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

### **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

#### 14 276 Superconductors – Materials, Properties and Applications Microwave Absorption by Vortices in Superconductors with a Washboard Pinning Potential <sup>15</sup>

taken into consideration because, as will be shown below, the absorption by vortices at an ac current substantially depends on both the magnitude of *�* and the frequency *ω*, and angle *α*. Moreover, the analysis shows that the resistive characteristics of a sample at a subcritical dc is independent of the value of the Hall constant. In other words, it is impossible to extract the value of *�* from experimental data at a dc, while *� can be determined from an analysis of the power absorption at an ac* (Sec. 4.3). The physical cause of such a behavior of the transverse resistive response at a subcritical dc current is associated with the suppression of the Hall response as a consequence of vortex pinning in the transverse with respect to the WPP channels direction of their possible motion.

The viscosity anisotropy is taken into consideration because the anisotropy in the *ab* plane is fairly large in most HTSC crystals: for example, for YBCO crystals, the magneto-resistivity as the vortices move along the *a* or *b* axis can differ by more than a factor of 2 [35].

### **4.2. The case of zero pinning strength and arbitrary currents**

### *4.2.1. Computing the dc resistivities*

Let the Hall constant now be arbitrary (*�* � 0), while the vortex viscosity is arbitrary and anisotropic (*γ* � 1). We first consider the 2*D* vortex motion in the absence of pinning, i.e., when **F***<sup>p</sup>* = 0. Note, that both the dc and the microwave ac currents are of arbitrary densities. The equation of motion for a vortex has the form

$$
\boldsymbol{\eta}\mathbf{v} + \boldsymbol{\kappa}\_H \mathbf{v} \times \mathbf{n} = \mathbf{F}.\tag{19}
$$

It should be pointed out that Eqs. (22) are independent of *j*0, i.e., the corresponding dc CVCs are *linear*. In other words, all three nonzero resistive responses correspond to the *flux-flow* regime of the vortex dynamics. The difference between them consists only in different

<sup>0</sup>�, *<sup>ρ</sup>*<sup>+</sup>

Let us now consider several simple, physically different limiting cases which follow from Eq. (22). If there is no viscosity anisotropy (*γ* = 1) and no Hall effect (*�* = 0), the vortex dynamics is isotropic, i.e., is independent of the angle *α*. In this case, the vortex dynamics corresponds to the flux-flow mode which is *independent of the field reversal*. As expected, the

However, if *γ* = 1 and only the Hall effect *�* �= 0 is to be considered, the vortex dynamics becomes anisotropic in the sense that the directions of **F**<sup>0</sup> and **v**<sup>0</sup> no longer coincide. This *odd-in-field* anisotropy is of a Hall origin and can be quantitatively characterized by the Hall

The presence of a viscosity anisotropy *γ* �= 1 even in the absence of the Hall effect (*�* = 0)

with respect to the inversion **B** → −**B** and, like the Hall effect, causes the vortex motion to be anisotropic, i.e., it causes the directions of **F**<sup>0</sup> and **v**<sup>0</sup> not to coincide. It is convenient to

By analogy with the appearance of directed motion of the vortices in the presence of a WPP, when there is no Hall effect and no viscosity anisotropy, the angle *β* defined by Eq. (24) can be

Carrying out an analysis for ac current in the same way like for dc current, one has *v*1*x*(*t*) =

*<sup>E</sup>*1<sup>⊥</sup> = [*<sup>ρ</sup> <sup>f</sup> <sup>j</sup>*1(*t*)/*γ*Δ](*γδ* + (<sup>1</sup> <sup>−</sup> *<sup>γ</sup>*2) sin *<sup>α</sup>* cos *<sup>α</sup>*) <sup>≡</sup> *<sup>j</sup>*1(*t*)*Z*⊥.

Longitudinal (�) and transverse (⊥) components are determined here with respect to the direction of **j**1. Separating the even and odd with respect to *n* components in Eqs. (25), one

*<sup>E</sup>*1� = [*<sup>ρ</sup> <sup>f</sup> <sup>j</sup>*1(*t*)/*γ*Δ](*γ*<sup>2</sup> sin2 *<sup>α</sup>* <sup>+</sup> cos<sup>2</sup> *<sup>α</sup>*) <sup>≡</sup> *<sup>j</sup>*1(*t*)*Z*�,

<sup>1</sup>*x*(*t*) = *<sup>F</sup>*1*<sup>x</sup>* <sup>−</sup> *γδF*1*y*, and *<sup>F</sup>*˜

<sup>0</sup>⊥/*ρ*<sup>+</sup>

<sup>0</sup><sup>⊥</sup> is the new transverse odd (Hall) component of the magneto-resistivity. If <sup>|</sup>*δ*| � 1, the Hall anisotropy is weak, and the vortex velocity **v**<sup>0</sup> is virtually perpendicular to the dc

tan *θ<sup>H</sup>* = *ρ*<sup>−</sup>

density **j**0, whereas, if |*δ*| � 1, the directions of **v**<sup>0</sup> and **j**<sup>0</sup> virtually coincide.

characterize the corresponding *even-in-field* anisotropy by the angle *β* defined as

<sup>0</sup>⊥/*ρ*<sup>+</sup>

treated similarly to that for the guiding effect in the problem with pinning.

<sup>1</sup>*y*(*t*)/*η*˜0, where *F*˜

results in the appearance of one more new magneto-resistivity *ρ*<sup>+</sup>

cot *<sup>β</sup>* <sup>=</sup> <sup>−</sup>*ρ*<sup>+</sup>

presence of pinning (see Sec. 4.3.1) substantially changes these final conclusions.

*4.2.2. Limiting cases of isotropic viscosity and/or zero Hall constant*

<sup>0</sup>⊥, and *<sup>ρ</sup>*<sup>−</sup>

Microwave Absorption by Vortices in Superconductors with a Washboard Pinning Potential 277

<sup>0</sup>� <sup>=</sup> *<sup>ρ</sup> <sup>f</sup>* which is even with respect to the change **<sup>B</sup>** → −**B**.

<sup>0</sup><sup>⊥</sup> on parameters *<sup>α</sup>*, *<sup>γ</sup>*, and *�*. The

<sup>0</sup>� <sup>=</sup> *<sup>δ</sup>*, (23)

� = (*γ*<sup>2</sup> <sup>−</sup> <sup>1</sup>)/(*γ*<sup>2</sup> tan *<sup>α</sup>* <sup>+</sup> cot *<sup>α</sup>*). (24)

<sup>0</sup>⊥. This component is even

<sup>1</sup>*y*(*t*) = *γ*2*F*1*<sup>y</sup>* + *γδF*1*x*.

(25)

dependences of the magnetoresistivities *ρ*<sup>+</sup>

only nonzero component is *ρ*<sup>+</sup>

*4.2.3. Computing the ac impedance*

⎧ ⎨ ⎩

<sup>1</sup>*x*(*t*)/*η*˜0 and *v*1*y*(*t*) = *F*˜

angle, *θH*, determined as

where *ρ*−

*F*˜

Then

In this case, projections of the vortex velocity on the *xy* axes at constant current are *v*0*<sup>x</sup>* = *F*˜ <sup>0</sup>*x*/*η*˜0 and *v*0*<sup>y</sup>* = *F*˜ <sup>0</sup>*y*/*η*˜0, where *η*˜0 = *η*0*γ*(1 + *δ*2), *F*˜ <sup>0</sup>*<sup>x</sup>* <sup>=</sup> *<sup>F</sup>*0*<sup>x</sup>* <sup>−</sup> *γδF*0*y*, and *<sup>F</sup>*˜ <sup>0</sup>*<sup>y</sup>* = *γ*2*F*0*<sup>y</sup>* + *γδF*0*x*. The main physical quantity that makes is possible to determine the resistive characteristics of the sample, i.e., its dc resistivity tensor *ρ*ˆ0 and the ac impedance tensor *Z*ˆ1 at frequency *ω*, is the electric field **E**(*t*) induced by the moving vortex system

$$\mathbf{E}(t) = \left[\mathbf{B} \times \mathbf{v}(t)\right]/c = (n\mathbf{B}/c)[-v\_{\mathcal{Y}}(t)\mathbf{x} + v\_{\mathbf{x}}(t)\mathbf{y}].\tag{20}$$

We note that **E**(*t*) = **E**<sup>0</sup> + **E**1(*t*), where **E**<sup>0</sup> is the dc electric field, while **E**1(*t*) = **E**<sup>1</sup> exp*iωt*, where **E**<sup>1</sup> is the complex amplitude of the ac electric field **E**1(*t*). We next recall that the experimentally measurable resistive responses (longitudinal *<sup>E</sup>*� and transverse *<sup>E</sup>*<sup>⊥</sup> with respect to the current direction) are associated with the responses *Ex* and *Ey* in the *xy* coordinate system by the relations *<sup>E</sup>*� <sup>=</sup> *Ex* sin *<sup>α</sup>* <sup>+</sup> *Ey* cos *<sup>α</sup>* and *<sup>E</sup>*<sup>⊥</sup> <sup>=</sup> <sup>−</sup>*Ex* cos *<sup>α</sup>* <sup>+</sup> *Ey* sin *<sup>α</sup>*, where *Ex* <sup>=</sup> <sup>−</sup>*n*(*B*/*c*)*vy* and *Ey* <sup>=</sup> *<sup>n</sup>*(*B*/*c*)*vx*. At dc current, *<sup>E</sup>*0� and *<sup>E</sup>*0<sup>⊥</sup> are respectively determined by

$$\begin{cases} E\_{0\parallel} = [\rho\_f j\_0 / \gamma \Delta](\gamma^2 \sin^2 \alpha + \cos^2 \alpha) \equiv j\_0 \rho\_{0\parallel}, \\ E\_{0\perp} = [\rho\_f j\_0 / \gamma \Delta](\gamma \delta + (1 - \gamma^2) \sin \alpha \cos \alpha) \equiv j\_0 \rho\_{0\perp}. \end{cases} \tag{21}$$

where *<sup>ρ</sup> <sup>f</sup>* <sup>≡</sup> *<sup>B</sup>*Φ0/*η*0*c*<sup>2</sup> is the flux-flow resistivity and <sup>Δ</sup> <sup>=</sup> <sup>1</sup> <sup>+</sup> *<sup>δ</sup>*2. Separating the even and odd in *n* components in Eqs. (21) one finally obtains

$$\begin{array}{ll} \rho\_{0\parallel}^{+} = (\rho\_f/\gamma\Delta)(\gamma^2\sin^2\alpha + \cos^2\alpha), & \rho\_{0,\parallel}^{-} = 0, \\ \rho\_{0\perp}^{+} = (\rho\_f/\gamma\Delta)(1-\gamma^2)\sin\alpha\cos\alpha, & \rho\_{0\perp}^{-} = \rho\_f\delta/\Delta. \end{array} \tag{22}$$

It should be pointed out that Eqs. (22) are independent of *j*0, i.e., the corresponding dc CVCs are *linear*. In other words, all three nonzero resistive responses correspond to the *flux-flow* regime of the vortex dynamics. The difference between them consists only in different dependences of the magnetoresistivities *ρ*<sup>+</sup> <sup>0</sup>�, *<sup>ρ</sup>*<sup>+</sup> <sup>0</sup>⊥, and *<sup>ρ</sup>*<sup>−</sup> <sup>0</sup><sup>⊥</sup> on parameters *<sup>α</sup>*, *<sup>γ</sup>*, and *�*. The presence of pinning (see Sec. 4.3.1) substantially changes these final conclusions.

