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Time-dependent universal conductance fluctuations and coherence in AuPd and Ag

A. Trionfi, S. Lee, and D. NatelsonDepartment of Physics and Astronomy, Rice University, 6100 Main St., Houston, TX 77005

(Dated: March 26, 2011)

Quantum transport phenomena allow experimental assessment of the phase coherence informationin metals. We report quantitative comparisons of coherence lengths inferred from weak localizationmagnetoresistance measurements and time-dependent universal conductance fluctuation data. Wedescribe these two measurements and their analysis. Strong agreement is observed in both quasi-2Dand quasi-1D AuPd samples, a metal known to have high spin-orbit scattering. However, quantita-tive disagreement is seen in quasi-1D Ag wires below 10 K, a material with intermediate spin-orbitscattering. We consider explanations of this discrepancy, with particular emphasis on the theoreticalexpressions used to analyze the field dependence of the conductance fluctuations. We also discussthe mechanism of the suppression of conductance fluctuations at high drive levels, and dephasingmechanisms at work in these systems.

PACS numbers: 73.23.-b,73.50.-h,72.70.+m,73.20.Fz

I. Introduction

Quantum phase coherence in normal metals givesrise to numerous corrections to the classically pre-dicted conductivity. These corrections are commonlyreferred to as quantum transport phenomena (QTP).The study of QTP has both fundamental physicalimportance[1] and possible implications in novel de-vice architectures[2, 3]. The typical experimental ap-plication of QTP is to infer quantum coherence timeand length scales. Weak localization magnetoresistance(WL)[4], magnetic field-dependent universal conductancefluctuations[5, 6] (MFUCF), time-dependent universalconductance fluctuations[7, 8](TDUCF), and Aharonov-Bohm oscillations[9] have all been used to examine co-herence in normal metals.

QTP in diffusive conductors arise due to interferencebetween possible electronic trajectories from one loca-tion to another. Interference, however, is only rele-vant when the phase of the conduction electrons’ partialwaves is well-defined. The coherence length is defined asthe distance scale over which the phase of a conductionelectron’s wave function remains correlated to its initialphase. This length can be related to a coherence time, τφ,

by Lφ =√

Dτφ where D is the diffusion constant of theelectron in the disordered solid. Decoherence or dephas-ing can occur when an electron experiences an interactionwith another dynamical degree of freedom. The threemost common dephasing processes are electron-electronscattering, electron-phonon scattering, and spin-flip in-teractions with magnetic impurities. The rates of theseprocesses have distinct temperature dependences, allow-ing QTP to be used to distinguish between these mecha-nisms in various metals.

Interesting questions have arisen from experimentalcharacterization of electron coherence. One question iswhether precisely the same coherence length is inferredfrom different QTP. This is a subtle issue because theprecise time scales and processes relevant to a partic-ular observable can be complicated, and the evolution

of electronic phase correlations is typically not a sim-ple single-time exponential decay. A previous test ofthis coherence length consistency led to equivocal re-sults in quasi-2D silver[10]. Another question is thecause of an observed low temperature saturation of thecoherence length in many materials[11]. An explana-tion with significant experimental support is scatteringfrom dilute concentrations of low Kondo temperaturemagnetic impurities[12], while others suggest intrinsicmechanisms[13]. These two questions are increasinglyrelated: Recent publications[14, 15] have compared ex-perimental results from different QTP when debating thecause of coherence saturation; it must be established,however that these analyses are truly comparing equiva-lent parameters.

In this paper we briefly review the physics underlyingWL, MFUCF, and TDUCF, and report measurementsof these effects in two different materials, Au0.6Pd0.4 andAg, over a broad temperature and field range. While wefind excellent quantitative agreement between LWL

φ (T )

and LTDUCFφ (T ) in all AuPd samples, we observe a di-

vergence between these two inferred coherence lengths inquasi-1d Ag samples, as seen previously in quasi-2D Agfilms[10]. We discuss candidate explanations, and sug-gest that a likely concern is the applicability of the theo-retical expressions used to analyze the TDUCF field de-pendence. We also show that the suppression of TDUCFamplitude at high drive currents is consistent with bias-induced energy averaging. Finally we discuss the impli-cations of these data on dephasing mechanisms at workin these systems.

Weak localization arises from the properties of elec-tronic trajectories under time-reversal symmetry. Manyelectronic paths in a diffusive conductor contain loops.Without a magnetic field, an electron circumnavigat-ing such a loop accumulates the same phase as onedoing so under time-reversed conditions. This phaseagreement causes constructive interference that enhancesback-scattering, leading to a conductivity lower than isclassically predicted. With strong spin-orbit interactions,

2

the sign of this interference is reversed and leads to en-hanced conduction. In the presence of a magnetic fieldnormal to the loop, the vector potential adds oppositephase shifts to each looped path and corresponding time-reversed conjugate. This eliminates the constructive (de-structive) interference when ∼ one quantum of flux isthreaded through a typical loop. The result is a mag-netoresistance with a field scale related to the area of atypical coherent loop, allowing inference of LWL

φ .

