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An experimental condensed matter physicist who, with Gene Dresselhaus and Charles Kittel, studied cyclotron resonance in silicon and germanium. Their 1955 work used single crystals near 4 K, light modulation and circularly polarized microwaves. Changing the orientation revealed direction-dependent band effective masses of electrons and electron holes, allowing measurements to reconstruct the shapes of constant-energy surfaces near the edges of energy bands.^[ref]

Band nonparabolicity is the departure of an energy band's dispersion relation from a quadratic function of crystal momentum near an extremum. A parabolic isotropic approximation writes $E={\hslash }^2{k}^2/(2m_b)$, measuring $E$ from the band edge. Band coupling can make the apparent effective mass depend on energy, especially when the separation from another band is small.^[ref]

A useful isotropic model iswhere $\alpha$ has units of inverse energy. A simple two-level illustration explains the extra factor: the eigenvalue above zero of a matrix with diagonal entries $0,-E_g$ and real off-diagonal entry $Pk$ satisfies $E(E+E_g)=P^2{k}^2$. Dividing by $E_g$ gives $\alpha =1/E_g$ and ${\hslash }^2/(2m_b)=P^2/E_g$. Real materials require a more detailed multiband model, including the appropriate symmetries and spin-orbit coupling.

Differentiating the model gives the group velocity and a momentum-to-velocity mass:This mass is distinct from the mass obtained from the local second derivative of $E(k)$. For $\alpha >0$, a state with parabolic estimate ${\hslash }^2{k}^2/(2m_b)=1/\alpha$ instead has $E=(\sqrt{5}-1)/(2\alpha )$. The energy grows more slowly than the band-edge parabola predicts. Evan O. Kane's 1957 InSb calculation connected such nonparabolic band coupling with measured optical absorption and cyclotron resonance data.^[ref]

A plasma whose linear wave response retains the equilibrium magnetic field and particle inertia but neglects pressure forces, thermal dispersion and collisions. Its applicability requires $k{v}_{\mathrm{th},s}\ll |\omega |$ and separate care near cyclotron resonances.

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.

By Gene Dresselhaus, Arthur F. Kip and Charles Kittel. Physical Review 98, 368–384.

Publisher: doi.org/10.1103/PhysRev.98.368

The abstract describes single-crystal cyclotron resonance measurements near 4 K, light modulation, circularly polarized microwaves and directional anisotropy of carrier energy surfaces.

Evan O. Kane developed influential descriptions of electronic band structure in semiconductors. His 1957 calculation of indium antimonide used k·p perturbation theory to treat the interaction of conduction and valence bands near their extrema. The small band gap made that interaction important rather than a negligible correction.^[ref]

A central consequence is band nonparabolicity: electron energy need not remain proportional to the square of wave number over the relevant range. A constant band-edge effective mass can therefore fail to describe electrons farther from the minimum. The calculation connects band coupling with a measurable change in the relation between energy and motion.

Kane also calculated the fundamental optical absorption using information from the cyclotron resonance mass and compared the prediction with experimental absorption and its dependence on impurity concentration. The agreement supplied an experimental test of the adopted band picture, rather than merely fitting an arbitrary energy curve.^[ref] This combination of a tractable coupled-band model and measurable consequences explains the continuing usefulness of his work in semiconductor theory.

Gene Dresselhaus, Arthur F. Kip and Charles Kittel used cyclotron resonance to investigate electrons and holes in silicon and germanium crystals. Their 1955 measurements near 4 kelvin combined light modulation with circularly polarized microwaves and changes in the crystal orientation relative to the magnetic field. The directional dependence of the resonance revealed anisotropic carrier effective mass values and the shape of constant-energy surfaces near the edges of the electronic band structure. The experiment turns the orbital motion of charge carriers into a probe of electronic band structure.^[ref]

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 $\hslash {\omega }_c$.

Use Gaussian units, plane waves proportional to $e^{i\mathbf{k}\cdot \mathbf{r}-i\omega t}$, and the signed cyclotron frequency ${\Omega }_s=q_s{B}_0/(m_{s}c)$. Write ${\Omega }_e=e{B}_0/(m_{e}c)>0$ for its electron magnitude, so the signed electron value is $-{\Omega }_e$. Define the species plasma frequency by ${\omega }_{ps}^2=4\pi n_{0s}{q}_s^2/m_s$. The linearization of the cold Lorentz force equations givesSolving this linear system and using the species current density ${\mathbf{j}}_s=n_{0s}{q}_s{\mathbf{u}}_s$ gives the conductivity tensorThe magnetized cold-plasma dielectric tensor is $\mathbf{ϵ}=\mathbf{I}+(4\pi i/\omega ){\sum }_s{\mathbf{\sigma }}_s$. ThusAt the stated electron frequency scale, the ion contribution is smaller by $O(Z{m}_e/m_i)$ for $q_i=Ze$ and quasineutrality. To this accuracy,These expressions require negligible collisions and the cold ordering $k{v}_{\mathrm{th},s}\ll |\omega |$; sufficiently close to a cyclotron resonance, thermal broadening or collisions must also be retained.