### *4.2.2. Limiting cases of isotropic viscosity and/or zero Hall constant*

14

*F*˜

of their possible motion.

<sup>0</sup>*x*/*η*˜0 and *v*0*<sup>y</sup>* = *F*˜

determined by

*ρ*+

*ρ*+

odd in *n* components in Eqs. (21) one finally obtains

*4.2.1. Computing the dc resistivities*

The equation of motion for a vortex has the form

taken into consideration because, as will be shown below, the absorption by vortices at an ac current substantially depends on both the magnitude of *�* and the frequency *ω*, and angle *α*. Moreover, the analysis shows that the resistive characteristics of a sample at a subcritical dc is independent of the value of the Hall constant. In other words, it is impossible to extract the value of *�* from experimental data at a dc, while *� can be determined from an analysis of the power absorption at an ac* (Sec. 4.3). The physical cause of such a behavior of the transverse resistive response at a subcritical dc current is associated with the suppression of the Hall response as a consequence of vortex pinning in the transverse with respect to the WPP channels direction

The viscosity anisotropy is taken into consideration because the anisotropy in the *ab* plane is fairly large in most HTSC crystals: for example, for YBCO crystals, the magneto-resistivity as

Let the Hall constant now be arbitrary (*�* � 0), while the vortex viscosity is arbitrary and anisotropic (*γ* � 1). We first consider the 2*D* vortex motion in the absence of pinning, i.e., when **F***<sup>p</sup>* = 0. Note, that both the dc and the microwave ac currents are of arbitrary densities.

In this case, projections of the vortex velocity on the *xy* axes at constant current are *v*0*<sup>x</sup>* =

*γ*2*F*0*<sup>y</sup>* + *γδF*0*x*. The main physical quantity that makes is possible to determine the resistive characteristics of the sample, i.e., its dc resistivity tensor *ρ*ˆ0 and the ac impedance tensor *Z*ˆ1 at

We note that **E**(*t*) = **E**<sup>0</sup> + **E**1(*t*), where **E**<sup>0</sup> is the dc electric field, while **E**1(*t*) = **E**<sup>1</sup> exp*iωt*, where **E**<sup>1</sup> is the complex amplitude of the ac electric field **E**1(*t*). We next recall that the experimentally measurable resistive responses (longitudinal *<sup>E</sup>*� and transverse *<sup>E</sup>*<sup>⊥</sup> with respect to the current direction) are associated with the responses *Ex* and *Ey* in the *xy* coordinate system by the relations *<sup>E</sup>*� <sup>=</sup> *Ex* sin *<sup>α</sup>* <sup>+</sup> *Ey* cos *<sup>α</sup>* and *<sup>E</sup>*<sup>⊥</sup> <sup>=</sup> <sup>−</sup>*Ex* cos *<sup>α</sup>* <sup>+</sup> *Ey* sin *<sup>α</sup>*, where *Ex* <sup>=</sup> <sup>−</sup>*n*(*B*/*c*)*vy* and *Ey* <sup>=</sup> *<sup>n</sup>*(*B*/*c*)*vx*. At dc current, *<sup>E</sup>*0� and *<sup>E</sup>*0<sup>⊥</sup> are respectively

*<sup>E</sup>*0� = [*<sup>ρ</sup> <sup>f</sup> <sup>j</sup>*0/*γ*Δ](*γ*<sup>2</sup> sin2 *<sup>α</sup>* <sup>+</sup> cos2 *<sup>α</sup>*) <sup>≡</sup> *<sup>j</sup>*0*ρ*0�,

<sup>0</sup>� = (*<sup>ρ</sup> <sup>f</sup>* /*γ*Δ)(*γ*<sup>2</sup> sin2 *<sup>α</sup>* <sup>+</sup> cos<sup>2</sup> *<sup>α</sup>*), *<sup>ρ</sup>*<sup>−</sup>

<sup>0</sup><sup>⊥</sup> = (*<sup>ρ</sup> <sup>f</sup>* /*γ*Δ)(<sup>1</sup> <sup>−</sup> *<sup>γ</sup>*2) sin *<sup>α</sup>* cos *<sup>α</sup>*, *<sup>ρ</sup>*<sup>−</sup>

where *<sup>ρ</sup> <sup>f</sup>* <sup>≡</sup> *<sup>B</sup>*Φ0/*η*0*c*<sup>2</sup> is the flux-flow resistivity and <sup>Δ</sup> <sup>=</sup> <sup>1</sup> <sup>+</sup> *<sup>δ</sup>*2. Separating the even and

<sup>0</sup>*y*/*η*˜0, where *η*˜0 = *η*0*γ*(1 + *δ*2), *F*˜

frequency *ω*, is the electric field **E**(*t*) induced by the moving vortex system

*η*ˆ**v** + *αH***v** × **n** = **F**. (19)

**E**(*t*)=[**B** × **v**(*t*)]/*c* = (*nB*/*c*)[−*vy*(*t*)**x** + *vx*(*t*)**y**]. (20)

*<sup>E</sup>*0<sup>⊥</sup> = [*<sup>ρ</sup> <sup>f</sup> <sup>j</sup>*0/*γ*Δ](*γδ* + (<sup>1</sup> <sup>−</sup> *<sup>γ</sup>*2) sin *<sup>α</sup>* cos *<sup>α</sup>*) <sup>≡</sup> *<sup>j</sup>*0*ρ*0⊥, (21)

0,� <sup>=</sup> 0,

<sup>0</sup><sup>⊥</sup> <sup>=</sup> *<sup>ρ</sup> <sup>f</sup> <sup>δ</sup>*/Δ. (22)

<sup>0</sup>*<sup>x</sup>* <sup>=</sup> *<sup>F</sup>*0*<sup>x</sup>* <sup>−</sup> *γδF*0*y*, and *<sup>F</sup>*˜

<sup>0</sup>*<sup>y</sup>* =

the vortices move along the *a* or *b* axis can differ by more than a factor of 2 [35].

**4.2. The case of zero pinning strength and arbitrary currents**

Let us now consider several simple, physically different limiting cases which follow from Eq. (22). If there is no viscosity anisotropy (*γ* = 1) and no Hall effect (*�* = 0), the vortex dynamics is isotropic, i.e., is independent of the angle *α*. In this case, the vortex dynamics corresponds to the flux-flow mode which is *independent of the field reversal*. As expected, the only nonzero component is *ρ*<sup>+</sup> <sup>0</sup>� <sup>=</sup> *<sup>ρ</sup> <sup>f</sup>* which is even with respect to the change **<sup>B</sup>** → −**B**.

However, if *γ* = 1 and only the Hall effect *�* �= 0 is to be considered, the vortex dynamics becomes anisotropic in the sense that the directions of **F**<sup>0</sup> and **v**<sup>0</sup> no longer coincide. This *odd-in-field* anisotropy is of a Hall origin and can be quantitatively characterized by the Hall angle, *θH*, determined as

$$
\tan \theta\_H = \rho\_{0\perp}^- / \rho\_{0\parallel}^+ = \delta\_\prime \tag{23}
$$

where *ρ*− <sup>0</sup><sup>⊥</sup> is the new transverse odd (Hall) component of the magneto-resistivity. If <sup>|</sup>*δ*| � 1, the Hall anisotropy is weak, and the vortex velocity **v**<sup>0</sup> is virtually perpendicular to the dc density **j**0, whereas, if |*δ*| � 1, the directions of **v**<sup>0</sup> and **j**<sup>0</sup> virtually coincide.

The presence of a viscosity anisotropy *γ* �= 1 even in the absence of the Hall effect (*�* = 0) results in the appearance of one more new magneto-resistivity *ρ*<sup>+</sup> <sup>0</sup>⊥. This component is even with respect to the inversion **B** → −**B** and, like the Hall effect, causes the vortex motion to be anisotropic, i.e., it causes the directions of **F**<sup>0</sup> and **v**<sup>0</sup> not to coincide. It is convenient to characterize the corresponding *even-in-field* anisotropy by the angle *β* defined as

$$\cot \beta = -\rho\_{0\perp}^+ / \rho\_{\parallel}^+ = (\gamma^2 - 1) / (\gamma^2 \tan \alpha + \cot \alpha). \tag{24}$$

By analogy with the appearance of directed motion of the vortices in the presence of a WPP, when there is no Hall effect and no viscosity anisotropy, the angle *β* defined by Eq. (24) can be treated similarly to that for the guiding effect in the problem with pinning.

