Thermal creep emerging from cooling a tilted vortex lattice in uniaxial superconductor

A perturbed system relaxes towards an equilibrium given by a minimum in the potential energy landscape. This often occurs by thermally activated jumps over metastable states. The corresponding dynamics is named creep and follows Arrhenius' law. Here we consider the situation where the equilibrium position depends on temperature. We show that this effect occurs in the vortex lattice of the anisotropic superconductor $2\mathrm{H}$-$\mathrm{Nb}\mathrm{Se}_{2}$ when the magnetic field is tilted away from the principal axes, and that it leads to the peculiar appearance of creep when cooling the sample. Temperature determines the system's ground state and at the same time brings the system back to equilibrium, playing a dual and antagonistic role. We expect that cooling induced creep occurs in correlated systems with many degrees of freedom allowing to tune the equilibrium state via heat treatment.

A perturbed system relaxes towards an equilibrium given by a minimum in the potential energy landscape. This often occurs by thermally activated jumps over metastable states. The corresponding dynamics is named creep and follows Arrhenius' law. Here we consider the situation where the equilibrium position depends on temperature. We show that this effect occurs in the vortex lattice of the anisotropic superconductor 2H-NbSe2 when the magnetic field is tilted away from the principal axes, and that it leads to the peculiar appearance of creep when cooling the sample. Temperature determines the system's ground state and at the same time brings the system back to equilibrium, playing a dual and antagonistic role. We expect that cooling induced creep occurs in correlated systems with many degrees of freedom allowing to tune the equilibrium state via heat treatment.
Superconducting vortices are lines of quantized magnetic flux Φ 0 = hc/2e where the superconducting order parameter is depressed at a length scale of order of the superconducting coherence length ξ and circular supercurrents are flowing at a length scale of order of the penetration depth λ. Vortices arrange in a lattice, which can be triangular, square or disordered due to the interactions of vortices with the crystalline environment 1-5 . Vortices tend to be pinned on material defects or inclusions consisting of places where superconductivity is depressed on length scales of order ξ. When varying the applied magnetic field, its strength or direction, vortices enter or exit the sample 1-3 . This produces vortex motion, which is often counteracted by pinning. The action of the pinning landscape results in long-lived out-of-equilibrium vortex distributions that relax through thermal creep over a manifold of barriers 1-3,6-10 . The phenomenon of creep has been observed in interacting systems of particles, such as colloids, polymers, solids consisting of mixtures or in lattices of entities formed through electronic interactions (domain walls or skyrmions) [11][12][13][14][15] . Vortex lattices are often considered as a model system. Most efforts to understand vortex creep have focused on trying to immobilize vortices and thereby increase the critical current j c for applications 16 . But the interaction between vortices and the underlying superconducting material is very rich and can produce counterintuitive phenomena. Here we find that the equilibrium state towards which the system creeps can be modified with the temperature, leading to a phenomenon that we term self-imposed creep. We study vortex creep in the the superconductor 2H-NbSe 2 and we observe that, as a consequence of self-imposed creep, the vortex lattice starts creeping when cooling, see Fig. 1.
The layered system 2H-NbSe 2 shows weak pinning and small creep rates. Vortex properties depend crucially on random disorder potential cooling creep fast relaxation 1. Schematic response of a particle confined in a potential subject to random disorder. The disorder potential is characterized by local maxima of size U0 and over a length scale ξ [the superconducting coherence length in the case of vortices in superconductors]. Upon warming, the particle rapidly moves to a newly defined minimum as the disorder is thermally smeared out (indicated by effectively smaller wells, red). When cooling, the potential changes and the particle increases its energy relative to the new minimum in V (x). As the motion is impeded by pinning barriers (blue), the particle creeps by thermal relaxation towards the new minimum.
the magnetic field direction with respect to the layers. The axis normal to the layers provides a distinguished direction and one speaks about a uniaxial superconductor. The current distribution for a magnetic field tilted towards the layers yields a misalignment between the field and the flux-lines by angle that depends on temperature through the first critical field ∝ H c1 (T ) 2,17-22 . The temperature induced variations in this misalignment angle modifies the equilibrium position for vortex creep.
