Anisotropy and penetration depth of MgB₂ from ¹¹B NMR

Bo Chen¹, Pratim Sengupta¹, W P Halperin¹, E E Sigmund², V F Mitrović³, M H Lee⁴, K H Kang⁴, B J Mean⁴, J Y Kim⁵ and B K Cho⁵

¹ Department of Physics and Astronomy, Northwestern University, Evanston, IL 60208, USA
² Department of Radiology, New York University, New York, NY 10016, USA
³ Department of Physics, Brown University, Providence, RI 02912, USA
⁴ Department of Physics, Konkuk University, Seoul, 143-701, Korea
⁵ Department of Materials Science and Engineering, Center for Frontier Materials, KJ-IST 500-712, Korea

Email: w-halperin@northwestern.edu

Received 23 October 2006
Published 16 November 2006

The discovery of unusually high superconductive transition temperatures of MgB₂, a simple bimetallic compound superconductor [1], has attracted considerable interest from both theory and experiment. Reconsideration and extension of BCS theory to two-band superconductivity has successfully accounted for experimental observations [2]–[7]. Nonetheless, the relation between anisotropy of the upper critical field and the penetration depth is still a controversial issue. Generally there are two points of view. One holds that there exist two different anisotropies at low temperatures, γ_H and γ_λ, for upper critical field and penetration depth respectively. They have different temperature dependence and merge at a common value at T_c [2, 8, 9]. The other perspective is that there is only one anisotropy parameter, and it is field dependent [4, 10, 11]. Moreover, small angle neutron scattering (SANS) gives different results on the penetration depth anisotropy on single crystal and powder MgB₂ samples [3, 12, 13]. In general, it is more of a challenge to determine the absolute value of the penetration depth as compared with its temperature dependence. Although muon spin resonance (μSR) [14] and nuclear magnetic resonance (NMR) methods [15] have often been used to obtain an absolute value of the penetration depth, the application of these resonance techniques to determine the penetration depth for an anisotropic superconductor with a sample consisting of a randomly oriented powder has never been attempted until now.

NMR and electron spin resonance (ESR) have been used previously to investigate the anisotropy of MgB₂. Two different components of the resonance signal have been identified in the superconductive state in a restricted range of magnetic field [16]–[18] and the anisotropy of the upper critical field has been deduced. Additionally, an attempt was made to determine the temperature dependent penetration depth from the NMR linewidth [17] assuming that MgB₂ is isotropic, which is clearly not the case.

Here, we report ¹¹B NMR measurements in the temperature range from 5 to 40 K on a powder sample of MgB₂ at two magnetic fields, 1.97 and 3.15 T. We find that on cooling the spectra acquire a broad asymmetric line below the superconductive transition temperature as shown in figure 1. The shape of the broad line suggests the expected lineshape from an inhomogeneous field distribution from vortices in their solid state. However, as we will see, this interpretation is too simplistic. The relative weight of this broad line, compared to the narrow normal component, increases with decreasing temperature. We associate this with the temperature and angular dependence of the upper critical field. From this behaviour we obtain the upper critical field anisotropy to be 5.4 at low temperature. We have also simulated the full spectrum at 8 K in a field of 1.97 T using this value for anisotropy and, by comparing with experiment, we have obtained the penetration depth λ  =  1152 ± 50 Å. Furthermore, we find that the penetration depth is isotropic for T  <  10 K even though the upper critical field and the coherence length are not. Our results support the theoretical claim that there are two different anisotropy parameters for upper critical field and penetration depth [2, 6, 8].

The polycrystalline MgB₂ sample was prepared by solid-state reaction techniques using a mixture of magnesium and boron powders. The superconductive transition temperature was measured to be 39.5 K for the onset of diamagnetism in a magnetic field of 1.0 mT and 39 K for zero resistance. A sample of 0.2 g randomly oriented MgB₂ powder was used in our experiments. NMR measurements were carried out in the temperature range between 5 and 40 K in magnetic fields of 1.97 and 3.15 T in a superconductive magnet. Broad spectra were obtained by summing Fourier transforms of echo signals for a suite of different frequencies that cover the NMR spectrum.