### *4.2.3. Computing the ac impedance*

Carrying out an analysis for ac current in the same way like for dc current, one has *v*1*x*(*t*) = *F*˜ <sup>1</sup>*x*(*t*)/*η*˜0 and *v*1*y*(*t*) = *F*˜ <sup>1</sup>*y*(*t*)/*η*˜0, where *F*˜ <sup>1</sup>*x*(*t*) = *<sup>F</sup>*1*<sup>x</sup>* <sup>−</sup> *γδF*1*y*, and *<sup>F</sup>*˜ <sup>1</sup>*y*(*t*) = *γ*2*F*1*<sup>y</sup>* + *γδF*1*x*. Then

$$\begin{cases} E\_{1\parallel} = [\rho\_f j\_1(t) / \gamma \Delta](\gamma^2 \sin^2 \alpha + \cos^2 \alpha) \equiv j\_1(t) Z\_{\parallel \prime} \\\\ E\_{1\perp} = [\rho\_f j\_1(t) / \gamma \Delta](\gamma \delta + (1 - \gamma^2) \sin \alpha \cos \alpha) \equiv j\_1(t) Z\_{\perp} \end{cases} \tag{25}$$

Longitudinal (�) and transverse (⊥) components are determined here with respect to the direction of **j**1. Separating the even and odd with respect to *n* components in Eqs. (25), one

#### 16 278 Superconductors – Materials, Properties and Applications Microwave Absorption by Vortices in Superconductors with a Washboard Pinning Potential <sup>17</sup>

finally gets

$$\begin{aligned} Z\_{\parallel}^+ &= (\rho\_f/\gamma \Delta)(\gamma^2 + \sin^2 \mathfrak{a} + \cos^2 \mathfrak{a}), \qquad Z\_{\parallel}^- = 0, \\ Z\_{\perp} &= (\rho\_f/\gamma \Delta)(1-\gamma^2)\sin \mathfrak{a} \cos \mathfrak{a}, \qquad Z\_{\perp}^- = \rho\_f \delta/\Delta. \end{aligned} \tag{26}$$

It should be pointed out that, as it can be seen from Eqs. (26) and (22), the relationships for the transverse and longitudinal resistive responses at dc current formally coincide with those for the corresponding impedances in the absence of pinning, i.e., the impedance components are *real*. However, it should be recalled that *<sup>ρ</sup>*�,<sup>⊥</sup> <sup>=</sup> Re[*Z*�,<sup>⊥</sup> exp *<sup>i</sup>ωt*].

### *4.2.4. Microwave absorption by vortices in the absence of pinning*

To compute the absorbed power *P* in the ac response per unit volume and averaged over the period of an ac cycle, we use the standard expression *P* = (1/2)Re(**E**<sup>1</sup> · **j**<sup>∗</sup> <sup>1</sup>), where **E**<sup>1</sup> and **j**<sup>1</sup> are the complex amplitudes of the ac electric field and the current density, respectively. Then, using Eqs. (25), it can be shown that

$$P = (\text{j}\_1^2/2)\overline{\rho} \equiv (\text{j}\_1^2/2)\text{Re}Z\_{\parallel} = P\_0(\gamma^2\sin^2a + \cos^2a)/\gamma\Delta,\tag{27}$$

**Figure 7.** Dependence of the absorbed power *P*/*P*<sup>0</sup> on the anisotropy parameter *γ* at the Hall parameter

<sup>0</sup>*<sup>x</sup>* + *Fpx*)/*η*˜0, *v*0*<sup>y</sup>* = *γF*0*y*/*η*˜0 + *γδ*(*F*˜

*x*<sup>0</sup> = (1/*k*) arcsin(*F*˜

and *T* = 2*π*/*ω*. We note that *v*0*<sup>x</sup>* = 0 when *j*0*<sup>y</sup>* < *jc*. As a result, one gets

**<sup>E</sup>**<sup>0</sup> <sup>=</sup> *nB c*

Here *F*0*<sup>y</sup>* = −*n*(Φ0/*c*)*j*0*x*, and therefore **E**<sup>0</sup> = −*γρ <sup>f</sup> j*0*x***x** and *E*0*<sup>x</sup>* = −*γρ <sup>f</sup> j*0*x*.

<sup>0</sup>*x*/*Fc* = ˜*j*0*y*/*jc*, ˜*j*0*<sup>y</sup>* = *n*(*j*0*<sup>y</sup>* + *γδj*0*x*), and *jc* is the critical current when *α* = 0.

We now add a small ac signal *j*1(*t*) with frequency *ω* and consider how a small ac force *F*1(*t*) affects the vortex dynamics in the subcritical dc current regime, i.e., when ˜*j*0*<sup>y</sup>* < *jc*. It follows from Eq. (20) that, when *j*<sup>1</sup> = 0, the value of *E*<sup>0</sup> can be obtained by averaging **E**1(*t*) over time, taking into account the periodicity of **<sup>F</sup>**1(*t*). Then **<sup>E</sup>** <sup>=</sup> �**E**(*t*)�, where �...� <sup>=</sup> 1/*<sup>T</sup>* � *<sup>t</sup>*0+*<sup>T</sup>*

> *<sup>v</sup>*0*y***<sup>x</sup>** <sup>=</sup> *nB c*

*<sup>E</sup>*0� <sup>=</sup> *<sup>E</sup>*0*<sup>x</sup>* sin *<sup>α</sup>* <sup>=</sup> <sup>−</sup>*γρ <sup>f</sup> <sup>j</sup>*<sup>0</sup> sin2 *<sup>α</sup>* <sup>≡</sup> *<sup>j</sup>*0*ρ*0�,

*<sup>E</sup>*0<sup>⊥</sup> = −*E*0*<sup>x</sup>* cos *<sup>α</sup>* = *γρ <sup>f</sup> <sup>j</sup>*<sup>0</sup> sin *<sup>α</sup>* cos *<sup>α</sup>* ≡ *<sup>j</sup>*0*ρ*0⊥.

*γF*0*<sup>y</sup> η*0

*<sup>ρ</sup>*0� <sup>=</sup> <sup>−</sup>*γρ <sup>f</sup>* sin2 *<sup>α</sup>*, *<sup>ρ</sup>*0<sup>⊥</sup> <sup>=</sup> *γρ <sup>f</sup>* sin *<sup>α</sup>* cos *<sup>α</sup>*, (33)

The motion of a vortex along the *x* axis will differ, depending on the values of the anisotropy

this direction, i.e., *V*0*<sup>x</sup>* = 0. As follows from Fig. 4, the vortex's rest coordinate *x*<sup>0</sup> in this case

<sup>0</sup>*x*. If *F*˜

Microwave Absorption by Vortices in Superconductors with a Washboard Pinning Potential 279

<sup>0</sup>*x*. It then follows from Eq. (29) that, to determine the dependence

<sup>0</sup>*<sup>x</sup>* + *Fpx*)/*η*˜0. (29)

*<sup>t</sup>*<sup>0</sup> ... *dt*,

(32)

<sup>0</sup>*<sup>x</sup>* < *Fc*, the vortex comes to rest in

<sup>0</sup>*<sup>x</sup>* + *Fpx* = 0. For the WPP given by Eq. (3), the

<sup>0</sup>*x*/*Fc*), (30)

**x**. (31)

*δ* = 0.1 for a series of values of the angle *α*, as indicated.

parameter *γ*, the Hall coefficient *�*, and the force *F*˜

<sup>0</sup>*x*), it is necessary to solve the equation *F*˜

The solution of this system of equation is

*v*0*<sup>x</sup>* = (*F*˜

depends on the value of *F*˜

It follows from Eq. (20) that

⎧ ⎨ ⎩

*x*0(*F*˜

solution is

where *F*˜

Finally,

where *P*<sup>0</sup> = *ρ <sup>f</sup>*(*j* 2 <sup>1</sup>/2). When *γ* = 1, the absorbed power becomes isotropic and depends only on the dimensionless Hall constant *�*, i.e., *P* = *P*0/Δ with *P* decreasing as *�* increases. This is physically associated with the already established fact that, as *�* increases, the direction of vector **v**<sup>1</sup> approaches closer and closer to the direction of vector **j**1, so that the corresponding component of the longitudinal ac electric field *<sup>E</sup>*1� decreases in amplitude as *<sup>θ</sup><sup>H</sup>* increases, while the absorbed power falls off. According to the physical picture and as it follows from Eq. (27), the power absorption is maximal and equals *P*<sup>0</sup> when *γ* = 1 and *�* = 0.

However, if *γ* �= 1 and *�* �= 0, one has *P* = *P*(*α*, *γ*, *�*), i.e., the absorbed power is anisotropic. Figure 7 shows the dependence of *P*/*P*<sup>0</sup> as a function of the anisotropy parameter *γ* at the Hall parameter *δ* = 0.1 for various values of the angle *α*. It follows from Eq. (27) that the influence of parameter *�* on *P*(*α*, *γ*, *�*) for any *α* and *γ* reduces to a reduction of the absorption with increasing *�*, as well as the fact that the absorption anisotropy when *γ* �= 1 is determined by the value of the nonlinear with respect to *α* and *γ* combination *γ* sin2 *α* + (1/*γ*) cos<sup>2</sup> *α*. The latter implies that the term (1/*γ*) cos2 *<sup>α</sup>* increases as *<sup>α</sup>* <sup>→</sup> 0 and *<sup>γ</sup>* <sup>→</sup> 0, and that the term *<sup>γ</sup>* sin2 *<sup>α</sup>* increases as *<sup>α</sup>* <sup>→</sup> *<sup>π</sup>*/2 and *<sup>γ</sup>* � 1. All these features are easy to see in Fig. (7).

### **4.3. The case of arbitrary pinning strength and subcritical currents**

### *4.3.1. Computing the dc resistivities*

Let us first consider the case in which there is no ac current, i.e., *j*<sup>1</sup> = 0. It then follows from Eq. (4) that

$$\begin{cases} \gamma v\_{0\chi} + \delta v\_{0y} = (F\_{0\chi} + F\_{px}) / \eta\_{0\prime} \\\\ (1/\gamma)v\_{0y} - \delta v\_{0x} = F\_{0y} / \eta\_{0} \end{cases} \tag{28}$$

**Figure 7.** Dependence of the absorbed power *P*/*P*<sup>0</sup> on the anisotropy parameter *γ* at the Hall parameter *δ* = 0.1 for a series of values of the angle *α*, as indicated.

