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Superconductivity in non-centrosymmetric materials
ARTICLE IN PRESS
Journal of Magnetism and Magnetic Materials 310 (2007) 536–540 www.elsevier.com/locate/jmmm
Superconductivity in non-centrosymmetric materials Manfred Sigrista,, D.F. Agterbergb, P.A. Frigeria, N. Hayashia, R.P. Kaurb, A. Kogac, I. Milata, K. Wakabayashia,d, Y. Yanasea a Theoretische Physik, ETH Zu¨rich, 8093 Zu¨rich, Switzerland Department of Physics, University of Wisconsin-Milwaukee, Milwaukee WI 53201, USA c Department of Applied Physics, Osaka University, Suita, Osaka 565-0871, Japan d Department of Quantum Matter, AdSM, Hiroshima University, Higashi-Hiroshima 739-8530, Japan b
Available online 10 November 2006
Abstract Superconductivity in non-centrosymmetric materials can display various intriguing properties. Using the example of the CePt3 Si a phenomenological description of this non-centrosymmetric superconductor is given, in an attempt to identify the symmetry of the Cooper pairing state. A short overview on other recently discovered non-centrosymmetric superconductors mainly, in strongly correlated electron systems, is given. r 2006 Elsevier B.V. All rights reserved. PACS: 74.20.z; 74.25.Ha; 74.70.b Keywords: Unconventional superconductivity; Spin–orbit coupling; Non-centrosymmetric materials
1. Basic aspects The formation of Cooper pairs in a superconductor relies on two essential symmetries: time-reversal and inversion symmetry. If both are present the complete set of possible Cooper pairs can be classified into even and odd-parity states which correspond to spin singlet and spin triplet configuration, respectively, due to the Pauli exclusion principle [1]. According to Anderson, missing time reversal symmetry is detrimental to the spin singlet pairing, while inversion is indispensable for spin triplet pairing. While the removal of time reversal symmetry is probed rather easily by applying a magnetic field, the equivalent possibility in case of inversion symmetry using an electric field is not straightforward, since the electric field is strongly screened in a metal. Thus, only if the crystal symmetry has no inversion center, we can study corresponding effects on bulk superconductors. Since the recent discovery of superconductivity in the heavy Fermion compound CePt3 Si [2,3] interest in this problem is growing Corresponding author. Tel.: +41 44 633 2584; fax: +41 44 633 1115.
E-mail address: [email protected] (M. Sigrist). 0304-8853/$ - see front matter r 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.jmmm.2006.10.141
and meanwhile a number of new interesting non-centrosymmetric superconductors have been identified, such as UIr, Li2 Pd3 B and Li2 Pt3 B, CeRhSi3 and possibly KOs2 O6 whose structure is still debated. Both the violation of time reversal and inversion symmetries lift the spin degeneracies in the electron bands. The most straightforward description is given by the following single-particle Hamiltonian [4–6]: n o X XX H¼ xk cyks cks þ kk cyks rss0 cks0 , (1) k;s
k
s;s0
where cks ðcyks Þ annihilates (creates) an electron with momentum k and spin s, xk is the bare kinetic energy measured relative to the chemical potential. The second term has the form of generalized Zeeman coupling with a momentum dependent ‘‘magnetic field’’ kk . Time reversal (inversion) symmetry is conserved, if kk ¼ kk ðkk ¼ kk Þ. Their influence on superconductivity can be probed in a perturbative approach assuming that the spin–orbit coupling is smaller than the characteristic energy scale (cutoff energy c ) of the pairing interaction. Then we find for the transition temperatures T c of spin-singlet ^ k ¼ cðkÞis^ y Þ and of spin-triplet pairing pairing states ðD
ARTICLE IN PRESS M. Sigrist et al. / Journal of Magnetism and Magnetic Materials 310 (2007) 536–540
^ s^ y Þ: states ðD^ k ¼ ðdðkÞ rÞi n oE X D Tc ln jcðkÞj2 f ðrnk Þ 1 þ nl^ k l^ k , ¼ k T c0 n¼þ; and n XD Tc f ðrnk Þ 2nðl^ k d ðkÞÞðl^ k dðkÞÞ ¼ ln T c0 n¼þ; oE þjdðkÞj2 ð1 nl^ k l^ k Þ
(2)
ð3Þ
k
with l^ k ¼ kk =jkk j; r k ¼ fjkk j jkk jg=2pkB T c in 1 X 1 1 f ðrÞ ¼ Re 2n 1 þ ir 2n 1 n¼1
which belong to the same irreducible representations of C 4v . These functions represent examples of basic functions of the irreducible representations. Since both the ‘‘even’’ and the ‘‘odd’’ parity forms are in the same representation, the gap function is in general a superposition of the two forms ^ k ¼ fD1 cðkÞ þ D2 dðkÞ rgi ^ s^ y . Note that of gap function: D ^ k , i.e., this leads in general to a non-unitary gap matrix D ^Dþ D^ k is not proportional to the 2 2-unit matrix. k The electronic spectrum of the Hamiltonian (1) gives two spin split bands with energies E k; ¼ xk jkk j.
