Current Applied Physics 16 (2016) 1046e1051
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Current Applied Physics
journal homepage: www.elsevier.com/locate/cap
Effect of vortex-vortex interactions on the critical current density of
single-crystalline MgB2 thin films
Duong Pham a, Soon-Gil Jung a, K.J. Song b, M. Ranot c, J.H. Lee a, N.H. Lee a, W.N. Kang a, *
a
BK21 Physics Division and Department of Physics, Sungkyunkwan University, Suwon 440-746, Republic of Korea
Division of Science Education (Physics), Institute of Fusion Science, Chonbuk National University, Jeonju 561-756, Republic of Korea
c
Materials Deformation Department, Korea Institute of Materials Science (KIMS), Changwon, Gayeongnam 51508, Republic of Korea
b
a r t i c l e i n f o
a b s t r a c t
Article history:
Received 6 April 2016
Received in revised form
2 June 2016
Accepted 9 June 2016
Available online 11 June 2016
We have investigated the influence of inter-vortex interactions on the critical current density (Jc) and flux
pinning properties of single-crystalline MgB2 thin films grown using a hybrid physicalechemical vapor
deposition system. Magnetic field dependences of Jc were measured over a 5e35 K range of temperatures
for fields applied parallel and perpendicular to the ab-plane of the film’s surface. Two regions along the
Jc(H) curve, the low-field plateau region and the high-field slope region, which are respectively known as
the single-vortex and collective pinning regimes, are clearly distinguishable. A ratio a0/l between the
inter-vortex spacing a0 and the penetration depth l was calculated at each temperature, to investigate
the effect of vortex-vortex interactions on the Jc(H). We found that the a0/l dependences of the
normalized Jc(H,T) data tend to fall onto one curve irrespective of the temperature, if only we use a
certain average value of l. Furthermore, the flux pinning mechanism shows a crossover from dTc-pinning
to dl-pinning with increasing magnetic fields, indicating the coexistence of different pinning mechanisms
in MgB2.
© 2016 Elsevier B.V. All rights reserved.
Keywords:
Single-crystalline MgB2 thin film
Flux pinning
Vortex-vortex interactions
HPCVD
1. Introduction
The critical current density (Jc) in a superconductor is the
maximum current density that superconducting materials can
carry without power dissipation, and it is one of the most important factors for engineering applications of superconductors. In
type-II superconductors, the Jc in magnetic fields usually shows an
angular dependence due to the anisotropy of materials. This
angular dependence provides primary information for the anisotropy (g) of superconducting electron’s effective mass (m*), which
leads to the anisotropy of penetration depth (l), coherence length
(x), and the upper critical field (m0Hc2) [1].
When an external magnetic field is applied that is higher than
the lower critical field (m0Hc1) of type-II superconductors, magnetic
flux can penetrate into superconductors in the form of quantized
magnetic flux, through so-called vortex, each with an integer
number of flux quanta of magnitude 4 0 ¼ h/2e where h is Plank’s
constant and e is the electron charge. Each vortex has a normal core
with a radius given by x. Electrical supercurrent circulates around
* Corresponding author.
E-mail address: wnkang@skku.edu (W.N. Kang).
http://dx.doi.org/10.1016/j.cap.2016.06.005
1567-1739/© 2016 Elsevier B.V. All rights reserved.
the vortex core, exponentially decaying in magnitude over a length
given by l [1,2]. Power dissipation in type-II superconductors is
generated by vortex motion, which can be controlled by flux
pinning because the vortex motion is mainly from the competition
between the flux pinning force Fp and the Lorentz driving force FL.
Generally, the flux pinning strength is determined by the kinds of
defects and their densities in the superconductor [2,3], which is
extrinsic pinning. But intrinsic pinning has been reported in high-Tc
cuprate superconductors (HTSs) due to their layered structure and
large anisotropy [1,4].
