Variation in superconducting transition temperature due to tetragonal domains in two-dimensionally doped ${\mathrm{SrTiO}}_3$

Strontium titanate is a low-temperature, non–Bardeen-Cooper-Schrieffer superconductor that superconducts to carrier concentrations lower than in any other system and exhibits avoided ferroelectricity at low temperatures. Neither the mechanism of superconductivity in strontium titanate nor the importance of the structure and dielectric properties for the superconductivity are well understood. We studied the effects of twin structure on superconductivity in a 5.5-nm-thick layer of niobium-doped ${\mathrm{SrTiO}}_3$ embedded in undoped ${\mathrm{SrTiO}}_3$. We used a scanning superconducting quantum interference device susceptometer to image the local diamagnetic response of the sample as a function of temperature. We observed regions that exhibited a superconducting transition temperature $T_c\gtrsim 10$% higher than the temperature at which the sample was fully superconducting. The pattern of these regions varied spatially in a manner characteristic of structural twin domains. Some regions are too wide to originate on twin boundaries; therefore, we propose that the orientation of the tetragonal unit cell with respect to the doped plane affects $T_c$. Our results suggest that the anisotropic dielectric properties of ${\mathrm{SrTiO}}_3$ are important for its superconductivity and need to be considered in any theory of the mechanism of the superconductivity.

Figure 1

In $\delta$-doped STO, long, narrow regions oriented along cubic [100] and [010] are superconducting at temperatures at which their surroundings are no longer superconducting. (a)–(c) Maps of magnetic susceptibility in different areas of a single $\delta$-doped sample reveal patterns of superconductivity along [010] (a), (b) and [100] (c). Negative susceptibility (in purple and blue) indicates that a region is superconducting. Yellow regions are in the normal state but become superconducting at lower temperatures. Scans were taken at (a) 330 mK, (b) 300 mK, and (c) 320 mK. The schematics of the SQUID pickup loop (red) and field coil (blue) are to scale and are oriented as they were during data acquisition.

Figure 2

Superconducting features change spatially after thermal cycling, indicating that they are most likely due to the cubic-to-tetragonal transition in STO. The same region of a single sample was imaged at 340 mK before (a) and after (b) cycling to room temperature. Distinct features are apparent in (b) in a region that lacked sharp features in (a). The box in (b) indicates the approximate location of the image taken in (a). During sample growth, a clip masked the paramagnetic region in the lower right of the image.

Figure 3

The spatial variation in susceptibility near $T_c$ is much weaker at lower temperatures. (a) Repeated scans at a single location. (b) Repeated scans at a second location. White lines in (a) and (b) indicate the positions of line cuts displayed in (c) and (d), respectively. (c), (d) Averaged line cuts taken at 330 mK and at base temperature (c) or at 330 and 100 mK (d) demonstrate that the relative amplitude of modulation of the superconducting response is larger at temperatures near $T_c$ than at the lowest temperatures.

Figure 4

The temperature dependence of the susceptibility indicates a $\approx 10$% spatial variation in $T_c$. (a) Representative image at 350 mK from a series of images taken in the same area at 15 temperatures. The scan area presented here is different from the ones shown in Figs. 1, 2, 3. (b) Susceptibility as a function of temperature at two locations marked by a blue diamond and a red triangle in (a). The transition temperatures are $\approx 320$ and $\approx 360$ mK.

Figure 5

Tilt map of the 1 at. % Nb $\delta$-doped STO reveals domain structure below 105 K. Changes in color indicate changes in the local orientation of the crystal lattice relative to the incident x-ray beam. Red, green, and blue correspond to the $x,y$, and $z$ components, respectively, of displacements of the $c$* reciprocal lattice vector relative to a reference position. The intensity of a particular color reflects the magnitude of the displacement relative to the maximum displacement in that channel in the entire scan area. The scan plane was parallel to the surface of the sample. (a) At room temperature, the lattice is relatively uniform. (b) In contrast, at 80 K, below the cubic-to-tetragonal structural transition, the sample displays features whose orientation and size are consistent with tetragonal domain structure. The white box in (b) indicates the position of the room temperature image in (a). The horizontal feature at the top of (a) and (b) did not change with thermal cycling and is likely due to physical damage to the sample.

Figure 6

We detected considerable variation in the widths of the regions that had $T_c$ higher than their surroundings, indicating that at least some of the features are not resolution limited. (a) Averaged line cuts of the vertical features in (b) and (c). (d) Averaged line cuts of the horizontal features in (e) and (f). The curves in (a) and (d) are offset by intervals of 0.$8{\mathrm{\Phi }}_0$/A for clarity. Dotted lines indicate zero susceptibility for each curve. The schematics of the SQUID pickup loop (red) and field coil (blue) in (a) and (d) are to scale and are oriented relative to the line cuts as they were during data acquisition. All images were taken at 320 mK and are of the $\delta$-doped sample with 1 at. % Nb doping.

Figure 7

Allowed orientations of boundaries between tetragonal domains as seen in a cubic (001) plane. Arrows indicate the former cubic [100] and [010] directions. Boundaries between a domain with its tetragonal $c$ axis parallel to the plane and a domain with $c$ axis perpendicular to the plane are at (a) $0^{\circ }$ or (b) ${90}^{\circ }$ to the cubic [100] direction. (c) Boundaries between two domains with $c$ axis parallel to the plane are at ${45}^{\circ }$ to the cubic [100] direction.

Figure 8

A single feature oriented at $-{45}^{\circ }$ to the former cubic [100] axis exhibits higher $T_c$ than its surroundings. Repeated scans at a single location on the 1 at. % Nb $\delta$-doped sample at the temperatures indicated on the images.

Figure 9

Regions of modulated $T_c$ in 0.2 at. % Nb $\delta$-doped STO. The scans shown in (a) and (b) were taken in different areas of the same sample.

Figure 10

Comparison of superconducting transitions measured in global transport and local susceptibility for the $\delta$-doped sample with $N_D=1$ at. % Nb and $d=5.5$ nm shows that zero resistance occurs just below the lower $T_c$ determined from local susceptibility. Resistance as a function of temperature measured in a separate cooldown is shown in black dots; ranges of lower and upper transition temperatures inferred from maps of susceptibility (and reported in Table 1) are shown by blue (left) and red (right) vertical bands, respectively. Inset: resistance as a function of temperature for the two $\delta$-doped samples studied shows that the overall temperature scale for superconductivity in the sample with $N_D=0.2$ at. %, $d=36.9$ nm is lower than for the $N_D=1$ at. % Nb, $d=5.5$ nm sample.

Figure 11

Susceptibility images of bulk 1 at. % Nb-doped STO near $T_c$ reveal faint regions of enhanced diamagnetic response oriented along cubic axes. (a), (b) Scans in the same area at (a) 325 mK and (b) 335 mK. At 325 mK, the entire image area still is superconducting. The band of faint features in the upper half of (a) is oriented along cubic [100] but does not persist above the bulk $T_c$ of its surroundings.