**Joule Heating Induced by Vortex Motion in a Type-II Superconductor**

Z. L. Xiao and E. Y. Andrei_{}

Department of Physics and
Astronomy, Rutgers University, Piscataway, New Jersey 08855

P. Shuk and M. Greenblatt

Department of Chemistry,
Rutgers University, Piscataway, New Jersey 08855

We present
experiments that determine the temperature increase in a type II superconductor due to Joule heating induced
by vortex motion. The effect of Joule heating is detected by comparing the
response of the vortex lattice to fixed amplitude current steps of short (10ms) and long (4s) duration, where the Joule
heating is negligible and saturates respectively. The thermometry is based on
the temperature dependence of the voltage response of the vortex lattice to a
driving current. By monitoring the temperature increase in well thermalized
NbSe2 samples adhered on a sapphire substrate with GE varnish we obtain the
heat transfer coefficient between the sample and the bath and show that the
heating is primarily due to the power dissipated by the vortex motion.

PACS number: 65.90.+i,
66.70.+f and 74.60.Ec

** **

** **

** **

** **

In the mixed state of a type-II superconductor
vortex motion induces dissipation. Due to the finite heat-removal rate from the
sample to the bath, Joule heating associated with the vortex motion leads to an
increase in sample temperature. While this heating can be accompanied by
interesting physical phenomena such as the hotspot effects [1-4], it also
causes difficulties in analyzing transport experiments and in determining the
physical properties related to the vortex motion, especially at high
dissipation levels [5-13]. Joule heating can be reduced significantly by using
short pulsed currents [5-7,14] or by applying the current at high ramping rates
[8,12]. In typical transport measurements however, the current is applied
continuously which leads to uncertainties in the temperature of the system.
Because most properties of the superconductor are temperature dependent it is
important to determine the temperature increase in the presence of a current.
This is usually estimated from the heat flow equations by using the heat
transfer coefficient *h* (or the
thermal boundary resistance *R*_{bd}=*h*^{-1}) between the sample and
the bath as determined from photoresponse [15-17] or by comparing the
experimental data with theoretical models [1-4,11,18]. However both methods
have limited applicability since the heat transfer coefficient derived from the
experimental data is model dependent. Another approach is to place the
thermometer close to or on the sample [13,19] but even in this case the
measured temperature is not necessarily that of the vortex lattice. In this
paper we introduce a method to obtain a direct measure of the temperature
increase of the moving vortex lattice by using the temperature dependence of
the vortex response to a driving current as a thermometer. The measurement
principle is based on the fact that for sufficiently short current pulses Joule
heating is negligible. Thus, by
comparing the vortex response to short and long current pulses we obtain a
direct and independent measure of Joule heating.

The experiment was carried out in the low-*T*_{c} superconductor 2H-NbSe_{2}.
This material exhibits a pronounced peak in the temperature dependence of the
critical current just below *T*_{c}.
Interesting physical phenomena such as plastic flow, metastabilities, and flow
induced organization were found to accompany the vortex motion in this system
[20-22]. Because of the strong
temperature dependence of these properties, small amounts of Joule heating can
induce significant deviations of the experimental data from those without
heating. For example it can produce results similar to those expected from
another mechanism - a peak in the current dependence of the differential
resistance *dV/dI* [23]- which is predicted to signal a dynamic phase
transition in the moving vortex system [24,25]. Thus, in order to correctly
interpret the transport results it is necessary and important to determine the
temperature increase due to Joule heating.

