letters to nature
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Vortex dynamics in superconducting
MgB2 and prospects for applications
Y. Bugoslavsky, G. K. Perkins, X. Qi, L. F. Cohen & A. D. Caplin
Centre for High Temperature Superconductivity, Blackett Laboratory,
Imperial College, London SW7 2BZ, UK
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The recently discovered1 superconductor magnesium diboride,
MgB2, has a transition temperature, Tc, approaching 40 K, placing
it intermediate between the families of low- and high-temperature
superconductors. In practical applications, superconductors are
permeated by quantized vortices of magnetic ¯ux. When a supercurrent ¯ows, there is dissipation of energy unless these vortices
are `pinned' in some way, and so inhibited from moving under
the in¯uence of the Lorentz force. Such vortex motion ultimately determines the critical current density, Jc, which the
superconductor can support. Vortex behaviour has proved to be
more complicated in high-temperature superconductors than in
low-temperature superconductors and, although this has stimulated extensive theoretical and experimental research2, it has
also impeded applications. Here we describe the vortex behaviour in MgB2, as re¯ected in Jc and in the vortex creep rate, S,
the latter being a measure of how fast the `persistent' supercurrents decay. Our results show that naturally occurring grain
boundaries are highly transparent to supercurrents, a desirable
property which contrasts with the behaviour of the high-temperature superconductors. On the other hand, we observe a
steep, practically deleterious decline in Jc with increasing magnetic ®eld, which is likely to re¯ect the high degree of crystalline
perfection in our samples, and hence a low vortex pinning
energy.
–0.2
b
1.5
TC= 37.5 K
–0.4
–0.6
1.0
106
0.5
–0.8
∆ = 8 meV
–1.0
10
20
30
40
0.0
50
–20
10
20
Jc (A cm–2)
m (10–6 A m–2)
0
V (meV)
T (K)
c
105
–10
0.1
0
104
10 K
4
103
µ Hirr (T)
°
0.0
σ/σn
m (a.u.)
a
Measurements of Jc on polycrystalline MgB2 Ðthe only form
available so farÐhave to be analysed carefully, because experience
of the high-temperature superconductor (HTS) phases has shown
that their grain boundaries impede the ¯ow of supercurrent. This
problem is particularly severe in the YBa2Cu3O7 phase, which
intrinsically has the highest Jc of the HTS oxides, but behaves
poorly in bulk polycrystalline form, with Jc depressed by several
orders of magnitude at even modest-angled grain boundaries3.
We extracted from MgB2 powder (Alfa Aesar Co., 98% purity)
several fragments of above 100 mm size; they were somewhat
irregular in shape, and X-ray diffraction indicated no obvious
crystalline texture. The superconducting transition (Fig. 1a) of a
single particle is sharp with a Tc of 37.5 K, and particles have wellde®ned tunnelling characteristics (Fig. 1b) with a gap 2¢ of ,5 kBTc
(for a weakly coupled BCS superconductor this factor is 3.5, for the
HTS phases Bi2Sr2CaCu2O8 and YBa2Cu3O7, it is 10 and 5.5,
respectively4,5). The key data are derived from magnetization
loops that have excellent signal-to-noise ratios (Fig. 1c). Before
proceeding further, we applied a number of cross-checks that have
been developed in the context of HTS studies6. These measurements
yield for this fragment a superconducting screened volume of
7:4 6 0:4† 3 10 2 12 m3 , independent of ®eld and temperature. In
a scanning electron microscope, the fragment appeared roughly
ellipsoidal in shape, with cross-section axes of 450 and 250 mm, and
between 100 and 150 mm in the third direction; its physical volume
­ is therefore …5±7† 3 10 2 12 m3 . The screened volume might be up
to a factor of 1.5 greater than this because of the demagnetizing
effects, so within the uncertainties involved, there is excellent
agreement with that obtained magnetically. Thus, the straightforward conclusion is that screening currents ¯ow in a wellconnected fashion around this small polycrystalline fragment, and
that over the ®eld and temperature range studied, the grain
boundaries are transparent to supercurrent, just as in low-temperature superconductor (LTS) materials. In contrast, a polycrystalline
2
20 K
Tc
0
35 K
102
–0.1
20
–1
0
1
2
3
4
µ H (T)
°
0.01
Figure 1 Characterization of superconductivity. a, Magnetic moment of a single small
MgB2 fragment as a function of temperature in a ®eld of 10 mT (the sample was previously
cooled in zero ®eld), showing a sharp superconducting transition at 37.5 K. b, Typical
tunnelling characteristic (differential conductance, j …V † ˆ dI=dV , normalized to its
normal-state value jn ˆ jn …0†† of a similar fragment; the parabola is an estimate of the
background conductivity. c, Magnetization loop (obtained using a vibrating sample
magnetometer; Model TVSM5, Oxford Instruments) at 20 K of the fragment shown in a;
data were taken at a ramp rate of 13.3 mT s-1. The initial and `reverse leg' slopes (dashed
lines) of the loop are a measure of the volume from which superconducting screening
currents exclude changes in magnetic ¯ux. If the sample is a single well-connected entity,
these slopes should be equal to the actual sample volume (apart from an enhancement
associated with classical demagnetizing effects).
