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Heat capacity of YNi2B2C in the vortex state: evidence of the extended s-wave state
Physica C 357±360 (2001) 513±516
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Heat capacity of YNi2B2C in the vortex state: evidence of the extended s-wave state K. Izawa a,*, A. Shibata a, Y. Matsuda a, Y. Kato b, H. Takeya c, K. Hirata c, C.J. van der Beek d, M. Konczykowski d a
Institute for Solid State Physics, University of Tokyo, Kashiwanoha 5-1-5, Kashiwa, Chiba 277-8581, Japan b Department of Applied Physics, University of Tokyo, Tokyo 113-0033, Japan c National Research Institute for Metals, Tsukuba, Ibaraki 305-0047, Japan d Laboratoire des Solides Irradies, Ecole Polytechnique, Palaiseau 91128, France Received 16 October 2000; accepted 16 November 2000
Abstract The low temperature heat capacity Cp and microwave surface impedance Zs in the vortex state of YNi2 B2 C have been studied. In contrast to conventional s-wave superconductors, Cp shows a nearly H 1=2 -dependence. This H 1=2 -dependence is little aected by the introduction of the columnar defects. On the other hand, ¯ux ¯ow resistivity obtained from Zs increases linearly with H. These results indicate that in the vortex state of YNi2 B2 C the delocalized quasiparticle states around the vortex core play a much more important role, similar to d-wave superconductors. Ó 2001 Elsevier Science B.V. All rights reserved. PACS: 74.72.Ny; 74.60.Ec; 74.25.Nf; 74.25.Jb Keywords: YNi2 B2 C; Heat capacity; Columnar defect; Quasiparticle structure
1. Introduction The borocarbide superconductors LnNi2 B2 C (Ln Y, Lu, Tm, Er, Ho and Dy) have attracted much attention because of their interesting physical properties [1±6]. Although considerable studies have been made on these subjects, one of the most fundamental properties in the superconducting state, namely the quasiparticle (QP) structure in the mixed state remains to be clari®ed. In fact, the heat capacity Cp of LuNi2 B2 C and YNi2 B2 C ex-
*
Corresponding author. Fax: +81-471-36-3243. E-mail address: [email protected] (K. Izawa).
hibits a nearly H 1=2 -dependence in the mixed state [7] in contrast to H-linear dependence of conventional s-wave superconductors. This unusual Hdependent Cp have been discussed in terms of some intriguing models; the shrinking of the vortex core with H [7±10] and the ®eld-induced gap nodes [11]. In the former model, however, physical origin behind this phenomenon is not clear and the latter is beyond the applicability of the original argument [12]. Moreover, the extended QP states outside the vortex core owing to a presupposed d-wave symmetry is invoked [13]. However, no corroborative evidence for d-wave pairing has been reported. Therefore, clari®cation of the QP structure in the mixed state is strongly required.
0921-4534/01/$ - see front matter Ó 2001 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 1 - 4 5 3 4 ( 0 1 ) 0 0 3 3 8 - 0
514
K. Izawa et al. / Physica C 357±360 (2001) 513±516
2. Experimental We have measured Cp of the single crystalline YNi2 B2 C in the mixed state by the thermal relaxation method. The surface impedance Zs Rs iXs was measured at 28.5 GHz by using a copper cavity resonator with Q 25,000. Here, Rs and Xs are the surface resistance and the reactance, respectively. The single crystal was placed at the center of the cavity in which the microwave magnetic ®eld is parallel to the c axis. The columnar defects (CD) were randomly introduced into some crystals by a heavy-ion (Pb) irradiation along the c axis. The density of the CD corresponds to the matching ®eld B/ 2 T. The inside of the CD is semiconducting [14] and the CD act as a strong pining center of the vortices; most of the vortices are trapped inside the CD below B/ [15]. In fact, as shown in the inset of Fig. 1(a), we have con®rmed a peak eect in the magnetization M H around B/ for the irradiated sample.