Time-dependent UCF are due to the enhanced sen-sitivity of the conductance to the motion of individualscatters. Unlike weak localization, all interfering pathscontribute to this phenomenon. When a scattering sitemoves, it changes the interference pattern of all inter-secting electronic paths within a coherent volume of thescattering site, leading to a conductance change. A sin-gle moving scatter can change the conductance within acoherent volume at zero temperature by roughly e2/h. Ifthe relaxation times of the scatters are appropriately dis-tributed, the TDUCF exhibit a 1/f power spectrum[16],which is the case in many normal metals. Much likeweak localization, the noise power of the 1/f noise issensitive to a perpendicular magnetic field. The time-reversed loop contribution (the cooperon) will be sup-pressed as the field is increased while the sensitivity dueto all remaining paths, known as the diffuson, remainunchanged[17]. This leads to a factor-of-two decrease inthe noise power as the field is ramped up, and allowsextraction of LTDUCF

φ .

Magnetic field-dependent UCF are closely related tothe time-dependent form. The explanation of this phe-nomenon comes from the ergodic hypothesis[18, 19],which implies that other effects that randomize the in-terference of electronic paths are equivalent to scatter-ing site motion. Since a perpendicular magnetic fieldintroduces an Aharonov-Bohm phase shift particular toeach electronic path, varying such a field leads to conduc-tance fluctuations of the universal size of ∼ e2/h. Theresult is a completely reproducible magnetoresistive pat-tern that is sample-specific, commonly referred to as themagnetofingerprint.

Although the expected size of the conductance fluc-tuations is universal, the measured effect may be muchsmaller. Samples much longer and wider than Lφ maybe treated as uncorrelated fluctuators in series and par-allel. The measured noise power is therefore reduced bya factor N , the number of coherent volumes between theends of the measured sample. Further averaging occurswhen the energy range of accessible single-particle statesexceeds the correlation energy[20], Ec = ~D/L2

φ. In thiscase the relevant states can be subdivided by energy intocoherent sub-bands, each nominally uncorrelated withthe others, leading to further ensemble averaging. Oneway to increase available energy levels is via thermal en-ergy. The thermal length, LT ≡ (~D/kBT )1/2, is the dis-tance two initially in-phase electrons separated in energyby kBT may move before their phases differ by ∼ 1. Thecondition LT < Lφ is equivalent to kBT > Ec, leading to

ensemble averaging by LT /Lφ to some power. Similarly,when eVc > Ec, where Vc is the voltage dropped across acoherence length, ensemble averaging will also occur[6].

The observed magnitude of the TDUCF may also besmaller than the universal limit if the conductance fluc-tuations are not “saturated”. A sample is said to be inthe saturated regime if the conductance variance withina coherent volume has reached the limiting ∼ e2/h am-plitude. How close a sample is to this condition de-pends on the microscopic nature of the fluctuators. SinceMFUCF should always exhibit conductance fluctuationson order e2/h within a coherent volume, if a sample isin the saturated regime, the TDUCF noise power inte-grated over the bandwidth of the fluctuators should equalthe MFUCF magnitude. That is,

∫

∞

0 SG(f)df should =

(δGMFUCF)2, where SG is the conductance noise power, fis the frequency, and G is the conductance. If the fluctua-tors are typical tunneling two-level systems (TLS) of thetype ubiquitous in disordered solids[21], their relaxationrates are estimated to span ∼ 20 decades[7]. TDUCFmeasurements in the literature are all thought to be non-saturated. This issue is important, as it determines thefunctional form appropriate for analysis of the field de-pendence of the TDUCF.

For WL and TDUCF, the number of the sample di-mensions longer than Lφ determines the effective dimen-sionality of the system with regard to coherence effects.Thus a quasi-2d sample is achieved when t << Lφ << wand a quasi-1d sample when w, t << Lφ. Here t and ware sample thickness and width, respectively.

Previous comparisons between LWLφ (T ) and

LTDUCFφ (T ) have shown a disagreement between

these parameters at low temperatures in quasi-2d Agsamples[10]. This was interpreted as evidence sup-porting a theoretical treatment[22] that argued thatthe Nyquist or electron-electron dephasing rate wouldlimit the coherence in WL while the out-scattering ratewould limit the coherence in universal conductancefluctuations. The out-scattering rate is the rate atwhich an electron will change its momentum state in theBoltzmann formalism. It was shown the two rates havedifferent temperature dependencies at low temperatureso a divergence between the coherence lengths inferredfrom WL and TDUCF was expected. Recent correctionsto the theory[23] show that, as long as electron-electronscattering is the only small-energy-transfer inelasticprocess, LWL

φ is expected to equal LTDUCFφ . This leaves

the experimental results in Ag without an explanation.We discuss this further below.

II. Procedure

Samples were patterned on undoped GaAs substratesusing standard electron beam lithography, as discussedin Ref. [24]. Distances between consecutive leads (volt-age or current) were 10 µm in AuPd samples and 20 µmin Ag samples. Ag (0.99999 purity) samples were made

3

TABLE I: Samples used in magnetotransport and noise mea-surements. Free electron density of states for Au and Ag usedto calculate D for AuPd and Ag samples: 1 × 1047 m−3J−1,from Ref. [42]. Diffusion constants calculated via the Einstienrelations. Sample D was deliberately contaminated with addi-tional ferromagnetic impurities as described in the text. Theeffective dimensionality d for coherence effects is determinedby the relative size of Lφ and the sample dimensions.