To study vortex creep we use a dilution refrigerator Scanning Tunneling Microscope (STM) in a three axis vector magnet 23 . We use a gold tip, sharpened and cleaned in-situ 24 and study a cleaved 2H-NbSe 2 sample (∼ 1×1×0.2 mm 3 ) grown with iodine vapor deposition. For the field strength H = 0.85 T, the magnetization is practically reversible so that no vortex motion is measured within days. After field cooling to the base temperature (T 0 = 150 mK), we rotate the magnetic field from perpendicular to the layers (c axis) towards the direction of the layers. We chose a tilt angle of 70 • , sufficient to install an out-of-equilibrium vortex state and observe creep while keeping tilt induced distortions in the symmetry of the vortex lattice or in vortex cores small. Distorted lattices in tilted fields in 2H-NbSe 2 have been discussed in detail in Refs. [25][26][27][28][29][30][31][32] and are related to the uniaxial anisotropy, ε ≡ H c2,ab /H c2,c ≈ 1/3. The direction of the induction B differs from the direction of H by an angle θ B − θ H , with θ H and θ B being the angle of H and B relative to the c axis. Minimizing the free energy with respect to θ B for fixed H and θ H yields up to a logarithmic correction of order unity 2,33-35 . The temperature dependence of the equilibrium angle θ B , which depends on the current distribution and on λ(T ), is encoded in H c1 (T ). For small changes of H c1 (T ) = H c1 (T 0 ) + δH c1 , large fields H H c1 and a tilt angle θ B away from 0 or π/2 around T = T 0 , the angle changes to In the regime where pinning is weak, i.e., where the Bean length 36 B ≈ cB/4πj c is larger than the sample width w and thickness d w, vortices in this critical state 36,37 are straight and oriented along the angle θ c Fig. 2(a). With j c ∼ 10 3 A/cm 2 this critical angle deviates from the equilibrium angle θ B as θ B − θ c B ≈ 0.5 • . Using H c1 (T = 0) ≈ 200 G for the lower critical field, we find θ B − θ H ≈ 0.8 • , i.e., vortices are more inclined towards the ab-plane than the external field, Fig. 2(a). Finally, the Ginzburg-Landau scaling H c1 (T ) ≈ H c1 (0)(1 − T /T c ), provides a relative change in H c1 between the experiment's low (T 0 = 150 mK) and To measure vortex motion we make consecutive STM images as a function of time [each one taken in 23 minutes, the average over eight consecutive images is shown in Fig. 2(b)]. We extract the vortex displacements from the images and show these in Fig. 2(c). We note that vortices always move along the direction of the component of the magnetic field within the layers. We also note that vortex motion shows a weak modulation at distances which correspond to multiples of the inter-vortex distance. This self-matching effect has been reported earlier in 2H-NbSe 2 and in disordered thin films [38][39][40][41] and evidences that the lattice moves as a whole in the creep regime. Other than that, no decay of vortex creep velocity is observed within the measurement time (∼ 5 h). The vortex displacement (160 nm) during this time translates to an angular velocity 10 −3 deg/h of vortices tilting to new equilibrium positions, see Fig. 2(a). The large time scale for thermal decay is consistent with the misalignment between magnetic field and vortex axis discussed Imaging is repeated at different temperatures. It is important to stress here that no reinitialization occurs. Rather the system is kept at finite field strength and angle, and solely the temperature is changed. Hence it is temperature that determines the state preparation, whose relevance has been previously discussed in vortex physics [42][43][44] . In Figs. 3(a,b) we show the temperature dependence of the vortex velocity along the creep direction, the average creep velocity v(T ), and the vor- tex motion between consecutive images along random directions, the jitter motion ∆x(T ), for a set of vortices, respectively. Each data point is obtained from a series of STM frames. To find v(T ) we determine the position r i j of vortex j in frame i and evaluate the average displacement for each vortex per frame, given by δr j = |r nj j −r 1 j |/(n j −1), where n j denotes the number of frames where the jth vortex appears. Averaging over all N v vortices for a given temperature, we arrive at the average creep velocity v(T ) = 1 t f Nv Nv j=1 δr j with t f = 23 min the time for measuring one frame. To quantify the jitter motion, we evaluate the average jitter displacement δs j = 1 nj −1 nj i=2 |δr i j | −δr j , with |δr i j | the vortex displacement between two subsequent frames i − 1 and i. The average over all frames at a fixed temperature now provides the jitter motion ∆x(T ) = 1 Nv Nv j=1 δs j . The average creep velocity decreases upon warming and vanishes above 2 K. Upon cooling, however, a finite velocity reappears. If the vortices were to reach a temperature-independent minimum upon warming, the jitter motion would decrease upon cooling without a reappearance of creep motion. The reversible directed vortex motion upon thermal cycling is therefore a clear signature of self-imposed creep.
