APPLIED PHYSICS LETTERS
VOLUME 78, NUMBER 13
26 MARCH 2001
Thermal instability near planar defects in superconductors
A. Gurevicha)
Applied Superconductivity Center University of Wisconsin, Madison, Madison Wisconsin 53706
共Received 1 December 2000; accepted for publication 1 February 2001兲
It is shown that the local Joule heating due to planar defects, such as grain boundaries, microcracks,
etc., can cause thermal instabilities, which limit the current-carrying capability of
YBa2Cu3O7-coated conductors. Explicit instability criteria are obtained for a planar defect in a film
and for a grain boundary. Thermal instabilities can be triggered by low-angle grain boundaries or
planar defects, which block only a small fraction of the sample cross section. Hot spots near small
defects and overheating of grain boundaries are essential for interpretation of experimental data on
ac losses and E – J curves of polycrystals. © 2001 American Institute of Physics.
关DOI: 10.1063/1.1358361兴
Recent
progress
in
the
development
of
YBa2Cu3O7-coated conductors 共CC兲 with record high critical
current densities J c 共Refs. 1–5兲 has focused more attention
on mechanisms which limit the current-carrying capability of
high-J c superconductors. Magneto-optical studies have
shown that among the strongest current-blocking defects in
CC are networks of extended planar obstacles, such as grain
boundaries 共GB兲, microcracks, or defects in the buffer layer.6
At the same time, recent calculations of current flow around
planar defects have shown that the highly nonlinear electricfield current-density characteristics, E⫽E c (J/J c ) with nⰇ1
causes strong disturbances of the electric field and dissipation, whose spatial extent ⬃na is much larger than the defect
size a. 7,8 The resulting local hot spots near defects can trigger thermal instability, which limits the apparent critical current and gives rise to its variation along the conductor.
An example of current-blocking defects is an edge planar obstacle of length a 共microcrack or a high-angle GB兲 in
a film of width d, or a GB in a bicrystal 共Fig. 1兲. The film is
connected to a dc power supply, which provides a uniform
electric field E 0 away from the defect. The film is on a substrate kept at the temperature T 0 in a strong magnetic field,
so that self-field effects are negligible. Detailed calculations
of E(x,y) for this geometry7,8 have shown that the electric
field and dissipation are strongly enhanced in a narrow ‘‘flux
jet,’’ which extends over the length ⬃an from a defect with
aⰆd/n. Even such small defects at the film edges can cause
local flux jumps,9 which are amplified by singularities of
E(x,y) near the end of the planar defect.7 In this letter, I
consider more ‘‘dangerous’’ larger defects, a⬎d/n, which
cause
a
significant
excess
voltage
⌬V⬃ 关 d/(d
⫺a) 兴 n E 0 d/ 冑n localized in a narrow ⬃d/ 冑n flux jet which
connects the defect and the opposite side of the film.8 For
nⰇ1, the hot spot near the flux jet can make the currentcarrying state unstable, even if the defect is much smaller
than the film width.
To obtain the instability criterion, we calculate a steadystate temperature distribution T(x) along the film, using the
approach which has been developed for inhomogeneous lowT c superconductors.10 For aⰆd, the excess voltage ⌬V is
a兲
Electronic mail: agourevi@facstaff.wisc.edu
constant along the flux jet,8 so we can neglect variations of
T(x,y) across the film and consider a one-dimensional temperature profile, T(x), whose spatial extent along the film is
determined by the thermal decay length L⫽ 冑s ␬ /h. Here, s
⫽A/ P, A is the cross-sectional area, P is the cooled perimeter of the film, h is the heat transfer coefficient to the coolant kept at T⫽T 0 , and ␬ is the thermal conductivity of the
film. We consider the case for which the width d/ 冑n of the
flux jet in Fig. 1 is smaller than L, so that T(x)⬇T m is
nearly constant over the region of strong variation of
E(x,y). Thus, we can use the isothermal solutions for
E(x,y) of Refs. 7 and 8, in which all parameters are now
taken at T m ⬎T 0 . In turn, both the stationary distribution
T(x) and the maximum temperature T m are determined selfconsistently from the following heat balance equations:
⳵ ⳵T h
␬ ⫺ 共 T⫺T 0 兲 ⫹J 0 E 0 共 T 兲 ⫹Q ␦ 共 x 兲 ⫽0,
⳵x ⳵x s
Q⫽
2dJ 0 E 0
冑en
冉 冊
d
d⫺a
n⫺1
E 0 ⫽E c
,
冉 冊
J0
Jc
共1兲
n
.
共2兲
Here, we use the conventional power-law E⫺J characteristic
with nⰇ1, the critical current density J c (T) is defined at the
electric field E c , Q(T m ) is the excess power dissipation due
to defect as calculated in Ref. 8, and e⬇2.718. The delta
function ␦ (x) in Eq. 共1兲 implies that the width d/ 冑n of the
flux jet in Fig. 1 is smaller than the thermal decay length L
⫽ 冑s ␬ /h, as discussed above. We will also use Eq. 共1兲, but
with a different source term Q, to describe overheating of a
GB, which blocks the entire cross section of a bicrystal.