Sample metal d w t R/L [Ω/µm] (1d) D

[nm] [nm] R/ [Ω] (2d) [m2/s]

A AuPd 1 43 9 722 1.34 ×10−3

B AuPd 1 35 9 857 1.34 ×10−3

C AuPd 2 500 6.5 84.5 7.9 ×10−4

D AuPd 2 500 8.5 47.5 9.6 ×10−4

E Ag 1 115 12 49 5.65 ×10−3

F Ag 1 140 12 35 6.70 ×10−3

G Ag 1 130 12 45 5.63 ×10−3

H Ag 1 70 20 42 6.50 ×10−3

I Ag 1 125 12 43 5.91 ×10−3

J Ag 1 100 12 88.5 3.61 ×10−3

using a single lithography/deposition step with the wireand leads all silver, and no adhesion layer. Au0.6Pd0.4

samples were made using two lithography steps, the firstfor the AuPd wire and the second for the Ti/Au leads(1.5 nm Ti, 25 nm Au). The AuPd source material is ex-pected to be free of ferromagnetic impurities to the 10−5

level. To minimize the contact resistance between theAuPd wires and the Ti/Au leads, samples were exposedto oxygen plasma for 30 seconds prior to the Ti/Au de-position to remove any resist residue. Typical contactresistances in the AuPd samples were less than 30 Ω. Alldepositions were performed using an electron beam evap-orator at 5 × 10−7 mB. To test the effects of magneticimpurities on the consistency of LWL

φ and LTDUCFφ , one

AuPd sample was deliberately contaminated with traceimpurities by evaporating 2.5 nm of Ni0.8Fe0.2 with thesample shutter closed, prior to the AuPd evaporation.

Measurements between 2 and 20 K were performed ina 4He cryostat while lower temperatures for two samples(G,H) were achieved in a dilution refrigerator. All sam-ples were initially characterized with standard ac fourterminal resistance measurements and tested as a func-tion of temperature at various drive currents to checkfor Joule heating. Upturns in the resistance at low tem-peratures were consistent with electron-electron interac-tion corrections. All subsequent measurements were per-formed at or below the limiting current set by the Jouleheating tests.

WL magnetoresistance measurements were made us-ing standard four terminal techniques. The applied fieldwas swept between ±1.25 T for AuPd samples while thefield range for the Ag samples was ± 0.9 T. The TDUCFmeasurement employed an ac five terminal bridge tech-nique developed by Scofield[25]. The ac bridge technique

renders the noise measurement insensitive to noise in thevoltage source. The bridge was measured with a lownoise differential pre-amplifier (1.5 nV/

√Hz, NF Elec-

tronics LI-75A). Signal frequencies ranged from 600 Hzfor AuPd samples to 1 kHz for Ag samples, chosen tooptimize the noise contours of the preamplifier. Both thein-phase and out-of-phase demodulated signals were fedinto a two-channel dynamic signal analyzer (SRS SR785)to transform the data into the frequency domain. Strong1/f dependent spectra were consistently observed fromthe in-phase channel while the out-of-phase channel pro-vided a measure of the white background noise of themeasurement circuit. By subtracting away this back-ground, the sample-induced noise could be isolated. Atypical frequency span ranged from 78 mHz to 1.5 Hzfor the AuPd samples and to 3 Hz in Ag samples. Thelow temperature noise power in all samples show the ex-pected amplitude increase with decreasing temperatureas well as the factor of two drop in the presence of a largeperpendicular magnetic field (except when local interfer-ence noise becomes non-negligible). This observation isconsistent with the expected TDUCF behavior.

∆Ω

2 Κ

8 Κ

14 Κ

FIG. 1: “Magnetofingerprint” measurements on sample Imade using the five terminal bridge technique. The curvesfrom top to bottom are at 2, 8, and 14 K. The curves areoffset for clarity.

The bridge technique was also employed to make theMFUCF measurements. By using this method instead ofthe standard four terminal resistance measurement, theWL magnetoresistance is nulled away since both sidesof the bridge will have identical resistance changes dueto this effect. The nulling of the WL magnetoresistanceallows the magnetofingerprint to be observed down tozero field. An example of the magnetofingerprint is givenin Figure 1.

4

III. Analytical Approach

A. Weak localization

Values of LWLφ were inferred from the AuPd WL mag-

netoresistance using the following equations for one andtwo dimensions respectively[12, 26]:

∆R

R

∣

∣

1d= − e2

2π~

R

L

[

1

L2φ

+1

12

(

w

L2B

)2]

−1/2

(1)

∆R

R

∣

∣

2d=

e2

4π2~R

[

ψ

(

1

2+

1

2

L2B

L2φ

)

− ln

(

1

2

L2B

L2φ

)]

(2)

ψ is the digamma function and LB ≡√

~/2eB. R isthe sheet resistance of the sample, R is R(B = ∞), wis the sample dimension transverse to both the appliedfield and current flow, and L is the length of the sampleparallel to current flow. These equations apply in thelimit of strong spin-orbit scattering (τSO << τφ). AuPdhas long been established as a strong spin-orbit scatteringmaterial[26].

