**1.1. The essential physical background**

It is well-known that a type-II superconductor, while exposed to a magnetic field **B** whose magnitude is between the lower and upper critical field, is penetrated by a flux-line array of Abrikosov vortices, or *fluxons* [1–3]. Each vortex contains one magnetic flux quantum, <sup>Φ</sup><sup>0</sup> <sup>=</sup> 2.07 <sup>×</sup> <sup>10</sup>−<sup>15</sup> Wb, and the repulsive interaction between vortices makes them to arrange in a triangular lattice, with the vortex lattice parameter *aL* � <sup>√</sup>Φ0/*<sup>B</sup>* where *<sup>B</sup>* <sup>=</sup> <sup>|</sup>**B**|. A vortex is often simplified by the hard-core model [4], where the core is a cylinder of normal material with a diameter of the order of the coherence length. In this model, the magnetic field is constant in the core but decays exponentially outside the core over a distance of the order of the effective magnetic penetration depth.

In an ideal material, the vortex array would move with average velocity **v** under the action of the Lorentz force **F***<sup>L</sup>* essentially perpendicular to the transport current. Due to the nonzero viscosity experienced by the vortices when moving through a superconductor, a faster vortex motion corresponds to a larger dissipation. In experiments, inhomogeneities are usually present or can intentionally be introduced in a sample [5] which may give rise to local variations of the superconducting order parameter. This may cause the vortices to be pinned. By this way, the resistive properties of a type-II superconductor are determined by the vortex dynamics, which due to the presence of pinning centers can be described as the motion of vortices in some *pinning potential* (PP) [6]. In particular, randomly arranged and chaotically distributed point-like pinning sites give rise to an ubiquitous, *isotropic* (*i*) pinning contribution, as said of the "background nature". Depending on the relative strength between the Lorentz and pinning forces, the vortex lattice can be either pinned or on move, with a nonlinear transition between these regimes. Thus, the current-voltage characteristics (CVC) of such a sample is strongly nonlinear.

©2012 Shklovskij and Dobrovolskiy, licensee InTech. This is an open access chapter distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. ©2012 Shklovskij and Dobrovolskiy, licensee InTech. This is a paper distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### 2 264 Superconductors – Materials, Properties and Applications Microwave Absorption by Vortices in Superconductors with a Washboard Pinning Potential <sup>3</sup>

The importance of flux-line pinning in preserving superconductivity in a magnetic field and the reduction of dissipation via control of the vortex motion has been in general recognized since the discovery of type-II superconductivity [1, 2, 7, 8]. Later on, it has been found that the dissipation by vortices can be suppressed to a large degree if the intervortex spacing *aL* geometrically matches the period length of the PP [9]. Moreover, artificially created linearly-extended pinning sites are known to be very effective for the reduction of the dissipation by vortices in one [10, 11] or several particular directions. Indeed, if the PP ensued in a superconductor is *anisotropic* (*a*), the direction of vortex motion can be deflected away from the direction of the Lorentz force. In this case, the nonlinear vortex dynamics becomes two-dimensional (2*D*) so that **v** ∦ **F***L*. The non-collinearity between **v** and **F***<sup>L</sup>* is evidently more drastic the weaker the background *i* pinning is [12], which can otherwise mask this effect [13]. The most important manifestation of the pinning anisotropy is known as guided vortex motion, or the *guiding* effect [14], meaning that vortices tend to move along the PP channels rather than to overcome the PP barriers. As a consequence of the guided vortex motion, an *even-in-field* reversal transverse resistivity component appears, unlike the ordinary Hall resistivity which is *odd* regarding the field reversal. A guiding of vortices can be achieved with different sorts of PP landscapes [15, 16] though it is more strongly enhanced and can be more easily treated theoretically when using PPs of the washboard type (WPP).

which Nb films have been grown. It has been demonstrated that pronounced guiding of vortices occurs. Experimental data [10] were in good agreement with theoretical results [12]

Microwave Absorption by Vortices in Superconductors with a Washboard Pinning Potential 265

From the viewpoint of theoretical modeling, saw-tooth and harmonic PPs represent the most simple forms of the WPP. On the one hand, these simple forms of the WPP allow one to explicitly calculate the dc magneto-resistivity and the ac impedance tensor as the physical quantities of interest in the problem. On the other hand, these WPP's forms are highly realistic in the sense of appropriate experimental realizations which range from naturally occurring pinning sites in high temperature superconductors (HTSCs) to artificially created linearly-extended pinning sites in high-*Tc* and conventional superconductors, more often in thin films. Some experimental systems exhibiting a WPP are exemplified in Fig. 1. The experimental geometry of the model discussed below implies the standard four-point bridge of a thin-film superconductor with a WPP placed into a small perpendicular magnetic field with a magnitude *B* � *Bc*<sup>2</sup> such that our theoretical treatment can be performed in the

Summarizing what has been said so far, by tuning the intensity, form, and asymmetry of the WPP in a superconductor, one can manipulate the fluxons via dynamical, directional, and orientational control of their motion. Evidently, a number of more sophisticated phenomena arise due to the variety of the dynamical regimes which the vortex ensemble passes through. In the next sections we will consider a particular problem in the vortex dynamics when the vortices are subjected to superimposed subcritical dc *j*<sup>0</sup> < *jc* and small ac *j*<sup>1</sup> → 0 current drives at frequencies *ω* in the microwave range. What is discussed below can be directly employed to the wide class of thin superconductors with a WPP, including but not limited to those examples shown in Fig. 1. In particular, by looking for the dc magneto-resistivity and the ac impedance responses, we will elucidate: a) how to derive the absorbed power by vortices in such a superconductor as function of all the driving parameters of the problem and b) how to solve the inverse problem, i.e., to reconstruct the coordinate dependence of a PP from the dc current-induced shift in the *depinning* frequency *ωp* [30], deduced from the

curves *P*(*ω*|*j*0). The importance and further aspects of this issue are detailed next.

**1.3. Which information can be deduced from microwave measurements?**

The measurement of the complex impedance response accompanied by its power absorption *P*(*ω*) in the radiofrequency and microwave range represents a powerful approach to investigate pinning mechanisms and the vortex dynamics in type-II superconductors. The reason for this is that at frequencies *ω* � *ωB*, substantially smaller than those invoking the breakdown of the zero-temperature energy gap (*ω<sup>B</sup>* = 2Δ(0)/¯*h* ≈ 100 GHz for a superconductor with a critical temperature *Tc* of 10 K), high-frequency and microwave impedance measurements of the mixed state yield information about flux pinning mechanisms, peculiarities in the vortex dynamics, and dissipative processes in a superconductor. It should be stressed that this information can not be extracted from the dc resistivity data obtained in the steady state regime when pinning is strong in the sample. This is due to the fact that in the last case when the critical current density *jc* is rather large, the realization of the dissipative mode, in which the flux-flow resistivity *ρ <sup>f</sup>* can be measured, requires *j*<sup>0</sup> *jc*. This is commonly accompanied by a non-negligible electron overheating in

that allowed to estimate both, the *i* and *a* PP parameters.

*single-vortex approximation*.

One more intriguing effect appears when the PP profile is asymmetric. In this case the reflection symmetry of the pinning force is broken and thus, the critical currents measured under current reversal are not equal. As a result, while subjected to an ac current drive of zero mean a net rectified motion of vortices occurs. This is known as a rocking *ratchet* effect and has been widely used for studying the basics of mixed-state physics, e.g., by removing the vortices from conventional superconductors [17], as well as to verify ideas of a number of nanoscale systems, both solid state and biological [18, 19].

## **1.2. Experimental systems with a washboard pinning potential**

The first experimental realization of a WPP used a periodic modulation of the thickness of cold-rolled sheets of a Nb-Ta alloy [14]. In this work the influence of isotropic pointlike disorder on the guiding of vortices was discussed for the first time. Later on, lithographic techniques have been routinely employed to create periodic pinning arrays consisting of practically identical nanostructures in the form of, e.g, microholes [20, 21], magnetic dots [22], and stripes [23]. The main idea all these works share is to suppress periodically the superconducting order parameter.

With regard to a theoretical description it has to be stated that a full and exact account of the nonlinear vortex dynamics in superconducting devices proposed in these works [20–23] in a wide range of external parameters is not available due to the complexity of the periodic PP used in these references. Due to this reason it has been proposed by the authors in a number of articles to study a simpler case, such as a WPP periodic in one direction [12, 24–27] or bianisotropic [28]. The main advantage of these approaches implies the possibility to describe the phenomenon of guided vortex motion along the WPP channels, i.e., the directional anisotropy of the vortex velocity, if the transport current is applied under various in-plane angles. For instance, self-organization has been used [10, 29] to provide semi-periodic, linearly extended pinning "sites" by spontaneous facetting of m-plane sapphire substrate surfaces on which Nb films have been grown. It has been demonstrated that pronounced guiding of vortices occurs. Experimental data [10] were in good agreement with theoretical results [12] that allowed to estimate both, the *i* and *a* PP parameters.

2

The importance of flux-line pinning in preserving superconductivity in a magnetic field and the reduction of dissipation via control of the vortex motion has been in general recognized since the discovery of type-II superconductivity [1, 2, 7, 8]. Later on, it has been found that the dissipation by vortices can be suppressed to a large degree if the intervortex spacing *aL* geometrically matches the period length of the PP [9]. Moreover, artificially created linearly-extended pinning sites are known to be very effective for the reduction of the dissipation by vortices in one [10, 11] or several particular directions. Indeed, if the PP ensued in a superconductor is *anisotropic* (*a*), the direction of vortex motion can be deflected away from the direction of the Lorentz force. In this case, the nonlinear vortex dynamics becomes two-dimensional (2*D*) so that **v** ∦ **F***L*. The non-collinearity between **v** and **F***<sup>L</sup>* is evidently more drastic the weaker the background *i* pinning is [12], which can otherwise mask this effect [13]. The most important manifestation of the pinning anisotropy is known as guided vortex motion, or the *guiding* effect [14], meaning that vortices tend to move along the PP channels rather than to overcome the PP barriers. As a consequence of the guided vortex motion, an *even-in-field* reversal transverse resistivity component appears, unlike the ordinary Hall resistivity which is *odd* regarding the field reversal. A guiding of vortices can be achieved with different sorts of PP landscapes [15, 16] though it is more strongly enhanced and can be

more easily treated theoretically when using PPs of the washboard type (WPP).

nanoscale systems, both solid state and biological [18, 19].

superconducting order parameter.

