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Decrease of soft vibrational modes with increasing density of two-level states in the metallic glass Zr-Rh.

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Ann. Physik 5 (1996) 553-558
der Physik
0 Johann Ambrosius Barth 1996
Decrease of soft vibrational modes with increasing
density of two-level states in the metallic glass Zr-Rh
B. Junge* and W. Gey
Institut fiir Technische Physik der Technischen Universitiit Braunschweig, D-38023 Braunschweig.
Received 9 February 1996, revision 3 July 1996, accepted 5 July 1996
Dedicated to Peter Fulde on the occasion of his 60th birthday
Abstract. Experiments are reported on the specific heat below 1 K of 13 melt quenched ZrRh alloys in the superconducting state (TpI.2 K) where electronic degrees of freedom are frozen out.
Large differences of the quasilinear (two-level system, TLS) contributions given by ym occur with
no correlation to the rhodium concentration. The coefficient o of the cubic, phonon-type term is
found to decrease with the density of TLS for this metallic glass. This is counter-intuitive to the
Soft-potential model and differs from the relation found for insulating glasses.
Keywords: Disordered systems; Metals; Heat capacity; Phonons.
The anomalous low-temperature properties of insulating glasses below 1 K are phenomenologically well described by additional low lying excitations (two-level tunneling systems, TLS) with nearly constant densities of states. Above 1 K further anomalies are
known in the literature. As far as the specific heat C, is concerned there is an additional
quasi-Debye p-term, followed by a peak in C$r3 due to a peak in the vibrational density of states (boson peak) [l]. It has been shown that these glassy anomalies can consistently be described by a phenomenological soft-potential model [2].
With respect to glassy metals it is generally accepted that the low-energy excitations are also present, although information is less consistent than for dielectrics. The
heat capacity below 0.5 K obeys a form
C p ( T )= yTLs - T v QT’
the exponent v being close to unity.
It was shown recently by measurements in the normal state in the temperature regime between 1.5 and 12 K that for glassy Zr&h,PdzS.,-alloys a boson peak also
exists [3]. It was found that this peak is strongest for alloys where the mass density
assumes its minimum (x z 13). The purpose of the present paper is to report on re-
* Present address:
FOrschung-Mel3technik. Volkswagen AG, D-38436 Wolfsburg
AM. Physik 5 (1996)
sults of the low temperature specific heat C(T) (170 mKcTcl000 mK) for a-Zrlw,
Rh, alloys in the superconducting state. This series was chosen for two reasons: (i)
Investigations of both, mass density p and average nearest neighbour distance from
x-ray data had revealed large scatter for the Zr-Rh system (Ae/e=f2%)which was
absent in the a-Zr-Pd system within experimental resolution (M.05%) [3]. Such nonsystematic fluctuation of the density reveals a strong sensitivity of the packing density of amorphous Zr-Rh alloys on preparation conditions, thus favouring variations
of the density of TLS. (ii) For all compositions of a-ZrRh superconductivity occurs
at temperatures T,>4.2 K, thus contributions of electronic excitations can safely be
ignored below 0.7 K. Data on y m for an amorphous system with fluctuating mass
density may therefore be isolated. We find - at variance with common knowledge for
dielectric glasses - that at low temperatures the phononic term a does not scale with
The amorphous ZrRh samples were prepared by repeated arc melting of pure Zr
(99.99% Materials Research Corp., France) and Rh (99.9% Ventron Corp., U.S.A.)
under a high-purity argon atmosphere of 0.2 bar on a polished copper plate (anvil)
and were finally quenched by passing a polished copper hammer at a speed of 40 to
50 ms-' straight through the arc onto the molten ingot. Splats of 50 to 80 pm thickness and 1.5 cm2 area were thus produced. No crystalline inclusions were detected
by inspection with an electron microscope and by X-ray analysis. The low-temperature specific heat of the small samples (mass about 50 mg) was measured with a
transient heat-pulse technique. The length of the heating pulse was 10 ms. Characteristic decay times were between 0.8 and 1.5 s. Time dependent heat capacities on our
time scale were not observed for the metglasses, as expected. The thermal bath was
realized by a 100 mK dilution refrigerator. The quality of the calorimeter was tested
by heat capacity measurements of cadmium, gold, and two copper probes in the temperature interval 170 mK<T<1500 mK. Comparison of this calibration with published
data yielded an absolute accuracy of the calorimeter of It3% below 500 mK and &4%
above 500 mK, respectively.
