Physical Review Letters 100, 167002 (2008)

Spontaneous Transverse Response and Amplified Switching in Superconductors with Honeycomb Pinning Arrays

C. Reichhardt and C.J. Olson Reichhardt
Theoretical Division, Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA

(Received 13 September 2007; published 25 April 2008)

Using numerical simulations, we show that a novel spontaneous transverse response can appear when a longitudinal drive is applied to type-II superconductors with honeycomb pinning arrays in a magnetic field near certain filling fractions. This response is generated by dynamical symmetry breaking that occurs at fields away from commensurability. We find a coherent strongly amplified transverse switching effect when an additional transverse ac drive is applied. The transverse ac drive can also be used to control switching in the longitudinal velocity response. We discuss how these effects could be used to create new types of devices such as current effect transistors.
DOI: 10.1103/PhysRevLett.100.167002
PACS numbers: 74.25.Qt


There have been extensive studies on superconducting systems with patterned pinning arrays which show commensurability effects when the vortex density matches the pinning site density [1,2,3,4,5]. Various properties of the vortex transport can be controlled by adjusting the shape and geometry of the pinning arrays, producing vortex channeling [6], dynamical transitions [7,8], fluxon ratchets [9], reversible ratchets [10,11], and vortex cellular automata [12]. These studies indicate that vortices interacting with patterned substrates may lead to a new field of fluxtronics or microelectronic devices based on the controlled motion of vortices. Further, vortices interacting with periodic pinning arrays also exhibit a number of collective dynamic behaviors that are important to the broader field of nonequilibrium physics.
Few studies have been performed on vortex dynamics in honeycomb pinning arrays since it was assumed that the vortex behavior in this pinning geometry would be similar to that found in square and triangular pinning arrays. In this Letter we demonstrate that, in fact, honeycomb pinning arrays produce new types of phenomena that do not occur in triangular or square pinning arrays. In particular, an effective dimerization or higher order n-merization of the interstitial vortices in the honeycomb pinning arrays for certain field ranges results in a novel spontaneous transverse response (STR) to a longitudinal drive. Normally, an applied current produces a perpendicular Lorentz force on superconducting vortices, which move and generate a voltage drop parallel to the current. We show that the honeycomb system can also produce a voltage drop perpendicular to the applied current. For magnetic fields away from commensuration, the vortex flow dynamically organizes to a STR state. At the matching fields, the STR arises due to symmetry breaking in the ground state. The vortex n-mer states are very similar to the colloidal molecular crystal states studied for colloids interacting with periodic optical trap arrays [13]. We specifically consider vortices in superconductors; however, we expect similar phases to occur whenever there is a spontaneous symmetry breaking of effective n-mer states for particles on periodic substrates.
We simulate a two-dimensional system containing Nv vortices and Np pinning sites with periodic boundary conditions in the x and y directions. The vortex density is B=Nvϕ0/L2, where L=24λ is the system size in units of the penetration depth λ and ϕ0=h/2e is the elementary flux quantum. The motion of a single vortex is given by the following overdamped equation:
η dRi
dt
 = Fvvi + Fpi + Fdc + Fac + FTi .
(1)
Here Ri is the location of vortex i, η = ϕ02d/2πξ2ρN is the damping constant, d is the thickness of the superconducting sample, ξ is the coherence length, and ρN is the normal state resistivity. The pairwise vortex-vortex interaction force is Fvvi = ∑Nvi ≠ jf0K1(Rij/λ)∧Rij where K1 is the modified Bessel function, Rij=|Ri−Rj|, ∧Rij=(Ri−Rj)/Rij, and f0 = ϕ02/(2πμ0λ3). The pinning force Fpi arises from parabolic traps of radius rp=0.3λ and strength fp=1.0f0 arranged in a honeycomb lattice with Fpi=−∑k=1NpfpRikrp−1Θ((Rp−Rik)/λ)∧Rik. Here Rik=|Ri−Rk| is the distance between vortex i and pin k, ∧Rik=(Ri−Rk)/Rik, and Θ is the Heaviside step function. The dc longitudinal driving force Fdc = Fdcf0∧RL, where we take the longitudinal direction ∧RL to be ∧R=∧x. This force mimics the Lorentz force created by an applied current. We also consider the effect of adding a transverse ac drive Fac = Facf0sin(ωt)∧RTr, where ω = 10−7f0/η and ∧RTr=∧y is the transverse direction. The thermal force FiT arises from Langevin kicks and has the properties 〈FTi(t)〉 = 0 and 〈FTi(t)FTj(t′)〉 = 2ηkBTδijδ(t − t′). Vortex positions are initialized with simulated annealing.
