CHINESE JOURNAL OF PHYSICS

VOL. 45 , NO. 2-II

APRIL 2007

Anisotropic Nature of Superconductivity in Hexagonal Rubidium Tungsten Bronze Rb0.23 WO3+δ L.C. Ting,1 J. J. Pan,1 T. W. Liao,1 H. H. Hsieh,2 F. Z. Chien,1 P.H. Hor,3 and D. C. Ling1, ∗ 1

Department of Physics, Tamkang University, Tamsui 251, Taiwan 2 Department of Electrical Engineering, National Defense University, Tahsi 335 Taiwan 3 Department of Physics, University of Houston, TX 77004, USA (Received July 31, 2006)

Low-field M (H), transport, and magnetization measurements have been performed on hexagonal rubidium tungsten bronze Rb0.23 WO3+δ . Various thermodynamic parameters, such as Hc1 (0), Hc2 (0), λab (0), λc (0), ξ(0), anisotropy parameter γ, and Ginzburg-Landau parameter κ are reported. The irreversibility line Hirr (T) follows (1 − T /Tc)n with exponent n = 1.51 for sample with δ ∼ -0.10 and 2.11 for sample with δ ∼ 0.02, indicating that the vortex dimensionality of Rb0.23 WO2.90 is anisotropic three-dimensional (3D) and that of Rb0.23 WO3.02 is quasi two-dimensional (2D). In addition, the magnetization curves in high-field regime near Tc collapse onto a universal curve 4πM/(TH) β =F[A((TT c (H))/(TH) β )]with exponent β = 2/3 for sample with δ ∼ -0.10 and 1/2 for sample with δ ∼ 0.02, indicative of an anisotropic 3D and a quasi-2D scaling behavior for Rb0.23 WO2.90 and Rb0.23 WO3.02 , respectively. The observed results strongly suggest that oxygen content has a significant influence on the nature of the superconducting properties of Rb0.23 WO3+δ . PACS numbers: 74.25.Ha, 74.62.Dh, 74.70.-b

I. INTRODUCTION

Tungsten bronzes Mx WO3 (M = alkali atoms) have long been the subject of investigations for their interesting structural, electronic, and electrochromic properties [1]. These materials crystallize a variety of structures including tetragonal tungsten bronze (TTB), hexagonal tungsten bronze (HTB), and intergrowth tungsten bronze (ITB). Most of the earlier studies have concentrated on the tetragonal tungsten bronzes. However, the hexagonal structure has been shown to be more favorable to superconductivity than other structures [2-3]. The structure of the HTBs consists of layers of corner-linked distorted WO6 octahedra and hexagonal tunnels formed by the linkage along the c-axis where alkali atoms are accommodated. The structure is stabilized by larger alkali atoms such as K, Rb, and Cs if they occupy more than a half of the tunnel sites. R.K. Stanley et al. showed that Rbx WO3 has the highest superconducting transition temperature Tc among the HTBs [4]. For HTB Rbx WO3 with 0.19 ≤ x ≤ 0.33, Tc monotonically decreases with increasing x in the range of 0.19 < x < 0.25 and then increases with increasing x in the range of 0.25 < x < 0.33 [5]. It has been suggested that the rubidium-ordering-induced structural instability along the c-direction might be responsible for the anomalous Tc (x). Nevertheless, the origin of

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the anomalous Tc (x) dependence is still under debate. Even though many studies have been devoted to superconducting properties of HTB Rbx WO3 , the basic superconducting parameters of HTB Rbx WO3 such as the critical fields Hc1 (T ) and Hc2 (T ), the penetration depth λab (0) and λc (0), the coherence length ξ(0), the anisotropy parameter γ, and the Ginzburg-Landau parameter κ have not been unambiguously determined yet. It is mainly due to the fact that a delicate and complicated process is compulsory for synthesizing singlephase samples as well as the presence of inevitable impurity phases arising from intergrowth tungsten bronze for 0.19 < x < 0.215 [7]. Recently, Ting et al. studied the oxygen concentration dependence of the superconducting transition temperature for Rbx WOy with x = 0.19, 0.23, 0.27 and 2.80 ≤ y ≤ 3.07 and argued that the anomalous Tc (x) dependence for Rbx WO3 reported earlier is likely associated with a small deviation of the oxygen concentration from 3.00 [8]. Therefore, it is desirable to perform extensive investigations on superconducting properties and associated parameters of oxygen-varied Rbx WO3+δ . In this paper, we report low-field M-H, resistivity, and magnetization of Rb0.23 WO3+δ to determine various superconducting parameters mentioned above and study the effect of oxygen content on the nature of the superconducting properties of Rb0.23 WO3+δ .

