New Journal of Physics 20, 023005 (2018)

Structural transitions in vortex systems with anisotropic interactions

M. W. Olszewski^1,2, M. R. Eskildsen², C. Reichhardt¹ and C. J. O. Reichhardt¹

¹ Theoretical Division, Los Alamos National Laboratory, Los Alamos, New Mexico 87545 USA

² Department of Physics, University of Notre Dame, Notre Dame, Indiana 46656, USA

Keywords: superconducting vortex, anisotropic interactions, structural transitions
Received 12 September 2017
Revised 30 September 2017
Accepted for publication 29 December 2017
Published 5 February 2018

Abstract

We introduce a model of vortices in type-II superconductors with a four-fold anisotropy in the vortex-vortex interaction potential. Using numerical simulations we show that the vortex lattice undergoes structural transitions as the anisotropy is increased, with a triangular lattice at low anisotropy, a rhombic intermediate state, and a square lattice for high anisotropy. In some cases we observe a multi-q state consisting of an Archimedean tiling that combines square and triangular local ordering. At very high anisotropy, domains of vortex chain states appear. We discuss how this model can be generalized to higher order anisotropy as well as its applicability to other particle-based systems with anisotropic particle-particle interactions.

1. Introduction
2. Methods
3. Results and discussion
4. Summary
References

1 Introduction

A type-II superconductor subjected to a magnetic field forms vortices that each carry one quantum of magnetic flux. Due to their mutual repulsion, these vortices arrange themselves in an ordered vortex lattice (VL) as long as the vortex-vortex interactions dominate external influences such as pinning by impurities or thermal disordering. The VL is triangular in isotropic superconductors [1]; however, anisotropy in the vortex-vortex interactions can cause changes in the VL symmetry [2]. One of the most studied systems exhibiting a transition to a square VL is borocarbide materials [3,4,5,6,7,8,9,10]. Transitions to square lattices have also been observed in high temperature superconductors [11,12,13,14], heavy fermion materials [15,16,17,18], and spin triplet superconductors such as Sr₂RuO₄ [19,20,21]. Additionally, transitions from triangular to square VLs can arise in other superfluid systems including Bose-Einstein condensates [22] and dense nuclear matter in extreme conditions [23].

In superconducting systems, theoretical approaches used to study the VL structural transitions include modifications to the London model [24,25], addition of four-fold symmetric terms to the Ginzburg-Landau free energy [26], Eilenberger theory [27], modified Ginzburg-Landau approaches [28], and modifications to the vortex interactions produced by strain fields [29]. Notably absent from this list are molecular dynamics (MD) simulations, which treat the vortices as point particles with bulk Bessel function interactions or thin film Pearl interactions. Until now, MD methods have only been applied to isotropic pairwise potentials, which produce a triangular VL in clean systems [30,31,32,33,34,35,36,37,38,39,40]. One of the issues is that simply adding a stronger radial repulsive force along certain directions does not capture the triangular to square transitions in the vortex system. Instead, it is essential to consider the full anisotropic potential in which nonradial forces also arise. In addition to the equilibrium VL configurations, MD simulations give access to the statics and dynamics of a large number of vortices over long times.

Here we introduce a model for vortices in anisotropic superconductors where the vortex-vortex interaction potential is extended to include a fourfold anisotropy. Using MD simulations we show that as the anisotropy is increased, the VL symmetry changes from triangular to rhombic to square. Additionally, in some cases we find a multi−q lattice state consisting of an Archimedean tiling with a combination of square and triangular local ordering. For very large anisotropy, domains of vortex chains appear. Our model can be generalized to any anisotropy and provides a basis for modeling VL reorientation or other vortex symmetry transitions such as those observed in multigap superconductors [41]. The model can be applied not only to vortices in type-II superconductors, but also to other particle-based systems with anisotropic interactions where triangular to square transitions can arise. These include skyrmion lattices, where triangular to square transitions have recently been observed [42,43], as well as colloidal particles with anisotropic interactions [44,45].

