Subsurface bending and reorientation of tilted vortex lattices in the bulk due to Coulomb-like repulsion at the surface

TL;DR

This study investigates how vortex lattices in superconducting crystals bend and reorient beneath the surface due to Coulomb-like repulsion effects, revealing surface-bulk interactions in tilted magnetic fields.

Contribution

It demonstrates that vortices exit perpendicularly and are bent beneath the surface, with surface Coulomb repulsion significantly influencing bulk vortex structure at large tilt angles.

Findings

Vortices exit perpendicular to the surface.

Vortex structures are bent beneath the surface.

Coulomb-like repulsion affects vortex orientation at large tilt angles.

Abstract

We study vortex lattices (VLs) in superconducting weak-pinning platelet-like crystals of $\beta$-Bi$_{2}$Pd in tilted fields with a Scanning Tunneling Microscope. We show that vortices exit the sample perpendicular to the surface and are thus bent beneath the surface. The structure and orientation of tilted VL in the bulk are, for large tilt angles, strongly affected by Coulomb-type intervortex repulsion at the surface due to stray fields.

Equations

$${\mathbf{u}}_1=[\cos \alpha ,\sin \alpha ],{\mathbf{u}}_2=\left[\cos \left(\alpha +\frac{\pi }{3}\right),\sin \left(\alpha +\frac{\pi }{3}\right)\right].$$

$${\mathbf{v}}_1=\left[\frac{\cos \alpha }{\cos \theta },\sin \alpha \right],{\mathbf{v}}_2=\left[\frac{\cos (\alpha +\pi /3)}{\cos \theta },\sin \left(\alpha +\frac{\pi }{3}\right)\right].$$

$$\frac{{\varphi }_0^2}{4{\pi }^2}\sum _{m,n}{}^{\prime }\frac{1}{R_{mn}},$$

$$F_s$$

$$S$$

$$(0,1),(1,0),(0,-1),(-1,0),(-1,1),(1,-1).$$

Full text

Vortices in superconductors are predicted to bend in order to orient perpendicular to boundaries between superconducting and normal areas or to the interface with vacuum Brandt [1993], Martynovich [1994]. Physically, the bending is due to the vortex supercurrent loops which, on one hand, should be parallel to the surface, while on the other, tend to be in plane perpendicular to the vortex axis (in isotropic materials). To the best of our knowledge, such bending has never been verified experimentally. We provide Scanning Tunneling Microscopy (STM) data that cannot be interpreted in any other way—vortices exit the sample perpendicular to the surface. Our data also show that in weak-pinning crystals the VL structure and orientation are affected not only by the intervortex interactions in the bulk, but also by interactions at the sample surface.

We study vortex lattices (VLs) at the plane surface of platelet-like single crystals of $\beta$-Bi2Pd with $T_{c}=5\,$K Imai et al. [2012], Alekseevski et al. [1954]. The crystal is tetragonal, although the Fermi surface has sheets of mixed orbital character that lead to a near-isotropic macroscopic behavior Shein and Ivanovskii [2013], Sakano and et al [2015], Coldea [2016]. Upper critical fields along the basal plane and perpendicular to it differ by barely 25% from $H_{c2,ab}=$ 0.7 T to $H_{c2,c}=$ 0.53 T (at low temperatures), leading to coherence lengths $\xi_{c}=19$ nm and $\xi_{ab}=24$ nm Kačmarčík et al. [2016], Herrera et al. [2015]. Estimates of the penetration depths from the data on the lower critical field give $\lambda_{ab}=105$ nm and $\lambda_{c}=132$ nm Kačmarčík et al. [2016].

We use a home-built STM/S attached to the dilution refrigerator inserted in a three axis vector magnet reaching 5 T in the $z$ direction and 1.2 T for $x$ and $y$ Suderow et al. [2011], Galvis et al. [2015]. $\beta$-Bi2Pd crystals (3 $\times$ 3 $\times$ 0.5 $mm^{3}$) are mounted with the $c$-axis along the $z$ direction of the magnet. The other two crystalline orientations with respect to $x$ and $y$ of the magnet are found by scanning the surface with atomic resolution to find the square Bi lattice as outlined in Herrera et al. [2015], where the crystal growth is also described. The surface consists of large atomically flat areas of several hundreds nm in size, separated by step edges Herrera et al. [2015]. We use an Au tip cleaned and atomically sharpened in-situ by repeated indentations onto Au sample Rodrigo et al. [2004]. VL images are obtained by mapping the zero-bias conductance normalized to voltages above the superconducting gap Guillamón et al. [2008]. All measurements are done at $T=150$ mK. Data are usually taken within field-cooled protocol, although, due to weak vortex pinning of this material, we find the same results when changing the magnetic field at low temperatures. No filtering or image treatment is applied to the conductance maps shown below.

