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|Anomalous thermal expansion in superconducting Mg1-xAlxB2 system|
V. Palmisano1, S. Agrestini1, G. Campi1, M. Filippi1, L. Simonelli1, M. Fratini1, A. Bianconi1
S. De Negri2, M. Giovannini2 A. Saccone2, A. N. Fitch3, M. Brunelli3, I. Margiolaki3,
1Dipartimento di Fisica, Università di Roma "La Sapienza", P. le Aldo Moro 2, 00185 Roma, Italy
2Dipartimento di Chimica e Chimica Industriale, Università di Genova, Via Dodecaneso 31, 16146 Genova, Italy
3European Synchrotron Radiation Facility, B.P. 220, F-38043 Grenoble Cedex, France
The thermal lattice expansion in the superconducting Mg1-xAlxB2 system (x=0, 0.13 and 0.59) has been measured using high-resolution X-ray powder diffraction. An unusual large negative thermal expansion (NTE) appears for temperatures below T*=60 K in the MgB2. The NTE effect increases in Mg0.87Al0.13B2 and disappears at high Al content in the Mg0.59Al0.41B2 where the temperature dependence of volume follows a standard Einstein model in the whole temperature range. The anomalous behavior of the thermal expansion provides a direct evidence in the physics of diborides for the relevance of the proximity to the 2.5 Lifshitz electronic topological transition where the Fermi surface of the band changes from a two-dimensional 2D to a three-dimensional 3D topology.
PACS No. 74.70.Ad, 61.10.Nz, 74.78.Fk
KEY WORDS: diborides, electronic topological transition, lattice instability, negative thermal expansion.
In the superconductivity of multiband materials  the configuration interaction between pairs in different bands opens a non BCS pairing channel, known as interband pairing, where the critical temperature (Tc) could increase via a repulsive interband pairing interaction . The characteristic feature of the multiband superconductivity is the formation of an anisotropic superconductivity with an electronic state dependent gap that is known also as multigap superconductivity. While the theory of multigap superconductivity was proposed 46 years ago and more than 200 papers on the subject [see ref. 2 for the reference list] have been published these phenomena do not appear in the experimental investigation of homogeneous superconducting elements and alloys since inelastic interband scattering  suppresses the multigap superconductivity. From the experimental investigation of the nanoscale heterogeneous striped phase of the CuO2 plane in cuprate high Tc perovskites [4,5] it was proposed that the 2D Fermi surface of superconducting plane is broken by the lattice heterogeneity into Fermi surface arcs associated with different subbands. In this scenario the key mechanism for high Tc is driven by the interband pairing between the subbands in multiband superconductivity [6-11]. The idea is that multiband superconductivity can be realized in metallic superlattices since disparity of the wavefunctions and the small overlap between the electronic states in different spatially separated subbands reduces drastically the electronic inelastic interband scattering rate σπ=1/σπ. Therefore the heterogeneous structure provides unique systems where the very strict condition for the “clean limit” is met (the mean free path for interband scattering , where av is the average superconducting gap) and the multigap superconducting phase can be realized.
The high Tc is realized in superconducting heterostructures at atomic limit showing multiband superconductivity by tuning the chemical potential at a “Feshbach shape resonance” [2,6-12] of the interband pairing occurring in the proximity of a quantum critical point where the topology of one Fermi surface of one of the subbands shows a dimensional cross over. This is known as a 2.5 Lifshitz electronic topological transition (ETT)  driven by the pressure or the charge density.
The superconducting MgB2, with the critical temperature, Tc=39.5K  provides a very simple realization of the process of increasing the critical temperature Tc of a bulk superconductor by making metal heterostructures at the atomic limit [10,11]. In fact it is actually a composite intermetallic formed by a multilayer of first planes of a bulk low Tc superconductor, boron that shows Tc=11K under pressure : graphene B monolayers, intercalated by second metallic planes of a non superconducting element, magnesium : hcp Mg monolayers. Therefore following the discovery of Nagamatsu et al.  it was first proposed by the Rome group [15-17] that MgB2 shows mutigap superconductivity. The multigap superconductivity in MgB2 was quickly confirmed by experiments on the upper critical field Hc2(T) , tunneling  and angular resolved photoemission (ARPES)  and many other measurements . The two gaps but a single Tc show directly the key role of interband pairing (IP) and there is now a wide scientific agreement that IP in MgB2 increases Tc by about a factor two. The theoretical interest is now focusing in understanding the physics of interband pairing by tuning the chemical potential in the proximity to the 2D-3D topology cross-over in the band (the 2D-3D ETT) [21-26]. The chemical potential can be tuned experimentally toward the 2D-3D ETT by chemical substitutions of aluminum or scandium for magnesium [28-30] or carbon for boron .
