THE RADIO-FREQUENCY SIZE EFFECT IN COPPER WHISKER BLADES : EXPERIMENTAL TECHNIQUE, LINE SHAPES

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THE RADIO-FREQUENCY SIZE EFFECT IN

COPPER WHISKER BLADES : EXPERIMENTAL

TECHNIQUE, LINE SHAPES

G. Thummes, H. Mende

To cite this version:

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JOURNAL DE PHYSIQUE Colloque C6, supplPment au no 8, Tome 39, aotit 1978, page _C6-1130

THE RADIO-FREQUENCY S I Z E EFFECT I N COPPER WHISKER BLADES

:

EXPERIMENTAL TECHNIQUE, L I N E SHAPES

G. Thummes and H.H. Mende

I n s t i t u t fZlr Angewandte Physik d e r U n i v e r s i t i f t RoxeZer S t r . 70/72, 0-4400 Milnster, G e m n y

Rlsum6.- L'effet de taille en ra$iofrSquence, a ltl mesurl sur des filaments lamellaires de cuivre orientss en <110> avec un champ B parallsle 2 <001>. La technique explrimentale se distingue des te- chniques habituelles. On a calcull num6riquement les formes des raies et on a trouvl une bonne con- cordance pour la forme, la largeur et l'amplitude des rales. De plus, nous avons calculg l'amplitude en fonction du libre parcours des llectrons.

Abstract.- The parallel field RFSE

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1<001>) has been measured in <110> copper whisker blades. The experimental technique differs from the conventional methods. For comparison the line shapes have been calculated by a numerical. method. Good agreement is found as to the shape, width and amplitude of the lines. In addition the mean free path dependence of the amplitude has been calculated.

INTRODUCTION.- In the past decade the radio-frequen-

cy size effect (RFSE)/I/ has been often used for PHASE LOCK

measuring Fermi Surface (FS) caliper dimensions and POWERSPLI~I L O C K - I N

electron mean free paths (m.f.p.). Concerning the _A

_-

line shape of the effect, however, there only exist

7 '

few experimental and theoretical works. Here we pre- sent RFSE measurements in the parallel magnetic field geometry performed on copper whiskers with blade-li-

ke shapes/2/ together with line shape calculations. CURRENT

EXPERIMENTAL PROCEDURE.- From the single-crystalline copper whiskers grown by the reduction of Cu 5/31 we selected whisker-blades with <110> growth axes and broad sides corresponding to

11

10) planes. The electron mean free path 1 was enlarged by oxygen annealing/3/. Due to the small dimensions (thick- ness d = 30 to 75 pm, width = 200 to 300 pm, length < 1 cm) and the geometry of the whiskers we use a measuring technique differing from the conventional methods/l/.

A

block diagram of the experimental set up is shown in figure 1. The impedance Z F R+iX of

the whi'skers is measured using a four wire technique in which the current and potential leads are balanced transmission lines. The voltage drop across the whisker is phase-sensitively detected using a rf heterodyne lock-in system similar to that descri- bed by Miller and Pierce/4/. The main voltage drop produced by the ordinary inductance of the whisker, is compensated using two small coils with variable mutual inductance, thus increasing the effective dynamic range of the lock-in amplifier. Measurements are carried out in the frequency range 0.5 to 8 MUz. The obtained signals are proportional to the change

F 1: Block diagram of the experimental set-up (Bfi<OOI>, rf currents

1

)<l 10,).

of the whisker resistance AR and reactance AX with magnetic field and hence to the change in resistive

(Ab,) and reactive (Abx) skin depth. Additional dc measurements can be made in the same experiment, e. g., the effect of magnetoresistance anisotropy (mi-

+-

nimum for

B I

1<001>) was used to aligne the samples in the magnetic field.

