Magnetosonic waves in antimony

Solid State Communications, Vol. 25, pp. 105—108, 1978.

Pergamon Press.

Printed in Great Britain

MAGNETOSONIC WAVES IN ANTIMONY W. Braune and R. Kuhi Sektion Physik der Humboldt-Universität zu Berlin, Bereich Tieftemperatur-Festkorperphysik, DDR-104 Berlin, E. Germany. J. Lebech and K. Saermark Physics Laboratory I, The Technical University of Denmark, DK-2800 Lyngby, Denmark (Received 18 July 1977 byL.Hedin) A report is given on the observation ofmagnetosonic waves propagating through antimony in the Voigt-configuration. The waves were observed from a cut-off field and up to field of the order of 60 kG. The usual Alfvén-wave behavior as well as the influence of non-local effects were observed, the latter at the lower magnetic field values. The non-local effects are analysed in terms of the expressions given by Yokota and by Guthmann et al. It turns out, however, that these expressions arethe notlower able dispersion observed at to accountfield for values. the non-linear in B1 magnetic —

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EXPERIMENTAL investigations on the propagation of magnetoplasma waves through semi-metals have been performed nearly exclusively on samples of bismuth and graphite. Due to a small carrier concentration in these substances the wave velocity in comparatively low magnetic fields is larger than the Fermi-velocity of the charge carriers and collisionless damping wifi in general be small or negligible. For antimony propagation of Alfvén-waves was noticed by Williams [1] in a conference abstract and was recently observed by Suematsu and Tanuma [2] in more detail for the Faradayconfiguration (k II B). The magnetic field values used in the latter experiments were of the order of 60—150 kG and the experimental results were used to deduce values for the mass-density function defined as in [3] corresponding to the three crystallographic axes. It is the purpose of the present paper to report on the first observations of propagation of magnetosonic waves through antimony in the Voigt-configuration (k I B). The antimony sample used had a thickness d = 0.8 mm, and was grown in a demountable quartz mould [4]. This method makes it possible to produce samples which, without further surface preparation, can be used in experiments aiming at a study of the propagation characteristics of magnetoplasma waves. In the magnetic field range, B <60kG, used in the present experiments, the wavelength of magnetoplasma waves in antimony is small, and rather severe requirements as to the planeness and plane-parallelity of the sample surfaces must be fulfilled. The results obtained indicate that this was the case here. The experimental set-up consists of a —

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superheterodyne-spectrometer containing two tuneable cavity-resonators arranged in a transmission configuration with the sample functioning as a common end-wall between the two cavities. The microwave frequency applied was 45.4 GHz and the experiments were performed both at 4.2 K and at 1.9K. A conventional magnetic field modulation technique was used in order to record the magnetic field derivative of the transmitted signal, however, also the directly transmitted signal could be recorded simultaneously. Finally, although the reflected as well as the transmitted signal was observed, we shall here only discuss the latter type of signal. In Fig. I we show a typical recording of the signal transmitted through the sample for k IC 3, B IC2 and Eli C1, where as usual C1, C2 and C3 indicates the bisectric, binary and trigonal axis, respectively. In Fig. 1 one notes a structure, at the field value 16.5 kG, which possibly may be interpreted as a cut-off edge, above which the wave propagation takes place; in the present experiments, however, this edge was only pronounced for temperatures well below 4.2 K. The field value at which the edge occurs can apparently not be correlated with a cyclotron- or a hybrid-resonance for this fielddirection. In Fig. 2 we show the dispersion of the waves observed by means of a plot of the order of the oscillations, n, vs the reciprocal magnetic field valueB’. For large magnetic field values (to around n ~ 33) one recognizes the classical dispersion relation for the fast magnetosonic wave often denoted as the Alfvén-wave in the Voigt-configuration whereas for smaller values of the magnetic field considerable deviations occur. In the former region it was possible to determine the

105

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MAGNETOSONIC WAVES IN ANTIMONY

Vol. 25, No.2 -~-

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10

H

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20

~1

-~

30

- -~

40

50

B/KG

Fig. 1. Magnetic field-derivative of the signal transmitted through antimony in the Voigt-configuration. BIIC2, EIIC1, kllC3.f=45.37GHz,d = 0.8 mm and T= 1.9K.

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Fig. 2. Reciprocal magnetic field values for the extrema 1/KG1 of Fig. 1 vs the order n of the oscillations. The arrow 50 40 30the 20 10 0 waves. .02 B .04,//~o°° 0000 0 .06 indicates the cut-off field for

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absolute order of the oscillations by means of the sample thickness, On the basis of the linear part of the dispersion curve shownmass-density in Fig. 2, onefunction. may deduce a value appropriate In doing so, for onethe may for the magnetic field values considered here neglect the influence of quantum effects on the dispersion of the waves observable as a modulation of the amplitudes of the oscillations and also the influence of the lattice dielectric constant, which according to [5] amounts to CL = 85 ±5. One then finds for the mass-density function 8 18 ( 3 —

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.‘22

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~

x

10

m0 ~

)

where we have used the same notation as in [6] such that the C 1-axis (II E) is indicated by the subscript 22, and the C2 -axis (II B)3by[7]theone superscript For Ne = then finds1.an average Nh = 5.5 mass x 1019 effective of cm 1 0 16 mOV,22 m~. The above experimentally determined value for the average effective mass may be compared with a calculated value. Thus, using the cyclotron resonance results of Datars and Vanderkooy [8] one finds, on the basis of an ellipsoidal approximation for the Fermi-surfaces, = 0.26 m 0 which should be compared with the value determined from the dispersion of the — —

