Magnetic field dependence of dephasing rate due to diluted Kondo impurities T. Micklitz,1 T. A. Costi,2 and A. Rosch1 1 Institute for Theoretical Physics, University of Cologne, 50937 Cologne, Germany 2 Institut fu¨r Festk¨orperforschung, Forschungszentrum Ju¨lich, 52425 Ju¨lich, Germany. (Dated: February6, 2008) 7 0 Weinvestigatethedephasingrate,1/τϕ,ofweaklydisorderedelectronsduetoscatteringfromdiluted 0 dynamical impurities. Our previous result for the weak-localization dephasing rate is generalized from 2 dilutedKondoimpuritiestoarbitrarydynamicaldefectswithtypicalenergytransferlargerthan1/τϕ. For magneticimpurities,westudytheinfluenceofmagneticfieldsonthedephasingofAharonov-Bohmoscil- n lationsanduniversalconductancefluctuationsbothanalyticallyandusingthenumericalrenormalization a group. These results are compared torecent experiments. J 1 PACSnumbers: 72.15.Lh,72.15.Qm,72.15.Rn 3 l] I. INTRODUCTION ions is surprisinglywell described15 by the theoreticallypre- al dicteddephasingrateforspin-1/2Kondoimpurities13 down h Decoherenceisthefundamentalprocessleadingtoasup- totemperaturesof0.1TK. Atthelowesttemperature,again - a plateau in the dephasing rate has been observed propor- s pression of quantum mechanical interference and therefore e is indispensable for our understanding of the appearance of tionaltothenumberofimplantedFeions. Theoriginofthis m puzzling behavior is still unclear but may arise from further the classical world. The destruction of phase coherence in dynamical defects created during the implantation process . a quantum system occurs due to interactions with its en- t a vironment and can be studied, e.g., in mesoscopic metals orbyrareFeionswithadifferentchemicalenvironmentand m strongly reduced magnetic screening. and semiconductors where the quantum-mechanical wave - nature of the electrons leads to a variety of novel transport An obvious option to study the influence of magnetic d phenomena at low temperatures. impurities on the dephasing rate is to measure its depen- n Although the concrete definition of the dephasing rate, dence on an externally applied magneticfield. The applica- o c 1/τϕ, depends on the experiment used to determine it, the tionofsufficientlylargemagneticfieldsfreezesoutinelastic [ electron-electron interactions are thought to be the dom- spin-flip processes as discussed theoretically in Refs. [12], inant mechanism for the destruction of phase coherence and therefore one expects the dephasing rate to return 3 in metals without dynamical impurities below about 1 K. to the value predicted by AAK for dephasing induced by v 4 The dephasing rate for interacting electrons in a diffusive Coulombinteractionsinadiffusiveenvironment. Theorbital 0 environment was first calculated by Altshuler, Aronov and contribution of the magnetic field does, however, destroy 3 Khmelnitsky (AAK) and vanishes at low temperatures, T, the weak-localization (WL) contribution to the magneto- 0 with some power of T, depending on the dimensionality of resistance, as the joint propagation of an electron and a 1 the system.1 hole along time-reversed trajectories (the Cooperon) picks 6 up extra (random) Aharonov-Bohm phases in the presence In the last decade several independent groups performed 0 / magnetoresistance experiments2,3,4,5 to probe the influence ofexternalmagneticfields. MeasuringtheB-dependentde- t phasing rate in a WL experiment is therefore only possible a of dephasing on weak localization in disordered metallic m wires. Irritatingly, a saturationofthe dephasingrate, 1/τ , in strictly one- or two-dimensional systems using magnetic ϕ fields almost exactly parallel to such a structure, requiring has been observed at the lowest experimentally accessible - d temperatures. This observation has triggered an intense an accurate alignment of magnetic fields. n discussion on the mechanism responsible for the excess of Universalconductancefluctuations(UCF)andAharonov- o dephasing.6,7,8 The most promising candidates to explain Bohm (AB) oscillations with a periodicity of h/e, on the c the saturation of 1/τ are extremely low concentrations of other hand, are not suppressed by orbital effects and can : ϕ v dynamical impurities, such as atomic two-level systems9,10 beusedratherdirectlytodeterminethefielddependenceof Xi ormagneticimpurities3,4,5,11,12,13,14. Thishasbeencorrobo- the dephasing rate.17 ratedontheonehandbyexperiments3,5 onextremelyclean UCFscanbeobservedascharacteristicfluctuationsofthe r a Ag and Au samples where the dephasing rate continues to conductanceasafunctionofthemagneticfield. Theexter- decreasewellbelow100mK andonthe otherhand bydop- nalmagneticfieldentersthemetalandchangesthepattern ing studies with magnetic impurities. 