The temperature dependences of the sound velocity v and attenuation alpha of high-frequency (50–160 MHz) sound in the bulk amorphous alloy Zr41.2Ti13.8Cu12.5Ni10Be22.5 are studied at helium temperatures in the normal and superconducting states. The alloy is characterized by a relatively small constant C determining the intensity of interaction between an elastic wave and two-level systems. The density of states of the latter systems is estimated. The peculiarities in the variation of v during the superconducting transition point to the possibility of a gapless superconductivity in a narrow temperature interval near Tc

Bezuglyi, E. V.

Fil, V. D.

Gaiduk, A. L.

Johnson, W. L.

English

Caltech Authors - Main

LETTERS TO THE EDITORLow-temperature acoustic characteristics of the amorphous alloy
Zr41.2Ti13.8Cu12.5Ni10Be22.5
A. L. Gaiduk, E. V. Bezuglyi, and V. D. Fil
B. Verkin Institute for Low Temperature Physics and Engineering, National Academy of Sciences
of the Ukraine, 310164 Kharkov, Ukraine*
W. L. Johnson
California Institute of Technology, Pasadena, CA 91125, USA
~Submitted July 22, 1997!
Fiz. Nizk. Temp. 23, 1139–1143 ~October 1997!
The temperature dependences of the sound velocity v and attenuation a of high-frequency
~50–160 MHz! sound in the bulk amorphous alloy Zr41.2Ti13.8Cu12.5Ni10Be22.5 are studied at helium
temperatures in the normal and superconducting states. The alloy is characterized by a
relatively small constant C determining the intensity of interaction between an elastic wave and
two-level systems. The density of states of the latter systems is estimated. The peculiarities
in the variation of v during the superconducting transition point to the possibility of a gapless
superconductivity in a narrow temperature interval near Tc . © 1997 American Institute
of Physics. @S1063-777X~97!01810-0#
Low-temperature kinetics of amorphous materials is nel levels with the energy splitting E5AD21D025v by/1mainly determined by specific low-energy quasilocal excita-
tions, viz., two-level systems ~TLS!. The model concept of
TLS developed in Refs. 1 and 2 proved to be very fruitful for
explaining the thermal, elastic, and electromagnetic proper-
ties of insulating glasses and metglasses.3 A simplest model
of a TLS is a heavy tunneling entity in an asymmetric
double-well potential. Both the initial model of TLS and its
subsequent versions4 are universal and are not connected
with specific features associated with the microscopic origin
of double-well potentials.
The properties of a large number of various glasses have
been studied extensively. One of the most remarkable results
in this field is the conclusion concerning virtual indepen-
dence of the TLS density of states of the composition and the
method of obtaining of glass. As regards metglasses, it
should be noted, however, that all the compositions that were
investigated before had been obtained at very high cooling
rates (>103 K/s), which probably determines the above-
mentioned universality.
New classes of glass-forming systems obtained recently
are stable over a wide temperature region of supercooled
liquid state. Their high resistance to crystallization makes it
possible to obtain bulk amorphous samples at very low
quenching rates (<10 K/s). It would be interesting to esti-
mate the density of states of TLS for a typical representative
of this family of metglasses. In this connection, we studied
the acoustic properties of the amorphous alloy
Zr41.2Ti13.8Cu12.5Ni10Be22.55 at temperatures near 1 K.
Acoustic measurements are known to be one of the most
informative methods of studying TLS.3 The interaction of
TLS with sound mainly follows the two principal mecha-
nisms: resonance and relaxation. The former mechanism cor-
responds to the excitation of direct transitions between tun-
857 Low Temp. Phys. 23 (10), October 1997 1063-777X/97sound of frequency v ~D is the asymmetry of a double-well
potential and D0 the tunnel splitting of energy levels for
D50!. In the available frequency range (v!T), the reso-
nant contribution to variations of the sound velocity is deter-
