WDS'07 Proceedings of Contributed Papers, Part III, 48–53, 2007.
ISBN 978-80-7378-025-8 © MATFYZPRESS
Critical State in Nb Thin Film
A. E. Youssef and Z. Švindrych
Charles University in Prague, Faculty of Mathematics and Physics, Ke Karlovu 3, 121 16 Prague 2, Czech Republic.
J. Hadač
Czech Technical University in Prague, Faculty of Nuclear Sciences and Physical Engineering,
Břehová 7, 115 19 Prague 1, Czech Republic.
Z. Janů
Institute of Physics ASCR v.v.i., Na Slovance 2, 182 21, Prague, Czech Republic.
Abstract. We present high resolution measurements of temperature dependence of low
frequency ac magnetic susceptibility of superconducting Nb thin film in perpendicular
time-varying periodic applied magnetic fields. We map this data, including harmonics,
to the ac susceptibility calculated on basis of the complete analytical model of
magnetization of thin superconducting disk in critical state. We find an excellent
agreement between the measured and model data for temperatures up to 0.9997Tc.
Introduction
Niobium has the highest critical temperature of elemental superconductors: 9.2 K. Also it has the largest λL
(London penetration length) of any element. It is one of only three elementary superconductors which are type-II
thus it may sustain mixed state with quantized vortices. Nb thin films are starting material for superconducting
electronics. SQUIDs, chips for Josephson voltage standard, electromagnetic cavity resonators, filters, etc. are
made by lithographic processes on these films.
A great advance in theoretical modeling of superconductors was achieved in 1957 by Abrikosov et al.
[1957], who presented theory for magnetization of hard (type-II) superconductors, in which he showed that, flux
enters specimen in quantized flux lines, starting at very low field Hc1 and completing its penetration at a much
higher field Hc2. Several years afterwards, Bean [1962,1964] introduced a model, later referred to as the critical
state model (CSM), in order to obtain the virgin magnetization curves in hard superconductors, which exhibited
magnetic hysteresis.
In type II superconductors, in field exceeding the lower critical field Hc1, the vortices nucleate at sample
surface and penetrate into sample interior. If there are pinning centers present, distance to which the vortices
penetrate depends on strength of vortex pinning. When Lorentz force acting on vortices in current density j is
balanced with the pinning force, the critical state occurs, which defines the critical depinning current density jc.
If ⏐j⏐ > jc the depinned vortices move and dissipate an energy. According to the critical state model, in a quasisteady state the vortices penetrate only as far as necessary to reduce the magnitude of the local current density j
to the level of the critical-current density jc.
Experiment
The Nb film of thickness of 250 nm was deposited by the dc magnetron sputtering in Ar gas on the 400 nm
thick silicon dioxide layer which was grown by a thermal oxidation of the silicon single crystal wafer
[May,1999]. The film is polycrystalline with texture of the preferred orientation in the (110) direction and is
highly tensile. The grain size is typically around 100 nm. The square samples of 5×5 mm2 in dimensions were
cut out from the wafer of diameter of 3 inches.
Magnetic moment of the sample was induced in applied homogeneous field and was detected by a
noncommercial continuously reading SQUID magnetometer, [Janu et al.,2006]. In this magnetometer, unlike
commercial ones, sample is stationary in gradiometer coils. The sample is placed in one of these coils and
cooled or warmed slowly. The SQUID output signal is proportional to the difference of magnetic fluxes in the
gradiometer coils caused by the sample's magnetic moment. The applied field is created by a superconducting
solenoid which operates in non-persistent mode. The solenoid is supplied from the voltage driven current source.
The SQUID output voltage and voltage monitoring the current fed to the solenoid as well as the voltages driving
the current supply are converted by over-sampling delta-sigma modulating converters. Delta-sigma converters
are inherently linear, low total harmonic distortion, and high signal-to-noise ratio. These features allow to
acquire or generate signals with high accuracy and fidelity and consequently perform the high resolution spectral
(harmonic) analysis.
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YOUSSEF ET AL.: CRITICAL STATE IN NB THIN FILM
The sample was mounted by Apiezon grease on a flat surface of the cylindrical sapphire holder. The
sapphire is an excellent thermal conductor but electrical insulator. The GaAlAs diode temperature sensor was
mounted on the opposite surface. The holder is suspended on the both thermally and electrically insulating
support which is connected to the stainless steel tube. This insert is placed in the anti-cryostat in 4He exchange
gas at atmospheric pressure. Temperature of the anti-cryostat is controlled by the Si diode temperature sensor,
Lake Shore temperature controller, and resistive wire heater.