2,3,5,15,16 As expected of the electrons wave functions and therefore the conduc- theoretically,12 theseexperimentsshowasaturationof1/τ tance in a random but reproducible way (“magnetofinger- ϕ above the Kondo temperature, T , the characteristic scale print”). In AB experiments performed onmesoscopicrings, K of screening of the magnetic moment, and a suppression thesesamplefluctuationsarefurthermodulatedbyperiodic of 1/τ below this scale. Recent highly controlled exper- h/e-oscillations resulting from the magnetic flux piercing ϕ iments,15,16 in which a few ppm (parts per million) of Fe the ring. Both UCF and AB oscillations rely on the con- ionshavebeenimplantedbyionbeamlithographyintovery structive interference occurring in the collective propaga- clean Ag samples, showed that the screening of these Fe tion of electrons and holes traveling along the same path (the diffuson). These are robust against the breaking of see Fig. 1. In the absence of dephasing by inelastic pro- time-reversal invariance (while Cooperon contributions are cesses C (q) is the bare Cooperon, Ω rapidly suppressed by small fields) but are sensitive to de- phasing by inelastic processes. Indeed, Benoit et al.18 and C0(q)= 1 , (2) Pierre et al.5 have shown that the amplitude of Aharonov- Ω Dq2+iΩ+1/τB Bohmoscillationsincreasesbyalmostanorderofmagnitude as diagrammatically depicted in Fig. 2 and21 for increasing magnetic fields clearly showing a suppression of dephasing by magnetic (Zeeman) fields. This leads to the conclusion that the main mechanism of dephasing in 1 the investigated low-temperature regime is the scattering =4DeB, d=2 (n B), (3) τ || B from magnetic impurities. 1 D Previously we have studied the zero-field dephasing rate = (eBL )2, d=1,2 (n B), (4) τB 3 ⊥ ⊥ due to diluted Kondo impurities as measured from the WL experiment.13 Weshowedthatthedephasingrateforallex- is the dephasing rate due to the applied magnetic field, B. perimentallyrelevanttemperaturesisproportionaltothein- Here D is the diffusion constant, d is the dimension of the elasticcrosssection,14 whichitselfcanbeexpressedinterms diffusion process, τ denotes the mean scattering time cor- of the T-matrix describing the scattering of the electrons responding to a mean free path l =v τ, n is a unit vector F from a single magnetic impurity. Such a relation has been orthogonaltotheprobeind=2andpointingalongthewire proposedpreviouslybySchwaband Eckern19 in the context in d=1 and L is the transverse dimension ofthe sample. of UCFs. As can be seen⊥from Eq. (1) the WL corrections depend on the strength of the applied magnetic field, B, and di- Inthispaperwegeneralizeourpreviousworktoarbitrary, verge in low dimensions, d = 1,2 for B = 0, reflecting the dilutedimpurityscattererswithtypicalenergytransferlarger fact that WL corrections in low-dimensional systems may than 1/τ . Furthermore we supply the magnetic field de- ϕ become strong and lead to strong Anderson localization. pendence of 1/τ as measured in the AB experiment (and ϕ Takingintoaccountinteractions(as e.g. providedbydy- compare our results to the AB experiments performed by Pierre and Birge.20) The outline of this paper is as follows: namicalimpurities)the bareCooperondresseswitha mass, 1/τ , i.e. (if purely exponential decay is guaranteed), In Sec. II we discuss the dephasing rate due to generic dy- ϕ namical scatterers as measured in the WL experiment. We 1 briefly review our previous results13 and generalize them to C (q)= . (5) Ω=0 Dq2+1/τ +1/τ arbitrary dynamical impurities with typical energy transfer ϕ B larger than 1/τ . In Sec. III we turn to the dephasing rate ϕ For weak magnetic fields (τ τ ) the WL corrections asmeasuredfromtheUCFandtheamplitudeoftheABos- ϕ ≪ B are therefore cut off by 1/τ , allowing to determine the cillations. We brieflyreview the mainconceptsentering the ϕ dephasing rate from fitting the magnetoresistance. analysisoftheseexperimentsanddiscusshowthedephasing Inthissectionwestudythedephasingrateduetogeneric rate measured from the UCF differs from that measured in diluted, dynamical scatterers with typical energy transfer theWL experiment. ThemaingoalofSec.IIIistogivethe larger than 1/τ as measured from WL. To be specific, we dephasing rate as measured from the amplitude of the AB ϕ consider a Hamiltonian of the general form oscillations. Results for the dephasing rate obtained using tinheSencu.mIVer.icSaelc.reVnosrummamlizaartizioens wgritohupa d(NisRcuGs)sioanre. described Himp =Hi0mp+ c†kσck′σ′fkαk′σσ′Xˆαei(k−k′)xi, (6) i X whereH0 istheHamiltonianoftheisolatedimpurity,c ,c imp † are creation and annihilation operators of conduction band electrons,x denotesthepositionoftheimpurities,andthe II. DEPHASING RATE FROM WEAK-LOCALIZATION i CORRECTIONS TO THE CONDUCTIVITY ε netTichefiemldosstisavcicaurtahtee WwaLyctoorreecxttiroancst t1o/τtϕheatDrlouwdemcoang-- p+ (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1) p− p p ductivity, which result from coherent back scattering of an − + electron-holepairtravelingalongtime-reversedpathsinthe ε disorderedenvironment. Technically,thecoherentpropaga- (cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1) tion of the electron-hole pair is described by the Cooperon, C (q), and the WL correction is given by (cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1) Ω FIG. 1: Diagrammatic representation of the Cooperon C0 (q) which enters the WL corrections to the