We present phase coherence time measurements in quasi-one-dimensional Ag
wires doped with Fe Kondo impurities of different concentrations $n_s$. Due to
the relatively high Kondo temperature $T_{K}\approx 4.3K$ of this system, we
are able to explore a temperature range from above $T_{K}$ down to below $0.01
T_{K}$. We show that the magnetic contribution to the dephasing rate $\gamma_m$
per impurity is described by a single, universal curve when plotted as a
function of $(T/T_K)$. For $T>0.1 T_K$, the dephasing rate is remarkably well
described by recent numerical results for spin $S=1/2$ impurities. At lower
temperature, we observe deviations from this theory. Based on a comparison with
theoretical calculations for $S>1/2$, we discuss possible explanations for the
observed deviations.Comment: 4 pages, 3 figures, submitted to Phys. Rev. Let

A. D. Wieck

A. Melnikov

A. Rosch

A. C. Hewson

C. Bäuerle

D. Mailly

D. Reuter

E. Akkermans

F. Mallet

J. Ericsson

L. Saminadayar

P. Mohanty

S. Ünlübayir

T. A. Costi

T. Micklitz

English

arXiv

Crossref

Scaling of the Low-Temperature Dephasing Rate in Kondo Systems

We present phase coherence time measurements in quasi-one-dimensional Ag wires doped with Fe Kondo impurities of different concentrations n(s). Because of the relatively high Kondo temperature T-K approximate to 4.3 K of this system, we are able to explore a temperature range from above T-K down to below 0.01T(K). We show that the magnetic contribution to the dephasing rate gamma(m) per impurity is described by a single, universal curve when plotted as a function of T/T-K. For T > 0.1T(K), the dephasing rate is remarkably well described by recent numerical results for spin S=1/2 impurities. At lower temperature, we observe deviations from this theory. Based on a comparison with theoretical calculations for S > 1/2, we discuss possible explanations for the observed deviations

Mallet, F.

Ericsson, J.

Saminadayar, L.

Bäuerle, C.

Mailly, D.

Janssen, S.

Reuter, D.

Melnikov, A.

Wieck, A. D.

Micklitz, T.

Rosch, A.

Costi, T. A.

