Strong pinning theory of thermal vortex creep in type II superconductors

Strong pinning theory of thermal vortex creep in type II superconductors

M. Buchacek Institute for Theoretical Physics, ETH Zurich, 8093 Zurich Switzerland    R. Willa Materials Science Division, Argonne National Laboratory, Lemont, IL 60439, USA    V.B. Geshkenbein Institute for Theoretical Physics, ETH Zurich, 8093 Zurich Switzerland    G. Blatter Institute for Theoretical Physics, ETH Zurich, 8093 Zurich Switzerland
August 24, 2019
Abstract

We study thermal effects on pinning and creep in type-II superconductors where vortices interact with a low density of strong point-like defects with pinning energy and extension , the vortex core size. Defects are classified as strong if the interaction between a single pin and an individual vortex leads to the appearance of bistable solutions describing pinned and free vortex configurations. Extending the strong pinning theory to account for thermal fluctuations, we provide a quantitative analysis of vortex depinning and creep. We determine the thermally activated transitions between bistable states using Kramer’s rate theory and find the non-equilibrium steady-state occupation of vortex states. The latter depends on the temperature and vortex velocity and determines the current–voltage (or force–velocity) characteristic of the superconductor at finite temperatures. We find that the linear excess-current characteristic with its sharp transition at the critical current density , keeps its overall shape but is modified in three ways due to thermal creep: a downward renormalization of to the thermal depinning current density , a smooth rounding of the characteristic around , and the appearance of thermally assisted flux flow (TAFF) at small drive , with the activation barrier defined through the energy landscape at the intersection of free and pinned branches. This characteristic emphasizes the persistence of pinning of creep at current densities beyond critical.

I Introduction

The properties of numerous materials are determined by the presence of topological excitations in the ordered states of matter; examples include vortices in type-II superconductors [1, 2], domain walls in ferroic materials [3, 4], or dislocations in metals [5, 6]. The motion of such objects within the host material has a significant effect on its response, e.g., the onset of finite resistivity in superconductors or the loss of coercivity in magnets. Immobilizing these excitations, usually by pinning onto material defects, is thus of great technological relevance. The dynamics of topological objects then exhibits a transition between a static or pinned phase and a sliding or unpinned phase upon exceeding a threshold or critical force . Understanding the pinned-to-sliding transition, optimizing the pinning threshold, and stabilizing it against thermal fluctuations present vital tasks at the cross-roads of disordered statistical physics and non-equilibrium phenomena. The complete description of the material’s response is captured by the force–velocity () characteristic of topological excitations; here, we extend the strong pinning theory to include effects of finite temperatures and calculate the response characteristic of vortex matter in type-II superconductors subject to a low density of point-like strong defects.

Figure 1: Current–velocity characteristic of a type II superconductor with strongly pinned vortices and including effects of thermal fluctuation. The linear excess-current characteristic (in blue) with its critical current density is modified due to thermal creep (red curves with ): i) finite temperatures shift the linear branch to the left as described by a thermal reduction of to the depinning current density , ii) the characteristic is rounded at the onset of vortex motion near , and iii) the characteristic turns ohmic at small current densities , see inset, an effect commonly known as thermally assisted flux flow or TAFF.

Vortex pinning is described by either of two frameworks: within weak pinning theory [7, 1], the pinning force due to an individual defect vanishes and it is the collective action of many defects which generates the average pinning-force density. On the contrary, for strong pinning [8, 7], individual defects induce substantial deformations that lead to bistable behavior and generate an average non-zero pinning force on the vortex lattice that scales linearly in the (small) density of defects. Weak collective pinning has been fully developed in the wake of the high- discovery [1, 9], although results have remained qualitative. On the other hand, the theory of strong pinning provides quantitative results, but its development is less advanced. The critical currents [10], current–voltage characteristics [11, 12], -response [13, 14, 15], and the overall pinning diagram [16, 17] have been analyzed and augmented by numerical simulations [18, 19, 20]. However, so far no systematic theory including thermal fluctuations has been developed, although important understanding of the creep mechanism can be derived from the work on charge-density wave pinning by Brazovskii, Larkin, and Nattermann [21, 22], see also early work by Fisher [23, 24].