Drude con- 2e2D ddq Ω=0 ∆σ0 = C (q), (1) ductivity. p± =q/2 p and wavy lines denote current oper- WL − π (2π)d Ω=0 ators. ± Z p ε+Ω/2 p’ p ε+Ω/2 p’ ε+Ω/2 ε′+Ω/2 + + + + R C = C + C Γ C = + 0 0 C0(q, Ω ) C0(q, Ω ) ε−Ω/2 ε′−Ω/2 (a) A p− ε−Ω/2 p−’ p− ε−Ω/2 p−’ Γ = + + + FIG. 2: Bethe-Salpeter equation for the bare Cooperon DC¯0a(sqh,eΩd)li=nes2πdν1eτn2oCt0e(qsc,aΩt)t,erpi±ng=frqo/m2±stapt,icanimdppu′±ri=tieqs/a2n±dpR′., (b) A denotes theparticle and hole lines, respectively. FIG. 3: Bethe-Salpeter equation for the Cooperon C¯ in the presence of dilute dynamical impurities to linear order in n . i momentum and spin dependent function fα parametrizes C¯0 isthebareCooperon in theabsenceof interactions andΓ the coupling to some operator Xˆ describing transitions of theirreduciblevertexobtainedbyaddingself-energy,elastic-, α andinelastic-vertexcontributions. Thecrosseswithattached the internal states of the dynamical impurity. Eq. (6), e.g., dashedlinesdenotetheaveragingoverimpuritypositions x , describes the coupling of the conduction band electrons to i the squares the inelastic scattering from a single impurity to Kondo impurities, two-level systems, etc. In order to find arbitrary order. the dephasing rate due to such generic impurities one has to compute the “self energy” or “mass” of the Cooperon generated by the operators of Eq. (6). term). Only this inelastic vertex makes the Bethe-Salpeter Twoassumptionsallow to reduce this problem to that of equation a true integral equation as it mixes frequencies summing up a simple geometric series. First, we assume but, fortunately, this term can be neglected1 if the typical that the concentration of dynamical impurities n is small, i energy, ∆E, exchanged between electrons and holes during and,second,thattheelasticmeanfree pathl is large. How an interaction process greatly exceeds the dephasing rate small n and how large l has to be, depends on both, the i due to the dynamical impurities, 1/τ , i.e., ∆Eτ 1. dynamics and the extension of the impurity as briefly dis- Physically,1 the suppression of the inelϕastic vertex aϕris≫es as cussed below and – for Kondo impurities – in more detail an exchange of energy ∆E leads to a phase mis-match in Ref.[13] (the influence of stronger disorder on the dis- of order ei∆Eτϕ between electron and hole destroying in- tribution of Kondo temperatures has recently been investi- terference completely for ∆Eτ 1. Technically, one gatedbyKettemannandMucciolo22). Notethatinrelevant ϕ ≫ can confirm this argument by estimating corrections to the experimental systems in the WL regime2,3,4,5 the two as- WL contributions due to the inelastic vertex, as, e.g., de- sumptions are well justified13. The first observation is that pictedinFig.4.13 Moreimportantly,however,thecondition quantum interference corrections to the inelastic scattering ∆Eτ 1 always holds for sufficiently small concentra- rate are smallwhen a diffusing electronis unlikelyto return ϕ ≫ tions n , since 1/τ n . In the case of Kondo impurities to the same dynamical impurity (i.e., when the weak local- i ϕ ∝ i discussed below, for example, this condition translates to ization corrections are weak). Technically speaking, this is n νT . reflected in the fact that diagrams mixing scattering from i ≪ K dynamical and static impurities are suppressed by factors Restrictingto the self-energyand elasticvertex contribu- of 1/N and a/l [or 1/(k l) for a < 1/k ], where N is tions, the Bethe-Salpeter equation is easily solved: Since F F the num⊥ber of transverse channels in a quasi 1 or 2 dim⊥en- self-energyandelasticvertexcontributionsconservetheen- sionalsystemandaisthetypicaldiameterofthedynamical ergy of single electron lines, the solution of the reduced impurity. This effect and further system dependent factors Bethe-Salpeterequationamountstoastraightforwardsum- relevantforthesuppressionofquantuminterferencecorrec- mation of a geometric series. Setting the center-of-mass tions are discussed in Ref.[13]. Only at lowest, experimen- tally unprobed temperatures, does the enhanced infrared singularity,causedbythepresenceofextradiffusionmodes, Γ overcompensatethisphasespacesuppressionfactor,asdis- cussedindetailinRef.[13]. Thesmallparameter1/(k l)or F a/l thereforereducesthe problemtocomputethe’mass’of C C theCooperontothatofsolvingtheBethe-Salpeterequation diagrammatically depicted in Fig. 3(a). For small n , one can furthermore restrict the analysis of i the irreducible vertex Γ to terms linear in n as shown in i Fig. 3(b). Γ can be separated into three distinct contribu- tions: self-energydiagrams[thefirsttwotermsinFig.3(b)], FIG. 4: Lowest order correction to WL contributions due to an ‘elastic’ vertex correction with no energy transfer be- inelastic vertex. The Cooperon, C¯, is dressed with a mass tween upper and lower line (third term), and an ‘inelastic’ resulting from summation of the elastic part of the vertex Γ, vertex where interaction lines connect the two lines (last i.e., theself energy and theelastic vertex contribution. frequency Ω to 0 (see Fig. 1, and Fig. 3), the