The LWLφ values inferred from the quasi-1d Ag wires

used the more general form[12],

∆R

R

∣

∣

1d= − e2

2π~

RL ×

3(

1

L2φ

+ 4

3L2SO

+ 112

(

w

L2B

)2)1/2 − 1(

1

L2φ

+ 112

(

w

L2B

)2)1/2

(3)

Here LSO ≡√DτSO is the spin-orbit length. ∆R for

Eq. (2) is defined as ∆R = R(B) − R(B = 0), while itis defined as ∆R = R(B) − R(B = ∞) in Eqs. (1) and(3). The only fitting parameter for the AuPd curves isLφ, while both Lφ and LSO are free in the fits for the Agcurves. The width is left free at 2 K in all one-dimensionalfits to confirm sample size and is then fixed for all sub-sequent fits. Spin-orbit lengths are also fixed above 10 Kto the average value found from lower temperature fits.

B. TDUCF noise

LTDUCFφ values were inferred from fits of the noise

power as a function of perpendicular field to the ap-propriate crossover function, ν(B), the theoretically ex-pected functional form. There are two methods of cal-culating ν(B); we have used both approaches and com-pared the results. First, analytical expressions for theone-dimensional and two-dimensional crossover functionswith large spin-orbit interaction have been derived re-cently by Aleiner[27]:

ν1d(B) = 1 − x

2

(

Ai(x)

Ai′(x)

)2

(4)

where x ≡ L2φ/(3(~/Bew)2), and

ν2d(B) =1

2+L2

B

4L2φ

ψ′

(

1

2+L2

B

2L2φ

)

. (5)

These functional forms are strictly valid when ~/τφ <<kBT . Here Ai(x) is the Airy function, and ψ′(x) is thederivative of the digamma function.

Prior to the derivation of these analytical results, thenoise crossover function was calculated from the theoret-ical expression for the magnetic field correlation functionF (∆E,∆B,B) ≡ 〈δg(EF, B)δg(EF + ∆E,B + ∆B)〉 ofthe MFUCF[17, 28, 29]. Here g is the conductance inunits of e2/h. An approximation of this correlation func-tion has been derived by Beenakker and van Houten[30]for quasi-1d samples. We also analyze the data withthis method and compare with the analytical expressionsabove. The samples are assumed to be in the unsatu-rated regime, so that the derivative of F with respect toτ−1φ must be computed, as per the explanation given by

Stone[17]. The resulting derivative has the form:

F ′(B) =L5

φB

(

1 +3L2

φB(B)

2πL2T

)

(

1 +9L2

φB(B)

2πL2T

) (6)

where

L2φB(B) =

3L2φ

(BeLφw/~)2 + 3. (7)

Spin-orbit interactions may be accomodated by changingEq. (6) to[31]:

F ′(B) =L5

φB

(

1 +3L2

φB(B)

2πL2T

)

4(

1 +9L2

φB(B)

2πL2T

) +3L5

φBt

(

1 +3L2

φBt(B)

2πL2T

)

4(

1 +9L2

φBt(B)

2πL2T

)

(8)where

L2φBt(B) =

3L2φL

2SO

(BeLφLSOw/~)2 + 3L2SO + 4L2

φ

. (9)

The approximate crossover function is therefore

ν(B) =1

2+

F ′(B)

2F ′(B = 0). (10)

The exact crossover function in quasi-2d systems wasreported by Stone[17] and was also used to infer LTDUCF

φin the quasi-2d AuPd samples. The fitting method em-ployed was that described in Ref. [32]. A comparison be-tween the crossover functions computed from the correla-tion functions and the analytic expressions of Eqs. (4,5)finds the following results. In the quasi-1d case, theLTDUCF

φ values inferred using the correlation functionssystematically exceed those extracted using the analyti-cal expressions by roughly 10%. Similarly, in the quasi-2d

5

case, the correlation function-based values exceed thosefrom Eq. (5) by 3%.

The actual fitting functions used to analyze the nor-malized noise power data included an additional fit-ting parameter to account for the local interferencenoise[33, 34, 35] that increases to non-negligible mag-nitudes at higher temperatures. Since local interferencenoise has no low order field dependence, the noise powerwill not drop by a full factor of two at higher tem-peratures. The corrected fitting function has the formf(B) = (1− z)+ zν(B) where z represents the fractionalsize of the UCF enhanced noise. Values of z were in-distinguishable from 1 for all data sets except at 20 K inthe quasi-2d AuPd samples and quasi-1d Ag samples. Allfitting was performed using standard χ2 minimization.

C. Role of magnetic impurity scattering

As was discussed extensively in Ref. [12], magnetic im-purity scattering can affect weak localization and UCFcoherence corrections differently or identically dependingon the temperature scale and impurity concentration. Attemperatures higher than a crossover temperature, theKorringa time for impurity spins to relax back to thermalequilibrium with the lattice is short compared to the spin-flip scattering time. In this regime (T > T ∗ ≡∼ 40 mK×the ppm concentration of magnetic impurities for typi-cal host noble metals), spin-flip scattering should involvelarge energy transfers[27], and affect WL and TDUCFidentically. At temperatures below this cutoff, spin scat-tering is more rapid than the relaxation of the impu-rity spins; under these conditions, the spin-flip scatteringtime is predicted to affect TDUCF and WL differently.

An estimate of the decoherence rate due to mag-netic impurities may be obtained from the Nagaoka-Suhlexpression[36]:

1

τsf=

cmag

π~ν(EF)

π2S(S + 1)

π2S(S + 1) + ln2(T/TK), (11)

where ν(EF) is the density of states at the Fermi level ofthe host metal, S is the spin of the impurity, and TK isthe Kondo temperature of the impurity in the host metal.

