Size Dependence in Flux-Flow Hall Effect using Time-Dependent Ginzburg-Landau Equations
Vineet Punyamoorty, Aditya Malusare, Shamashis Sengupta, Sumiran Pujari, Kasturi Saha
SSize Dependence in Flux-Flow Hall Effect using Time Dependent Ginzburg-LandauEquations
Vineet Punyamoorty, Aditya Malusare, Shamashis Sengupta, Sumiran Pujari, and Kasturi Saha Department of Electrical Engineering, Indian Institute of Technology Bombay, Mumbai 400076, India Department of Physics and Astronomy, Purdue University, West Lafayette, IN 47907 Université Paris-Saclay, CNRS/IN2P3, IJCLab, 91405 Orsay, France Department of Physics, Indian Institute of Technology Bombay, Mumbai 400076, India (Dated: 2 February 2021)We study the Hall effect in square, planar type-II superconductors using numerical simulations oftime dependent Ginzburg-Landau (TDGL) equations. The Hall field in some type-II superconduc-tors displays sign-change behavior at some magnetic fields due to the induced field of vortex flow,when its contribution is strong enough to reverse the field direction. In this work, we use modifiedTDGL equations which couple an externally applied current, and also incorporate normal-state andflux-flow Hall effects. We obtain the profile of Hall angle as a function of applied magnetic fieldfor four different sizes ( l × l ) of the superconductor: l/ξ ∈ { , , , } . We obtain vastly differentprofiles for each size, proving that size is an important parameter that determines Hall behavior. Wefind that electric field dynamics provides an insight into several anomalous features including sign-change of Hall angle, and leads us to the precise transient behavior of order parameter responsiblefor them. I. INTRODUCTION
One of the most interesting aspects of superconduct-ing systems is the physics of vortices. Advances in ex-perimental methods have made it possible to probe type-II superconductors at small lengthscales where the be-haviour of individual vortices becomes visible. Vorticeshave been imaged with electron spins in diamonds [1, 2],scanning superconducting quantum-interference devices(SQUIDs) [3, 4] and Hall-probe magnetometry [5, 6].The behaviour of a superconducting system is expectedto vary significantly if its dimensions are reduced to becomparable to the coherence length. In this work, we the-oretically investigate the effect of finite size on the prop-erties of vortices with numerical simulations of solutionsof the time dependent Ginzburg-Landau (TDGL) equa-tions. The flux-flow Hall effect in square planar type-IIsuperconductors are studied for different sample sizes,given by l/ξ ratio of 3, 5, 15 and 20 (where l is thelength of the square and ξ is the superconducting co-herence length). It will be seen that the electric fieldand hall angle profiles under a unidirectional current arewidely different depending upon the dimension of the su-perconductor.Hall effect in superconductors has been observed todisplay anomalous properties below the critical temper-ature, most significant of which is the sign-reversal ofHall voltage at certain magnetic fields [7–9]. This couldnot be explained by either the phenomenological modelsof vortex flow of the time, namely the Bardeen-Stephen(BS) model [10] and the Nozières-Vinen (NV) model [11],or by the microscopic theory [12]. It was later proposedthat the induced electric field of magnetic vortex flowcould contribute to the Hall field and cause anomalousbehavior [9]. Dorsey [12] and Kopnin et al. [13] provedusing analytical approximations of a modified time de-pendent Ginzburg-Landau (TDGL) system, that indeed sign reversal of Hall effect is possible under some circum-stances. These theories make use of microscopic quan-tities (related to the electronic structure) to define theregimes of sign-reversal. Alternatively, one may numeri-cally compute the Hall effect in a superconducting sam-ple governed by the modified TDGL system of Dorsey[12] and Kopnin et al. [13], and find the magnetic fieldregimes of sign-reversal. This could