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Observation of a crossover of the inelastic electron scattering in Sc100-xAgx thick films

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Observation of a crossover of the inelastic electron scattering in Sc

100Àx

Ag

x

thick films

T. C. Lee

Department of Electrophysics, National Chiao Tung University, Hsinchu 300, Taiwan J. J. Lin*and S. F. Chang

Institute of Physics, National Chiao Tung University, Hsinchu 300, Taiwan 共Received 2 April 2003; published 29 August 2003兲

We have made a series of thick Sc films doped with different amounts of Ag, which results in a systematic decrease in the resistivities, i.e., disorder, of the films. From measurements of the low-field magnetoresistances and comparison with three-dimensional weak-localization theoretical predictions, the electron dephasing times are extracted in every film. We find a crossover of the inelastic electron process from the critical electron-electron scattering to the electron-electron-phonon scattering as the disorder decreases and the system progressively moves away from the Anderson localization.

DOI: 10.1103/PhysRevB.68.073407 PACS number共s兲: 72.15.Lh, 72.15.Rn, 72.10.Di

I. INTRODUCTION

Since the theoretical and experimental realization of weak localization, namely, the coherent backscattering effects in disordered conductors, the quantum-interference transport has been extensively investigated in mesoscopic structures.1,2 It is now established that one of the key physical quantities governing all kinds of the quantum-interference transport phenomena 共weak localization, universal conductance fluc-tuations, Aharonov-Bohm oscillations, etc.兲 is the electron dephasing time ␶. In fact, our understanding of the under-lying physics and microscopic mechanisms of electron dephasing has advanced significantly over the last two decades.3 In practice, the temperature and disorder depen-dences of␶ can be written as2,3

1 ␶␸共T,l兲⫽ 1 ␶␸0共l兲 1 i共T,l兲 , 共1兲

where l is the electron elastic mean free path,0 is presumed to be a temperature-independent constant, and ␶i is the rel-evant inelastic electron scattering time共s兲 in question. The problem whether␶0 should reach a finite or an infinite value as T→0 has been intensively studied very recently.3In this work, we shall focus our discussion on the second term, i.e., the temperature-dependent inelastic scattering time ␶i. In general, the inelastic scattering time can be effectively ex-pressed as 1/␶i⬀Tpover the temperature range accessible for a typical experiment, where p is an exponent of temperature. The knowledge of the value of p is indispensable for identi-fying the responsible inelastic process 共electron-electron scattering, electron-phonon scattering, etc.兲 in the mesos-copic conductors under study.

The inelastic electron scattering in reduced dimensions in the weakly disordered regime has been extensively studied and it is now understood that the Nyquist electron-electron interaction is responsible for the dephasing at low temperatures.4 On the other hand, it is established that the inelastic electron scattering in three dimensions in the weakly disordered regime is due to the electron-phonon

interaction.5 Electron-phonon scattering times have been measured in many bulk metals and alloys.3 What is much less studied is the inelastic electron process in three-dimensional conductors near the Anderson localization or the mobility edge. Under such latter circumstances, Belitz and Wysokinski6have proposed that the electron-electron scatter-ing should be very sensitive to the critical, as opposed to diffusive, current dynamics, resulting in an electron-electron scattering rate dominating over the electron-phonon scatter-ing rate in three dimensions. 共In the present work, this Belitz-Wysokinski process will be referred to as the ‘‘criti-cal’’ electron-electron scattering, so as to be distinguished from the more familiar quasielastic, Nyquist electron-electron scattering originally proposed by Altshuler, Aronov, and Khmelnitskii.7兲 Experimentally, the critical electron-electron scattering time ␶EE has been observed in several systems having very low values of electron diffusion constant

D, including thick granular aluminum films,8 doped semiconductors,9 heavily doped conjugated polymers,10 polycrystalline disordered Sc thick films,11 and RuO2 and IrO2 thick films.12A very low value of D basically signifies that the conductor is not very far away from the mobility edge.

