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New high field state of flux line lattice in CeCoIn5
Physica C 426–431 (2005) 36–40 www.elsevier.com/locate/physc
New high field state of flux line lattice in CeCoIn5 Y. Kasahara a,*, T. Watanabe a, K. Izawa a, T. Sakakibara a, C.J. van der Beek b, T. Hanaguri c, M. Nohara d, H. Takagi d, H. Shishido e, R. Settai e, Y. Onuki e, Y. Matsuda a,f a
d
Matsuda Laboratory, Division of New Material Science, Institute for Solid State Physics, University of Tokyo, Kashiwanoha, Kashiwa, Chiba 277-8581, Japan b Laboratoire des Solides Irradie´s, CNRS-UMR 7642, Ecole Polytechnique, 91128 Palaiseau, France c The Institute of Physical and Chemical Research (RIKEN), Saitama 351-0198, Japan Department of Advanced Materials Science and Applied Chemistry, University of Tokyo, Kashiwa, Chiba 277-8565, Japan e Graduate School of Science, Osaka University, Toyonaka, Osaka 560-0043, Japan f Department of Physics, Kyoto University, Kyoto 606-8502, Japan Received 23 November 2004; accepted 15 February 2005 Available online 20 June 2005
Abstract We have measured the ultrasound velocity of quasi-two-dimensional superconductor CeCoIn5 with extremely large Pauli paramagnetic susceptibility. The results indicate that the new high field superconducting phase, which is revealed by the recent heat capacity measurements, is characterized by the unusual softening of flux line lattice. The softening is most likely due to the collapse of the flux line lattice tilt modulus and transition to quasi-two-dimensional vortex state. These results provide a strong evidence that the high field phase is the Fulde–Ferrell–Larkin–Ovchinnikov phase, in which the order parameter is spatially modulated and has planar nodes aligned perpendicularly to the vortices. 2005 Elsevier B.V. All rights reserved. PACS: 74.25.Dw; 74.25.Jb; 74.25.Ld; 74.70.Tx Keywords: FFLO state; Flux line lattice; Ultrasound velocity; CeCoIn5
1. Introduction * Corresponding author. Tel.: +81 4 7136 3242; fax: +81 4 7136 3243. E-mail address: [email protected] (Y. Kasahara).
In spin-singlet superconductors, superconductivity is suppressed by the orbital effect (vortices) and by the Zeeman effect of the conduction
0921-4534/$ - see front matter 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.physc.2005.02.018
Y. Kasahara et al. / Physica C 426–431 (2005) 36–40
electron spins (Pauli paramagnetism). In most of the superconductors, the upper critical field Hc2 is determined by the orbital effect. In this case the transition at Hc2 is always second order. On the other hand, when the orbital effect is absent and Hc2 is determined by the Pauli effect, which can be realized in thin films in parallel field, the phase transition from superconducting (SC) state to normal state is known to become first order at Hc2 [1]. Moreover in such a case, the appearance of a new inhomogeneous SC state in the vicinity of Hc2 was proposed by Fulde and Ferrell and Larkin and Ovchinnikov (FFLO) [2–9]. In the FFLO state, a new pairing state with non-zero total momentum (k", k + q#) is formed in contrast to (k", k#)-pairing in ordinary superconductors. As a result, the SC order parameter exhibits spatial modulation with planar nodes which locate perpendicular to the applied field with a spatial modulation length 2p/jqj. Quasi-particles around these nodes develop a spin polarization in the direction of the applied magnetic field to reduce the Pauli paramagnetic pair breaking. In spite of enormous efforts to find the FFLO state, it has never been observed in conventional superconductors. Heavy fermion UPd2Al3 and intermediate valence compound CeRu2 have been suggested to have the FFLO state, but subsequent research has called the interpretation of the data for these materials in terms of an FFLO state into question [10–12]. Recently a new heavy fermion superconductor CeCoIn5 (Tc = 2.3 K) with quasi-two-dimensional (2D) electronic structure has been discovered [13]. The SC gap symmetry determined by the angle resolved thermal magnetotransport and heat capacity measurements [14–20] revealed that the symmetry of CeCoIn5 is most likely to be d-wave with line nodes perpendicular to the 2D planes. One of the most striking feature in the superconducting state of CeCoIn5 is that the first order transition occurs at Hc2 below T* Tc/2 both in Hkab and Hkc, indicating that the superconductivity is limited by the Pauli paramagnetism [17,21,22]. More excitingly, very recent heat capacity measurements revealed the presence of the second order transition in the vortex state, which branches from the first order transition line at