NATURE | VOL 410 | 29 MARCH 2001 | www.nature.com
30
T (K)
30 K
25 K
40
0.1
1
10
µ H (T)
°
Figure 2 Critical current density Jc of a MgB2 single fragment as a function of magnetic
®eld. In order to convert the irreversible magnetic moment to Jc, we model the fragment as
a disk of radius a ˆ 300 mm and thickness t ˆ 100 mmÐthat is, a volume ­ of
7 3 10 2 12 m3 , at the upper bound of our geometrical estimateÐso as to yield
conservative values for Jc. The standard Bean model13 gives J c ˆ 3Dm=2­a (in SI units,
but in keeping with standard practice in the ®eld, we report Jc in units of A cm-2 rather
than A m-2), where Dm is the full-width of the m(H ) loop. Inset, the irreversibility ®eld Hirr,
above which irreversible (that is, screening) currents cannot be sustained, as a function of
temperature; here we de®ne Hirr as the ®eld where Jc drops below the noise level,
,103 A cm-2.
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563
letters to nature
100
0
35K
YB
a
30K
2
Cu
10–1
O
7
χIn
S
3
–5
25K
35K
30K
25K
self-field
10–2
0
1
2
3
0.1
0.2
0.3
0.4
S
Figure 3 Vortex creep rate, S, as a function of ®eld. We use the `dynamic relaxation'
Ç to a
method10 in which the full-width of the m(H ) loops depends on the ®eld ramp rate H;
Ç These data are derived from the differences of
good approximation, S ˆ ] ln m=] ln H.
irreversible magnetic moment between loops taken at ramp rates of 13 and 3 mT s-1.
Because the differences are small, the signal has been increased by using a sample of ten
MgB2 fragments. The oscillations in the curves, which become more severe as the
temperature is reduced, are artefacts arising from small ¯uctuations in sample
temperature, whose effects are ampli®ed by the differencing of the data used to extract S.
In the region below the dotted line, the self-®eld (the ®eld generated by the screening
currents themselves) is comparable to or larger than the applied ®eld, and so complicates
the interpretation.
sample of YBa2Cu3O7 (unless very well textured crystallographically) would behave as an assembly of disconnected grains at all
but the lowest applied ®elds.
The values of Jc (Fig. 2) are high; as a benchmark, at 20 K and 1 T,
Jc is 1:0 3 105 A cm 2 2 . However, Jc decreases quite rapidly with
applied ®eld, and an `irreversibility ®eld' Hirr (Fig. 2, inset) can be
de®ned above which supercurrents cannot be sustained; that is, Jc
becomes unmeasurably small. Because we have established that the
sample is well connected, the Jc values shown in Fig. 2 are a reliable
measure of the intragranular critical current density in MgB2.
There have already been several magnetic studies of pressed and
sintered pellets of MgB2 (refs 7±9). All the irreversibility ®elds are
similar to the values shown in Fig. 2, but in two cases7,8 the Jc values
are more than an order of magnitude less than reported here.
