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Strong superconducting strength in ε-PbBi microcubes
Journal of Magnetism and Magnetic Materials 407 (2016) 155–159
Contents lists available at ScienceDirect
Journal of Magnetism and Magnetic Materials journal homepage: www.elsevier.com/locate/jmmm
Strong superconducting strength in ε-PbBi microcubes Ashish Chhaganlal Gandhi, Sheng Yun Wu n Department of Physics, National Dong Hwa University, Hualien 97401, Taiwan
art ic l e i nf o Article history: Received 28 August 2015 Received in revised form 2 December 2015 Accepted 24 January 2016 Available online 25 January 2016 Keywords: Binary compounds Superconductor Electron–phonon coupling Microcubes
a b s t r a c t Single phase ε-PbBi microcubes were synthesized using a simple thermal evaporation method. Synchrotron x-ray measurement of the crystal structure of the ε-PbBi microcubes revealed a space group of P63/mmc. Enhanced superconducting transitions were observed from the temperature dependent magnetization, showing a main diamagnetic Meissner state below a TC of 8.66(2) K. An extremely strong superconducting strength (α ¼2.51(1)) and electron-phonon constant (λEP ¼ 2.25) are obtained from the modified Allen and Dynes theory, which give rise to higher TC superconductivity in this type of structure. The electron–phonon coupling to low lying phonons is found to be the leading mechanism for the observed strong-coupling superconductivity in the PbBi system. & 2016 Elsevier B.V. All rights reserved.
1. Introduction Nanosized post-transition binary superconductors, such as InSn, PbBi and InBi, feature the highest electron density of all metallic elements, and the pronounced higher critical fields can cause an enhancement of the superconducting transition temperature [1–3]. The interesting superconducting properties of these materials also appear in their physical behavior. For example, electron–phonon coupled bimetallic superconductors with enhanced superconducting transitions have been discovered, which has renewed interest in conventional phonon-mediated superconductivity [4,5]. During the 1980s, long filaments of Pb–Bi and Pb–Sn alloys were synthesized and their unusual higher critical temperature reported. For example, Pb45Bi35Sn20 and Pb45Bi40Te15 systems with high TCs of 10.1 K and 10.2 K, respectively were obtained [6,7]. Although much work has already been done, more studies need to be conducted to ascertain the effects of different structural phases on the superconductivity. Observations show the significant properties of a few binary lead-bismuth compounds which exhibit anomalous superconductivity. This has motivated the current search for a new inter-metallic phase of PbBi. In this study, ε-PbBi microcubes (MCs) of a single phase were fabricated from high purity Pb and Bi ingots (99.99%, mass ratio of 7.3) using the thermal evaporation method2 in a tungsten boat. The resultant samples were in powdered form, and consisted of a macroscopic amount of individual ε-PbBi microcubes. The purpose of this current study is to investigate the structure, superconductivity, and magnetic properties under various applied n
Corresponding author. E-mail address: [email protected] (S.Y. Wu).
http://dx.doi.org/10.1016/j.jmmm.2016.01.078 0304-8853/& 2016 Elsevier B.V. All rights reserved.
magnetic field of ε-PbBi MCs. This procedure has two main advantages: (1) the composition of the alloys can be controlled correctly to produce a single phase; (2) the process is efficient and easy to carry out. Our primary results are: (1) an extremely strong superconducting strength α ¼2.51(1) and electron–phonon constant λEP ¼2.25 were obtained with the modified Allen and Dynes theory which is what gives rise to the higher TC superconductivity in this type of structure. (2) The electron–phonon coupling to low lying phonons is found to be the leading mechanism for the observed strong-coupling superconductivity in the ε-PbBi system.
2. Experimental details
Τhe ε-PbBi MCs were fabricated using the thermal evaporation method [8]. The evaporation of high purity Pb and Bi ingots (99.99%) mounted in a tungsten boat was initiated by heating using a power supply (90 A/220 V). The morphology and size of the resultant ε-PbBi MCs was controlled by varying the pressure in the range of 0.02–2 Torr and the flow rates of argon gas was kept constant at 10 sccm (sccm denotes cubic centimeter per minute at STP). The ε-PbBi MCs were collected on quartz plates (1 cm 1 cm2), which were mounted about 12 cm above the heating source, and maintained at the temperature of liquid nitrogen. The resultant samples appeared in the form of dried powder, comprised of a macroscopic amount of individual ε-PbBi MCs. The critical Ar pressure region was found to be about 0.05– 0.08 Torr, leading to the formation of microcubes. Moreover, further synthetic studies reveal that, by controlling the Ar pressure and Pb/Bi ratio in the precursors, various size of PbBi nanparticles can be prepared at relatively high pressure (Ar ¼2–5 Torr, data not shown), which suggests that the morphology of the final products
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is strongly dependent on the Ar pressure of the reactants. The morphology and structures of the prepared samples were characterized using field emission scanning electron microscopic (FESEM) analysis and high energy synchrotron radiation X-ray diffraction (SR-XRD). Analysis of the crystalline properties was carried out by XRD at the National Synchrotron Radiation Research center (NSRRC) in Hsinchu, Taiwan (λ ¼0.7749 Å) using an SR-XRD with a BL01C2 beam line. The magnetic properties of the samples were measured with a superconducting quantum interference device (Quantum Design, MPMS-SQUID-VSM) magnetometer using a maximum applied field of 7 T.
