## Electron renormalization of sound interaction with two-level systems in superconducting met

Electron renormalization of sound interaction with two-level systems in superconducting metglasses

E.V. Bezuglyi1 , A.L. Gaiduk1 , V.D. Fil1 , W.L. Johnson2 , G. Bruls3 , B. L¨ thi3 , B. Wolf3 , and S. Zherlitsyn1,3 u

B.Verkin Institute for Low Temperature Physics and Engineering, 47 Lenin Avenue, Kharkov 310164, Ukraine 2 California Institute of Technology, Pasadena, CA 91125, USA 3 Physikalisches Institut, Universit¨t Frankfurt, Robert Mayerstr.2-4, 60054 Frankfurt, Germany a

1

arXiv:cond-mat/9802286v1 26 Feb 1998

The crossing of temperature dependences of the sound velocity in normal and superconducting state of metallic glasses points out unambiguously the renormalization of the intensity of sound interaction with two-level systems (TLS), caused by their coupling with electrons. We propose a simple scenario which allows us to describe qualitatively the in?uence of the superconducting transition on the sound velocity and attenuation in superconducting metglasses and to make quantative estimations of renormalization parameters. PACS numbers: 61.43.Fs, 62.65+k, 74.25.Ld

Acoustic measurements in superconducting metglasses Pd30 Zr70 [1,2], Cu30 Zr70 [3], and (Mo1?x Rux )0.8 P0.2 [4], carried out about a decade ago, have discovered, on the face of it, considerable contradictions with the predictions of a well-known tunneling model (TM) [5]. Below we summarize the experimental facts which lead the authors of Refs. [1–4] to such a conclusion. i) The sound velocity v in superconducting (s) phase just below Tc is less than in normal (n) state. This e?ect was observed both in low-frequency (LF) vibrating-reed measurements [1,3,4], and in high-frequency (HF) experiments [2]. According to the original TM, the sound velocity should always increase below Tc due to freezing out of the negative relaxation contribution to v. ii) The sound attenuation Γ reveals an analogous anomaly: Γs exceeds Γn over some temperature interval, which is about of T /Tc ≥ 0.8 in HF measurements [2], and extends up to T /Tc ≥ 0.05 in LF experiments [1,3], where dΓ/dT has a large negative value. In contrast, TM predicts the attenuation to be nearly independent of T (with a small dΓ/dT > 0) until the maximum relaxation rate exceeds the sound frequency ω. Thus, the attenuation in LF experiments should be insensitive at all to superconducting transition, whereas in HF measurements Γs should either be temperature-independent just below Tc , or decrease at low enough Tc . iii) The slope of linear dependence v(ln T ) in normal phase at HF is 4 times less than in superconducting state at T ? Tc [2], whereas TM yields the ratio of 1:2. iv) At least in HF experiments, the normal-state line vn (T ) crosses the superconducting dependence vs (T ) at T ? Tc [2]. From the point of view of TM, this is impossible in principle (see below).

v) Within a wide temperature range, the sound attenuation in the normal state reveals a noticeable temperature dependence with dΓ/dT > 0, which signi?cantly exceeds the changes calculated on the basis of TM. vi) In contrast to TM predictions, dv/dT < 0 in LF experiments [1,3] in normal state at T < 1K, where the phonon relaxation can be neglected. On the other hand, in HF experiments [2] dv/dT > 0 within the same temperature interval. It was supposed in Refs. [1,2] that all (or most of all) deviations from TM are related to the electron renormalization of the constant C = pγ 2 /ρv 2 (where p is the TLS density of states, γ is the deformation potential, ρ is the mass density), which determines the scale of changes of v and Γ in the presence of TLS. However, it was not proposed any consistent scheme to trace, even qualitatively, the genesis of peculiarities of superconducting metglasses mentioned above. Also, there were no attempts to estimate either the value of renormalization of C, or its possible energy dependence. In this work, we have investigated low-temperature HF acoustic properties of superconducting amorphous alloy Zr41.2 Ti13.8 Cu12.5 Ni10 Be22.5 [6] and found similar discrepances with the TM. Following the suggestion of [1,2] we propose a simple scenario, which allows us to explain most of the puzzling experimental facts within the framework of TM by introducing an energy-dependent electron renormalization of C. It is supposed to be signi?cant up to some characteristic energy Ek and to be suppressed in the superconducting state. The alloy under investigation has a high resistance for crystallization in the state of supercooled melt and remains amorphous at extremely low cooling rate (< 10 K/s) [6]. This makes it possible to obtain bulk homogeneous samples, which suit perfectly the acoustic measurements. Ultrasonic experimental technique is described elsewhere [7]. The experimental dependences δv(ln T )/v of the transverse sound for normal and superconducting phases are presented in Figure 1 by ?lled and open symbols, correspondingly. The normal-state measurements were carried out in the magnetic ?eld B = 2.5 T. In accordance with TM, the plots v(ln T ) represent almost straight lines both in the normal and in the deep superconducting state (here and below all the experimental data are normalized on the slope value in the deep superconducting phase, 1

