloch state> in an isolated band,
$$
\mathbf v_g=\frac1\hbar\nabla_{\mathbf k}\epsilon(\mathbf k),\qquad
\hbar\dot{\mathbf k}=\mathbf F.
$$
Differentiation gives
$$
a_i=\sum_j(m^{*-1})_{ij}F_j,\qquad
\boxed{(m^{*-1})_{ij}=\frac1{\hbar^2}\frac{\partial^2\epsilon}{\partial k_i\partial k_j}}.
$$
In one dimension this becomes $\boxed{m^*=\hbar^2/(d^2\epsilon/dk^2)}$. A locally parabolic isotropic band minimum has $\epsilon\simeq\epsilon_c+\hbar^2|\mathbf k-\mathbf k_c|^2/(2m^*)$ with positive $m^*$. A flatter band has a larger <effective mass>. At a band maximum the <Electron> curvature is negative, so the <Electron> accelerates oppositely to the applied <force>. Describing the missing <Electron> as a positive-charge <electron hole> with positive <mass> is often simpler.
In the scalar relaxation-time approximation, a conduction <Electron> in an electric field reaches $\mathbf v_d=-e\tau\mathbf E/m^*$. Thus
$$
\boxed{\mu_e=\frac{e\tau}{m^*}},\qquad
\boxed{\sigma=ne\mu_e=\frac{ne^2\tau}{m^*}}.
$$
For a crystal with different directional curvatures, conductivity is a tensor and involves the occupied states and their scattering. Contributions from <Electrons> and holes add: $\sigma=e(n\mu_e+p\mu_h)$ in an isotropic two-carrier description. A filled band contributes no ordinary longitudinal current because opposite group velocities cancel; a partially filled band or holes permit a current response.
<Effective mass> therefore connects measured electrical transport to <electronic band structure>, but <carrier density> and <momentum relaxation time> matter as much as curvature. Ionised impurities and <phonons> alter $\tau$. A smaller <mass> does not guarantee a higher conductivity if carrier population or scattering changes. Nor are all reported masses identical: density-of-states, cyclotron and conductivity masses weight anisotropic bands differently. The simple semiclassical formula assumes an isolated band, slowly varying weak fields and negligible interband transitions; near degeneracies or strong fields a single constant scalar <mass> is inadequate.
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