In all of those experiments just mentioned共except Ref. 8, see below兲, the critical electron-electron scattering time was observed, but the electron-phonon scattering time was not. This is mainly because that, in most three-dimensional con-ductors, it is extremely difficult to make the resistivities suf-ficiently high and vary over a wide range, while keeping the disorder microscopically homogeneous. Thus, most real ex-periments can only reveal either the critical electron-electron or the electron-phonon process, depending on the amount of randomness and the microscopic quality of disorder in the sample. In this work, using a single material system, namely, Ag doped Sc thick films, we are able to observe a crossover from the critical electron scattering to the electron-phonon scattering. By controlling the amount of Ag addition, we gradually adjust the resistivities of the films, and hence move the films from being close to being away from the mobility edge. Our results are reported below.

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II. EXPERIMENTAL METHODS

We have fabricated a series of thick 共i.e., three-dimensional兲 Sc100⫺xAgxfilms with the nominal composition

x ranging from 1 to 15. Appropriate amounts of Sc共99.99%

pure兲 and Ag 共99.995% pure兲 were first arc melted in a high-purity Ar atmosphere to form ingots of 2 to 3 g. Appropriate amounts of the ingots were then placed in a vacuum chamber to make thick film samples, using the standard thermal-flash deposition technique. A background pressure of 3

⫻10⫺6 Torr was reached before the evaporation-deposition process was initiated. The films were deposited on glass sub-strates held at room temperature, and they were patterned into meanders of 0.46 mm wide and 18 mm long using a mechanical metal mask. The thickness of the films varied somewhat around 5000 Å. Table I lists the relevant param-eters for our samples studied in this work.

The homogeneity of our Sc100⫺xAgx films for a given composition x was checked by both energy dispersive x-ray spectroscopy and resistivity ␳ measurement. We found that, for films with xⱗ 15, the measured values of x and␳ were similar for all the films made from the same batch with the same nominal composition. In addition, the resistivity de-creased monotonically with increasing composition x 共see Table I兲. For x⬎15, however, we found that the film resis-tivities were much less reproducible from batch to batch. Thus, we will focus our discussion on those films with x

ⱗ15 in this work.

The resistances and magnetoresistances of our films were measured by the standard four-probe technique at liquid-helium temperatures. Measurement temperatures down to 0.3 K were achieved using a 3He fridge. A calibrated carbon glass thermometer was used for monitoring temperatures above 7 K, while a calibrated RuO2 thermometer was used for monitoring temperatures below 7 K. In this work, we shall concentrate our discussion on the magnetoresistance data measured at various temperatures, from which the elec-tron dephasing times are inferred. The measured magnetore-sistance curves were fitted with three-dimensional weak-localization theoretical predictions13,14since our films were made sufficiently thick, as mentioned above. In addition, the phonons are three dimensional in our case. The absence of phonon confinement effects makes our extraction of the in-elastic electron-phonon scattering time have a much higher degree of accuracy than that usually achieved in the studies of low-dimensional structures.3,15 In the latter case, the

electron-phonon interactions are often complicated by the film thickness, the substrate material, and the acoustic trans-parency of the film-substrate interface.

III. RESULTS AND DISCUSSIONS

Figure 1 shows the measured, normalized magnetoresis-tivities 䉭␳(B)/␳2(0)⫽关␳(B)⫺␳(0)兴/␳2(0) 共symbols兲 and the three-dimensional weak-localization theoretical predic-tions 共solid curves兲 for a representative film Sc85Ag15, at three measuring temperatures as indicated in the caption to Fig. 1. The inset shows the 䉭␳(B)/␳2(0) for another film Sc99Ag1, at three measuring temperatures as indicated in the caption to Fig. 1. It is clearly seen that the theoretical pre-dictions agree very well with the experimental data in low magnetic fields. The discrepancies in higher magnetic fields especially at low measuring temperatures are expected and are only drawn for reference.16The fitted values of the spin-TABLE I. Values of relevant parameters for several thick Sc100⫺xAgxfilms studied in this work. x is the

actual composition determined by energy dispersive x-ray spectroscopy. t is the film thickness, D is the electron diffusion constant at 10 K, p is the fitted effective exponent of temperature in 1/i⬀Tp, and␶sois the

spin-orbit scattering time. The resistivity␳ is in␮⍀ cm.