The measurements were done in time-varying field, H(t) = Hac sin (ω0t), oscillating continually while
temperature was ramped up or down. Applied frequencies, ω0/2π, range from 1.5625 to 12.5 Hz. The
magnetization signal, M(t), and signal monitoring applied field, H(t), are digitized and Fourier transformed in
real time (via discrete fast Fourier transform) to complex frequency spectra, M(ω) and H*(ω). In general, because
of a phase shift in the applied field H(t) and analyzed data segment, the fundamental frequency and harmonics of
the complex susceptibility is computed as
χn =
H ∗ (nω0 ).M (nω0 )
(1)
H ∗ (nω0 ).H (nω0 )
The asterisks mark the complex conjugated numbers. The coefficient of n-th harmonic of the field spectrum
is defined as
H(nω0) = ⏐H(nω0)⏐(arg H(nω0))n = Hac⏐(arg H(nω0)) n
(2)
Critical state model and calculation
The Bean's model of the critical state, on which the model is constructed, posses essential simplifications :
a) The critical state is isotropic,⏐ j⏐ = jc in flux penetrated regions, where ,⏐B⏐≠ 0, while j = 0 in non-penetrated
regions. b) jc is independent from the local flux density B. c) The surface does not constitute barrier. d) The
critical state is quasistatic.
Mikheenko and Kuzovlev [1993] constructed the model in 2D disk-shaped superconductor by analogy to
that of a long cylinder and found the complete analytical solutions of field and current patterns in the thin film
disk-shaped type-II superconductors in perpendicular time-varying periodic applied magnetic fields. This model
was extended and corrected by Zhu et al., [1993] who showed that the magnitude of the current density
decreases continuously in vortex-free annulus from jc to zero in the center of the disk. They gave complete
analytical solution to current density j(r) and magnetization M(r) profiles in 2D thin film disk for H0/jcd. Finally,
they gave the analytical expression for the magnetic moment m(H0/jcd), for both initial (virgin) magnetization
and for time-varying field, and the effective magnetic susceptibility. Because of strong demagnetizing effects,
the flux density bends around the disk, and shielding currents flow over the entire surface of the disk. The
magnitude ∂Br/∂z is much larger than magnitude of ∂Bz/∂r for weak external fields except at the center of the
disk; the shielding current mainly comes from the term ∂Br/∂z, in contrast to the case of a long cylindrical
sample. The model is constrained for the film thickness d ≤ 2λ, where ¸ λ is the London flux penetration length,
to assure that the circulating currents in the film plane may be treated as having uniform density in the thickness
direction. Also, the external field is assumed to be weak enough so that the critical current density in the film is
independent of the local density of trapped vortices, i.e. jcd = const. A characteristic field for disk geometry is
defined as Hd = jcd/2.
In 2D disk-shaped superconductors, as the external field is decreased from the maximum field H0, the
current density flowing in the outer annulus b < r < R is the critical current density +jc but the current density
j(r) flowing in the inner annulus a < r < b is a function of location r. In any case the critical current density must
satisfy constraint ⏐j(r)⏐ < jc. Considerable shielding currents also flow in the vortex-free circular region r < a,
which is very different, except of the axis. The latter behavior contrasts strongly with that of in an infinitely long
superconducting cylinder. In a 2D disk-shaped superconductor, an abrupt jump in the current density is unphysical.
Clem and Sanchez [1994] showed that the model may be extended for either d ≥ λ or, if d < λ, that Λ =
2 λ2/d << R, where Λ is the two-dimensional screening length. They calculated analytical solution to hysteretic
current density and magnetic flux-density profiles in the critical state, hysteretic magnetization curves, and
complex ac-susceptibility (fundamental frequency and harmonics) components χ(H0/jcd) on basis of this model
and gave an approximative behavior. They concluded that for finite applied fields the annular region where the
current density is jc never fills the entire disk, and the critical-state flux-density profiles never penetrate all the
way to the center, where Bz remains equal to zero. In real thin films, however, the above 2D approach breaks
down and Bz becomes nonzero when the vortex-free radius a approaches the largest of the quantities d, λ and Λ.
