Determination of the coordinate dependence of a pinning potential from the microwave experiment with vortices
The measurement of the complex impedance response and accompanied power absorption P(ω) in the radiofrequency and microwave ranges represents a most popular experimental method to investigate pinning mechanisms and the vortex dynamics in type-II superconductors. In the theory, the pinning potentia...
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Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України
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| Цитувати: | Determination of the coordinate dependence of a pinning potential from the microwave experiment with vortices / V.A. Shklovskij, O.V. Dobrovolskiy // Физика низких температур. — 2013. — Т. 39, № 2. — С. 162–167. — Бібліогр.: 20 назв. — англ. |
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irk-123456789-1182602017-05-30T03:05:14Z Determination of the coordinate dependence of a pinning potential from the microwave experiment with vortices Shklovskij, V.A. Dobrovolskiy, O.V. Свеpхпpоводимость, в том числе высокотемпеpатуpная The measurement of the complex impedance response and accompanied power absorption P(ω) in the radiofrequency and microwave ranges represents a most popular experimental method to investigate pinning mechanisms and the 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 = 0 K and exemplify how the coordinate dependence of a PP can be determined from a set of experimental curves P(ω|j₀) measured at subcritical dc currents 0 < j₀ < jc under a small microwave excitation j₁ << jc with frequency ω. We furthermore elucidate how and why the depinning frequency ωp, which separates the non-dissipative (quasi-adiabatic) and the dissipative (high-frequency) regimes of small vortex oscillations in the PP, is reduced with the increase of j₀. The results can be directly applied to a wide range of conventional superconductors with a PP subjected to superimposed dc and small microwave ac currents at T << Tc. 2013 Article Determination of the coordinate dependence of a pinning potential from the microwave experiment with vortices / V.A. Shklovskij, O.V. Dobrovolskiy // Физика низких температур. — 2013. — Т. 39, № 2. — С. 162–167. — Бібліогр.: 20 назв. — англ. 0132-6414 PACS: 74.25.Wx, 74.25.F–, 74.25.nn http://dspace.nbuv.gov.ua/handle/123456789/118260 en Физика низких температур Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
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Свеpхпpоводимость, в том числе высокотемпеpатуpная Свеpхпpоводимость, в том числе высокотемпеpатуpная |
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Свеpхпpоводимость, в том числе высокотемпеpатуpная Свеpхпpоводимость, в том числе высокотемпеpатуpная Shklovskij, V.A. Dobrovolskiy, O.V. Determination of the coordinate dependence of a pinning potential from the microwave experiment with vortices Физика низких температур |
| description |
The measurement of the complex impedance response and accompanied power absorption P(ω) in the radiofrequency and microwave ranges represents a most popular experimental method to investigate pinning mechanisms and the 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 = 0 K and exemplify how the coordinate dependence of a PP can be determined from a set of experimental curves P(ω|j₀) measured at subcritical dc currents 0 < j₀ < jc under a small microwave excitation j₁ << jc with frequency ω. We furthermore elucidate how and why the depinning frequency ωp, which separates the non-dissipative (quasi-adiabatic) and the dissipative (high-frequency) regimes of small vortex oscillations in the PP, is reduced with the increase of j₀. The results can be directly applied to a wide range of conventional superconductors with a PP subjected to superimposed dc and small microwave ac currents at T << Tc. |
| format |
Article |
| author |
Shklovskij, V.A. Dobrovolskiy, O.V. |
| author_facet |
Shklovskij, V.A. Dobrovolskiy, O.V. |
| author_sort |
Shklovskij, V.A. |
| title |
Determination of the coordinate dependence of a pinning potential from the microwave experiment with vortices |
| title_short |
Determination of the coordinate dependence of a pinning potential from the microwave experiment with vortices |
| title_full |
Determination of the coordinate dependence of a pinning potential from the microwave experiment with vortices |