Takano et al.9, who sintered their material at high temperatures
and pressures, report a rather larger Jc, ,4 3 104 A cm 2 2 at 20 K
and 1 T, but still a factor of 2 or 3 less than we measure. Overall, these
results can be understood straightforwardly: in sintered pellets of
MgB2 the intergrain contact is less perfect than in our polycrystalline as-grown fragments, to a degree dependent upon processing.
The reduced cross-sectional contact area limits the magnitude of the
macroscopic critical current, but its ®eld and temperature dependences appear to be essentially independent of the source or
preparation of the materialÐin sharp contrast to the sensitivity
of the HTS phases to processing.
The other key aspect of the vortex physics is that because of
thermal activation, the vortices are never totally immobile, and the
screening current and associated magnetic moment decay with time
in a quasi-logarithmic fashion. The relevant parameter is the vortex
creep rate, S, de®ned as the logarithmic derivative -] ln m/] ln t. S is
small at low ®elds (Fig. 3), but increases quasi-exponentially as the
®eld approaches Hirr. The creep rate is related directly to the
current±voltage (or equivalently, current density J±electric ®eld
E) characteristic of the material10, with E ~ J 1=S ; thus in MgB2 the E±J
characteristics at low ®elds are very steep, which is desirable for
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–10
0.0
µ H (T)
°
Figure 4 Comparison of the vortex behaviour in MgB2 and YBa2Cu3O7. Within the context
of standard vortex pinning theory, a plot (at ®xed temperature) of xln Ðthe dimensionless
slope of the ln Jc ±ln H curves of Fig. 2Ðagainst the creep rate S, with H as implicit
parameter, should have some simple form11. The oscillations in the data are instrumental
artefacts, as described in Fig. 3 legend. The dotted line represents the behaviour of the
HTS superconductor YBa2Cu3O7 at temperatures not too close to Tc (ref. 11).
applications. At ®elds above Hirr, the vortex response becomes
linear, E ~ J; consequently, it is to be expected that at ®elds close
to Hirr, S should approach unity.
The two quantities Jc and S, measured over a range of ®eld and
temperature, together provide the essential information for an
understanding of the vortex behaviour. Rather than use Jc itself, it
is better to consider the dimensionless derivative ] ln Jc/] ln H; that
is, the slope of the curves of Fig. 2. This quantity is identical to
] ln m/] ln H, and so is a logarithmic susceptibility, which we denote
xln. A plot of xln (at a ®xed temperature) against S with H as the
implicit parameter has a direct physical interpretation11: in
YBa2Cu3O7, these plots are linear, with slopes almost independent
of T (except close to Tc). This linearity indicates simple power-law
dependences on H of two fundamental quantities: the current
density J0 that would be attained if there were no thermal activation
of the vortices, and the energy U0, which measures the vortex
pinning as modi®ed by vortex±vortex interactions. In MgB2,
these plots (Fig. 4) are again close to linear, and at 35 K (that is,
close to Tc) have almost the same slope as that seen in YBa2Cu3O7.
However, as the temperature drops, they become considerably
steeper, which indicates that in MgB2 at temperatures not too
close to Tc, U0 decreases very rapidly with increasing ®eld. Detailed
measurements of xln in conventional LTS superconductors are
technically dif®cult, so that no comparison of this kind can be
made at present.
MgB2 is certainly a promising material in some respectsÐJc is
high and the grain boundaries appear benign in the context of
supercurrent ¯owÐbut the irreversibility ®eld is rather low, only
about half of the upper critical ®eld Hc2 (refs 7, 8). The sharp rise
in the vortex creep rate S as the ®eld approaches Hirr, and the steep
xln ±S plots, suggest that the vortex pinning interactions weaken
catastrophically with increasing ®eld. Perhaps there is a paucity of
pinning defects in the `as-grown' material, which would be consistent with the high degree of crystalline perfection of this
compoundÐindicated by the very low residual resistivity12, and
by the reproducibility of Tc and Hirr between samples from
different sources. For MgB2 to become more useful for applications, it will be essential to ®nd ways to enhance the vortex pinning
energy.
M
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letters to nature
Received 20 February; accepted 6 March 2001.
1. Nagamatsu, J., Nakagawa, N., Muranaka, T., Zenitani, Y. & Akimitsu, J. Superconductivity at 39 K in
magnesium diboride. Nature 410, 63±64 (2001).