mined ~to within a constant term! by the formula
S dv iv i D res5
P¯g i
2
rv i
2 ln T5Ci ln T , ~1!
where P¯ is the TLS density of states, g the deformation
potential, r the mass density, and i denotes the longitudinal
(l) or transverse (t) polarization.
The resonant contribution to the absorption coefficient
a/q ~q is the wave number! is small in the parameter v/T on
the Ci scale.
Elastic deformation of the wave also shifts the spectrum
of tunnel levels. If there exist the relaxation mechanism
which makes it possible for TLS to tune the occupancy of the
levels to a new set of E rapidly ~on the v21 scale!, the
attenuation and the velocity of sound acquire a relaxation
component:
S a iqidv i
v i
D 5 Ci2 S pln vtminD , ~2!
where tmin is the minimum relaxation time for D50 and
E;T . Expressions ~2! are valid for vtmin!1; otherwise, the
relaxation interaction can be neglected. In the general case,
the TLS relaxation in metglasses is due to interaction with
85700857-04$10.00 © 1997 American Institute of Physics
sponse x of the sample recorded for the ac field amplitude
˜electrons as well as with phonons, but the contribution of the
latter can be neglected at T,2 – 3 K, and the relaxation fre-
quency can be written in the form3
~tmin
21 !e5
p
2 h
2T , ~3!
where h is the dimensionless parameter of coupling between
electrons and the TLS.
The sample with an acoustic path length of 0.74 cm was
cut from an oval ingot.5 The values of v and a were mea-
sured in the pulsed mode by the phase-sensitive bridge
method at frequencies close to 54.3 and 162.9 MHz. The
diamagnetic response x was measured at a frequency of 22.5
Hz with the concentric arrangement of the sample, the re-
ceiving and exciting coils. Note the high homogeneity of the
alloy: 8–10 well-defined reflexes were observed at an acous-
tic wavelength ;15 mm. In order to avoid possible errors of
interference origin, main measurements were made by com-
paring phases and amplitudes of first and second or third
pulses. At 77 K, the velocities were v l5(5.1760.03)
3105, v t5(2.4960.01)3105 cm/s.
Figure 1 shows the relative change in the velocity of
transverse sound in the alloy under investigation in the nor-
mal (n) and superconducting (sc) states. In the former case,
the field H51.5 T(Hiq) was applied. It should be noted that
above Tc, such fields did not affect the behavior of v and a.
In the normal phase, the temperature variation of the velocity
is correctly approximated by a logarithmic dependence with
the coefficient Ci/2 appearing when expressions ~1! and ~2!
are added. To within the error of measurements, the values of
the constants Ci ~Cl5(0.5860.02)31025, Ct5(1.42
60.03)31025! were independent of frequency.
In the superconducting phase, freezing out of normal
excitations leads to an increase in tmin6 and to a sharp de-
crease in the relaxation contributions to the velocity and ab-
sorption of sound, which is reflected in a sharp kink on the
v(T) curve in Fig. 1 at the temperature Ton50.85 K. The
inset to Fig. 1 shows the changes in the velocity near Ton
measured simultaneously (H50) with the diamagnetic re-
FIG. 1. emperature dependences of relative changes in the velocity of trans-
verse sound in the amorphous alloy Zr41.2Ti13.8Cu12.5Ni10Be22.5 at a fre-
quency of 54.3 MHz. The inset shows a fragment of dv/v(T) dependence
and the diamagnetic response x in the superconducting transition region.
858 Low Temp. Phys. 23 (10), October 1997H.4 mOe. The sample obviously contains two phases with
the phase-transition temperatures Tc1.0.9 K and
Tc2.1.02 K. The value of Tc2 probably corresponds to a
surface phase since this transition does not affect the sound
velocity in any way. Besides, an increase in H˜ to 40 mOe
leads to virtually complete suppression of the jump in x at
Tc2 , while the peculiarity at Tc1 is preserved up to H˜51 Oe.
The non-coincidence between the resistive and magnetic su-
perconducting transitions in a glass of the same composition
observed by Gerber et al.7 is apparently also connected with
the existence of two phases.