3. Results and discussions Fig. 1(a) shows the ®eld dependent part of the heat capacity, DCp H =T Cp H Cp 0=T for the pristine sample plotted as a function of H 1=2 . At high ®elds, H 1=2 P 0:25T 1=2 H P 70 mT), DCp H =T increases linearly with respect to H 1=2 . At low ®elds, however, DCp H =T deviates from the H 1=2 -dependence and shows an upward curvature. Fig. 1(b) shows the same plot for the irradiated crystal with B/ 2 T. The unusual H-dependence endures even after the irradiation. Surprisingly, DCp H =T is little aected by the introduction of the CD (see the inset of Fig. 1(b)). Fuller discussion will be presented later. Fig. 2 shows H-dependence of Rs and Xs at 1.5 K. With increasing H, Rs increases as H 1=2 , indicating that the ¯ux ¯ow resistivity qf / H . To discuss qf H quantitatively, we analyzed the data by means of the expression of Zs in the ¯ux ¯ow state by Coey and Clem [16], " #1=2 1 i=2d2f =k2L Zs il0 xkL 1 1 2ik2L =d2qp
Fig. 1. (a) Field dependent part of the heat capacity DCp H=T Cp H Cp 0=T for the pristine YNi2 B2 C at low temperatures. Inset: M±H curves for pristine and irradiated (B/ 2 T) crystals at 5 K. While in the pristine crystal M H is reversible in a wide H-range, showing very weak pinning centers, M H show a broad peak at B B/ in the irradiated crystal. (b) The same plot for the irradiated crystal with B/ 2 T. Inset: The H-dependence of DCp =T before and after the irradiation.
where dqp 2qqp =l0 x1=2 with the QP resistivity qqp is the normal-¯uid skin depth and df 2qf = 1=2 l0 x . We considered two dierent H-dependences of N H , namely N H / H (case I) and N H / H 1=2 (case II). Case I corresponds to the Bardeen±Stephen relation, qf H =Hc2 qn while case II corresponds to the shrinking of the vortex
K. Izawa et al. / Physica C 357±360 (2001) 513±516
Fig. 2. Rs and Xs plotted as a function of (H =Hc2 1=2 at 1.5 K. The solid and dashed lines represent the result of the theoretical calculations of Rs and Xs by Eq. (1), respectively, assuming two dierent H-dependences of N(H); N H / H (case I) and N H / H 1=2 (case II). For the detail, see text.
1=2
core qf H =Hc2 qn . For both calculations, we which is obtained from Xs . used kL 0 500 A, Comparing the two cases, case I obviously describes the data much better than case II. This result shows that the number of QP localized in the each vortex core does not depend on H because the ¯ux ¯ow dissipation mainly comes from the localized QP inside the core and is proportional to the number of the vortices. Therefore, we can exclude the scenario of the shrinking of the vortex core as an origin of the nonlinear H dependence of Cp , which proposed by several groups [7±10]. Further important clues for understanding the QP structure is provided by the eect of the CD on Cp . Because the presence of the CD with a comparable size with the radius of the coherence length n strongly changes the electronic structure inside the vortex core, little in¯uence of the CD on Cp implies that the QPs within the core radius n is not important for the total heat capacity. From the results of Cp and Zs , we come to the conclusion that the extended QP states around the vortex core play an important role in determining the superconducting properties in YNi2 B2 C, similar to dwave superconductors. Moreover, we can address
515
