Directional-dependent Berezinskii-Kosterlitz-Thouless transition at EuO/KTaO$_3$(111) interfaces

Abstract: In two dimensions, a phase-coherent superconducting state is established via a Berezinskii-Kosterlitz-Thouless (BKT) transition, whose critical temperature $T_{\rm BKT}$ is determined by the global superfluid stiffness in uniform superconducting systems. We report that at the interface between (111)-oriented KTaO$3$ and ferromagnetic EuO, the two-dimensional superconducting state exhibits a BKT transition relying on the direction of in-plane bias current. The highest $T{\rm BKT}$ occurs when current is applied along one of the [11$\bar{2}$] axes of KTaO$3$, underscoring a spontaneous breaking of the threefold lattice rotational symmetry. Such directional dependence of $T{\rm BKT}$ is consistently reflected in the nonreciprocal signals stemming from superconducting fluctuations above the transition. We attribute this phenomenon to an interfacial phase segregation; the phase with higher $T_{\rm BKT}$ self-organizes into quasi-one-dimensional textures that stretch along one of the [11$\bar{2}$] directions. Our results point toward the emergence of exotic phases of matter beyond the description of conventional BKT physics at a superconducting interface that is subjected to ferromagnetic proximity.

• The paper demonstrates that the BKT transition temperature (T_BKT) varies with the in-plane current direction at the EuO/KTaO₃ (111) interface.
• Comprehensive transport measurements and Halperin-Nelson analysis confirm anisotropic superconducting behavior, with higher T_BKT along the [11̄2] axis.
• The study suggests that ferromagnetism, strong Rashba SOC, and multiorbital physics induce a spontaneous nematic stripe phase that breaks rotational symmetry.

Directional-Dependent BKT Transition at the EuO/KTaO$_3$(111) Interface

Introduction and Context

This work systematically investigates the emergence of a directional-dependent Berezinskii-Kosterlitz-Thouless (BKT) transition in superconductivity at the ferromagnetic EuO/KTaO$_3$(111) interface. In two-dimensional superconducting systems, phase coherence is established via the BKT mechanism, characterized by the dissociation of bound vortex-antivortex pairs, with $T_{\rm BKT}$ set by the global superfluid stiffness. The present study reveals that for the EuO/KTaO$_3$(111) interface, $T_{\rm BKT}$ not only departs from isotropy but exhibits a pronounced dependence on the direction of in-plane current, which is inconsistently explained by standard BKT theory.

Figure 1: Transport and crystalline structure characterization of EuO/KTaO$_3$(111) interfaces and evidence for directional disparity in superconducting transitions.

The system displays the highest $T_{\rm BKT}$ when current is applied along a specific [11$\bar{2}$] axis, indicating spontaneous breaking of the crystalline threefold rotational ($C_3$) symmetry and the emergence of unconventional electronic phases at oxide interfaces proximitized by ferromagnetism.

Experimental Evidence for Anisotropic BKT Transition

Comprehensive transport measurements (van der Pauw and Hall-bar geometries) were carried out on a series of EuO/KTaO$_3$(111) heterointerfaces with varying 2D carrier density ($_3$0). In all superconducting 2DEGs with $_3$1 cm$_3$2, the zero-resistance ($_3$3), BKT ($_3$4), and mean-field ($_3$5) transition temperatures are direction-dependent, with all three systematically higher for $_3$6\,[11$_3$7] compared to $_3$8\,[1$_3$90]. Halperin-Nelson analysis of paraconductivity validates BKT criticality but unambiguously reveals directionality of $T_{\rm BKT}$0. Moreover, corresponding anisotropy is observed in the upper critical field $T_{\rm BKT}$1.

Rotational Symmetry Breaking in the Superconducting State

To directly probe the correspondence between superconducting anisotropy and lattice symmetry, “double-tri-beam” and radial six-beam Hall-bar devices were fabricated, enabling resistance measurements along all symmetry-equivalent axes. The data show that only one of the three [11$T_{\rm BKT}$2] channels consistently manifests the highest $T_{\rm BKT}$3, with its perpendicular [1$T_{\rm BKT}$40] exhibiting the lowest $T_{\rm BKT}$5. Thus, emergent twofold (rather than threefold) anisotropy arises in the superconducting state, manifesting as a nematicity not dictated by the underlying $T_{\rm BKT}$6 electronic structure.