II. EXPEREMENTAL

Samples investigated were prepared by the solid-state reaction method. The proper stoichiometric amounts of high purity powders of Rb2 WO4 , WO3 , and W were thoroughly mixed and grounded, then pressed into pellets. Subsequently, the pellets were sealed in quartz tubes with ambient pressure of 10−2 Torr. The sealed pellets were calcined at 650◦ for 5 hours followed by sintered at 850◦ for 20 hours. X-ray diffraction patterns were obtained using Cukα radiation from Rigaku 12 kW RU200 x-ray generator. Transport properties were made by a standard four-probe method. Magnetic properties of the samples were performed by PPMS.

III. RESULTS AND DISCUSSION

The XRD patterns of Rb0.23 WO3+δ clearly reveal that the predominant phase of the sample investigated is hexagonal tungsten bronze with space group P 63 /mcm reported by Magneli [6]. Detailed XRD results will be published elsewhere [8]. The representative low-field superconducting M-H isotherms for oxygen-deficient (Rb0.23 WO2.90 ) and oxygen-rich (Rb0.23 WO3.02 ) samples are shown in Figs. 1(a) and 1(b), respectively. The average lower critical field Hc1 (T ) is determined from the peaks of magnetization curves. Hc1 (T ) decreases from 30.7 G at 2 K to 3.5 G at 2.8 K for Rb0.23 WO2.90 and from 110 G at 1.8 K to 20 G at 3.8 K for Rb0.23 WO3.02 . It should be noted that a broad peak structure similar to what was observed in high-Tc cuprates exhibits in M-H isomer of Rb0.23 WO3.02 . The average lower critical field at zero temperature Hc1 (0) can be extracted from fitting data with an empirical expression Hc1 (T ) = Hc1 (0)[1 − (T /Tc )2 ] . It turns

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FIG. 1: (a) Low-field M-H curves for Rb0.23 WO2.90 at superconducting state in the range of 2.0 K < T < 2.8 K. (b) Low-field M-H curves for Rb0.23 WO3.02 at superconducting state in the range of 1.8 K < T < 3.8 K.

out that Hc1 (0) = 53 G for Rb0.23 WO2.90 and 140 G for Rb0.23 WO3.02 , respectively. The zero-temperature average penetration depth λ(0) of 2450 ˚ A for Rb0.23 WO2.90 and 1506 ˚ A 1/2 for Rb0.23 WO3.02 is derived from λ(0)= [Φ0 /2πHc1 (0)] , where Φ0 is flux quantum. In addition, the M-H isotherm reveals a linear region where full Meissner signals are present below a field smaller than Hc1 (T ). The field is likely corresponding to the anisotropic lower ab with extrapolated H ab (0) about 37 G for Rb critical field in the ab-plane Hc1 0.23 WO2.90 c1 and 20 G for Rb0.23 WO3.02 , respectively. By using of anisotropic Ginzburg-Landau formula ab + H c ]/3, the zero-temperature lower critical field along the c-axis, H c (0), Hc1 = [2Hc1 c1 c1 is calculated to be 85 G for Rb0.23 WO2.90 and 380 G for Rb0.23 WO3.02 . The anisotropy c (0)/H ab (0) is estimated to be about 2.3 for Rb parameter γ = Hc1 0.23 WO2.90 and 19 for c1 Rb0.23 WO3.02 ,respectively.Furthermore, λab (0) = 1940 ˚ A for Rb0.23 WO2.90 and 920 ˚ A for Rb0.23 WO3.02 , λc (0) = 4450 ˚ A for Rb0.23 WO2.90 and 17400 ˚ A for Rb0.23 WO3.02 are estic = Φ /2πλ2 and H ab = Φ /2πλ λ . The shorter penetration depth and mated from Hc1 0 0 ab c c1 ab more anisotropic nature of Rb0.23 WO3.02 are likely responsible for a broad peak structure present in the M-H curve due to the fact that shorter penetration depth makes screening

