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arXiv:cond-mat/0410356v1 [cond-mat.supr-con] 14 Oct 2004

The low-frequency response in the surface superconducting state of ZrB12single

crystal

Grigory I. Leviev, Valery M. Genkin, Menachem I. Tsindlekht, and Israel Felner

The Racah Institute of Physics, The Hebrew University of Jerusalem, 91904 Jerusalem, Israel

Yurii B. Paderno and Vladimir B. Filippov

Institute for Problems of Materials Science, National Academy of Sciences of Ukraine, 03680 Kiev, Ukraine

(Dated: February 2, 2008)

The large nonlinear response of a single crystal ZrB12 to an ac field (frequency 40 - 2500 Hz) for

H0 > Hc2 has been observed. Direct measurements of the ac wave form and the exact numerical

solution of the Ginzburg-Landau equations, as well as phenomenological relaxation equation, permit

the study of the surface superconducting states dynamics. It is shown, that the low frequency

response is defined by transitions between the metastable superconducting states under the action

of an ac field. The relaxation rate which determines such transitions dynamics, is found.

PACS numbers: 74.25.Nf; 74.60.Ec

I.INTRODUCTION

Recently high-quality superconducting ZrB12 single

crystals with transition temperatures Tc= 6.06 K have

been grown. The investigation of their physical proper-

ties, including electron transport, tunnel characteristics,

and critical fields, have shown that the Ginzburg-Landau

parameter κ is only slightly larger than the boundary be-

tween the type-I - type-II superconductor value [1, 2].

In this paper we concern the low-frequency response

of a ZrB12 crystal when the dc external magnetic field

H0 > Hc2 is parallel to the sample surface.

of the fact that the sample is in the surface supercon-

ducting state (SSS) [3], no static magnetic moment is

observed, while the ac response in this regime is large

and nonlinear even for an ac amplitude h0<< H0. In-

deed, at equilibrium the total surface current equals zero

in SSS, the internal dc magnetic field in the bulk equals

H0, and the magnetic moment of the specimen is small.

On the other hand, the ac magnetic field drives the sam-

ple into a metastable SSS where the total surface cur-

rent reaches a finite value. The internal magnetic field

deviates from the external one, and as a results the ac

response becomes large. The low-frequency response of

superconductors in SSS was the focus of intensive exper-

imental investigations [4, 5, 6] since the first prediction

of the existence of SSS in [3]. The observed [4] wave

form of the ac response, corresponding to the flux pass-

ing through the specimen, explicitly invoked a model sim-

ilar to the Bean model [7]. Recently SSS attracted re-

newed interest from various directions as described in

Ref. [8, 9, 10, 11, 12, 13]. Paramagnetic effect in a super-

conducting disk [8], stochastic resonance [9], , the per-

colation transition in the field H0 = 0.81Hc3 [10] have

been observed. It was proposed to use low-frequency

response for testing the quality of superconducting res-

onators in accelerators [11]. Surface states were observed

also in single crystals of MgB2 [12]. Our experimental

results presented here show that in ZrB12single crystal,

In spite

the Bean critical model of surface sheath does not give

an adequate description of the observed wave form which

corresponds to the flux passing through the sample. In

the framework of Ginzburg-Landau, theory we calculated

the surface current in metastable SSS’s which exists un-

der an ac magnetic field. Observing the wave forms, we

studied the metastable SSS dynamics and determined the

relaxation rate under an ac field. We found that the re-

laxation time for transition to the equilibrium state is not

constant and depends on the surplus of the free energy.

This relaxation time is decreased with the dc magnetic

field, and depends on the driving field frequency.

II. EXPERIMENTAL

The measurements were carried out at T = 5 K on

ZrB12 single crystal. The sample was grown in the In-

stitute for Problems of Materials Science, Ukraine. Its

dimensions are 10.3 × 3.2 × 1.2 mm3and it was cut by

an electric spark from a large crystal of 6 mm diame-

ter and 40 mm length. The surface of the sample was

polished mechanically and then chemical etched in boil-

ing HNO3/H2O (1:1) for 10 minutes was used. X-ray

pictures showed that a sample was single-phase material

with the UB12structure (space group Fm3m, a = 7.407

˚ A [14]. The tunnel characteristics of this sample were

described earlier [2]. The dc-magnetic moment was mea-

sured using a SQUID magnetometer. A block diagram

of the ac linear and nonlinear setup is shown in Fig. 1.

