Observations of the effect of strong Pauli paramagnetism on the vortex lattice in superconducting ${\mathrm{CeCu}}_2{\mathrm{Si}}_2$

We present the results of a study of the vortex lattice in the heavy fermion superconductor ${\mathrm{CeCu}}_2{\mathrm{Si}}_2$, using small-angle neutron scattering (SANS). In this material at temperatures well below $T_{\mathrm{c}}\sim 0.6$ K, the value of the upper critical field $B_{\mathrm{c}2}\sim 2.2$ T is strongly limited by the Pauli paramagnetism of the heavy fermions. In this temperature region, our SANS data show an increase in the magnetization of the flux line cores with field, followed by a rapid fall near $B_{\mathrm{c}2}$. This behavior is the effect of Pauli paramagnetism and we present a theory-based model, which can be used to describe this effect in a range of materials. The pairing in ${\mathrm{CeCu}}_2{\mathrm{Si}}_2$ appears to arise from the effect of magnetic fluctuations, but the evidence for a $d$-wave order parameter is rather weak. We find that the vortex lattice structure in ${\mathrm{CeCu}}_2{\mathrm{Si}}_2$ is close to regular hexagonal. There are no phase transitions to square or rhombic structures; such transitions are expected for $d$-wave superconductors and observed in ${\mathrm{CeCoIn}}_5$; however, the temperature dependence of the SANS intensity indicates that both large and small gap values are present, most likely due to multiband $s$-wave superconductivity, rather than a nodal gap structure.

Figure 1

Rocking curve in $\omega$ at 130 mK and 1.5 T for the reflection labeled $\vec{A}$ in the inset, which shows the diffraction pattern of the vortex lattice in ${\mathrm{CeCu}}_2{\mathrm{Si}}_2$ obtained by summing $\omega$ and $\varphi$ rocking scans with the field applied along $\mathbf{c}$. One pair of spots is aligned with the crystal $\left[100\right]$ direction.

Figure 2

(a) $\theta$ angle dependence of the opening angle $\beta$ between the bottom diffraction spot, which lies on the $\left[1\overline{1}0\right]$ direction, and the bottom right spot. The data were obtained at 1.5 T and base temperature. The curve is a fit to Eq. (9). The $\diamond$ point corresponds to the average of all $\beta$ angles measured at 40 mK. The $◃$ point at $\theta =0^{\circ }$ is the average of all $\alpha$ angles measured at 130 mK [Fig. 5(d)], and is not included in the fit as it corresponds to a different VL orientation. (b) Schematic of the setup where $\theta$ is the angle between the incoming magnetic field and the c axis of the sample. (c) Diffraction pattern of the vortex lattice in ${\mathrm{CeCu}}_2{\mathrm{Si}}_2$ obtained at $\theta ={30}^{\circ }$, 50 mK and 1.5 T. (d) Field dependence of the opening angle $\alpha$ between the directions $\left[100\right]$ and $\vec{A}$ at 130 mK. The green squares represent the anisotropy measured in previous STM experiments [18].

Figure 3

(a) Field dependence of the squared form factor calculated from the first order diffraction spots with a well-defined rocking curve, for temperatures between 40 mK and 350 mK. The symbol ($▵$) corresponds to the data obtained at ILL in 2016, ($\circ$) corresponds to data obtained at ILL in 2018, and ($\diamond$) data were obtained at PSI in 2018. $\diamond$ data were measured with the sample rotated away from the B $\parallel$ c orientation [see Fig. 2] but the integrated intensity is not affected by this change, as we can observe from the points at 1.6 T, 250 mK and 130 mK, that match with measurements from other experiments. The cyan arrow corresponds to the value of $B_{\mathrm{c}2}\left(T\right)$ for $T=350$ mK and the wide black arrow covers all the upper critical fields for $T=40,130$, and 250 mK obtained from Ref. [5]. Solid lines are guides for the eye and the crossing of the lines is not real [simulated dependence at low fields shown in Fig. 4]. (b) Comparison between the normalized squared form factor of ${\mathrm{CeCu}}_2{\mathrm{Si}}_2$ at 40 mK and ${\mathrm{CeCoIn}}_5$ at 50 mK [22]. The lines are the results of fits using a model for PPE described in the next section.

Figure 4

(a) Field and (b) temperature dependence of the squared form factor with the results of the fits using our model for paramagnetic effects, with a single $s$-wave gap (dashed line) and two-gap $s$-wave model (dotted line). For all fits $\lambda \left(0\right)=7000$ Å [33].

Figure 5

$B-T$ phase diagram for superconductivity in ${\mathrm{CeCu}}_2{\mathrm{Si}}_2$ with $\mathbf{B}\parallel \mathbf{c}$. The different colors highlight the BCS region and where the Pauli paramagnetism is dominant at low temperatures and high fields and the FFLO region appearing close to the upper critical field at low temperatures. The dashed line indicates the limit of the FFLO region proposed by NMR studies [5]. Arrows show the field- and temperature scans covered by the SANS experiments. Shaded points show the highest field or temperature at which the vortex lattice was visible in each scan. The black line represents the upper critical field $B_{\mathrm{c}2}$ as a function of temperature for $\mathbf{B}\parallel \mathbf{c}$ with the green-solid points representing experimental data obtained by ac susceptibility [5].