Fig1.png
Figure 1: Vortex positions (black dots) and pinning site locations (open circles) in a 20λ×20λ section of the sample at Fdc = 0. (a),(b),(c): Three different realizations of B/Bϕ=2.0 with interstitial dimers aligned (a) in the y-direction; (b) at +30° to the x-direction; (c) at −30° to the x-direction. The dashed lines indicate the alignment direction. (d) B/Bϕ = 1.77 with no overall dimer alignment.
In Fig. 1 we illustrate the vortex and pinning site positions in a sample with Fdc=0, Fac=0, and T = 0. At B/Bϕ=2.0, shown in Figs. 1(a)-1(c), where Bϕ is the field at which Nv=Np, the large interstitial space of each plaquette in the honeycomb pinning lattice traps two interstitial vortices that form an effective rigid dimer with a director field that can point in one of three degenerate directions, as indicated by the dashed lines. Neighboring dimers interact through an effective quadrupole interaction that causes the dimers to align into one of three degenerate ground states [14]. For fillings 1.5 < B/Bϕ < 2.0 and 2.0 < B/Bϕ < 2.5, the dimer alignment is disrupted and there is no global symmetry breaking in the ground state. Figure 1(d) shows an example at B/Bϕ = 1.77 where the dimer ordering is lost.
Fig2.png
Figure 2: (a) Transverse velocities VTr± and longitudinal velocity VL vs Fdc for the sample in Fig. 1 at B/Bϕ=2.0. (b) |VTr| vs Fdc for the same system at B/Bϕ = 1.58, 1.61, 1.67, 1.72, 1.77, and 1.81, from bottom to top. (c) VTrmax versus B/Bϕ. (d) |VTr|/VL versus B/Bϕ at fixed Fdc = 0.225. (e) Time series of VTr for Fdc=0.3 at B/Bϕ=2.0 (top curve) and B/Bϕ=1.81 (two lower curves). (f) VTr during consecutive cycles of Fdc from 0 to Fdc=0.3 at B/Bϕ=2.0 (open squares) and B/Bϕ=1.81 (filled circles).
We next apply a dc drive Fdc in the longitudinal direction and measure the vortex velocity v to obtain the longitudinal velocity VL=∑iNvvi ·∧RL and the transverse velocity VTr=∑iNvvi ·∧RTr. In Fig. 2(a) we plot VL and VTr versus Fdc for the system in Fig. 1 at B/Bϕ=2.0. For Fdc < 0.14 the system is pinned, while for 0.14 ≤ Fdc < 0.37 there is a finite longitudinal velocity VL accompanied by a finite transverse response VTr that can be in either the positive (VTr+) or the negative (VTr−) transverse direction. The trajectories of the moving vortices for the VTr+ and VTr− states are shown in Figs. 3(a) and 3(b), respectively. When Fdc > Fdcc, where Fdcc is the critical drive at which vortices in the pinning sites begin to depin, the symmetry breaking is lost. At B/Bϕ=2.0, Fdcc=0.37, and above this drive, |VTr| drops abruptly to zero and VL rapidly increases. The curve with VTr+ was obtained by starting from the ground state in Fig. 1(a) at T = 0 while the curve with VTr− was generated starting from the same ground state but applying a finite but small temperature T=0.2Tm, where Tm is the vortex lattice melting temperature. If we repeat the finite temperature simulation with different random seeds, we are equally likely to observe VTr+ or VTr−. If the initial ground state already has a global symmetry preferred positive or negative orientation with respect to the transverse direction, as in Figs. 1(b) and 1(c), the transverse response is in the same direction indicated by the dashed lines in Figs. 1(b) and 1(c). The appearance of a transverse velocity at B/Bϕ = 2.0 is not a manifestation of a dynamical symmetry breaking since the dimer alignment symmetry is already broken as in Fig. 1. In contrast, for the fields 1.5 < B/Bϕ < 2.0 and 2.0 < B/Bϕ < 2.5 there is no global symmetry breaking in the ground states; however, a dynamical symmetry breaking occurs when the vortices organize into a dynamical phase resembling the states in Figs. 3(a) and 3(b) after a transient period of time during which the vortices move in both the VTr+ and VTr− directions.
