= Solution
An <intrinsic semiconductor> has its carrier populations set primarily by thermal excitation across the <band gap>; in the ideal pure, neutral case its conduction-electron and hole densities are equal, $n=p=n_i$. An <extrinsic semiconductor> has carrier populations significantly controlled by donor or acceptor impurities, producing predominantly <n-type semiconductor> or <p-type semiconductor> behavior. Chemical purity alone does not fix conductivity independently of <temperature> and gap size.
The positive <charge carrier mobility> is the magnitude of mean drift speed per unit weak <electric field>. For <electrons> $\mathbf v_e=-\mu_e\mathbf E$, for holes $\mathbf v_h=\mu_h\mathbf E$, and its SI unit is $\mathrm{m^2\,V^{-1}\,s^{-1}}$. In an isotropic relaxation-time model, $\mu=|q|\tau/m^*$.
A <band effective mass> characterizes how an <electron> wave packet accelerates in an <energy band>, rather than replacing its vacuum rest <mass>. In one dimension and its tensor generalization,
$$
\boxed{\frac1{m^*}=\frac1{\hbar^2}\frac{d^2E}{dk^2},\qquad
(M^{*-1})_{ij}=\frac1{\hbar^2}\frac{\partial^2E}{\partial k_i\partial k_j}.}
$$
These follow by differentiating $v_i=\hbar^{-1}\partial E/\partial k_i$ and using $\hbar\dot k_j=F_j$. Effective masses can be anisotropic and <electron> band curvature can be negative.
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