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A METHOD OF MEASURING THE MOMENTUM OF ELECTRONS IN A METAL
V. F. GANTMAKHE R Institute of Physics Problems, Academy of Sciences, U.S.S.R. Submitted to JET P editor March 12, 1962 J. Exptl. Theoret. Phys. (U.S.S.R. ) 42, 1416-1418 (May, 1962) A series of size effects , which are determined by the relationships between the size of electron orbits and the size of a metallic sample, can be observed when a sample having a long fre e path is placed in a constant magnetic field. Phenomena of similar type were discovered a relatively long time ago in dc measurements of conductivity. They can be observed, however, in muc h mor e distinct form in measurements of high-frequency impedance, owing to the presence of a supplementary paramete r with the dimensions of length--the skin depth d. A similar effec t was firs t foun d by Khaikin , [1] who discovered the disappearance of cyclotron resonance upon decrease of the field, starting with that field fo r which the diameter of the electron orbit is comparable with the thickness of the sample. For observation of the size effects , however, one can also use frequencies muc h lower than in Khaikin's experiment, on the order of 106 cps, at whic h there is no cyclotron resonance, bu t the condition d » d is well satisfied (d is the diameter of the electron orbit) . Let us consider a flat slab with a constant magnetic field parallel to its surface. The electrons mov e along helices with axes parallel to the surfac e of the metal ; the majo r part of the electro n trajectorie s passes deep in the metal , wher e there is no high frequenc y field ; on return -

ing to the skin layer, the electrons find there a high frequency field in the same phase as it was durin g the preceding passage through the skin layer. The reason for this is that the field does not have time to change significantly durin g the time of rotation of the electron in orbit (~ 10 - 9 sec) . The dependence of the impedance on the field does not have, however, a resonance character, since the condition of constant phase of the electric field fo r all passages of the electron through the skin layer is fulfille d fo r all fields. When the field is increased the radius of the electron orbit decreases, and the numbe r of return s of the electron throug h the skin layer durin g the fre e path time increases. However , the electron returns to the skin layer only if the diameter of its orbit is less than the thickness of the sample. In the contrary case, it is scattered on the surfac e of the crystal. Thus fo r that value of the field at which the orbit diameter of the electron on the extremal section of the Fermi surfac e becomes equal to the thickness of the slab, a certain singularity should be observed in the field dependence of the impedance. The character of this singularity depends on the variation of the dispersion in the vicinity of the extremal section. For example, using a method analogous to that used by Heine, [2] it is easy to show that fo r a quadratic dispersion law the curve has a kink, and if the part of the Fermi surfac e under consideration is a tub e of constant cross section, then the singularity is a discontinuity. We observed this phenomenon experimentally. A flat sample was inserted into the coil of the tank circuit of an oscillator. The cross section of the coil was an elongated rectangle. A constant magnetic field was applied parallel to the surfac e of the sample. The oscillation frequency varied with the magnitude of field, owing to the variation of the reactive component of the impedance of the sample. The dependence of the frequency on the magnitude of the field was measured by a modulation method; the modulation frequency was 20 cps. The samples were highly purifie d monocrystalline tin (about 10 - 4 % impurities) , grown fro m a melt in dismountable crysta l molds . At heliu m temperature s the mea n fre e path of the electrons in the samples reached apparently (1--3 ) x 10 -1 cm : the skin depth at 1--5 Mc s was about 10 - 4 cm . One of the curves obtained is shown in the figure . The sample had a thickness of 0.54 mm . The [100] axis was perpendicular to the surfac e of the slab; a high-frequenc y and a constant magnetic field were directed along [001] axis. The temper-


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ature of this experiment was 3.75°K and the fre quency was f = 2.8 Mcs. As expected, the position of the line depends on the thickness of the sample and depends neither on the frequenc y nor on the inclination of the constant field relative to the surface , within a limit of several degrees. The intensity of the effec t increases with decreasing temperature , approximately doubling in the interval fro m 4.2 to 2.9°K. If the line is recorded mor e slowly and the modu lation amplitude decreased, a fine structur e can be observed in the line, bu t its reproducibilit y fro m sample to sample has not yet been investigated. Generally speaking the shape of the line may depend on many factors . For example, it should reflec t the fact that when the diameter of the orbit is somewhat smaller than the thickness of the sample the electron passes through the skin layer on both sides of the slab during each revolu tion, thereby making a supplementary contribution to the conductivity. In principle, the same phenomenon can be observed at high "cyclotron" frequencies , and it was actually observed by Khaikin [3] at ~ 1010 cps ("non-resonant orbit cutoff") . In this case, however , the singularity in the dependence of the impedance on the field is connected with the scattering of electrons returning to the skin layer at a phase differen t fro m the phase of the field. This aggravates the observation conditions. The phenomenon described yields, generally speaking, the same information about the Fermi surfac e as the cyclotron resonance cutof f gives --the magnitude of the Fermi momentum in the extremal cross section in a direction normal to

both the surfac e and the magnetic field vector . By rotating the field in the plane of the sample we can plot the cross section of the Fermi surfac e in this plane. In ou r case, however , the phenomeno n takes place against a background of a smoother variation of the impedance. That is unquestionably a favorabl e circumstanc e and offer s hope of obtaining some additional informatio n about the behavior of electrons in a metal. On the other hand, we do not obtain here the effectiv e mass fo r this cross-section simultaneously with the value of the momentu m of the electron. This makes the interpretation of the results and comparison with cyclotron-resonance data mor e difficult . The numerica l values we obtained agree almost completely with the data of Khaikin. In particular, we obtained a value 5.7 x 10 -2 0 g-cm/se c fo r the diameter of the orbit in momentu m space in the [100] direction. The part of the surfac e referre d to is a tube of almost constant cross section, which favor s the observation of the effect . However , under the same conditions, we were able to observe other muc h weaker singularities in the plot of the dependence of df/d H on H, in fields of abou t 90 and 105 Oe, which are probably connected with other portions of the Fermi surface . Thus , the dimensional effec t under anomalous skin-effec t conditions presents one mor e convenient way of studying the Fermi surface s of metals. The author offer s deep gratitude to Yu . V. Sharvin fo r daily counsel, aid, and support , to A. I. Shal'nikov fo r continuous interes t in the research, and to L. P. Pitaevskii fo r usefu l discus sion.
1 M. S. Khaikin , JET P 41, 1773 (1961), Soviet Phys. JET P 14, 1260 (1962). 2 V. Heine , Phys. Rev. 107, 431 (1957) . 3 M. S. Khaikin, JET P 43, 66 (1962) , Soviet Phys. JET P 16, in press.

Translated by B. V. Bronk 235