Abstract
Finding a physical approach for increasing the superconducting transition temperature (T_{c}) is a challenge in the field of material science. Shear strain effects on the superconductivity of rhenium were investigated using magnetic measurements, Xray diffraction, transmission electron microscopy, and firstprinciples calculations. A large shear strain reduces the grain size and simultaneously expands the unit cells, resulting in an increase in T_{c}. Here we show that this shear strain approach is a new method for enhancing T_{c} and differs from that using hydrostatic strain. The enhancement of T_{c} is explained by an increase in net electron–electron coupling rather than a change in the density of states near the Fermi level. The shear strain effect in rhenium could be a successful example of manipulating Bardeen–Cooper–Schrieffertype Cooper pairing, in which the unit cell volumes are indeed a key parameter.
Introduction
Singleelement superconductors are Bardeen–Cooper–Schrieffer (BCS)type superconductors, and volume shrinkage under applied pressure (P) is known to be an effective approach to manipulating the superconducting transition temperature (T_{c})^{1}. When the superconductors are subjected to high pressure, important changes occur in both the conduction electron state and lattice vibration. The change in T_{c} under hydrostatic contraction, in which the strain tensor has only diagonal components, is not uniform; in simple metals such as Al, Zn, Ga, Cd, In, Sn, Hg, and Pb, T_{c} decreases under pressure, whereas in those such as V, Zr, and Tl, T_{c} increases under pressure^{1,2}. According to the McMillan–Allen–Dynes formula, T_{c} can be expressed on a quantitative basis as
where ω_{ln} is the phonon frequency, λ is the electron lattice coupling, and μ* is the effective Coulomb potential^{3,4}. However, the change in T_{c} under pressure in simple metals can be understood more concretely using the following formula,
where M is the mass of the ions, k is the spring constant, N(E_{F}) is the electronicdensity of states at the Fermi level E_{F}, and <I^{2}> is the average square electronic matrix element^{1,2}. The product of N(E_{F}) and <I^{2}> is an electronic term called as the Hopfield parameter η^{5}. In equation (2), any change in the prefactor is negligible compared to the terms in the exponent. The relative magnitudes of the increases in k and η determine whether T_{c} will increase or decrease under pressure. Thus, the decrease in T_{c} with pressure in simple metals originates mainly from the increase in k, i.e., lattice stiffening. Thus, in simple metals with d T_{c}/dP < 0, the hydrostatic strain effects are understood to be governed by stiffening in the lattice vibration spectrum rather than the changes in the electronic properties such as broadening of the density of states near the Fermi level^{1,2}. Here, the electronlattice coupling parameter λ in equation (1) is expressed as λ = N(E_{F}) <I^{2}>/M <ω^{2}>, where <ω^{2}> is the average square phonon frequency. Further, it is also thought that the decrease in T_{c} with pressure in simple metals results from a weakening of λ due to the shift of the phonon spectrum to higher frequencies^{1}. Interestingly, the hydrostatic pressure effect in rhenium (Re) is unique in that the T_{c} exhibits a decrease followed by an increase with a minimum at approximately 0.6 GPa^{6,7}.
In highpressure studies, the exploitation of the isotropic features of stress, i.e., hydrostatic pressure, has been viewed as an ideal structural manipulation; the intrinsic response to stress has been studied, and firstprinciples calculations have been performed. As mentioned above, under hydrostatic pressure conditions, only diagonal components exist in the strain tensor. Indeed, hydrostatic pressure is especially important in the development of organic superconductors, in which nondiagonal components are detrimental to the straininduced superconductivity^{8}. However, in this study of Re, we present a new approach for greatly increasing T_{c}, in which shear stress instead of hydrostatic pressure is a key parameter affecting the T_{c} value of superconductors.
