Weak localization has a strong influence on both the normal and
superconducting properties of metals. In particular, since weak localization
leads to the decoupling of electrons and phonons, the temperature dependence of
resistance (i.e., $\lambda_{tr}$) is decreasing with increasing disorder, as
manifested by Mooij's empirical rule. In addition, Testardi's universal
correlation of $T_{c}$ (i.e., $\lambda$) and the resistance ratio (i.e.,
$\lambda_{tr}$) follows. This understanding provides a new means to probe the
phonon mechanism in superconductors including MgB$_{2}$. The merits of this
method are its applicability to any superconductors and its reliability because
the McMillan's electron-phonon coupling constant $\lambda$ and $\lambda_{tr}$
change in a broad range, from finite values to zero, due to weak localization.
Karkin et al's preliminary data of irradiated MgB$_{2}$ show the Testardi
correlation, indicating that the dominant pairing mechanism in MgB$_{2}$ is the
phonon-mediated interaction.Comment: 9 pages, latex, 3 figure

Ahn J S

Akimitsu J

Bouquet F

Dynes C

Economou E N

Grimvall G

Karkin A E

Kerim Savran

Kong Y

Liu A Y

McMillan W L

Mi-Ae Park

Mooij J H

Schmidt H

Sharoni A

Yildirim Y

Yong-Jihn Kim

English

arXiv

Weak localization has a strong influence on both the normal and superconducting properties of metals. In particular, since weak localization leads to the decoupling of electrons and phonons, the temperature dependence of resistance (i.e. λtr) decreases with increasing disorder, as manifested by Mooij's empirical rule. In addition, Testardi's universal correlation of Tc (i.e. λ) and the resistance ratio (i.e. λtr) follows. This understanding provides a new means to probe the phonon mechanism in superconductors, including MgB2. The merits of this method are its applicability to any superconductor and its reliability because the McMillan's electron-phonon coupling constant λ and λtr change in a broad range, from finite values to zero, due to weak localization. Karkin et al's preliminary data of irradiated MgB2 show the Testardi correlation, indicating that the dominant pairing mechanism in MgB2 is a phonon-mediated interaction

Park, M.-A.

Savran, K.

Kim, Y.-J.