**1.2. Experimental systems with a washboard pinning potential**

One more intriguing effect appears when the PP profile is asymmetric. In this case the reflection symmetry of the pinning force is broken and thus, the critical currents measured under current reversal are not equal. As a result, while subjected to an ac current drive of zero mean a net rectified motion of vortices occurs. This is known as a rocking *ratchet* effect and has been widely used for studying the basics of mixed-state physics, e.g., by removing the vortices from conventional superconductors [17], as well as to verify ideas of a number of

The first experimental realization of a WPP used a periodic modulation of the thickness of cold-rolled sheets of a Nb-Ta alloy [14]. In this work the influence of isotropic pointlike disorder on the guiding of vortices was discussed for the first time. Later on, lithographic techniques have been routinely employed to create periodic pinning arrays consisting of practically identical nanostructures in the form of, e.g, microholes [20, 21], magnetic dots [22], and stripes [23]. The main idea all these works share is to suppress periodically the

With regard to a theoretical description it has to be stated that a full and exact account of the nonlinear vortex dynamics in superconducting devices proposed in these works [20–23] in a wide range of external parameters is not available due to the complexity of the periodic PP used in these references. Due to this reason it has been proposed by the authors in a number of articles to study a simpler case, such as a WPP periodic in one direction [12, 24–27] or bianisotropic [28]. The main advantage of these approaches implies the possibility to describe the phenomenon of guided vortex motion along the WPP channels, i.e., the directional anisotropy of the vortex velocity, if the transport current is applied under various in-plane angles. For instance, self-organization has been used [10, 29] to provide semi-periodic, linearly extended pinning "sites" by spontaneous facetting of m-plane sapphire substrate surfaces on

From the viewpoint of theoretical modeling, saw-tooth and harmonic PPs represent the most simple forms of the WPP. On the one hand, these simple forms of the WPP allow one to explicitly calculate the dc magneto-resistivity and the ac impedance tensor as the physical quantities of interest in the problem. On the other hand, these WPP's forms are highly realistic in the sense of appropriate experimental realizations which range from naturally occurring pinning sites in high temperature superconductors (HTSCs) to artificially created linearly-extended pinning sites in high-*Tc* and conventional superconductors, more often in thin films. Some experimental systems exhibiting a WPP are exemplified in Fig. 1. The experimental geometry of the model discussed below implies the standard four-point bridge of a thin-film superconductor with a WPP placed into a small perpendicular magnetic field with a magnitude *B* � *Bc*<sup>2</sup> such that our theoretical treatment can be performed in the *single-vortex approximation*.

Summarizing what has been said so far, by tuning the intensity, form, and asymmetry of the WPP in a superconductor, one can manipulate the fluxons via dynamical, directional, and orientational control of their motion. Evidently, a number of more sophisticated phenomena arise due to the variety of the dynamical regimes which the vortex ensemble passes through. In the next sections we will consider a particular problem in the vortex dynamics when the vortices are subjected to superimposed subcritical dc *j*<sup>0</sup> < *jc* and small ac *j*<sup>1</sup> → 0 current drives at frequencies *ω* in the microwave range. What is discussed below can be directly employed to the wide class of thin superconductors with a WPP, including but not limited to those examples shown in Fig. 1. In particular, by looking for the dc magneto-resistivity and the ac impedance responses, we will elucidate: a) how to derive the absorbed power by vortices in such a superconductor as function of all the driving parameters of the problem and b) how to solve the inverse problem, i.e., to reconstruct the coordinate dependence of a PP from the dc current-induced shift in the *depinning* frequency *ωp* [30], deduced from the curves *P*(*ω*|*j*0). The importance and further aspects of this issue are detailed next.

### **1.3. Which information can be deduced from microwave measurements?**

The measurement of the complex impedance response accompanied by its power absorption *P*(*ω*) in the radiofrequency and microwave range represents a powerful approach to investigate pinning mechanisms and the vortex dynamics in type-II superconductors. The reason for this is that at frequencies *ω* � *ωB*, substantially smaller than those invoking the breakdown of the zero-temperature energy gap (*ω<sup>B</sup>* = 2Δ(0)/¯*h* ≈ 100 GHz for a superconductor with a critical temperature *Tc* of 10 K), high-frequency and microwave impedance measurements of the mixed state yield information about flux pinning mechanisms, peculiarities in the vortex dynamics, and dissipative processes in a superconductor. It should be stressed that this information can not be extracted from the dc resistivity data obtained in the steady state regime when pinning is strong in the sample. This is due to the fact that in the last case when the critical current density *jc* is rather large, the realization of the dissipative mode, in which the flux-flow resistivity *ρ <sup>f</sup>* can be measured, requires *j*<sup>0</sup> *jc*. This is commonly accompanied by a non-negligible electron overheating in

the sample [40, 41] which changes the value of the sought *ρ <sup>f</sup>* . At the same time, measurements of the absorbed power by vortices from an ac current with amplitude *j*<sup>1</sup> � *jc* allow one to

frequency *ω* probe the pinning forces virtually in the absence of overheating effects which

The multitude of experimental works published recently utilizing the usual four-point scheme [42], strip-line coplanar waveguides (CPWs) [43], the Corbino geometry [44, 45], or the cavity method [46] to investigate the microwave vortex response in as-grown thin-film superconductors or in those containing some nano-tailored PP landscape reflects the explosively growing interest in the subject. In connection with this, from the microwave power absorption further insight into the pinning mechanisms can be gained. In particular, artificially fabricated pinning nanostructures provide a PP of unknown shape that requires certain assumptions concerning its coordinate dependence in order to fit the measured data. At the same time, in a real sample a certain amount of disorder is always presented, acting as pinning sites for a vortex as well. By this way, an approach how to reconstruct the form of the PP experimentally realized in a sample is of self-evident importance for both, application-related and fundamental reasons. A scheme how to reconstruct the coordinate dependence of a PP has been recently proposed by the authors [47] and will be elucidated in

A very early model to describe the absorbed power by vortices refers to the work of Gittleman and Rosenblum (GR) [30]. GR measured the power absorption by vortices in PbIn and NbTa films over a wide range of frequencies *ω* and successfully analyzed their data on the basis of a simple model for a 1*D* parabolic PP. In their pioneering work, a small ac excitation of vortices in the absence of a dc current was considered. Later on, GR have supplemented their equation of motion for a vortex with a dc current and have introduced a cosine PP [48]. The GR results have been obtained at *T* = 0 in a linear approximation for the pinning force and will be presented here not only for their historical importance but rather to provide the foundation

Later on, the theory accounting also for vortex creep at non-zero temperature in a 1*D* cosine PP has been extended by Coffey and Clem (CC) [49]. In the following, the CC theory has been experimentally proved to be very successful [16] to describe the high-frequency electromagnetic properties of superconductors. However, it had been developed for a small

Recently, the CC results have been substantially generalized by the authors [25, 27] for a 2*D* cosine WPP. The washboard form of the PP allowed for an exact theoretical description of the 2*D* anisotropic nonlinear vortex dynamics for any arbitrary values of the ac and dc current amplitudes, temperature, and the angle between the transport current direction with respect to the guiding direction of the WPP. The influence of the Hall effect and anisotropy of the vortex viscosity on the absorbed power by vortices has also been analyzed [50, 51]. Among other nontrivial results obtained, an enhancement [25] and a sign change [27] in the power

are otherwise unavoidable at overcritical steady-state dc current densities.

2

<sup>0</sup>. Consequently, measurements of the complex ac response versus

Microwave Absorption by Vortices in Superconductors with a Washboard Pinning Potential 267

<sup>1</sup> which can be many orders of magnitude

determine *ρ <sup>f</sup>* at a dissipative power level of *P*1∼ *ρ <sup>f</sup> j*

2

**1.4. Development of the theory in the field**

for the subsequent generalization of the model.

microwave current and in the absence of a dc drive.

absorption for *j*<sup>0</sup> *jc* have been predicted.

smaller than *P*0∼ *ρ <sup>f</sup> j*

Sec. 3.3.

**Figure 1.** Examples of selected experimental systems exhibiting a washboard pinning potential re-printed after original research papers: a) In-2% Bi foil imprinted with diffraction grating [31]. b) Parallel lines of Ni prepared by electron-beam lithography on a Si substrate onto which a Nb film was sputtered [32]. c) Superconducting microbridge (1) with an overlaying magnetic tape (2) containing a pre-recorded magnetization distribution [33]. d) Nb film deposited onto faceted *α* − Al2O3 substrate surface [10]. e) Nb film surface with an array of ferromagnetic Co stripes fabricated by focused electron beam-induced deposition [11]. f) Nb film surface with an array of grooves etched by focused electron beam milling [34]. In addition, there is a large number of HTSC-based experimental systems with a WPP, ranging from uniaxially twinned films to the usage of the intrinsic layers within a HTSC [35–39].

the sample [40, 41] which changes the value of the sought *ρ <sup>f</sup>* . At the same time, measurements of the absorbed power by vortices from an ac current with amplitude *j*<sup>1</sup> � *jc* allow one to determine *ρ <sup>f</sup>* at a dissipative power level of *P*1∼ *ρ <sup>f</sup> j* 2 <sup>1</sup> which can be many orders of magnitude smaller than *P*0∼ *ρ <sup>f</sup> j* 2 <sup>0</sup>. Consequently, measurements of the complex ac response versus frequency *ω* probe the pinning forces virtually in the absence of overheating effects which are otherwise unavoidable at overcritical steady-state dc current densities.