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FIG. 2: The temperature dependence of the upper critical field for Rb0.23 WO2.90 and Rb0.23 WO3.02 . The dashed line corresponds to a fit the phenomenological relation Hc2 (T ) = Hc2 (0)[1 − (T /Tc )2 ]α . The insets of (a) and (b) display resistivity as a function of temperature at different fields for Rb0.23 WO2.90 and Rb0.23 WO3.02 , respectively.

more effective and more anisotropic nature makes thermal fluctuations more pronounced. Resistivity as a function of temperature at different fields for Rb0.23 WO3+δ is present in the inset of Fig. 2. It is interesting to note that oxygen-deficient sample Rb0.23 WO2.90 has lower superconducting transition temperature (Tc ) and sharper superconducting transition width (∆Tc ) comparing with oxygen-rich sample Rb0.23 WO3.02 . In fact, the lower Tc accompanied by the sharper ∆Tc is commonly observed in Rbx WOy with oxygen concentration y < 3.0 and rubidium concentration 0.19 < x< 0.27 [8]. The ordering of the rubidium atoms [5], induced by the oxygen vacancy in the WO6 matrix, may play an important role in the superconductivity of Rbx WOy with y < 3.0. For Rbx WOy with y > 3.0, it is speculated that the ordering of the rubidium atoms along the c-axis would be influenced by the excess oxygen atoms sliding into the rubidium vacancies. This in turn would induce a local structure distortion and make oxygen-rich sample have higher Tc and wider ∆Tc . More remarkably, normal-state resistivity of the oxygen-rich sample Rb0.23 WO3.02 is three orders of magnitude larger than that of the oxygen-deficient sample Rb0.23 WO2.90 . Furthermore, metallic behavior with dρ/dT > 0 is observed in the normal-state of Rbx WOy with

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FIG. 3: Irreversibility lines of Rb0.23 WO2.90 and Rb0.23 WO3.02 .The insets of (a) and (b) show a log-log plot of Hirr versus 1 − T /Tc for Rb0.23 WO2.90 and Rb0.23 WO3.02 , respectively. The solid line is the fitting curve Hirr (T ) = H0 (1 − T /Tc )n .

y < 3.0 and semiconducting-like feature with dρ/dT < 0 is observed in the normal-stare of Rbx WOy with y > 3.0 [8], indicating that the normal-state transport properties of the oxygen-deficient and oxygen-rich samples Rb0.23 WO3+δ are distinctly different. In order to have better understanding of nature of superconducting properties of rubidium tungsten bronze, the temperature dependence of the upper critical field of Rb0.23 WO2.90 and Rb0.23 WO3.02 is determined from resistivity measurements. The (Hc2 , T ) pairs in the phase diagram illustrated in Figs. 2(a) and 2(b) are extracted from 10% drop of normal-state resistivity. The dashed line corresponds to a fit the phenomenological relation Hc2 (T ) = Hc2 (0)[1 − (T /Tc )2 ]α . The fitted Hc2 (0) is about 1100 G for Rb0.23 WO2.90 and 12000 G for Rb0.23 WO3.02 , respectively. Also, the exponent α is about 1.20 close to the value of two-fluid model for Rb0.23 WO2.90 and 1.97 similar to observed values in high-Tc cuprates for Rb0.23 WO3.02 . It should be mentioned that the upper critical field of oxygenrich samples is consistently one order of magnitude larger than that of oxygen-deficient samples. Based upon the relation Hc2 (0) = Φ0 /2πξ 2 (0), an average zero-temperature superconducting coherence length ξ(0) of 538 ˚ A for Rb0.23 WO2.90 and 163 ˚ A for Rb0.23 WO3.02

