Solid State Communications, Vol. 33, pp. 523—526. Pergamon Press Ltd. 1980. Printed in Great Britain. ULTRASONIC ATTENUATION IN SUPERCONDUCTING Nb20Zr~ Neil Thomas,* W. Arnoldt and G. Weiss Max-Planck-Institut für Festkorperforschung, Heisenbergstrasse 1, D-7000 Stuttgart 80, FRG and H.v. Lohneysen 2. Physikalisches Institut der RWTH Aachen, Templergraben 55, D-5 100 Aachen, FRG (Received 5 October 1979 by M. Cardona) We have measured the low.temperature ultrasonic attenuation in quenched polycrystalline Nb20Zr~at 30 and 90 MHz. The quenching process creates tunnelling states due to metastable regions of the athermal wphase. Our results can be explained on the basis of the interaction between electrons and tunnelling states. SOME TIME AGO, Lou  showed that quenched polycrystallinity. The low-temperature ultrasonic attencrystals of Nb20Zr~exhibit “glassy” properties attriuation at 30 and 90 MHz was measured using a Matec butable to atomic tunnelling states. More recently, a RF pulse generator and attenuation recorder. Some good deal of attention has been given to tunnelling mesurements were also made with a 50 kG magnetic states in amorphous metals owing to the importance of field roughly parallel to the sound propagation direction. their interaction with conduction electrons (see Black Goasdoue et a!.  have found a large internal  and references therein). In particular, Black and friction peak at 110 K for a quenched crystal of Nb20 Fulde  have considered the influence of superconZr~.We must emphasize that this peak was not present ductivity on the ultrasonic attenuation due to tunnelling in our sample, owing to too slow a quenching rate, states in an amorphous metal and they predicted that which produces larger w domains and fewer tunnelling the attenuation should decrease very rapidly just below states, as Lou  showed by annealing his quenched Ti,. However, Weiss et al. [41found that the attenuation sample. l’his means that our sample may contain conin superconducting amorphous Pd~Zr70did not begin siderably fewer tunnelling states than the properly to decrease significantly until well below T~.Since quenched single crystal studied by Lou. Although the Nb20Zr~is superconducting below 8.6 K, contains 110 K peak was absent, we saw a very shallow peak at tunnelling states and is available in bulk samples (in about 15 K, which may correspond to the lowcontrast to the splat-cooled discs of Pd30Zr~),it is an temperature peak seen by Nelson eta!. . excellent material in which to study the influence of Our results for the attenuation of 30 MHz longisuperconductivity on the tunnelling states. tudinal waves are shown in Fig. 1(a). A residual attenA polycrystalline sample of Nb20Zr~was first uation of 0.64 dB cm’ (mainly attributable to diffracquenched from within the b.c.c. 13-field of the Nb~Zr1_~ tion losses) was estimated by extrapolating the data to phase diagram  a procedure which produces lowabsolute zero and has been subtracted from the results. temperature tunnelling states due to very small metastEvidently there is no sudden decrease in the attenuation able regions of the athermal c~,-phase. Etching the just below T~.Rather the attenuation both above and sample showed crystal grains with an average diameter below 7’, is slowly decreasing with temperature and of about 200 pm. The end faces were polished flat and even shows a plateau between 3 and 4.5 K. Only below parallel, and a 30 MHz X-cut quartz transducer was 3 K does the attenuation fall off rapidly. These results bonded to one face with Nonaq stopcock grease. Good are similar to those obtained by Weiss eta!. [41in echoes were obtained at 30 MHz despite the amorphous Pd30Zr70 at 740 MHz, although at the lowest temperatures their data decreased more rapidly. Our results for the magnetic-field dependence shown in * Present address: Physics Department, Birmingham Fig. 1(a) are also simil in the region of the plateau up University, Birmmgham B15 2TT England. until just below T~the attenuation shows a small reduct Present address: Brown Boveri Research Centre, tion in a 50 kG field and there is no field dependence CH-5405 Baden-Dlittwil, Switzerland. above T~, At the lowest temperatures, however, ,
ULTRASONIC ATTENUATION IN SUPERCONDUCTING Nb~Zr~ 0
Nb20 Zr~ 0.3-
Some part of the ultrasonic attenuation in metals at low temperatures is always attributable to the direct
electron—phonon interaction  We think that this is not the dominant attenuation process in our sample for the following reasons:
30MHz longitudinaL •
(i) the electron mean free path is very short, per-
H 50 kG
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Fig. 1. Ultrasonic attenuation as a function of temperature. The attenuation does not decrease at T~but only below 3 K for 30 MHz longitudinal waves and below 3.5 K at 90 MHz. The residual loss has been subtracted from the measured attenuation as described in the text. there is a strong enhancement of the attenuation in the magnetic field. We think H~2for our sample is about 100 kG, so that except near T~the sample at 50 kG i~ in the mixed state. Near T~we found that the attenuation as a function of field behaved very similarly to that in PdZr, i.e. a linear rise saturating in the region of what is presumed to be H~2.The magnetization of our sample was slightly irreversible: there was no change in the attenuation until the applied field was about 5 kG, and we found that a small offset attributable to flux trapping remained after reducing the applied field to zero. offset disappeared after warming aboveThis T~.Measurements of the specific heatthe andsample susceptibility showed that there was no second superconducting transition between 2 K and T~. Figure 1(b) shows our attenuation results at 90 MHz. Once again the residual attenuation (5.6 dB cm’) was estimated by extrapolation and subtracted from the results. The large residual attenuation is probably attributable to scattering by grain boundaries, which is much stronger at higher frequencies. The 90 MHz results are substantially similar to those obtained at 30 MHz, except that the plateau is less pronounced and the attenuation falls off at a slightly higher temperature,1about K. at 303.5 MHz The 0.95 heightdBofcm~at the plateau (0.265 dB roughly cm and 90 MHz) scales with the ultrasonic frequency, allowing for the uncertainty in detennining the background losses.