Previous studies of the strain effects in Re are briefly reviewed here. According to an earlier study of Re by Hulm and Goodman in 1957, an arcmelted powder without strain has a sharp superconducting transition at T_{c} = 1.70 K^{9}. The Meissner effect reveals the magneticfielddependent characteristics of typical typeI superconductors. They also reported that Re ground to a cylinder has a T_{c} value of 2.7 K, suggesting that some shear strain effect could work positively to increase T_{c}. However, when pure Re crystals with T_{c} of 1.69–1.70 K are placed under hydrostatic pressure, T_{c} decreases with a slope of dT_{c}/dP ~ –2 × 10^{–2} K/GPa for P < 0.4 GPa, and the initial slope corresponds to approximately −1 × 10^{–2} K at a volume shrinkage of 0.1%^{6,7}. The change in T_{c} is found to increase at around 0.6 GPa, where T_{c} has the decrease of approximately 0.01 K. For P > 1.6 GPa, the increase in T_{c} tends to saturate at T_{c} ~ 1.69 K. Thus, the effect of shear strain in Re is completely different from the hydrostatic pressure effects described above, but is instead qualitatively consistent with the effects of doping with Os or W^{10}. Generally, the shear strain reduces the domain size; hence it is not considered a promising factor for physical manipulation. Here, we consider the shear strain effect in Re using the proposed new approach.
The experimental apparatus for our highpressure experiments was a diamond anvil cell^{11}. A gasket was clamped between two diamond anvils. The gasket material must be hard and resistant to any stress. In fact, Re is often used as a gasket material. From our experience, we have found that a Re gasket often has a T_{c} above 2.0 K, and T_{c} depends on the degree of prior treatment (that is, it is related to the magnitude of internal strain in the gasket). Indeed, this phenomenon is the same as the strain effect reported by Hulm and Goodman^{9}. Given the above background, we systematically study the effect of shear strain on the superconductivity of Re over a wide strain range.
Severe plastic deformation can be used to apply a large strain to materials, resulting in bulk nanostructured materials with ultrafinegrained structures^{12}. Examples of such deformation include equalchannel angular pressing^{13}, accumulative rollbonding^{14}, and highpressure torsion (HPT)^{15}. In 1935, Bridgman introduced the HPT method combined with hydrostatic pressure as a method for obtaining high shearing stress^{16}. In particular, this HPT method can introduce intense shear strain so that the materials have ultrafine grains on the submicrometer or nanometer scale^{15,17,18,19}. In the present study, the HPT method is adopted to apply intense shear strain to Re. The shear strain effects are physically elucidated using structural analysis and firstprinciples calculations.
Results
Magnetic measurements
The inphase and outofphase ac magnetic susceptibility (denoted by 4πm’/h and 4πm”/h, respectively) as a function of temperature (T) for the arcmelted sample are shown in Fig. 1(a,b), respectively, and Fig. 2(a) shows the magnetization (M) as a function of the dc magnetic field (H). At around 1.7 K, a sharp Meissner signal appeared within a temperature width of 20 mK, and T_{c} was estimated to be 1.7 K. We confirmed that in a spherical sample of arcmelted material, an almost perfect Meissner signal appeared. In the disk samples, a Meissner signal with a similar magnitude was observed. The magnetization M(H) exhibited a sharp jump at around H = 170 ± 30 Oe, indicating that the unstrained sample was a typeI superconductor. The results are consistent with those reported by Hulm and Goodman^{9}.
Figure 1 also shows the temperature dependence of the inphase and the outofphase ac susceptibility for HPTprocessed Re. In the 4πm’/h data, the Meissner signal shifted toward the hightemperature side with increasing strain. When we evaluated T_{c} according to the onset of 4πm’/h, T_{c} was estimated to be 2.5, 2.9, and 3.0 K for N = 0, 1, and 10, respectively, where N is the number of revolutions in the HPT process. For reference, the T_{c} value of the asreceived sample was 2.2 K. We recognize a large enhancement of T_{c} caused by torsion. The susceptibility 4πm”/h reveals the energy loss against the ac magnetic field and the distribution of T_{c}. The peak of 4πm”/h corresponds to the midpoint of a large decrease in 4πm’/h. The upper limit of a finite m” corresponds to a possible optimal T_{c}. Even at N = 0, the optimal T_{c} of about 2.8 K implies the possibility of increasing T_{c}. The possible T_{c} for N = 10 is 3.2 K. We confirmed that the possible T_{c} of Re was enhanced to 3.4 K when the sample was filed using sandpapers as shown in Fig. S4, suggesting the importance of using shear strain instead of hydrostatic compression.