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FIG. 4: Magnetization as a function of temperature for Rb0.23 WO2.90 and Rb0.23 WO3.02 at various magnetic fields. The insets of (a) and (b) show 3D and 2D scaling of the magnetization around Tc (H).

is obtained. Combining ξ(0) and λ(0), the Ginzburg-Landau parameter κ is determined to be 4.6 for Rb0.23 WO2.90 and 9.2 for Rb0.23 WO3.02 , respectively. Figures 3(a) and 3(b) display irreversibility lines Hirr of Rb0.23 WO2.90 and Rb0.23 WO3.02 obtained from magnetization measurements. The irreversibility temperature, Tirr , is determined from a simple criterion MF C /MZF C = 0.98. A linear dependence is observed in a log-log plot of Hirr versus 1−T /Tc shown in the insets of Figs. 3(a) and 3(b), indicating that Hirr has a power-law behavior. The solid line is the fitting curve Hirr (T ) = H0 (1 − T /Tc )n with exponent n = 1.51 for Rb0.23 WO2.90 and 2.11 for Rb0.23 WO3.02 . It has been reported that Hirr is proportional to (1 − T /Tc )2 for quasi-2D superconductor Bi-2212 and (1-T/Tc)1.5 for anisotropic 3D superconductor LSCO, respectively [9]. Therefore, the results suggest that the vortex dimensionality of Rb0.23 WO2.90 is anisotropic three dimensional and that of Rb0.23 WO3.02 is quasi-two dimensional. To further elucidate the anisotropic nature of rubidium tungsten bronze, the scaling behavior of the fluctuation-induced magnetization in high-field regime is also investigated. According to Ullah and Dorsey [10], the scaling form for the magnetization near Tc is given by 4πM/(TH) β =F[A((T-T c (H))/(TH) β )], where F is a scaling function, A is a temperature-

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and field-dependent coefficient. The exponent β is 2/3 for 3D and 1/2 for 2D superconductors, respectively. The magnetization as a function of temperature near Tc at different fields for Rb0.23 WO2.90 and Rb0.23 WO3.02 is shown in Figs. 4(a) and 4(b), respectively. It should be noted that all M (T ) curves measured at different fields collapse onto a universal curve around Tc (H) plotted in the insets of Figs. 3(a) and 3(b). For Rb0.23 WO2.90 , a 3D scaling behavior is followed. Whereas for Rb0.23 WO3.02 , a 2D scaling behavior is observed. The analysis of dimensionality of the superconducting fluctuation along with the irreversibility line studies concluded that the anisotropy of the oxygen-deficient Rb0.23 WO2.90 is relatively small and the vortex system has essentially 3D character, whereas the anisotropy of the oxygen-rich Rb0.23 WO3.02 is more pronounced and the vortex system has quasi-2D character.

IV. CONCLUTION

In summary, low-field M (H), transport, and magnetic measurements have been performed on hexagonal rubidium tungsten bronze Rb0.23 WO3+δ . Various thermodynamic parameters, such as Hc1 (0), Hc2 (0), λab (0), λc (0), ξ(0), γ, and κ are reported. The irreversibility line and the scaling behavior of magnetization around Tc (H) strongly suggest that oxygen content has a significant impact on the nature of the superconducting properties of Rb0.23 WO3+δ .

Acknowledgment We thank Profs. C.M. Ho and Wei Ku for fruitful discussions. This work was supported by National Science Council of Taiwan under contract No. NSC94-2112-M-032-003.

