Determination of coordinate dependence of a pinning potential from a microwave experiment with vortices
V. A. Shklovskij and O. V. Dobrovolskiy Citation: Low Temp. Phys. 39, 120 (2013); doi: 10.1063/1.4791773 View online: http://dx.doi.org/10.1063/1.4791773 View Table of Contents: http://ltp.aip.org/resource/1/LTPHEG/v39/i2 Published by the American Institute of Physics.

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LOW TEMPERATURE PHYSICS

VOLUME 39, NUMBER 2

FEBRUARY 2013

Determination of coordinate dependence of a pinning potential from a microwave experiment with vortices
V. A. Shklovskij
Institute of Theoretical Physics, NSC-KIPT, Kharkiv 61108, Ukraine and Physical Department, Kharkiv National University, Kharkiv 61077, Ukraine

O. V. Dobrovolskiya)
Physikalisches Institut, Goethe-University, Frankfurt am Main 60438, Germany

(Submitted July 22, 2012; revised August 30, 2012) Fiz. Nizk. Temp. 39, 162–167 (February 2013) The measurement of the complex impedance response accompanied by power absorption P (x) in the radiofrequency and microwave ranges represents a most popular experimental method for the investigation of pinning mechanisms and vortex dynamics in type-II superconductors. In the theory, the pinning potential (PP) well for a vortex must be a priori speciﬁed in order to subsequently analyze the measured data. We have theoretically solved the inverse problem at T ¼ 0 K and exemplify how the coordinate dependence of a PP can be determined from a set of experimental curves P (xjj0) measured at subcritical dc currents 0 < j0 < jc under a small microwave excitation j1 ( jc with frequency x. We furthermore elucidate how and why the depinning frequency xp, which separates the non-dissipative (quasi-adiabatic) and the dissipative (high-frequency) regimes of small vortex oscillations in the PP, is reduced with increasing j0. The results can be directly applied to a wide range of conventional superconductors with a PP subjected to C 2013 American Institute of superimposed dc and small microwave ac currents at T ( Tc. V Physics. [http://dx.doi.org/10.1063/1.4791773]

1. Introduction

One of the most popular experimental methods for the investigation of vortex dynamics in type-II superconductors is the measurement of the complex ac response in the radiofrequency and microwave ranges.1 The reason for this is that at frequencies substantially smaller than those invoking the breakdown of the energy gap, the high-frequency and microwave impedance measurements of a mixed state contain information about ﬂux pinning mechanisms and vortex dynamics accompanied by dissipative processes in a superconductor. It should be noted that this information cannot be extracted from the dc resistivity data obtained in the steady state regime when pinning in the sample is strong. In fact, in the latter, when the critical current densities jc are rather large, the realization of the dissipative mode, in which the ﬂux-ﬂow resistivity qf can be measured, requires that j0 տ jc. This is commonly accompanied by non-negligible electron overheating in the sample,2,3 which changes the desired value of qf. At the same time, measurements of power absorbed by the vortices from an ac current with the amplitude j1 ( jc allow one to determine qf at a dissipative power P 1 $ q f j2 1 , which can be many orders of magnitude less than P 0 $ q f j2 0 . Consequently, measurements of the complex ac response versus frequency x practically probe the pinning forces in the absence of overheating effects, otherwise unavoidable at overcritical steady-state dc current densities. The appearance of experimental works utilizing the usual four-point scheme,4 strip-line coplanar waveguides (CPWs),5 Corbino geometry,6,7 or the cavity method8 to investigate the microwave vortex response in as-grown thin-ﬁlm superconductors (or in those containing some
1063-777X/2013/39(2)/5/$32.00 120

nano-tailored pinning potential (PP) landscape) in the recent years reﬂects the explosively growing interest in the subject. In fact, such 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 present, acting as pinning sites for vortices as well. Therefore, an approach to experimentally reconstructing the form of the PP ensued in the sample is of great demand for both application-related and fundamental reasons. An early scheme to reconstructing the coordinate dependence of the pinning force from measurements implying a small ripple magnetic ﬁeld superposed on a larger dc magnetic ﬁeld was reported in Ref. 9. Similar problems in the reconstruction of a speciﬁc form of the potential subjected to a superimposed constant and small alternating signals arise not only in vortex physics, but also in a number of other ﬁelds. Mainly as the closest mathematical analogy, we would like to mention the Josephson junction problem, wherein plenty of non-sine forms of the current-phase relation are known to occur,11 and which could in turn beneﬁt from the results reported here. Turning back to the development of the theory of our problem, the very early model describing the power absorbed by vortices refers to the work of Gittleman and Rosenblum (GR),12 where a small ac excitation of vortices in the absence of dc current is considered. The GR results were obtained at T ¼ 0 K in the linear approximation for the pinning force. We will brieﬂy present their results in the present work since subsequent description of our new results requires these as essential background. The theory, accounting also for the vortex creep at non-zero temperature in a
C 2013 American Institute of Physics V

