Measurement of the penetration depth and coherence length of $\mathrm{MgB}{}_{2}$ in all directions using transmission electron microscopy

Measurement of the penetration depth and coherence length of $MgB_{2}$ in all directions using transmission electron microscopy

J. C. Loudon, S. Yazdi, T. Kasama, N. D. Zhigadlo, and J. Karpinski

Phys. Rev. B 91, 054505 – Published 5 February 2015

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 $MgB_{2}$ , 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 $MgB_{2}$ single crystal cut in the $a c$ plane by focused ion beam milling and tilted to $45^{\circ}$ 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 $Λ_{a b} = 100 \pm 35$ nm and $Λ_{c} = 120 \pm 15$ nm at 10.8 K in a field of 4.8 mT. The large error in $Λ_{a b}$ is a consequence of tilting the sample about $a$ and had it been tilted about $c$ , the errors on $Λ_{a b}$ and $Λ_{c}$ 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 $Λ_{a b}$ using a sample cut in the $a b$ plane. Using this value gives $Λ_{a b} = 107 \pm 8$ nm, $Λ_{c} = 120 \pm 15$ nm, $ξ_{a b} = 39 \pm 11$ nm, and $ξ_{c} = 35 \pm 10$ 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, $30 \times 15 μ m$ 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.

Received 16 July 2014
Revised 30 November 2014

DOI:https://doi.org/10.1103/PhysRevB.91.054505

Authors & Affiliations

J. C. Loudon^1,*, S. Yazdi², T. Kasama², N. D. Zhigadlo³, and J. Karpinski^3,4

¹Department of Materials Science and Metallurgy, 27 Charles Babbage Road, Cambridge, CB3 0FS, United Kingdom
²Centre for Electron Nanoscopy, Technical University of Denmark, DK-2800 Kongens Lyngby, Denmark
³Laboratory for Solid State Physics, ETH Zurich, Otto-Stern-Weg 1, CH-8093, Zurich, Switzerland
⁴Institute of Condensed Matter Physics, EPFL, 1015-Lausanne, Switzerland

^*j.c.loudon@gmail.com

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Vol. 91, Iss. 5 — 1 February 2015

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Figure 1
(a) Flux vortex with its axis normal to the $a c$ plane. The core (shaded) is elliptical with dimensions $ξ_{a b}$ and $ξ_{c}$ . Supercurrents $J$ follow ellipses with the same aspect ratio, diminishing on length scales $Λ_{c}$ and $Λ_{a b}$ . Streamlines of $J$ are also contours of magnetic flux density $B$ . (b) Experimental arrangement for imaging flux vortices. The electron beam is deflected by the component of $B$ normal to the beam so vortices appear as black-white features in an out-of-focus image. (c)–(f) Simulated images with defocus $Δ f = 7.15$ mm for a flux vortex in a 180-nm-thick specimen in the orientation shown in (a) but tilted $45^{\circ}$ about $a$ . Contrast values (see text) are shown below each image. (c) $Λ_{a b} = 100$ nm, $Λ_{c} = 120$ nm, $ξ_{V} = 34$ nm, (d) $Λ_{a b} = 100$ nm, $Λ_{c} = 120$ nm, $ξ_{V} = 1$ nm, (e) $Λ_{a b} = 200$ nm, $Λ_{c} = 120$ nm, $ξ_{V} = 34$ nm, and (f) $Λ_{a b} = 100$ nm, $Λ_{c} = 200$ nm, $ξ_{V} = 34$ nm.
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Figure 2
The relationship between the coordinates $X, Y, Z$ referring to the specimen and the microscope coordinates $x, y, z$ . The $y$ and $Y$ axes are normal to the other two axes and point in the direction given by the right-hand rule. The axis of the vortex is parallel to $Z$ and the $B$ field enters the specimen at the bottom and exits at the top. The electron beam is parallel to $z$ .
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Figure 3
The relationship between the coordinates $X, Y, Z$ , referring to the specimen (shown by the grey rectangle) and the microscope coordinates $x, y, z$ .
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Figure 4
Contours of the phase shift spaced by 0.2 rad illustrating the projected $B$ field from a flux vortex with $Λ_{a b} = 100$ nm and $Λ_{c} = 120$ nm and (a) $ξ_{V} = 1$ nm and (b) $ξ_{V} = 36$ nm. The red lines indicate the surface of the specimen, which is 128-nm thick. The $a$ axis is parallel to $x$ with $c$ pointing into the page.
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Figure 5
(a)–(f) Defocus series showing flux vortices in $MgB_{2}$ at 10.8 K in a field of 4.8 mT with defoci: (a) $Δ f = - 21.4$ , (b) $- 14.3$ , (c) $- 7.15$ , (d) $7.15$ , (e) $14.3$ , and (f) $21.4$ mm. (g) Average vortex images at each defocus level. (h) Average simulated images. (i) Difference images between experiment and simulation. (j) Line scans across the images from the region shown by the red box in (h). Black lines show experimental data and red lines the fit.
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Figure 6
Difference images between the average defocus series and simulated images as each parameter is varied. (a) Difference images for the best-fit values of the parameters. (b) Difference images when $Λ_{a b}$ is varied: the top row shows difference images when $Λ_{a b}$ is set at two error bars below the best-fit value. The next row is for $Λ_{a b}$ set one error bar below. The next two rows are for $Λ_{a b}$ increased one and two error bars above the best-fit value, respectively. (c) A similar series showing the effect of changing $Λ_{c}$ . (d) Series showing the effect of changing $ξ_{V}$ . (e) Reduced $χ^{2}$ values as $Λ_{a b}, Λ_{c}$ , and $ξ_{V}$ are adjusted. Acceptable values of $χ^{2}$ lie between the dashed lines.
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Physical Review B

covering condensed matter and materials physics

Measurement of the penetration depth and coherence length of $MgB_{2}$ in all directions using transmission electron microscopy

J. C. Loudon, S. Yazdi, T. Kasama, N. D. Zhigadlo, and J. Karpinski

Phys. Rev. B 91, 054505 – Published 5 February 2015

Abstract

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Physical Review B

covering condensed matter and materials physics

Measurement of the penetration depth and coherence length of MgB2 in all directions using transmission electron microscopy

J. C. Loudon, S. Yazdi, T. Kasama, N. D. Zhigadlo, and J. Karpinski

Phys. Rev. B 91, 054505 – Published 5 February 2015

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Measurement of the penetration depth and coherence length of $MgB_{2}$ in all directions using transmission electron microscopy