Anisotropy and Penetration Depth of MgB from 11B NMR 2 Bo Chen1, Pratim Sengupta1, W. P. Halperin1, E. E. Sigmund2, V. F. Mitrovi´c3, M. H. Lee4, K. H. Kang4, B. J. Mean4, J. Y. Kim5, B. K. Cho5 1Department of Physics and Astronomy, Northwestern University, Evanston, Illinois 60208 2Department of Radiology, New York University, New York, New York 10016 7 3Department of Physics, Brown University, Providence, Rhode Island, 02912 0 4Department of Physics, Konkuk University, Seoul 143-701, Korea 0 5Center for Frontier Materials, Department of Materials Science and Engineering, KJ-IST 500-712, Korea 2 (Dated: Version February 6, 2008) n a The11BNMRspectrainpolycrystallineMgB2 weremeasuredforseveralmagneticfields(1.97 T J and 3.15 T) as a function of temperature from 5 K to 40 K. The composite spectra in the super- 4 conducting state can be understood in terms of anisotropy of the upper critical field, γH, which is 2 determined to be 5.4 at low temperature. Using Brandt’s algorithm1 the full spectrum, including satellites, was simulated for the temperature 8 K and a magnetic field of 1.97 T. The penetration ] depth λ was determined to be 1,152±50 ˚A, and the anisotropy of the penetration depth, γλ, was n estimated to be close to one at low temperature. Therefore, our findings establish that there are o two different anisotropies for uppercritical field and penetration depth at low temperatures. c - r p The discoveryofunusually highsuperconductivetran- u sition temperatures of MgB , a simple bimetallic com- 70 2 s pound superconductor2, has attracted considerable in- 60 . at terest from both theory and experiment. Reconsidera- 50 m tion and extension of BCS theory to two-band super- conductivity hassuccessfully accountedfor experimental 40 T d- observations3,4,5,6,7,8. Nonetheless, the relation between (K 30 ) n anisotropyof the upper criticalfield andthe penetration o 20 depth is still a controversial issue. Generally there are c [ two points of view. One holds that there exist two dif- 10 ferent anisotropies at low temperatures, γ and γ , for 2 upper critical field and penetration depthHrespectλively. 26.60 26.80 27.0042.8 43.0 43.02 v They have different temperature dependence and merge Frequency (MHz) 3 at a common value at T 3,9,10. The other perspective 7 c 4 is that there is only one anisotropy parameter, and it FIG. 1: 11B NMR spectra of the central transition in mag- 1 is field dependent5,11,12. Moreover, small angle neutron neticfieldsof1.97T(left)and3.15T(right)obtainedfroma 1 scattering (SANS) gives different results on the penetra- frequency sweep described in the text. The (±3/2 ↔ ±1/2) 6 tiondepthanisotropyonsinglecrystalandpowderMgB satellites areabout 340 kHzaway from thecentraltransition 2 0 samples4,13,14. Ingeneral,itismoreofachallengeto de- and, although they are not shown here, they are shown in / Fig.6. t termine the absolute value of the penetration depth as a compared with its temperature dependence. Although m muon spin resonance (µSR)15 and nuclear magnetic res- - onance(NMR)methods16 haveoftenbeenusedtoobtain d perature range from 5 K to 40 K on a powder sample of n an absolute value of the penetration depth, the applica- MgB attwomagneticfields,1.97Tand3.15T.We find o tion of these resonance techniques to determine the pen- 2 that on cooling the spectra acquire a broad asymmetric c etration depth for an anisotropic superconductor with linebelowthesuperconductivetransitiontemperatureas v: a sample consisting of a randomly oriented powder has shownin Fig.1. The shape ofthe broadline suggeststhe i never been attempted until now. X expected lineshape from an inhomogeneous field distri- NMR and electron spin resonance (ESR) have been bution from vortices in their solid state. However, as we r a used previously to investigate the anisotropy of MgB2. willsee,thisinterpretationistoosimplistic. Therelative Two different components of the resonance signal have weightofthisbroadline,comparedtothenarrownormal been identified in the superconductive state in a re- component, increases with