3. Results and discussion The partly magnified SEM image, shown in Fig. 1(a), exhibits satisfactory two-dimensional crystal lattices with a mean length of d 1 μm. An EDS (Inca x-sight model 7557, Oxford Instrument, UK) mapping technique was used to measure a single ε-PbBi MC. The typical EDS elemental spectra revealed two peaks (shown in Fig. 1(b)) associated with the elemental Pb and Bi, which can be assigned to Pb-Mα1 and Bi-Mα1, verifying that the MCs contain only Pb and Bi elements. Moreover, the Pb/Bi atomic ratio is estimated to be 7/3, which is close to the stoichiometric composition of Pb7Bi3, indicating the high purity of the microcubes and the existence of lead vacancies. EDS mapping generates a two-dimensional image indicating the abundance of an element. The intensity of the image allows direct visualization of the spatial distribution of any element, such as lead, bismuth, or oxygen. Fig. 1 (c) depicts an EDS map of the selected single ε-PbBi MC, where the distribution of elements is presented using lock-in energies of
Pb-Mα1 (2.2 to 2.4 keV) and Bi-Mα1 (2.4 to 2.6 eV). The red and green dots indicate the existence of the elements Pb (Fig. 1(d)) and Bi (Fig. 1(e)), respectively. The EDS mapping results reveal a homogeneous distribution on a single ε-PbBi MC. More evidence regarding the PbBi structure of the microcubes can be obtained by high energy synchrotron radiation X-ray diffraction. As seen in the phase diagram for the Pb-Bi system [9], there are two additional stable crystalline phases, namely α-Pb and ε-Pb which are solid solutions of Pb and Bi [10], respectively. Here α-Pb has a similar crystal structure to that of the bulk Pb i.e., face centered cubic (fcc) (c/a ¼1). The inter-metallic ε-Pb phases are hexagonal A3 type structures with c/a ¼1.654. This indicates that depending on the initial composition of Pb and Bi, the Pb–Bi system will behave like a single crystalline phase (α-Pb or ε-Pb) or a composite alloy (α-Pb þ ε-Pb or Bi þ ε-Pb). The x-ray and Rietveld refined diffraction patterns of the ε-PbBi MC taken at room temperature are shown in Fig. 2. The Rietveld analysis [11], where the preferred orientation has been taking into account, was performed employing the General Structure Analysis System (GSAS) package [12]. The solid curves shown in Fig. 2 indicates the fitted pattern with the differences between them plotted at the bottom. Good agreement in the factors was reached (Rp(%)¼ 1.8, Rwp(%)¼ 3.56, and χ2 ¼ 0.885). The refined lattice parameters at 320 K were a¼ b¼3.5009(1) Å and c¼ 5.7897(3) Å. The value of c/a ¼ 1.653 that we obtained for the P63/mmc structure agrees very well with that obtained for the hexagonal closed packed (hcp) structure (c/ a¼ 1.654) with a statistical distribution of Pb and Bi atoms [13], as shown in the inset to Fig. 2. The superconducting properties of αPb and ε-Pb are reported to be a type II strongly coupled s-wave superconductor with a superconducting transition temperature TC higher than that of bulk Pb [10,14]. The temperature dependence
Fig. 1. (a) Plots of a partly magnified SEM image. (b) Typical EDS elemental spectra, revealing a series of peaks associated with elemental Pb and Bi, verifying that the MCs contain only Pb and Bi with an atomic ratio of 7/3; (c) depicts an EDS map of the selected single ε-PbBi MC, where the distribution of elements is presented using the following lock-in energies: Pb-Mα1 (2.2 to 2.4 keV) and Bi-Mα1 (2.4 to 2.6 eV), respectively; (d) and (e) the red and green dots reflect the existence of elemental Pb and Bi elements. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
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157
Fig. 2. X-ray diffraction pattern taken at room temperature, where the solid red curves indicate the fitted pattern and the differences between the observed and the fitted pattern (blue curve) are plotted at the bottom. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
of the magnetization as shown in Fig. 3(a), reveals a single superconducting transition near 8.66(2) K, which is consistent with that expected for a ε-PbBi system. The step-like magnetization shows a good fit (solid curve) to the results obtained using the following modified London equation [15] for small size particles. The effects originating from thermal fluctuation of the order parameter and the penetration of the field have been accounted for, with fitted values for the critical temperature and penetration depth at zero temperature of TC(0) ¼8.29(1) K and λ(0)¼ 181 (1) nm, respectively. Furthermore, although the small size effect has an influence on superconducting hysteresis, the size is far too small to accommodate even a single flux vortex. Under an applied magnetic field, the vortex penetration is delayed significantly due to the presence of a potential barrier of geometrical origin, resulting in an irreversible hysteretic magnetization [16]. The type-II like hysteresis loop M(Ha) then reflects the characteristic of the penetration of magnetic flux into the MCs. Fig. 3(b) shows the M(Ha) loop which displays an asymmetric profile on the field-increasing and decreasing branches and a large penetration field Hp 125 Oe was obtained at 3 K. The irreversibility of the entry and exit of the magnetic flux lines through the MC surfaces shows that it is mainly the geometric edge barriers that control the movements of flux lines at the surface. A Bean-Livingston [17] surface barrier effect has been