(4)
and T c0 is the transition temperature, if both symmetries are conserved. If only one of the two symmetries is present, then r k ¼ 0. The absence of time reversal symmetry, e.g., the coupling of a finite magnetic field kk ¼ mB H ðkk ¼ kk Þ, leads immediately to the result, that T c of singlet pairing is suppressed, while T c of triplet pairing remains unaffected, if dðkÞ ? H for all k, i.e., equal-spin pairing with the spins parallel to H. On the other hand, the lack of inversion symmetry ðkk ¼ kk Þ, does not affect the singlet pairing phase, and among the spin triplet pairing states the one with dðkÞkkk has an unchanged T c . This clear results hold, if _hjkk jik 5c . The last result is of particular importance in view of superconductivity in materials without inversion center. 2. Symmetry aspects With the absence of inversion symmetry the classification of superconducting phases into even and odd-parity pairing phases becomes obsolete. Instead a classification not relying on parity is appropriate. As an example we consider CePt3 Si with the space group P4mm and the generating point group C 4v . This compound lakes the mirror symmetry z ! z. By symmetry arguments this leads to the basic form kk ¼ aðk z^Þ corresponding to a Rashba type of spin–orbit coupling (Table 1). It is easy to see that we can find pairing states represented by the scalar and the vector gap functions, cðkÞ and dðkÞ, respectively,
G
cðkÞ
dðkÞ
A1 A2
1 kx ky ðk2x k2y Þ
kk kx ky ðk2x k2y Þkk
B1
k2x k2y k x ky fkz kx ; kz ky g
ðk2x k2y Þkk k x k y kk fkz kx kk ; kz ky kk g
(5)
The quasiparticle gap is different on the two Fermi surfaces: jDk j ¼ D1 jcðkÞj D2 jdðkÞj. Thus, one of the two components appears with opposite sign on the two Fermi surfaces, leading to gaps of different size. 3. Symmetry of pairing state in CePt3 Si Several experiments provide information on the pairing symmetry of CePt3 Si. Power laws in the low-temperature behavior of the London penetration depth lðTÞ [7] and the heat conductance [8] are compatible with line nodes in the gap. On the other hand, 1=T 1 displays a Hebel–Slichter peak which suggests the presence of a finite coherence factor [10]. The latter can only be accounted for by the phase which belongs to the A1 -representation (Table 1) and provides a finite average hDk ik . This seems incompatible with the presence of line nodes. However, the structure of the gap function provides the possibility of accidental line nodes, if the spin triplet component is dominant [9]: D ðyk Þ ¼ D1 D2 sin yk
(6)
for yk as the angle between z^ and k (Fig. 1). Under this condition line nodes appear (horizontal to the basal plane for a spherical Fermi surface) and giving rise to the observed power laws [10,11]. While this is one possible explanation for the presence of line nodes, it was pointed out that line nodes can be induced by the antiferromagnetic
z
Table 1 Symmetry classification of possible ‘‘even’’ and ‘‘odd’’ parity states according to the irreducible representations of the point group C 4v relevant for CePt3 Si
B2 E
537
θ
z
θ
+
Fig. 1. Gap structure on the two Fermi surfaces leading to horizontal line nodes.
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order occurring in CePt3 Si with T N ¼ 2:2 K [12], or by special band structure effects [13].
