Superconducting Transition and Phase Diagram of Single Crystal MgB2
U. Welp, G. Karapetrov, W. K. Kwok, G. W. Crabtree
Materials Science Division, Argonne National Laboratory, Argonne, IL 60439
Ch. Marcenat, L. Paulius*
Département de Recherche Fondamentale sur la Matière Condensée, Service de
Physique, Magnétisme et Superconductivité, CEA-Grenoble, 38054 Grenoble, France
T. Klein, J. Marcus
Laboratoire d’Etudes des Propriétés des Solides, CNRS, BP 166, 38042 Grenoble, France
K. H. P. Kim, C. U. Jung, H.-S. Lee, B. Kang, S.-I. Lee
NCRICS and Dept. of Physics, Pohang University of Science and Technology, Pohang
790-784, Republic of Korea
The superconducting phase diagram of MgB2 was determined from magnetization,
magneto-transport and the first single-crystal specific heat measurements. A zero-
temperature in-plane coherence length of 8 nm is determined. The superconducting
anisotropy γ increases from a value around 2 near Tc to above 4.5 at 22 K. For H || c a
pronounced peak effect in the critical current occurs at the upper critical field. Evidence
for a surface superconducting state is presented for H || c, which might account for the
wide spread in reported values of γ.
MgB2 has emerged as a fascinating new superconducting material  which in addition
to its surprisingly high value of Tc displays a variety of unusual properties. Electronic
structure calculations  indicate a highly anisotropic, complex Fermi surface consisting
of two disconnected sections: a three dimensional tubular network of mostly boron π-
states and two dimensional cylindrical sheets derived mostly from boron σ-states. Some
of these features have been observed in recent de Haas-van Alphen experiments . The
appearance of multiple superconducting gaps was predicted , with a large gap residing
on the 2D sheets and a small gap on the 3D network. Specific heat  and spectroscopic
measurements [6,7] give evidence for this scenario. In addition, calculations within the
anisotropic Eliashberg formalism  indicate a strongly anisotropic electron-phonon
interaction. However, one of the basic parameters describing an anisotropic
superconductor, the anisotropy coefficient γ = Hc2
c = ξab/ξc, is not well established
for MgB2. Here, Hc2
c, ξab and ξc are the in-plane and out-of-plane upper critical
fields and Ginsburg-Landau coherence lengths, respectively. Reported values vary
widely ranging from 1.1 to 6 depending on the measurement technique and on sample
type, i.e., single crystals [10-15], oriented films , aligned crystallites , or powders
[18,19]. Recent torque  and thermal conductivity  measurements on single
crystals  as well as magnetization measurements on powders  indicate that the
anisotropy coefficient is temperature dependent increasing strongly with decreasing
Here we present a detailed study of the superconducting phase diagram of MgB2
combining magnetization, M(T), magneto-transport and the first single-crystal specific
heat, Cp(T), measurements. The transport and magnetization data were taken on the same
crystal. The upper critical fields for in- and out-of-plane fields were determined from
M(T) and Cp(T) data. A coherence length of ξab (0) = 8 nm is obtained. Transport data
reveal a pronounced peak-effect in the critical current density at Hc2
c. For fields above
c extending up to 1.66xHc2
c we observe strongly non-ohmic transport behavior which
we attribute to surface superconductivity. Upward curvature in Hc2
ab(T) results in a
temperature dependent anisotropy that increases from about 2 near Tc to above 4.5 at 22
K. We note that the occurrence of surface superconductivity could account for the wide
variation in reported values for the anisotropy constant.
The MgB2 crystals were prepared by heat-treating a 1:1 mixture of Mg and B under high
pressure conditions . The crystals are well shaped with straight hexagonal facets and
smooth faces (see picture in inset of Fig. 1b) with typical size of 50 µm. The
magnetization was measured in a commercial SQUID magnetometer. The specific heat
was measured in an ac-specific heat calorimeter  optimized to detect signals from
minute crystals (of the order of 50 ng).
Fig. 1 shows M(T) measured on warming after field cooling the sample for (a) H || c and
(b) H || ab, respectively. Breaks in the slope of the temperature dependence of the
magnetization indicated by the vertical dotted lines are clearly seen and mark the onset of
superconductivity. With increasing field, there is an essentially parallel shift of the
superconducting transition to lower temperatures. This shift is much more pronounced
for H || c (note the different temperature scales in panels (a) and (b)) indicating a strong
superconducting anisotropy of MgB2 as discussed below. Fig. 1c) shows the heat
capacity of a second crystal from the same batch. In zero field a clear step in Cp(T) with
a width of about 2 K is observed. With increasing field the step stays well defined and the
step height decreases as is expected. However, in contrast to polycrystalline samples ,
the transition width remains essentially constant. Using an entropy conserving
construction for defining Tc a phase boundary is obtained that agrees with that determined
from M(T) as discussed below. Thus, the data shown in Fig. 1 represent the
thermodynamic bulk transition of MgB2 into the superconducting state.