haps of order of 10 A, which reduces the absorption to be expected from the electron—phonon interaction to a value much smaller than our observations; (ii) our results vary linearly rather than quadratically with frequency; (iii) we see no drop in the attenuation at T~ (iv) our magnetic-field dependence is very different from that produced by the electron—phonon interaction. Clearly the attenuation is affected strongly by the magnetic field at the lowestsotemperatures hardly all at higher temperatures, that electronsbut may play at some role other than in the direct electron—phonon process. Given the work of Lou , it seems reasonable to suggest that relaxation absorption by systems in doublewell potentials may be important, as in amorphous insulators . Such systems can produce two types of relaxation asborption. At low temperatures relaxation arises from the tunnelling states, or two-level systems, whose relaxation time T1 is expected to be strongly influenced by the electrons  and hence by magnetic fields in the superconducting state. At higher temperatures relaxation occurs by thermal activation over a potential barrier, and the relaxation time for this should not be influenced by electrons. In both cases, a broad distribution of relaxation times arises. A relaxation peak or shoulder at an ultrasonic frequency w/21r generally occurs when wT1 1 for the dominant systems and produces a linear frequency dependence of the attenuation, in agreement with our observations. The relaxational absorption a due to tunnelling states of energy E can be by  dE, 2 described wTj sech2 (1) a = 4kTp1 3 nD 1 + w2T~ 0v where Po is the density, v is the sound velocity, n is the tunnelling density of states, and D is a deformation potential for the tunnelling states. This is, in fact, a simplified formula, which ignores the distribution of relaxation times at a given energy but which is adequate for our purposes. A similar formula describes the relaxational absorption due to thermally activated processes  In the region where wT 1 1 for the dominant relaxing systems, we can write (2) a “~ 4po~ Applying this to the plateau at 30 MHz and taking —~
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ULTRASONIC ATTENUATION IN SUPERCONDUCTING Nb20Zr~ 525 3 and v = 4.18 x iO~cm sec’ ,we below T Po = 6.76 g2cm 7 x 108 erg cm3 This is somewhat shoulder0atasabout it is, 3forK.30WeMHz therefore we seepropose a plateauthat with thea estimate nDin SiO lower than 2  or PdSiCu  but comparplateau arises owing to the increase in T1 below T0. For able to that observed in PdZr  and some other low-energy tunnelling states below T~,the relaxation metallic glasses. Although some of the absorption rate due to inelastic scattering of quasiparticles as at the plateau is probably attributable to the tail of the derived by Black and Fulde  is thermally activated peak at 15 K (rather than tunnelling kT 1 = ~ v alone), our estimate of the approximate magnitude of Tç 1)~~~/kT + 1’ (3) seems quite reasonable. e Equation (1) is also useful in understanding the where pV1 is an effective coupling constant between the magnetic-field dependence of the attenuation. In the electrons and tunnelling states and 2i~is the superconspirit of Black and Fulde’s theory  we assume that ducting energy gap. From our data in Figs. 1(a) and (b) T1 in the normal state is much shorter than T1 in the we take wT1 = 1 for 30 MHz at 3 K and for 90 MHz at superconductor. We have supposed that wT1 I for the 3.5 K. Using equation (3) with the zero-temperature gap dominant systems at the plateau, so that decreasing Ti 2~= 3.52kT0, we find the ratio of the relaxation rates with the magnetic field should decrease the attenuation, at 3.5 and 3 K is about 2.4. This estimate ignores the as observed. At lower temperatures we enter the region phonon contribution to T~in the superconducting where wT~> 1 in the superconductor. In this case, state, which must be present here as in insulating glasses. decreasing T1 with the magnetic field should increase However, it appears that in our temperature range this the relaxational absorption. (We do not expect wT~ 1 is less important than the quasiparticle relaxation. If in the normal state until extremely low temperatures.) phonons were the dominant relaxation process for the The linear magnetic-field dependence of the attenuation tunnelling states, we should expect the shoulder in the can be explained on the basis of localized tunnelling attenuation to vary more rapidly with frequency, movsystems in the mixed state of a superconductor: a tuning from 3 K at 30 MHz to about 4.3 K at 90 MHz. The nelling state