References Electronic address: [email protected] [1] C.J. Raub, A.R. Sweedler, M.A. Jensen, S. Broadston, and B.T. Matthias, Phys. Rev. Lett. 13, 746 (1964). [2] H.B. Krause, R. Vincent, and J.W. Steeds, Solid State Commun. 68, 937 (1988). [3] M. Sato, B.H. Grier, G. Shirane, and H. Fujishita, Phys. Rev. B 25, 501 (1982). [4] R.K. Stanley, R.C. Morris, and W.G. Moulton, Phys. Rev. B 20, 1903 (1979). [5] R. Brusetti, P. Haen, and J. Marcus, Phys. Rev. B 65, 144528 (2002). [6] A. Magneli, Acta Chem. Scand. 7, 315 (1953). [7] A. Hussain, L. Kihlborg, Acta Crystallogr. A 30, 129 (1974). [8] L. C. Ting, H. H. Hsieh, H. H. Kang, D. C. Ling, H. L. Liu, W. F. Pong, F. Z. Chien and P. H. Hor, J. Supercond. Novel Mag. 20, 249 (2007). [9] A. Schilling, R. Jin, J.D. Guo, and H.R. Ott, Phys. Rev. Lett. 71, 1899 (1993). [10] S. Ullah and A.T. Dorsey, Phys. Rev. lett. 65, 2066 (1990). ∗

VOL. 45 , NO. 2-II

APRIL 2007

Anisotropic Nature of Superconductivity in Hexagonal Rubidium Tungsten Bronze Rb0.23 WO3+δ L.C. Ting,1 J. J. Pan,1 T. W. Liao,1 H. H. Hsieh,2 F. Z. Chien,1 P.H. Hor,3 and D. C. Ling1, ∗ 1

Department of Physics, Tamkang University, Tamsui 251, Taiwan 2 Department of Electrical Engineering, National Defense University, Tahsi 335 Taiwan 3 Department of Physics, University of Houston, TX 77004, USA (Received July 31, 2006)

Low-field M (H), transport, and magnetization measurements have been performed on hexagonal rubidium tungsten bronze Rb0.23 WO3+δ . Various thermodynamic parameters, such as Hc1 (0), Hc2 (0), λab (0), λc (0), ξ(0), anisotropy parameter γ, and Ginzburg-Landau parameter κ are reported. The irreversibility line Hirr (T) follows (1 − T /Tc)n with exponent n = 1.51 for sample with δ ∼ -0.10 and 2.11 for sample with δ ∼ 0.02, indicating that the vortex dimensionality of Rb0.23 WO2.90 is anisotropic three-dimensional (3D) and that of Rb0.23 WO3.02 is quasi two-dimensional (2D). In addition, the magnetization curves in high-field regime near Tc collapse onto a universal curve 4πM/(TH) β =F[A((TT c (H))/(TH) β )]with exponent β = 2/3 for sample with δ ∼ -0.10 and 1/2 for sample with δ ∼ 0.02, indicative of an anisotropic 3D and a quasi-2D scaling behavior for Rb0.23 WO2.90 and Rb0.23 WO3.02 , respectively. The observed results strongly suggest that oxygen content has a significant influence on the nature of the superconducting properties of Rb0.23 WO3+δ . PACS numbers: 74.25.Ha, 74.62.Dh, 74.70.-b

I. INTRODUCTION

Tungsten bronzes Mx WO3 (M = alkali atoms) have long been the subject of investigations for their interesting structural, electronic, and electrochromic properties [1]. These materials crystallize a variety of structures including tetragonal tungsten bronze (TTB), hexagonal tungsten bronze (HTB), and intergrowth tungsten bronze (ITB). Most of the earlier studies have concentrated on the tetragonal tungsten bronzes. However, the hexagonal structure has been shown to be more favorable to superconductivity than other structures [2-3]. The structure of the HTBs consists of layers of corner-linked distorted WO6 octahedra and hexagonal tunnels formed by the linkage along the c-axis where alkali atoms are accommodated. The structure is stabilized by larger alkali atoms such as K, Rb, and Cs if they occupy more than a half of the tunnel sites. R.K. Stanley et al. showed that Rbx WO3 has the highest superconducting transition temperature Tc among the HTBs [4]. For HTB Rbx WO3 with 0.19 ≤ x ≤ 0.33, Tc monotonically decreases with increasing x in the range of 0.19 < x < 0.25 and then increases with increasing x in the range of 0.25 < x < 0.33 [5]. It has been suggested that the rubidium-ordering-induced structural instability along the c-direction might be responsible for the anomalous Tc (x). Nevertheless, the origin of