37
low temperature and high field, indicating the presence of a new SC phase [23,24]. It has been pointed out that this new SC phase may be the FFLO state. In fact, CeCoIn5 seems to satisfy several requirements for the formation of the FFLO state. For instance, it is in the very clean regime l n (l is quasi-particle mean free path, n is coherence length), and has very large Ginzburg–Landau parameter j = k/n 10 (k is penetration length). However it is premature to conclude the FFLO phase solely from the heat capacity measurements, since a possibility of some sort of magnetic transition cannot be ruled out. In order to establish the existence of the FFLO state, it is particularly important to elucidate the structure of flux line lattice (FLL) because it is predicted that FLL is divided into segments and becomes quasi-two-dimensional (Q2D) structure in FFLO state [5]. In this paper, we present the results of ultrasound velocity measurements which provide direct information on the FLL structure in CeCoIn5.
2. Experimental The ultrasound velocity measurements were performed on high quality single crystals of CeCoIn5 with tetragonal symmetry, grown by the self-flux method. We measured the relative change of the sound velocity by pulse echo method with a phase comparison technique. Ultrasonic waves with a frequency of 90 MHz were generated and detected by a pair of LiNbO3 transducers glued on the polished sample surfaces. The resolution of the relative velocity measurements was 1 part in 106. In all measurements, we measured the transverse sound velocity with propagation kk[1 0 0] and polarization uk[0 1 0], which corresponds to the shear modulus C c66 of the crystal lattice, and the magnetic field H was applied parallel to the plane (Hk[1 0 0] or [0 1 0]).
3. Results and discussion Fig. 1 displays the relative change of the transverse sound velocity Dv0t =v0t as a function of
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Y. Kasahara et al. / Physica C 426–431 (2005) 36–40
Fig. 1. The relative change of the transverse sound velocity as a function of temperature in applied magnetic field 10.5 T in two different configurations; the Lorentz mode (H ? u) and the non-Lorentz mode (Hku).
temperature at 10.5 T in two different configurations with respect to the polarization vector u, the sound propagation vector k, and magnetic field H. We have performed the measurements in two different configurations; the Lorentz mode (Fig. 1, u ? H) where the sound wave couples to the vortices and the crystal lattice, and the non-Lorentz mode (Fig. 1, ukH) where the sound wave couples only to the crystal lattice. As shown in Fig. 1, the transverse sound velocity in the Lorentz mode is strongly enhanced compared to that in the non-Lorentz mode. This indicates that the transverse ultrasound strongly couples to the FLL in the Lorentz mode. In the Lorentz mode the ultrasound velocity increases rapidly with decreasing T, indicating the hardening of the FLL. In the Lorentz mode, the ultrasound shows a distinct kink at T*, whereas there is no anomaly in the non-Lorentz mode. This suggests that the anomaly purely originates from the FLL. Fig. 2 displays several curves of the relative shift of the FLL contribution to the transverse ultrasound velocity Dvt/vt below Hc2 as a function of temperature, which are obtained as the difference between Dv0t =v0t in the Lorentz mode and that in the non-Lorentz mode. As the temperature is low-
Fig. 2. The relative shift of the transverse sound velocity arising from the contribution of the FLL, Dvt/vt, as a function of temperature, below upper critical field Hc2. The solid and dashed lines indicate T* and Tc(H), respectively. For clarity, the zero levels of the different curves are vertically shifted.