reported in high TC superconductors [18]. Moreover, similar M(Ha) loops are measured at higher temperatures but the loop width and penetration field Hp decreases as the temperature increases, as shown in the inset to Fig. 3(b). Although the diamagnetic screening effects are reduced when an external magnetic field Ha is applied, it is still clearly evident at Ha ¼5 kOe, as shown in Fig. 4(a), because of an increase in the penetration length λL with a reduction of the superconducting volume fraction and the shielding capability. For the critical temperature TC determination in a MCs superconductor, M(T) is a step function with a transition width obtained by crystal imperfections and varies with applied magnetic fields. The temperature dependence of magnetization M(T) at various applied magnetic fields are presented on Fig. 4(a). It is clear that the critical temperature TC at which the Meissner diamagnetic screening signal appears shifts to a lower temperature as the applied magnetic field is increased. Fig. 4(b) shows the critical temperature obtained under various applied magnetic fields Ha and fitted to the expression
Fig. 3. Plots of (a) the temperature dependence of the magnetization taken at Ha ¼ 100 Oe; (b) a series of asymmetric profiles of M(Ha) taken at various temperatures. An anomalous penetrating field Hp 125 Oe was observed at T ¼3 K. (b) Inset showing the temperature dependence of Hp as deduced from the hysteretic loops. Initially, Hp drops with increasing temperature and can be extrapolated to 50 Oe at 8 K for ε-PbBi MCs. The solid lines are guides for the eye only.
TC = TC (0)[1 − (Ha/HC (0))]γ [19], where TC(0) is the zero field critical temperature, HC(0) is the zero temperature critical field, and γ is related to the magnetic energy of the superconductivity. It can be seen that the experimental data fit the case of γ ¼0.398(5). The fitted values were HC(0) ¼13(1) kOe, which is a factor of about 0.37 times lower than the bulk PbBi ( 35 kOe) [20–22]. In this study, an unusual reduction of TC is seen as Ha increases that is found to be extremely sensitive to the applied magnetic field. The reduction of the diamagnetic screening by an Ha is known to be caused by the increase of λL, which in turn reduces the superconducting volume fraction, hence the shielding capability of the micro crystals [23]. Furthermore the magnetic flux is expected to penetrate appreciably into the micro crystals, so that the increase in the internal energy due to the presence of an applied field Ha is much smaller than in bulk systems. In the ε-PbBi MCs system, the obtained γ value can be used to examine the fraction of the applied γ magnetic energy (Ha) that affects the superconductivity. In this study, a smaller γ value was obtained, only 66%( ¼ (HC(0))0⊡398 /(HC(0))2x%). Taking HC(0)¼13(1) kOe) of the applied magnetic energy may affect the superconductivity and an appreciable amount of the magnetic field penetrates into the ε-PbBi MCs. Therefore, a significant increase in the critical field HC(T) can be anticipated for the microcubes. Padamsee et al. [24] introduced
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which can be revealed by the deviation of HC (T ) /HC (0) from the parabolic dependency of 1 − (T /TC )2. It is known that strongly coupled systems yield positive deviations; while weakly coupled systems yield negative ones [23]. Such positive deviations for the present samples are shown in Fig. 5, with the data fitted to the HC (T ) /HC (0) = [1 − (T /TC )α ]. The fitting following expression: parameter is determined to be α ¼ 2.51(1). The positive deviation is larger than the BCS prediction ( HC (T ) /HC (0)~[1 − 1.06 (T /TC )2]), indicating that the micro-scale coupling strength is stronger than the BCS expectations α ¼1.764, mean field α ¼2, and bulk Pb α ¼2.15. The strong coupling strength for the ε-PbBi MC, as compared with the bulk material, could be mediated by low lying phonons resulting in an enhanced TC. In the study, the obtained 2α is close to 5.02, revealing an anomalous and extremely strong coupling strength. The related average value of the logarithmic phonon energy ωln and the dimensionless electron–phonon coupling constant λEP can be calculated with the Eliashberg theory. The correction of the BCS values by strong electron–phonon interactions can be deduced using the approximate analytic formula that links ωln /TC to experimental thermodynamic quantities: α= 1.764 [1 + 12.5 (TC (0) /ωln )2 × ln (ωln /2TC (0))] . The calculated value of ωln ¼ 47.2 K obtained with the above equation reveals a low lying phonon frequency. In general, the dimensionless electron– phonon constant λEP ¼ 2.25 can be estimated from the McMillan formulation [25] although this formulation is useful only for intermediate coupled superconductors at λEP o1. Analytic expressions for TC in terms of the electron–phonon coupling constant λEP and the phonon frequency can be obtained from the McMillan formula modified by Allen and Dynes [5], which leads to where TC /ωln = f1 f2 /1.2 exp [−1.04 (1 + λEP ) /λEP − μ* (1 + 0.62λEP )], f1 = [1 + (λEP /Λ1 )3/2]1/3 is a strong coupling correction function; 2 2 f2 = 1 + (ϖ 2/ωln − 1) λEP /(λEP + Λ22 ) is a shape correction function; 1/2
Fig. 4. Plots of (a) the temperature dependence of magnetization at various applied magnetic fields Ha. (b) the effects of the applied field Ha on TC for the ε-PbBi MCs, fitted to the expression TC ¼ TC(0)[1–(Ha/HC(0))]γ.