4. Paramagnetic limiting and spin susceptibility The observation of the rather large upper critical field H c2 has motivated the study of the spin susceptibility for the possible superconducting phases. The comparison between the magnetic energy gain due to spin polarization in a magnetic field and the condensation energy can be used to estimate the so-called paramagnetic limiting field H p at which depairing would be caused by spin polarization. Naturally an equal-spin pairing state (spin triplet) with its spins parallel to the magnetic field would not be limited in this way at all. The analysis of the spin susceptibility leads to following approximative form in the case of strong spin–orbit coupling ðabkB T c Þ: D E o n wmn wP dmn l^ km l^ kn ½1 Y ðk; TÞ , (7) k
where Y ðk; TÞ is the momentum dependent Yosida function which changes monotonically from 1 at T ¼ T c to 0 for T ! 0 whereby its detailed form depends on the gap structure (wP : Pauli susceptibility) [14]. The basic behavior of wðTÞ is determined by kk for all states. For CePt3 Si we find that fields parallel to the z-axis leave the susceptibility unchanged in the superconducting state, while it drops to wP =2 for fields perpendicular to the z-axis. The estimate of H p , the paramagnetic limiting field, at T ¼ 0 by means of the energy argument leads to jDð0Þj mB H pn pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi . 2ð1 wnn ð0Þ=wP Þ
(8)
This suggests that for fields parallel to the z-axis no limiting exists, while for perpendicular fields an intermediate limiting is expected. Recent measurements of H c2 in single crystals of CePt3 Si display no limiting behavior for any field direction in contradiction with (7) [15]. In addition, NMR-Knight shift data show that for both z-axis and inplane fields the spin susceptibility does not change in the superconducting phase, consistent with the behavior of H c2 [16]. There are various proposals to reconcile the discrepancy between theoretical and experimental results. It was found that impurity scattering would give rise to a more isotropic spin susceptibility. However, in order to arrive at an isotropic susceptibility the disorder would have to be substantial [17]. Further possible mechanisms to cause an isotropically constant spin susceptibility are based on band structure and correlation effects [18], or on the presence of antiferromagnetism [13]. Finally the non-centrosymmetricity incorporates also the possibility to generate a so-called helical phase in a magnetic field, which we will consider here closer [19].
5. Helical phase A Zeeman-like field yields a shift of the Fermi surfaces, different for the two types of quasiparticles. Unlike in the usual case it generates a shift of the center of the two Fermi surfaces obtained in the non-centrosymmetric system modeled by (1). This is easily seen in the spin dependent part of (1): X ðakk mB HÞ cyks rss0 cks0 (9) k;s;s0
which leads to the electron spectrum: E k xk ajkk j kk mB H.
(10)
The center is shifted by q / ð^z HÞ for field components perpendicular to the z-axis. This is the origin of the depairing effect indicated by the drop of susceptibility above. The shifted Fermi surfaces do not provide the phase space to produce zero-momentum Cooper pairs as only few electron pairs with jksi and j ks0 i lie on the shifted Fermi surface. The superconductor can however escape this annoying situation by forming Cooper pairs with the finite total momentum q lying entirely on the Fermi surface: jk þ q; si and j k þ q; s0 i. This leads to a superconducting order parameter Z with a phase gradient: ZðrÞ ¼ f ðrÞeiqr
with
q / z^ H,
(11)
where f ðrÞ corresponds to the usual space dependent order parameter in the mixed phase [19,20]. This phase is called helical phase and is an analog to the Fulde–Ferrel state (and the related Larkin–Ovchinikov phase which has order parameter magnitude modulations). However, the mechanisms yielding these phases are different. We can also formulate this property within the Ginzburg–Landau theory introducing an additional term to the free energy expansion: F ¼ F GL þ gð^z BÞ Z DZ þ ðDZÞ Z (12) with D ¼ i_= þ ið2e=cÞA. This term corresponds to the direct coupling of the magnetic field and the supercurrent: ð^z BÞ J s . Note that despite the presence of a phase gradient the gauge invariance ensures that there is no net current flowing (apart from the vortex current of the mixed phase) [19]. It is obvious that the possibility to generate a helical phase would increase the spin susceptibility. It was shown also by Kaur et al. that the helical phase could account for the absence of paramagnetic limiting in the upper critical field for H ? z^ [19]. 6. Other non-centrosymmetric superconductors There is a large number of known non-centrosymmetric superconductors. Not all of them display extraordinary physical properties. After the discovery of CePt3 Si several other compounds have been found within the class of strongly correlated electron systems, such as the heavy
ARTICLE IN PRESS M. Sigrist et al. / Journal of Magnetism and Magnetic Materials 310 (2007) 536–540