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Low Temp. Phys. 39 (2), February 2013

V. A. Shklovskij and O. V. Dobrovolskiy

121

one-dimensional cosine PP, was extended by Coffey and Clem (CC) later.13 However, this theory was developed for a small microwave current in the absence of a dc current. We have recently substantially generalized the CC results14,15 for a two-dimensional cosine washboard pinning potential (WPP). The washboard form of the PP enables exact theoretical description of two-dimensional anisotropic nonlinear vortex dynamics for any arbitrary values of ac and dc amplitudes, temperature, the Hall constant, and the angle between the direction of the transport current and the guiding direction of the WPP. Among other nontrivial results obtained, an enhancement14 and a sign change15 in the power absorption for j0 տ jc have been predicted. Whereas the general solution of the problem in Refs. 14 and 15 has been obtained in terms of a matrix continued fraction and is suitable for the analysis mainly in the form of a data ﬁgure due to a large number of variable parameters, an analytical implementation of the solution at T ¼ 0 K, j0 < jc, and j1 ﬁ 0 has been performed in Refs. 16 and 17, also taking into account the anisotropy of the vortex viscosity and an arbitrary Hall constant. In the present work, we report on the possibility of reconstructing the coordinate dependence of a PP if a set of P 0(x) curves is measured at different dc current amplitudes in the entire range 0 j0 Շ jc at a small microwave amplitude j1 ! 0. Whereas a preliminary communication on this matter can be found in Ref. 18, here we provide a detailed description of the PP reconstruction procedure. Geometry of the problem implies a standard four-point thin-ﬁlm superconductor microstrip bridge placed in a small perpendicular magnetic ﬁeld with a magnitude B ( Bc2 at T ( Tc. The sample is assumed to have at least one pinning site, and dc and ac currents are directed collinearly. Theoretical treatment of the problem is described in detail below.
2. Dynamics of pinned vortices in a small microwave current

EðxÞ ¼

q f j1  Z ðx Þj 1 : 1 À ixp =x

(3)

Here, qf ¼ BU0/gc2 is the ﬂux-ﬂow resistivity, and Z(x)  qf/ (1 À ixp/x) is the microwave impedance of the sample. In order to calculate the power P absorbed per unit volume and averaged over a period of an ac cycle, the standard relation P ¼ (1/2)Re(EJ*) is used, where E and J are the complex amplitudes of the ac electric ﬁeld and the current density, respectively. The asterisk denotes the complex conjugate. Then from Eq. (3) it follows that q f j2 1 1 1 : PðxÞ ¼ Re Z ðxÞj2 1 ¼ 2 2 1 þ ðxp =xÞ2 (4)

For subsequent analysis, it is convenient to write out the real and imaginary parts of the impedance Z ¼ ReZ þ iImZ, namely ReZ ðxÞ ¼ qf 1 þ ðxp =xÞ
2

;

ImZðxÞ ¼

qf ðx=xp Þ 1 þ ðx=xp Þ2

: (5)

The frequency dependences (5) are plotted in dimensionless units Z/qf and x/xp in Fig. 1 (see the curve for j0 ¼ 0). From Eqs. (1), (2), and (4) it follows that pinning forces dominate at low frequencies (x ( xp), where Z(x) is non-dissipative with ReZ(x) % (x/xp)2, whereas at higher frequencies (x ) xp) frictional forces dominate, and Z(x) is dissipative with ReZ(x) % qf [1 À (xp/x)2]. In other words, due to the reduction of the amplitude of vortex displacement with the increase in ac frequency, the pinning force does not inﬂuence the vortex. This can be seen from Eq. (2), where x $ 1/x for x ) xp; however, this is accompanied by the independence of vortex velocity of x in this regime, in accordance with Eq. (3).
3. Influence of dc current on depinning frequency

The GR model12 considers oscillations of damped vortices in a parabolic PP. Absorption of power by vortices in PbIn and NbTa ﬁlms was measured over a wide range of frequencies x, and the data was successfully analyzed on the basis of a simple equation for a vortex moving with velocity t(t) along the x-axis _ þ kp x ¼ f L ; gx (1)