Figure 2(b–d) shows M(H) for the HPTprocessed Re at T < T_{c}. The field region of the Meissner signal was greatly enhanced with increasing N, and the type of superconductivity, determined by extracting the magnetic flux, changed from typeI to typeII. At T = 1.8 K, the lower critical field H_{c1} is less than 30, 90, and 130 Oe for N = 0, 1, and 10, respectively. Furthermore, the upper critical field H_{c2} at T = 1.8 K is 0.7, 1.3, and >1.5 kOe for N = 0, 1, and 10, respectively. Thus, a reduction in the coherence length ξ was experimentally suggested by the change in the magnetic field dependence of the Meissner signal as well as the change in T_{c} (∝ξ^{−1}). The positive effect of the shear strain appears in the increases in H_{c1} and H_{c2} as well as T_{c}.
Structural characterization
Figure S2 shows the Xray diffraction profiles of the asreceived and HPTprocessed samples. The shift in a series of diffraction peaks yields information about the lattice parameters. The halfwidth of the diffraction peaks indicates the average grain size and internal strain. Figure 3 shows the relation between T_{c} and several structural parameters; the crystalline size D (a), crystalline strain ε (b), lattice parameters a (c) and c (d), and unit cell volume V (e). These figures include data for the asreceived specimen, the HPTprocessed ones (N = 0, 1, and 10 at P = 24 GPa), and a filed specimen (#11 in Table S1). The filing was performed under a slight stress along the direction perpendicular to the disk, and the sample was subjected to strong strain caused by rotation. The shear strain accompanying rotation on the cplane was thought to expand the size of the cplane and decrease the crystalline size. As the shear strain increased, expansion along the aaxis also occurred, resulting in an increase in the unit cell volume. Indeed, the largest expansion along the aaxis was observed in the filed Re rather than the HPTprocessed Re with N = 10. According to the results for the asreceived, HPTprocessed, and filed specimens, D, a, and V are promising parameters for scaling T_{c}. Figure S3 shows a transmission electron microscopy micrograph of the sample processed by HPT at P = 24 GPa and N = 10. The grain size was reduced to the order of ~100 nm, suggesting a reduction in the coherence length. As shown in Fig. 1, a prominent Meissner signal was observed, and the present materials were considered to be strongly coupled superconductinggrain systems.
Band structure calculations
To see the effect of changes in the electronic structures, we performed band structure calculations for Re with different configurations. Figure 4(a) shows the calculated abinitio band structure for an experimental structure with a = 2.758 Å and c = 4.447 Å, and Fig. 4(b) compares the densities of states for different lattice parameters. A slight lattice expansion (around 1%) gave rise to appreciable overall shrinkage of the density of states toward to the Fermi level (E = 0) (thin red to thick blue curves). In particular, a shift of the peak around 5 eV was noticeable. These observations indicate a lowering of the excitation energy caused by volume expansion. In manybody perturbation theory^{20,21,22}, the lowering of the excitation energy is generally associated with enhancement of the polarization function. This polarization enhancement, i.e., the large dielectric screening, can weaken the effective repulsion between electrons. We thus expect the electronic pseudopotential μ and also μ* to reduce with volume expansion. In addition, we see in the inset in panel (b) that the density of states at the Fermi level increases with volume expansion. In 5d metals, the increase in the density of states at the Fermi level can generally lead to enhancement of the electronphonon coupling λ^{3}. The change in these parameters supports reasonably well the presented experimental observation that volume expansion (i.e., the shear strain) leads to an enhancement of T_{c}. We note that the same tendency was observed in alkalidoped C_{60} systems^{23}. Moreover, it is known that Re uniquely exhibits an increase in T_{c} (at maximum, 2.2 K) when some Re is replaced with Os or W, in which the electronic density of states increases just below the Fermi level^{10}. According to Chu et al., the change in the T_{c} value of Re with alloying can be understood in terms of a change in λ^{10}. The present results are reasonably consistent with the effect of replacing some Re with Os or W, where the former exhibits a more drastic increase in T_{c}.
