We have measured the amplitude and the phase of an electromagnetic (EM) field
radiated from superconductor (amorphous ZrTiCuNiBe alloy) in the mixed state
due to interaction of the flux lattice with an elastic wave. The results
undoubtedly point to an essential contribution of a surface pinning into the
flux lattice dynamics. We propose a model that describes radiation of EM field
from superconductors with non-uniform pinning. The model allows to reconstruct
the viscosity and the Labush parameters from the experimental data. The
behavior of the Labush parameter can be qualitatively explained in terms of the
collective pinning theory with the allowance of thermal fluctuations.Comment: 4 pages, 4 figure

A. A. Abrikosov

A. L. Gaiduk

D. Dominguez

D. V. Fil

H. Haneda

V. D. Fil

W. L. Johnson

Yu. A. Avramenko

English

arXiv

Crossref

Surface pinning in an amorphous ZrTiCuNiBe alloy

We have measured the amplitude and the phase of an electromagnetic (EM) field radiated from a superconductor (amorphous ZrTiCuNiBe alloy) in the mixed state due to the interaction of the flux lattice with an elastic wave. The results point to an essential contribution of a surface pinning into the flux lattice dynamics. We propose a model that describes radiation of the EM field from superconductors with nonuniform pinning. The model allows us to reconstruct the viscosity and the Labusch parameters from the experimental data. The behavior of the Labusch parameter can be qualitatively explained in terms of the collective pinning theory with the allowance of thermal fluctuations

Fil, V. D.

Fil, D. V.

Avramenko, Yu. A.

Gaiduk, A. L.

Johnson, W. L.

