Upper critical field Hc2 in Bechgaard salts (TMTSF)2PF6
© Folgueras et al 2008
Received: 30 November 2007
Accepted: 09 December 2008
Published: 09 December 2008
The symmetry of the superconductivity in Bechgaard salts is still unknown, though the triplet pairing has been established by Hc2 and NMR for (TMTSF)2PF6. The large upper critical field at T = 0K (Hc2 ~ 5 Tesla) both for and also indicates strongly the triplet pairing.
Here we start with a low energy effective Hamiltonian and study the temperature dependence of the corresponding Hc2(T)'s.
The present analysis suggests that one chiral f-wave superconductor should be the most likely candidate near the upper critical field.
PACS Codes: 74.70.Kn ; 74.20.Rp; 74.25.Op.
The Bechgaard salt (TMTSF)2 PF6 is the first organic superconductor discovered in 1980 . Until very recently the superconductivity was believed to be conventional s-wave . More recently the symmetry of the superconductivity has become one of the central issues . The upper critical field at T = 0K in (TMTSF)2PF6 and (TMTSF)2ClO4 are clearly beyond the Pauli limit [4–7], suggesting triplet pairing. Recent NMR data [8, 9] from (TMTSF)2PF6 supports triplet superconductivity.
Here we shall first derive Hc2(T) for a variety of p-wave and f-wave superconductors . Later, we will discuss the relation between the nuclear spin relaxation rate and the nodal lines.
with v : v b : v c ~ 1 : 1/10 : 1/300 and v = v a , v b = and v c = ; for example, P. M. Grant  gives v c ~ 1 meV, t b ~ 26.2 meV and t a ~ 365 meV.
There are earlier analysis of Hc2 of Bechgaard salts starting from the one dimensional models [13, 14]. However, those models predict diverging Hc2(T) for T → 0K or the reentrance behaviour, which have not been observed in the experiments [4, 5]. The one dimensional model, like the one proposed by Lebed [13, 15] is valid only when 2t c < 2.14T c ~ 3 K, in Bechgaard salts it is believed that the transfer integral in the c direction is 2t c ~ 10 – 30 K while the superconducting transition temperature is T c ~ 1.2 K, so the 1D model is unrealistic. Also, the quasilinear T dependence of Hc2(T) in both (TMTSF)2PF6 and (TMTSF)2ClO4 is very unusual.
We consider a 3D model, though strongly anisotropic. We start with a continuum model, where the cristal anisotropy is incorporated only through the great anisotropies of the Fermi velocities. We have considered chiral superconductors because these symmetries have been shown to lead to higher Hc2s. In the absence of an applied magnetic field, we could obtain one of those chiral states as a combination of two different order parameters (with two different transition temperatures), but the external magnetic field breaks the time reversal symmetry, allowing the formation of a chiral state in the superconducting phase (see ).
Among the symmetries we have considered, the chiral f'-wave superconductor with , looks most promising, where χ1 = and χ2 = are and the cristal vectors.
Moreover, if the superconductor belongs to one of the nodal superconductors [17, 18] and if nodes lay parallel to within the two sheets of the Fermi surface, the angle dependent nuclear spin relaxation rate in a magnetic field rotated within the b' - c* plane will tell the nodal directions.
1 Results and discussion
Upper critical field for
In the following we neglect the spin component of . Most likely the equal spin pairing is realised in Bechgaard salts as in Sr2RuO4 . In this case the spin component is characterized by a unit vector . Also is most likely oriented parallel to . Let's assume , though Hc2(T) is independent of as long as the spin orbit interaction is negligible. Experimental data from both UPt3 and Sr2RuO4 indicate that the spin-orbit interactions in these systems are not negligible but extremely small . We consider a variety of triplet superconductors (see some of them in Fig. 1), most of them chiral variants, as we find in general that the chiral variant has larger Hc2 than the non-chiral one:
Simple p-wave SC:
and , and ⟨...⟩ means average over χ2. Here v a , v c are the Fermi velocities parallel to the a axis and the c axis respectively.
where is the Abrikosov state , C n the occupancy of the n th Landau level (we assume there is only one occupied Landau level) and is the raising operator.