When an arbitrary dc current is superimposed on a small microwave signal, the GR model can be generalized for an arbitrary PP. For deﬁniteness we consider a subcritical dc current with the density j0 < jc, where jc is the critical current

where x is vortex displacement, g is vortex viscosity, and kp is the constant that characterizes the restoring force fp in the PP well Up(x) ¼ (1/2)kpx2 and fp ¼ ÀdUp/dx ¼ Àkpx. In Eq. (1) fL ¼ (U0/c) j1(t) is the Lorentz force acting on the vortex, U0 is the magnetic ﬂux quantum, c is the speed of light, and j1(t) ¼ j1 exp (ixt) is the density of a small microwave current with the amplitude j1. Looking for a solution of Eq. (1) in the form x(t) ¼ x exp (ixt), where x is the complex amplitude of vortex displacement, one immediately gets _ (t) ¼ ixx(t) and x x¼ ðU0 =gcÞj1 ; ixþxp (2)
FIG. 1. Frequency dependences of real (a) and imaginary (b) parts of the ac impedance calculated for a cosine pinning potential Up(x) ¼ (Up/2)(1 À 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. (5).

where xp  kp/g is the depinning frequency. To calculate the magnitude of the complex electric ﬁeld arising due to the _ /c. Then vortex on the move, one takes E ¼ Bx

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122

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density in the absence of a microwave current. Our goal now is to determine the changes in PP parameters of the superimposition of dc current leads. In the presence of j0 6¼ 0, the ~ (x)  Up(x) À xf0, where Up(x) is the effective PP becomes U x-coordinate dependence of the PP when j0 ¼ 0. Note also that f0 < fc, where f0 and fc are the Lorentz forces corresponding to the current densities j0 and jc, respectively. In the presence of a dc current, the equation of motion for a vortex has the form gtðtÞ ¼ f ðtÞ þ fp ; (6)

f~ðx À x0 Þ ’ f~ðx0 Þ þ f~0 ðx0 Þu þ Á Á Á

(9)

Then, taking into account that f~(x0) ¼ 0 and f~0 (x0) ¼ Up* (x0), Eq. (7) takes on the form ~p u ¼ f1 ; _1 þ k gu (10)

where f(t) ¼ (U0/c) j(t) is the Lorentz force with j(t) ¼ j0 þ j1(t), where j1(t) ¼ j1 exp(ixt), and j1 is the amplitude of a small microwave current. Due to the fact that f(t) ¼ f0 þ f1(t), where f0 ¼ (U0/c) j0 and f1(t) ¼ (U0/c)j1(t) are the Lorentz forces for the subcritical dc and microwave currents, respectively, one can naturally assume that t(t) ¼ t0 þ t1(t), where t0 does not depend on the time, whereas t1(t) ¼ t1 exp(ixt). In Eq. (6) the pinning force is fp ¼ ÀdUp (x)/dx, where Up(x) is a PP of some form. Our goal is to determine t(t) from Eq. (6) which, taking into account the considerations above, acquires the following form: g½t0 þ t1 ðtÞ ¼ f0 þ fp þ f1 ðtÞ: (7)

Let us consider the case when j1 ¼ 0. If j0 < jc, i.e., f0 < fc, where fc is the maximal value of the pinning force, then t0 ¼ 0, i.e., the vortex is at rest. As seen in Fig. 2, the rest coordinate x0 of the vortex in this case depends on f0 and is determined from the condition of equality to zero of the ~ (x)/dx ¼ fp(x) þ f0, which effective pinning force f~(x) ¼ ÀdU reduces to the equation fp(x0) þ f0 ¼ 0, or  dUp ðxÞ   ; (8) f0 ¼ dx  x¼x0 the solution of which is the function x0(f0). Now add a small oscillation of the vortex in the vicinity of x0 under the action of a small external alternating force f1(t) with the frequency x. For this we expand the effective pinning force f~(x) in the vicinity of x ¼ x0 into a series of small displacements u  x À x0, which gives

