Magnetic fields above the surface of a superconductor with internal magnetism

The author presents a method for calculating the magnetic fields near a planar surface of a superconductor with a given intrinsic magnetization in the London limit. He computes solutions for various magnetic domain boundary configurations and derives relations between the spectral densities of the magnetization and the resulting field in the vacuum half space, which are useful if the magnetization can be considered as a statistical quantity and its features are too small to be resolved individually. The results are useful for analyzing and designing magnetic scanning experiments. Application to existing data from such experiments on ${\mathrm{Sr}}_2\mathrm{Ru}{\mathrm{O}}_4$ shows that a domain wall would have been detectable, but the magnetic field of randomly oriented small domains and small defects may have been smaller than the experimental noise level.

Figure 1

(Color online) Illustration of different one-dimensional domain structures for $\mid M\mid =1$. (a) Correlator of $M$, and small sample of the corresponding real space domain structure (inset). (b) Corresponding spectral functions. The domain width in the red (lower) structure is exponentially distributed with mean 1, which gives $⟨M\left(x\right)M(x+x^{\prime })⟩=e^{-2\mid x^{\prime }\mid }$ and $S_M\left(k\right)=1∕(1+k^2∕4)$. The domain width in the blue (upper) structure has a Gaussian distribution with mean 1 and standard deviation 0.3, leading to an oscillatory correlator and peaks in the spectral function at $k=\pm \pi$. The curves for this case were obtained numerically from an ensemble of ${10}^5$ domains. The dashed lines in (b) represent the propagation factors $e^{-\mid k\mid z}$ for heights $z=2$ and $z=0.2$. For large $z$, only $S_M(k\approx 0)$ is relevant.

Figure 2

(Color online) Field profile at different heights $z$ above a domain wall were $M_z$ changes sign. The curves labeled “smooth” and “sharp domain wall” were obtained from the full solution with $M_z\left(x\right)=M\mathrm{sgn}\left(x\right)(1-e^{-\mid x\mid ∕\left(0.7\lambda \right)})$ and $M_z\left(x\right)=M\mathrm{sgn}\left(x\right)$, respectively. The latter solution and the corresponding large $z$ approximation as shown are given by Table , case (2). The value of the domain boundary width was chosen according to the fit to the numerical results for ${\mathrm{Sr}}_2\mathrm{Ru}{\mathrm{O}}_4$ from Ref. 6. $z=8\lambda$ corresponds to the Hall probe scans from Ref. 15.