ered below the SC transition temperature in magnetic field Tc(H), Dvt/vt starts to increase. At fields of 9.5 T and higher, Dvt/vt shows the anomaly, and reduces its value smaller than that extrapolated above T* while no anomaly is observed at 8.5 T. These results indicate a softening of the FLL below T*. Fig. 3 displays H–T phase diagram obtained by ultrasonic measurements. In this temperature range, the transition at Hc2 is of first order. T* coincide well with the reported second order transition line within SC phase [23]. On the basis of our results, we conclude that the second order phase transition within SC phase is characterized by the softening of the FLL [25]. We here discuss several possible origins of the softening of the FLL. We can rule out the possibility of the so-called peak effect because we have never observed the magnetization peak at T* [22]. In addition, peak effect should give rise to the hardening of the FLL, which is opposite to the present effect [26,27]. A lowering of the
Y. Kasahara et al. / Physica C 426–431 (2005) 36–40
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nodes. The schematic figure of the structural transition of flux lines at T* is illustrated in Fig. 3. We next analyze the softening of the FLL quantitatively by using the collective pinning theory in type-II superconductors [30]. The pinning strength W in the single vortex limit for three dimensional superconductors is written as [31] 7=2
W ¼ ðU0 B3=2 j3c =4p2 l0 k2 Þ
1=2
ð1Þ
;
where U0 = h/2e and jc is the critical current density. Experimental value of the critical current density jc is obtained from magnetic field dependence of the FLL contribution to ultrasound velocity, which is written as [32] Fig. 3. Experimental H–T phase diagram for CeCoIn5 below 0.7 K for (Hk[1 0 0]). The transition temperature T*, indicated by open circles, were obtained from ultrasonic measurements as indicated by the arrows in Fig. 2. Solid circles and solid line depict the upper critical field Hc2 determined by the ultrasonic experiments. The dashed line is the second order transition line determined by the heat capacity reported in Ref. [14]. The region shown by dark gray is in the FFLO state. The schematic figures are sketches of structural transition of flux lines at T*.
ultrasound velocity occurs at melting transition of the FLL [28]. However we can rule out this scenario, because the entropy jump at T* reported in Refs. [23,24] is significantly larger than that expected in melting transition. Moreover, Dvt/vt should be abruptly to zero at T* because pinning vanishes at melting transition in contrast to our results. We can also rule out the possibility of the magnetic transition at T* because we observed no anomaly in non-Lorentz mode at T*. We point out that the observed phenomena is qualitatively consistent with the FFLO state in which the segmentation of the FLL occurs by the formation of the planar nodes perpendicular to the FLL [5]. In this sense the FFLO phase resembles to high-Tc and organic superconductors in which the flux line are divided into two dimensional pancake vortices and they are coupled by Josephson effect [29]. In the conventional flux line the ultrasound can propagate easily along the flux line. On the other hand, in FFLO state, the propagation of ultrasound is expected to be seriously prevented because the line tension of the flux line becomes very small at the position of the planar
Dvt ¼ ðB2 =l0 qvt Þð1=½1 þ fðU0 BÞ
1=2
2
jkj =l0 jc gÞ; ð2Þ
where q is mass density. As shown in Fig. 4, by fitting to Eq. (2), we obtain a field-independent critical current density, jc = 7.0 · 108 A m2, below approximately 9 T. Then, we deduce W above T*, 5 · 104 N2 m3 from Eq. (1). In the quasi-2D state of FLL, jc is given by the expression for 1=2 1=2 f a 2D layer of thickness d, j2D c66 d. c ¼ W =U0 B At the transition, j2D and W is must be equal to c the experimental values 7.0 · 108 A m2 and 5 · 104 N2 m3, respectively. We then obtain a layer of thickness d of 4 · 108 m. This value is close to the modulation length 2p/jqj = (me/m*)(hkF/ eB) = 3.5 · 108 m expected for the order parameter structure associated with the formation of
Fig. 4. The relative shift of the FLL contribution to the transverse ultrasound velocity Dvt/vt as a function of applied magnetic field at 0.2 K. Solid line is obtained by Eq. (2).
