Observation of signatures of subresolution defects in two-dimensional superconductors with a scanning SQUID

The diamagnetic susceptibility of a superconductor is directly related to its superfluid density. Mutual inductance is a highly sensitive method for characterizing thin films, however, in traditional mutual inductance measurements, the measured response is a nontrivial average over the area of the mutual inductance coils, which are typically of millimeter size. Here we measure localized, isolated features in the diamagnetic susceptibility of Nb superconducting thin films with lithographically defined through holes, $\delta$-doped ${\mathrm{SrTiO}}_3$, and the two-dimensional electron system at the interface between ${\mathrm{LaAlO}}_3$ and ${\mathrm{SrTiO}}_3$, using scanning superconducting quantum interference device susceptometry, with spatial resolution as fine as $0.7\mu \mathrm{m}$. We show that these features can be modeled as locally suppressed superfluid density, with a single parameter that characterizes the strength of each feature. This method provides a systematic means of finding and quantifying submicron defects in two-dimensional superconductors.

Figure 1

Assumed geometry. A superconducting film of thickness $d$ and infinite extent in the $x$ and $y$ directions is centered on the plane $z=0$. The susceptometer field coil and pickup loop are modeled as coplanar, infinitely thin circular loops of radius $a$ and $b$ respectively, oriented parallel to the thin film in the plane $z=z_0$. A point defect is at $x=x_0,y=0,z=0$.

Figure 2

In-phase susceptibility of defects in thin-film superconductors. (a) Plot of normalized in-phase defect response susceptibility $-2{\mathrm{\Phi }}_0{a}^2{\varphi }_{r,1}/{\gamma }^3{\mu }_0$ vs lateral spacing $x_0/a$ for various reduced Pearl lengths ${\mathrm{\Lambda }}_0/a$, for spacing $z_0/a=0.25$ between the sample surface and the field coil/pickup loop (see Fig. 1), and $b/a=0.46$. (b) Peak value of the normalized in-phase defect response susceptibility ${\varphi }_1^r$ as a function of ${\mathrm{\Lambda }}_0/a$, for various values of $b/a$, with $z/a=0.25$. For large values of the Pearl length ${\varphi }_1^r$ is approximately proportional to $1/{\mathrm{\Lambda }}_0^2$; for small ${\mathrm{\Lambda }}_0$'s ${\varphi }_1^r$ is nearly proportional to $1/a^2$. In all cases the defect response susceptibility scales with the cube of the defect strength parameter $\gamma$.

Figure 3

Experimental susceptibilities of lithographically defined holes through a $0.2-\mu \text{m}$ niobium film. (a) Pickup loop (black)/field coil (blue) layout for the susceptometer used. The red layer is a superconducting shield for the pickup loop. (b)–(d) Susceptibility images for square holes with sizes as labeled. Field coil current 1 mA at 2.024 kHz, $T=5\mathrm{K}$. The SQUID substrate and sample were touching during the scan, such that $z_0$, the spacing between the sample surface and the pickup loop layer, was about $0.5\mu \mathrm{m}$. The dashed line in (b) shows the location of the cross section through the data displayed in Fig. 4.

Figure 4

Theoretically predicted susceptibilities. (a) The symbols are a cross section as indicated in Fig. 3. The solid line is from the model [12] described above [Eq. (11)], with $b=0.79\mu \text{m}$ (white circle) and $a=0.22\mu \text{m}$ (yellow circle) in (b). The fitting parameter $\gamma =-0.5\mu \mathrm{m}$. (b)–(d) Calculated susceptibilities for square holes in a niobium film with the sizes as labeled, obtained by convolving the point spread function from (b) with the hole shapes.

Figure 5

Calculated susceptibilities using an experimental point spread function. (a) Point spread function inferred from experimental susceptibility of a $0.5\mu \text{m}\times 0.5\mu \text{m}$ hole in niobium. (b)–(d) Calculated susceptibilities for rectangular holes with sizes as labeled.

Figure 6

We observe “halos,” approximately circular features of reduced susceptibility, in several two-dimensional superconductors. In-phase susceptibility images of (a) $\delta$-doped ${\mathrm{SrTiO}}_3$ with 5.5 nm thick, 1 at. % Nb doping layer (studied in detail in Ref. [14]) and a set temperature of 150 mK, and (b) ${\mathrm{LaAlO}}_3/{\mathrm{SrTiO}}_3$ (sample G of Ref. [30, Chap. 3]), where the temperature as measured at the sample thermometer was 54 mK before and 84 mK after the scan.