2. Methods

In an isotropic bulk superconductor, the vortex-vortex interaction potential is typically isotropically repulsive and takes the form of a zeroth order Bessel function, U(r) = K₀(r) [46]. For anisotropic materials, we propose the following modified pair potential for the vortex interactions:

U(r,θ) = A_v K₀(r)

⎡
⎣

1 + A_a cos²

⎛
⎝

n_a (θ− ϕ)

⎞
⎠

⎤
⎦

(1)

where r=|r_i−r_j| is the distance between two vortices at positions r_i and r_j. The angle between the two vortices with respect to the positive x axis is given by θ = tan⁻¹(r_y/r_x) where r=r_i−r_j, r_x=r·∧x, and r_y=r·∧y. The rotation angle of the anisotropic axes is defined to be ϕ, and n_a is the order of the anisotropic interaction. The prefactor A_v represents the strength of the isotropic component of the vortex-vortex pair interaction force, and A_a controls the amplitude of the anisotropic interaction.

Figure 1: Equipotential lines (a,c,e) and force fields (b,d,f) for the vortex-vortex interaction potential in Eq. (1) with n_a=4 and ϕ = 45^°. (a,b) Isotropic case with anisotropy strength A_a = 0. (c,d) At A_a = 0.1, nonradial forces begin to appear. (e,f) At A_a = 0.25 the nonradial forces are stronger.

To fully understand the manner in which Eq. (1) produces nonradial interactions, we examine the nature of the anisotropic forces generated by U(r). For nonzero values of A_a, the potential is stronger along certain interaction directions and weaker between these directions. The anisotropic order n_a determines the number of strong interaction directions, which are evenly spaced in θ. For example, for n_a=6 and ϕ = 0, the interaction potential passes through local maxima at θ = 0^°, 60^°, 120^°, 180^°, 240^°, and 300^°. Setting ϕ = 15^°, the local maxima of the interaction potential are shifted to θ = 15^°, 75^°, 135^°, 195^°, 255^°, and 315^°. Between these values of θ, the interaction strength varies as a cosine. The force field from the potential is F_vv = −∇U = (− ∂U/∂x, − ∂U/∂y) with components

\fl F_x = A_v

⎡
⎣

cos(θ) K₁(r)

⎛
⎝

1+ A_a cos²

⎛
⎝

n_a(θ− ϕ)

⎞
⎠

−

sin(θ)

K₀(r)

n_a A_a

sin( n_a (θ− ϕ))

⎤
⎦

(2)

\fl F_y = A_v

⎡
⎣

sin(θ) K₁(r)

⎛
⎝

1 + A_a cos²

⎛
⎝

n_a(θ− ϕ)

⎞
⎠

cos(θ)

K₀(r)

n_a A_a

sin(n_a(θ− ϕ))

⎤
⎦

(3)

In the present work we focus on the n_a = 4 interaction potential with fourfold anisotropy; however, by changing n_a our model can be applied to twofold (n_a = 2), sixfold (n_a = 6), or other degrees of anisotropy. In Fig. 1(a,b) we plot the equipotential lines and force field for U(r) from Eq. 1 for the isotropic case with A_a = 0, where the equipotential lines are circularly symmetric and the forces are strictly radial. At A_a = 0.1 in Fig. 1(c,d), the fourfold symmetry is apparent and weak nonradial forces appear. In Fig. 1(e,f) at A_a = 0.25, the fourfold symmetry is much more pronounced and the nonradial forces are clearly visible.

To investigate the VL ground states that emerge as the anisotropy of the potential increases, we perform MD simulations of N=441 vortices in a two-dimensional system of size L ×L with L=36λ and periodic boundary conditions in the x and y directions. Distances are measured in units of the London penetration depth λ. The dynamics of vortex i is governed by an overdamped equation of motion:

d r_i

= Fⁱ_vv + Fⁱ_T.

(4)

Here η is the damping constant which we set equal to unity. Thermal forces are modeled by Langevin kicks F_Tⁱ which have the properties 〈F_T〉 = 0.0 and 〈Fⁱ_T(t)F^j_T(t^′)〉 = 2ηk_BT δ_ijδ(t − t^′) where k_B is the Boltzmann constant. We perform simulated annealing by starting in a high temperature molten state and gradually cooling the system to T = 0. We use an initial temperature of F^T = 4.0 and decrement the temperature by ∆F^T = −0.01 every 40,000 simulation time steps, which is long enough to ensure that the system reaches an equilibrium state.