The spectra displayed in figure 1 are the central transition (–1/2 ↔ 1/2) of ¹¹B. At high temperature, the sample is metallic in the normal state and this spectrum consists of a single narrow and symmetric line. As the temperature is lowered, a broad and asymmetric line appears. We associate this with the inhomogeneous field distribution from vortices in the superconductive state in addition to diamagnetic screening currents [20]. The weight of the broad line increases with decreasing temperature, while that of the narrow line decreases. The two lines coexist to a temperature of 5 K at 3.15 T, whereas only the broad line survives below 17 K at 1.97 T. This can be explained by anisotropy of the upper critical field in MgB₂.

Due to the temperature dependence and anisotropy of the two gap parameters in MgB₂[2, 6, 7], its upper critical field has a temperature and angular dependence:

where θ is the angle between the applied magnetic field and the c-axis of a crystal. The temperature dependence of H^c_c2 and H^ab_c2 is sketched in figure 2. For temperatures below T_c(H), the upper critical field H_c2 will be equal to the applied magnetic field for crystals oriented at a certain angle θ_cr(T). The crystals with θ larger than θ_cr(T) have their H_c2 greater than the applied field, and are superconductive. Due to the random distribution of the orientation of the crystals, the superconductive fraction in the sample simply equals cos θ_cr(T). As the temperature decreases further, H^c_c2 increases and, if it crosses the applied magnetic field, the whole sample becomes superconductive. In figure 1, for H  =  1.97 T, the narrow line disappears below 17 K leaving only the broad line. However, in H  =  3.15 T the c-axis upper critical field, H^c_c2, is always smaller than the applied field. Therefore, part of the sample remains in the normal state in this field and contributes to the narrow line in the spectrum, even at the lowest temperatures.

In figure 1, the position of the narrow peak is almost temperature independent and has a gaussian shape. Therefore, the contribution of crystals in the normal state can be deconvolved from the composite spectra. The ratio of the remaining area to the whole spectrum gives the superconductive fraction cos θ_cr(T), plotted in figure 3. Furthermore, with H^ab_c2(0) taken to be 16 T from Bud'ko and Canfield [19], and the upper critical field at θ_cr at 5 K equal to 3.15 T, the external applied magnetic field, the upper critical field anisotropy γ_H can be obtained from equation (1) and we find this to be 5.4 at low temperature. This value is consistent with previous reports [2]–[4], [6], [8]–[11], [16, 21, 22].

Figure 3. Temperature dependence of the fraction of superconductive crystallites in the sample determined from the composite spectra plotted versus temperature for 1.97 T () and 3.15 T ().

Assuming γ_H to be temperature independent, H^ab_c2 at each temperature point can be obtained following equation (1). The temperature dependence for this analysis is plotted in figure 4 where it is compared with results for H^ab_c2 from other groups [18, 19]. The discrepancy grows with increasing temperature. However, it is now accepted that γ_H decreases with increasing temperature [2, 6]. Therefore, in our derivation at high temperatures we have used a value for γ_H that is too large which will produce a larger H^ab_c2 and consequently an overestimate of the transition temperature in a given field. In contrast, figure 4 shows that the critical field curve deduced from our data and equation (1) is too low. In fact, it extrapolates to a zero field transition temperature around 30 K. The principal reason for this discrepancy is vortex dynamics. At high temperatures vortices are in a liquid state [15, 23] and their dynamics on the NMR time scale average the local fields to zero at the ¹¹B nucleus. This transfers spectral weight from the broad line to the narrow line and reduces the apparent superconductive fraction obtained from NMR. At low temperatures, in the vortex solid state our analysis of the superconductive fraction is reliable and, as can be seen in figure 4, our results match H^ab_c2(T) below 10 K.

Figure 4. Habc2. The circles are results from Simon et al  [18] () and diamonds are from Bud'ko and Canfield [19]. Analysis of our data using equation (1) is plotted assuming a constant γH  =  5.4 for 1.97 T () and 3.15 T (). We infer that our interpretation of the NMR signal as a vortex-broadened, inhomogeneous magnetic field distribution, is valid only below 10 K.