The solution of this system of equation is

$$
\upsilon\_{0\mathbf{x}} = \left(\tilde{F}\_{0\mathbf{x}} + F\_{\mathbf{p}\mathbf{x}}\right) / \tilde{\eta}\_0, \qquad \upsilon\_{0y} = \gamma F\_{0y} / \tilde{\eta}\_0 + \gamma \delta(\tilde{F}\_{0\mathbf{x}} + F\_{\mathbf{p}\mathbf{x}}) / \tilde{\eta}\_0. \tag{29}
$$

The motion of a vortex along the *x* axis will differ, depending on the values of the anisotropy parameter *γ*, the Hall coefficient *�*, and the force *F*˜ <sup>0</sup>*x*. If *F*˜ <sup>0</sup>*<sup>x</sup>* < *Fc*, the vortex comes to rest in this direction, i.e., *V*0*<sup>x</sup>* = 0. As follows from Fig. 4, the vortex's rest coordinate *x*<sup>0</sup> in this case depends on the value of *F*˜ <sup>0</sup>*x*. It then follows from Eq. (29) that, to determine the dependence *x*0(*F*˜ <sup>0</sup>*x*), it is necessary to solve the equation *F*˜ <sup>0</sup>*<sup>x</sup>* + *Fpx* = 0. For the WPP given by Eq. (3), the solution is

$$\mathbf{x}\_0 = (1/k)\arcsin(\tilde{F}\_{0\mathbf{x}}/F\_{\mathbf{c}})\_\prime \tag{30}$$

where *F*˜ <sup>0</sup>*x*/*Fc* = ˜*j*0*y*/*jc*, ˜*j*0*<sup>y</sup>* = *n*(*j*0*<sup>y</sup>* + *γδj*0*x*), and *jc* is the critical current when *α* = 0.

We now add a small ac signal *j*1(*t*) with frequency *ω* and consider how a small ac force *F*1(*t*) affects the vortex dynamics in the subcritical dc current regime, i.e., when ˜*j*0*<sup>y</sup>* < *jc*. It follows from Eq. (20) that, when *j*<sup>1</sup> = 0, the value of *E*<sup>0</sup> can be obtained by averaging **E**1(*t*) over time, taking into account the periodicity of **<sup>F</sup>**1(*t*). Then **<sup>E</sup>** <sup>=</sup> �**E**(*t*)�, where �...� <sup>=</sup> 1/*<sup>T</sup>* � *<sup>t</sup>*0+*<sup>T</sup> <sup>t</sup>*<sup>0</sup> ... *dt*, and *T* = 2*π*/*ω*. We note that *v*0*<sup>x</sup>* = 0 when *j*0*<sup>y</sup>* < *jc*. As a result, one gets

$$\mathbf{E}\_0 = \frac{nB}{c} \upsilon\_{0y} \mathbf{x} = \frac{nB}{c} \frac{\gamma F\_{0y}}{\eta\_0} \mathbf{x}.\tag{31}$$

Here *F*0*<sup>y</sup>* = −*n*(Φ0/*c*)*j*0*x*, and therefore **E**<sup>0</sup> = −*γρ <sup>f</sup> j*0*x***x** and *E*0*<sup>x</sup>* = −*γρ <sup>f</sup> j*0*x*. It follows from Eq. (20) that

$$\begin{cases} E\_{0\parallel} = E\_{0\mathbf{x}} \sin \mathfrak{a} = -\gamma \rho\_f j\_0 \sin^2 \mathfrak{a} \equiv j\_0 \rho\_{0\parallel}, \\\\ E\_{0\perp} = -E\_{0\mathbf{x}} \cos \mathfrak{a} = \gamma \rho\_f j\_0 \sin \mathfrak{a} \cos \mathfrak{a} \equiv j\_0 \rho\_{0\perp}. \end{cases} \tag{32}$$

Finally,

16

finally gets

*Z*<sup>+</sup>

using Eqs. (25), it can be shown that

2

*4.3.1. Computing the dc resistivities*

Eq. (4) that

where *P*<sup>0</sup> = *ρ <sup>f</sup>*(*j*

*P* = (*j* 2

� = (*<sup>ρ</sup> <sup>f</sup>* /*γ*Δ)(*γ*<sup>2</sup> <sup>+</sup> sin2 *<sup>α</sup>* <sup>+</sup> cos2 *<sup>α</sup>*), *<sup>Z</sup>*<sup>−</sup>

It should be pointed out that, as it can be seen from Eqs. (26) and (22), the relationships for the transverse and longitudinal resistive responses at dc current formally coincide with those for the corresponding impedances in the absence of pinning, i.e., the impedance components are

To compute the absorbed power *P* in the ac response per unit volume and averaged over the

are the complex amplitudes of the ac electric field and the current density, respectively. Then,

on the dimensionless Hall constant *�*, i.e., *P* = *P*0/Δ with *P* decreasing as *�* increases. This is physically associated with the already established fact that, as *�* increases, the direction of vector **v**<sup>1</sup> approaches closer and closer to the direction of vector **j**1, so that the corresponding component of the longitudinal ac electric field *<sup>E</sup>*1� decreases in amplitude as *<sup>θ</sup><sup>H</sup>* increases, while the absorbed power falls off. According to the physical picture and as it follows from

However, if *γ* �= 1 and *�* �= 0, one has *P* = *P*(*α*, *γ*, *�*), i.e., the absorbed power is anisotropic. Figure 7 shows the dependence of *P*/*P*<sup>0</sup> as a function of the anisotropy parameter *γ* at the Hall parameter *δ* = 0.1 for various values of the angle *α*. It follows from Eq. (27) that the influence of parameter *�* on *P*(*α*, *γ*, *�*) for any *α* and *γ* reduces to a reduction of the absorption with increasing *�*, as well as the fact that the absorption anisotropy when *γ* �= 1 is determined by the value of the nonlinear with respect to *α* and *γ* combination *γ* sin2 *α* + (1/*γ*) cos<sup>2</sup> *α*. The latter implies that the term (1/*γ*) cos2 *<sup>α</sup>* increases as *<sup>α</sup>* <sup>→</sup> 0 and *<sup>γ</sup>* <sup>→</sup> 0, and that the term

Let us first consider the case in which there is no ac current, i.e., *j*<sup>1</sup> = 0. It then follows from

*γv*0*<sup>x</sup>* + *δv*0*<sup>y</sup>* = (*F*0*<sup>x</sup>* + *Fpx*)/*η*0,

(1/*γ*)*v*0*<sup>y</sup>* − *δv*0*<sup>x</sup>* = *F*0*y*/*η*0.

*<sup>Z</sup>*<sup>⊥</sup> = (*<sup>ρ</sup> <sup>f</sup>* /*γ*Δ)(<sup>1</sup> <sup>−</sup> *<sup>γ</sup>*2) sin *<sup>α</sup>* cos *<sup>α</sup>*, *<sup>Z</sup>*<sup>−</sup>

*real*. However, it should be recalled that *<sup>ρ</sup>*�,<sup>⊥</sup> <sup>=</sup> Re[*Z*�,<sup>⊥</sup> exp *<sup>i</sup>ωt*].

period of an ac cycle, we use the standard expression *P* = (1/2)Re(**E**<sup>1</sup> · **j**<sup>∗</sup>

2

Eq. (27), the power absorption is maximal and equals *P*<sup>0</sup> when *γ* = 1 and *�* = 0.

*<sup>γ</sup>* sin2 *<sup>α</sup>* increases as *<sup>α</sup>* <sup>→</sup> *<sup>π</sup>*/2 and *<sup>γ</sup>* � 1. All these features are easy to see in Fig. (7).

**4.3. The case of arbitrary pinning strength and subcritical currents**

⎧ ⎨ ⎩

*4.2.4. Microwave absorption by vortices in the absence of pinning*

<sup>1</sup>/2)*ρ*¯ ≡ (*j*

� <sup>=</sup> 0,

(26)

(28)

<sup>1</sup>), where **E**<sup>1</sup> and **j**<sup>1</sup>

<sup>⊥</sup> <sup>=</sup> *<sup>ρ</sup> <sup>f</sup> <sup>δ</sup>*/Δ.

<sup>1</sup>/2)Re*Z*� <sup>=</sup> *<sup>P</sup>*0(*γ*<sup>2</sup> sin2 *<sup>α</sup>* <sup>+</sup> cos2 *<sup>α</sup>*)/*γ*Δ, (27)

<sup>1</sup>/2). When *γ* = 1, the absorbed power becomes isotropic and depends only

$$
\rho\_{0\parallel} = -\gamma \rho\_f \sin^2 \mathfrak{a}\_{\prime} \qquad \rho\_{0\perp} = \gamma \rho\_f \sin \mathfrak{a} \cos \mathfrak{a}\_{\prime} \tag{33}
$$

#### 18 280 Superconductors – Materials, Properties and Applications Microwave Absorption by Vortices in Superconductors with a Washboard Pinning Potential <sup>19</sup>

from which it immediately follows that these responses are independent of the **B**-reversal. The only information which can be extracted from Eq. (33) is concerning the angle *α* for the given sample, from the relation tan *<sup>α</sup>* <sup>=</sup> <sup>−</sup>*ρ*0�/*ρ*0<sup>⊥</sup> and the value of the product *γρ <sup>f</sup>* . From Eq. (33) it also follows that the longitudinal and transverse responses are nondissipative only when *α* = 0; this is caused by the subcritical nature of the transport current. A dissipation arises when *α* �= 0 due to the appearance of a component of the driving force *F*0*<sup>y</sup>* that does not contain the Hall constant [see Eq. (29) for *V*0*y*, taking into account that *V*0*<sup>x</sup>* = 0] and is directed along the WPP channels. Thus, when *F*˜ <sup>0</sup>*<sup>x</sup>* < *Fc*, the vortex motion and the resistive response of the sample are independent of *�*, i.e., *the Hall parameter can not be determined from experiment at a constant subcritical current*, unlike the case already described in Sec. 4.2.1.

### *4.3.2. Computing the ac impedance tensor*

Let us now proceed to an analysis of the responses to an ac current, using the relationship **E**1(*t*) = **E**(*t*) − **E**<sup>0</sup> = **E**(*t*) − �**E**(*t*)�. From this and from Eq. (4) one has that

$$\begin{cases} \gamma v\_{1x}(t) + \delta v\_{1y}(t) = \left[ \tilde{F}\_{0x} + F\_{1x}(t) + F\_{px} \right] / \eta\_{0\prime} \\\\ v\_{1y}(t) / \gamma - \delta v\_{1x}(t) = F\_{1y}(t) / \eta\_{0\prime} \end{cases} \tag{34}$$

where *Z*<sup>1</sup> ≡ Δ/Δ*<sup>γ</sup>* = 1/(1 − *iωq*/*ω*), and

⎧

*Z*<sup>+</sup>

*Z*−

*Z*<sup>+</sup>

*Z*−

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

⎧ ⎨ ⎩

<sup>1</sup> = 1/[1 + (*ω*¯ *<sup>q</sup>*/*ω*)2].