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the FFLO state (the Fermi wave vector kF 8 · 109 m1). Based on these results, we conclude that the elastic properties of the FLL in the new high field phase of CeCoIn5 is quantitatively consistent with that expected in the FFLO phase. We note that the very recent NMR measurements also support the FFLO phase [33].
4. Summary In summary, ultrasound velocity measurements of CeCoIn5 reveal the unusual softening of the FLL at the second order phase transition within the SC phase. The FLL softening is most likely characterized by the collapse of the FLL tilt modulus and transition to quasi-2D state. Our results provide a strong evidence that the high field phase is the FFLO phase, in which the order parameter is spatially modulated and has planar nodes aligned perpendicular to the vortices.
References [1] D. Saint-James, G. Sarma, E.J. Thomas (Eds.), Type II Superconductivity, Pergamon Press, Oxford, 1969 (Chapter 6). [2] P. Fulde, R.A. Ferrell, Phys. Rev. A 135 (1964) 550. [3] A.I. Larkin, Y.N. Ovchinnikov, Sov. Phys. JETP 20 (1965) 762. [4] L.W. Gruenberg, L. Gunther, Phys. Rev. Lett. 16 (1996) 996. [5] M. Tachiki, S. Takahashi, P. Gegenwart, M. Weiden, M. Lang, C. Geibel, F. Steglich, R. Modler, C. Paulsen, Y. Onuki, Z. Phys. B 100 (1996) 369. [6] H. Adachi, R. Ikeda, Phys. Rev. B 68 (2003) 184510. [7] H. Shimahara, Phys. Rev. B 50 (1994) 12760. [8] U. Klein, D. Rainer, H. Shimahara, J. Low Temp. Phys. 118 (2000) 91. [9] A.I. Buzdin, H. Kachkachi, Phys. Lett. A 225 (1997) 341. [10] K. Gloos, R. Modler, H. Schimanski, C.D. Bredl, C. Geibel, F. Steglich, A.I. Buzdin, N. Sato, T. Komatsubara, Phys. Rev. Lett. 70 (1993) 501. [11] M.R. Norman, Phys. Rev. Lett. 71 (1993) 3391. [12] R. Modler, P. Gegenwart, M. Lang, M. Deppe, M. Weiden, T. Lu¨hmann, C. Geibel, F. Steglich, Phys. Rev. Lett. 76 (1996) 1292.