Λ1 = 2.36 (1 + 3.8μ*), Λ2 = 1.82 (1 + 6.3μ*)(ϖ 2/ωln ) and ϖ 2 = ω22 is the square root average logarithm of the phonon frequency introduced by Allen and Dynes; and μ* is the Coulomb pseudopotential. A Coulomb pseudopotential of μ* ¼0.15 represents the repulsive part of the pairing interaction that can be obtained from the Eliashberg equations. The obtained electron–phonon constant value is in agreement with previous predictions [14]. In this earlier work it is revealed that the hcp phase of this PbBi alloys system which occurs at a Bi concentration of about 30 at% is an extremely strong coupling superconductor with λEP ¼ 2.34 and lower lying phonon frequency ωln ¼ 52 K.
4. Conclusion
Fig. 5. The effects of temperature T on HC(T) fitted to the relation HC(T)/HC (0)¼ [1–(TC/TC(0))α], with α ¼ 2.51(1) indicating a strong coupled superconducting system.
the so-called α-model using the relative coupling strength, α = Δ (0) /kB TC (where Δ(0) is the superconducting energy gap at 0 K and kB is the Boltzmann constant) of a superconducting system
In summary, ε-PbBi microcubes (MCs) of a single phase were successfully prepared with an average size of d 1 μm from high purity Pb and Bi ingots using the thermal evaporation method. EDS spectrum result reveals that the MCs contain only Pb and Bi elements and the Pb/Bi atomic ratio is estimated to be 7/3, which is close to the stoichiometric composition of Pb7Bi3, indicating the high purity of the microcubes and the existence of lead vacancies. Magnetization measurements show that the superconducting transitions temperature are sensitive to applied magnetic field and can be determined using modified London equation. A lower critical field HC(0) ¼1.3(1) T was obtained which can be understood as due to the reduction of the fraction of the applied magnetic energy, affecting superconductivity. The estimated upper critical field can be well described by a power law with a α ¼2.51(1) value higher than the mean field α ¼ 2, which is consistent with the strong electron–phonon coupling. The related average value of the logarithmic phonon energy ωln ¼ 47.2 K and the dimensionless
A.C. Gandhi, S.Y. Wu / Journal of Magnetism and Magnetic Materials 407 (2016) 155–159
electron–phonon coupling constant λEP ¼2.25 can be calculated with the McMillan formula modified by Allen and Dynes [5]. The electron–phonon coupling to low lying phonons is found to be the leading mechanism for the observed strong-coupling superconductivity in ε-PbBi microcubes.
Acknowledgments We would like to thank the Ministry of Science and Technology (MOST) of the Republic of China for their financial support of this research through project numbers: MOST-103-2112-M-259-005 and MOST-104-2112-M-259-001.