2. Cohen, L. F. & Jensen, H. J. Open questions in the magnetic behaviour of high temperature
superconductors. Rep. Prog. Phys. 60, 1581±1672 (1997).
3. Mannhart, J., Chaudhari, P., Dimos, D., Tsuei, C. C. & McGuire, T. R. Critical currents in [001] grains
and across their tilt boundaries in YBa2Cu3O7 ®lms. Phys. Rev. Lett. 61, 2476±2479 (1988).
4. Miyakawa, N. et al. Predominantly superconducting origin of large energy gaps in underdoped
Bi2Sr2CaCu2O8+ from tunneling spectroscopy. Phys. Rev. Lett. 83, 1018±1021 (1999).
5. Cucolo, A. M. et al. Quasiparticle tunneling properties of planar YBa2Cu3O7-d/PrBa2Cu3O7-d/
HoBa2Cu3O7-d heterostructures. Phys. Rev. Lett. 76, 1920±1923 (1996).
6. Caplin, A. D., Cohen, L. F., Perkins, G. K. & Zhukov, A. A. The electric-®eld within high-temperature
superconductorsÐmapping the E-J-B surface. Supercond. Sci. Technol. 7, 412±422 (1994).
7. Larbalestier, D. C. et al. Strongly linked current ¯ow in polycrystalline forms of the superconductor
MgB2. Nature 410, 186±189 (2001).
8. Finnemore, D. K., Ostenson, J. E., Bud'ko, S. L., Lapertot, G. & Can®eld, P. C. Thermodynamic and
transport properties of superconducting MgB2. Preprint cond-mat/0102114 at hhttp://xxx.lanl.govi
(2001).
9. Takano, Y. et al. Superconducting properties of MgB2 bulk materials prepared by high pressure
sintering. Preprint cond-mat/0102167 at hhttp://xxx.lanl.govi (2001).
10. Pust, L., Jirsa, M. & Durcok, S. Correlation between magnetic hysteresis and magnetic-relaxation in
YBaCuO single-crystals. J. Low Temp. Phys. 78, 179±186 (1990).
11. Perkins, G. K. & Caplin, A. D. Collective pinning theory and the observed vortex dynamics in
(RE)Ba2Cu3O7-d crystals. Phys. Rev. B 54, 12551±12556 (1996).
12. Can®eld, P. C. et al. Superconductivity in dense MgB2 wires. Preprint cond-mat/0102289 at
hhttp://xxx.lanl.govi (2001).
13. Bean, C. P. Magnetization of high-®eld superconductors. Rev. Mod. Phys. 36, 31±36 (1964).
Acknowledgements
Y.B. is on leave of absence from the General Physics Institute, Moscow, Russia. We thank
D. Cardwell and N. H. Babu for the supply of MgB2 powder, and A. J. P. White for
crystallographic advice. This work was supported by the UK Engineering and Physical
Sciences Research Council.
Correspondence and requests for materials should be addressed to Y.B.
(e-mail: y.bugoslav@ic.ac.uk).
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Extreme damping in composite
materials with negativestiffness inclusions
R. S. Lakes*²³§k, T. Lee³, A. Bersie*² & Y. C. Wang*²
* Department of Engineering Physics; ² Engineering Mechanics Program;
³ Biomedical Engineering Department; § Materials Science Program; and
k Rheology Research Center, University of Wisconsin-Madison,
147 Engineering Research Building, 1500 Engineering Drive, Madison,
Wisconsin 53706-1687, USA
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When a force deforms an elastic object, practical experience
suggests that the resulting displacement will be in the same
direction as the force. This property is known as positive
stiffness1. Less familiar is the concept of negative stiffness,
where the deforming force and the resulting displacement are in
opposite directions. (Negative stiffness is distinct from negative
Poisson's ratio2±6, which refers to the occurrence of lateral expansion upon stretching an object.) Negative stiffness can occur, for
example, when the deforming object has stored7 (or is supplied8
with) energy. This property is usually unstable, but it has been
shown theoretically9 that inclusions of negative stiffness can be
stabilized within a positive-stiffness matrix. Here we describe the
experimental realization of this composite approach by embedding negative-stiffness inclusions of ferroelastic vanadium dioxide in a pure tin matrix. The resulting composites exhibit extreme
mechanical damping and large anomalies in stiffness, as a consequence of the high local strains that result from the inclusions
deforming more than the composite as a whole. Moreover, for
NATURE | VOL 410 | 29 MARCH 2001 | www.nature.com
certain temperature ranges, the negative-stiffness inclusions are
more effective than diamond inclusions for increasing the overall
composite stiffness. We expect that such composites could be
useful as high damping materials, as stiff structural elements or
for actuator-type applications.