provide insights intoHall effect behaviour of a superconductor, as a functionof macroscopic quantities alone (e.g. GL parameter κ ,sample size, etc.).In this work, we follow the alternative numerical routementioned above, which is direct and does not resortto any analytical approximations. We first use stan-dard time dependent Ginzburg-Landau (TDGL) equa-tions [14–16] to numerically compute the time-varyingorder parameter of a planar superconductor in the vortexstate. We benchmark our simulations by comparing thenumerically obtained fluxoid value of each vortex against Φ o , the superconducting flux quantum, as a rigorous test(III A). With these benchmarks in hand, we simulate amodified TDGL system that includes an externally ap-plied current to probe the Hall effect in these systems.Importantly, we go beyond existing literature by incor-porating the normal-state Hall conductivity and flux-flowterms into the dynamical equations that we numericallysimulate. In this, we have taken inspiration from theanalytical works of Dorsey [12] and Kopnin et al. [13](III B). Next, in III C we study the resultant changes influx-flow and resultant induced electric fields based on thenumerical simulation of our modified TDGL system. Wecompute the Hall angle profile for various sizes of the su-perconductor, and find vastly different profiles. We findthat transient electric fields and related order parameterbehavior give us good insight into explaining the anoma-lous Hall behavior. a r X i v : . [ c ond - m a t . s up r- c on ] F e b II. THEORETICAL MODEL
The TDGL equations are [14–16]: γ (cid:18) ¯ h ∂ψ∂t + ie s Φ ψ (cid:19) + 12 m s (cid:16) i ¯ h ∇ + e s c A (cid:17) ψ + αψ + β | ψ | ψ = 0 (1a) ν (cid:18) c ∂ A ∂t + ∇ Φ (cid:19) + ie s ¯ h m s ( ψ ∗ ∇ ψ − ψ ∇ ψ ∗ ) + e s m s c | ψ | A + c π ∇ × ∇ × A = 0 (1b)where, ψ is the complex-valued order parameter, α and β are the phenomenological parameters of Ginzburg-Landau theory [17, 18] and φ , A are the electric andmagnetic potentials respectively. ν is the normal-stateconductivity and γ is a relaxation constant for the or-der parameter. The charge and mass of Cooper pairs aredenoted by e s = 2 e and m s = 2 m e respectively. A. Gauge invariance and normalization
For an arbitrary function χ ( x, y, t ) with well-definedspatial and time derivatives, the gauge invariance canbe expressed as ( ψ → ψe iκχ , A → A + ∇ χ , φ → φ − ∂χ∂t ) . We choose the zero electric potential gauge (i.e. φ = 0 ) [14, 15, 19]. The physical quantities in (1) arerenormalized as follows: xξ → x, tγ ¯ h/ ( − α ) → t, A √ H c ξ → A , ψ (cid:112) − α/β → ψ (2)where H c = (cid:113) πα β and ξ is the GL coherence length [19].In the chosen gauge, the resulting dimensionless equa-tions applicable over the superconducting domain are: ∂ψ∂t + (cid:18) i ∇ + 1 κ A (cid:19) ψ − (cid:0) − | ψ | (cid:1) ψ = 0 (3a) σ ∂ A ∂t + ∇ × ∇ × A + i κ ( ψ ∗ ∇ ψ − ψ ∇ ψ ∗ ) +1 κ | ψ | A = 0 (3b)where, κ is the GL parameter and σ is normalizednormal-state conductivity. When the sample is sur-rounded by vacuum on all sides and no external currentis passed, the following boundary conditions (BCs) apply[14, 19–21]: ∇ × A = H ext , (cid:18) i ∇ + 1 κ A (cid:19) ψ · n = 0 , − σ ∂ A ∂t · n = 0 (4) (where H ext is the externally applied magnetic field, per-pendicular to the sample). The first condition imposescontinuity of transverse magnetic field across the bound-ary, while the second and third ensure that neither su-percurrent nor normal current crosses the boundary, re-spectively.In principle, one also needs to solve Maxwell’s equation ∇ × ∇ × A = 0 (considering vacuum) in the surroundingdomain [19]. However, in the case of two-dimensionalsamples (i.e. infinitely long cylinders) with perpendic-ularly applied magnetic field, as in our system, solvingthe “interior problem” (3) alone is sufficient to a goodapproximation [15, 22]. For reference, the “full problem”and its boundary conditions are discussed in Refs. [14]and [19]. B. Inclusion of externally applied current
