Derivation of cyclotron effective mass in Landau quantization

Question

The shortest derivation of Landau quantization in the completely free electron gas starts with the minimal-coupling Hamiltonian $$ H = \frac{1}{2m} \left( \vec p + \frac{e}{c}\vec A \right) ^2 \equiv \frac{1}{2m} \vec P^2 $$ and due to $[P_x, P_y]=-i \hbar e B / c$ then proceeds in complete analogy to the derivation of a harmonic oscillator's spectrum: Introduce ladder operators $a \propto P_x + i P_y$ such that $[a^\dagger, a] = 1$, then express the Hamiltonian using these operators to arrive at $$ H = \hbar \omega_c \left( n + \frac{1}{2} \right) + \frac{\hbar^2 k_z^2}{2m} $$ with $n = a^\dagger a$ and $\omega_c = e B /m$.

What this derivation is missing, however, is that in a solid, you get the cyclotron effective mass

$$m_c = \frac{\hbar^2}{2\pi}\frac{\partial S(\epsilon, k_z)}{\partial \epsilon}$$

instead of the plain electron mass $m$, where $S$ is the area of a semiclassical cyclotron orbit.

Is there a way to amend it so the effective mass does show up, while retaining its simplicity?

Also, is there a relation between the cyclotron effective mass and the inertial effective mass known from the semiclassical equations of motion in a solid? I read that they are equal for parabolic bands, yet people seem to conflate them no matter the shape - e.g. when discussing Landau and Pauli susceptibilities $$\chi_{Landau} \propto \left(\frac{m}{m^*}\right)^2 \chi_{Pauli},$$ $m^*$ is usually just taken to be the ordinary, inertial effective mass.

Answer 1

In solids, the electron dynamics is usually treated in the framework of the semiclassical equations of motion using the $E(\vec{k})$ energy-wave vector dispersion relations for Bloch wave solutions of the Schrödinger equation in the periodic potential of the crystal lattice. These equations of motion are $$\vec{F}=\frac{d\vec{p}}{dt}$$ where is the force on an electron and $\vec{p}=\hbar \vec{k}$ is the crystal momentum. The force $\vec{F}$ includes external electric and magnetic fields (Lorentz force). Thus the movement in k-space is fully determined, which also determines the energy of the electron according to the dispersion relation $E(\vec{k})$. If there is only a magnetic field, there is no energy change in the k-space movement and the electron will move in k-space on a closed path (not necessarily a circle) which lies on the intersection of an constant-energy surface with a plane normal to the magnetic field. For the movement of an electron in real space, the group velocity $\vec{v}=\frac{∂E/∂\vec{k}}{\hbar}$ of a superposition of Bloch waves has to be considered giving the expectation value of velocity, and the expectation value of acceleration $\vec{a}=M^{-1}\vec{F}$, where $M^{-1}$ is the tensor of the inverse effective mass. If this tensor is diagonal in the chosen cartesian coordinate system, the diagonal elements are $1/m_i=\frac{∂^2E}{\hbar^2∂k_i^2}$, $i=x,y,z$, where $m_i$ are the effective masses. These effective masses appear in different expressions for the so called conductivity effective mass, the density-of-states effective mass, the optical mass and also the cyclotron effective mass $m_c = \frac{\hbar^2}{2\pi}\frac{\partial S(\epsilon, k_z)}{\partial \epsilon}$, where S is the area enclosed by the path in k-space. This $m_c$ is an effective mass related to the tensor of inverse mass along the closed path in k-space.