IV. Results and Discussion

Magnetoresistance curves for both AuPd and Ag sam-ples are given in Figure 2. The data are fit very wellby Eqs. (1,2,3). Sample widths inferred for the quasi-1dwires via the fitting procedure are consistent with SEMimages and estimates based on resistances of codepositedfilms. Including LSO as a fit parameter in the AuPd dataleads to LSO . 10 nm, with little impact on Lφ. We findLSO ≈ 290 nm for Ag, and ∼ 9 nm for AuPd.

Similarly, examples of the measured normalized noisepower (SR(B)/SR(B = 0)) versus field are shown in Fig-ure 3 for sample F, a quasi-1d silver wire. As is clear from

∆

2 Κ

4 Κ

6 Κ 8 Κ10 Κ

2 Κ

4 Κ6 Κ8 Κ10 Κ14 Κ20 Κ2 Κ4 Κ6 Κ8 Κ

14 Κ20 Κ

Sample A

Sample C

Sample F

FIG. 2: WL magnetoresistance curves at various tempera-tures for a 43 nm and 500 nm AuPd wire (samples A and C)and a 140 nm Ag wire (sample F). Quasi-1D data shifted topass through origin.

the graph, these data are fit well by the Beenakker/vanHouten correlation function approach (Eq. (8) and fol-lowing).

2 Κ

14 Κ

20 Κ

FIG. 3: Noise power data from Ag sample F at 2, 14 and20 K. The crossover field becomes larger as the coherencelength diminishes. The 20 K data does not drop by a fullfactor of 2 due to local interference noise.

A. AuPd comparison of WL and TDUCF

Figure 4a shows the resulting coherence lengths in-ferred from both QTP for the two quasi-1d AuPd sam-ples, as originally reported in Ref. [24]. The inferred LWL

φ

and LTDUCFφ are in strong quantitative agreement for the

AuPd samples over the temperature range measured.As shown in Ref. [24], this agreement remains strong

even in the presence of magnetic impurity scattering sig-nificant enough to suppress the coherence length by more

6

than a factor of two. This strongly supports the theoret-ical statement[23] that weak localization and UCF mea-surements probe precisely the same coherence physics,even in the presence of strong spin-orbit and magneticimpurity scattering over this temperature range. Fur-thermore, the agreement persists even though ~/τφ isnever << kBT , suggesting that Eqs. (4,5) are robust evenwhen that constraint is somewhat relaxed.

B. Ag comparison of WL and TDUCF

Figure 4b shows the equivalent Lφ(T ) data for theAg samples over the same temperature range. Notethat LWL

φ (T ) shows no saturation, and at low temper-atures approaches the Nyquist predicted value with no

adjustable parameters. This strongly suggests that e-einteractions are the only non-negligible dephasing mech-anism in the Ag samples at temperatures near 2 K.

φ φ

(a) (b)

FIG. 4: (a) Coherence lengths as a function of T for the AuPdsamples (lettered accordingly). Triangles points are inferredfrom Eq. (4), circles are inferred from Eq. (6), and squaresare inferred from WL. Only one UCF fit is shown in quasi-2D since the two results only differ by 3%. The dashed linesshow the predicted dephasing length due to Nyquist scatter-ing calculated from sample parameters. The solid line rep-resents the calculated thermal length. (b) Coherence lengthsas a function of T for Ag samples E and F (lettered accord-ingly). Circle points are inferred from Eq. (8), and squaresare inferred from WL. The dashed lines show the predicteddephasing length due to Nyquist scattering calculated fromsample parameters. The solid line represents the calculatedthermal length. Unlike the AuPd case, there is a statisticallysignificant discrepancy between LWL

φ and LTDUCF

φ in thesesamples.

Comparing Fig. 4b with Fig. 4a highlights a dramaticdifference between the Ag and AuPd data: There is sub-stantial disagreement between LWL

φ and LTDUCFφ in these

quasi-1D Ag samples. In particular, below 8 K, LTDUCFφ

is shorter and has a significantly weaker temperature de-pendence than LWL

φ (T ). This difference is very similar

to that observed previously in quasi-2D Ag films[10].

The reason for this disagreement is unknown. A possi-bility put forward by Aleiner and Blanter is that a subtleeffect due to triplet channel electrons is responsible[23].This suggestion reflects the observation in the quasi-2dAg data that the disagreement appears at temperaturesbelow Lφ ≈ LSO. Our AuPd data, however, appearsinconsistent with such an explanation. The inferred co-herence lengths in a strong spin-orbit scattering mate-rial (AuPd) should resemble the low-temperature limit-ing behavior of a material with intermediate spin-orbitscattering (Ag). The WL/TDUCF agreement seen inthe AuPd would imply that the coherence lengths in-ferred from the Ag should agree as the temperature isreduced. To the lowest temperatures measured, no suchconvergence is observed. A related prediction[27] wouldbe for a signature of unusual triplet effects in R(T ) whenkBT <∼ ~/τSO. Since τSO ≈ 3.5×10−11 s, this crossoverwould be predicted at ∼ 200 mK. No change in the R(T )properties is observed down to 50 mK. TDUCF noisemeasurements at these lower temperatures have yet tobe performed successfully and are very challenging dueto Joule heating concerns. On the basis of the AuPd dataat hand, a triplet channel effect seems extremely unlikelyto explain the differences between Fig. 4a and b.

