Anisotropic Enhancement of Lower Critical Field in Ultraclean Crystals of Spin-Triplet Superconductor UTe₂

The superconducting state of UTe₂ has been intensively investigated because of its spin-triplet nature with a paramagnetic normal state Ran et al. (2019a); Aoki et al. (2022a). The spin-triplet pairing state induces an extremely high upper critical field ($H_{\rm c2}$) beyond the Pauli limit Ran et al. (2019a, b) and multiple superconducting phases under hydrostatic pressure or magnetic field Braithwaite et al. (2019); Thomas et al. (2020); Aoki et al. (2020); Kinjo et al. ; Rosuel et al. ; Sakai et al. . Furthermore, chiral spin-triplet states are discussed in scanning tunneling microscopy Jiao et al. (2020), polar Kerr effect Hayes et al. (2021), surface impedance Bae et al. (2021), and penetration depth studies Ishihara et al. . As for the pairing interactions, the anisotropic $H_{\rm c2}$ and reentrant superconductivity Ran et al. (2019b); Knebel et al. (2019) suggest that ferromagnetic fluctuations are at play analogous to the ferromagnetic superconductors Aoki et al. (2019), which is supported by the nuclear magnetic resonance (NMR) and muon spin rotation ($\mu$SR) measurements Tokunaga et al. (2019); Sundar et al. (2019); Tokunaga et al. (2022). On the other hand, recent neutron scattering studies revealed the presence of antiferromagnetic fluctuations related to superconductivity Duan et al. (2020); Knafo et al. (2021a); Duan et al. (2021). Thus, the nature of magnetic fluctuations and their relationship to superconductivity are still unclear.

What has complicated systematic understandings of the superconducting state in UTe₂ are the inhomogeneity of crystals and the presence of magnetic impurities. Single crystals grown by the conventional chemical vapor transport (CVT) method sometimes show double superconducting transitions likely induced by inhomogeneity Hayes et al. (2021); Thomas et al. (2021); Rosa et al. (2022) and a small increase in magnetic susceptibility at low temperatures, which reflects the presence of U vacancies acting as magnetic impurities Ran et al. (2019a); Rosa et al. (2022); Knafo et al. (2021b); Tokunaga et al. (2022); Haga et al. (2022). Recently, Sakai et al. developed another crystal growth method called the molten salt flux (MSF) method and succeeded in obtaining high-quality single crystals of UTe₂ with a transition temperature $T_{\rm c}=2.1$ K, a residual resistivity ratio as high as 1,000, and low magnetic impurity density Sakai et al. (2022). In these ultraclean crystals, novel superconducting and normal state properties have been reported in addition to the observation of de Haas-van Alphen oscillations Sakai et al. ; Tokiwa et al. ; Aoki et al. (2022b). Therefore, measurements on the high-quality single crystals are crucial to clarify the superconducting nature in UTe₂.

In this study, we investigate the thermodynamic critical field ($H_{\rm c}$), the lower critical field ($H_{\rm c1}$), and $H_{\rm c2}$ in high-quality MSF single crystals. Regarding the Ginzburg-Landau (GL) theory, the anisotropies in $H_{\rm c1}$ and $H_{\rm c2}$ should be opposed because the critical fields satisfy $H_{\rm c1}H_{\rm c2}=H_{\rm c}^{2}(\ln\kappa+0.5)$, where $\kappa$ is the GL parameter, and $H_{\rm c}$ is independent of field directions. In contrast, we find that the anisotropy of $H_{\rm c1}$ does not follow the expectations from that of $H_{\rm c2}$. Quantitatively, the above GL relation holds for ${\bm{H}}\parallel{\bm{a}}$, while it is obviously violated for ${\bm{H}}\parallel{\bm{b}}$. Furthermore, $H_{\rm c1}$ for ${\bm{H}}\parallel{\bm{b}}$ and ${\bm{H}}\parallel{\bm{c}}$ show unusual upturns at low temperatures. To explain these anomalous behaviors of the critical fields, we propose an anisotropic enhancement of $H_{\rm c1}$ induced by the Ising-like ferromagnetic fluctuations in UTe₂.

