Observations of Pauli Paramagnetic Effects on the Flux Line Lattice in CeCoIn5

From small-angle neutron scattering studies of the flux line lattice (FLL) in CeCoIn5, with magnetic field applied parallel to the crystal c-axis, we obtain the field- and temperature-dependence of the FLL form factor, which is a measure of the spatial variation of the field in the mixed state. We extend our earlier work [A.D. Bianchi et al. 2008 Science 319, 177] to temperatures up to 1250 mK. Over the entire temperature range, paramagnetism in the flux line cores results in an increase of the form factor with field. Near H_c2 the form factor decreases again, and our results indicate that this fall-off extends outside the proposed FFLO region. Instead, we attribute the decrease to a paramagnetic suppression of Cooper pairing. At higher temperatures, a gradual crossover towards more conventional mixed state behavior is observed.


Introduction
The heavy-fermion superconductor CeCoIn 5 continues to excite great interest, because it shows strong Pauli paramagnetic effects [1,2] and also the close proximity of superconductivity to a quantum critical point [3,4]. It has a superconducting transition temperature, T c ∼ 2.3 K in zero field [5], with a d x 2 −y 2 order-parameter [6,7]. At low temperature, the transition to the normal state is first-order [1,2], showing that the superconductivity is suppressed by coupling of the field to the anti-parallel spins of the singlet Cooper pair (the Pauli effect) rather than the more common coupling to the orbital motion of Cooper pairs in the mixed state (the orbital effect) [8,9]. The dominance of the Pauli effect, combined with the quasi-2D structure and superclean crystal properties (ℓ ∼ 1000ξ [10]), means that the stringent requirements for the stabilisation of the inhomogeneous Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) state are satisfied [11][12][13]. Numerous studies report experimental signatures that are compatible with the formation of FFLO [14][15][16][17][18], though an unambiguous microscopic observation remains elusive. Recently, both nuclear magnetic resonance (NMR) [19] and neutron diffraction studies [20] have provided microscopic evidence for the stabilisation of field-induced long range antiferromagnetic order for fields applied within the basal plane. This magnetically ordered phase (termed 'Q-phase' in [20]) exists only within the superconducting mixed state, and occupies the same high field, low temperature region of the superconducting (H, T ) phase diagram as proposed for FFLO. It remains unclear whether this ordered phase replaces, or coexists with, a non-standard FFLO state.
Small-angle neutron scattering (SANS) studies of the superconducting mixed state of CeCoIn 5 have determined the flux line lattice (FLL) structure and orientation for the major part of the superconducting phase diagram with magnetic field applied parallel to the crystal c-axis (H c) [21][22][23][24]. In addition, SANS can be used to determine the FLL form factor (FF) which is a measure of the field contrast in the mixed state, i.e. the difference between the local magnetic induction in the flux line cores and at the field-minima between them [21][22][23][24]. Most notably, an anomalous fielddriven enhancement of the FF was observed at 50 mK upon approaching H c2 (T = 0) = 4.95 T [23]. We proposed that the increasing FF arises from paramagnetic moments induced in the flux line cores where the antiparallel alignment of electron spins in Cooper pairs is suppressed, a phenomenon that is consistent with the predictions of recent numerical calculations [25]. Just before entering the normal state, an abrupt fall of the FF was observed, which we speculated was associated with the formation of the FFLO state [23]. However, the numerical study [25] reproduced a similar reduction of the FF at high field, apparently unrelated to the onset of the FFLO state. Hence, the physical origin of the decreasing FF just below H c2 remains unclear. To address this question, we have used SANS to investigate the field-dependence of the FLL FF in CeCoIn 5 for temperatures up to 1250 mK. For all temperatures in this range, the FF initially rises with increasing field, reaching a maximum before it starts to fall prior to entering the normal state. Our results show that while we are unable to rule out a contribution to the field-dependence of the FF due to a FFLO-type state, the dominant contribution to the high field fall in the FF at low temperature is due to a paramagnetic suppression of the Cooper pairing.