In this paper, we adopt the strong pinning paradigm and present quantitative results on the pinning and creep of vortices in type II superconductors in the presence of thermal fluctuations. Such vortices result from a magnetic field penetrating the superconductor [25, 26, 27], each vortex carrying a quantum of flux and together forming a lattice of density inducing the average magnetic field . The resulting vortex matter is pushed by the current density via the Lorentz-force density . The resulting force–velocity characteristic follows from the dissipative force-balance equation

(1)

where denotes the Bardeen-Stephen viscosity (per unit volume, is the upper-critical field and the normal-state resistivity) and is the average pinning force density, the quantity of central importance in this paper.

The weak pinning approach provides estimates for the dynamical properties of vortices: the pinning force density has been calculated by Larkin and Ovchinnikov [28] and by Schmid and Hauger [29] using a high-velocity perturbative expansion, while the depinning dynamics around has been studied via renormalization group techniques by Narayan and Fisher [30] and by Chauve et al. [31]. On the contrary, the strong pinning scenario produces quantitative results, provided that the density of defects is small such that they act independently, i.e., the pinning-force density is linear in the density of defects (but scales non-trivially in the force of individual defects). Pinning is strong if the largest (negative) curvature of the pinning potential wins over the effective vortex stiffness ; this is expressed in the Labusch criterion [8] characterizing strong pins. The vortex deformation due to an individual pin then exhibits bistable solutions defining pinned and free vortex branches (Fig. 3). Their asymmetric occupation is at the origin of a finite average pinning-force density exerted on the vortex lattice by the randomly positioned defects. The determination of its maximal value provides the critical force density [8, 7, 16] . The calculation of its dynamical variant in the absence of thermal fluctuations [11, 12] gives access to the full characteristic; this turns out to be of an excess-current form, i.e., the linear flux-flow characteristic of the defect-free superconductor is shifted by a finite critical current density , see Fig. 1. Note that, although pinning by an individual defect is strong, the small defect density results in small or moderate pinning forces; hence, strong pinning theory is not necessarily the theory producing highest critical current densities.

In order to account for thermal fluctuations in the average pinning-force density , we follow Brazovskii and Larkin [21] and use Kramer’s rate theory [32] to describe transitions between the pinned and free vortex branches and determine the branch occupation at finite temperatures and velocities. At large velocities but below the (thermal) velocity , finite temperature assists the motion of vortices, diminishes the asymmetry in the vortex branch occupation, and thus reduces the pinning-force density to lie below the critical value ; here, the depinning velocity characterizes the dissipative motion in the defect potential. The critical current density is reduced to a depinning current density separating flat and steep regions of the characteristic, see Fig. 1; to leading order, the relative shift depends on temperature as and is logarithmically dependent on the density of defects, see Sec. III.2. Beyond depinning, we find a weak dependence of on the velocity and thus recover a close to linear excess-current characteristic, shifted downward in current with respect to the result. Hence, contrary to usual expectations, a large pinning-force density as well as thermal creep remain present far beyond the critical current density , see Ref. [33]. Finally, writing the vortex velocity in the form , reminiscent of its thermal origin with the activation barrier, we find a decreasing activation barrier when approaching the depinning region from below. However, the barrier persists well beyond , where it is characterized by a slow logarithmic variation with the current density , , consistent with a linear force–velocity characteristic.

Weak drives are characterized by a nearly symmetric occupation of branches, more precisely, an occupation that is shifted linearly in with respect to the thermal equilibrium occupation. This results in an ohmic response with exponentially small velocities , usually known as TAFF, thermally assisted flux-flow [34], a specific form of vortex creep at low drive. As implied by its name, the resistivity is thermally assisted, i.e., , the flux-flow resistivity, with the finite activation barrier derived directly from the bistable solutions, see Sec. III.3 below. As a result, within the framework of strong pinning, the superconductor loses its defining property of dissipation-free current transport. This is quite different as compared with the result of weak collective pinning theory, where barriers diverge , thereby establishing a truly superconducting (glass) state at low drives .

Below, we start with a brief review of the strong pinning formalism and show how the interaction of independent defects with the vortex lattice is reduced to a single-pin–single-defect problem involving the effective vortex elasticity , see Sec. II.1. In Sec. III, we extend the analysis to include thermal fluctuations; we discuss creep effects at large drives and velocities in Sec. III.2 and find the depinning current density and the relevant creep barriers in its vicinity. In section III.3, we focus on small drives and low velocities and find the ohmic TAFF characteristics with a quantitative prediction of the activation barrier. Finally, in the appendices we provide details of the energy landscape in the marginally- and very strong pinning regime and other technical details of calculations omitted in the main text. In our analytic work, we focus on the relevant limiting cases, small () and large () drives as well as the limits of marginally strong pinning with and very strong pinning, . The new insights on the persistence of pinning and creep beyond the critical drive has been published in a short format in Ref. [33].