Altogether 13 runs on the heat capacity of 10 different samples of a-Zrl~,Rh, with
7 different compositions have been carried out. In all cases the data are well represented
by Eq. (1). Fig. 1 gives an example for x=26. Open circles for ?>0.35 K2 are original
data for C R in the superconducting state; they have been corrected (full dots) for incomplete fkezing of quasiparticles in the superconducting state by subtracting C,,=
yT,.a.exp (-bTJT) where {a, b} = ( 12.0, 1.65) which for all concentrations is a valid
procedure for Tc0.7 T, [4] and where 4.20 K ~ T ~ c 4 . 4K.2 Obviously such corrections
are small and introduce negligible errors for Tc0.7 K. Although corrected data above
0.7 K lie quite close to a straight line, in order not to generate possible systematic errors,
they have been omitted for evaluating y m and a.A possible quadrupolar nuclear con'tribution a.T2,due to 91Zrnuclei which was shown to be related to the TLS specific
heat anomaly [5] is estimated to be at least two orders of magnitude less than Y ~ S 'for
D 1 7 0 mK. It can therefore be neglected in the data analysis. As for the choice of the
exponent v, we note that a consistent analysis for all specimens requires v to be close to
unity. Results of the data analysis of y m with v= 1 for all runs are displayed in Fig. 2,
where the numbers at the data points denote rhodium concentrations x. The errors indicated are the sum of the sample's and the sample holder's standard error and the standard deviations of the C,(T)-plots. As the holder was not changed, this is a pessimistic
estimate of the total error. We first discuss a possible variation of the first term y m with
composition. There are large variations of ym with no apparent correlation to compo-
B. Junge, W.Gey, Decrease of soft vibrational modes with increasing density
Fig. 1 Low temperature specific heat data
typical for one of the
amorphous BRh-alloys investigated.
Open circles: original
data for c/T in the
superconducting state.
Full dots: data corrected for incomplete
freezing of quasi-particles in the superconducting state (see
3 - 3/ns = (0.505M.06)p.T/gK2
a = (2.64M.3)
T,= (4.33M.01)K
, .%
Fig. 2 Negative correlation between twolevel states and the
phonon-type heat
capacity term for
amOqhoUS Zr-Rh
metals. Numbers
denote rhodium concentrations x. filled
circles denote results
on identical samples.
sition x. As examples, significant differences occur at x=28 which are far outside experimental error, and for x=29 where two runs on identical samples agree within errors
(full circles), while a different sample of the same nominal composition (open circle)
Ann. Physik 5 (1996)
differs. Other runs on identical samples at x=23 and x=27 show agreement within experimental error. We thus conclude that the low-temperature anomaly does not depend
primarily on the Rh-concentration. Rather, we must attribute the variations of yTLs to
differences in the microstructure that occur during the process of forming the amorphous state of ZrRh. Measurements of the angle of the x-ray diffraction maxima for estimates of the nearest neighbor distance and direct measurements of the mass density
were made earlier. Both types of measurement yield characteristic differences for the
packing density of a-Zrl mxRhx
and a-ZrlaxPdxalloys when plotted versus concentration x [6].
We now discuss the coefficient a of the cubic term of the specific heat, which
formally may be expressed by a “Debye temperature” 0 in K via a=12/5.n . R e 3
=1944K3 J mole-’ K4 (R gas constant, 1 mole varies from 93.7 to 94.6 g for
x=21 to 29, respectively). We recall that in insulating glasses this coefficient is
usually much larger than the phononic value determined by ultrasonic measurements,
i.e. there is an excess caloric quasi-Debye contribution [7]. The results of data analysis with v = l are shown in Fig. 2. Similar as for y n s , no systematic variation with
concentration can be detected. Rather, the changes of a, which occur within the composition range, are comparable in magnitude with changes of a for, e.g., the fixed
composition x = 29. Thus, similar as for y- we are led to attribute the variations of
a to differences in the microstructure that occur during the process of forming the
amorphous state.
