= Solution
A <magnetic field> quantizes a charged carrier's cyclotron motion into <Landau levels>. For an isotropic parabolic band, neglecting <spin> splitting,
$$
\varepsilon_{\ell,k_z}=\hbar\omega_c\left(\ell+\frac12\right)+\frac{\hbar^2k_z^2}{2m^*},\qquad \omega_c=\frac{eB}{m^*},\qquad \ell=0,1,2,\ldots.
$$
The degeneracy of each orbital level per unit perpendicular area is $eB/h$ per <spin>. Increasing the field changes both the level spacing and the number of states in each level. Successive levels cross the <Fermi energy>, producing an oscillatory <density of states> and hence <quantum oscillations> in thermodynamic and transport quantities.
For a general closed semiclassical orbit, <Onsager quantization> relates its wavevector-space area to field:
$$
A_k(\varepsilon_\ell)=\frac{2\pi eB}{\hbar}(\ell+\gamma).
$$
Setting this area to an extremal Fermi-surface cross-section gives successive inverse-field crossings separated by
$$
\boxed{\Delta\left(\frac1B\right)=\frac{2\pi e}{\hbar A_{\rm ext}},\qquad F=\frac{\hbar A_{\rm ext}}{2\pi e}.}
$$
A typical oscillatory contribution is proportional to $\cos(2\pi F/B+\varphi)$. The oscillations are periodic in $1/B$, not in $B$. In three dimensions, extremal cross-sections dominate because contributions from other longitudinal momenta cancel by phase averaging. Rotating the field therefore maps the shape of the <Fermi surface>.
\Image[/past-exam-of-the-quantum-condensed-matter-course-of-the-university-of-cambridge/2016-quantum-oscillations.png]
{title=Quantized orbital crossings and their periodicity in inverse magnetic field}
Oscillations in <magnetization> are the <de Haas-van Alphen effect>; oscillations in longitudinal <electrical resistivity> are the <Shubnikov-de Haas effect>. They are most readily observed in clean samples at low <temperature>. The thermal <damping> factor is
$$
R_T=\frac{X}{\sinh X},\qquad X=\frac{2\pi^2k_BT}{\hbar\omega_c},
$$
and a common first-harmonic disorder factor is $R_D=\exp[-\pi/(\omega_c\tau_q)]$. Thus the Landau spacing must be sufficiently large compared with thermal broadening and disorder broadening, approximately $k_BT\lesssim\hbar\omega_c$ and $\omega_c\tau_q\gtrsim1$, with the displayed factors giving the quantitative suppression.
The <temperature> dependence measures the <cyclotron effective mass>, and the field-dependent <damping> probes the <quantum lifetime> $\tau_q$. That lifetime need not equal the <momentum relaxation time> controlling <electrical conductivity>: small-angle scattering can broaden <Landau levels> while relaxing <electric current> inefficiently. <Spin> splitting can generate further modulation or resolved <spin> structure. Oscillations are not the same as the ordinary smooth <magnetoresistance> or <cyclotron resonance>; they probe quantized orbital state counting near the <Fermi energy>.
Back to article page