decreasing temperature. We strictedrangeofmagneticfield17,18,19 andtheanisotropy associate this with the temperature and angular depen- of the upper critical field has been deduced. Addition- dence of the upper critical field. From this behavior we ally,anattemptwasmadeto determinethe temperature obtainthe uppercriticalfieldanisotropytobe 5.4atlow dependent penetrationdepth fromthe NMR linewidth18 temperature. We have also simulated the full spectrum assumingthatMgB2 isisotropic,whichisclearlynotthe at8 K in a field of1.97 T using this value for anisotropy case. and, by comparing with experiment, we have obtained Here we report 11B NMR measurements in the tem- thepenetrationdepthλ=1,152±50˚A.Furthermore,we 2 field has a temperature and angular dependence: 20 Hacb2 Hc2(θ,T)=Hca2b(T)/q1+(γH2 −1)cos2θ (1) 15 where θ is the angle between the applied magnetic field andthec−axisofacrystal. Thetemperaturedependence H (T) 10 bofelHowcc2Tacn(dH)H,act2bheisuspkpetecrhcerditiicnalFfiige.ld2H. cF2owritllembepeerqautaulrteos the applied magnetic field for crystals oriented at a cer- tainangleθ (T). Thecrystalswithθ largerthanθ (T) cr cr 5 Hcc2 havetheir Hc2 greaterthanthe appliedfield, andare su- perconductive. Due to the random distribution of the orientation of the crystals, the superconductive fraction 0 in the sample simply equals cos θ (T). As the tempera- 0 5 10 15 20 25 30 35 40 cr turedecreasesfurther,Hc increasesand,ifitcrossesthe T (K) c2 appliedmagnetic field, the whole sample becomes super- conductive. In Fig. 1, for H = 1.97 T, the narrow line FIG. 2: Schematic Hc2 diagram of MgB2. The upper dotted disappearsbelow17Kleavingonly the broadline. How- linedenotes3.15T,andthelowerlinecorrespondsto1.97T. The valuefor Hc2(0) is taken from Bud’ko and Canfield20. ever, in H = 3.15 T the c−axis upper critical field, Hcc2, is always smaller than the applied field. Therefore, part of the sample remains in the normal state in this field andcontributes to the narrowline in the spectrum, even at the lowest temperatures. findthatthepenetrationdepthis isotropicforT <10 K In Fig. 1, the position of the narrow peak is al- even though the upper critical field and the coherence mosttemperatureindependentandhasagaussianshape. lengtharenot. Ourresults supportthe theoreticalclaim Therefore,thecontributionofcrystalsinthenormalstate that there are two different anisotropy parameters for upper critical field and penetration depth3,7,9. can be deconvolvedfrom the composite spectra. The ra- tio of the remaining area to the whole spectrum gives The polycrystalline MgB2 sample was prepared by the superconductive fraction cos θcr(T), plotted in Fig. solidstatereactiontechniquesusingamixtureofmagne- 3. Furthermore, with Hab(0) taken to be 16 T from c2 siumandboronpowders. Thesuperconductivetransition Bud’ko and Canfield20, and the upper critical field at temperature was measured to be 39.5 K for the onset of θ at 5 K equal to 3.15 T, the external applied mag- cr diamagnetism in a magnetic field of 1.0 mT and 39 K netic field, the upper critical field anisotropy γ can H for zero resistance. A sample of 0.2 gram randomly ori- be obtained from Eq. 1 and we find this to be 5.4 at ented MgB2 powder was used in our experiments. NMR low temperature. This value is consistent with previous measurementswerecarriedoutinthe temperaturerange reports3,4,5,7,9,10,11,12,17,21,22. between 5 K and 40 K in magnetic fields of 1.97 T and Assuming γ to be temperature independent, Hab at H c2 3.15Tinasuperconductivemagnet. Broadspectrawere eachtemperature point canbe obtainedfollowingEq. 