Conduction in Solids

My doubts are:

• What is effective mass $m^*$ (real meaning apart from explained in this text) and how can we calculate it/or what is it's value? and why?
• In actual situation $\lambda$ would not be $\infty$. yes? and if yes then how can we make it very large by changing temperature and frequency, i.e. which range of these parameters (high or low) is suitable?

In a crystal, each electron polarizes its vicinity, meaning positive charges prefer to stay nearer to the electron while negative charges move away from it. As the electron traverses through the crystal, it has to drag the polarization cloud with it, which can effectively increase or decrease its mass. Actually, the system "electron+polarization cloud" can be considered as a quasiparticle and it is the mass of this quasiparticle that we usually refer to as the effective mass of the electron (or hole). There are many ways to determine $m^*$, but perhaps the simplest is to do a band structure calculation and find the second derivative of the conduction band minimum at the $\Gamma$ point, i.e. $1/m^* = \hbar^{-2} \frac{\partial^2 E}{\partial q^2}$. Alternatively, in a cyclotron resonance experiment, $m^*$ is determined from $\omega = eB/m^*$. The effective masses generally range from the order of a 100th of the free electron mass to almost infinity. The latter can refer to the case of self-trapping, where the electron or hole remains immobile regardless of the force applied.
In an ideal crystal without lattice vibrations, the mean free path would be infinite. In a real crystal there is always some scattering off defects, which include structural defects, impurities, and phonons. By lowering the temperature, $\lambda$ can be increased, but not to infinity. Also, even if you produce a perfectly pure sample, there will always be some contribution from the phonon zero-point motion, which leads to some scattering and hence a finite $\lambda$.
• can we get $m^*$ in terms of $\omega,m,q,B,E$? and why does lowering temperature help? and which frequency makes $\lambda$ $\infty$? Sorry but I find your answer a little missing, sorry, but (+1) and accepted for the moment. – RE60K May 13 '15 at 12:24
• @ADG I modified the post to show explicitly the relation between $m^*$, $q$, $E$, when doing a band structure calculation, and $m^*$, $\omega$, $B$, when doing a cyclotron resonance experiment. As for the mean free path, lowering the temperature means lowering the number of phonons and thus lowering the rate of electron-phonon scattering, which in turn translates into a longer free path. $\lambda$ is a property of the material and does not depend on the frequency of the external magnetic field. It is not possible to make $\lambda$ infinite by tuning $\omega$. – Raul Laasner May 13 '15 at 13:32
• There's no $\lambda$ in that expression. By varying $B$ you vary $\omega$, that's it. The other quantities remain constant. – Raul Laasner May 13 '15 at 14:24