Fig. 2 shows the resistive transitions in various fields along the c-axis. The sample is
characterized by a resistivity of ρ =1.6 µΩcm at 40 K and a negligibly small normal state
magneto-resistance. With increasing field the resistive transition moves to lower
temperature and broadens significantly. Similar broadening has been observed in
previous studies on single crystals [10-13]. However, here we show that the broadening
is strongly current dependent. Non-ohmic behavior appears at the onset of the transition,
labeled Ton. With increasing current a steep resistive drop emerges at a lower almost
current independent temperature. At even higher currents a non-monotonic, hysteretic
resistivity behavior arises that is reminiscent of the peak-effect. Peak-effects, that is,
sharp maxima in the temperature and/or field dependence of the critical current and the
corresponding suppression of the resistivity, have been observed just below Hc2(T) in a
variety of low pinning superconductors . The peak-effect occurs also right below the
melting transition separating the vortex lattice from the vortex liquid state in clean,
untwinned YBa2Cu3O7 crystals . However, the vortex liquid state in high-Tc
superconductors is essentially ohmic in contrast to the MgB2-data shown in Fig. 2. These
results will be discussed further below in conjunction with the phase diagram. For H || ab
(not shown here) the resistive transitions do not broaden with field in agreement with
previous reports [10-13], and the peak-effect is largely suppressed.
Fig. 3 summarizes the transport behavior in the peak-effect region at 1.5 T || c. The
current-voltage (I-V) characteristics after field-cooling to 20 K display pronounced
hysteretic behavior. On first increasing the current, a sharp onset of dissipation occurs
near a critical current of 10 mA whereas for decreasing current zero-dissipation is
approached near 5.5 mA. All subsequent current ramps and also the I-Vs taken after
zero-field cooling follow this curve. These results are a manifestation of a current
induced transition from a meta-stable high-Ic vortex phase into a stable low-Ic phase .
As the sample is field cooled through Tc(H) a high-pinning vortex phase nucleates and
stays in equilibrium until the peak-effect temperature is reached (see inset of Fig. 3). At
lower temperatures this phase may survive as supercooled meta-stable state. The
application of a strong enough current dislodges vortices from their pinned meta-stable
configuration and triggers a transition into the stable low-pinning state which does not
change on subsequent current ramps. In zero-field cooled measurements the initial
vortex configuration is the result of flux-gradient driven motion of vortices across the
sample and a low-pinning state analogous to the current-induced state is created.
The magnetic, calorimetric and transport data are summarized in the phase diagram
shown in Fig. 4. For H || c the onset of superconductivity determined from M(T) and
Cp(T) coincide with each other and with the location of the peak effect within the
experimental uncertainty. We identify this line with the upper critical field for the c-axis,
c(T). The observed peak-effect is the one occurring just below Hc2
c. A zero-
temperature value of Hc2
c(0) ≈ 3.5 T can be estimated which, using the WHH relation 
c(0) = 0.7 Φ0/2πξab(0)2, yields the zero-temperature coherence length ξab(0) ≈ 8 nm.
From the resistivity data in Fig. 2 and the Drude relation l = 3/[ρ N(0) vF e2] an electron
mean free path of l ≈ 98 nm can be estimated using the density of states N(0) = 0.7/(eV
unit cell)  and in-plane Fermi velocity vF = 4.8x107 cm/sec . The BCS coherence
length is given by ξ0 = 0.18 h / vF/(kB Tc) ≈ 19 nm indicating that our MgB2 crystals are in
the clean limit, in agreement with previous reports [10-13].
The onset of non-ohmic transport with decreasing temperature defines a line in the phase
diagram lying a factor 1.66 above the Hc2
c-line. This suggests that the resistive onset is a
manifestation of the onset of surface superconductivity  at Hc3
which for a flat surface
in parallel magnetic field occurs at 1.7xHc2. The surface superconducting state supports a
finite critical current which induces non-ohmic transport behavior, however, there is no
diamagnetic signal in the magnetization . For H || c the surface-superconducting
currents are flowing on the vertical side faces of the plate-like crystals. Although
resistive transitions as shown in Fig. 2 could in principle result from filamentary
conduction along impurity phases the observation of a single sharp, current-independent
superconducting transition in zero field indicates an intrinsic mechanism. Within the
experimental resolution there is no feature in the magnetization and specific heat data that
would indicate a second superconducting phase. In addition, nearly identical resistivity
results were obtained on a second, smaller crystal. We also note that the limits of the
resistive broadening reported earlier  encompass the same coefficient, 1.7.