Collective pinning theory is often used to describe the flux
pinning strength, which can simply be determined by the powerlaw relation Jc(H) ∝ Hb, where the exponent b depends on the
pinning strength. For instance, if the flux pinning force becomes
weaker, the b value should be larger because the Jc will be rapidly
decreased with increases in the magnetic field, and vice versa [3].
MgB2 discovered in 2001 by Nagamatsu et al. [5] is a metallic
superconductor with two superconducting energy gaps (Dp and
Ds) which lead to an unusually strong temperature dependence in
the anisotropy of the upper critical field (m0Hc2) [6]. In addition, this
compound is a promising material for superconducting device
applications due to its superior properties such as its simple crystal
D. Pham et al. / Current Applied Physics 16 (2016) 1046e1051
structure, relatively high critical temperature Tc, absence of weak
links, and so on [7,8]. For these reasons, both fundamental research
on physical properties and applied research such as on improvements of Jc have been conducted on this material. Numerous results
have been reported to enhance the Jc in MgB2 wires, tapes, and
single crystals through heavy-ion irradiation and chemical doping
[9,10]. These methods were also carried out on MgB2 thin films, but
even high energy ion irradiation could not improve the Jc significantly [11], carbon doping was shown to decrease the self-field Jc,
although it is effective at high fields [12]. Therefore, a complete
understanding of the flux pinning properties in high-quality MgB2
thin films should be done, which could be a big aid in improving its
Jc. In addition, it has been agreed that vortex-vortex interactions
play a crucial role in the Jc performance of superconducting thin
films [13,14] but more experiments should be done to clarify the
relationship between them. In this work, we study the effect of
vortex-vortex interactions on the Jc, considering the inter-vortex
distance as well as the flux pinning properties of singlecrystalline MgB2 thin films. The flux pinning mechanism was
investigated by means of the generalized inversion scheme (GIS).
2. Experiment
Single-crystalline MgB2 thin films were fabricated by a hybrid
physical-chemical vapor deposition (HPCVD) system, which is
known to be one of the best methods by which to deposit highquality MgB2 thin films. The detailed fabrication process of the
HPCVD system has been described elsewhere [15]. One of the most
important requirements for MgB2 film deposition is providing a
sufficiently high Mg vapor pressure around the substrate, as the Mg
1047
to B ratio should be larger than 1 to 2 in order to produce the
stoichiometry of the film’s composition [16]. In addition, a clean
environment without oxygen and pure Mg and B sources are other
requirements for the growth of high-quality MgB2 thin films. The ccut Al2O3 (10 mm 10 mm) substrates were used for the growth of
single-crystalline MgB2 thin films, which provides a good lattice
match with hexagonal MgB2 [17].
The crystal structure and epitaxial growth of the MgB2 thin films
were investigated by x-ray diffraction (XRD). A detailed study on
the quality of the MgB2 thin films was reported elsewhere [18]. The
thickness and surface morphology of the films were measured
using a scanning electron microscope (SEM), and films with
thicknesses of around 1 mm were used for this study. The temperature dependence of the resistivity under an applied field up to 9
Tesla was measured by using a physical property measurement
system (PPMS 9T, Quantum Design). The magnetization hysteresis
(M-H) loops were measured by using a magnetic property measurement system (MPMS 5T, Quantum Design). The critical current
density (Jc) was estimated from the M-H loops by using Bean’s
critical state model (Jc ¼ 30DM/r), where DM is the height of the MH loops and r is the radius corresponding to the total area of the
film’s surface normal to the field.