Caltech Authors - Main

Surface pinning in an amorphous ZrTiCuNiBe alloy
V. D. Fil,1 D. V. Fil,2,3 Yu. A. Avramenko,1 A. L. Gaiduk,1 and W. L. Johnson4
1B. Verkin Institute for Low Temperature Physics and Engineering, National Academy of Sciences of Ukraine, Lenin av. 47,
Kharkov 61103, Ukraine
2Institute for Single Crystals, National Academy of Sciences of Ukraine, Lenin av. 60, Kharkov 61001, Ukraine
3Ukrainian State Academy of Railway Transport, Feyerbakh Sq. 7, Kharkov 61050, Ukraine
4California Institute of Technology, Pasadena, California 91125, USA
sReceived 23 December 2004; published 24 March 2005d
We have measured the amplitude and the phase of an electromagnetic sEMd field radiated from a supercon-
ductor samorphous ZrTiCuNiBe alloyd in the mixed state due to the interaction of the flux lattice with an elastic
wave. The results point to an essential contribution of a surface pinning into the flux lattice dynamics. We
propose a model that describes radiation of the EM field from superconductors with nonuniform pinning. The
model allows us to reconstruct the viscosity and the Labusch parameters from the experimental data. The
behavior of the Labusch parameter can be qualitatively explained in terms of the collective pinning theory with
the allowance of thermal fluctuations.
DOI: 10.1103/PhysRevB.71.092504 PACS numberssd: 74.25.Qt, 74.70.Ad
Soft type-II superconductors were studied intensively dur-
ing last four decades with the goal to investigate various
aspects of vortex matter dynamics ssee the comprehensive
review by Brandt1d. Nevertheless, many questions that re-
quire further investigation remain. Among these questions is
the problem of relation between the surface and the bulk
pinning. Up to now the dynamics of the vortex state in such
an exclusively nonuniform situation was studied by the sur-
face impedance method.2 In this paper, on the example of
amorphous Zr41.2Ti13.8Cu12.5Ni10Be22.5 alloy we demonstrate
abilities of a method based on excitation of vortex lattice
oscillations by a high-frequency sound wave.
The essence of the method is the following. A supercon-
ductor situated in the lower half space sz,0d is subjected by
a constant magnetic field H i z. A transverse elastic wave
propagating along H and polarized in x direction produces
transverse swith respect to Hd oscillations of the vortex lat-
tice caused by pinning and viscous friction, and, conse-
quently, induces an electromagnetic sEMd field. An antenna
receives the EM field swith Ey and Hx componentsd radiated
through the elastically free surface of the sample sthe surface
perpendicular to the direction of propagation of the elastic
waved. While a similar experimental setup was already ap-
plied for the study of type-II superconductors,3,4 the key
point is measuring both the amplitude and the phase of the
EM field smore accurately, the changes of these quantitiesd.
In the uniform case and in the local limit sq! l−1, q is the
wave number and l is the mean-free pathd the components of
the EM field at z=0 are given by a simple expression,5,6
Hx = Ey =
u˙s0d
c
H
k2
q2 + k2
, s1d
where uszd=u0 cos qzeivt is the displacement in the elastic
wave, k2 is the square of the complex wave number of the
EM field in a conductor, and c is the light velocity.
Equation s1d is applicable for the normal as well as the
superconducting state of the metal. In the normal state k2
=kn
2
=4pivs0 /c2 ss0 is the static conductivity in the normal
stated is imaginary valued quantity. In the superconducting
state at small magnetic fields sH,Hc1d k2=ks
2
=lL
−2 slL is the
London penetration lengthd is real valued. At such fields the
phase of the EM signal w is a constant and we use its value
as the reference point. In the well-developed Shubnikov state
k2=km
2
=4psivh+aLd /H2, where h.0 is the viscosity pa-
rameter and aL.0 is the Labusch “spring” parameter. As
follows from Eq. s1d, the phase is positive at all H.Hc1 and
it approaches wn,90° sthe phase in the normal stated at H
→Hc2.
The initial goal of our experiment was to obtain experi-
mentally the dependencies hsHd and aLsHd using Eq. s1d, as
it was done before for MgB2.7 We use the working frequency
v /2p,55 MGz. Details of the measuring procedure are de-
scribed in Ref. 8. The results of our first experiments are
presented in Fig. 1.
Prior to discussing these results let us give the values of
parameters for the material under study. The dc resistivity