LINE-SHAPE CALCULATIONS.- In our experimental geo-

i

metry with

B I

1

<001> in a [ 100) plane the electron orbits contributing to the RFSE signal are situated on a nearly cylindrical part of the FS' between the necks. For comparison we calculated the line shapes for a cylindrical FS and bilateral symmetric mode corresponding to the experimental situation. The computations are made by use of the self-consistent numerical method given by Juras/5/. The integrodif- ferential equation for the electric field inside the sample at a finite number of points (41 or 51) is

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solved. The surface impedance z =pow(di-idx) is then calculated from the field distribution near the surface. The height of the Fermi cylinder is adjusted with the correct number of conduction electrons in Cu, i.e. n = 8.55 X 1 0 ~ ~ c m - ~ . This gives calculated line amplitudes which are too high, sin- ce the relative number of electrons taking part in the resonance is only about n/no

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0.5. In the calculations we assumed diffuse surface scattering of the conduction electrons which is reasonable in the case of oxygen annealed Cu-whiskersl31. For the parameters 1, d and f in the calculations we used the actual values from the experiment.

RESULTS.- The measurements and calculations were performed for samples with different thicknesses down to d = 30 um. Figure 2 shows examples of measured RFSE-Lines.

-

A R *

. ... A X s

Fig. 2 : RFSE lines measured on a <110> whisker blade at 4.2 K f = 2.01 MHz, d = 73.1 pm, 1 = 85 pm

(from dc resistance ratio).

The lines exhibit the characteristic relation X ~R/~B/I/. The measured frequency dependence of the fundamental line width is proportional to f- J3 121, as one expects under the condition of the anomalous skin effect. From this the characteristic frequency independent field balue B = e.p<I10,d/2 can be deduced (e : elementary charge, p : electron momentum). An explicit comparison with the calcula-

ted curves for the fundamental resonance is given in figure 3. The amplitude of the experimental line is magnified by a factor of two which is consistent with the estimated relative number of effective elec-

trons. The B value, obtained from a fit of the experimental and theoretical line width in the frequen-

Fig. 3 : Calculated and experimental lines, parameters see fig. 2, B = 0.238 T.

cy range 1 to 6 MHz, is about 3 % higher than the value determined from the f-lP-dependence, in accor- dance with the calculated frequency dependence of the extremal field values (figure 2). The total line widths agree within 5 %. A value of

p<lIO~pF = 0.96 (+ 3 %) is obtained agreeing well with other measurements e.g./6/ (pF : Fermi momentum). For zero magnetic field the calculated values

6 = 1.276 pm, 6x = 2.06 um = 1.62)conform with those calculated from the anomalous skin effect

theory/7/. The line BB-B. corresponds to a change in trajectory diameter of 7.9. (2 6r) according with/5/. Similar good agreement in the line shapes as in figure 3 was found for other samples. The same change of the line amplitude ratio Ar/Ax with sample thick- ness was found in the experimental and theoretical

lines varying from 1.2 with d = 73.1 pm to 0.9 with d = 30 pm. As to the second resonance, only observed in thick samples, there are differences between calculations and experiment ; the behaviour with respect to amplitude and width of the experimental curves is between that for an ideal cylindrical and a spheri- cal FS/5/. In addition we calculated the m.f.p. dependence of the amplitude of the fundamental line and found in the range 0.5 < I/d 2 (&Id 0.035) a dependence proportional to exp(-c.d/l) with c =

1.35 differing from the expected c = +/2. The dif- ference can be explained in terms of the finite skin depth.

To our knowledge the only existing quantita- tive comparison between theoretical and experimental line shapes is that Wagner and Cochran/8/ made in their RFSE experiments on potassium in the transmission geometry. From their calculations they also

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References

/l/ Gantmakher,V.F., Prog. Low Temp. Phys.

5

(1967) 181 /2/ Thummes,G. and Mende,H.H., Phys. Stat. Sol. (b)

75 (1976) K 61

-

131 Thummes,G. and Mende,H.H., Phys. Stat. Sol. (a)

26 (1974) 243

-

/4/ Miller,J.R. and Pierce,J.M., Rev. Sci. Instr.

43 (1972) 1721

/5/ Juras,G.E., Phys. Rev.

187

(1969) 784

161 Gasparov,V.A., Harutunian,M.H., Phys. Stat. Sol. (b)

74 (1976) K 107

-

/7/ Dingle,R.B., Physica,

2

(1953) 311

/8/ Wagner,D.K. and Cochran,R., J. Low Temp. Phys.