To capture the main observations we discuss a simple, yet quite generic, model for creep motion in a disordered landscape. First, let us note that motion is driven by thermal fluctuations, through an Arrheniustype activation process across the pinning barriers. In such a case, the timescale t ∼ τ exp(U 0 /k B T ) for thermal activation is determined by (i) the temperature T , (ii) the energy barrier U 0 , and (iii) a microscopic time scale τ = ω −1 (ω is an attempt frequency to overcome the barrier). Creep is observed when the time scale of the experiment is such that U 0 ∼ k B T ln(ωt). To see the consequences of modifying the equilibrium, let us consider the problem of a particle confined in one-dimensional parabolic trap V 0 (x) = kx 2 /2 and subject to a driving force V d (x) = −F x. In the context of a many-body system, the variable x represents an observable parametrization, such as the angle between the magnetic field and the layers for the vortex lattice of a uniaxial superconductor. The force produces a drive towards the equilibrium positionx = F/k. In addition, let us consider a disorder landscape V p (x) characterized by a typical depth U 0 and width ξ, with kξ 2 /U 0 1, see Fig. 1. We approximate the bare potential between neighboring minima The thermally activated motion in the opposite direction, i.e. from x + to x − is penalized by an additional energy −2(x −x)kξ > 0. A particle initially far from the minimumx will glide down the potential until reachingx − 2U 0 /kξ from where it will be thermally activated across ever-growing barriers. After a time t, the particle has reached a position x T satisfying Arrhenius' condition as smaller barriers have been overcome in exponentially shorter activation times. Inserting Eq. (3) into the condition (4), we obtain If x T is still far fromx in the sensex−x T k B T ln(ωt)/kξ [translating to T T b ≡ (U 0 /k B )/ ln(ωt)], the particle moves with an average velocity v ≈ 2ξ/t. If however the particle has relaxed in the vicinity of the global minimum, the thermal activation becomes almost equally probable in both directions. Accounting for this bidirectional, yet asymmetric, motion we find a net average creep velocity The temperature dependence of v is shown in Fig. 4(a), for different values ωt. It is interesting to note that although the thermal activation energy increases upon warming, the creep velocity decreases. The validity of the above result is limited to temperatures T < T b . For larger temperatures, the disorder landscape becomes irrelevant, as the particle relaxes within ∼ ω −1 . When temperature affects both the activation dynamics and the global potential minimum, see Fig. 1, the velocity profile is traced reversibly upon warming and cooling. (c) Standard deviation ∆x of the mean particle displacement, quantifying the isotropic thermal motion, or jitter, see Eq. (7).
The directed relaxation via creep, Eq. (6), is blurred by an isotropic contribution, or jitter motion with zero mean and amplitude (standard deviation) given by This result is obtained by describing the (forwardbackward) activation across the barriers as a stochastic process (random-walk motion), where the variance (∆x) 2 = ξ 2 ω rw t of the displacement grows linearly in time and is determined by the random-walk attempt The exponent only depends on the difference between the barrier heights for forward and backward motion. This jitter motion persists beyond the disappearance of creep motion and reaches a similar magnitude when k B T ln(ωt) ∼ U 0 , as shown in Fig. 4(b). Now consider what occurs if the temperature is modified. We have to distinguish two scenarios: one where the global minimumx is constant, and one wherex(T ) depends on T through a temperature-dependent force F (T ). In the first case, the average creep velocity is given by the local disorder landscape seen by the particle and hence follows Eq. (6). For T > T b , the system is fully relaxed and the particle reachesx. Meanwhile the magnitude of the jitter motion continuously increases, see Fig. 4(b). Subsequent cooling lowers the thermal energy and the particle's motion freezes in place atx. This is the conventional response expected for creep. In the second case, whenx depends on T through a temperaturedependent force F (T ), the situation is drastically different. ∆T imposes a shiftx(T + ∆T ) −x(T ) ξ of the global equilibrium position. Upon warming, the velocity and jitter motion are similar to the previous case. However, the values of x(t) are different and, upon cooling, triggers particle motion tracing back (in magnitude) the warming curve, as schematically shown in Fig. 1.