the microscopic origin of H 1=2 dependence of Cp . Two sources for the H 1=2 dependence has been pointed out in the presence of line nodes [17]. One arises from the localized QP in the node directions. Since the wave function of the localized QP is cut o by the adjacent vortices, the area per each vortex is determined by the intervortex distance R / 1=H 1=2 and then N(H) is given as N H / nRH / H 1=2 . The other arises from the delocalized QP. In the presence of the supercurrent ¯ow, the energy spectrum of the delocalized QP is shifted by the Doppler eect. In the superconductors in which the DOS is proportional to the energy around the Fermi level EF , the Doppler-shifted QP gives rise to the ®nite DOS at EF . This eect leads to a H 1=2 dependence of N(H). Little in¯uence of the CD on Cp shows that the Doppler shift of the delocalized QP is mainly responsible for the H 1=2 dependence of Cp because the contribution from the localized QP in the node directions is strongly aected by the destruction of the vortex lattice. Turning now to the upward curvature of Cp at low ®elds l0 Hcr 6 70 mT. In the superconductors with line nodes, generally, both the thermally excited QP and the Doppler-shifted QP contribute to Cp . Since the number of the Doppler-shifted QP ND is larger than that of the thermally excited QP Nth , Cp shows a H 1=2 dependence at high ®elds. With decreasing H, however, Nth exceeds ND at a crossover ®eld and then Cp approaches the zero ®eld value Cp = cn T kB T =D, where cn and D are the Sommerfeld coecient in the normal state and the superconducting gap, respectively [18]. The crossover ®eld at 1.5 K estimated from the relation 1=2 H =Hc2 kB T =D with l0 Hc2 5:5 T and Tc 13:4 K is about 24 mT, which is the same order of l0 Hcr . Therefore, the deviation from the H 1=2 dependence of Cp =T below Hcr is quantitatively consistent with the existence of the line nodes. Similar behavior is also observed in YBa2 Cu3 O7 d [19]. It follows from the present results that the in¯uence of the delocalized QP, which many authors have ignored [20±22], should be taken into account in the discussion of the vortex lattice structure, the fourfold anisotropy of Hc2 and M(H). Finally, we will mention the symmetry of the superconductivity. It is tempting to relate the observed
516
K. Izawa et al. / Physica C 357±360 (2001) 513±516
relevance of the delocalized QPs to a d-wave superconductivity [13]. However, because of the robustness of the superconductivity against impurities [7,23,24], a d-wave state is unlikely. Therefore, we believe that an extended s-wave state is most likely for YNi2 B2 C. Acknowledgements We thank N. Chikumoto, J. Clem, T. Hanaguri, R.P. Huebener, K. Maki, H. Takagi, A. Tanaka and H. Yoshizawa for helpful discussions. References [1] R.J Cava, et al., Nature 367 (1994) 146. [2] P.C. Can®eld, et al., Phys. Today 51 (1998) 40 and references therein.
[3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24]
M. Yethiraj, et al., Phys. Rev. Lett. 78 (1997) 4849. Y. De Wilde, et al., Phys. Rev. Lett. 78 (1997) 4273. M. Eskildsen, et al., Phys. Rev. Lett. 79 (1997) 487. H. Sakata, et al., Phys. Rev. Lett. 84 (2000) 1583. M. Nohara, et al., J. Phys. Soc. Jpn. 68 (1999) 1078. J.E. Sonier, et al., Phys. Rev. Lett. 82 (1999) 4914. J.E. Sonier, et al., Phys. Rev. Lett. 79 (1997) 1742. M. Ichioka, et al., Phys. Rev. B 59 (1999) 8902. M. Hedo, et al., J. Phys. Soc. Jpn. 67 (1998) 272. U. Brandt, et al., Z. Phys. 201 (1967) 209. G. Wang, K. Maki, Phys. Rev. B 58 (1998) 6493. N. Nishida, et al., Physica B 284±288 (2000) 967. M.V. Indenbom, et al., Phys. Rev. Lett. 84 (2000) 1792. M.W. Coey, R. Clem, Phys. Rev. Lett. 67 (1991) 386. G.E. Volovik, JETP Lett. 58 (1993) 469. N.B. Kopnin, G.E. Volovik, JETP Lett. 64 (1996) 690. A. Junod, et al., Physica B 284 (2000) 1043. V.G. Kogan, et al., Phys Rev. Lett. 79 (1997) 741. S.B. Dugdale, et al., Phys Rev. Lett. 83 (1999) 4824. L. Civale, et al., Phys Rev. Lett. 83 (1999) 3920. K.O. Cheon, et al., Phys. Rev. B 58 (1998) 6463. L.S. Borkowski, P.J. Hirschfeld, Phys. Rev. B 49 (1994) 15404.