The multitude of experimental works published recently utilizing the usual four-point scheme [42], strip-line coplanar waveguides (CPWs) [43], the Corbino geometry [44, 45], or the cavity method [46] to investigate the microwave vortex response in as-grown thin-film superconductors or in those containing some nano-tailored PP landscape reflects the explosively growing interest in the subject. In connection with this, from the microwave power absorption further insight into the pinning mechanisms can be gained. In particular, artificially fabricated pinning nanostructures provide a PP of unknown shape that requires certain assumptions concerning its coordinate dependence in order to fit the measured data. At the same time, in a real sample a certain amount of disorder is always presented, acting as pinning sites for a vortex as well. By this way, an approach how to reconstruct the form of the PP experimentally realized in a sample is of self-evident importance for both, application-related and fundamental reasons. A scheme how to reconstruct the coordinate dependence of a PP has been recently proposed by the authors [47] and will be elucidated in Sec. 3.3.

## **1.4. Development of the theory in the field**

4

**Figure 1.** Examples of selected experimental systems exhibiting a washboard pinning potential re-printed after original research papers: a) In-2% Bi foil imprinted with diffraction grating [31]. b) Parallel lines of Ni prepared by electron-beam lithography on a Si substrate onto which a Nb film was sputtered [32]. c) Superconducting microbridge (1) with an overlaying magnetic tape (2) containing a pre-recorded magnetization distribution [33]. d) Nb film deposited onto faceted *α* − Al2O3 substrate surface [10]. e) Nb film surface with an array of ferromagnetic Co stripes fabricated by focused electron beam-induced deposition [11]. f) Nb film surface with an array of grooves etched by focused electron beam milling [34]. In addition, there is a large number of HTSC-based experimental systems with a WPP, ranging from uniaxially twinned films to the usage of the intrinsic layers within a HTSC [35–39].

A very early model to describe the absorbed power by vortices refers to the work of Gittleman and Rosenblum (GR) [30]. GR measured the power absorption by vortices in PbIn and NbTa films over a wide range of frequencies *ω* and successfully analyzed their data on the basis of a simple model for a 1*D* parabolic PP. In their pioneering work, a small ac excitation of vortices in the absence of a dc current was considered. Later on, GR have supplemented their equation of motion for a vortex with a dc current and have introduced a cosine PP [48]. The GR results have been obtained at *T* = 0 in a linear approximation for the pinning force and will be presented here not only for their historical importance but rather to provide the foundation for the subsequent generalization of the model.

Later on, the theory accounting also for vortex creep at non-zero temperature in a 1*D* cosine PP has been extended by Coffey and Clem (CC) [49]. In the following, the CC theory has been experimentally proved to be very successful [16] to describe the high-frequency electromagnetic properties of superconductors. However, it had been developed for a small microwave current and in the absence of a dc drive.

Recently, the CC results have been substantially generalized by the authors [25, 27] for a 2*D* cosine WPP. The washboard form of the PP allowed for an exact theoretical description of the 2*D* anisotropic nonlinear vortex dynamics for any arbitrary values of the ac and dc current amplitudes, temperature, and the angle between the transport current direction with respect to the guiding direction of the WPP. The influence of the Hall effect and anisotropy of the vortex viscosity on the absorbed power by vortices has also been analyzed [50, 51]. Among other nontrivial results obtained, an enhancement [25] and a sign change [27] in the power absorption for *j*<sup>0</sup> *jc* have been predicted.

Whereas the general *exact* solution of the problem [25, 27] has been obtained for non-zero temperature in terms of a matrix continued fraction [52], here we treat the problem analytically in terms of only elementary functions which allow a more intuitive description of the main effects. Solving the equation of motion for a vortex at *T* = 0, *j*<sup>0</sup> < *jc*, and *j*<sup>1</sup> → 0 in the general case, we also consider some important limiting cases of isotropic vortex viscosity and zero Hall constant provided it substantially helps us to elucidate the physical picture. The theoretical treatment of the problem is provided next.

### **2. General formulation of the problem to be solved**

Let the *x* axis with the unit vector **x** (see Fig. 2) be directed perpendicular to the washboard channels, while the *y* axis with the unit vector **y** is along these channels. The equation of motion for a vortex moving with velocity **v** in a magnetic field **B** = *B***n**, where *B* ≡ |**B**|, **n** = *n***z**, **z** is the unit vector in the *z* direction,and *n* ± 1, has the form

$$
\hbar \frac{\partial \mathbf{v} + \boldsymbol{\kappa}\_H \mathbf{v} \times \mathbf{n}}{} = \mathbf{F} + \mathbf{F}\_{p\prime} \tag{1}
$$

If *x* and *y* are the coordinates along and across the anisotropy axis, respectively, tensor *η*ˆ is diagonal in the *xy* representation, and it is convenient to define *η*<sup>0</sup> and *γ* by the formulas

Since *Up*(*x*) depends only on the *x* coordinate and is periodic, i.e., *Up*(*x*) = *Up*(*x* + *a*), where *a* is the period of the PP, the pinning force **F***p* is directed always along the anisotropy axis *x* and has no component along the *y* axis, i.e., *Fpy* = 0. As usually [25, 27, 48, 49, 53, 54], we use

where *k* = 2*π*/*a*, **F***<sup>p</sup>* = −(*dUp*/*dx*)**x** = *Fpx***x**, and *Fpx* = −*Fc* sin *kx*, where *Fc* = *Upk*/2 is the maximum value of the pinning force. Because **F** ≡ **F**(*t*) = **F**<sup>0</sup> + **F**1(*t*), where **F**<sup>0</sup> = (Φ0/*c*)**j**<sup>0</sup> × **n** is the Lorentz force invoked by the dc current and **F**<sup>1</sup> = (Φ0/*c*)**j**1(*t*) × **n** is the Lorentz force invoked by the small ac current, we assume that **v**(*t*) = **v**<sup>0</sup> + **v**1(*t*), where **v**<sup>0</sup> is

Our goal is to determine **v** from Eq. (1) and to substitute it then in the expression for the electric field. To accomplish this, Eq. (1) can be rewritten in projections on the coordinate axes

*γ*[*v*0*<sup>x</sup>* + *v*1*x*(*t*)] + *δ*[*v*0*<sup>y</sup>* + *v*1*y*(*t*)] = [*F*0*<sup>x</sup>* + *F*1*x*(*t*) + *Fpx*]/*η*0,

(1/*γ*)[*v*0*<sup>y</sup>* + *v*1*y*(*t*)] − *δ*[*v*0*<sup>x</sup>* + *v*1*x*(*t*)] = [*F*0*<sup>y</sup>* + *F*1*y*(*t*)]/*η*0,

However, instead of to straightforwardly proceed with the solution of Eqs. (4), we first consider some physically important limiting cases in which Eq. (1) is substantially simplified.

Let us consider the case of an isotropic vortex viscosity, i.e., *γ* = 1 while *η*<sup>0</sup> = *η* in the absence of Hall effect, i.e., *�* = 0. We restrict our analysis to the consideration of the vortex motion with velocity *v*(*t*) only along the *x*-axis. This case corresponds to *α* = 0 when the vortices move across the PP barriers. Furthermore, we first assume that *j*<sup>0</sup> = 0. Then Eq. (1) can be

where *x* is the vortex displacement, *η* is the vortex viscosity, *kp* is the constant which characterizes the restoring force *fp* in the PP well *Up*(*x*)=(1/2)*kp <sup>x</sup>*<sup>2</sup> and *fp* <sup>=</sup> <sup>−</sup>*dUp*/*dx* <sup>=</sup> −*kpx*. In Eq. (5) *fL* = (Φ0/*c*)*j*1(*t*) is the Lorentz force acting on a vortex, and *j*1(*t*) = *j*<sup>1</sup> exp *iωt* is the density of a small microwave current with the amplitude *j*1. Looking for the solution

Later on, in Sec. 4.3, Eq. (1) will be dealt with in its general form.

**3.1. Dynamics of pinned vortices on a small microwave current**

rewritten in the form originally used [30] for a parabolic pinning potential

where *η*<sup>0</sup> is the averaged viscous friction coefficient, and *γ* is the anisotropy parameter.

*ηxx*/*ηyy*, *ηxx* = *γη*0, *ηyy* = *η*0/*γ*, (2)

Microwave Absorption by Vortices in Superconductors with a Washboard Pinning Potential 269

*Up*(*x*)=(*Up*/2)(1 − cos *kx*), (3)

*ηx*˙ + *kpx* = *fL*, (5)

(4)

�

*<sup>η</sup>*<sup>0</sup> = √*ηxxηyy*, *<sup>γ</sup>* =

time-independent, while **v**1(*t*) = **v**<sup>1</sup> exp *iωt*.

**3. The Gittleman-Rosenblum model**

⎧ ⎨ ⎩

where *�* = *αH*/*η*<sup>0</sup> and *δ* = *n�*.

a WPP of the cosine form

where **F** = (Φ0/*c*)**j** × **n** is the Lorentz force, **j** = **j**<sup>0</sup> + **j**1(*t*), and **j**1(*t*) = **j**<sup>1</sup> exp*iωt*, where **j**<sup>0</sup> and **j**<sup>1</sup> are the densities of dc and small ac currents, respectively, and *ω* is the ac frequency. Φ<sup>0</sup> is the magnetic flux quantum and *c* is the speed of light. *η*ˆ is the vortex viscosity tensor and *α<sup>H</sup>* is the Hall coefficient. In Eq. (1) **F***<sup>p</sup>* = −∇*Up*(*x*) is the anisotropic pinning force, where *Up*(*x*) is some periodic pinning potential (PP).