r<200 mV cm is practically temperature independent. The
Alfer-Rubin effect9 sthe dependence of the sound attenuation
coefficient on the magnetic fieldd allows us to determine the
parameter b= uq2 /kn
2u=91±1, in excellent coincidence with
the value of r measured sq=1400 cm−1, see Ref. 10d. The
FIG. 1. Amplitude and phase of the EM field measured for the
sample with imperfect surface. Inset—behavior of uEu /H near Hc2
f1—experiment, 2—computed from Eq. s7d with aL
ef f ssee belowdg.
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Hall constant, measured at room temperature, is positive and
very small: RH=1/enc= s3.2±0.4d310−25 cgs units, that
yields a rather high density of the carriers n<2.231023
cm−3. Taking m=me, we obtain the relaxation time t<0.8
310−16 s, and for the Fermi velocity vF,108 cm/s we find
l,10−8 cm. It was shown before11 that the alloy under study
belongs to the family of weak-coupling superconductors with
the standard BCS energy gap Ds0d<1.75Tc sTc<0.85 Kd. In
the dirty limit the formula for the penetration depth12 can be
rewritten as lL
−2
=kn
2sD / ivdtanhsD /2Td, that yields lL=3
310−4 cm at T=0.4 K. The coherence length, calculated
from Hc2, is js0.4 Kd=1.4310−6 cm.
As follows from Eq. s1d, for b given above one could
expect wn<90° that contradicts the results presented in Fig.
1 swn<120° d. Let us show that this discrepancy can be ac-
counted for a lowering of the conductivity near the surface of
the sample caused by defects.
We imply the following model dependence of the conduc-
tivity on z: s0szd=s0vf1− p expsz /zsdg swith pł1 and zs
.0d. Then, in the normal state, the electrodynamic equation
for the EM field has the form6 fcompare with Eq. s4d belowg
d2E˜
dz2
− aszdE˜ = aszdcosszd , s2d
where aszd=af1− p expsz /z0dg, a=knv
2 /q2, knv
2
=4pivs0v /c2,
z0=qzs. Here we use the dimensionless variables E˜
=Ec / sivu0Hd and z=qz. The boundary conditions for Eq. s2d
are that uE˜ szdu is finite at all z and udE˜ /dzuz=0=0. The latter
condition is due to continuity of the EM field on the
conductor-vacuum interface and is valid with the accuracy
d /lEM &10−4 sd is the skin depth and lEM is the wavelength
of the EM field in vacuumd.
The solution of Eq. s2d can be expressed through the
Bessel functions of the first kind of complex order n
=2z0˛a on the complex variable tszd=2z0˛pa expsz /2z0d.
The field at the conductor-vacuum interface is
E˜ sz = 0d = SUdJnftszdgdz Uz=0D
−1E
−‘
0
dzJnftszdgaszdcos z .
s3d
One can evaluate Eq. s3d using the expansion of Jnstd in
series in t and integrating each term analytically. For the case
of interest it is enough to take into account the first two terms
of the expansion.
Analysis of Eq. s3d shows that the phase wn increases
under increasing p. In our case of the extremely dirty con-
ductor and for z0,1 szs<7 mmd we find that a rather small
lowering of the surface conductivity sp,0.1d results in in-
creasing of wn up to 120°.
Removing the layer ,50 mm from the surface of the
sample by additional polishing with a fine powder we man-
aged to eliminate the effect caused by the lowering of surface
conductivity. The results of measurements sat various Td are
shown in Fig. 2. One can see that in this case wn<90°.
The main peculiarity of the dependencies presented in
Fig. 2 is that the phase w becomes negative at intermediate
magnetic fields. This effect cannot be described by Eq. s1d
under any reasonable variation of h and aL with H.
In what follows we will argue that the effect observed can
be explained by nonuniform pinning near the surface of the
sample that takes place even for uniform s0 sand, conse-
quently, for uniform hd.
The EM field in the mixed state is described by the sys-
tem of equations, namely, the Maxwell equation, the matter
equation, that determines the value of the current, and the
equation of motion for the vortex lattice,6
d2E
dz2
=
4piv
c2
j = ks2SE + 1
c
u˙v 3 HD , s4d
1
c
j 3 H + ivhsu − uvd + aLsu − uvd = 0, s5d
where uv is the displacement of the vortex lattice. Here we
neglect the normal component of the current.
Nonuniform pinning can be modeled by the z-dependent
Labusch parameter. We specify this dependence as aLszd
=aLv+aLs expsz /zpd. Using Eqs. s4d and s5d we obtain the
equation for the EM field in the mixed state. At the magnetic
field that satisfies the inequality ks
2sH2 /4pd@ uivh+aLu, this
equation coincides in form with Eq. s2d with the same
boundary conditions and the quantity aszd defined as
aszd =