Juelich Shared Electronic Resources

Scaling of the Low-Temperature Dephasing Rate in Kondo SystemsF. Mallet,1,2 J. Ericsson,1,2 D. Mailly,4 S. U¨ nlu¨bayir,5 D. Reuter,5 A. Melnikov,5 A. D. Wieck,5 T. Micklitz,6 A. Rosch,6T. A. Costi,7 L. Saminadayar,1,2,3 and C. Ba¨uerle1,21Centre de Recherches sur les Tre`s Basses Tempe´ratures, B.P. 166 X, 38042 Grenoble Cedex 09, France2Institut Ne´el, B.P. 166 X, 38042 Grenoble Cedex 09, France3Universite´ Joseph Fourier, B.P. 53, 38041 Grenoble Cedex 09, France4Laboratoire de Photonique et Nanostructures, route de Nozay, 91460 Marcoussis, France5Lehrstuhl fu¨r Angewandte Festko¨rperphysik, Ruhr-Universita¨t, Universita¨tsstr. 150, 44780 Bochum, Germany6Institute for Theoretical Physics, University of Cologne, 50937 Cologne, Germany7Institut fu¨r Festko¨rperforschung, Forschungszentrum Ju¨lich, 52425 Ju¨lich, Germany(Received 6 July 2006; published 30 November 2006)We present phase coherence time measurements in quasi-one-dimensional Ag wires doped with FeKondo impurities of different concentrations ns. Because of the relatively high Kondo temperature TK 4:3 K of this system, we are able to explore a temperature range from above TK down to below 0:01TK.We show that the magnetic contribution to the dephasing rate m per impurity is described by a single,universal curve when plotted as a function of T=TK . For T > 0:1TK , the dephasing rate is remarkably welldescribed by recent numerical results for spin S  1=2 impurities. At lower temperature, we observedeviations from this theory. Based on a comparison with theoretical calculations for S > 1=2, we discusspossible explanations for the observed deviations.DOI: 10.1103/PhysRevLett.97.226804 PACS numbers: 73.23.b, 72.15.Qm, 73.20.Fz, 75.20.HrThe Kondo effect has been a central theme in solid statephysics for several decades. The fascination it still arousesis due to the fact that it represents a paradigm of the genericproblem of dynamical impurities in metals [1]. Its mostwell-known manifestation is the logarithmic increase ofthe low-temperature resistivity of metals containing mag-netic impurities [2]. The phenomenon is accompanied by aprogressive screening of the spin of the impurities by thesurrounding conduction electrons, resulting, in the case ofcomplete screening, in a Fermi liquid ground state [3]. Inthe last decade, a number of experiments have also dem-onstrated [4–7] that Kondo impurities provide an efficientmechanism for electronic decoherence at low temperature.Conversely, measurement of the phase coherence time is apowerful tool to probe the ground state of a Kondo system[8].On the theoretical side, it has been shown very recently[9] that the dephasing rate of electrons scattering fromdiluted Kondo impurities can be calculated exactly fromthe inelastic scattering cross section [10] by usingNumerical Renormalization Group (NRG) techniques.The calculated full temperature dependence of the dephas-ing rate [9] thereby allows for a quantitative comparisonbetween experimental data and theoretical predictions,bridging the gap between the low-temperature Fermi liquidtheory and the ‘‘high-’’temperature Nagaoka-Suhl expan-sion. Indeed, recent experiments [11] have confirmedqualitative aspects of these NRG calculations; in particular,the dephasing rate has been found to be linear with tem-perature over almost one decade below TK. The relativelylow Kondo temperature of the system used in that work, onthe other hand, did not allow to perform measurementswell below 0:1TK where the Fermi liquid regime shouldappear. In this context, a natural challenge is to investigatethe very low-temperature limit of the magnetic contribu-tion to the dephasing rate m.In this Letter, we present measurements of the phasecoherence time  of AgFe Kondo wires with differentmagnetic impurity concentrations and down to tempera-tures of 0:01TK. We show that the magnetic contribution tothe dephasing rate m per magnetic impurity presents auniversal scaling when plotted as a function of T=TK. Thisdephasing rate is remarkably well described by the recentnumerical result for spin S  1=2 impurities [9,10]. On theother hand, at very low temperatures (T < 0:1TK), weobserve deviations from this result.In order to be able to explore the very low-temperaturelimit of Kondo systems, several severe experimental re-quirements have to be met. First of all, a suitable Kondosystem with a relatively high Kondo temperature is neededin order to be able to attain temperatures of the order of0:01TK. At the same time, one would like to avoid that theKondo maximum of the dephasing rate is masked by thephonon contribution. Secondly, one has to be able to startwith a very pure sample which already shows good agree-ment with theory in the absence of magnetic impurities. Inaddition, one would like to be able to use identical samplesand add magnetic impurities with a very high precision.A Kondo system which satisfies the above conditions isAg=Fe with a Kondo temperature of around 3 K [12]. Forthis system, we can easily attain temperatures well below0:1TK. In addition, we are able to prepare extremely puresilver samples [4]. The idea is hence to fabricate severalidentical quantum wires (see inset of Fig. 1) on the samePRL 97, 226804 (2006) P H Y S I C A L R E V I E W L E T T E R S week ending1 DECEMBER 20060031-9007=06=97(22)=226804(4) 226804-1 © 2006 The American Physical Societywafer. Subsequently, the samples are implanted in a verycontrolled manner with magnetic impurities via FocusedIon Beam (FIB) technology. The Fe ion implantation isdone with an energy of 100 keV and the implanted ionconcentration ns is determined directly via the ion current[13]. In Table I, we summarize the parameters of oursamples.We define TK as the Kondo temperature extracted fromresistivity measurements. The electrical resistance of aquasi one-dimensional metallic wire containing magneticimpurities at low temperature is given by RT  R0  =Tp  ns  fT=TK (1)where R0 is the residual resistance and ns is the magneticimpurity concentration. The second term corresponds tothe electron-electron interaction term [14,15] and the thirdterm to the Kondo contribution. The electron-electroninteraction contribution can be determined independentlyby measuring the resistivity of a clean sample (Ag 2)containing no magnetic impurities. Fitting the tempera-ture dependence of the resistance variation to RT exp=Tp, we obtain exp  0:081 K1=2, in goodagreement with the theoretical value theo  2R2=RK=L@D=kBp  0:082 K1=2, where RK  h=e2 is thequantum of resistance. The Kondo contribution to theresistivity per impurity concentration is shown in Fig. 1for sample AgFe2 and AgFe3. Both data sets scale nicelywith impurity concentration, and one observes typicalfeatures of a Kondo system: a logarithmic increase belowthe Kondo minimum and a saturation at the lowest tem-peratures. Fitting the resistivity data to the NRG resultsfor S  1=2 [16,17], we obtain a Kondo temperature ofTK  3:0 0:3 K.Phase coherence measurements, on the other hand, aremuch more sensitive to the presence of magnetic impuri-ties. Here we determine the phase coherence time  asdisplayed in Fig. 2 by fitting the low field magnetoresis-tance to standard weak localization theory [14,18].The experimentally measured phase coherence time for a metallic quantum wire containing magnetic impuri-ties can be described by the following expression [19] 1 1e-e 1e-ph 1m(2)where 1e-e atheoT2=3 2p RRKkB@DpL2=3T2=3 (3)corresponds to the electron-electron interaction term [15],1=e-ph  bT3 to the electron-phonon interaction, while1=m  m corresponds to the magnetic contribution tothe dephasing. In absence of magnetic scattering, the tem-perature dependence of the phase coherence time is deter-mined by the first two terms in Eq. (2) [14]. Our data on theclean wire (sample Ag2) follow nicely the expected tem-perature dependence down to 40 mK (see Fig. 2). Fromfitting the data, we extract aexp  0:42 ns1 K2=3 and b 0:04 ns1 K3 [22]. At very high temperatures (T > 5K),all samples follow essentially the same temperature depen-dence. In this regime, the phase coherence time is mainlydetermined by electron-phonon scattering.The phase coherence time  for samples containingmagnetic Fe impurities shows a quite different behavior.One first observes a plateau around 2 K and subsequently adesaturation of the phase coherence time. This can beunderstood within the framework of Kondo physics. Attemperatures above the Kondo scale TK spin flip scatteringis dominant and leads to a plateau in . At temperaturesbelow TK, the magnetic impurity spins get screened and thephase coherence time increases. Using the NRG results forS  1=2 [9] to determine the magnetic contribution to thedephasing in Eq. (2), we can fit the experimental data (seeFig. 2) and extract TK as well as the magnetic impurityconcentration ns using the value   1:03	 1047 J1 m3TABLE I. Sample characteristics: l, R, , and D correspond tothe length, electrical resistance, resistivity, and diffusion coeffi-cient. ns corresponds to the nominal impurity concentrationmeasured via the ion current during the implantation. ns is theimpurity concentration extracted from fitting m to the NRGS  1=2 theory. All samples have a width of w  150 nm andthickness of t  50 nm.Samplel(m)R()(cm) Dns(ppm)ns(ppm)AgFe1 745 877 0.88 429 2.7 1.3AgFe2 405 517 0.96 400 27 13AgFe3 160 222 1.04 360 67.5 33Ag2 765 1305 1.28 295 0    FIG. 1 (color online). Resistivity per magnetic impurity con-centration as a function of temperature for sample AgFe2and AgFe3. The experimental data have been superposed atlow temperatures and the electron-electron contribution hasbeen subtracted. The solid curved lines are the NRG resultsfor S  1=2, S  1, and S  3=2.PRL 97, 226804 (2006) P H Y S I C A L R E V I E W L E T T E R S week ending1 DECEMBER 2006226804-2for the density of states [4]. We obtain ns  1:3, 13, and33 ppm for sample AgFe1, AgFe2, and AgFe3, respec-tively. The Kondo temperature for all samples are on theorder of TK  4:3 0:2 K.To extract specifically the dephasing rate due to mag-netic impurities m, we subtract the experimentally mea-sured electron-electron and electron-phonon contributionof the clean sample. We then plot our data scaled permagnetic impurity as a function of T=TK as shown inFig. 