Ii Formalism

ii.1 Strong Pinning

In the absence of defects (pins) and thermal fluctuations, our vortex array, aligned along the -axis, is arranged in a two-dimensional (2D) hexagonal lattice with equilibrium positions . The presence of strong defects results in deformations of the lattice described by the planar displacement field . We consider a representative single defect placed in the origin and characterised by a radially symmetric bare pinning potential , with decaying on the length and the maximal pinning energy. For a point-like defect, the pinning potential extends over a distance ; for a defect of size , the energy is determined by the condensation energy, , see Ref. [14] for more details. Furthermore, we consider a situation where the repulsion between vortices prevents two of them from occupying the same defect [35], limiting the interaction between vortices and the defect to the single reference vortex closest to the origin. Such a situation is realized at small and intermediate fields with and a not too large pinning energy . The free energy of this setup then takes the form

(2)

where the first term describes the vortex–defect interaction and the second contributes the elastic energy, expressed through the (symmetric and real) reciprocal-space elastic matrix [8] . Here, denotes the tip position of our reference vortex pinned to the defect at the origin. The displacement fields in real and reciprocal space are related through [we decompose and ]

(3)
(4)

The integration over is restricted to the 2D Brillouin zone of the vortex lattice, in the circularized approximation, while is subject to the cutoff .

The variation of Eq. (2) with respect to the displacement field provides us with the response

(5)

where denotes the pinning force. Expressing the real-space perturbation in the first term through the Fourier modes , this reads

(6)

For a lattice moving with a steady drift velocity , the asymptotic vortex positions are given by . The full dynamical response of the vortex lattice including time-dependent displacements has to be calculated from the dissipative dynamical equation of motion and is addressed in Refs. [11, 12]. In the present work, we neglect dynamical effects and assume that the drift velocity is sufficiently small such that the vortex displacement field locally minimizes the energy (2) at any moment of time, . The displacement field then depends on time only through the boundary condition, the asymptotic position of the reference vortex , and relates to the pinning force via

(7)

with the Green’s function . Using Eq. (7), we can first solve for , the -component of , and then express the complete displacement field through the tip position of the reference vortex. After transformation back to real space, we obtain

(8)

We express the last integral through the effective elasticity ,

(9)

(with summation over implied) and make use of the self-consistent solution of Eq. (8) to obtain the displacement field expressed through the amplitude ,

(10)

Inserting this result back into Eq. (2), we obtain a simple expression for the free energy for our specific configuration with the tip at of the vortex displaced by from its asymptotic position due to the action of the defect,

(11)
(12)
Figure 2: Effective pinning energy versus for four different asymptotic vortex positions , . We assume a Lorentzian defect with a potential and a Labusch parameter . Two local minima at and appear for and are separated by the local maximum at defining the barrier for depinning (the barrier to escape the pin) and the barrier for pinning (or jumping into the pin). Bottom: Minimizing with respect to the tip position at fixed asymptotic position corresponds to solving the self-consistency equation , see (15), here done graphically. Multiple solutions , , show up if the slope is smaller than the maximum slope of , what corresponds to a Labusch parameter .

Next, we choose the vortex position along the -axis, and assume a radially symmetric pin, implying that ; expressing the effective free energy in Eq. (12) in terms of the vortex tip coordinate , we arrive at the simplified effective pinning energy

(13)

The effective pinning energy involves the bare pinning potential augmented by the elastic deformation energy of the vortex in the form of a parabolic potential centered at , see Fig. 2, with a curvature given by the effective vortex lattice elasticity . The latter can be expressed through the compression, tilt and shear elastic moduli known from elasticity theory [1, 14], ; the numerical depends on the chosen approximations, see Refs. [12, 35]. The simple estimate , where , is governed by the value of the vortex line energy .

If pinning is sufficiently strong, i.e., is sufficiently deep, the total free energy has two minima within a finite range of asymptotic vortex positions . Minima appear or vanish whenever the total pinning energy develops an inflection point, . This requires that the condition

(14)

the so-called Labusch criterion[8], is fulfilled, see bottom of Fig. 2. The value marks the transition between weak pinning with a unique vortex configuration and the strong-pinning situation where the vortex can choose between two alternative configurations, a pinned and a free one, whenever its asymptotic position resides in the interval . The two local minima and are obtained from minimizing with respect to at fixed ,

(15)

The local maximum at is an unstable solution that plays an important role in the context of creep, see below.