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Heat capacity of YNi2B2C in the vortex state: evidence of the extended s-wave state K. Izawa a,*, A. Shibata a, Y. Matsuda a, Y. Kato b, H. Takeya c, K. Hirata c, C.J. van der Beek d, M. Konczykowski d a
Institute for Solid State Physics, University of Tokyo, Kashiwanoha 5-1-5, Kashiwa, Chiba 277-8581, Japan b Department of Applied Physics, University of Tokyo, Tokyo 113-0033, Japan c National Research Institute for Metals, Tsukuba, Ibaraki 305-0047, Japan d Laboratoire des Solides Irradies, Ecole Polytechnique, Palaiseau 91128, France Received 16 October 2000; accepted 16 November 2000
Abstract The low temperature heat capacity Cp and microwave surface impedance Zs in the vortex state of YNi2 B2 C have been studied. In contrast to conventional s-wave superconductors, Cp shows a nearly H 1=2 -dependence. This H 1=2 -dependence is little aected by the introduction of the columnar defects. On the other hand, ¯ux ¯ow resistivity obtained from Zs increases linearly with H. These results indicate that in the vortex state of YNi2 B2 C the delocalized quasiparticle states around the vortex core play a much more important role, similar to d-wave superconductors. Ó 2001 Elsevier Science B.V. All rights reserved. PACS: 74.72.Ny; 74.60.Ec; 74.25.Nf; 74.25.Jb Keywords: YNi2 B2 C; Heat capacity; Columnar defect; Quasiparticle structure
1. Introduction The borocarbide superconductors LnNi2 B2 C (Ln Y, Lu, Tm, Er, Ho and Dy) have attracted much attention because of their interesting physical properties [1±6]. Although considerable studies have been made on these subjects, one of the most fundamental properties in the superconducting state, namely the quasiparticle (QP) structure in the mixed state remains to be clari®ed. In fact, the heat capacity Cp of LuNi2 B2 C and YNi2 B2 C ex-
*
Corresponding author. Fax: +81-471-36-3243. E-mail address: [email protected] (K. Izawa).
hibits a nearly H 1=2 -dependence in the mixed state [7] in contrast to H-linear dependence of conventional s-wave superconductors. This unusual Hdependent Cp have been discussed in terms of some intriguing models; the shrinking of the vortex core with H [7±10] and the ®eld-induced gap nodes [11]. In the former model, however, physical origin behind this phenomenon is not clear and the latter is beyond the applicability of the original argument [12]. Moreover, the extended QP states outside the vortex core owing to a presupposed d-wave symmetry is invoked [13]. However, no corroborative evidence for d-wave pairing has been reported. Therefore, clari®cation of the QP structure in the mixed state is strongly required.
0921-4534/01/$ - see front matter Ó 2001 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 1 - 4 5 3 4 ( 0 1 ) 0 0 3 3 8 - 0
514
K. Izawa et al. / Physica C 357±360 (2001) 513±516
2. Experimental We have measured Cp of the single crystalline YNi2 B2 C in the mixed state by the thermal relaxation method. The surface impedance Zs Rs iXs was measured at 28.5 GHz by using a copper cavity resonator with Q 25,000. Here, Rs and Xs are the surface resistance and the reactance, respectively. The single crystal was placed at the center of the cavity in which the microwave magnetic ®eld is parallel to the c axis. The columnar defects (CD) were randomly introduced into some crystals by a heavy-ion (Pb) irradiation along the c axis. The density of the CD corresponds to the matching ®eld B/ 2 T. The inside of the CD is semiconducting [14] and the CD act as a strong pining center of the vortices; most of the vortices are trapped inside the CD below B/ [15]. In fact, as shown in the inset of Fig. 1(a), we have con®rmed a peak eect in the magnetization M H around B/ for the irradiated sample.