VLs in fields along $c$ are hexagonal up to $H_{c2}$ with one of the VL vectors along $a$ or $b$ of the tetragonal crystal Herrera et al. [2015]. This gives a two-fold degeneracy in the VL orientation Gränz et al. [2016]. Hence we observe domains of differently oriented VLs; the spatial distribution of domains is random and determined by the pinning landscape. In some tetragonal materials in fields along $c$, the four-fold-symmetric nonlocal corrections to the London theory modify the isotropic repulsion and lead to two degenerate rhombic VLs which at large fields transform to the square VL Eskildsen et al. [1998], Yethiraj et al. [1999]. One of the requirements for the nonlocal corrections to work is a large Ginzburg-Landau parameter $\kappa$ Kogan et al. [1997]. In our crystals, $\kappa\sim 4$ and we do not observe VL transitions.

Let us consider the vortex core shapes at the surface. At $\bm{H}\parallel\bm{c}$, the cores have a circular shape of a size $\xi_{ab}\approx 24$ nm at 0.3 T (see discussions Herrera et al. [2015], Fente et al. [2016]) shown in Fig. 1 as a white circle. If the vortex in a tilted field would have arrived to the surface being straight without any bending, the expected core shape at the surface would be an ellipse with the minor and major semiaxes of 24 nm and $24/\cos\theta$  nm ($\theta$ is the angle between $\bm{H}$ and $\bm{c}$). For $\theta=80^{\circ}$ we would obtain the white ellipse shown in the right panel of Fig. 1. Instead we find the vortex core of the same shape and size as for $\bm{H}$ normal to the surface as shown by the circle at the right panel of Fig. 1. Thus, our images show that vortices must bend under the surface to exit the sample being perpendicular to the surface.

Vortex bending is expected to occur over a length of the order of the penetration depth $\lambda\approx 100$ nm Brandt [1993], Martynovich [1994], which is large relative to the core size of $24$ nm. Hence, we do not expect that the electronic density of states at the surface is influenced by the bent part of the vortex deep underneath.

The surface VLs are shown in Fig. 2(a) for a few tilts $\theta$. The panel 2(b) shows that the density of vortices at the surface goes as $\cos\theta$, as expected.

As mentioned, the material of our interest is nearly isotropic. The model we offer to describe VLs in tilted fields is isotropic. The model predictions, by and large, agree with the STM data. Within this model, the VL in the vortex frame of an infinite sample is hexagonal and degenerate: the angle $\alpha$ shown on the left of Fig. 2(c) can be taken as the degeneracy parameter. The circle where all nearest neighbors are situated has a radius $a$ fixed by the flux quantization, $a^{2}=2\phi_{0}/\sqrt{3}H$. We use the vortex coordinate frame $(x,y,z)$ with $z$ along the vortex direction and the $x$ axis in the tilt plane. For a given $\alpha$, the VL unit cell vectors (in units of $a$) are

[TABLE]

In a sample much thicker than $\lambda$, the VL structure in the bulk is dominated by the bulk interactions, i.e. the VL is still hexagonal. However, the degeneracy is removed in tilted fields by the surface contribution to the interaction.

To evaluate this contribution, we note that due to subsurface bending the point of vortex exit is shifted relative to the exit point of a straight vortex. In small fields when the vortices are well separated, each one will experience the same shift. We assume that shifts are the same also in fields of our interest. In particular, this implies that the density of bent vortices at the surface is the same as if vortices were straight; this is consistent with the macroscopic boundary condition for the normal component of the magnetic induction $H\cos\theta$. Hence, the arrangement of vortices at the surface is just shifted relative to the VL which would have been there without subsurface bending. Then, considering the VL structure, one can disregard the bending and the bulk nearest neighbors will be situated at the cros-section of a circular cylinder of radius $a$ with the flat surface, i.e. at the ellipse with semi-axes $a/\cos\theta$ and $a$, the right panel of Fig. 2c. Taking again the $x$ axis of the surface frame in the tilt plane, one obtains new unit cell vectors at the surface (in units of $a$):