They summarized that independently of geometry of the sample the hysteretic critical-state behavior is that
during quasistatic changes of an applied magnetic field, vortices move and thus the local flux density B changes
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YOUSSEF ET AL.: CRITICAL STATE IN NB THIN FILM
wherever the magnitude of the current density j (assumed perpendicular to the vortices) exceeds the critical value
jc. On the other hand, in long samples the local current density j can change only where the flux density changes
while in a film changes in the applied field induce screening currents j to flow not just at the edges but
throughout the film. However, these currents do not cause flux motion or a change in the perpendicular
component of the magnetic flux density unless the magnitude of j exceeds jc.
As temperature approaches Tc, both Hc1 and jc turn to zero. Then even in weak applied fields may be H >>
H*c1 and one may put B = µ0H. In this case the quasistatic Bean's model fails and dynamic model should be used
instead.
The magnetization was computed using the complete analytical solution to the magnetization of the 2D
disk-shaped superconductor in time-varying applied field [Zhu et al., 1993; Clem et al.,1994]. The M(φ) and
H(φ) arrays were computed for discrete φ values, for the oscillatory external field, H(φ) = Hac sinφ, and for ln
(Hac/Hd) varied slowly in comparison with the period of the external field. The data length per period was the
same for both experimental and model data. Nonlinear complex ac susceptibilities were calculated from the
hysteresis loops of the magnetization using coefficients of Fourier transformed M(φ) and H(φ) arrays in the same
way the experimental susceptibility was calculated.
Discussion of results
Primarily we assess meeting of the model assumptions: a) The film thickness, d = 250 nm, is larger than is
the flux penetration length for Nb, λ(0) = 40nm, thus the first Clem and Sanchez condition, d ≥ λ is fulfilled.
However, the λ may become larger than the thickness near Tc. Considering Gorter-Casimir or Ginzburg-Landau
model for temperature dependence of λ¸the λ(T) ≈ d for T≈ 0.99-0.999 Tc. However, in this case, the second
Clem and Sanchez assumption on the two-dimensional screening length, Λ, is fulfilled as λ2 << dR/2 ≈ 0.5×2.5
mm ×250 nm, i.e. λ<< 18 µm. b) Althogh the model is for disk geometry and ours are of the square geometry but
according to Brandt's calculations the difference between both geometries is only 1%. [Brand, 1998a; Brand et
al. 1998b; Herzog et al., 1997] c) The ac susceptibilities were measured in low frequency field for frequencies
from 1.5625 up to 12.5 Hz. As the susceptibilities do not change with frequency, field may be considered quasistatic. d) The appropriateness of the model may be reviewed briefly looking on the peak value of the normalized
measured χ″, as the model gives value χ″ ≈ 0.241, which occurs for Hac/Hd = 1.942. This value is independent on
any particular temperature dependence of the critical current density. Using this criterion, the model fits
experimental data for µ0Hac ≥ 5 µΤ. For µ0Hac < 5 µΤ, the peak is smaller than 0.241 and disappears with
decreasing amplitude of the ac field. The model ac susceptibility is plotted versus 2Hac/jcd [Clem et al.,1994;
Gencer, 2002]. This may be interpreted as the dependence on Hac for fixed jc or the dependence on jc for fixed
Hac. The later case meets a common experimental technique: measuring of the temperature dependence of the ac
susceptibility. To match the experimental and model susceptibilities, we plot the model χ versus negative
reciprocal value of 2Hac/jcd scaled by parameter c, i.e. -c(jcd/2Hac), and experimental χ in a convenient
temperature scale, (T - Tc) /Tc. Such scaling, jc(T) α ((Tc - T) /Tc) α, where α > 0, meets expectation that jc → 0 as
T→Tc.
The fundamental frequency experimental ac susceptibility for frequency 1.5625 Hz and model ac
susceptibility are shown in Figure. 1. The experimental susceptibilities were measured in sequence: warmed in
field µ0Hac = 10 µT; cooled in field µ0Hac = 5 µT; cooled in field µ0Hac = 2 µT; warmed in field µ0Hac = 1 µT.
The temperature rate was 0.1 K/min. We have mapped the model ac susceptibilities to the experimental ones and
we have found that the linear dependence (α = 1) with two only tuning parameters, c and Tc, gives an excellent
agreement for µ0Hac = 10 and 5 µT after all. The Tc shifts the experimental susceptibility horizontally and c
makes the model transition wider or narrower. The traced temperature dependence of the critical depinning
current density is jc(T) = jc0 (Tc - T) /Tc, where jc0 = 2Hac/cd. Since the film thickness, d = 250 nm, and ac field
amplitude Hac are given and the parameter c is obtained by mapping the experimental and model data one may
calculate jc0. The found values of Tc, c, temperature Tp at which a maximum χ″1p of χ″1 occurs, and calculated jc0,
for set of amplitudes Hac are shown in Table I.