within about a coherence length of the rather weak frequency dependence of the shoulder temcentre of a flux line will behave as though it were in the perature suggests that T1 varies rapidly with temperanormal metal, so that the attenuation should be proture, in line with equation (3). From this equation and portional to the number of flux-lines, which varies as taking wT1 = I ~or 30 MHz at 3 K, we estimate that the the applied field up to H02 in a dirty material, coupling constant p V1 0.15, compared to 0.2 assumed The above argument is rather general, assuming by Black and Fulde. only that T1 is shorter in the normal state. The crux of In summary, we have presented our attenuation the matter is: what is determining T1 below T0 and why results in superconducting polycrystalline Nb2oZrse do our results (and those for Pd3oZr~)bear little resem- which we believe to contain tunnelling states. Our blance to the attenuation predicted by Black and Fulde? results are remarkably similar to those obtained by In the first place we must say that their numerical calWeiss et al.  in amorphous Pd30Zr~.Therefore, our culation of the attenuation was for a frequency of 500 Nb20Zr~results not only support their data but also To [MHz] which is much higher than that used here: suggest that these results may be representative of all dirty for NbwZr~the frequency would be 4.3 GHz, whilst superconductors containing tunnelling states. We feel for Pd30Zr~it would be 1.3 GHz, somewhat higher that there is no genuine conflict with the theory of than the 740 MHz used by Weiss eta!.  Particularly Black and Fulde  although the interpretation of the for our Nb20Zr80 we have to consider the Black and data is far from straightforward. We suggest that in the Fulde model at low frequencies. Assuming that the low-frequency regime studied so far wT1 ~ I for the electrons couple to the tunnelling states with approxitunnelling states at Ti,, so that these only contribute to mately the same strength as in PdSiCu, we expect that the relaxational absorption well below T0 when wT1 above T~we are in the regime where wT1 1. The 1. This absorption is superimposed on the lowrapidly relaxing tunnelling states will accordingly contemperature tail of a thermally activated attenuation tribute very little to the relaxation absorption, which is peak which is not influenced by the electrons. The then mainly due to the tail of the thermally activated superposition of both absorption mechanisms leads to peak. Below T0, however, the electron relaxation the plateau which we see below ~‘0, The magnetic-field “freezes out”. The relaxation time T1 therefore becomes dependence of the attenuation in the mixed state can longer, and at some temperature below T0 we must be understood on the basis of T1 being shorter in the reach the condition wT~ I. But for the tail of the normal state. The change in attenuation is caused by thermally activated peak (which is unaffected by the those tunnelling states within about a coherence length electrons), we would expect to see a relaxation peak of the centre of a flux line, which explains both the .
ULTRASONIC ATTENUATION IN SUPERCONDUCTING Nb2oZrse
linear field dependence and the saturation of the absorption at H02. Acknowledgements We are grateful to K. Dransfeld and S. Hunklinger for their interest and encouragement during this work. We also thank R. Schweizer for the susceptibility measurements and K. Guckelsberger and E. Gmelin for measuring the specific heat. We are particularly grateful to S. Roth for the use of his superconducting magnet. One of us (N.T.) is grateful to the Royal Society for a research fellowship under the European Exchange Programme and wishes to thank the technical and academic staff of MPI—FKF for their assistance and hospitality. —
REFERENCES 1. 2.
L.F. Lou,Solid State Commun. 19, 335 (1976). J.L. Black, Proc. 3rd mt. Conf on Phonon Scat-
3. 4. 5. 6. 7. 8. 9. 10. 11.
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tering in Condensed Matter. Plenum Press (1979) (to be published). J.L. Black & P. Fulde,Phys. Rev. Lett. 43,453 (1979). G. Weiss, W. Arnold, K. Dmansfeld & H.J. Güntherodt, Solid State Commun. (1979) (to be published). S.L. Sass, J. Less-Common Metals 28, 157 (1972). C. Goasdoue, P.S. Ho & S.L. Sass, Acta Met. 20, 725 (1972). C.W. Nelson, D.G. Gibbons & R.F. Hehemann, J. App!. Phys. 37, 4677 (1966). Above T0 we found ~a a(H = 50 kG) a(H = 0) = o.ooi ±0.00 1 dB cnf’. W.P. Mason,Phys. Acoust. 4A, 299 (1966). 5. Hunklinger & W. Arnold, Phys. Acoust. 12, 155 (1976). B. Golding, i.E. Graebner, A.B. Kane & J.L. Black,Phys. Rev. Lett. 41, 1487 (1978). —