Fig. 3 shows the multi-valued energy landscape with the three branches , , and corresponding to the extremal solutions, ; three of them coexist in the two intervals where , while outside those regions only one solution is realized. The total pinning force exerted on a moving vortex is derived from this energy landscape, with the pinning force acting on a vortex given by . As this force differs when evaluated in the pinned and free branches, it is the occupation of these branches that determines the total pinning-force density acting on the vortex system. The pinned and free branches in the pinning-energy landscape are separated by an energy barrier. The depinning barrier has to be overcome for transitions to the free branch, while the pinning barrier is relevant for jumps into the pin, i.e., the transitions to the pinned branch.

Figure 3: Multivalued energy landscape for a strong defect. Within the regions , pinned and free equilibrium positions show up with different locations and of the vortex tips either close to or away from the defect; they define the energies and of the pinned (nearly parabolic) and free (nearly flat) branches. In addition, an unstable extremum at , see Fig. 2, maximises the pinning energy (dotted) unstable branch and defines the energy barriers connecting pinned and free branches. Thick blue curves denote the occupation of branches in the critical state realised for . The pinning force is proportional to the sum of jumps , in the energy landscape.

Unfortunately, no closed expressions for the branches can be given since the equilibrium equation (15) fixing is in general not solvable analytically. Progress can be made in the limits of marginally strong pinning with or for very strong pinning . In the first case the slope is close to the maximum slope of and the energy branches can be derived from a cubic expansion of around the point of maximum slope, . For very strong pinning, the slope is small compared to and the pinned and unstable solutions are obtained by analyzing the tail of . We will assume an algebraically decaying potential, such that the pinning force with , in order to make analytic progress in this situation.

The boundaries of the multi-valued interval are found by first determining the critical tip positions from the condition for the appearance of inflection points in and then deriving the associated asymptotic position from the equilibrium condition Eq. (15), see also Fig. 2. For marginally strong pinning, we find that (see Ref. [16], Appendix A and B for the derivation)

(16)
(17)

where . Note that coincides with the branch crossing point , . For very strong pinning, resides on the tail of the pinning potential, while is located near the maximum force,

(18)

The associated asymptotic positions are largely different, see also Fig. 2,

(19)

Finally, the branch crossing point is located at a position where the free and pinned branches have the same energies, , implying that .

The free-energy landscape in Fig. 3 has much in common with the one appearing in the phenomenological theory of a first-order phase transition in thermodynamic systems, e.g., the Gibb’s energy of the Van der Waal’s theory of the gas–liquid transition or the energy of a magnetic transition. In developing this analogy, we can identify with the volume and (or the inverse Labusch parameter ) with the reduced temperature . Expanding , we can identify the energy with the free energy . If is identified with pressure , then is (up to the constant term ) equivalent to the Gibb’s energy . Minimizing with respect to for fixed and corresponds to minimizing with respect to for fixed and and provides the (metastable) equilibrium states (note that and play the roles of constraint parameters). The barrier separating the minima in the thermodynamic system are relevant in the description of the hysteretic transition and nucleation phenomena—here, the analogous barriers describe thermal transitions between pinned and free states and thus are relevant in the description of thermal creep.

ii.2 Pinning force

The average pinning force per defect acting on the vortex system is obtained by position-averaging the force between a defect and its nearest vortex while accounting for the random positions of the defects in the material. Driving the vortices in the positive -direction results in an average force per defect (in accordance with (1), we choose and hence to be positive). The instantaneous force acting on a vortex with asymptotic position is different for pinned and free states. Let be the occupation probability of the pinned branch; the occupation probability for the free branch then is . For a vortex passing centrally through the defect, the average pinning force is given by the position and occupation average

(20)

with denoting the pinning forces on the pinned and free branches. For only the pinned branch is available, while for the occupation is restricted to the free branch; hence, we set and , respectively, in those two regions. The integration is restricted to due to the vortex lattice periodicity. The antisymmetric force with allows to write the previous equation in the form

(21)

with and the integration restricted to the multivalued intervals .

The branch occupation for vortices driven along the positive -axis is shown in Fig. 3, see blue solid lines. An individual vortex approaches the defect on the free branch and remains there until , even though the pinned branch becomes energetically more favorable at the branch crossing point . The vortex becomes pinned at when the pinning barrier vanishes and stays on the pinned branch until , where it jumps again to the unpinned branch, this time due to the vanishing of the depinning barrier . The occupation then can be written through the Heaviside step function ,

(22)

and using in Eq. (20), we find that

(23)

with and . Note that and thus corresponds to the sum of the energy jumps in the multivalued energy landscape evaluated at the end points of the multivalued intervals.