We suggest another possible resolution to this discrep-ancy between the AuPd and Ag data: the applicabilityof the noise crossover expressions used in analyzing thedata. Due to microscopic differences in the (unknown)fluctuators responsible for the TDUCF noise, the AuPdand Ag samples may be in different regimes. In partic-ular, if the Ag samples were transitioning into the sat-

urated noise limit, then the fitting functions based onF ′(B) above used to infer the coherence length wouldbe inappropriate. In the saturated limit, F (B) ratherthan its derivative with respect to the inelastic rate isthe correct function from which to derive ν(B).

Comparing the results of LTDUCFφ inferred using the

saturated noise crossover function shows that incorrectlyusing the unsaturated noise crossover function will re-sult in inferred coherence lengths less than the actualvalue. A comparison of integrated noise power amplitudeto MFUCF resistance variances calculated from mag-netofingerprint data confirms that the Ag noise is not yetsaturated. The TDUCF noise in sample F would need tobe integrated over 190 decades in frequency (completelyunphysical) to achieve the conductance fluctuation sizeseen in MFUCF.

Fig. 5 shows the results of trying to infer LTDUCFφ (T )

using the crossover function appropriate for saturatedTDUCF, in comparison with the weak localization dataand the LTDUCF

φ values calculated using the unsaturatedTDUCF expression. Clearly the system is not in thesaturated regime over the observed temperature range.However, the saturated crossover function data becomes a

better match to the WL data as T decreases. It seems rea-sonable that an interpolating crossover function betweenthe saturated and unsaturated crossover functions couldbe necessary. If an unsaturated/saturated transition is

7

1 10100

1000

L φ [nm

]

T [K]

FIG. 5: Coherence lengths from sample F. Squares are fromWL, circles are inferred via the unsaturated TDUCF crossoverfunction, and triangles are calculated with the saturatedTDUCF crossover function.

at work, the coherence lengths extracted via a correctlyderived interpolating crossover function could agree withthose inferred from WL for all temperatures. A defini-tive test of this hypothesis is to measure TDUCF andMFUCF in extremely thin Ag samples down to dilutionrefrigerator temperatures and compare their magnitudesand correlation fields. These attempts are ongoing.

C. Drive dependence of the TDUCF

We have also considered whether the nonequilibriumnature of the transport measurements could result in thediscrepancy seen in Fig. 4. As has been discussed ex-tensively in Ref. [37], once a system is driven out ofequilibrium, it is a subtle question whether one shouldexpect consistency between, e.g., LWL

φ (T ), LTDUCFφ (T ),

and R(T ). We have examined the drive dependence ofour WL and TDUCF measurements, and return to thisissue below.

Increasing the measurement current has a significanteffect on the measured noise power amplitude long beforeany change is observed in R(T ), LWL

φ (T ), or LTDUCFφ (T ).

This was seen previously in quasi-2D Ag films[10]. A plotof noise power versus applied current is given in Fig. 6 forseveral different temperatures. It is clear that the noisepower begins to drop at a different applied current foreach temperature.

We reemphasize that while the TDUCF amplitude isstrongly affected by drive level, the field-dependence usedto infer LTDUCF

φ is essentially unchanged (within the er-

ror bars) below drive levels where heating is clearly man-ifest. Figure 7 shows the experimentally observed noisecrossover data for sample J at 2 K for three differentdrive currents. The left inset indicates that the noisepower amplitude shows a distinct drive current depen-

1 1010-13

10-12

10-11

S(B

)/S(B

=0)

Iac

[µA]

2 K, 0 T 4 K, 0 T 6 K, 0 T 8 K, 0 T 10 K, 0 T 8 K, 2 T

FIG. 6: Sensitivity of the noise power of sample E to appliedcurrent. Enhanced suppression of the noise occurs when eVc

becomes larger than the correlation energy. The drop-off cur-rent at 8 K, 2 T is the same as 8 K, 0 T implying that thecoherence length is not sensitive to large applied fields.

dence while the crossover field of the three curves remainsunchanged (within the error bars). Over the same rangeof drive currents, neither LTDUCF

φ or LWLφ are altered,

to within the error bars on those quantities. This resultimplies that the coherence length differences observed inthe Ag samples (as in Fig. 4b) are not a result of thetype of out-of-equilibrium dephasing effects described inRef. [37].

0.01 0.1 10.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

0 1 22

3

4

50.1 1

10-12

10-11

S(B)

/S(B

=0)

B [T]

500 nA 1 µA 2 µA

S V/(I

R)2 [1

0-12 H

z -1]

I [µA]

S V/(I

R)2 [H

z-1]

f [Hz]

0 T 0.016 T 0.512 T

FIG. 7: Normalized noise power of sample J at 2 K as a func-tion of field for three different values of drive current (rms500 nA, 1 µA, and 2 µA). The unchanging crossover fielddemonstrates explicitly that the inferred LTDUCF

φ is not af-fected strongly by drive level. Left inset: dimensionless con-ductance noise power as a function of drive current at zerofield, showing that the higher drive currents do suppress themagnitude of the TDUCF. Right inset: log-log plot of thenormalized noise power as a function of frequency at 2 K,2 µA drive, for three different values of B, showing the 1/fdependence typical for all the TDUCF data in this work.