3. Results and discussion

Figure 2: Measures P_n of VL ordering versus temperature F^T during the annealing procedure for P₇ (blue), P₆ (red), P₅ (green), and P₄ (maroon) in samples with A_v = 2.0. (a) In the isotropic system with A_a = 0, the vortices form a triangular lattice with P₆ = 1.0. (b) For A_a = 0.039, rhombic ordering appears. (d) At A_a = 0.099, the vortices form a square lattice with P₄ = 1.0.

We use a Voronoi construction to obtain the local coordination number z_i of each vortex, and compute the fraction P_n of vortices with coordination number n using P_n=N⁻¹∑_i=1^Nδ(z_i−n) for n=4, 5, 6, and 7. Figure 2(a) shows P₄, P₅, P₆, and P₇ versus F^T obtained during the annealing process in an isotropic system with A_a = 0 and A_v = 2.0. Initially the system is in a high temperature molten state, and as F^T is reduced the vortices order into a triangular lattice with P₆ = 1.0. In Fig. 2(b) at A_a = 0.039, the vortices freeze into a state with rhombic ordering, while in Fig. 2(c) at A_a = 0.099, the vortices form a square lattice with P₄ = 1.0. Together, these results show how the equilibrium (T = 0) value of P₆ is suppressed and the value of P₄ grows as the anisotropy is increased. Compared to the drop in P₆, the rise of P₄ is more gradual and happens at a lower F^T.

Figure 3: Real space VL configurations (a,c,e) and corresponding heightfield plots of the structure factor |S(k)| (b,d,f) after annealing in samples with A_v = 2.0. (a,b) The isotropic system with A_a = 0 has triangular ordering and a lattice angle θ_l (marked in white) of θ_l ≈ 60^°. (c,d) At A_a = 0.039, the vortices form a rhombic lattice with θ_l ≈ 76^°. (e,f) At A_a = 0.099, the vortices form a square lattice with θ_l=90^°.

We can further characterize the final F^T=0 state using the structure factor S(k)=N⁻¹|∑_i^Nexp(−ik ·r_i)|² . In Fig. 3(a) we illustrate the final real space vortex positions in a sample with A_a = 0 and A_v = 2.0, and in Fig. 3(b) we plot the corresponding |S(k)| as a heightfield. The lattice angle θ_l is defined to be the angle between adjacent first-order peaks in |S(k)|, as illustrated in Fig. 3(b), and for the A_a = 0 triangular lattice, θ_l ≈ 60^°. The triangular lattice is slightly distorted by our perfectly square simulation box, which gives us the minor deviation from the ideal value θ_l = 60^°. In Fig. 3(c,d), at A_a = 0.039 the vortices form a rhombic lattice with θ_l ≈ 76^°. At A_a = 0.099 in Fig. 3(e,f), a square lattice appears with θ_l = 90^°. In Fig. 4 we plot the lattice angle θ_l versus A_a for samples with A_v = 1.0, 1.5, 2.0, and 2.5. In all cases, at low A_a <~0.3 the vortex ordering is triangular, at intermediate A_a a rhombic lattice structure appears, and at large A_a >~0.5 a square VL emerges. In Fig. 5 we plot a structural phase diagram indicating where the triangular, rhombic, and square VLs appear as a function of A_a versus A_v. Increasing the vortex-vortex interaction through the parameters A_v and/or A_a is equivalent to increasing the vortex density and thus the magnetic field. We therefore define an effective magnetic field as proportional to the two-dimensional integration of the interaction potential in Eq. (1)

B_eff ∝

⌠
⌡

∞

0

r dr

⌠
⌡

2 π

0

dθ U(r,θ) = 2πA_v

⎛
⎝

1 +

A_a

⎞
⎠

(5)

Increasing the magnetic field correspond to an upward trajectory in the phase diagram, perpendicular to the equal field contours given by A_v ∝ (1 + A_a/2)⁻¹ and shown in Fig. 5. This gives rise to a triangular-rhombic-square transition with increasing field, in agreement with experimental observations.

Figure 4: Lattice angle θ_l versus A_a for systems with (a) A_v = 1.0, (b) A_v = 1.5, (c) A_v = 2.0, and (d) A_v = 2.5. In all cases, the vortices form a triangular VL at small A_a with θ_l ≈ 60^°, pass through an intermediate rhombic state with θ_l ≈ 75^°, and then form a square lattice with θ_l ≈ 90^° at large A_a.