We note that jc0 is assumed to change with temperature but not with Hac otherwise the Bean's model would
be inapplicable. As the field is increased, the Tc lowers, in agreement with expected suppression of the
superconductivity by applied field. Concurrently, the transition region broadens and the absorption peak shifts to
lower temperature. The value of jc0 ≈ 2 MA cm-2 sounds reasonably. The dependence Tc(Hac) is approximately
linear and Tc → 8.986 K for zero field. We note that the model ac susceptibility fits only the real part of the
experimental one for Hac < 5 µT as the experimentally observed absorption is smaller than the model predicts.
The experimental and model ac susceptibility plotted in log x-axis scale for µ0Hac = 10 µT are shown in Figure.2.
The model evidently fits the experimental data up to temperature only 3 × 10-4 Tc below the critical temperature.
This interval represents ≈ 3 mK using common temperature scale. For Hac = 0.5 µT, by increasing frequency
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YOUSSEF ET AL.: CRITICAL STATE IN NB THIN FILM
from 1.5625, 3.125, 6.25, and 12.5 Hz both Tp and Tc change only within temperature resolution. In normal state,
the absorption increases linearly with the applied field frequency ω/2π. In bias dc field Hdc = 100 µT, Tc
decreases by 5 mK (0.0015 on relative temperature scale) in comparison with zero field.
Figure 1. Real (χ′1) and imaginary (χ″1 ) parts of the fundamental frequency complex susceptibility. The marks
show the measured temperature dependence for µ0Hac = 10, 5, 2, and 1 µT plotted as [(Tc - T) /Tc, χ(T)] and the
solid curves show the model data plotted as [-c(Hd/Hac), χ(Hac/Hd)]. Both dependencies are mapped by tuning
only c and Tc.
Figure 2. Real (χ′1) and imaginary (χ″1 ) parts of the fundamental frequency complex ac susceptibility for
µ0Hac=10 µT. The marks show measurd dependence and the curves show model dependence. log plot shows
region near the critical temperature in detail.
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YOUSSEF ET AL.: CRITICAL STATE IN NB THIN FILM
Table I. Dependence of the characteristic parameters on applied field amplitude Hac.
Tc (K)
c
Tp (K)
µ0Hac (µT)
χ"p
10
5
2
1
8.965
8.975
8.980
8.985
0.0025
0.0018
0.0012
0.001
8.955
8.964
8.973
8.978
0.239
0.224
0.205
0.141
Jc0 (MA cm-2)
2.6
1.8
1.1
0.64
Figure 3. Real and imaginary parts of odd (3rd and 5th) harmonics of complex ac susceptibility. The marks show
measured temperature dependence for µ0Hac = 10 µT. The curves show model dependence.
Figure 4. Real and imaginary parts of even (2rd and 4th) harmonics of complex ac susceptibility. The marks
show measured temperature dependence for µ0Hac = 10 µT. curves.
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YOUSSEF ET AL.: CRITICAL STATE IN NB THIN FILM
The nonlinear response of the superconductor when pinning is present is reflected in appearance of higher
harmonics in the susceptibility. The odd harmonics, 3rd and 5th, of both experimental and model susceptibilities
are shown in Figure. 3. As the symmetry of magnetization hysteresis curves is odd, M(H) = -M(-H), the even
harmonics of ac susceptibility must be zero. The 2nd and 4th harmonics of the experimental susceptibility are
shown in Figure. 4. For easy confrontation, the vertical scales were left intentionally the same as in Figure.3. The
2nd harmonic is smaller than 3rd and 4th is smaller than 5th.
Conclusion
We have measured temperature dependence of magnetization of Nb thin film in slowly varying
perpendicular applied field. Both fundamental-frequency and harmonics of the experimental ac susceptibility
agree well with ac susceptibility computed on basis of the complete analytical model with Bean's critical state in
2D superconducting disk. Even though this model ignores surface barriers, thermal activation, and flux dynamics
(diffusion, creep, and flow) it works in case of measured Nb thin film up to 0.9997Tc.
Acknowledgments. The authors are grateful to T. May, M. Grajcar, F. Soukup, and R. Tichy for stimulating discussion
and help with experiment. This work was supported by the Czech Science Foundation under Contract No. 102/05/0942,
Research project No. AVOZ10100520, and MSM 0021620834.
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