Characteristics of bloch electron in a priodic potential

Question

Effective mass of a Bloch electron in a periodic potential is negative why ?

Answer 1

It's negative because the effective mass is given by $$ \left(\frac{1}{m_{\rm eff}}\right)_{ij} = \frac{1}{\hbar^2} \frac{\partial^2 {\mathcal E}}{\partial k_i\,\partial k_j} $$ It's the second derivative of the energy with respect to the wave number (i.e. momentum). Near the minima, the second derivative is positive, near the maxima, it's negative. It's the same relationship as one knows from quantum field theory except that in a QFT, we differentiate the potential energy of the field $V$ and the second derivative is the squared mass and not just mass.

Click the first link in this answer if you need some justification of the formula above. In effect, one "fits" the shape of the energy as a function of $p$ to the usual form of the kinetic energy, $p^2/2m_{\rm eff}$. Near a minimum, the required effective mass is positive, near the maximum, it's negative.