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For a closed semiclassical orbit with area $S(E)$ in a plane of wave vector space, the cyclotron effective mass is defined by the displayed derivative. It need not equal the local curvature-based band effective mass.

A massless-cone description permits a nonzero cyclotron effective mass. For a circular constant-energy contour of area $S(E)=\pi E^2/(\hslash v_F{)}^2$, the semiclassical definition givesThe vanishing mass parameter in a Dirac Hamiltonian and the finite cyclotron response away from neutrality are different statements. This makes the temperature dependence of Shubnikov-de Haas oscillations a useful test of the dispersion rather than a measurement of an electron's vacuum rest mass.

Cyclotron resonance is resonant absorption by charged particles in a magnetic field when an applied oscillating electric field drives their cyclotron motion. For a parabolic band, the resonance frequency is ${\omega }_c=|q|B/m^{\ast }$, so measuring it determines the cyclotron effective mass. In the quantum description, transverse electric dipole transitions connect adjacent Landau levels, whose separation is $\hslash {\omega }_c$. For an electron and a field along $z$, only the circular polarization rotating in the electron's orbital sense drives the positive-frequency resonance. Scattering broadens the absorption peak; resolving it requires an orbital period short compared with the relaxation time.

Andre Geim, Konstantin Novoselov and collaborators mechanically cleaved graphite, identified thin flakes, and fabricated multi-terminal Hall bars on oxidized silicon. A substrate gate changed the carrier density; longitudinal electrical resistance and the Hall coefficient supplied different checks. The 2004 report included one-, two- and three-layer devices and described a few-layer semimetal interpretation. Layer identification and electrical control were separate experimental achievements.^[ref]

The 2005 monolayer study used an optical microscope, electron microscopy and atomic force microscopy to select monolayers. It then measured Shubnikov-de Haas oscillations as temperature, gate voltage and magnetic field varied. Their damping inferred a density-dependent cyclotron effective mass; comparison with oscillation frequency tested a linear dispersion relation. A comparison with two-layer devices also distinguished their quantum Hall effect sequences. These are complementary tests of the monolayer model, rather than consequences of finding a thin-looking flake alone.^[ref]

A strongly correlated material whose low-energy electronic quasiparticles have a very large thermodynamic effective mass. An enhanced electronic heat capacity coefficient, spin susceptibility and quantum-oscillation cyclotron effective mass are complementary signatures; they need not all be related by a single isotropic band mass. A Kondo lattice is an important setting for such behaviour. In CeAl3, low-temperature specific heat and quadratic resistivity provided an early quantitative example.^[ref]

In a low-carrier-density semiconductor use a thin, cold sample in a transmission spectrometer. Direct microwave or far-infrared radiation through it, commonly with propagation parallel to $\mathbf{B}$ and electric field transverse to it. Sweep $B$ at fixed frequency and detect transmitted power. The active circular polarization shows an absorption peak, or transmission dip, near $B_{\mathrm{res}}=m^{\ast }\omega /e$. Resolving it requires a long enough scattering time, ${\omega }_c\tau \gtrsim 1$.

A metal's large plasma frequency usually prevents that radiation from sampling its bulk directly. In surface cyclotron resonance in a metal, put $\mathbf{B}$ approximately parallel to a clean metal surface and excite an rf electric field in the skin depth, transverse to $\mathbf{B}$. Electron orbits extend into the bulk and repeatedly return to that layer. Coherent energy absorption occurs when the rf phase repeats on a return, including harmonics $\omega =n{\omega }_c$. Measure cavity loss, reflected power, or surface impedance rather than attempting ordinary bulk transmission. The characteristic regime has cyclotron radius larger than the skin depth and sufficiently specular, long-lived orbits.

For the spherical parabolic band the metal harmonic fields areThey are evenly spaced in $1/B$, with amplitudes and widths determined by surface geometry and scattering. A semiconductor generally gives the simple bulk fundamental, whereas the metal can show this harmonic series in a surface-loss signal. For an anisotropic real Fermi surface the measured mass is the appropriate cyclotron effective mass rather than necessarily every component of a transport effective-mass tensor. These classical resonance harmonics are distinct from the Landau-level quantum oscillations in c.

For $\mathbf{B}\parallel [111]$, its squared projection onto the long axis of every valley is $1/3$. All six valleys therefore have the same cyclotron effective mass:At the fixed photon energy,Thus the two-peak spectrum is replaced by one peak near $1.4\mathrm{T}$. The source does not determine the linewidth or absolute oscillator strength for the new direction; the sketch below shows only a normalized illustrative line at the computed position.