**Figure 2.** The system of coordinates *xy* with the unit vectors **x** and **y** is associated with the WPP channels which are parallel to the vector **y**. The transport current density vector **j** = **j**<sup>0</sup> + **j**<sup>1</sup> exp *iωt* is directed at an angle *α* with respect to **y**. *β* is the angle between the average velocity vector **v** and **j**. **F***<sup>p</sup>* is the average pinning force provided by the WPP and **F***<sup>L</sup>* is the Lorenz force for a vortex. Inset: a schematic sample configuration in the general case. A thin type-II superconductor (foil, thin film, or thin layer of crystal) is placed into a small perpendicular magnetic field **B**. A WPP is formed in the sample and the direction of the WPP channels is shown by hatching. Experimentally deducible values are the dc voltages *<sup>E</sup>*� and *<sup>E</sup>*<sup>⊥</sup> as well as the ac impedances *<sup>Z</sup>*� and *<sup>Z</sup>*⊥.

If *x* and *y* are the coordinates along and across the anisotropy axis, respectively, tensor *η*ˆ is diagonal in the *xy* representation, and it is convenient to define *η*<sup>0</sup> and *γ* by the formulas

$$
\eta\_0 = \sqrt{\eta\_{\text{xx}} \eta\_{yy}} \qquad \gamma = \sqrt{\eta\_{\text{xx}} / \eta\_{yy}} \qquad \eta\_{\text{xx}} = \gamma \eta\_{0} \qquad \eta\_{yy} = \eta\_0 / \gamma \tag{2}
$$

where *η*<sup>0</sup> is the averaged viscous friction coefficient, and *γ* is the anisotropy parameter.

Since *Up*(*x*) depends only on the *x* coordinate and is periodic, i.e., *Up*(*x*) = *Up*(*x* + *a*), where *a* is the period of the PP, the pinning force **F***p* is directed always along the anisotropy axis *x* and has no component along the *y* axis, i.e., *Fpy* = 0. As usually [25, 27, 48, 49, 53, 54], we use a WPP of the cosine form

$$\mathcal{U}\_p(\mathbf{x}) = (\mathcal{U}\_p/2)(1 - \cos k\mathbf{x}),\tag{3}$$

where *k* = 2*π*/*a*, **F***<sup>p</sup>* = −(*dUp*/*dx*)**x** = *Fpx***x**, and *Fpx* = −*Fc* sin *kx*, where *Fc* = *Upk*/2 is the maximum value of the pinning force. Because **F** ≡ **F**(*t*) = **F**<sup>0</sup> + **F**1(*t*), where **F**<sup>0</sup> = (Φ0/*c*)**j**<sup>0</sup> × **n** is the Lorentz force invoked by the dc current and **F**<sup>1</sup> = (Φ0/*c*)**j**1(*t*) × **n** is the Lorentz force invoked by the small ac current, we assume that **v**(*t*) = **v**<sup>0</sup> + **v**1(*t*), where **v**<sup>0</sup> is time-independent, while **v**1(*t*) = **v**<sup>1</sup> exp *iωt*.

Our goal is to determine **v** from Eq. (1) and to substitute it then in the expression for the electric field. To accomplish this, Eq. (1) can be rewritten in projections on the coordinate axes

$$\begin{cases} \gamma [v\_{0x} + v\_{1x}(t)] + \delta [v\_{0y} + v\_{1y}(t)] = [F\_{0x} + F\_{1x}(t) + F\_{px}] / \eta\_{0\prime} \\\\ (1/\gamma)[v\_{0y} + v\_{1y}(t)] - \delta [v\_{0x} + v\_{1x}(t)] = [F\_{0y} + F\_{1y}(t)] / \eta\_{0\prime} \\\\ \text{we and } \delta - w\varepsilon \end{cases} \tag{4}$$

where *�* = *αH*/*η*<sup>0</sup> and *δ* = *n�*.

6

Whereas the general *exact* solution of the problem [25, 27] has been obtained for non-zero temperature in terms of a matrix continued fraction [52], here we treat the problem analytically in terms of only elementary functions which allow a more intuitive description of the main effects. Solving the equation of motion for a vortex at *T* = 0, *j*<sup>0</sup> < *jc*, and *j*<sup>1</sup> → 0 in the general case, we also consider some important limiting cases of isotropic vortex viscosity and zero Hall constant provided it substantially helps us to elucidate the physical picture. The

Let the *x* axis with the unit vector **x** (see Fig. 2) be directed perpendicular to the washboard channels, while the *y* axis with the unit vector **y** is along these channels. The equation of motion for a vortex moving with velocity **v** in a magnetic field **B** = *B***n**, where *B* ≡ |**B**|,

where **F** = (Φ0/*c*)**j** × **n** is the Lorentz force, **j** = **j**<sup>0</sup> + **j**1(*t*), and **j**1(*t*) = **j**<sup>1</sup> exp*iωt*, where **j**<sup>0</sup> and **j**<sup>1</sup> are the densities of dc and small ac currents, respectively, and *ω* is the ac frequency. Φ<sup>0</sup> is the magnetic flux quantum and *c* is the speed of light. *η*ˆ is the vortex viscosity tensor and *α<sup>H</sup>* is the Hall coefficient. In Eq. (1) **F***<sup>p</sup>* = −∇*Up*(*x*) is the anisotropic pinning force, where

**Figure 2.** The system of coordinates *xy* with the unit vectors **x** and **y** is associated with the WPP channels which are parallel to the vector **y**. The transport current density vector **j** = **j**<sup>0</sup> + **j**<sup>1</sup> exp *iωt* is directed at an angle *α* with respect to **y**. *β* is the angle between the average velocity vector **v** and **j**. **F***<sup>p</sup>* is the average pinning force provided by the WPP and **F***<sup>L</sup>* is the Lorenz force for a vortex. Inset: a

schematic sample configuration in the general case. A thin type-II superconductor (foil, thin film, or thin layer of crystal) is placed into a small perpendicular magnetic field **B**. A WPP is formed in the sample and the direction of the WPP channels is shown by hatching. Experimentally deducible values are the dc

*η*ˆ**v** + *αH***v** × **n** = **F** + **F***p*, (1)

theoretical treatment of the problem is provided next.

*Up*(*x*) is some periodic pinning potential (PP).

voltages *<sup>E</sup>*� and *<sup>E</sup>*<sup>⊥</sup> as well as the ac impedances *<sup>Z</sup>*� and *<sup>Z</sup>*⊥.

**2. General formulation of the problem to be solved**

**n** = *n***z**, **z** is the unit vector in the *z* direction,and *n* ± 1, has the form

However, instead of to straightforwardly proceed with the solution of Eqs. (4), we first consider some physically important limiting cases in which Eq. (1) is substantially simplified. Later on, in Sec. 4.3, Eq. (1) will be dealt with in its general form.

### **3. The Gittleman-Rosenblum model**

### **3.1. Dynamics of pinned vortices on a small microwave current**

Let us consider the case of an isotropic vortex viscosity, i.e., *γ* = 1 while *η*<sup>0</sup> = *η* in the absence of Hall effect, i.e., *�* = 0. We restrict our analysis to the consideration of the vortex motion with velocity *v*(*t*) only along the *x*-axis. This case corresponds to *α* = 0 when the vortices move across the PP barriers. Furthermore, we first assume that *j*<sup>0</sup> = 0. Then Eq. (1) can be rewritten in the form originally used [30] for a parabolic pinning potential

$$
\eta \dot{\mathbf{x}} + k\_p \mathbf{x} = f\_{\mathbf{L}} \tag{5}
$$

where *x* is the vortex displacement, *η* is the vortex viscosity, *kp* is the constant which characterizes the restoring force *fp* in the PP well *Up*(*x*)=(1/2)*kp <sup>x</sup>*<sup>2</sup> and *fp* <sup>=</sup> <sup>−</sup>*dUp*/*dx* <sup>=</sup> −*kpx*. In Eq. (5) *fL* = (Φ0/*c*)*j*1(*t*) is the Lorentz force acting on a vortex, and *j*1(*t*) = *j*<sup>1</sup> exp *iωt* is the density of a small microwave current with the amplitude *j*1. Looking for the solution

**Figure 3.** The frequency dependences of a) real and b) imaginary parts of the ac impedance calculated for a cosine pinning potential *Up*(*x*)=(*Up*/2)(1 − cos *kx*) at a series of dc current densities, as indicated. In the absence of a dc current, the GR results are revealed in accordance with Eqs. (9).

of Eq. (5) in the form *x*(*t*) = *x* exp *iωt*, where *x* is the complex amplitude of the vortex displacement, one immediately gets *x*˙(*t*) = *iωx*(*t*) and

$$\mathbf{x} = \frac{(\Phi\_0/\eta\mathbf{c})j\_1}{i\omega + \omega\_p} \mathbf{'} \tag{6}$$

**Figure 4.** Modification of the effective pinning potential *<sup>U</sup>*˜*i*(*x*) <sup>≡</sup> *Up*(*x*) <sup>−</sup> *<sup>f</sup>*0*ix*, where *Up*(*x*)=(*Up*/2)(1 − cos *kx*) is the WPP, with the gradual increase of *f*<sup>0</sup> such as

**3.2. Influence of a dc current on the depinning frequency**

correspond to the current densities *j*<sup>0</sup> and *jc*, respectively.

the vicinity of the rest coordinate *x*0*i*.

this regime in accordance with Eq. (7).

