Visualizing the morphology of vortex lattice domains in a bulk type-II superconductor

Alike materials in the solid state, the phase diagram of type-II superconductors exhibit crystalline, amorphous, liquid and spatially inhomogeneous phases. The multitude of different phases of vortex matter has thence proven to act as almost ideal model system for the study of both the underlying properties of superconductivity but also of general phenomena such as domain nucleation and morphology. Here we show how neutron grating interferometry yields detailed information on the vortex lattice and its domain structure in the intermediate mixed state of a type-II niobium superconductor. In particular, we identify the nucleation regions, how the intermediate mixed state expands, and where it finally evolves into the Shubnikov phase. Moreover, we complement the results obtained from neutron grating interferometry by small-angle neutron scattering that confirm the spatially resolved morphology found in the intermediate mixed state, and very small-angle neutron scattering that confirm the domain structure of the vortex lattice.

: TI and DFI results of an ultra-pure Niobium rod as a function of increasing magnetic field after ZFC to T = 4 K. The field was applied perpendicular to the neutron beam. The contour of the sample is indicated by the yellow dashed boxes. The TI (top row) and DFI (bottom row) results are normalized with the results with µ 0 H = 0 mT, hence only the pure magnetic contribution from the VL is visualized. As the VL is rotated away from the Bragg condition no TI contrast arises. However, USANS scattering at the IMS domains, still causes a pronounced DFI contrast. The scale bar corresponds to 5 mm.

Supplementary Note 1: On the origin of the contrast variation in the transmission image
In the following paragraphs we explain how a slight bending of the VL due to demagnetization effects and Lorentz forces acting on the vortices leads to the line-shaped contrast which is observed in the transmission image (TI) presented in the top row of Figure 5.
The TI data of the Nb rod shown in Fig. 5 were taken in different magnetic fields between 0 and 205 mT after ZFC to 4 K. They reveal the emergence of a vertically aligned, line-shaped contrast variation in magnetic fields above 89 mT which slightly changes its morphology in fields higher than 123 mT.
The TI for fields above 0 mT were normalized to the data for 0 mT. Hence, the observed TI contrast variation is of magnetic origin only. It could either be generated by USANS scattering off the domain structure of the IMS (typical domain size of approx. 10 µm) or by SANS scattering off the VL within the IMS domains (lattice spacing 160 nm). For the employed SDD of 300 mm, USANS and SANS scattering on the length scales mentioned above lead to a neutron deflection at the detector position of 0.01 mm and 0.77 mm, respectively. A possible USANS scattering contribution of 0.01mm resulting from the IMS domain structure lies much below the detector resolution of 0.5 mm and hence cannot be resolved. This proves that the TI contrast originates solely from SANS scattering off the VL.
However, SANS scattering will only occur if the angle enclosed by the incoming neutron beam and the orientation of the VL fulfils the Bragg condition. In this case, neutrons which are locally deflected away from their original direction will lead to a reduced intensity (TI <1) at the corresponding detector position. As direct consequence, some parts of the detector will measure a higher intensity (TI>1) due to the additional number of neutrons scattered to these positions (see Supplementary Figure 1

a).
For the case of a VL, strictly parallel aligned to the external magnetic field direction over the entire sample volume, a TI signal different than unity would only arise if the VL is rotated to the Bragg angle.
In this case, the TI would decrease at one sample edge and increase at the other as the intensity is just slightly shifted in the horizontal plane. This is obviously not the case for the TI of the Nb rod presented in Fig. 5. To generate the distinct TI contrast variations of high and low intensity streaks seen between 89 and 112 mT, a spatial variation of the VL orientation (mosaic) is required in addition.
The TI pattern can be concisely explained introducing a symmetrical bending of the VL in the Nb rod peak structure was obtained for both VL Bragg peaks. This is in agreement with the proposed symmetrical bending of the VL within the horizontal plane. The double peak structure in the rocking curve evolves to a single peak in fields above 123 mT. This suggests a straightening of the VL in increasing fields, which can be seen in the change of the TI morphology at this field value as well. It coincides with the IMS to Shubnikov transition. The observed distortion effect is of pure intrinsic nature, as field cooling does not essentially change the rocking curves.

Supplementary Note 2: nGI data processing
Both, the TI and the DFI were processed out of a series of 17 images each taken with an exposure time of 90 seconds. The series of 17 images corresponds to an equidistant stepping of G 0 over one period along the x-direction perpendicular to the grating lines (compare Fig. 4). Thereby the intensity I(m,n,x) in each pixel (m,n) in the detector plane oscillates sinusoidally [41]. To analyse these changes quantitatively, we write the intensity oscillation for each detector pixel in a Fourier series I(m,n,x) = Σ i a i (m,n)cos(ikx g +Φ 1 (m,n))≈ a 0 (m,n) + a 1 (m,n) cos(kx g +Φ 1 (m,n)) (1) where a i are the amplitude coefficients, Φ i the corresponding phase shifts, k = 2π/p 2 , with p 2 being the periodicity of G 2 .
The TI of the specimen in each detector pixel is given by TI(m n) = a 0 (sample) / a 0 (reference). (2) The DFI of the sample in each detector pixel is given by the normalized visibility where the visibility V(m,n) is defined by V(m,n) = (I max -I min ) / (I max + I min ) = a 1 / a 0 (4).
Hence the normalized DFI is given by DFI(m,n) = a 1 (sample) · a 0 (reference) / (a 0 (sample) · a 1 (reference)) Simply speaking, the zeroth order Fourier component (offset of the oscillations) provides the TI signal, whereas the first Fourier component (amplitude of the oscillations) provides the DFI signal.