A completely filled valence band carries no net current: its occupied crystal-momentum states have velocities that cancel. Near a valence-band maximum, an almost full band is conveniently described by the few missing electrons, or electron holes. Removing an electron of charge $-e_0$, with $e_0>0$, changes the charge relative to the filled band by $+e_0$. This is a quasiparticle description of electronic motion, not a mobile positive ion or a Positron.

For an isotropic parabolic maximum, ${ϵ}_v(\mathbf{k})=E_v-{\hslash }^2{k}^2/(2m_h^{\ast })$. Assign the missing-electron state momentum ${\mathbf{k}}_h=-{\mathbf{k}}_e$ and hole energyIt has positive effective mass $m_h^{\ast }$. Its velocity equals the velocity of the missing electron state, and its positive-charge current replaces the negative of that state's electronic current. Semiclassical motion is thus $\hslash {\dot{\mathbf{k}}}_h=+e_0(\mathbf{E}+{\mathbf{v}}_h\times \mathbf{B})$. With momentum relaxation time ${\tau }_h$, the low-field mobility is ${\mu }_h=e_0{\tau }_h/m_h^{\ast }$ andIf electrons also contribute, $\sigma =e_0(p{\mu }_h+n{\mu }_e)$: the two species' longitudinal conductivities add.

A Hall bar separates carrier concentration from mobility. Apply current density $j_x$, magnetic field $B_z$, and measure transverse electric field $E_y$. For one isotropic hole population, transverse force balance gives $E_y=v_x{B}_z$, soFor thickness $t$, the Hall-voltage magnitude is $|V_H|=IB/(e_{0}pt)$. A longitudinal four-terminal resistance measurement supplies $\sigma$ independently. Use ohmic contacts, known conducting thickness, controlled temperature and low enough current to avoid self-heating. Reversing magnetic field removes even-in-field longitudinal offsets from contact misalignment; reversing current also helps reject thermoelectric offsets. These are central controls, rather than merely taking one transverse voltage reading.^[ref]

The simple inference assumes one carrier type and a known Hall scattering factor. More generally $R_H=r_H/(e_{0}p)$, so $\sigma R_H$ measures Hall mobility rather than automatically the drift mobility. With two carrier populations at weak field,Its sign alone need not identify the more numerous carrier. A Hall experiment also cannot determine $m_h^{\ast }$ and ${\tau }_h$ separately: their ratio sets mobility. cyclotron resonance supplies an independent mass measurement by sweeping frequency or magnetic field through $\omega =e_{0}B/m_c$ and detecting resonant absorption. The measured cyclotron effective mass can depend on band anisotropy and need not equal every curvature mass; the isotropic parabolic model identifies them. Its linewidth provides a scattering-time estimate only with a specified damping model.

semiconductor doping generates holes by adding acceptor impurities. In silicon, a substitutional group-III atom such as boron provides one fewer valence electron than silicon. A shallow acceptor level lies slightly above the valence-band edge: accepting a valence electron makes the dopant negatively ionized and leaves a mobile hole in the valence band. The impurity stays fixed in the lattice. charge neutrality and the nondegenerate equilibrium mass-action law giveFor fully ionized, uncompensated acceptors in the extrinsic regime, $p\simeq N_A$ and $n\simeq n_i^2/N_A$; with fully ionized donor compensation, $p\simeq N_A-N_D$ while holes remain dominant. The Fermi level moves towards the valence band. At low temperature, semiconductor carrier freeze-out suppresses mobile electron holes; at high temperature intrinsic electron-hole generation can dominate. Increasing dopant density also changes scattering, so a conductivity increase alone cannot distinguish increased carrier density from changed mobility. Temperature-dependent Hall and conductivity measurements separate these effects, subject to the carrier and scattering qualifications above.

In a heavy fermion material, correlated low-energy quasiparticles have a large electronic density of states and effective mass. A Kondo lattice with hybridization between localized f-electron states and conduction states is an important setting. The large mass is a low-energy dynamical property, not a change in the electron's bare mass or the ionic mass.

Low-temperature calorimetry measures $C/T\simeq \gamma +\beta T^2$ in a regime where an electronic linear term and a Debye lattice term are appropriate. The large intercept $\gamma =({\pi }^2{k}_B^2/3)D(E_F)$ measures the enhanced quasiparticle density of states, with volume or molar conventions kept consistent. Apply a known heat pulse to a thermally characterized sample platform and measure the temperature rise, then vary the base temperature to separate contributions.