Experimental
Our experiments were performed on the SANS-I instrument at the Paul Scherrer Institut. The samples were mosaics of co-aligned CeCoIn 5 plate-like single crystals (with small dimension along the c-axis) grown from an excess indium flux [5]. Most results reported here were obtained using a sample of total volume 0.2 × 11 × 14 mm 3 and of total mass 250 mg. Thin samples were necessary due to the strong neutron absorption of indium. The sample was mounted in a dilution refrigerator with base temperature ∼ 50 mK which was inserted into a horizontal field cryomagnet, and oriented so that the c-axis was 2 • from the field direction. Neutrons of wavelengths λ n = 5 − 6Å were selected with a wavelength spread ∆λ n /λ n = 10%, and were incident approximately parallel to the field; the diffracted neutrons were collected by a position-sensitive multidetector. At all fields the FLL was prepared by field-cooling through T c . Diffraction measurements were performed as the sample and cryomagnet were rotated together to carry the FLL diffraction spots through the Bragg condition. Background subtraction was performed using measurements taken either above T c or after zero-field cooling, with no noticeable dependence on which method was used.

Results and Discussion
The local field in the mixed state may be expressed as a sum over its spatial Fourier components with indices (h, k), and scattering vectors q hk belonging to the twodimensional FLL reciprocal lattice. The form factor (FF) at wavevector q hk is the magnitude of the Fourier component F (q hk ). The value of the FF is obtained from the integrated intensity of a FLL Bragg reflection as the FLL is rotated through the diffraction condition. This integrated intensity, I hk , is related to the modulus squared of the FF, |F (q hk )| 2 , by [26] Here, V is the illuminated sample volume, φ is the incident neutron flux density, λ n is the neutron wavelength, γ is the magnetic moment of the neutron in nuclear magnetons, and Φ 0 = h/2e is the flux quantum. General predictions from Ginzburg-Landau (GL) theory, whether by numerical methods or algebraic approximations, are for a monotonic decrease of the FLL FF with increasing field [27][28][29][30]. Although strictly valid only close to T c , these models have been widely used and give a good description of results from many superconductors at lower temperatures [31][32][33][34][35][36][37]. Quasiclassical theory, which is valid well away from T c , gives qualitatively similar results for both sand d-wave orbitally-limited superconductors [38]. We will see that the conventional picture provided by all of these theories lies in stark contrast to the results that we now report.
In figure 1 we show the field-dependence of the FLL FF for first-order reflections, at various temperatures between 50 and 1250 mK. At all temperatures, the FF initially rises with increasing field before reaching a maximum, and then begins to fall again on approaching H c2 . At temperatures of 500 mK and below, the FF remains finite all the way up to H c2 , where the FLL signal disappears abruptly upon entering the normal state. This is consistent with the first-order nature of the superconducting transition seen in thermodynamic measurements at low temperatures [1,2], and predicted theoretically for strong Pauli-limiting [9]. Above 500 mK, the field-dependence of the FF becomes increasingly conventional, smoothly approaching zero at H c2 , as expected for a second-order transition to the normal state. However, even at these higher temperatures, Pauli-paramagnetic effects remain important, leading to a maximum in the FF at intermediate fields. Figure 2 shows the regions within the superconducting phase diagram occupied by the various FLL structures [23]. For the hexagonal structures, two of the six Bragg spots in each of the two domains lie along 110 directions. In the high-field region, we find that the value of |F (q 10 )| 2 for these spots is ∼ 20% larger than for any of the other four spots. This FF anisotropy is consistent with that expected to arise due to the slight distortion of the hexagonal structure away from the isotropic hexagon [23]. As a consequence of this, the magnitude of q 10 is always slightly less for the two 110 aligned Bragg spots than for the other four spots. To accommodate this variation in intensity, the FF values shown in figure 1 are an average over all six spots. Whenever the FLL structure changes, both the magnitudes and directions of q hk alter; however, figure 1 shows that any changes in the value of |F (q 10 )| 2 at the FLL structure phase boundaries are no bigger than the errors. Finally in figure 1, we see that at low field there is a common value of FF, independent of temperature, showing the differences between the data at different temperatures occur at higher fields. We may interpret this behavior as showing that the effects of Pauli limiting are small at low fields, and that for T T c /2, orbital-limiting effects are fairly temperature-independent. However, at high fields, Pauli-limiting is dominant and the proximity of H c2 (T ) causes major changes in the behaviour of the FF.