Discussion
Here we discuss some factors that could be promising for increasing T_{c}. According to the BCS theory, T_{c} increases as the density of states at the Fermi level increases, because the number of electron pairs contributing to the superconductivity increases there. Furthermore, according to the MAD formula in equation (1), T_{c} increases as the phonon frequency becomes higher and the electronphonon interaction becomes larger. In previous highpressure experiments on singleelement superconductors, much attention was given to changes in the phonon stiffening as well as the density of states. In the present work, the most important factor is the electronphonon interaction. This tendency is unique in a series of highstrain experiments on singleelement superconductors, suggesting a new approach to modifying T_{c}. Under hydrostatic pressure, T_{c} decreases with a slope of approximately −1 × 10^{–2} K for a volume shrinkage of 0.1%^{6,7}. However, in the HPT process including shear strain, T_{c} increases with a slope of 1.5 K for a volume expansion of 0.7%, which is more than 20 times the change rate under hydrostatic pressure^{6,7}. Thus, the effect of shear strain is expected to be a promising factor in enhancing T_{c}.
In conclusion, Re subjected to HPT has an ultrafine structure in which the grain size decreases to below the submicrometer level and the lattice expands. The superconducting transition temperature T_{c} increases with increasing strain, and an important strain component here is shear instead of hydrostatic compression. According to firstprinciples calculations, the density of states at the Fermi energy increases only slightly, but the vanHovesingularity shifts toward the lowenergy side, suggesting enhanced screening. This means that electrons are affected by fluctuation of the lattice, resulting in an increase in the electronphonon coupling. Thus, a change in the electronphonon coupling instead of the density of states at the Fermi level gives rise to an increase in T_{c}.
Methods
Materials
Highpurity Re disks (99.97%, Johnson Mathey) with 4 mm in diameter and 0.25 mm in thickness were subjected to HPT processing under a pressure of P = 24 GPa for the revolution number of HPT process N = 0, 1, and 10 turns, as shown in Fig. S1. Tool steel was used for the anvils for N = 0 and 1, and tungsten carbide was used for N = 10. The pressure of 24 GPa is the value given by the applied load, 50 tons, divided by the total area of the central shallow hole made on the anvils, (2.5 mm)^{2}π. This pressure may be slightly overestimated if a burr is formed during the processing. When we used tungsten carbide anvils, processing was feasible without breaking them. Note that not only the strength of the anvils but also the geometry is important for maintaining feasible processing. The strain imposed on the sample was estimated by ε = 2π r N /t (where r is the radius and t is the thickness) as the equivalent strain^{24,25}. The shear strain generated by HPT processing typically decreases the grain size and thus increases the grain boundary area. As a reference sample, a balllike sample, whose residual strain was removed, was prepared by sufficient arc melting. Furthermore, for another reference, a Re plate was filed by sandpapers to introduce random shear strain under a small stress perpendicular to the disk. The number of times the plate was filed is denoted by n (the detailed experimental data are shown in the supplemental material). Indeed, the HPT effect has already been studied for other singleelement superconductors; a typeII superconductor, niobium (Nb), with dT_{c}/dP = –16 to –25 mK/GPa at low pressures, has a maximum T_{c} of approximately 10 K at around 10 GPa^{26}. However, Nb prepared by HPT (6 GPa) exhibits an increase in T_{c} of only ~0.10 K at maximum^{27}. On the other hand, a thin film with a 100 nm thickness under hydrostatic pressure exhibits an increase in T_{c} with dT_{c}/dP = 73 mK/GPa even at a low pressure^{28}. The HPT effect in Nb is qualitatively consistent with the hydrostatic pressure effect in a Nb film. Given the above background, the HPT effect in Re with T_{c} > 1.5 K reported herein is relatively large; hence, we need to understand the effects of the nondiagonal as well as diagonal components of the strain tensor.