[13] C. Petrovic, P.G. Pagliuso, M.F. Hundley, R. Movshovich, J.L. Sarrao, J.D. Thompson, Z. Fisk, P. Monthoux, Europhys. Lett. 53 (2001) 354. [14] K. Izawa, H. Takahashi, H. Yamaguchi, Y. Matsuda, M. Suzuki, T. Sasaki, T. Fukase, Y. Yoshida, R. Settai, Y. Onuki, Phys. Rev. Lett. 86 (2001) 2653. [15] K. Izawa, H. Yamaguchi, T. Sasaki, Y. Matsuda, Phys. Rev. Lett. 88 (2002) 027002. [16] K. Izawa, K. Kamata, Y. Nakajima, Y. Matsuda, T. Watanabe, M. Nohara, H. Takagi, P. Thalmeier, K. Maki, Phys. Rev. Lett. 89 (2002) 137006. [17] K. Izawa, Y. Nakajima, J. Goryo, Y. Matsuda, S. Osaki, H. Sugawara, H. Sato, P. Thalmeier, K. Maki, Phys. Rev. Lett. 90 (2003) 117001. [18] K. Izawa, H. Yamaguchi, Y. Matsuda, H. Shishido, R. Settai, Y. Onuki, Phys. Rev. Lett. 87 (2001) 057002. [19] T. Watanabe, K. Izawa, Y. Kasahara, Y. Haga, Y. Onuki, P. Thalmeier, K. Maki, Y. Matsuda, Phys. Rev. B 70 (2004) 184502. [20] H. Aoki, T. Sakakibara, H. Shishido, R. Settai, Y. Onuki, P. Miranovic´, K. Machida, J. Phys. C 16 (2004) L13. [21] A. Bianchi, R. Movshovich, N. Oeschler, P. Gegenwart, F. Steglich, J.D. Thompson, P.G. Pagliuso, J.L. Sarrao, Phys. Rev. Lett. 89 (2002) 137002. [22] T. Tayama, A. Harita, T. Sakakibara, Y. Haga, H. Shishido, R. Settai, Y. Onuki, Phys. Rev. B 65 (2002) 180504. [23] A. Bianchi, R. Movshovich, C. Capan, P.G. Pagliuso, J.L. Sarrao, Phys. Rev. Lett. 91 (2003) 187004. [24] H.A. Radovan, N.A. Fortune, T.P. Murphy, S.T. Hannahs, E.C. Palm, S.W. Tozer, D. Hall, Nature 425 (2003) 51. [25] T. Watanabe, K. Kasahara, K. Izawa, T. Sakakibara, Y. Matsuda, C.J. van der Beek, T. Hanaguri, H. Shishido, R. Settai, Y. Onuki, Phys. Rev. B 70 (2004) 020506(R). [26] B. Wolf, C. Hinkel, S. Holtmeier, D. Wichert, I. Kouroudis, G. Bruls, B. Lu¨thi, M. Hedo, Y. Inada, E. Yamamoto, Y. Haga, Y. Onuki, J. Low Temp. Phys. 107 (1997) 421. [27] M.B. Gaifullin, Y. Matsuda, N. Chikumoto, J. Shimoyama, K. Kishio, Phys. Rev. Lett. 84 (2000) 2945. [28] E. Zeldov, D. Majer, M. Konczykowski, V.B. Geshkenbein, V.M. Vinokur, H. Shtrikman, Nature 375 (1995) 373. [29] Y. Matsuda, M.B. Gaifullin, K. Kumagai, K. Kadowaki, T. Mochiku, Phys. Rev. Lett. 75 (1995) 4512. [30] A.I. Larkin, Y.N. Ovchinnikov, J. Low Temp. Phys. 34 (1979) 409. [31] R. Wo¨denweber, P.H. Kes, Phys. Rev. B 34 (1986) 494. [32] D. Domı´nguez, L. Bulaevskii, B. Ivlev, M. Maley, R. Bishop, Phys. Rev. B 53 (1996) 6682. [33] K. Kakuyanagi, M. Saito, K. Kumagai, S. Takashina, M. Nohara, H. Takagi, Y. Matsuda, cond-mat/0405661.