The ac magnetic field h(t) = h0sin(ωt) was supplied

by the magnetometer copper solenoid. The ac response

was measured by an inductive pick-up coil method [15].

The sample was put into one coil of a balanced pair of

pick-up coils and the induced voltage V (t) ∝ dM(t)/dt

was measured with an oscilloscope. Here M is the mag-

netic moment of the sample. The lock-in amplifier was

used in order to measure simultaneously in-phase and

out-of-phase signals of the first and third harmonics of

the driving frequency. An oscilloscope measured the wave

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LCK

Secondary

Coils

dc?magnet

Driving?ac

LFG

?

OSC

Sample

FIG. 1: (Color online) Block diagram of the experimental

setup. LFG - low frequency generator, LCK - lock-in amplifier

and OSC - oscillograph.

form of the signal in one channel. The second channel

of the oscilloscope measured the time derivative of the

excitation field.

III.EXPERIMENTAL RESULTS

Fig. 2a shows the real part of the ac susceptibility at

the fundamental frequency, χ

(ZFC) dc susceptibility, χdc= M/H0as a function of dc

field H0. The inset to Fig. 2a presents the ZFC mag-

netization curve at 5 K. Field dependencies of the ac

susceptibility imaginary part at fundamental frequency,

χ

on Fig. 2b and Fig. 2c respectively. Amplitude depen-

dence of the third harmonic, A3ω(h0) for H0= 180 Oe

is presented on inset to Fig. 2c. It is clear that this de-

pendence is far to be cubic as the perturbation theory

predicts. Experiment shows that amplitude dependence

of A3ω(h0) is not cubic at any dc field H0.

It is clear that the observed large signal of the A3ω

and maximum of the χ located in a magnetic field

Hc2 < H0 < Hc3, e.g.in a surface superconducting

state, although the zero dc signal indicates that bulk of

the sample is in the normal state. The absorption in the

SSS exceeds losses in the mixed and normal states.

Fig. 3 shows the time derivative of the magnetic mo-

ment of a sample at different applied magnetic fields at

T = 5 K. Note (i) that only in the SSS, the signal does

not have the sine-form. (ii) The amplitude dependence

of the third harmonic, A3ω(h0), presented in the inset to

Fig. 2c does not exhibit any cubic dependence.

The experimental data presented in Figs. 2 and 3 are

complex and the theoretical model which explains these

observations is given in the next section.

′, and the zero-field cooled

′′, and amplitude of the third harmonic, A3ωare shown

′′

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FIG. 2: (Color online) (a) χ′and χdc= M/H0 magnetic field

dependencies at T = 5 K. Inset: magnetization curve after

ZFC.

(b) Magnetic field dependence of χ′′.

(c) Field dependence of the amplitude of the third har-

monic, A3ω.Inset: amplitude dependence of A3ω(h0) at

H0 = 180 Oe.

ac measurements were carried out at frequency ω/2π =

170 Hz and h0 = 0.4 Oe.

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FIG. 3: (Color online) Oscillogram of dM/dt for three differ-

ent magnetic fields in the Meissner state (H0 = 0), in surface

superconducting state, (H0 = 180 Oe < Hc3), and for a nor-

mal state (H0 = 300 Oe > Hc3) at h0 = 4.75 Oe.

IV.THEORETICAL MODEL

Our theoretical approach is based on the numerical

solution of the two Ginzburg-Landau equations [18] for

the order parameter and vector potential, which in the

normalized form are as follows:

−(i∇/κ +− →

−curlcurl− →

The order parameter, Ψ, is normalized with respect

to its value in zero magnetic field, the distances with re-

spect to the London penetration length λ, and the vector

potential− →

modynamic critical field and κ = λ/ξ is the Ginzburg-

Landau parameter, and ξ is the correlation length. In

the second equation for the vector potential we neglected

the normal current, assuming that the skin depth ex-

ceeds both the London penetration length and sample

thickness. It is assumed that the sample form is a slab

with 2d thickness and that the external magnetic field is

parallel to its surface. The chosen coordinates are: the

x-axis normal to the slab, (thus the symmetry plane is

x = 0), and the z-axis is directed along magnetic field.