0 = *f*<sup>0</sup> < *f*<sup>01</sup> < *f*<sup>02</sup> *f*<sup>03</sup> = *fc*, i.e., a vortex is oscillating in the gradually tilting pinning potential well in

Microwave Absorption by Vortices in Superconductors with a Washboard Pinning Potential 271

*at low frequencies*, i.e., when *ω* � *ω<sup>p</sup>* and *Z*(*ω*) is mainly nondissipative with Re*Z*(*ω*) ≈ (*ω*/*ωp*)<sup>2</sup> � 1, whereas *frictional forces dominate at higher frequencies*, i.e., when *<sup>ω</sup>* � *<sup>ω</sup><sup>p</sup>* and *<sup>Z</sup>*(*ω*) is dissipative with Re*Z*(*ω*) <sup>≈</sup> *<sup>ρ</sup> <sup>f</sup>* [<sup>1</sup> <sup>−</sup> (*ωp*/*ω*)2]. In other words, due to the reduction of the amplitude of the vortex displacement with the increase of the ac frequency, a vortex is getting not influenced by the pinning force. This can be seen from Eq. (6) where *x* ∼ 1/*ω* for *ω* � *ωp*; this is accompanied, however, with the independence of the vortex velocity of *ω* in

The GR model can be generalized for an arbitrary PP and can also account for an arbitrary dc current superimposed on a small microwave signal. For determinacy, let us consider a subcritical dc current with the density *j*<sup>0</sup> < *jc*, where *jc* is the critical current density in the absence of a microwave current. Our aim now is to determine to which changes in the effective PP parameters the superimposition of the dc current leads, because *<sup>U</sup>*˜ (*x*) <sup>≡</sup> *Up*(*x*) <sup>−</sup> *x f*<sup>0</sup> in the presence of *j*<sup>0</sup> �= 0. Here *Up*(*x*) is the *x*-coordinate dependence of the PP when *j*<sup>0</sup> = 0. The modification of the effective PP with the gradual increase of *f*<sup>0</sup> is illustrated in Fig. 4 for the WPP by Eq. (3). Note also that *f*<sup>0</sup> < *fc*, where *f*<sup>0</sup> and *fc* are the Lorentz forces which

In the presence of an arbitrary dc current, the equation of motion for a vortex (5) has the form

where *f*(*t*)=(Φ0/*c*)*j*(*t*) is the Lorentz force with *j*(*t*) = *j*<sup>0</sup> + *j*1(*t*), where *j*1(*t*) = *j*<sup>1</sup> exp*iωt*, and *j*<sup>1</sup> is the amplitude of a small microwave current. Due to the fact that *f*(*t*) = *f*<sup>0</sup> + *f*1(*t*), where *f*<sup>0</sup> = (Φ0/*c*)*j*<sup>0</sup> and *f*1(*t*)=(Φ0/*c*)*j*1(*t*) are the Lorentz forces for the subcritical dc and microwave currents, respectively, one can naturally assume that *v*(*t*) = *v*<sup>0</sup> + *v*1(*t*), where *v*<sup>0</sup> is time-independent, whereas *v*1(*t*) = *v*<sup>1</sup> exp *iωt*. In Eq. (10) the pinning force is *fp* =

*ηv*(*t*) = *f*(*t*) + *fp*, (10)

where *ω<sup>p</sup>* ≡ *kp*/*η* is the depinning frequency. This frequency *ω<sup>p</sup>* determines the transition from the non-dissipative to dissipative regimes in the vortex dynamics in response to a small microwave signal, as will be elucidated in the text later. To calculate the magnitude of the complex electric field arising due to the vortex on move, one takes *E* = *Bx*˙/*c*. Then

$$E(\omega) = \frac{\rho\_f \dot{\mathbf{j}} \mathbf{1}}{1 - i\omega\_p/\omega} \equiv Z(\omega) \dot{\mathbf{j}}\_1. \tag{7}$$

Here *<sup>ρ</sup> <sup>f</sup>* <sup>=</sup> *<sup>B</sup>*Φ0/*ηc*<sup>2</sup> is the flux-flow resistivity and *<sup>Z</sup>*(*ω*) <sup>≡</sup> *<sup>ρ</sup> <sup>f</sup>* /(<sup>1</sup> <sup>−</sup> *<sup>i</sup>ωp*/*ω*) is the microwave impedance of the sample.

In order to calculate the power *P* absorbed per unit volume and averaged over the period of an ac cycle, the standard relation *P* = (1/2)Re(*E* · *J*∗) is used, where *E* and *J* are the complex amplitudes of the ac electric field and current density, respectively. The asterisk denotes the complex conjugate. Then, from Eq. (7) it follows

$$P(\omega) = (1/2) \text{Re}Z(\omega) j\_1^2 = (1/2)\rho\_f j\_1^2 / [1 + (\omega\_p/\omega)^2]. \tag{8}$$

For the subsequent analysis, it is convenient to write out real and imaginary parts of the impedance *Z* = Re*Z* + *i*Im*Z*, namely

$$\text{ReZ}(\omega) = \rho\_f / [1 + (\omega\_p/\omega)^2], \qquad \text{ImZ}(\omega) = \rho\_f(\omega/\omega\_p) / [1 + (\omega/\omega\_p)^2]. \tag{9}$$

The frequency dependences (9) are plotted in dimensionless units *Z*/*ρ <sup>f</sup>* and *ω*/*ω<sup>p</sup>* in Fig. 3 (see the curve for *j*0/*jc* = 0). From Eqs. (5), (6), and (8) it follows that *pinning forces dominate*

**Figure 4.** Modification of the effective pinning potential *<sup>U</sup>*˜*i*(*x*) <sup>≡</sup> *Up*(*x*) <sup>−</sup> *<sup>f</sup>*0*ix*, where *Up*(*x*)=(*Up*/2)(1 − cos *kx*) is the WPP, with the gradual increase of *f*<sup>0</sup> such as 0 = *f*<sup>0</sup> < *f*<sup>01</sup> < *f*<sup>02</sup> *f*<sup>03</sup> = *fc*, i.e., a vortex is oscillating in the gradually tilting pinning potential well in the vicinity of the rest coordinate *x*0*i*.

*at low frequencies*, i.e., when *ω* � *ω<sup>p</sup>* and *Z*(*ω*) is mainly nondissipative with Re*Z*(*ω*) ≈ (*ω*/*ωp*)<sup>2</sup> � 1, whereas *frictional forces dominate at higher frequencies*, i.e., when *<sup>ω</sup>* � *<sup>ω</sup><sup>p</sup>* and *<sup>Z</sup>*(*ω*) is dissipative with Re*Z*(*ω*) <sup>≈</sup> *<sup>ρ</sup> <sup>f</sup>* [<sup>1</sup> <sup>−</sup> (*ωp*/*ω*)2]. In other words, due to the reduction of the amplitude of the vortex displacement with the increase of the ac frequency, a vortex is getting not influenced by the pinning force. This can be seen from Eq. (6) where *x* ∼ 1/*ω* for *ω* � *ωp*; this is accompanied, however, with the independence of the vortex velocity of *ω* in this regime in accordance with Eq. (7).

### **3.2. Influence of a dc current on the depinning frequency**

8

**Figure 3.** The frequency dependences of a) real and b) imaginary parts of the ac impedance calculated for a cosine pinning potential *Up*(*x*)=(*Up*/2)(1 − cos *kx*) at a series of dc current densities, as indicated.

of Eq. (5) in the form *x*(*t*) = *x* exp *iωt*, where *x* is the complex amplitude of the vortex

*<sup>x</sup>* <sup>=</sup> (Φ0/*ηc*)*j*<sup>1</sup> *iω* + *ω<sup>p</sup>*

where *ω<sup>p</sup>* ≡ *kp*/*η* is the depinning frequency. This frequency *ω<sup>p</sup>* determines the transition from the non-dissipative to dissipative regimes in the vortex dynamics in response to a small microwave signal, as will be elucidated in the text later. To calculate the magnitude of the

Here *<sup>ρ</sup> <sup>f</sup>* <sup>=</sup> *<sup>B</sup>*Φ0/*ηc*<sup>2</sup> is the flux-flow resistivity and *<sup>Z</sup>*(*ω*) <sup>≡</sup> *<sup>ρ</sup> <sup>f</sup>* /(<sup>1</sup> <sup>−</sup> *<sup>i</sup>ωp*/*ω*) is the microwave

In order to calculate the power *P* absorbed per unit volume and averaged over the period of an ac cycle, the standard relation *P* = (1/2)Re(*E* · *J*∗) is used, where *E* and *J* are the complex amplitudes of the ac electric field and current density, respectively. The asterisk denotes the

<sup>1</sup> = (1/2)*ρ <sup>f</sup> j*

For the subsequent analysis, it is convenient to write out real and imaginary parts of the

Re*Z*(*ω*) = *ρ <sup>f</sup>* /[1 + (*ωp*/*ω*)2], Im*Z*(*ω*) = *ρ <sup>f</sup>*(*ω*/*ωp*)/[1 + (*ω*/*ωp*)

The frequency dependences (9) are plotted in dimensionless units *Z*/*ρ <sup>f</sup>* and *ω*/*ω<sup>p</sup>* in Fig. 3 (see the curve for *j*0/*jc* = 0). From Eqs. (5), (6), and (8) it follows that *pinning forces dominate*

2

<sup>1</sup>/[1 + (*ωp*/*ω*)

2

complex electric field arising due to the vortex on move, one takes *E* = *Bx*˙/*c*. Then

*<sup>E</sup>*(*ω*) = *<sup>ρ</sup> <sup>f</sup> <sup>j</sup>*<sup>1</sup>

, (6)

<sup>2</sup>]. (8)

<sup>2</sup>]. (9)

<sup>1</sup> <sup>−</sup> *<sup>i</sup>ωp*/*<sup>ω</sup>* <sup>≡</sup> *<sup>Z</sup>*(*ω*)*j*1. (7)

In the absence of a dc current, the GR results are revealed in accordance with Eqs. (9).

displacement, one immediately gets *x*˙(*t*) = *iωx*(*t*) and

complex conjugate. Then, from Eq. (7) it follows

impedance *Z* = Re*Z* + *i*Im*Z*, namely

*P*(*ω*)=(1/2)Re*Z*(*ω*)*j*

impedance of the sample.