1. obtained by summing Fourier transforms of echo signals The temperature dependence for this analysis is plotted for a suite of different frequencies that cover the NMR in Fig. 4 where it is compared with results for Hab from c2 spectrum. other groups19,20. The discrepancy grows with increas- ing temperature. However, it is now accepted that γ The spectra displayed in Fig. 1 are the central tran- H decreases with increasing temperature3,7. Therefore, in sition (−1/2↔1/2) of 11B. At high temperature, the ourderivationathightemperatureswehaveusedavalue sample is metallic in the normalstate and this spectrum for γ that is too large which will produce a larger Hab consists of a single narrow and symmetric line. As the H c2 and consequently an overestimate of the transition tem- temperatureis lowered,a broadandasymmetricline ap- perature in a given field. In contrast, Fig. 4 shows that pears. We associate this with the inhomogeneous field the critical field curve deduced from our data and Eq. 1 distributionfromvorticesinthesuperconductivestatein is too low. In fact, it extrapolates to a zero field transi- addition to diamagnetic screening currents1. The weight tion temperature around 30 K. The principal reason for of the broad line increases with decreasing temperature, this discrepancy is vortex dynamics. At high tempera- while that of the narrow line decreases. The two lines tures vortices are in a liquid state16,23 and their dynam- coexist to a temperature of 5 K at 3.15 T, whereas only icsontheNMRtimescaleaveragethelocalfieldstozero the broad line survives below 17 K at 1.97 T. This can at the 11B nucleus. This transfers spectral weight from be explained by anisotropy of the upper critical field in the broadline to the narrowline and reduces the appar- MgB . 2 entsuperconductivefractionobtainedfromNMR.Atlow Due to the temperature dependence and anisotropyof temperatures,inthevortexsolidstateouranalysisofthe the two gap parameters in MgB 3,7,8, its upper critical superconductive fraction is reliable and, as can be seen 2 3 citations in different parts of the vortex structure as has 1.2 been reported25 for YBCO.However,our spin-lattice re- laxationresultsserveasaguidetohelpusavoidselective 1.0 saturation,particularlyatlowfrequencieswheretherate n o is small, allowing us to obtain a faithful representation cti 0.8 of the spectrum. a g fr The absolute value ofthe penetrationdepth λ is a key uctin 0.6 pyeatraimt eisterdifffiorcuclhtartaoctmereiazsautrioenaocfcusruapteerlcyo.ndUuscintigvitaytaunnd- nd nel diode oscillator technique Fletcher et al.10 found a o 0.4 c penetration depth of MgB between 800 and 1,200 ˚A. r 2 pe Finnemoreetal.26 determinedthatλabwas1,400˚Afrom Su 0.2 transport measurements. Using ESR, Simon et al.19 re- ported a value of the penetration depth between 1,100 ˚A and 1,400 ˚A. From analysis of the second moment of 0 4 8 12 16 20 24 28 the measured NMR linewidth, Lee et al. calculated the T (K) penetration depth to be 2,100 ˚A. But, as we mentioned earlier, the resonance methods cannot obtain a reliable FIG.3: Temperaturedependenceofthefraction ofsupercon- measureofthepenetrationdepth,ifitisassumedintheir ductive crystallites in the sample determined from the com- interpretationthatthe superconductoris isotropic. Here positespectraplotted vstemperaturefor1.97 T(uptriangle) and 3.15 T (downtriangle). wedeterminethepenetrationdepthbycomparisonofour measured spectrum with a simulation of the local fields in the mixed state for an anisotropic random powder at in Fig. 4, our results match Hab(T) below 10 K. 8 K in a magnetic field of 1.97 T using the penetration c2 depth as a variational parameter. The NMR spectrum is a local magnetic field map. 1.0 At low temperature, the vortices are in the solid state and contribute to the associated field distribution of the 0.8 NMRspectrum. Thefielddistributionofthemixedstate can be calculated by solving the Ginzburg-Landau (GL) abH (0)c2 0.6 eaqnuiatteiroant.ivFeo,rqtuhiicskplyurcpoonsveewrgeinadgomptetBhroadn.dtT’shaelgsoorliuthtimon1 abH (T) /c2 0.4 glianivtttehisceetshauenpdecrutchrorenedndituacmatniandggnfiseettlaidctefid.eiTsldthrsiebfrruoetqmiounisrcefrdreoeimnnipnutghtsecuavrroreerttnhetxes externalfield,coherencelengthξ,andpenetrationdepth 0.2 λ. We calculate the coherence length, ξ = 108 ˚A from ab the upper criticalfield20 andwe take its anisotropyfrom Eq. 1, 0.0 0 10 20 30 40 T (K) ξ(θ)=ξ / 1+(γ2 −1)cos2θ (2) ab q H FIG. 4: Hac2b. The circles are results from F. Simon et al.19 (circle) and diamonds