Solution of Eq. 共1兲 gives the following general equation
for the maximum temperature T m :
冕 冋
Tm
T0
册
h
Q 2共 T m 兲
,
共 T⫺T 0 兲 ⫺J 0 E 0 共 T 兲 ␬ dT⫽
s
8
共3兲
which was analyzed in detail in Ref. 10. Equation 共3兲 has
stable solutions only below a threshold current density J m or
electric field E m . Thus, the stable temperature profile shown
in Fig. 1 exists only if E 0 ⬍E m . For E 0 ⬎E m , the defect
causes a local thermal runaway.
We obtain the instability criterion for the case when the
Joule dissipation is mostly localized in the flux jet, so that
0003-6951/2001/78(13)/1891/3/$18.00
1891
© 2001 American Institute of Physics
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1892
Appl. Phys. Lett., Vol. 78, No. 13, 26 March 2001
A. Gurevich
FIG. 2. Graphic solution of Eq. 共4兲, where the solid curve shows the function ␪ (1⫺ ␪ ) n for n⫽10. The points s and u correspond to stable and
unstable states, respectively.
FIG. 1. Current flow, electric field, and temperature distributions in the flux
jet near a planar defect of length a⬎d/n in a film. Bottom: current streamlines as calculated in Ref. 8, where the dark region shows the flux jet of high
electric field, and the light gray marks the region of elevated temperature.
Top: distributions of T(x) and E(x) along the side of the film opposite to
the defect. Similar T(x) and E(x) profiles are characteristic of a GB in a
thin-film bicrystal.
the bulk term E 0 J 0 , in Eqs. 共1兲 and 共3兲 is negligible. As
shown below, the instability occurs for small overheating,
T m ⫺T 0 ⰆT 0 , so we can neglect all temperature dependencies of ␬, h, and n in Eqs. 共1兲–共3兲, except for the most
essential dependence of J c (T), which can be linearized as
J c (T)⫽J 0 ⫺J ⬘ (T⫺T 0 ), where J 0 ⫽J c (T 0 ), and J ⬘
⫽ 兩 ⳵ J c / ⳵ T 兩 T 0 . Then, the integration of Eq. 共3兲 yields the
equation T m ⫺T 0 ⫽(s/ ␬ h) 1/2Q(T m ,E 0 )/2, which can be conveniently expressed in terms of the dimensionless temperature, ␪ ⫽(T m ⫺T 0 )J ⬘ /J 0 :
␪ 共 1⫺ ␪ 兲 n ⫽
E0
,
Ep
E p⫽
冑h ␬ en
dJ ⬘ 冑s
冉 冊
1⫺
a
d
1
dJ ⬘
冑 ␬冉 冊
h
a
1⫺
sen
d
.
共4兲
n⫺1
,
a⬎a c ⫽d 关 1⫺ 共 dJ ⬘ E c 冑sen/h ␬ 兲 1/共 n⫺1 兲 兴 .
共6兲
For the above values of the parameters, we get a c ⯝0.17d for
the liquid cooling (h⯝1 W/cm2 K), and a c ⯝0.1d for the gas
cooling (h⬃10⫺2 W/cm2 K). Therefore, even small defects
(aⰆd) can limit the current-carrying capability of high-J c
superconductors.
The runaway current density J m ⫽J c (E m /E c ) 1/n normalized to the cross-sectional area away from the defect can be
obtained from Eq. 共5兲. For aⰆd, this yields,
J m ⫽J c
冉 冊
a
1⫺ g,
d
g⫽
冉
冑␬ h
dJ ⬘ E c 冑sen
冊
1/n
.
共7兲
n⫺1
Equation 共4兲 determines the maximum temperature T m (E 0 )
as a function of the applied electric field E 0 . As seen from
Fig. 2, the stable root of Eq. 共4兲 共point s兲 exists only if E 0
⬍Em , where the threshold field E m corresponds to ␪ m
⫽1/(1⫹n). Hence, E m ⫽E p ␪ m (1⫺ ␪ m ) n takes the form
E m⫽
results in the nearly exponential drop of E m , well below the
conventional criterion E c ⬃1 ␮V/cm for J c . The thermal instability becomes a dominant current-limiting mechanism, if
E m ⬍E c , when the defect size a exceeds a critical value a c :
共5兲
where the identity (1⫹1/n) n ⫽e for nⰇ1 was used. For
characteristic of CC values, ␬ ⯝0.2 W/cm K, 11 h
⯝1 W/cm2 K 共cooling by liquid nitrogen兲 J 0 ⫽2 MA/cm2,
J ⬘ ⫽0.2 MA/cm2 K, n⫽30, s⫽1 ␮ m, and d⫽1 mm, we obtain that E m ⯝12 ␮V/cm for a⫽0.1d and E m ⯝0.4 ␮V/cm for
a⫽0.2d. For nⰇ1, further increase of the obstacle size a
Therefore, J m is the product of the geometrical factor (1
⫺a/d), which accounts for the reduction of the currentcarrying cross section near the defect 共Fig. 1兲, and the thermal factor g, which is determined by the dissipation distribution. In the adiabatic limit, h→0, both g and J m vanish.
