homework and exercises - What the heck is negative effective mass?

I am reading this book:Solid State Electronic Devices by Ben G Streetman and Sanjay Kumar Banerjee.
I have some doubts in the article 3.2.2 Effective mass.

In this the aythors say that $E=\dfrac{1}{2}mv^2=\dfrac{1}{2}\dfrac{P^2}{m}=\dfrac{{\hbar} ^2}{2m}k^2$.

• Electrons usually have thermal velocity of order $10^7$m/s. So Shouldn't we use $E=mc^2$ rather than $E=\frac{1}{2}mv^2$.

The author further says that electron mass is related to curvature of (E,k) relationship as $\dfrac{d^2E}{dk^2}=\dfrac{\hbar^2}{m}\tag{3.2(d)}$. then the author says that the effective mass is given by $m^{*}={{\hbar^2}/{{\dfrac{d^2E}{dk^2}}}}\tag{3.3}$

• Is equation $3.3$ derived from equation $3.2(b)$. In equation $3.2(d)$ we have $m$ the original mass and in eqn $3.3$ we have $m^{*}$ the effective mass which is completely different from $m$.

In the end of the article the author says that the total force on the electron is given by $F_{tot}=ma$. After this the author says the external force applied on an electron is related to effective mass as: $F_{ext}=m^{*}a$.

• From where this equation comes? Sometimes the effective mass $m^{*}$ is negative then the equation implies that if we apply certain external force on electrons in the crystal they will accelerate in the opposite direction, how this is possible? What's going on?