C = 2.85 · 10?5 ). The growth of v below Tc re?ects the freezing out of its relaxation component and agrees with the TM conception. There are also obvious deviations from the TM: the ratio of slopes in the normal and the superconducting phases di?ers from its canonical value 1:2 and is close to 1:4, and the curves v(T ) for both phases intersect at some frequency-dependent temperatures Tcr . Such e?ects have been observed before in Pd30 Zr70 [2]. In general, the “anomalous” slope ratio can be explained by the energy dependence of C, which results in di?erent contributions into the resonant and relaxation components of v. However, within such a simple approach, the crossing remains unaccountable, because it would mean the change of sign of the relaxation contribution below Tcr , that contradicts the physical sense of a relaxation process. The only possibility to explain the crossing is to assume the growth of C in the superconducting phase, as the result of the suppression of the electron renormalization at energies less than the superconducting gap. To validate this assumption, we use the expression for the resonant contribution of TLS into the sound velocity with account of energy dependence of C [8]: δv(T ) v

∞

The similar dependence for “renormalized” constant C ′ = C(1 ? R) < C, also plotted in Figure 2 (line 2), appears to be located above the ?rst one for T < Ek . Here we use the simplest step-like model of renormalization: R = R0 > 0 for E < Ek , and R = 0 in opposite case. Thus, the sound velocity increases when C ′ reduces (at T < Ek ), and vice versa; this basic conclusion can not be derived from Eq. (2), where the “dc component is lost”. A numerical calculation shows, that the relaxation contribution (δv(T )/v)rel at T < 1K, where only the elec? tron relaxation is signi?cant, is also well approximated by piece-linear dependence δv(T ) Cv =

rel

1?R 2

0 T < ω/η 2 2 ln(ω/T η ) T > ω/η 2

(4)

=P

res 0

C(ω ′ )ω ′ tanh(ω ′ /2T ) ′ dω ω2 ? ω′2

(1)

Here and below the energy, the temperature and the frequency are expressed in the same units. In a simple case of energy-independent C, to avoid the logarithmic divergence of Eq. (1) at the upper limit, this formula is conventionally used to obtain an expression for the relative variations of v counted o? its value at some arbitrary reference temperature T0 : (δv(T )/v)res = C ln (T /T 0 ), T >ω (2)

Obviously, if the value of C varies with energy and/or temperature, Eq. (2) is inapplicable even for qualitative estimates, since the reference value of δv/v may also change with C. To account correctly for the changes of C, it is necessary to analyze the complete integral of Eq. (1) by introducing a cut-o? energy Em (say, of the order of melting temperature). It describes a real negative contribution of resonant transitions in the TLS system into the sound velocity. Within the logarithmic accuracy, in the case of C = const Eq. (1) can be approximated by the following piece-linear dependence (see Figure 2, line 1): δv(T ) Cv =

res

ln(ω/Em ) T < ω ln(T /Em ) T > ω

(3)

Here we neglect small variations in v at T < ω: a ? quadratic fall near T = 0 and a shallow minimum at ω = 2.2T [9].

with a dimensionless parameter η < 1 of coupling be? tween the electrons and TLS [5]. The resulting δv/v plot (line 3) shows the velocity variation in normal state. It is clear, that the intersection of lines 3 and 1 (the latter represents a sketch of the velocity change in the deep superconducting state) is possible only under the condition of C ′ < C in normal state. Now we can explain qualitatively the behavior of v and Γ at the superconducting transition. Below Tc the electron renormalization reduces rapidly, and the e?ective C ′ grows, providing the decrease in v and the increase in Γ. However, a competitive e?ect arises simultaneously: the time τ of relaxation of TLS on electrons grows and, therefore, changes v and Γ into the opposite direction. Thus, if the phonon relaxation predominates near Tc , the e?ective τ changes weakly, and the sound velocity will decrease (correspondingly, Γ will increase) below Tc , as it was observed in [1–4]. If, on the contrary, the electron relaxation prevails (for materials with lower Tc like our system), the changes of v and Γ near Tc may have any sign, depending upon the relation between Tc and Ek . Following our approach, the observed slope ratio of v(ln T ) plots might be treated as the decrease of C ′ down to 0.5C (R0 = 0.5) at E < Ek . Making use of known values of Tcr (see Figure 1) and the elementary geometry of Figure 2, we obtain the value of Ek = 0.2 K, which, however, seems to be too low to reduce su?ciently the TM ratio of slopes in normal state at T ? 1K and in deep superconducting state. In fact, these values of R0 and Ek are to be interpreted as tentative estimates for a numerical analysis of Eq. (1), implemented by us to compare quantitatively the proposed model with the experimental data. For this purpose, we need ?rst the value of η which can be found from the temperature dependence of sound attenuation in superconducting state, Γs (T )/Γn (Tc ), plotted in Figure 3, under assumption of validity of conventional BCS theory [10]. According to TM and the theoretical dependence of τ in the superconducting phase [12], the low-temperature (T ? ?) behavior of attenuation is described by the exponential dependence 2