Cooperon is where ... denotesanangularaverageweightedby1/v (p) F h i given by to take into account that fast electrons are scattered more frequently from elastic impurities. According to Eq. (10), τ is nothing but the average time needed (in a semiclassi- 1 ϕ C (ǫ,q)= , (7) cal picture) to scatter from an impurity with cross section Ω=0 Dq2+1/τ (ǫ,T)+1/τ ϕ B σ . Note that the vanishing of 1/τ for static impurities inel ϕ is guaranteed by the optical theorem. with the T and ǫ dependent dephasing rate From Eq. (9) we can read off the dephasing rate for di- 1 2n d3p 1 luted dynamical isotropic s-wave scatterers13,19 = i g (p) TA (ǫ) TR (ǫ) τϕ(ǫ,T) πν "Z (2π)3 ǫ 2i −p,−p − pp d3p d3p (cid:2) (cid:3) 1 = 2ni πνlocIm TA(ǫ) πνlocTR(ǫ)2 , − (2π)3 (2π)′3gǫ(p)gǫ(p′)TpRp′(ǫ)T−Ap,−p′(ǫ)#. τϕ(ǫ,T) πν h h i−| | i(11) Z Z (8) where νloc is the local density of states at the Fermi en- Here g (p) = π/2τ restricts the electrons mo- ǫ [ǫ(p) ǫ]2+ 1 ergy at the site of the impurity which can differ from − 4τ2 menta, p, and energies, ǫ, to the Fermi-surface, ǫ(p) is the thermodynamic density of states entering the prefac- thedispersionrelationoftheconductionband,TA,R arethe tor. Note that in the case of Kondo impurities discussed advanced/retarded T-matrices, defined by the Green func- below (and in Ref. [13]), the combination νlocTR/A(ǫ) = tion Gxx′(ǫ) = G0xx′(ǫ)+G0x0(ǫ)T(ǫ)G00x′(ǫ), and ν de- f(ǫ/TK,T/TK,B/TK)isanuniversaldimensionlessfunction notes the densityofstates per spin. Eq.(8) generalizes our of the ratios ǫ/TK,T/TK,B/TK. If the assumptions under- result of Ref. [13] to arbitrary shaped diluted impurities. lying the derivation of Eq. (11) are valid, one can therefore Notice that also forward scattering processes enter 1/τ , predictwithoutanyfreeparameterthedephasingrateifthe ϕ which do not contribute to the transport scattering rate. concentration of spin-1/2 impurities, the Kondo tempera- WestressthatEq.(8)isthegeneralresultforthedephasing ture and the thermodynamic density of states are known rate for a weakly disordered metal due to a low concentra- (see, e.g., Ref. [15]). However, one of the assumptions un- tionofgenericdynamicalimpuritiesforwhichthecondition derlying the derivationof the prefactorofEq.(11) maynot ∆E 1/τ holds. In the opposite limit, ∆E 1/τ , bevalidinrealisticmaterials: weassumedthatthestaticim- ϕ ϕ ≫ ≫ vertexcorrectionsbecomeimportant,ashasbeendiscussed puritiesarecompletelyuncorrelatedandlocalsuchthatelec- earlier on23,24 in the context of magnetic impurities. trons are scattered uniformly overthe Fermi surface. While As we assumed that a l, Eq. (8) can be further sim- thisshouldbeagoodassumptionindopedsemiconductors, ≪ plified, thismaynotbevalidinmetalswithcomplexFermisurfaces andstronglyvaryingFermi velocities. Under thelattercon- 1 2n d2p 1 ditions, we expect that the prefactor of Eq. (11) becomes = i Im πTA (ǫ) τ (ǫ,T) πν (2π)3 v (p) pp nonuniversal, yielding temperature-independent corrections ϕ "ZSFǫ | F | oforderone,whichmaybe importantfortheinterpretation (cid:2) (cid:3) d2p d2p 1 1 of high-precision experiments.15,16 −ZSFǫ (2π)3 ZSFǫ (2π)′3|vF(p)||vF(p′)||πTpRp′(ǫ)|2#, In Ref. [13] we have calculated the leading corrections to Eq. (11) arising from mixed diagrams involving com- (9) bined scattering from static and dynamical impurities and whereSFǫ istheFermi-surface(ormorepreciselythesurface fromdiagramsincludinghigherprocessesinni. Weshowed with ǫk =ǫ). Here we also assumed a time-reversal invari- that, suppressed by the small parameter 1/(kFl), their con- ant system with TR (ǫ) = TR (ǫ) and employed the tributions are negligible at all experimentally relevant tem- pp′ p′, p identity TpRp′(ǫ) ∗ = TpA′p(ǫ).−Al−so Kettemann and Muc- pteemrapteurraetsu,rTes. dOontlyheastetchoerrleocwteiosntsexbpeecroimmeenitmalplyoritrarenltevdaunet ciolo have generalized the dephasing rate for a momentum (cid:2) (cid:3) to infrared singularities of the dressed interaction potential independent T matrix to Eq. (9) independently in a recent report.22 The dephasing rate given in Eq. (9) has a simple (dressed by coherent backscattering processes). The esti- interpretation14: Since the Fermi-surface integrated imagi- mates of subleading corrections presented in Ref. [13] can be generalized to extended dynamical impurities by replac- nary part of the T-matrixis proportional to the total cross- section and TR 2 (integrated over the Fermi surface) is ing 1/(kFl) by a/l for kFa>1. proportional|toptph′e| elastic crosssection,its difference is,by Theexperimentallymeasureddephasingratedoesnotre- definition, proportional to the inelastic cross section, σ , solve the dependence on the electrons energy ǫ. We de- inel introducedinRef.[14]. Therefore,Eq.