J Supercond Nov Magn (2012) 25:1283–1287 DOI 10.1007/s10948-012-1595-0
ORIGINAL PAPER
Field-Direction Dependence of the Upper Critical Field in Organic Superconductors
M.D. Croitoru · M. Houzet · A.I. Buzdin
Received: 6 April 2012 / Accepted: 11 April 2012 / Published online: 10 May 2012 © Springer Science+Business Media, LLC 2012
Abstract In this work, we investigate the anisotropy of and singlet pairing is mainly limited by the Zeeman effect. the in-plane critical field in conventional and spatially Nevertheless, this limitation can be surmounted by allow- modulated phases of layered superconductors within the ing a spin-triplet state or a spatially modulated spin-singlet- quasi-classical approach, taking into account the interlayer state (limit is upshifted) [15, 16]. Whether a spin part of Josephson coupling. We show that the anisotropy of the on- the superconducting order parameter singlet or triplet is cur- set of superconductivity may change dramatically in the rently under discussion because the NMR experiment with FFLO state as compared with the conventional supercon- (TMTSF)2PF6 salt at Tc and under pressure showed the ab- ducting phase. sence of the Knight shift, supporting the triplet scenario of pairing [17], while the 77Se NMR Knight shift in recent ex- Keywords FFLO state · Anisotropy · Critical field periment with (TMTSF)2ClO4 reveals a decrease in spin susceptibility χs consistent with singlet spin pairing [18]. 13 C NMR measurements with κ-(BEDT-TTF)2Cu(NCS)4 In the last decades, the quasi low-dimensional correlated evidenced for a Zeeman-driven transition within the super- electron systems have been the focus of the theoretical conducting state and stabilization of inhomogeneous, spa- and experimental investigations due to their rich variety of tially modulated phase [19]. normal and superconducting properties not found in three- Recently the field-amplitude and field-angle dependence dimensional materials [1–8]. Amongst them, layered super- of the superconducting transition temperature Tc(H ) of the conductors subjected to an external magnetic field have at- organic superconductor (TMTSF)2ClO4 in magnetic field tracted much attention [1, 9]. Indeed, in organic layered su- applied along the conduction planes have been reported [14]. perconductors, (TMTSF)2X, where anion X is PF6,ClO4, The authors observed an upturn of the curve of the up- etc., a very large upper critical fields, exceeding the Pauli per critical field at low temperatures. This behavior has of- paramagnetic limit, for a magnetic field aligned parallel to ten been discussed in connection with the possibility of the their conducting layers were reported [10–14]. The reason is FFLO state formation [20–23]. Moreover, an unusual in- that, in layered conductors, the spatial orbital motion of elec- plane anisotropy of Hc2 in the high-field regime was ob- trons is mostly restricted to the conducting layers and the or- served, which authors interpret as an evidence of the FFLO bital depairing due to the in-plane magnetic field is strongly state formation. Motivated by the experimental findings, in quenched. Hence, the magnetic field only influences spins this work we investigate the influence of the modulating su- perconducting phase on the in-plane anisotropy of Hc2 in layered conductors. M.D. Croitoru () · A.I. Buzdin We consider here a quasi-two-dimensional superconduc- Université Bordeaux I, LOMA, UMR 5798, 33405 Talence, tor in magnetic field applied parallel to the plane of the most France e-mail: [email protected] conductive layers (xy-plane). The system single-electron (2) spectrum is approximated by ξp = Ep + ε(pz), where M. Houzet (2) px py Ep = + , and the tight-binding approximation is DRFMC/SPSMS/LCP, CEA, Grenoble, 38054 Grenoble Cedex 9, 2mx 2my France used to describe the electron motion along the z-direction: 1284 J Supercond Nov Magn (2012) 25:1283–1287