| title_fullStr |
Determination of the coordinate dependence of a pinning potential from the microwave experiment with vortices |
| title_full_unstemmed |
Determination of the coordinate dependence of a pinning potential from the microwave experiment with vortices |
| title_sort |
determination of the coordinate dependence of a pinning potential from the microwave experiment with vortices |
| publisher |
Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
| publishDate |
2013 |
| topic_facet |
Свеpхпpоводимость, в том числе высокотемпеpатуpная |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/118260 |
| citation_txt |
Determination of the coordinate dependence of a pinning
potential from the microwave experiment with vortices / V.A. Shklovskij, O.V. Dobrovolskiy // Физика низких температур. — 2013. — Т. 39, № 2. — С. 162–167. — Бібліогр.: 20 назв. — англ. |
| series |
Физика низких температур |
| work_keys_str_mv |
AT shklovskijva determinationofthecoordinatedependenceofapinningpotentialfromthemicrowaveexperimentwithvortices AT dobrovolskiyov determinationofthecoordinatedependenceofapinningpotentialfromthemicrowaveexperimentwithvortices |
| first_indexed |
2025-07-08T13:38:30Z |
| last_indexed |
2025-07-08T13:38:30Z |
| _version_ |
1837086184903278592 |
| fulltext |
© V.A. Shklovskij and O.V. Dobrovolskiy, 2013
Low Temperature Physics/Fizika Nizkikh Temperatur, 2013, v. 39, No. 2, pp. 162–167
Determination of the coordinate dependence of a pinning
potential from the microwave experiment with vortices
V.A. Shklovskij1,2 and O.V. Dobrovolskiy3
1Institute of Theoretical Physics, NSC-KIPT, Kharkiv 61108, Ukraine
2Physical Department, Kharkiv National University, Kharkiv 61077, Ukraine
3Physikalisches Institut, Goethe-University, Frankfurt am Main 60438, Germany
E-mail: Dobrovolskiy@physik.uni-frankfurt.de
Received July 22, 2012, revised August 30, 2012
The measurement of the complex impedance response and accompanied power absorption P(ω) in the radio-
frequency and microwave ranges represents a most popular experimental method to investigate pinning mechan-
isms and the vortex dynamics in type-II superconductors. In the theory, the pinning potential (PP) well for a vor-
tex must be a priori specified in order to subsequently analyze the measured data. We have theoretically solved
the inverse problem at T = 0 K and exemplify how the coordinate dependence of a PP can be determined from a set
of experimental curves P(ω|j0) measured at subcritical dc currents 0 < j0 < jc under a small microwave excitation
j1 << jc with frequency ω. We furthermore elucidate how and why the depinning frequency ωp, which sepa-
rates the non-dissipative (quasi-adiabatic) and the dissipative (high-frequency) regimes of small vortex oscil-
lations in the PP, is reduced with the increase of j0. The results can be directly applied to a wide range of conven-
tional superconductors with a PP subjected to superimposed dc and small microwave ac currents at T << Tc.
PACS: 74.25.Wx Vortex pinning;
74.25.F– Transport properties;
74.25.nn Surface impedance.
Keywords: microwave power absorption, pinning potential, Abrikosov vortices.
1. Introduction
One of the most popular experimental methods for the
investigation of the vortex dynamics in type-II supercon-
ductors 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 me-
chanisms, the vortex dynamics, and accompanied with it
dissipative processes in a superconductor. It should be
noted that this information can not be extracted from the dc
resistivity data obtained in the steady state regime when
pinning in the sample is strong. In fact, in the last case
when the critical current densities cj are rather large, the
realization of the dissipative mode, in which the flux-flow
resistivity fρ can be measured, requires 0 .cj j This is
commonly accompanied by a non-negligible electron
overheating in the sample [2,3] which changes the value of
the desired fρ . At the same time, measurements of the
absorbed by vortices power from an ac current with the
amplitude 1 cj j allow one to determine fρ at a dissip-
ative power 1P
2
1f jρ∼ which can be many orders of
magnitude less than 0P
2
0 .f jρ∼
Consequently, measure-
ments of the complex ac response versus frequency ω
practically probe the pinning forces in the absence of over-
heating effects, otherwise unavoidable at overcritical
steady-state dc current densities.