3. In addition, we also plot data of sample AuFe2 ofRef. [11] having a different Kondo temperature. All datapoints fall on a universal curve. For comparison, we plotthe NRG results for S  1=2 [9] as indicated by the arrow.We find remarkable agreement with this theory down totemperatures of 0:1TK. At lower temperatures, however,we observe deviations from this theory, as emphasized onthe log-log plot in the inset of Fig. 3. The fact that weobserve a universal curve for different impurity concen-trations which depends only on T=TK is a strong indicationthat we are in the single impurity limit. On the other hand,it is clear that our data are not consistent with the Fermiliquid prediction of the spin 1=2 impurity model where aT2 [23] temperature dependence of the magnetic scatteringtime is expected at low temperature.It is quite remarkable that the temperature dependenceof m above 0:1TK is this well described by the S  1=2,single channel model. Fe is expected to be characterized byboth an orbital degree of freedom and a much larger spin,S  2, coupling to electrons in up to five angular momen-tum channels. The relevant theoretical model depends onthe position of the Fe within the Ag crystal structure,crystal field effects, and the strength of the spin-orbitcoupling and is presently not known. An obvious candidateto explain the weak temperature dependence of m for lowT is to assume that the large spin of Fe is only partiallyscreened. This occurs when the effective number of chan-nels nc is smaller than 2S (for cubic symmetry nc  2 or 3is expected).We have therefore computed m and the Kondo contri-bution to the resistance RT for 1=2  S  5=2 in thesingle channel, nc  1, limit as shown in Figs. 1 and 3 forS  3=2 [24]. Already for S  1, the temperature depen-dence of both, the resistivity as well as the dephasing rate ismuch slower than for the case of S  1=2. By screening asingle channel, one can only reduce the size of the spin by1=2. The resulting spin of size S 1=2 remains, however,so strongly coupled to the electrons that m remains largeeven considerably below TK. The corresponding verybroad maximum around TK is clearly inconsistent withthe experimental result already for S  1 as is shown inFig. 3. While the resistivity is much less sensitive at low T,similar conclusions can be drawn by comparing our resultsfor T to the NRG theory in Fig. 1.The above analysis shows that the experimental data areindeed extremely close to a spin 1=2 model and we candefinitely rule out any underscreened scenario. Also anoverscreened Kondo model with nc > 2S can not explainthe experimental data as it would lead to an even largerdephasing rate at low T [26]. Our data strongly indicate aperfect screening of almost all of the Fe impurities. Such ascreening may, however, involve more than one channeland a spin larger than 1=2 and also spin-orbit coupling.This may explain the differences between TK and TK aswell as nS and ns. Note that there is also a theoreticaluncertainty in the prefactor of the dephasing rate andtherefore in nS in systems with strong variations of theFermi velocity along the Fermi surface (as in Ag) asdiscussed in Ref. [27]. FIG. 3 (color online). Dephasing rate per magnetic impurity asa function of (T=TK). The NRG results, keeping 1900 states foreach NRG iteration and using the discretization parameter  1:50, for S  1=2, S  1, and S  3=2 have been scaled intemperature such that the maxima coincide with the experimen-tal data. Inset: Same data on a log-log scale. The thick solid linecorresponds to S  1=2 with a constant background added. FIG. 2 (color online). Phase coherence time of Ag wires con-taining different amounts of Fe impurities. The solid line corre-sponds to the assumption that only electron-electron andelectron-phonon interaction contribute to dephasing. The dottedline is a fit to the experimental data taking into account themagnetic contribution to the dephasing for S  1=2 impurities.PRL 97, 226804 (2006) P H Y S I C A L R E V I E W L E T T E R S week ending1 DECEMBER 2006226804-3What could then be the origin of this puzzling tempera-ture dependence of m below 0:1TK? One could argue thatthe relatively weak temperature dependence below 0:1TKoriginates from some other source, different from magneticimpurities. As m scales perfectly with the magnetic im-purity concentration, we can definitely rule out any extrin-sic effects. The curious temperature dependence of m hasto originate from the magnetic impurities themselves orfrom the implantation process. A small fraction (2%) ofimplanted Fe impurities may, for example, end up close toa lattice defect or in a grain boundary, ‘‘accidentially’’giving rise to either a much lower Kondo