ε(pz) = 2t cos(pzd) [24]. The coupling between layers, t, Here, one can see that T>Tct , where Tct is introduced via 2 7ζ(3) t2 Tc0 the relation ln(T /T ) = . satisfies the condition t T .HereT is the criti- c0 ct 4π2 T 2 EF c0 c0 c0 cal temperature of the system at H = 0 with neglected inter- The previous treatment can be extended to extremely layer coupling and EF is the Fermi energy. The Eilenberger large magnetic fields, provided that one can omit the as- equation describing such systems acquires the form [25] sumption that the characteristic scale of the order parameter variations is much smaller than ξ . For this purpose, we need L f (n , r,p ) = Δg (n , r,p ), (1) 0 2 ω p z ω p z to reconsider the solution of the Eilenberger equation. Ac- where gω and fω are the quasiclassical Green functions, op- cording to the 2D Eilenberger equation, the magnetic field = + 1 ∇ + [ iQr − −iQr] erator L2 Ωn 2 vF x,y t sin(pzd) e e with induced potential has the form V(r) = 2it sin(pzd)sin(Qr), πd Ωn ≡ ωn − ih and Q = (−Hy,Hx) with φ = π c/e. i.e., it is periodic in real space. Since the system under con- φ0 0 Here h ≡ μB H is the Zeeman energy, vF is the 2D Fermi sideration is in an arbitrary magnetic field, the solution of velocity and d is the interlayer distance. In the following, Eq. (1) can be written without any loss of generality as a we assume that the system is near the transition, therefore Bloch function g (n , r,p ) sign(ω ), and that the mean free path inside ω p z n f (n , r,p ) = eiqr eimQrf (ω , n ,p ), (5) the layers are much larger than the corresponding coherence ω p z m n p z m length. = iqr Restricting ourselves first to the case when paramagnetic or fω(np, r,pz) e fω(np, r,pz), where fω(np, r,pz) limit (FFLO state) is absent, performing averaging over the has the periodicity of 2π/Q.InEq.(5), we take into account + q ↑; the possibility for the formation of the pairing state (k 2 Fermi surface, and using the iterative procedure with re- q −k + ↓) with the modulation vector q ∼ 2μB H/ vF .Be- spect to fω(np, r,pz) we obtain the extended Lowerence– 2 Doniach equation (in the sequel, MLD equation) cause of the form for fω(np, r,pz) of Eq. (5), one can write Δ(r) as T = iqr i2mQr Δ(r) ln πT LMLD(ωn)Δ(r) (2) Δ(r) = e e Δ2m. (6) Tc0 n m ≡ 3 { 2 2 +2 2}− = with LMLD(ωn) 1/(4ωn) vFx ∂x vFy ∂y From symmetry considerations, it follows that Δ−2m Δ2m. 2[ − ] 3 +[ 2 2 + 2 2]× 2[ − Substituting Eqs. (5) and (6) into Eq. (1) and retaining in the t 1 cos(2Qr) /ωn vFx Qx vFy Qy t 1 7 cos(2Qr)]/4ω5. In order to obtain the iterative scheme sum only the zero and the first term, since we want to inves- n Tc0 tigate the regime H φ0 (Tc0 vF Q), when q = 0 convergent, we have required Δ(r) vF ∇x,yΔ(r)/2πTc0, πdvF one gets after averaging over the Fermi surface which implies that the characteristic scale of the order 2 −1/2 parameter variations should be much smaller than ξ0 = T t2 v Q ln c0 = πT ω2 + F . (7) vF /2πTc0. The MLD equation contains the term propor- T ω2 n 2 2 2 n n tional to (vF Q) t , which is absent in the Lowerence– Doniach equation. In the case of v Q T , one may For extremely large values of the external magnetic field, F c0 ∼ neglect the last term and obtain the standard Lowerence– we can use ωn Tc0 vF Q and obtain Eq. (3) with 2 π t φ0 Hc ( )|κ = . Therefore, summarizing the Doniach equation. 2 2 IV 2dvF (Tc0−T) The angle resolved highest magnetic field at which su- above discussion, we can conclude that in the non-modulated perconductivity can nucleate in a sample is obtained as phase the in-plane anisotropy of the upper critical magnetic π | π | field can be defined by Eq. (3) and Hc2( 2 ) lim should be Hc2( 2 ) lim Hc2(α, T )|lim = , (3) taken accordingly, as described in Table 1. sin2(α) + mx cos2(α) my As it is known [23, 28], the FFLO state only appears at T
Table 1 The in-plane upper π Regime Limits of validity Hc2( )|lim critical magnetic field Hc2(π/2) 2
φ0 mx vF Tc0 φ0 − T I H πdv t dt 2π (1 T ) F c0 √ 2 φ0 φ0 7ζ(3)t mx II t H tTc0 3 − φ0 πdvF πdvF 8π dTc0 T Tct √ √ √ − III φ0 tT H φ0 T 28 ζ(3) Tct T Tct 2mx φ πdvF c0 πdvF c0 31 ζ(5) dt 0 2 φ0 t φ0 IV H Tc πdvF 0 2dvF (Tc0−T)
where the following notation is introduced: Ln(q) = ωn + + vF q ihsign(ωn) i 2 . We assume that the optimum direction of the FFLO modulation vector is fixed by the symmetry of the Fermi surface and is solely determined in the paramag- netic limit [29, 30]. The magnitude of the FFLO modulation vector is defined according to [31]. Having in our disposal the absolute value and the direction of the FFLO modulation vector, we can calculate the contribution of the orbital ef- fect. Introducing the temperature of the Pauli limited case by ln( Tc0 ) = πT { 1 − 1 }, we can write the equa- TcP cP n ωn Ln(q) tion for the onset of the superconductivity as T ln cP = AπT T (ω, q, +Q) + T (ω, q, −Q), (9) T n n n