It is assumed that external magnetic field H > Hc2= κ.

Assuming the surface solutions have the form of

A)2Ψ2+ Ψ− | Ψ |2Ψ = 0

A =− →

A | Ψ |2+i/2κ(Ψ∗∇Ψ − Ψ∇Ψ∗)

(1)

A with respect to√2Hcλ, where Hcis the ther-

Ψ(x,y) = f(x)exp(iκky) (2)

therefore, Eqs. 1 reduce to

−1

κ

∂2f

∂x2+ (A − k)2f − f + f3= 0

∂2A

∂x2 = f2(A − k),

(3)

where k is constant. The boundary conditions at x =

±d are: ∂f(±d)/∂x = 0, ∂A(±d)/∂x = H ; at x = 0

f(0) = 0 and A(0)=0; H = H0+h0sin(ωt). An additional

condition for the surface states ∂f(0)/∂x = 0 is satisfied

only asymptotically for d −→ ∞. For the equilibrium

state the value of k = keqcan be obtained by minimizing

the Gibbs free energy defined as:

˜F =?dV {1

2|Ψ|4− |Ψ|2+ |i∇Ψ/κ + AΨ|2+

+B2− 2BH},

(4)

where B is the magnetic induction. Using Eq. 3 one then

obtains

˜F = −HA(d) −?d

0dx{1

−k]f2(x)}

2f4(x) + A(x)[A(x)−

(5)

The two coupled Eqs. 3 could be solved by numerical

methods. The order parameter for surface solutions de-

viates from zero only near the sample boundary, and we

could consider comparatively small d/λ ≤ 10. The actual

sample thickness exceeds λ by 4 or 5 orders of magnitude.

The solutions for large d could be found from the ones

for small d by transformation k = ks+Hi(d−ds), where

Hi= Hs(0,H,K) is the magnetic field at x = 0 in the

problem for d = ds∼= 10λ. The index s corresponds to

the solution for this small d. This choice of dsis sufficient

for numerical calculations and provides the solutions with

fs(0) = 0, ∂fs(0)/∂x = 0. The free energy transformed

as

˜F =˜ Fs− Hi(d − ds)(H −

?

[As(x) − ks]f2

s(x)dx) (6)

To simplify the calculations we use below variable ks

and omit index s.The properties of the equilibrium

solutions for a semi-infinite half-space have been dis-

cussed in [16]. The order parameter, the supercurrent,

and the internal magnetic field were calculated. In these

states the total surface current equal zero and free energy

reaches a minimum value. The ac magnetic field drives

the superconductor into a metastable state. These states

correspond to the solutions of Eqs. 3 for k ?= keq. The so-

lution of Eqs. 3 shows that surface states exist in a wide

range of k near keqas shown in the upper panel of Fig. 4 ,

but only for |k−keq| << keqfree energy of these states is

lower than the energy of the normal state. Moreover, this

range shrinks with increasing sample thickness (Fig. 5).

This is due to the increase of the contribution of the

first term in Eq. 5 which is the order of the Gibbs en-

ergy of the normal state˜ F0= −H2d. The total surface

current equals zero for equilibrium k and increases with

increasing |k − keq| as is shown in the lower panel of

Fig. 4. For a given k, the free energy versus magnetic

field does not exhibit any minimum in the equilibrium.

Only the difference between the Gibbs energy of the su-

perconducting and normal states exhibits minimum near

equilibrium field as was discussed in [4], but for a such

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FIG. 4: Upper panel. The maximal value of the order pa-

rameter, fmax, as a function of k. Lower panel. Total surface

current, J, as a function of the k parameter. The equilibrium

value of k = 8.8 corresponds to zero surface current.

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FIG. 5: (Color online) Free energy of the surface supercon-

ducting states relative to the free energy of normal state (F0)

in magnetic field H0 = 1 and k = 0.75 for two values of the

sample thickness.

representation the reference point moved with changing

field.

