High-Temperature Superconductivity
Introduction
Cuprate High-temperature superconductivity remains a crucial but as yet unsolved problem in condensed Matter Physics. High-temperature superconductivity was first discovered in 1986 and has since been the focus of much experimental work. Neutron scattering can be used probe the magnetic excitation spectrum. This is important because high-Tc materials are dervied from antiferromagnetic insulators via hole doping and magnetism is evident even in the superconducting phase and is thought to play an important role in the superconducting mechanism. Our measurements show that antiferromagnetic order and fluctautions in La2-xSrxCuO4 are enhanced by an applied magnetic field, and a summary of the highlights is given below.
Background
Some of the simplest high-Tcs are derived from the parent compound La2CuO4. The Cu2+ ions possess spin-½ and give rise to long-range magnetic order below a Néel temperature of TN=325K. The magnetism is essentially two-dimensional with strong antiferromagnetic exchange interactions between nearest neighbours within the CuO2 planes that form within the material and weak interactions between these planes. Introduction of Sr doping, where the Sr replaces the La and releases charge carriers, produces a rich phase diagram. For Sr dopings, 0.06<x<0.25, <span class="Wichtig">La2-xSrxCuO4 is superconducting and the long-range commensurate magnetic order of the parent is weakened and replaced by incommensurate magnetism visible as four peaks surrounding the original Bragg peak position. There are in fact two regimes within the superconducting doping range. The underdoped regime, 0.06<x<0.125, where long-range order continues to exist at the peak positions, and the optimally doped and overdoped regime, 0.125<x<0.25 where long-range order is lost but magnetic fluctuations persist at the peak positions and are visible in inelastic neutron scattering.
When a magnetic field is applied to La2-xSrxCuO4 it behaves like a type-II superconductor and magnetic flux is able to penetrate the material due to the formation of vorticies. These are cylindrical regions of normal state material embedded within the bulk superconductor each carrying one fluxon and lying parallel to the field direction. In conventional superconductors the vortices form below the zero field superconducting transition temperature, initially they are mobile but as temperature is lowered they freeze into a lattice. The vortex freezing temperature is also the temperature below which phase coherent superconductivity characterised by zero resistance occurs for a given applied field, and in this article this is called the irreversibility temperature, Tirr,. The vortex density increases with applied field and for H=7.5T the separation between the vorticies is av(7.5T)=166Å. The size of the vortex cores is typically given by the superconducting pair coherence length ξ which for La2-xSrxCuO4 is ξ~20Å.
Figure 1: (a) shows the phase diagram of La2-xSrxCuO4 as a function of Sr doping x and temperature, the system passes through the phases of antiferro-magnetism, spin-glass/stripe behaviour and superconductivity with increasing x. (b) shows the reciprocal space of the two-dimensional copper oxide planes, in the undoped material, the commensurate antiferromagnetism gives rise to a Bragg peak at the (½,½) position (blue dot). As doping is increased the strength of the magnetism is reduced and the original Bragg peak splits into four incommensurate peaks, the red dots in (c) show the positions of the magnetism typically found in the superconducting phase.
Figure 2: (a) shows the reciprocal vortex lattice (small blue dots) for a magnetic field of 7.5T applied perpendicular to the CuO2 plane. The magnetic peak positions for optimally doped La2-xSrxCuO4, x=0.16, are also shown (red dots). (b) shows the irreversibility line for La2-xSrxCuO4, x=0.16, as a function of temperature and applied magnetic field.
Field-induced magnetic fluctuations in optimally doped La2-xSrxCuO4
Science 291 1759 (2001)
The experiments on optimally doped La2-xSrxCuO4, x=0.16, took place on the RITA-1 triple-axis spectrometer at Risø National Laboratory, Denmark. In zero field the main feature is the spin gap that appears in the superconducting state. The spin-gap has similar origins to the superconducting energy gap observed in the quasi-particle excitation spectrum. The latter is the energy required to break up the charge pairing that occurs in the superconducting state while the spin gap refers to the pairing energy of the spins of these quasi-particles.
Figure 3 shows the magnetic susceptibility at the incommensurate peak position as a function of energy for various fields and temperatures. The red circles give the magnetic signal at 5K in zero field and show that there is complete loss of signal below the spin gap energy in the superconducting state, compared to the normal state at these energies (red triangles). If the measurement is now repeated at T=5K in an applied field of 7.5T, where the field direction is perpendicular to the copper oxide planes, (blue circles) the data reveal magnetic signal induced below the spin-gap energy. Analysis of the field-induced signal reveals inverse lifetime which is much less than the normal state value suggesting slower magnetic fluctuations. Further measurements of the wavevector-dependence reveal a correlation length which is considerably greater than the normal state value. These two results - an enhanced lifetime and an enhanced correlation length - suggest a greater tendency towards long-range magnetic order for La2-xSrxCuO4 in an applied field compared to its normal state.