ivh + aLv
q2
H2
4p
+
aLs
q2
H2
4p
exp
z
z0
; A + C exp
z
z0
, s6d
where z0=qzp.
Taking into account the first two terms in the expansion of
the Bessel function we obtain from Eq. s3d the approximate
expression
FIG. 2. Amplitude ssolid curves 1, 2, 3d and phase sdotted
curves 1a, 2a, 3ad of the EM field measured for the sample with a
perfect surface at T=0.4,0.69,0.8 K, respectively sfor T=0.8 K the
results for the normal state are not shownd. Circles—the depen-
dence uEsHdu measured under decreasing of H. Inset—the same de-
pendencies in the interval of small magnetic fields sall notations are
the same as in the main plotd.
BRIEF REPORTS PHYSICAL REVIEW B 71, 092504 s2005d
092504-2
E˜ s0d =
˛A A
1 + A
+ C
z0
1 + z0
2
˛A + Cz0
. s7d
In contrast to Eq. s1d, that requires h,0 s“negative viscos-
ity”d to describe negative phases of the EM signal, Eq. s7d
may yield negative w at positive h.
Equation s7d can be used to reconstruct the dependencies
hsHd and aLsHd from the experimental data presented in Fig.
2, if the parameters z0 and aLv /aLs are specified. These pa-
rameters are not known, but there are limitations on the
choice of them. We require the functions hsHd and aLsHd to
be smooth and real valued. Numerical analysis shows that
these requirements satisfy for aLv!aLs sit means that the
bulk pinning can be neglectedd and z0&0.5. At such condi-
tions the amplitude and the phase of the EM signal are, in
fact, determined by two parameters: h and z0aLs, and the
procedure of reconstruction of hsHd and z0aLssHd can be
done without any additional assumption.
The dependencies hsHd are shown in Fig. 3. In a rather
wide interval of H these dependencies are linear with the
slope S~Hc2 sFig. 3, insetd and are described by the formula
hsHd= s2±0.05dHHc2s0 /c2. Such type of dependencies are
typical for the mixed state and we consider it as a confirma-
tion of the model proposed.
The dependencies z0aLssHd are shown in Fig. 4. They
demonstrate two peculiarities: the strong asymmetry of the
function aLssHd at T=0.4 K and a rather slow decreasing of
aLs under lowering of Hc2 susually aL~Hc2
n
, where n=2–3d.
We will show below that such a behavior can be explained at
least semiquantitatively in the framework of the accepted
theories of the vortex state dynamics.
Down to the lowest temperatures available in our experi-
ment we did not find ssee Fig. 2d any signs of freezing of the
magnetic flux sthe irreversibility lined. It means that the pin-
ning is quite weak and it can be described by the collective
pinning sCPd theory.13 The CP theory contains an additional
parameter connected with quenched disorder of the vortex
lattice caused by a random pinning potential. It is convenient
to choose a field Hsvł0.5Hc2 as such a parameter.14 This
field separates the regime of single vortex pinning sSVPd
from the regime of bundle vortex pinning sBVPd. In the SVP
regime sHłHsv or HøHc2−Hsvd the value of aL is given
by the expression aL<C66av
−2sHsvd, where C66
= fF0H / s8plLd2gf1− sH /Hc2dg2 is the shear modulus of the
vortex lattice, avsHd=˛F0 /H is the vortex lattice constant,
and F0 is the quantum of magnetic flux. In the BVP regime
sHsvłHłHc2−Hsvd the Labusch parameter is aL
<C66Rc
−2sHd, where the collective pinning radius Rc is given
by the formula15
RcsHd < avsHdexpH12FS HsHc2 − HdHsvsHc2 − HsvdD3/2 − 1GJ . s8d
We note that Eq. s8d determines the dependence of aLsHd
almost symmetric relative to the H=0.5Hc2 axis.
One can see that the scale of aL is determined mainly by
the value of C66. But one finds from the data in Fig. 4 that
even for maximum z0 sz0=0.5, and zp=3.5 mmd the scale of
measured aL exceeds significantly the estimate that follows
from the value of lL given above. Probably, it is connected
with the fact that the relation for lL overestimates this quan-
tity. Unfortunately, in our experiment we cannot measure the
field Hc1 directly with an accuracy sufficient to refine the
value of lL. Assuming z0=0.5, we model aLsHd, taking C66
in six times larger than it follows from the estimate of Ref.
12 and implying their standard temperature dependence fthat
corresponds to lLs0.4 Kd=1.2310−4 cmg.
Since the system is in a weak pinning regime we must
take into account temperature fluctuations. To evaluate the
effect of thermal fluctuations one should add a random force
fLsTd into Eq. s5d.16 Strictly speaking, the returning force
jH /c= sH2 /4pd]2uv /]z2 in Eq. s5d has to be found from a