New high field state of flux line lattice in CeCoIn5 Y. Kasahara a,*, T. Watanabe a, K. Izawa a, T. Sakakibara a, C.J. van der Beek b, T. Hanaguri c, M. Nohara d, H. Takagi d, H. Shishido e, R. Settai e, Y. Onuki e, Y. Matsuda a,f a
d
Matsuda Laboratory, Division of New Material Science, Institute for Solid State Physics, University of Tokyo, Kashiwanoha, Kashiwa, Chiba 277-8581, Japan b Laboratoire des Solides Irradie´s, CNRS-UMR 7642, Ecole Polytechnique, 91128 Palaiseau, France c The Institute of Physical and Chemical Research (RIKEN), Saitama 351-0198, Japan Department of Advanced Materials Science and Applied Chemistry, University of Tokyo, Kashiwa, Chiba 277-8565, Japan e Graduate School of Science, Osaka University, Toyonaka, Osaka 560-0043, Japan f Department of Physics, Kyoto University, Kyoto 606-8502, Japan Received 23 November 2004; accepted 15 February 2005 Available online 20 June 2005
Abstract We have measured the ultrasound velocity of quasi-two-dimensional superconductor CeCoIn5 with extremely large Pauli paramagnetic susceptibility. The results indicate that the new high field superconducting phase, which is revealed by the recent heat capacity measurements, is characterized by the unusual softening of flux line lattice. The softening is most likely due to the collapse of the flux line lattice tilt modulus and transition to quasi-two-dimensional vortex state. These results provide a strong evidence that the high field phase is the Fulde–Ferrell–Larkin–Ovchinnikov phase, in which the order parameter is spatially modulated and has planar nodes aligned perpendicularly to the vortices. 2005 Elsevier B.V. All rights reserved. PACS: 74.25.Dw; 74.25.Jb; 74.25.Ld; 74.70.Tx Keywords: FFLO state; Flux line lattice; Ultrasound velocity; CeCoIn5
1. Introduction * Corresponding author. Tel.: +81 4 7136 3242; fax: +81 4 7136 3243. E-mail address: [email protected] (Y. Kasahara).
In spin-singlet superconductors, superconductivity is suppressed by the orbital effect (vortices) and by the Zeeman effect of the conduction
0921-4534/$ - see front matter 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.physc.2005.02.018
Y. Kasahara et al. / Physica C 426–431 (2005) 36–40
electron spins (Pauli paramagnetism). In most of the superconductors, the upper critical field Hc2 is determined by the orbital effect. In this case the transition at Hc2 is always second order. On the other hand, when the orbital effect is absent and Hc2 is determined by the Pauli effect, which can be realized in thin films in parallel field, the phase transition from superconducting (SC) state to normal state is known to become first order at Hc2 [1]. Moreover in such a case, the appearance of a new inhomogeneous SC state in the vicinity of Hc2 was proposed by Fulde and Ferrell and Larkin and Ovchinnikov (FFLO) [2–9]. In the FFLO state, a new pairing state with non-zero total momentum (k", k + q#) is formed in contrast to (k", k#)-pairing in ordinary superconductors. As a result, the SC order parameter exhibits spatial modulation with planar nodes which locate perpendicular to the applied field with a spatial modulation length 2p/jqj. Quasi-particles around these nodes develop a spin polarization in the direction of the applied magnetic field to reduce the Pauli paramagnetic pair breaking. In spite of enormous efforts to find the FFLO state, it has never been observed in conventional superconductors. Heavy fermion UPd2Al3 and intermediate valence compound CeRu2 have been suggested to have the FFLO state, but subsequent research has called the interpretation of the data for these materials in terms of an FFLO state into question [10–12]. Recently a new heavy fermion superconductor CeCoIn5 (Tc = 2.3 K) with quasi-two-dimensional (2D) electronic structure has been discovered [13]. The SC gap symmetry determined by the angle resolved thermal magnetotransport and heat capacity measurements [14–20] revealed that the symmetry of CeCoIn5 is most likely to be d-wave with line nodes perpendicular to the 2D planes. One of the most striking feature in the superconducting state of CeCoIn5 is that the first order transition occurs at Hc2 below T* Tc/2 both in Hkab and Hkc, indicating that the superconductivity is limited by the Pauli paramagnetism [17,21,22]. More excitingly, very recent heat capacity measurements revealed the presence of the second order transition in the vortex state, which branches from the first order transition line at
37
low temperature and high field, indicating the presence of a new SC phase [23,24]. It has been pointed out that this new SC phase may be the FFLO state. In fact, CeCoIn5 seems to satisfy several requirements for the formation of the FFLO state. For instance, it is in the very clean regime l n (l is quasi-particle mean free path, n is coherence length), and has very large Ginzburg–Landau parameter j = k/n 10 (k is penetration length). However it is premature to conclude the FFLO phase solely from the heat capacity measurements, since a possibility of some sort of magnetic transition cannot be ruled out. In order to establish the existence of the FFLO state, it is particularly important to elucidate the structure of flux line lattice (FLL) because it is predicted that FLL is divided into segments and becomes quasi-two-dimensional (Q2D) structure in FFLO state [5]. In this paper, we present the results of ultrasound velocity measurements which provide direct information on the FLL structure in CeCoIn5.