(9)canberewritten scribed in Ref. [13] that to allow for a comparison with in the form the ǫ–independent dephasing rate, τϕ−1(T), extracted from the WL experiment, the energy–resolved representations of 1 =ni vF(p)σinel(p,ǫ) , (10) 1/τϕ, Eq. (9)-(11), still require an average over energies τϕ(ǫ) h i according to For wires of length L L = D/T, the fluctuations T ≫ of the conductance, δgδg, are determined25 by 2 p 1 − dǫfF′(ǫ)τϕ(ǫ,T)2−2d d−2 d=1,3, τϕ(T) = hτ1 exRp dǫfF′(ǫ)lnτϕ(τǫ,Ti) d=2, δgδg = (32πe2TDL)42 dǫ1dǫ2fF′(ǫ1)fF′(ǫ2)  −R dǫhfRF′(ǫ)/τϕ(ǫ,T) i τϕ/τB ≫(11.2) Z dx1dx2|Pǫ1,ǫ2(x1,x2)|2. (16) Z Here the lastlineappliesto acase wherea relativelystrong magnetic field, B, is present. Here P (x ,x ) is the amplitude for an electron-hole ǫ1,ǫ2 1 2 Specifying to a situation where the coupling of the con- pair, with energies ǫ ,ǫ respectively, to diffusively travel 1 2 duction band electrons to the (diluted) dynamical impuri- from x to x along the same trajectory (diffuson, see 1 2 ties is described by the spin-1/2 Kondo effect, the general Fig. 5). The overbar denotes the ensemble average, which Hamiltonian of Eq. (6) takes the form is experimentally realized by changing the magnetic field. Eq. (16) assumes the large ring diameters, L L , such ϕ ≫ that the dephasing rate, 1/τ , controls the magnitude of ϕ HS =J Sˆic†σ(xi)σσσ′cσ′(xi), (13) the fluctuations. Furthermore we assume temperatures Tτ 1. Noticethat generallythere is alsoa contribution Xi ϕ ≫ from the Cooperon, which is, however, suppressed already where J is the exchange constant. An external magnetic for small magnetic fields. field, B, causes Zeeman splitting, ǫ , of the conduction z Wechangetomomentumrepresentationandseparatethe band electron spin states, two-particle propagator P into its spin-singlet and -triplet components, P = P(i), where (following the no- i=1,..,4 tation of Ref. 12) ǫ =g µ B, (14) P z e B and couples to the impurity spins according to P(i) (q)= ǫ1ǫ2 1 . HB =gSµBB Sˆzi. (15) Dq2+i(ǫ1−ǫ2+ζiǫz)+1/τS(iO) +1/τϕ(i,)S(ǫ1,ǫ2,B) (17) i X g , g are the electrons and the magnetic impurities gyro- e S The modes i = 1,2,3 describe the spin triplet state with magnetic factors, respectively. As already mentioned in the S component equal to 1, 1, and 0, respectively. i = 4 introduction, measuring the B-field dependence of the de- z − denotes the spin singlet channel. The Zeeman splitting en- phasing rate due to Kondo impurities in a WL experiment ters only the triplet-diffuson with nonvanishing projection is a highly delicate task. In view of this difficulty it is more S = 1, i.e. ζ = 1 for i = 1,2 and zero otherwise. feasibletomeasuretheB-dependenceof1/τ fromtheam- z i ϕ ± ± plitude of the AB oscillations as discussed in the following 1/τ(i) is the spin-orbit scattering rate. 1/τ(i) is identical SO SO section. for the three spin triplet-diffuson (i = 1,2,3) and zero for thespinsingletmode(i=4). Forstrongspin-orbitscatter- ing only the singlet diffusion contributes (otherwise 1/τ(i) SO III. DEPHASING RATE FROM UNIVERSAL isanadditionalfittingparameter). Finally1/τ(i) isthede- ϕ,S CONDUCTANCE FLUCTUATIONS AND phasing rate for the ith diffuson mode due to the presence AHARANOV-BOHM OSCILLATIONS Let us first begin with a brief discussion of the universal p ε+Ω/2 p’ p ε+Ω/2 p’ + + + + conductance fluctuations (UCF) and their dependence on R 1/τ and then turn to the experiment on Aharonov-Bohm = + ringϕs. D0(q, Ω ) D0(q, Ω ) A Tobespecific,weconsiderawireofnon-interactingelec- trons, scattering elastically from static impurities, and in- p− ε−Ω/2 p−’ p− ε−Ω/2 p−’ elastically off a low concentration, n , of Kondo impuri- i ties, where the coupling of the conduction band electrons FIG. 5: Bethe-Salpeter equation for the bare diffuson tothedynamicalimpuritiesisdescribedbytheHamiltonian D¯0(q,Ω)= 2πν1τ2D0(q,Ω). p± =p±q/2 and p′± =p′±q/2. Eq.(13)andtheinfluenceofthemagneticfieldisaccounted Dashed lines denote scattering from static impurities and R, for by Eqs. (14) and (15). A denotes theparticle and hole lines, respectively. R D = D + D Γ D 0 0 (a) A R Γ = + + + A (b) FIG. 6: Diagram giving the main contribution to the UCF. Dashed lines represent coherent impurity scattering of elec- FIG. 7: (a) Bethe-Salpeter equation for the diffuson, D¯, in tron(R)hole(A)pair,i.e. thebarediffusonwhereinteraction the presence of (dilute) magnetic impurities to linear order due to scattering from magnetic impurities is not yet taken in ni. D¯0 is the bare diffuson in the absence of interactions. into account. (b) Diagrammatic representation of the irreducible interac- tion vertex, Γ, consisting of the self-energy (represented by the first two contributions), the elastic vertex (third contri- of diluted magnetic impurities which has the structure bution). The inelastic vertex (the fourth contribution) does not enter thediffuson as measured in theUCF. 1 2n πν = i T(i,a)(ǫ ,B) T(i,b)(ǫ ,B) τ(i) (ǫ ,ǫ ) πν 2i 2 − 1 ϕ,S 1 2 (cid:2) (cid:3) (πν)2T(i,c)(ǫ ,B)T(i,d)(ǫ ,B) , (18) We point out the following differences for 1/τ mea- 1 2 ϕ,S − ! sured from the UCF experiment, Eq. (18), compared to that found from the WL, Eq. (11). First, the T-matrices where the proper combination of T-matrices for the vari- entering Eq. (18) depend on the spin configuration of the ous channels can be read off by comparison with Table I. diffuson-mode and have acquired a B-dependence due to Eq.