Γs (T )/T Γn (Tc ) ? exp(??(0)/T ), con?rmed by the experimental data plotted in the inset of Figure 3. Note, that the slope of the linear part of this curve yields the BCS ratio of ?(0)/Tc = 1.73±0.1 (Tc = 0.9K). A numerical calculation of Γs (T ) with the ?tting value η = 0.84 describes satisfactorily the evolution of attenuation over the actual temperature region below Tc as shown by solid line in Fig.3. In our numerical calculations, we have used the model of a more realistic smoothed energy dependence of C ′ : C ′ (E)/C = 1 ? R0 1 ? tanh2 (E/Ek ) (5)

The two ?tting parameters 0 < R0 < 1 and Ek were found to be determined by the slopes of δv(ln T ) in the normal state and by the crossing temperatures Tcr at two frequences. In order to extend our model over the superconducting state, we have accounted for the freezing out of normal electron excitations with additional multiplying of the renormalization parameter R0 by the freezing factor 2f (?/T ) ( f (x) is the Fermi function) at energies less than the superconducting gap 2?. Of course, this approximation ignores the energy dispersion of the electron density of states in the superconductor, but, nevertheless, it reproduces perfectly the experimental velocity dependence, as it is obvious from Figure 1. The calculated velocity dependences, with account of relaxation contribution, deviate from experimental data at most of noise level within a narrow interval of Ek = 1.75 ± 0.25K and R0 = 0.27 ± 0.01. Note that, according to Eqs. (1), (3), (δv/v)res does not depend on frequency at T ? ω . This allowed us to compare the results at di?erent frequences for a given C by aligning the low-temperature parts of the experimental curves, where only the resonant component is present. The changes of relaxation and resonant contributions into the sound velocity almost compensate each other near Tc for our R0 and Ek , with the creation of a shallow minimum (of the noise level). Apparently, this causes the mismatch between critical temperatures, found in [11] by the magnetic (0.9 K) and acoustic (0.85 K) measurements (see also Figure 1). For larger Tc > 2÷3K, the relaxation ? component of v, determined mainly by the interaction of TLS with phonons, changes more slowly, and the sound velocity below Tc should decrease much stronger. As regards the relaxation contribution to sound attenuation, in the normal state the system “scans” the relation of Eq. (5), therefore Γn (T ) decreases with temperature, and the TM plateau is absent in fact. The evolution of Γs (T ) at the superconducting transition is determined ?rst by the suppression of renormalization, caused by freezing out of normal excitations, that results in the growth of attenuation (? ?2 (T )). Such a behavior agrees qualitatively with the experiments of Refs. [1–4]. However, in our case the experimental changes of attenuation are noticeably less than calculated ones, as it is 3