The GR model can be generalized for an arbitrary PP and can also account for an arbitrary dc current superimposed on a small microwave signal. For determinacy, let us consider a subcritical dc current with the density *j*<sup>0</sup> < *jc*, where *jc* is the critical current density in the absence of a microwave current. Our aim now is to determine to which changes in the effective PP parameters the superimposition of the dc current leads, because *<sup>U</sup>*˜ (*x*) <sup>≡</sup> *Up*(*x*) <sup>−</sup> *x f*<sup>0</sup> in the presence of *j*<sup>0</sup> �= 0. Here *Up*(*x*) is the *x*-coordinate dependence of the PP when *j*<sup>0</sup> = 0. The modification of the effective PP with the gradual increase of *f*<sup>0</sup> is illustrated in Fig. 4 for the WPP by Eq. (3). Note also that *f*<sup>0</sup> < *fc*, where *f*<sup>0</sup> and *fc* are the Lorentz forces which correspond to the current densities *j*<sup>0</sup> and *jc*, respectively.

In the presence of an arbitrary dc current, the equation of motion for a vortex (5) has the form

$$
\eta v(t) = f(t) + f\_{p\_f} \tag{10}
$$

where *f*(*t*)=(Φ0/*c*)*j*(*t*) is the Lorentz force with *j*(*t*) = *j*<sup>0</sup> + *j*1(*t*), where *j*1(*t*) = *j*<sup>1</sup> exp*iωt*, and *j*<sup>1</sup> is the amplitude of a small microwave current. Due to the fact that *f*(*t*) = *f*<sup>0</sup> + *f*1(*t*), where *f*<sup>0</sup> = (Φ0/*c*)*j*<sup>0</sup> and *f*1(*t*)=(Φ0/*c*)*j*1(*t*) are the Lorentz forces for the subcritical dc and microwave currents, respectively, one can naturally assume that *v*(*t*) = *v*<sup>0</sup> + *v*1(*t*), where *v*<sup>0</sup> is time-independent, whereas *v*1(*t*) = *v*<sup>1</sup> exp *iωt*. In Eq. (10) the pinning force is *fp* =

#### 10 272 Superconductors – Materials, Properties and Applications Microwave Absorption by Vortices in Superconductors with a Washboard Pinning Potential <sup>11</sup>

−*dUp*(*x*)/*dx*, where *Up*(*x*) is some PP. Our aim is to determine *v*(*t*) from Eq. (10) which, taking into account the considerations above, acquires the next form

$$
\eta[v\_0 + v\_1(t)] = f\_0 + f\_p + f\_1(t),
\tag{11}
$$

of the nonlinear transition in Re*Z*(*ω*). It should be noted that in the presence of *j*<sup>0</sup> �= 0 *the dissipation remains non-zero* even at *T* = 0, though it is very small at very low frequencies.

Microwave Absorption by Vortices in Superconductors with a Washboard Pinning Potential 273

**3.3. Reconstruction of a pinning potential from the microwave absorption data**

We now turn to the detailed analytical description how to reconstruct the coordinate dependence of a PP experimentally ensued in the sample, on the basis of microwave power absorption data in the presence of a subcritical dc transport current. It will be shown that from the dependence of the depinning frequency *ω*˜ *<sup>p</sup>*(*j*0) as a function of the dc transport current *j*<sup>0</sup> one can determine the coordinate dependence of the PP *Up*(*x*). The physical background for the possibility to solve such a problem is Eq. (12) which gives the correlation of the vortex rest coordinate *x*<sup>0</sup> with the value of the static force *f*<sup>0</sup> acting on the vortex and arising due to the

From Eq. (12) it follows that while increasing *f*<sup>0</sup> from zero up to its critical value *fc* one in fact "probes" all the points in the dependence *Up*(*x*). Taking the *x*0-coordinate derivative in

> <sup>=</sup> <sup>1</sup> *ηω*˜ *<sup>p</sup>*(*f*0)

If the dependence *ω*˜(*f*0) has been deduced from the experimental data, i.e., fitted by a known

*η f*<sup>0</sup> 0

Then, having calculated the inverse function *f*0(*x*0) to *x*0(*f*0) and using the relation *f*0(*x*0) =

0

Here we would like to support the above-mentioned considerations by giving an example of the reconstruction procedure for a WPP. Let us suppose that a series of power absorption curves *P*(*ω*) has been measured for a set of subcritical dc currents *j*0. Then for determinacy, let us imagine that each *i*−curve of *P*(*ω*|*j*0) like those shown in Fig. 3 has been fitted with its fitting parameter *ω*˜ *<sup>p</sup>* so that one could map the points [(*ω*˜ *<sup>p</sup>*/*ωp*)*i*,(*j*0/*jc*)*i*], as shown by

*<sup>p</sup>* (*x*0) = 1/˜

*d f ω*˜ *<sup>p</sup>*(*f*)

*kp*(*x*0) has been used [see Eq. (14) and the text below]. By

*kp*(*x*0), (15)

. (16)

. (17)

*dx*<sup>0</sup> *f*0(*x*0). (18)

*kp*[*x*0(*f*0)], and thus,

*dx*0/*d f*<sup>0</sup> = 1/*U*��

*dx*<sup>0</sup> *d f*<sup>0</sup>

*<sup>x</sup>*0(*f*0) = <sup>1</sup>

*Up*(*x*) = *<sup>x</sup>*

substituting *x*<sup>0</sup> = *x*0(*f*0), Eq. (15) can be rewritten as *dx*0/*d f*<sup>0</sup> = 1/˜

function, then Eq. (16) allows one to derive *x*0(*f*) by integrating

*3.3.1. General scheme of the pinning potential reconstruction*

dc current *j*0.

*U*�

triangles in Fig. 5.

Eq. (12), one obtains

where the relation *U*��(*x*0) = ˜

*<sup>p</sup>*(*x*0), i.e., Eq. (12), one finally obtains

*3.3.2. Example procedure to reconstruct a pinning potential*

Let us consider the case when *j*<sup>1</sup> = 0. If *j*<sup>0</sup> < *jc*, i.e., *f*<sup>0</sup> < *fc*, where *fc* is the maximal value of the pinning force, then *v*<sup>0</sup> = 0, i.e., the vortex is in rest. As it is seen from Fig. 4 the rest coordinate *x*<sup>0</sup> of the vortex in this case depends on *f*<sup>0</sup> and is determined from the condition of equality to zero of the effective pinning force ˜ *<sup>f</sup>*(*x*) = <sup>−</sup>*dU*˜ (*x*)/*dx* <sup>=</sup> *fp*(*x*) + *<sup>f</sup>*0, which reduces to the equation *fp*(*x*0) + *f*<sup>0</sup> = 0, or

$$f\_0 = \frac{d\mathcal{U}\_p(\mathbf{x})}{d\mathbf{x}}|\_{\mathbf{x} = \mathbf{x}\_0} \tag{12}$$

the solution of which is the function *x*0(*f*0).

Let us now add a small oscillation of the vortex in the vicinity of *x*<sup>0</sup> under the action of the small external alternating force *f*1(*t*) with the frequency *ω*. For this we expand the effective pinning force ˜ *f*(*x*) in the vicinity of *x* = *x*<sup>0</sup> into a series in terms of small displacements *<sup>u</sup>* <sup>≡</sup> *<sup>x</sup>* <sup>−</sup> *<sup>x</sup>*<sup>0</sup> which gives ˜

$$
\tilde{f}(\mathbf{x} - \mathbf{x}\_0) \simeq \tilde{f}(\mathbf{x}\_0) + \tilde{f}'(\mathbf{x}\_0)\mu + \dots \tag{13}
$$

Then, taking into account that ˜ *f*(*x*0) = 0 and ˜ *f* � (*x*0) = *U*�� *<sup>p</sup>* (*x*0), Eq. (11) acquires the form

$$
\hbar \eta \dot{u}\_1 + \tilde{k}\_p u = f\_{1\prime} \tag{14}
$$

where ˜ *kp*(*x*0) = *U*�� *<sup>p</sup>* (*x*0) is the effective constant characterizing the restoring force ˜ *f*(*u*) at small oscillations of a vortex in the effective PP *U*˜ (*x*) close by *x*0(*f*0), and *v*<sup>1</sup> = *u*˙ = *iωu*. Equation (14) for the determination of *v*<sup>1</sup> is physically equivalent to GR Eq. (5) with the only distinction that the vortex depinning frequency *<sup>ω</sup>*˜ *<sup>p</sup>* <sup>≡</sup> ˜ *kp*/*η* now depends on *f*<sup>0</sup> through Eq. (12), i.e., on the dc transport current density *j*0. Thereby, all the results of Sec. 3.1 [see Eqs. (6)-(9)] can be repeated here with the changes *x* → *u* and *ω<sup>p</sup>* → *ω*˜ *<sup>p</sup>*. It should be noted, that all the described till now did not require one to know the actual form of the PP. In order to discuss the changes in the dependences Re*Z*(*ω*) and Im*Z*(*ω*) caused by the dc current, the PP must be specified. We take the cosine WPP determined by Eq. (3); though any other non-periodic PP can also be used. For the curves Re*Z*(*ω*|*j*0) and Im*Z*(*ω*|*j*0) plotted in Fig. 3, the dependence *ω*˜ *<sup>p</sup>*(*j*0/*jc*) = *ω<sup>p</sup>* <sup>1</sup> <sup>−</sup> (*j*0/*jc*)<sup>2</sup> for the cosine WPP is used [50]. Its derivation will also be outlined in Sec. 3.3.