(18)isevaluatedfromsummingupself-energyandelas- the coupling of the impurity spin to B, Eq. (15). Second, tic vertex contributions. Notice that in contrast to the 1/τ dependsontwoenergies. Thisresultsfromthefact, ϕ,S WL experiment the electron and hole lines (i.e., the inner that in the UCF experiment electron and hole lines consti- andouterrings)inFig.6representdifferentmeasurements. tuting the diffuson are produced in different measurements Therefore there are no correlations between dynamical im- of the conductance (see Fig. 6). Therefore their energies purities residingon different ringsand interactionlines may are individually averaged as can be seen in Eq. (16). only be drawn within the same ring. Consequently the in- elasticvertexcontributionsdonotentertheBethe-Salpeter From Eq. (16) and (17) the amplitude of the UCFs is equation for the diffuson, see Fig. 7. Notice that there are obtained to be proportionalto √τϕ,S. Especiallycompared inelastic vertex contributions, as, e.g., depicted in Fig. 8, to the AB oscillations, discussed below, the 1/τ depen- ϕ,S whichbecomeimportantinthecontextofelectron-electron dence of the UCFs is rather weak. In the following, we will interactions.26 It is instructive to compare those to the in- therefore focus our discussion on AB experiments. elastic vertex corrections relevant for WL depicted in Fig. 4. In the latter case, the sum of the incoming momenta of Aharonov-Bohm oscillations are measured in a ring ge- the vertex is small due to the Cooperon in Fig. 4. Conse- ometry,20,28 where the conductance oscillates periodically quently, the inelastic vertex corrections to the WL dephas- as a function of B piercing the ring, and the amplitude of ingratearenotsuppressedbypowersof1/(k l)butonlyby F these AB-oscillations decrease exponentially with 1/τ . ϕ,S powersof1/(∆Eτ ). In contrast,the relevant momentain ϕ The periodic oscillations result from the change of bound- Fig.8areuncorrelated(i.e.,particleandholearefarapart), ary conditions, due to flux lines piercing the ring and can leading to an suppressionboth by powers of 1/(k l) and of F 1/(∆Eτ ).27 ϕ i S,M Combinations of T-matrices 1 |S =1i,M =1 T1= 1 TA TR TRTA D D 2i` ↓ − ↑´− ↓ ↑ 2 S =1,M = 1 T2= 1 TA TR TRTA Γ − 2i` ↑ − ↓´− ↑ ↓ 3 S =1,M =0 T3= 1Im TA+TA 1 TRTA TRTA 4 S =0 T4= 21Im`T↑A+T↓A´−12`T↑RT↑A−T↓RT↓A´ D D 2 ` ↑ ↓´−2` ↑ ↑ − ↓ ↓´ TABLEI:CombinationofT-matricesenteringthedephasing FIG. 8: Diagrammatic representation of lowest order correc- rates for spin-triplet and spin-singlet diffusons. S denotes tions to UCF due to inelastic vertex contributions. Notice the total spin and M its z component. T↑,T↓ denotes the that for a local interaction these are not only suppressed by T-matrix for spin-up and spin-down electrons, respectively. powers of 1/(∆Eτϕ) but also small in powers 1/(kFl). be calculated from describes the typical ’hitting rate’. As in the WL experiment discussed above, fits to exper- (2e2D)2 δg(φ)δg(φ+∆φ)= dǫ dǫ f (ǫ )f (ǫ ) imental data have to be done with the ǫ-independent de- 3πTL4 1 2 F′ 1 F′ 2 phasing rate. For a comparison with experiment we there- Z dx dx P∆φ (x ,x )2. fore have to give the ǫ-independent dephasing rate which 1 2| ǫ1,ǫ2 1 2 | for k =1 is obtained by solving the equation Z (19) Here P∆φ is again given by Eq. (17) but now the con- Lϕ(T,B,L) L e−Lϕ(T,B,L) tinuous q have to be replaced25 by discrete momenta, L q = q (∆φ) = 2π(m + ∆φ), depending on the differ- L m L φ0 3 e−Lϕ(ǫ,T,B) Lϕ(ǫ,T,B) ence of the magnetic flux during the individual measure- = dǫ , (25) 8T cosh4(ǫ/2T) L ments of g. The fluctuations are a periodic function in Z ∆φ/φ , where φ = 2π/e is the elementary flux-quantum 0 0 where L = Dτ is the dephasing length. Notice that and ∆φ=∆BL2/(4π). Therefore an expansion in its har- ϕ ϕ theactuallymeasureddephasingratedependsonthelength monics can be made,25 p of the ring. It also differs from the WL result due to the different energy averages. Ce4 ∞ ∆φ δg(φ)δg(φ+∆φ)= (B)cos 2πk , π2 Ak φ k=0 (cid:20) 0 (cid:21) IV. NUMERICAL RESULTS FOR DEPHASING RATES X (20) FROM AHARANOV-BOHM OSCILLATIONS where C is a factor of order 1, depending on the sample In order to evaluate the dependence of the dephasing geometry in the vicinity of the ring. Restricting to the situation of strong spin-orbit scattering29 where the spin rate on magnetic field, temperature, and ring length from Eq.(27), we require the T-matrix for the single impurity singlet-diffuson gives the leading contributions to Eq. (19), Kondo model defined in Sec. II. By using the equation of one finds that (for L L ) ≫ ϕ motion method this can be expressed as30 kL (2π)3D3/2 e−√Dτϕ(ǫ) Ak(B)= T2L3 dǫcosh4(ǫ/2T) τϕ(ǫ), (21) Tσ(ω,T,B) = JhSzi+J2 Sc†ασασ;Sσσα′cα′ (26) Z q where ... denotes a retarded(cid:10)(cid:10)correlation function(cid:11)(cid:11)and σ where [τ =τ(4) in the notation of Eq. (18)] hh ii ϕ ϕ,S arethePaulispinmatrices. WecalculateEq.