However, for CC, we typically have g⬎1, nⰇ1, thus the
dependence of g on the cooling conditions and the sample
geometry is very week. For ␬ ⫽0.2 W/cm K, J ⬘
⫽0.2 MA/cm2 K, n⫽30, d⫽1 mm, and E c ⫽1 ␮ V/cm, we
obtain g⬇1.21 for the liquid-nitrogen cooling (h
⯝1 W/cm2 K), and g⬇1.12 for the gas cooling (h
⬃10⫺2 W/cm2 K).
Now, we address the GB overheating in a thin-film bicrystal, regarding the GB as an extra dissipation source, Q
⫽J 关 J⫺J b (T) 兴 R. It is assumed that the GB switches into a
flux flow state with the resistance R, if J exceeds the GB
critical current density J b (T), below the bulk J c . Then, T m
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Appl. Phys. Lett., Vol. 78, No. 13, 26 March 2001
A. Gurevich
on a GB can be calculated from Eq. 共3兲, where the bulk
power dissipation JE for J⬍J c is again negligible. The
equation for T m takes the form
2 共 ␬ h/s 兲 1/2共 T m ⫺T 0 兲 ⫽J 关 J⫺J b 共 T m 兲兴 R.
共8兲
For T m ⫺T 0 ⰆT 0 , we linearize J b (T)⫽J b0 ⫺J ⬘b (T⫺T 0 ), and
obtain
T m ⫽T 0 ⫹
J 共 J⫺J b0 兲 R
2 共 ␬ h/s 兲 1/2⫺RJJ b⬘
共9兲
.
Substituting this back to V⫽ 关 J⫺J b0 ⫹J ⬘b (T m ⫺T 0 ) 兴 R, we
obtain the nonisothermal V – J characteristics of a GB,
V⫽
共 J⫺J b0 兲 R
,
1⫺ ␤ 共 J 兲
␤⫽
RJJ b⬘
2
冑␬
s
,
h
1893
the current-carrying capability of GBs. This effect was observed in Ref. 12.
In summary, thermal instabilities can be caused by small
planar defects or low-angle grain boundaries in high-J c superconductors. These instabilities are more pronounced at
lower magnetic fields and temperatures, or under poorer gas
cooling when they can become a serious current-limiting factor. Even if the current flow is stable, the local hot spots near
small defects and overheating of GBs may be essential for
interpretation of ac losses and E – J curves in polycrystals.
This work was supported by the NSF MRSEC 共under
Grant No. DMR 9214707兲.
共10兲
where the parameter ␤ (J) quantifies the effect of the Joule
heating. For ␤ (J)⬎1, the GB triggers a thermal runaway.
For ␤ (J)⬍1, heating gives rise to an upturn of the V – J
curve and an increase of the effective GB resistance, R̃
⫽R/(1⫺ ␤ 0 ), where ␤ 0 ⫽ ␤ (J b0 ). We estimate ␤ 0 , taking
the data of Ref. 12 for a 7° YBCO bicrystal at 2 T and 77 K:
J b0 ⫽0.2 MA/cm2, J b⬘ ⫽0.05 MA/cm2 K, ␬ ⫽0.2 W/cm K, s
⫽260 nm, d⫽25 ␮ m, and R/sd⫽4 m⍀. For the liquid
cooling (h⯝1 W/cm2 K), we obtain that ␤ 0 ⯝1.5⫻10⫺2 ,
thus heating is negligible. However, ␤ 0 considerably increases under poorer gas cooling (h⬃10⫺2 W/cm2 K) at
lower T, where J b ⬃1 – 2 MA/cm2 at 4.2 K. For the above
numbers, we obtain that ␤ (J)⬎1 for J⬎J m ⫽2 冑␬ h/RJ b⬘ 冑s
⯝1.35 MA/cm2, which explains the thermal instabilities observed in Ref. 12 on a 7° YBCO bicrystal at lower T as
J b (T) increases above the 1 – 1.5 MA/cm2 level. A more
quantitative description of this effect requires the account of
the nonlinear V – I characteristics of the GB at J⬍J b .
Heating can, therefore, aggravate the weak-link behavior
of GBs, whose nondissipative critical current densities J b are
determined by the order parameter suppression due to charging and strain effects and local hole depletion on GBs.13
Recently, it has been shown that J b can be increased by Ca
overdoping of GBs.14 In this letter, we show another benefit
of Ca doping, which comes from the significant reduction of
the GB resistance R. 14 As a result, the thermal instability
parameter ␤ in Eq. 共10兲 decreases despite the increase in J b .
Thus, Ca doping also reduces the GB overheating, improving
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