Figure 2: Evidence for broken rotational symmetry in superconducting transitions—only one [11$T_{\rm BKT}$7] channel shows maximal $T_{\rm BKT}$8.

Analysis of the $T_{\rm BKT}$9-$_3$0 characteristics further corroborates directional criticality, as the nonlinear exponent $_3$1 satisfies $_3$2 (BKT criterion) at different $_3$3 for orthogonal channels, with $_3$4 remaining finite down to zero bias, ruling out nonequilibrium current effects.

Nonreciprocal Charge Transport and Superconducting Fluctuations

Nonreciprocal charge transport, a signature of inversion and time-reversal symmetry breaking, is probed via second-harmonic resistance ($_3$5) under in-plane magnetic field. An appreciable and highly anisotropic rectification coefficient $_3$6 emerges close to $_3$7, exhibiting a $_3$8 divergence as predicted for vortex-mediated nonreciprocity.

Figure 3: Strong, direction-sensitive nonreciprocal charge transport in the fluctuation regime; $_3$9 exhibits critical divergence above the BKT transition.

The direction with higher $T_{\rm BKT}$0 (and $T_{\rm BKT}$1), again [11$T_{\rm BKT}$2], is consistently distinguishable under these fluctuation-sensitive probes. The field dependence of $T_{\rm BKT}$3 shows complex sign reversals, demarcating multiple fluctuation regimes, with all critical fields and temperatures showing similar directional enhancements.

Figure 4: $T_{\rm BKT}$4-$T_{\rm BKT}$5 contour plots of nonreciprocal and linear resistance, revealing structured sign changes and their correspondence to the superconducting phase with pronounced directional asymmetry.

Theoretical Implications and Interpretation

The observed directionality in $T_{\rm BKT}$6 is distinct from conventional expectations: for a homogeneous 2D superconductor (XY model), $T_{\rm BKT}$7 is tied to a global superfluid stiffness and must be isotropic. The measured data are inconsistent with explanations based on direction-dependent vortex core energy or randomly inhomogeneous Josephson networks.

Instead, the results point toward a self-organized phase segregation at the interface, where the higher-$T_{\rm BKT}$8 phase organizes into quasi-one-dimensional “stripe” textures aligned preferentially along one [11$T_{\rm BKT}$9] direction. This microscopic nematicity emerges out of competition between ferromagnetism, strong Rashba SOC, and multiorbital 2DEG physics unique to KTaO$_3$0 interfaces. The widths of these filamentary regions substantially exceed the coherence length, excluding true 1D superconducting nanowire behavior.

Numerical fits and multilayer comparison across device geometries establish that such stripes do not act as isolated nanowires but as extended 2D segments whose connectivity ensures anisotropic global phase coherence without enforcing full isotropy. The orientation-specific criticality is reinforced by tight coupling to the underlying electronic band structure, which is inherently different between KTaO$_3$1(110) and (111) terminations.

Experimental Implications and Outlook

The ubiquity of anisotropic superconducting transitions in KTO-based interfaces with ferromagnetism, and their disappearance at high carrier density (where ferromagnetic signatures fade), suggest an intricate interplay between superconductivity, spontaneous spin order, and crystal symmetry. The data strongly suggest that the coexistence regime favors spatially modulated and directionally-selective superconductivity.

Given analogous findings in other oxide and nickelate heterointerfaces and the revelation of finite-momentum and mixed-parity pairing in the presence of SOC and Zeeman fields, these results raise the possibility that the observed anisotropy is a general feature whenever electronic nematicity and magnetic proximity are combined in strongly 2D interfaces.

Conclusion

This study reveals directional-dependent BKT criticality at the EuO/KTaO$_3$2(111) interface, which cannot be captured by scalar order parameter models or uniform superfluid stiffness. The emergence of a preferential [11$_3$3]-aligned superconducting stripe phase demonstrates an electronically-driven spontaneous rotational symmetry breaking. The interplay of interfacial ferromagnetism, strong Rashba SOC, and KTO band structure produces an exotic interfacial superconducting state, representative of a broader class of unconventional 2D superconductors.

Theoretical models addressing the cooperative roles of magnetic proximity, spatial phase segregation, and inter-stripe Josephson coupling are needed to fully characterize the collective phase dynamics and vortex structure in such systems. This work lays a foundation for future studies on the control of superconducting nematicity, vortical excitations, and nonreciprocal transport in engineered oxide interfaces and related 2D quantum materials.