3. Results and discussions Fig. 1(a) shows the ®eld dependent part of the heat capacity, DCp H =T Cp H Cp 0=T for the pristine sample plotted as a function of H 1=2 . At high ®elds, H 1=2 P 0:25T 1=2 H P 70 mT), DCp H =T increases linearly with respect to H 1=2 . At low ®elds, however, DCp H =T deviates from the H 1=2 -dependence and shows an upward curvature. Fig. 1(b) shows the same plot for the irradiated crystal with B/ 2 T. The unusual H-dependence endures even after the irradiation. Surprisingly, DCp H =T is little aected by the introduction of the CD (see the inset of Fig. 1(b)). Fuller discussion will be presented later. Fig. 2 shows H-dependence of Rs and Xs at 1.5 K. With increasing H, Rs increases as H 1=2 , indicating that the ¯ux ¯ow resistivity qf / H . To discuss qf H quantitatively, we analyzed the data by means of the expression of Zs in the ¯ux ¯ow state by Coey and Clem [16], " #1=2 1 i=2d2f =k2L Zs il0 xkL 1 1 2ik2L =d2qp
Fig. 1. (a) Field dependent part of the heat capacity DCp H=T Cp H Cp 0=T for the pristine YNi2 B2 C at low temperatures. Inset: M±H curves for pristine and irradiated (B/ 2 T) crystals at 5 K. While in the pristine crystal M H is reversible in a wide H-range, showing very weak pinning centers, M H show a broad peak at B B/ in the irradiated crystal. (b) The same plot for the irradiated crystal with B/ 2 T. Inset: The H-dependence of DCp =T before and after the irradiation.
where dqp 2qqp =l0 x1=2 with the QP resistivity qqp is the normal-¯uid skin depth and df 2qf = 1=2 l0 x . We considered two dierent H-dependences of N H , namely N H / H (case I) and N H / H 1=2 (case II). Case I corresponds to the Bardeen±Stephen relation, qf H =Hc2 qn while case II corresponds to the shrinking of the vortex
K. Izawa et al. / Physica C 357±360 (2001) 513±516
Fig. 2. Rs and Xs plotted as a function of (H =Hc2 1=2 at 1.5 K. The solid and dashed lines represent the result of the theoretical calculations of Rs and Xs by Eq. (1), respectively, assuming two dierent H-dependences of N(H); N H / H (case I) and N H / H 1=2 (case II). For the detail, see text.
1=2
core qf H =Hc2 qn . For both calculations, we which is obtained from Xs . used kL 0 500 A, Comparing the two cases, case I obviously describes the data much better than case II. This result shows that the number of QP localized in the each vortex core does not depend on H because the ¯ux ¯ow dissipation mainly comes from the localized QP inside the core and is proportional to the number of the vortices. Therefore, we can exclude the scenario of the shrinking of the vortex core as an origin of the nonlinear H dependence of Cp , which proposed by several groups [7±10]. Further important clues for understanding the QP structure is provided by the eect of the CD on Cp . Because the presence of the CD with a comparable size with the radius of the coherence length n strongly changes the electronic structure inside the vortex core, little in¯uence of the CD on Cp implies that the QPs within the core radius n is not important for the total heat capacity. From the results of Cp and Zs , we come to the conclusion that the extended QP states around the vortex core play an important role in determining the superconducting properties in YNi2 B2 C, similar to dwave superconductors. Moreover, we can address
515
