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Soon after the discovery of cuprate superconductivity, it was recognized that their high transition temperature (Tc) would mean that thermal fluctuations can produce a liquid vortex state9. In fact, the thermodynamic transition to superconductivity in a magnetic field occurs between a thermally fluctuating liquid vortex phase to one or more solid phases10. The liquid vortex state is inherently unstable with non-zero electrical resistance. For extremely anisotropic materials, such as Bi-2212, the liquid phase covers a wide range of temperature; but it is not known exactly how wide this is in high magnetic fields. For fields perpendicular to the conducting planes, H∥c-axis, the transition temperature between liquid and solid vortex phases, Tm(H), is principally controlled by vortex–vortex interactions that get stronger as the density of vortices increases, proportional to the field. The supercurrents that form each vortex are mainly confined to the conducting planes and, in high field, they lose their coherence from one plane to the next so that vortices are expected to become two dimensional. It is remarkable that the theory3 for the two-dimensional vortex melting transition has only one parameter, the magnetic field penetration depth, and that this simple picture has not yet been experimentally confirmed.

We use nuclear magnetic resonance (NMR) of 17O to detect the melting transition of vortices as a function of temperature and magnetic field. NMR can be carried out on selected elements in site-specific locations in the structure11,12 as can be seen in the 17O spectra in Fig. 1. There are three stoichiometric positions for oxygen in Bi-2212: in atomic planes containing Cu, the O(1) site; Sr planes, the O(2) site; and Bi planes. The last oxygen atoms are unobserved11 owing to disorder in this plane. In addition, there is a small amount of non-stoichiometric oxygen, δ-oxygen, too small to be observed directly by NMR and whose location in the structure is not yet established13. The parts of the NMR spectra in the middle range of field in Fig. 1 are the central transitions from the O(1) (broad) and O(2) (narrow) sites. The electronic coupling to 17O is much stronger for O(1) compared with O(2), as is apparent from the temperature dependences in Fig. 1. This is confirmed by measurements of the spin–lattice relaxation rate, which are one order of magnitude larger for O(1) compared with O(2). The narrowness of the O(2) resonance indicates a homogeneous electronic environment with negligible spin shift (Knight shift) and uniform electric field gradient. For fast pulse repetition, as is the case for our measurements in Fig. 1, O(2) is partially saturated and comprises less than 6% of the spectrum at 60 K, decreasing with decreasing temperature to 2% at 40 K. We remove it numerically in this range of temperature and below this, it has negligible contribution to the spectrum. The O(2) resonance serves as a useful marker for the zero spin-shift position of 17O for both O(1) and O(2) sites. Here, we focus on 17O(1) NMR as a probe of the local magnetic field in the CuO2 plane.

Figure 1: Spectra of 17O NMR in Bi-2212 for magnetic field parallel to the c axis at fixed frequency.
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The sample is an overdoped, 28 mg, single crystal of Bi2Sr2CaCu2O8+δ (Bi-2212), with Tc=75 K and with ≈60% of the 16O exchanged for 17O. The optimally doped Tc=93 K. In field-sweep experiments (T=20, 40, 60, 100 K), a decreasing Knight shift moves the NMR spectrum to the left. At high temperatures, two oxygen sites can be distinguished. The central transition, for oxygen in the copper–oxygen plane, O(1), is the wide line near 13.49 T at 100 K. The central transition for the oxygen in the strontium–oxygen plane, O(2), is the narrow line near the zero Knight shift position at 13.51 T. The other two peaks are quadrupolar satellites of the O(1) resonance. For T=4 and 12 K, we show the Fourier transform of the echo at fixed field, having a smaller spectral bandwidth so that satellites are not observed here. Our separate measurement of the satellite intensity at T/Tm=0.67 compared with the central transition is the same as that at T/Tm=1.33, confirming this picture. The spectra at different temperatures are normalized to have the same peak intensity of the O(1) central transition.

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Decreasing the temperature below the superconducting transition temperature, we find that the 17O(1) resonance peaks move to the left, that is, to a higher field at a fixed NMR frequency, (lower frequency at fixed field) approaching the zero spin-shift position. Simultaneously, the central NMR line narrows. The decrease of the Knight shift to zero in the superconducting state is a characteristic signature for spin-singlet pairing. The line broadening with decreasing temperature in the normal state, Figs 1 and 2, can be associated with a Knight shift distribution introduced by the δ-oxygen. Similar behaviour has been observed14 for chemical impurities, such as Ni, Zn or Li, substituted for Cu in the CuO2 plane in yttrium–barium–copper oxide (YBCO), and which form a local moment. Their contribution to the 17O NMR linewidth is given by a Curie law, proportional to the ratio of applied magnetic field to temperature. In addition, we find that below Tc this linewidth is proportional to the temperature-dependent Knight shift, which we measure independently, thereby accounting for the decrease with decreasing temperature in Fig. 2 as shown by the smooth curve, assuming a temperature-independent residual contribution of 1.2 kHz T−1. In this region, liquid vortex dynamics effectively average to zero their associated local magnetic fields.