2. Experimental The ultrasound velocity measurements were performed on high quality single crystals of CeCoIn5 with tetragonal symmetry, grown by the self-flux method. We measured the relative change of the sound velocity by pulse echo method with a phase comparison technique. Ultrasonic waves with a frequency of 90 MHz were generated and detected by a pair of LiNbO3 transducers glued on the polished sample surfaces. The resolution of the relative velocity measurements was 1 part in 106. In all measurements, we measured the transverse sound velocity with propagation kk[1 0 0] and polarization uk[0 1 0], which corresponds to the shear modulus C c66 of the crystal lattice, and the magnetic field H was applied parallel to the plane (Hk[1 0 0] or [0 1 0]).
3. Results and discussion Fig. 1 displays the relative change of the transverse sound velocity Dv0t =v0t as a function of
38
Y. Kasahara et al. / Physica C 426–431 (2005) 36–40
Fig. 1. The relative change of the transverse sound velocity as a function of temperature in applied magnetic field 10.5 T in two different configurations; the Lorentz mode (H ? u) and the non-Lorentz mode (Hku).
temperature at 10.5 T in two different configurations with respect to the polarization vector u, the sound propagation vector k, and magnetic field H. We have performed the measurements in two different configurations; the Lorentz mode (Fig. 1, u ? H) where the sound wave couples to the vortices and the crystal lattice, and the non-Lorentz mode (Fig. 1, ukH) where the sound wave couples only to the crystal lattice. As shown in Fig. 1, the transverse sound velocity in the Lorentz mode is strongly enhanced compared to that in the non-Lorentz mode. This indicates that the transverse ultrasound strongly couples to the FLL in the Lorentz mode. In the Lorentz mode the ultrasound velocity increases rapidly with decreasing T, indicating the hardening of the FLL. In the Lorentz mode, the ultrasound shows a distinct kink at T*, whereas there is no anomaly in the non-Lorentz mode. This suggests that the anomaly purely originates from the FLL. Fig. 2 displays several curves of the relative shift of the FLL contribution to the transverse ultrasound velocity Dvt/vt below Hc2 as a function of temperature, which are obtained as the difference between Dv0t =v0t in the Lorentz mode and that in the non-Lorentz mode. As the temperature is low-
Fig. 2. The relative shift of the transverse sound velocity arising from the contribution of the FLL, Dvt/vt, as a function of temperature, below upper critical field Hc2. The solid and dashed lines indicate T* and Tc(H), respectively. For clarity, the zero levels of the different curves are vertically shifted.