It can be seen from Fig. 1 that the temperatures Ton and
Tc1 do not coincide. Naturally, we can assume that the
sample is not homogeneous as regards Tc1 , but the small
width of the jump in x at Tc1 renders this assumption hardly
plausible. Probably, the regime of so-called gapless super-
conductivity is realized in the temperature interval
Tc12Ton .8
An analytic expression for tmin in a superconductor can
be derived only for E!Dsc ~Dsc is the superconducting
gap!.6 Since the main contribution to the measured param-
eters comes from TLS with E<T , the above limitation boils
down to T!Dsc . In this case, we can write
~tmin
21 !e sc5
p
2 h
2Te2Dsc /T. ~4!
Since the velocity in the superconducting phase contin-
ues to grow up to T;0.4 K, the condition vtmin,1 obvi-
ously remains valid, and expression ~2! for the relaxation
contribution to dv/v still holds. Assuming that the resonant
contribution in the normal and superconducting phases is the
same, we have the following expression for the difference in
the velocities of sound in the sc- and n-phases from ~1-4!:
Dv
v
5S dvv D
sc
2S dvv D
n
;2
Dsc
T .
In actual practice, the strong inequality between T and Dsc is
not satisfied in our experiments. Nevertheless, we can expect
that the experimental dependences Dv/v(T21) become lin-
ear in a certain temperature interval T,Tc , where
Dsc(T)'Dsc(0).
A dependence of this type is shown in Fig. 2a. For com-
parison with theoretical dependences, the temperature scale
in Fig. 2 is given in reduced form, and the superconducting
transition temperature is assumed to be equal to Ton . The
velocity scale is also normalized to Ct/2. It was found that
the dependence Dv/v(T21) is close to linear in the entire
temperature range below Ton , but its slope is twice the BCS
value of Dsc(0). In all probability, such a behavior of
Dv/v(T21) is the result of joint operation of several
temperature-dependent mechanisms. For comparison, Fig. 2a
presents the results of complete calculation of Dv/v for the
BCS model ~h50.85, see below! on the basis of equations
from Ref. 6. The agreement between the theoretical and ex-
perimental dependences is obviously unsatisfactory over vir-
tually the entire temperature range, although the scale of
variation is reproduced correctly on the whole. The agree-
ment can probably be improved in the vicinity of Tc by
858Gaiduk et al.
the electron relaxation time becomes equal to the phonontaking into account magnetic scattering effects which elimi-
nate the divergence in the derivatives of the order parameter
and energy gap at Tc ,8 but a rigorous theory of interaction
between sound and TLS in the presence of magnetic impu-
rities has not yet been constructed. The assumption concern-
ing the equality of the resonant contributions in the sc- and
n-phases might be incorrect due to a considerable narrowing
of the tunnel levels upon a decrease in (tmin21 )e sc ,9 which can
affect the behavior of Dv/v at lower temperatures, e.g., can
change the normalization of the ordinate axis in Fig. 2a.
Figure 2b shows the temperature dependences of varia-
tions in sound absorption, which is also normalized by pCt/2
in accordance with ~2!. Further, the normalized absorption in
the normal phase was assumed to be equal to unity in con-
formity with ~2! since the real value of a cannot be deter-
mined in the temperature interval under investigation ~this
can be done only from the difference in absorption in the
normal and deep superconducting states!. The theoretical de-
pendences in the BCS model correctly reproduce the differ-
ence in the behavior of absorption at two frequencies, al-
though the steepness of the variation of a/q for T/Ton;0.45
is higher than the theoretical value. It should be recalled,
however, that the steepness of experimental dependences in
Fig. 2b is also determined by the choice of the normalization
constant Ct . The estimated value of h'0.85 naturally de-
pends on the value of Dsc : the value of h decreases with
Dsc . In this case, theoretical dependences become still more