Use the time dependence actually specified, $e^{i\omega t}$, and $\mathbf{B}=B\hat{\mathbf{z}}$. The component equations areAdding the second equation multiplied by $\pm i$ diagonalizes the response. With the literal definitions $v_{\pm }=v_x\pm i{v}_y$, $E_{\pm }=E_x\pm i{E}_y$, and $J_{\pm }=-ne{v}_{\pm }$,The source's printed $\omega \pm {\omega }_c$ reverses the helicity labels for these definitions. It is recovered by naming the two circular components oppositely, reversing $B$, or changing the matching time/polarization convention. As a concrete check, with $E_{+}\ne 0$, $E_{-}=0$ and positive $B$, the collisionless component equation resonates at $\omega ={\omega }_c$. The printed plus-labeled denominator would instead remain proportional to $2i{\omega }_c$. The physical result is unambiguous: only the circular polarization rotating with the electron orbit has positive-frequency cyclotron resonance.

For the convention used here, the dielectric eigenvalues are ${ϵ}_{\pm }={ϵ}_{\infty }+{\sigma }_{\pm }/(i{ϵ}_0\omega )$. In the lossless limit,These eigenvalues determine propagation and reflection in the Faraday geometry; changing the helicity names swaps their subscripts without changing any optical prediction.

At low density, the plasma frequency and free-carrier dielectric screening are smaller, so radiation can penetrate a suitable semiconductor sample. In a high field with ${\omega }_c\tau \gg 1$, the active circular component has a resolved cyclotron absorption line near $\omega ={\omega }_c$, with width of order $1/\tau$. The opposite circular component lacks that positive-frequency resonance.

Different complex refractive indices for the two components produce Faraday rotation and circular dichroism. Transmission with controlled circular polarization therefore isolates the cyclotron resonance and measures $m^{\ast }=e{B}_{\mathrm{res}}/\omega$. The low-density/high-field labels do not by themselves imply complete transparency at every frequency; one must still avoid the plasma stop bands, strong interband absorption and excessive sample thickness.

For the electron-doped sample, sketch six closed ellipsoids centred on the $\Gamma$--X axes, just inside the six square faces of the face-centred-cubic first Brillouin zone, with long axes pointing radially toward X. In the parabolic approximation,The radial-to-transverse semiaxis ratio is $\sqrt{m_2/m_1}\simeq 2.2$. The readable conduction minimum near X and its crossing of $\mu$ support these electron Fermi surface pockets. The two $[001]$ cyclotron resonance fields arise from the two distinct orientations of the six ellipsoids, rather than two unrelated carrier species. The left panel of the original sketch includes axes, $\Gamma$, and the three positive-axis X points; the negative-axis X points are symmetry-equivalent.

The surviving valence maximum lies roughly $1.2\mathrm{eV}$ below the original $\mu$. Lowering $\mu$ by $1.5\mathrm{eV}$ therefore places it in the upper valence band in the conventional silicon band interpretation. The electron pockets disappear and one sketches a hole pocket centred at $\Gamma$, as in the middle panel. Detailed valence-band warping and the number of spin-orbit-resolved sheets cannot be extracted from the damaged figure, and the single drawn surface is schematic rather than a claim that all such sheets coincide.

A discriminating physical property is the Hall coefficient: electron-dominated transport gives a negative value, whereas hole-dominated transport gives a positive value in the ordinary single-dominant-carrier regime. One should not infer a numerical resistivity change without the carrier densities and mobilities.

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.

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.

Use positive angular frequencies and abbreviate $p={\omega }_{pe}$, $b={\omega }_{ce}$. The left-handed index is $n_L^2=1-p^2/[\omega (\omega +b)]$. It is negative below its positive zero,For the right-handed mode, $n_R^2=1-p^2/[\omega (\omega -b)]$. Below the cyclotron resonance $0<\omega <b$ its denominator is negative and the index square is positive. Just above the resonance it is negative untilThis is the lossless electron-only cold-plasma model. Ions, collisions and resonance absorption matter outside its applicable range.

For a whistler with $\omega \ll b$ and $p^2/(\omega b)\gg 1$,The travel time through a uniform slab is $t-t_s=Lp{\omega }^{-1/2}/(2c\sqrt{b})$. In terms of ordinary frequency,An arbitrary emission time changes the intercept, not the slope. The original figure labels a dispersion constant $D=81\pm 0.5\mathrm{H}{\mathrm{z}}^{1/2}\mathrm{s}$: it is the reciprocal of the plotted slope, not the slope itself. ThereforeFor $B=80$ nT and $L=6.0\times 10^7$ m this gives $b\simeq 1.41\times 10^4{\mathrm{s}}^{-1}$, $p\simeq 2.41\times 10^5{\mathrm{s}}^{-1}$, andThe resulting forbidden right-handed frequency width isThe lower edge is about $2.24$ kHz and the upper edge about $39.5$ kHz. In the real nonuniform plasma, the travel-time coefficient measures a path integral proportional to $\int {\omega }_{pe}/\sqrt{{\omega }_{ce}}ds$. The inferred density is consequently an effective uniform-slab estimate, not a unique local density at the spacecraft. Uncertainty in the path and magnetic field likely exceeds the formal error in $D$.