Fermion superconductors UIr, CeRhSi3 and CeIrSi3 which show unusual behavior. In this context it is interesting to compare CePt3 Si with its sister compound LaPt3 Si which has the identical structure and is superconducting with T c ¼ 0:6 K, only slightly lower than CePt3 Si (T c ¼ 0:75 KÞ. Surprising this (non-heavy Fermion) compound shows conventional behavior in the superconducting phase. We may guess that LaPt3 Si is a conventional electron–phonon interaction mediated superconductor with dominant ‘‘spinsinglet’’ s-wave component. CePt3 Si, on the other hand, has strong magnetic fluctuations which likely stabilize a dominant ‘‘spin-triplet’’ p-wave component [13]. The fact that coherence length is much shorter in CePt3 Si due to the heavy mass of the quasiparticles makes it possible to investigate here the effect of paramagnetic limiting, while in LaPt3 Si the depairing corresponds to the standard orbitally driven mechanism. UIr: The heavy Fermion compound UIr is a ferromagnet without inversion center (monoclinic). The application of pressure leads to the gradual suppression of magnetic order, and eventually at the quantum critical point to the paramagnetic phase a ‘‘dome’’ of a superconducting phase appears with maximal T c 0:1 K [21,22]. Due to low transition temperature and the high pressure no detailed characterization of the superconducting phase has been possible so far. Nevertheless, the aspect that superconductivity may coexist here with a ferromagnetic phase may be very interesting in the context of non-centrosymmetricity. In particular, the ferromagnetism may induce a helical type of phase [23]. CeRhSi 3 and CeIrSi3 : These compounds are antiferromagnetic heavy Fermion systems [24,26]. Also here pressure is needed to suppress the Nee´l temperature and induce superconductivity. The crystal symmetries are similar to CePt3 Si giving rise to same Rashba-type of spin–orbit coupling. In both cases the upper critical field is remarkably high [24,26]. CeRhSi3 shows the anisotropy expected from our above spin susceptibility analysis [25]. Li2 Pd 3 B and Li2 Pt3 B: These two compounds have the same symmetry and both are superconducting [27]. Recently, it was shown that the superconducting phase of the two behave rather different. In the measurement of the London penetration depth Li2 Pd3 B was found to have a full gap, while Li2 Pt3 B behaves like a superconductor with line nodes [28]. This contrast was attributed to the different strength of the spin–orbit coupling as Pt is heavier than Pd. An interesting opportunity is opened by the fact that the alloy Li2 Pd3x Ptx B is apparently superconducting throughout the whole range of x and T c ðxÞ is continuous [29]. Therefore, it may possible to observe the evolution of the gap as Pd is replaced gradually by Pt. KOs2 O6 : The pyrochlor superconductor KOs2 O6 with the rather high T c ¼ 9:6 K [30] has been reported to have a non-centrosymmetric crystal structure [31]. Recent measurements of the upper critical field show
539
unusual temperature dependence and a very high zerotemperature value which probably exceeds the paramagnetic limiting [32,33]. 7. Conclusion In summary, superconductivity in non-centrosymmetric materials displays a number of novel properties, in particular, if it occurs in materials with strong electron correlations. An especially interesting aspect is the behavior in magnetic fields when spin–orbit coupling gives rise unusual response to Zeeman coupling. Other aspects such as the vortex physics [34], surface states [35] and Josephson effect [36] may contain novel intriguing features. It is conceivable that artificially structured superconductors (multi-layered systems) may be created which by design show some of these novel behaviors. Acknowledgments We are grateful to E. Bauer, I. Bonalde, S. Curnoe, S. Fujimoto, N. Kimura, V.P. Mineev, K. Samokhin, E.W. Scheidt, I. Sergenko, S.K. Yip and H.Q. Yuan for enlightening discussions. We are financially supported by the Swiss Nationalfonds (Nr. 200020-101726/200020109467) , the NCCR MaNEP, the Center for Theoretical Studies of ETH Zurich and the grant DMR-0318665 of the National Science Foundation. N. H. is supported by the Japan Society for the Promotion of Science and Y.Y. by the Nishina Memorial Foundation. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26]
M. Sigrist, K. Ueda, Rev. Mod. Phys. 63 (1991) 239. E. Bauer, et al., Phys. Rev. Lett. 92 (2004) 027003. E. Bauer, et al., Low Temp. Phys. 31 (2005) 748. L.P. Gor’kov, E.I. Rashba, Phys. Rev. Lett. 87 (2001) 037004. P.A. Frigeri, et al., Phys. Rev. Lett. 92 (2004) 097001. K.V. Samokhin, et al., Phys. Rev. B 69 (2004) 094514. I. Bonalde, et al., Phys. Rev. Lett. 94 (2005) 207002. K. Izawa, et al., Phys. Rev. Lett. 94 (2005) 197002. P.A. Frigeri, cond-mat/0505108. N. Hayashi, et al., Phys. Rev. B 73 (2006) 092508. N. Hayashi, et al., Phys. Rev. B 73 (2006) 024504. S. Fujimoto, J. Phys. Soc. Japan 75 (2006) 083704. Y. Yanase, in preparation. P.A. Frigeri, D.F. Agterberg, M. Sigrist, New. J. Phys. 6 (2004) 115. T. Yasuda, et al., J. Phys. Soc. Japan 73 (2004) 1657. M. Yogi, et al., J. Phys. Soc. Japan 75 (2006) 013709. P.A. Frigeri, Ph.D. Thesis (2006). S. Fujimoto, cond-mat/0605290. R.P. Kaur, D.F. Agterberg, M. Sigrist, Phys. Rev. Lett. 94 (2005) 137002. O.V. Dimitrova, M.V. Feigelman, JETP Lett. 78 (2003) 637. T. Akazawa, et al., J. Phys. Soc. Japan 73 (2004) 3129. T. Akazawa, et al., J. Phys. Condens. Matter 16 (2004) L29. M. Sigrist, News and Comments, J. Phys. Soc. Japan, November 2004. N. Kimura, et al, Phys. Rev. Lett. 95 (2005) 247004. N. Kimura, private communications. I. Sugitani, et al., J. Phys. Soc. Japan 75 (2006) 043703.