The data presented here were acquired in two pure
single crystals sample A and sample B.
The corresponding dimensions are 3(*L*)´0.65(*W*)´0.025(*D*) mm^{3}
and 5´1.65´ 0.020 mm^{3}.
In zero magnetic field the critical temperatures are 7.1 K and 7.2 K
for sample A and sample B respectively. As shown in the inset of Fig.1(a) the
samples were glued on sapphire substrates with a thin layer of GE (7031)
varnish. The sapphire substrate was thermally anchored to a regulated pumped
helium bath through a Cu holder. A RhFe calibrated thermometer mounted on the
sapphire substrate was used to monitor the substrate temperature. In order to
achieve good thermal contact and avoid mechanical stress, current and voltage
pads were made by depositing a layer of gold (5-10mm) with a thin buffer layer of titanium on
both the sapphire substrate and the sample and then soldered together with Ag_{0.1}In_{0.9}.
The same solder was used to attach current and voltage leads to the gold pads
on the sapphire substrate. The solder for the current leads was wrapped around
the sample edges to improve the homogeneity of current injection and to
minimize contact resistance which was typically less than 0.5W at room temperature. Our
measurements employed a standard four-probe technique. The distance between the
two voltage contacts was *l *= 1.5 mm
and 1.4 mm for sample A and sample B respectively. A waveform generator was
used to apply the current steps and the corresponding voltage response was
amplified with a low noise (1nV/Hz^{1/2}) fast amplifier and recorded with a 100MHz digital
oscilloscope. For the DC resistance and differential resistance measurements we
used a commercial current source, a nanovoltmeter and a low frequency lock-in
detector. The magnetic field was kept along the *c* axis of the sample and the current flow was in the *a*-*b*
plane. In order to avoid the possible
metastabilities of the vortex system the data reported here were recorded from
the annealed vortex lattice [21,22].

The effect of
Joule heating are clearly seen in Fig.1 by comparing the current-voltage (*I*-*V*)
curves obtained with DC and short pulsed currents. The pulses consisted of
intervals with current “on” followed by cooling intervals without current. The
duration of the pulses was varied to find the optimal conditions of no heating
while minimizing the noise level. We determine the presence of heating from the
*I*-*V*
curves at high currents. At high currents *I>>I _{c}*,
if no heating is present one observes the expected free flux flow behavior [26]

The temperature
increase due to Joule heating grows with pulse duration until, for sufficiently
long pulses (~4s) it is
indistinguishable from DC measurements as illustrated in Fig.1(b). The time
scale for which heating in the pulsed and DC data become comparable is given by
the heat diffusion time, *t* =S_{i} *L*_{i}^{2 }/*D*_{i} ~10ms, between sample and thermal anchoring point
through the various substrate layers- GE varnish and sapphire. Here the
summation is over the two substrate layers, *L*_{i}
is a characteristic layer thickness, *D*_{i}=*C*_{pi}/*k*_{i} the thermal diffusion constant with *C*_{pi }and *k*_{i} the
specific heat and thermal conductivity respectively. Thus, in the limit of long pulses of duration *t*>>*t*,
the temperature increase no longer depends on the pulse length and approaches
that of a DC current. Indeed, as shown in Fig.1, the *I-V* curves obtained with a pulsed currents of *t*_{0}=4s are nearly identical to those obtained with a DC
current. Based on this result we used 4s pulses to simulate heating in DC
currents in order to expedite data collection and simplify its analysis. Thus
our experimental procedure consists of monitoring the voltage response to a
short (10ms) current pulse at bath
temperature *T*_{1 }followed by
a long (4s) pulse at the same bath temperature. Since heating can be ignored at 10 ms and saturates at 4s, the first measurement
gives the voltage response at sample temperature *T*_{1, }while the second is the response at the heated
sample temperature *T*_{1}*+dT*. In order to determine *dT*, we heat the sample to reach a bath
temperature *T*_{2 }at which
the voltage response to the 10ms pulses is equal to the DC
response at bath temperature *T*_{1}.
The difference of the two bath temperatures *T*_{2}-*T*_{1}=*dT* is thus the temperature change due to the Joule heating of the
DC current at bath temperature *T*_{1}.

The response to long and short pulsed currents is
shown in the left and right panels of Fig.2 respectively. Panels (a1) and (a2)
represent calibration curves obtained by measuring the voltage response to a
current step in the normal phase (*T>T*_{c})
where the resistance is almost temperature independent. From this calibration
we ascertain that the response time of the voltage amplifier is sufficiently
short (<2ms) to ensure good temporal
resolution of the pulsed measurements. In panels (b1) and (b2) we present the
response to short and long 70mA current pulses below the peak effect region, at
a field of 1T and bath temperature *T*_{1 }= 4.60K.
Since in this region d*I*_{c}/d*T*<0, Joule heating reduces the
critical current and should result in an increased voltage response. Indeed the
data show that the long pulse voltage response is larger than the 10ms response, as expected of Joule heating. By
raising the bath temperature to *T*_{2}=4.677K
the 10ms response becomes equal to
the 4s response at *T*_{1}=4.60K.
It follows that for bath temperature 4.60K, the sample temperature increases to
4.677K in the presence of a DC current of amplitude 70mA. This gives a
temperature increase *dT*=77mK in the
presence of a 70mA DC current. In panels (c1) and (c2) we present data in
the lower part of the peak where d*I*_{c}/d*T*>0, for a field of 1.8T and
temperature *T*_{1}=4.30K.
Using a similar procedure we find that in the presence of a 29mA DC current the
sample temperature is *T*_{2}=4.318K
and the corresponding temperature increase is *dT *=18mK.