are from S. L. Bud’ko and Canfield20. Withtheseinputs,thefielddistributionforacrystalata specificangleisgeneratedbyBrandt’salgorithm1includ- Analysis of our data using Eq. 1 is plotted assuming a con- stant γH = 5.4 for 1.97 T (uptriangle) and 3.15 T (downtri- ing the central transition and its quadrupolar satellites. angle). We infer that our interpretation of the NMR signal We convolute the spectrum with a broadening function, as a vortex-broadened, inhomogeneous magnetic field distri- exp−2(H/δ)2, which will also include the effect of the bution, is valid only below 10 K. finite width of the NMR line in the normal state. In a powder sample, which we assume to be composed of We have found that the spin-lattice relaxation rate of single crystal ellipsoids of revolution, we must consider the broad line is much slower than that of the narrow the shiftsofmagnetizationowingto demagnetizationac- line. This agrees with a previous report24. Additionally, cording to the shape and orientation distribution of the we have found that the rate increases smoothly with in- grains27. For simplicity we characterize this distribu- creasingfrequencywithinthespectrum,risinginthehigh tion by an average demagnetization factor, D. This as- frequency tail of the central transition. Owing to the sumptionwouldbepreciseifthegrainshapedistribution inherent inhomogeneity of the field distribution, which is uncorrelated with the crystal structure. The demag- we will discuss later, it is not possible to deconvolute netization effect gives a relative shift of the magnetiza- spin-lattice relaxationsignals to searchfor electronic ex- tionwhichitselfdepends onthe orientationofthe grains 4 since the diamagnetic moment from screening currents of the first moment of the distribution, as can be seen is strongly angular dependent. Simulations of spectra at in Fig. 5, must be correctly handled since it contributes three different, but representative, angles are presented significantly to the overall lineshape. Earlier work on in Fig. 5. The spectrum for the whole sample is then other superconductors analyzing the field distribution in obtained as the integral of 91 spectra with orientation the mixedstate has been directedatthe moments of the uniformly distributed between 0 and π/2, weighted by distribution, so we have calculated the first three mo- a factor sin θ appropriate for a random distribution of ments for an anisotropic superconductor with randomly grain orientations. oriented grains, as a function of the penetration depth, restricted to the case of γ =5.4 and γ =1. H λ 140 2500 120 2000 100 y sit n 80 e y 1500 Int 60 nsit e 40 Int 1000 20 500 0 1.90 1.95 2.00 2.05 H (T) 0 1.90 1.95 2.00 2.05 H (T) FIG. 5: Simulated spectra for crystals at different orienta- tions. Thesolid, dashedand dottedcurvesare thespectraof crystals with c−axis at π/2, π/4 and 0 angles to the applied FIG. 6: The spectrum at a temperature of 8 K and a mag- field,respectively. Ademagnetization factor, D=1/3, anda netic field of 1.97 T. The blue solid line is the experimental gaussian broadening parameter, δ=5.2 mT, were chosen for spectrum. The dotted line is the simulation described in the these spectra. text. The second moment of the magnetic field distribution There are three variational parameters, λ, D, and δ. of a spectrum from a vortex lattice can be related1 to We then carry out a χ2 minimization of the difference itspenetrationdepth λforlowmagneticfieldscompared between the simulated spectrum and the experimental to H by the Pincus’ formula1,16 where the second mo- one, taking their areas to be equal. The simulated spec- c2 mentvariesastheinversefourthpowerofthepenetration trum is shown in Fig. 6 together with the experimental depth, B2 = (0.0609φ )2/λ4. In the present case the spectrum. The numerical results provide an excellent (cid:10) (cid:11) 0 simulated spectrum is the superposition of spectra with representation of the complex measured spectrum with anisotropic coherence lengths and upper critical fields. values for the variationalparameters