Effect of screening current induced pair breaking on magnetization of superconducting lead nanoparticles

Article abstract of DOI:10.1016/j.physc.2008.04.006

Magnetization measurements on lead nanoparticles in the size range 35-45 nm are presented. It is shown that the critical fields in these nanoparticles are enhanced significantly above their bulk values with temperature dependence also distinct from that o

Full text of DOI:10.1016/j.physc.2008.04.006

Physica C 468 (2008) 944–947
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Physica C
journal homepage: www.elsevier.com/locate/physc
Effect of screening current induced pair breaking on magnetization
of superconducting lead nanoparticles
G. Yashwant a, C.L. Prajapat a, O.D. Jayakumar b, M.R. Singh a, S.K. Gupta a, A.K. Tyagi b, G. Ravikumar a,
*
a Technical Physics and Prototype Engineering Division, Bhabha Atomic Research Centre, Mumbai, Maharashtra 400 085, India
b Chemistry Division, Bhabha Atomic Research Centre, Mumbai, Maharashtra 400 085, India
a r t i c l e i n f o
a b s t r a c t
Article history:
Magnetization measurements on lead nanoparticles in the size range 35–45 nm are presented. It is
shown that the critical fields in these nanoparticles are enhanced significantly above their bulk values
with temperature dependence also distinct from that of bulk. The observed ‘‘type II” like shape of the
magnetization curves is explained on the basis of the Ginzburg–Landau phenomenology by invoking
the pair breaking effect of the London screening currents, which makes the effective penetration depth
an increasing function of the field. The temperature dependence of critical field is found to be consistent
with our explanation.
Received 24 March 2008
Accepted 9 April 2008
Available online 15 April 2008
PACS:
74.40.+k
74.20.De
Ó 2008 Elsevier B.V. All rights reserved.
Keywords:
Nanoparticle
Screening current
Magnetization
1. Introduction
diamagnetism eventually vanishes, M(H) is the field dependent
magnetization. It is well known that the magnitude of the zero field
Studying various phenomena in nano-scale materials and in
particular superconducting particles [1–4] and nano-wires [5,6]
of dimensions much smaller or comparable to the superconducting
penetration depth k and coherence length n is of current interest.
According to the Anderson criterion [7], Cooper pairing is possible
as long as the separation of the electronic levels d (ꢀ1/N(0)V where
N(0) is the density of states at the Fermi surface and V is the vol-
ume of the particle) at the Fermi surface in the normal state is
smaller than the superconducting energy gap D. In lead and other
elemental superconductors, the critical particle size rc below which
this criterion breaks down is approximately an order of magnitude
smaller than the coherence length n. It is indeed experimentally
confirmed that the critical temperature Tc of nanoparticles of lead
(Pb) remains unaffected down to about 5–10 nm of size, while
n ꢀ 83 nm [8]. Further, experiments suggest that the condensation
energy given by the integral
slope of M–H curve decreases with decreasing particle size. It is
therefore easy to imagine that Hc should increase with decreasing
particle size in order that the above integral is particle size indepen-
dent. But the integral gives no details about the shape of the M(H)
curves.
It is commonly observed that M(H) curves in both nanoparticles
and nano-wires are ‘‘type II” like [5,6], in the sense that the dia-
magnetism increases at low fields and then decreases gradually
after attaining a maxima. According to Michotte et al. [6], the lead
nano-wire transforms into type II superconductor due to a sharp
reduction in the electronic mean free path. We argue that the ob-
served magnetic behavior is of an origin basically different from
the samples turning to type II. Even in micro-sized superconduc-
tors, vortex formation above a first critical field is expected to be
highly energy intensive and is accompanied by sharp discontinu-
ities in magnetization [9,10]. It is therefore difficult to envisage
that the nano-sized superconductors exhibit type II behavior.
Other possible reasons for the type II like magnetic behavior in
nanoparticles is order parameter fluctuations. Effectiveness of the
order parameter fluctuations depends on the ratio of thermal en-
ergy kT to the condensation energy VH2c0=2l0 where Hc0 is the ther-
modynamic critical field. This ratio is usually negligible for particles
of size comparable to the London penetration depth k or n (such
as in our experiments) for most elemental superconductors, and
Z
Hc
ꢁl0
MðHÞdH
ð1Þ
0
remains independent of particle size provided it is above the critical
size rc [1]. In the above expression Hc is the critical field at which
* Corresponding author. Tel.: +91 22 25591659; fax: +91 22 25505151.
E-mail address: gurazada@barc.gov.in (G. Ravikumar).
0921-4534/$ - see front matter Ó 2008 Elsevier B.V. All rights reserved.
doi:10.1016/j.physc.2008.04.006