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K. Togano, et al., Phys. Rev. Lett. 93 (2004) 247004. H.Q. Yuan, et al., Phys. Rev. Lett. 97 (2006) 017006. P. Badica, T. Kondo, K. Togano, J. Phys. Soc. Japan 74 (2005) 1014. S. Yonezawa, Y. Muraoka, Z. Hiroi, J. Phys. Soc. Japan 73 (2004) 1655. [31] G. Schuck, et al., Phys. Rev. B 73 (2006) 144506.
[32] M. Bru¨hwiler, et al., Phys. Rev. B 73 (2006) 094518. [33] T. Shibauchi, et al., cond-mat/0603309. [34] Y. Nagai, Y. Kato, N. Hayashi, J. Phys. Soc. Japan 75 (2006) 043706. [35] U. May, Diploma Thesis, ETH, 2005. [36] A. Sugiyama, et al., J. Phys. Soc. Japan 74 (2005) 3041.
Journal of Magnetism and Magnetic Materials 310 (2007) 536–540 www.elsevier.com/locate/jmmm
Superconductivity in non-centrosymmetric materials Manfred Sigrista,, D.F. Agterbergb, P.A. Frigeria, N. Hayashia, R.P. Kaurb, A. Kogac, I. Milata, K. Wakabayashia,d, Y. Yanasea a Theoretische Physik, ETH Zu¨rich, 8093 Zu¨rich, Switzerland Department of Physics, University of Wisconsin-Milwaukee, Milwaukee WI 53201, USA c Department of Applied Physics, Osaka University, Suita, Osaka 565-0871, Japan d Department of Quantum Matter, AdSM, Hiroshima University, Higashi-Hiroshima 739-8530, Japan b
Available online 10 November 2006
Abstract Superconductivity in non-centrosymmetric materials can display various intriguing properties. Using the example of the CePt3 Si a phenomenological description of this non-centrosymmetric superconductor is given, in an attempt to identify the symmetry of the Cooper pairing state. A short overview on other recently discovered non-centrosymmetric superconductors mainly, in strongly correlated electron systems, is given. r 2006 Elsevier B.V. All rights reserved. PACS: 74.20.z; 74.25.Ha; 74.70.b Keywords: Unconventional superconductivity; Spin–orbit coupling; Non-centrosymmetric materials
1. Basic aspects The formation of Cooper pairs in a superconductor relies on two essential symmetries: time-reversal and inversion symmetry. If both are present the complete set of possible Cooper pairs can be classified into even and odd-parity states which correspond to spin singlet and spin triplet configuration, respectively, due to the Pauli exclusion principle [1]. According to Anderson, missing time reversal symmetry is detrimental to the spin singlet pairing, while inversion is indispensable for spin triplet pairing. While the removal of time reversal symmetry is probed rather easily by applying a magnetic field, the equivalent possibility in case of inversion symmetry using an electric field is not straightforward, since the electric field is strongly screened in a metal. Thus, only if the crystal symmetry has no inversion center, we can study corresponding effects on bulk superconductors. Since the recent discovery of superconductivity in the heavy Fermion compound CePt3 Si [2,3] interest in this problem is growing Corresponding author. Tel.: +41 44 633 2584; fax: +41 44 633 1115.