shown in Figure 4. This is obviously a consequence of roughness of our model, and we shall outline a way to remove this imperfection. Following the results of Ref. [13], the renormalization of constant C seems to be dependent not only on the energy E, but also on the splitting parameter u = ?0 /E, where ?0 is the tunnel splitting in symmetric double-well potential. Both of them determine the relaxation parameter ωτ ? (ω/u2 E) tanh(E/2T ), which is of the order of ωτ ? (ω/u2 T ) for all actual energies E ≤ T . The relaxation contribution into the sound velocity is formed by TLS with ωτ ? 1, or, equivalently, u2 ? ω/T ? 10?2 , that corresponds to nearly symmetric TLS. In contrary, the attenuation is contributed by asymmetric TLS with ωτ ? 1, or u2 ? ω/T . It was shown in [13] that the in?uence of electrons on TLS is mainly signi?cant in the symmetric case. Thus, the summary of our analysis can be treated as the di?erence between the e?ective con′ ′ stants Cv and CΓ , describing the renormalization of symmetric and asymmetric TLS, correspondingly. After the ′ averaging over u, the constant Cv is well described by the model expression of Eq. (5) with ?tting parameters ′ found above. Apparently, CΓ can be described by an analogous formula, but with much smaller renormalization parameter R0 ? 0.05. In that way, the proposed scenario allows us to explain qualitatively all discrepancies between the real experimental pattern and the original TM, mentioned in the preamble, except the item vi). Indeed, in our model the derivative dv/dT is always positive at T < 1K (though less than in simple TM). At present time, within a scope of TM or its modi?cations, there are no mechanism to explain the change of sign of dv/dT upon decreasing of frequency, when the condition ωτ ? 1 still holds. This problem needs further experimental investigations over the intermediate frequency range. In summary, the crossing of the temperature dependences of sound velocity in normal and superconducting states of superconducting metglasses at T ? ? unambiguously con?rms the existence of electron renormalization of the constant C = pγ 2 /ρv 2 towards smaller values. An assumption about the energy-dependent renormalization concentrated within a low-energy region allows us to describe perfectly the behavior of sound velocity in normal and superconducting phases by only two ?tting parameters within the framework of TM. This research was partially supported by Ukrainian State Foundation for Fundamental Research (Grant No 2.4/153) and the Deutsche Forschungsgemeinschaft via SFB 252. W.L.J. wishes to acknowlege the U.S. Dept. of Energy for support under Grant No. DE-FG0386ER45242. S.Z. would like to thank Alexander von Humboldt-Foundation for support.

[1] H. Neckel et al., Sol. St. Comm 57, 151 (1986). [2] P. Esquinazi et al, Z. Phys. B: Cond. Matter 64, 81 (1986). [3] P. Esquinazi, and J. Luzuriaga, Phys. Rev B37, 7819 (1988). [4] F. Lichtenberg et al, in: Phonons 89. S. Hunklinger, W. Ludwig, G. Weiss (eds.), p. 471. Singapore: World Scienti?c (1990). [5] S. Hunklinger, A.K. Raychaudhuri, in: Progress in low temperature physics. D.F. Brewer (ed.) Vol IX. Amsterdam: North-Holland (1986). [6] A. Pecker, and W.L. Johnson, Appl. Phys. Lett. 63, 2342 (1993). [7] B. L¨thi et al, J. Low Temp. Phys. 95, 257 (1994). u [8] L. Pich` et al, Phys. Rev. Lett., 32, 1426 (1974). e [9] B. Golding, J. E. Groebner, and A.B. Kane, Phys. Rev. Lett. 37, 1248 (1976). [10] The latter, generally, was not obvious, since the critical temperature of studied alloy found from AC magnetic measurements does not coincide with a sharp bend on v(T ) [11], that may be attributed to the e?ect of paramagnetic depairing. [11] A.L. Gaiduk et al, Low Temp. Phys. 23, 857 (1997). [12] J.L. Black, and P. Fulde, Phys. Rev. Lett 43, 453 (1979). [13] K. Vl?dar, and A. Zawadowski, Phys. Rev. B28, 1564, a 1582, 1596 (1983).

FIG. 1. Experimental (symbols) and calculated (solid curves) temperature dependences of transverse sound velocity in Zr41.2 Ti13.8 Cu12.5 Ni10 Be22.5 for two frequencies: circles 62 MHz, triangles - 186 MHz. Open symbols represent the result for the superconducting state, ?lled symbols -for the normal state, C = 2.85 ? 10?5 . Arrows indicate the critical temperature of the superconducting transition Tc found from magnetic measurements [11] and the crossing temperatures Tcr .

FIG. 2. Diagram of the temperature dependences of the sound velocity in superconducting metglass: 1 - (δv/v)res in the superconducting phase, 2 - (δv/v)res in the normal phase, 3 - the total δv/v in the normal phase.

FIG. 3. Temperature dependence of sound attenuation in superconducting state of Zr41.2 Ti13.8 Cu12.5 Ni10 Be22.5 : circles - experiment at 62 MHz, solid curve - calculation with η = 0.84. Inset: to determination of the superconducting energy gap.

FIG. 4. Comparison of the sound attenuation at 54 MHz with model calculation: open and solid symbols - experimental dependences in normal and superconducting phases, correspondingly; lines - the related calculation for Ek = 1.75 K, R0 = 0.27. Note that the vertical scale is strongly enlarged.

4

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