Including an externally applied transport current inthe TDGL system entails two tasks: (a) accounting forthe magnetic field induced by the transport current and(b) modifying the boundary conditions (4) to account forflow of normal current across boundaries. The former canbe achieved by modifying the first boundary condition to ∇ × A = H tot where H tot = H ext + H c is the sum ofapplied ( H ext ) and induced ( H c ) magnetic fields [20]. H c is to be computed from the current profile (a function of ψ , A ), making the system self-consistent. However, anapproximation is frequently used in literature [20, 23–26]to simplify the computation of H c : the current profileis assumed to be a uniform band, which reduces H c toa simple expression involving J a , the applied (uniform)current density. In this paper, J a is assumed to be in the + ˆx direction (fig. 1), which gives us H c along all fourboundaries as H top, bottom c = ± W J a / ˆz (where W is thelength of the sample along ˆy ) and H left, right c vary linearlybetween the bottom and top edges.Secondly, to account for flow of normal current acrossthe boundary, the “vacuum-superconductor” BCs (4) arenow replaced by “metal-superconductor” BCs (5b) [20,26] at the left and right edges, while retaining the formerBCs (5a) at the top and bottom edges: ∇ × A = H tot , (cid:18) i ∇ + 1 κ A (cid:19) ψ · n = 0 , − σ ∂ A ∂t · n = 0 (5a) ∇ × A = H tot , ψ = 0 , − σ ∂ A ∂t · n = J a · n (5b)In (5b), the third BC accounts for flow of current acrossthe boundary, and the second condition ensures that thedensity of superconducting electrons is zero at the edges.This ensures that the injected normal current transitionsto supercurrent gradually, rather than abruptly, insidethe superconductor. NormalConductor NormalConductorSuperconductorCurrent in Current outVacuumVacuum ˆ x ˆ y Figure 1: Schematic representation of our simulationdomain when an external current is applied – l × l square superconductor with metallic contacts on the leftand right, and vacuum at top and bottom. Current J a is applied along ˆ x and magnetic field H ext along ˆ z . C. Inclusion of Hall effect
Normal state hall effect is a result of the conduction ofelectrons transverse to applied electric field. This can beincorporated in the TDGL system (3), (5) by rewritingthe normal-state conductivity σ as a tensor: σ = (cid:18) σ xx σ xy σ yx σ yy (cid:19) (6)We assume an isotropic sample ( σ xx = σ yy ), and by sym-metry σ yx = − σ xy . In order to determine σ xx and σ xy ,we use the following as the model for the normal-stateconductivity [12, 27]: σ xx = σ
11 + ω c τ , σ xy = σ ω c τ ω c τ (7)where ω c is the cyclotron frequency e s ( ∇ × A ) /m s and τ is the electron scattering time. Under typical condi-tions, we have ω c τ (cid:28) (low-field limit) [12, 27]. Thus, σ xx ≈ σ and σ xy ≈ ω c τ σ . Due to the spatially varyingmagnetic field, cyclotron frequency ω c and consequently, σ xy are also spatially varying. In order to enforce thelow-field limit, we take ω c τ = 10 − ( ∇ × A ) , where thepre-factor of − ensures that ω c τ (cid:28) .Josephson [28] proved that macroscopically, a vortexmoving at velocity v L gives rise to an induced electricfield E = − c v L × H ,where all the quantities are spatiallyand temporally averaged [9, 12, 13]. In the Bardeen-Stephen (BS) model [10], under an applied current J a ,vortices experience a Lorentz force ∼ J a × H [9, 12].This force gives rise to velocity v L along J a × H andin a system such as ours (fig. 1), this results in vortexmotion along − ˆ y . This velocity v L along − ˆ y producesa field E = − c v L × H along ˆ x , the same direction as J a , thereby causing dissipation. However, if we were tohave an additional component of v L along ˆ x (the samedirection as J a ), this would create a field contribution inthe Hall direction. Dorsey [12] and Kopnin et al. [13]proved that adding a non-zero imaginary part to the re-laxation parameter γ in the TDGL system (1) producessuch flux-flow contribution to the Hall field by lendingthe vortices a velocity component parallel to the appliedcurrent. Thus, we write γ = γ + iγ . In the microscopicpicture, the value of γ /γ has been shown to dependupon the electronic structure of the material [12, 13]. Thesign of γ /γ determines whether vortices travel along J a or against, and therefore crucially affects the sign of flux-flow contribution to Hall field. Kopnin et al. [13] provedthat a sign reversal in Hall effect would be observed fornegative values of γ /γ . III. RESULTS AND DISCUSSION