Film x t 共Å兲 ␳(300 K) ␳(10 K) D(cm2/s) p so 共ps兲 Sc99Ag1 1.1 5950 740 641 0.11 1.2⫾0.1 48 Sc97Ag3 3.0 4900 400 348 0.20 1.1⫾0.1 17 Sc95Ag5 3.7 5060 356 332 0.21 1.6⫾0.2 4.1 Sc90Ag10 11.0 5170 273 250 0.28 1.8⫾0.1 3.1 Sc85Ag15 15.7 5400 224 197 0.36 1.9⫾0.1 2.7

FIG. 1. Normalized magnetoresistivity as a function of magnetic field for the Sc85Ag15thick film at共from top down兲 3.0, 6.0, and

10.0 K, respectively. The symbols are the experimental results, and the solid curves are the three-dimensional weak-localization theo-retical predictions. Inset: Normalized magnetoresistivity as a func-tion of magnetic field for the Sc99Ag1thick film at共from top down兲

3.0, 6.0, and 10.0 K, respectively.

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orbit scattering times ␶SO, also involved in the weak-localization theory, are listed in Table I. As expected, the magnitude of ␶SO decreases with increasing Ag addition, which can be readily attributed to the large atomic weight of Ag in comparison with that of relatively light Sc. This result provides a consistency check of our experimental method

共samples fabrication and data analysis兲.

Figure 2 shows the extracted electron dephasing time␶ as a function of temperature for five Sc100⫺xAgx samples. For clarity, the values of␶have been scaled by multiplying a factor of 1, 1, 2, 3, and 4 for Sc99Ag1, Sc97Ag3, Sc95Ag5, Sc90Ag10, and Sc85Ag15, respectively. The basic feature of this figure shows that the magnitude of ␶ increases with decreasing temperature, indicating that inelastic processes are responsible for the electron dephasing, i.e., 1/␶⬇1/␶i共at least at not too low temperatures17兲. Physically, the inelastic electron scattering time over the finite temperature range for a typical experiment is effectively given by 1/␶i⬀Tp, where the exponent of temperature p is of order unity. Figure 2 shows the main result of this work, namely, the value of p increases systematically with increasing Ag concentration in these Sc100⫺xAgx thick films. With only 1% of Ag addition, the electron dephasing time reveals a fairly linear tempera-ture dependence over the wide temperatempera-ture range from 0.3 to above 10 K. This linear behavior of␶(T) is extremely simi-lar to that previously observed in pure Sc thick films.11 In fact, the measured ␶ in this sample basically overlaps the value of ␶⬇(1.3⫻10⫺10)T⫺1 s found in those pure Sc films11 just mentioned. This observation provides a consis-tency check of our experimental method and data analysis.

As the Ag addition increases, one sees that the tempera-ture behavior of␶varies systematically. More precisely, the value of p increases with increasing Ag concentration.共Our

fitted value of p for each sample is listed in Table I.兲 Inspec-tion of Fig. 2 demonstrates that, from 1 up to 15 % of Ag addition, the temperature behavior of our experimental ␶ evolves progressively from a linear dependence in Sc99Ag1

共our most resistive sample兲 to a more or less quadratic

de-pendence in Sc85Ag15 共our least resistive sample兲. This sys-tematic change in the temperature dependence of ␶ is un-derstood as follows.

In the Sc99Ag1thick film, the linear temperature behavior of ␶ can be successfully ascribed to the ‘‘critical’’ electron-electron scattering in three-dimensional conductors near the mobility edge.6,12 This is due to the fact that the electron diffusion constant is extremely low (⬃0.1 cm2/s) in Sc99Ag1, signifying that this film is much closer to the mo-bility edge than ordinary metals are. In contrast, D