the microscopic origin of H 1=2 dependence of Cp . Two sources for the H 1=2 dependence has been pointed out in the presence of line nodes [17]. One arises from the localized QP in the node directions. Since the wave function of the localized QP is cut o by the adjacent vortices, the area per each vortex is determined by the intervortex distance R / 1=H 1=2 and then N(H) is given as N H / nRH / H 1=2 . The other arises from the delocalized QP. In the presence of the supercurrent ¯ow, the energy spectrum of the delocalized QP is shifted by the Doppler eect. In the superconductors in which the DOS is proportional to the energy around the Fermi level EF , the Doppler-shifted QP gives rise to the ®nite DOS at EF . This eect leads to a H 1=2 dependence of N(H). Little in¯uence of the CD on Cp shows that the Doppler shift of the delocalized QP is mainly responsible for the H 1=2 dependence of Cp because the contribution from the localized QP in the node directions is strongly aected by the destruction of the vortex lattice. Turning now to the upward curvature of Cp at low ®elds l0 Hcr 6 70 mT. In the superconductors with line nodes, generally, both the thermally excited QP and the Doppler-shifted QP contribute to Cp . Since the number of the Doppler-shifted QP ND is larger than that of the thermally excited QP Nth , Cp shows a H 1=2 dependence at high ®elds. With decreasing H, however, Nth exceeds ND at a crossover ®eld and then Cp approaches the zero ®eld value Cp = cn T kB T =D, where cn and D are the Sommerfeld coecient in the normal state and the superconducting gap, respectively [18]. The crossover ®eld at 1.5 K estimated from the relation 1=2 H =Hc2 kB T =D with l0 Hc2 5:5 T and Tc 13:4 K is about 24 mT, which is the same order of l0 Hcr . Therefore, the deviation from the H 1=2 dependence of Cp =T below Hcr is quantitatively consistent with the existence of the line nodes. Similar behavior is also observed in YBa2 Cu3 O7 d [19]. It follows from the present results that the in¯uence of the delocalized QP, which many authors have ignored [20±22], should be taken into account in the discussion of the vortex lattice structure, the fourfold anisotropy of Hc2 and M(H). Finally, we will mention the symmetry of the superconductivity. It is tempting to relate the observed
516
K. Izawa et al. / Physica C 357±360 (2001) 513±516
relevance of the delocalized QPs to a d-wave superconductivity [13]. However, because of the robustness of the superconductivity against impurities [7,23,24], a d-wave state is unlikely. Therefore, we believe that an extended s-wave state is most likely for YNi2 B2 C. Acknowledgements We thank N. Chikumoto, J. Clem, T. Hanaguri, R.P. Huebener, K. Maki, H. Takagi, A. Tanaka and H. Yoshizawa for helpful discussions. References [1] R.J Cava, et al., Nature 367 (1994) 146. [2] P.C. Can®eld, et al., Phys. Today 51 (1998) 40 and references therein.
[3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24]
M. Yethiraj, et al., Phys. Rev. Lett. 78 (1997) 4849. Y. De Wilde, et al., Phys. Rev. Lett. 78 (1997) 4273. M. Eskildsen, et al., Phys. Rev. Lett. 79 (1997) 487. H. Sakata, et al., Phys. Rev. Lett. 84 (2000) 1583. M. Nohara, et al., J. Phys. Soc. Jpn. 68 (1999) 1078. J.E. Sonier, et al., Phys. Rev. Lett. 82 (1999) 4914. J.E. Sonier, et al., Phys. Rev. Lett. 79 (1997) 1742. M. Ichioka, et al., Phys. Rev. B 59 (1999) 8902. M. Hedo, et al., J. Phys. Soc. Jpn. 67 (1998) 272. U. Brandt, et al., Z. Phys. 201 (1967) 209. G. Wang, K. Maki, Phys. Rev. B 58 (1998) 6493. N. Nishida, et al., Physica B 284±288 (2000) 967. M.V. Indenbom, et al., Phys. Rev. Lett. 84 (2000) 1792. M.W. Coey, R. Clem, Phys. Rev. Lett. 67 (1991) 386. G.E. Volovik, JETP Lett. 58 (1993) 469. N.B. Kopnin, G.E. Volovik, JETP Lett. 64 (1996) 690. A. Junod, et al., Physica B 284 (2000) 1043. V.G. Kogan, et al., Phys Rev. Lett. 79 (1997) 741. S.B. Dugdale, et al., Phys Rev. Lett. 83 (1999) 4824. L. Civale, et al., Phys Rev. Lett. 83 (1999) 3920. K.O. Cheon, et al., Phys. Rev. B 58 (1998) 6463. L.S. Borkowski, P.J. Hirschfeld, Phys. Rev. B 49 (1994) 15404.