In figure 3 we show the results of investigating the detailed temperaturedependence of the FF in the high field hexagonal FLL structure phase. For the fields of 4.60 T, 4.85 T and 4.90 T, we have recorded the temperature-dependence of the diffracted intensity of a first-order Bragg reflection. At the two higher fields, these temperature scans pass through H c2 almost parallel to the upper critical field phase boundary; nonetheless, there is a sharp fall in intensity as the first-order boundary is crossed, with perhaps a slight smearing arising from tiny differences between crystallites in the mosaic. We estimate that the contribution due to demagnetisation effects to the width of the discontinuity at T c2 is unimportant. At the lower field, a slower variation with temperature of the normalised intensity is seen, reflecting the second-order transition at H c2 . However, as shown by the absolute data in the inset, the FF is at its maximum at the lower field, deeper within the mixed state, and falls as either field or temperature is increased towards H c2 .
Previous discussion of the field-induced amplification of the FF emphasised the inability of conventional theory to explain our results at low temperatures [23]. Here, we consider the FF behavior in the context of an extension [25] to the quasi-classical Eilenberger theory [39]. In this work, Pauli paramagnetic effects are incorporated by adding a Zeeman energy term µB(r) to the Eilenberger equations [25], where the parameter µ represents the relative strength of the paramagnetism [40]. This theory was first used to successfully describe the similar, but less extreme, field dependence of the FF in TmNi 2 B 2 C [41]. At low temperatures, and for large values of µ, the model predicts not only an initial increase of the FF with increasing field, but also a decrease close to H c2 [25]. To obtain a reasonable qualitative agreement between the theory and the observed FF field-dependence in CeCoIn 5 requires µ ∼ 2.6. However, experimentally determined material parameters for CeCoIn 5 imply a µ ≃ 1.7 for H c. Hence, for a quantitative agreement with our data, some refinement of the theory is required. For instance, it might be necessary to incorporate a more three-dimensional character of the Fermi surface of CeCoIn 5 [42], or account for the non-Fermi liquid behavior which is most prominent near H c2 . We further mention that heat capacity measurements [43] reveal a field-dependence to the low energy density of states (DOS), which further needs to be accounted for [44]. The inclusion of a field-dependent DOS would be consistent with the observation by thermal conductivity studies of twoband superconductivity in CeCoIn 5 [45], which imply that some of the supercarriers become depaired at very low fields. These depaired carriers would not only be available to contribute to the paramagnetic response at high fields, but their absence at low fields might also explain why the form factor at very low fields is not given by the expected expression (see [23]) using the zero-field value of the London penetration depth, λ L [10,46]. An independent bulk measurement of λ L at finite fields would test this suggestion.
It was previously suggested that the FF maximum at 50 mK could mark the onset of the FFLO state [23]. Fig. 2 shows the temperature dependence of the field at which a maximum in the FF is observed, superposed onto the FLL structure phase diagram. It is clear that the FF maximum does not follow the FFLO phase boundary proposed experimentally [14,18] or predicted theoretically [47,48]. Furthermore the FF maximum does not appear to be correlated with the high-field rhombic to hexagonal FLL symmetry transition. To understand the observed FF behavior close to H c2 , we therefore return to the the theory of Ichioka and Machida [25], which reproduces the observed drop of the FF without including any effects of an FFLO state. Beginning at low fields, they predict an increase in the magnitude and spatial extent of the paramagnetic moment in the flux line cores, arising from the spin-split quasiparticle states located there. The spatially varying paramagnetic moment adds to the orbital contribution to the field variation in the mixed state to give an increase in the field contrast and hence in F (q 10 ). The size of the flux line cores, as measured by the extent of the region with a suppressed order parameter, also increases, showing the effects of paramagnetic depairing in these regions. However, at high fields where the width of the expanded flux line cores becomes comparable with the flux line spacing, the suppression of the order parameter allows the paramagnetic moment to spread further through the entire flux lattice unit cell. This leads to an additional paramagnetic moment of the whole crystal which is observed in magnetisation measurements [2], but to a reduction in the field contrast, as measured by SANS, and hence to a reduction in F (q 10 ). We emphasise that these effects are a consequence of paramagnetic effects dominating pairing in the near-core regions where the order parameter is suppressed, and that these effects spread out from there. Thus the decrease of the FF close to H c2 is quite different in origin from the behavior seen in conventional materials.
