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The de Haas-van Alphen effect is an oscillation of a metal's magnetization as changing magnetic field moves Landau levels through the Fermi energy. The associated Onsager quantization gives inverse-field period $\Delta (1/B)=2\pi e/(\hslash S_F)$ for an extremal Fermi-surface cross-section $S_F$. Oscillatory resistance is instead the Shubnikov-de Haas effect.

A uniform magnetic field quantizes the transverse cyclotron motion of an Electron. With Electron charge $-e$ and the Landau gauge for a uniform magnetic field $\mathbf{A}=(0,Bx,0)$, minimal coupling givesFor an eigenstate of $p_y$ with eigenvalue $\hslash k_y$, this is a quantum harmonic oscillator centered at $x_c=-\hslash k_y/(eB)$, with cyclotron frequency ${\omega }_c=eB/m$. Its Landau levels areThe oscillator width is set by the magnetic length ${\ell }_B=\sqrt{\hslash /(eB)}$. In three dimensions, the free motion parallel to the magnetic field adds ${\hslash }^2{k}_z^2/(2m)$. The Electron spin adds a Zeeman splitting $\pm g_s{\mu }_{B}B/2$. In a crystal the appropriate band effective mass and effective spin factor replace the free-electron values.

Changing $k_y$ moves the orbit center without changing the energy. In a rectangle $L_x{L}_y$, adjacent allowed centers differ by $2\pi {\ell }_B^2/L_y$. Requiring the center to lie in the sample gives the degeneracy of a Landau level, per spin,This large state degeneracy, together with the increasing level spacing, rearranges the occupied states as the magnetic field changes. To resolve the Landau levels, thermal broadening $k_{B}T$ and disorder broadening should both be small compared with $\hslash {\omega }_c$.

The de Haas-van Alphen effect is an oscillation of magnetization as successive Landau levels pass the Fermi energy. The Shubnikov-de Haas effect is the corresponding oscillation in longitudinal magnetoresistance. For a simple orbit, successive crossings occur periodically in $1/B$; the general relation to the extremal orbit area is Onsager quantization. These measurements reveal the geometry and effective masses of a Fermi surface.

In a two-dimensional Electron system, filling an integer number $\nu$ of resolved levels leads to the integer quantum Hall effect, withand very small longitudinal resistivity on a plateau. Disorder localization helps produce finite plateau widths. Cyclotron resonance provides another direct signature: radiation is absorbed at the adjacent-level spacing $\hslash {\omega }_c$.

Different measurements reveal different aspects of electronic band structure; a convincing determination combines their constraints.

Angle-resolved photoemission. In angle-resolved photoemission spectroscopy, photons of known energy eject occupied Electrons. Measuring their kinetic energy and emission angle reconstructs binding energy and the component of crystal momentum parallel to the surface:Here $\Phi$ is the sample work function when kinetic energy is referenced to the nearby vacuum; an analyzer energy scale requires its own work-function calibration. Momentum is conserved modulo a surface reciprocal lattice vector, neglecting photon momentum. Tracking the peaks maps occupied dispersion relations and the Fermi surface. The measured intensity also contains the matrix element, the Fermi-Dirac distribution and the spectral function, so an absent bright feature does not establish an absent band. For a three-dimensional solid, the perpendicular momentum requires a final-state model. Inverse photoemission spectroscopy probes unoccupied states by injecting Electrons and detecting emitted photons.

Optical spectroscopy. Photon absorption and ellipsometry measure interband transitions, optical selection rules and the complex dielectric response. Because optical photon momentum is small on the scale of the Brillouin zone, direct transitions approximately preserve crystal momentum. Their onset constrains a direct band gap; a threshold involving a phonon can reveal an indirect band gap. Peak positions constrain energy differences between bands, but the response also depends on the joint density of states and optical matrix elements, so it does not uniquely determine every $E_n(\mathbf{k})$.

Quantum oscillations and cyclotron resonance. In clean samples at low temperature, the de Haas-van Alphen effect measures oscillations in magnetization and the Shubnikov-de Haas effect measures oscillations in electrical resistivity as the inverse magnetic field is varied. Their period in inverse magnetic field is $1/F$, where Onsager quantization relates $F$ to an extremal Fermi-surface area perpendicular to the field:Rotating the magnetic field maps the geometry of extremal orbits. The thermal damping determines the cyclotron effective mass,with the appropriate magnitude for a hole orbit. Cyclotron resonance supplies the corresponding dynamical mass through ${\omega }_c=eB/m_c$. These are detailed probes near the Fermi level, rather than measurements of all occupied bands. Diffraction supplies the reciprocal lattice needed to label the measured momenta, but diffraction by itself is not a measurement of electronic dispersion.

A magnetic field quantizes a charged carrier's cyclotron motion into Landau levels. For an isotropic parabolic band, neglecting spin splitting,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:Setting this area to an extremal Fermi-surface cross-section gives successive inverse-field crossings separated byA typical oscillatory contribution is proportional to $\cos (2\pi F/B+\phi )$. 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.

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 isand 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 $k_{B}T\lesssim \hslash {\omega }_c$ and ${\omega }_c{\tau }_q\gtrsim 1$, 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.

Energy and momentum measurement. In angle-resolved photoemission spectroscopy, monochromatic photons illuminate a clean surface in vacuum, and an electron energy analyzer measures the outgoing kinetic energies and emission directions. Measuring both gives the occupied electronic band structure, rather than only a momentum-integrated distribution of removal energies.

