Has the FFLO state been observed in the organic superconductor $\kappa-$(BEDT-TTF$)_2$Cu(NCS$)_2$ ?

Abstract

We compare the theoretical anisotropic upper critical field $H_{C}(\Theta,T)$ of a quasi-two-dimensional d-wave superconductor with recent $H_{c2}$ data for the layered organic superconductor $\kappa-(BEDT-TTF)_2Cu(NCS)_2$. We find agreement both with regard to the angular and the temperature dependence of $H_{C}$. This supports the suggestion that the Fulde-Ferrell-Larkin-Ovchinnikov state (FFLO state) exists in this material for exactly plane-parallel orientation of the magnetic field. Indications of precursor states, occurring for small deviations from the plane-parallel field direction, are also pointed out and further measurements for confirming the existence of the FFLO state are proposed.

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arXiv:cond-mat/0006327v1 [cond-mat.supr-con] 21 Jun 2000
Has the FFLO state been observed in the organic
superconductor κ−(BEDT-TTF)2Cu(NCS)2?
S Manalo and U Klein§
Johannes Kepler Universit¨ at Linz, Institut f¨ ur Theoretische Physik, A-4040 Linz,
Austria
Abstract.
We compare the theoretical anisotropic upper critical field HC(Θ,T) of
a quasi-two-dimensional d-wave superconductor with recent Hc2data for the layered
organic superconductor κ − (BEDT − TTF)2Cu(NCS)2. We find agreement both
with regard to the angular and the temperature dependence of HC. This supports
the suggestion that the Fulde-Ferrell-Larkin-Ovchinnikov state (FFLO state) exists in
this material for exactly plane-parallel orientation of the magnetic field. Indications of
precursor states, occurring for small deviations from the plane-parallel field direction,
are also pointed out and further measurements for confirming the existence of the
FFLO state are proposed.
PACS numbers: 74.25.Ha,74.70.Kn,74.80.Dm
If the superconducting state in high magnetic fields is limited by paramagnetic
pair breaking alone, the transition from the homogeneous superconducting state to the
normal conducting state may proceed either directly, at the Pauli limiting field HP, or
via an interposed inhomogeneous superconducting state predicted in 1964 by Fulde and
Ferrell [1] and by Larkin and Ovchinnikov [2] (FFLO state). The latter state, which
stabilizes as a consequence of spin polarization, has attracted considerable interest over
the years, but has, despite of this, not yet been definitely verified experimentally.
Favourable conditions for observing the FFLO state are found in clean
superconductors with orbital critical fields much larger than HP.
seems always necessary to reduce the orbital pair breaking effect by using layered
superconductors with nearly decoupled planes or extremely thin films (quasi-two-
dimensional superconductors) and applying the magnetic field in a direction parallel
to the conducting planes. Several classes of superconducting materials with favourable
conditions for observing the FFLO state do exist.
intercalated transition metal dichalcogenides as well as more exotic materials like High-
Tccompounds and organic superconductors.
In this Letter we refer to a recent measurement of the upper critical field Hc2in
the organic superconductor κ−(BEDT −TTF)2Cu(NCS)2[3]. This layered material
shows strong anisotropy of the superconducting properties with regard to out-of-plane
directions. In addition, a number of experiments listed in reference [3] are interpreted
In practice, it
These include the “classical”
§ To whom correspondence should be addressed.