Figure 7

The profile of a halo along a high-symmetry direction is reproduced by our model for SQUID susceptometry of a pointlike defect. (a) In-phase susceptibility image taken, with field coil current $0.25{\mathrm{mA}}_{\mathrm{rms}}$ at 1472 Hz, on 5.5 nm, 1 at. % Nb $\delta$-doped STO [cropped version of Fig. 6]. Superimposed on the image are the positions of three line cuts, and the layout of the pickup loop (black) and field coil (blue) region of the susceptometer used for this image [10]. (b) Average of line cuts (dots) and fit to Eq. (11) plus linear background (dashed line). (c) Simulated image calculated with same parameters as in (b) and a second-order background determined by fitting a surface to the data far from the circular feature in (a). The field coil and pickup loop (blue and black overlays, respectively) are represented by concentric, coplanar circles of $8.4-$ and $2.7-\mu \mathrm{m}$ radii, respectively.

Figure 8

Temperature dependence of defect image in $\delta$-doped STO. Images of in-phase susceptibility (a) without and (b) with background subtraction and (c) out-of-phase susceptibility without background subtraction in the region of the $\delta$-doped STO sample plotted in Fig. 7, as a function of temperature. Temperatures noted on the figure were measured at the mixing chamber at the beginning of each scan. Images and associated measurements of susceptibility as a function of height were taken at a field coil current of $0.25{\mathrm{mA}}_{\mathrm{rms}}$, 1472 Hz. (d) Line cuts through the in-phase (${\varphi }_1$) images as indicated by the dashed lines in (a). (e) Line cuts through the out-of-phase (${\varphi }_2$) images as indicated by the dashed lines in (c). The zeros of each curve (dashed lines) are offset by $-0.02{\mathrm{\Phi }}_0/A$ as the temperature is lowered. (f),(g) Best-fit values for the Pearl length ${\mathrm{\Lambda }}_0$ and the defect strength parameter $\gamma$ from fits to the in-phase susceptibility ${\varphi }_1$ as described in the text.

Figure 9

Temperature dependence of defect images in LAO/STO. Images of (a) in-phase and (b) out-of-phase susceptibility taken as a function of temperature on the same 5 unit cell LAO/STO sample as the data shown in Fig. 6. The labels at the top of the images display the temperatures at the mixing chamber at the beginning and end of the scans. During this temperature series, a backgate voltage of 0 V was applied to the LAO/STO. Images and associated susceptibility vs height measurements taken at field coil current of 1 ${\mathrm{mA}}_{\mathrm{rms}}$, 1863.3 Hz. (b) Line cuts through the in-phase susceptibility images (${\varphi }_1$) as indicated by the dashed lines in (a). (d) Line cuts through the out-of-phase images (${\varphi }_2$) as indicated by the dashed lines in (b). (e),(f) Best-fit values for the uniform Pearl length ${\mathrm{\Lambda }}_0$ and defect strength parameter $\gamma$ from fits to the in-phase susceptibility as described in the text.

Figure 10

Field coil current dependence of defect images in LAO/STO. Susceptibility images as a function of field coil current at temperatures of $\sim 0.4–0.5T/T_c$ on the same 5 u.c. LAO/STO sample as the data shown in Figs. 6 and 9. (a) In-phase and (b) out-of-phase susceptibility. During this field coil series, a backgate voltage of $-30$ V was applied to the LAO/STO. rms field coil currents as noted on images; excitation frequency of 1863.3 Hz for all images and associated measurements of susceptibility vs height. (c) Line cuts through the in-phase susceptibility (${\varphi }_1$) images as indicated by the dashed lines in (a). (d) Line cuts through the out-of-phase susceptibility (${\varphi }_2$) images as indicated by the dashed lines in (b). Successive curves are offset by $-0.05{\mathrm{\Phi }}_0/A$ as the field coil current is increased. (e),(f) Best-fit values for the uniform Pearl length ${\mathrm{\Lambda }}_0$ and defect strength parameter $\gamma$ as a function of field coil current $I$, obtained by fitting the in-phase susceptibility of (c).