E-mail address: [email protected] (M. Sigrist). 0304-8853/$ - see front matter r 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.jmmm.2006.10.141
and meanwhile a number of new interesting non-centrosymmetric superconductors have been identified, such as UIr, Li2 Pd3 B and Li2 Pt3 B, CeRhSi3 and possibly KOs2 O6 whose structure is still debated. Both the violation of time reversal and inversion symmetries lift the spin degeneracies in the electron bands. The most straightforward description is given by the following single-particle Hamiltonian [4–6]: n o X XX H¼ xk cyks cks þ kk cyks rss0 cks0 , (1) k;s
k
s;s0
where cks ðcyks Þ annihilates (creates) an electron with momentum k and spin s, xk is the bare kinetic energy measured relative to the chemical potential. The second term has the form of generalized Zeeman coupling with a momentum dependent ‘‘magnetic field’’ kk . Time reversal (inversion) symmetry is conserved, if kk ¼ kk ðkk ¼ kk Þ. Their influence on superconductivity can be probed in a perturbative approach assuming that the spin–orbit coupling is smaller than the characteristic energy scale (cutoff energy c ) of the pairing interaction. Then we find for the transition temperatures T c of spin-singlet ^ k ¼ cðkÞis^ y Þ and of spin-triplet pairing pairing states ðD
ARTICLE IN PRESS M. Sigrist et al. / Journal of Magnetism and Magnetic Materials 310 (2007) 536–540
^ s^ y Þ: states ðD^ k ¼ ðdðkÞ rÞi n oE X D Tc ln jcðkÞj2 f ðrnk Þ 1 þ nl^ k l^ k , ¼ k T c0 n¼þ; and n XD Tc f ðrnk Þ 2nðl^ k d ðkÞÞðl^ k dðkÞÞ ¼ ln T c0 n¼þ; oE þjdðkÞj2 ð1 nl^ k l^ k Þ
(2)
ð3Þ
k
with l^ k ¼ kk =jkk j; r k ¼ fjkk j jkk jg=2pkB T c in 1 X 1 1 f ðrÞ ¼ Re 2n 1 þ ir 2n 1 n¼1
which belong to the same irreducible representations of C 4v . These functions represent examples of basic functions of the irreducible representations. Since both the ‘‘even’’ and the ‘‘odd’’ parity forms are in the same representation, the gap function is in general a superposition of the two forms ^ k ¼ fD1 cðkÞ þ D2 dðkÞ rgi ^ s^ y . Note that of gap function: D ^ k , i.e., this leads in general to a non-unitary gap matrix D ^Dþ D^ k is not proportional to the 2 2-unit matrix. k The electronic spectrum of the Hamiltonian (1) gives two spin split bands with energies E k; ¼ xk jkk j.
(4)
and T c0 is the transition temperature, if both symmetries are conserved. If only one of the two symmetries is present, then r k ¼ 0. The absence of time reversal symmetry, e.g., the coupling of a finite magnetic field kk ¼ mB H ðkk ¼ kk Þ, leads immediately to the result, that T c of singlet pairing is suppressed, while T c of triplet pairing remains unaffected, if dðkÞ ? H for all k, i.e., equal-spin pairing with the spins parallel to H. On the other hand, the lack of inversion symmetry ðkk ¼ kk Þ, does not affect the singlet pairing phase, and among the spin triplet pairing states the one with dðkÞkkk has an unchanged T c . This clear results hold, if _hjkk jik 5c . The last result is of particular importance in view of superconductivity in materials without inversion center. 2. Symmetry aspects With the absence of inversion symmetry the classification of superconducting phases into even and odd-parity pairing phases becomes obsolete. Instead a classification not relying on parity is appropriate. As an example we consider CePt3 Si with the space group P4mm and the generating point group C 4v . This compound lakes the mirror symmetry z ! z. By symmetry arguments this leads to the basic form kk ¼ aðk z^Þ corresponding to a Rashba type of spin–orbit coupling (Table 1). It is easy to see that we can find pairing states represented by the scalar and the vector gap functions, cðkÞ and dðkÞ, respectively,
G
cðkÞ
dðkÞ
A1 A2
1 kx ky ðk2x k2y Þ
kk kx ky ðk2x k2y Þkk
B1
k2x k2y k x ky fkz kx ; kz ky g
ðk2x k2y Þkk k x k y kk fkz kx kk ; kz ky kk g
(5)
The quasiparticle gap is different on the two Fermi surfaces: jDk j ¼ D1 jcðkÞj D2 jdðkÞj. Thus, one of the two components appears with opposite sign on the two Fermi surfaces, leading to gaps of different size. 3. Symmetry of pairing state in CePt3 Si Several experiments provide information on the pairing symmetry of CePt3 Si. Power laws in the low-temperature behavior of the London penetration depth lðTÞ [7] and the heat conductance [8] are compatible with line nodes in the gap. On the other hand, 1=T 1 displays a Hebel–Slichter peak which suggests the presence of a finite coherence factor [10]. The latter can only be accounted for by the phase which belongs to the A1 -representation (Table 1) and provides a finite average hDk ik . This seems incompatible with the presence of line nodes. However, the structure of the gap function provides the possibility of accidental line nodes, if the spin triplet component is dominant [9]: D ðyk Þ ¼ D1 D2 sin yk
(6)
for yk as the angle between z^ and k (Fig. 1). Under this condition line nodes appear (horizontal to the basal plane for a spherical Fermi surface) and giving rise to the observed power laws [10,11]. While this is one possible explanation for the presence of line nodes, it was pointed out that line nodes can be induced by the antiferromagnetic
z
Table 1 Symmetry classification of possible ‘‘even’’ and ‘‘odd’’ parity states according to the irreducible representations of the point group C 4v relevant for CePt3 Si
B2 E
537
θ
z
θ
+
Fig. 1. Gap structure on the two Fermi surfaces leading to horizontal line nodes.