We compare our observations in 2H-NbSe 2 with the above model. The typical barrier to overcome during pinning by thermal fluctuations is given by Arrhenius' law U 0 = k B T ln(ωt). To observe both the equilibrium phase at high temperature and reentrant creep at low temperature it is important that the temperature of the experiment is of order of (U 0 /k B )/ ln(ωt). In contrast to the pinning energy of one defect site, here U 0 denotes the energy barrier for vortex creep 45,46 . Weak collective pinning theory 1,2,47 provides the estimate U 0 ∼ k B T c [(j c /j dp )(B/H c1 ) 3 /Gi] 1/2 , with the depairing current j dp = cΦ 0 /12 √ 3π 2 λ 2 ξ, the Ginzburg-Levanyuk 48,49 number Gi ∼ [T c /H c (0) 2 ξ 3 ] 2 , and the condensation energy H c (0) 2 ξ 3 = Φ 2 0 ξ/8π 2 λ 2 . From Refs. [50][51][52][53][54][55], we infer j c /j dp ∼ 10 −6 and Gi ∼ 10 −4 and obtain U 0 ∼ 10k B T c . This estimate is compatible with U 0 k B T c ln(ωt), provided ωt ≈ 2 × 10 4 ; somewhat larger than the values considered above (Fig. 4). Given the simplicity of the one-dimensional model, the agreement is still remarkable. All important features predicted by the model-the disappearance and reappearance of the directed motion, together with the temperature-evolution of the jitter motion [behaviors of v(T ) and ∆x(T ), shown in Figs. 3(a,b) and in Figs. 4(a,b)]-are found in the experiment.
Given that the experimental time scale spans several minutes, our observation ωt ∼ 10 4 suggest a value for ω of order of one Hz. While a route for accurate determination of this attempt frequency is still lacking, the estimate ω = α L /η for a single vortex depends on the vortex viscosity η and on the Labusch parameter 56 α L (which in turn relates to the averaged potential curvature 57 ). While values in the range 10 6 -10 10 Hz have been reported 58 , the analysis assumes vibrations with large k-vectors. In our case, vortices are not isolated, but rather interact non-locally with many vortices 2,3,59-61 . Low k-vectors, or wave-lengths comparable to the sample size, leads to highly dispersive elastic moduli which modify the attempt frequency by orders of magnitude [60][61][62] . Similar to our observation, previous measurements of slow vortex dynamics have reported 63,64 very low frequency values for thermal motion and creep. Creep rates observed in layered cuprate superconductors involve extremely large time scales, indicating the relevance of col-lective creep 2,3,60,62 . Thus, even if the attempt rate of individual vortices is large, the dynamics as a lattice involves rates that are many orders of magnitude smaller. The temperature is far from melting, thus favoring collective rather than a single-vortex dynamics 65,66 . It is this near-equilibrium configuration with ultra-small collective dynamics that allows for the observed cooling imposed creep in our experiments.
The creep discussed here is very slow and no decay in the vortex velocity is observed within our experimental time. However, the creep rate S = d ln(j)/d ln(t) can assume a seizeable value compatible with the suggested lower bound 67 S > (T /T c )Gi 1/2 . Actually, 2H-NbSe 2 is among the materials with lowest creep rates, close to MgB 2 67 . Creep between metastable vortex states that occur near the order-disorder transition of the vortex lattice in 2H-NbSe 2 or related to domain formation of lattices with different orientations in MgB 2 has been reported [68][69][70] . Motion then appears when modifying the relative strength of competing interactions, and it might well occur that the equilibrium configuration at some particular locations is influenced by temperature. Collective motion is also found in stochastic behavior of particle arrangements 11 . Depending on particle interactions, the dynamics transits from individual random motion to flocking. The time scale related to flocking motion shows a divergent behavior with increasing interaction. Other long term dynamical behavior should appear in thermal effects and might lead self-imposed creep whenever there are two or more parameters influencing the behavior of the system. Mixtures, such as alloys, concrete or rocks 71 , liquids re-solidifying under stress 12 , steel under stress 13 , colloidal systems, magnetic domain walls or skyrmions 15 are cases of complex systems where self-imposed creep may be induced from temperaturedependent interactions.
The model of self-imposed creep explains the critical state dynamics in 2H-NbSe 2 at tilted magnetic fields; in particular the commonly unexpected appearance of vortex motion when cooling. Likely, the balanced thermal activation dynamics and the temperature-dependent equilibrium could be matched in other uniaxial superconductors with weak pinning and in complex systems.