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in terms of a strong in-plane anisotropy, of d-wave type, of the gap parameter. The
coherence length ξ⊥perpendicular to the layers of this clean material is smaller than
the interlayer spacing d and one expects an extreme reduction of orbital pair breaking
for plane-parallel applied field. The angular dependence of Hc2was measured in Ref [3]
both with respect to the angle Θ between applied magnetic field and the direction normal
to the conducting planes and with respect to the azimuthal angle φ, which denotes the
direction of magnetic field lying within the plane. The results showed, as expected, a
strong variation of Hc2with Θ. On the other hand, no dependence on φ was observed.
The maximal value of Hc2at the plane-parallel position Θ = π/2 was of the order of,
but 50% higher than, the Pauli paramagnetic limit HP. These facts led the authors of
reference [3] to propose that their in-plane critical field is the phase boundary between
the normal-conducting state and the FFLO state of a d-wave superconductor.
We examined this question by calculating the angular and temperature dependence
of the theoretical phase boundary HC(Θ,T) between the normal-conducting state
and the superconducting states of a quasi-two-dimensional d-wave superconductor.
Comparison with the data of reference [3] showed good agreement, supporting the
hypothesis that in this experiment the phase boundary of a FFLO state (for d-wave
superconductors) has been observed for the first time.
We assume that the coupling between the conducting planes of κ − (BEDT −
TTF)2Cu(NCS)2can be neglected, making our problem effectively two-dimensional.
Then, if the field has both a perpendicular and a parallel component,
superconducting state is limited by both orbital and paramagnetic pair breaking. For
exactly plane-parallel magnetic field the FFLO phase should be realized below a reduced
temperature (t = T/Tc) of t ≈ 0.4 [4]. Such a situation, with competition between both
pair breaking effects, has been studied first by Bulaevskii [5]. His treatment was later
generalized to arbitrary temperatures and d-wave superconductors by Shimahara and
Rainer [6]. Below the stability limit of the normal-conducting state, which will be
referred to as HC(Θ,T), a series of different inhomogeneous superconducting states,
depending on Θ, appear. For s-wave superconductors each one of these states belongs
to a particular value of Landau’s quantum number n, which takes integer values
n = 0,1,2,... These states are (for s-wave) the following:
the
• The vortex state for small Θ belongs to n = 0.
• A series of inhomogeneous states for Θ near π/2, each one characterized by a single
value n > 0, with n increasing with increasing Θ.
• The FFLO state for Θ = π/2, which may be characterized by n → ∞.
The structure of the higher Landau level states, for n > 0, has been calculated recently
for s-wave superconductors by minimizing the quasiclassical free energy [7]. For d-wave
superconductors [6] a state below HC(Θ,T) is no longer characterized by a single value
of n but rather by an infinite subset {n0,n0± 4,n0± 8,...}. However, the dominant
contribution may still be characterized by a single number n, which increases again
with increasing Θ and approaches infinity in the FFLO limit. Thus, basically the

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above classification scheme remains valid for d-wave symmetry. The phase boundary
of the ‘pure’ FFLO state for d-wave superconductors [the curve HC(π/2,T)] has been
calculated by Maki and Won [4].
The linearized gap equation to be solved is given by [5,6]
− log(T
−cos[s{µ0H −1
Tc)∆(? r)) = πkBT
?∞
0
ds
sinh(πkBTs)
?2π
0
dφ′
2π
?
γ(ˆ p′)2]
?
[1
2? v′
F?Π}]]∆(? r). (1)
We consider a cylindrical Fermi surface, appropriate for the present two-dimensional
problem, with the Fermi velocity ? vF= vF(? excosφ+? eysinφ). The gap parameter is given
by ∆(? r, ˆ p) = ∆(? r)γ(ˆ p), where γ(ˆ p) = 1 for s-wave, and γ(ˆ p) =√2(ˆ p2
for d-wave pairing.The canonical momentum is defined by?Π =
The magnetic field?H is assumed to lie in the yz-plane, with Hy = H? = H sinΘ
and Hz = H⊥ = H cosΘ.We use the following gauge for the vector potential:
Ax= H?z − H⊥y, and Ay= Az= 0. Paramagnetic pair breaking enters via the term
µ0H in (1); the electron’s magnetic moment is µ0= −gLµB/2, with the Lande factor
gL≈ 2 and Bohr‘s magneton µB= ¯ he/(2mc). The method of reducing equation (1) to
a set of algebraic equations follows exactly reference [6] and need not be repeated here.
As a first check of our numerical method we compared our results with reference [6] and
found complete agreement whenever the same set of input parameters was used.
Let us proceed to a comparison of the solutions HC of equation (1) with the Hc2
data reported in reference [3]. The first point to address is the independence of Hc2on
the azimuthal angle φ. Such a dependence may easily be incorporated in the present
model by replacing ? v′
pair-breaking is present for a parallel field component, the symmetry-breaking term
Ax = H?z has to be kept in (1). If, on the other hand, complete decoupling of the
(infinitely thin) conducting planes can be assumed, the term H?z can be dropped and
HC becomes independent of φ, as can be explicitely confirmed numerically. In this
context, we recall that a three-dimensional d-wave superconductor shows anisotropy of
the upper critical field [8]. Thus, the observed independence of Hc2on φ is in agreement
with the present model and, in particular, with the assumption of quasi-two-dimensional
superconductivity.
If T is measured in units of Tc and H in units of µ∆0 (where ∆0 is the BCS-
gap at T = 0), the present model requires only one single parameter kBTc/EF to be
fitted. This parameter is proportional to the “bulk” ratio of the orbital and spin critical
fields ¯ hc/(2eξ2
ratio of spin and orbital pair-breaking depends on Θ and may be written in the form
kBTc/(EFcosΘ). The best fit to the Θ-dependence of Hc2at T = 1.45K (see figure
4 of reference [3]) has been obtained for kBTc/EF = 0.058. This value of kBTc/EF is
consistent with a critical temperature Tc= 10.4K of the sample studied in reference
[3], and a Fermi energy of the order of 100K as estimated from several experiments [9].
Using this value we found very good agreement between HCand the data of reference [3],
x−ˆ p2
y) = 1+cos(4φ)
¯ h
ı(∇ − ı2e
¯ hc?A).
Fin (1) by vF[? excos(φ′+ φ) +? eysin(φ′+ φ)]. As long as orbital
0) and kBTc/µ0respectively. In the present anisotropic model the actual