We use COMSOL Multiphysics ® [29], a commercialfinite element tool to numerically simulate TDGL equa-tions (3)–(5). Throughout the paper, we take GL pa-rameter κ = 2 and normalized normal-state conductivity σ = 1 . In section III A, we obtain a vortex state solutionand propose a procedure to rigorously verify the solutionto help identify any numerical errors or artifacts. In sec-tions III B–III C, we study a system with externally ap-plied current and Hall effect included. Throughout thesesimulations, we apply an external current J a = 0 .
04 ˆ x .We consider four geometries: l × l square superconduc-tors for l/ξ ∈ { , , , } . We apply an external mag-netic field H ext along ˆ z , whose value is swept between 0and H c = κ = 2 in steps of 0.05. For each combinationof size l and H ext , we solve the modified TDGL system(3), (5) and compute the time-varying order parameter ψ ( r , t ) and vector potential A ( r , t ) . This gives us a com-plete insight into the dynamic vortex motion and electricfields E x = − ∂A x /∂t and E y = − ∂A y /∂t , which formthe basis for much of our analysis.Since we use normalized units throughout the paper,we compute the order of magnitude for these units tounderstand the typical physical values. We chose κ = 2 ,which broadly corresponds to Niobium [30]. Using thephysical parameters of Nb, we compute the time units as ∼ − s [31], current density units as ∼ − [32, 33]and the applied magnetic field units as ∼ kOe [33]. A. Vortex state and verification
We first solve TDGL equations (3) on a ξ × ξ pla-nar sample (GL parameter κ = 2 ) with no applied cur-rent or Hall effect enabled. Thus, in this case we use thevacuum-superconductor BCs (4) on all four sides, andobserve a vortex state solution at H ext = 0 . z (fig. 2).We perform a thorough verification of our simulation asdescribed in the following. First, we confirm that the y (a) t = 20 . t = 33 . x y (c) t = 40 . x (d) t = 500 . . . . . | ψ | Figure 2: | ψ | at various instants of time, depictingmajor events in the formation of a vortex state in the ξ × ξ sample. Applied magnetic field is | H ext | = 0 . .simulation results are stable with respect to mesh size.We then sweep | H ext | widely and observe the existenceof both upper and lower critical fields marked by the van-ishing of vortices, as expected from a type-II supercon-ductor ( κ > / √ ). We also observe that for the same κ , the number of vortices increases (decreases) when theexternal field | H ext | is increased (decreased). This is anexpected qualitative behaviour of the superconductor tolet more (less) incident flux pass through the supercon-ductor. One of the hallmarks of superconducting vorticesunder Ginzburg-Landau theory is fluxoid quantization.It states that the fluxoid value Φ (cid:48) associated with eachvortex is quantized by Φ = hc e [18]. Φ (cid:48) = Φ + 4 πc (cid:73) λ J s · dl (8)where Φ is the magnetic flux (cid:82) B · d s associated with thevortex, and the integral term involving super current J s and penetration depth λ is performed on a closed contourenclosing the vortex. We compute the fluxoid value, andobtain Φ (cid:48) ≈ . for each vortex, confirming that oursimulations firmly uphold fluxoid quantization (fig. 3). B. Flux flow under applied current with Hall effectincorporated
In the Bardeen-Stephen (BS) model [10], magnetic vor-tices experience a Lorentz force under applied externalcurrent along J a × H [9, 12]. We apply an external cur-rent J a = 0 .