Re*Z*¯

�,<sup>⊥</sup> are determined by

is unmeasurable, since Re*Z*¯

*η*<sup>0</sup> = *B*Φ<sup>0</sup> sin2 *α*/Re*Z*¯

Re*Z*¯

where Re*Z*¯

Re*Z*¯ <sup>−</sup>

�

replacement *n* → −*n*, i.e., *δ* → −*δ* because the Hall effect is present.

experimentally deducible, field orientation-independent quantities are

� = (*<sup>ρ</sup> <sup>f</sup>* /*γ*Δ)*Z*<sup>−</sup>

<sup>⊥</sup> = (*<sup>ρ</sup> <sup>f</sup>* /*γ*Δ){*Z*<sup>+</sup>

*4.3.3. Determination of the Hall constant from microwave measurements*

Then *<sup>Z</sup>*¯ <sup>1</sup> <sup>≡</sup> *<sup>Z</sup>*1(*<sup>γ</sup>* <sup>=</sup> <sup>1</sup>) = 1/(<sup>1</sup> <sup>−</sup> *<sup>i</sup>ω*¯ *<sup>q</sup>*/*ω*), where *<sup>ω</sup>*¯ *<sup>q</sup>* <sup>≡</sup> (*ωp*/Δ)

the main approximation with respect to 1/*ω* one has Re*Z*¯

� <sup>=</sup> 0), and Re*Z*¯<sup>⊥</sup> <sup>=</sup> *<sup>ρ</sup> <sup>f</sup> <sup>δ</sup>*/Δ, i.e., Re*Z*¯ <sup>+</sup>

�*c*2.

<sup>⊥</sup> = (*<sup>ρ</sup> <sup>f</sup>* /*γ*Δ){*δγZ*<sup>+</sup>

The culmination of this subsection is an analysis of the dependence Re*Z*¯ <sup>±</sup>

Moreover, as *<sup>ω</sup>* <sup>→</sup> <sup>∞</sup>, for any *<sup>α</sup>*, there is a relationship of the form *<sup>δ</sup>* <sup>=</sup> Re*Z*¯ <sup>−</sup>

i.e., *δ* has been canceled out of the results. It follows from this that *ρ <sup>f</sup>* = Re*Z*¯

� = (*<sup>ρ</sup> <sup>f</sup>* /*γ*Δ)[*γ*2(<sup>Δ</sup> <sup>−</sup> *<sup>δ</sup>*2*Z*<sup>+</sup>

<sup>1</sup> <sup>−</sup> (˜*j*0*y*/*jc*)<sup>2</sup> = [*ωp*/*γ*Δ]

The quantity *ω<sup>q</sup>* in Eq. (39) a generalization to the case *γ* �= 1 and *�* �= 0, and is physically analogous to the depinning frequency *ωp* introduced in Seq. 3.1 and dependent on the subcritical transport current. However, it should be emphasized that, unlike the depinning frequency *ωp* which is independent on the **B**-inversion, the value of the *ωq* changes with the

Having separated even and odd parts in the impedance components, finally the

�

Microwave Absorption by Vortices in Superconductors with a Washboard Pinning Potential 281

<sup>1</sup> ) sin2 *<sup>α</sup>* <sup>+</sup> *<sup>Z</sup>*<sup>+</sup>

<sup>1</sup> (cos<sup>2</sup> *<sup>α</sup>* <sup>−</sup> *<sup>δ</sup>*2*γ*<sup>2</sup> sin2 *<sup>α</sup>*),

<sup>1</sup> <sup>−</sup> *<sup>γ</sup>*2(<sup>Δ</sup> <sup>−</sup> *<sup>δ</sup>*2*Z*<sup>+</sup>

<sup>1</sup> + *Z*<sup>−</sup>

In this case, the expressions for the real part of the longitudinal and transverse impedances

� = (*<sup>ρ</sup> <sup>f</sup>* /Δ)[Re*Z*¯ <sup>1</sup> <sup>+</sup> <sup>Δ</sup>(<sup>1</sup> <sup>−</sup> Re*Z*¯ <sup>1</sup>) sin2 *<sup>α</sup>*],

Re*Z*¯<sup>⊥</sup> = (*<sup>ρ</sup> <sup>f</sup>* /Δ)[*δ*Re*Z*¯ <sup>1</sup> <sup>−</sup> <sup>Δ</sup>(<sup>1</sup> <sup>−</sup> Re*Z*¯ <sup>1</sup>) sin *<sup>α</sup>* cos *<sup>α</sup>*],

*<sup>ω</sup>* at large or small frequencies. If *<sup>ω</sup>* <sup>→</sup> <sup>∞</sup>, one has (*ω*¯ *<sup>q</sup>*/*ω*) <sup>→</sup> 0, i.e., Re*Z*¯ <sup>1</sup> <sup>=</sup> 1. Then in

constant *�* increases (i.e., *δ* increases), the absorbed power *P* decreases as *ω* → ∞ (*P* = *P*0/Δ).

let *<sup>ω</sup>* <sup>→</sup> 0 (i.e., Re*Z*¯ <sup>1</sup> <sup>=</sup> 1). Then, in the main approximation with respect to *<sup>ω</sup>*, the Hall effect

<sup>⊥</sup> <sup>=</sup> 0 and Re*Z*¯ <sup>−</sup>

� <sup>=</sup> *<sup>ρ</sup> <sup>f</sup>* sin2 *<sup>α</sup>* and *<sup>P</sup>* <sup>=</sup> *<sup>P</sup>*<sup>0</sup> sin2 *<sup>α</sup>*, while Re*Z*¯<sup>⊥</sup> <sup>=</sup> <sup>−</sup>*<sup>ρ</sup> <sup>f</sup>* sin *<sup>α</sup>* cos *<sup>α</sup>*;

Let us assume that *γ* = 1 and the Hall constant is arbitrary but satisfies the condition *F*˜

<sup>1</sup> cos<sup>2</sup> *<sup>α</sup>*],

� <sup>=</sup> *<sup>ρ</sup> <sup>f</sup>* /Δ, (i.e., Re*Z*¯ <sup>+</sup>

<sup>1</sup> )] sin *α* cos *α*},

<sup>1</sup> (<sup>1</sup> <sup>+</sup> *<sup>δ</sup>*2*γ*) sin *<sup>α</sup>* cos *<sup>α</sup>*},

<sup>1</sup> − (*j*0*y*/*jc*)2(cos *<sup>α</sup>* + *δγ* sin *<sup>α</sup>*)<sup>2</sup> (39)

(40)

<sup>0</sup>*<sup>x</sup>* < *Fc*.

(41)

� and

� . Now

�/ sin2 *<sup>α</sup>*; i.e.,

�,<sup>⊥</sup> as a function of

� <sup>=</sup> Re*Z*¯

⊥/Re*Z*¯ <sup>−</sup>

<sup>⊥</sup> <sup>=</sup> Re*Z*¯⊥. Thus, as the Hall

�<sup>1</sup> <sup>−</sup> (*j*0/*jc*)2(cos *<sup>α</sup>* <sup>+</sup> *<sup>δ</sup>* sin *<sup>α</sup>*)2.

*ω<sup>q</sup>* = *ω*˜ *<sup>p</sup>γ*/Δ = [*ωp*/*γ*Δ]

where *F*1*x*(*t*) ≡ (*n*Φ0/*c*)*j*1*y*(*t*) and *F*1*y*(*t*) ≡ −(*n*Φ0/*c*)*j*1*x*(*t*), where *j*1*x*(*t*) = *j*1(*t*) sin *α*, *j*1*y*(*t*) = *j*1(*t*) cos *α*, and *j*1(*t*) = *j*<sup>1</sup> exp *iωt*. We use the fact that *F*˜ <sup>0</sup>*<sup>x</sup>* <sup>+</sup> *Fpx* <sup>≡</sup> *<sup>F</sup>*˜ *px* = <sup>−</sup>*dU*˜ *<sup>p</sup>*(*x*)/*dx*, where *<sup>U</sup>*˜ *<sup>p</sup>*(*x*) <sup>≡</sup> *Up*(*x*) <sup>−</sup> *xF*˜ <sup>0</sup>*<sup>x</sup>* is the effective PP, taking into account the driving force component along the *x* axis (see Fig. 4). In this case, the rest coordinate for a vortex is given by Eq. (30). The effective PP *U*˜ *<sup>p</sup>*(*x*) can be expanded in series in the small difference (*x* − *x*0) like it was done in Sec. 3.2, and using Eq. (30) one gets

$$\mathcal{F}\_{\rm px}/\eta\_0 = -\tilde{\omega}\_p (\mathbf{x} - \mathbf{x}\_0), \quad \text{where} \quad \tilde{\omega}\_p = \omega\_p \sqrt{1 - (\tilde{f}\_{0y}/j\_c)^2}, \quad \text{and} \quad \omega\_p \equiv k\_p/\eta\_0. \tag{35}$$

Now it is possible to rewrite Eq. (34) as

$$\begin{cases} \left(\gamma + \tilde{\omega}\_p / i\omega\right) v\_{1x}(t) + \delta v\_{1y}(t) = F\_{1x}(t) / \eta\_0, \\\\ -\delta v\_{1x}(t) + (1/\gamma) v\_{1y}(t) - \delta v\_{1x}(t) = F\_{1y}(t) / \eta\_0. \end{cases} \tag{36}$$

The solution of this system of equation is

$$v\_{1x}(t) = (F\_{1x}/\gamma - \delta F\_{1y})/(\eta\_0 \Delta\_\gamma), \qquad v\_{1y}(t) = \left[\gamma F\_{1y}(1 + \tilde{\omega}\_{p\gamma}/i\omega) + \delta F\_{1x}\right]/(\eta\_0 \Delta\_\gamma), \tag{37}$$

where *<sup>ω</sup>*˜ *<sup>p</sup><sup>γ</sup>* <sup>≡</sup> *<sup>ω</sup>*˜ *<sup>p</sup>*/*<sup>γ</sup>* and <sup>Δ</sup>*<sup>γ</sup>* <sup>≡</sup> <sup>Δ</sup> <sup>+</sup> *<sup>ω</sup>*˜ *<sup>p</sup>γ*/*iω*. From the relation **<sup>E</sup>**1(*t*) = *<sup>Z</sup>*ˆ**j**1(*t*), where the components of the ac impedance tensor *Z*ˆ are measured in the *xy* coordinate system (see Fig. 2), Eqs. (4), and (37), the longitudinal and transverse (with respect to the direction of **j**1) impedances *<sup>Z</sup>*� and *<sup>Z</sup>*<sup>⊥</sup> are determined as

$$\begin{cases} Z\_{\parallel} = (\rho\_f / \gamma \Delta) [\gamma^2 (\Delta - \delta^2 Z\_1) \sin^2 a + Z\_1 \cos^2 a] \, \end{cases} \tag{38}$$
 
$$Z\_{\perp} = (\rho\_f / \gamma \Delta) \{\delta \gamma Z\_1 + [Z\_1 - \gamma^2 (\Delta - \delta^2 Z\_1) \sin a \cos a] \} \,,$$

where *Z*<sup>1</sup> ≡ Δ/Δ*<sup>γ</sup>* = 1/(1 − *iωq*/*ω*), and

18

along the WPP channels. Thus, when *F*˜

*4.3.2. Computing the ac impedance tensor*

⎧ ⎨ ⎩

<sup>−</sup>*dU*˜ *<sup>p</sup>*(*x*)/*dx*, where *<sup>U</sup>*˜ *<sup>p</sup>*(*x*) <sup>≡</sup> *Up*(*x*) <sup>−</sup> *xF*˜