High-quality single crystals are grown by the MSF method as described in Ref. Sakai et al. (2022). Crystals #A1 and #A2 in this study are the same samples as crystal #A1 in Ref. Ishihara et al. and crystal #M7-1 in Ref. Sakai et al. (2022), respectively, and crystal #A3 for the $H_{\rm c2}$ measurement is picked up from the same batch with the crystal used in the de Haas-van Alphen experiments Aoki et al. (2022b). Crystals #A1 and #A2 have a cuboid shape with dimensions $455\times 250\times 95$ $\mu$m³ and $1400\times 285\times 165$ $\mu$m³, respectively. $H_{\rm c}$ is calculated from the specific heat measured by the long-relaxation method where a Cernox resistor is used as a thermometer, a heater, and a sample stage Tanaka et al. (2022). $H_{\rm c1}$ is measured by a miniature Hall-sensor array probe tailored in a GaAs/AlGaAs heterostructure Shibauchi et al. (2007); Okazaki et al. (2009, 2010); Putzke et al. (2014). The distance of neighboring sensers is 20 $\mu$m so that we can measure the local magnetic induction and the first flux penetration field $H_{\rm p}$ near crystal edges. $H_{\rm c2}$ is estimated from specific heat data $C(T)$ near $T_{\rm c}$ measured in the Physical Property Measurement System (Quantum Design). The same single crystal piece appropriately shaped relative to the principal axes was used for all the field directions relative to the crystal orientation.

Figure 1(a) shows the temperature dependence of $C/T$ of crystal #A1. A sharp peak at 2.1 K and a small residual value at low temperatures confirm that this crystal is ultraclean. $H_{\rm c}$ can be calculated via $H_{\rm c}^{2}(T)=2\mu_{0}\int_{T}^{T_{\rm c}}(S_{\rm n}-S_{\rm sc})dT$, where $\mu_{0}$ is the permeability of vacuum and $S_{\rm sc}$ and $S_{\rm n}$ are the electronic entropy in the superconducting and normal states, respectively, calculated from the specific heat. Note that, to satisfy the entropy balance, we assume linear temperature dependence of the electronic contribution of $C/T$ in the normal state below $T_{\rm c}$ (see Supplemental Material). The obtained $\mu_{0}H_{\rm c}(0)=77.6$ mT shown in Fig. 1(b) is higher than the reported value in a low-$T_{\rm c}$ crystal Paulsen et al. (2021), reflecting the higher quality of our crystal.

Figure 2 represents external magnetic-field dependence of the local magnetic induction $B_{\rm edge}$ near the crystal edges of crystal #A1. The typical behaviors of the Hall resistance of the Hall-array probe near the crystal edges are shown in the insets in Fig. 2. The linear dependence at the low field region originates from the finite distance between the Hall sensor and the crystal. By subtracting the linear function, we can derive the local magnetic induction $B_{\rm edge}$. At lower fields, $B_{\rm edge}=0$ because of the Meissner state. As the magnetic field increases, the $B_{\rm edge}$ value deviates from zero at $H_{\rm p}$ depicted as black triangles in Fig. 2. We found that the $H_{\rm p}$ value does not depend on the crystal positions (see Supplemental Material), confirming that our measurements are free from surface pinning or geometrical barriers of superconducting vortices. Here, we note that $H_{\rm p}$ is lower than the bulk $H_{\rm c1}$ because of the demagnetization effect. The relation between $H_{\rm p}$ and $H_{\rm c1}$ has been precisely studied Brandt (1999), and we can evaluate $H_{\rm c1}$ from $H_{\rm p}$ through

where $t$ and $w$ are the sample thickness and width, respectively.

The obtained $H_{\rm c1}^{i}$ with the magnetic field along $i$-axis is shown in Fig. 3. Considering the usual relation

$H_{\rm c1}(T)$ is expected to scale with the normalized superfluid density $\rho_{s}(T)\propto\lambda^{-2}(T)$ when the temperature dependence of the GL parameter, $\kappa$, is small. The black solid line in Fig. 3(a) is the normalized superfluid density for ${\bm{H}}\parallel{\bm{a}}$, $\rho_{s}^{a}=(\lambda_{b}^{2}(0)/\lambda_{b}^{2}(T)+\lambda_{c}^{2}(0)/\lambda_{c}^{2}(T))/2$, calculated from the anisotropic penetration depth measurements Ishihara et al. . It is obvious that $H_{\rm c1}$ and $\rho_{s}^{a}$ show similar $T$ dependence, confirming the consistency of $H_{\rm c1}$ and penetration depth measurements. On the other hand, the $H_{\rm c1}$ data along $b$- and $c$-axes show an anomalous concave temperature dependence around 0.5 K, which is not detected in the penetration depth measurements. The origin of this unusual $T$ dependence will be discussed later. We note that, although the absolute value of $H_{\rm c1}^{c}$ in crystal #A1 may have an estimation error of the demagnetization effect because of its plate-like shape, we obtain very similar $H_{\rm c1}^{c}(T)$ in another crystal #A2 [Fig. 3(c)], confirming the correct estimation of $H_{\rm c1}^{c}$ values in these crystals.