At last years, the appearance of experimental works uti-
lizing the usual four-point scheme [4], strip-line coplanar
waveguides (CPWs) [5], the Corbino geometry [6,7], or
the cavity method [8] to investigate the microwave vortex
response in as-grown thin-film superconductors (or in
those containing some nano-tailored pinning potential (PP)
landscape) reflects the explosively growing interest to the
subject. In fact, such artificially fabricated pinning nano-
structures 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. Therefore, an
Determination of the coordinate dependence of a pinning potential from the microwave experiment with vortices
Low Temperature Physics/Fizika Nizkikh Temperatur, 2013, v. 39, No. 2 163
approach how to reconstruct the form of the PP experimen-
tally ensued in the sample is of great demand for both, ap-
plication-related and fundamental reasons. 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 was reported in Ref. 9. Similar problems to recon-
struct specific form of a potential subjected to superim-
posed constant and small alternating signals arise not only
in the vortex physics but also in a number of other fields.
Mainly due to the closest mathematical analogy we would
like to mention the Josephson junction problem wherein a
plenty of non-sine forms of the current-phase relation is
known to occur [11] and which could in turn benefit from
our results reported here.
Turning back to the development of theory of our prob-
lem, the very early model to describe the absorbed power
by vortices refers to the work of Gittleman and Rosenblum
(GR) [12] where a small ac excitation of vortices in the
absence of a dc current has been considered. The GR re-
sults have been obtained at = 0T K in the linear approxi-
mation for the pinning force. We will present briefly their
results in the present work since the subsequent description
of our new results requires these as the essential back-
ground. Later on, the theory accounting also for the vortex
creep at non-zero temperature in a one-dimensional cosine
PP has been extended by Coffey and Clem (CC) [13].
However, this theory has been developed for a small mi-
crowave current in the absence of a dc. Recently, the CC
results have been substantially generalized by us [14,15]
for a two-dimensional cosine washboard pinning potential
(WPP). The washboard form of the PP has enabled an ex-
act theoretical description of the two-dimensional aniso-
tropic nonlinear vortex dynamics for any arbitrary values
of ac and dc amplitudes, temperature, the Hall constant,
and the angle between the transport current direction with
respect to the guiding direction of the WPP. Among other
nontrivial results obtained, an enhancement [14] and a sign
change [15] in the power absorption for 0 cj j have been
predicted. Whereas the general solution of the problem in
Refs. 14, 15 has been obtained in terms of a matrix contin-
ued fraction and is suitable for the analysis mainly in the
form of figure data due to a large number of variable pa-
rameters, an analytical implementation of the solution at
= 0T K, 0 < cj j , and 1 0j → has been performed in
Refs. 16, 17, taking the anisotropy of the vortex viscosity
and an arbitrary Hall constant also into account.
In the present work, we report the possibility of recon-
struction the coordinate dependence of a PP if a set of
0 ( )ωP curves has been measured at different dc current
amplitudes in the whole range 00 cj j≤ at a small mi-
crowave amplitude 1 0.j → Whereas a preliminary com-
munication on this matter can be found in Ref. 18, here we
provide a detailed description of the PP reconstruction pro-
cedure. The geometry of the problem implies a standard
four-point microstrip bridge of a thin-film superconductor
placed into a small perpendicular magnetic field with a
magnitude 2cB B at .cT T The sample is assumed to
have at least one pinning site and dc and ac currents are
directed collinearly. The theoretical treatment of the prob-
lem is detailed next.