3. Results and discussion
Fig. 1(a) presents the x-ray diffraction (XRD) 4-scan curve of
single-crystalline MgB2 thin films, which shows sharp peaks every
60 , indicating the hexagonal symmetry. The cross-sectional and
surface image were observed with a scanning electron microscope
(SEM), which show very dense MgB2 films without any grain
Fig. 1. (a) The XRD 4-scan curve of the (000l) Bragg reflection shows significant intensities every 60 , which confirms the hexagonal symmetry of the MgB2 thin films. (b) Crosssectional SEM image of single-crystalline MgB2 thin films grown on Al2O3 substrate, which shows a very dense structure without any grain boundaries. (c) Temperature dependence
of resistivity r for MgB2 thin films. Inset shows a sharp superconducting transition at 40.2 K with a DTc ¼ Tc,90% Tc,10% ¼ 0.1 K. The residual resistivity ratio (RRR ¼ r300K/r41K) was
about 37. (d) The surface SEM image of MgB2 thin films, which shows a smooth surface morphology and free of grain boundaries.
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D. Pham et al. / Current Applied Physics 16 (2016) 1046e1051
boundaries, as shown in Fig. 1(b) and Fig. 1(d), respectively. Fig. 1(c)
shows the temperature dependence of the resistivity (r) for singlecrystalline MgB2 thin films. The high residual resistivity ratio
(RRR ¼ r300K/r41K z 37) indicates a high purity of our films [19]. The
high superconducting critical temperature (Tc,onset ¼ 40.2 K) of the
films compared with bulk MgB2 probably came about due to
epitaxial strain generated during the growth process [20].
jjab
Fig. 2(a) shows Jc(H) curves for H⊥ab ðJc⊥ab Þ and H||ab ðJc Þ at 5
and 20 K, for single-crystalline MgB2 thin films, calculated from MH loops data. In both directions of applied magnetic field Jc(H)
clearly exhibits two regions: the H-independent plateau region at
low fields, which is usually attributed to a single-vortex pinning
regime; and, the high field region that follows the power-law
behavior Jc(H) ∝ Hb, where collective pinning occurs owing to
the b value in the sloped region being over 1 (b will be further
jjab
discussed below). The single-vortex pinning regime in the Jc
is
significantly wider compared to that of the Jc⊥ab , as presented in the
inset of Fig. 2(a). That is often observed in superconducting materials of high anisotropy, especially in HTSs [21]. The wide plateau
jjab
region of Jc in MgB2 films may originate from numerous factors,
but we mainly consider the two reasons, intrinsic pinning and
surface pinning (a kind of extrinsic pinning).
Intrinsic pinning could occur in MgB2 due to the layered structure of the material, as in HTSs. Intrinsically, the vortices can be
pinned to the region between CuO2 planes in HTSs, because the
region between the CuO2 layers is non-superconducting and the
distance of separation is longer than x, which usually induces an
angular dependence of Jc(H) [4,22]. Similarly, MgB2 has a layered
structure with two superconducting energy gaps of Ds and Dp
originating from the px,y and pz orbitals of the boron layer,
respectively [23]. But the possibility of intrinsic pinning in MgB2 is
still subject to debate because the coherence length is much larger
than the distance between superconducting boron planes [24].
However, some experimental evidence for intrinsic pinning in
MgB2 single crystals was reported, and also in MgB2 films [25,26].
As shown in Fig. 1(b), the samples are free of grain boundaries,
suggesting that the pinning by grain boundaries, which is one of the
most effective sources of flux pinning in MgB2, can be ignored in
these films. Therefore, we believe that the intrinsic pinning contributes to the anisotropy of Jc in single-crystalline MgB2 thin films.
In parallel configuration, the contact of vortex with the ab-plane
surface is significant compared to the perpendicular case. The
vortex bends and lengthens its core when crossing an elevation of
the surface and then produces an energy barrier against the vortex
motion [27]. Therefore, surface pinning would be predictable in our
films. An additional pinning effect caused by the strain of the filmjjab
substrate interface might be the reason for the broadening of Jc .
That would be quite consistent with the idea that the high Tc of the
film comes from epitaxial strain generated in the growth process
[20].
Although the anisotropy of Jc(H) in MgB2 thin films is quite clear,
the estimation of the exact Jc value from M-H loops is difficult
because of the demagnetization factor and the geometrical effect
produced by the surface barrier [28]. However, in this study, the Jc
behavior as a function of magnetic field is of major importance,
more than the absolute Jc value.