Now we turn to the discussion of the figure data from which it is evident that with increase of *j*<sup>0</sup> the curves Re*Z*(*ω*|*j*0) and Im*Z*(*ω*|*j*0) shift to the left. The reason for this is that with increase of *j*<sup>0</sup> the PP well while tilted is broadening, as illustrated in Fig. 3. Thus, during the times shorter than *τ<sup>p</sup>* = 1/*ωp*, i.e., for *ω* > *ωp*, a vortex can no longer non-dissipatively oscillate in the PP's well. As a consequence, the enhancement of Re*Z*(*ω*) occurs at lower frequencies. At the same time, the curves in Fig. 3 maintain their original shape. Thus, *the only universal parameter to be found experimentally is the depinning frequency ωp*. For a fixed frequency and different *j*0, real part of *Z*(*ω*) always acquires larger values for larger *j*0, whereas the maximum in imaginary part of *Z*(*ω*) precisely corresponds to the middle point of the nonlinear transition in Re*Z*(*ω*). It should be noted that in the presence of *j*<sup>0</sup> �= 0 *the dissipation remains non-zero* even at *T* = 0, though it is very small at very low frequencies.

### **3.3. Reconstruction of a pinning potential from the microwave absorption data**

### *3.3.1. General scheme of the pinning potential reconstruction*

10

−*dUp*(*x*)/*dx*, where *Up*(*x*) is some PP. Our aim is to determine *v*(*t*) from Eq. (10) which,

Let us consider the case when *j*<sup>1</sup> = 0. If *j*<sup>0</sup> < *jc*, i.e., *f*<sup>0</sup> < *fc*, where *fc* is the maximal value of the pinning force, then *v*<sup>0</sup> = 0, i.e., the vortex is in rest. As it is seen from Fig. 4 the rest coordinate *x*<sup>0</sup> of the vortex in this case depends on *f*<sup>0</sup> and is determined from the condition of

*<sup>f</sup>*<sup>0</sup> <sup>=</sup> *dUp*(*x*)

*<sup>f</sup>*(*<sup>x</sup>* <sup>−</sup> *<sup>x</sup>*0) � ˜

distinction that the vortex depinning frequency *<sup>ω</sup>*˜ *<sup>p</sup>* <sup>≡</sup> ˜

*f*(*x*0) = 0 and ˜

*ηu*˙ <sup>1</sup> + ˜

Let us now add a small oscillation of the vortex in the vicinity of *x*<sup>0</sup> under the action of the small external alternating force *f*1(*t*) with the frequency *ω*. For this we expand the effective

> *f*(*x*0) + ˜ *f* �

> > *f* �

small oscillations of a vortex in the effective PP *U*˜ (*x*) close by *x*0(*f*0), and *v*<sup>1</sup> = *u*˙ = *iωu*. Equation (14) for the determination of *v*<sup>1</sup> is physically equivalent to GR Eq. (5) with the only

Eq. (12), i.e., on the dc transport current density *j*0. Thereby, all the results of Sec. 3.1 [see Eqs. (6)-(9)] can be repeated here with the changes *x* → *u* and *ω<sup>p</sup>* → *ω*˜ *<sup>p</sup>*. It should be noted, that all the described till now did not require one to know the actual form of the PP. In order to discuss the changes in the dependences Re*Z*(*ω*) and Im*Z*(*ω*) caused by the dc current, the PP must be specified. We take the cosine WPP determined by Eq. (3); though any other non-periodic PP can also be used. For the curves Re*Z*(*ω*|*j*0) and Im*Z*(*ω*|*j*0) plotted in Fig. 3,

Now we turn to the discussion of the figure data from which it is evident that with increase of *j*<sup>0</sup> the curves Re*Z*(*ω*|*j*0) and Im*Z*(*ω*|*j*0) shift to the left. The reason for this is that with increase of *j*<sup>0</sup> the PP well while tilted is broadening, as illustrated in Fig. 3. Thus, during the times shorter than *τ<sup>p</sup>* = 1/*ωp*, i.e., for *ω* > *ωp*, a vortex can no longer non-dissipatively oscillate in the PP's well. As a consequence, the enhancement of Re*Z*(*ω*) occurs at lower frequencies. At the same time, the curves in Fig. 3 maintain their original shape. Thus, *the only universal parameter to be found experimentally is the depinning frequency ωp*. For a fixed frequency and different *j*0, real part of *Z*(*ω*) always acquires larger values for larger *j*0, whereas the maximum in imaginary part of *Z*(*ω*) precisely corresponds to the middle point

*f*(*x*) in the vicinity of *x* = *x*<sup>0</sup> into a series in terms of small displacements

(*x*0) = *U*��

*<sup>p</sup>* (*x*0) is the effective constant characterizing the restoring force ˜

*η*[*v*<sup>0</sup> + *v*1(*t*)] = *f*<sup>0</sup> + *fp* + *f*1(*t*), (11)

*<sup>f</sup>*(*x*) = <sup>−</sup>*dU*˜ (*x*)/*dx* <sup>=</sup> *fp*(*x*) + *<sup>f</sup>*0, which reduces

*dx* <sup>|</sup>*x*=*x*<sup>0</sup> , (12)

(*x*0)*u* + ... (13)

*<sup>p</sup>* (*x*0), Eq. (11) acquires the form

*kp*/*η* now depends on *f*<sup>0</sup> through

*f*(*u*) at

*kpu* = *f*1, (14)

<sup>1</sup> <sup>−</sup> (*j*0/*jc*)<sup>2</sup> for the cosine WPP is used [50]. Its derivation

taking into account the considerations above, acquires the next form

equality to zero of the effective pinning force ˜

the solution of which is the function *x*0(*f*0).

to the equation *fp*(*x*0) + *f*<sup>0</sup> = 0, or

*<sup>u</sup>* <sup>≡</sup> *<sup>x</sup>* <sup>−</sup> *<sup>x</sup>*<sup>0</sup> which gives ˜

Then, taking into account that ˜

*kp*(*x*0) = *U*��

the dependence *ω*˜ *<sup>p</sup>*(*j*0/*jc*) = *ω<sup>p</sup>*

will also be outlined in Sec. 3.3.

pinning force ˜

where ˜

We now turn to the detailed analytical description how to reconstruct the coordinate dependence of a PP experimentally ensued in the sample, on the basis of microwave power absorption data in the presence of a subcritical dc transport current. It will be shown that from the dependence of the depinning frequency *ω*˜ *<sup>p</sup>*(*j*0) as a function of the dc transport current *j*<sup>0</sup> one can determine the coordinate dependence of the PP *Up*(*x*). The physical background for the possibility to solve such a problem is Eq. (12) which gives the correlation of the vortex rest coordinate *x*<sup>0</sup> with the value of the static force *f*<sup>0</sup> acting on the vortex and arising due to the dc current *j*0.

From Eq. (12) it follows that while increasing *f*<sup>0</sup> from zero up to its critical value *fc* one in fact "probes" all the points in the dependence *Up*(*x*). Taking the *x*0-coordinate derivative in Eq. (12), one obtains

$$d\mathbf{x}\_0/df\_0 = 1/\mathcal{U}\_p^{\prime\prime}(\mathbf{x}\_0) = 1/\bar{k}\_p(\mathbf{x}\_0),\tag{15}$$

where the relation *U*��(*x*0) = ˜ *kp*(*x*0) has been used [see Eq. (14) and the text below]. By substituting *x*<sup>0</sup> = *x*0(*f*0), Eq. (15) can be rewritten as *dx*0/*d f*<sup>0</sup> = 1/˜ *kp*[*x*0(*f*0)], and thus,

$$\frac{d\mathbf{x}\_0}{df\_0} = \frac{1}{\eta \tilde{\omega}\_p(f\_0)}.\tag{16}$$

If the dependence *ω*˜(*f*0) has been deduced from the experimental data, i.e., fitted by a known function, then Eq. (16) allows one to derive *x*0(*f*) by integrating

$$\alpha\_0(f\_0) = \frac{1}{\eta} \int\_0^{f\_0} \frac{df}{\tilde{\omega}\_P(f)}.\tag{17}$$

Then, having calculated the inverse function *f*0(*x*0) to *x*0(*f*0) and using the relation *f*0(*x*0) = *U*� *<sup>p</sup>*(*x*0), i.e., Eq. (12), one finally obtains

$$\mathcal{U}I\_p(\mathbf{x}) = \int\_0^\mathbf{x} d\mathbf{x}\_0 f\_0(\mathbf{x}\_0). \tag{18}$$

### *3.3.2. Example procedure to reconstruct a pinning potential*

Here we would like to support the above-mentioned considerations by giving an example of the reconstruction procedure for a WPP. Let us suppose that a series of power absorption curves *P*(*ω*) has been measured for a set of subcritical dc currents *j*0. Then for determinacy, let us imagine that each *i*−curve of *P*(*ω*|*j*0) like those shown in Fig. 3 has been fitted with its fitting parameter *ω*˜ *<sup>p</sup>* so that one could map the points [(*ω*˜ *<sup>p</sup>*/*ωp*)*i*,(*j*0/*jc*)*i*], as shown by triangles in Fig. 5.

non-sine current-phase relations is known to occur [56] and which could in turn benefit from

Microwave Absorption by Vortices in Superconductors with a Washboard Pinning Potential 275

A respective experiment can be carried out at *T* � *Tc* and implies a small microwave current density *j*<sup>1</sup> � *jc*. Though the potential reconstruction scheme has been exemplified for a cosine WPP, i.e., for a periodic and symmetric PP, single PP wells [43] can also be proven in accordance with the provided approach. In the general case, *the elucidated here procedure does not require periodicity of the potential and can account also for asymmetric ones*. If this is the case, one has to perform the reconstruction procedure under the dc current reversal, i.e., two

