# Cyclotron Resonance

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## cyclotron resonance

[′sī·klə‚trän ′rez·ən·əns]
(physics)
Resonance absorption of energy from an alternating-current electric field by electrons or ions in a uniform magnetic field when the frequency of the electric field equals the cyclotron frequency, or the cyclotron frequency corresponds to the effective mass of electrons in a solid. Also known as diamagnetic resonance.

## Cyclotron Resonance

the selective absorption of electromagnetic energy by charge carriers in a conductor situated in a magnetic field; the absorption occurs at frequencies that are equal to the cyclotron frequency of the charge carriers or are multiples of the cyclotron frequency. During cyclotron resonance, a sharp increase in the electrical conductivity of the conductor is observed.

In a constant electric field E and a constant magnetic field H, charge carriers—that is, charged particles—move along helical orbits with the axes of the helices parallel to the magnetic field (Figure 1,a). In the plane perpendicular to the magnetic field, the particles undergo periodic motion at the cyclotron frequency Ω. If a uniform alternating electric field E of frequency Ω acts on a particle, the energy absorbed by the particle turns out to be a periodic function of time t with an angular frequency equal to the frequency difference Ω – ω. Therefore, the average energy absorbed over a long period of time increases markedly in the case where ω = Ω. An increase in the particle energy leads to an increase in the diameter of the particle orbit and to the occurrence of an increment Δv of the mean particle velocity v, that is, to an increase in electrical conductivity. The conductivity is proportional to Nev/E, where N is the concentration of charge carriers and e is the elementary charge.

The occurrence of discrete allowed states called Landau levels corresponds to the periodic motion of charge carriers in a magnetic field. The Landou levels are described by the quantization condition Φ = (n + ½) Φ0, where Φ is the magnetic flux encompassed by a moving charge, Φ0 = ch/2e is the flux quantum (c is the speed of light and h is Planck’s constant), and n is an integer. The frequency of the quantum transitions between equidistant neighboring levels is the cyclotron frequency. Thus, cyclotron resonance may be interpreted as the excitation of charge-carrier transitions between Landau levels by an applied alternating field.

Cyclotron resonance may be observed if the charge carriers undergo many revolutions before experiencing a collision with other particles and being scattered. This condition has the form Ωτ > 1, where τ is the mean time between collisions, which is approximately equal to the relaxation time and is determined by the physical properties of the conductor. For example, in a gaseous plasma, τ is the time between collisions of free electrons with other electrons or with ions or neutral particles. In a solid conductor, collisions of electrons in the conductor with crystal defects (for which τ ≈ 10–9–10–11 sec) and scattering by thermal lattice vibrations (the electron-phonon interaction) play a decisive role. The electron-phonon interaction restricts the range in which cyclotron resonance may be observed to low temperatures (~ 1–10°K). The maximum relaxation times achievable in practice place a lower limit on the frequency range (v = ω/2π > 109hertz [Hz]) in which cyclotron resonance may be observed in solid conductors.

Cyclotron resonance may be observed in various conductors. Examples include the cyclotron resonance of electrons or ions in gaseous plasmas, the cyclotron resonance of conduction electrons in metals, the cyclotron resonance of excess carriers generated by light or by heating in semiconductors and dielectrics, and cyclotron resonance in two-dimensional systems (see below). However, the term “cyclotron resonance” has come to be used primarily in solid-state physics in cases where there is no radiation from a medium as a result of quantum transitions between Landau levels.

Cyclotron resonance in semiconductors was predicted in 1951 by la. G. Dorfman (USSR) and, independently, R. Dingle (Great Britain); it was discovered in 1953 by G. Dresselhaus, A. F. Kip, and C. Kittel (USA). It is observed at frequencies of ~ 1010–1011 Hz in magnetic fields of 1–10 kilo-oersteds (kOe). Since the concentration of free charge carriers generated, for example, by light or by heating usually does not exceed 1014–1015cm–3, cyclotron resonance is observed at frequencies of ω ≫ ωp, where ωp is the plasma frequency of the carriers and m* is the effective mass of the carriers. The medium is practically transparent to waves of such frequencies, and the refractive

Figure 1. Electron trajectories: (a) in a uniform constant magnetic field H under the action of an alternating electric field E ⊥ H, (b) in a metal in a magnetic field H parallel to the surface of the metal, and (c) in the case of specular reflection from the surface of a metal

index of the medium is close to 1. Since the wavelength λ ~ 1 cm at such frequencies and the diameters of the electron orbits are of the order of micrometers, the charge carriers move in a virtually uniform electromagnetic field. The cyclotron resonance observed in a uniform electromagnetic field is also called diamagnetic resonance, because the cyclotron motion of the charge carriers results in diamagnetism of the electron gas (seeLANDAU DIAMAGNETISM).