Fig. 2(b) shows the temperature dependence of the upper critical field (m0Hc2) for H⊥ab and H||ab, where the m0Hc2(T) was
determined by a 90% criterion from r-T curves at each magnetic
field. In order to estimate m0Hc2(0), we extrapolated the m0Hc2(T)
curves to zero Kelvin using a linear fit to experiment data at the
higher fields above 4 T. If we consider an anisotropic superconductor in an applied field H⊥ab, the supercurrent circulates in the
Fig. 2. (a) Magnetic field dependences of the critical current density (Jc) of singlecrystalline MgB2 thin films for H⊥ab and H||ab at 5 and 20 K. Inset shows the
normalized Jc(H) for H⊥ab and H||ab at low fields and at 5 K. (b) The upper critical field
(m0Hc2) for single-crystalline MgB2 thin films determined from resistivity measurements for H⊥ab and H||ab, where the m0Hc2 was determined by a 90% criterion from r-T
curves. The m0Hc2(0) was estimated by a linear extrapolation.
basal ab-plane. Hence, the penetration depth only consists of lab
where the a and b axis are equivalent. But for H||ab, the penetration
depth will include both lab and lc, so that the vortices have ellipselike shape due to the anisotropy. Thus, it is convenient to estimate
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
3
2
an average value of penetration depth as lavg ðTÞ ¼ lab ðTÞlc ðTÞ for
further investigation. The details of this expression with corresponding to effective mass tensor were shown in Refs. [29,30]. In
this study, we use lab(0) ¼ 100 nm [31], and the relation of
lab(T) ¼ lab(0)/[1 e (T/Tc)2]0.5 to obtain the lab (T), which was
Table 1
The lab(T) and lc(T) are calculated by using lab(T) ¼ l(0)/[1 (T/Tc)2]0.5 (Ref. [32]),
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
3
and gl (Ref. [33]). The average l are determined by: lavg ðTÞ ¼ l2ab ðTÞlc ðTÞ
(Refs. [29,30]).
T (K)
gl ¼ lc/lab
lab (nm)
lc (nm)
lavg (nm)
5
10
15
20
25
30
35
1.13
1.28
1.51
1.77
2.00
2.22
2.40
101
103
108
115
128
151
207
114
132
163
204
256
335
497
105
112
124
140
161
197
277
D. Pham et al. / Current Applied Physics 16 (2016) 1046e1051
1049
Fig. 3. (a) Jc(H) for H||ab at various temperatures. Normalized Jc(H,T) as functions of (b) a0/lab, (c) a0/lc and (d) a0/lavg, where lab, lc and lavg are penetration depths along the abqffiffiffiffiffiffiffiffiffiffiffi
3
l2ab lc and a0 z 1.07(40/B)0.5.
plane, c-axis and the average penetration depth of MgB2, respectively, and a0 is the inter-vortex distance. Here, we used the relations lavg ¼
reported to be adequate for MgB2 [32]. The anisotropy of l (gl ¼ lc/
lab) in MgB2 is inconsistent with the anisotropy of Hc2
jjab
⊥ab Þ due to the existence of two superconducting
ðgHc2 ¼ Hc2 =Hc2
energy gaps. When the temperature increases, the gl increases as
well, because of the rapid reduction of the supercurrent density on
the p-band, small gap, compared to that on the s-band. Therefore,
we considered the temperature dependence of gl for anisotropic
multi-gap superconductors to calculate lc [33]. The detailed values
of gl(T), lab(T), lc(T), and lavg(T) are summarized in Table 1.
jjab
Fig. 3(a) shows magnetic field dependences of Jc for the singlecrystalline MgB2 thin films at various temperatures. The exponent b
in the sloped region is 1.30 and 1.33 at 5 and 30 K, respectively,
whereas b ¼ 1.0 and 1.15 were observed for the Jc⊥ab at those temperatures in our previous work [34], reflecting that the thermal
jjab
fluctuation effect in vortex pinning is probably weak for the Jc .