04 ˆ x and H ext along +ˆ z , and accordingly ob-serve vortex flow along − ˆ y with vortices entering at thetop edge and leaving at the bottom (fig. 4). Consistentwith the BS model, we observe faster (slower) motion x y . . . . . . | H | Figure 3: We verify that the fluxoid value of each of ourvortices is ≈ Φ . To compute the fluxoid value we takea circular area centred at the vortex-centre defined bythe local maxima in magnetic field, with a radius . ξ (shown with a dotted hatch on the top-left vortex).Figure shown for | H ext | = 0 . .of vortices with increased (decreased) magnitudes of J a .Choosing a larger value of | J a | would increase the mag-nitude of current-induced magnetic field H c , driving thesuperconductor into normal state (after some time) foreven low values of applied field H ext . With our chosenvalue of | J a | = 0 . , we find that the system is ultimatelydriven into normal state for all values of | H ext | ≥ . inthe case of sizes l/ξ ∈ { , } . For the other two smallersizes, the system is driven into normal state for all valuesof | H ext | .For sizes l/ξ ∈ { , } , we observe vortices and theirmovement, before the system eventually goes into normalstate ( ψ ( r ) = 0 ), whereas for sizes l/ξ ∈ { , } , we donot observe vortices for any value of | H ext | . To under-stand further, we solve the standard TDGL system (3),(4) (with no applied current) for l/ξ ∈ { , } and sweepthe magnetic field | H ext | as earlier. We do not find a vor-tex state solution for any value of | H ext | . We concludethat this is a size-effect: the system is smaller than acritical size, forbidding the possibility of a vortex statesolution.Next, we also enable normal-state and flux-flow Halleffects following the discussion in II C, with γ /γ = 0 . and − . as separate cases. The choice of | γ /γ | wasmade in such a way that the vortices gain a perceivableamount of velocity in the direction of J a , but not sig-nificantly enough, so that vortices still primarily movealong − ˆ y . With Hall effect enabled, we observe complexvortex motion (fig. 5). Vortices enter the system at thetop edge and traverse smooth but irregular trajectoriesthrough the domain, due to complex vortex-vortex andvortex-boundary interactions. The key difference is thatthey also obtain velocity in the ± ˆ x direction, unlike whenHall effect is not enabled. We seek to capture the effect ofthis complex motion on the longitudinal and Hall electricfields and explain the observed behavior. y (a) t = 50 (b) t = 210 x y (c) t = 270 x (d) t = 320 . . . . . | ψ | Figure 4: We observe vortex motion under an appliedexternal current in the transverse direction. Vorticesenter at the top edge and leave at the bottom. Thismotion is followed by the superconductor being driveninto normal state. ( | H ext | = 0 . ) C. Analysis of electric fields and Hall angle
We obtain the spatially averaged electric field by aver-aging across the entire domain. We find that the so ob-tained E profiles can be grouped into three distinct types(fig. 6) of behaviour based on the l/ξ ratio and value of | H ext | . First, at a large l/ξ of 20 (fig. 6a), the longi-tudinal field saturates to σ J a (normal state) with theobservation of a series of spikes prior to that. This spik-ing occurs due to vortex entry and exit, as discussed inRef. [23]. The Hall field E y also exhibits similar spikingas a result of the vortex velocity component in the direc-tion of J a . Such behaviour occurs for sizes l/ξ ∈ { , } and applied field | H ext | ≥ . . For fields lower than . , the system rapidly evolves to the superconductingMeissner state with E x reaching zero as shown in fig. 