We have found that the spin-lattice relaxation rate of the broad line is much slower than that of the narrow line. This agrees with a previous report [24]. Additionally, we have found that the rate increases smoothly with increasing frequency within the spectrum, rising in the high frequency tail of the central transition. Owing to the inherent inhomogeneity of the field distribution, which we will discuss later, it is not possible to deconvolute spin-lattice relaxation signals to search for electronic excitations in different parts of the vortex structure as has been reported [25] for YBa₂Cu₃O₇. However, our spin-lattice relaxation results serve as a guide to help us avoid selective saturation, particularly at low frequencies where the rate is small, allowing us to obtain a faithful representation of the spectrum.

The absolute value of the penetration depth λ is a key parameter for characterization of superconductivity and yet it is difficult to measure accurately. Using a tunnel diode oscillator technique Fletcher et al  [9] found a penetration depth of MgB₂ between 800 and 1200 Å. Finnemore et al  [26] determined that λ_ab was 1400 Å from transport measurements. Using ESR, Simon et al  [18] reported a value of the penetration depth between 1100 and 1400 Å. From analysis of the second moment of the measured NMR linewidth, Lee et al  [17] calculated the penetration depth to be 2100 Å. But, as we mentioned earlier, the resonance methods cannot obtain a reliable measure of the penetration depth, if it is assumed in their interpretation that the superconductor is isotropic. Here we determine the penetration depth by comparison of our measured spectrum with a simulation of the local fields in the mixed state for an anisotropic random powder at 8 K in a magnetic field of 1.97 T using the penetration depth as a variational parameter.

The NMR spectrum is a local magnetic field map. At low temperature, the vortices are in the solid state and contribute to the associated field distribution of the NMR spectrum. The field distribution of the mixed state can be calculated by solving the Ginzburg–Landau (GL) equation. For this purpose we adopt Brandt's algorithm [20] an iterative, quickly converging method. The solution gives the current and field distribution from the vortex lattice and the diamagnetic fields from screening currents in the superconducting state. The required inputs are the external field, coherence length ξ, and penetration depth λ. We calculate the coherence length, ξ_ab  =  108 Å from the upper critical field [19] and we take its anisotropy from equation (1),

With these inputs, the field distribution for a crystal at a specific angle is generated by Brandt's algorithm [20] including the central transition and its quadrupolar satellites. We convolute the spectrum with a broadening function, exp(–2(H/δ)²), which will also include the effect of the finite width of the NMR line in the normal state. In a powder sample, which we assume to be composed of single crystal ellipsoids of revolution, we must consider the shifts of magnetization owing to demagnetization according to the shape and orientation distribution of the grains [27]. For simplicity we characterize this distribution by an average demagnetization factor, D. This assumption would be precise if the grain shape distribution is uncorrelated with the crystal structure. The demagnetization effect gives a relative shift of the magnetization which itself depends on the orientation of the grains since the diamagnetic moment from screening currents is strongly angular dependent. Simulations of spectra at three different, but representative, angles are presented in figure 5. The spectrum for the whole sample is then obtained as the integral of 91 spectra with orientation uniformly distributed between 0 and π/2, weighted by a factor sinθ appropriate for a random distribution of grain orientations.

There are three variational parameters, λ, D and δ. We then carry out a χ² minimization of the difference between the simulated spectrum and the experimental one, taking their areas to be equal. The simulated spectrum is shown in figure 6 together with the experimental spectrum. The numerical results provide an excellent representation of the complex measured spectrum with values for the variational parameters for the penetration depth λ  =  1152 ± 50Å, average demagnetization factor, D  =  0.31 ± 0.01 and the Gaussian broadening, δ  =  5.2 mT, which is larger than, but of the same order as, the normal state linewidth, 2 mT. The quoted accuracy is statistical. This value for D is rather close to that anticipated for a spherical geometry, D_sphere  =  1/3, and it is reasonable to expect this value for the average demagnetization factor for a large ensemble of grains. Earlier reports for the value of the penetration depth [9, 18, 26, 28] are similar to ours although our accuracy is higher. Our simulation and its comparison with experiment, as represented in figure 6, is the most precise such comparison obtained by resonance methods and it is the first time that such a simulation has been attempted for a strongly anisotropic supercondcutor. We emphasize that previous study has generally focused on moments of the measured spectrum, often restricting attention to the second moment. For an anisotropic superconductor the angular dependence of the first moment of the distribution, as can be seen in figure 5, must be correctly handled since it contributes significantly to the overall lineshape. Earlier study on other superconductors analysing the field distribution in the mixed state has been directed at the moments of the distribution, so we have calculated the first three moments for an anisotropic superconductor with randomly oriented grains, as a function of the penetration depth, restricted to the case of γ_H  =  5.4 and γ_λ  =  1.