2. Dynamics of pinned vortices on a small microwave
current
The GR model [12] considers oscillations of damped
vortex in a parabolic PP. They measured the power absorp-
tion 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 equation for a vortex moving with the
velocity ( )tv along the x axis
= ,p Lx k x fη + (1)
where x is the vortex displacement, η is the vortex viscosi-
ty, pk is the constant which characterizes the restoring force
pf in the PP well 2( ) = (1/2)p pU x k x and =pf
/ = .p pdU dx k x= − − In Eq. (1) 0 1= ( / ) ( )Lf c j tΦ is the
Lorentz force acting on a vortex, 0Φ is the magnetic flux
quantum, c is the speed of light, and 1 1( ) = exp ( )j t j i tω is
the density of a small microwave current with the amplitude
1.j Looking for the solution of Eq. (1) in the form
( ) = exp ( ),x t x i tω where x is the complex amplitude of
the vortex displacement, one immediately gets ( ) =x t
( )i x t= ω and
0 1( / )
= ,
p
c j
x
i
Φ η
ω+ω
(2)
where /p pkω ≡ η is the depinning frequency. To calculate
the magnitude of the complex electric field arising due to
the vortex on move, one takes = / .E Bx c Then
1
1( ) = ( ) .
1 /
f
p
j
E Z j
i
ρ
ω ≡ ω
− ω ω
(3)
Here 2
0= /f B cρ Φ η is the flux-flow resistivity and
( ) /(1 / )f pZ iω ≡ ρ − ω ω is the microwave impedance of the
sample.
In order to calculate the power P absorbed per unit vo-
lume and averaged over the period of an ac cycle, the stan-
dard relation = (1/2)Re( )EJ∗P is used, where E and J
are the complex amplitudes of the ac electric field and cur-
rent density, respectively. The asterisk denotes the com-
plex conjugate. Then, from Eq. (3) it follows
2
12
1 2
1 1( ) = Re ( ) = .
2 2 1 ( / )
f
p
j
Z j
ρ
ω ω
+ ω ω
P (4)
V.A. Shklovskij and O.V. Dobrovolskiy
164 Low Temperature Physics/Fizika Nizkikh Temperatur, 2013, v. 39, No. 2
For the subsequent analysis, it is convenient to write out
real and imaginary parts of the impedance
= Re Im ,Z Z i Z+ namely
2 2
( / )
Re ( ) = , Im ( ) = .
1 ( / ) 1 ( / )
f f p
p p
Z Z
ρ ρ ω ω
ω ω
+ ω ω + ω ω
(5)
The frequency dependences (5) are plotted in dimensionless
units / fZ ρ and / pω ω in Fig. 1 (see the curve for 0 = 0).j
From Eqs. (1), (2), and (4) it follows that pinning forces do-
minate at low frequencies ( ),pω ω where ( )Z ω is nondis-
sipative with 2Re ( ) ( / ) ,pZ ω ≈ ω ω whereas at higher fre-
quencies ( pω ω ) frictional forces dominate and ( )Z ω is
dissipative with 2Re ( ) [1 ( / ) ].f pZ ω ≈ ρ − ω ω 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. (2) where 1/x ω∼ for ;pω ω this is ac-
companied, however, with the independence of the vortex
velocity of ω in this regime in accordance with Eq. (3).
3. Influence of a dc current on the depinning frequency
When an arbitrary dc current is superimposed on a
small microwave signal, the GR model can be generalized,
for an arbitrary PP. For definiteness sake, let us consider a
subcritical dc current with the density 0 < ,cj j where cj
is the critical current density in the absence of a microwave
current. Our aim now is to determine the changes in the PP
parameters the superimposition of the dc current leads. In
the presence of 0 0,j ≠ the effective PP becomes ( )U x ≡
0( ) ,pU x xf≡ − where ( )pU x is the x-coordinate depen-
dence of the PP when 0 = 0.j Note also that 0 < cf f
where 0f and cf are the Lorentz forces which correspond
to the current densities 0j and ,cj respectively.