8

A reasonable explanation for the decrease in TDUCFamplitude with increasing drive current is energetic en-semble averaging as eVc becomes larger than the cor-relation energy of the samples. As T is increased, thedecrease in Lφ and corresponding increase in Ec wouldrequire larger drives to observe such averaging, consistentwith what is observed. The inability to observe such asuppression of noise at high drives in the AuPd samples,where Lφ is much shorter, further supports this expla-nation. An estimate of the current required such thateVc ∼ Ec in AuPd leads to a current greater than thatneeded empiricially to heat the samples significantly.

The detailed above-threshold dependence of the noisepower on the applied current, however, is surprising.Properly normalized noise power is not proportional to1/Iac above the critical current where e × I(R/L)Lφ ∼Ec, as a simple treatment would predict. Instead thenoise power decreases like I−0.5

ac above the threshold. Forcompleteness, the drive dependence at 8 K was repeatedat 2 T. The suppression of the noise amplitude started atthe same threshold drive current, further evidence thatthe coherence length in Ag does not change at high fields(i.e. dephasing in Ag is not magnetic impurity scatteringin origin).

D. Dephasing mechanisms

The LWLφ and LTDUCF

φ data have implications for thedecoherence mechanisms at work in these materials. Aswe argue below, the data strongly support that magneticimpurity scattering is relevant in AuPd samples, and thatscattering from dynamical defects such as tunneling two-level systems (TLS) are unlikely to be significant in thesematerials.

We first consider AuPd, in which the Lφ values at lowtemperatures are significantly lower than those predictedfrom Nyquist scattering. Exact quantitative agreementbetween the theoretical Nyquist length and experimentalcoherence length is not necessarily expected since AuPdis not a simple metal, i.e. it does not have the typicalspherical Fermi surface. However, if the Nyquist dephas-ing mechanism is at work, one should expect that the ex-perimental coherence lengths at low temperature wouldshow the predicted power law dependence with the tem-perature. Deviations from this power law are particu-larly clear as apparent low-T saturation of Lφ in samplesC and D, shown in more detail in Ref. [24].

These deviations are due at least in part to detectableconcentrations of magnetic impurities in all AuPd sam-ples. The presence of such impurities is strongly sup-ported by high magnetic field noise power data. Figure 8ashows the normalized noise power versus applied field upto 8 T for three samples, the AuPd sample intentionallycontaminated with magnetic impurities (D), one nomi-nally clean AuPd sample (C), and one Ag sample (F). Anoticeable upturn in the noise power at high fields is seenin both AuPd samples, while no such upturn is seen in

(a)

(b)

FIG. 8: (a) High field noise data for samples C, D, and F, allat 2 K. The upturn at high fields implies that non-negligibleconcentrations of magnetic impurities exist in the AuPd sam-ples. (b) High field noise data for sample B at several tem-peratures, showing the upturn in this nominally clean AuPdsample. The lines connecting the points for each temperatureare a guide to the eye. Note that the upturn is smaller andhappens at higher fields for higher temperatures, consistentwith Zeeman splitting of magnetic impurities.

the Ag at the lowest measured temperature.

The upturn is caused by a suppression of spin-flip de-coherence as the Zeeman splitting of the magnetic im-purities exceeds kBT . An analogous upturn has beenobserved in investigations of Li wires[38] and in recentAharonov-Bohm measurements in Cu rings[14]. The in-creased coherence length leads to an increase in the mag-nitude of the noise power via reduced ensemble averag-ing. Some upturn is visible at the highest B/T ratioin all AuPd samples, consistent with some magnetic im-purities even in nominally “clean” devices. Figure 8bshows the large field noise upturn at multiple tempera-tures in sample B. The characteristic field scale for thenoise upturn increases with increasing temperature, con-sistent with Zeeman splitting splitting of magnetic scat-terers.

9

One can use Eq. (11) to estimate the magnetic impu-rity concentration required to produce the observed Lφ

saturation values. Assuming that the 2 K data representcomplete saturation (as appears to be the case for, e.g.,sample C), that TK ∼ 2 K, and a typical ν(EF) for thenoble metals, one finds cmag ∼ 17 ppm for sample C.This value is surprisingly high and not consistent withthe starting purity of the source material. One possi-ble explanation for this would be an enhancement of thespin-flip scattering process due to the strong paramag-netism of the Pd component of the host alloy. Note thateven if this concentration of magnetic impurities is accu-rate, the inferred T ∗ below which spin-flips should affectUCF and WL differently is < 1 K, outside the range ofthese measurements.

Comparing the saturated values of Lφ for two sam-ples allows the relative concentrations of magnetic im-purities to be computed, independent of possible para-magnetic enhancement or density of states uncertainties,

sincecmag,2

cmag,1= D2

D1

(

Lφ,1

Lφ,2

)2

. Comparing sample C (quasi-

2d) and D (quasi-2d, “spiked” with additional magneticimpurities), one finds that cmag,D/cmag,C ≈ 5. We notethat R(T ) for these two samples shows no discernabledifference beyond what would be expected from theirdiffusion constants. This also supports the hypothesisthat the absolute concentration above is an overestimate,since ∼ 100 ppm magnetic impurities in sample D wouldlikely cause other discernable effects in addition to en-hanced dephasing. Furthermore, at such a concentrationthe crossover temperature T ∗ ∼ 4 K; however, no changein LWL

φ vs. LTDUCFφ is apparent there, again suggesting

that this concentration is an overestimate.