The magnetic moment of the sample actually depends

on the total surface current J, because the current is

localized in a thin surface layer. This current is a function

of the external magnetic field H and k, J = J(H,k). The

response of the sample to the ac magnetic field depends

on the dynamics of k. A priori, one can assume that the

equation that governs the dynamics is

dk

dt= −ν[k − keq(H)], (7)

where keq(H) is the equilibrium k in instantaneous

value of external magnetic field and ν = ν(k−keq) is the

relaxation rate. Function keq(H) has to be found from

Eqs. 3 and for |H−H0| << H0is well approximated by a

polynomial of the third order of h = H − H0. Using the

function J(h,k) calculated from Eqs. 3 and Eq. 7 one

can obtain the time evolution of the surface current in

an ac field and compare with observed wave forms. The

time derivative of the surface current is proportional to

the observed signal V . The coefficient α =

on the experimental apparatus parameters. Actually we

could obtain k(t) directly from experimental data and

test the correctness of Eq. 7. We may write

1

V

dJ

dtdepends

∂J(h,k)

∂h

dh

dt+∂J(h,k)

∂k

dk

dt= αV (t) (8)

This expression permits us to obtain k(t) from the ob-

served wave form. It is a first order differential equation

for k(t). To evaluate k(t) we have to know k at t = 0

and α, since the derivatives ∂J/∂h and ∂J/∂k can be cal-

culated from Eqs. 3. The k(0) value can be found from

the condition when the maximal current value during the

period is minimal. In order to find α we calculated J(t)

assuming that ν in Eq. 7 is constant. Then, we choose

the ν value in order to minimize difference between the

calculated and experimental data. This procedure gives

both ν and α. To be sure that ν is actually constant,

one has to collect the weak ac field data. The observed

signal during one period of the ac field and the result of

a simulation with Eqs. 3 and 7 are shown in Fig. 4. The

data in this figure were collected in a dc field of 130 Oe,

and an ac field with amplitude 1.78 Oe and frequency

ω/2π = 733 Hz. In our calculations we took ν/ω = 0.05

and the Ginzburg-Landau parameter κ = 0.75 [2]. The

good correlation between the calculated and experimen-

tal data permits one to find the scale coefficient α which

is used below.

V.DISCUSSION

As was shown above the losses are small both in

the mixed and normal states and have a maximum at

H0 > Hc2 (see Fig. 2a). The Hc2 = 126 Oe is deter-

mined from the dc magnetization curve (inset to Fig. 2a).

The oscillogram, Fig. 3, in both the Meissner and nor-

mal states (H0 = 0 and 300 Oe) has a sine shape, and

for Hc2< H0< Hc3the wave form deviates from a sine

shape. We do not observe any clear plateau for dM/dt

in the ac period. Such a plateau is a peculiarity of the

Bean model when it is applied to surface currents [4, 17].

Using the experimental data and the model developed

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FIG. 6: (Color online) The time derivative dk/dt plotted as

a function of k − keq for at H0 = 130 Oe and ω/2π = 733 Hz

(a): h0 = 1.78 Oe and (b): h0 = 5.9 Oe. The linear fit of

the experimental curve (a) gives ν/ω = 0.051. The hysteresis

at large amplitudes (b) shows that, generally speaking, dk/dt

depends on k and the instantaneous magnetic field not only

through k = keq.

in previous section one can calculate dk/dt as a function

of k − keq. Fig. 6 shows dk/dt plotted as a function of

k−keqobtained from the wave forms that were observed

for H0 = 130 Oe and ω/2π = 733 Hz. The linear fit

of dk/dt at h0 = 1.78 Oe yields ν(0)/ω = 0.051 which

agrees well with the ν/ω = 0.05 used in Eq. 7 when

the scale coefficient has been found. Visible hysteresis in

Fig. 6b indicates that at a high amplitude of excitation

the relaxation rate in Eq. 7 ν depends on k and on the

instantaneous value of h(t) not only through k − keq(h).