In the presence of a dc current, the equation of motion
for a vortex has the form
( ) = ( ) ,pt f t fη +v (6)
where 0( ) = ( / ) ( )f t c j tΦ is the Lorentz force with
0 1( ) = ( ),j t j j t+ where 1 1( ) = exp( )j t j i tω , and 1j is the
amplitude of a small microwave current. Due to the fact
that 0 1( ) = ( ),f t f f t+ where 0 0 0= ( / )f c jΦ and 1( ) =f t
0 1( / ) ( )c j t= Φ are the Lorentz forces for the subcritical dc
and microwave currents, respectively, one can naturally
assume that 0 1( ) = ( ),t t+v v v where 0v does not depend
on the time, whereas 1 1( ) = exp( ).t i tωv v In Eq. (6) the
pinning force is = ( )/ ,p pf dU x dx− where ( )pU x is a PP
of some form. Our aim is to determine ( )tv from Eq. (6)
which, taking into account the considerations above, ac-
quires the following form:
0 1 0 1[ ( )] = ( ).pt f f f tη + + +v v (7)
Let us consider the case when 1 = 0.j If 0 < ,cj j i.e.,
0 < ,cf f where cf is the maximal value of the pinning
force, then 0 = 0,v i.e., the vortex is in rest. As it is seen
from Fig. 2 the rest coordinate 0x of the vortex in this case
depends on 0f and is determined from the condition of
equality to zero of the effective pinning force
0( ) = ( )/ = ( ) ,pf x dU x dx f x f− + which reduces to the
equation 0 0( ) = 0,pf x f+ or
0 = 0
( )
= | ,p
x x
dU x
f
dx
(8)
the solution of which is the function x0(f0).
Let us now add a small oscillation of the vortex in the
vicinity of 0x under the action of the small external alter-
nating force 1( )f t with the frequency ω . For this we ex-
pand the effective pinning force ( )f x in the vicinity of
Fig. 1. The frequency dependences of real (a) and imaginary (b)
parts of the ac impedance calculated for a cosine pinning poten-
tial ( ) = ( /2)(1 cos )p pU x U 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).
1.0
0.5
0
0.6
0.3
0
0.01 0.1 1 10
� �/ p
R
e
Z
(
)/
�
�
f
Im
Z
(
)/
�
�
f
(a)
(b)
j j =0/ c
0.
99
9
0.
99
7
0.
99
0.
96 0.
9
0.
7 0.
3
0
0.
99
9
0.
99
7
0.
99
0.
96 0.
9
0.
7 0.
3
0
0.4
0.2
0
–0.2
–0.4
–0.2 0 0.2 0.4 0.6
U xp( )
x0 x01 x02 x03
x, a
U x2( )
�
U x1( )
�
U x3( )
�
U
x
U
(
),
p
Fig. 2. Modification of the effective PP 0( ) ( )i p iU x U x f x≡ −
where ( ) = ( /2)(1 cos )p pU x U kx− is the WPP, with the gradual
increase of f0 such as 0 01 02 030 = < < = ,cf f f f f i.e., a vortex
is oscillating in the gradually tilting pinning potential well in the
vicinity of the rest coordinate x0i.
Determination of the coordinate dependence of a pinning potential from the microwave experiment with vortices
Low Temperature Physics/Fizika Nizkikh Temperatur, 2013, v. 39, No. 2 165
0=x x into a series in terms of small displacements
0u x x≡ − which gives
0 0 0( ) ( ) ( )f x x f x f x u′− + +… (9)
Then, taking into account that 0( ) = 0f x and 0( ) =f x′
0( ),pU x′′= Eq. (7) acquires the form
1 1= ,pu k u fη + (10)
where 0 0( ) = ( )p pk x U x′′ is the effective constant characte-
rizing the restoring force ( )f u at small oscillations of a
vortex in the effective PP ( )U x close by 0 0( )x f and
1 = = .u i uωv Equation (10) for the determination of 1v is
physically equivalent to GR Eq. (1) with the only distinc-
tion that the vortex depinning frequency /p pkω ≡ η now
depends on 0f through Eq. (8), i.e., on the dc transport
current density 0j . Thereby, all the results of the previous
section [see Eqs. (2)–(5)] can be repeated here with the
changes x u→ and .p pω → ω
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. As usually [13–15], we take a cosine
WPP of the form ( ) = ( /2)(1 cos ),p pU x U kx− where
= 2 /k aπ and a is the period; though any other non-
periodic PP can also be used. Then, as it has been pre-
viously shown for the cosine WPP [16], 0( / ) =p cj jω
2
01 ( / )p cj j= ω − and the appropriate series of the curves
0( | )jωP is plotted in Fig. 1. As evident from the figure
data, the curves shift to the left with the increase of 0.j
The reason for this is that with the increase of 0j the PP
well while tilted is broadening, as evident from Fig. 2.
