Scanning SQUID susceptometry of a paramagnetic superconductor

Scanning SQUID susceptometry images the local magnetization and susceptibility of a sample. By accurately modeling the SQUID signal we can determine physical properties such as the penetration depth and permeability of superconducting samples. We calculate the scanning SQUID susceptometry signal for a superconducting slab of arbitrary thickness with isotropic London penetration depth $\lambda$ on a nonsuperconducting substrate, where both slab and substrate can have a paramagnetic response that is linear in the applied field. We derive analytical approximations to our general expression in a number of limits. Using our results, we fit experimental susceptibility data as a function of the sample-sensor spacing for three samples: (1) $\delta$-doped SrTiO${}_3$, which has a predominantly diamagnetic response, (2) a thin film of LaNiO${}_3$, which has a predominantly paramagnetic response, and (3) the two-dimensional electron layer at a SrTiO${}_3$/LaAlO${}_3$ interface, which exhibits both types of response. These formulas will allow the determination of the concentrations of paramagnetic spins and superconducting carriers from fits to scanning SQUID susceptibility measurements.

Figure 1

Model geometry. (a) The layout of the field coil/pickup loop region of the SQUID susceptometers used in this paper. (b) Approximations to this layout are considered in Appendices 1a and 1b. (c) For this paper we consider a slab of thickness $t$ with magnetic permeability ${\mu }_2$ and superconducting penetration depth $\lambda$ on a nonsuperconducting substrate with magnetic permeability ${\mu }_3$. We define $z=0$ at the sample surface, with a pickup loop (radius b) and concentric field coil (radius a) located at a height $z=z_0$ in a plane parallel to the sample surface.

Figure 2

Theoretical height dependence of the scanning SQUID susceptibility, divided by the maximum of the absolute value of the susceptibility, of a paramagnetic superconductor in various limits. The letters correspond to the entries in Table .

Figure 3

(a) Susceptometry image of a $\delta$-doped SrTiO${}_3$ sample. (b) Susceptibility as a function of $\Delta V_z$, the change in $z$ piezovoltage from contact between the SQUID substrate and the sample surface, at the position of the gray circle in (a). The circles are data, the solid line is a fit using the thin diamagnetic limit expression in Table e with a linear slope added, resulting in a best fit $\Lambda =954$ $\mu$m. The dashed line is the same fit without an added linear slope, resulting in a best fit $\Lambda =829$ $\mu$m. (c) Scanning susceptibility image of a patterned nonsuperconducting thin film of LaNiO${}_3$. (d) Susceptibility approach curve for the LaNiO${}_3$ film at the position of the square symbol in (c). The dots are data, the line is a fit using the thin paramagnetic limit expression [Table b], with ${\chi }_{2}t=1.3\times {10}^{-5}\mu \text{m}$.

Figure 4

Scanning SQUID susceptometry image of a patterned LAO/STO interface at 0.087 K. The labels indicate where the data in Fig. 5 was taken.

Figure 5

Scanning SQUID susceptibility as a function of sensor-sample spacing at two positions on the patterned LAO/STO sample of Fig. 4: P1, in a region showing predominantly diamagnetic shielding; and P2, in a region showing predominantly paramagnetic response. The symbols are data and the lines are fits as described in the text.

Figure 6

Calculated $z$ component of the field $B_z$, divided by the current $I$ for the model of Fig. 1b, for a circular loop of the same radius (dot-dashed curve), the contribution from the incomplete circular loop (dotted curve), the contribution from the leads (dashed curve) and the sum of the previous two (solid line). The $x$ axis is oriented along the leads, with the leads coming in along the positive $x$ direction. This calculation assumes the pickup loop and the field coil are in the same plane. The field at the center is 7.5$\%$ larger for the circular loop model than for the incomplete circle plus leads model.

Figure 7

Calculated magnetic scalar potential ${\varphi }_s(\vec{r},0)$, divided by the current through the loop, for the model of Fig. 1b with $a=8.4\mu \text{m}$, $s=7.4\mu \text{m}$, and $z_0=1.5\mu \text{m}$.

Figure 8

$z$ component of the source field, divided by the current, with $a=8.4\mu \text{m}$, $s=7.4\mu \text{m,}$ and $z_0=1.5\mu \text{m}$, calculated using Biot-Savart vs taking the gradient of the scalar potential. Also shown for comparison is a cross section through the response field image of Fig. 9.

Figure 9

$z$ component of the response field, divided by the current, with $a=8.4\mu \text{m}$, $s=7.4\mu \text{m}$, $z_0=1.5\mu \text{m}$, $t=10\mu \text{m}$, and $\lambda =0.1\mu \text{m}$.

Figure 10

(a) Fractional error $|{\phi }_{\mathrm{exact}}-{\phi }_{\mathrm{analytical}}|/{\phi }_{\mathrm{exact}}$ associated with using the thin paramagnetic (Table b) analytical expression instead of the exact expression Eq. (7), assuming $\lambda \to \infty$ and ${\mu }_3={\mu }_0$. (b) Fractional error for the thin diamagnetic (Table e) analytical expression, assuming ${\mu }_2={\mu }_3={\mu }_0$ and $z_0/a=0.2$.

Figure 11

(a) Fractional error $|{\phi }_{\mathrm{exact}}-{\phi }_{\mathrm{analytical}}|/{\phi }_{\mathrm{exact}}$ associated with using the bulk paramagnetic analytic expression (Table a), assuming ${\mu }_3={\mu }_0$ and $\lambda \to \infty$. (b) Fractional error for the bulk strong diamagnetic expression (Table c), assuming $t\to \infty$ and ${\mu }_2={\mu }_3={\mu }_0$. (c) Fractional error for the bulk strong diamagnetic expression (Table d), assuming $z_0/a=0.2$ and ${\mu }_2={\mu }_3={\mu }_0$.

Figure 12

Plot of the sum of the errors squared, divided by the global minimum (${\Xi }^2/{\Xi }_{\min }^2$), for the $\delta$-doped SrTiO${}_3$ sample data of Fig. 3b, fit to the thin diamagnetic expression (Table e), projecting the minimum value in the $\Lambda$, $z_0$, $dz/dV$ parameter space, taken along the third axis, onto the $\Lambda -z_0$ plane (a), the $\Lambda -dz/dV$ plane (b), and the $z_0-dz/dV$ plane (c), taking fixed values $a=8.4$ $\mu$m and $b=2.7$ $\mu$m. The global best fit values are $\Lambda =954\mu \text{m}$, $z_0=1.7\mu \text{m}$, and $dz/dV=2.8\mu \text{m}/\mathrm{V}$. The solid symbols and lines are the best fit and 95$\%$ confidence limits for the parameters from a statistical bootstrap analysis.

Figure 13

Plot of the sum of squares error, divided by the global minimum (${\Xi }^2/{\Xi }_{\min }^2$), for fits of the LaNiO${}_3$ data of Fig. 3d to the thin paramagnetic limit expression (Table b), projecting the minimum value in the ${\chi }_{2}t$, $z_0$, $dz/dV$ parameter space, taken along the third axis, onto the ${\chi }_{2}t-z_0$ plane (a), the ${\chi }_{2}t-dz/dV$ plane (b), and the $z_0-dz/dV$ plane (c), taking fixed values $a=8.4\mu$m and $b=2.7$ $\mu$m. The global best fit values are ${\chi }_{2}t=1.3\times {10}^{-5}\mu$m, $z_0=1.9\mu$m, and $dz/dV=2.6\mu$m/V. The solid symbols and lines are most probable values and 95$\%$ confidence limits for the parameters from a statistical bootstrap analysis.