Low-energy excitations and thermal conductivity of CuxSn100−x films at low temperatures

Low-energy excitations and thermal conductivity of CuxSn100−x films at low temperatures

Journal of Non-Crystalline Solids 250ÿ252 (1999) 811ÿ814 www.elsevier.com/locate/jnoncrysol Low-energy excitations and thermal conductivity of Cux S...

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Journal of Non-Crystalline Solids 250ÿ252 (1999) 811ÿ814


Low-energy excitations and thermal conductivity of Cux Sn100ÿx ®lms at low temperatures R. Schmidt, Th. Franke, P. H aussler


TU Chemnitz, Institut f ur Physik, 09107 Chemnitz, Germany

Abstract We report on the thermal conductivity k…T † and the resistivity q…T † of amorphous and polycrystalline Cux Sn100ÿx …0 < x < 100†. The measurements were performed with an improved steady-state technique by preparing the samples on a microchip. Both, k…T † and q…T † were measured after preparation at T ˆ 5 K in the amorphous and, after annealing, in the crystalline state in the temperature range from 1.2 to 360 K. We separate the electronic part from the total thermal conductivity with the WiedemannÿFranz law. The thermal conductivity is discussed in relation to the atomic structure. The phonon-thermal conductivity of the amorphous state has a plateau region due to low-energy excitations at temperatures between T ˆ 2 K and 30 K. The plateau shifts with increasing x to lower temperatures. It can be observed also after crystallization for alloys with x P 60 at.% Cu due to a low degree of crystallinity but vanishes after further annealing. The plateau relates to structural data at scattering vectors K ˆ Kpe  2kF . Ó 1999 Elsevier Science B.V. All rights reserved.

1. Introduction The thermal conductivity k…T † of amorphous materials is still not yet fully understood. The socalled plateau (a region of a smaller temperature dependence of k…T † at low temperatures) that was found in dielectrics [1] as well as in the phononthermal conductivity of amorphous metals [2] is still a subject of many investigations. It is explained by additional scattering processes due to low-energy excitations [2]. Among several models for them [3ÿ5] we prefer the phononÿroton model [6ÿ8] because it allows a discussion of k in relation to anomalies of electronic transport properties such as the thermopower as well as structural data

* Corresponding author. Tel.: +49 371 531 3140; fax: +49 371 531 3555; e-mail: [email protected]

and the phonon-dispersion relation [9,10]. By varying the composition of the alloy, the structure factor S…K† related to k…T † changes and the temperature at which the plateau in k…T † occurs, changes too. 2. Experimental procedure For the measurement of k, we use a substrate consisting of a silicon microchip with a free-standing membrane of a thickness of about dm ˆ 800 nm [11]. On top of the membrane, a bolometer stripe … 5 lm) is evaporated serving as a heater and thermometer simultaneously. It consists of Al for T P 25 K and Au20 Ge80 for temperatures < 25 K. Onto the bottom of the liquid-He cooled membrane, the sample was prepared by ¯ash evaporation. First km dm of the membrane is determined from

0022-3093/99/$ ÿ see front matter Ó 1999 Elsevier Science B.V. All rights reserved. PII: S 0 0 2 2 - 3 0 9 3 ( 9 9 ) 0 0 1 8 3 - 0


km dm ˆ

R. Schmidt et al. / Journal of Non-Crystalline Solids 250ÿ252 (1999) 811ÿ814

…Ph ÿ P0 †b dR : 4…Rh ÿ R0 †l dT


Rh and R0 are the resistances of the bolometer at the heating rates Ph and P0 , respectively. b is the width and l the length of the membrane. After the deposition, kd of the system membrane/®lm is measured once more and we get kf of the ®lm from kf ˆ …kd ÿ km dm †=df :


The resistivity is measured at the same time at a separate thin-®lm sample which is evaporated at the same time. Below T ˆ 12 K we use a steadystate method. Above a slight heating (<100 mK/ min) is allowed. The calculated radiation losses are smaller than 1% for T 6 150 K. The ®lms had a thickness of 65 nm 6 df 6 250 nm. So the system is three-dimensional. The accuracy is about 6 ‹10% for k…T † and about 6 ‹2% in temperature. 3. Results In Fig. 1 the thermal conductivity in the amorphous state is plotted for three di€erent compositions of Cux Sn100ÿx . The amorphous alloys become superconducting below the transition temperature Tca and crystallize at the crystallization temperature Tk . Tca and Tk are given in Table 1. With increasing Cu-concentration, Tca decreases because of decreasing electronÿphononÿcoupling [12]. The greatest stability of the amorphous samples is found at about 70 at.% Cu [12,13]. After crystallization (Fig. 2) the transition temperature, Tccr , decreases. All samples were annealed in situ to 360 K. Thus, the samples with lowest Tk have the largest degree of crystallinity and hence, the largest kcr . A region …Tpa † with a smaller temperature dependence of k…T † was found for the amorphous samples as well as for the crystallized samples …Tpcr † with 60 and 70 at.% Cu due to their crystallinity after the annealing procedure (Tp is de®ned from the point of in¯ection in k…T †). Below Tc , the rate at which k decreases with decreasing temperature increases because the electrons partly combine to form Cooper pairs [14]. These electrons do not contribute to the electronic part of k [14]. On the other hand, the electronÿphonon interaction be-