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the anomalous Tc (x) dependence is still under debate. Even though many studies have been devoted to superconducting properties of HTB Rbx WO3 , the basic superconducting parameters of HTB Rbx WO3 such as the critical fields Hc1 (T ) and Hc2 (T ), the penetration depth λab (0) and λc (0), the coherence length ξ(0), the anisotropy parameter γ, and the Ginzburg-Landau parameter κ have not been unambiguously determined yet. It is mainly due to the fact that a delicate and complicated process is compulsory for synthesizing singlephase samples as well as the presence of inevitable impurity phases arising from intergrowth tungsten bronze for 0.19 < x < 0.215 [7]. Recently, Ting et al. studied the oxygen concentration dependence of the superconducting transition temperature for Rbx WOy with x = 0.19, 0.23, 0.27 and 2.80 ≤ y ≤ 3.07 and argued that the anomalous Tc (x) dependence for Rbx WO3 reported earlier is likely associated with a small deviation of the oxygen concentration from 3.00 [8]. Therefore, it is desirable to perform extensive investigations on superconducting properties and associated parameters of oxygen-varied Rbx WO3+δ . In this paper, we report low-field M-H, resistivity, and magnetization of Rb0.23 WO3+δ to determine various superconducting parameters mentioned above and study the effect of oxygen content on the nature of the superconducting properties of Rb0.23 WO3+δ .

II. EXPEREMENTAL

Samples investigated were prepared by the solid-state reaction method. The proper stoichiometric amounts of high purity powders of Rb2 WO4 , WO3 , and W were thoroughly mixed and grounded, then pressed into pellets. Subsequently, the pellets were sealed in quartz tubes with ambient pressure of 10−2 Torr. The sealed pellets were calcined at 650◦ for 5 hours followed by sintered at 850◦ for 20 hours. X-ray diffraction patterns were obtained using Cukα radiation from Rigaku 12 kW RU200 x-ray generator. Transport properties were made by a standard four-probe method. Magnetic properties of the samples were performed by PPMS.

III. RESULTS AND DISCUSSION

The XRD patterns of Rb0.23 WO3+δ clearly reveal that the predominant phase of the sample investigated is hexagonal tungsten bronze with space group P 63 /mcm reported by Magneli [6]. Detailed XRD results will be published elsewhere [8]. The representative low-field superconducting M-H isotherms for oxygen-deficient (Rb0.23 WO2.90 ) and oxygen-rich (Rb0.23 WO3.02 ) samples are shown in Figs. 1(a) and 1(b), respectively. The average lower critical field Hc1 (T ) is determined from the peaks of magnetization curves. Hc1 (T ) decreases from 30.7 G at 2 K to 3.5 G at 2.8 K for Rb0.23 WO2.90 and from 110 G at 1.8 K to 20 G at 3.8 K for Rb0.23 WO3.02 . It should be noted that a broad peak structure similar to what was observed in high-Tc cuprates exhibits in M-H isomer of Rb0.23 WO3.02 . The average lower critical field at zero temperature Hc1 (0) can be extracted from fitting data with an empirical expression Hc1 (T ) = Hc1 (0)[1 − (T /Tc )2 ] . It turns

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FIG. 1: (a) Low-field M-H curves for Rb0.23 WO2.90 at superconducting state in the range of 2.0 K < T < 2.8 K. (b) Low-field M-H curves for Rb0.23 WO3.02 at superconducting state in the range of 1.8 K < T < 3.8 K.