Now it is possible to rewrite Eq. (34) as

The solution of this system of equation is

impedances *<sup>Z</sup>*� and *<sup>Z</sup>*<sup>⊥</sup> are determined as

⎧ ⎨ ⎩ ⎧ ⎨ ⎩

*F*˜

from which it immediately follows that these responses are independent of the **B**-reversal. The only information which can be extracted from Eq. (33) is concerning the angle *α* for the given sample, from the relation tan *<sup>α</sup>* <sup>=</sup> <sup>−</sup>*ρ*0�/*ρ*0<sup>⊥</sup> and the value of the product *γρ <sup>f</sup>* . From Eq. (33) it also follows that the longitudinal and transverse responses are nondissipative only when *α* = 0; this is caused by the subcritical nature of the transport current. A dissipation arises when *α* �= 0 due to the appearance of a component of the driving force *F*0*<sup>y</sup>* that does not contain the Hall constant [see Eq. (29) for *V*0*y*, taking into account that *V*0*<sup>x</sup>* = 0] and is directed

of the sample are independent of *�*, i.e., *the Hall parameter can not be determined from experiment*

Let us now proceed to an analysis of the responses to an ac current, using the relationship

where *F*1*x*(*t*) ≡ (*n*Φ0/*c*)*j*1*y*(*t*) and *F*1*y*(*t*) ≡ −(*n*Φ0/*c*)*j*1*x*(*t*), where *j*1*x*(*t*) = *j*1(*t*) sin *α*,

force component along the *x* axis (see Fig. 4). In this case, the rest coordinate for a vortex is given by Eq. (30). The effective PP *U*˜ *<sup>p</sup>*(*x*) can be expanded in series in the small difference

(*γ* + *ω*˜ *<sup>p</sup>*/*iω*)*v*1*x*(*t*) + *δv*1*y*(*t*) = *F*1*x*(*t*)/*η*0,

−*δv*1*x*(*t*)+(1/*γ*)*v*1*y*(*t*) − *δv*1*x*(*t*) = *F*1*y*(*t*)/*η*0.

*v*1*x*(*t*)=(*F*1*x*/*γ* − *δF*1*y*)/(*η*0Δ*γ*), *v*1*y*(*t*)=[*γF*1*y*(1 + *ω*˜ *<sup>p</sup>γ*/*iω*) + *δF*1*x*]/(*η*0Δ*γ*), (37)

where *<sup>ω</sup>*˜ *<sup>p</sup><sup>γ</sup>* <sup>≡</sup> *<sup>ω</sup>*˜ *<sup>p</sup>*/*<sup>γ</sup>* and <sup>Δ</sup>*<sup>γ</sup>* <sup>≡</sup> <sup>Δ</sup> <sup>+</sup> *<sup>ω</sup>*˜ *<sup>p</sup>γ*/*iω*. From the relation **<sup>E</sup>**1(*t*) = *<sup>Z</sup>*ˆ**j**1(*t*), where the components of the ac impedance tensor *Z*ˆ are measured in the *xy* coordinate system (see Fig. 2), Eqs. (4), and (37), the longitudinal and transverse (with respect to the direction of **j**1)

*<sup>Z</sup>*<sup>⊥</sup> = (*<sup>ρ</sup> <sup>f</sup>* /*γ*Δ){*δγZ*<sup>1</sup> + [*Z*<sup>1</sup> <sup>−</sup> *<sup>γ</sup>*2(<sup>Δ</sup> <sup>−</sup> *<sup>δ</sup>*2*Z*1) sin *<sup>α</sup>* cos *<sup>α</sup>*]},

*<sup>Z</sup>*� = (*<sup>ρ</sup> <sup>f</sup>* /*γ*Δ)[*γ*2(<sup>Δ</sup> <sup>−</sup> *<sup>δ</sup>*2*Z*1) sin2 *<sup>α</sup>* <sup>+</sup> *<sup>Z</sup>*<sup>1</sup> cos<sup>2</sup> *<sup>α</sup>*],

�

*at a constant subcritical current*, unlike the case already described in Sec. 4.2.1.

**E**1(*t*) = **E**(*t*) − **E**<sup>0</sup> = **E**(*t*) − �**E**(*t*)�. From this and from Eq. (4) one has that

*γv*1*x*(*t*) + *δv*1*y*(*t*)=[*F*˜

*j*1*y*(*t*) = *j*1(*t*) cos *α*, and *j*1(*t*) = *j*<sup>1</sup> exp *iωt*. We use the fact that *F*˜

(*x* − *x*0) like it was done in Sec. 3.2, and using Eq. (30) one gets

*px*/*η*<sup>0</sup> = −*ω*˜ *<sup>p</sup>*(*x* − *x*0), where *ω*˜ *<sup>p</sup>* = *ω<sup>p</sup>*

*v*1*y*(*t*)/*γ* − *δv*1*x*(*t*) = *F*1*y*(*t*)/*η*0,

<sup>0</sup>*<sup>x</sup>* < *Fc*, the vortex motion and the resistive response

<sup>0</sup>*<sup>x</sup>* is the effective PP, taking into account the driving

<sup>1</sup> <sup>−</sup> (˜*j*0*y*/*jc*)2, and *<sup>ω</sup><sup>p</sup>* <sup>≡</sup> *kp*/*η*0. (35)

(34)

*px* =

(36)

(38)

<sup>0</sup>*<sup>x</sup>* <sup>+</sup> *Fpx* <sup>≡</sup> *<sup>F</sup>*˜

<sup>0</sup>*<sup>x</sup>* + *F*1*x*(*t*) + *Fpx*]/*η*0,

$$\omega\_q = \tilde{\omega}\_{p\gamma}/\Delta = \left[\omega\_p/\gamma\Delta\right]\sqrt{1 - (\tilde{\jmath}\_{0y}/j\_c)^2} = \left[\omega\_p/\gamma\Delta\right]\sqrt{1 - (\jmath\_{0y}/j\_c)^2(\cos\alpha + \delta\gamma\sin\alpha)^2} \tag{39}$$

The quantity *ω<sup>q</sup>* in Eq. (39) a generalization to the case *γ* �= 1 and *�* �= 0, and is physically analogous to the depinning frequency *ωp* introduced in Seq. 3.1 and dependent on the subcritical transport current. However, it should be emphasized that, unlike the depinning frequency *ωp* which is independent on the **B**-inversion, the value of the *ωq* changes with the replacement *n* → −*n*, i.e., *δ* → −*δ* because the Hall effect is present.

Having separated even and odd parts in the impedance components, finally the experimentally deducible, field orientation-independent quantities are

$$\begin{cases} Z\_{\parallel}^{+} = (\rho\_f/\gamma\Delta)[\gamma^2(\Delta-\delta^2Z\_1^+)\sin^2a + Z\_1^+\cos^2a], \\\\ Z\_{\parallel}^{-} = (\rho\_f/\gamma\Delta)Z\_1^-(\cos^2a - \delta^2\gamma^2\sin^2a), \\\\ Z\_{\perp}^{+} = (\rho\_f/\gamma\Delta)\{Z\_1^+ - \gamma^2(\Delta-\delta^2Z\_1^+)\}\sin a\cos a \}, \\\\ Z\_{\perp}^{-} = (\rho\_f/\gamma\Delta)\{\delta\gamma Z\_1^+ + Z\_1^-(1+\delta^2\gamma)\sin a\cos a\}, \end{cases} \tag{40}$$

### *4.3.3. Determination of the Hall constant from microwave measurements*

Let us assume that *γ* = 1 and the Hall constant is arbitrary but satisfies the condition *F*˜ <sup>0</sup>*<sup>x</sup>* < *Fc*. Then *<sup>Z</sup>*¯ <sup>1</sup> <sup>≡</sup> *<sup>Z</sup>*1(*<sup>γ</sup>* <sup>=</sup> <sup>1</sup>) = 1/(<sup>1</sup> <sup>−</sup> *<sup>i</sup>ω*¯ *<sup>q</sup>*/*ω*), where *<sup>ω</sup>*¯ *<sup>q</sup>* <sup>≡</sup> (*ωp*/Δ) �<sup>1</sup> <sup>−</sup> (*j*0/*jc*)2(cos *<sup>α</sup>* <sup>+</sup> *<sup>δ</sup>* sin *<sup>α</sup>*)2. In this case, the expressions for the real part of the longitudinal and transverse impedances Re*Z*¯ �,<sup>⊥</sup> are determined by

$$\begin{cases} \text{Re}\bar{\mathbf{Z}}\_{\parallel} = (\rho\_f/\Delta)[\text{Re}\bar{\mathbf{Z}}\_1 + \Delta(1 - \text{Re}\bar{\mathbf{Z}}\_1)\sin^2 a] \\\\ \text{Re}\bar{\mathbf{Z}}\_{\perp} = (\rho\_f/\Delta)[\delta \text{Re}\bar{\mathbf{Z}}\_1 - \Delta(1 - \text{Re}\bar{\mathbf{Z}}\_1)\sin a \cos a] \end{cases} \tag{41}$$

where Re*Z*¯ <sup>1</sup> = 1/[1 + (*ω*¯ *<sup>q</sup>*/*ω*)2].