Next we discuss the abrupt fall of the FF seen at H c2 and low temperature. The finite value of the FF at H c2 can be directly related to the jump in the magnetisation at the upper critical field [2]. First, we note that the spatial field modulation B(r) = F (q hk ) e iq hk ·r where the sum is over all FLL reciprocal lattice indices h and k, and F (q = 0) is equal to the average induction B . Close to H c2 it is a good approximation to include in the sum only the dominant Fourier component F (q {10} ), which is the FF measured in our SANS experiments. In the case of CeCoIn 5 it is important to repeat that the field modulation contains contributions from both the orbital screening currents and the polarisation of the unpaired electron spins. However, just below H c2 , the spin magnetisation is by far the dominant contribution to the field modulation, such that B(r) = µ 0 [M (r) + H appl. ] [44]. Assuming that the magnetisation in the "normal" flux line cores is equal to M normal just above H c2 , the average magnetisation in the mixed state is easily found to be µ 0 M = M normal − 6|F (q 10 )|. This is illustrated in Fig. 4. Our measured value of the FF of 0.4 mT just below H c2 at 50 mK would thus correspond to a jump in magnetisation of ∆(µ 0 M ) = 2.4 mT as the field is increased through the upper critical field. This is in very satisfactory agreement with the results of bulk magnetisation measurements, which give a jump at H c2 of 3 mT [2].
We now consider the effects of a flux line core paramagnetism on the sequence of FLL structures shown in Fig. 2, for fields H > 0.5H c2 and low temperature. We note that this sequence of transitions from square to rhombic and then to high-field distorted hexagonal, is the exact reverse of what is observed for H < 0.5H c2 . In the central square phase, the FLL nearest neighbor directions are aligned with the nodes of the d x 2 −y 2 gap; this is consistent with the low-field sequence of transitions being driven by flux line core anisotropy, which reflects that of the order parameter. However, on the approach to H c2 , paramagnetic depairing causes the flux line cores to begin to expand and also become more isotropic [44]. We expect that this suppresses the anisotropy which stabilised the square phase, and that core expansion is the driving mechanism for the high-field sequence of FLL structures. This may also explain the similar sequence of FLL transitions observed in TmNi 2 B 2 C [49], where Pauli paramagnetic effects have also been demonstrated [41]. At higher temperatures, although the high field effects of paramagnetic depairing on both the FLL FF and the FLL structure become suppressed, figure 2 shows that the re-entrant square to rhombic transition persists to temperatures above those for which H c2 is first-order. This is a further demonstration that paramagnetic effects remain important at higher temperatures where H c2 (T ) is becoming orbitally-limited. These deductions are consistent with those of another recent study based on a GL type theory, that also reproduces the observed sequence of high-field transitions and ascribes them primarily to the effects of paramagnetic depairing and Fermi surface anisotropy [50]. However, we point out that GL theory is not expected to be numerically accurate over this region of the (H, T ) phase diagram.

Conclusion
In conclusion, we have studied the detailed field-and temperature-dependence of the diffracted neutron intensity from the flux line lattice in CeCoIn 5 . At high fields and low temperatures, we most clearly observe the effects of Pauli paramagnetism in the flux line cores, which results in an increase of the FLL form factor that extends well into the temperature region where H c2 becomes second order. Close to H c2 the flux line lattice form factor decreases, which we attribute to the effects of a paramagnetic suppression of the Cooper pairing. We suggest that further consequences of this paramagnetic de-pairing are manifested in the observed high field sequence of FLL structure transitions.