In the single-vortex pinning regime, the inter-vortex spacing
a0 z 1.07(4 0/B)0.5 is large enough to make the vortex-vortex interactions ignorable so that the vortices can be pinned without
power dissipation. Hence, Jc is independent of the magnetic field. As
the magnetic field increases, the vortex density increases as well.
The interaction between the vortices becomes stronger and leads to
the reduction of Jc [3]. While the inter-vortex interactions seem to
be the most important factor affecting the Jc for H⊥ab, the vortex
energy for H||ab is quite complicated due to the influence of other
factors. For instance, the vortices near the film’s surface will be
influenced by the London screening currents. In addition, for H||ab,
the interaction between vortices depends not only on the spacing of
the vortices but also on vortex positions in the film. A vortex located
at the edge of the film does not interact with other vortices because
the magnetic flux inside a core is zero [14,28]. However, within
thick films (d [ l), such as our samples, the vortex-vortex interactions can be considered similar to those of the bulk case.
Therefore, the vortex-vortex interaction energy (Eint) can be
roughly estimated by the following relation [13]:
Eint ¼
a 420 d
K0 0 ;
2
l
2pm0 l
(1)
where K0(x) is the modified Bessel function of the second kind of
zeroth order. Qualitatively, there are two main forces which act on a
vortex in the surface layer in a parallel field, i.e., repulsion between
vortices making them tend to move away from each other, and the
force acting on a vortex due to screening currents which is directed
toward the film’s center.
The effect of inter-vortex interactions on the Jc is investigated by
considering the temperature dependence of the a0/l ratio as shown
in Fig. 3(b)e(d). Here, the Jc’s are normalized by the zero-field Jc and
the inter-vortex distances are divided by lab, lc and lavg at each
temperature. In order to investigate the overall relationships,
jjab
including the temperature dependencies of Jc and a0/l, we have
evaluated the average l at each temperature. According to equation
(1), the Eint will rapidly increase at a0 z 2l, so Jc will start to
decrease at a0/l z 2 which is consistent with the previous results
obtained for H⊥ab [34]. But for H||ab, Jc starts to decrease at a0/
lavg(T) z 1.3, as shown in Fig. 3(d). This broad single-vortex pinning
jjab
regime of Jc is caused by an increment of pinning sources and
surface screening currents, as mentioned above.
The comparison of inter-vortex interactions on Jc along the abplane and c-axis is indicated in Fig. 3(b) and (c), respectively. While
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D. Pham et al. / Current Applied Physics 16 (2016) 1046e1051
density of state [36]. The coherence length is proportional to the
mean free path (l) of the carriers, hence, the applied pressure can
enhance the mean free path fluctuations in MgB2 [36]. In addition,
T. Klein et al. had reported a significant reduction of a coherence
length by increasing applied magnetic fields in single-crystal
quality of MgB2 [37]. Therefore, the crossover pinning mechanism
from dTc- to dl-pinning in single-crystalline MgB2 films is believed
due to a reduction of the mean free path, leading to the dominance
of dl-pinning. These results on the crossover of pinning mechanism
will provide useful information for comprehending the pinning
mechanism and improving the performance of Jc(T,H) for MgB2.
4. Conclusions
Fig. 4. Normalized Jc as a function of temperature at various magnetic fields. The solid
lines are fitting curves for the flux pinning mechanism obtained from the generalized
inversion scheme (GIS) with l(t) ∝(1t4)1/2 for high-Tc cuprate superconductors,
while the dotted lines are obtained from the modified GIS using a reasonable temperature dependences of penetration depth l(t) ∝(1t2)1/2 for MgB2. The modified
GIS results in a suitable description for the flux pinning mechanism of high-quality
MgB2 thin films.
the Jc’s start to decrease at a0/lab(T) z 1.7, this value is around 1.1 for
a0/lc(T), reflecting the strong effect of surface currents along the caxis compared to that of currents along the ab-plane. The tendency
of Jc with respect to temperature is reversed in Fig. 3(b) compared
to Fig. 3(c). In addition, there is a large deviation of Jc as function of
a0/lc(T). That can be explained by the rapid increases of lc(T) rather
than lab(T), leading the exponentially increase of the denominator
in equation (1) in which the Eint is reduced at high temperature.