The culmination of this subsection is an analysis of the dependence Re*Z*¯ <sup>±</sup> �,<sup>⊥</sup> as a function of *<sup>ω</sup>* at large or small frequencies. If *<sup>ω</sup>* <sup>→</sup> <sup>∞</sup>, one has (*ω*¯ *<sup>q</sup>*/*ω*) <sup>→</sup> 0, i.e., Re*Z*¯ <sup>1</sup> <sup>=</sup> 1. Then in the main approximation with respect to 1/*ω* one has Re*Z*¯ � <sup>=</sup> *<sup>ρ</sup> <sup>f</sup>* /Δ, (i.e., Re*Z*¯ <sup>+</sup> � <sup>=</sup> Re*Z*¯ � and Re*Z*¯ <sup>−</sup> � <sup>=</sup> 0), and Re*Z*¯<sup>⊥</sup> <sup>=</sup> *<sup>ρ</sup> <sup>f</sup> <sup>δ</sup>*/Δ, i.e., Re*Z*¯ <sup>+</sup> <sup>⊥</sup> <sup>=</sup> 0 and Re*Z*¯ <sup>−</sup> <sup>⊥</sup> <sup>=</sup> Re*Z*¯⊥. Thus, as the Hall constant *�* increases (i.e., *δ* increases), the absorbed power *P* decreases as *ω* → ∞ (*P* = *P*0/Δ). Moreover, as *<sup>ω</sup>* <sup>→</sup> <sup>∞</sup>, for any *<sup>α</sup>*, there is a relationship of the form *<sup>δ</sup>* <sup>=</sup> Re*Z*¯ <sup>−</sup> ⊥/Re*Z*¯ <sup>−</sup> � . Now let *<sup>ω</sup>* <sup>→</sup> 0 (i.e., Re*Z*¯ <sup>1</sup> <sup>=</sup> 1). Then, in the main approximation with respect to *<sup>ω</sup>*, the Hall effect is unmeasurable, since Re*Z*¯ � <sup>=</sup> *<sup>ρ</sup> <sup>f</sup>* sin2 *<sup>α</sup>* and *<sup>P</sup>* <sup>=</sup> *<sup>P</sup>*<sup>0</sup> sin2 *<sup>α</sup>*, while Re*Z*¯<sup>⊥</sup> <sup>=</sup> <sup>−</sup>*<sup>ρ</sup> <sup>f</sup>* sin *<sup>α</sup>* cos *<sup>α</sup>*; i.e., *δ* has been canceled out of the results. It follows from this that *ρ <sup>f</sup>* = Re*Z*¯ �/ sin2 *<sup>α</sup>*; i.e., *η*<sup>0</sup> = *B*Φ<sup>0</sup> sin2 *α*/Re*Z*¯ �*c*2.

#### 20 282 Superconductors – Materials, Properties and Applications Microwave Absorption by Vortices in Superconductors with a Washboard Pinning Potential <sup>21</sup>

Thus, the high-frequency limit *ω* � *ω*¯ *<sup>q</sup>* (*ω* → ∞) is needed to determine *�* = *αH*/*η*0, whereas the low-frequency limit *ω* � *ω*¯ *<sup>q</sup>* (*ω* → 0) is sufficient to determine *η*0. Because of the dependence *ω*¯ *<sup>q</sup>*(*j*), appropriate measurements can be performed *even at one fixed frequency ω ωp*. Thus, for any *�* and *α* = *π*/2, the Hall constant in a periodic PP from microwave absorption by vortices is determined by

$$\mathfrak{a}\_{H} = \left(\mathcal{B}\Phi\_{0}/\bar{\rho}(0)c^{2}\sqrt{\bar{\rho}(0)/\bar{\rho}(\infty) - 1}\right) \tag{42}$$

of a pinning potential can be reconstructed from the microwave absorption data measured at a set of subcritical dc currents. Second, the dependences of the longitudinal and transverse dc resistivity and ac impedance tensors, as well as of the absorbed power on the subcritical constant current density *j*0, the ac frequency *ω*, the dimensionless Hall parameter *δ*, the anisotropy coefficient *γ*, and the angle *α* between the direction of the collinear currents *j*<sup>0</sup> and *j*1(*t*) with respect to the channels of the WPP have been derived for the general case. The physics of the vortex motion in a pinning potential subjected to superimposed dc an ac drives has been elucidated. In particular, it has been shown that the results are most substantially affected not by the value of *γ* but by the value of the Hall parameter *�*. At a constant subcritical current *j*<sup>0</sup> < *jc*, it turns out that *�* does not appear in the resistive responses, whereas, at a small ac current, two new effects result from the presence of *�*, namely (i) a falloff of the absorbtion as *�* increases at any subcritical current *j*<sup>0</sup> < *jc*, and (ii) the appearance of an odd-in-field

Microwave Absorption by Vortices in Superconductors with a Washboard Pinning Potential 283

**Figure 8.** Dependence of the absorbed power *P*/*P*<sup>0</sup> on the dimensionless frequency *ω*/*ω<sup>p</sup>* calculated on the basis of a stochastic model [25]. In the presence of a dc current *j*<sup>0</sup> = 1 and a series of dimensionless inverse temperatures *g* = *Up*/2*T*, as indicated, *P*(*ω*) demonstrates (i) an enhanced power absorption by vortices at low frequencies, (ii) a pronounced temperature-dependent minimum at intermediate frequencies and (iii) a sign change at certain conditions [27]. Experimentally, *U*<sup>0</sup> � 1000 ÷ 5000 K typically [10], and *ω<sup>p</sup>* usually ensues in the microwave range [16, 30, 42]. In the limit of zero dc current *j*<sup>0</sup> = 0, the curves coincide with the well-known results of Coffey and Clem (dashed line) [49].

Our discussion has been limited by the consideration of *j*<sup>0</sup> < *jc* and *j*<sup>1</sup> → 0 mainly for two

significant and because of this the *overheating* of the sample is unavoidable and as a result the heat release in the film should be properly analyzed. We note, however, this difficulty can be overcome provided high-speed current sweeps [59] or short-pulse measurements [60] are employed. The second reason is that at *j*<sup>0</sup> > *jc* a *running* mode in the vortex dynamics appears [25], when the vortex moves in a tilted WPP with instantaneous velocity oscillating with frequency *ωi*. Due to the presence of the two frequencies, i.e., *intrinsic ω<sup>i</sup>* and *external ω*, the problem of their synchronization arises. Though an analytical treatment of this issue could be presented, to work out a clear physical picture for this problem would be more complicate.

2 <sup>0</sup> may be

reasons. First, at a steady-state dc current *j*<sup>0</sup> > *jc* the dissipated power *P*<sup>0</sup> ∼ *ρ <sup>f</sup> j*

component *P*−(*ω*) when *α* �= 0; 90, and this increases with increasing *�*.

where *<sup>ρ</sup>*¯(0) = *<sup>ρ</sup> <sup>f</sup>* sin2 *<sup>α</sup>* for *<sup>ω</sup>* <sup>→</sup> 0 and *<sup>ρ</sup>*¯(∞) = *<sup>ρ</sup> <sup>f</sup>* /<sup>Δ</sup> for *<sup>ω</sup>* <sup>→</sup> <sup>∞</sup>.

### *4.3.4. Microwave absorption by vortices in a washboard pinning potential*

By analogy with Sec. 4.2.4, the absorbed power is *P* = (*j* 2 <sup>1</sup>/2)Re*Z*� <sup>≡</sup> (*<sup>j</sup>* 2 <sup>1</sup>/2)*ρ*¯, where now

$$\bar{\rho} = (\rho\_f/\gamma \Delta) \{ \Delta \gamma^2 \sin^2 a + [1 - (1 + \delta^2 \gamma^2) \sin^2 a] \text{Re}\mathbf{Z}\_1 \} \tag{43}$$

If *δ* = 0 and *γ* = 1, Eq. (43) reduces to *ρ*¯ = *ρ <sup>f</sup>*(sin2 *α* + Re*Z*<sup>1</sup> cos<sup>2</sup> *α*) which has been dealt with previously [50]. Let *Z*<sup>1</sup> ≡ 1 − *iG*1, where *G*<sup>1</sup> = −(*ωq*/*ω*)/(1 − *iωq*/*ω*), and consequently Re*Z*<sup>1</sup> = 1 − Re(*iG*1) = 1 + Im*G*1. Then from Eq. (43) one has that

$$\bar{\rho} = (\rho\_f/\gamma \Delta) \{ \gamma^2 \sin^2 \alpha + \cos^2 \alpha + [1 - (1 + \delta^2 \gamma^2) \sin^2 \alpha] \text{Im} \mathcal{G}\_1 \} \tag{44}$$

where Im*G*<sup>1</sup> <sup>=</sup> <sup>−</sup>1/[<sup>1</sup> + (*ω*/*ωq*)2]. When *<sup>γ</sup>* <sup>=</sup> 1, Eq. (44) reduces to Eq. (85) in our previous work [25]. Finally,

$$P = P\_0 \{ 1 + (\gamma^2 - 1)\sin\mathfrak{a} + [1 - (1 + \delta^2 \gamma^2)\sin^2\mathfrak{a}] \text{Im}\!G\_1\}/\gamma\Delta,\tag{45}$$

where *P*<sup>0</sup> = *ρ <sup>f</sup>*(*j* 2 <sup>1</sup>/2)/ Unlike the case in which is no pinning (Sec. 4.2.4), the absorbed power *P* in this case not only depends on angle *α*, anisotropy parameter *γ*, and Hall constant *�*, but also depends on frequency *ω* and current density *j*0.

It is essential to point out, that in that case under consideration, the absorbed power contains both even and odd parts with regards to the change **B** → −**B**; this is because of the dependence of Im*G*<sup>1</sup> through *ω<sup>q</sup>* on *n* [see Eq. (39)]. Thus, the experimentally observed *P*(*B*) changes under the reversal of the direction of **B**. Therefore, it is convenient to represent the absorbed power as *<sup>P</sup>*(*B*) = *<sup>P</sup>*+(*B*) + *<sup>P</sup>*−(*B*), where *<sup>P</sup>*±(*B*) <sup>≡</sup> [*P*(*B*) <sup>±</sup> *<sup>P</sup>*(−*B*)]/2 are moduli that do not change their quantities under inversion of **B**.