~p(x0) ¼ Up00 (x0) is the effective constant characterizwhere k ing the restoring force f~(u) at small oscillations of the vortex ~ (x) near x0(f0), and t1 ¼ u _ ¼ ixu. Equain the effective PP U tion (10) used to determine t1 is physically equivalent to the GR Eq. (1) with the only distinction that the vortex depin~p/g now depends on f0 through Eq. (8), ~pk ning frequency x i.e., on the dc transport current density j0. Thereby, all the results of the previous section [see Eqs. (2)–(5)] can be ~ p. repeated here with the changes x ! u and xp ! x In order to discuss the changes in the dependences ReZ(x) and ImZ(x) caused by the dc current, the PP must be speciﬁed. As usual,13–15 we take a cosine WPP of the form Up(x) ¼ (Up/2)(1 À cos kx), where k ¼ 2p/a, and a is the period; though any other non-periodic PP can also be used. Then, as it has been previously shown for the cosine WPP,16 qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ~ p(j0/jc) ¼ xp 1 À ðj0 =jc Þ2 and the appropriate series of x curves P (xjj0) are plotted in Fig. 1. As evident from the ﬁgure, the curves shift to the left with increasing j0. The reason for this is that with the increase of j0 the PP well broadens while tilted, as is evident from Fig. 2. Thus, for the times shorter than sp ¼ 1/xp (i.e., for x > xp) a vortex can no longer non-dissipatively oscillate in the PP well. As a consequence, the enhancement of ReZ(x) occurs at lower frequencies. At the same time, the curves in Fig. 1 maintain their original shape. Thus, the only universal parameter to be found experimentally is the depinning frequency xp. For a ﬁxed frequency and variable j0, the real part of Z(x) always acquires larger values for larger j0, whereas the maximum in the imaginary part of Z(x) corresponds precisely to the middle point of the nonlinear transition in ReZ(x). It should be noted that even for T ¼ 0 K the dissipation, though small, is non-zero even at very low frequencies.
4. Reconstruction of a pinning potential from microwave absorption data

We now turn to a detailed analytical description of the process of reconstruction of 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 ~ p(j0) as a function dependence of the depinning frequency x of dc transport current j0 one can determine the coordinate dependence of the PP Up(x). The physical basis for the possibility of solving this problem is Eq. (8), which gives the ratio of the vortex rest coordinate x0 to the value of the static force f0 acting on the vortex and arising due to the dc current j0.
4.1. General scheme of the reconstruction
~ i(x)  Up(x) À f0ix, where FIG. 2. Modiﬁcation of the effective PP U Up(x) ¼ (Up/2)(1 À cos kx) is the WPP, with the gradual increase of f0 such as 0 ¼ f0 < f01 < f02 Շ f03 ¼ fc, i.e., a vortex is oscillating in the gradually tilting pinning potential well in the vicinity of the rest coordinate x0i.

From Eq. (8) it follows that when increasing f0 from zero to its critical value fc one in fact “probes” all the points of the dependence Up(x). Taking the x0-coordinate derivative in Eq. (8), one obtains

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Low Temp. Phys. 39 (2), February 2013

V. A. Shklovskij and O. V. Dobrovolskiy

123

dx0 1 1 ; ¼ ¼ 00 ~p ðx0 Þ df0 Up ðx 0 Þ k

(11)

~p(x0) is used [see Eq. (10) and where the relation U00 (x0) ¼ k the text below]. By substituting x0 ¼ x0(f0), Eq. (11) can be ~p[x0(f0)], and thus, rewritten as dx0/df0 ¼ 1/k dx0 1 : ¼ ~ p ðf0 Þ df0 gx (12)

~ (f0) has been deduced from the experiIf the dependence x mental data, i.e., ﬁtted by a known function, then Eq. (12) allows one to derive x0(f) by integrating
f0 ð 1 df : x 0 ðf 0 Þ ¼ ~ p ðf Þ g x 0

(13)

FIG. 4. The pinning potential reconstruction procedure: step 2. The inverse function to x0(f0) is f0(x0) ¼ fc sin(x0kp/fc) (dashed line). Then according to Eq. (14) Up(x) ¼ (Up/2)(1 À cos kx) is the PP sought (solid line).

Then, having calculated the function f0(x0), inverse to x0(f0), 0 (x0), i.e., Eq. (8), one and using the relation f0(x0) ¼ Up ﬁnally obtains Up ðxÞ ¼ dx0 f0 ðx0 Þ:
0 x ð

Evidently, the inverse function is f0(x0) ¼ fc sin(x0kp/fc) with the period a ¼ 2pfc/kp (see also Fig. 4). Taking the integral of Eq. (14) one ﬁnally gets Up(x) ¼ (Up/2)(1 À cos kx), where k ¼ 2p/a and Up ¼ 2 fc2/kp.
5. Conclusion

(14)

4.2. Sample WPP reconstruction procedure

Here we would like to support the above-mentioned considerations by giving an example of the reconstruction procedure for a WPP. Suppose that a series of power absorption curves P (x) was measured for a set of subcritical dc currents j0. Then, to be speciﬁc, let’s say that each i-curve of P (xjj0), like those shown in Fig. 1, has been ﬁtted with its ﬁtting ~ p/xp)i, ~ p so that one could map the points [(x parameter x (j0/jc)i], as shown by triangles in Fig. 3. ~ p/ We ﬁt the data in Fig. 3 to the function x qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ xp ¼ 1 À ðj0 =jc Þ2 and then substitute it into Eq. (13), from which one can calculate x0(f0). In this case, the function has a simple analytical form, namely, x0(f0) ¼ (fc/kp) arcsin (f0/fc).