Now, as is evident from Fig. 2, there is clearly no positive correlation of the overall behaviour of a with that of y- as it has commonly been observed for glasses.
Since the cubic term contains phononic contributions which depend on conventional
interatomic forces and on contributions due to low-frequency soft modes, the correlation between its coefficient a and the density of the low energy excitations expressed
by y m is of particular interest. Here the glassy behaviour seems to be influenced by
the degree of disorder. Contrary to the action of TLS in insulating glasses where an
increase of a together with a rise of yns is commonly observed [7], in metallic
ZrRh glasses a is significantly reduced upon increasing y m . A linear regression of
the data yields
with ao=4.20 p J g-’ K4 and c=2.52 K-2; the correlation coefficient of that fit is
r=-0.725. The observed data scatter beyond the estimated errors. This also indicates
that the preparation process partly influences both y m and a in a random manner.
Thus the linear regression only describes the overall negative correlation between the
tunneling and the soft mode parameters. To our knowledge, such negative correlation
has not yet been observed for any amorphous system including metallic glasses [8].
In our view it would have far-reaching consequences. Therefore we tested carefully
whether the negative correlation between y n s and CY is of physical origin or an artifact of data processing. We checked as follows: First, it was assumed that v = l in
Eq. (1). An incorrect determination of y- would then result in a wrong slope a and
vice versa, with a similar tendency as represented by Rel. (2). It turned out, however,
that at most 25% of the actual range of variation of a ( y m ) contained in Fig. 2 can
stem from such artifacts. We can thus rule out this error source. Second, the exponent v in Eq. (1) is not necessarily unity. A wrong choice would of course influence
the cubic term and could in principle produce an artificial correlation. To test this it
B. Junge, W. Gey, Decrease of soft vibrational modes with increasing density
was assumed that the coefficient a would not depend on y-.
A corresponding aver= 2.55 pJg-' K4 ((0) =201 K) were then to be defined. Using this constant
age (a)
( a )for all measurements and keeping y m and the exponent v as free fit parameters,
a very large range of variation -0.4 <v<+2.0 results. Within this scatter of v none of
the ambiguities of Fig. 2 disappear, instead new ones appear. This leads to the conclusion that a constant cubic term produces unphysical variations of v and is thus excluded by the data.
As stated above, such negative correlation between y n s and a has not yet been
observed for any other amorphous system. Searching for its origin we have investigated a number of parameters that are possibly relevant for glasses, i.e.: glass temperature Tg, crystallization temperature TK, electrical resistivity at T = 5 K and 300 K,
T, of superconductivity, upper critical field BC2(T),spin orbit scattering time zso, electronic specific heat coefficient ye, and specific heat discontinuity ACIyT, at T,. All of
them vary continuously with rhodium concentration or are nearly constant. Thus, no
correlation of these parameters with the observed variation of yms and a can be constructed. Consequently, the observed correlation between YTLS and a can hardly be
connected with the above glass parameters. As mentioned above, there is, however, a
large fluctuation of both, the mass density p and the nearest neighbour distance in
the a-ZrRh system, as compared, e.g. for a-ZrPd. Here appears to be the key to the
mysterious behaviour described above, as may be seen indirectly as follows:
In a recent paper heat capacity data above T= 1.5 K in the normal state have been
reported for amorphous Zrl~,,Rh,Pdy alloys with x+y=25 [3]. This ternary system
also offers an opportunity of "tuning" the degree of disorder because of a minimum
of the mass density near x=y z 13 (see Fig. 1 of Ref. 3). It was found that a peak of
Cfl' near T = 7 K developed upon introducing disorder by approaching the densit
minimum (see Fig. 3 of Ref. 3, where an effective Debye temperature 0 (T) a a-Y
is plotted for 1.5 K>T>12 K. The peak in this Fig. is thus a dip. The most disordered
sample is Zr75Rh13Pd12). This peak, having much in common, even quantitatively,
with the boson peak reported for vitreous silica [ll, has one peculiarity radically different from the dielectric glass: upon its formation by introducing disorder there are
fewer excitations at temperatures below and also around the peak, i.e. the peak near
7 K develops at the expense of the excess Debye type contribution to Q (near 2 K).
In more detail, it can be shown that both experimental C,(T) findings - in the superconducting state and in the normal state - quantitatively agree in the extremes of
their dependencies on disorder. In the superconducting case (i.e. for T<1 K, present
paper) these.may be parametrized by y"=0.39
pJg-' K-3 and $'=0.96
@g-' K-'.