In low-temperature calorimetry and resistivity of CeAl3, measurements below $0.2\mathrm{K}$ gave $\gamma =1.62\mathrm{J}\mathrm{mo}{\mathrm{l}}^{-1}{\mathrm{K}}^{-2}$ and $\rho ={\rho }_0+A{T}^2$ with $A=35\mu \Omega \mathrm{cm}{\mathrm{K}}^{-2}$. The large thermal coefficient and enhanced quadratic transport coefficient are complementary evidence for a strongly renormalized Fermi liquid.^[ref] Enhanced spin susceptibility provides another probe. Where coherent quantum oscillations can be resolved, their thermal damping determines a cyclotron effective mass, giving a band-sensitive comparison with the thermodynamic mass inferred from heat capacity.

Angle-resolved photoemission spectroscopy maps occupied energy bands: ultraviolet or x-ray photons eject electrons, and an energy- and angle-resolving analyzer measures $E_{\mathrm{kin}}$ and emission direction. The photoelectric effect gives the binding energy $E_B=h\nu -\Phi -E_{\mathrm{kin}}$; the parallel momentum follows from $k_{\parallel }=\sqrt{2m{E}_{\mathrm{kin}}}\sin \theta /\hslash$, up to surface reciprocal vectors. Clean surfaces and controlled photon energy are needed, and an occupied-state measurement alone does not map an empty conduction band.

Scanning tunneling spectroscopy instead brings a conducting tip close to a surface and modulates the bias while recording the differential tunnel current. At low temperature with a smooth tip density of states and tunnelling matrix element, $dI/dV$ probes the local density of states near energy $E_F+eV$. The signal is thermally broadened and depends on tunnelling selection, so it is not an unqualified map of the whole bulk band structure.

Bulk probes complement these surface measurements. In the de Haas-van Alphen effect, a low-temperature magnetization or torque measurement versus magnetic field gives quantum-oscillation frequencies $F=\hslash A_{\mathrm{ext}}/(2\pi e)$, measuring extremal Fermi surface sections; temperature damping gives a cyclotron effective mass. The linear electronic heat capacity gives the density of low-energy excitations at the Fermi level. Optical absorption measures transition thresholds and matrix-element-weighted joint densities, rather than a unique full dispersion by itself.

In a heavy fermion material, interactions of narrow, often $f$-derived states with conduction electrons produce low-energy quasiparticles with very large thermodynamic effective mass. A coherent Kondo lattice can form a narrow low-energy band, so the large mass is an interaction-renormalized response rather than a larger bare electron mass.

Two concrete experimental signatures are an enormous linear electronic heat capacity coefficient and a strongly enhanced quadratic low-temperature electrical resistivity. Calorimetry separates $C=\gamma T+\beta T^3$ by plotting $C/T$ against $T^2$; the intercept measures the electronic coefficient rather than a lattice contribution. Four-terminal transport tests the Fermi liquid law $\rho ={\rho }_0+A{T}^2$. The low-temperature calorimetry and resistivity of CeAl3 found $\gamma =1.62\mathrm{J}\mathrm{mo}{\mathrm{l}}^{-1}{\mathrm{K}}^{-2}$ and $A=35\mu \mathrm{\Omega }\mathrm{cm}{\mathrm{K}}^{-2}$ below $0.2\mathrm{K}$, making both signatures quantitative.^[ref]

An enhanced nearly temperature-independent spin susceptibility and a large quantum-oscillation cyclotron effective mass provide additional tests. Not every heavy-fermion compound has the same ordered ground state; magnetism or superconductivity can intervene before a simple normal Fermi-liquid limit is reached.

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.

The Shubnikov-de Haas effect is an oscillatory variation of electrical resistance with magnetic field caused by Landau levels moving through the Fermi energy. As the density of accessible states near that energy changes, the conductivity and scattering response oscillate.

A measurement sweeps field at low temperature while recording a longitudinal voltage for a controlled current. Oscillations are commonly approximately periodic in inverse field for a fixed closed Fermi-surface orbit. Their period probes the orbit area, while temperature-dependent amplitude constrains a cyclotron effective mass. Disorder and thermal broadening can wash out the oscillations even when a Fermi surface exists. The effect is a transport measurement, distinct from the oscillating magnetization measured in the thermodynamic quantum-oscillation counterpart.

A finite-temperature measurement averages the oscillatory response over the derivative of the Fermi-Dirac distribution. For the fundamental harmonic and a locally constant cyclotron effective mass, this gives $R_T=X/\sinh X$ with $X=2{\pi }^2{k}_{B}T/(\hslash {\omega }_c)$. The oscillations become small when thermal smearing substantially exceeds their Landau-level spacing.