Supplementary Note 3: The sensitivity of the nGI setup
The dark field contrast as defined in Equation 5 describes the local degradation of the nGI interference pattern caused by neutrons been scattered within the sample. In the following we explain how the measured DFI contrast is linked to the samples microstructure and its scattering function S(q).
Furthermore, based on simplified considerations we will define the sensitivity of nGI to different sized structures which is necessary to understand the origin of the DFI contrast in the superconducting Nb sample.
Several attempts exist to determine how the nGI interference pattern is influenced by the change of They found, that the visibility exponentially decreases with a dark-field extinction coefficient (DFEC) which depends on the autocorrelation function of the refraction index (or the scattering length density) within the material.
A more general and instructive description has recently been given by Strobl [3] which we will summarize in the following. In this model the details of wave propagation are disregarded. Instead, the visibility on the detector decreases due to a superposition of i) the intensity modulation generated by neutrons which pass the sample without scattering and ii) by intensity modulations which are phase shifted due to scattering within the sample. In small angle approximation, the phase shift Δω of the latter components is directly proportional to the momentum transfer q caused by the scattering event: with the neutron wavelength λ, an effective sample to detector distance L S and the periodicity of the interference fringes p. The parameter ξ GI hereby defines a correlation length, which is a specific constant for every nGI setup.
For scattering of the whole neutron beam involving only one specific q the visibility would decrease as: However, if the whole scattering function S(q) is taken into account a general solution is derived by integration over all possible q values: Finally, if multiple scattering is taken into account, a general solution of the DFI contrast is obtained as: Therefore, the DFI contrast depends only on the macroscopic scattering cross section of the material Σ, the sample thickness t, and a real space correlation function G. The latter one is the cosine Fourier transformation of the scattering function S(q). It has to be emphasize that despite using different approaches Lynn et al. and Strobl achieve similar results.
Note, that nGI measures a real space correlation at one specific correlation distance ξ GI , instead of a reciprocal space scattering function S(q). Furthermore, these results reveal that there is no direct connection between the structure size and the dark field contrast. Instead, some a priori assumptions about the microstructure (and hence about G) are necessary for a complete modelling of the DFI contrast.
Nonetheless, for the specific case of a diluted systems of identical particles a quantitative definition of the sensitivity to different sized structures is possible. It has been proposed by Lynn et al. to use spherical particles which have a well-known correlation function [4] as reference material. (Note that the term particle-size selectivity is used in [1]). This approach is warranted, as the correlation function G(r) for all diluted systems decays to zero at the longest distance that characterizes the microstructure, which is the diameter in the spherical case. Furthermore the difference in the macroscopic scattering cross section of the particles only multiplies the curve with a constant, leaving its shape unchanged. A more detailed geometrical interpretation of G can be found in [5].
Supplementary Figure 2 shows the calculated DFI contrast of a reference sample consisting a 10 vol% suspension of spherical particles as function of the particle diameter. The contrast was calculated for the used nGI setup parameter based on the formula derived by Lynch [1]. A homogenous sample thickness of 5 mm was assumed. This DFI curve reveals a maximal sensitivity to particles having a size in the range of 1-50 µm, corresponding to the expected size of IMS domains.
The definition of particle-size selectivity via a comparison to microspheres is valid, if no long range order exists in the sample and the systems inter particle distance is large in comparison to ξ GI . We have insofar no evidence that this assumption is not valid within the IMS domain structure as the inter domain distance probed be VSANS seems to be larger than ξ GI and the DFI contrast arises already at 89 mT where a small IMS domain concentration is expected.
In contrast long range order exists within the VL which means that the correlation function has discrete peaks. However, the VL spacing is in the range of 100 nm, thus far away from the probed correlation length ξ GI~3 µm. Hence, no contribution to the DFI besides the cross talk effect discussed below is expected.

Supplementary Note 4: Crosstalk between TI and DFI
For purely absorbing samples the TI shows contrasts smaller than unity. However, for samples that introduce coherent scattering such as the flux line lattice in superconductors, the TI signal locally shows values larger than unity as shown in Fig. 5 (white lines).
Referring to Eq. 5, the DFI remains smaller than unity as long as the zeroth order Fourier coefficient for the sample is smaller than unity, which is the case if no coherent scattering is present in the sample. However, it is obvious that coherently scattered neutrons locally lead to a larger amplitude a 0 (sample). This larger value directly influences the calculated DFI signal as seen in Eq. 5. Under the assumption that the absorption of the sample is negligible, which holds in our case as the data are zero-field normalized, as well as no USANS scattering in the sample is present (a 1 = const.) , the TI and the DFI signals are related by DFI Crosstalk = (a 1 (sample) / a 1 (reference) )·( a 0 (reference) / a 0 (sample) ) = 1 / TI.
According to Eqn. 10 the cross talk of the TI to the DFI can therefore be quantified by DFI Crosstalk = 1/TI, namely as the inverted signal of the TI.
In case of the nGI data as presented in Fig. 5 the crosstalk from the TI to the DFI can be better seen by taking horizontal line profiles through the individual TI and DFI images for each magnetic field.
These line profiles show that the crosstalk can be directly compared as shown in Supplementary    The presented data further supports the proposed peculiar IMS distribution as other contributions to the DFI contrast than USANS scattering are ruled out. Hence, a perpendicular geometry should always be considered, if one is interested in the domain distribution only and not in the morphology of the underlying VL.