[TABLE]

In particular, the angle $\alpha_{s}$ between $\bm{v}_{1}$ and ${\bm{\hat{x}}}$ is related to the parameter $\alpha$ by $\tan\alpha_{s}=\tan\alpha\cos\theta$. All vortex positions at the surface ${\bm{R}}_{mn}=m{\bm{v}}_{1}+n{\bm{v}}_{2}$ ($m,n$ are integers) can be expressed in terms of $\theta$ and $\alpha$ (or $\alpha_{s}$).

Interaction of vortices at the surface is due to stray fields out of the sample, which can be approximated by point “monopoles” producing the magnetic flux $\phi_{0}$ in the free space within the solid angle $2\pi$. The interaction energy of the vortex at the origin with the rest is

[TABLE]

where $\sum^{\prime}$ is over all $m,n$ except $m=n=0$. The surface vortex density is $H\cos\theta/\phi_{0}$, so that surface interaction per cm2 is

[TABLE]

It is readily checked that $\partial S/\partial\alpha=0$ at $\alpha=-\pi/6$ and $\pi/3$. The corresponding structures in the vortex frame are hexagons, called hereafter $A$ and $A^{\prime}$; in $A$ two out of six nearest neighbors are at $y$ axis, in $A^{\prime}$ they are at $x$.

The sum $S$ is divergent and as such depends on the summation domain. We, however, are interested only in the angular dependence of $S(\alpha)$, because of its role in removing the VL degeneracy in the bulk. The angular dependence arises mostly due to vortices in the vicinity of the central one, because the number of far-away vortices grows with the distance and their contribution to the interaction is nearly isotropic. Our strategy for evaluation of $S(\alpha)$ is based on the fact that the Coulomb interaction out of the sample is isotropic and therefore we can do the summation within a circle $R_{mn}<L$, where $L$ is large enough to include a few “nearest-neighbor shells” of vortices surrounding the one at the origin. To provide a smooth truncation we add a factor $e^{-R^{2}_{mn}/L^{2}}$ to the summand of Eq. (5).

Results of these calculations are given in Fig. 3 for $\theta\approx 80^{\circ},70^{\circ}$ and $60^{\circ}$; smaller tilt angles are discussed in the supplemental material.

Clearly, the structure $A$ ($\alpha=-\pi/6$) is unstable. The minimum energy for $\theta=70^{\circ}$ and $60^{\circ}$ is at $\alpha=0$, so that the preferred structure is $A^{\prime}$. For $\theta=80^{\circ}$ the minimum is shifted to $\alpha\approx-0.1$. These qualitative conclusions do not change if one takes a larger radius $L$ of the summation domain, notwithstanding the increase of the calculated surface energy $F_{s}$.

To show that our model describes the data well, and in particular to check again that the bulk hexagonal VL projects onto the surface as if the vortices were straight, one can follow the bulk nearest neighbors and their surface projections. The six nearest neighbors in the vortex frame correspond to the pairs $(m,n)$:

[TABLE]

At the surface, these pairs mark six vortex positions situated at $\pm\bm{v}_{1}$, $\pm\bm{v}_{2}$, and $\pm(\bm{v}_{1}-\bm{v}_{2})$, see the sketch in Fig. 2c. These positions at the surface are not necessarily nearest, because at large tilt angles, they form a strongly stretched hexagon, whereas the position $(1,1)$, moves closer to the center.

Let us consider $\theta=80^{\circ}$ for which according to Fig. 3 $\alpha\approx-0.1$ and evaluate distances $d_{1}=|\bm{v}_{1}|$, $d_{2}=|\bm{v}_{2}|$, and $d_{3}=|\bm{v}_{1}-\bm{v}_{2}|$. Skipping algebra, we provide formulas for these distances in the supplemental material. In units of $a$ we obtain $(d_{1},d_{2},d_{3})=(5.68,3.56,2.63)$. Direct measurements at the corresponding image at Fig. 1 give $(d_{1},d_{2},d_{3})\approx(6,3.4,2.4)$ in a good agreement with calculated values. Hence, the nearly degenerate hexagonal VL in the bulk bends as a whole when reaching the surface, just shifting the geometric projection of the tilted hexagonal bulk VL onto the surface.