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as shown in figure 1. The theoretical curves in figure 1 have been calculated assuming
d-wave symmetry; the difference between d-wave and s-wave was found to be small
except very close to T = 0K and Θ = 90 deg.
0
20
40
60
80
100
120
40 4550 556065 70 7580 8590
H2
c2(θ)/H2
c2(0)
θ [deg]
κ-(BEDT-TTF)2Cu(NCS)2 kBTc/EF=0.058
theory: t=0.14
experiment: t=0.14
theory: t=0.8
Figure 1: Square of upper critical field, normalized to its value at Θ = 0, as a function of
Θ. Full squares: data of reference [3] at t = T/Tc= 0.14. Full line: theoretical result for
the Bulaevskii-Shimahara-Rainer phase boundary for t = 0.14. Dashed line: theoretical
result for t = 0.8.
The Hc2 data show a small but clearly visible kink near the plane-parallel
orientation, at Θ ≈ 87 deg. A similar feature is also found in the theoretical phase
boundary HC(Θ), as shown in figure 1 (where the square of the critical fields has been
plotted in order to make this discontinuous change in slope better visible). This kink
indicates the transition from the vortex state, with n = 0, to the first of the above
mentioned FFLO-precursor states, with n = 1. Still closer to Θ ≈ 90 deg equation (1)
yields additional transitions corresponding to n = 2,3, which are still visible in figure 1 in
the theoretical curve but not in the data points. The HC-curves describing the n = 0,1,2
transitions, for the same material but higher T, are shown on a larger scale in figure
2. The order parameter structure of the precursor phases, where pairing takes place in
Landau levels n > 0, has been investigated recently for s-wave superconductors [7]. Two
types of such precursor states have been found: (i) quasi-one-dimensional states, which
may be considered as a mixture of rows of vortices and one-dimensional FFLO-type
oscillations, and (ii) two-dimensional lattices with several zeros of the order parameter
with different vorticity [7]. Such unusual states [of type (ii)] have been predicted to
occur in the extremely high field region where quantization of single electron levels

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becomes important [10]. The present arrangement might provide a relatively feasible
way of observing such vortex structures. It should be mentioned, however, that for
d-wave superconductors neither the equilibrium structure of the FFLO state nor that
of the n > 0 states has been calculated so far.
74
76
78
80
82
84
88.6 88.88989.2
θ [deg]
89.489.6
H2
c2(θ)/H2
c2(0)
κ-(BEDT-TTF)2Cu(NCS)2 kBTc/EF=0.058, t=0.4
n0 = 0
n0 = 1
n0 = 2
Figure 2: The branches n0= 0,1,2 of the upper critical field, as calculated from equation
(1), are plotted in more detail, using the same value of kBTC/EF as in figure 1 but a
higher temperature t = 0.4.
The shape of the HC(Θ) curve depends distinctively on temperature, as shown
by the plot (dashed line) of HC(Θ)2at t = 0.8 in figure 1.
enhancement at higher T is, of course, that paramagnetic pair breaking becomes less
effective at higher temperature. Data at higher T have not been reported in reference
[3] but would be useful in order to check the present interpretation. Looking for similar
measurements we found old data by Morris and Coleman [11] for intercalated transition
metal dichalcogenide TaS2− (pyridine) samples. In this material, which represents a
nearly perfect realization of two-dimensional superconductivity, an unexplained anomaly
with regard to the behavior of the upper critical field at different T has been reported
(figure 10 of reference [11]). We find excellent agreement (see figure 3) comparing the
data of reference [11] with the solutions of equation (1) for an s-wave superconductor.
Again, a single parameter has been adjusted (kBTc/EF= 0.024) to obtain both of the
theoretical curves shown in figure 3. The resistance data reported in reference [11]
show a non-monotonic behavior near the plane-parallel field orientation (see figure 3 of
reference [11]), which may be due to transitions to the n > 0 states. The latter states
are discussed in more detail in reference [7].
The reason for this