Fig. 1. Thermal conductivity k…T † in the amorphous state (curves are shifted each by one decade in relation to Cu30 Sn70 ). Table 1 Characteristic temperatures of the samples (explanation in text) x (at.%)

Tca (K)

Tccr (K)

Tk (K)

Tpa (K)

Tpcr (K)

0 20 30 40 50 60 70 80 100

4.46 5.71 4.91 4.19 3.26 2.26 1.20 ÿ ÿ

3.84 3.27 2.57 2.04 ÿ ÿ ÿ ÿ ÿ

ÿ 174 194 212 228 261 322 (275) ÿ

ÿ 23.5 10.0 7.3 8.2 3.7 2.0 2.6 ÿ

ÿ ÿ ÿ ÿ ÿ 1.9 1.2 ÿ ÿ

comes smaller and so it is possible, that the phonon part of k which dominates at low temperature …< 10 K† increases with decreasing temperature [2] before it decreases due to the decreasing number of phonon states (Fig. 1 Cu30 Sn70 ). 4. Discussion Cux Sn100ÿx is a metallic alloy. Accordingly, electrons as well as phonons contribute to the thermal conductivity. The common way to separate the

R. Schmidt et al. / Journal of Non-Crystalline Solids 250ÿ252 (1999) 811ÿ814


the so-called phononÿroton states [7,8] (Fig. 3). A smaller peak at Kpe results in a minimum at larger energies, hx0 . These localized modes can be excited above an excitation energy, hx0 , that corresponds to a temperature, T0 . Phononÿrotons do not contribute to k. They act as additional scatterers for Debyeÿphonons [6,18]. Accordingly, k…T † shows a smaller temperature dependence around T0 . For CuÿSn an increasing amplitude in

Fig. 2. Thermal conductivity k…T † in the crystalline state (curves are shifted each by one decade in relation to Cu30 Sn70 ).

electronic contribution is the use of the WiedemannÿFranz law (WFL) [2]. The electronic part in the superconducting state can be calculated by the BRT-theory (Bardeen, Rickayzen and Tewordt) [14]. Below Tca where the phonon part dominates we found a T 1:9 ÿ dependence known for amorphous materials [1,15]. Above Tca the electronÿphonon interaction must be taken into account and we get k  T . Due to additional inelastic scattering processes at low and middle temperatures …< 20 K† and a larger electronÿphonon coupling [12] the WFL is not valid using the theoretical Lorenz-number, L0 [16]. This problem might be overcome if one assumes a smaller Lorenz-number at low temperatures. In addition it is not known, if there is a contribution from the electronic part to the plateau-region. So we discuss the total thermal conductivity, k…T †, in the plateau region. We found a shift of Tp with increasing Cuconcentration at lower temperatures. This shift indicates a relation to structure because the electronic-induced structure factor at the scattering vector Kpe produces a minimum in the phonon dispersion relation at the wavenumber Qpe  Kpe ,

Fig. 3. Structure factor [17] and schematically its in¯uence on phononÿroton states.

Fig. 4. Correlation between the plateau temperature Tp and the peak at Kpe in the structure factor.


R. Schmidt et al. / Journal of Non-Crystalline Solids 250ÿ252 (1999) 811ÿ814

the structure factor at Kpe with increasing Cu-concentration was observed [17] with the consequence that T0 decreases with increasing Cuconcentration and thus the plateau shifts to lower temperatures. In Fig. 4 the correlation between Tp and T0  …S…Kpe † ÿ 1†ÿ1 is shown. …S…K† ÿ 1† describes the deviation of S…K† from a homogeneous medium. After crystallization, the structure factor becomes larger and the plateau is shifted to lower temperatures. So it is possible to discuss k in the phononÿroton model and to understand the anomalies in relation to structural data. 5. Conclusion The thermal conductivity of Cux Sn100ÿx can be explained by common theories. The separation of the electronic part of the thermal conductivity by the WiedemannÿFranz law causes diculties which will be overcome if one assumes a smaller, temperature-depending Lorenz-number. It has been shown that a plateau-region occurs in k…T † of the amorphous state. This plateau shifts with increasing Cu-concentration to lower temperatures. This e€ect can be understood by the phononÿroton model. If a large peak in the structure factor at Kpe is measured, the plateau appears at

low temperatures. So the plateau region is directly related to the structure. For alloys with more than 60 at.% Cu, the plateau also exists in the crystalline state but shifted to lower temperatures.

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