The second moment of the magnetic field distribution of a spectrum from a vortex lattice can be related [20] to its penetration depth λ for low magnetic fields compared to H_c2 by the Pincus' formula [15, 20] where the second moment varies as the inverse fourth power of the penetration depth, B²  =  (0.0609 ₀)²/λ⁴. In the present case the simulated spectrum is the superposition of spectra with anisotropic coherence lengths and upper critical fields. Nonetheless, we find the 1st, 2nd and 3rd moments of the spectrum can be similarly related to inverse, even powers of the penetration depth in the following elegant way:

where A₁, A₂, A₃ are numerical constants. The Gaussian broadening factor is δ and D is the demagnetization factor. We find A₁  =  1.415 × 10⁴ T Å², A₂  =  2.621 × 10⁷ T² Å⁴, A₃  =  1.524 × 10¹¹ T³ Å⁶. However we caution that anisotropy and field dependence of the local field distributions mean that these numerical constants hold only in a limited range which we have explored for MgB₂ with γ_H  =  5.4, γ_λ  =  1 and H  =  1.97 T.

We have also investigated the effect of the penetration depth anisotropy γ_λ at low temperature. In actuality, the vortex structure for arbitrary angle θ is found as the solution to the anisotropic GL equations [29]. However, as a reasonable approximation for an almost isotropic penetration depth, we introduce another variational parameter, γ_λ, and continue to adopt the solution of the isotropic GL equation for each crystallite. The anisotropy of H_c1 is the inverse of H_c2, therefore the penetration depth has the inverse angular dependence of the coherence length,

With the same approach as before we generate the spectrum for the powder sample with γ_λ larger than one. The central transition is found to decrease with increase of γ_λ and the spectra become more asymmetric as figure 7 illustrates. This suggests that γ_λ at low temperatures should be close to one. Magnetization measurements show that γ_λ is around 1.7 between 20 and 27 K [28]. SANS experiments on a powder MgB₂ sample [12] give an upper limit of γ_λ to be around 1.5 and essentially magnetic field independent. Our result is consistent with these values. However, SANS measurements on a single crystal [3, 13] indicate that γ_λ is close to one at T  =  2 K and at low field, H  <  0.5 T, and that it increases with external field reaching ≈3.5 in a field of 0.8 T. The lower γ_λ in the powder sample is believed to be caused by a limiting crystallite size effect [12]. Further study will be required to elucidate this phenomenon.

In conclusion, we measure the ¹¹B NMR spectra of a random powder sample of MgB₂ in magnetic fields of 1.97 and 3.15 T. The evolution of the spectra through the temperature range can be explained by the anisotropy of the upper critical field γ_H, which is determined to be 5.4 at low temperature. We find from our simulation that the penetration depth carries a different anisotropy from the upper critical field and that at low temperatures it is almost isotropic similar to that reported from SANS [12] for a powder sample. The value of the penetration depth that we have obtained for MgB₂ is 1152 ± 50 Å at 8 K in a magnetic field of 1.97 T. From our numerical studies we have found simple expressions for the penetration depth dependence of the moments of the field distribution in a random powder of an anisotropic superconductor.

Acknowledgments

This study was supported by the DOE: DE-FG02-05ER46248. Two of us (ML and BKC) acknowledge financial support from the Korean Research Foundation through, respectively, Grant 2003-015-C00161 and ABRL program at Ehwa Woman University.

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