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[6] T. Goto, M. Hayashi, Jpn. J. Appl. Phys. 25 (1986) 44. [7] T. Goto, A. Mori, J. Mater. Sci. 22 (1987) 356. [8] Jhong-Yi Ji, Po-Hsun Shih, Chun Chuen Yang, Ting Shan Chan, Yuan-Ron Ma, Sheng Yun Wu, Nanotechnology 21 (2010) 045603. [9] J.E. Evetts, J.M.A. Wade, J. Phys. Chem. Solids 31 (1970) 973. [10] M.M. Ozer, Y. Jia, Z. Zhang, J.R. Thompson, H.H. Weitering, Science 316 (2007) 1594. [11] H.M. Rietveld, J. Appl. Cryst. 2 (1969) 65. [12] A.C. Larson, R.B. Von Dreele, General Structure Analysis System, Report LA-UR86-748, Los Alamos National Laboratory, Los Alamos, NM, 1990. [13] S. Rasmussen, B. Lundtoft, Powder Diffr. 2 (1987) 28. [14] J.G. Adler, T.T. Chen, Solid State Commun. 9 (1971) 1961. [15] F. London, H. London, Proceedings of the Royal Society of London, Series A-Math. Phys. Sci. 149 (1935) 71. [16] E. Zeldov, A.I. Larkin, V.B. Geshkenbein, M. Konczykowski, D. Majer, B. Khakovich, V.M. Vinokur, H. Shtrikman, Phys. Rev. Lett. 73 (1994) 1428. [17] C.P. Bean, J.D. Livingston, Phys. Rev. Lett. 12 (1964) 14. [18] M. Konczykowski, L.I. Burlachkov, Y. Yeshurun, F. Holtzberg, Phys. Rev. B 43 (1991) 13707. [19] J.P. Hurault, K. Maki, M.T. Béal-Monod, Phys. Rev. B 3 (1971) 762. [20] V.Z. Kresin, H. Morawitz, S. Wolf, “Mechanisms of conventional and high Tc superconductivity”, Oxford University Press, New York, 1993. [21] J.D. Leslie, J.T. Chen, T.T. Chen, Can J. Phys. 48 (1970) 2783. [22] J.G. Adler, S.C. Ng, Can. J. Phys. 43 (1965) 594. [23] A.G. Sivakov, A.M. Glukhov, A.N. Omelyanchoul, Y. Koval, P. Muller, A. V. Ustinov, Phys. Rev. Lett. 91 (2003) 267001. [24] H. Padamsee, J.E. Neighbor, C.A. Shiffman, J. Low Temp. Phys. 12 (1973) 387. [25] W.L. McMillan, J.M. Rowell, Phys. Rev. Lett. 14 (1965) 108.
Contents lists available at ScienceDirect
Journal of Magnetism and Magnetic Materials journal homepage: www.elsevier.com/locate/jmmm
Strong superconducting strength in ε-PbBi microcubes Ashish Chhaganlal Gandhi, Sheng Yun Wu n Department of Physics, National Dong Hwa University, Hualien 97401, Taiwan
art ic l e i nf o Article history: Received 28 August 2015 Received in revised form 2 December 2015 Accepted 24 January 2016 Available online 25 January 2016 Keywords: Binary compounds Superconductor Electron–phonon coupling Microcubes
a b s t r a c t Single phase ε-PbBi microcubes were synthesized using a simple thermal evaporation method. Synchrotron x-ray measurement of the crystal structure of the ε-PbBi microcubes revealed a space group of P63/mmc. Enhanced superconducting transitions were observed from the temperature dependent magnetization, showing a main diamagnetic Meissner state below a TC of 8.66(2) K. An extremely strong superconducting strength (α ¼2.51(1)) and electron-phonon constant (λEP ¼ 2.25) are obtained from the modified Allen and Dynes theory, which give rise to higher TC superconductivity in this type of structure. The electron–phonon coupling to low lying phonons is found to be the leading mechanism for the observed strong-coupling superconductivity in the PbBi system. & 2016 Elsevier B.V. All rights reserved.
1. Introduction Nanosized post-transition binary superconductors, such as InSn, PbBi and InBi, feature the highest electron density of all metallic elements, and the pronounced higher critical fields can cause an enhancement of the superconducting transition temperature [1–3]. The interesting superconducting properties of these materials also appear in their physical behavior. For example, electron–phonon coupled bimetallic superconductors with enhanced superconducting transitions have been discovered, which has renewed interest in conventional phonon-mediated superconductivity [4,5]. During the 1980s, long filaments of Pb–Bi and Pb–Sn alloys were synthesized and their unusual higher critical temperature reported. For example, Pb45Bi35Sn20 and Pb45Bi40Te15 systems with high TCs of 10.1 K and 10.2 K, respectively were obtained [6,7]. Although much work has already been done, more studies need to be conducted to ascertain the effects of different structural phases on the superconductivity. Observations show the significant properties of a few binary lead-bismuth compounds which exhibit anomalous superconductivity. This has motivated the current search for a new inter-metallic phase of PbBi. In this study, ε-PbBi microcubes (MCs) of a single phase were fabricated from high purity Pb and Bi ingots (99.99%, mass ratio of 7.3) using the thermal evaporation method2 in a tungsten boat. The resultant samples were in powdered form, and consisted of a macroscopic amount of individual ε-PbBi microcubes. The purpose of this current study is to investigate the structure, superconductivity, and magnetic properties under various applied n
Corresponding author. E-mail address: [email protected] (S.Y. Wu).
http://dx.doi.org/10.1016/j.jmmm.2016.01.078 0304-8853/& 2016 Elsevier B.V. All rights reserved.
magnetic field of ε-PbBi MCs. This procedure has two main advantages: (1) the composition of the alloys can be controlled correctly to produce a single phase; (2) the process is efficient and easy to carry out. Our primary results are: (1) an extremely strong superconducting strength α ¼2.51(1) and electron–phonon constant λEP ¼2.25 were obtained with the modified Allen and Dynes theory which is what gives rise to the higher TC superconductivity in this type of structure. (2) The electron–phonon coupling to low lying phonons is found to be the leading mechanism for the observed strong-coupling superconductivity in the ε-PbBi system.