G. Yashwant et al. / Physica C 468 (2008) 944–947
945
therefore order parameter fluctuations are unlikely to be responsi-
ble for any magnetic effects, except at temperatures very close to
the Tc.
In this paper, we present magnetization measurements on lead
nanoparticles, reproducing the type II like behavior observed by
others. Further, temperature dependence of the critical field Hc
exhibits interesting deviations from the bulk behavior. We present
an understanding based on the Ginzburg–Landau theory for both
these observations.
2. Experiment
Pb nanopartcles of different sizes have been synthesized using
Polyol method. In a typical synthesis of 40–42 nm (sample code
Pb-1) particles, a mixture of 500 mg lead acetate and 20 ml of gly-
col was held at 160° for 4 h while refluxing. Ethylene glycol served
both as a solvent and a reducing agent when heated to 160°. To
prepare particles less than 40 nm size, (sample code Pb-2) ethylene
glycol along with ascorbic acid and oleic acid were used as sol-
vents. The mixture has been heated at 160° for 4 h while refluxing.
While, both ethylene glycol and ascorbic acid served as reducing
agents, oleic acid plays the role of a capping ligand. These solutions
were allowed to cool to room temperature and the centrifuged par-
ticles were washed using absolute ethanol followed by vacuum
drying. Same method has been followed to prepare 42–45 nm size
(sample code Pb-3) lead nanoparticles except that no oleic acid has
been added in this case. Phase purity and the structure of the sam-
ples were analyzed using Cu Ka radiation by employing a Philips
Diffractometer (model PW 1071) fitted with graphite crystal
monochromater. The average crystallite size was determined from
the extra broadening of the X-ray diffraction peaks (Fig. 1) of the
sample using Scherer’s formula applied to the strongest peak.
Magnetization measurements are carried out using Quantum
Design MPMS5 SQUID magnetometer. Temperature dependent
magnetization measurement in an applied field of 50 Oe yielded
Tc = 7.2 K (Fig. 2) similar to that of bulk lead [8]. In Fig. 3, we pres-
ent the field dependent magnetization curves at different temper-
atures, which exhibit a hysteresis (see the inset of Fig. 3), albeit
small. The origin of the hysteresis in such small particles is not well
understood but may be due to the sharp corners in the particles
and/or weak coupling between the various particles in the con-
glomerates taken. As the focus of the paper is on the shape of the
Fig. 2. ZFC Magnetization of Pb-1 particles showing the superconducting transition.
Fig. 3. Reversible part of magnetic moment m vs. H data of Pb-1 at different tem-
peratures. Inset shows hysteresis loop.
equilibrium magnetization curves and the highest field Hc upto
which the diamagnetism persists, we have drawn the reversible
part of the magnetization in Fig. 3, obtained by averaging the
increasing and decreasing field branches of the magnetization
curves thereby eliminating the irreversible part.
We note the following important features of the magnetization
data generally observed in all the three samples studied. First, the
nature of the M–H curve significantly deviates from the typical
type I behavior, i.e., after attaining a maximum, the diamagnetism
gradually decreases towards zero. The field Hc upto which super-
conductivity persists is significantly larger compared to the critical
field in the bulk. The zero temperature critical field in lead is
known to be about 800 Oe [8] while the Hc in the samples studied
are about three to four times larger. The critical field Hecxt is ex-
tracted from the M–H curves by drawing a tangent at the field
where the slope dM/dH is maximum as shown in the inset of
Fig. 4. The intersection of the tangent on the field axis gives Hcext
,
which is a lower limit on the actual critical field Hc. The reversible
part of M–H curves at different temperatures obey a simple scaling
as depicted in Fig. 5. We therefore rule out the role of thermal
Fig. 1. X-ray diffraction of the sample Pb-1. Inset blows up the strongest peak. The
continuous and dotted lines are the Gaussian fits to the peak and the instrument
resolution, respectively.