(26)byapply- ing the numerical renormalization group (NRG) method31 1 for finite temperature dynamics32. At finite magnetic field, = it is also important to use the reduced density matrix33 to τ (ǫ,T,B) ϕ evaluate the above dynamical quantity. For all calculations 2ni πνIm TA (ǫ,B) (πν)TR (ǫ,B)2 . presented here we used a discretization parameter for the πν (4) −| (4) | ! conductionbandofΛ=1.5andweretained 960states per h i NRG iteration. We checked that this number of states was (22) sufficient to maintain particle-hole symmetry of the spec- Hereǫ=ǫ +ǫ andweusedthatrelevant contributionsto traldensitiesImT (ω,T,B)=ImT ( ω,T,B)atthisrel- 1 2 theintegraloverenergydifferences,ǫ¯=ǫ ǫ ,resultfrom atively small valu↑e of the discretiza↓ti−on parameter. The 1 2 − energies ¯ǫ 1/τ to eliminate the ¯ǫ dependence. Notice FriedelsumrulefortheT =0spectraldensitywassatisfied ϕ ≪ thatsuchareductiontoasingleenergy-integralcanonlybe to more than 1% accuracy in our calculations. doneinaone-dimensionalsystemwheretheq-integralover Fig.9showsthenumericalevaluationof (B)forvar- 1 A thesquareofthediffuson,Eq.(17),isdominatedbyinfrared ious T at a given length L= 10L where L =√Dτ . hit p hit hit divergences. In a 2-d system, e.g., relevant energies extend Notice that, if lengths are measured in units of L , the hit to ¯ǫ T. For the following, it is convenient to rewrite amplitude (A τ T )1/2 becomes a universal function of ∼ 1 hit K Eq. (22) [see also Eq. (10)] as B/T and T/T . K K Fig. 10 gives the magnetic field dependence of the de- 1 1 σ = h ineli, (23) phasingrateatvariousT andafixedringlengthL=10Lhit. τ τ σ ϕ hit max For large magnetic fields, B T,TK, the dephasing rate isexpected12 tovanishpropor≫tionalto(T/B)2/ln4[B/T ], where σ =4π/k2 is the cross section of a unitary scat- K max F consistent with the numerical results (the precise form also terer, σ /σ is the conditional probability of inelastic inel max depends on L/L , see below). scattering if an electron hits the impurity, and ϕ Fig.11showstheresultsfor1/τ asafunctionofT for ϕ,S 1 2n different strengths of the magnetic field B at a fixed ring i = (24) τhit πν length L = 10Lhit. While the maximal dephasing occurs 1 B=0T 3/2 B=0.5KT ) K π 0.25 B=1T 2 K ( B=2T / K 1/2 0.1 hit 0.2 BB==48TTK ) 0.2T τ K TK 0.4TKK τ φ 0.15 B=16TK τ hit0.01 00..68TTK 1/ 0.1 ( K 2 1T 1/) 2TK 0.05 A1 K 0.001 5T ( K 0 0 2 B/T 4 6 0 1 2 3 4 T/T5 6 7 8 9 10 K K FIG. 9: Amplitude of the Aharonov-Bohm oscillations [in FIG.11: NRGresults for thedephasingrate asa function of unitsof((2π)3/(τhitTK))1/2]asafunctionoftheappliedmag- T and for different values of B. τhit = 2πnνi as defined above, netic field, B (in units TK, µB/kB =1), for different temper- T and B are given in units of TK34 (we set µB/kB =1). The atures T (in units TK) obtained from NRG calculations for values are for a ring of length, L, L = 10Lhit where Lhit = L/Lhit = 10. The rapid rise of the oscillation amplitude re- √Dτhit. Whilethemaximaldephasingoccursfor T ∼TK for sults from the suppression of dephasing (see Fig. 10) by po- small fields B TK, it shifts to larger values (T B) for ≪ ∼ larizing the spins. Here, and in the remaining figures in this B TK. ≫ section, thesymbols representthediscretevaluesof(B,T) at which NRGcalculations were carried out. can be seen from Fig. 12 the dephasing rate only changes by a factor 1/4 on increasing the ring length by a factor 200. We included curves for the rather academic cases 0.25 2T L = 100 1000L (the amplitude is too strongly sup- K hit 1T pressed to−be observed for such large ring lengths) in order K 0.2 0.8T to show this weak L-dependence. Fig. 12 also shows that K 0.6T for temperatures T . T the dephasing rate becomes en- K K hit0.15 0.4TK tirely L-independent. This reflects the fact that the energy τ 0.2T resolved dephasing rate, 1/τ (ǫ,T), establishes a deep φ K ϕ,S τ 1/ 0.1 0.05 0.25 L=5L hit L=20L hit 00 2 4 6 L=100L B/T 0.2 hit L=200L K hit FIG. 10: The dephasing rate as a function of the magnetic τhit 0.15 LL==510000L0Lhit hit φ field,B (inunitsTK) forvarioustemperatures, T,andaring τ / of length L = 10Lhit. Note the rapid suppression of 1/τϕ 1 0.1 especially for low temperatures. 0.05 for T T for small fields B T , it shifts to larger val- K K ues (T∼∼ B) for B ≫ TK. F≪or high temperatures, small 00 1 2 3 4 T/T5 6 7 8 9 10 fields and not too large L/Lϕ, see below, T TK,B the K ≫ dephasing rate is well described by the Nagaoka-Suhl for- mula,13 1/τ (T)= ni π23/4 . FIG. 12: Temperature dependent dephasing rate for various Fig. 12 shϕows the2πdνepπe2n3/d4e+nlcne2To/fT1K/τϕ,S(T) on the ring ring lengths L, measured in units Lhit = qD2nπiν and calcu- length, L, at zero magnetic field. As pointed out above, lated via NRG for B =0. The logarithmic suppression with the L-dependence of the experimentally measured dephas- Lfor larger T arises duetotheinterference of electrons with ing rate enters through the energy average of Eq. (25). As energies larger than T. minimum at ǫ = 0 for T . T and the integral on the K right hand side of Eq. (25) is therefore well approximated by setting ǫ=0. To be precise, in the limit of ring lengths 0.08 L Lϕ theintegralontheright handsideofEq.