Thus, during the times shorter than = 1/p pτ ω (i.e., for
> )pω ω a vortex can no longer non-dissipatively oscillate
in the PP well. As a consequence, the enhancement of
Re ( )Z ω 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 .pω For a fixed frequency and differ-
ent 0j , real part of ( )Z ω always acquires larger values for
larger 0j , whereas the maximum in imaginary part of
( )Z ω corresponds precisely to the middle point of the non-
linear transition in Re ( ).Z ω It should be noted that even
for = 0T K the dissipation, though is small, still remains
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 how to
reconstruct the coordinate dependence of a PP experimen-
tally ensued in the sample, on the basis of microwave pow-
er absorption data in the presence of a subcritical dc trans-
port current. It will be shown that from the dependence of
the depinning frequency 0( )p jω as a function of the dc
transport current 0j one can determine the coordinate de-
pendence of the PP ( ).pU x The physical background for
the possibility to solve such a problem is Eq. (8) which
gives the correlation of the vortex rest coordinate 0x with
the value of the static force 0f acting on the vortex and
arising due to the dc current 0.j
4.1. General scheme of the reconstruction
From Eq. (8) it follows that while increasing 0f from
zero to its critical value cf one in fact “probes” all the
points of the dependence ( ).pU x Taking the x0-coordinate
derivative in Eq. (8), one obtains
0
0 0 0
1 1= = ,
( ) ( )p p
dx
df U x k x′′
(11)
where the relation 0 0( ) = ( )pU x k x′′ has been used [see
Eq. (10) and the text below]. By substituting 0 0 0= ( ),x x f
Eq. (11) can be rewritten as 0 0 0 0/ = 1/ [ ( )],pdx df k x f and
thus,
0
0 0
1= .
( )p
dx
df fηω
(12)
If the dependence 0( )fω has been deduced from the expe-
rimental data, i.e., fitted by a known function, then Eq.
(12) allows one to derive 0 ( )x f by integrating
0
0 0
0
1( ) = .
( )
f
p
dfx f
fη ω∫ (13)
Then, having calculated the inverse function 0 0( )f x to
0 0( )x f and using the relation 0 0 0( ) = ( ),pf x U x′ i.e.,
Eq. (8), one finally obtains
0 0 0
0
( ) = ( ).
x
pU x dx f x∫ (14)
4.2. Example procedure to reconstruct a WPP
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 ( )ωP has been measured for a set of
subcritical dc currents 0.j Then for determinacy, let us
imagine that each i-curve of 0( | )jωP like those shown in
Fig. 1 has been fitted with its fitting parameter pω so that
one could map the points 0[( / ) , ( / ) ],p p i c ij jω ω as shown
by triangles in Fig. 3.