Figure 2: Temperature dependence of the O(1) linewidth.
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Temperature dependence of the linewidth of the 17O NMR central transition presented as the square root of the second moment of the frequency spectrum, 17Δν, divided by magnetic field, H, in the range 3 to 22 T. Fields above 8 T were obtained at the National High Magnetic Field Laboratory in Tallahassee, Florida. Measurements at 27 and 29 T are included in Figs 3 and 4, but omitted here for clarity. On decreasing the temperature, there is a common behaviour independent of magnetic field, represented by the smooth curve and its continuation to low temperature (dotted curve), that can be understood in terms of defects in the copper–oxygen plane induced by oxygen doping. This curve is given by Δν=Δν0+K H D/T, where Δν0 is a background magnetic contribution to the linewidth possibly associated with regions of the sample where superconductivity is suppressed, K(T) is the measured Knight shift, H/T is the field-to-temperature ratio and D is a Curie constant. The key feature of the data is the systematic break from this curve that we identify with the transition from a liquid to a solid vortex state.

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On further cooling in the superconducting state there is a sharp onset for a new contribution to the linewidth that is not proportional to applied magnetic field. The well-defined temperature at which this line broadening occurs is shown in Fig. 3, decreasing, but progressively more slowly, with increasing magnetic field in our range between 3 and 29 T. We identify this behaviour with the formation of a solid vortex structure. More precisely, the extra linewidth corresponds to an inhomogeneous magnetic field distribution that is static on the NMR timescale, ≈0.1 ms, and is asymmetric; see the T=4 K spectrum in Fig. 1. This behaviour is characteristic of the transition from liquid to solid vortex matter such as has been observed by muon spin relaxation7 (μSR) and small-angle neutron scattering8 (SANS) in Bi-2212, and NMR in YBCO (ref. 15). Our observations are qualitatively consistent with extension to high magnetic fields of the vortex melting phase diagram, H<0.1 T, determined by Hall probe16,17 and transport measurements18. Khaykovich et al.16 explore this behaviour as a function of oxygen doping (anisotropy). NMR is complementary to these methods with definite advantages for detecting vortex melting at very high magnetic fields. The NMR or μSR spectrum is a direct map of the magnetic field distribution from vortex supercurrents. Specifically, NMR with 17O affords excellent resolution as a magnetometer on the atomic scale and has been exploited in previous work19 to spatially resolve and study excitations in the vortex core.

Figure 3: The magnetic-field–temperature phase diagram for vortex melting in Bi-2212 for H∥c.
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The transition temperatures, Tm(H), are identified from the data, as shown in Fig. 2. The two-dimensional vortex melting transition temperature, Tm2D (vertical dashed line), is determined from a fit to the data. Deviations of the fit from the data are expected3 for H≈Hcr.

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The strong upward curvature of the phase diagram in Fig. 3 has been anticipated theoretically3,10. Torque magnetometry in fields H≤5 T gave similar indications20. For highly anisotropic superconductors, the electromagnetic interaction between vortices dominates the Josephson coupling between planes. In this high field limit, the simplest picture for vortex melting is a first-order thermodynamic transition3 given by

for H larger than a crossover field,

The limiting two-dimensional melting temperature in equation (1) is

where 0.605≤A≤0.615 comes from numerical calculations (A. E. Koshelev, private communication). This spread in A reflects the existence of an intermediate phase because the ideal two-dimensional melting scenario proceeds in two steps with an intervening hexatic phase bounded by continuous transitions10. The melting temperature, equation (3), depends on the layer spacing21 d=1.5 nm, the flux quantum φ0, Boltzmann’s constant kB and the single superconductive parameter λa b, which is the penetration depth for supercurrents in the CuO2 plane. The crossover field, Hcr, depends mainly on the product of d and the mass anisotropy factor, γ=λc/λa b=ξa b/ξc. Out-of-plane components are denoted by a subscript c and ξ is the coherence length. The numerical constants in the theory, equation (1), are b≈1 and the exponent ν=0.37.