± = t2 2π dϕ −1 ± −2 where Tn(ω, q, Q) 2 0 2π Ln (q Q)Ln (q), and A is a function taken at TcP (H ) and it will be displayed else- where [32]. This equation determines H–T phase diagram for layered superconductors in the presence of the param- 2 agnetic and the orbital effects in the limits: tTc0 (vF Q) and t Tc0. The summation over the Matsubara frequencies of the orbital terms were performed numerically. In the sum over the Matsubara frequencies, N = 104 terms were used. This number suffices for the convergence. The results presented in this section are in the dimensionless units. Let us start with a far-fetched situation and investigate the anisotropy of the upper critical field in the absence of the FFLO modu- lation vector for H>H∗. The correction to the pure Pauli limited (PL) H –T phase diagram due to the orbital effects is then described by 2 + 2 − 2 TcP 2 ωn(a ωn 3h ) Fig. 1 The in-plane field-direction dependence of Tc/TcP at ln = AπT t , = = = T (ω2 + h2)z H/Hc0 0.96 (red line)andH/Hc0 0.9(green line)formx 5my , n n q = 0(upper panel), and q = 0(lower panel), respectively. The Fermi 4 2 2 2 2 2 2 velocity parameter η = 3.45, and the q-vector is along the x-axis with z ≡ a + 2a (ω − h ) + (ω + h ) and a = vF Q/2. n n (Color figure online) At low temperature, the sum in r.h.s can become neg- ative. However, coefficient A is also negative, meaning = × 7 that TcP >T. This is seen in the polar plot in the upper velocity vF 1.25 10 cm/s, or introducing dimension- panel in Fig. 1, which illustrates the magnetic field an- less velocity parameter η = vF πd/φ0μB ,forη = 3.45, gular dependence of the normalized onset temperature in and for the in-plane mass-anisotropic situation, when mx = the non-modulated superconducting phase, Tc/TcP , taken 5my . Hereinafter, t/Tc0 = 0.2. In the polar plot, the direc- at H/Hc0 = 0.96 and H/Hc0 = 0.9, where Hc0 is the tion of each point seen from the origin corresponds to the critical magnetic field at T = 0 in Pauli limited 2D- field direction and the distance from the origin illustrates to superconductors. Calculations were performed for the Fermi the normalized critical temperature. In the case of uniform 1286 J Supercond Nov Magn (2012) 25:1283–1287
anisotropic. Namely, the position of the maximum of the on- set temperature is rotated from α = 0◦, and 180◦ typical for ◦ ◦ q = 0, to α =±70 , ±110 for q = 0 and H/Hc0 = 0.9. ◦ The minimum in Tc(α) is rotated from α =±90 for q = 0 ◦ ◦ to α =±50 , ±130 for q = 0 and H/Hc0 = 0.9. Let us now investigate the influence of the absolute value of the Fermi velocity on the in-plane anisotropy of the up- per critical field in layered superconductors. The upper and lower panels of Fig. 2 display Tc(α) for the Fermi velocity parameter η = 0.31 and η = 0.63, respectively. One can im- mediately infer that absolute value of the Fermi velocity pa- rameter essentially influences the field-direction dependence of the superconducting onset temperature. It is seen that the position of the maximum value of Tc(α) in the case of the smallest considered Fermi velocity parameter is shifted back towards α =±90◦. The same tendency is observed for η>4 (not shown here). In conclusion, in this work we have analyzed the field- amplitude and the field-direction dependence of the onset temperature of the superconductivity in layered conductors. In such superconductors, the interplay of the system dimen- sionality, orbital pair-breaking, Pauli paramagnetism, and a possible formation of the FFLO state can lead to an anoma- lous in-plane anisotropy of the upper critical field.
Acknowledgements We acknowledge the support by the European Community under a Marie Curie IEF Action (Grant Agreement No. PIEF-GA-2009-235486-ScQSR), the French Project “SINUS” ANR- 09-BLAN-0146, and thank A. S. Mel’nikov for fruitful discussions.
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