### **5. Conclusion**

### **5.1. Summary**

The microwave absorbtion by vortices in a superconductor with a periodic (washboard-type) pinning potential in the presence of the Hall effect and viscosity anisotropy has been studied theoretically. Two groups of results have been discussed. First, it has been shown how the Gittleman-Rosenblum model can be generalized for the case when a subcritical dc current is superimposed on a weak ac current. It has been elucidated how the coordinate dependence of a pinning potential can be reconstructed from the microwave absorption data measured at a set of subcritical dc currents. Second, the dependences of the longitudinal and transverse dc resistivity and ac impedance tensors, as well as of the absorbed power on the subcritical constant current density *j*0, the ac frequency *ω*, the dimensionless Hall parameter *δ*, the anisotropy coefficient *γ*, and the angle *α* between the direction of the collinear currents *j*<sup>0</sup> and *j*1(*t*) with respect to the channels of the WPP have been derived for the general case. The physics of the vortex motion in a pinning potential subjected to superimposed dc an ac drives has been elucidated. In particular, it has been shown that the results are most substantially affected not by the value of *γ* but by the value of the Hall parameter *�*. At a constant subcritical current *j*<sup>0</sup> < *jc*, it turns out that *�* does not appear in the resistive responses, whereas, at a small ac current, two new effects result from the presence of *�*, namely (i) a falloff of the absorbtion as *�* increases at any subcritical current *j*<sup>0</sup> < *jc*, and (ii) the appearance of an odd-in-field component *P*−(*ω*) when *α* �= 0; 90, and this increases with increasing *�*.

20

absorption by vortices is determined by

work [25]. Finally,

where *P*<sup>0</sup> = *ρ <sup>f</sup>*(*j*

**5. Conclusion**

**5.1. Summary**

2

also depends on frequency *ω* and current density *j*0.

that do not change their quantities under inversion of **B**.

Thus, the high-frequency limit *ω* � *ω*¯ *<sup>q</sup>* (*ω* → ∞) is needed to determine *�* = *αH*/*η*0, whereas the low-frequency limit *ω* � *ω*¯ *<sup>q</sup>* (*ω* → 0) is sufficient to determine *η*0. Because of the dependence *ω*¯ *<sup>q</sup>*(*j*), appropriate measurements can be performed *even at one fixed frequency ω ωp*. Thus, for any *�* and *α* = *π*/2, the Hall constant in a periodic PP from microwave

> 2

If *δ* = 0 and *γ* = 1, Eq. (43) reduces to *ρ*¯ = *ρ <sup>f</sup>*(sin2 *α* + Re*Z*<sup>1</sup> cos<sup>2</sup> *α*) which has been dealt with previously [50]. Let *Z*<sup>1</sup> ≡ 1 − *iG*1, where *G*<sup>1</sup> = −(*ωq*/*ω*)/(1 − *iωq*/*ω*), and consequently

where Im*G*<sup>1</sup> <sup>=</sup> <sup>−</sup>1/[<sup>1</sup> + (*ω*/*ωq*)2]. When *<sup>γ</sup>* <sup>=</sup> 1, Eq. (44) reduces to Eq. (85) in our previous

*P* in this case not only depends on angle *α*, anisotropy parameter *γ*, and Hall constant *�*, but

It is essential to point out, that in that case under consideration, the absorbed power contains both even and odd parts with regards to the change **B** → −**B**; this is because of the dependence of Im*G*<sup>1</sup> through *ω<sup>q</sup>* on *n* [see Eq. (39)]. Thus, the experimentally observed *P*(*B*) changes under the reversal of the direction of **B**. Therefore, it is convenient to represent the absorbed power as *<sup>P</sup>*(*B*) = *<sup>P</sup>*+(*B*) + *<sup>P</sup>*−(*B*), where *<sup>P</sup>*±(*B*) <sup>≡</sup> [*P*(*B*) <sup>±</sup> *<sup>P</sup>*(−*B*)]/2 are moduli

The microwave absorbtion by vortices in a superconductor with a periodic (washboard-type) pinning potential in the presence of the Hall effect and viscosity anisotropy has been studied theoretically. Two groups of results have been discussed. First, it has been shown how the Gittleman-Rosenblum model can be generalized for the case when a subcritical dc current is superimposed on a weak ac current. It has been elucidated how the coordinate dependence

2

*<sup>ρ</sup>*¯ = (*<sup>ρ</sup> <sup>f</sup>* /*γ*Δ){Δ*γ*<sup>2</sup> sin2 *<sup>α</sup>* + [<sup>1</sup> <sup>−</sup> (<sup>1</sup> <sup>+</sup> *<sup>δ</sup>*2*γ*2) sin2 *<sup>α</sup>*]Re*Z*1} (43)

*<sup>ρ</sup>*¯ = (*<sup>ρ</sup> <sup>f</sup>* /*γ*Δ){*γ*<sup>2</sup> sin2 *<sup>α</sup>* <sup>+</sup> cos<sup>2</sup> *<sup>α</sup>* + [<sup>1</sup> <sup>−</sup> (<sup>1</sup> <sup>+</sup> *<sup>δ</sup>*2*γ*2) sin2 *<sup>α</sup>*]Im*G*1} (44)

*<sup>P</sup>* <sup>=</sup> *<sup>P</sup>*0{<sup>1</sup> + (*γ*<sup>2</sup> <sup>−</sup> <sup>1</sup>) sin *<sup>α</sup>* + [<sup>1</sup> <sup>−</sup> (<sup>1</sup> <sup>+</sup> *<sup>δ</sup>*2*γ*2) sin2 *<sup>α</sup>*]Im*G*1}/*γ*Δ, (45)

<sup>1</sup>/2)/ Unlike the case in which is no pinning (Sec. 4.2.4), the absorbed power

<sup>1</sup>/2)Re*Z*� <sup>≡</sup> (*<sup>j</sup>*

*ρ*¯(0)/*ρ*¯(∞) − 1, (42)

2

<sup>1</sup>/2)*ρ*¯, where now

*α<sup>H</sup>* = (*B*Φ0/*ρ*¯(0)*c*

where *<sup>ρ</sup>*¯(0) = *<sup>ρ</sup> <sup>f</sup>* sin2 *<sup>α</sup>* for *<sup>ω</sup>* <sup>→</sup> 0 and *<sup>ρ</sup>*¯(∞) = *<sup>ρ</sup> <sup>f</sup>* /<sup>Δ</sup> for *<sup>ω</sup>* <sup>→</sup> <sup>∞</sup>.

By analogy with Sec. 4.2.4, the absorbed power is *P* = (*j*

*4.3.4. Microwave absorption by vortices in a washboard pinning potential*

Re*Z*<sup>1</sup> = 1 − Re(*iG*1) = 1 + Im*G*1. Then from Eq. (43) one has that

**Figure 8.** Dependence of the absorbed power *P*/*P*<sup>0</sup> on the dimensionless frequency *ω*/*ω<sup>p</sup>* calculated on the basis of a stochastic model [25]. In the presence of a dc current *j*<sup>0</sup> = 1 and a series of dimensionless inverse temperatures *g* = *Up*/2*T*, as indicated, *P*(*ω*) demonstrates (i) an enhanced power absorption by vortices at low frequencies, (ii) a pronounced temperature-dependent minimum at intermediate frequencies and (iii) a sign change at certain conditions [27]. Experimentally, *U*<sup>0</sup> � 1000 ÷ 5000 K typically [10], and *ω<sup>p</sup>* usually ensues in the microwave range [16, 30, 42]. In the limit of zero dc current *j*<sup>0</sup> = 0, the curves coincide with the well-known results of Coffey and Clem (dashed line) [49].

Our discussion has been limited by the consideration of *j*<sup>0</sup> < *jc* and *j*<sup>1</sup> → 0 mainly for two reasons. First, at a steady-state dc current *j*<sup>0</sup> > *jc* the dissipated power *P*<sup>0</sup> ∼ *ρ <sup>f</sup> j* 2 <sup>0</sup> may be significant and because of this the *overheating* of the sample is unavoidable and as a result the heat release in the film should be properly analyzed. We note, however, this difficulty can be overcome provided high-speed current sweeps [59] or short-pulse measurements [60] are employed. The second reason is that at *j*<sup>0</sup> > *jc* a *running* mode in the vortex dynamics appears [25], when the vortex moves in a tilted WPP with instantaneous velocity oscillating with frequency *ωi*. Due to the presence of the two frequencies, i.e., *intrinsic ω<sup>i</sup>* and *external ω*, the problem of their synchronization arises. Though an analytical treatment of this issue could be presented, to work out a clear physical picture for this problem would be more complicate.

### **5.2. Extension of the theory for non-zero temperature**

Finally, let us compare the results presented in the chapter with the analogous but more general results obtained by the authors [25] on the basis of a stochastic model for arbitrary temperature *T* and densities *j*<sup>0</sup> and *j*1. In that work, the Langevin equation (1), supplemented with a thermofluctuation term, has been *exactly* solved for *γ* = 1 in terms of a matrix continued fraction [52] and, depending on the WPP's tilt caused by the dc current, two substantially different modes in the vortex motion have been predicted. In more detail, at low temperatures and relatively high frequencies in a *nontilted* pinning potential each pinned vortex is *confined* to its pinning potential well during the *ac* period. In the case of superimposed strong ac and dc driving currents a *running* state of the vortex may appear when it can visit several (or many) potential wells during the ac period. As a result, two branches of new findings have been elucidated [25, 27]. First, the influence of an ac current on the usual *E*0(*j*0) and ratchet *E*0(*j*1) CVCs has been analyzed. Second, the influence of a dc current on the ac nonlinear impedance response and nonlinear power absorption has been investigated. In particular, the appearance of Shapiro-like steps in the usual CVC and the appearance of phase-locking regions in the ratchet CVC has been predicted. At the same time, it has been shown that an anomalous power absorption in the ac response is expected at close-to critical currents *j*<sup>0</sup> *jc* and relatively low frequencies *ω ωp*. Figure 8 shows the main predictions of these works. Namely, predicted are (i) an enhanced power absorption at low frequencies, (ii) a temperature- and current-dependent minimum at intermediate frequencies. (iii) At substantially low temperatures, the absorption can acquire negative values which physically corresponds to the generation by vortices. However, a more general and formally precise solution of the problem in terms of a matrix-continued fraction does not allow the main physical results of the problem to be investigated in the form of explicit analytical functions of the main physical quantities (*j*0, *j*1, *ω*, *α*, *T*, *�*, and *γ*) which, we believe, has helped us greatly to elucidate the physics in the problem under consideration.

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## **Acknowledgements**

The authors are very grateful to Michael Huth for useful comments and critical reading. O.V.D. gratefully acknowledges financial support by the Deutsche Forschungsgemeinschaft (DFG) through Grant No. DO 1511/2-1.