First, we compare the anisotropies in $H_{\rm c1}$ and $H_{\rm c2}$. As mentioned before, the GL theory implies that the anisotropies in these critical fields are opposite as long as we discuss the initial slope near $T_{\rm c}$ where the GL formalism is valid. Figure 4 depicts $H_{\rm c2}^{i}(T)$ with the magnetic field along $i$-axis near $T_{\rm c}=2.1$ K. The slope of $H_{\rm c2}(T)$ along $a$-axis significantly changes with decreasing temperature in the low-field region, which has been already reported in previous studies Rosuel et al. ; Kittaka et al. (2020). Focusing on the initial slope, we find the anisotropy of $H_{\rm c2}^{b}>H_{\rm c2}^{a}>H_{\rm c2}^{c}$. In contrast, from the initial slope of $H_{\rm c1}$ data (see Fig. 3), we obtain the anisotropy $H_{\rm c1}^{c}>H_{\rm c1}^{b}>H_{\rm c1}^{a}$, the order of which is not as expected from the anisotropy of $H_{\rm c2}$. We note that, while the anisotropy in $H_{\rm c1}$ is similar to the previous study of global magnetization measurements in a lower-quality sample Paulsen et al. (2021), the absolute values of $H_{\rm c1}$ in this study are considerably larger. This point is important to discuss the origin of the discrepancy between the anisotropies of $H_{\rm c1}$ and $H_{\rm c2}$ as described below.

To discuss possible origins of this apparent inconsistency in critical field anisotropies, we calculate the lower critical field $H_{\rm c1}^{\rm cal}$ expected from $H_{\rm c}$ and $H_{\rm c2}$ values via the following GL relations,

The results are shown in Fig. 3 as the black broken lines. We can clearly find in Fig. 3(a) that $H_{\rm c1}^{\rm cal}$ along $a$-axis matches well with the experimental $H_{\rm c1}$ value, meaning that the experimental critical fields ($H_{\rm c}$, $H_{\rm c1}$, and $H_{\rm c2}$) along $a$-axis satisfy the usual thermodynamic relations of Eqs. (3,4). This result suggests that the significant change in the slope of $H_{\rm c2}^{a}$ near $T_{\rm c}$ shown in Fig. 4 is an intrinsic property of UTe₂ possibly induced by the reduction of ferromagnetic fluctuations with ${\bm{H}}\parallel{\bm{a}}$ Rosuel et al. . In the previous study, because the critical fields violated the above GL relations, the authors assumed a magnetic field effect on $T_{\rm c}$, $dT_{\rm c}/dH$ Paulsen et al. (2021). However, our results consistent with the GL relations for $H\parallel a$ indicate that the $dT_{\rm c}/dH$ effect is negligible as long as we focus on the initial slopes of the critical fields.

In contrast to $H_{\rm c1}^{a}$, however, the initial slope of $H_{\rm c1}^{b}$ is significantly larger than the expected $H_{\rm c1}^{\rm cal}$ value. In addition to the steep initial slope, $H_{\rm c1}^{b}$ and $H_{\rm c1}^{c}$ in ultraclean UTe₂ show an unusual increase below 0.5 K, which has not been observed previously. A similar $H_{\rm c1}(T)$ behavior was reported in multi-band iron-based superconductors Song et al. (2011); Adamski et al. (2017) and the chiral superconductor candidates, UPt₃ Vincent et al. (1991); Amann et al. (1998) and PrOs₄Sb₁₂ Cichorek et al. (2005). We emphasize that the increase of $H_{\rm c1}(T)$ at low temperatures in these materials was observed regardless of the magnetic field directions, which is clearly different from the case in UTe₂ where the increase is discernible only in ${\bm{H}}\parallel{\bm{b}}$ and ${\bm{H}}\parallel{\bm{c}}$. Moreover, magnetic penetration depth measurements in PrOs₄Sb₁₂ also show a kink at lower temperatures Chia et al. (2003), while the anisotropic penetration depth in UTe₂ shows smooth $T$ dependence around 0.5 K Ishihara et al. . Thus, while the enhancement of $H_{\rm c1}$ at low temperatures in the previous studies on the various superconductors reflects the increase of superfluid density, the anomalous $H_{\rm c1}(T)$ in UTe₂ likely has a different origin which induces a large anisotropy.

