The anomalous skin effect

1952 ◽  
Vol 215 (1123) ◽  
pp. 481-497 ◽  
Keyword(s):  
Free Path ◽  
Low Temperatures ◽  
Skin Effect ◽  
Mean Free Path ◽  
Skin Depth ◽  
Surface Conductance ◽  
High Frequencies ◽  
Surface Imperfections ◽  
Anomalous Skin Effect ◽  
Cadmium Lead

The anomalous skin effect arises in good conductors at low temperatures and high frequencies when the electronic mean free path becomes comparable with or greater than the classically calculated skin depth. Measurements have been made on a number of metals at frequencies of 1200 and 3600 Mc/s, and the form of variation of r. f. surface conductance with d. c. conductivity agrees well with that predicted theoretically by Reuter & Sondheimer, assuming that the electrons are scattered diffusely when they hit the surface of the metal. From the results, estimates are made of the effective value of σ/ l , the ratio of d. c. conductivity to mean free path, and hence of the free surface area of the occupied region of k -space. The estimate for copper agrees well with that expected theoretically; those for silver and gold are rather lower than the theoretical values. For the other metals investigated, tin, cadmium, lead and aluminium, no theoretical estimates are available. The results are very sensitive to the presence of surface imperfections; the effect of these is discussed.

scholarly journals The surface impedance of superconductors and normal metals at high frequencies II. The anomalous skin effect in normal metals

1947 ◽  
Vol 191 (1026) ◽  
pp. 385-399 ◽  
Keyword(s):  
Free Path ◽  
Skin Effect ◽  
Classical Model ◽  
Mean Free Path ◽  
Skin Depth ◽  
Normal Metals ◽  
Anomalous Skin Effect ◽  
Skin Conductivity ◽  
The Mean ◽  
Dimensional Argument

Measurements on the skin conductivity of the normal metals silver, gold, and tin show that at low temperatures the skin conductivity tends to become independent of the d. c. conductivity, which is at variance with the predictions of classical skin effect theory. Following a suggestion of H. London that this anomalous behaviour is due to the mean free path of the electrons becoming much greater than the skin depth, an attempt is made to calculate the effect for a semi-classical model of a metal. Although a rigorous solution has not been found, it is shown that the model predicts constancy of skin conductivity when the mean free path becomes very long. Moreover, there is reason to suppose that under these conditions only a small proportion of the conduction electrons contribute effectively to the high-frequency current, and an exact solution is given for a model based on this concept, which also predicts that the skin conductivity should be independent of the d. c. conductivity. A simple dimensional argument may be applied to enable values of the mean free path in copper, gold, aluminium and tin, relative to the value in silver, to be deduced from the experimental results. These values are not in good agreement with theoretical estimates by Mott and Jones. The behaviour of mercury is different from that of the other metals in­vestigated, in that the skin conductivity does not tend to a constant value. It is suggested that the theory based on a crude classical model is inapplicable to a metal such as mercury, in which the anomalous skin effect appears at such temperatures that the ideal resistance is still many times greater than the residual resistance.

scholarly journals The theory of the anomalous skin effect in metals

1948 ◽  
Vol 195 (1042) ◽  
pp. 336-364 ◽  
Keyword(s):  
Surface Impedance ◽  
Low Temperatures ◽  
Skin Effect ◽  
Frequency Variation ◽  
High Frequencies ◽  
Microwave Region ◽  
Conduction Electrons ◽  
Relaxation Effects ◽  
Infra Red ◽  
Anomalous Skin Effect

The problem of metallic conduction at high frequencies and low temperatures, recently discussed by Pippard, is reformulated using the general methods of the theory of metals, and exact solutions are obtained which are valid for all frequencies and temperatures. It is shown that, for large values of the free path of the conduction electrons, the electric field is propagated through the metal as a ‘surface wave’ which differs considerably from the classical exponential solution. The temperature variation of the surface impedance in the microwave region is considered in detail. Pippard’s simplified theory is shown to be qualitatively correct, and a quantitative discussion of his experimental results is given. The frequency variation of the surface impedance at low temperatures is also discussed, and it is shown that relaxation effects are negligible in the microwave region but become important in the infra-red and eventually restore the validity of the classical theory. The theory predicts that, as the frequency is increased, the reflexion coefficient of metals passes through a minimum in the far infra-red.