6bfor | H ext | of 0.2. Sizes l/ξ ∈ { , } are always driven intonormal state. This results in the third type, with E x saturating to σ J a (fig.6c). These differences in electricfield behavior produce significantly different Hall effectprofiles, which we address next.We characterize the Hall effect using Hall angle, theratio of effective transverse to longitudinal conductivity: tan θ H = σ (cid:48) xy /σ (cid:48) xx [12, 13]. These macroscopic effec-tive conductivties are marked with a prime to distinguishthem from the normal-state quantities σ xx and σ xy (7).From the macroscopic equation J a = σ (cid:48) · E , we have: (cid:18) J a (cid:19) = (cid:18) σ (cid:48) xx σ (cid:48) xy − σ (cid:48) xy σ (cid:48) xx (cid:19) · (cid:18) E x E y (cid:19) (9)Thus, we get tan θ H = σ (cid:48) xy /σ (cid:48) xx = E y /E x , where thefields are both spatially and temporally averaged. We y (a) t = 200 (b) t = 500 y (c) t = 900 (d) t = 1500 x y (e) t = 2500 x (f) t = 5500 . . . . . | ψ | Figure 5: Complex vortex flow under an appliedexternal current with Hall effect enabled. The setting inof normal state from the right edge starting at t = 1500 induces a gradual transient field largely affecting Hallbehavior (fig. 8). ( | H ext | = 0 . and γ /γ = 0 . )obtain the tangent of Hall angle tan θ H for each com-bination of applied magnetic field | H ext | and size l/ξ ∈{ , , , } (fig. 7). We first note that the Hall angleprofiles resulting from our choice of | γ /γ | closely re-semble the experimental data with respect to orders ofmagnitude ( tan θ H ∼ − ) [8, 34].We find that as | H ext | → H c , when the superconduc-tor is rapidly driven into normal state, Hall angle varieslinearly with | H ext | , as in the case of normal metals. Inthis regime, flux-flow contribution is negligible and allsizes l/ξ ∈ { , , , } have the same profile. This isin agreement with expected behavior because we do nothave any size-effect for normal metals. When | H ext | ≈ ,the Hall angle approaches 0. At these low fields, sizes l/ξ ∈ { , } are driven to normal-state. As normal met-als, they have negligible Hall fields at | H ext | ≈ , andconsequently a very small Hall angle. On the other hand,sizes l/ξ ∈ { , } are in the superconducting state atthese low fields, and therefore have negligible E y and E x .The small non-zero contribution is a result of transientbehavior as seen in fig. 6b. Thus, although the Hall an-gle is small for both groups of sizes at | H ext | ≈ , the t . . . M ag n i t ud e o f E x , E y (a) E x E y t . . . M ag n i t ud e o f E x , E y (b) E x E y t . . . . M ag n i t ud e o f E x , E y (c) E x E y Figure 6: The three different types of electric fieldprofiles observed in our simulations across all sizes andmagnitudes of applied field | H ext | . (a) Saturates to σ J a , exhibits spiking behavior corresponding to entryand exit of vortices from sample of size l/ξ = 20 , and | H ext | = 0 . . (b) Saturates to , of sample size l/ξ = 20 ,and | H ext | = 0 . ). (c) Saturates to σ J a , for sample size l/ξ = 5 , and | H ext | = 1 . .underlying states are different.We next note that the Hall angle profile is completelylinear for size l/ξ = 3 . This is a result of it behaving asa normal metal with no vortex state, and electric field E x ( t ) saturating to σ J a for all values of | H ext | (thirdtype shown in fig. 6c). However, size l/ξ = 5 also ex-hibits identical behavior, but we find significant devia-tions from linearity. To understand further, we look atthe difference in transient fields between the two sizes. . . . . T a n g e n t o f H a ll a n g l e (a) γ /γ = 0 . . . . . . . . . . . . | H ext |− . − . . . .