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order occurring in CePt3 Si with T N ¼ 2:2 K [12], or by special band structure effects [13].
4. Paramagnetic limiting and spin susceptibility The observation of the rather large upper critical field H c2 has motivated the study of the spin susceptibility for the possible superconducting phases. The comparison between the magnetic energy gain due to spin polarization in a magnetic field and the condensation energy can be used to estimate the so-called paramagnetic limiting field H p at which depairing would be caused by spin polarization. Naturally an equal-spin pairing state (spin triplet) with its spins parallel to the magnetic field would not be limited in this way at all. The analysis of the spin susceptibility leads to following approximative form in the case of strong spin–orbit coupling ðabkB T c Þ: D E o n wmn wP dmn l^ km l^ kn ½1 Y ðk; TÞ , (7) k
where Y ðk; TÞ is the momentum dependent Yosida function which changes monotonically from 1 at T ¼ T c to 0 for T ! 0 whereby its detailed form depends on the gap structure (wP : Pauli susceptibility) [14]. The basic behavior of wðTÞ is determined by kk for all states. For CePt3 Si we find that fields parallel to the z-axis leave the susceptibility unchanged in the superconducting state, while it drops to wP =2 for fields perpendicular to the z-axis. The estimate of H p , the paramagnetic limiting field, at T ¼ 0 by means of the energy argument leads to jDð0Þj mB H pn pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi . 2ð1 wnn ð0Þ=wP Þ
(8)
This suggests that for fields parallel to the z-axis no limiting exists, while for perpendicular fields an intermediate limiting is expected. Recent measurements of H c2 in single crystals of CePt3 Si display no limiting behavior for any field direction in contradiction with (7) [15]. In addition, NMR-Knight shift data show that for both z-axis and inplane fields the spin susceptibility does not change in the superconducting phase, consistent with the behavior of H c2 [16]. There are various proposals to reconcile the discrepancy between theoretical and experimental results. It was found that impurity scattering would give rise to a more isotropic spin susceptibility. However, in order to arrive at an isotropic susceptibility the disorder would have to be substantial [17]. Further possible mechanisms to cause an isotropically constant spin susceptibility are based on band structure and correlation effects [18], or on the presence of antiferromagnetism [13]. Finally the non-centrosymmetricity incorporates also the possibility to generate a so-called helical phase in a magnetic field, which we will consider here closer [19].
5. Helical phase A Zeeman-like field yields a shift of the Fermi surfaces, different for the two types of quasiparticles. Unlike in the usual case it generates a shift of the center of the two Fermi surfaces obtained in the non-centrosymmetric system modeled by (1). This is easily seen in the spin dependent part of (1): X ðakk mB HÞ cyks rss0 cks0 (9) k;s;s0
which leads to the electron spectrum: E k xk ajkk j kk mB H.