for the penetration Nonetheless, we find the 1st, 2nd and 3rd moments of depthλ=1,152±50˚A,averagedemagnetizationfactor, the spectrum can be similarly related to inverse, even D=0.31±0.01andthegaussianbroadening,δ =5.2mT, powers of the penetration depth in the following elegant whichislargerthan,butofthesameorderas,thenormal way: state linewidth, 2 mT. The quoted accuracy is statisti- cal. This value for D is rather close to that anticipated for asphericalgeometry,Dsphere =1/3,anditis reason- hBi=−(1−D)·A /λ2 (3) 1 abletoexpectthisvaluefortheaveragedemagnetization factor for a large ensemble of grains. Earlier reports for the value of the penetration depth10,19,26,28 are similar B2 =δ2/4+A /λ4 (4) 2 to ours although our accuracy is higher. Our simulation (cid:10) (cid:11) and its comparison with experiment, as represented in Fig. 6, is the most precise such comparison obtained by B3 =A /λ6, (5) 3 (cid:10) (cid:11) resonance methods and it is the first time that such a simulationhasbeenattemptedforastronglyanisotropic where A , A , A are numerical constants. The gaus- 1 2 3 supercondcutor. We emphasize that previous work has sian broadening factor is δ and D is the demagne- generallyfocusedonmomentsofthemeasuredspectrum, tization factor. We find A = 1.415 × 104 T ˚A2, 1 often restricting attention to the second moment. For A = 2.621 × 107 T2 ˚A4, A = 1.524 × 1011 T3 ˚A6. 2 3 an anisotropic superconductor the angular dependence Howeverwecautionthatanisotropyandfielddependence 5 2000 isthe inverseofH ,thereforethe penetrationdepthhas c2 the inverse angular dependence of the coherence length, γ = 1 λ 1000 λ(θ)=λ 1+(γ2−1)cos2θ (6) abq λ 0 With the same approach as before we generate the 2000 spectrumforthepowdersamplewithγ largerthanone. λ The central transition is found to decrease with increase y ensit 1000 γ λ= 1.2 iolfluγsλtraanteds.theTshpisecsturaggbeesctosmtheamtoγrλeaastymlowmettermicpaesraFtiugr.e7s nt should be close to one. Magnetization measurements I showthatγ isaround1.7between20and27K28. SANS λ experiments on a powder MgB sample13 give an upper 2 0 limitofγ tobearound1.5andessentiallymagneticfield λ 2000 independent. Our result is consistent with these values. However,SANS measurements on a single crystal4,14 in- γ = 1.5 λ dicatethatγλ isclosetooneatT=2Kandatlowfiled, 1000 H<0.5T,andthatitincreaseswithexternalfieldreach- ing ≈ 3.5 in a field of 0.8 T. The lower γ in the powder λ sample is believed to be caused by a limiting crystallite size effect13. Further work will be required to elucidate 0 this phenomenon. 1.90 1.94 1.98 2.02 In conclusion, we measure the 11B NMR spectra of H (T) a random powder sample of MgB in magnetic fields 2 of 1.97 T and 3.15 T. The evolution of the spectra FIG.7: Simulationwithdifferentγλplottedtogetherwithex- through the temperature range can be explained by the perimentalspectrumatatemperatureof8Kandamagnetic anisotropy of the upper critical field γ , which is deter- field of 1.97 T. The dashed lines are simulations. It is clear H mined to be 5.4 at low temperature. We find from our thatthebestcomparisonbetweenexperimentandsimulation holds for γλ ≈1. simulation that the penetration depth carries a differ- ent anisotropy from the upper critical field and that at lowtemperaturesitisalmostisotropicsimilartothatre- of the local field distributions mean that these numeri- ported from SANS13 for a powder sample. The value of calconstants hold only in a limited rangewhich we have thepenetrationdepththatwehaveobtainedforMgB2 is explored for MgB with γ = 5.4, γ = 1 and H = 1.97 1,152±50˚Aat8Kinamagneticfieldof1.97T.Fromour 2 H λ T. numerical studies we have found simple expressions for We havealsoinvestigatedthe effect ofthe penetration thepenetrationdepthdependence ofthe momentsofthe depth anisotropy γ at low temperature. In actuality, field distribution in a random powder of an anisotropic λ the vortex structure for arbitraryangle θ is found as the superconductor. solution to the anisotropic GL equations29. 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