Fig3.png
Figure 3: Vortex positions (black dots), pinning site locations (open circles), and trajectories (lines) over a constant time interval. (a) The VTr+ dynamically broken symmetry state from Fig. 2(a) at Fdc=0.225. (b) The VTr− state at Fdc = 0.225. (c) A sample with B/Bϕ = 2.11 at Fdc=0.4 where complete symmetry breaking is lost. (d) The VTr+ state in a sample with B/Bϕ=4.5 and Fdc = 0.15.
In Fig. 2(b) we plot |VTr| as a function of Fdc at B/Bϕ = 1.58, 1.61, 1.67, 1.72, 1.77, and 1.81 showing the appearance of the dynamical broken symmetry state. In Fig. 2(e) we plot VTr vs time for Fdc = 0.3 at B/Bϕ = 2.0 and B/Bϕ = 1.81, where the latter curves were generated from samples in slightly different initial states. At B/Bϕ = 2.0, the vortices immediately move in the broken symmetry direction of the ground state, which in this case is VTr+ shown in Fig. 1(b). For B/Bϕ = 1.81, the initial motion is symmetric and only develops into a VTr+ or VTr− broken symmetry state over time. The system may even fluctuate significantly in one direction before locking into the other direction. This shows that the moving states can have global symmetry breaking even when the equilibrium ground state does not. Because of the initial symmetric transient moving state, the memory of the pinned state configuration is lost at incommensurate fields but retained at commensurate fields when Fdc is repeatedly cycled from zero into the moving broken symmetry state and back again. In Fig. 2(f), the plot of VTr at B/Bϕ = 2.0 and 1.81 during consecutive cycles of Fdc shows that the sign of the symmetry breaking switches randomly between cycles at the incommensurate field.
In Fig. 2(c) we plot VTrmax, the maximum value of |VTr| obtained when varying Fdc, vs B/Bϕ. The overall maximum amount of transverse motion occurs at B/Bϕ=2.0. The fraction of the velocity that is in the longitudinal direction |VTr|/VL is shown in Fig. 2(d) for fixed Fdc=0.225 as a function of B/Bϕ. For complete symmetry breaking, the vortices move at ±30° from the longitudinal axis so we expect |VTr|/VL=tan(30°)=0.577, as seen for 1.5 < B/Bϕ ≤ 2.0 in Fig. 2(d). For B/Bϕ > 2.0, both VTrmax and |VTr|/VL drop rapidly.
The same type of symmetry breaking flow occurs at higher commensurate states such as B/Bϕ=4.5 where elongated trimers can form, as illustrated in Fig. 3(d). This is accompanied by spontaneous dynamical symmetry breaking over a range of fields. We have also considered the role of temperature. Previous work [14] indicated that the interstitial vortices melt at a well defined temperature Tm; using parameters appropriate for Nb crystals gives Tm ≈ 8.5 K. We find that our results are robust for all T < Tm. In Fig. 4(d) we plot |VTr| vs T/Tm at B/Bϕ = 2.0 and Fdc = 0.3, showing that VTr drops sharply with increasing T just below T/Tm = 1.0.
Fig4.png
Figure 4: (a) VTr(t) (thick lower line) and VL(t) (thick upper line), with the analytic values of Vac(t) (thin lower line) and Vdc(t) (thin upper line) shown for comparison. VL and Vdc have been shifted up by 0.2 for presentation purposes. Time is given in units of 2π/ω. Here B/Bϕ=2.0, Fac=0.125, and Fdc=0.28. (b) Time series of VTr for a system with B/Bϕ=1.77, Fdc = 0.28, and varied ac drive amplitude of Fac=0.15, 0.1, 0.075, 0.05, 0.035, and 0.02, from top to bottom on the positive response side. (c) The amplification factor A versus Fac for samples at Fdc = 0.28 with B/Bϕ = 1.61 (filled circles), 2.0 (open squares), and 2.11 (filled triangles). (d) |VTr| vs T/Tm at B/Bϕ = 2.0 and Fdc = 0.3.