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0
200
400
600
800
1000
1200
8082 8486 88 90
H2
c2(θ)/H2
c2(0)
θ [deg]
TaS2-(pyridine) kBTc/EF=0.024
experiment: 2.86 K
theory: 2.86 K
experiment: 1.4 K
theory: 1.4 K
Figure 3: Comparison of the Θ-dependence of the upper critical field of TaS2−
(pyridine), as reported in reference [11], with the solutions of equation (1) for s-wave
superconductivity at t = 2.86 and t = 1.4.
0
5
10
15
20
25
30
35
0123456789 10
µ0H2
c2 [T]
T [K]
κ-(BEDT-TTF)2Cu(NCS)2
theory Maki, Won
experiment
Figure 4: Comparison of the temperature dependence of the plane-parallel upper critical
field reported in reference [3] with the theoretical result reported in reference [4].

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Finally, let us compare the measured temperature-dependence of Hc2for the plane-
parallel field orientation (figure 3 of reference [3]) with HC(π/2,T). According to our
interpretation of these data, the states below Hc2(π/2,T) should be a d-wave version
of the FFLO state for T < T∗ ∼= 0.4Tc, and the homogeneous superconducting state
for T > T∗. This phase boundary has been calculated first by Maki and Won [4].
The Shimahara Rainer d-wave phase boundary must agree with reference [4] in the
limit Θ → π/2. We found agreement, except for the steep rise of HC(π/2,T) below
0.05Tc reported in reference [4] (a possible reason for this discrepancy may be slow
convergence of our numerical method for low T and high n). The comparison between
theory [4] and experiment [3] depicted in figure 4 shows again fairly good agreement. A
characteristic difference in temperature variation above and below T∗is visible in the
data points, although it is less pronounced than in the theoretical curve. The difference
in critical fields between s-wave and d-wave is again rather small; at T = 0.05Tc, the
lowest temperature, where measurements for κ − (BEDT − TTF)2Cu(NCS)2 have
been reported [3], both are approximately given by the standard 2D-result [12] for the
FFLO state, µ0HC = ∆0, which exceeds the Pauli limiting field by∼= 40%. Thus,
these numbers do also fit well into the FFLO interpretation of the phase boundary for
plane-parallel field orientation.
Summarizing, the proposal of Nam et al. [3] that upper critical field data for a
plane-parallel field orientation in the layered organic superconductor κ − (BEDT −
TTF)2Cu(NCS)2should be interpreted in terms of a FFLO state, has been supported
by our calculations. The data agree with the predictions of a model of a quasi-two-
dimensional superconductor both with regard to the angular and the temperature
dependence of the critical field. Further confirmation of this interpretation could
be obtained by means of measurements at higher temperatures, where paramagnetic
pair-breaking is strongly reduced. If this interpretation is correct, precursor states
with interesting properties should appear for applied fields close to the plane-parallel
orientation.
References
[1] Fulde P and Ferrell R A 1964 Phys. Rev. 135 A550
[2] Larkin A I and Ovchinnikov Y N 1965 Sov. Phys. JETP 20 762
[3] Nam M S, Symington J A, Singleton J, Blundell S J, Ardavan A, Perenboom J A A J, Kurmoo
M and Day P 1999 J.Phys.:Condens.Matter 11 L477
[4] Maki K and Won H 1996 Czech. J. Phys. 46 1035
[5] Bulaevskii L N 1974 Sov. Phys. JETP 38 634
[6] Shimahara H and Rainer D 1997 J. Phys. Soc. Jpn. 66 3591
[7] Klein U, Rainer D and Shimahara H 2000 J. Low Temp. Phys. 118 91
[8] Won H and Maki K 1994 Physica B199&200 354
[9] McKenzie R H 1998 Comments Cond. Matt. Phys. 18 309
[10] Akera H, MacDonald A H, Girvin S M, and Norman M R 1991 Phys. Rev. Lett 67 2375
[11] Morris R C and Coleman R V 1973 Phys.Rev. B7 991
[12] Burkhardt H and Rainer D 1994 Ann. Physik 3 181