946
G. Yashwant et al. / Physica C 468 (2008) 944–947
the increasing applied field resulting in a linear relation between
magnetization M and applied field H. For particles of size compara-
ble to k, Meissner screening currents pervade all over the volume
of the particle and therefore diminish the overall phase space of
zero momentum pairs (k",ꢁk;) resulting in suppression of the
Ginzburg–Landau order parameter w. We therefore argue that
the pair breaking effect results in an increase in the penetration
depth and thus weaker magnetic screening at higher magnetic
fields. Therefore, k must be replaced by an effective penetration
depth keff renormalized by the field dependent order parameter
as given by
k2
k2eff
¼
ð3Þ
2
hjwj i
2
such that hjwj i tends to unity at zero applied field and decreases at
higher magnetic field. The triangle brackets indicate averaging over
the volume of the superconducting particle. We believe that this ef-
fect becomes very prominent in samples whose length scales are
smaller than or comparable to k.
Fig. 4. Log–log plot of Hecxt and (1 ꢁ t) for the samples Pb-1, Pb-2 and Pb-3. The inset
shows the criterion (discussed in the text) used to extract Hcext
.
Now we apply the Ginzburg–Landau theory to small particles
by idealizing them to be spherical in shape with radius R, in order
to determine the field dependence of hjwj i and thereby understand
2
the peculiar magnetic behavior and the temperature dependence
of the critical field observed experimentally. In a simply connected
system, such as a sphere, with an appropriate choice of gauge for
the vector potential A, the order parameter w is real. Further, for
a particle size small compared to the coherence length n, we can
neglect the spatial derivatives of the order parameter w and con-
sider it to be a constant over the volume of the sphere [11]. The
first GL equation then becomes
ꢀ
ꢁ
2
2pn
u0
2
A2w ꢁ w þ jwj w ¼ 0
ð4Þ
which gives
ꢀ
ꢁ
ꢀ
ꢁ
2
2
2pn
u0
1
2
jwj ¼ 1 ꢁ
A2 ¼ 1 ꢁ
A2
ð5Þ
H
c0k
with critical field in the bulk Hc0 = u0/2pkn. In the above equation,
we must emphasize that k is the zero field penetration depth and
is merely a constant for a given temperature. Eq. (5) describes the
pair breaking effect of the magnetic field, which eventually results
in a decrease in the magnetic screening (increase in penetration
depth) which is accounted by Eq. (3). To understand the magnetiza-
tion of a superconducting nanoparticle at non-zero field, we con-
Fig. 5. Data in Fig. 3 re-plotted in scaled form, i.e., m/mmin vs. H=Hext
.
c
2
2
sider the solution of the London equation r A ¼ ð1=keff ÞA for a
fluctuations of the order parameter as an explanation for the ob-
served behavior.
spherical particle given by A ¼ u^Au where [12]
ꢀ
ꢁ
In the bulk, critical field derived from the Ginzurg–Landau the-
ory is given by the expression Hc0 = u0/2pkn, which gives the tem-
perature dependence Hc0(t) ꢂ (1 ꢁ t) where t = T/Tc. In Fig. 4 we
plot the temperature dependence of Hecxt. In contrast to the temper-
ature dependence of the bulk critical field, HecxtðtÞ obeys the
relation
3H
2b2r2
R
Au
¼
½sin hbr ꢁ br cos hbrꢃ sin h
ð6Þ
sin hbR
^
where u is the azimuthal unit vector in the spherical polar coordi-
nate system, r and h are the radial coordinate and polar angle,
respectively. H is the applied field and b = 1/keff. Upon applying
the limit bR < p and br < p on Eq. (6) and averaging over the volume
of the sphere gives
a
HecxtðtÞ ꢀ Hcextð0Þð1 ꢁ tÞ
ð2Þ
2
hA2i ꢀ 0:1ðHR=2Þ ½1 ꢁ b2R2=3ꢃ:
ð7Þ
where the exponent a ꢀ 0.6–0.7, significantly less than 1 as ex-
pected in the case of bulk superconductors. We now present a pos-
sible explanation for the observed M–H curves and the temperature
dependence of the critical field in the nanoparticle regime.
We note that the above inequalities are valid as R ꢂ 20 nm while
pk ꢂ 100 nm. Combining the above equation with Eq. (5), we obtain
"
#
ꢀ
ꢁ
2
HR
2Hc0k
b2R2
3
2
hjwj i ¼ 1 ꢁ ð0:1Þ ꢄ
1 ꢁ
:
ð8Þ
3. Discussion
2
In bulk type I superconductors, magnitude of induced London
screening currents in Meissner state is governed by the London
equation Js = A/k2, where Js is the supercurrent density and A is
the vector potential. The screening current increases linearly with
Critical field Hc is defined as the field H where hjwj i = 0 [11] and
we obtain,
pffiffiffiffiffiffi
Hc ¼ 10 ꢄ Hc0k=R
ð9Þ