Here, our consideration was limited to *T* = 0, *j*<sup>0</sup> < *jc*, and *j*<sup>1</sup> → 0 because this has allowed us to provide a clear reconstruction procedure in terms of elementary functions accompanying with a simple physical interpretation. Experimentally, adequate measurements can be performed, i.e., on conventional thin-film superconductors (e.g., Nb, NbN) at *T* � *Tc*. These are suitable due to substantially low temperatures of the superconducting state and that relatively strong pinning in these materials allows one to neglect thermal fluctuations of a vortex with regard to the PP's depth *Up* � 1000 ÷ 5000 K [10]. It should be stressed that due to the universal form of the dependences *P*(*ω*|*j*0), the depinning frequency *ω<sup>p</sup>* plays a role of the only fitting parameter for each of the curves *P*(*ω*|*j*0), thus fitting of the measured data seems to be uncomplicated. However, one of most crucial issues for the experiment is to adequately superimpose the applied currents and then, to uncouple the picked-up dc and microwave signals maintaining the matching of the impedances of the line and the sample. Quantitatively, experimentally estimated values of the depinning frequency in the absence of a dc current at a temperature of about 0.6*Tc* are *ω<sup>p</sup>* ≈ 7 GHz for a 20 nm-thick [45] and a 40 nm-thick [46] Nb films. This value is strongly suppressed with increase of both, the field

Concerning the general validity of the results obtained, one circumstance should be recalled. The theoretical consideration here has been performed in the single-vortex approximation, i.e., is valid only at small magnetic fields *B* � *Bc*2, when the distance between two neighboring vortices, i.e., the period of a PP is larger as compared with the effective magnetic field

**4.1. General remarks on the Hall parameter and anisotropy of the vortex viscosity** Now, the extent to which the Hall term in the equation of motion of the vortex and a possible anisotropy of the viscous term affect the 2*D* dynamics and the resistive properties of the vortex ensemble both, at a direct (subcritical) current and at a small microwave ac current, will be investigated. It should be pointed out that, even though the Hall angle *θ<sup>H</sup>* and, consequently, the dimensionless Hall coefficient *�* is small for most superconductors, i.e., *�* � 1, anomalously large values of *�* are observed in YBCO, NbSe2, and Nb films in a number of cases at sufficiently low temperatures [58]; i.e., tan *θ<sup>H</sup>* ≥ 1. In the absence of pinning (Sec. 4.2), this means that the vortex velocity **v** in this case is directed predominantly along the direction of **j**1(*t*), whereas, with a small Hall angle (tan *θH*), the directions of **v** and **j**1(*t*) are virtually orthogonal. The influence of the Hall term on the vortex dynamics is

the results reported here.

times: for +*j*<sup>0</sup> and −*j*0.

magnitude and the film's thickness.

**4. Solution of the problem in the general case**

penetration depth, *a λ*.

**Figure 5.** The pinning potential reconstruction procedure: step 1. A set of [(*ω*˜ *<sup>p</sup>*/*ωp*)*i*,(*j*0/*jc* )*i*] points () has been deduced from the supposed measured data and fitted as *<sup>ω</sup>*˜ *<sup>p</sup>*/*ω<sup>p</sup>* <sup>=</sup> <sup>1</sup> <sup>−</sup> (*j*0/*jc* )<sup>2</sup> (solid line). Then by Eq. (17) *x*0(*f*0)=(*fc*/*kp*) arcsin(*f*0/ *fc*) (dashed line).

**Figure 6.** The pinning potential reconstruction procedure: step 2. The inverse function to *x*0(*f*0) is *f*0(*x*0) = *fc* sin(*x*0*kp*/ *fc*) (dashed line). Then by Eq. (18) *Up*(*x*)=(*Up*/2)(1 − cos *kx*) is the PP sought (solid line).

We fit the figure data in Fig. 5 by the function *<sup>ω</sup>*˜ *<sup>p</sup>*/*ω<sup>p</sup>* <sup>=</sup> <sup>1</sup> <sup>−</sup> (*j*0/*jc*)<sup>2</sup> and then substitute it into Eq. (17) from which one calculates *x*0(*f*0). In the case, the function has a simple analytical form, namely *x*0(*f*0)=(*fc*/*kp*) arcsin(*f*0/ *fc*). Evidently, the inverse to it function is *f*0(*x*0) = *fc* sin(*x*0*kp*/ *fc*) with the period *a* = 2*π fc*/*kp* (see also Fig. 6). By taking the integral (18) one finally gets *Up*(*x*)=(*Up*/2)(<sup>1</sup> <sup>−</sup> cos *kx*), where *<sup>k</sup>* <sup>=</sup> <sup>2</sup>*π*/*<sup>a</sup>* and *Up* <sup>=</sup> <sup>2</sup> *<sup>f</sup>* <sup>2</sup> *<sup>c</sup>* /*kp*.

### **3.4. Concluding remarks on the reconstruction scheme of a pinning potential**

The problem to reconstruct the actual form of a potential subjected to superimposed constant and small alternation perturbations arises not only in the vortex physics but also in related fields such as charge-density-wave pinning [55] and Josephson junctions [56]. In the vortex physics, an early scheme how to reconstruct the coordinate dependence of the pinning force from measurements implying a small ripple magnetic field superposed on a larger dc magnetic field had been previously reported [57]. Also, due to the closest mathematical analogy should be also mentioned the Josephson junction problem wherein a plenty of non-sine current-phase relations is known to occur [56] and which could in turn benefit from the results reported here.

12

(solid line).

**Figure 5.** The pinning potential reconstruction procedure: step 1. A set of [(*ω*˜ *<sup>p</sup>*/*ωp*)*i*,(*j*0/*jc* )*i*] points () has been deduced from the supposed measured data and fitted as *<sup>ω</sup>*˜ *<sup>p</sup>*/*ω<sup>p</sup>* <sup>=</sup> <sup>1</sup> <sup>−</sup> (*j*0/*jc* )<sup>2</sup> (solid line).

**Figure 6.** The pinning potential reconstruction procedure: step 2. The inverse function to *x*0(*f*0) is *f*0(*x*0) = *fc* sin(*x*0*kp*/ *fc*) (dashed line). Then by Eq. (18) *Up*(*x*)=(*Up*/2)(1 − cos *kx*) is the PP sought

**3.4. Concluding remarks on the reconstruction scheme of a pinning potential**

The problem to reconstruct the actual form of a potential subjected to superimposed constant and small alternation perturbations arises not only in the vortex physics but also in related fields such as charge-density-wave pinning [55] and Josephson junctions [56]. In the vortex physics, an early scheme how to reconstruct the coordinate dependence of the pinning force from measurements implying a small ripple magnetic field superposed on a larger dc magnetic field had been previously reported [57]. Also, due to the closest mathematical analogy should be also mentioned the Josephson junction problem wherein a plenty of

finally gets *Up*(*x*)=(*Up*/2)(<sup>1</sup> <sup>−</sup> cos *kx*), where *<sup>k</sup>* <sup>=</sup> <sup>2</sup>*π*/*<sup>a</sup>* and *Up* <sup>=</sup> <sup>2</sup> *<sup>f</sup>* <sup>2</sup>

We fit the figure data in Fig. 5 by the function *<sup>ω</sup>*˜ *<sup>p</sup>*/*ω<sup>p</sup>* <sup>=</sup> <sup>1</sup> <sup>−</sup> (*j*0/*jc*)<sup>2</sup> and then substitute it into Eq. (17) from which one calculates *x*0(*f*0). In the case, the function has a simple analytical form, namely *x*0(*f*0)=(*fc*/*kp*) arcsin(*f*0/ *fc*). Evidently, the inverse to it function is *f*0(*x*0) = *fc* sin(*x*0*kp*/ *fc*) with the period *a* = 2*π fc*/*kp* (see also Fig. 6). By taking the integral (18) one

*<sup>c</sup>* /*kp*.

Then by Eq. (17) *x*0(*f*0)=(*fc*/*kp*) arcsin(*f*0/ *fc*) (dashed line).

A respective experiment can be carried out at *T* � *Tc* and implies a small microwave current density *j*<sup>1</sup> � *jc*. Though the potential reconstruction scheme has been exemplified for a cosine WPP, i.e., for a periodic and symmetric PP, single PP wells [43] can also be proven in accordance with the provided approach. In the general case, *the elucidated here procedure does not require periodicity of the potential and can account also for asymmetric ones*. If this is the case, one has to perform the reconstruction procedure under the dc current reversal, i.e., two times: for +*j*<sup>0</sup> and −*j*0.

Here, our consideration was limited to *T* = 0, *j*<sup>0</sup> < *jc*, and *j*<sup>1</sup> → 0 because this has allowed us to provide a clear reconstruction procedure in terms of elementary functions accompanying with a simple physical interpretation. Experimentally, adequate measurements can be performed, i.e., on conventional thin-film superconductors (e.g., Nb, NbN) at *T* � *Tc*. These are suitable due to substantially low temperatures of the superconducting state and that relatively strong pinning in these materials allows one to neglect thermal fluctuations of a vortex with regard to the PP's depth *Up* � 1000 ÷ 5000 K [10]. It should be stressed that due to the universal form of the dependences *P*(*ω*|*j*0), the depinning frequency *ω<sup>p</sup>* plays a role of the only fitting parameter for each of the curves *P*(*ω*|*j*0), thus fitting of the measured data seems to be uncomplicated. However, one of most crucial issues for the experiment is to adequately superimpose the applied currents and then, to uncouple the picked-up dc and microwave signals maintaining the matching of the impedances of the line and the sample. Quantitatively, experimentally estimated values of the depinning frequency in the absence of a dc current at a temperature of about 0.6*Tc* are *ω<sup>p</sup>* ≈ 7 GHz for a 20 nm-thick [45] and a 40 nm-thick [46] Nb films. This value is strongly suppressed with increase of both, the field magnitude and the film's thickness.

Concerning the general validity of the results obtained, one circumstance should be recalled. The theoretical consideration here has been performed in the single-vortex approximation, i.e., is valid only at small magnetic fields *B* � *Bc*2, when the distance between two neighboring vortices, i.e., the period of a PP is larger as compared with the effective magnetic field penetration depth, *a λ*.