Thermal Conductivity of Single-Crystal and Sintered RBa2Cu3O7 at low Temperatures

1987 ◽  
Vol 99 ◽  
Author(s):  
J. E. Graebner ◽  
L. F. Schneemeyer ◽  
R. J. Cava ◽  
J. V. Waszczak ◽  
E. A. Rietman
Keyword(s):  
Thermal Conductivity ◽  
Particle Size ◽  
Single Crystals ◽  
Free Path ◽  
Low Temperatures ◽  
Rayleigh Scattering ◽  
Resonant Scattering ◽  
Sintered Sample ◽  
Mean Free Path ◽  
Strong Scattering

ABSTRACTThe thermal conductivity k of micro-twinned single crystals of YBa2Cu3O7 and HoBa2Cu3O7 and a sintered sample of YBa2Cu3O7 has been measured for temperatures 0.03<T<5K. For the single crystals, k is small and varies as T1.8-1.9 This behavior resembles k for glassy insulators except for the lack of a plateau above IK. It is concluded that the thermal carriers are phonons with their mean free path limited by resonant scattering from tunneling entities, as in glasses. Suggestions for the location of tunneling systems are given. For the sinter, k is still smaller but does not follow a power law T-dependence. It is similar to other sintered ceramics with the same particle size, where the phonon mean free path is dominated by Rayleigh scattering from the particles. This strong scattering from the microstructure presumably masks the scattering from TS within each particle.

scholarly journals The mean free path of electrons in polar crystals

1939 ◽  
Vol 171 (947) ◽  
pp. 496-504 ◽  
Keyword(s):  
Free Path ◽  
Conduction Band ◽  
Thermal Energy ◽  
Low Temperatures ◽  
Energy Levels ◽  
Stoichiometric Composition ◽  
Mean Free Path ◽  
Polar Crystals ◽  
Absorption Of Light ◽  
The Mean

Polar crystals of stoichiometric composition at low temperatures are insulators of electricity. If, however, electrons are raised into the normally empty conduction band of energy levels, either through the absorption of light or the thermal energy of surrounding atoms, the crystal can conduct. The purpose of this note is to calculate the mean free path of such electrons, and hence their mobility (velocity of drift in unit field). The results obtained will be compared with experimental material obtained from semi-conductors and from substances which show photoconductivity.

The surface impedance of superconductors and normal metals at high frequencies. IV. Impedance at 9400 Mc. /sec. of single crystals of normal and superconducting tin

1950 ◽  
Vol 203 (1072) ◽  
pp. 98-118 ◽  
Keyword(s):  
Single Crystals ◽  
Surface Impedance ◽  
Surface Resistance ◽  
Skin Effect ◽  
Scaling Factors ◽  
High Frequencies ◽  
Normal Metals ◽  
Anomalous Skin Effect ◽  
The Difference ◽  
Fundamental Equations

The measurements described in the earlier papers of this series have been extended to 9400 Mc. /sec., a resonance technique being employed to determine the surface resistance of normal and superconducting tin, and the difference between the surface reactances of the material in the two states. Measurements on single crystals of different orientations have brought to light a marked anisotropy of all these quantities, of a kind which shows clearly the non-tensorial nature of the fundamental equations relating the field vectors. The prediction of the theory of the anomalous skin effect in normal metals, that the surface resistance should vary with frequency as ω ⅔ , is confirmed. The temperature variation of the resistance and reactance of superconducting tin has been studied in detail for a number of specimens of different orientations, and it has been found that over certain ranges of temperature the shapes of corresponding curves for different specimens are similar, apart from scaling factors depending on the orientation; the values of these scaling factors are used to characterize the surface impedance of each orientation.

Heat flow in liquid helium in the transition region

1970 ◽  
Vol 315 (1521) ◽  
pp. 239-247 ◽  
Keyword(s):  
Heat Flow ◽  
Liquid Helium ◽  
Free Path ◽  
Transition Region ◽  
Cross Section ◽  
Low Temperatures ◽  
Circular Cross Section ◽  
Mean Free Path ◽  
Theoretical Treatment ◽  
Tube Radius

A theoretical treatment is given of heat flow at low temperatures in liquid helium, contained in a tube of circular cross-section. Results are obtained for arbitrary ratio of phonon–phonon collision mean free path to tube radius, and agree with experiment to within a few parts per cent.