Estimates for the jumps are derived in Appendix A and B, see also Refs. [8, 16]. For marginally strong pinning, one finds that

(24)

while for very strong pinning

(25)

in particular, using Eq. (19), we find that is large compared to .

Figure 4: Pinning of vortices passing the defect at transverse distances (a), (b), and (c) for (blue, left) and finite temperatures (red, right). The corresponding pinning energy landscapes are plotted for the Lorentzian pinning potential and . At , vortices are pinned whenever they enter the defect trapping area (along the grey half circle of radius , where the free and unstable branches merge and disappear such that . In case (c) the vortex trajectory lies outside the trapping area and vortices always stay on the free branch, which is available all the way along the vortex trajectory; the pinning force vanishes in this case. At finite temperature , the transitions between the free and pinned branches are realized at a radial distance from the pinning centre. Due to thermal activation, vortices can overcome the barrier separating free and pinned branches and occupy the pinned branch even for , resulting in a non-vanishing average pinning force in the extended region (light red).

Above, we have considered the situation where the vortex impacts straight on the defect center. For those vortices passing the defect at a finite transverse distance (Fig. 4), the effective pinning energy is given by , where and denote the asymptotic- and tip-position of the vortex line, see Eq. (12) (we drop the index ). The equilibrium condition yields the solutions , for the free, pinned, and unstable branches. For a radially symmetric pinning potential, we have and the equilibrium condition is satisfied for the radial geometry . The energy is then brought to the same form as in Eq. (13), albeit with the replacement . Evaluating provides us with the energies of the various branches in the multivalued energy landscape, ,

(26)

The energy landscape is plotted in Fig. 4 for three impact parameters , , and . Eq. (26) shows that the shape of the energy landscape at finite impact parameter is similar to the one at with an excluded region . In particular, the minimal energy of the pinned branch satisfies . For a large impact parameter (the situation with is shown in Fig. 4) the free branch never terminates, implying that such a vortex is never trapped at . On the other hand, vortices hitting the defect with a finite impact parameter are trapped onto the defect at the radial distance and released back to the free branch at a radial distance , hence the vortex remains pinned over the finite interval , with the direction of drive, and for all ; we call the transverse trapping length. The average pinning force along the -direction is once more given by Eq. (20), but with the pinning forces replaced by and the jumps in the occupation (22) now appearing at and . A simple calculation then shows, that the average force contributed by a vortex with an impact parameter is identical with the result (23). While vortices passing the defect at larger distances cannot get trapped at zero temperature, fluctuations at finite temperature will render such processes statistically possible, see Sec. III.2.3 below.

Combining the above results, we can determine the average pinning force density for a finite density of defects by multiplying the average pinning force (23) with the fraction of trajectories that are trapped by one defect and the density of independently acting defects; including a minus sign in order to respect our definition of pinning force density in the equation of motion (1), we obtain the critical force density

(27)

Collecting the various factors from above, we obtain the estimates

(28)

and

(29)

in the marginally strong and very strong pinning limits, respectively. Quite often, these results are written through the trapping area[10, 16] ,

(30)

which assumes values and at marginally strong and very strong pinning. For a rapidly decaying pinning potential (with a large value of ), the trapping area, critical force density, and critical current density scale like , , and ; such a field dependence (cut off at small fields when strong pinning becomes 1D, single-vortex type, see Ref. [16]) is often taken as a signature for strong pinning.

The critical state occupation in (22) is the one maximizing the pinning force. If the applied force density exceeds , the vortex lattice moves with drift velocity as given through the dissipative force balance equation . The dynamical pinning force changes on a scale and has been calculated at in Refs. [11, 12]; below, we focus on the calculation of at finite temperatures but small velocities , where the dynamical motion of the vortex through the pin can be neglected, and derive the thermally renormalized force–velocity characteristic.