out that Hc1 (0) = 53 G for Rb0.23 WO2.90 and 140 G for Rb0.23 WO3.02 , respectively. The zero-temperature average penetration depth λ(0) of 2450 ˚ A for Rb0.23 WO2.90 and 1506 ˚ A 1/2 for Rb0.23 WO3.02 is derived from λ(0)= [Φ0 /2πHc1 (0)] , where Φ0 is flux quantum. In addition, the M-H isotherm reveals a linear region where full Meissner signals are present below a field smaller than Hc1 (T ). The field is likely corresponding to the anisotropic lower ab with extrapolated H ab (0) about 37 G for Rb critical field in the ab-plane Hc1 0.23 WO2.90 c1 and 20 G for Rb0.23 WO3.02 , respectively. By using of anisotropic Ginzburg-Landau formula ab + H c ]/3, the zero-temperature lower critical field along the c-axis, H c (0), Hc1 = [2Hc1 c1 c1 is calculated to be 85 G for Rb0.23 WO2.90 and 380 G for Rb0.23 WO3.02 . The anisotropy c (0)/H ab (0) is estimated to be about 2.3 for Rb parameter γ = Hc1 0.23 WO2.90 and 19 for c1 Rb0.23 WO3.02 ,respectively.Furthermore, λab (0) = 1940 ˚ A for Rb0.23 WO2.90 and 920 ˚ A for Rb0.23 WO3.02 , λc (0) = 4450 ˚ A for Rb0.23 WO2.90 and 17400 ˚ A for Rb0.23 WO3.02 are estic = Φ /2πλ2 and H ab = Φ /2πλ λ . The shorter penetration depth and mated from Hc1 0 0 ab c c1 ab more anisotropic nature of Rb0.23 WO3.02 are likely responsible for a broad peak structure present in the M-H curve due to the fact that shorter penetration depth makes screening

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FIG. 2: The temperature dependence of the upper critical field for Rb0.23 WO2.90 and Rb0.23 WO3.02 . The dashed line corresponds to a fit the phenomenological relation Hc2 (T ) = Hc2 (0)[1 − (T /Tc )2 ]α . The insets of (a) and (b) display resistivity as a function of temperature at different fields for Rb0.23 WO2.90 and Rb0.23 WO3.02 , respectively.

more effective and more anisotropic nature makes thermal fluctuations more pronounced. Resistivity as a function of temperature at different fields for Rb0.23 WO3+δ is present in the inset of Fig. 2. It is interesting to note that oxygen-deficient sample Rb0.23 WO2.90 has lower superconducting transition temperature (Tc ) and sharper superconducting transition width (∆Tc ) comparing with oxygen-rich sample Rb0.23 WO3.02 . In fact, the lower Tc accompanied by the sharper ∆Tc is commonly observed in Rbx WOy with oxygen concentration y < 3.0 and rubidium concentration 0.19 < x< 0.27 [8]. The ordering of the rubidium atoms [5], induced by the oxygen vacancy in the WO6 matrix, may play an important role in the superconductivity of Rbx WOy with y < 3.0. For Rbx WOy with y > 3.0, it is speculated that the ordering of the rubidium atoms along the c-axis would be influenced by the excess oxygen atoms sliding into the rubidium vacancies. This in turn would induce a local structure distortion and make oxygen-rich sample have higher Tc and wider ∆Tc . More remarkably, normal-state resistivity of the oxygen-rich sample Rb0.23 WO3.02 is three orders of magnitude larger than that of the oxygen-deficient sample Rb0.23 WO2.90 . Furthermore, metallic behavior with dρ/dT > 0 is observed in the normal-state of Rbx WOy with

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FIG. 3: Irreversibility lines of Rb0.23 WO2.90 and Rb0.23 WO3.02 .The insets of (a) and (b) show a log-log plot of Hirr versus 1 − T /Tc for Rb0.23 WO2.90 and Rb0.23 WO3.02 , respectively. The solid line is the fitting curve Hirr (T ) = H0 (1 − T /Tc )n .

y < 3.0 and semiconducting-like feature with dρ/dT < 0 is observed in the normal-stare of Rbx WOy with y > 3.0 [8], indicating that the normal-state transport properties of the oxygen-deficient and oxygen-rich samples Rb0.23 WO3+δ are distinctly different. In order to have better understanding of nature of superconducting properties of rubidium tungsten bronze, the temperature dependence of the upper critical field of Rb0.23 WO2.90 and Rb0.23 WO3.02 is determined from resistivity measurements. The (Hc2 , T ) pairs in the phase diagram illustrated in Figs. 2(a) and 2(b) are extracted from 10% drop of normal-state resistivity. The dashed line corresponds to a fit the phenomenological relation Hc2 (T ) = Hc2 (0)[1 − (T /Tc )2 ]α . The fitted Hc2 (0) is about 1100 G for Rb0.23 WO2.90 and 12000 G for Rb0.23 WO3.02 , respectively. Also, the exponent α is about 1.20 close to the value of two-fluid model for Rb0.23 WO2.90 and 1.97 similar to observed values in high-Tc cuprates for Rb0.23 WO3.02 . It should be mentioned that the upper critical field of oxygenrich samples is consistently one order of magnitude larger than that of oxygen-deficient samples. Based upon the relation Hc2 (0) = Φ0 /2πξ 2 (0), an average zero-temperature superconducting coherence length ξ(0) of 538 ˚ A for Rb0.23 WO2.90 and 163 ˚ A for Rb0.23 WO3.02