joint solution of the system s4d and s5d, but for the estimates
one can replace ]2 /]z2 with −q2. In this case Eq. s5d can be
rewritten in the form of an equation of diffusion of the
Brownian particle
h˜
]w
]t
= −
]
]w
sV˜ 0 + V˜u + V˜ pd + fLsTd , s9d
where w=u−uv, the “tildes” indicate that the corresponding
quantities are given per one diffusing “particle:” y˜
=yF0Lc /H sLc is the collective pinning lengthd, V0
= sq2H2 /8pdw2, Vu=−sq2H2 /4pduw, and Vp is the pinning
potential. Taking into account equal probabilities of diffuse
FIG. 3. Values of vh reconstructed from the data in Fig. 2 using
Eq. s7d. Inset—the slope S of vh for intervals of linear increase of
vh with H scirclesd; solid line—linear fit.
FIG. 4. Solid curves—values of z0aLs reconstructed from the
data in Fig. 2 using Eq. s7d at z0=0.5 and z0=0.1 sonly for T
=0.4 Kd; dashed curves—computed dependence aL
ef fsHd at z0=0.5
ssee textd. 1 and 1a curves are for T=0.4 K; 2 and 2a for T
=0.69 K; and 3 and 3a for T=0.8 K.
BRIEF REPORTS PHYSICAL REVIEW B 71, 092504 s2005d
092504-3
motion of a vortex in opposite directions we model Vp by a
three-well potential
Vp =
aL
2 5sw + dd
2 w ł − d/2
w2 uwu ł d/2
sw − dd2 w ø d/2 6 , s10d
where d is the distance between the minima of the pinning
potential. The linearized Fokker-Planck equation that corre-
sponds to Eq. s9d with the pinning potential s10d has the
exact solution in terms of hypergeometric functions. At
aL / fq2sH2 /4pdg!1 the following simple estimate for the
averaged displacement of the vortex lattice is found kuvl
<u ·4psivh+aL
ef fd / sq2H2d, where aL
ef f
=aLf1− s4/˛pdce−c
2
g
and c2= sq2HF0Lc /2pTdd2. One can see from the compari-
son of this formula with Eq. s5d that the thermal fluctuations
reduce the effective Labusch parameter, but the viscosity pa-
rameter remains unchanged.
For the general case the dependencies aL
ef fsHd obtained
from the solution of the Fokker-Planck equation are shown
in Fig. 4. We use the following parameters for the computa-
tions. To achieve semiquantitative agreement between the
theoretical results and the experimental data at T=0.4 K we
set Hsvs0.4d<0.15Hc2s0.4d and ds0.4d<5310−7 cm sas ex-
pected, d is of order of jd. We assume that the temperature
dependence of Hsv is determined by the dTc pinning:
Hsv /Hc2~ f1− sT /Tcd2g−1/3.14 It looks quite reasonable, be-
cause there are inclusions of another superconducting phase
with higher Tc on the surface of our sample.11 The tempera-
ture dependence dsTd~jsTd is implied. We set Lc=avsHd.
One can see from Fig. 4 that the computed temperature
and field dependencies of aL are in qualitative agreement
with the experimental results.
It is interesting to note that if we substitute aL
ef f into Eq.
s7d we obtain uEu /H that has a local minimum near Hc2. Such
a minimum is observed sometimes in our experiment sFig. 1,
insetd as well as in Ref. 3. Physically this minimum, as well
as negative w, is caused by that the surface layer with a
reduced penetration depth screens the EM field radiated from
deeper regions of the sample. Reminiscence of this minimum
is also present in Fig. 2: under transition from the normal to
mixed state the increase of uEu begins only after a substantial
lowering of w.
In conclusion, we propose a method of investigating the
dynamical characteristics of the vortex matter that consists in
measuring the amplitude and the phase of the EM field radi-
ated from the sample under excitation of the vortex lattice
oscillations by the elastic wave. It is established that unusual
behavior of the amplitude and the phase of the EM field are
accounted for the nonuniform pinning. The parameters mea-
sured sthe viscosity coefficient and the Labusch parameterd
are in qualitative agreement with the estimates obtained from
the accepted theories of vortex matter dynamics.
This study was supported in part by CRDF sGrant No.
UP1-2566-KH-03d, by INTAS sGrant No. 03-51-3036d, and
by Ukrainian Government Foundation for Basic Research
sGrant No. 0207/00359d. We would like to thank G. P.
Mikitik and K. V. Maslov for helpful discussions.
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Ignatova, and S.-I. Lee, Low Temp. Phys. 29, 72 s2003d.
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BRIEF REPORTS PHYSICAL REVIEW B 71, 092504 s2005d
092504-4


Surface pinning in amorphous ZrTiCuNiBe alloy

Surface pinning in amorphous ZrTiCuNiBe alloy

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