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MAGNETOSONIC WAVES IN ANTIMONY I

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Fig. 3. The logarithm of the ratio of the oscillation-amplitude to the order n plotted vs n. From the slope of the straight line one obtains wr 17 at 1.9 K and wr 14 at 4.2 K. Open circles: T= 4.2 K; crosses: T= 1.9 K. magnetoplasma waves. The rather large difference between the two values is presumably to be ascribed to a failure of the ellipsoidal approximation for the electron- and hole-part of the Fermi-surface for antimony. In this connection we note, that similar deviations between calculated and experimentally determined average effective masses were found also in the experiments performed in the Faraday-configuration [2] We remark, further, that the value of the mass density function f, which depends on the field configuration considered, in the present case and also for kIIC3 and BIIC1, EIIC2 assumes a value which is smaller than the ones found in [2] for the Faradayconfigurations considered there. This appears understandable in view of the position of the Fermi-surfaces and is in marked contrast to the situation found in bismuth. We now turn to a consideration of the behavior of the dispersion curvedeviations of Fig. 2 at lowa magnetic field values, where large from linear dependence .

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va,, of the magnetosonic wave should be given by I

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V

—

VA

\21 1/211

J

L’

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to

/n\21 1/2

‘.PhI’-’) J

where the symbols B,,, B~,and VA have the following meaning, B,, is the magnetic field corresponding to the hybrid resonance, and B~is an acoustic coupling field defined as that magnetic field value which yields equality of the magnetic and thermal pressure. Finally, for large magnetic field values, B ~‘ B~,one finds from (1) that the phase velocity of the waves approaches the fast Alfvén-wave velocity VA, which in the case considered here is given by VA

6

=

3.16x10 Bcmsec

-i

(BinkG).

For small magnetic field values, B ~B~,,the phase velocity of the waves approach the magnetic field independent phase velocity v8 of the magnetosonic wave given in first approximation 2 + ~2 \inby ~2 V v 8 = me Fe m~Fh) \5 me + m~ In the transition region B B~,finally, the situation is more complex, however, the equation (1) is thought [II] to be applicable. The present experimental results on the dispersion of the waves propagating through the antimony sample, as summarized in Fig. 2, have been examined in terms of equation (I). It turned out, however, to be impossible to find a value forB~,(~10kG) which yields a completely satisfactory fit of the experimental data to this equation. In particular, the structure observed at 16 kG appears not to be explicable in terms of equation (1). This indicates, that for antimony a more —

occur. For such field values, Yokota [9] has shown that the thermal motion of the charge carriers can modify the velocity of the Alfvén waves when they propagate perpendicular to the magnetic field direction. Experimentally this effect was first observed by Lupatkin and Nanney [10] in bismuth as an acoustic-like wave propagating in magnetic fields such that the magnetic pressure was smaller than the thermal pressure of the charge carriers. Finally, Guthmann etal. [Ii] have performed systematic experimental and theoretical studies of the magnetosonic wave. One result of the analysis given by Guthmann et al. [11] is that the magnetic field dependent phase velocity

/

,-

~1+ ~B~1B,

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detailed analysis of the influence ofnon-local effects on plasma waves in the Voigt-configuration may therefore the dispersion relation should be performed, including provide for a valuable extension of the knowledge of also the effects of a finite value of car, r being the the energy-band structure, and the possibility of examinrelaxation time. In this connection we remark, that an ing the wave propagation under the influence of spatial analysis of the magnitude of the amplitudes of the and temporal dispersion including the cut-off field oscillations leads to the estimate wr 17 and 14 for position appears to be a useful means of determining T = 1.9 and 4.2 K, respectively (see Fig. 3). This estimate the various Fermi-velocities entering in the analysis. is in good agreement with the result of a similar analysis However, this requires at least for antimony a of the cyclotron resonance signals observed for the same detailed examination of the validity of equation (1). sample at low magnetic field values. Due to the very pronounced anisotropy of the Acknowledgement One of us (W. Braune) gratefully Fermi-surface for antimony, the value of the massacknowledges the financial support extended to him density function for the different crystallographic direc- from the Scientific Exchange Program between tions differ very much. Investigations of the magnetoE. Germany and Denmark —

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REFERENCES 1.

WILLIAMS G.A., Bull. Amer. Phys. Soc. 8, 353 (1963).

2.

SUEMATSU M. & TANUMA S., J. Phys. Soc. Japan 41,496 (1976).

3. 4.

ISAACSON R.T. & WILLIAMS G.A.,Phys. Rev. 177, 738 (1969). KUHL R., Unpublished thesis, Humboldt University Berlin (1976).

5.

NANNEY C.A.,Phys. Rev. 129, 109 (1963).

6.

FAUGHNAN B.W.,J. Phys. Soc. Japan 20, 574 (1965).

7. 8.

WINDMILLER L.R., Phys. Rev. 149,472(1966). DATARS W.R. & VANDERI(OOY J., IBMJ. Res. Develop. 8,247(1964).

9.

YOKOTA I.,J. Phys. Soc. Japan 21, 1851 (1966).

10.

LUPATKIN W.L. & NANNEY C.A.,Phys. Rev. Lett. 20,212(1968).

11.

GUTHMANN C., D’HAENENS J.P. & LIBCHABER A., Phys. Rev. B4, 1538 (1971).

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