2. Experimental details
Τhe ε-PbBi MCs were fabricated using the thermal evaporation method [8]. The evaporation of high purity Pb and Bi ingots (99.99%) mounted in a tungsten boat was initiated by heating using a power supply (90 A/220 V). The morphology and size of the resultant ε-PbBi MCs was controlled by varying the pressure in the range of 0.02–2 Torr and the flow rates of argon gas was kept constant at 10 sccm (sccm denotes cubic centimeter per minute at STP). The ε-PbBi MCs were collected on quartz plates (1 cm 1 cm2), which were mounted about 12 cm above the heating source, and maintained at the temperature of liquid nitrogen. The resultant samples appeared in the form of dried powder, comprised of a macroscopic amount of individual ε-PbBi MCs. The critical Ar pressure region was found to be about 0.05– 0.08 Torr, leading to the formation of microcubes. Moreover, further synthetic studies reveal that, by controlling the Ar pressure and Pb/Bi ratio in the precursors, various size of PbBi nanparticles can be prepared at relatively high pressure (Ar ¼2–5 Torr, data not shown), which suggests that the morphology of the final products
156
A.C. Gandhi, S.Y. Wu / Journal of Magnetism and Magnetic Materials 407 (2016) 155–159
is strongly dependent on the Ar pressure of the reactants. The morphology and structures of the prepared samples were characterized using field emission scanning electron microscopic (FESEM) analysis and high energy synchrotron radiation X-ray diffraction (SR-XRD). Analysis of the crystalline properties was carried out by XRD at the National Synchrotron Radiation Research center (NSRRC) in Hsinchu, Taiwan (λ ¼0.7749 Å) using an SR-XRD with a BL01C2 beam line. The magnetic properties of the samples were measured with a superconducting quantum interference device (Quantum Design, MPMS-SQUID-VSM) magnetometer using a maximum applied field of 7 T.
3. Results and discussion The partly magnified SEM image, shown in Fig. 1(a), exhibits satisfactory two-dimensional crystal lattices with a mean length of d 1 μm. An EDS (Inca x-sight model 7557, Oxford Instrument, UK) mapping technique was used to measure a single ε-PbBi MC. The typical EDS elemental spectra revealed two peaks (shown in Fig. 1(b)) associated with the elemental Pb and Bi, which can be assigned to Pb-Mα1 and Bi-Mα1, verifying that the MCs contain only Pb and Bi elements. Moreover, the Pb/Bi atomic ratio is estimated to be 7/3, which is close to the stoichiometric composition of Pb7Bi3, indicating the high purity of the microcubes and the existence of lead vacancies. EDS mapping generates a two-dimensional image indicating the abundance of an element. The intensity of the image allows direct visualization of the spatial distribution of any element, such as lead, bismuth, or oxygen. Fig. 1 (c) depicts an EDS map of the selected single ε-PbBi MC, where the distribution of elements is presented using lock-in energies of
Pb-Mα1 (2.2 to 2.4 keV) and Bi-Mα1 (2.4 to 2.6 eV). The red and green dots indicate the existence of the elements Pb (Fig. 1(d)) and Bi (Fig. 1(e)), respectively. The EDS mapping results reveal a homogeneous distribution on a single ε-PbBi MC. More evidence regarding the PbBi structure of the microcubes can be obtained by high energy synchrotron radiation X-ray diffraction. As seen in the phase diagram for the Pb-Bi system [9], there are two additional stable crystalline phases, namely α-Pb and ε-Pb which are solid solutions of Pb and Bi [10], respectively. Here α-Pb has a similar crystal structure to that of the bulk Pb i.e., face centered cubic (fcc) (c/a ¼1). The inter-metallic ε-Pb phases are hexagonal A3 type structures with c/a ¼1.654. This indicates that depending on the initial composition of Pb and Bi, the Pb–Bi system will behave like a single crystalline phase (α-Pb or ε-Pb) or a composite alloy (α-Pb þ ε-Pb or Bi þ ε-Pb). The x-ray and Rietveld refined diffraction patterns of the ε-PbBi MC taken at room temperature are shown in Fig. 2. The Rietveld analysis [11], where the preferred orientation has been taking into account, was performed employing the General Structure Analysis System (GSAS) package [12]. The solid curves shown in Fig. 2 indicates the fitted pattern with the differences between them plotted at the bottom. Good agreement in the factors was reached (Rp(%)¼ 1.8, Rwp(%)¼ 3.56, and χ2 ¼ 0.885). The refined lattice parameters at 320 K were a¼ b¼3.5009(1) Å and c¼ 5.7897(3) Å. The value of c/a ¼ 1.653 that we obtained for the P63/mmc structure agrees very well with that obtained for the hexagonal closed packed (hcp) structure (c/ a¼ 1.654) with a statistical distribution of Pb and Bi atoms [13], as shown in the inset to Fig. 2. The superconducting properties of αPb and ε-Pb are reported to be a type II strongly coupled s-wave superconductor with a superconducting transition temperature TC higher than that of bulk Pb [10,14]. The temperature dependence