We fit the data in Fig. 3 by the function / =p pω ω
2
01 ( / )cj j= − and then substitute it into Eq. (13) from
which one calculates 0 0( ).x f In this case, the function
has a simple analytical form, namely 0 0( ) =x f
0( / ) arcsin ( / )c p cf k f f= . Evidently, the inverse to it func-
V.A. Shklovskij and O.V. Dobrovolskiy
166 Low Temperature Physics/Fizika Nizkikh Temperatur, 2013, v. 39, No. 2
tion is 0 0 0( ) = sin( / )c p cf x f x k f with the period
= 2 /c pa f kπ (see also Fig. 4). By taking the integral (14) one
finally gets ( ) = ( /2)(1 cos ),p pU x U kx− where = 2 /k aπ
and 2= 2 / .p c pU f k
5. Conclusion
In this paper we have shown how from data on the mi-
crowave power absorption by vortices in the presence of a
subcritical dc transport current the coordinate dependence
of the PP in the sample can be determined. The proposed
procedure can be used at cT T and implies a small mi-
crowave current density 1 .cj j In order to keep the
transport current distribution in the sample as homogene-
ous as possible the pinning potential is assumed to be not
very “strong” in the sense that vortex pinning is caused by,
e.g., the vortex length reduction rather than the supercon-
ducting order parameter suppression. Though the potential
reconstruction scheme has been exemplified for a cosine
WPP, i.e., for a periodic and symmetric PP, the elucidated
procedure in the general case 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 pro-
cedure under the dc current reversal, i.e., two times: for 0j+
and 0.j− The scheme to reconstruct the WPP ( )pU x from
the experimental data on 0( )p jω can be briefly summarized
as follows: a) by using the data 0( / ( ))p jω ωP to find
0( )p jω ; b) taking the integral (13) to calculate 0 0( );x f
c) then from 0 0( )x f to find the inverse to it function
0 0( );f x and finally d) to integrate 0 0( )f x and by using
Eq. (14) to recover the PP ( ).pU x
Theoretically, we have limited our consideration by
= 0T K, 0 < ,cj j and 1 0j → because this has 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, i.e., on conventional thin-film superconduc-
tors (e.g., Nb, NbN) at .cT T 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 depth pU 1000–5000 K [19,20]. It should be
stressed that due to the universal form of the dependences
0( | ),jωP the depinning frequency pω plays a role of the
only fitting parameter for each of the curves 0( | ),jωP
thus fitting of the measured data seems to be uncompli-
cated. However, one of most crucial issues for the experi-
ment is to superimpose adequately 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 val-
ues of the depinning frequency in the absence of a dc cur-
rent and a temperature of about 0.6 cT are 7pω ≈ GHz for
a 20 nm-thick [7] and a 40 nm-thick [8] 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 figure data
have been provided here for a cosine WPP as for the most
commonly used potential, the coordinate dependence can
be reconstructed for not only periodic potentials. In fact,
single PP wells, like one used in Ref. 5, can also be proven
in accordance with the provided approach. Second, if a PP
is periodic, however, it should be noted that the theoretical
consideration here has been performed in the single-vortex
approximation, i.e., is valid only at small magnetic fields
2 ,cB B 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.
Fig. 3. The pinning potential reconstruction procedure: step 1.
A set of 0[( / ) ,( / ) ]p p i c ij jω ω points ( ) has been deduced from
the supposed measured data and fitted as 2
0/ = 1 ( / )p p cj jω ω −
(solid line). Then by Eq. (13) 0 0 0( ) = ( / )arcsin ( / )c p cx f f k f f
(dashed line).
1.5
1.2
0.9
0.6
0.3
0
0 0.2 0.4 0.6 0.8 1.0
supposed rf data
fit / = [1 – ( / ) ]� �p p cj j0
2 1/2�
k x f fp c0 0( )/ by Eq. (13)
2 ( )/ by Eq. (14)U x Up p
f x f0 0( )/ c
2
1
0
–1
0 0.2 0.4 0.6 0.8 1.0
x, a
Fig. 4. The pinning potential reconstruction procedure: step 2.
The inverse function to 0 0( )x f is 0 0 0( ) = sin( / )c p cf x f x k f
(dashed line). Then by Eq. (14) ( ) = ( /2)(1 cos ),p pU x U kx− is
the PP sought (solid line).
Determination of the coordinate dependence of a pinning potential from the microwave experiment with vortices
Low Temperature Physics/Fizika Nizkikh Temperatur, 2013, v. 39, No. 2 167
Finally, the results can be directly verified in, e.g., the
microstrip geometry for combined microwave and dc elect-
rical transport measurements. Whereas experimental works
on this matter have started to appear [4–7], we hope to
have stimulated further developments in the field. Fur-
thermore, 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 the re-
construction can also be adopted for that case.
Acknowledgments
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
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