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.

Figures (15)

• Figure 1: Transport measurements on 2D electron gas (2DEG) at the EuO/KTaO$_3$ (111) interfaces.a, A schematic illustration representing the lattice structure of cubic KTaO$_3$; an expanded view of the (111) plane is shown with the in-plane high-symmetry directions marked by dashed arrows: three equivalent [11$\bar{2}$] axes (blue) and three equivalent [1$\bar{1}$0] axes. b, Superconducting transition temperature $T_{\rm c0}$ measured in a van der Pauw geometry on ten interface samples (see Supplementary Fig. 4 for raw data), plotted against the carrier density $n_{\rm 2D}$ of the 2DEGs determined from Hall data (details are listed in Supplementary Table 1). $T_{\rm c0}$ is defined to be when the sheet resistance $R_{\rm s}$ reaches 0.1$\%$ of its normal state value $R_{\rm N}$ at $T$ = 3 K. Red squares and black circles denote $T_{\rm c0}$ measured with current $I$ applied along the crystallographic [11$\bar{2}$] and [1$\bar{1}$0] directions, respectively. Blue and yellow shaded area highlights the presence and absence of the directional disparity of $T_{\rm c0}$, respectively. c, $R_{s}$ as a function of $T$ measured on the Hall-bar Device 1 (illustrated in the inset) which permits a simultaneous detection of $R_{s}$ along the [11$\bar{2}$] (blue) and [1$\bar{1}$0] (red) channels. d,e, Low-$T$$R_{\rm s}$ ($T <$ 2.5 K) signifying the superconducting transitions for $I \parallel$[1$\bar{1}$0] (d) and [11$\bar{2}$] (e), respectively. Solid lines are fits to the Halperin-Nelson (HN) formula; vertical dotted lines indicate the yielded $T_{\rm BKT}$ (BKT transition temperature). Insets show $dR_{\rm s}/dT$ versus $T$ and its maximum (i.e., the inflection point) implies the position of mean-field critical temperature $T_{\rm c}^{\rm MF}$ (black arrows) BenfattoLa214.
• Figure 2: Broken in-plane rotational symmetry in the superconducting state.a,b, Normalized temperature-dependent resistance $R(T)$ curves showing different superconducting transition temperatures along three equivalent [11$\bar{2}$] (a) and [1$\bar{1}$0] (b) directions, respectively. Data were measured on Device 2 with the carrier density $n_{\rm 2D}$ = 5.24$\times$10$^{13}$ cm$^{-2}$); the [11$\bar{2}$] and [1$\bar{1}$0] Hall-bar channels (colored correspondingly to the curves) are illustrated in the insets. $T_{\rm BKT}$ (BKT transition temperature) attained from Halperin-Nelson (HN) formula fits (Supplementary Fig. 6) are noted beside the curves. The maximum (minimum) $T_{\rm BKT}$ = 1.41 (1.14) K is observed for the [11$\bar{2}$] ([1$\bar{1}$0]) channel marked by cyan (red) color in a (b); these two channels are orthogonal to each other. c, Temperature dependent exponent $\alpha(T)$ extracted from the $I-V$ (current-voltage) characteristics: $V = I^{\alpha}$ (Supplementary Fig. 8). Blue and red symbols denote the data for the [11$\bar{2}$] channel with the highest $T_{\rm BKT}$ in a and the [1$\bar{1}$0] channel with the lowest $T_{\rm BKT}$ in b, respectively; the $I-V$ analysis yields a $T_{\rm BKT}$ (at which $\alpha$ = 3) 0.01-0.02 K lower than the HN fitting result for each direction. d, The directional disparity of superconducting transitions, $\Delta T_{\rm c}$, plotted against the applied current $I$ for the two channels referred to in c. $\Delta T_{\rm c}$ were calculated for $T_{\rm c0}$ and $T_{\rm c}^{\rm mid}$ ($T_{\rm c0}$ is defined to be when the sheet resistance $R_{\rm s}$ reaches 0.1$\%$ of its normal state value $R_{\rm N}$ at $T =$3 K and the latter is defined to be where $R_{\rm s}$ reaches 0.5$R_{\rm N}$). Inset shows the superconducting transitions in a logarithmic scale of $R(T)$ measured with the smallest current in practice, $I$ = 10 nA. $\Delta T_{\rm c}$ remains finite at the $T$ = 0 limit.