According to Lifshitz  in the proximity of an ETT point it is possible that the compressibility becomes negative and the system shows a lattice instability and phase separation. Multiphase behavior has already been reported for a series of Mg1-xAlxB2 (x≥ 0.025) systems studied by means of synchrotron X-ray powder diffraction and 11B NMR techniques , in Mg1-xScxB2 (x<0.12) , as well as an extensive series of MgB2-xCx systems (x ≥ .06) .
In the case of the Mg1-xAlxB2 system the chemical potential is tuned at the 2D/3D ETT point for x=0.33 . It has been shown  that the interband pairing term shows a broad resonance as expected for a superconducting Feshbach shape resonance. Experimental evidence for the 2D-3D topology cross-over in the band is indicated a frequency hardening and a line-width narrowing of the Raman E2g mode indicating a strong reduction of the electron-phonon interaction going from 2D regime to the 3D regime through the ETT point [29,30].
In this work we have addressed our interest to probe the proximity of the chemical potential to the 2D-3D ETT point by measuring the thermal expansion in the low temperature range. The lattice instability in the proximity of an ETT according with Lifshitz  can induces a negative thermal expansion (NTE) of the lattice volume at very low temperature and it is expected to change by changing the proximity to the ETT point. The thermal expansion of the MgB2 has been measured via neutron diffraction  and a negative thermal expansion below Tc has been observed . We have studied the system Al1-xMgxB2 in order to characterize the correlation between the thermal expansion of the MgB2 and its electronic and superconducting properties.
2. Experiment and data processing
We have measured the low-temperature thermal expansion, using high-resolution powder X-ray powder diffraction, as a function of Al substitution for Mg in the AlxMg1-xB2 system. Samples of AlxMg1-xB2 have been synthesized by direct reaction of the elemental magnesium, aluminium (rod, 99.9 mass% nominal purity) and boron (99.5% pure, <60 mesh powder) enclosed in a stoichiometry ratio in tantalum (Ta) crucibles, sealed by arc welding under argon atmosphere. The Ta crucibles have been sealed in a heavy iron cylinder and heated for 1 h at 800 ºC and 2 h at 900 ºC. We have characterized the Al/Mg lattice disorder of several samples at room temperature with high-resolution X-ray powder diffraction using synchrotron radiation. We have investigated the samples with Al content x=0.00, x=0.13 and x=0.57. The Al content x has been determined by the measured volume of the crystallographic unit cell of each sample at room temperature The critical temperatures of the three samples have been determined by measuring the temperature dependence of the susceptibility.
High-resolution X-ray powder diffraction experiments were carried out on the beam-line ID31 [34,35] at the European Synchrotron Radiation Facility (ESRF), France. The high resolution powder diffraction beam line ID31 (ESRF, Grenoble) was employed for a detailed crystallographic study of an extensive compositional series. Samples were ground and introduced into a 1.0 mm diameter borosilicate glass capillary. The diffraction profiles have been collected in continuous scanning mode using nine Si (111) analyser crystals and the capillaries have been continuously spun during data acquisition. The data were rebinned in the 2range 4-45º with a step of 0.005º. The measurements have been carried out using the x-ray beam with wavelength =0.50180(3) Å for the x=0,0 and x=0.13 samples while the wavelength =0.33637(3) Å has been used for the x=0.57 sample. The data have been collected at 20-30 different temperatures in the range between 4 K and 295 K. The x=0 sample has been slowly cooled from 295K to 50K in 8 hours and from 50K to 5 K in 14 hours. The x=0.13 sample has been slowly cooled from 295K to 100K in 4 hours and from 100K to 5 K in 15 hours. The x=0.57sample has been measured at 300K, rapidly quenched at 4.5K and the data have been collected by slowly heating up to 200K in 8 hours. The structural model has been refined using the Rietvield method  provided by GSAS refining program .
3. Results and discussion
Figure 1 shows the X-ray powder diffraction profiles measured on the three samples of AlxMg1-xB2 (x= 0,0.13,0.59) at room temperature. Inspection of the synchrotron X-ray powder diffraction profiles shows that the majority of reflections index to hexagonal symmetry (with space group P6/mmm), implying the absence of any structural phase transitions with the crystal structure of Mg1-xAlxB2 remaining hexagonal down to low temperatures. Concerning the sample purity only a very small quantity of Mg and Al metal are observed as minority phases. The data have a high signal-to-noise ratio due to the high brilliance of synchrotron beam and the utilization of nine detectors. The diffraction patterns were collected in the Q range between 1 Å-1 and 9 Å-1. The high quality powder diffraction data reflect the enhanced crystallinity and purity of the samples and the very low instrumental contribution to the diffraction lines due to the parallel beam optics geometry of ID31.