Thermal conduction in artificial sapphire crystals at low temperatures I. Nearly perfect crystals

1955 ◽  
Vol 231 (1184) ◽  
pp. 130-144 ◽  
Keyword(s):  
Free Path ◽  
Low Temperatures ◽  
Thermal Conduction ◽  
Rough Surfaces ◽  
Mean Free Path ◽  
Boundary Scattering ◽  
Crystal Diameter ◽  
Different Sources

In order to obtain a detailed verification of the theory of thermal conduction in dielectric crystals, measurements have been made on a number of artificial sapphire crystals between 2° and 100° K. In the region of the maximum there are variations in conductivity between crystals from different sources. The highest conductivities measured are about 140 W/cm deg., which suggests that estimates of several hundred watts for the maxima of ideal sapphire crystals are not unreasonable. At sufficiently low temperatures the conductivity of a very perfect, long crystal with rough surfaces is observed, in agreement with Casimir’s theory of boundary scattering, to be proportional to T 3 and to the radius; the phonon mean free path is then nearly equal to the crystal diameter. Imperfect crystals show some anomalous effects. The extension of Casimir’s theory to apply to short specimens has been verified. Perfect crystals with smooth surfaces exhibit some specular reflexion of phonons; a statistical description of the surface is proposed which leads to the observed variation of this effect with temperature and is compatible with the results of interferometric examination of the surface.

scholarly journals Anomalous skin effect for linearly and circularly polarized intense laser light

2002 ◽  
Vol 20 (1) ◽  
pp. 101-107 ◽  
Author(s):  
TATSUFUMI NAKAMURA ◽  
SUSUMU KATO ◽  
TOMOKAZU KATO
Keyword(s):  
Distribution Function ◽  
Absorption Coefficient ◽  
Electron Distribution ◽  
Skin Effect ◽  
Electron Distribution Function ◽  
Stochastic Theory ◽  
Plasma Boundary ◽  
Skin Depth ◽  
Intense Laser ◽  
Anomalous Skin Effect

Anomalous absorption of an intense short laser pulse in overdense plasmas is analyzed with a stochastic theory. A diffusion equation describing a time evolution of the electron distribution function is derived. From the equation it is shown that the electron distribution function becomes anisotropic in the momentum space, which gives rise to the absorption of the energy. The diffusion is not dominant in the pz direction, which is longitudinal to the vacuum-plasma boundary, rather it is dominant in the px direction, which is transverse to the boundary. However, the diffusion in pz enhances the absorption. Analytical expressions of the absorption coefficient and skin depth are obtained for the anomalous skin effect regime (ω02c2 << ωp2ve2), which evolves in time as the electron distribution becomes anisotropic. The asymptotic value of the absorption coefficient is proportional to [square root]I. The temperature and density dependence of the absorption coefficient is also discussed.

scholarly journals The surface impedance of superconductors and normal metals at high frequencies I. Resistance of superconducting tin and mercury at 1200 Mcyc. /sec

1947 ◽  
Vol 191 (1026) ◽  
pp. 370-384 ◽  
Keyword(s):  
Low Temperatures ◽  
Electrical Impedance ◽  
Skin Effect ◽  
Marked Effect ◽  
High Frequencies ◽  
Normal Metals ◽  
Effect Theory ◽  
Very High Frequencies ◽  
Very High ◽  
Good Agreement

This paper is the first in a series in which the behaviour of the electrical impedance of metals at low temperatures and very high frequencies will be considered from experimental and theoretical standpoints. The technique of resonator measurements at 1200 Mcyc. /sec. is described in detail, and experimental curves are given showing the variation with temperature of the r. f. resistivities of superconducting tin and mercury. In contrast to the behaviour of superconductors in static fields, a finite resistance is present at all temperatures, tending as the absolute zero is approached to a very low value, which is probably zero for mercury but not for tin. The experimental results are in good agreement with London’s measurements on tin by a different method. The latter’s observation, that the r. f. resistance of normal tin above the transition point is much greater than that predicted from the classical skin-effect theory, is confirmed, and a similar, though less marked, effect is found also for mercury.

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