Data for am" and am=may be calculated using Rel. (2). In the normal state (i.e. for
n 1 . 8 K, Ref. 3) the extremes of disorder occur at rhodium concentrations xmln=7.5
and xmm= 13.0 respectively. The corresponding phononic C,-date (the electronic contributions Cf=yT can be subtracted consistently because of their almost equal values
y=4.98kO.04 J mole-' K2 for all alloys [31) merge across the temperature window
(1 K c T d . 8 K) in a quantitative manner, showing a crossover near 0.7 K. Therefore
the quasi-Debye term a can increase with increasing number of TLS in the superconducting state and at the same time can decrease with increasing disorder in the normal state while the boson peak, i.e. essentially the coefficient of the p-term in C,
still positively correlates with increasing disorder [3, 91. This will be discussed in a
separate paper.
In conclusion for amorphous ZrRh alloys which show large fluctuations of their
mass density it has been observed that at low temperatures the phononic (quasi De-
Ann. Physik 5 (1996)
bye) term of the specific heat decreases with increasing disorder. This is counter-intuitive to the soft-potential model and differs from the relation between the phononic
coefficient a and the two-level contribution yTLs found in insulating glasses. Data
are consistent with results on ZrRhPd alloys which also possess large mass density
variations. To our knowledge there is no explanation at present for these findings in
the literature. This includes recent reports on a simulation procedure which are able
to describe the low-energy excitations in a glass on a microscopic level [lo] and
where connections to the soft-potential model are discussed.
We would like to acknowledge helpful conversations with Yu.M. Galperin. Work supported by
Deutsche Forschungsgemeinschaft.
[l] For reviews see: R.O. Pohl, in: Amorphous Solids: Low Temperature Properties, ed. by W.A. Phillips, Springer-Verlag, Berlin Heidelberg New-York 1981, p. 27; W.A. Phillips, Rep. Prog. Phys. 50
(1987) 1657; U. Buchenau, Yu.M. Galperin. V.L. Gurevich, H.R. Schober. Phys. Rev. B 43 (1991)
5039; U. Buchenau, Europhys. News 27 (1993) 77; H. v. Lijhneysen, in: Rapidly Solidified Alloys,
ed. by H.H. Liebemann, Marcel Dekker Inc.. New York Base1 Hong Kong 1993, p. 461
[2] V.G.Karpov, M.I. Klinger, EN. Ignat'iev, Solid State Comm. 44 (1982) 333, Sov. Phys. JETP 57
(1983) 439; Yu. M. Galperin, V.G. Karpov, V. I. Kozub, Advances in Physics 38 (1990) 669
[3] W. Gey, W.Eschner, Yu.M. Galperin, Phys. Rev. B 48 (1993) 15666
[4] N. Kluge, Diploma thesis, Univ. Braunschweig 1983
[5] J.C. Lasjaunias, A. Ravex, J. Phys. F Met. Phys. 13 (1983) LlOl
[6] P. Engel, Diploma thesis, Univ. Braunschweig 1982
[7] R.B. Stephens, Phys. Rev. B 8 (1973) 2896, R.O. Pohl in Ref. 1
[8] C. Siirgers, H. v. Uhneysen, Z. Phys. B - Condensed Matter 70 (1988) 361
[9] A positve Ts-term in C, has also been reported for glassy Pdl,,SixCuy
alloys, and interpreted as
being due to excess specific heat in the temperature range 1.6 KeTe3.5 K. U. Mizutani, K.T. Hartwig, T.B. Massalski. R.W. Hopper, Phys. Rev. Lett 41 (1978) 661
[lo] A. Heuer, R.J. Silbey, Phys. Rev. Lett.70 (1993) 3911
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