FIG. 3. The pinning potential reconstruction procedure: step 1. A set of ~ p/xp)i, (j0/jc)i] points (᭝) has been deduced from the supposed measured [(x qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ~ p/xp ¼ 1 À ðj0 =jc Þ2 (solid line). Then according to Eq. data and ﬁtted as x (13) x0(f0) ¼ (fc/kp) arcsin (f0/fc) (dashed line).

In this paper we show how coordinate dependence of the PP in the sample can be determined from the data on microwave power absorption by vortices in the presence of a subcritical dc transport current. The proposed procedure can be used at T ( Tc and implies a small microwave current density j1 ( jc. In order to keep the transport current distribution in the sample as homogeneous as possible, the pinning potential is assumed to be not very “strong” in the sense that vortex pinning is caused, for example, by the reduction of vortex length rather than by suppression of the superconducting order parameter. Although the potential reconstruction scheme has been exempliﬁed for a cosine WPP, that is, for the periodic and symmetric PP, the elucidated procedure in general does not require periodicity of the potential and can also account for an asymmetric potential. If this is the case, one has to perform the reconstruction under dc current reversal, i.e., two times: for þj0 and Àj0. The scheme of reconstruction of the WPP Up(x) from the experimental data on ~ p (j0) can be brieﬂy summarized as follows: (a) from the x ~ p(j0); (b) take the integral (13) to ~ p (j0)) ﬁnd x data on P(x/x calculate x0(f0); (c) then from x0(f0) ﬁnd the inverse function f0(x0); and ﬁnally (d) integrate f0(x0), and by using Eq. (14) recover the PP Up(x). Theoretically, we have limited our analysis to T ¼ 0 K, j0 < jc, and j1 ! 0, as this allowed us to provide a clear reconstruction procedure in terms of elementary functions accompanied by a simple physical interpretation. Experimentally, adequate measurements can be performed at T ( Tc, i.e., on conventional thin-ﬁlm superconductors (e.g., Nb, NbN). These are suitable due to signiﬁcantly lower temperatures of the superconducting state, and relatively strong pinning in these materials allows one to neglect thermal ﬂuctuations of the vortex due to the PP depth Up ’ 1000–5000 K.19,20 It should be emphasized that due to the universal form of the dependence P (xjj0), the depinning

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Low Temp. Phys. 39 (2), February 2013

V. A. Shklovskij and O. V. Dobrovolskiy

frequency xp plays a role of the only ﬁtting parameter for each of the curves P (xjj0), thus ﬁtting the measured data seems simple. However, for the experiment it is crucial to adequately superimpose the applied currents and then to uncouple the picked-up dc and microwave signals maintaining that the line and the sample impedances be matching. Quantitatively, experimentally estimated values of the depinning frequency in the absence of a dc current and at a temperature of about 0.6Tc are xp % 7 GHz for a 20 nm-thick7 and a 40 nm-thick8 Nb ﬁlms. This value is strongly suppressed with the increase of both the ﬁeld magnitude and the ﬁlm thickness. Concerning the general validity of the results obtained, three remarks should be given. First, though the data ﬁgure has been provided here for a cosine WPP as for the most commonly used potential, the coordinate dependence can be reconstructed not only for periodic potentials. In fact, single PP wells, like the one used in Ref. 5, can also be proven to be in accordance with the provided approach. Secondly, it should be noted that if a PP is periodic then the theoretical consideration here has been performed in the single-vortex approximation, i.e., is valid only at small magnetic ﬁelds B ( Bc2, when the distance between two neighboring vortices, i.e., the period of a PP, is larger than the effective magnetic ﬁeld penetration depth. Finally, the results can be directly veriﬁed, for example, in the microstrip geometry for combined microwave and dc electrical transport measurements. While the experimental work on the subject has begun to appear,4–7 we hope to have stimulated further developments in the ﬁeld. Furthermore, due to the mathematical analogy between the equation of motion for a vortex used in this work and the equation for the phase difference in the Josephson junction problem, we believe that the proposed scheme of reconstruction can also be adopted for that case. V.A.S. thanks partial ﬁnancial support from the STCU project BNL-T2-368-UA through Grant No. P-424. O.V.D. gratefully acknowledges ﬁnancial support by the Deutsche

Forschungsgemeinschaft (DFG) through Grant No. DO 1511/2-1.
a)

Email: Dobrovolskiy@physik.uni-frankfurt.de

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1

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