Indications for the development of a surface superconducting state have also been
obtained from the comparison of electrical and thermal transport data . Furthermore,
in a recent study  on the low-current resistive transitions of NbSe2, results closely
resembling those in Fig. 2 have been obtained and interpreted as signature of surface
superconductivity. In that study it was also observed that for reasons not entirely
understood the surface effects are strongly suppressed for in-plane magnetic fields just as
is the case of MgB2 presented here. For H || ab the phase boundaries determined from the
resistive and magnetic onsets coincide within the experimental uncertainty as shown in
Fig. 4. The nature of the surfaces of MgB2 and their effect on superconductivity has
attracted recent interest since surface electronic states on Mg as well as on B terminated
surfaces were observed in bandstructure calculations [27,28] as well as in angle resolved
photoemission experiments . Although their influence on suprconductivity is still
controversial it has been suggested that superconductivity at the ab-surfaces is suppressed
. This might account for the strongly reduced surface superconductivity for H || ab.
Reported γ-values determined from resistivity measurements (usually the resistive onset
is identified with Hc2) on crystals [10-13] as well as on c-axis oriented films  are
generally low, in the range of 2 to 3. In contrast, magnetic measurements on either
powder samples [18,19] or on single crystals  as well as thermal conductivity
measurements  yield γ-values around 4 to 6 at low temperatures. Since surface
superconductivity does not contribute to the magnetization nor the thermal conductivity
but does induce non-linear response in the resistivity a discrepancy between both
determinations by a factor of order 1.7 might be expected which is actually in reasonable
agreement with the spread of the reported values.
While the upper critical field for H || c follows a conventional temperature dependence
for the ab-directions a pronounced upward curvature of Hc2(T). As a result the
superconducting anisotropy is temperature dependent as shown in the inset of Fig. 4.
Similar results have recently been obtained from torque , magnetization on powder
samples  and thermal conductivity  measurements. At high temperatures (i.e.
low fields) γ has a value between 1.5 and 2. At temperatures below 32 to 33 K (fields
around 0.5 T) γ increases rapidly and reaches values above 4.5 near 22 K. An upward
curvature of the Hc2(T)-line can arise in clean superconductors due to non-local effects as
seen for example in borocarbides . However, in those materials the upward curvature
occurs in all crystal directions, and the out-of-plane anisotropy is essentially temperature
independent. An alternative origin of the temperature dependent anisotropy could lie in
the two-gap structure of MgB2. Since the small 3D gap is readily suppressed in applied
fields [5.6] MgB2 behaves like a quasi 2D superconductor in sufficiently high parallel
fields. Thus a steep Hc2
ab(T)-line can be expected. The cross-over between
predominantly 3D to 2D behavior occurs around 0.5 T.
In conclusion, the superconducting phase diagram of MgB2 has been determined using
magnetization, magneto-transport and the first single-crystal caloric measurements. The
in-plane coherence length is 8 nm corresponding to Hc2
c(0) ≈ 3.5 T. The superconducting
anisotropy increases with decreasing temperature from a value around 2 near Tc to above
4.5 at 22 K. For H || c a pronounced peak effect in the critical current occurs at the upper
critical field. Evidence for a surface superconducting state is presented for H || c which
might account for the wide spread of reported values for the anisotropy coefficient.
This work was supported by the U.S. Department of Energy, BES, Materials Science
under contract W-31-109-ENG-38, by the National Science Foundation under grant No.
0072880 and by the Ministry of Science and Technology of Korea through the Creative
Research Initiative Program.
* permanent address: Dept. of Physics, Western Michigan University, Kalamazoo,
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Fig.1 a) and b) Temperature dependence of the magnetic moment measured on
warming after cooling in the indicated fields. Breaks in the slope of M(T) mark
the onset of superconductivity as indicated by the dotted lines. The inset in b)
shows a photo of the crystal. The smooth faces and the Au-contacts are seen.
The irregular shaped features on the surface are glue residues. c) Temperature
dependence of the heat capacity in several fields. The 3T data was used to
subtract the background signal.
Fig.2 Resistive transition measured on cooling in various fields and with various current
densities, 1 mA corresponds to a current density of 360 A/cm2. The onset of non-
ohmic behavior and peak-effect are indicated. At 1.5 T and 10 mA the hysteresis
arising between cooling and warming is indicated by arrows.
Fig.3 I-V characteristics at 20 K and 1.5 T || c measured for increasing and decreasing
current after field cooling (solid arrows) and after zero-field cooling (dashed
arrows). The inset shows the temperature dependence of the critical current in 1.5
T || c.
Fig. 4 Superconducting phase diagram of MgB2 as determined from the magnetization,
specific heat and transport measurements.
H || c
2.52.01.51.0 0.7 0.4 0.1 T
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H || ab3.02.0 1.0 0.1 T0.4
H || c
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3.0 1.50.5 0.0 T
H || c
U. Welp et al.
810 12 14
T = 20 K
H = 1.5 T
fc, I increasing
fc, I decreasing
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5.0 10152025 303540
H || c H || ab
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