⬎1 cm2/s in ordinary 共weakly disordered兲 metals. There-fore, the electron-electron interaction is very sensitive to the critical, as opposed to diffusive, current dynamics. According to Belitz and Wysokinski,6 the inelastic electron-electron scattering rate 1/␶EE under such circumstances should pos-sess a linear dependence on temperature and be independent of disorder. Indeed, besides a linear temperature dependence, a mean-free-path independence of 1/␶EEhas previously been confirmed in experiments.11,12 It should be noted that this 1/␶⬇1/␶EE⬀T is not due to the two-dimensional Nyquist electron-electron scattering usually operating in weakly

dis-ordered thin metal films in which an exponent p⫽1 has been

widely observed.18

In the Sc85Ag15 thick film, the film resistivity drops sig-nificantly from that in Sc99Ag1 共see Table I兲. As a conse-quence, one sees that the inelastic electron process illustrates a somewhat quadratic temperature dependence共Fig. 2兲. This quadratic temperature behavior of 1/␶ can be understood in terms of the electron-phonon scattering in the presence of disorder, i.e., 1/␶⬇1/␶e p, where␶e pis the electron-phonon scattering time. It has been established over the past two decades that the inelastic electron scattering in weakly disor-dered conductors is dominated by electron-phonon interac-tion in three dimensions,5 while the inelastic scattering is dominated by the Nyquist electron-electron scattering in re-duced dimensions.4Moreover, it is now clear共especially ex-perimentally兲 that the electron-phonon scattering rate fre-quently changes from the T3dependence for the pure case to a T2 dependence for the disordered case.15,19–22We believe this T2law for the electron-phonon scattering in impure con-ductors is exactly what is observed in the case of the Sc85Ag15 thick film. Quantitatively, the measured value of ␶␸(10 K)⬇␶e p(10 K)⬇1⫻10⫺11 s in this sample is compatible with the electron-phonon scattering time (⬃10⫺10⫺10⫺12s at 10 K兲 measured in, e.g., TiAl alloys,19,20 and AgPd 共Ref. 21兲 and AuPd 共Ref. 15兲 thick films. However, Fig. 2 indicates that a quadratic temperature dependence is not yet fully developed in this sample at our lowest measuring temperatures. We think that this is due to the fact that the Sc85Ag15 film is still too resistive for the electron-phonon scattering to fully dominate the total inelas-tic scattering rate. That is, the criinelas-tical electron-electron scat-tering rate is probably not yet completely suppressed even in this sample. On the other hand, it may be possible that the

FIG. 2. Experimental electron dephasing time as a function of temperature for five Sc100⫺xAgx thick films. For clarity, the values

of␶have been scaled by multiplying a factor of 1, 1, 2, 3, and 4 for Sc99Ag1, Sc97Ag3, Sc95Ag5, Sc90Ag10, and Sc85Ag15,

respec-tively. The solid line is drawn proportional to T⫺1, and the dashed line is drawn proportional to T⫺2. They are guides to the eye.

BRIEF REPORTS PHYSICAL REVIEW B 68, 073407 共2003兲

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deviation from a T2law at low temperatures is a signature of the occurrence of a saturation in␶ as T→0.3,17

Between 1 and 15 % doping of Ag, the film resistivity decreases gradually with increasing amount of Ag concentra-tion共see Table I兲. As a result, the effective exponent of tem-perature p in 1/ increases systematically from 1 共i.e., 1/␶⬇1/␶EE) toward 2共i.e., 1/␶␸⬇1/␶e p).

In conclusion, using a single material system with differ-ent levels of disorder, we demonstrate a crossover of the inelastic electron process from the ‘‘critical’’ electron-electron scattering to the electron-electron-phonon scattering as the randomness of the system decreases, and the system moves away from the mobility edge. Previously, a similar crossover of the inelastic process has already been observed in a series

of high-resistivity, thick granular aluminum films by Mui

et al.,8 where most of the resistivities are contributed from the grain boundaries while the metal grains might only be weakly disordered. However, pertinent theoretical under-standing was unavailable then and their experimental result could not have been satisfactorily explained until recently.

ACKNOWLEDGMENT

The authors are grateful to N. Giordano for valuable dis-cussions which initiated them to undertake this experiment. This work was supported by the Taiwan National Science Council through Grant No. NSC 91-2112-M-009-028.

*Electronic address: jjlin@mail.nctu.edu.tw

1G. Bergmann, Phys. Rep. 107, 1共1984兲.