Since the dynamical symmetry-broken states are bistable, it is natural to ask whether it is possible to induce a switching behavior between the two states. To address this, we consider the effect of adding a transverse oscillating drive Fac to a system with B/Bϕ=2.0 moving under a fixed longitudinal drive Fdc=0.28. Experiments have already been conducted in superconductors with periodic pinning arrays where two orthogonal driving currents were simultaneously applied and the transverse and longitudinal responses were simultaneously measured [15]. In Fig. 4(a) we plot the time series VTr(t) and VL(t) in a system with Fac=0.125. Here the ac drive induces a periodic switching between the positive and negative transverse response states VTr+ and VTr−. We find that this abrupt switching effect persists for 1.5 < B/Bϕ ≤ 2.0. Similar switching appears over a wide range of ac driving amplitudes, as illustrated in Fig. 4(b) where we plot VTr(t) at B/Bϕ = 1.77 and Fdc=0.28 for different values of Fac. We observe a novel response in the longitudinal direction in which the value of VL(t) varies by up to 50% during each ac drive period. There is also a switching effect in VL(t) that accompanies each switch in VTr(t). This phenomenon is reminiscent of the current effect transistor found in transversely driven charge density waves where a transverse force can be used to control the longitudinal response [16].
There is an amplification of the transverse ac response VTr(t) over the value expected based only on the magnitude of the ac input driving signal. To illustrate this, in Fig. 4(a) we plot Vac(t) ≡ NmFac(t)/η, the transverse ac velocity produced when the Nm=Nv−Np interstitial vortices move only in response to the ac driving force, along with Vdc ≡ NmFdc/η, the longitudinal velocity of the interstitial vortices under only the dc drive. Figure 4(a) indicates that |VTr(t)| > |Vac(t)|. The relative amplification VTr(t)/Vac(t) can be increased by lowering Fac; however, for a finite Fdc there is a threshold value Facc below which the system no longer switches between VTr+ and VTr−. For example, in Fig. 4(b), Fac=0.02 is below the threshold Facc, and thus the response stays locked in the VTr+ direction. We quantify the amplification of the transverse ac response using A=VTrη/(NmFac), where VTr is the amplitude of the transverse response, such that for A=1.0, there is no amplification. The plot of A vs Fac in Fig. 4(c) for B/Bϕ = 1.61, 2.0, and 2.11 shows that for a fixed dc drive, the amount of ac amplification that occurs depends on both Fac−Facc and Fdcc−Fdc. The closer the ac and dc drives are to the respective critical thresholds, the larger the amplification. Since Fdcc is maximized at B/Bϕ = 2.0, for fixed Fdc=0.28 a larger amplification can be obtained at B/Bϕ ≠ 2.0 than at B/Bϕ=2.0, as shown in Fig. 4(c). For the parameters used in Fig. 4, the frequencies for a typical superconductor are between 10 and 30 Hz. If ω is increased, the sharp switching response can still be achieved by increasing Fac until ω∼ 20 to 60 kHz, at which point Facc crosses above the depinning threshold for the pinned vortices and the dynamical symmetry breaking is completely lost.
Many of the switching features, including bistability in the transverse response and the abrupt switching effect induced by the ac drive, are analogous to semiconductors and may be useful for creating logic devices. The strong coupling between the longitudinal and transverse responses indicates that a transverse ac drive could be used to control longitudinal switching, while the amplification effect implies that very small ac inputs can induce large response changes in certain regimes. Our results should apply to any system of repulsively interacting particles confined by honeycomb pinning arrays. We have observed a similar dynamical symmetry breaking flow for colloids on triangular substrates [17]; however, the dynamical symmetry breaking for the vortex system presented here is much more robust and occurs for a considerably wider range of parameters and particle fillings.
In summary, we have demonstrated that vortices in superconductors with honeycomb pinning arrays can exhibit a novel transverse response when a longitudinal drive is applied for certain ranges of fields where the vortices in the large interstitial regions of the pinning array form effective dimer or higher order n-mer states. At commensurate fields, the symmetry is broken in the ground state, while at incommensurate fields, there is a dynamical symmetry breaking. If a transverse ac drive is added to the system, a pronounced transverse switching response occurs, as well as an amplification of the transverse ac signal. There is a threshold ac drive required to induce the switching, which becomes very small when the longitudinal drive is close to the value above which the transverse response disappears. The transverse ac drive can be used to modulate the longitudinal response of the system as well. We discuss how this effect may be useful for creating new types of fluxon based devices.
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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