Because the $H_{\rm c1}$ value is determined by the vortex line energy, the unusual $H_{\rm c1}^{b}(T)$ and $H_{\rm c1}^{c}(T)$ behaviors can be related to an exotic vortex state. As for the vortices in UTe₂, recent scanning SQUID measurements on the (011)-plane observed pinned vortices and antivortices even in a zero-field cooling condition, which can be related to the presence of strong and slow magnetic fluctuations detected in $\mu$SR and NMR studies Iguchi et al. . Moreover, optical Kerr effect measurements on the (001)-plane suggest the presence of magnetized vortices induced by strong magnetic fluctuations. From these results, we consider that the enhancement of $H_{\rm c1}^{b}(T)$ and $H_{\rm c1}^{c}(T)$ is related to the strong Ising-like ferromagnetic fluctuations along $a$-axis inducing an exotic vortex state in UTe₂. We note that anomalous low-$T$ behavior has been also detected in $1/T_{1}T$ of NMR measurements Nakamine et al. (2019) and zero-field relaxation rate of $\mu$SR measurements Sundar et al. (2019, ).

The relationship between strong magnetic fluctuations and the enhancement in $H_{\rm c1}$ has been investigated in BaFe₂(As_1-xP_x)₂ Putzke et al. (2014). In this system, an antiferromagnetic quantum critical point (QCP) appears at $x=0.3$ where $\lambda(0)$ shows a sharp peak reflecting the strong mass enhancement Hashimoto et al. (2012). In contrast, $H_{\rm c1}$ also shows a peak at QCP, contradicting with the expected behavior from Eq. (2). This discrepancy suggests that the vortex line energy is significantly enhanced by the strong magnetic fluctuations near QCP. Thus, considering the presence of strong anisotropic ferromagnetic fluctuations in UTe₂, the vortex line energy can be anisotropically enhanced by the Ising-like magnetic fluctuations. Indeed, the normal-state NMR measurements in UTe₂ Tokunaga et al. (2019) reported that $1/T_{1}T$ shows a pronounced low-$T$ enhancement for ${\bm{H}}\parallel{\bm{b}}$ and ${\bm{H}}\parallel{\bm{c}}$, which is absent for ${\bm{H}}\parallel{\bm{a}}$. This shows a good correspondence with the anisotropic $H_{\rm c1}$ enhancement.

In conclusion, we measured the critical fields, $H_{\rm c}(T)$, $H_{\rm c1}(T)$, and $H_{\rm c2}(T)$, in high-quality single crystals of UTe₂ with $T_{\rm c}=2.1$ K. While the three critical fields for ${\bm{H}}\parallel{\bm{a}}$ satisfy the usual GL relations near $T_{\rm c}$, the experimental $H_{\rm c1}^{b}$ and $H_{\rm c1}^{c}$ are larger than the expected values, which become more apparent below 0.5 K. These results indicate that the anisotropic ferromagnetic fluctuations in UTe₂ significantly enhance the vortex line energy with ${\bm{H}}\parallel{\bm{b}}$ and ${\bm{H}}\parallel{\bm{c}}$. Our experimental results not only suggest that an exotic vortex state caused by anisotropic ferromagnetic fluctuations is realized in UTe₂, but also promote further studies on magnetic fluctuations in high-quality single crystals of UTe₂ with low magnetic impurity density.

We thank J.-P. Brison for fruitful discussions. This work was supported by Grants-in-Aid for Scientific Research (KAKENHI) (Nos. JP22H00105, JP22K20349, JP21H01793, JP19H00649, JP18H05227), Grant-in-Aid for Scientific Research on innovative areas “Quantum Liquid Crystals” (No. JP19H05824), Grant-in-Aid for Scientific Research for Transformative Research Areas (A) “Condensed Conjugation” (No. JP20H05869) from Japan Society for the Promotion of Science (JSPS), and CREST (No. JPMJCR19T5) from Japan Science and Technology (JST).

Supplemental Material