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FIG. 4: Magnetization as a function of temperature for Rb0.23 WO2.90 and Rb0.23 WO3.02 at various magnetic fields. The insets of (a) and (b) show 3D and 2D scaling of the magnetization around Tc (H).

is obtained. Combining ξ(0) and λ(0), the Ginzburg-Landau parameter κ is determined to be 4.6 for Rb0.23 WO2.90 and 9.2 for Rb0.23 WO3.02 , respectively. Figures 3(a) and 3(b) display irreversibility lines Hirr of Rb0.23 WO2.90 and Rb0.23 WO3.02 obtained from magnetization measurements. The irreversibility temperature, Tirr , is determined from a simple criterion MF C /MZF C = 0.98. A linear dependence is observed in a log-log plot of Hirr versus 1−T /Tc shown in the insets of Figs. 3(a) and 3(b), indicating that Hirr has a power-law behavior. The solid line is the fitting curve Hirr (T ) = H0 (1 − T /Tc )n with exponent n = 1.51 for Rb0.23 WO2.90 and 2.11 for Rb0.23 WO3.02 . It has been reported that Hirr is proportional to (1 − T /Tc )2 for quasi-2D superconductor Bi-2212 and (1-T/Tc)1.5 for anisotropic 3D superconductor LSCO, respectively [9]. Therefore, the results suggest that the vortex dimensionality of Rb0.23 WO2.90 is anisotropic three dimensional and that of Rb0.23 WO3.02 is quasi-two dimensional. To further elucidate the anisotropic nature of rubidium tungsten bronze, the scaling behavior of the fluctuation-induced magnetization in high-field regime is also investigated. According to Ullah and Dorsey [10], the scaling form for the magnetization near Tc is given by 4πM/(TH) β =F[A((T-T c (H))/(TH) β )], where F is a scaling function, A is a temperature-

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and field-dependent coefficient. The exponent β is 2/3 for 3D and 1/2 for 2D superconductors, respectively. The magnetization as a function of temperature near Tc at different fields for Rb0.23 WO2.90 and Rb0.23 WO3.02 is shown in Figs. 4(a) and 4(b), respectively. It should be noted that all M (T ) curves measured at different fields collapse onto a universal curve around Tc (H) plotted in the insets of Figs. 3(a) and 3(b). For Rb0.23 WO2.90 , a 3D scaling behavior is followed. Whereas for Rb0.23 WO3.02 , a 2D scaling behavior is observed. The analysis of dimensionality of the superconducting fluctuation along with the irreversibility line studies concluded that the anisotropy of the oxygen-deficient Rb0.23 WO2.90 is relatively small and the vortex system has essentially 3D character, whereas the anisotropy of the oxygen-rich Rb0.23 WO3.02 is more pronounced and the vortex system has quasi-2D character.

IV. CONCLUTION

In summary, low-field M (H), transport, and magnetic measurements have been performed on hexagonal rubidium tungsten bronze Rb0.23 WO3+δ . Various thermodynamic parameters, such as Hc1 (0), Hc2 (0), λab (0), λc (0), ξ(0), γ, and κ are reported. The irreversibility line and the scaling behavior of magnetization around Tc (H) strongly suggest that oxygen content has a significant impact on the nature of the superconducting properties of Rb0.23 WO3+δ .

Acknowledgment We thank Profs. C.M. Ho and Wei Ku for fruitful discussions. This work was supported by National Science Council of Taiwan under contract No. NSC94-2112-M-032-003.

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