In Fig. 4a we show results for a fixed polar angle (tilt) $\theta=80^{\circ}$ and several azimuthal angles $\varphi$ of the field $\bm{B}$. Interestingly, the surface vortex lattice shows different arrangements depending on $\varphi$. For $\varphi=72^{\circ}$, the surface VL is a nearly perfect square, whereas for $\varphi=317^{\circ}$ it is nearly hexagonal. As we show in the supplementary material, the $\theta$ and $\varphi$ where we expect such a square vortex lattice within our model are close to what we observe experimentally.

In Fig. Fig. 4b we show $\alpha$ for experiments changing the azimuthal angle $\varphi$ (for a polar angle of $\theta=80^{\circ}$). We observe that $\alpha$ varies around the orientation A’. The variations might be caused by the tetragonal symmetry, not included in our model or by weak pinning.

Tilted VLs have been studied using STM and neutron scattering in anisotropic 2H-NbSe2 Bending [1999], Buzdin and Baladié [2002], Koshelev [2005], Hess et al. [1992, 1994] and in materials with extreme anisotropies such as Bi2Sr2CaCu2Oδ. In the latter, Josephson and pancake vortices form crossing lattices and VL properties are more involved than in our case Bending [1999], Buzdin and Baladié [2002], Koshelev [2005]. The tilted VLs in 2H-NbSe2 are distorted hexagons Bolle et al. [1993], Gammel et al. [1994], Hess et al. [1992, 1994], Fridman et al. [2011, 2013] whose orientation, however, disagrees with theoretical expectations. It was proposed that vortex-induced strain of the crystal might explain the data Kogan et al. [1995]. At high tilt angles, buckling transitions produce superlattices with chain-like vortex arrangements Hess et al. [1992, 1994]. Besides, vortex core STM images have star-like shapes in fields along $c$ and acquire comet-like tails in tilted fields with no evidence for vortices exiting the sample perpendicular to the surface Hess et al. [1990], et. al. [(2016]. Clearly, the results shown here are different, most probably due to the fact that $\beta-$Bi2Pd is close to being isotropic, other than those systems.

We note also that high purity Nb crystals were studied in tilted fields Mühlbauer et al. [2009]. The bulk VL shows a variety of transitions, including two-fold structures breaking the crystalline four-fold rotation symmetry and scalene unit cells. In pure Nb, however, the Ginzburg-Landau $\kappa\sim 1$, the London approach does not apply, the microscopic theory becomes a necessity, and interpretation of VL structures is difficult.

In summary, we have studied VLs in tilted fields in $\beta$-Bi2Pd, a nearly isotropic superconductor. We demonstrate that vortices exit the sample being perpendicular to the surface, that necessitates the subsurface bending of vortex lines. We find that intervortex Coulomb-like repulsion at the surface due to stray fields removes the degeneracy of the bulk hexagonal VLs thus fixing the bulk VL orientation. It is quite surprising to have a highly ordered VLs at the surface whereas under the surface all vortices are bent.

Authors are grateful to P.C. Canfield for discussions and for having proposed growth of single crystals of $\beta-$Bi2Pd and shown how to do that. E.H. is supported by the Departamento Administrativo de Ciencia, Tecnología e Innovación, COLCIENCIAS (Colombia) Programa Doctorados en el Exterior Convocatoria 568-2012. I.G. is supported by the ERC (grant agreement 679080). This work was also supported by the Spanish MINECO (FIS2014-54498-R, MAT2014-52405-C2-02), by the Comunidad de Madrid through program NANOFRONTMAG-CM (S2013/MIT-2850) and by Axa Research Funds. We also acknowledge SEGAINVEX workshop of UAM, Banco Santander and COST MP1201 action, and the EU through grant agreements FP7-PEOPLE-2013-CIG 618321, 604391 and Nanopyme FP7-NMP-2012-SMALL-6 NMP3-SL 2012-310516. V.K. is supported by the U.S. Department of Energy, Office of Science, Basic Energy Sciences, Materials Sciences and Engineering Division. The Ames Laboratory is operated for the U.S. DOE by Iowa State University under Contract No. DE-AC02-07CH11358.