2B.L. Altshuler, A.G. Aronov, M.E. Gershenson, and Yu.V.

Shar-vin, Sov. Sci. Rev., Sect. A 9, 223共1987兲.

3For a recent review, see J.J. Lin and J.P. Bird, J. Phys.: Condens.

Matter 14, R501共2002兲.

4B. L. Altshuler and A. G. Aronov, Electron-Electron Interactions

in Disordered Systems, edited by A. L. Efros and M. Pollark 共Elsevier, Amsterdam, 1985兲.

5J. Rammer and A. Schmid, Phys. Rev. B 34, 1352共1986兲. 6D. Belitz and K.I. Wysokinski, Phys. Rev. B 36, 9333共1987兲. 7B.L. Altshuler, A.G. Aronov, and D.E. Khmelnitskii, J. Phys. C

15, 7376共1982兲.

8K.C. Mui, P. Lindenfeld, and W.L. McLean, Phys. Rev. B 30,

2951共1984兲.

9P. Dai, Y. Zhang, and M.P. Sarachik, Phys. Rev. B 46, 6724

共1992兲.

10M. Ghosh, A. Barman, A. Das, A.K. Meikap, S.K. De, and S.

Chatterjee, J. Appl. Phys. 83, 4230共1998兲.

11T.J. Li and J.J. Lin, Phys. Rev. B 56, 8032共1997兲.

12J.J. Lin, W. Xu, Y.L. Zhong, J.H. Huang, and Y.S. Huang, Phys.

Rev. B 59, 344共1999兲.

13H. Fukuyama and K. Hoshino, J. Phys. Soc. Jpn. 50, 2131共1981兲.

14C.Y. Wu and J.J. Lin, Phys. Rev. B 50, 385共1994兲. 15Y.L. Zhong and J.J. Lin, Phys. Rev. Lett. 80, 588共1998兲. 16It is known that, in higher magnetic fields, electron-electron

in-teraction effects关P. A. Lee and T.V. Ramakrishnan, Phys. Rev. B

26, 4009共1982兲兴 would also contribute to the

magnetoresistivi-ties which need be included in order to fully account for the experimental data.

17At very low temperatures, the experimentally measured ␸ in

many metal and semiconductor mesoscopic structures reveals a ‘‘saturation.’’ The reason for the occurrence of a saturation in␶ as T→0 is currently under intensive theoretical and experimen-tal investigations. For example, see the recent review in Ref. 3; P. Mohanty, E.M.Q. Jariwala, and R.A. Webb, Phys. Rev. Lett.

78, 3366 共1997兲; F. Pierre and N.O. Birge, ibid. 89, 206804

共2002兲; and J.J. Lin, T.J. Li, and Y.L. Zhong, J. Phys. Soc. Jpn.

72, 7共2003兲, Suppl. A.

18F. Komori, S. Kobayashi, and W. Sasaki, J. Phys. Soc. Jpn. 52,

368共1983兲, Suppl. A.

19J.J. Lin and C.Y. Wu, Europhys. Lett. 29, 141共1995兲.

20S.Y. Hsu, P.J. Sheng, and J.J. Lin, Phys. Rev. B 60, 3940共1999兲. 21Y.L. Zhong, J.J. Lin, and L.Y. Kao, Phys. Rev. B 66, 132202

共2002兲.

22A. Sergeev and V. Mitin, Phys. Rev. B 61, 6041共2000兲;

Euro-phys. Lett. 51, 641共2000兲.

BRIEF REPORTS PHYSICAL REVIEW B 68, 073407 共2003兲

數據

Figure 1 shows the measured, normalized magnetoresis- magnetoresis-tivities 䉭 ␳ (B)/ ␳ 2 (0) ⫽关 ␳ (B) ⫺ ␳ (0) 兴/ ␳ 2 (0) 共symbols兲 and the three-dimensional weak-localization theoretical  predic-tions 共solid curves兲 for a representative film Sc 85 Ag 15 ,
Figure 2 shows the extracted electron dephasing time ␶ ␸ as a function of temperature for five Sc 100 ⫺x Ag x samples

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