(25)(for T=100mK ≫ NRG, T/T = 9.1 fixed T and B) is dominated by the saddle points of the K PT, T = 100 mK function 0.06 T=40mK h) NRG1, T/TK = 4.1 2/ NRG2, T/TK = 4.1 L L(ϕ4)(ǫ) ǫ (e0.04 PT, T = 40 mK f(ǫ)= ln +4ln cosh . L(ϕ4)(ǫ) − " L # 2T ∆G h (cid:16) (cid:17)i For temperatures T . max B,T f has a saddle point 0.02 K { } at ǫ = 0, which for temperatures T & T becomes unsta- K ble. At very large ring diameters L/L 102, a second hit saddle point at ǫ = L T starts≫to dominate the 0 -100 0 100 2Lhitln2h2LLhiti gB/TK integral for T & T . Here L = √Dτ is the diffusive K hit hit length scale correspondingto the time τhit = 2πnνi and intro- FIG.13: AmplitudeoftheABoscillations inunitsofe2/has duced above. Although this limit is rather academic it is afunction of T/T for T =40mK (∆) and100mK ( ) mea- K interesting that for such big rings dephasing is dominated sured by Piere et al.5,20, assuming T = 10mK. So⋄lid and K by rare events of highly excited thermal electrons scatter- dot-dashed lines are the numerically calculated amplitudes ing from the magnetic impurities. This originates from the with fitting parameters described in the main text. As for fact that high-energy electrons scatter less effectively from very high magnetic fields, B 100TK, numerical errors in- ≫ Kondo spins, as Kondo renormalization becomes less effec- crease when the dephasing rate becomes very small, we used tive for ǫ TK. Inserting this second saddle point into an extrapolation of the numerical results, 1/τϕ ∝ 1/B2, in ≫ this regime. The saturation of the amplitude at these high Eq. (25) one finds that the length dependence of 1/τ for ϕ fields arises as the dephasing due to electron-electron inter- high temperatures follows action dominates. The data is equally well described by the 1 1 fits used in Refs. [20] and [5] (dashed lines), see main text. (27) Forthesolid lines weused thesamevaluesfor thedephasing τ (T,L) ∼ ϕ ln2 L rates(τee =5.4nsand9.9nsforT =100mKandT =40mK, (cid:20)2Lhitln2h2LLhiti(cid:21) respectively) as in Ref. [5], where τee T−2/3 was assumed. ∝ For the dot-dashed curve we use instead τ = 13.5ns for ee explaining the weak suppression of 1/τϕ for large ring T = 40mK since one expects theoretically35 that τ 1/T ee lengths shown in Fig. 12. for L L (note, however, that L L in this regim∝e ex- φ φ ≫ ∼ plaining therather weak dependenceon τ ). ee V. DISCUSSION AND CONCLUSIONS (determined from weak localization) at low temperatures In this paper we generalized previous results for the de- (30mK. T . 1K), it was suspected that tiny concen- phasing rate due to diluted Kondo impurities as measured trations of magnetic impurities with Kondo temperatures in the weak localization experiment to describe dephasing below 30mK may be at the origin of the observed satu- duetoarbitrarydilutedimpurities. Furthermore,weinvesti- ration. This picture could be confirmed as measurements gatedhowmagneticfieldsmodifythedephasingratedueto of the amplitude of Aharonov-Bohm oscillation displayed a Kondo spins as can be measured in mesoscopic Aharonov- dramatic rise by almost an order of magnitude in moderate Bohm rings. We give results for the numerically evaluated magnetic fields (see Fig. 13), proving the magnetic origin dephasing rate as a function of the magnetic field, temper- of the low-B, low-T dephasing. ature, and the ring length. Asneithertheconcentrationsnorthetype(s)ofmagnetic The influence of magnetic impurities on dephasing has impurities are known, a parameter-free comparison to our been studied in a number of magneto-resistance experi- predictions is not possible for these systems. Assuming Mn ments in Cu, Ag or Au wires doped with magnetic impuri- impurities,believedtobecharacterizedbyaKondotemper- ties.2,3,5,15,16 More recently, high-precision experiments us- ature ofthe orderof 10mK,37 and, using the same dephas- ingion-implantedFeimpuritiesin Agwires alloweda quan- ing rates due to electron-electron interactions as in Ref. [5] titative comparison with our theory for spin-1/2impurities, (τ = 5.4ns and 9.9ns for T = 100mK and T = 40mK, ee seeRefs.[15]and[16]foracriticaldiscussion. Suchstudies respectively)weobtainthefitsshowninFig.13fora g fac- using samples doped with magnetic impurities have, to our tor of g 1.4 and an impurity concentration of 2.7ppm. knowledge, only been performed in the spin-glass regime36 Wehave≈alsoaddedacurveatT =40mK(dot-dashedline) using magnetic ions with tiny Kondo temperatures. Pierre whichusesτ =13.5ns(keepingallotherparametersiden- ee etal.5,20 studiedringsmadefrom nominallycleanCu wires. tical) to take into account that one expects theoretically35 As these wires show a saturation of the dephasing rate τ 1/T. The fits and the extracted parameters are not ee ∝ very reliable as can be seen from the observation that the the physics of strongly correlated dynamical impurities and datahasbeenequallywelldescribedinRefs.[5]and[20]by their interactions. the simple perturbative formula τϕ,S(B=0) = gµB/kBT We thank Ch. B¨auerle, N. Birge, J. v. Delft, L. Glaz- τϕ,S(B) sinh(gµB/kBT) with g =1.08, see Fig. 13. man, S. Kettemann, S. 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