Figure 3: (a) gives the magnetic susceptibility as a function of energy at the magnetic peak position for different fields and temperatures; red triangles – normal state, T=29K, H=0T; red circles – superconducting state, T=5K, H=0T; blue circles – superconducting state with applied field, T=7K, H=7.5T. (b) shows the field-induced signal – low temperature signal measured
for H=7.5T minus zero-field signal.
As a first guess one might assume that the field-induced magnetism originates from the vortex cores however the correlation length (75Å) which gives the size of the magnetic regions in the material is considerably larger than the diameter of the vorticies (ξ~20Å). Furthermore the size of the ordered spin moment obtained by a phonon normalization is 0.22 μΒ/Cu2+ in absolute units; this value is averaged over the Cu2+ ions throughout the material and is clearly too large to come from the vortex cores alone which make up less than 5% of the total sample for a field of 7.5T. A potential scenario is for vortices to nucleate magnetism but where these magnetic regions extend beyond the vortex cores and spill out into the surrounding superconducting regions. This result reveals the highly complex interaction between superconductivity and magnetism and a definitive interpretation has not yet been achieved.
Field-induced magnetic order in underdoped doped La2-xSrxCuO4
Nature 415 299 (2002)
As the previous work showed an applied magnetic field induces magnetic fluctuations in optimally doped La2-xSrxCuO4, the question then arises as to what would be the effect of field on underdoped La2-xSrxCuO4 which has long-range magnetic order in zero field. Elastic neutron scattering measurements were performed on underdoped La2-xSrxCuO4 x=0.10. These measurements took place on the V2/FLEX triple-axis spectrometer at the Helmholtz Centre Berlin. Figure 4 shows an elastic scan through one of the incommensurate peaks in zero field and for H=14.5T. In the normal state at Tc =29K (blue circles) no signal is observed, but on cooling down below Tc in zero field (red circles), signal appears in the superconducting state. If a magnetic field is now applied perpendicular to the CuO2 planes at low temperatures, an enhancement of the elastic magnetic signal is observed. This enhancement is strong, a factor of three larger than the zero field signal for an applied field of H=14.5T. The magnetic peaks are resolution limited in both zero and non-zero field putting a lower limit of 400Å on the magnetic correlation length. This large correlation length is interesting because it suggests that the magnetic regions are not only greater than the vortex cores (ξ~20Å) but are also, unlike the case of the optimally doped sample, greater than the separation of the vorticies (av(5T)=200Å for H=5T). The large correlation length coupled with the large value of the average ordered spin moment per Cu2+ site of 0.24μΒ/Cu2+ at H=5T, imply that as for the optimally doped sample many more sites are involved in the magnetic ordering than the 3% of Cu2+ ions that form the vortex cores for this field. At this point it is not known whether there is coexistence between superconducivity and magnetism in the same regions of the material or whether they phase separate and exclude one another. However given the large size of the magnetic correlation length a phase separation scenario would mean that the superconducting regions would be pushed into the pockets between the magnetic regions.
Figure 4: The elastic magnetic scattering in underdoped La2-xSrxCuO4, x=0.10, measured in the normal and superconducting states for zero field and H=14.5T. The data is plotted as a function of wavevector through one of the magnetic peaks (see inset diagram). (a) shows the data collected in zero field, the signal appears in the superconducting state measured at T=2K (red circles) but is absent in the normal state at Tc=29K (blue circles). (b) shows the data collected in an applied field of 14.5T, the signal in the superconducting state has increased by a factor of three while there is again no signal at Tc=29K.
Three-dimensionality of field-induced magnetism in a high-temperature superconductor
Nature Materials 4, 658-662 (2005).
In all previous experiments to characterize the field-induced order in high-temperature superconductors, the magnetism was probed within the CuO2 planes. Well-defined peaks were observed when scanning with in-plane wavevector suggesting sizeable in-plane correlation lengths. Due to the structure of the La2-xSrxCuO4 only weak magnetic interactions exist between the layers and we might therefore expect weaker correlation between the layers. In these next experiments we probed these interlayer correlations. Figure 7 shows the measured signal for various fields and temperatures as a function of in-plane wavevector through the incommensurate magnetic peak (y-axis) and out-of-plane wavevector (x-axis). In zero field the signal shows little variation with out-of-plane wavevector suggesting that the zero field magnetism is two-dimensional, while the in-field magnetism is peaked at the reciprocal lattice point suggesting that the field-induced magnetism is three-dimensional with increased correlations along the interplanar direction.
Figure 5: Magnetic signal as a function of wavevectors in the superconducting plane ([h,0,0.11h] or [0.11xl,0,l]) and perpen-dicular to the superconducting plane ([0,k,0]); the colors indicate the strength of the magnetic scattering with red being strong intensity and blue being background. a gives the data in the superconducting state in zero field (T=2K, H=0T). b shows the scattering in the same reciprocal space region in an applied magnetic field (T=2K, H=6T//b).