Fig. 1. (a) Plots of a partly magnified SEM image. (b) Typical EDS elemental spectra, revealing a series of peaks associated with elemental Pb and Bi, verifying that the MCs contain only Pb and Bi with an atomic ratio of 7/3; (c) depicts an EDS map of the selected single ε-PbBi MC, where the distribution of elements is presented using the following lock-in energies: Pb-Mα1 (2.2 to 2.4 keV) and Bi-Mα1 (2.4 to 2.6 eV), respectively; (d) and (e) the red and green dots reflect the existence of elemental Pb and Bi elements. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
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Fig. 2. X-ray diffraction pattern taken at room temperature, where the solid red curves indicate the fitted pattern and the differences between the observed and the fitted pattern (blue curve) are plotted at the bottom. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
of the magnetization as shown in Fig. 3(a), reveals a single superconducting transition near 8.66(2) K, which is consistent with that expected for a ε-PbBi system. The step-like magnetization shows a good fit (solid curve) to the results obtained using the following modified London equation [15] for small size particles. The effects originating from thermal fluctuation of the order parameter and the penetration of the field have been accounted for, with fitted values for the critical temperature and penetration depth at zero temperature of TC(0) ¼8.29(1) K and λ(0)¼ 181 (1) nm, respectively. Furthermore, although the small size effect has an influence on superconducting hysteresis, the size is far too small to accommodate even a single flux vortex. Under an applied magnetic field, the vortex penetration is delayed significantly due to the presence of a potential barrier of geometrical origin, resulting in an irreversible hysteretic magnetization [16]. The type-II like hysteresis loop M(Ha) then reflects the characteristic of the penetration of magnetic flux into the MCs. Fig. 3(b) shows the M(Ha) loop which displays an asymmetric profile on the field-increasing and decreasing branches and a large penetration field Hp 125 Oe was obtained at 3 K. The irreversibility of the entry and exit of the magnetic flux lines through the MC surfaces shows that it is mainly the geometric edge barriers that control the movements of flux lines at the surface. A Bean-Livingston [17] surface barrier effect has been reported in high TC superconductors [18]. Moreover, similar M(Ha) loops are measured at higher temperatures but the loop width and penetration field Hp decreases as the temperature increases, as shown in the inset to Fig. 3(b). Although the diamagnetic screening effects are reduced when an external magnetic field Ha is applied, it is still clearly evident at Ha ¼5 kOe, as shown in Fig. 4(a), because of an increase in the penetration length λL with a reduction of the superconducting volume fraction and the shielding capability. For the critical temperature TC determination in a MCs superconductor, M(T) is a step function with a transition width obtained by crystal imperfections and varies with applied magnetic fields. The temperature dependence of magnetization M(T) at various applied magnetic fields are presented on Fig. 4(a). It is clear that the critical temperature TC at which the Meissner diamagnetic screening signal appears shifts to a lower temperature as the applied magnetic field is increased. Fig. 4(b) shows the critical temperature obtained under various applied magnetic fields Ha and fitted to the expression
Fig. 3. Plots of (a) the temperature dependence of the magnetization taken at Ha ¼ 100 Oe; (b) a series of asymmetric profiles of M(Ha) taken at various temperatures. An anomalous penetrating field Hp 125 Oe was observed at T ¼3 K. (b) Inset showing the temperature dependence of Hp as deduced from the hysteretic loops. Initially, Hp drops with increasing temperature and can be extrapolated to 50 Oe at 8 K for ε-PbBi MCs. The solid lines are guides for the eye only.