Interestingly, while the Jc’s as functions of a0/lab(T) and a0/lc(T)
show large deviations in the slope region, they can be considered to
be a single curve when we use a0/lavg(T). This clear overlapping
presented in Fig. 3(d) probably demonstrates that the effect of the
temperature on vortex-vortex interactions can be negligible by
considering the temperature dependence of lavg.
The generalized inversion scheme (GIS) was used to investigate
the flux pinning mechanism for MgB2 thin films in a parallel field
[35]. The temperature-dependent normalized critical current densities, Jc(T)/Jc(5 K), at various fields were plotted to understand the
pinning mechanism, as presented in Fig. 4. We slightly modified
pinning by spatial variations of Tc (dTc-pinning) and pinning by
variations of the mean free path (dl-pinning) of charge carrier by
considering l(t) ∝(1 e t2)1/2 instead of l(t) ∝(1 e t4)1/2 which has
been generally used for HTSs in GIS [35], where t ¼ T/Tc. Using this
reasonable l(t) for MgB2 [32], we induced Jc(t) ∝ (1 þ t2)3/2
(1 e t2)5/2 and Jc(t) ∝ (1 þ t2)1/6 (1 e t2)7/6 for dl-pinning and dTcpinning, respectively, indicated on Fig. 4 by dotted lines. The solid
lines are Jc(t) ∝ (1 þ t2)1/2 (1 e t2)5/2 and Jc(t) ∝ (1 þ t2)5/6 (1 e t2)7/6
for dl-pinning and dTc-pinning, respectively, using l(t) ∝ (1 e t4)1/2.
The temperature dependence of Jc(T)/Jc(5K) shows a crossover from
dTc- to dl-pinning with increasing magnetic field, indicating the
possibility of co-existence of different pinning mechanisms, while
dTc-pinning has been mostly reported to be dominant in MgB2 [24].
Since a similar behavior has been observed for H⊥ab [34], we
consider that the practical pinning centers of these pinning
mechanism may come from the point defects in the films [18].
Recently, the crossover pinning mechanism from dTc- to dlpinning has also been reported in MgB2 by applying a high pressure, which reduces a coherence length, and lead to a change of
In conclusion, we have investigated Jc(H) for H⊥ab and H||ab of
single-crystalline MgB2 thin films. The better field performance of
Jc(H) for H||ab compared to Jc(H) for H⊥ab was explained by the
presence of intrinsic pinning and surface pinning. In addition, we
clarified the relationship between inter-vortex interactions and
Jc(H) via the a0/l ratio, where l at various temperatures was estimated from a formula suitable for MgB2. When Jc values are plotted
versus a0/lavg(T) the points fall very near to a single smooth curve,
regardless of the temperature, which clearly shows that vortexvortex interactions are one of the principal determinants of Jc
performance. A modified GIS was used to study the flux pinning
mechanism of MgB2 thin films, which shows a crossover from dTcto dl-pinning with increasing magnetic fields.
Acknowledgements
This work was supported by Mid-career Researcher Program
through National Research Foundation of Korea (NRF) grant funded
by the Ministry of Education, Science and Technology (MEST) (No.
2010-0029136). S.-G. Jung was supported by the National Research
Foundation (NRF) of Korea grant funded by the Korean Ministry of
Science, ICT and Planning (No. 2012R1A3A2048816) and Basic
Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (NRF2015R1D1A1A01060382).
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