Bilkent University Institutional Repository

Superconductor Science and TechnologyRAPID COMMUNICATIONA new method of probing the phonon mechanismin superconductors, including MgB2To cite this article: Mi-Ae Park et al 2001 Supercond. Sci. Technol. 14 L31 View the article online for updates and enhancements.Related contentTopical reviewCristina Buzea and Tsutomu Yamashita-Acoustic Debye temperature and the roleof phonons in cuprates and relatedsuperconductorsR Abd-Shukor-Microwave surface resistance in MgB2A A Zhukov, L F Cohen, K Yates et al.-Recent citationsEnhancing the pinning strengths inpolycrystalline MgB2I. M. Obaidat et al-Weak localization and the Mooij rule indisordered metalsKerim Savran et al-Three-dimensional Anderson model oflocalization with binary random potentialR. A. Römer et al-This content was downloaded from IP address 139.179.72.98 on 03/01/2019 at 14:57INSTITUTE OF PHYSICS PUBLISHING SUPERCONDUCTOR SCIENCE AND TECHNOLOGYSupercond. Sci. Technol. 14 (2001) L31–L35 www.iop.org/Journals/su PII: S0953-2048(01)24608-9RAPID COMMUNICATIONA new method of probing the phononmechanism in superconductors, includingMgB2Mi-Ae Park1, Kerim Savran2 and Yong-Jihn Kim21 Department of Physics, University of Puerto Rico at Humacao, Humacao, PR 00791,Puerto Rico2 Department of Physics, Bilkent University, 06533 Bilkent, Ankara, TurkeyReceived 1 May 2001AbstractWeak localization has a strong influence on both the normal andsuperconducting properties of metals. In particular, since weak localizationleads to the decoupling of electrons and phonons, the temperaturedependence of resistance (i.e. λtr ) decreases with increasing disorder, asmanifested by Mooij’s empirical rule. In addition, Testardi’s universalcorrelation of Tc (i.e. λ) and the resistance ratio (i.e. λtr ) follows. Thisunderstanding provides a new means to probe the phonon mechanism insuperconductors, including MgB2. The merits of this method are itsapplicability to any superconductor and its reliability because theMcMillan’s electron–phonon coupling constant λ and λtr change in a broadrange, from finite values to zero, due to weak localization. Karkin et al’spreliminary data of irradiated MgB2 show the Testardi correlation,indicating that the dominant pairing mechanism in MgB2 is aphonon-mediated interaction.1. IntroductionThe recent discovery of superconductivity in MgB2 at 39 Kby Akimitsu and co-workers [1] has renewed our interest insuperconductivity. There are already many experimental andtheoretical investigations [2–15]. For instance, the isotopeeffect coefficient of Boron is measured to be about 0.26–0.3 [2, 3] and temperature-dependent resistivity measurementsindicate that MgB2 is a highly conducting material withλtr  0.6 and room-temperature resistance ratio (RRR) ofRRR = 25.3 [4]. The electron–phonon coupling constant λis estimated to be about 0.6–0.7 based on low-temperaturespecific heat measurements [5, 6], whereas the phonon density-of-states measurements suggest λ ∼ 0.9 [7]. Tunnellingmeasurements of the energy gap show the BCS form withsome variations of the maximum gap values [8–10]. Onthe other hand, electronic and phononic structures have beencomputed by numerical methods [11–15]. It has been foundthat the (possibly anharmonic [15]) Boron bond stretchingmodes are strongly coupled to the px,y electronic bands. TheMcMillan’s electron–phonon coupling constant λ is calculatedto be about λ ∼ 0.7–0.9 [11–15]. These investigations seem tobe consistent with the BCS phonon-mediated superconductingbehaviour.However, there is no definite experimental evidence asyet. It is also not clear whether MgB2 is an intermediate-coupling or strong-coupling superconductor [11–15], eventhough it is plausible that the high-frequency boron phononmodes may lead to strong electron–phonon coupling and thehigh Tc [11–15]. From a fundamental point of view it remainsto be clarified whether the conventional strong-coupling theorycan explain the high Tc = 39 K. In other words, what is themaximum Tc which can be produced by the phonon-mediatedinteraction [16, 17]? In this context it is clear that MgB2 willlead us to refine our understanding of superconductivity.There are basically two methods to probe the phononmechanism directly. One is the isotope effect measurement[2, 3] and the other is the tunnelling measurement of theelectron–phonon spectral density [18]. Although the existenceof the boron isotope effect on Tc is strong evidence for theimportance of the phonon mechanism [2, 3], the observedreduced isotope effect requires more investigation on the0953-2048/01/070031+05$30.00 © 2001 IOP Publishing Ltd Printed in the UK L31Rapid communicationpairing mechanism [3]. Unfortunately, the tunnelling data arenot available at this moment.In this rapid communication we introduce a new methodof probing the phonon mechanism in superconductors,including MgB2. This method is based on the correlationof the McMillan’s electron–phonon coupling constant λ insuperconductivity and λtr in the phonon-limited resistivityof the normal transport phenomena [19, 20]. In most sp-orbital metals λ and λtr are almost the same in magnitude[19, 20]. Testardi et al [21, 22] found that disorder significantlydecreases both quantities and leads to the universal correlationof Tc and the resistance