Then in the vicinity of t → 1 we find and .
Chiral p-wave SC:
Here is the analogue of e ιϕ if in the 3D systems in the quasi 1D system.
and the same expressions for t, ρ,...
For t → 1 we find and ρ = 0.3838(-ln t).
On the other hand, for t → 0 we obtain C = -0.3660 and ρ0 = 0.27343.
Chiral f-wave SC:
Here now means the average over both χ1 and χ2. As in previous sections, (s depends on the direction of the magnetic field). Then it is easy to see that the chiral f-wave SC has the same Hc2(t) and C(t) as the chiral p-wave SC, since the variable χ1 is readily integrated out.
Chiral f'-wave SC:
Now we have a set of equations similar to the chiral f-wave except (1 + cos 2χ1) in both eqs. 13 has to be replaced by (1 + cos 2χ1). We obtain, for t → 1, C = -0.2247 and ρ = 0.5181(-ln t). On the other hand, for t → 0 we find C = -0.3660 and ρ0 = 0.3734.
Note that C(t) is the same for three chiral states (chiral p-wave, chiral f-wave and chiral f'-wave) as well as chiral p-wave studied in .
Therefore for the magnetic field , the chiral f'-wave have the largest Hc2(t) if we assume T c and v, v c are the same. Also Hc2(t) of these states are closest to the observation.
Upper critical field for
In this section, we assume the applied magnetic field runs parallel to the direction defined by . We calculate the upper critical field in these circunstances for different symmetries of the order parameter, following the same procedure as the one we used in previous section.
Simple p-wave SC:
where and s = (sin χ1 + ι sin χ2) with χ1 = and χ2 = .
Then for t → 1, we find and . While for t → 0 and , where α0 = -⟨ln|s|2⟩ = 0.220051 and . From these we obtain h(0) = 0.73673.
Chiral p-wave SC:
Now Hc2(t) is determined by a similar set of equations as Ec. 10–11. Now, s = (sin χ1 + ι sin χ2). In particular we find for t → 1 C = -0.027735 and ρ = 0.212598(ln t) while for t → 0 C = -0.067684 and ρ0 = 0.139672. We obtain h(0) = 0.6566. We show h(t) and C(t) in Fig. 3a) and 3b) respectively.
Chiral f-wave SC:
Again we use a similar set of equations as Ec. 12–13, with s = (sin χ1 + ι sin χ2), we find for t → 1 C = -0.0356236 and ρ = 0.2744495(ln t) while for t → 0 C = 0.066 and ρ0 = 0.1920 and h(0) = 0.6997. Both h(t) and C(t) are evaluated numerically and shown in Fig. 3a) and 3b).
Chiral f'-wave SC:
Now we find for t → 1 C = -0.05 and ρ = -0.2910(ln t), while for t → 0 C = -0.1019 and ρ0 = 0.2090.
Summary of results. Here and
Nodal lines in Δ()
We have seen that from the temperature dependence of Hc2(T), we can deduce the chiral f-wave and chiral f'-wave superconductors are the most favourable candidates. They have nodal lines on the Fermi surface (i.e. the χ1 - χ2 plane), the chiral f-wave SC at χ1 = , while chiral f'-wave SC at χ2 = .
These nodal lines may be detected if the nuclear spin relaxation rate is measured in a magnetic field rotated within the b' - c* plane.
for the chiral f-wave SC.
a) Assuming all these superconductors have the same T c , the chiral f'-wave SC appears to be the most favourable with largest Hc2's for both and .
b) However, non of these states exhibit the quasilinear temperature dependence of Hc2(T) as observed in .
c) Also the present theory predicts Hc2(0) ~ (v a v c )-1 and (v b v c )-1 for and respectively. This means Hc2(0) for is about 5 time larger than the one for contrary to observation.
d) From Hc2(0) ~ 5T and T c = 1.5 K we can extract v2 = ~ 1.5104 cm s-1, consistent with the known values of v a , v c .