If a wave that is circularly polarized in the plane perpendicular to H is used to observe cyclotron resonance, charged particles rotating in the same direction as the polarization vector will absorb electromagnetic energy. The determination of the sign of charge carriers in semiconductors is based on this effect.

Metals, in which the concentration of charge carriers is N ≈ 1022cm–3, have a high electrical conductivity. In metals, cyclotron resonance has been observed at frequencies Ω ≪ ωp. Electromagnetic waves are reflected almost totally from the surface of a specimen, penetrating the metal to a skin depth δ of approximately 10–5 cm (seeSKIN EFFECT). As a result, conduction electrons move in a highly nonuniform electromagnetic field; as a rule, the diameter of the electron orbits D ≫ δ. If a constant magnetic field H is parallel to the surface, some of the electrons—although moving most of the time in the depth of the metal, where there is no electric field—return briefly to the surface layer, where they interact with the electromagnetic wave (Figure l,b). In this case, the mechanism of energy transfer from the wave to the charge carriers is analogous to the operation of a cyclotron. Resonance occurs if an electron enters the surface layer each time at the same phase of the electric field, which is possible when nΩ = ω. This condition corresponds to resonances that recur periodically as the quantity 1/H varies (Figure 2).

Figure 2. Cyclotron resonance in a single-crystal metal plate: (X) the reactive portion of the surface impedance of the metal

If H is at an angle to the surface of the metal, an electron cannot return repeatedly to the surface layer and a Doppler frequency shift occurs which is associated with electron drift along the field (seeDOPPLER EFFECT). As a result, the resonance lines are broadened and the amplitude of the lines is reduced, so that even at small angles of inclination (10”–100”) the cyclotron resonance corresponding to the condition nΩ = ω is not observed in the general case.

In metals, an effect similar to cyclotron resonance—namely, a resonant variation of surface conductance owing to quantum transitions between surface magnetic levels—may be observed under the same conditions as is cyclotron resonance. The effect was discovered in 1960 by M. S. Khaikin (USSR), and the theory of the effect was developed in 1967 by T.-W. Nee and R. E. Prange (USA). Surface magnetic levels arise if electrons, while moving in a magnetic field, can be specularly reflected from the surface of a specimen, thus undergoing periodic orbital motion (Figure l,c). The periodic motion is quantized, and orbits for which the magnetic flux Ф through a section formed by an arc of the trajectory and the surface of the specimen (the shaded area in Figure l,c) is equal to Φ = (n + ¼)Φ0 turn out to be allowed orbits.

Cyclotron resonance is also observed in two-dimensional systems. If a constant electric field is applied perpendicular to the surface of a semiconductor, an excess concentration of charge carriers that can move freely only along the surface arises in the surface layer, which has a thickness of ~ 10–100 angstroms. A conducting layer of electrons may be formed in a similar manner above the surface of a dielectric in a vacuum when the dielectric is bombarded by an electron beam. Resonance absorption of the energy of an electromagnetic wave with a frequency of ω = eH/mc is observed in a magnetic field in such two-dimensional systems. Cyclotron resonance of electrons localized above a liquid helium surface was observed at a frequency of ~1010 Hz by T. R. Brown and C. C. Grimes (USA) in 1972. Cyclotron resonance of electrons at the surface of a semiconductor has been observed at a frequency of ~1012Hz.

Cyclotron resonance is usually studied by the methods of radio-frequency spectroscopy and infrared optics.

Cyclotron resonance is widely used in solid-state physics to study the energy spectrum of conduction electrons. It is employed primarily for the precise measurement of the effective mass m* of such electrons. Studies of cyclotron resonance have established that the effective mass of conduction electrons is anisotropic and that typical values of m* are approximately (10–3–10–1),m0—where m0 is the mass of a free electron—in semiconductors and semimetals, (10–1–10)m0 in good metals, and greater than 10m0 in dielectrics. Cyclotron resonance may also be used to determine the sign of charge carriers, to study the processes of charge-carrier scattering in metals, and to investigate the electron-phonon interaction in metals. The components of the effective-mass tensor may be determined by varying the direction of the constant magnetic field with respect to the crystallographic axes. Cyclotron resonance may be used in microwave engineering to generate and amplify electromagnetic oscillations, for example, in the cyclotron-resonance maser.

### REFERENCES

Ziman, J. M. Elektrony i fonony. Moscow, 1962. (Translated from English.)
Abrikosov, A. A. Vvedenie v teoriiu normal’nykh metallov. Moscow, 1972.
Khaikin, M. S. “Magnitnye poverkhnostnye urovni.” Uspekhi fizicheskikh nauk, 1968, vol. 96, issue 3.

V. S. EDEL’MAN

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