Composite materials were prepared with inclusions of vanadium
dioxide (VO2) in a pure (99.99%) tin matrix. Particulate inclusions
150 mm in size or smaller were incorporated into a tin matrix by
rolling sheets of tin with particles, followed by casting into a
cylindrical mould 3 mm in diameter. Vanadium dioxide is a ferroelastic material10±12 that undergoes a transformation from monoclinic to tetragonal at transformation temperature T c ˆ 67 8C; the
transition exhibits hysteresis. It is similar in structure10 to the stiff
mineral rutile (TiO2) with elastic moduli13 C11 ˆ 271 GPa and
C33 ˆ 483 GPa. By contrast, steel has a Young's modulus of
E ˆ 200 GPa (C 11 ˆ C 33 ˆ 269 GPa); for tin E ˆ 50 GPa.
The material properties were measured using a broadband
viscoelastic spectroscopy apparatus14,15, capable of a range of
eleven decades of time and frequency in torsion. Specimens were
mounted using cyanoacrylate cement: at the top to its support rod,
and at the bottom, to a high magnetic intensity neodymium iron
boron magnet. The dynamic experiments were conducted by
applying a sinusoidal voltage at 100 Hz, from a digital function
generator to a Helmholtz coil. The lowest specimen resonance was
at several kilohertz. The coil imposed a magnetic ®eld on the
permanent magnet and transmitted an axial torque to the specimen.
The angular displacement of the specimen was measured using laser
light re¯ected from a mirror mounted on the magnet to a splitdiode light detector. The detector signal was ampli®ed with a wideband differential ampli®er. Torque was inferred from the Helmholtz
coil current, supported by calibrations using the well characterized
6061 Al alloy29. Input and output voltages and the phase between
them were recorded using a lock-in ampli®er (SRS 850, Stanford
Research Systems). Data were reduced using the analytical relationship for the torsional rigidity of a viscoelastic cylinder. Constituent
stiffness was tuned by varying the temperature. An insulated inner
chamber was heated to above 85 8C and the temperature was
maintained for at least 30 minutes. The heater was shut off and
the chamber allowed to cool. Cooling rate over the temperature
range of interest was less than 1 8C per minute.
Material properties, stiffness and damping (tand, where d is the
phase between stress and strain) as a function of temperature of a
particulate VO2 ±tin composite with 1% by volume of inclusions are
shown in Fig. 1. Although the concentration of inclusions is dilute,
large anomalies are observed in the mechanical damping, tand, as
well as in the stiffness. For comparison, tand of pure Sn is 0.019 and
varies little with temperature, as shown. The sharp dependence on
temperature we attribute to the fact that the inclusions are much
stiffer than the matrix (away from the transition temperature) and
can only balance the matrix stiffness near the transition. The peaks
observed in the present 1% VO2 composite are larger than in 100%
VO2, which had a peak12 of 0.04 to 0.05 in tand and a dip of 8% in
stiffness at the transition. In the composite, there is interplay
between constituents of positive and negative stiffness; by contrast,
100% VO2 has positive stiffness at all temperatures due to the
formation of domains.
A negative-stiffness lumped element (a structure) can be stabilized by a hard constraint. For an unconstrained block (or, surface
traction boundary condition, in the language of elasticity1) of
isotropic material to be stable, the shear modulus G and the bulk
modulus B, as stiffness quantities, must be positive. This corresponds to a range of Poisson's ratio n (de®ned as the negative
transverse strain of a stretched or compressed body divided by its
longitudinal of strain) 2 1 , n , 0:5. Materials of negative Poisson's ratio, while exhibiting the unusual property of transverse
expansion when stretched, can have all positive-stiffness values,
and hence stability. For most solids1, n is between 0.25 and 0.33.
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