The NMR vortex lineshape is asymmetric but less so than for a perfect line-vortex lattice suggesting that the vortex structure is somewhat disordered. In fact, μSR and SANS show that at low temperature and H>0.1 T, it is a vortex glass. We will assume that the difference in energy between disordered and perfect vortex structures, at least in high magnetic field, is small compared with the energy for condensation from liquid to solid, and we use the framework of equations (1)–(3) to analyse the freezing that we have observed. From a fit to the data, dashed curve in Fig. 3, we find the crossover field Hcr=2.5 T and γ≈78 and the two-dimensional melting transition temperature Tm2D=12±1 K for which the penetration depth is λa b=220±10 nm. Precision measurements of the absolute value of the penetration depth at low temperature are notoriously difficult. From earlier reports for Bi-2212, λa b is ≈210 nm from magnetization data21, 269±15 nm from cavity resonance methods22 and ≈180 nm from μSR (ref. 7). These results are in agreement with what we report here. From torque measurements, Iye et al. 23 found γ≥200 for nearly optimally doped samples, and Watauchi et al. 24 obtained a value of 91 on overdoped material using resistivity methods. Although γ≈78 is in this range, it must be considered approximate because the theory is imprecise in the crossover field region. If we constrain γ, to any value over this wide range, our fit for Tm2D is unaffected. For a less overdoped crystal, Tc=85 K, we find a slightly higher value, Tm2D=16 K.

Experiments consistent with the two-dimensional melting theory have basic significance. But they are also relevant to applications of superconductivity at very high magnetic field. One of the most promising materials for magnet wire4,5 to make a 30 T superconducting NMR magnet, as was recently recommended by the National Research Council6, is Bi-2212. To achieve this goal, there will be an upper limit on the operational temperature of the magnet, determined by supercurrents that are more easily stabilized in the vortex solid phase.

At low magnetic fields, H≪Hcr, it is well established that there is a complex phase diagram for Bi-2212 with transitions on cooling from liquid vortex matter to a solid. At low temperature, an increasing magnetic field drives a transition at H*≈0.1 T from Bragg glass25 to vortex glass, indicated by an abrupt increase in the symmetry of the μSR spectrum7, a decrease in its second moment7 and the disappearance of Bragg peaks in SANS (ref. 8). At this transition, vortices lose coherence between planes. The resulting destructive interference between vortices on adjacent planes averages out the magnetic field distribution26,27,28 that is expected for straight line vortices such as those seen in the Bragg glass phase7. This interference reduces both asymmetry and linewidth in the μSR spectrum. With even larger fields, the second moment of the 17O NMR spectrum, σ(H), (≈linewidth squared) progressively increases and asymmetry is restored, until the spectrum linewidth collapses at 5 T. There are two possibilities for this anomalous behaviour at 5 T. Either the transition is field-induced ordering (ordering in-plane) or field-induced disordering (further disorder between planes). The significant asymmetry in the spectrum in the high field phase compared with that at our lowest fields favours vortex ordering with increasing field. This scenario might follow if interplanar coupling is weakened with increasing field compared with intraplanar interactions, reducing frustration that originated from vortices in adjacent planes. As a consequence, vortices order two dimensionally and we observe that the second moment decreases abruptly. At much larger fields, we can see that the two-dimensional vortex solid becomes progressively more disordered. On theoretical grounds29, arbitrarily small amounts of impurity will lead to disorder of the vortex lattice. Sensitivity to quenched-in disorder should increase progressively with increasing magnetic field as the lattice spacing decreases with fixed vortex size.

To make a comparison with straight line vortices, we have carried out a Ginzburg–Landau (GL) calculation30 of the corresponding magnetic field distribution, Fig. 4. Our measurement is of a similar magnitude as expected from this calculation, although the GL approximation does not capture the details of the observed field dependence. In fact, the transition from a liquid to a solid in high magnetic fields is predicted to occur first to a supersolid phase and then at lower temperatures to a decoupled solid phase where defects in the vortex lattice become zero dimensional10,31. NMR techniques may be helpful in exploring these fascinating aspects of vortex behaviour in strongly anisotropic superconductors.

Figure 4: Magnetic field dependence of the vortex linewidth and asymmetry.
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a, The vortex contribution to the second moment of the NMR spectra, σ(H), at low temperature, T≈4 K, given as the difference between the data shown in Fig. 3 and the background contribution, smooth curve in Fig. 2, expressed as a second moment. A comparison with that expected from rigid-line vortices is calculated from GL theory30, taking the upper critical field to be 230 T. Low-field μSR measurements26,27 up to 1.5 T are shown as triangles with the lowest field less than H* in the Bragg glass phase, consistent with three-dimensional ordering there, but dropping dramatically in the vortex glass phase. The linewidth data are normalized to the theoretical value at low field. b, The skewness, , where ν1 is the first moment of the spectrum and the average is over the spectrum. A decrease in the second moment H≈5 T with no drop in asymmetry suggests intraplanar vortex ordering. The error bars are statistical, calculated from the NMR spectra.

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