G. Yashwant et al. / Physica C 468 (2008) 944–947
947
field range between zero and Hc, where the superconductor exhib-
its diamagnetism. In bulk materials, m/H is expected to be a con-
stant. The analysis presented in Fig. 6 clearly indicates the
London screening current induced pair breaking and the whole
field range falls under a single regime of field dependent penetra-
tion depth as discussed above, and the type II like description is
perhaps not necessary in the case of nano-scale superconductors.
In conclusion, we presented magnetization measurements on
lead nanoparticles in the size range 35–45 nm with critical tem-
perature same as that in the bulk. It is shown that the critical
fields in these nanoparticles are enhanced significantly above
their bulk values. Further the critical fields are higher for smaller
particles. Moreover, they exhibit magnetic behavior reminiscent
of type II superconductors. The results are interpreted by invoking
the pair breaking effect of the London screening currents which
pervade all over the volume of the particles. This makes the pen-
etration depth increase with field. Further, the enhancement in
the critical field Hc and its temperature dependence in the nano-
particles are shown to be consistent with the Ginzburg–Landau
theory.
Fig. 6. Data in Fig. 3 re-plotted in the form m/H vs. H2.
clearly suggesting that the critical field increases with reducing
particle size. Further Hc0 ꢂ (1 ꢁ t) and k ꢂ (1 ꢁ t)ꢁ1/2 therefore
Hc ꢂ (1 ꢁ t)1/2 where the exponent 0.5 is close to the observed
exponent 0.6–0.7.
To understand the field dependence of the magnetization, let us
consider k2eff to the lowest significant order in R/k by combining
Eqs. (3) and (8) to obtain
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"
#
ꢀ
ꢁ
2
HR
2Hc0k
k2eff ¼ k2 1 þ 0:1
:
ð10Þ
[7] P.W. Anderson, J. Phys. Chem. Solids 11 (1959) 26.
[8] C. Kittel, Introduction to Solid State Physics, seventh ed., John Wiley and Sons
Inc., New York, 1995. Chapter 12.
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Springer-Verlag, Berlin, Heidelberg, 1997 (Chapter 3).
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(Chapter B).
We now calculate the magnetization of a sphere of radius R fol-
lowing London [10] in the small particle limit, i.e., R/keff < p to
obtain
"
#
ꢀ ꢁ
ꢀ
ꢁ
2
2
3
8
R
k
HR
2Hc0k
M ꢀ ꢁ
1 ꢁ 0:1
H:
ð11Þ
Following above equation, we re-plot the M–H data in Fig. 6 in
the form M/H vs. H2, which exhibits linear behavior in the entire