In figure 2 we have plotted the susceptibility of the samples as a function of temperature. The narrow superconducting transitions again indicate the high quality of the superconducting samples used for the present work. The Tc determined from the susceptibility curve were found to be 39.6 K, 29.5 K and 2.3 K for the three samples with increasing Al content respectively.
In figure 3a) the refined unit cell volume of the three samples versus temperature is shown. The temperature dependance of the unit cell volume as given by diffraction data between 300 K and 60 K can be well described by the Einstein model using a single effective phonon frequency (solid line). The refined phonon frequencies seem to be nearly constant in 2D regime and decrease in 3D regime: 504 K, 502 K and 442 K respectively with increasing Al content. A large NTE is observed in the MgB2 at low temperatures, in qualitative agreement with previous neutron diffraction data . However, a very high “signal to noise ratio” combined with a small instrumental contribution to the peak width of our diffraction profile permits to reveal that the NTE start to appear below T*=60 K, and not at Tc as previously believed . The NTE is observed also in the compound x=0.13. It occurs in the same temperature range (T< T*=60 K) and it is much higher than the one revealed in pure MgB2. This is the first experimental measure of the very strong NTE below T* in the compound with x=0.13. As for the temperature dependence of the unit cell volume in Al0.57Mg0.43B2 there is no evidence of NTE and the volume follows the standard Einstein model in the whole temperature range (4-300 K).
Figure 3b) shows an expanded view of the low-temperature behavior of the thermal expansion. The NTE (fitted through a weighted curve) is clearly stronger in the sample with Al content 0.13, which is the closest to the ETT point. The values of NTE at 10K are 2.5*K-1, 4.0*K-1and 0.0*K-1 for the samples with Al content 0.00, 0.13 and 0.50 respectively.
The smaller NTE measured by Jorgensen et al  could be due assigned to the difference between the slow cooling measurements (present case) and slow heating measurements . Further experimental investigation is required to verify this point.
An evident anomaly in the thermal expansion of the MgB2 at Tc indicates coupling of the superconducting condensate with the lattice. This effect is commonly associated with the change in the specific heat.
The NTE can be explained in terms of the proximity of MgB2 to the 2D-3D ETT point. According to Lifshitz’s theory a system may show lattice instability when the chemical potential is slightly higher than its own value at the ETT point. In the case of Mg1-xAlxB2 this situation occurs when the band is 2D i.e., for x<0.33. This instability should disappear when the chemical potential shifts beyond the ETT point that is when the band becomes 3D. The results obtained are the first experimental demonstration of both the proximity of the MgB2 to a 2.5 Lifshitz transition and the consequent instability of its lattice structure.
In this work we have studied the correlation between the electronic properties and the lattice structure in the superconducting system Mg1-xAlxB2 through measurements of the low-temperature negative thermal expansion as a function of Al content. We have observed that the MgB2 shows a lattice instability evidenced by a strong negative thermal expansion. This instability grows strongly with the chemical substitution and disappears in the 3D regime. In particular for the first time we have evidenced the strong NTE of Mg0.87Al0.13B2.
This work is supported by MIUR in the frame of the project Cofin 2003 "Leghe e composti intermetallici: stabilità termodinamica, proprietà fisiche e reattività" on the "synthesis and properties of new borides" and by European project 517039 "Controlling Mesoscopic Phase Separation" (COMEPHS) (2005). We thank the ESRF for provision of synchrotron radiation time.
Figure 1: Diffraction profiles of the samples of MgB2, Al0.13Mg0.87B2 and Al0.59Mg0.41B2 at room temperature.
Figure 2: Magnetic susceptibility as a function of temperature for the three samples of Mg1-xAlxB2.
Figure 3: a) Temperature dependence of unit cell volume for the samples of MgB2, Al0.13Mg0.87B2 and Al0.59Mg0.41B2: the corresponding Einstein’s model fits are plotted as solid lines. Al0.59Mg0.49B2 does not show negative thermal expansion while an anomalous behavior is evident in the samples of MgB2 and Al0.13Mg0.87B2 in low temperature regime.
b) Low temperature behavior of the samples of MgB2 Al0.13Mg0.87B2 and Al0.59Mg0.41B2: the last one shows a standard thermal expansion; MgB2 present negative thermal expansion below T*=60K and a clear anomaly at the Tc; Al0.13Mg0.87B2 has a much stronger negative thermal expansion below the T*.