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T O TH E EDITO R

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DIMENSIONAL EFFECT IN A METAL IN MULTIPLES OF A CERTAIN MAGNETIC FIELD
V. F. GANTMAKHE R Institute for Physics Problems, Academy of Sciences, U.S.S.R. Submitted to JETP editor May 18, 1962 J. Exptl. Theoret. Phys. (U.S.S.R.) 43, 345-347 (July, 1962) new dimensional effect has been discovered on measurement of the dependence of the surfac e impedance of tin on a 1--5 MHz magnetic field at helium temperatures. A flat sample was placed in a coil of rectangular cross section which was part of an oscillating circuit . A constant magnetic field was applied along the plane of the sample. The frequency of the oscillator f varied with the magnitude of the field because of variation of the reactance X of the sample. The dependence of the frequenc y on the field was measured by a modulation method; the field modulation frequency was 20 cps. The sample was a single crystal of high-purity tin (abou t 10 -4 % impurities ) grown fro m the melt in a demountable quart z mold. The sample surface was perpendicular to the [100] axis. The thickness of the plate was 0.39 mm , the electron mean free path reached (1--3 ) x 10-1 cm at helium temperatures, and the skin-effec t depth was 10 - 4 cm at 1--5 MHz. In a field H0 = 2cp/ed (p is the half-width of the extremal electron orbit in the momentum space along a direction at right angles to the magnetic field and to the sample-surfac e normal; d is the plate thickness ), when the width of the electron trajectory on the extremal cross section of the Fermi surfac e becomes equal to the plate thickness, a singularity [1] appears on the X(H) curve and this singularity can be used to measur e the Fermi surfac e cross section . Further experiments have shown that singularities on the X(H) curve appear also in fields that are multiples of H0 (w e found them up to a field 5 H 0 ) when the thickness of the plate is equal respectively to 2, 3, or more widths of the electron trajectory . Figure 1 shows curve s on which the singularities are clearly visible in fields of 2 H 0 and 3 H 0 . The reason fo r the appearance of these singularities in multiple fields is as follows. Electrons in an orbit passing through the skin layer experience a systematic increase in velocity Dv due

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to interaction with the electric field ; this governs the participation of such electrons in conduction. As they move along the trajectory , these electrons reach after a time a depth equal to the trajector y width 2p and the increase of their velocity reverses sign (fo r the sake of simplicity we assume that the electric field is at right angles to the magnetic field and that the electron mean fre e path is muc h greater than the perimeter of its trajectory) . Thus we have in the interior of the metal a curren t equal in magnitude and opposite in direction to the curren t in the surface layer. The density of this curren t is very low since electrons participating in conduction belong to differen t cross sections of the Fermi surfac e and therefor e move away fro m the surfac e to differen t depths. However, near the extremal cross sections of the Fermi surface, and correspondingly at the extrema l widths of the electron trajectories in the plate, the number of electrons increases strongly and therefor e the curren t density at the depth equal to the width of the extremal trajector y rises sharply. The natur e of the dispersion law near the extremal cross section may tend to intensify the effect . This happened in the experiment referre d to here : the effec t was observed on the very nearly cylindrical part of the Fermi surfac e [2,3] (th e experimental points in Fig. 2 indicate that the Fermi surfac e diameter in this cross section varies only 2--3%) . Therefor e all the surfac e electrons contribute to the effect. Moreover the form of the orbit is such that the curvatur e of the trajector y is small where the electrons are farthest fro m the surfac e and the electrons move parallel to the surfac e fo r a relatively long time.

In this experiment the skin layers are on both sides of the surfac e and the electric fiel d vectors at the two surfaces are in antiphase. In the 2 H 0 field the trajectorie s of electrons passing through the skin layers touch in the center of the plate and this produces an interaction between the skin layers and a change in the impedance. In the 3 H 0 field the trajector y width is d/3 so that the coupling between the skin layers is via a chain of three trajectories ; in the 4 H 0 field a chain of fou r trajectorie s is involved, and so on. There was no differenc e between the form of the lines corresponding to the even and odd numbers of consecutive trajectories . The curves of Fig. 1 show also singularities in fields close to H0+ H1 and H0 + H2. These singularities can be explained by assuming that one of the trajectorie s of a chain which couples the skin layers is replaced by the trajector y of an electron belonging to the Fermi surfac e of another zone. The first-orde r effec t of the trajectorie s with


LETTER S T O TH E EDITO R smaller widths is clearly visible in the left-hand parts of the curves in Fig. 1 (th e fields H1 and H2). Local penetrations of the electromagnetic fiel d into the interior of a metal were predicted theoretically by Azbel';[4] however , he discussed only the high-frequenc y case when the electrons that contribute to this effec t take part in cyclotron resonance. Therefor e Azbel's theory is not directly applicable to our experiments although the phenomenon dealt with by him and our effec t are very similar. The author is deeply gratefu l to Yu . V. Sharvin fo r constant guidance, to M. S. Khaikin and R. T. Mina for useful discussions, and to A. I. Shal'nikov for his interest.
1 V. F. Gantmakher, JET P 42 , 1416 (1962), Soviet Phys. JET P 15, 982 (1962). 2 M. S. Khaikin , JET P 41, 1773 (1961), Soviet Phys. JET P 14, 1260 (1962). 3 M. S. Khaikin , JET P 43, 59 (1962) , this issu e p . 42 . 4 M . Ya . Azbel' , JET P 39 , 40 0 (1960) , Soviet Phys. JET P 12, 283 (1961).

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Translated by A. Tybulewicz 59