Measurement of the penetration depth and coherence length ofMgB2in all directions using transmission electron microscopy

Abstract

We demonstrate that images of flux vortices in a superconductor taken with a transmission electron microscope can be used to measure the penetration depth and coherence length in all directions at the same temperature and magnetic field. This is particularly useful for $\mathrm{MgB}{}_{2}$, where these quantities vary with the applied magnetic field and values are difficult to obtain at low field or in the $c$ direction. We obtained images of flux vortices from a $\mathrm{MgB}{}_{2}$ single crystal cut in the $ac$ plane by focused ion beam milling and tilted to ${45}^{\ensuremath{\circ}}$ with respect to the electron beam about the crystallographic $a$ axis. A new method was developed to simulate these images that accounted for vortices with a nonzero core in a thin, anisotropic superconductor and a simplex algorithm was used to make a quantitative comparison between the images and simulations to measure the penetration depths and coherence lengths. This gave penetration depths ${\ensuremath{\Lambda}}_{ab}=100\ifmmode\pm\else\textpm\fi{}35$ nm and ${\ensuremath{\Lambda}}_{c}=120\ifmmode\pm\else\textpm\fi{}15$ nm at 10.8 K in a field of 4.8 mT. The large error in ${\ensuremath{\Lambda}}_{ab}$ is a consequence of tilting the sample about $a$ and had it been tilted about $c$, the errors on ${\ensuremath{\Lambda}}_{ab}$ and ${\ensuremath{\Lambda}}_{c}$ would be reversed. Thus obtaining the most precise values requires taking images of the flux lattice with the sample tilted in more than one direction. In a previous paper [J. C. Loudon et al., Phys. Rev. B 87, 144515 (2013)], we obtained a more precise value for ${\ensuremath{\Lambda}}_{ab}$ using a sample cut in the $ab$ plane. Using this value gives ${\ensuremath{\Lambda}}_{ab}=107\ifmmode\pm\else\textpm\fi{}8$ nm, ${\ensuremath{\Lambda}}_{c}=120\ifmmode\pm\else\textpm\fi{}15$ nm, ${\ensuremath{\xi}}_{ab}=39\ifmmode\pm\else\textpm\fi{}11$ nm, and ${\ensuremath{\xi}}_{c}=35\ifmmode\pm\else\textpm\fi{}10$ nm, which agree well with measurements made using other techniques. The experiment required two days to conduct and does not require large-scale facilities. It was performed on a very small sample, $30\ifmmode\times\else\texttimes\fi{}15 \ensuremath{\mu}\mathrm{m}$ and 200-nm thick, so this method could prove useful for superconductors where only small single crystals are available, as is the case for some iron-based superconductors.