While the above data show that spin-flip scatteringis definitely relevant in AuPd samples, unconventional(non-Nyquist, non-spin-flip) dephasing must also be con-sidered as an alternative explanation for the suppressionof low-T coherence lengths as compared to those expectedfrom Nyquist theory. Coherence lengths inferred via WLin Ag samples (G & H) in dilution refrigerator mea-surements show some evidence of saturation at temper-atures much less than 1 K. (Other Ag samples were notmeasured at dilution temperatures, and clearly had notsaturated down to 1.7 K.) The cause of this saturationcannot be dismissed easily as spin-flip scattering at thisstage, since we have yet to perform measurements (e.g.TDUCF, Aharonov-Bohm) that would directly probe formagnetic impurities in these samples at those tempera-tures.

Another proposed mechanism put forth to explain ob-served coherence saturation is a dephasing mechanismcaused by the same dynamic two-level systems that causeTDUCF[39, 40]. The data presented herein imply thatsuch a mechanism is unlikely to be the cause of the co-herence saturation seen in the AuPd. Using the TDUCFnoise amplitude of the two materials at a given temper-ature, it is possible to use the results of Feng, Lee, andStone[16] and the measured sample parameters to esti-

TABLE II: LWL

φ (max) normalized by that for sample H,compared with the expected ratio from Eq. (12) for zero-temperature saturation of τφ, using sample parameters fromTable I.

Sample ratio Measured Uncertainty Expected

C/H 0.0597 0.0017 0.0148

D/H 0.0221 0.0006 0.0218

G/H 0.6561 0.0186 0.7502

mate the ratio of TLS in the two materials. If the micro-scopic scattering properties of the TLS are assumed to beidentical in AuPd (sample A) and Ag (sample J), usingthe appropriately normalized TDUCF magnitude at 2 K,we find that the density of TLS is ∼ three times larger inAg than in AuPd. This number should be considered arough estimate since the scattering cross-sections of theTLS in the two materials may be different. A conclusionthat the mobile defect density is roughly the same in bothmaterials is reasonable. If TLS-induced dephasing trulyis significant in AuPd at ∼ 2 K, one would therefore ex-pect it to be similarly important in Ag samples. Giventhe excellent agreement at that temperature of LWL

φ inAg with the Nyquist prediction, this seems unlikely.

For completeness, we compare maximum (lowestT ) coherence times observed for both AuPd and Agwith a prediction of zero temperature coherence timesaturation[41]. The predicted zero-temperature dephas-ing time is

1

τφ,sat=

√2ρe2

3hπ√D

(

b

τe

)3/2

. (12)

Here, τe is the elastic relaxation time of the conductionelectrons, ρ is the resistivity, and b is a constant of orderone. Since a strict value for b is not known, we selectone sample as a reference, and compare the ratio of eachmaximum LWL

φ to the ratio expected from Eq. (12) andthe measured resistivities. We acquired WL data on Agsamples G and H in the dilution refrigerator, with someindications of LWL

φ saturation; for AuPd samples C andD, such saturation has an onset at temperatures above2 K. Since the Ag shows the least magnetic contamina-tion, we choose sample H as the reference. In Table II, weshow the predicted ratios from applying Eq. (12), whichfall well outside the errors in the experimentally mea-sured ratios for two of the three samples. In fact, the onesample that shows good agreement with (12) is sampleD, the sample intentionally contaminated with additional

magnetic impurities known to significantly impact coher-ence. These data do not appear to support Eq. (12).Given the demonstrated effectiveness[12] of trace mag-netic impurities to affect LWL

φ , one must view analyses oflow temperature coherence saturation with appropriatecaution.

10

V. Conclusion

We have performed detailed measurements of the WLmagnetoresistance, TDUCF amplitude as a function offield, and the magnetofingerprint of both AuPd andAg mesoscopic wires. A comparison of the coherencelengths inferred from WL and TDUCF measurements arein strong quantitative agreement for the AuPd sampleswhile a disagreement was observed in the Ag samplessimilar to that previously seen in Ag films. We hypothe-size that the reason for this discrepancy is that Ag maybe approaching the saturated noise limit, and further ex-periments to test this are ongoing. The observed sup-pression of TDUCF magnitude at high drive currents inAg agrees qualitatively with ensemble averaging related

to the correlation energy scale, though a quantitative un-derstanding is still lacking. Finally, we have consideredthe coherence saturation seen in our quasi-2D AuPd sam-ples, and discussed the important influence of magneticimpurities in such systems. The subtle physics, rich phe-nomenology, and continued presence of surprises (suchas the disagreement between WL and TDUCF in Ag)demonstrate why electronic coherence in solids remainsa lively area for investigation.

We would like to thank N.O. Birge for his helpful ad-vice concerning noise measurements, and I.L. Aleiner andA.D. Stone for discussions of the theory. This work wassupported by DOE grant DE-FG03-01ER45946/A001and the David and Lucille Packard Foundation.

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