ered below the SC transition temperature in magnetic field Tc(H), Dvt/vt starts to increase. At fields of 9.5 T and higher, Dvt/vt shows the anomaly, and reduces its value smaller than that extrapolated above T* while no anomaly is observed at 8.5 T. These results indicate a softening of the FLL below T*. Fig. 3 displays H–T phase diagram obtained by ultrasonic measurements. In this temperature range, the transition at Hc2 is of first order. T* coincide well with the reported second order transition line within SC phase [23]. On the basis of our results, we conclude that the second order phase transition within SC phase is characterized by the softening of the FLL [25]. We here discuss several possible origins of the softening of the FLL. We can rule out the possibility of the so-called peak effect because we have never observed the magnetization peak at T* [22]. In addition, peak effect should give rise to the hardening of the FLL, which is opposite to the present effect [26,27]. A lowering of the
Y. Kasahara et al. / Physica C 426–431 (2005) 36–40
39
nodes. The schematic figure of the structural transition of flux lines at T* is illustrated in Fig. 3. We next analyze the softening of the FLL quantitatively by using the collective pinning theory in type-II superconductors [30]. The pinning strength W in the single vortex limit for three dimensional superconductors is written as [31] 7=2
W ¼ ðU0 B3=2 j3c =4p2 l0 k2 Þ
1=2
ð1Þ
;
where U0 = h/2e and jc is the critical current density. Experimental value of the critical current density jc is obtained from magnetic field dependence of the FLL contribution to ultrasound velocity, which is written as [32] Fig. 3. Experimental H–T phase diagram for CeCoIn5 below 0.7 K for (Hk[1 0 0]). The transition temperature T*, indicated by open circles, were obtained from ultrasonic measurements as indicated by the arrows in Fig. 2. Solid circles and solid line depict the upper critical field Hc2 determined by the ultrasonic experiments. The dashed line is the second order transition line determined by the heat capacity reported in Ref. [14]. The region shown by dark gray is in the FFLO state. The schematic figures are sketches of structural transition of flux lines at T*.
ultrasound velocity occurs at melting transition of the FLL [28]. However we can rule out this scenario, because the entropy jump at T* reported in Refs. [23,24] is significantly larger than that expected in melting transition. Moreover, Dvt/vt should be abruptly to zero at T* because pinning vanishes at melting transition in contrast to our results. We can also rule out the possibility of the magnetic transition at T* because we observed no anomaly in non-Lorentz mode at T*. We point out that the observed phenomena is qualitatively consistent with the FFLO state in which the segmentation of the FLL occurs by the formation of the planar nodes perpendicular to the FLL [5]. In this sense the FFLO phase resembles to high-Tc and organic superconductors in which the flux line are divided into two dimensional pancake vortices and they are coupled by Josephson effect [29]. In the conventional flux line the ultrasound can propagate easily along the flux line. On the other hand, in FFLO state, the propagation of ultrasound is expected to be seriously prevented because the line tension of the flux line becomes very small at the position of the planar
Dvt ¼ ðB2 =l0 qvt Þð1=½1 þ fðU0 BÞ
1=2
2
jkj =l0 jc gÞ; ð2Þ
where q is mass density. As shown in Fig. 4, by fitting to Eq. (2), we obtain a field-independent critical current density, jc = 7.0 · 108 A m2, below approximately 9 T. Then, we deduce W above T*, 5 · 104 N2 m3 from Eq. (1). In the quasi-2D state of FLL, jc is given by the expression for 1=2 1=2 f a 2D layer of thickness d, j2D c66 d. c ¼ W =U0 B At the transition, j2D and W is must be equal to c the experimental values 7.0 · 108 A m2 and 5 · 104 N2 m3, respectively. We then obtain a layer of thickness d of 4 · 108 m. This value is close to the modulation length 2p/jqj = (me/m*)(hkF/ eB) = 3.5 · 108 m expected for the order parameter structure associated with the formation of
Fig. 4. The relative shift of the FLL contribution to the transverse ultrasound velocity Dvt/vt as a function of applied magnetic field at 0.2 K. Solid line is obtained by Eq. (2).
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the FFLO state (the Fermi wave vector kF 8 · 109 m1). Based on these results, we conclude that the elastic properties of the FLL in the new high field phase of CeCoIn5 is quantitatively consistent with that expected in the FFLO phase. We note that the very recent NMR measurements also support the FFLO phase [33].
4. Summary In summary, ultrasound velocity measurements of CeCoIn5 reveal the unusual softening of the FLL at the second order phase transition within the SC phase. The FLL softening is most likely characterized by the collapse of the FLL tilt modulus and transition to quasi-2D state. Our results provide a strong evidence that the high field phase is the FFLO phase, in which the order parameter is spatially modulated and has planar nodes aligned perpendicular to the vortices.
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