Repeating this procedure at various driving currents
we map out the current dependence of the temperature increase for various
temperatures and magnetic fields, as shown in Fig.3(a). For all the data sets
we find that *dT* is linear in *P*=*IV*,
the power dissipated due the vortex motion, suggesting that the Joule heating
induced by the vortex motion is the primary contribution to the temperature
increase. The slope of this data gives the heat transfer coefficient*, h=P/lWdT, *with *lW* the sample area between the voltage leads in contact with the
substrate. By plotting *dT* against the
reduced power *P/lWh* all the data in
Fig.3(a) collapse onto a single straight line as shown in Fig.3(b). The values
of *h* shown in the inset of Fig.3(b)
are in the range 40 to 100mW/cm^{2}K. This indicates that for our
samples which were adhered on the sapphire substrate with GE varnish, the heat
transfer coefficient is at least three orders of magnitude below the values
reported for samples directly grown on sapphire or SrTiO_{3} substrates
[4,15-18].

As is the case in other transport measurements in
superconducting crystals [27], the resistance of our sample (~0.4-0.8W at room temperature) is
comparable to the contact resistance. The contact resistance is expected to
drop significantly with decreasing temperature, but its value at our
measurement temperatures is unknown. This leaves the possibility that part of
the observed Joule heating is due to the power dissipated in the current
contacts. If heating is generated by power dissipated due to the lead
resistance as well as to vortex motion the temperature increase is: *dT* =*aP+a _{c}R_{c}I*

The temperature
sensitivity of the *I*-*V* characteristics becomes strikingly
evident when the experiment is carried out at a bath temperature that places
the system in the lower part of the peak regime *T<T*_{p}, and then heating brings it above the peak *T>T*_{p}. In this case the
response to a DC current initially diminishes with increasing current
amplitude, since *dI*_{c}/*dT*>0 [lower right inset of Fig.1(b)],
but once the temperature exceeds *T*_{p} and *dI*_{c}/*dT<*0,
the response grows with current amplitude. This leads to the non-monotonic
current dependence of the differential resistance shown in Fig.5. The minimum
of the differential resistance at high current corresponds to the current *I*_{0} at which the sample
temperature reaches the peak temperature *T*_{p}=4.345K
[see lower right inset of Fig.1(b)]. It follows that the temperature increase
at *I*_{0} is *dT=45mK*. The temperature increase at various
applied currents obtained with the pulsed method is also shown in Fig.5. From
it we derive a temperature increase of 42mK*
*at* I*_{0}. Within our
experimental resolution the effects of the Joule heating determined by these
two methods are in good agreement.

In conclusion, we have introduced a new method to
determine the temperature increase due to Joule heating in a superconductor by
comparing the voltage response to short and long current steps. In the
experiments presented here we find that the temperature increase in the
presence of an applied current is due to dissipation associated with the vortex
motion.

Work supported by DOE DE-FG02-99ER45742.

** **

** **

** **

** **

** **

** **

** **

** **

** **

** **

**References**

[1] W. J. Skocpol, M. R.
Beasley, and M. Tinkham, J. Appl. Phys. **45**,
4054 (1974).

[2] G. Dharmadurai, Phys.
Stat. Sol. (a) **62**, 11 (1980).

[3] A. VI. Gurevich and R.
G. Mints, Rev. Mod. Phys. **59**, 941
(1987).

[4] Z. L. Xiao, E. Y.
Andrei, and P. Ziemann, Phys. Rev. B **58**,
11185 (1998).

[5]
J. W. Bremer and V. I. Newhouse, Phys. Rev. **116**, 309 (1959).

[6]
M. N. Kunchur, D. K. Christen, and J. M. Phillips, Phys. Rev. Lett. **70**, 998 (1993).

[7] S.
K. Gupta et al., Physica C** 206**, 335
(1993).

[8] S. G. Doettinger
et al., Phys. Rev. Lett. **73**, 1691
(1994).