Figure 5: Lattice ordering phase diagram as a function of A_a versus A_v. Blue: triangular order; red: square order; yellow and green shades: rhombic order, which is centered around A_a = 0.04. The white lines are equal field contours as described in the text.

Figure 6: Multi-q Voronoi constructions for the real space vortex positions (a,c) and the corresponding structure factor |S(k)| (b,d) in samples with A_a = 0.04 at A_v = 2.0 (a,b) and A_v = 2.8 (c,d). The Voronoi polygons indicate that there is a combination of square and triangular ordering in the form of an Archimedean tiling, producing multiple peaks in |S(k)|.

For some combinations of A_a and A_v what we term a multi-q state appears in which the vortices exhibit simultaneous square and triangular ordering. Figure 6 shows both the real space Voronoi polygons and the corresponding |S(k)| for multi-q states in samples with A_a = 0.04. For both A_v=2.0 in Fig. 6(a,b) and A_v=2.8 in Fig. 6(c,d) we find the same combination of square and triangular ordering in the Voronoi polygons, while |S(k)| exhibits multiple sets of peaks. As illustrated in Fig. 6, the multi-q ordering can be oriented along either the x or the y direction. The multi-q VL structure closely resembles an Archimedean tiling in which space is filled with a combination of square and triangular tiles [47]. Archimedean ordering of this type has also been observed for colloidal assemblies driven over quasiperiodic substrates, where it arises due to the competition between the ordering imposed by the substrate and the triangular ordering that minimizes the colloid-colloid interaction energy [48,49] . In our system the competition responsible for producing this structure is between the square and triangular orderings favored by the anisotropy. We note that the multi-q state only appears occasionally in regions of A_a and A_v that are dominated by the rhombic state, suggesting that it could be metastable. Additionally, the two different orientations of the multi-q state that we find indicate that in diffraction experiments on macroscopic samples, domains of different orientations will most likely coexist. This would smear out the peaks in |S(k)|, making it difficult to deconvolute the signal from an individual domain orientation. As a result, local probe imaging techniques may be the best method for observing multi-q states.

Figure 7: Real space vortex configurations showing the emergence of chain states in systems with A_v=2.0. At A_a = 0.1 (a) the VL is square. For A_a = 0.6 (b) the square lattice develops some local distortions and dislocations. At A_a = 1.3 (c) chain state domains appear, and these become more pronounced for A_a = 3.0 (d).

For anisotropies A_a larger than those discussed above, we find that the square lattice gradually transforms into domains of vortex chains. This process is illustrated in Fig. 7 for systems with A_v = 2.0 where A_a is increased from A_a=0.1 to A_a=3.0. Although such levels of anisotropy may seem unphysically large, there have been several observations of vortex chain states, including domains of chains in different borocarbides and in Sr₂RuO₄ at low fields [50,51,52]. While these observations are typically attributed to an attractive interaction between vortices at intermediate range, it may still be possible to model these chain states by introducing strongly anisotropic vortex-vortex interactions.

4. Summary

We have introduced a model for vortices with anisotropic pairwise interactions, focusing on the case of four-fold symmetry. Using MD simulations we show that this model captures a transition from a triangular lattice at low anisotropy to a square VL at high anisotropy, with an intermediate rhombic phase. We also find that in some cases a multi-q state with Archimedean ordering appears in which the vortices have both square and triangular local ordering. For the highest anisotropy values, domains of vortex chains form. Our model could be applied to study the dynamics near the VL transitions, where nonequilibrium phenomena can arise [53,54]. Additionally, it can be generalized for higher order anisotropy in order to capture other types of symmetry and reorientation transitions in VLs [55,56,57,58,59]. It is also possible to use the model with different isotropic pairwise interactions to investigate hexagonal to square transitions in other particle-based systems, such as skyrmions or colloids with anisotropic interactions.

Acknowledgments

We are grateful to D. Green, M. Lamichhane, X. Ma, D. McDermott and K. Newman for assistance and discussions. This research was supported in part by the Notre Dame Center for Research Computing. M.R.E. was supported by the U.S. Department of Energy, Office of Basic Energy Sciences, under Award No. DE-SC0005051. This work was carried out under the auspices of the NNSA of the U.S. DoE at LANL under Contract No. DE-AC52-06NA25396.

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