• Figure 3: Nonreciprocal charge transport in the superconducting fluctuation regimea, $\gamma = \frac{2R^{2\omega}}{\mu_0 HI R^{\omega}}$ as a function of $T$ (plotted with a sign reversal). Here $\gamma$ is a coefficient characterizing the current rectification, $R^{\omega}$ and $R^{2\omega}$ are linear resistance and second harmonic resistance, respectively. Data were collected under a magnetic field $\mu_0H$ = 0.2 T with $I$ = 1 $\mu$A. Blue (red) symbols are data taken on the [11$\bar{2}$] ([1$\bar{1}$0]) channel of Device 3 (Supplementary Fig. 9a, $n_{\rm 2D} = 5.59\times10^{13}$ cm$^{-2}$). Inset is a sketch of the measurement geometry for the second-harmonic nonreciprocal signal $R^{2\omega}$. b,c, Comparisons of the $T$ dependence of $\gamma$ and $R^{\omega}$ measured along the [1$\bar{1}$0] direction (b) and the [11$\bar{2}$] direction (c), respectively. Approaching $T_{\rm BKT}$ (BKT transition temperature), $\gamma(T)$ is traced by a divergence $(T-T_{\rm BKT})^{-3/2}$ for both directions, which reinforces the identification of a directional-dependent $T_{\rm BKT}$ (orange vertical dashed lines). Green solid lines are fits of $R(T)$ to the Aslamazov-Lakin model (Supplementary Note 4); resulted $T_{\rm c}^{\rm MF}$ (mean-field critical temperature) are marked by olive vertical dashed lines. d,e, $H$ dependence of $R^{2\omega}$ at varying $T$ for $I \parallel$ [1$\bar{1}$0] (d) and $I \parallel$ [11$\bar{2}$] (e), respectively. $R^{2\omega}(\mu_0H)$ curves are vertically shifted and data measured above 1.5 K are re-scaled for clearance. Symbols denote characteristic magnetic fields: $H_0^*$ (purple diamonds) and $H_N$ (orange arrows) are the onset and termination of the finite $R^{2\omega}$ due to superconductivity WakatsukiMoS2, respectively; $H_1^*$ (blue arrowheads) and $H_2^*$ (green up triangles) denote two characteristic fields at which $R^{2\omega}(H)$ changes its sign.
• Figure 4: Directional-dependent contour plots $R^{2\omega}$ and $R^{\omega}$ in the $H-T$ plane.a,b, Color contour plots of $R^{2\omega}(T,H)$ for $I$ along [1$\bar{1}$0] (a) and [11$\bar{2}$] (b), respectively, constructed based on the data shown in Figs. 3d and e. $T_{\rm BKT}$ and $T_{\rm c}^{\rm MF}$ (BKT transition temperature and mean-field critical temperature) for each direction are denoted by black arrows on the vertical axis. $R^{\omega}$ and $R^{2\omega}$ are linear resistance and second harmonic resistance, respectively. $H_0^*$ (purple triangles), $H_1^*$ (blue pentagons), $H_2^*$ (green diamonds) and $H_N$ (orange squares) are represented by symbols overlaid to the contour plots. c,d, Contour maps of the normalized magnetoresistance $\Delta R(H)/R_{\rm N}$ [$\Delta R(H) = R(H)-R(H =0)$] for $I \parallel$ [1$\bar{1}$0] (c) and $I \parallel$ [11$\bar{2}$] (d), respectively (see Supplementary Fig. 9 for raw data). Orange and yellow short-dashed lines are contour lines of $R$ = 0.8 $R_{\rm N}$ and (1/3) $R_{\rm N}$, respectively. The boundary of the white area denotes the contour line of $R$ = 0.01 $R_{\rm N}$ . These three contour lines are reproduced in a and b to underline their correspondence with the characteristic fields.
• Figure S1: Structural characterization of the EuO/KTaO$_3$(111) interfaces.a, Illustrations of the KTO(111) plane and the EuO(001) plane showing the positions of different ions. The distance between nearest Ta ions along $[1\bar{1}0]$ ($[11\bar{2}]$) is 0.564 (0.977 nm). The distance between nearest Eu ions is 0.515 nm. b, $\theta-2\theta$ X-ray diffraction (XRD) patterns of a (111)-oriented KTO substrate (black) and a Ge/EuO/KTO heterostructure (red). The EuO film does not contribute any visible Bragg peak. c, The ring-shaped reflective high-energy electrons diffraction (RHEED) pattern of the EuO epitaxial film demonstrates the polycrystalline nature of the overlayer. d-f, Atomic force microscope (AFM) images measured on two samples with $n_{\rm 2D} = 9.79\times10^{13}$ cm$^{-2}$ (d) and 7.67$\times$10$^{13}$ cm$^{-2}$ (e), respectively, as well as an annealed KTO substrate (f). Scanned areas have the size of 5$\times$5 $\mu$m$^{2}$. The surface roughness (see curves in the bottom panels) was measured along the horizontal yellow bar in the AFM images; the typical root mean square roughness $R_{\rm q}$ is 3.5, 4.4 and 2.7 Å for (d), (e) and (f), respectively.