Iii Thermal creep

We start with a short qualitative overview of thermal creep effects at large and small velocities before deriving precise expressions for the two limits.

iii.1 Qualitative overview

At finite temperatures , one has to account for thermal fluctuations in the determination of the branch occupation as vortices can jump between branches by overcoming the activation barrier; the same physics appears in the context of pinned charge density waves, see Refs. [21, 22]. We find the pinned branch occupation through solving the rate equation derived from Kramers’ theory [32] (we set from now on),

(31)

This rate equation accounts for the depinning of vortices via the activation barrier as well as the filling of the pinned branch due to transitions over the barrier . The steady-state probabilities depend on the time only through the coordinate and thus we have replaced the total derivative by . The frequencies and can be understood as the number of attempts per unit of time made by a vortex to escape from its current, pinned or free, state. The success probability of such attempts is exponentially small in the activation barrier. We calculate the barriers and attempt frequencies later in Sec. III.2 from a ‘microscopic’ theory.

Focusing on the high- and low-velocity regimes with qualitatively distinct solutions of the rate equation (31) provides us with a first understanding of the problem. The velocity derived below, see Eqs. (36) and (62), sets the scale below which thermal effects modify the excess-current characteristic; above , the characteristic smoothly joins the one at . The high-velocity regime , with the maximal activation barrier located at the branch crossing point , see Fig. 3, is characterized by an occupation of a shape similar to the one of the critical state, but with the transitions between branches realized close to the thermally renormalized jump points , see Fig. 6. Ignoring the finite width and of these jumps, we can write and express the pinning force density through the thermally renormalized jumps in the energy landscape

(32)

where

(33)

The jump location follows from the following consideration (a corresponding analysis provides the location , see below): Close to the jump at , the occupation dynamics is dominated by the smaller depinning barrier and the second term on the right hand side of the rate equation (31) can be ignored. The rate equation then takes the simple form , with

(34)

defining the depinning length at the position , telling us over what distance the vortex will transit from the pinned to the free branch. The depinning length is large near where the barrier is large and decreases rapidly with increasing due to the decreasing barrier . The transition to the lower (free) state appears at the position where the vortex can escape the pin while itself moving a distance , implying that the relative change in over the distance should be of order unity. With the help of Eq. (34), we can reexpress the corresponding condition in the form

(35)

where we focus on the main -dependence in the exponent and denote the space derivative with a prime, . At the maximal value , the barrier vanishes and we reach the maximal velocity ,

(36)

where the thermal characteristic goes over into the excess-current characteristic. From the condition (35) and using Eq. (36), we find that the relevant depinning barrier can be written in the form

(37)

where we have approximated by its value at . Similar results apply for the jump at and are quantitatively derived below, see Sec. III.2. Given the barriers (and ) for a specific defect potential, we can solve Eq. (37) for (and similar for ) and using the results in the definition of the energy jump Eq. (33) leads to the velocity- and temperature-dependent pinning force density. We cast the final result (see Sec. III.2 for details) in the form

(38)

with a factor of order unity that can be derived as a function of pinning strength for any given defect potential .

A different approach has to be used in solving the rate equation for small velocities . Starting from the above analysis and decreasing the velocity , the jump positions approach the branch crossing point and the activation barriers increase towards their maximum , see Fig. 3. At , the renormalized energy jumps vanish and Eq. (32), providing a vanishing pinning force, can no longer be used. In this limit, a good starting point for our analysis is the equilibrium distribution obtained by setting in the rate equation (31).

(39)

with defining the local relaxation distance for the case of pinning. The expression Eq. (39) is valid away from the endpoints of the multi-valued interval where barriers vanish. The rate equation then can be cast into the form

(40)

where the equilibrium relaxation distance ,

(41)

includes processes that connect both pinned and free branches. Treating as a small parameter, we find that the solution of the rate equation is given by the shifted equilibrium distribution, . Assuming similar scales and , we obtain a simple estimate for the equilibrium relaxation length in the form and the condition defining the low-velocity regime is then equivalent to implying that the shift is small compared to the scale of variations in . Our low-velocity analysis improves on the work by Brazovskii, Larkin, and Nattermann (BLN) [21, 22] discussing thermal effects on the pinning of charged density waves that exhibits similar bistable solutions as found here. In their analysis, the smooth variation in the equilibrium distribution is ignored, what results in a different shift scale , see Sec. III.3 for further details.

The equilibrium occupation is symmetric and thus yields no average pinning force, allowing us to rewrite Eq. (21) as

(42)
(43)

Hence, the pinning force depends linearly on for small velocities. A detailed analysis (see Sec. III.3) shows that the average pinning-force has a non-trivial dependence on the transverse distance, reaching its maximum value given by Eq. (43) at and vanishing at . This results in an additional numerical prefactor in the formula for the average pinning force density