ratio, which may be called the Testardicorrelation. In fact, this experimental result is a manifestationof the weak localization effect on the electron–phononinteraction [23]. More precisely, weak localization leads to thedecoupling of electrons and phonons and thereby gives rise tothe Testardi correlation. Therefore, if MgB2 shows the Testardicorrelation we may say that the dominant pairing mechanismof MgB2 is the phonon-mediated interaction. This correlationhas already been confirmed in A-15 compounds [21, 22]and ternary superconductors [24]. Another experimentalmanifestation of the weak localization effect on the electron–phonon interaction is the Mooij rule [25]. This rule states thatas the system becomes disordered the temperature dependenceof the resistivity decreases, that is the coupling betweenelectrons and phonons weakens. This provides another test forthe importance of the phonon mechanism in superconductors.The main advantages of this new method are its wideapplicability to any superconductor and its reliability, becausethe McMillan’s coupling constant λ and λtr can be variedfrom finite values to zero. Since the temperature dependenceof the resistivity at room temperature is dominated bythe electron–phonon interaction, the decrease of λtr clearlysignals the reduction of the electron–phonon interaction andthereby probes the importance of the phonon mechanismin superconductors. This method may also providecrucial information on the pairing mechanism in exoticsuperconductors, such as fullerene superconductors, organicsuperconductors, heavy fermion superconductors, high-Tccuprates, and Sr2RuO4.2. Manifestations of weak localization effect on theelectron–phonon interactionIn this section we point out that Testardi’s correlation of Tc andthe resistance ratio and the Mooij rule are caused by the weaklocalization of electrons in disordered systems.2.1. Testardi’s correlation of Tc and the resistance ratioIn the 1970s Testardi et al [21, 22] found the universalcorrelation of Tc and the resistance ratio in A-15 compounds,such as Nb–Ge, V3Si, and V3Ge. Since Tc and the resistanceratio are determined by McMillan’s electron–phonon couplingconstant λ and λtr respectively, this means the correlationbetween λ and λtr . The McMillan’s coupling constant λ isdefined by [16]λ = 2∫α2(ω)F (ω)ωdω = N0 〈I2〉M〈ω2〉 (1)Figure 1. Tc/Tc0 against the resistance ratio. The shaded regionrepresents the Tc–resistance-ratio correlation band for A-15compounds from Testardi et al [21, 22]. The circles are for ErRh4B4and the triangles for LuRh4B4; the data are from Dynes et al [24].where F(ω) is the phonon density of states and M is the ionicmass. 〈I 2〉 and 〈ω2〉 are the average over the Fermi surfaceof the square of the electronic matrix element and the phononfrequency. The RRR is given asρ(300 K)ρ0= ρ0 + ρph(300 K)ρ0(2)where ρ0 and ρph denote the residual resistivity and thephonon-limited resistivity. The phonon-limited resistivity ρphat high temperature is defined byρph(T ) = 4πmkBTne2h̄∫α2trF (ω)ωdω = 2πmkBTne2h̄λtr . (3)Hereαtr includes an average of a geometrical factor 1−cos θkk′ .Inserting (3) into (2) we obtainρ(300 K)ρ0= 1 + 2πτkB × 300 Kh̄λtr . (4)Figure 1 shows the correlation ofTc and the resistance ratiofor A-15 compounds and ternary superconductors. Data arefrom Testardi et al [21, 22] and Dynes et al [23]. The shadedregion denotes the correlation band for A-15 compounds. Itis clear that as the system becomes disordered by radiationdamage or substitutional alloying, both λ and λtr decrease andthereby reduce Tc and the resistance ratio. This behaviourexemplifies strong correlations between the physical propertiesin normal and superconducting states.L32Rapid communication0 100 200 300 400T (K)050100150200250ρ (µΩcm)RRR=1~1.19~1.57~2.28~300361133% AlFigure 2. Resistivity against temperature for Ti and TiAl alloyscontaining 0, 3, 6, 11, and 33% Al; the data are from Mooij [25].The dashed lines were used to estimate the RRR.2.2. The Mooij ruleMooij [25] pointed out that the size and sign of the temperaturecoefficient of resistivity (TCR) in many disordered systemscorrelate with its residual resistivity ρ0 as follows:dρ/dT > 0 if ρ0 < ρMdρ/dT < 0 if ρ0 > ρM. (5)Thus, the TCR changes the sign when ρ0 reaches the Mooijresistivity ρM ∼ 150 µ! cm. In other words, as the systembecomes disordered, the TCR decreases.Now we show that the Testardi correlation is equivalent tothe Mooij rule. Since the resistivity at temperature T is givenbyρ(T ) = ρ0 + ρph(T ) (6)the TCR is determined mainly by ρph, i.e.dρ/dT = dρph/dT ∼= 2πmkBne2h̄λtr . (7)Note that since the TCR is controlled by λtr , the decrease ofthe TCR due to disorder means the reduction of λtr , whichis the essence of the Testardi correlation. Therefore, boththe Testardi correlation and the Mooij rule are manifestationsof the weak localization correction to the electron–phononinteraction; that is to λ and λtr .Figure 2 shows the resistivity as a function of temperaturefor pure Ti and TiAl alloys containing 3, 6, 11, and 33%Al; the data are from Mooij [25]. The TCR decreases as theresidual resistivity increases due to disorder; note that the RRRdecreases accordingly. The dashed lines are our conjectureddata points to estimate RRR. Rough estimated values are ∼30,∼2.28, ∼1.57, and ∼1.19 for Al concentrations of 0, 3, 6, and11%. When the RRR is about one, equation (3) tells us thatλtr (and λ) is zero. If the system shows superconductivity, Tcshould drop to zero at this point, which is in agreement withthe Testardi correlation of Tc and the resistance ratio.2.3. Weak localization correction to McMillan’s couplingconstant λ and λtrWe briefly review the derivation by Park and Kim [23].Since the equivalent electron–electron potential in the electron-phonon problem is determined by the phonon Green’s function,D(x − x ′), the Fröhlich interaction at a finite temperature foran Einstein model is given byVnn′(ω, ω′) = I20Mω2D∫ ∫dr dr′ψ∗n′(r)ψ∗n̄′(r′)×D(r − r′, ω − ω′)ψn̄(r′)ψn(r)= I2oMω2D∫|ψn′(r)|2|ψn(r)|2 dr ω2Dω2D + (ω − ω′)2= Vnn′ ω2Dω2D + (ω − ω′)2(8)whereD(r − r′, ω − ω′) =∑qω2D(ω − ω′)2 + ω2Deiq·(r−r′)= ω2D(ω − ω′)2 + ω2Dδ(r − r′). (9)Here ω means the Matsubara frequency and ψn denotes thescattered state. I0 is the electronic matrix element for the planewave states. Accordingly, the McMillan’s electron–phononinteraction coupling constant λ is given byλ = N0〈Vnn′(0, 0)〉 = N0 I2oMω2D〈 ∫|ψn(r)|2|ψn′(r)|2 dr〉.(10)This expression shows that the McMillan’s coupling constantis basically determined by the short time density correlationfunction [23, 26], since the phonon-mediated interaction isretarded for tret ∼ 1/ωD:λ = N0 I20Mω2D[1 − 3(kF ()2(1 − (L)]. (11)Here ( andL denote the elastic mean free path and the inelasticdiffusion length, respectively. Subsequently, one findsλtr = 2∫α2tr (ω)F (ω)ω dω ∼= N0I 20Mω2D[1− 3(kF ()2]. (12)We have used the fact that L is effectively infinite at T = 0.It is worth noting that the weak localization correction term isthe same as that of the conductivity.L33Rapid communicationFigure 3. Tc–resistance-ratio correlation band for A-15 compoundswith the data of MgB2 superimposed; the data are from Canfieldet al [4] (sample 1), Finnemore et al [30] (sample 2), Jung et al [31](sample 3), and Karkin et al [29] (samples 4 and 5).3. Using weak localization to probe the phononmechanism in magnesium diborideSince weak localization of the electrons occurs for the meanfree path of the order of 10 Å [23], heavy doses of radiation orhigh concentrations of impurities are required to see the effectof weak localization. At the same time the disordered samplesshould be macroscopically homogeneous. Consequently,recent impurity doping experiments [27, 28] in MgB2 werenot successful in seeing this effect, while Karkin et al [29]observed the decrease of Tc (onset temperature) from 39 to 5 Kby neutron irradiation. This behaviour is very similar to the Tcdecrease of A-15 compounds and ternary superconductors dueto radiation damage, which has been explained by the weaklocalization effect [23]. Now we check whether or not MgB2data satisfy the Testardi correlation and the Mooij rule.Figure 3 shows Tc/Tc0 against the RRR for MgB2(points/samples 1,3,4 and 5) and Mg10B2 (point/sample 2).The points/samples lie within the correlation band of A-15compounds, although sample 4 shows some deviation,presumably due to intergrain resistivities [29]. Note that thiscorrelation is universal for phonon-mediated superconductors.For MgB2 Tc0 was assumed to be 39.4 K, corresponding to theTc of MgB2 wire [4], whereas the Tc0 of Mg10B2 was chosen tobe 40.2 K [30]. The data are from Canfield et al [4] (sample 1),Finnemore et al [30] (sample 2), and Jung et al [31], and Karkinet al [29] (samples 4 and 5). Since samples 4 and 5 show abroad transition width (∼8 K), a criterion of a 50% drop of theresistivity was used to determine Tc for samples 4 and 5. Themeasured values of RRR are: sample 1, 25.3 [4]; sample 2, 19.7[30]; sample 3, 3 [31]; sample 4, ∼1.30 [29]; and sample 5,∼1.076 [29]. For sample 3 the disorder of the sample maybe due to the high-pressure sintering at high temperature [31].Overall, the Mooij rule is also satisfied, approximately.It seems that the preliminary data support the phononmechanism in MgB2. It is highly desirable to performirradiation experiments using a better quality sample to confirmthis result. The Mooij rule can also be confirmed separately,although the Testardi correlation would invariably lead to theMooij rule.4. ConclusionsWe introduce a new method of probing the phonon mechanismin superconductors, including MgB2. Weak localizationdecreases both λ in superconductivity and λtr in the phonon-limited resistivity at the same rate, as manifested by the Testardicorrelation of the Tc and the resistance ratio. Above Tc theMooij rule follows accordingly. Preliminary data of MgB2show the Testardi correlation and thereby support the phononmechanism in this newly discovered superconductor. Morethorough experimental investigations are required using bettersamples to clarify the details of the pairing mechanism.AcknowledgmentsWe are grateful to Professors Ceyhun Bulutay, B Tanatar, andA E Karkin for discussions. MP thanks NSF-EPSCOR (grantNo. EPS9874782) for financial support.References[1] Akimitsu J 2001 Proc. Symp. on Transition Metal Oxides(Sendai, January 10, 2001)Nagamatsu J, Nakagawa N, Muranaka T, Zenitani Y andAkimitsu J 2001 Nature 410 63[2] Bud’ko S L, Lapertot G, Petrovic C, Cunningham C E,Anderson N and Canfield P C 2001 Phys. Rev. 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