temperatureand/or a different crystal field or spin-orbit split statewith much less screening and consequently an enhanceddephasing rate at the relevant low temperatures. One couldalso imagine that the ion implantation creates additionallydynamical defects, like two level systems (TLS) which canlead to a much slower decay of the dephasing rate at lowtemperatures [26]. Assuming a temperature independentbackground, arising from such slow TLS, as indicated bythe thick solid line in the inset of Fig. 3, results in a verygood agreement between the experimental data and theS  1=2 result. Let us mention, however, that electron spinresonance experiments, where Cu implantations have beenperformed into a Cu host, did not see any difference beforeand after implantation down to temperatures of 3 K [28].In conclusion, we have measured the phase coherencetime of AgFe quantum wires down to temperatures of0:01TK. We have shown that the magnetic contribution tothe dephasing rate m per magnetic impurity is a universalfunction of T=TK. Its temperature dependence is in remark-able agreement with recent NRG studies for S  1=2,down to temperatures of 0:1TK. At lower temperatures,we observe deviations from this theory. The comparison ofthe experimental data with our NRG calculations for higherspins allows us to rule out a purely underscreened Kondoeffect to explain the deviations at the lowest temperatures.Ion implantation of non magnetic impurities and furthertests of scaling should allow us to shed light onto thispuzzling temperature dependence of m in the very low-temperature limit.We acknowledge helpful discussions with P. Simon,G. Zara´nd, L. Borda, L. Glazman, A. Zawadowski,S. Kettemann, J-L. Tholence, O. Laborde, K. Matho,A. Altland, P. H. Dederichs, A. Liebsch, H. Bouchiat,M. Lavagna, and D. Feinberg. We are indebted to theQuantronics group for the silver evaporation. This workhas been supported by the European Commission FP6NMP-3 Project 505457-1 Ultra-1D and ANR-PNANOQuSpin and by SFB 608 and TR12 of the DFG.Note added.—We recently became aware of a relatedwork on the same topic [21].[1] A. C. Hewson, The Kondo Problem to Heavy Fermions(Cambridge Studies in Magnetism, Cambridge, UK,1997).[2] J. Kondo, Prog. Theor. Phys. 32, 37 (1964).[3] P. Nozie`res, J. Low Temp. Phys. 17, 31 (1974).[4] F. Pierre et al., Phys. Rev. B 68, 085413 (2003).[5] P. Mohanty and R. A. Webb, Phys. Rev. Lett. 84, 4481(2000); and , 91, 066604 (2003).[6] F. Pierre and N. O. Birge, Phys. Rev. Lett. 89, 206804(2002).[7] F. Schopfer, C. Ba¨uerle, W. Rabaud, and L. Saminadayar,Phys. Rev. Lett. 90, 056801 (2003).[8] R. P. Peters, G. Bergmann, and R. M. Mueller, Phys. Rev.Lett. 58, 1964 (1987); C. Van Haesendonck, J. Vranken,and Y. Bruynseraede, Phys. Rev. Lett. 58, 1968 (1987).[9] T. Micklitz, A. Altland, T. A. Costi, and A. Rosch, Phys.Rev. Lett. 96, 226601 (2006).[10] G. Zara´nd, L. Borda, J. v. Delft, and N. Andrei, Phys. Rev.Lett. 93, 107204 (2004).[11] C. Ba¨uerle, F. Mallet, F. Schopfer, D. Mailly, G. Eska, andL. Saminadayar, Phys. Rev. Lett. 95, 266805 (2005).[12] C. Rizutto, Rep. Prog. Phys. 37, 147 (1974).[13] The SRIM simulations indicate that 22% of the ions pene-trate into the substrate. This correction is taken intoaccount for the determination of ns (Table I).[14] B. L. Altshuler, A. G. Aronov, and D. E. Khmelnitzky,J. Phys. C 15, 7367 (1982).[15] E. Akkermans and G. Montambaux, Mesoscopic Physicsof Electrons and Photons (Cambridge University Press,Cambridge, UK, 2007).[16] T. A. Costi, A. C. Hewson, and V. Zlatic, J. Phys. Condens.Matter 6, 2519 (1994).[17] T. A. Costi, Phys. Rev. Lett. 85, 1504 (2000).[18] S. Hikami, A. I. Larkin, and Y. Nagaoka, Prog. Theor.Phys. 63, 707 (1980).[19] Note that the prefactor of the magentic part may differ bya factor of 2 [8,11] depending on whether the spin relaxesfaster or slower than m [20,21].[20] M. G. Vavilov, L. I. Glazman, and A. I. Larkin, Phys.Rev. B 68, 075119 (2003).[21] G. M. Alzoubi and N. O. Birge, Phys. Rev. Lett. 97,226803 (2006).[22] This number is about 2 times larger than atheo 0:22 ns1 K2=3.[23] At very low T, one expects 1=m  T3=2 due to weaklocalization corrections to 1=m[9].[24] Similar calculations for the frequency-dependence of theinelastic cross section at T  0 can be found in Ref. [25],analytical results for m at low and high T are discussed in[20]).[25] W. Koller, A. C. Hewson, and D. Meyer, Phys. Rev. B 72,045117 (2005).[26] A. Zawadowski, J. v. Delft, and D. C. Ralph, Phys. Rev.Lett. 83, 2632 (1999).[27] T. Micklitz, T. A. Costi, and A. Rosch, cond-mat/0610304.[28] P. Monod, H. Hurdequint, A Janossy, J. Obert, andJ. Chaument, Phys. Rev. Lett. 29, 1327 (1972).PRL 97, 226804 (2006) P H Y S I C A L R E V I E W L E T T E R S week ending1 DECEMBER 2006226804-4

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Scaling of the low temperature dephasing rate in Kondo systems

Scaling of the low temperature dephasing rate in Kondo systems

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