We have also shown that the nodal lines should be visible through the angle dependent in NMR with the magnetic field rotating in the c*-b' plane.
We thank S. Brown, P. Chaikin, S. Haas and H. Won for useful discussion. ADF also acknowledges gratefully the discussion with J. Ferrer and F. Guinea. The authors would also like to aknowledge the useful comments of the reviewers during the correction process.
- Jerome D, Mazard A, Ribault M, Bechgaard K: J Phys Lett (Paris). 1980, 47: L95-View ArticleGoogle Scholar
- Ishiguro T, Yamaji K, Saito G: Organic Superconductors. 1998, Berlin: Springer-VerlagView ArticleGoogle Scholar
- Maki K, Haas S, Parker D, Won H: Chinease J Phys. 2005, 43: 532-ADSGoogle Scholar
- Lee IJ, Chaikin PM, Naughton MJ: Phys Rev B. 2002, 65: R180502-10.1103/PhysRevB.65.180502.View ArticleADSGoogle Scholar
- Oh JI, Naughton MJ: Phys Rev Lett. 2004, 92: 067001-10.1103/PhysRevLett.92.067001.View ArticleADSGoogle Scholar
- Clogston AM: Phys Rev Lett. 1962, 9: 266-10.1103/PhysRevLett.9.266.View ArticleADSGoogle Scholar
- Chandrasekhar BS: Appl Phys Lett. 1962, 1: 7-10.1063/1.1777362.View ArticleADSGoogle Scholar
- Lee IJ, Brown SE, Clark WG, Strouse MJ, Naughton MJ, Kang W, Chaikin PM: Phys Rev Lett. 2002, 88: 017004-10.1103/PhysRevLett.88.017004.View ArticleADSGoogle Scholar
- Lee IJ, Brown S, Naughton MJ: J Phys Soc Jpn. 2006, 75: 051011-10.1143/JPSJ.75.051011.View ArticleADSGoogle Scholar
- Gor'kov LP: Soviet Phys JETP. 1960, 10: 59-Google Scholar
- Luk'yanchuk I, Mineev VP: Soviet Phys JETP. 1987, 66: 1168-Google Scholar
- Grant PM: J Phys. 1983, 44 C3: 847-Google Scholar
- Lebed AG: JETP Lett. 1986, 44: 114-ADSGoogle Scholar
- Dupuis N, Montambaux G, Sá de Mello EAR: Phys Rev Lett. 1993, 70: 2613-10.1103/PhysRevLett.70.2613.View ArticleADSGoogle Scholar
- Kuroki K, Arita R, Aoki H: Phys Rev B. 2001, 63: 094509-10.1103/PhysRevB.63.094509.View ArticleADSGoogle Scholar
- Sigrist M, Ueda K: Rev Mod Phys. 1991, 63: 239-10.1103/RevModPhys.63.239.View ArticleADSGoogle Scholar
- Won H, Maki K: Europhys Lett. 1995, 30: 421-10.1209/0295-5075/30/7/008.View ArticleADSGoogle Scholar
- Won H, Maki K: Phys Rev B. 1996, 53: 5927-10.1103/PhysRevB.53.5927.View ArticleADSGoogle Scholar
- Abrikosov AA: Soviet Phys JETP. 1957, 5: 1174-Google Scholar
- Wang GF, Maki K: Europhysic Lett. 1999, 45: 71-10.1209/epl/i1999-00133-6.View ArticleADSGoogle Scholar
- Wong H, Haas S, Parker D, Telang S, Vanyolos A, Maki K: BCS theory of nodal superconductors. Lectures on the Physics of Highly Correlated Electron Systems IX. Edited by: Avella A, Mancini F. 2005, AIPGoogle Scholar
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