Choose a common energy reference for initial and final states. Let $E_i$ be the initial electron energy, $E_F$ the Fermi energy, and $E_{\mathrm{vac}}=E_F+\varphi$ the vacuum level immediately outside the sample. The binding energy is $E_B=E_F-E_i\ge 0$. For a direct emission event without additional losses, conservation of energy givesHere the work function $\varphi$ is the energy from the Fermi level to the local vacuum. The initial and final energies are total single-electron energies on the same scale, while $E_{\mathrm{kin}}$ is measured above vacuum. The diagram illustrates each interval independently.

Neglecting ultraviolet photon momentum, photoemission momentum conversion giveswhere $\theta$ is measured from the surface normal. Parallel crystal momentum is conserved modulo a surface reciprocal lattice vector; the surface breaks perpendicular translation symmetry. A three-dimensional sample therefore generally needs a final-state model to infer perpendicular momentum. The stated two-dimensional sheet avoids that ambiguity. The kinetic energy used in this formula is the sample-vacuum value; if the analyzer and sample have different work functions, their contact potential must be included when converting the analyzer's reading, as in photoelectron binding energy reference.

Spectral information. Define $ϵ=E_i-E_F=-E_B$ and the electron spectral function by $A(\mathbf{k},ϵ)=-\mathrm{Im}{G}^R(\mathbf{k},ϵ)/\pi$. In the sudden approximation, the basic photoemission intensity and spectral function relation isbefore instrumental broadening and background are included. Here $f$ is the Fermi-Dirac distribution, and $\mathcal{M}$ is the transition matrix element. The electron removal spectral function contains the occupied part of the spectrum. In a noninteracting band it has a delta peak at the band energy; interactions shift and broaden peaks and can add incoherent weight. Thus peak trajectories reveal dispersion, while line shapes can constrain lifetimes after resolution effects are accounted for. Dividing by known matrix and occupation factors can estimate $A$ where those factors are appreciable; intensity alone is not identical to $A$, and a missing peak can reflect a matrix-element suppression. States well above $E_F$ are not accessible to ordinary low-temperature photoemission.

Two independent band-structure probes. First, the de Haas-van Alphen effect measures oscillatory magnetization as Landau levels cross the Fermi level when a magnetic field changes. The inverse-field frequency iswhere $S_{\mathrm{ext}}$ is an extremal Fermi surface cross-sectional area perpendicular to the field. Changing field orientation maps its shape. This reveals the momentum-space geometry of occupied bands; the related Shubnikov-de Haas effect measures oscillatory resistance.

Second, absorption or reflectance spectroscopy detects interband transitions. Photon energies at absorption edges and peaks constrain the band gap and interband energy differences. For a direct band gap, negligible photon momentum selects approximately vertical transitions in the band diagram, weighted by the joint density of states and transition matrix elements. An indirect transition needs a phonon to carry the extra crystal momentum. These measurements provide evidence beyond a free-electron parabola, although an optical spectrum alone does not reconstruct every band's dispersion.

Reading the graphite spectra. For each direction, follow a moving peak through successive angles and convert each peak's own kinetic energy to $k_{\parallel }$; angle is not itself proportional to momentum over this wide range. The prominent dispersing valence feature starts at roughly $7$–$9\mathrm{eV}$ binding energy near $\Gamma$ and moves towards lower binding energy as $k_{\parallel }$ increases. Along $\Gamma K$ it approaches the Fermi level at the corner, then turns back to larger binding energy on continuing beyond the corner. Along $\Gamma M$ it reaches a finite minimum binding energy of roughly $3$–$4\mathrm{eV}$ near the edge midpoint and turns back; it does not reach the Fermi level there. Additional deeper peaks describe other occupied valence branches, with visibility varying across the spectra.

A useful qualitative description is the upper occupied $\pi$ band of the nearest-neighbour model of graphene: $E_{\pi }=-t|f(\mathbf{k})|$. It rises from $-3t$ at $\Gamma$ to zero at $K$, but only to $-t$ at $M$. The schematic curve in the figure captures those two different behaviours. It is not a full multi-orbital assignment or a quantitative fit to graphite, whose deeper valence features are also visible in the supplied spectra.

Angle needed to reach the corner. The corner's parallel wavevector isUse the near-zero binding energy of the upper valence band at that point. With the sample work function interpreted as the stated $3.8\mathrm{eV}$ separation below vacuum,HenceThe energy and angle readings in the PDF are approximate. If a peak at the corner were instead assigned a small binding energy $E_B$, the estimate becomesFor $E_B=0$–$1\mathrm{eV}$ this gives about $57$–$60^{\circ }$, consistent with the near-Fermi feature in that angular range. Substituting the full photon energy for the kinetic energy would omit the work-function cost and underestimate the required angle.

As $B$ changes, each quantized orbit area grows linearly with field while the Fermi surface cross-section stays approximately fixed. Landau energies repeatedly cross the Fermi energy, making the density of states there oscillate. If the relevant extremal orbit has area $S_F$, its crossings satisfyThus the oscillations are periodic in inverse field rather than field. In three dimensions integration over longitudinal momentum makes extremal cross-sections dominate.

One observable is the de Haas-van Alphen effect: the field-dependent occupation changes the electronic free energy, so its field derivative, the magnetization, oscillates. The corresponding oscillations of longitudinal electrical resistance are the Shubnikov-de Haas effect. Both probe Fermi-surface geometry.

A quantum oscillation is an oscillatory contribution to a metal's response as a magnetic field moves quantized orbital levels past the Fermi energy. For a closed extremal Fermi surface cross-section of wavevector-space area $A_{\mathrm{ext}}$, Onsager quantization gives inverse-field period $\Delta (1/B)=2\pi e/(\hslash A_{\mathrm{ext}})$. Oscillations of magnetization and electrical resistivity are respectively the de Haas-van Alphen effect and Shubnikov-de Haas effect. Low temperature and long quantum lifetime are needed so thermal smearing and disorder do not wash out the Landau levels.