The expression for ν(k − keq) could be found from fit-

ting of dk/dt by the polynomial of k − keq. The approx-

imation expression which has a form

ν(x)/ω = 0.051− 0.117x+

+1.323x2+ 0.184x3− 0.747x4

provides the calculated data, which with an accuracy

of better than 10%, reproduces the experimental data for

a dc field of 130 Oe and a frequency of 733 Hz as is shown

in Fig. 7.

Increasing the dc field leads to the increasing of the

relaxation rate ν. We found that for H0= 138 and 180

Oe, the relaxation parameter for weak ac amplitudes are

ν(0)/ω = 0.145 and 4.725 respectively (see Fig. 8).

The calculated wave forms with the help of the pro-

posed model reproduce experimental data only for a weak

ac field as shown in Fig. 9. This is due to the increase

in the difference between the two values of dk/dt for the

same k−keqat larger ac amplitudes, ( see at Fig. 8b, d).

(9)

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FIG. 7: (Color online) The observed and calculated (solid

lines) oscillogram for H0 = 130 Oe, ω/2π = 733 Hz at h0 =

1.78 Oe and h0 = 5.9 Oe.

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FIG. 8: (Color online)dk/dt as a function of k−keq at ω/2π =

733 Hz. (a) - H0 = 138 Oe, h0 = 0.59 Oe; (b) - H0 = 138 Oe,

h0 = 5.9 Oe; (c) - H0 = 180 Oe, h0 = 0.59 Oe; (d) - H0 =

180 Oe, h0 = 5.9 Oe. Linear fit at low amplitude of excitation

( (a) and (c) panels) gives ν/ω = 0.144 for H0 = 138 Oe and

ν/ω = 4.73 for H0 = 180 Oe.

Fig. 10 shows the dk/dt(k−keq) dependence for differ-

ent frequencies at H0= 130 Oe and h0= 4.75 Oe. One

may conclude from this figure that the relaxation rate

ν (if Eq. 7 could be applied) increases with excitation

frequency ω.

It is clear that the model equation 7, where the relax-

ation rate ν depends on the one variable k −keq, is valid

only for small ac amplitudes. The transition from the

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FIG. 9: Calculated (solid lines) and experimental (triangle)

oscillograms at ω/2π = 733 Hz. (a) - H0 = 138 Oe, h0 =

0.59 Oe; (b) - H0 = 138 Oe, h0 = 5.9 Oe; (c) - H0 = 180 Oe,

h0 = 0.59 Oe; (d) - H0 = 180 Oe, h0 = 5.9 Oe.

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FIG. 10: (Color online) dk/dt as a function of k − keq at

H0 = 130 Oe and h0 = 4.75 Oe for different frequencies.

surface state with one k to another k, requires changing

the order parameter in the whole sample. Possibly, it

happens through the nucleation of a new phase. This

process is governed by excess energy. In general the re-

laxation constant in Eq. 7 may depend on the energy of

the state. Approximately it could be taken into account

by the assumption that the relaxation constant depends

on k−keq. We see that for small ac fields and not far away

from Hc2, this assumption is correct. But increasing the

ac amplitude and/or the dc field results in the explicit

dependence on both: k and the instantaneous value of

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FIG. 11: Calculated dependence of the Gibbs energy as a

function of the k − keq parameter during the ac cycle for dif-

ferent values of the dc field H0 and amplitude of excitation,

h0 at frequency ω/2π = 733 Hz.

the magnetic field H0. The straightforward calculations

of the Gibbs energy F, exhibited in Fig. 11, shows that

when this energy is a single-valued function of k − keq,

the simulation with Eq. 7 gives acceptable results.

When F becomes a multivalued function, the calcu-

lated wave forms differ from the experimental data. In

order to obtain a proper theoretical description, one has

to take into account that the energy of SSS is not ex-

pressed only through k − keq.

VI.CONCLUSION

We experimentally investigated the dynamics of the

surface metastable superconducting states of ZrB12in ac

fields at low frequencies (40 - 2500 Hz). It was shown that

for low ac amplitudes of excitation these dynamics are

governed by a simple relaxation equation. The relaxation

rate depends on the deviation from the equilibrium state.

Decreasing of the frequency of the applied ac field results

in increasing the relaxation time.

VII.ACKNOWLEDGMENTS

This work was supported by the INTAS program under

the project No. 2001-0617.

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