(10)
The center is shifted by q / ð^z HÞ for field components perpendicular to the z-axis. This is the origin of the depairing effect indicated by the drop of susceptibility above. The shifted Fermi surfaces do not provide the phase space to produce zero-momentum Cooper pairs as only few electron pairs with jksi and j ks0 i lie on the shifted Fermi surface. The superconductor can however escape this annoying situation by forming Cooper pairs with the finite total momentum q lying entirely on the Fermi surface: jk þ q; si and j k þ q; s0 i. This leads to a superconducting order parameter Z with a phase gradient: ZðrÞ ¼ f ðrÞeiqr
with
q / z^ H,
(11)
where f ðrÞ corresponds to the usual space dependent order parameter in the mixed phase [19,20]. This phase is called helical phase and is an analog to the Fulde–Ferrel state (and the related Larkin–Ovchinikov phase which has order parameter magnitude modulations). However, the mechanisms yielding these phases are different. We can also formulate this property within the Ginzburg–Landau theory introducing an additional term to the free energy expansion: F ¼ F GL þ gð^z BÞ Z DZ þ ðDZÞ Z (12) with D ¼ i_= þ ið2e=cÞA. This term corresponds to the direct coupling of the magnetic field and the supercurrent: ð^z BÞ J s . Note that despite the presence of a phase gradient the gauge invariance ensures that there is no net current flowing (apart from the vortex current of the mixed phase) [19]. It is obvious that the possibility to generate a helical phase would increase the spin susceptibility. It was shown also by Kaur et al. that the helical phase could account for the absence of paramagnetic limiting in the upper critical field for H ? z^ [19]. 6. Other non-centrosymmetric superconductors There is a large number of known non-centrosymmetric superconductors. Not all of them display extraordinary physical properties. After the discovery of CePt3 Si several other compounds have been found within the class of strongly correlated electron systems, such as the heavy
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Fermion superconductors UIr, CeRhSi3 and CeIrSi3 which show unusual behavior. In this context it is interesting to compare CePt3 Si with its sister compound LaPt3 Si which has the identical structure and is superconducting with T c ¼ 0:6 K, only slightly lower than CePt3 Si (T c ¼ 0:75 KÞ. Surprising this (non-heavy Fermion) compound shows conventional behavior in the superconducting phase. We may guess that LaPt3 Si is a conventional electron–phonon interaction mediated superconductor with dominant ‘‘spinsinglet’’ s-wave component. CePt3 Si, on the other hand, has strong magnetic fluctuations which likely stabilize a dominant ‘‘spin-triplet’’ p-wave component [13]. The fact that coherence length is much shorter in CePt3 Si due to the heavy mass of the quasiparticles makes it possible to investigate here the effect of paramagnetic limiting, while in LaPt3 Si the depairing corresponds to the standard orbitally driven mechanism. UIr: The heavy Fermion compound UIr is a ferromagnet without inversion center (monoclinic). The application of pressure leads to the gradual suppression of magnetic order, and eventually at the quantum critical point to the paramagnetic phase a ‘‘dome’’ of a superconducting phase appears with maximal T c 0:1 K [21,22]. Due to low transition temperature and the high pressure no detailed characterization of the superconducting phase has been possible so far. Nevertheless, the aspect that superconductivity may coexist here with a ferromagnetic phase may be very interesting in the context of non-centrosymmetricity. In particular, the ferromagnetism may induce a helical type of phase [23]. CeRhSi 3 and CeIrSi3 : These compounds are antiferromagnetic heavy Fermion systems [24,26]. Also here pressure is needed to suppress the Nee´l temperature and induce superconductivity. The crystal symmetries are similar to CePt3 Si giving rise to same Rashba-type of spin–orbit coupling. In both cases the upper critical field is remarkably high [24,26]. CeRhSi3 shows the anisotropy expected from our above spin susceptibility analysis [25]. Li2 Pd 3 B and Li2 Pt3 B: These two compounds have the same symmetry and both are superconducting [27]. Recently, it was shown that the superconducting phase of the two behave rather different. In the measurement of the London penetration depth Li2 Pd3 B was found to have a full gap, while Li2 Pt3 B behaves like a superconductor with line nodes [28]. This contrast was attributed to the different strength of the spin–orbit coupling as Pt is heavier than Pd. An interesting opportunity is opened by the fact that the alloy Li2 Pd3x Ptx B is apparently superconducting throughout the whole range of x and T c ðxÞ is continuous [29]. Therefore, it may possible to observe the evolution of the gap as Pd is replaced gradually by Pt. KOs2 O6 : The pyrochlor superconductor KOs2 O6 with the rather high T c ¼ 9:6 K [30] has been reported to have a non-centrosymmetric crystal structure [31]. Recent measurements of the upper critical field show
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unusual temperature dependence and a very high zerotemperature value which probably exceeds the paramagnetic limiting [32,33]. 7. Conclusion In summary, superconductivity in non-centrosymmetric materials displays a number of novel properties, in particular, if it occurs in materials with strong electron correlations. An especially interesting aspect is the behavior in magnetic fields when spin–orbit coupling gives rise unusual response to Zeeman coupling. Other aspects such as the vortex physics [34], surface states [35] and Josephson effect [36] may contain novel intriguing features. It is conceivable that artificially structured superconductors (multi-layered systems) may be created which by design show some of these novel behaviors. Acknowledgments We are grateful to E. Bauer, I. Bonalde, S. Curnoe, S. Fujimoto, N. Kimura, V.P. Mineev, K. Samokhin, E.W. Scheidt, I. Sergenko, S.K. Yip and H.Q. Yuan for enlightening discussions. We are financially supported by the Swiss Nationalfonds (Nr. 200020-101726/200020109467) , the NCCR MaNEP, the Center for Theoretical Studies of ETH Zurich and the grant DMR-0318665 of the National Science Foundation. N. H. is supported by the Japan Society for the Promotion of Science and Y.Y. by the Nishina Memorial Foundation. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26]
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