[9] Z. L. Xiao and P.
Ziemann, Phys. Rev. B **53**, 15265
(1996).

[10] L.
Antognazza et al., J. Low Temp. Phys. **117**,
1543 (1999).

[11] Z.
L. Xiao, P. Voss-de Haan, G. Jakob, and H. Adrian, Phys. Rev. B **57**, R736 (1998).

[12]
Maxime Pauly et al., Physica B **284-288**,
721 (2000).

[13] M.
Decroux et al., Physica B **284-288**,
2099 (2000).

[14]
M. N. Kunchur et al., Phys. Rev. Lett. **84**,
5204 (2000).

[15] M. Nahum et al., Appl. Phys. Lett. **59**, 2034 (1991).

[16] M. Danerud et al, J. Appl. Phys. **76**, 1902 (1994).

[17] A.
V. Sergeev et al., Phys. Rev. B **49**,
9091 (1994).

[18]
Z. L. Xiao et al., Phys. Rev. B **59**,
1481 (1999).

[19]
J. A. Fendrich et al. , Phys. Rev. Lett. **77**,
2073 (1996).

[20]
S. Bhattarchaya and M. J. Higgins, Phys. Rev. Lett. **70**, 2617 (1993).

[21]
W. Henderson et al., Phys. Rev. Lett. **77**,
2077 (1996).

[22] Z.
L. Xiao, E. Y. Andrei, and M. J. Higgins, Phys. Rev. Lett. **83**, 1664 (1999).

[23]
D. M. Kroeger, Solid State Comm. **7**,
843 (1969).

[24]
A. E. Koshelev and V. M. Vinokur, Phys. Rev. Lett. **73**, 3580 (1994).

[25]
C. J. Olson, C. Reichhardt, and F. Nori, Phys. Rev. Lett. **81**, 3757 (1998).

[26]
Y. B. Kim et al., Phys. Rev. **14**0,
1197 (1967).

[27]
B. Sas et al., Phys. Rev. B **61**, 9118
(2000).

** **

** **

** **

** **

** **

** **

** **

** **

** **

** **

** **

** **

** **

** **

** **

** **

**Figure Captions**

Fig.1 *I*-*V*
curves obtained in sample A with DC, short (10ms) and long (4s) pulsed currents:
(a) *B*=0.1T , *T*=6.6K (below the peak region); (b) *B*=1.8T , *T*=4.30K (in the
lower part of the peak). Dashed line- calculated free flux flow response. Lower
right insets show the temperature dependence of the critical current (defined
with a 5mV criterion) and the
temperature at which the I-V curves were taken (dotted lines) . The upper left
inset in (a) shows illustrates the thermal anchoring of the sample (TH
represents the thermometer).

Fig.2 Determination of
the temperature increase by comparing the voltage response to short (10ms, right panels) and long (4s, left panels)
current steps. The data in (a1,a2), (b1,b2) and (c1,c2) represent typical
results obtained in sample A for the normal state (*T*>*T*_{c}),
below the peak region and in the lower part of peak region respectively. The dashed lines in the right panels are
traced at the voltage values in the left panels. The temperature increase at a
bath temperature *T*_{1} in the
presence of a DC current is given by *dT*=*T*_{2}-*T*_{1} (see text for details).

Fig.3 (a) Current
dependence of temperature increase in sample A. The solid curves are fits of
the data to *dT**~** I*^{n}, with n=2.642, 3.090, 2.372
and 2.172 for the curves from left to right; (b) temperature increase versus
normalized power dissipated by vortex motion, showing linear dependence (solid
line). The temperature dependence of the heat transfer coefficient is shown in
the inset of (b).

Fig.4
Temperature increase versus current amplitude obtained in sample B at *T*=6.0K for *B*=0.2, 0.3 and 0.5T. The solid curves are a fit of *dT*~*I*^{n}, with n=2.411, 2.533 and
2.667 for 0.2, 0.3 and 0.5T respectively. The upper inset shows the collapse of
the data when plotted as *dT*/H against
the applied current. The lower inset shows the temperature increase versus the
power dissipated in the vortex motion.
The line is a guide to the eye showing the expected linear behavior.

Fig.5 Current
dependence of the differential resistances (*dV/dI*)
(open circles) and the temperature increase (solid circles) obtained in sample
A. *I*_{0} is the current for
which the sample temperature coincides with the temperature of the peak in
critical current.