Magnetic measurements
The superconducting transition was investigated by observing the Meissner signal in the ac magnetic susceptibility using a superconducting quantum interference device magnetometer (Quantum Design Inc.) equipped with an ac measurement option and a ^{3}He refrigerator option (IQUANTUM)^{29}. The frequency and amplitude (h) of the ac magnetic field were 10 Hz and 4 Oe, respectively. In the ac magnetic susceptibility measurements, the data for the magnetization M were gathered as a function of time t. Fourier analysis of the M(t) data yields the amplitudes of both the inphase component (m’) and that of the outofcomponent (m”). Both m’ and m” are divided by the amplitude of the ac field (h) to obtain the ac magnetic susceptibility. For the spherical sample of arcmelted material, the demagnetization field effect was calibrated using a demagnetization coefficient of 1/3, and we obtained a value of 99.4% as the volume fraction of the superconductor. To reduce the diamagnetic effect, the disks of the HPTtreated and filed specimens were placed so that the normal vectors of their round surfaces were perpendicular to the dc magnetic field. The stability of superconductivity in a magnetic field was investigated using the dc magnetic measurement.
Xray diffraction (XRD)
The grain size, strain, and unit cell parameters were investigated in XRD experiments using an Xray diffractometer (Rigaku, SmartLab) with CuKα radiation at 45 kV and 200 mA. The grain size and strain were calculated from the integrated intensities of the diffracted peaks using WilliamsonHall analysis^{30}.
Transmission electron microscopy (TEM)
Microstructural observation was performed using TEM, and the grain size was measured using darkfield images. The grain size estimated by XRD is generally smaller than that measured by TEM.
Band calculation
Densityfunctional band structure calculations were performed for several lattice configurations of Re to study the structural change effect on the lowenergy electronic structures. Then, firstprinciples calculations were performed using the Tokyo Abinitio Program Package^{31} with planewave basis sets, where normconserving pseudopotentials^{32,33} and the generalized gradient approximation of the exchangecorrelation potential^{34} with partial core correction were employed. The cutoff energies for the wave function and charge densities are 49 and 196 Ry, respectively, and 41 × 41 × 41 kpoint sampling was employed. The integral over the Brillouin zone was evaluated by the tetrahedron method with a broadening of 0.01 eV. The density of states for Re at the Fermi energy (E = 0) is estimated to be 1.486/eV, which is equivalent to 0.372 (eV·spin·atom)^{−1}. This value is consistent with the value of 0.33 (eV·spin·atom) ^{−1} in McMillan’s study^{3}. We also calculated the Fermi surface^{35} to see the volume change effect on the electronic structure.
Additional Information
How to cite this article: Mito, M. et al. Large enhancement of superconducting transition temperature in singleelement superconducting rhenium by shear strain. Sci. Rep. 6, 36337; doi: 10.1038/srep36337 (2016).
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Acknowledgements
We acknowledge Mr. N. Minagawa for his collaboration in the experiments of filed samples. This work was supported by MEXT KAKENHI [GrantinAid for Scientific Research on Innovative Areas “Bulk Nanostructured Metals” No. 25102709] and [GrantinAid for Scientific Research (B) No. 26289091].
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Contributions
M.M., H.M., K.T. and H.D. carried out the magnetic measurements above 1.8 K, and prepared the filed specimens. H.I., Y.I. and Z.H. prepared HPT specimens and performed the transmission electron microscopy measurements. N.S. performed the magnetic measurements below 1.8 K. H.A. and T. Yamasaki carried out the structural determination. T. Yamaguchi conducted the arc melting of the filed specimens. K.N. performed the firstprinciples calculations. M.M. prepared the manuscript with editing by K.N., H.A., H.D. and Z.H. All the authors discussed the results and commented on the manuscript. M.M. and Z.H. supervised all of the work.
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Mito, M., Matsui, H., Tsuruta, K. et al. Large enhancement of superconducting transition temperature in singleelement superconducting rhenium by shear strain. Sci Rep 6, 36337 (2016). https://doi.org/10.1038/srep36337
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DOI: https://doi.org/10.1038/srep36337
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