TC = TC (0)[1 − (Ha/HC (0))]γ [19], where TC(0) is the zero field critical temperature, HC(0) is the zero temperature critical field, and γ is related to the magnetic energy of the superconductivity. It can be seen that the experimental data fit the case of γ ¼0.398(5). The fitted values were HC(0) ¼13(1) kOe, which is a factor of about 0.37 times lower than the bulk PbBi ( 35 kOe) [20–22]. In this study, an unusual reduction of TC is seen as Ha increases that is found to be extremely sensitive to the applied magnetic field. The reduction of the diamagnetic screening by an Ha is known to be caused by the increase of λL, which in turn reduces the superconducting volume fraction, hence the shielding capability of the micro crystals [23]. Furthermore the magnetic flux is expected to penetrate appreciably into the micro crystals, so that the increase in the internal energy due to the presence of an applied field Ha is much smaller than in bulk systems. In the ε-PbBi MCs system, the obtained γ value can be used to examine the fraction of the applied γ magnetic energy (Ha) that affects the superconductivity. In this study, a smaller γ value was obtained, only 66%( ¼ (HC(0))0⊡398 /(HC(0))2x%). Taking HC(0)¼13(1) kOe) of the applied magnetic energy may affect the superconductivity and an appreciable amount of the magnetic field penetrates into the ε-PbBi MCs. Therefore, a significant increase in the critical field HC(T) can be anticipated for the microcubes. Padamsee et al. [24] introduced
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which can be revealed by the deviation of HC (T ) /HC (0) from the parabolic dependency of 1 − (T /TC )2. It is known that strongly coupled systems yield positive deviations; while weakly coupled systems yield negative ones [23]. Such positive deviations for the present samples are shown in Fig. 5, with the data fitted to the HC (T ) /HC (0) = [1 − (T /TC )α ]. The fitting following expression: parameter is determined to be α ¼ 2.51(1). The positive deviation is larger than the BCS prediction ( HC (T ) /HC (0)~[1 − 1.06 (T /TC )2]), indicating that the micro-scale coupling strength is stronger than the BCS expectations α ¼1.764, mean field α ¼2, and bulk Pb α ¼2.15. The strong coupling strength for the ε-PbBi MC, as compared with the bulk material, could be mediated by low lying phonons resulting in an enhanced TC. In the study, the obtained 2α is close to 5.02, revealing an anomalous and extremely strong coupling strength. The related average value of the logarithmic phonon energy ωln and the dimensionless electron–phonon coupling constant λEP can be calculated with the Eliashberg theory. The correction of the BCS values by strong electron–phonon interactions can be deduced using the approximate analytic formula that links ωln /TC to experimental thermodynamic quantities: α= 1.764 [1 + 12.5 (TC (0) /ωln )2 × ln (ωln /2TC (0))] . The calculated value of ωln ¼ 47.2 K obtained with the above equation reveals a low lying phonon frequency. In general, the dimensionless electron– phonon constant λEP ¼ 2.25 can be estimated from the McMillan formulation [25] although this formulation is useful only for intermediate coupled superconductors at λEP o1. Analytic expressions for TC in terms of the electron–phonon coupling constant λEP and the phonon frequency can be obtained from the McMillan formula modified by Allen and Dynes [5], which leads to where TC /ωln = f1 f2 /1.2 exp [−1.04 (1 + λEP ) /λEP − μ* (1 + 0.62λEP )], f1 = [1 + (λEP /Λ1 )3/2]1/3 is a strong coupling correction function; 2 2 f2 = 1 + (ϖ 2/ωln − 1) λEP /(λEP + Λ22 ) is a shape correction function; 1/2
Fig. 4. Plots of (a) the temperature dependence of magnetization at various applied magnetic fields Ha. (b) the effects of the applied field Ha on TC for the ε-PbBi MCs, fitted to the expression TC ¼ TC(0)[1–(Ha/HC(0))]γ.
Λ1 = 2.36 (1 + 3.8μ*), Λ2 = 1.82 (1 + 6.3μ*)(ϖ 2/ωln ) and ϖ 2 = ω22 is the square root average logarithm of the phonon frequency introduced by Allen and Dynes; and μ* is the Coulomb pseudopotential. A Coulomb pseudopotential of μ* ¼0.15 represents the repulsive part of the pairing interaction that can be obtained from the Eliashberg equations. The obtained electron–phonon constant value is in agreement with previous predictions [14]. In this earlier work it is revealed that the hcp phase of this PbBi alloys system which occurs at a Bi concentration of about 30 at% is an extremely strong coupling superconductor with λEP ¼ 2.34 and lower lying phonon frequency ωln ¼ 52 K.
4. Conclusion
Fig. 5. The effects of temperature T on HC(T) fitted to the relation HC(T)/HC (0)¼ [1–(TC/TC(0))α], with α ¼ 2.51(1) indicating a strong coupled superconducting system.
the so-called α-model using the relative coupling strength, α = Δ (0) /kB TC (where Δ(0) is the superconducting energy gap at 0 K and kB is the Boltzmann constant) of a superconducting system
In summary, ε-PbBi microcubes (MCs) of a single phase were successfully prepared with an average size of d 1 μm from high purity Pb and Bi ingots using the thermal evaporation method. EDS spectrum result reveals that the MCs contain only Pb and Bi elements and the Pb/Bi atomic ratio is estimated to be 7/3, which is close to the stoichiometric composition of Pb7Bi3, indicating the high purity of the microcubes and the existence of lead vacancies. Magnetization measurements show that the superconducting transitions temperature are sensitive to applied magnetic field and can be determined using modified London equation. A lower critical field HC(0) ¼1.3(1) T was obtained which can be understood as due to the reduction of the fraction of the applied magnetic energy, affecting superconductivity. The estimated upper critical field can be well described by a power law with a α ¼2.51(1) value higher than the mean field α ¼ 2, which is consistent with the strong electron–phonon coupling. The related average value of the logarithmic phonon energy ωln ¼ 47.2 K and the dimensionless
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electron–phonon coupling constant λEP ¼2.25 can be calculated with the McMillan formula modified by Allen and Dynes [5]. The electron–phonon coupling to low lying phonons is found to be the leading mechanism for the observed strong-coupling superconductivity in ε-PbBi microcubes.
Acknowledgments We would like to thank the Ministry of Science and Technology (MOST) of the Republic of China for their financial support of this research through project numbers: MOST-103-2112-M-259-005 and MOST-104-2112-M-259-001.
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