02 (b) γ /γ = − . Size l/ξ
35 1520
Figure 7: (a)Hall angle profiles for γ /γ = +0 . .(b)Hall angle profiles for γ /γ = − . . Evidently, size isan important parameter in determining Hall behavior,with the smallest size l/ξ = 3 behaving identical to anormal conductor. Sign-reversal is only seen for l/ξ = 20 , in which case γ /γ = 0 . and − . showopposite behavior. Fields were averaged from t = 0 to .Large transients in E x ( t ) are common for size l/ξ = 5 ,such as the one shown in fig. 6c, whereas the transient ishighly suppressed in size l/ξ = 3 . In fact, across all valuesof | H ext | , the maximum percentage change in E x ( t ) (rel-ative to saturation value σ J a ) is limited to ≈ . forsize l/ξ = 3 , and to ≈ in the case of size l/ξ = 5 .We conclude that this enormous difference in transientlevels leads to deviations from linearity in size l/ξ = 5 ,in spite of non-superconducting behavior.We next look at sign-reversal of Hall angle, a keyanomaly in the Hall behavior of some superconductors.For size l/ξ = 20 , we observe a region of sharp devia-tion from linear, “normal-conductor” behavior for inter-mediate values of applied field, i.e. around | H ext | ≈ . (fig. 7). This is the region where flux-flow contribu-tion to the Hall effect is most significant, because forhigher fields, normal state sets in rapidly, and for lowerfields, there are no vortices (Meissner state). We findthat strong, negative flux-flow contribution leads to sign-reversal for γ /γ = − . , and positive contribution leadsto a peak for γ /γ = 0 . , in agreement with Kopnin etal. [13]. In order to understand the precise transient be- t − . − . − . . . . . E y γ /γ . − . Figure 8: E y fields for size l/ξ = 20 at | H ext | = 0 . The gradual transients starting at t ≈ are responsiblefor the vastly different Hall angle for γ /γ = 0 . and − . .Similar transients are seen for | H ext | ≈ . , leading to theHall angle profile in fig. 7. havior leading to this vast difference, we look at the spa-tially averaged fields E x and E y as functions of time (for | H ext | = 0 . ). First, we find that for both γ /γ = 0 . and − . , E x has an identical profile. This is expectedbecause γ only influences the motion of vortices along J a , thus affecting only E y (ref. II C). On the other hand,we find significant difference in E y ( t ) profiles (fig. 8). Itis evident that the difference in Hall behavior between γ /γ = 0 . and − . can be attributed almost entirelyto the transients starting at t ≈ . Although spikingbehavior results in peaks in the opposite direction, theyhave negligible contribution to the average. Instead, thegradual transient starting at t ≈ leads to differentprofiles for γ /γ = 0 . and − . , giving a strong positiveand negative contribution to Hall field, respectively. Wetry to relate this transient with order parameter (fig. 5)in order to determine the precise behavior causing such transients. We find that this transient is the inducedfield resulting from variation in A during the onset ofnormal state (fig. 5). This onset occurs from the rightboundary for γ /γ = 0 . (as seen in fig. 5), and fromthe left for γ /γ = − . (fig. 8). For both γ /γ = 0 . and − . , this transient leads ultimately to saturation of E y to the same, small positive value. This is the non-flux-flow, or normal-state contribution to the Hall field.Interestingly, we find that although size l/ξ = 15 ex-hibits vortex behavior similar to l/ξ = 20 , we do not findsign-reversal in Hall angle. This is due to the suppressedgradual transient fields in l/ξ = 15 , leading to a muchweaker contribution of flux-flow Hall effect. Therefore,it is evident that along with γ (whose value depends onthe electronic structure) and several other parameters,size alters Hall behavior significantly, adding to the rea-soning behind the observation of a diverse variety of Hallangle profiles in various materials.In summary, we have simulated the anomalous Halleffect using the modified TDGL equations in COMSOLMultiphysics and shown that the solutions provide in-sights into the precise temporal dynamics of transientfields and vortex behavior that scale with the samplesize. We have explored theoretically how features of theanomalous Hall effect evolve with a variation of the lin-ear dimensions when the lengths are only a few times thecoherence length. The Hall effect behaviour predicted bythese simulations may be probed with advanced experi-mental techniques that have already been applied to im-age vortices [1–6, 35]. Such studies would be importantfor understanding finite-size effects in superconductorsand their evolution at microscopic lengthscales. ACKNOWLEDGMENTS
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