gently sloping, and the estimated value of h can not be re-
duced by a factor of more than two for the same normaliza-
tion constant Ct .
Let us now estimate the TLS density of states. This can
be done if we know the crossover temperature Tcr at which
FIG. 2. Difference of the relative changes in the transverse sound velocity in
the superconducting and normal states as a function of reduced temperature.
Smooth curves correspond to the results of calculation (h50.85) ~a!; ab-
sorption in the superconducting phase: h50.89 ~curve 1! and 0.81 ~curve 2!
~b!. The temperature is given on reciprocal scale.
859 Low Temp. Phys. 23 (10), October 1997one: (te)min5(tph)min . Usually it is assumed that the cross-
over occurs at a temperature at which the velocity of sound
passes through its maximum.3 In our case, Tcr'3.3 K. The
phonon relaxation rate is mainly determined by transverse
phonons:3
~tmin
21 !ph5
8g t
2
prv t
5 T3. ~5!
Combining ~1!, ~3!, and ~5!, we obtain
P¯58.4~Tcr /h!2Ct1032 erg21cm3 ~6!
~in this equation, Tcr is in degrees!. Substituting Tcr53.3 K,
h50.85, Ct51.421025, we obtain P¯51.81029
erg21cm23. This value is two–three orders of magnitude
smaller than values known from literature.4
In formula ~6!, we specially singled out the quantities
whose values are not known exactly. In actual practice, a
noticeable contribution ~proportional to T2! to the velocity of
sound is observed in the sample under investigation ~we are
planning to publish the corresponding results later!; the com-
bination of this contribution with the logarithmic dependence
can also lead to the formation of a velocity peak. For this
reason, the value of Tcr can be slightly higher. The fitting
value of h ~see Fig. 2b! depends on the energy gap for which
the BCS value was used. If our assumption concerning the
effect of magnetic scattering is confirmed, the energy gap
will be smaller,8 which can reduce the estimated value of h2
by a factor of several times. The exact value of Ct is not
known to us either; it can be obtained from the logarithmic
decrease in the velocity of sound in the deep superconduct-
ing state. On account all these remarks, the estimated value
of P¯ in the alloy under investigation can be an order of
magnitude higher than the value given above, but still con-
siderably smaller than the known values from the literature
sources.
In summary, the values of the TLS density of states in
the amorphous alloy Zr41.2Ti13.8Cu12.5Ni10Be22.5 estimated on
the basis of acoustic measurements are much smaller than for
other metglasses. This is probably due to the very low rate of
cooling of the melt (;10 K/s). The observed peculiarities of
superconducting ordering indicate, on one hand, the two-
phase structure of the amorphous alloy under investigation,
and on the other hand, the possible realization of a gapless
superconductivity in this material in a narrow temperature
range.
The authors are grateful to V. Z. Bengus, E. D. Tabach-
nikova, and A. S. Bakai for fruitful discussions and to E. A.
Masalitin for his assistance in preparing the apparatus.
*E-mail: fil@ilt.kharkov.ua
1 P. W. Anderson, B. I. Halperin, and C. M. Varma, Philos. Mag. 25, 1
~1972!.
2 W. A. Phillips, J. Low Temp. Phys. 7, 351 ~1972!.
3 S. Hunklinger and A. K. Raychaudhuri, Progr. Low Temp. Phys. 9, 266
~1986!.
4 D. A. Parshin, Phys. Rev. B 49, 9400 ~1994!.
859Gaiduk et al.
5 A. Pecker and W. L. Johnson, Appl. Phys. Lett. 63, 2342 ~1993!.
6 J. L. Black and P. Fulde, Phys. Rev. Lett. 43, 453 ~1979!.
@ Sov. Phys. JETP 12, 1243 ~1960!#.
9 N. Thomas, Philos. Mag. B 48, 297 ~1983!.
87 A. Gerber, A. Milner, A. Voronel et al., Physica C 275, 333 ~1997!.
8 A. A. Abrikosov and L. P. Gor’kov, Zh. E´ ksp. Teor. Fiz. 39, 1781 ~1960!60 Low Temp. Phys. 23 (10), October 1997Translated by R. S. Wadhwa860Gaiduk et al.


Low-temperature acoustic characteristics of the amorphous alloy Zr41.2Ti13.8Cu12.5Ni10Be22.5

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Low-temperature acoustic characteristics of the amorphous alloy Zr41.2Ti13.8Cu12.5Ni10Be22.5

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